Bij het verankeren van staal aan beton worden doorgaans ankerbouten of chemische ankers gebruikt om constructieve stabiliteit en draagvermogen in bouwprojecten te waarborgen. Het model wordt geanalyseerd in de IDEA StatiCa Detail app.
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"value": "<p>Beste keuze voor staven met D-gebieden, openingen en consoles, en algemeen gevormde 2D-staven </p>\n<ul>\n <li>Niet-lineaire CSFM methode </li>\n <li>Spanning en rek van beton en alle wapeningsstaven</li>\n <li>Scheurwijdte normtoetsing</li>\n <li>Spanningsbegrenzing normtoetsing</li>\n <li>Niet-lineaire doorbuiging</li>\n <li>Importeer gegevens vanuit de Beam applicatie</li>\n</ul>"
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"value": "<p>Het ontwerp en de toetsing van betonelementen worden normaal gesproken uitgevoerd op doorsnede- (1D-element) of puntniveau (2D-element). Deze werkwijze is beschreven in alle normen voor constructief ontwerp, bijv. in (EN 1992-1-1 of ACI 318-19), en wordt dagelijks toegepast in de constructiepraktijk. Het is echter niet altijd bekend of in acht genomen dat deze werkwijze alleen toelaatbaar is in gebieden waar de hypothese van Bernoulli-Navier voor een vlakke rekverdeling geldt (B-gebieden genoemd). De plaatsen waar deze hypothese niet geldt, worden discontinuïteitsgebieden of verstoorde gebieden genoemd (D-gebieden). Voorbeelden van B- en D-gebieden van 1D-elementen zijn weergegeven in (Fig. 1). Dit zijn bijv. opleggebieden, delen waar geconcentreerde belastingen worden aangebracht, locaties met een plotselinge verandering van de doorsnede, openingen, enz. Bij het ontwerpen van betonconstructies komen we veel andere D-gebieden tegen, zoals wanden, brugdiafragma's, consoles, enz. </p>\n<figure data-asset-id=\"874c8092-fb41-44c6-804d-52727044d470\" data-image-id=\"874c8092-fb41-44c6-804d-52727044d470\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/dc96c2fd-25aa-43fd-b6d5-556b5242b9cf/Discontinuity%20regions.png\" data-asset-id=\"874c8092-fb41-44c6-804d-52727044d470\" data-image-id=\"874c8092-fb41-44c6-804d-52727044d470\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 1\\qquad Discontinuity regions (Navrátil et al. 2017)}}}\\]</em></p>\n<p>In het verleden werden semi-empirische ontwerpregels gebruikt voor het dimensioneren van discontinuïteitsgebieden. Gelukkig zijn deze regels de afgelopen decennia grotendeels vervangen door staafwerkmodellen (Schlaich et al., 1987) en spanningsvelden (Marti 1985), die zijn opgenomen in de huidige normen en tegenwoordig vaak door ontwerpers worden gebruikt. Deze modellen zijn mechanisch consistent en krachtige hulpmiddelen. Houd er rekening mee dat spanningsvelden in het algemeen continu of discontinu kunnen zijn en dat staafwerkmodellen een speciaal geval van discontinue spanningsvelden zijn.</p>\n<p>Ondanks de evolutie van rekentools in de afgelopen decennia worden staafwerkmodellen in wezen nog steeds als handberekeningen gebruikt. Toepassing ervan op reële constructies is omslachtig en tijdrovend, omdat iteraties nodig zijn en meerdere belastingsgevallen moeten worden beschouwd. Bovendien is deze methode niet geschikt voor het toetsen van bruikbaarheidscriteria (vervormingen, scheurwijdten, enz.).</p>\n<p>De interesse van constructeurs in een betrouwbare en snelle tool voor het ontwerpen van D-gebieden leidde tot het besluit om de nieuwe Compatible Stress Field Method te ontwikkelen, een methode voor computerondersteund ontwerpen met spanningsvelden die het automatisch ontwerpen en toetsen mogelijk maakt van constructieve betonstaven die worden belast in het vlak.</p>\n<p>De Compatible Stress Field Method (CSFM) is een continue, op de Eindige Elementen Methode gebaseerde analysemethode voor spanningsvelden, waarin klassieke spanningsveldoplossingen worden aangevuld met kinematische beschouwingen, d.w.z. de rektoestand wordt over de gehele constructie bepaald. Hierdoor kan de effectieve druksterkte van beton automatisch worden berekend op basis van de rektoestand in dwarsrichting, op een vergelijkbare wijze als bij drukveldanalyses die rekening houden met compression softening (Vecchio en Collins 1986; Kaufmann en Marti 1998) en de EPSF-methode (Fernández Ruiz en Muttoni 2007). Bovendien houdt de CSFM rekening met tension stiffening, waardoor de elementen realistische stijfheden krijgen, en dekt zij alle normvoorschriften (inclusief aspecten van bruikbaarheid en vervormingscapaciteit) die in eerdere benaderingen niet consistent werden behandeld. De CSFM gebruikt gangbare uniaxiale constitutieve relaties uit de normen voor beton en wapening. Deze zijn bekend in de ontwerpfase, waardoor de partiëlefactorenmethode kan worden toegepast. Ontwerpers hoeven dus geen aanvullende, vaak willekeurige materiaaleigenschappen op te geven, zoals doorgaans vereist is voor niet-lineaire EEM-analyses, waardoor de methode uitermate geschikt is voor de praktijk.</p>\n<p>Om het gebruik van computerondersteunde spanningsvelden door constructeurs te bevorderen, moeten deze methoden worden geïmplementeerd in gebruiksvriendelijke softwareomgevingen. Hiertoe is de CSFM geïmplementeerd in <em>IDEA StatiCa Detail</em>; een nieuwe, gebruiksvriendelijke commerciële software, gezamenlijk ontwikkeld door ETH Zürich en het softwarebedrijf IDEA StatiCa in het kader van het DR-Design Eurostars-10571-project.</p>"
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"value": "<h3>Werkwijze en doelen</h3>\n<p>Het doel van de ontwerptools voor wapening in de <a data-item-id=\"42ce7f6b-6491-4224-a01e-c4c0072ed1cd\" href=\"\">CSFM</a> is ontwerpers te helpen om de positie en de benodigde hoeveelheid wapeningsstaven efficiënt te bepalen. De volgende tools ondersteunen en begeleiden de gebruiker bij dit proces: lineaire berekening en <a data-item-id=\"decdf07d-a46b-5894-9a22-793436e318c7\" href=\"\">topologie-optimalisatie</a>.</p>\n<p>Ontwerptools voor wapening gebruiken meer vereenvoudigde constitutieve modellen dan de modellen die worden gebruikt voor de definitieve toetsing van de constructie. De wapeningsdefinitie in deze stap moet daarom worden beschouwd als een voorontwerp dat tijdens de definitieve toetsing wordt bevestigd of verfijnd. Het gebruik van de verschillende ontwerptools voor wapening wordt toegelicht aan de hand van het model in Fig. 3. Dit model bestaat uit één uiteinde van een vrij opgelegde ligger met variabele hoogte, belast door een gelijkmatig verdeelde belasting.</p>\n<figure data-asset-id=\"eee2b9e4-83cd-4b9c-98e7-f575b2ff9cff\" data-image-id=\"eee2b9e4-83cd-4b9c-98e7-f575b2ff9cff\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/9b0c4840-5a55-46f3-95ba-86a9baabbf0c/Model%20used%20to%20illustrate%20the%20use%20of%20the%20reinforcement%20design%20tools.png\" data-asset-id=\"eee2b9e4-83cd-4b9c-98e7-f575b2ff9cff\" data-image-id=\"eee2b9e4-83cd-4b9c-98e7-f575b2ff9cff\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 3\\qquad Model used to illustrate the use of the reinforcement design tools.}}}\\]</em></p>\n<h3>Lineaire analyse</h3>\n<p>De lineaire analyse gaat uit van lineair-elastische materiaaleigenschappen en verwaarloost de wapening in het betongebied. Het is daarom een zeer snelle berekening die een eerste inzicht geeft in de locaties van trek- en drukzones. Een voorbeeld van een dergelijke berekening is weergegeven in Fig. 4.</p>\n<figure data-asset-id=\"f6c14a09-4d2b-40e6-ac82-5ff08c10439a\" data-image-id=\"f6c14a09-4d2b-40e6-ac82-5ff08c10439a\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/ea7896d1-8276-4d08-b811-066cca73b455/Results%20from%20the%20linear%20analysis%20tool.jpg\" data-asset-id=\"f6c14a09-4d2b-40e6-ac82-5ff08c10439a\" data-image-id=\"f6c14a09-4d2b-40e6-ac82-5ff08c10439a\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 4\\qquad Results from the linear analysis tool for defining reinforcement layout}}}\\]</em></p>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{(red: areas in compression, blue: areas in tension).}}}\\]</em></p>\n<h3>Topologie-optimalisatie</h3>\n<p>Topologie-optimalisatie is een methode die de optimale verdeling van materiaal in een gegeven volume zoekt voor een bepaalde belastingconfiguratie. De topologie-optimalisatie in <em>Idea StatiCa Detail</em> maakt gebruik van een lineair eindig-elementenmodel. Elk eindig element kan een relatieve dichtheid hebben van 0 tot 100 %, die de relatieve hoeveelheid gebruikt materiaal weergeeft. Deze elementdichtheden zijn de optimalisatieparameters in het optimalisatieprobleem. De resulterende materiaalverdeling is optimaal voor de gegeven belastingen als de totale rekenergie van het systeem minimaal is. Per definitie is de optimale verdeling ook de geometrie met de grootst mogelijke stijfheid voor de gegeven belastingen.</p>\n<p>Het iteratieve optimalisatieproces start met een homogene dichtheidsverdeling.<em> </em>De berekening wordt uitgevoerd voor meerdere totale volumefracties (20%, 40%, 60% en 80%), zodat de gebruiker het meest praktische resultaat kan kiezen. De resulterende vorm bestaat uit vakwerken met drukdiagonalen en trekstaven en vertegenwoordigt de optimale vorm voor de gegeven belastinggevallen (Fig. 5).</p>\n<figure data-asset-id=\"f4f47d5e-3196-4a88-96ca-7162b0c8c271\" data-image-id=\"f4f47d5e-3196-4a88-96ca-7162b0c8c271\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/f4d37064-76c7-4413-b1aa-87455a32852c/Results%20from%20the%20topology%20optimization%201.jpg\" data-asset-id=\"f4f47d5e-3196-4a88-96ca-7162b0c8c271\" data-image-id=\"f4f47d5e-3196-4a88-96ca-7162b0c8c271\" alt=\"\"></figure>\n<figure data-asset-id=\"7ddd1329-64ea-4a47-be5d-64994439e729\" data-image-id=\"7ddd1329-64ea-4a47-be5d-64994439e729\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/d81f2841-8274-414a-8f30-b55427216169/Results%20from%20the%20topology%20optimization%202.png\" data-asset-id=\"7ddd1329-64ea-4a47-be5d-64994439e729\" data-image-id=\"7ddd1329-64ea-4a47-be5d-64994439e729\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 5\\qquad Results from the topology optimization design tool with 20\\% and 40\\% effective volume}}}\\]</em></p>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{(red: areas in compression, blue: areas in tension).}}}\\]</em></p>\n<p><br></p>"
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"value": "<p>De CSFM beschouwt continue spanningsvelden in het beton (2D-eindige elementen), aangevuld met discrete \"staaf\"-elementen die de wapening representeren (1D-eindige elementen). Daarom is de wapening niet diffuus ingebed in de 2D-eindige elementen van het beton, maar expliciet gemodelleerd en met deze elementen verbonden. In het rekenmodel wordt uitgegaan van een vlakspanningstoestand.</p>\n<figure data-asset-id=\"9e86fe68-36a5-433d-9451-40d2b5078b86\" data-image-id=\"9e86fe68-36a5-433d-9451-40d2b5078b86\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/3f70008c-0c34-4dbe-8219-4d8aa7079bb5/Visualization%20of%20the%20calculation%20model.png\" data-asset-id=\"9e86fe68-36a5-433d-9451-40d2b5078b86\" data-image-id=\"9e86fe68-36a5-433d-9451-40d2b5078b86\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 6\\qquad Visualization of the calculation model of a structural element (trimmed beam) in Idea StatiCa Detail.}}}\\]</em></p>\n<p>Zowel complete <a data-item-id=\"a11adc2d-9c84-4667-8061-600660e1ad87\" href=\"\">wanden</a> en liggers als details (delen) van liggers (geïsoleerd discontinuïteitsgebied, ook wel afgekapt uiteinde genoemd) kunnen worden gemodelleerd. Bij wanden en complete liggers moeten de steunpunten zo worden gedefinieerd dat een (uitwendig) isostatische (statisch bepaalde) of hyperstatische (statisch onbepaalde) constructie ontstaat. De krachtsinleiding aan de afgekapte uiteinden van liggers wordt gerealiseerd door middel van een speciale Saint-Venant-overgangszone, die zorgt voor een realistische spanningsverdeling in het beschouwde detailgebied.</p>"
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"value": "<p>De beoordeling van de constructie met de CSFM gebeurt met twee verschillende analyses: één voor de belastingcombinaties in de bruikbaarheidsgrenstoestand en één voor die in de uiterste grenstoestand. De bruikbaarheidsanalyse gaat ervan uit dat het uiterste gedrag van het element voldoende is en dat de vloeivoorwaarden van het materiaal bij belastingen in de bruikbaarheidsgrenstoestand niet worden bereikt. Deze benadering maakt het mogelijk om voor de bruikbaarheidsanalyse vereenvoudigde constitutieve modellen te gebruiken (met een lineaire tak van het spanning-rekdiagram van beton). Dit verbetert de numerieke stabiliteit en de rekensnelheid. Daarom wordt aanbevolen de hieronder beschreven werkwijze te volgen, waarbij de analyse in de uiterste grenstoestand als eerste stap wordt uitgevoerd.</p>\n<h3>Analyse in de uiterste grenstoestand</h3>\n<p>De verschillende toetsingen die door specifieke normen worden vereist, worden beoordeeld op basis van de directe resultaten van het model. De UGT-toetsingen worden uitgevoerd voor de betonsterkte, de sterkte van de wapening en de verankering (afschuifspanningen in de aanhechting).</p>\n<p>Om een efficiënt ontwerp van een constructief element te waarborgen, wordt sterk aanbevolen een voorafgaande analyse uit te voeren met de volgende stappen:</p>\n<ul>\n <li>Kies een selectie van de meest kritieke belastingcombinaties.</li>\n <li>Bereken alleen de belastingcombinaties voor de uiterste grenstoestand (UGT).</li>\n <li>Gebruik een grof mesh (door de vermenigvuldigingsfactor van de standaard meshgrootte in Setup te verhogen (Fig. 19)).</li>\n</ul>\n<figure data-asset-id=\"8c27dc0f-1cfe-4026-bbf5-4b51604c3558\" data-image-id=\"8c27dc0f-1cfe-4026-bbf5-4b51604c3558\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/aabe4d74-d599-4c9d-a62d-8e448a66360a/Mesh%20multiplier.PNG\" data-asset-id=\"8c27dc0f-1cfe-4026-bbf5-4b51604c3558\" data-image-id=\"8c27dc0f-1cfe-4026-bbf5-4b51604c3558\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 19\\qquad Mesh multiplier.}}}\\]</em></p>\n<p>Een dergelijk model rekent zeer snel. Ontwerpers kunnen hiermee de detaillering van het constructieve element efficiënt beoordelen en de analyse opnieuw uitvoeren totdat aan alle toetsingseisen voor de meest kritieke belastingcombinaties is voldaan. Zodra aan alle toetsingseisen van deze voorafgaande analyse is voldaan, wordt aanbevolen de volledige belastingcombinaties voor de uiterste grenstoestand mee te nemen en een fijne meshgrootte te gebruiken (de meshgrootte die door het programma wordt aanbevolen). De gebruiker kan de meshgrootte wijzigen met de vermenigvuldigingsfactor, die waarden van 0,5 tot 5 kan aannemen (Fig. 19).</p>\n<p>De basisresultaten en toetsingen (spanning, rek en benuttingsgraad (d.w.z. de berekende waarde/grenswaarde uit de norm), evenals de richting van de hoofdspanningen bij betonnen elementen) worden weergegeven in verschillende grafieken, waarin druk over het algemeen rood en trek blauw wordt weergegeven. Globale minimum- en maximumwaarden voor de gehele constructie kunnen worden gemarkeerd, evenals minimum- en maximumwaarden voor elk door de gebruiker gedefinieerd deel. In een apart tabblad van het programma kunnen geavanceerde resultaten worden weergegeven, zoals tensorwaarden, vervormingen van de constructie en wapeningspercentages (effectief en geometrisch) die worden gebruikt voor het berekenen van de tension stiffening van wapeningsstaven. Verder kunnen belastingen en reacties voor geselecteerde combinaties of belastinggevallen worden getoond.</p>\n<h3>Analyse in de bruikbaarheidsgrenstoestand</h3>\n<p>De BGT-beoordelingen worden uitgevoerd voor de spanningsbegrenzing, de scheurwijdte en de grenswaarden voor doorbuiging. De spanningen in beton en wapening worden getoetst volgens de van toepassing zijnde norm, op een vergelijkbare manier als voor de UGT is gespecificeerd.</p>\n<p>De bruikbaarheidsanalyse bevat bepaalde vereenvoudigingen van de constitutieve modellen die worden gebruikt voor de analyse in de uiterste grenstoestand. Er wordt uitgegaan van een perfecte aanhechting, d.w.z. de verankeringslengte wordt in de bruikbaarheidsgrenstoestand niet getoetst. Verder wordt de plastische tak van de spanning-rekkromme van beton op druk buiten beschouwing gelaten, terwijl de elastische tak lineair en oneindig is. Deze vereenvoudigingen verbeteren de numerieke stabiliteit en de rekensnelheid en doen geen afbreuk aan de algemeenheid van de oplossing, zolang de resulterende materiaalspanningen in de bruikbaarheidsgrenstoestand duidelijk onder hun vloeigrens liggen (zoals vereist door de normen). 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"value": "<p>Het niet-lineaire (inelastische) eindige-elementenmodel wordt opgebouwd uit verschillende typen eindige elementen die worden gebruikt om beton, wapening en de aanhechting daartussen te modelleren. Beton- en wapeningselementen worden eerst onafhankelijk van elkaar gemesht en daarna met elkaar verbonden via multi-point randvoorwaarden (MPC-elementen). Hierdoor kan de wapening een willekeurige, relatieve positie ten opzichte van het beton innemen. Als de verankeringslengte moet worden getoetst, worden tussen de wapening en de MPC-elementen aanhechtings- en verankeringseindveerelementen ingevoegd.</p>\n<figure data-asset-id=\"03fd72f4-b362-492a-8885-349785eaa70a\" data-image-id=\"03fd72f4-b362-492a-8885-349785eaa70a\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/511cc4d5-618a-4542-ac53-52a29549070f/Finite%20element%20model.png\" data-asset-id=\"03fd72f4-b362-492a-8885-349785eaa70a\" data-image-id=\"03fd72f4-b362-492a-8885-349785eaa70a\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 13\\qquad Finite element model: reinforcement elements mapped to concrete mesh using MPC elements and bond elements.}}}\\]</em></p>\n<h3>Beton</h3>\n<p>Beton wordt gemodelleerd met vierhoekige en driehoekige schaalelementen, CQUAD4 en CTRIA3. Deze worden respectievelijk gedefinieerd door vier of drie knopen. In deze elementen wordt alleen een vlakspanningstoestand aangenomen, d.w.z. spanningen of rekken in de z-richting worden niet in rekening gebracht.</p>\n<p>Elk element heeft vier of drie integratiepunten, die op ongeveer 1/4 van de elementafmeting zijn geplaatst. In elk integratiepunt van elk element worden de richtingen van de hoofdrekken α<sub>1</sub>, α<sub>2</sub> berekend. In beide richtingen worden de hoofdspanningen σ<em><sub>c</sub></em><sub>1</sub>, σ<em><sub>c</sub></em><sub>2</sub> en de stijfheden <em>E</em><sub>1</sub>, <em>E</em><sub>2</sub> bepaald volgens het opgegeven spanning-rekdiagram van beton, zoals in Fig. 2. Houd er rekening mee dat het compression softening effect het gedrag in de hoofddrukrichting koppelt aan de actuele toestand in de andere hoofdrichting.</p>\n<h3>Wapening</h3>\n<p>Wapeningsstaven worden gemodelleerd met 1D-“staaf”-elementen met twee knopen (CROD), die alleen axiale stijfheid hebben. Deze elementen zijn verbonden met speciale “aanhechtingselementen”, die zijn ontwikkeld om het slipgedrag tussen een wapeningsstaaf en het omringende beton te modelleren. Deze aanhechtingselementen worden vervolgens met MPC-elementen (multi-point randvoorwaarden) verbonden met de mesh die het beton voorstelt. Deze aanpak maakt het onafhankelijk meshen van wapening en beton mogelijk, terwijl hun onderlinge verbinding achteraf wordt gewaarborgd.</p>\n<h3>Aanhechtingselementen</h3>\n<p>De verankeringslengte wordt getoetst door in het eindige-elementenmodel de aanhechtingsschuifspanningen tussen betonelementen (2D) en wapeningsstaafelementen (1D) te implementeren. Hiervoor is een eindig element van het type “aanhechting” ontwikkeld.</p>\n<p>De definitie van het aanhechtingselement is vergelijkbaar met die van een schaalelement (CQUAD4). Het wordt eveneens door 4 knopen gedefinieerd, maar heeft, in tegenstelling tot een schaalelement, alleen een stijfheid ongelijk aan nul in afschuiving tussen de twee bovenste en de twee onderste knopen. In het model zijn de bovenste knopen verbonden met de elementen die de wapening voorstellen en de onderste knopen met de elementen die het beton voorstellen. Het gedrag van dit element wordt beschreven door de aanhechtingsspanning, τ<em><sub>b</sub></em>, als bilineaire functie van de slip tussen de bovenste en onderste knopen, δ<em><sub>u</sub></em>, zie Fig. 14.</p>\n<figure data-asset-id=\"a031a0ff-a5a7-4a37-b59f-cb1c408f080b\" data-image-id=\"a031a0ff-a5a7-4a37-b59f-cb1c408f080b\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/1cc20fd2-92d7-42dc-ac17-24f318cbd45c/Bond.PNG\" data-asset-id=\"a031a0ff-a5a7-4a37-b59f-cb1c408f080b\" data-image-id=\"a031a0ff-a5a7-4a37-b59f-cb1c408f080b\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 14\\qquad (a) conceptual illustration of the deformation of a bond element; (b) a stress-deformation function.}}}\\]</em></p>\n<p><br></p>\n<p>De elastische stijfheidsmodulus van de aanhechting-slip-relatie, <em>G</em><em><sub>b</sub></em>, wordt als volgt gedefinieerd:</p>\n<p>\\[G_b = k_g \\cdot \\frac{E_c}{Ø}\\]</p>\n<p>waarin:</p>\n<p><em>k</em><em><sub>g</sub></em> coëfficiënt afhankelijk van het oppervlak van de wapeningsstaaf (standaard <em>k</em><em><sub>g</sub></em><sub> </sub>= 0,2)</p>\n<p><em>E</em><em><sub>c</sub></em> elasticiteitsmodulus van beton (genomen als <em>E</em><em><sub>cm</sub></em> bij EN)</p>\n<p>Ø de diameter van de wapeningsstaaf</p>\n<p>De rekenwaarden (gefactoreerde waarden) van de uiterste aanhechtingsschuifspanning, <em>f</em><em><sub>bd</sub></em>, uit de geselecteerde rekennormen EN 1992-1-1 of ACI 318-19 worden gebruikt om de verankeringslengte te toetsen. De verharding van de plastische tak wordt standaard berekend als <em>G</em><em><sub>b</sub></em>/10<sup>5</sup>.</p>\n<h3>Verankeringsveer</h3>\n<p>Het aanbrengen van verankeringseinden aan de wapeningsstaven (d.w.z. bochten, haken, lussen…), die voldoen aan de voorschriften van de rekennormen, maakt het mogelijk de basisverankeringslengte van de staven (<em>l</em><em><sub>b,net</sub></em>) met een bepaalde factor β te verminderen (hieronder de ‘verankeringscoëfficiënt’ genoemd). De rekenwaarde van de verankeringslengte (<em>l</em><em><sub>b</sub></em>) wordt dan als volgt berekend:</p>\n<p>\\[l_b = \\left(1 - \\beta\\right)l_{b,net}\\]</p>\n<p>De beoogde reductie van <em>l</em><em><sub>b,net</sub></em> is gelijkwaardig aan het activeren van de wapeningsstaaf aan zijn uiteinde tot een percentage van zijn maximale capaciteit, gegeven door de verankeringsreductiecoëfficiënt, zoals weergegeven in Fig. 15a.</p>\n<figure data-asset-id=\"6e05f6d3-2d4c-4c6c-90f0-89e34117415c\" data-image-id=\"6e05f6d3-2d4c-4c6c-90f0-89e34117415c\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/748b5346-4251-4154-b923-919c94d0c6d0/Model%20for%20the%20reduction%20of%20the%20anchorage%20length.PNG\" data-asset-id=\"6e05f6d3-2d4c-4c6c-90f0-89e34117415c\" data-image-id=\"6e05f6d3-2d4c-4c6c-90f0-89e34117415c\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 15\\qquad Model for the reduction of the anchorage length:}}}\\]</em></p>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{(a) anchorage force along the anchorage length of the reinforcing bar; (b) slip-anchorage force constitutive relationship.}}}\\]</em></p>\n<p>De reductie van de verankeringslengte wordt in het eindige-elementenmodel opgenomen met een veerelement aan het uiteinde van de staaf (Fig. 15), dat wordt gedefinieerd door het constitutieve model in Fig. 15b. De maximale kracht die door deze veer wordt overgedragen (<em>F</em><em><sub>au</sub></em>) is:</p>\n<p>\\[F_{au} = \\beta \\cdot A_s \\cdot f_{yd}\\]</p>\n<p>waarin :</p>\n<p><em>β</em> de verankeringscoëfficiënt op basis van het verankeringstype,</p>\n<p><em>A</em><em><sub>s</sub></em> de doorsnede van de wapeningsstaaf,</p>\n<p><em>f</em><em><sub>yd</sub></em><em> </em> de rekenwaarde (gefactoreerde waarde) van de vloeigrens van de wapening.</p>"
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"value": "<p><strong>CSFM houdt rekening met de maximale hoofdspanning in het beton op druk (σ</strong><em><strong><sub>c</sub></strong></em><strong><sub>2</sub></strong><em><strong><sub>r</sub></strong></em><strong>) en met de wapeningsspanningen (σ</strong><em><strong><sub>sr</sub></strong></em><strong>) ter plaatse van de scheuren, terwijl de treksterkte van het beton (σ</strong><em><strong><sub>c</sub></strong></em><strong><sub>1</sub></strong><em><strong><sub>r</sub></strong></em><strong> = 0) wordt verwaarloosd, behalve het stiffening effect ervan op de wapening.</strong> Door tension stiffening mee te nemen kunnen de gemiddelde wapeningsrekken (ε<em><sub>m</sub></em>) worden gesimuleerd. Er worden fictieve, roterende, spanningsvrije scheuren beschouwd die zonder slip openen (Fig. 2a), en het evenwicht ter plaatse van de scheuren wordt samen met de gemiddelde wapeningsrekken in rekening gebracht. </p>\n<figure data-asset-id=\"a5b4f7ac-3fc1-4050-9269-afdb9901a92e\" data-image-id=\"a5b4f7ac-3fc1-4050-9269-afdb9901a92e\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/70d687dc-a209-4d67-aeb9-c0bdabacd5c1/Fig.%202%20-%20Basic%20assumptions%20of%20CSFM.png\" data-asset-id=\"a5b4f7ac-3fc1-4050-9269-afdb9901a92e\" data-image-id=\"a5b4f7ac-3fc1-4050-9269-afdb9901a92e\" alt=\"\"></figure>\n<p><em>\\( \\textsf{\\textit{\\footnotesize{Fig. 2\\qquad Basic assumptions of the CSFM: (a) principal stresses in concrete; (b) stresses in the reinforcement direction;}}}\\) \\( \\textsf{\\textit{\\footnotesize{(c) stress-strain diagram of concrete in terms of maximum stresses with consideration of compression softening;}}}\\) \\( \\textsf{\\textit{\\footnotesize{(d) stress-strain diagram of reinforcement in terms of stresses at cracks and average strains; (e) compression softening}}}\\) \\( \\textsf{\\textit{\\footnotesize{law; (f) bond shear stress-slip relationship for anchorage length verifications.}}}\\)</em></p>\n<p><br></p>\n<p>Ondanks hun eenvoud is aangetoond dat vergelijkbare uitgangspunten nauwkeurige voorspellingen opleveren voor gewapende staven onder belasting in het vlak (Kaufmann 1998; Kaufmann en Marti 1998), mits de aangebrachte wapening bros bezwijken bij scheurvorming voorkomt. Bovendien is het niet in rekening brengen van enige bijdrage van de treksterkte van beton aan de uiterste belasting in overeenstemming met de principes van moderne normen, die grotendeels op de plasticiteitstheorie zijn gebaseerd.</p>\n<p>De CSFM is echter <strong>niet geschikt voor slanke elementen</strong> zonder dwarswapening, aangezien voor dergelijke elementen relevante mechanismen zoals vertanding van het toeslagmateriaal, resterende trekspanningen aan de scheurtip en deuvelwerking – die allemaal direct of indirect afhankelijk zijn van de treksterkte van het beton – worden verwaarloosd. Hoewel sommige normen het ontwerp van dergelijke elementen toestaan op basis van semi-empirische bepalingen, is de CSFM niet bedoeld voor dit type potentieel bros bezwijkende constructie.</p>\n<h4>Beton</h4>\n<p>Het betonmodel dat in de CSFM is geïmplementeerd, is gebaseerd op de uniaxiale constitutieve wetten voor druk die in normen zijn voorgeschreven voor het ontwerp van doorsneden en die alleen afhankelijk zijn van de drukvastheid. Het parabool-rechthoekdiagram (Fig. 2c) wordt standaard gebruikt in de CSFM, maar ontwerpers kunnen ook kiezen voor een eenvoudiger elastisch-ideaalplastisch verband. Bij toetsing volgens de ACI-norm kan alleen het parabool-rechthoek spanning-rekdiagram worden gebruikt. Zoals eerder vermeld, wordt de treksterkte verwaarloosd, zoals ook gebruikelijk is bij klassiek ontwerp van gewapend beton.</p>\n<p>De effectieve drukvastheid wordt automatisch bepaald voor gescheurd beton op basis van de hoofdrek in trek (ε<sub>1</sub>) met behulp van de reductiefactor <em>k</em><em><sub>c</sub></em><sub>2</sub>, zoals weergegeven in Fig. 2c en e. Het geïmplementeerde reductieverband (Fig. 2e) is een veralgemening van het voorstel uit <em>fib</em> Model Code 2010 voor afschuivingstoetsingen, dat een grenswaarde van 0,65 bevat voor de maximale verhouding tussen de effectieve betonsterkte en de drukvastheid van het beton, en die niet van toepassing is op andere belastingsgevallen.</p>\n<p>De CSFM in <a data-item-id=\"b4790cf9-a605-45b3-b41b-e36909ad4291\" href=\"\"><em>IDEA StatiCa Detail</em></a> hanteert geen expliciet bezwijkcriterium in termen van rekken voor beton op druk (het gaat dus uit van een oneindig plastische tak nadat de piekspanning is bereikt). Door deze vereenvoudiging kan het vervormingsvermogen van constructies die op druk bezwijken niet worden getoetst. Hun uiterste draagvermogen wordt echter correct voorspeld wanneer, naast de factor voor gescheurd beton (<em>k</em><em><sub>c</sub></em><sub>2</sub>) zoals gedefinieerd in (Fig. 2e), ook de toename van de brosheid van beton bij toenemende sterkte in rekening wordt gebracht door middel van de reductiefactor <em>\\( \\eta_{fc} \\)</em> zoals gedefinieerd in <em>fib</em> Model Code 2010:</p>\n<p>\\[f_{c,red} = k_c \\cdot f_{c} = \\eta _{fc} \\cdot k_{c2} \\cdot f_{c}\\]</p>\n<p>\\[{\\eta _{fc}} = {\\left( {\\frac{{30}}{{{f_{c}}}}} \\right)^{\\frac{1}{3}}} \\le 1\\]</p>\n<p>waarin:</p>\n<p><em>k</em><em><sub>c </sub></em>de globale reductiefactor van de drukvastheid is</p>\n<p><em>k</em><em><sub>c</sub></em><sub>2</sub> de reductiefactor is vanwege de aanwezigheid van dwarsscheuren</p>\n<p><em>f</em><em><sub>c</sub></em> de karakteristieke cilindersterkte van het beton is (in MPa voor de definitie van <em>\\( \\eta_{fc} \\)</em>).</p>\n<p>Er vindt ook een reductie van de factor <em>k</em><em><sub>c</sub></em><sub>2</sub> plaats vanwege de stabiliteit van de berekening. Deze reductie heeft geen invloed op de totale sterkte van staven. Uitgaande van <em>f</em><em><sub>cd</sub></em> als de rekenwaarde van de betonsterkte, wordt de waarde van <em>k</em><em><sub>c</sub></em><sub>2</sub> gereduceerd volgens de onderstaande regels.</p>\n<p>σ<em><sub>c</sub></em><sub>2</sub><em><sub>r</sub></em><em> < 0.11f</em><em><sub>cd</sub></em><em> k</em><em><sub>c</sub></em><sub>2</sub><em>=1.0<br>0.11f</em><em><sub>cd</sub></em><em> < </em>σ<em><sub>c</sub></em><sub>2</sub><em><sub>r</sub></em><em> < 0.37f</em><em><sub>cd</sub></em><em> k</em><em><sub>c</sub></em><sub>2</sub><em> </em>is een lineaire interpolatie tussen 1,0 en de waarde uit de<br> grafiek in Fig. 2f<em><br></em>σ<em><sub>c</sub></em><sub>2</sub><em><sub>r</sub></em><em> > 0.37f</em><em><sub>cd</sub></em><em> k</em><em><sub>c</sub></em><sub>2</sub><em> </em>wordt direct overgenomen uit de grafiek in Fig. 2f</p>\n<h4>Wapening</h4>\n<p>Het geïdealiseerde bilineaire spanning-rekdiagram voor blanke wapeningsstaven zoals dat doorgaans in normen is gedefinieerd (Fig. 2d), wordt toegepast. Voor de definitie van dit diagram zijn tijdens de ontwerpfase alleen de basiseigenschappen van de wapening nodig (sterkte- en ductiliteitsklasse). Er kan ook een door de gebruiker gedefinieerd spanning-rekverband worden ingevoerd.</p>\n<p>Tension stiffening wordt in rekening gebracht door het spanning-rekverband van de blanke wapeningsstaaf als invoer aan te passen, om de gemiddelde stijfheid van de in het beton ingebedde staven te beschrijven (ε<em><sub>m</sub></em>).</p>\n<h4>Aanhechtingsmodel</h4>\n<p>De aanhechting-slip tussen wapening en beton wordt in het eindige-elementenmodel geïntroduceerd door het vereenvoudigde stijf-ideaalplastische constitutieve verband te hanteren dat in Fig. 2f is weergegeven, waarbij <em>f</em><em><sub>bd</sub></em> de rekenwaarde is van de uiterste aanhechtingsspanning zoals die in de norm is gespecificeerd voor de betreffende aanhechtingsomstandigheden.</p>\n<p>Dit is een vereenvoudigd model met als enig doel het toetsen van de aanhechtingsvoorschriften volgens de normen (d.w.z. verankering van de wapening). De reductie van de verankeringslengte bij het gebruik van haken, lussen en vergelijkbare staafvormen kan in rekening worden gebracht door een bepaald draagvermogen aan het uiteinde van de wapening te definiëren, zoals verderop zal worden beschreven. </p>"
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"value": "<p>Om de meeste situaties tijdens het bouwproces te modelleren, zijn in de CSFM veel typen steunpunten (Fig. 7) en componenten voor het overdragen van belasting (Fig. 8) beschikbaar.</p>\n<h3>Steunpunten</h3>\n<p>Een puntsteun kan op verschillende manieren worden gemodelleerd, zodat de spanningen niet in één punt worden geconcentreerd, maar over een groter oppervlak worden verdeeld. De eerste optie is een verdeelde puntsteun (Fig. 7a), die de belasting op de rand van de staaf gelijkmatig verdeelt over de opgegeven breedte.</p>\n<figure data-asset-id=\"168a03f0-9bf7-4893-87d9-9744163d0453\" data-image-id=\"168a03f0-9bf7-4893-87d9-9744163d0453\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/e51c52f3-be54-4b55-bb4d-c4089b8239a5/Supports.png\" data-asset-id=\"168a03f0-9bf7-4893-87d9-9744163d0453\" data-image-id=\"168a03f0-9bf7-4893-87d9-9744163d0453\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 7\\qquad Various types of supports:}}}\\]</em></p>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{(a) point distributed; (b) bearing plate; (c) line support; (d) patch support; (e) hanging.}}}\\]</em></p>\n<p>Een patch support (Fig. 7d) kan daarentegen alleen binnen een betonvolume met een gedefinieerde effectieve straal worden geplaatst. Deze is vervolgens met stijve elementen verbonden met de knopen van het wapeningsmesh binnen deze straal. Daarom is het noodzakelijk om rond de patch support een wapeningskorf te definiëren.</p>\n<p>Voor een nauwkeurigere modellering van bepaalde praktijksituaties zijn er nog twee andere opties voor een puntsteun. Ten eerste is er een puntsteun met een oplegplaat met gedefinieerde breedte en dikte (Fig. 7b). Het materiaal van de oplegplaat kan worden opgegeven en de hele oplegplaat wordt afzonderlijk gemeshed. Ten tweede is er een ophangsteun beschikbaar (Fig. 7e), die kan worden gebruikt voor het modelleren van hijsankers of hijsdeuvels.</p>\n<p>Een lijnsteun (Fig. 7c) kan worden gedefinieerd op een rand (door de lengte op te geven) of binnen een element (met een polylijn). Het is ook mogelijk om de stijfheid en/of het niet-lineaire gedrag op te geven (steun op druk/trek of alleen op druk).</p>\n<ul>\n <li>Lees gedetailleerde beschrijvingen in<strong> </strong><a data-item-id=\"5a121972-f384-4f14-8788-9da298e1aae1\" href=\"\"><strong>Soorten steunpunten in IDEA StatiCa Detail</strong></a></li>\n</ul>\n<h3>Belastingoverdragende componenten</h3>\n<p>Het introduceren van belastingen in de constructie kan eveneens op verschillende manieren worden gemodelleerd. Voor puntlasten kan een oplegplaat (Fig. 8a) op vergelijkbare wijze als bij een puntsteun worden gebruikt, waarbij de geconcentreerde belasting wordt verdeeld over een groter oppervlak dankzij een stalen plaat met gedefinieerde breedte en dikte. </p>\n<figure data-asset-id=\"d0cdeffe-373f-419a-8e49-d714b8494a68\" data-image-id=\"d0cdeffe-373f-419a-8e49-d714b8494a68\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/069fe6fe-74e0-41a9-90ba-1aeeede8a0fb/Load%20transmitting%20devices.png\" data-asset-id=\"d0cdeffe-373f-419a-8e49-d714b8494a68\" data-image-id=\"d0cdeffe-373f-419a-8e49-d714b8494a68\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 8\\qquad Various types of load transfer components:}}}\\]</em></p>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{(a) bearing plate; (b) patch load; (c) hanging; (d) partially loaded area.}}}\\]</em></p>\n<p>De puntlast kan worden aangebracht direct op het oppervlak van de constructie met een gedefinieerde werkingsstraal (de belasting wordt aangebracht op de betonelementen) of via een speciaal overdrachtsmiddel, de patch load (Fig. 8b en Fig. 9). Een patch load maakt het mogelijk de belasting direct over te dragen naar de gedefinieerde wapening die zich binnen het gebied van de effectieve straal bevindt. Om de correcte werking van de patch load te waarborgen, moet een groep wapeningsstaven worden gedefinieerd (in de wapeningseigenschappen) die met de belasting wordt verbonden. Wanneer de verbonden wapening niet is gedefinieerd, is het belastingoverdrachtsmechanisme gelijk aan dat van een puntlast op het oppervlak van een staaf en wordt de belasting via de randvoorwaarden overgedragen naar de betonelementen, niet direct naar de wapening. </p>\n<figure data-asset-id=\"04324fc6-7d2d-43a7-9248-3056e9bcc513\" data-image-id=\"04324fc6-7d2d-43a7-9248-3056e9bcc513\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/38d4656d-6c90-445a-858b-cd97d4b29730/Patch%20support.png\" data-asset-id=\"04324fc6-7d2d-43a7-9248-3056e9bcc513\" data-image-id=\"04324fc6-7d2d-43a7-9248-3056e9bcc513\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 9\\qquad Patch load: (a) load application; (b) load transferred through rebars (a group of bars for the load transfer is defined);}}}\\]</em></p>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{(c) load transferred through concrete (a group of bars for the load transfer is not defined).}}}\\]</em></p>\n<p>Hijsankers of hijsdeuvels kunnen worden gemodelleerd met een ophangbelasting (Fig. 8c). De gebruiker kan een gedeeltelijk belast oppervlak gebruiken (Fig. 8d), waarmee de draagcapaciteit van beton op druk kan worden verhoogd volgens Eurocode (dit type belastingoverdragende component kan niet worden gebruikt wanneer ACI is ingesteld). De constructie kan ook worden belast met lijnlasten op de randen, met een algemene polylijn of met oppervlaktelasten. De Detail applicatie kan het eigen gewicht automatisch in de berekening meenemen.</p>\n<p><br></p>"
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"value": "<p>In veel gevallen hoeven we slechts een deel (detail) van een constructieve staaf te modelleren, zoals de oplegging van een ligger, een sparing midden in de ligger, enzovoort. Deze aanpak kan leiden tot opleggingsconfiguraties die instabiel zijn, maar toelaatbaar in <em>IDEA StatiCa Detail</em> (inclusief het geval zonder opleggingen). In dergelijke gevallen is het echter ook nodig om de doorsnede te modelleren die de verbinding met het aansluitende B-gebied vertegenwoordigt, inclusief de snedekrachten in deze doorsnede die aan het evenwicht voldoen. In bepaalde gevallen (bijvoorbeeld bij het modelleren van de oplegging van een ligger) kunnen deze snedekrachten automatisch door het programma worden bepaald.</p>\n<p>Tussen het B-gebied en het geanalyseerde discontinuïteitsgebied wordt automatisch een Saint-Venant overgangszone gecreëerd om een realistische spanningsverdeling in het geanalyseerde gebied te waarborgen. De breedte van de overgangszone wordt bepaald als de helft van de hoogte van de doorsnede. Omdat de Saint-Venant-zone uitsluitend dient om een correcte spanningsverdeling in de rest van het model te verkrijgen, worden er geen resultaten van dit gebied weergegeven in de toetsing en worden hier geen stopcriteria in beschouwd.</p>\n<p>De rand van de Saint-Venant-zone die het afgesneden uiteinde van de ligger vertegenwoordigt, wordt als stijf gemodelleerd, dat wil zeggen dat deze kan roteren, maar vlak moet blijven. Dit wordt bereikt door alle FEM-knopen van de rand te koppelen aan een afzonderlijke knoop in het zwaartepunt van de doorsnede met behulp van een stijf lichaamselement<em> </em>(RBE2). De snedekrachten van het element kunnen vervolgens op deze knoop worden aangebracht, zoals weergegeven in Fig. 10.</p>\n<figure data-asset-id=\"aa4c7293-3a3e-4c89-b88b-f6a84b0c457f\" data-image-id=\"aa4c7293-3a3e-4c89-b88b-f6a84b0c457f\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/a2eb228a-7276-410a-a213-edf91bcfb6e9/Saint-Venant%20zone.PNG\" data-asset-id=\"aa4c7293-3a3e-4c89-b88b-f6a84b0c457f\" data-image-id=\"aa4c7293-3a3e-4c89-b88b-f6a84b0c457f\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 10\\qquad Transfer of internal forces at a trimmed end.}}}\\]</em></p>"
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"value": "<p>Voor het oplossen van een niet-lineair EEM-probleem wordt een standaard volledig Newton-Raphson-algoritme (NR) gebruikt. </p>\n<p>Over het algemeen convergeert het NR-algoritme vaak niet wanneer de volledige belasting in één stap wordt aangebracht. Een gebruikelijke aanpak, die ook hier wordt toegepast, is de belasting achtereenvolgens in meerdere incrementen aan te brengen en het resultaat van het vorige belastingsincrement te gebruiken als startpunt voor de Newton-oplossing van het volgende increment. Hiervoor is bovenop het Newton-Raphson-algoritme een belastingsgestuurd algoritme geïmplementeerd. Als de NR-iteraties niet convergeren, wordt het huidige belastingsincrement gehalveerd en worden de NR-iteraties opnieuw uitgevoerd.</p>\n<p>Een tweede doel van het belastingsgestuurde algoritme is het bepalen van de kritieke belasting, die overeenkomt met bepaalde \"stopcriteria\". Dit zijn de maximale rek in het beton, de maximale slip in aanhechtingselementen, de maximale verplaatsing in verankeringselementen en de maximale rek in wapeningsstaven. De kritieke belasting wordt bepaald met de bisectiemethode. Als het stopcriterium ergens in het model wordt overschreden, worden de resultaten van het laatste belastingsincrement verworpen en wordt een nieuw increment berekend dat half zo groot is als het vorige. Dit proces wordt herhaald totdat de kritieke belasting is gevonden binnen een bepaalde foutentolerantie.</p>\n<p>Voor beton is het stopcriterium in de integratiepunten van schaalelementen ingesteld op een rek van 5% in druk (dus ongeveer een orde van grootte groter dan de werkelijke bezwijkrek van beton) en 7% in trek. In trek is de waarde zo gekozen dat eerst de grensrek van de wapening kan worden bereikt, die zonder rekening te houden met tension stiffening meestal rond 5% ligt. In druk is de waarde gekozen uit meerdere alternatieven als een waarde die groot genoeg is om de effecten van verbrijzelen zichtbaar te maken in de resultaten, maar klein genoeg om niet te veel problemen met de numerieke stabiliteit te veroorzaken.</p>\n<figure data-asset-id=\"883637b4-6077-43ff-b6e8-ac1e86785345\" data-image-id=\"883637b4-6077-43ff-b6e8-ac1e86785345\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/c1026dcf-91ed-47ab-af2e-705ca886a9ed/Constitutive%20relationship%20of%20bond%20and%20anchorage.PNG\" data-asset-id=\"883637b4-6077-43ff-b6e8-ac1e86785345\" data-image-id=\"883637b4-6077-43ff-b6e8-ac1e86785345\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 17\\qquad Constitutive relationship of bond and anchorage elements used for anchorage length verification:}}}\\]</em></p>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{(a) bond shear stress slip response of a bond element; (b) force-displacement response of an anchorage element.}}}\\]</em></p>\n<p>Voor wapening is het stopcriterium gedefinieerd in termen van spanningen. Omdat de spanningen ter plaatse van de scheur worden gemodelleerd, komt het criterium in trek overeen met de treksterkte van de wapening, rekening houdend met de veiligheidscoëfficiënt. Dezelfde waarde wordt gebruikt voor het criterium in druk.</p>\n<p>Het stopcriterium in aanhechtingselementen en verankeringsveren is α·δ<em>u</em><em><sub>max</sub></em>, waarbij δ<em>u</em><em><sub>max</sub></em> de maximale slip is die in normtoetsingen wordt gebruikt en α = 10.</p>"
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"value": "<h3>Beton - UGT</h3>\n<p>Het betonmodel dat in de CSFM is geïmplementeerd, is gebaseerd op de eenassige constitutieve wetten voor druk uit EN 1992-1-1 voor het ontwerp van doorsneden, die alleen afhangen van de druksterkte. In de CSFM wordt standaard het parabool-rechthoekdiagram gebruikt dat is gespecificeerd in EN 1992-1-1 Art. 3.1.7 (1) (Fig. 24a). Ontwerpers kunnen ook kiezen voor een vereenvoudigde elastisch-ideaal-plastische relatie volgens EN 1992-1-1 Art. 3.1.7 (2) (Fig. 24b). De treksterkte wordt verwaarloosd, zoals gebruikelijk is bij het klassieke ontwerp van gewapend beton.</p>\n<figure data-asset-id=\"d99ce820-6afd-4050-a438-c0bd6d3e5e29\" data-image-id=\"d99ce820-6afd-4050-a438-c0bd6d3e5e29\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/e72b03ac-c1db-4c39-bbc2-f4d87b7522f2/Concrete%20stress-strain%20diagram%20CSFM.PNG\" data-asset-id=\"d99ce820-6afd-4050-a438-c0bd6d3e5e29\" data-image-id=\"d99ce820-6afd-4050-a438-c0bd6d3e5e29\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 24\\qquad The stress-strain diagrams of concrete for ULS: a) parabola-rectangle diagram; b) bilinear diagram.}}}\\]</em></p>\n<p>De implementatie van de CSFM in <em>IDEA StatiCa Detail</em> houdt geen rekening met een expliciet bezwijkcriterium in termen van rek voor beton onder druk. Na het bereiken van de piekspanning wordt een plastische tak aangehouden met ε<em><sub>cu</sub></em><sub>2</sub> (ε<em><sub>cu</sub></em><sub>3</sub>) met een waarde van 5%, terwijl EN 1992-1-1 uitgaat van een uiterste rek kleiner dan 0,35%. Door deze vereenvoudiging kan het vervormingsvermogen van constructies die onder druk bezwijken niet worden getoetst. Hun uiterste capaciteit <em>f</em><em><sub>cd</sub></em> volgens EN 1992-1-1 3.1.3 wordt echter correct voorspeld wanneer, naast de factor voor gescheurd beton (<em>k</em><em><sub>c</sub></em><sub>2</sub>, gedefinieerd in (Fig. 25)), ook de toename van de brosheid van beton bij hogere sterkte in rekening wordt gebracht via de reductiefactor <em>\\(\\eta_{fc}\\)</em> uit <em>fib</em> Model Code 2010, als volgt:</p>\n<p>\\[f_{cd}={\\alpha_{cc}} \\cdot \\frac{f_{ck,red}}{γ_c} = {\\alpha_{cc}} \\cdot \\frac{k_c \\cdot f_{ck}}{γ_c} = {\\alpha_{cc}} \\cdot \\frac{\\eta _{fc} \\cdot k_{c2} \\cdot f_{ck}}{γ_c}\\]</p>\n<p>\\[{\\eta _{fc}} = {\\left( {\\frac{{30}}{{{f_{ck}}}}} \\right)^{\\frac{1}{3}}} \\le 1\\]</p>\n<p>waarin:</p>\n<p>α<em><sub>cc</sub></em> de coëfficiënt is die rekening houdt met langetermijneffecten op de druksterkte en met ongunstige effecten die voortvloeien uit de wijze van belasten. Deze is conform EN 1992-1-1 Art. 3.1.6 (1). De standaardwaarde is 1,0.</p>\n<p><em>k</em><em><sub>c </sub></em>de globale reductiefactor van de druksterkte is</p>\n<p><em>k</em><em><sub>c</sub></em><sub>2</sub> de reductiefactor is vanwege de aanwezigheid van dwarsscheuren</p>\n<p><em>f</em><em><sub>ck</sub></em> de karakteristieke cilindersterkte van het beton is (in MPa voor de definitie van <em>\\( \\eta_{fc} \\)</em>).</p>\n<figure data-asset-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\" data-image-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/085222c7-055a-4870-9bcb-8f18bd65620f/Compression%20softening%20CSFM.PNG\" data-asset-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\" data-image-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 25\\qquad The compression softening law.}}}\\]</em></p>\n<h3>Beton - BGT</h3>\n<p>De bruikbaarheidsanalyse bevat enkele vereenvoudigingen van de constitutieve modellen die worden gebruikt voor de analyse in de uiterste grenstoestand. De plastische tak van de spanning-rekcurve van beton onder druk wordt buiten beschouwing gelaten, terwijl de elastische tak lineair en oneindig is. De compression softening wet wordt niet in rekening gebracht. Deze vereenvoudigingen verbeteren de numerieke stabiliteit en de rekensnelheid en verminderen de algemeenheid van de oplossing niet, zolang de resulterende spanningsgrenzen van het materiaal in de bruikbaarheidsgrenstoestand duidelijk onder hun vloeigrens liggen (zoals vereist door Eurocode). De vereenvoudigde modellen voor de bruikbaarheid zijn daarom alleen geldig als aan alle toetsingseisen wordt voldaan.</p>\n<figure data-asset-id=\"78f0e024-ae44-4ec0-b939-6ac74688ae23\" data-image-id=\"78f0e024-ae44-4ec0-b939-6ac74688ae23\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/bca48b51-2839-4b96-8dac-078574e47c12/Fig.%2011%20-%20Concrete%20stress-strain%20for%20serviceability%20.png\" data-asset-id=\"78f0e024-ae44-4ec0-b939-6ac74688ae23\" data-image-id=\"78f0e024-ae44-4ec0-b939-6ac74688ae23\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 26\\qquad Concrete stress-strain diagrams implemented for serviceability analysis: short- and long-term verifications.}}}\\]</em></p>\n<p><br></p>\n<p><strong>Langetermijneffecten</strong></p>\n<p>In de bruikbaarheidsanalyse worden de langetermijneffecten van beton in rekening gebracht met een effectieve kruipcoëfficiënt voor oneindige tijd (\\(\\varphi\\), standaard genomen als 2,5). Deze past de secant elasticiteitsmodulus van beton (<em>E</em><em><sub>cm</sub></em>) als volgt aan, conform EN 1992-1-1, paragraaf 3.1.4 (3) resp. 7.4.3 (5):</p>\n<p>\\[E_{c,eff} = \\frac{E_{cm}}{1+\\varphi}\\]</p>\n<p>Bij het in rekening brengen van langetermijneffecten wordt eerst een belaststap met alle permanente belastingen berekend met de kruipcoëfficiënt (dus met de effectieve elasticiteitsmodulus van beton, <em>E</em><em><sub>c,eff</sub></em>), en daarna worden de aanvullende belastingen berekend zonder kruipcoëfficiënt (dus met <em>E</em><em><sub>cm</sub></em>). Om de kortetermijntoetsingen uit te voeren, wordt bovendien een aparte berekening uitgevoerd waarin alle belastingen zonder kruipcoëfficiënt worden berekend. Beide berekeningen voor de lange- en kortetermijntoetsingen zijn weergegeven in Fig. 26.</p>\n<p>Kruipfactoren worden door de gebruiker gedefinieerd in de materiaaleigenschappen en moeten worden berekend volgens EN 1992-1-1, Fig 3.1.</p>\n<h3>Wapening</h3>\n<p>Standaard wordt het geïdealiseerde bilineaire spanning-rekdiagram voor blanke wapeningsstaven gebruikt dat is gedefinieerd in EN 1992-1-1, paragraaf 3.2.7 (Fig. 27). Voor de definitie van dit diagram zijn tijdens de ontwerpfase alleen de basiseigenschappen van de wapening nodig (sterkte- en ductiliteitsklasse). Wanneer bekend, kan de werkelijke spanning-rekrelatie van de wapening (warmgewalst, koudvervormd, afgeschrikt en zelfontlaten, …) worden toegepast. Het spanning-rekdiagram van de wapening kan door de gebruiker worden gedefinieerd, maar in dat geval kan het tension stiffening effect niet worden aangehouden (de scheurwijdte kan dan niet worden berekend). Het gebruik van het spanning-rekdiagram met een horizontale bovenste tak maakt een toetsing van de constructieve duurzaamheid niet mogelijk. Daarom is een handmatige toetsing van de standaard ductiliteitseisen noodzakelijk.</p>\n<figure data-asset-id=\"ba3b27c3-ad63-46d8-b734-279c1a98639f\" data-image-id=\"ba3b27c3-ad63-46d8-b734-279c1a98639f\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/47fb26f0-9509-403c-ac42-7d68821d59d1/Steel%20stress-strain%20diagram%20CSFM.PNG\" data-asset-id=\"ba3b27c3-ad63-46d8-b734-279c1a98639f\" data-image-id=\"ba3b27c3-ad63-46d8-b734-279c1a98639f\" alt=\"\"></figure>\n<p><em>\\( \\textsf{\\textit{\\footnotesize{Fig. 27 \\qquad Stress-strain diagram of reinforcement: a) bilinear diagram with an inclined top branch; b) bilinear diagram}}}\\) \\( \\textsf{\\textit{\\footnotesize{with a horizontal top branch.}}}\\)</em></p>\n<p><br></p>\n<p>Tension stiffening (Fig. 28) wordt automatisch in rekening gebracht door de spanning-rekrelatie van de blanke wapeningsstaaf aan te passen, zodat de gemiddelde stijfheid van de in het beton ingebedde staven (ε<em><sub>m</sub></em>) wordt weergegeven.</p>\n<figure data-asset-id=\"4a23c310-98c5-488d-a3a0-2ec9064a2f61\" data-image-id=\"4a23c310-98c5-488d-a3a0-2ec9064a2f61\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/111ff130-8480-486a-adca-4c0068bcf66e/Tension%20stiffening%20CSFM.PNG\" data-asset-id=\"4a23c310-98c5-488d-a3a0-2ec9064a2f61\" data-image-id=\"4a23c310-98c5-488d-a3a0-2ec9064a2f61\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 28\\qquad Scheme of tension stiffening.}}}\\]</em></p>"
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"value": "<p>De Compatible Stress Field Method voldoet aan moderne normen. Omdat de rekenmodellen uitsluitend standaard materiaaleigenschappen gebruiken, kan het in de normen voorgeschreven formaat met partiële veiligheidsfactoren zonder aanpassing worden toegepast. Hierbij worden de invoerbelastingen vermenigvuldigd met factoren en worden de karakteristieke materiaaleigenschappen gereduceerd met de respectieve veiligheidscoëfficiënten uit de normen, precies zoals bij een conventionele betonberekening. De waarden van de materiaalveiligheidsfactoren die zijn voorgeschreven in EN 1992-1-1, hoofdstuk 2.4.2.4, zijn standaard ingesteld. De gebruiker kan de veiligheidsfactoren echter wijzigen in de Norm- en rekeninstellingen (Fig. 29).</p>\n<figure data-asset-id=\"7b26aa26-7ec4-4296-9296-645d3d6041b5\" data-image-id=\"7b26aa26-7ec4-4296-9296-645d3d6041b5\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/4cadae4a-9a8a-4f9b-935c-51395116ed4e/Material%20factors.png\" data-asset-id=\"7b26aa26-7ec4-4296-9296-645d3d6041b5\" data-image-id=\"7b26aa26-7ec4-4296-9296-645d3d6041b5\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 29\\qquad The setting of material safety factors in Idea StatiCa Detail.}}}\\]</em></p>\n<p><br></p>\n<p>De belastingveiligheidsfactoren moeten door de gebruiker worden gedefinieerd in de Combinatieregels voor elke niet-lineaire combinatie van belastingsgevallen (Fig. 30). Voor alle templates die in <a data-item-id=\"b4790cf9-a605-45b3-b41b-e36909ad4291\" href=\"\">Idea StatiCa Detail</a> zijn geïmplementeerd, zijn de partiële veiligheidsfactoren al vooraf gedefinieerd.</p>\n<figure data-asset-id=\"99632028-f378-4338-b74b-bef12aec3f6a\" data-image-id=\"99632028-f378-4338-b74b-bef12aec3f6a\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/2d2607d1-29e9-4dfd-80ef-db2ba7d172bf/Combination%20factors.png\" data-asset-id=\"99632028-f378-4338-b74b-bef12aec3f6a\" data-image-id=\"99632028-f378-4338-b74b-bef12aec3f6a\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 30\\qquad The setting of load partial factors in Idea StatiCa Detail.}}}\\]</em></p>\n<p><br></p>\n<p>Door geschikte, door de gebruiker gedefinieerde combinaties van partiële veiligheidsfactoren te gebruiken, kunnen gebruikers met de CSFM ook rekenen volgens de globale weerstandsfactormethode (Navrátil et al. 2017). Deze aanpak wordt in de ontwerppraktijk echter nauwelijks toegepast. Sommige richtlijnen bevelen de globale weerstandsfactormethode aan voor niet-lineaire berekeningen. Bij vereenvoudigde niet-lineaire berekeningen (zoals de CSFM), waarvoor alleen de materiaaleigenschappen nodig zijn die ook bij conventionele handberekeningen worden gebruikt, verdient het formaat met partiële veiligheidsfactoren nog steeds de voorkeur.</p>"
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"value": "<p>De verschillende toetsingen die EN 1992-1-1 vereist, worden beoordeeld op basis van de directe resultaten van het model. De UGT-toetsingen worden uitgevoerd voor de betonsterkte, de wapeningssterkte en de verankering (aanhechtingsschuifspanningen).</p>\n<p>De <strong>betonsterkte</strong> op druk wordt bepaald als de verhouding tussen de maximale hoofddrukspanning σ<em><sub>c </sub></em>= σ<em><sub>c</sub></em><sub>2</sub> uit de EEM-analyse en de grenswaarde σ<em><sub>c,lim</sub></em> = <em>f</em><em><sub>cd</sub></em>. </p>\n<p>De <strong>sterkte van de wapening</strong> wordt zowel op trek als op druk bepaald als de verhouding tussen de spanning in de wapening bij de scheuren σ<em><sub>sr</sub></em> en de opgegeven grenswaarde σ<em><sub>s,lim</sub></em>:</p>\n<p>\\(σ_{s,lim} = \\frac{k \\cdot f_{yk}}{γ_s}\\qquad\\qquad\\textsf{\\small{for bilinear diagram with inclined top branch}}\\)</p>\n<p>\\(σ_{s,lim} = \\frac{f_{yk}}{γ_s}\\qquad\\qquad\\,\\,\\,\\,\\textsf{\\small{for bilinear diagram with horizontal top branch}}\\)</p>\n<p>waarin:</p>\n<p><em>f</em><em><sub>yk</sub></em> de vloeigrens van de wapening volgens EN 1992-1-1 art. 3.2.3,</p>\n<p><em>k</em> de verhouding tussen de treksterkte <em>f</em><em><sub>tk</sub></em> en de vloeispanning,<br> \\(k = \\frac{f_{tk}}{f_{yk}}\\)</p>\n<p><em>γ</em><em><sub>s </sub></em><sub> </sub>de partiële veiligheidsfactor voor wapening is</p>\n<p>De <strong>aanhechtingsschuifspanning</strong> wordt afzonderlijk bepaald als de verhouding tussen de aanhechtingsspanning τ<em><sub>b</sub></em>, berekend met de EEM-analyse, en de uiterste aanhechtingssterkte <em>f</em><em><sub>bd</sub></em><sub>,</sub> volgens EN 1992-1-1 hoofdstuk 8.4.2:</p>\n<p>\\[\\frac{τ_{b}}{f_{bd}}\\]</p>\n<p>\\[f_{bd} = 2.25 \\cdot η_1\\cdot η_2\\cdot f_{ctd}\\]</p>\n<p>waarin:</p>\n<p><em>f</em><em><sub>ctd</sub></em><sub> </sub> de rekenwaarde van de betontreksterkte is volgens EN 1992-1-1 art. 3.1.6 (2). Vanwege de toenemende brosheid van beton met hogere sterkte wordt <em>f</em><em><sub>ctk,0.05</sub></em><sub> </sub>begrensd tot de waarde voor C60/75 volgens EN 1992-1-1 art. 8.4.2 (2)</p>\n<p>η<sub>1</sub> een coëfficiënt is die samenhangt met de kwaliteit van de aanhechtingsomstandigheden en de positie van de staaf tijdens het storten (Fig. 31).</p>\n<p>η<sub>1</sub> = 1,0 bij ‘goede’ omstandigheden en</p>\n<p>η<sub>1</sub> = 0,7 in alle andere gevallen en voor staven in constructieve elementen die met glijbekisting zijn vervaardigd, tenzij kan worden aangetoond dat ‘goede’ aanhechtingsomstandigheden bestaan</p>\n<p>η<sub>2</sub> samenhangt met de staafdiameter:</p>\n<p> η<sub>2</sub> = 1,0 voor Ø ≤ 32 mm</p>\n<p> η<sub>2</sub> = (132 - Ø)/100 voor Ø > 32 mm</p>\n<figure data-asset-id=\"c6ca9e31-4172-4034-a8b0-cdb2ad98d82a\" data-image-id=\"c6ca9e31-4172-4034-a8b0-cdb2ad98d82a\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/7aa307dc-3cd6-4d42-8dd8-d0ff97994677/Bond%20conditions.PNG\" data-asset-id=\"c6ca9e31-4172-4034-a8b0-cdb2ad98d82a\" data-image-id=\"c6ca9e31-4172-4034-a8b0-cdb2ad98d82a\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 31\\qquad EN 1992-1-1 Figure 8.2 - Description of bond conditions.}}}\\]</em></p>\n<p>In IDEA StatiCa Detail worden de aanhechtingsomstandigheden meegenomen volgens Fig. 31 c) en d). De stortrichting kan in de applicatie voor elk projectitem als volgt worden ingesteld.</p>\n<figure data-asset-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\" data-image-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/e00845bc-3d60-4315-a8b3-67d4a52666a4/Direction%20of%20concreting.png\" data-asset-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\" data-image-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\" alt=\"\"></figure>\n<p>Deze toetsingen worden uitgevoerd ten opzichte van de passende grenswaarden voor de betreffende delen van de constructie (d.w.z. ondanks dat er voor zowel het beton als het wapeningsmateriaal één enkele sterkteklasse geldt, verschillen de uiteindelijke spanning-rek-diagrammen per deel van de constructie door de effecten van tension stiffening en compression softening).</p>\n<p>Het is ook mogelijk om <strong>gladde wapeningsstaven</strong> te modelleren. Meer informatie vindt u hier: <a data-item-id=\"182f8ba8-899b-44fc-a1c7-59d562ef8c6c\" href=\"\">Gladde wapeningsstaven in Detail</a></p>\n<p><strong>Totale kracht </strong><em><strong>F</strong></em><em><strong><sub>tot</sub></strong></em><strong> en grenskracht </strong><em><strong>F</strong></em><em><strong><sub>lim</sub></strong></em></p>\n<p>De totale kracht <em><strong>F</strong></em><em><strong><sub>tot</sub></strong></em> is een resultaat van de eindige-elementenanalyse en kan op twee manieren worden bepaald.</p>\n<p>\\[F_{tot}=A_{s}\\cdot \\sigma_{s}\\]</p>\n<p>waarin <em>A</em><em><sub>s</sub></em> de doorsnede van de wapeningsstaaf is en <em>σ</em><em><sub>s</sub></em> de spanning in de staaf.</p>\n<p>Of als de som van de verankeringskracht <em>F</em><em><sub>a </sub></em>en de aanhechtingskracht <em>F</em><em><sub>bond</sub></em><em>.</em></p>\n<p>\\[F_{tot}=F_{a}+F_{bond}\\]</p>\n<p>waarin <em>F</em><em><sub>a</sub></em> de werkelijke kracht in de verankeringsveer is en <em>F</em><em><sub>bond</sub></em> de aanhechtingskracht, die kan worden verkregen door de aanhechtingsspanning <em>τ</em><em><sub>b</sub></em> te integreren over de lengte van de wapeningsstaaf <em>l.</em></p>\n<p>\\[F_{bond}=C_{s} \\cdot \\int_{0}^{l}\\tau_{b}\\left( x \\right)dx\\]</p>\n<p>C<sub>s</sub> is de omtrek van de wapeningsstaaf.</p>\n<p>De grenskracht <em><strong>F</strong></em><em><strong><sub>lim</sub></strong></em> is de maximale kracht in het element van de wapeningsstaaf, rekening houdend met de <strong>uiterste sterkte</strong> van de wapeningsstaaf en met de <strong>verankeringsomstandigheden </strong>(aanhechting tussen beton en wapening en verankeringshaken, lussen, enz.).</p>\n<p>\\[F_{lim}=min\\left( F_{lim,bond}+F_{au},F_{u} \\right)\\]</p>\n<p>\\[F_{u}=k\\cdot f_{yd}\\cdot A_{s}\\]</p>\n<p>\\[F_{au}=\\beta\\cdot k\\cdot f_{yd}\\cdot A_{s}\\]</p>\n<p>\\[F_{lim,bond}=C_{s}\\cdot l \\cdot f_{bd}\\]</p>\n<p>waarin C<sub>s</sub> de omtrek van de wapeningsstaaf is en <em>l</em> de lengte vanaf het begin van de wapeningsstaaf tot het beschouwde punt.</p>\n<figure data-asset-id=\"d3675eaf-0adb-4512-9366-58e4bdf171b1\" data-image-id=\"d3675eaf-0adb-4512-9366-58e4bdf171b1\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/1a6bbdca-e56b-47e1-a85f-00d4317689a8/Flim.png\" data-asset-id=\"d3675eaf-0adb-4512-9366-58e4bdf171b1\" data-image-id=\"d3675eaf-0adb-4512-9366-58e4bdf171b1\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 32\\qquad Definition of the limit force Flim}}}\\]</em></p>\n<p><br></p>\n<p>\\[F_{lim,2}=F_{lim,1}+F_{lim,add}\\]</p>\n<p>waarin <em>F</em><em><sub>lim,add</sub></em> de aanvullende kracht is, berekend uit de grootte van de hoek tussen naburige elementen. <em>F</em><em><sub>lim,2</sub></em> moet altijd lager zijn dan <em>F</em><em><sub>u</sub></em>.</p>\n<p><br></p>\n<p>De beschikbare <strong>verankeringstypen</strong> in de CSFM omvatten een rechte staaf (d.w.z. zonder reductie van het ankereinde), bocht, haak, lus, aangelaste dwarsstaaf, perfecte aanhechting en doorgaande staaf. Al deze typen, samen met de bijbehorende verankeringscoëfficiënten β, zijn weergegeven in Fig. 32 voor langswapening en in Fig. 33 voor beugels. De waarden van de toegepaste verankeringscoëfficiënten zijn in overeenstemming met EN 1992-1-1 paragraaf 8.4.4 Tab. 8.2. Opgemerkt moet worden dat de CSFM, ondanks de verschillende beschikbare opties, drie typen verankeringseinden onderscheidt: (i) geen reductie van de verankeringslengte, (ii) een reductie van 30 % van de verankeringslengte bij een genormaliseerde verankering en (iii) perfecte aanhechting.</p>\n<figure data-asset-id=\"a4b32213-4a43-4c1d-a3c3-21d42d5dfbad\" data-image-id=\"a4b32213-4a43-4c1d-a3c3-21d42d5dfbad\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/b16975dc-aeea-4e7e-bfc7-23a8f8b28c7e/Available%20anchorage%20types%20for%20longitudinal%20rebars.png\" data-asset-id=\"a4b32213-4a43-4c1d-a3c3-21d42d5dfbad\" data-image-id=\"a4b32213-4a43-4c1d-a3c3-21d42d5dfbad\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 33\\qquad Available anchorage types and respective anchorage coefficients for longitudinal reinforcing bars in the CSFM:}}}\\]</em></p>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{(a) straight bar; (b) bend; (c) hook; (d) loop; (e) welded transverse bar; (f) perfect bond; (g) continuous bar.}}}\\]</em></p>\n<p><br></p>\n<figure data-asset-id=\"ec5159ea-3a7f-43fa-a807-a217b79d6cc9\" data-image-id=\"ec5159ea-3a7f-43fa-a807-a217b79d6cc9\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/86ffb525-5912-4a7f-9576-fff17481b7a1/Available%20anchorage%20types%20for%20stirrups.png\" data-asset-id=\"ec5159ea-3a7f-43fa-a807-a217b79d6cc9\" data-image-id=\"ec5159ea-3a7f-43fa-a807-a217b79d6cc9\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 33\\qquad Available anchorage types and respective anchorage coefficients for stirrups.}}}\\]</em></p>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Closed stirrups: (a) hook; (b) bend; (c) overlap. Open stirrups: (d) hook; (e) continuous bar.}}}\\]</em></p>\n<p>Om aan EN 1992-1-1 te voldoen, moet de verankeringsveer in de berekening worden gebruikt. De verankeringsveer wordt aangepast met de coëfficiënt β, zodat de gebruiker bij het definiëren van de begin- en eindvoorwaarden van de wapening een van de beschikbare verankeringstypen moet kiezen. </p>"
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"value": "<p>Bij het ontwerpen van betonconstructies komen we twee grote groepen gedeeltelijk belaste oppervlakken (PLA) tegen. De eerste groep omvat oplegingen, de tweede groep verankeringsgebieden. Volgens de huidige normen voor het ontwerp van gewapende betonconstructies, EN 1992-1-1 hoofdstuk 6.7 (<em>Fig. 34</em>), moeten bij gedeeltelijk belaste oppervlakken het lokaal verbrijzelen van beton en dwarse trekkrachten worden beschouwd. Bij een gelijkmatig verdeelde belasting op een oppervlak, <em>A</em><em><sub>c0</sub></em>, mag de drukcapaciteit van het beton tot maximaal drie keer worden verhoogd, afhankelijk van het rekenkundige verdelingsoppervlak <em>A</em><em><sub>c1.</sub></em></p>\n<figure data-asset-id=\"d2ebd9b3-ebcd-4cb6-a090-4b0a9a1d2566\" data-image-id=\"d2ebd9b3-ebcd-4cb6-a090-4b0a9a1d2566\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/94ecb791-703a-44b7-8665-2f1526a20c1e/Partially%20loaded%20areas%20EC.PNG\" data-asset-id=\"d2ebd9b3-ebcd-4cb6-a090-4b0a9a1d2566\" data-image-id=\"d2ebd9b3-ebcd-4cb6-a090-4b0a9a1d2566\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 34\\qquad Partially loaded areas according to EN 1992-1-1.}}}\\]</em></p>\n<p>Het gedeeltelijk belaste oppervlak moet voldoende worden voorzien van dwarswapening, ontworpen om de optredende splijtkrachten over te dragen. Voor het ontwerp van de dwarswapening in gedeeltelijk belaste oppervlakken wordt volgens de Eurocode de Staafwerk-methode gebruikt. Zonder de vereiste dwarswapening is het niet toegestaan om een verhoging van de drukcapaciteit van het beton in rekening te brengen.</p>\n<p><br></p>\n<p><strong>Gedeeltelijk belaste oppervlakken in de CSFM</strong></p>\n<figure data-asset-id=\"77fdebe4-afac-4ee7-aee5-716984d4e6d3\" data-image-id=\"77fdebe4-afac-4ee7-aee5-716984d4e6d3\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/3dcea2b1-7700-46f3-a938-4c08204d52e8/Fictitious%20struts.PNG\" data-asset-id=\"77fdebe4-afac-4ee7-aee5-716984d4e6d3\" data-image-id=\"77fdebe4-afac-4ee7-aee5-716984d4e6d3\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 35\\qquad Fictitious struts with concrete finite element mesh.}}}\\]</em></p>\n<p>Met de CSFM is het mogelijk om gewapende betonconstructies te ontwerpen en te toetsen, waarbij ook de invloed van de toenemende drukweerstand van beton in gedeeltelijk belaste oppervlakken wordt meegenomen. Omdat de CSFM een wandmodel (2D) is en gedeeltelijk belaste oppervlakken een ruimtelijk (3D) vraagstuk vormen, was het nodig een oplossing te vinden die deze twee verschillende typen vraagstukken combineert (<em>Fig. 35</em>). Als de functie \"gedeeltelijk belaste oppervlakken\" is geactiveerd, wordt de toelaatbare kegelgeometrie opgebouwd volgens de Eurocode (<em>Fig. 34</em>). Alle geometrische botsingen worden volledig in 3D opgelost voor de opgegeven geometrie van de betonnen staaf en de afmetingen van elk PLA. Vervolgens wordt een rekenmodel van het gedeeltelijk belaste oppervlak gegenereerd.</p>\n<figure data-asset-id=\"05c2e193-bc14-42b5-bc07-da8610febda8\" data-image-id=\"05c2e193-bc14-42b5-bc07-da8610febda8\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/6ae87bd2-682b-4b92-ab1f-4b12e9d3a0df/Cone%20geometry.png\" data-asset-id=\"05c2e193-bc14-42b5-bc07-da8610febda8\" data-image-id=\"05c2e193-bc14-42b5-bc07-da8610febda8\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 36\\qquad Allowable cone geometries.}}}\\]</em></p>\n<p>Het aanpassen van het materiaalmodel bleek een ongeschikte aanpak, vooral omdat het toewijzen van eigenschappen aan het eindige-elementenmesh problematisch is. Vastgesteld werd dat een aanpak die onafhankelijk is van het eindige-elementenmesh een geschiktere oplossing is. Voor de bekende geometrie van de drukkegel worden volledig samenhangende fictieve drukdiagonalen gegenereerd (<em>Fig. 35</em> <em>en Fig. 37</em>). Deze drukdiagonalen hebben dezelfde materiaaleigenschappen als het beton dat in het model wordt gebruikt, inclusief het spanning-rek diagram. De vorm van de kegel bepaalt de richting van de drukdiagonalen, die de belasting geleidelijk vanaf het PLA over het rekenkundige verdelingsoppervlak spreiden. De oppervlaktedichtheid van de fictieve drukdiagonalen varieert in elk deel van de kegel en voegt een fictief betonoppervlak toe in de belastingsrichting. Op het niveau van het belaste oppervlak (<em>A</em><em><sub>c0</sub></em>) wordt een fictief betonoppervlak toegevoegd volgens de verhouding \\(\\sqrt{A_{c0} \\cdot A_{c1}} - A_{real}\\) (waarbij <em>A</em><em><sub>real</sub></em> het oppervlak van de oplegging is zoals aangenomen in het 2D-rekenmodel). Dit oppervlak neemt lineair af tot nul in de richting van het rekenkundige verdelingsoppervlak (<em>A</em><em><sub>c1</sub></em>). Deze oplossing zorgt ervoor dat de drukspanning in het beton constant is over het gehele kegelvolume.</p>\n<figure data-asset-id=\"47a5fe4b-0b51-4d87-a9cd-8e59e61835e4\" data-image-id=\"47a5fe4b-0b51-4d87-a9cd-8e59e61835e4\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/c4ff37a9-9d49-493b-946e-f048713b05cf/Partially%20loaded%20areas.PNG\" data-asset-id=\"47a5fe4b-0b51-4d87-a9cd-8e59e61835e4\" data-image-id=\"47a5fe4b-0b51-4d87-a9cd-8e59e61835e4\" alt=\"\"></figure>\n<p>\\[\\rho \\left( {\\beta ,z} \\right) = \\left( {\\sqrt {\\frac{A_{c1}}{A_{c0}}} - \\frac{A_{real}}{A_{c0}}} \\right)\\,\\cdot\\,\\left( {1 - \\frac{z}{h}} \\right)\\,\\cdot\\,\\frac{1}{{\\cos \\beta }}\\]</p>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 37\\qquad Fictitious struts in the computational model}}}\\]</em></p>\n<p>De weerstand van het gedeeltelijk belaste oppervlak wordt verhoogd volgens de verhouding tussen het rekenkundige verdelingsoppervlak en het belaste oppervlak, zoals vastgelegd in EN 1992-1-1 (6.7). Houd er rekening mee dat dit een rekenmodel is dat de spanningstoestand over een gedeeltelijk belast oppervlak, waarvan de werkelijke krachtsafdracht veel complexer is, niet exact kan beschrijven. Deze oplossing maakt echter een correcte verdeling van de belasting over het hele model mogelijk, met inachtneming van de verhoogde draagcapaciteit van het gedeeltelijk belaste oppervlak. Bovendien worden hiermee de dwarsspanningen in dit gebied correct geïntroduceerd.</p>\n<p>Wanneer de functie Gedeeltelijk belaste oppervlakken wordt gebruikt om de toename van de drukcapaciteit van beton te simuleren, moet de normtoetsing afzonderlijk worden uitgevoerd volgens EN 1992-1-1, paragraaf 6.7 (2). De dwarse trekkrachten (splijtkrachten) die door de wapening worden overgedragen, worden automatisch getoetst.</p>"
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"value": "<h3>Beton - Sterkte</h3>\n<p>Het betonmodel dat voor sterkteberekeningen in CSFM is geïmplementeerd, is gebaseerd op de parabolisch-plastische spanning-rek-curve voor beton van de Portland Cement Association, beschreven in PCA's Notes on ACI 318-99 Building Code Requirements for Structural Concrete, Figure 6-8. De treksterkte wordt verwaarloosd, zoals gebruikelijk is bij het klassieke ontwerp van gewapend beton.</p>\n<figure data-asset-id=\"a84d11ec-b1f2-431e-afad-b6e1b7e8a83c\" data-image-id=\"a84d11ec-b1f2-431e-afad-b6e1b7e8a83c\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/f578dd02-9167-45e0-b80f-4a1331dfe20a/Concrete%20stress-strain%20diagram%20CSFM%20-%20ACI.png\" data-asset-id=\"a84d11ec-b1f2-431e-afad-b6e1b7e8a83c\" data-image-id=\"a84d11ec-b1f2-431e-afad-b6e1b7e8a83c\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 38\\qquad The stress-strain diagram of concrete for Strength analysis}}}\\]</em></p>\n<p>De implementatie van CSFM in <em>IDEA StatiCa Detail</em> hanteert geen expliciet bezwijkcriterium in termen van rek voor beton onder druk (na het bereiken van de piekspanning wordt een plastische tak aangehouden met ε<em><sub>c</sub></em><sub>0</sub> met een maximumwaarde van 5%, terwijl ACI 318-19 Cl. 22.2.2.1 uitgaat van een uiterste rek van minder dan 0,3%). Door deze vereenvoudiging kan het vervormingsvermogen van constructies die door druk bezwijken niet worden geverifieerd. De sterkte wordt echter correct voorspeld wanneer, naast de factor voor gescheurd beton (<em>k</em><em><sub>c</sub></em><sub>2</sub>, gedefinieerd in (Fig. 39)), ook de toename van de brosheid van beton bij hogere sterkte in rekening wordt gebracht met de reductiefactor <em>\\(\\eta_{fc}\\)</em> uit de <em>fib</em> Model Code 2010, als volgt:</p>\n<p>\\[f'_{c,lim}=\\alpha_{1}\\cdot\\phi_{c}\\cdot k_{c}\\cdot f'_{c}\\]</p>\n<p>\\[k_{c}=\\eta_{fc}\\cdot k_{c2}\\]</p>\n<p>\\[{\\eta _{fc}} = {\\left( {\\frac{{30}}{{{f'_{c}}}}} \\right)^{\\frac{1}{3}}} \\le 1\\]</p>\n<p>waarbij:</p>\n<p><em>α</em><sub>1</sub> de reductiefactor is voor de drukspanning van beton, gedefinieerd in ACI 318-19 Cl. 22.2.2.4.1. Bij gebruik van een parabool-rechthoek spanning-rek-diagram moet de maximale drukspanning met deze factor worden verlaagd. Hierdoor wordt de spanningsverdeling in de drukzone zodanig gemiddeld dat de resulterende druksterkte kleiner is dan of gelijk is aan de druksterkte berekend met een spanning-rek-diagram met een afnemende plastische tak<em>.</em></p>\n<p><em>Φ</em><em><sub>c </sub></em>de sterktereductiefactor voor beton is. De standaardwaarde is ingesteld volgens ACI 318-19 Table 24.2.1 (b)(f).</p>\n<p><em>k</em><em><sub>c</sub></em><sub>2</sub> de reductiefactor is vanwege de aanwezigheid van dwarsscheuren.</p>\n<p><em>f'</em><em><sub>c</sub></em> de cilindersterkte van het beton is (in MPa voor de definitie van <em>\\( \\eta_{fc} \\)</em>).</p>\n<figure data-asset-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\" data-image-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/085222c7-055a-4870-9bcb-8f18bd65620f/Compression%20softening%20CSFM.PNG\" data-asset-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\" data-image-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 39\\qquad The compression softening law.}}}\\]</em></p>\n<p><em>k</em><em><sub>c</sub></em><sub>2</sub> is een reductiefactor die gebaseerd is op dezelfde uitgangspunten als de knoopzonecoëfficiënt <em>β</em><em><sub>n</sub></em> uit ACI 318-19 Table 23.9.2, met dien verstande dat in CSFM de aanwezigheid van een hoofdtrekspanning loodrecht op de hoofddrukspanning voor elk eindig element wordt getoetst (en niet alleen voor de knopen van het Staafwerkmodel).</p>\n<h3>Beton – Bruikbaarheid</h3>\n<p>De bruikbaarheidsanalyse bevat enkele vereenvoudigingen van de constitutieve modellen die voor de sterkteanalyse worden gebruikt. De plastische tak van de spanning-rek-curve van beton onder druk wordt buiten beschouwing gelaten, terwijl de elastische tak lineair en oneindig is. De compression softening law wordt niet in rekening gebracht. Deze vereenvoudigingen verbeteren de numerieke stabiliteit en de rekensnelheid en verminderen de algemeenheid van de oplossing niet, zolang de resulterende spanningsgrenzen van het materiaal in de bruikbaarheidsgrenstoestand duidelijk onder de vloeigrens blijven (zoals door ACI wordt vereist). De vereenvoudigde modellen voor de bruikbaarheid zijn daarom alleen geldig als aan alle toetsingseisen is voldaan.</p>\n<figure data-asset-id=\"0d015331-6ce6-4a70-b087-58766f33e248\" data-image-id=\"0d015331-6ce6-4a70-b087-58766f33e248\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/07b977ad-1511-48d6-b96e-12b3c67bb3b9/Concrete%20stress-strain%20for%20serviceability%20-%20ACI.png\" data-asset-id=\"0d015331-6ce6-4a70-b087-58766f33e248\" data-image-id=\"0d015331-6ce6-4a70-b087-58766f33e248\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 40\\qquad Concrete stress-strain diagrams implemented for serviceability analysis: short- and long-term verifications.}}}\\]</em></p>\n<p><br></p>\n<p><strong>Langetermijneffecten</strong></p>\n<p>Het langetermijngedrag van de constructie, zoals doorbuigingen op lange termijn of de berekening van scheurwijdten als gevolg van langdurige belastingen, wordt beïnvloed door de kruip van beton. ACI 318-19 definieert in paragraaf 24.2.4.1.3 de tijdsafhankelijke factor voor langdurige belastingen – ξ, die het kruipeffect weergeeft voor een gespecificeerde duur van de langdurige belasting.</p>\n<p>In de Detail applicatie wordt de elasticiteitsmodulus <em>E</em><em><sub>c</sub></em> aangepast om het langetermijngedrag van de constructie te bepalen met behulp van de factor ξ. De aangepaste elasticiteitsmodulus wordt aangeduid als <em>E</em><em><sub>c,eff</sub></em> – zie Figure 40.</p>\n<p>Als we aannemen dat de vervorming van het element wordt uitgedrukt als rek, kan worden geschreven:</p>\n<p>\\[\\epsilon_{tot} = \\epsilon_{0} + \\epsilon_{creep} = \\epsilon_{0} \\cdot (1+\\xi)\\]</p>\n<p>waarbij:</p>\n<p><em>ε</em><em><sub>0</sub></em> de rek op korte termijn is (zonder invloed van kruip) en <em>ε</em><em><sub>creep</sub></em> de rek als gevolg van kruip.</p>\n<p>Met de wet van Hooke kunnen we schrijven:</p>\n<p>\\[E_{c,eff} = \\frac{f_{c}}{\\epsilon_{tot}}\\]</p>\n<p>Door \\(\\epsilon_{tot} = \\epsilon_{0} \\cdot (1+\\xi)\\) en \\(\\epsilon_{0} = f_{c} / E_{c}\\) te substitueren, krijgen we:</p>\n<p>\\[E_{c,eff} = \\frac{E_{c}}{1+\\xi}\\]</p>\n<p>De duur van de langdurige belasting voor het bepalen van de factor ξ kan voor elke langetermijncombinatie in de bruikbaarheidsgrenstoestand afzonderlijk worden ingesteld.</p>\n<figure data-asset-id=\"f5a1e9f7-76c9-4bdf-9d6b-a28ade763397\" data-image-id=\"f5a1e9f7-76c9-4bdf-9d6b-a28ade763397\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/1bb4b6d8-065d-4c52-a7e0-66ed3c01281f/Sustained%20load%20duration%20-%20ACI.png\" data-asset-id=\"f5a1e9f7-76c9-4bdf-9d6b-a28ade763397\" data-image-id=\"f5a1e9f7-76c9-4bdf-9d6b-a28ade763397\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 41\\qquad Sustained load duration}}}\\]</em></p>\n<p>De tijdsafhankelijke doorbuigingen, spanningen en scheurwijdten worden vervolgens berekend met een aangepast materiaalmodel, waarin het effect van drukverfijning automatisch wordt meegenomen door de aard van de eindige-elementenanalyse. Het is daarom niet nodig deze verder te vermenigvuldigen met de factor die in 24.2.4.1.1 is gedefinieerd.</p>\n<p><strong>Korte-termijneffecten</strong></p>\n<p>Om toetsingen op korte termijn uit te voeren, wordt een tweede berekening gemaakt waarin alle belastingen zonder de tijdsafhankelijke factor voor langdurige belastingen worden berekend. Beide berekeningen, voor toetsingen op lange en korte termijn, zijn weergegeven in Fig. 40.</p>\n<h3>Wapening</h3>\n<p>Voor de niet-voorgespannen wapening wordt een perfect elastisch-plastisch spanning-rek-diagram met een gedefinieerde vloeigrens aangehouden, zie ACI 319-19 CL. 20.2.1. Voor de definitie van dit diagram hoeven alleen de basiseigenschappen van de wapening bekend te zijn – de sterkte en de elasticiteitsmodulus.</p>\n<p>Het spanning-rek-diagram van de wapening kan ook door de gebruiker worden gedefinieerd, maar in dat geval is het niet mogelijk het tension stiffening effect in rekening te brengen (het is niet mogelijk de scheurwijdte te berekenen). </p>\n<figure data-asset-id=\"2d9c6401-28af-4bfe-bc92-1d6f830f7c93\" data-image-id=\"2d9c6401-28af-4bfe-bc92-1d6f830f7c93\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/77dadff9-85d4-402e-94e5-a3725f908933/Steel%20stress-strain%20diagram%20CSFM%20-%20ACI.png\" data-asset-id=\"2d9c6401-28af-4bfe-bc92-1d6f830f7c93\" data-image-id=\"2d9c6401-28af-4bfe-bc92-1d6f830f7c93\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 42 \\qquad Stress-strain diagram of reinforcement}}}\\]</em></p>\n<p>waarbij:</p>\n<p><em>Φ</em><em><sub>s </sub></em>de sterktereductiefactor voor wapening is. De standaardwaarde is ingesteld volgens ACI 318-19 Table 24.2.1.</p>\n<p><em>f</em><em><sub>y</sub></em> de vloeigrens van de wapening is</p>\n<p><em>E</em><em><sub>s</sub></em> de elasticiteitsmodulus van de wapening is</p>\n<p>Als grensrek waarbij de berekening wordt gestopt, is 10% gekozen. Dit wordt als veilig beschouwd op basis van ASTM A955/A955M-20c Article 7.</p>\n<p>Tension stiffening (Fig. 43) wordt automatisch in rekening gebracht door de invoer van de spanning-rek-relatie van de blote wapeningsstaaf aan te passen, zodat de gemiddelde stijfheid van de in het beton ingebedde staven (ε<em><sub>m</sub></em>) wordt weergegeven.</p>\n<figure data-asset-id=\"c9add949-2ad5-4922-8e6c-0d75fb47cb70\" data-image-id=\"c9add949-2ad5-4922-8e6c-0d75fb47cb70\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/c045fcb6-32c6-4a92-aa15-24530fb11484/Tension%20stiffening%20CSFM%20-%20ACI.png\" data-asset-id=\"c9add949-2ad5-4922-8e6c-0d75fb47cb70\" data-image-id=\"c9add949-2ad5-4922-8e6c-0d75fb47cb70\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 43\\qquad Scheme of tension stiffening.}}}\\]</em></p>"
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"value": "<p>De Compatible Stress Field Method voldoet aan moderne normen. Omdat de rekenmodellen alleen standaard materiaaleigenschappen gebruiken, kan het in de normen voorgeschreven formaat met partiële veiligheidsfactoren zonder aanpassing worden toegepast. Op deze manier worden de invoerbelastingen met factoren vermenigvuldigd en de karakteristieke materiaaleigenschappen gereduceerd met de bijbehorende reductiefactoren voor sterkte, precies zoals bij een conventionele betonberekening.</p>\n<p>De waarden van de <strong>reductiefactoren voor sterkte</strong> zijn voorgeschreven in ACI 318-19 Cl. 21.2. De standaardwaarden voor beton en wapening zijn gekozen op basis van de aanname dat het typische voorbeeld in de applicatie door afschuiving wordt bepaald (conform Tabel 21.2.1 (b), (f), (g)). Het is echter mogelijk om elk type element te modelleren. Als een element wordt getoetst dat door druk of trek wordt bepaald, kan de gebruiker daarom de waarde van de reductiefactor voor sterkte wijzigen in de Voorkeuren.</p>\n<figure data-asset-id=\"1fa1394b-aa7d-4e35-ba1b-74d51ffa7f89\" data-image-id=\"1fa1394b-aa7d-4e35-ba1b-74d51ffa7f89\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/7f5c8c73-4050-4623-9f74-04bee16498f2/Strength%20reduction%20factors%20-%20ACI.png\" data-asset-id=\"1fa1394b-aa7d-4e35-ba1b-74d51ffa7f89\" data-image-id=\"1fa1394b-aa7d-4e35-ba1b-74d51ffa7f89\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 44\\qquad The setting of strength reduction factors in IDEA StatiCa Detail.}}}\\]</em></p>\n<p><br></p>\n<p><strong>Belastingfactoren</strong> voor sterktecombinaties moeten worden bepaald volgens ACI 318-19 Tabel 5.3.1.</p>\n<p>Behalve zoals vermeld in Hoofdstuk 34 zijn belastingcombinaties voor de bruikbaarheidsgrenstoestand niet gedefinieerd in ACI 318-19. Aanbevolen wordt om combinatieregels te gebruiken op basis van Appendix C van ASCE/SEI 7-16. Voor alle templates zijn de belastingfactoren al vooraf ingesteld.</p>\n<figure data-asset-id=\"fe8369c9-e929-4d00-b389-fa2c8d9c0cca\" data-image-id=\"fe8369c9-e929-4d00-b389-fa2c8d9c0cca\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/db9f1517-72eb-45bd-9f0c-6c748d7c9146/Load%20factors%20-%20ACI.png\" data-asset-id=\"fe8369c9-e929-4d00-b389-fa2c8d9c0cca\" data-image-id=\"fe8369c9-e929-4d00-b389-fa2c8d9c0cca\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 45\\qquad The setting of load factors in Idea StatiCa Detail.}}}\\]</em></p>"
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"value": "<p>De verschillende verificaties die ACI 318-19 vereist, worden getoetst op basis van de directe resultaten van het model. De verificaties worden uitgevoerd voor de betonsterkte, de wapeningssterkte en de verankering (afschuifspanningen van de aanhechting).</p>\n<p>De <strong>betonsterkte</strong> op druk wordt bepaald als de verhouding tussen de maximale hoofddrukspanning <em>f</em><em><sub>c</sub></em> (ook σ<sub>2</sub> in Hulpresultaten) uit de EM-analyse en de grenswaarde <em>f'</em><em><sub>c,lim</sub></em>.</p>\n<p>De <strong>sterkte van de wapening</strong> wordt zowel op trek als op druk bepaald als de verhouding tussen de spanning in de wapening ter plaatse van de scheuren <em>f</em><em><sub>s</sub></em> en de gespecificeerde grenswaarde <em>f</em><em><sub>y,lim</sub></em>.</p>\n<p>De <strong>afschuifspanning van de aanhechting</strong> wordt afzonderlijk bepaald als de verhouding tussen de aanhechtspanning τ<em><sub>b</sub></em>, berekend met de EM-analyse, en de aanhechtsterkte <em>f</em><em><sub>bu</sub></em>.</p>\n<p>De ACI-norm behandelt de aanhechtsterkte echter niet expliciet, maar werkt met de berekening van de zogenaamde verankeringslengte, die in paragraaf 25.4.2 wordt beschreven. Omdat de aanhechtsterkte een fundamentele invoerparameter is voor het bepalen van de verankeringslengte, zie R25.4.1.1 en ACI Committee 408 1966, kan de aanhechtsterkte als volgt worden berekend:</p>\n<p>Stel dat een wapeningsstaaf in een betonblok wordt verankerd over de verankeringslengte <em>l</em><em><sub>d</sub></em> of meer. Het uittrekken van de wapening leidt dan tot breuk van de wapening en niet tot het uittrekken uit het beton. Dit kan met de volgende formule worden uitgedrukt.</p>\n<p>\\[\\pi\\cdot d_{b} \\cdot l_{d} \\cdot f_{bu}=f_{y}\\cdot A_{s}\\]</p>\n<p>waarin:</p>\n<p><em>d</em><em><sub>b</sub></em> de diameter van de wapeningsstaaf is, <em>l</em><em><sub>d</sub></em> de verankeringslengte, <em>f</em><em><sub>bu</sub></em> de aanhechtsterkte, <em>f</em><em><sub>y</sub></em> de vloeigrens van de wapening en <em>A</em><em><sub>s</sub></em> het oppervlak van de wapeningsstaaf.</p>\n<p>Hieruit kan eenvoudig de formule voor de aanhechtsterkte worden afgeleid:</p>\n<p>\\[f_{bu}=\\frac{f_{y}\\cdot A_{s}}{\\pi\\cdot d_{b} \\cdot l_{d} }\\]</p>\n<p>De verankeringslengte <em>l</em><em><sub>d</sub></em> wordt vervolgens bepaald volgens ACI 318-19 Tabel 25.4.2.3 als volgt:</p>\n<p>\\[l_{d}=\\left( \\frac{f_{y}\\cdot\\psi_{t}\\cdot\\psi_{e}\\cdot\\psi_{g}}{C\\cdot\\lambda\\sqrt{f'_{c}}} \\right)\\cdot d_{b}\\]</p>\n<p>waarin:</p>\n<p><em>C = 25</em> (2,1 voor metrisch) voor staven nr. 6 en kleiner en geprofileerde draden, <em>C = 20</em> (1,7 voor metrisch) voor staven nr. 7 en groter, λ = 1,0 voor beton met normaal gewicht, en <em>ψ</em><em><sub>t</sub></em>, <em>ψ</em><em><sub>e</sub></em><sub>,</sub> <em>ψ</em><em><sub>g</sub></em> worden bepaald volgens ACI 318-19 Tabel 25.4.2.3. </p>\n<p>Alleen onbehandelde of verzinkte wapening wordt ondersteund, dus <em>ψ</em><em><sub>e</sub></em><em> = 1,0</em>. <em>ψ</em><em><sub>g</sub></em> wordt automatisch bepaald op basis van de wapeningskwaliteit en <em>ψ</em><em><sub>t</sub></em> wordt automatisch afgeleid uit de positie van de wapening in het model en uit de stortrichting, die in de applicatie voor elk projectonderdeel als volgt kan worden ingesteld.</p>\n<figure data-asset-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\" data-image-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/e00845bc-3d60-4315-a8b3-67d4a52666a4/Direction%20of%20concreting.png\" data-asset-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\" data-image-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 46\\qquad Direction of concreting}}}\\]</em></p>\n<p>Deze verificaties worden uitgevoerd ten opzichte van de passende grenswaarden voor de betreffende delen van de constructie (d.w.z. ondanks dat er voor beton en wapening slechts één kwaliteit wordt gebruikt, verschillen de uiteindelijke spanning-rek-diagrammen per deel van de constructie door de effecten van tension stiffening en compression softening).</p>\n<p>Er is ook een optie om <strong>gladde wapeningsstaven</strong> te modelleren. Meer informatie vindt u hier: <a data-item-id=\"182f8ba8-899b-44fc-a1c7-59d562ef8c6c\" href=\"\">Gladde wapeningsstaven in Detail</a></p>\n<p><strong>Totale kracht </strong><em><strong>F</strong></em><em><strong><sub>tot</sub></strong></em><strong> en grenskracht </strong><em><strong>F</strong></em><em><strong><sub>lim</sub></strong></em></p>\n<p>De totale kracht <em><strong>F</strong></em><em><strong><sub>tot</sub></strong></em> is een resultaat van de eindige-elementenanalyse en kan op twee manieren worden gedefinieerd.</p>\n<p>\\[F_{tot}=A_{s} \\cdot f_{s}\\]</p>\n<p>waarin <em>A</em><em><sub>s</sub></em> het oppervlak van de wapeningsstaaf is en <em>f</em><em><sub>s</sub></em> de spanning in de staaf.</p>\n<p>Of als de som van de verankeringskracht <em>F</em><em><sub>a </sub></em>en de aanhechtkracht <em>F</em><em><sub>bond</sub></em><em>.</em></p>\n<p>\\[F_{tot}=F_{a}+F_{bond}\\]</p>\n<p>waarin <em>F</em><em><sub>a</sub></em> de werkelijke kracht in de verankeringsveer is en <em>F</em><em><sub>bond</sub></em> de aanhechtkracht, die kan worden verkregen door de aanhechtspanning <em>τ</em><em><sub>b</sub></em> te integreren over de lengte van de wapeningsstaaf <em>l.</em></p>\n<p>\\[F_{bond}=C_{s} \\cdot \\int_{0}^{l}\\tau_{b}\\left( x \\right)dx\\]</p>\n<p>C<sub>s</sub> is de omtrek van de wapeningsstaaf.</p>\n<p>De grenskracht <em><strong>F</strong></em><em><strong><sub>lim</sub></strong></em> is de maximale kracht in het element van de wapeningsstaaf, rekening houdend met de <strong>sterkte</strong> van de wapeningsstaaf en ook met de <strong>verankeringscondities </strong>(aanhechting tussen beton en wapening en verankeringshaken, lussen, enz.).</p>\n<p>\\[F_{lim}=min\\left( F_{lim,bond}+F_{au},F_{u} \\right)\\]</p>\n<p>\\[F_{u}=f_{y,lim}\\cdot A_{s}\\]</p>\n<p>\\[F_{au}=\\beta\\cdot f_{y,lim}\\cdot A_{s}\\]</p>\n<p>\\[F_{lim,bond}=C_{s}\\cdot l \\cdot f_{bu}\\]</p>\n<p>waarin C<sub>s</sub> de omtrek van de wapeningsstaaf is en <em>l</em> de lengte vanaf het begin van de wapeningsstaaf tot het beschouwde punt.</p>\n<figure data-asset-id=\"d3675eaf-0adb-4512-9366-58e4bdf171b1\" data-image-id=\"d3675eaf-0adb-4512-9366-58e4bdf171b1\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/1a6bbdca-e56b-47e1-a85f-00d4317689a8/Flim.png\" data-asset-id=\"d3675eaf-0adb-4512-9366-58e4bdf171b1\" data-image-id=\"d3675eaf-0adb-4512-9366-58e4bdf171b1\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 47\\qquad Definition of the limit force Flim}}}\\]</em></p>\n<p><br></p>\n<p>\\[F_{lim,2}=F_{lim,1}+F_{lim,add}\\]</p>\n<p>waarin <em>F</em><em><sub>lim,add</sub></em> de aanvullende kracht is, berekend uit de grootte van de hoek tussen naburige elementen. <em>F</em><em><sub>lim,2</sub></em> moet altijd lager zijn dan <em>F</em><em><sub>u</sub></em>.</p>\n<p><br></p>\n<p>De beschikbare <strong>verankeringstypen</strong> in CSFM omvatten een rechte staaf (d.w.z. geen reductie voor het ankereinde), een haak van 90 graden, een haak van 180 graden, perfecte aanhechting en een doorlopende staaf. Al deze typen worden, samen met de bijbehorende verankeringscoëfficiënten β, voor langswapening weergegeven in Fig. 48. De waarden van de toegepaste verankeringscoëfficiënten zijn afgeleid uit de vergelijking van de formule uit paragraaf ACI 318-19 25.4.3.1 met de formules uit paragraaf ACI 318-19 25.4.2.3. Merk op dat CSFM, ondanks de verschillende beschikbare opties, drie typen verankeringseinden onderscheidt: (i) geen reductie van de verankeringslengte, (ii) een reductie van 30% van de verankeringslengte bij een genormaliseerde verankering, en (iii) perfecte aanhechting.</p>\n<figure data-asset-id=\"85c164c0-d864-4723-8c34-a84a426100b2\" data-image-id=\"85c164c0-d864-4723-8c34-a84a426100b2\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/b76bc446-995d-4d16-8ef9-4aa26671edda/Available%20anchorage%20types%20for%20longitudinal%20rebars.png\" data-asset-id=\"85c164c0-d864-4723-8c34-a84a426100b2\" data-image-id=\"85c164c0-d864-4723-8c34-a84a426100b2\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 48\\qquad Available anchorage types and respective anchorage coefficients for longitudinal reinforcing bars in CSFM:}}}\\]</em></p>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{(a) straight bar; (b) 90-degree hook; (c) 180-degree hook; (d) perfect bond; (e) continuous bar}}}\\]</em></p>\n<p>De verankeringscoëfficiënt voor beugels is altijd β = 1,0.</p>\n<p>Om aan ACI te voldoen, moet de verankeringsveer in de berekening worden gebruikt. 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"value": "<p>Bij het ontwerpen van betonconstructies komen we twee grote groepen gedeeltelijk belaste oppervlakken (PLA) tegen. De eerste groep omvat <strong>opleggingen</strong>, de tweede <strong>verankeringsgebieden</strong>. </p>\n<p>Volgens de momenteel geldende normen voor het ontwerp van gewapende betonconstructies, ACI 318-19 hoofdstuk 22.8, moeten bij <strong>opleggingen</strong> het plaatselijk verbrijzelen van het beton en dwarse trek krachten worden beschouwd. Bij een gelijkmatig verdeelde belasting op een oppervlak, <em>A</em><em><sub>c1</sub></em>, mag de drukcapaciteit van het beton met maximaal een factor twee worden verhoogd, afhankelijk van het rekenoppervlak voor de lastspreiding <em>A</em><em><sub>c2</sub></em>. Zie ACI 318-19 tabel 22.8.3.2.</p>\n<figure data-asset-id=\"0d1d9eab-8cca-488d-a1fc-a0e55a22ba6e\" data-image-id=\"0d1d9eab-8cca-488d-a1fc-a0e55a22ba6e\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/2d1db553-b91c-4327-8c20-396cc2144140/Partially%20loaded%20areas%20Bearings.png\" data-asset-id=\"0d1d9eab-8cca-488d-a1fc-a0e55a22ba6e\" data-image-id=\"0d1d9eab-8cca-488d-a1fc-a0e55a22ba6e\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 49\\qquad Partially loaded areas for bearings according to ACI 318-19}}}\\]</em></p>\n<p>Voor <strong>verankeringszones</strong> bij voorspanning met nagerekt staal moet ACI 318-19 hoofdstuk 25.9 worden gevolgd.</p>\n<p>Het gedeeltelijk belaste oppervlak moet voldoende worden voorzien van dwarswapening, die is ontworpen om de optredende splijtkrachten in het gebied over te dragen. Zonder de vereiste dwarswapening is het niet toegestaan om uit te gaan van een verhoogde drukcapaciteit van het beton.</p>\n<p><br></p>\n<p><strong>Gedeeltelijk belaste oppervlakken in CSFM</strong></p>\n<figure data-asset-id=\"77fdebe4-afac-4ee7-aee5-716984d4e6d3\" data-image-id=\"77fdebe4-afac-4ee7-aee5-716984d4e6d3\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/3dcea2b1-7700-46f3-a938-4c08204d52e8/Fictitious%20struts.PNG\" data-asset-id=\"77fdebe4-afac-4ee7-aee5-716984d4e6d3\" data-image-id=\"77fdebe4-afac-4ee7-aee5-716984d4e6d3\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 50\\qquad Fictitious struts with concrete finite element mesh.}}}\\]</em></p>\n<p>Met CSFM is het mogelijk om gewapende betonconstructies te ontwerpen en te toetsen, waarbij de invloed van de toenemende drukweerstand van het beton in gedeeltelijk belaste oppervlakken wordt meegenomen. Omdat CSFM een wandmodel (2D) is en gedeeltelijk belaste oppervlakken een ruimtelijk (3D) vraagstuk vormen, was het nodig een oplossing te vinden die deze twee verschillende typen vraagstukken combineert (<em>Fig. 50</em>). Wanneer de functie \"gedeeltelijk belaste oppervlakken\" is geactiveerd, wordt de toelaatbare conusgeometrie gegenereerd volgens ACI (<em>Fig. 49</em>). Alle geometrische botsingen worden volledig in 3D opgelost voor de opgegeven geometrie van de betonstaaf en de afmetingen van elk PLA. Vervolgens wordt een rekenmodel van het gedeeltelijk belaste oppervlak opgesteld.</p>\n<figure data-asset-id=\"05c2e193-bc14-42b5-bc07-da8610febda8\" data-image-id=\"05c2e193-bc14-42b5-bc07-da8610febda8\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/6ae87bd2-682b-4b92-ab1f-4b12e9d3a0df/Cone%20geometry.png\" data-asset-id=\"05c2e193-bc14-42b5-bc07-da8610febda8\" data-image-id=\"05c2e193-bc14-42b5-bc07-da8610febda8\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 51\\qquad Allowable cone geometries.}}}\\]</em></p>\n<p>Het aanpassen van het materiaalmodel bleek een ongeschikte aanpak, vooral omdat het toewijzen van eigenschappen aan het eindige-elementenmesh problematisch is. Een aanpak die onafhankelijk is van het eindige-elementenmesh bleek een geschiktere oplossing. Voor de bekende geometrie van de drukconus worden volledig samenhangende fictieve drukdiagonalen gegenereerd (<em>Fig. 51</em> <em>en Fig. 52</em>). Deze drukdiagonalen hebben dezelfde materiaaleigenschappen als het in het model gebruikte beton, inclusief het spanning-rek diagram. De vorm van de conus bepaalt de richting van de drukdiagonalen, die de belasting geleidelijk over het PLA verdelen naar het rekenoppervlak voor de lastspreiding. De oppervlaktedichtheid van de fictieve drukdiagonalen varieert in elk deel van de conus en voegt een fictief betonoppervlak toe in de belastingsrichting. Ter hoogte van het belaste oppervlak (<em>A</em><em><sub>c1</sub></em>) wordt een fictief betonoppervlak toegevoegd volgens \\(\\sqrt{A_{c1} \\cdot A_{c2}} - A_{real}\\) (waarbij <em>A</em><em><sub>real</sub></em> het oplegoppervlak is dat in het 2D-rekenmodel wordt aangenomen). Dit oppervlak neemt lineair af tot nul in de richting van het rekenoppervlak voor de lastspreiding (<em>A</em><em><sub>c2</sub></em>). Deze oplossing zorgt ervoor dat de drukspanning in het beton over het gehele conusvolume constant is.</p>\n<figure data-asset-id=\"aff079fa-74f7-4575-a46b-8e589950238a\" data-image-id=\"aff079fa-74f7-4575-a46b-8e589950238a\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/1dae350c-2f3a-445d-930f-f383e991dcca/Partially%20loaded%20areas%20-%20ACI.png\" data-asset-id=\"aff079fa-74f7-4575-a46b-8e589950238a\" data-image-id=\"aff079fa-74f7-4575-a46b-8e589950238a\" alt=\"\"></figure>\n<p>\\[\\rho \\left( {\\beta ,z} \\right) = \\left( {\\sqrt {\\frac{A_{c2}}{A_{c1}}} - \\frac{A_{real}}{A_{c1}}} \\right)\\,\\cdot\\,\\left( {1 - \\frac{z}{h}} \\right)\\,\\cdot\\,\\frac{1}{{\\cos \\beta }}\\]</p>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 52\\qquad Fictitious struts in the computational model}}}\\]</em></p>\n<p>De weerstand van het gedeeltelijk belaste oppervlak wordt verhoogd volgens de verhouding tussen het rekenoppervlak voor de lastspreiding en het belaste oppervlak, zoals vastgelegd in ACI 318-19 hoofdstuk 22.8. Houd er rekening mee dat dit een rekenmodel is dat de spanningstoestand over een gedeeltelijk belast oppervlak, waarvan het werkelijke krachtenverloop veel complexer is, niet exact kan beschrijven. Deze oplossing zorgt echter voor een correcte verdeling van de belasting over het gehele model, waarbij de verhoogde draagcapaciteit van het gedeeltelijk belaste oppervlak in acht wordt genomen. Bovendien worden in dit gebied de dwarsspanningen correct geïntroduceerd, zodat de wapening voor splijtkrachten correct kan worden ontworpen.</p>\n<p>De toelaatbare <strong>oplegspanning</strong> van <em>0.85f</em><em><sub>c</sub></em><em>'</em> staat in tabel 22.8.3.2. De dichtheid wordt zodanig begrensd dat de maximale dubbele capaciteit uit de formule in tabel 22.8.3.2(b) niet wordt overschreden. </p>\n<p>Voor de <strong>verankeringszones</strong> wordt PLA in de applicatie op dezelfde wijze gebruikt als voor opleggingen. Daarom moeten de lokale zones, gedefinieerd in ACI 318-19 hoofdstuk 25.9, handmatig worden getoetst volgens ACI 318-19 25.9.3. PLA wordt dus alleen gebruikt om te voorkomen dat het rekcriterium in de lokale zone wordt overschreden en de berekening daardoor voortijdig wordt gestopt. Daarentegen kan, volgens ACI 318-19, Cl. 25.9.4.3.1 (b), de wapening die weerstand biedt aan de splijt- en afspatspanningen in het vlak, direct en voordelig in de applicatie worden geverifieerd.</p>"
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"value": "<p>Bruikbaarheidsverificaties worden uitgevoerd voor spanningsbeperking, scheurwijdte en doorbuigingsgrenzen. De spanningen in beton en wapening worden getoetst volgens ACI 318-19, op een vergelijkbare wijze als voorgeschreven voor de Sterkte.</p>\n<h3>Spanningsbeperking</h3>\n<p>De toelaatbare drukspanningen in het beton onder gebruiksbelasting moeten worden geverifieerd voor voorgespannen staven van klasse U en T. Volgens Tabel R24.5.2.1 is voor beton dat als gescheurd wordt beschouwd geen toetsing van de spanningsbeperking vereist. De gebruiker moet de klasse van de voorgespannen staaf instellen in de instellingen van de rekenstaaf.</p>\n<figure data-asset-id=\"aebd4701-afaa-4f1f-b7f6-e531c65ed403\" data-image-id=\"aebd4701-afaa-4f1f-b7f6-e531c65ed403\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/5dff4f86-fd02-432a-812c-cf520aabe92b/Prestressed%20member%20class.png\" data-asset-id=\"aebd4701-afaa-4f1f-b7f6-e531c65ed403\" data-image-id=\"aebd4701-afaa-4f1f-b7f6-e531c65ed403\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 53\\qquad Prestressed flexural member class selection}}}\\]</em></p>\n<p>De toelaatbare drukspanning voor staven die aan tijdelijke belastingen worden blootgesteld, wordt in ACI 318-19 24.5.4.1 gegeven als <em>0.6f</em><em><sub>c</sub></em><em>'. </em>De drukspanningslimiet van <em>0.45f</em><em><sub>c</sub></em><em>'</em> is vastgesteld om de kans op bezwijken van voorgespannen betonstaven door herhaalde belastingen te verkleinen. Deze limiet bleek ook redelijk om overmatige kruipvervorming te voorkomen. Bij hogere spanningswaarden nemen de kruiprekken sneller toe naarmate de aangebrachte spanning toeneemt.</p>\n<p>De betonspanning in druk wordt bepaald als de verhouding tussen de maximale hoofddrukspanning <em>f</em><em><sub>c</sub></em> <em>= σ</em><em><sub>c</sub></em><sub>2</sub><em><sub> </sub></em>uit de EEM-analyse voor de bruikbaarheid en de limietwaarde, die is vastgesteld op basis van Tabel 24.5.4.1.</p>\n<figure data-asset-id=\"5f5abc59-7c83-43de-9aa6-045ba160e215\" data-image-id=\"5f5abc59-7c83-43de-9aa6-045ba160e215\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/26aa9ff8-a409-41a2-b69b-b28fc2841ec0/Concrete%20compressive%20stress%20limits%20at%20service%20loads%20-%20ACI.png\" data-asset-id=\"5f5abc59-7c83-43de-9aa6-045ba160e215\" data-image-id=\"5f5abc59-7c83-43de-9aa6-045ba160e215\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 54\\qquad Concrete compressive stress limits at service loads}}}\\]</em></p>\n<p>In de applicatie wordt <em>Voorspanning plus blijvende belasting</em> behandeld als een langetermijncombinatie en <em>Voorspanning plus totale belasting</em> als een kortetermijncombinatie.</p>\n<h3>Doorbuiging</h3>\n<p>Afhankelijk van het geselecteerde combinatietype (lange termijn of korte termijn) wordt de doorbuiging op lange of korte termijn beoordeeld. De maximaal toelaatbare doorbuiging moet door de gebruiker worden bepaald en moet worden beschouwd conform ACI 138-19 24.2. </p>\n<figure data-asset-id=\"977137a7-f1f0-4e67-8f44-06634328b1a4\" data-image-id=\"977137a7-f1f0-4e67-8f44-06634328b1a4\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/35ae9de1-6a34-4952-a6e7-ffc528e1e5aa/Deflection%20limit%20value%20selection.png\" data-asset-id=\"977137a7-f1f0-4e67-8f44-06634328b1a4\" data-image-id=\"977137a7-f1f0-4e67-8f44-06634328b1a4\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 55\\qquad Maximum allowable deflection value}}}\\]</em></p>\n<p>In de applicatie kunnen de doorbuigingen ten gevolge van het eigen gewicht <em>Δ</em><em><sub>DL</sub></em> en de veranderlijke belasting <em>Δ</em><em><sub>LL</sub></em> afzonderlijk worden weergegeven, evenals de totale doorbuiging <em>Δ</em><em><sub>Tot</sub></em><sub> </sub>(eigen gewicht + veranderlijke belasting), steeds samen met de vervormde vorm.</p>\n<p>Doorbuigingen bij ingekorte uiteinden kunnen niet worden getoetst.</p>\n<h3>Scheurwijdte</h3>\n<p><br></p>\n<p>Scheurwijdten en scheuroriëntaties worden berekend voor bruikbaarheidscombinaties op korte of lange termijn. Aangezien ACI geen grenswaarden voor scheurwijdten rechtstreeks voorschrijft, moet de gebruiker een grenswaarde voor de scheurwijdte <em>w</em><em><sub>lim</sub></em> opgeven.</p>\n<p>De verificaties worden als volgt weergegeven:</p>\n<p>\\[\\frac{w}{w_{lim}}\\]</p>\n<p>waarin:</p>\n<p><em>w</em> scheurwijdte op korte of lange termijn, berekend met de EEM-analyse,</p>\n<p><em>w</em><em><sub>lim</sub></em> grenswaarde van de scheurwijdte, gedefinieerd door de gebruiker.</p>\n<p>De in de applicatie gebruikte methode voor het berekenen van scheurwijdten, die ook in dit document nader wordt beschreven, is in overeenstemming met ACI 224R-01. Het is daarom mogelijk om Tabel 4.1 van ACI 224R-01 te gebruiken voor het bepalen van de grenswaarde van de scheurwijdten.</p>\n<figure data-asset-id=\"00675749-f338-4b86-80b7-14648ef6e0b5\" data-image-id=\"00675749-f338-4b86-80b7-14648ef6e0b5\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/4af498a4-6b3b-4043-be8f-f10522f5b188/Reasonable%20crack%20widths%20-%20ACI.png\" data-asset-id=\"00675749-f338-4b86-80b7-14648ef6e0b5\" data-image-id=\"00675749-f338-4b86-80b7-14648ef6e0b5\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 56\\qquad Reasonable crack widths for reinforced concrete under service load}}}\\]</em></p>\n<p>Er zijn twee manieren om scheurwijdten te berekenen (gestabiliseerde en niet-gestabiliseerde scheurvorming). In het algemene geval (gestabiliseerde scheurvorming) wordt de scheurwijdte berekend door de rekken te integreren over de 1D-elementen van de wapeningsstaven. 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"value": "<h3>Beton - Sterkte</h3>\n<p>Het betonmodel dat voor sterkteberekeningen in CSFM is geïmplementeerd, is gebaseerd op de uitgangspunten van AASHTO LRFD voor sterkteontwerp: evenwicht en rekcompatibiliteit. Conform AASHTO LRFD (2024) artikel 5.6.2.1 wordt de treksterkte van beton verwaarloosd.</p>\n<figure data-asset-id=\"a84d11ec-b1f2-431e-afad-b6e1b7e8a83c\" data-image-id=\"a84d11ec-b1f2-431e-afad-b6e1b7e8a83c\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/f578dd02-9167-45e0-b80f-4a1331dfe20a/Concrete%20stress-strain%20diagram%20CSFM%20-%20ACI.png\" data-asset-id=\"a84d11ec-b1f2-431e-afad-b6e1b7e8a83c\" data-image-id=\"a84d11ec-b1f2-431e-afad-b6e1b7e8a83c\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 57\\qquad The stress-strain diagram of concrete for Strength analysis}}}\\]</em></p>\n<p>De implementatie van CSFM in <em>IDEA StatiCa Detail</em> hanteert geen expliciet bezwijkcriterium in termen van rek voor beton op druk (na het bereiken van de piekspanning wordt een plastische tak aangenomen met ε<em><sub>c</sub></em><sub>0</sub> met een maximale waarde van 5%, terwijl AASHTO LRFD (2024) artikel 5.6.2.1 uitgaat van een uiterste rek kleiner dan 0,3%). Door deze vereenvoudiging kan de vervormingscapaciteit van constructies die op druk bezwijken niet worden geverifieerd. De sterkte wordt echter correct voorspeld wanneer, naast de factor voor gescheurd beton (<em>k</em><em><sub>c</sub></em><sub>2</sub> gedefinieerd in (Fig. 57)), ook de toename van de brosheid van beton bij toenemende sterkte wordt meegenomen met de reductiefactor <em>\\(\\eta_{fc}\\)</em> uit de <em>fib</em> Model Code 2010, als volgt:</p>\n<p>\\[f'_{c,lim}=\\alpha_{1}\\cdot\\phi_{c}\\cdot k_{c}\\cdot f'_{c}\\]</p>\n<p>\\[k_{c}=\\eta_{fc}\\cdot k_{c2}\\]</p>\n<p>\\[{\\eta _{fc}} = {\\left( {\\frac{{30}}{{{f'_{c}}}}} \\right)^{\\frac{1}{3}}} \\le 1\\]</p>\n<p>waarin:</p>\n<p><em>α</em><sub>1</sub> de reductiefactor is voor de druksterkte van beton, gedefinieerd in AASHTO LRFD (2024) artikel 5.6.2.2. Bij gebruik van een parabool-rechthoek spanning-rek diagram moet de maximale drukspanning met deze factor worden verlaagd. Hierdoor wordt de spanningsverdeling in de drukzone zodanig gemiddeld dat de resulterende druksterkte kleiner is dan of gelijk is aan de druksterkte berekend met een spanning-rek diagram met een dalende plastische tak<em>.</em></p>\n<p><em>Φ</em><em><sub>c </sub></em>de weerstandsfactor voor beton is. De standaardwaarde is ingesteld conform AASHTO LRFD (2024) artikel 5.5.4.2.</p>\n<p><em>k</em><em><sub>c</sub></em><sub>2</sub> de reductiefactor is vanwege de aanwezigheid van dwarsscheuren.</p>\n<p><em>f'</em><em><sub>c</sub></em> de cilinderdruksterkte van beton is (in MPa voor de definitie van <em>\\( \\eta_{fc} \\)</em>).</p>\n<figure data-asset-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\" data-image-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/085222c7-055a-4870-9bcb-8f18bd65620f/Compression%20softening%20CSFM.PNG\" data-asset-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\" data-image-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 58\\qquad The compression softening law.}}}\\]</em></p>\n<p><em>k</em><em><sub>c</sub></em><sub>2</sub> is een reductiefactor die op dezelfde uitgangspunten is gebaseerd als de efficiëntiefactor voor beton <em>ν</em> uit AASHTO LRFD (2024) 5.8.2.5.3a en Tabel 5.8.2.5.3a-1. Het verschil is dat in CSFM de aanwezigheid van een hoofdtrekspanning loodrecht op de hoofddrukspanning voor elk eindig element wordt gecontroleerd (en niet alleen voor de knopen van het Staafwerkmodel).</p>\n<h3>Beton – Bruikbaarheid</h3>\n<p>De bruikbaarheidsberekening bevat enkele vereenvoudigingen van de constitutieve modellen die voor de sterkteberekening worden gebruikt. De plastische tak van de spanning-rek curve van beton op druk wordt buiten beschouwing gelaten, terwijl de elastische tak lineair en oneindig is. De compression softening wet wordt niet toegepast. Deze vereenvoudigingen verbeteren de numerieke stabiliteit en de rekensnelheid en verminderen de algemeenheid van de oplossing niet, zolang de resulterende spanningsgrenzen van de materialen in de bruikbaarheidsgrenstoestand duidelijk onder hun vloeigrens liggen (in overeenstemming met de benadering voor de bruikbaarheidsgrenstoestand in AASHTO LRFD). De vereenvoudigde modellen voor bruikbaarheid zijn daarom alleen geldig als aan alle verificatie-eisen wordt voldaan.</p>\n<figure data-asset-id=\"fdcc5f99-090b-4af6-af2f-efa12840c367\" data-image-id=\"fdcc5f99-090b-4af6-af2f-efa12840c367\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/be2c4149-9e8e-4595-b5a5-7e9fa87c20f3/Concrete%20stress-strain%20for%20serviceability%20-%20AASHTO.png\" data-asset-id=\"fdcc5f99-090b-4af6-af2f-efa12840c367\" data-image-id=\"fdcc5f99-090b-4af6-af2f-efa12840c367\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 59\\qquad Concrete stress-strain diagrams implemented for serviceability analysis: short- and long-term verifications.}}}\\]</em></p>\n<p><br></p>\n<p><strong>Langetermijneffecten</strong></p>\n<p>De langetermijn constitutieve wet (de rode curve in Fig. 59) wordt gebruikt voor de berekening van de scheurwijdte, de totale doorbuiging en de spanningsbeperking van voorgespannen staven wanneer het langetermijneffect is geselecteerd in het bovenste lint. In de IDEA StatiCa Detail applicatie wordt de effectieve elasticiteitsmodulus gebruikt voor de verificatie van langetermijneffecten, zoals vermeld in AASHTO LRFD (2024) C5.12.5.3.6-1.</p>\n<p>\\[E_{eff} = \\frac{E_{c}}{1+\\psi}\\]</p>\n<p>waarin:<br><em>E</em><em><sub>c</sub></em> de elasticiteitsmodulus is, gedefinieerd in AASHTO LRFD (2024) artikel 5.4.2.4<br><em>ψ</em> de kruipcoëfficiënt is, gedefinieerd in AASHTO LRFD (2024) artikel 5.4.2.3.2</p>\n<p>Kruipfactoren worden door de gebruiker gedefinieerd in de materiaaleigenschappen.</p>\n<p><strong>Korte-termijneffecten</strong></p>\n<p>Voor de korte-termijnverificaties wordt een aparte berekening uitgevoerd waarin alle belastingen zonder kruipfactor worden berekend. Beide berekeningen, voor lange- en korte-termijnverificaties, zijn weergegeven in Fig. 59.</p>\n<h3>Wapening</h3>\n<p>Voor de niet-voorgespannen wapening wordt een perfect elastisch-plastisch spanning-rek diagram met een gedefinieerde vloeigrens aangenomen, zie AASHTO LRFD (2024) artikel 5.4.3. Voor de definitie van dit diagram zijn alleen de basiseigenschappen van de wapening nodig: de sterkte en de elasticiteitsmodulus.</p>\n<p>Het spanning-rek diagram van de wapening kan ook door de gebruiker worden gedefinieerd, maar in dat geval kan het tension stiffening effect niet worden meegenomen (het is dan niet mogelijk de scheurwijdte te berekenen). </p>\n<figure data-asset-id=\"2d9c6401-28af-4bfe-bc92-1d6f830f7c93\" data-image-id=\"2d9c6401-28af-4bfe-bc92-1d6f830f7c93\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/77dadff9-85d4-402e-94e5-a3725f908933/Steel%20stress-strain%20diagram%20CSFM%20-%20ACI.png\" data-asset-id=\"2d9c6401-28af-4bfe-bc92-1d6f830f7c93\" data-image-id=\"2d9c6401-28af-4bfe-bc92-1d6f830f7c93\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 60 \\qquad Stress-strain diagram of reinforcement}}}\\]</em></p>\n<p>waarin:</p>\n<p><em>Φ</em><em><sub>s </sub></em>de weerstandsfactor voor wapening is. De standaardwaarde is ingesteld conform AASHTO LRFD (2024) artikel 5.5.4.2.</p>\n<p><em>f</em><em><sub>y</sub></em> de vloeigrens van de wapening is</p>\n<p><em>E</em><em><sub>s</sub></em> de elasticiteitsmodulus van de wapening is</p>\n<p>Als grensrek waarbij de berekening wordt gestopt, is 10% geselecteerd. Dit wordt als veilig beschouwd op basis van ASTM A955/A955M-20c artikel 7.</p>\n<p>Tension stiffening (Fig. 61) wordt automatisch meegenomen door de invoer van de spanning-rek relatie van het blanke wapeningsstaal aan te passen, zodat de gemiddelde stijfheid van de in het beton ingebedde staven wordt vastgelegd (ε<em><sub>m</sub></em>).</p>\n<figure data-asset-id=\"c9add949-2ad5-4922-8e6c-0d75fb47cb70\" data-image-id=\"c9add949-2ad5-4922-8e6c-0d75fb47cb70\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/c045fcb6-32c6-4a92-aa15-24530fb11484/Tension%20stiffening%20CSFM%20-%20ACI.png\" data-asset-id=\"c9add949-2ad5-4922-8e6c-0d75fb47cb70\" data-image-id=\"c9add949-2ad5-4922-8e6c-0d75fb47cb70\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 61\\qquad Scheme of tension stiffening.}}}\\]</em></p>"
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"value": "<p>De Compatible Stress Field Method voldoet aan de moderne ontwerpnormen. Omdat de rekenmodellen uitsluitend gebruikmaken van standaard materiaaleigenschappen, kan het in de ontwerpnormen voorgeschreven formaat met partiële veiligheidsfactoren zonder aanpassing worden toegepast. Op deze manier worden de invoerbelastingen met factoren vermenigvuldigd en worden de karakteristieke materiaaleigenschappen verlaagd met de betreffende weerstandsfactoren, precies zoals bij conventionele betonberekeningen.</p>\n<p>De waarden van de <strong>weerstandsfactoren</strong> worden voorgeschreven in AASHTO LRFD (2024), Article 5.5.4. De standaardwaarden voor beton en wapening zijn conservatief gekozen, uitgaande van de aanname dat het typische uitgewerkte voorbeeld een D-gebied is - een typisch geval voor de SaT-methode. Het is echter mogelijk om elk type element te modelleren. Wanneer een door druk of trek gedomineerd element wordt getoetst, kan de gebruiker daarom de waarde van de reductiefactor voor de sterkte wijzigen in de Preferences.</p>\n<figure data-asset-id=\"4d2e1aae-3ec0-4e09-9461-ce5d5cb329b4\" data-image-id=\"4d2e1aae-3ec0-4e09-9461-ce5d5cb329b4\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/22c831e5-30cb-49da-9661-edf4748ca6aa/Resistance%20factors%20-%20AASHTO.png\" data-asset-id=\"4d2e1aae-3ec0-4e09-9461-ce5d5cb329b4\" data-image-id=\"4d2e1aae-3ec0-4e09-9461-ce5d5cb329b4\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 62\\qquad The setting of resistance factors in IDEA StatiCa Detail.}}}\\]</em></p>\n<p><br></p>\n<p>Belastingfactoren en belastingcombinaties moeten worden gedefinieerd volgens AASHTO LRFD Bridge Design Specifications (2024), Article 3.4.1 en Tables 3.4.1-1 tot en met 3.4.1-6. AASHTO LRFD specificeert expliciet de belastingcombinaties voor de grenstoestand van sterkte (Strength I tot en met Strength V) en de belastingcombinaties voor de gebruiksgrenstoestand (Service I tot en met Service IV), inclusief de bijbehorende belastingfactoren voor elk geval.</p>\n<p>Voor elk template bevat het programma vooraf gedefinieerde basiscombinaties die moeten worden aangevuld, afhankelijk van het element dat wordt verwerkt.</p>\n<figure data-asset-id=\"d3234a33-200e-46f1-99bd-ef3eb153e6ee\" data-image-id=\"d3234a33-200e-46f1-99bd-ef3eb153e6ee\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/b3a557f2-2e6a-4b2c-8ba8-bfd210e04c3f/Load%20factors%20AASHTO.png\" data-asset-id=\"d3234a33-200e-46f1-99bd-ef3eb153e6ee\" data-image-id=\"d3234a33-200e-46f1-99bd-ef3eb153e6ee\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 63\\qquad The setting of load factors in Idea StatiCa Detail.}}}\\]</em></p>"
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De toetsingen worden uitgevoerd voor de betonsterkte, de wapeningssterkte en de verankering (afschuifspanningen in de aanhechting).</p>\n<p>De <strong>betonsterkte</strong> op druk wordt bepaald als de verhouding tussen de maximale hoofddrukspanning <em>f</em><em><sub>c</sub></em> (ook σ<sub>2</sub> in Hulpresultaten) uit de EEM-analyse en de grenswaarde <em>f'</em><em><sub>c,lim</sub></em>.</p>\n<p>De <strong>sterkte van de wapening</strong> wordt zowel op trek als op druk bepaald als de verhouding tussen de spanning in de wapening ter plaatse van de scheuren <em>f</em><em><sub>s</sub></em> en de opgegeven grenswaarde <em>f</em><em><sub>y,lim</sub></em>.</p>\n<p>De <strong>afschuifspanning in de aanhechting</strong> wordt afzonderlijk bepaald als de verhouding tussen de aanhechtingsspanning τ<em><sub>b</sub></em> uit de EEM-analyse en de aanhechtingssterkte <em>f</em><em><sub>bu</sub></em>.</p>\n<p>Omdat de aanhechtingssterkte in AASHTO niet expliciet is gedefinieerd, moet de waarde ervan worden bepaald met de vergelijkingen voor de verankeringslengte. De aanhechtingssterkte is namelijk de belangrijkste invoer voor het bepalen van de verankeringslengte; zie bijvoorbeeld AASHTO LRFD (2024) Article C5.10.8.2 of NCHRP Report 733, Attachment E, pagina E-9.</p>\n<p>De berekening in AASHTO LRFD (2024) Article 5.10.8.2.1 en 5.10.8.2.2 vereist kennis van de maximale hart-op-hartafstand van de dwarswapening binnen <em>l</em><em><sub>d</sub></em>, het aantal staven of draden dat langs het splijtvlak wordt verankerd, de totale doorsnede van alle dwarswapening en andere geometrische grootheden. Deze kunnen in het model van de Detail applicatie niet betrouwbaar worden bepaald voor algemene invoer. Daarom is een benadering uit AASHTO LRFD (2014) Article 5.11.2.1.1 toegepast, op de volgende wijze:</p>\n<p>We nemen aan dat bij verankering van de wapeningsstaaf in een betonblok over de verankeringslengte <em>l</em><em><sub>d</sub></em> of meer, het uittrekken van de wapening leidt tot breuk van de wapening en niet tot het uittrekken uit het beton. Dit kan met de volgende formule worden uitgedrukt.</p>\n<p>\\[\\pi\\cdot d_{b} \\cdot l_{d} \\cdot f_{bu}=f_{y}\\cdot A_{b}\\]</p>\n<p>waarbij:</p>\n<ul>\n <li><em>d</em><em><sub>b</sub></em> de diameter van de wapeningsstaaf is</li>\n <li><em>l</em><em><sub>d</sub></em> de verankeringslengte is</li>\n <li><em>f</em><em><sub>bu</sub></em> de aanhechtingssterkte is</li>\n <li><em>f</em><em><sub>y</sub></em> de vloeigrens van de wapening is</li>\n <li><em>A</em><em><sub>b</sub></em> de doorsnede van de wapeningsstaaf is</li>\n</ul>\n<p>Hieruit kan eenvoudig de formule voor de aanhechtingssterkte worden afgeleid.</p>\n<p> \\[f_{bu}=\\frac{f_{y}\\cdot A_{b}}{\\pi\\cdot d_{b} \\cdot l_{d} }\\]</p>\n<p><br></p>\n<p>De basisverankeringslengte op trek <em>l</em><em><sub>db</sub></em> wordt in AASHTO LRFD (2014) Article 5.11.2.1.1 als volgt bepaald:</p>\n<p>Voor staven No. 11 en kleiner: \\(l_{bd}=\\max\\left(1.25\\cdot\\dfrac{A_{b}\\cdot f_{y}}{\\sqrt{f'_{c}}},\\ 0.4\\cdot d_{b}\\cdot f_{y}\\right)\\)</p>\n<p>Voor staven No. 14: \\(l_{bd}=\\dfrac{2.70\\cdot f_{y}}{\\sqrt{f'_{c}}}\\)</p>\n<p>Voor staven No. 18: \\(l_{bd}=\\dfrac{3.5\\cdot f_{y}}{\\sqrt{f'_{c}}}\\)</p>\n<p>waarbij:</p>\n<ul>\n <li><em>A</em><em><sub>b</sub></em> de doorsnede van de wapeningsstaaf is (in<sup>2</sup>)</li>\n <li><em>f</em><em><sub>y</sub></em> de opgegeven vloeigrens van de wapening is (ksi)</li>\n <li><em>f'</em><em><sub>c</sub></em> de opgegeven druksterkte van het beton na 28 dagen is, tenzij een andere leeftijd is opgegeven (ksi)</li>\n <li><em>d</em><em><sub>b</sub></em> de diameter van de wapeningsstaaf is (in)</li>\n</ul>\n<p>Door de basisverankeringslengte <em>l</em><em><sub>db</sub></em> te vermenigvuldigen met de factoren uit AASHTO LRFD (2014) Article 5.11.2.1.2 en 5.11.2.1.3 wordt de verankeringslengte <em>l</em><em><sub>d</sub></em> als invoer bepaald.</p>\n<p>De modificatiefactoren uit 5.11.2.1.3 die de verankeringslengte verkleinen, zijn in de applicatie altijd gelijk aan 1,0. De modificatiefactor voor bovenliggende horizontale of nagenoeg horizontale wapening is gelijk aan 1,4 bij 'slechte' aanhechtingsomstandigheden, volgens de onderstaande figuur:</p>\n<figure data-asset-id=\"bdffaabf-ad2b-43bf-b943-deeafb3d57b3\" data-image-id=\"bdffaabf-ad2b-43bf-b943-deeafb3d57b3\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/9daaedd3-8368-4677-b72f-1dbc0933690e/Bond%20conditions%20-%20AASHTO.png\" data-asset-id=\"bdffaabf-ad2b-43bf-b943-deeafb3d57b3\" data-image-id=\"bdffaabf-ad2b-43bf-b943-deeafb3d57b3\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 64\\qquad Description of bond conditions; a) b) ‘good’ bond conditions for all bars; c) d) unhatched zone – ‘good’ bond conditions, hatched zone – ‘poor’ bond conditions}}}\\]</em></p>\n<p>De stortrichting kan in de applicatie worden ingesteld.</p>\n<figure data-asset-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\" data-image-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/e00845bc-3d60-4315-a8b3-67d4a52666a4/Direction%20of%20concreting.png\" data-asset-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\" data-image-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 65\\qquad Direction of concreting}}}\\]</em></p>\n<p>Alle overige factoren uit 5.11.2.1.2 zijn gelijk aan 1,0, omdat alleen beton met normaal gewicht en alleen ongecoate wapening worden ondersteund.</p>\n<p>De afschuifspanning in de aanhechting en de aanhechtingssterkte van staven op druk worden analoog aan die van staven op trek berekend, maar daarbij worden de vergelijkingen uit AASHTO LRFD (2014) Article 5.11.2.2 gebruikt.</p>\n<p>Het is ook mogelijk om <strong>gladde wapeningsstaven</strong> te modelleren. Meer informatie vindt u hier: <a data-item-id=\"182f8ba8-899b-44fc-a1c7-59d562ef8c6c\" href=\"\">Gladde wapeningsstaven in Detail</a></p>\n<p><br></p>\n<p><strong>Totale kracht </strong><em><strong>F</strong></em><em><strong><sub>tot</sub></strong></em><strong> en grenskracht </strong><em><strong>F</strong></em><em><strong><sub>lim</sub></strong></em></p>\n<p>De totale kracht <em><strong>F</strong></em><em><strong><sub>tot</sub></strong></em> is een resultaat van de eindige-elementenanalyse en kan op twee manieren worden bepaald.</p>\n<p>\\[F_{tot}=A_{b} \\cdot f_{s}\\]</p>\n<p>waarbij <em>A</em><em><sub>b</sub></em> de doorsnede van de wapeningsstaaf is en <em>f</em><em><sub>s</sub></em> de spanning in de staaf.</p>\n<p>Of als de som van de verankeringskracht <em>F</em><em><sub>a </sub></em>en de aanhechtingskracht <em>F</em><em><sub>bond</sub></em><em>.</em></p>\n<p>\\[F_{tot}=F_{a}+F_{bond}\\]</p>\n<p>waarbij <em>F</em><em><sub>a</sub></em> de werkelijke kracht in de verankeringsveer is en <em>F</em><em><sub>bond</sub></em> de aanhechtingskracht, die kan worden verkregen door de aanhechtingsspanning <em>τ</em><em><sub>b</sub></em> te integreren over de lengte van de wapeningsstaaf <em>l.</em></p>\n<p>\\[F_{bond}=C_{s} \\cdot \\int_{0}^{l}\\tau_{b}\\left( x \\right)dx\\]</p>\n<p>C<sub>s</sub> is de omtrek van de wapeningsstaaf.</p>\n<p>De grenskracht <em><strong>F</strong></em><em><strong><sub>lim</sub></strong></em> is de maximale kracht in het element van de wapeningsstaaf, rekening houdend met de <strong>sterkte</strong> van de wapeningsstaaf en ook met de <strong>verankeringsomstandigheden </strong>(aanhechting tussen beton en wapening en verankeringshaken, lussen, enz.).</p>\n<p>\\[F_{lim}=min\\left( F_{lim,bond}+F_{au},F_{u} \\right)\\]</p>\n<p>\\[F_{u}=f_{y,lim}\\cdot A_{b}\\]</p>\n<p>\\[F_{au}=\\beta\\cdot f_{y,lim}\\cdot A_{b}\\]</p>\n<p>\\[F_{lim,bond}=C_{s}\\cdot l \\cdot f_{bu}\\]</p>\n<p>waarbij C<sub>s</sub> de omtrek van de wapeningsstaaf is en <em>l</em> de lengte vanaf het begin van de wapeningsstaaf tot het beschouwde punt.</p>\n<figure data-asset-id=\"d3675eaf-0adb-4512-9366-58e4bdf171b1\" data-image-id=\"d3675eaf-0adb-4512-9366-58e4bdf171b1\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/1a6bbdca-e56b-47e1-a85f-00d4317689a8/Flim.png\" data-asset-id=\"d3675eaf-0adb-4512-9366-58e4bdf171b1\" data-image-id=\"d3675eaf-0adb-4512-9366-58e4bdf171b1\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 66\\qquad Definition of the limit force Flim}}}\\]</em></p>\n<p><br></p>\n<p>\\[F_{lim,2}=F_{lim,1}+F_{lim,add}\\]</p>\n<p>waarbij <em>F</em><em><sub>lim,add</sub></em> de extra kracht is die wordt berekend uit de grootte van de hoek tussen naburige elementen. <em>F</em><em><sub>lim,2</sub></em> moet altijd kleiner zijn dan <em>F</em><em><sub>u</sub></em>.</p>\n<p><br></p>\n<p>De beschikbare <strong>verankeringstypen</strong> in CSFM zijn een rechte staaf (d.w.z. zonder reductie voor het ankereinde), een haak van 90 graden, een haak van 180 graden, perfecte aanhechting en een doorlopende staaf. Al deze typen zijn, samen met de bijbehorende verankeringscoëfficiënten β, voor langswapening weergegeven in Fig. 67. De waarden van de toegepaste verankeringscoëfficiënten zijn afgeleid uit de vergelijking van de formule uit AASHTO LRFD (2014) 5.11.2.1 met de formules uit AASHTO LRFD (2014) 5.11.2.4.1. Merk op dat CSFM, ondanks de verschillende beschikbare opties, drie typen verankeringseinden onderscheidt: (i) geen reductie van de verankeringslengte, (ii) een reductie van 30% van de verankeringslengte bij een genormaliseerde verankering en (iii) perfecte aanhechting.</p>\n<figure data-asset-id=\"85c164c0-d864-4723-8c34-a84a426100b2\" data-image-id=\"85c164c0-d864-4723-8c34-a84a426100b2\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/b76bc446-995d-4d16-8ef9-4aa26671edda/Available%20anchorage%20types%20for%20longitudinal%20rebars.png\" data-asset-id=\"85c164c0-d864-4723-8c34-a84a426100b2\" data-image-id=\"85c164c0-d864-4723-8c34-a84a426100b2\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 67\\qquad Available anchorage types and respective anchorage coefficients for longitudinal reinforcing bars in CSFM:}}}\\]</em></p>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{(a) straight bar; (b) 90-degree hook; (c) 180-degree hook; (d) perfect bond; (e) continuous bar}}}\\]</em></p>\n<p>De verankeringscoëfficiënt voor beugels (beschikbaar voor liggerelementen) is altijds β = 1,0.</p>\n<p>Om aan AASHTO te voldoen, moet de verankeringsveer in de berekening worden gebruikt. De verankeringsveer wordt aangepast met de coëfficiënt β, dus de gebruiker moet een van de beschikbare verankeringstypen kiezen bij het definiëren van de begin- en eindvoorwaarden van de wapening. </p>"
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"value": "<p>Bij het ontwerpen van betonconstructies komen we twee grote groepen gedeeltelijk belaste oppervlakken (PLA) tegen. De eerste groep bestaat uit <strong>opleggingen</strong>, de tweede uit <strong>verankeringsgebieden</strong>. </p>\n<p>Volgens de huidige normen voor het ontwerp van gewapende betonconstructies moeten bij <strong>opleggingen</strong> het lokaal verbrijzelen van het beton en de dwarstrek kracht in rekening worden gebracht. Bij een gelijkmatig verdeelde belasting op een oppervlak <em>A</em><em><sub>1</sub></em> kan de drukcapaciteit van het beton tot twee keer worden verhoogd, afhankelijk van het rekenkundige spreidingsoppervlak <em>A</em><em><sub>2</sub></em>. Zie AASHTO LRFD (2024) Article 5.6.5.<br></p>\n<figure data-asset-id=\"635e6258-ee56-41b1-8137-b791039b6b3b\" data-image-id=\"635e6258-ee56-41b1-8137-b791039b6b3b\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/e777285d-ac56-4c61-801e-d2edd9ae0318/PLA%20AASHTO.png\" data-asset-id=\"635e6258-ee56-41b1-8137-b791039b6b3b\" data-image-id=\"635e6258-ee56-41b1-8137-b791039b6b3b\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 68\\qquad Partially loaded areas for bearings according to AASHTO LRFD (2024) Article 5.6.5}}}\\]</em></p>\n<p>Voor <strong>verankeringszones</strong> bij voorspanning met nagerekt staal moet AASHTO LRFD (2024) Article 5.8.4.4 worden gevolgd.</p>\n<p>Het gedeeltelijk belaste oppervlak moet voldoende zijn voorzien van dwarswapening, die is ontworpen om de splijtkrachten in het gebied over te brengen. Zonder de vereiste dwarswapening is het niet mogelijk om rekening te houden met een verhoogde drukcapaciteit van het beton.</p>\n<p><br></p>\n<p><strong>Gedeeltelijk belaste oppervlakken in CSFM</strong></p>\n<figure data-asset-id=\"77fdebe4-afac-4ee7-aee5-716984d4e6d3\" data-image-id=\"77fdebe4-afac-4ee7-aee5-716984d4e6d3\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/3dcea2b1-7700-46f3-a938-4c08204d52e8/Fictitious%20struts.PNG\" data-asset-id=\"77fdebe4-afac-4ee7-aee5-716984d4e6d3\" data-image-id=\"77fdebe4-afac-4ee7-aee5-716984d4e6d3\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 69\\qquad Fictitious struts with concrete finite element mesh.}}}\\]</em></p>\n<p>Met CSFM is het mogelijk om gewapende betonconstructies te ontwerpen en te toetsen, waarbij de invloed van de toenemende drukweerstand van het beton in gedeeltelijk belaste oppervlakken wordt meegenomen. Omdat CSFM een wandmodel (2D) is en gedeeltelijk belaste oppervlakken een ruimtelijk (3D) vraagstuk vormen, was het nodig een oplossing te vinden die deze twee verschillende typen vraagstukken combineert (<em>Fig. 69</em>). Als de functie “gedeeltelijk belaste oppervlakken” is geactiveerd, wordt de toelaatbare conusgeometrie aangemaakt volgens ACI (<em>Fig. 68</em>). Alle geometrische botsingen worden volledig in 3D opgelost voor de opgegeven geometrie van de betonnen staaf en de afmetingen van elk PLA. Vervolgens wordt een rekenmodel van het gedeeltelijk belaste oppervlak aangemaakt.</p>\n<figure data-asset-id=\"05c2e193-bc14-42b5-bc07-da8610febda8\" data-image-id=\"05c2e193-bc14-42b5-bc07-da8610febda8\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/6ae87bd2-682b-4b92-ab1f-4b12e9d3a0df/Cone%20geometry.png\" data-asset-id=\"05c2e193-bc14-42b5-bc07-da8610febda8\" data-image-id=\"05c2e193-bc14-42b5-bc07-da8610febda8\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 70\\qquad Allowable cone geometries.}}}\\]</em></p>\n<p>Het aanpassen van het materiaalmodel bleek een ongeschikte aanpak, voornamelijk omdat het toewijzen van eigenschappen aan de eindige-elementen mesh problematisch is. Een aanpak die onafhankelijk is van de eindige-elementen mesh bleek een geschiktere oplossing. Voor de bekende geometrie van de drukconus worden volledig samenhangende fictieve drukdiagonalen aangemaakt (<em>Fig. 70</em> <em>en Fig. 71</em>). Deze drukdiagonalen hebben dezelfde materiaaleigenschappen als het beton in het model, inclusief het spanning-rekdiagram. De vorm van de conus bepaalt de richting van de drukdiagonalen, die de belasting over het PLA geleidelijk spreiden tot het rekenkundige spreidingsoppervlak. De oppervlaktedichtheid van de fictieve drukdiagonalen varieert in elk deel van de conus en voegt een fictief betonoppervlak toe in de belastingsrichting. Ter hoogte van het belaste oppervlak (<em>A</em><em><sub>1</sub></em>) wordt een fictief betonoppervlak toegevoegd volgens de verhouding \\(\\sqrt{A_{1} \\cdot A_{2}} - A_{real}\\) (waarbij <em>A</em><em><sub>real</sub></em> het oppervlak van de oplegging is dat in het 2D-rekenmodel wordt aangenomen). Dit oppervlak neemt lineair af tot nul richting het rekenkundige spreidingsoppervlak (<em>A</em><em><sub>2</sub></em>). Deze oplossing zorgt ervoor dat de drukspanning in het beton constant is over het volledige conusvolume.</p>\n<figure data-asset-id=\"aff079fa-74f7-4575-a46b-8e589950238a\" data-image-id=\"aff079fa-74f7-4575-a46b-8e589950238a\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/1dae350c-2f3a-445d-930f-f383e991dcca/Partially%20loaded%20areas%20-%20ACI.png\" data-asset-id=\"aff079fa-74f7-4575-a46b-8e589950238a\" data-image-id=\"aff079fa-74f7-4575-a46b-8e589950238a\" alt=\"\"></figure>\n<p>\\[\\rho \\left( {\\beta ,z} \\right) = \\left( {\\sqrt {\\frac{A_{2}}{A_{1}}} - \\frac{A_{real}}{A_{1}}} \\right)\\,\\cdot\\,\\left( {1 - \\frac{z}{h}} \\right)\\,\\cdot\\,\\frac{1}{{\\cos \\beta }}\\]</p>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 71\\qquad Fictitious struts in the computational model}}}\\]</em></p>\n<p>De weerstand van het gedeeltelijk belaste oppervlak wordt verhoogd volgens de verhouding tussen het rekenkundige spreidingsoppervlak en het belaste oppervlak, zoals vastgelegd in AASHTO LRFD (2024) Article 5.6.5. Houd er rekening mee dat dit een rekenmodel is dat de spanningstoestand over een gedeeltelijk belast oppervlak, waarvan de werkelijke krachtsafdracht veel complexer is, niet exact kan beschrijven. Deze oplossing zorgt echter voor een correcte verdeling van de belasting over het hele model, met inachtneming van de verhoogde draagkracht van het gedeeltelijk belaste oppervlak. Bovendien worden in dit gebied op correcte wijze dwarsspanningen geïntroduceerd, zodat de wapening voor splijtkrachten correct kan worden ontworpen.</p>\n<p>De toelaatbare <strong>oplegspanning</strong> van <em>0.85f</em><em><sub>c</sub></em><em>'</em> is opgenomen in AASHTO LRFD (2024) Article 5.8.4.4. De dichtheid wordt zodanig beperkt dat de maximale dubbele capaciteit uit formule 5.6.5-3 niet wordt overschreden. </p>\n<p>Voor de <strong>verankeringszones</strong> wordt PLA in de applicatie op dezelfde manier gebruikt als voor opleggingen. Daarom moeten de drukspanningen in de lokale en globale zones, zoals gedefinieerd in Article 5.8.4.4 en 5.8.4.5, handmatig worden getoetst. PLA wordt dan ook alleen gebruikt om te voorkomen dat het rekcriterium in de lokale zone wordt overschreden en de berekening daardoor voortijdig stopt. De wapening die weerstand biedt aan barstspanningen, afspattingsspanningen in het vlak en randtrekspanningen in de algemene zones (gedefinieerd in Article 5.8.4.5) kan daarentegen direct en met voordeel in de applicatie worden getoetst.</p>"
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"value": "<p>Bruikbaarheidstoetsingen worden uitgevoerd voor spanningsbeperking, scheurwijdte en doorbuigingsgrenzen. Spanningen worden in beton en wapeningselementen getoetst volgens AASHTO LRFD, op een vergelijkbare manier als voor de Strength.</p>\n<h3>Spanningsbeperking</h3>\n<p>De betondrukspanning wordt alleen beoordeeld voor voorgespannen staven (wanneer de belastinggeval Prestressing aanwezig is in het model), als de verhouding tussen de maximale hoofddrukspanning <em>f</em><em><sub>c</sub></em> <em>= σ</em><em><sub>c</sub></em><sub>2</sub><em><sub> </sub></em>uit de EEM-analyse voor de bruikbaarheidsgrenstoestand en de grenswaarden, die zijn vastgesteld op basis van AASHTO LRFD Tabel 5.9.2.3.2a-1.</p>\n<figure data-asset-id=\"0946a8a5-4fdf-4626-ad28-c49499d4d6eb\" data-image-id=\"0946a8a5-4fdf-4626-ad28-c49499d4d6eb\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/aacf656a-eb2c-4aae-a8b2-6b1c16cdc864/Compressive%20Stress%20Limits%20in%20Prestressed%20Concrete%20at%20Service%20Limit%20State%20-%20AASHTO.png\" data-asset-id=\"0946a8a5-4fdf-4626-ad28-c49499d4d6eb\" data-image-id=\"0946a8a5-4fdf-4626-ad28-c49499d4d6eb\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 72\\qquad Concrete compressive stress limits at service loads}}}\\]</em></p>\n<p>In de applicatie wordt <em>Voorspanning plus permanente belasting</em> behandeld als een Sustain load en <em>Voorspanning, permanente en veranderlijke belasting</em> als een Total load.</p>\n<figure data-asset-id=\"ac528856-0620-4e95-9877-ea4415ba38b5\" data-image-id=\"ac528856-0620-4e95-9877-ea4415ba38b5\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/bf637c0b-a972-4836-a119-45f59df9ed58/Combination%20types%20-%20AASHTO.png\" data-asset-id=\"ac528856-0620-4e95-9877-ea4415ba38b5\" data-image-id=\"ac528856-0620-4e95-9877-ea4415ba38b5\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 73\\qquad Serviceability combination types}}}\\]</em></p>\n<p>Daarnaast is het altijd mogelijk een analyse uit te voeren voor zowel kortetermijn- als langetermijneffecten, met materiaalmodellen die de kruipfactor wel of niet meenemen. Zie de paragraaf \"Materiaalmodellen (AASHTO)\".</p>\n<figure data-asset-id=\"ceefbba0-847a-4d2d-8c6c-5430e5e9c43d\" data-image-id=\"ceefbba0-847a-4d2d-8c6c-5430e5e9c43d\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/74bf96e4-46c2-4bb5-9b9f-2ab9ebf8f58b/Stress%20limitation%20model%20type%20-%20AASHTO.png\" data-asset-id=\"ceefbba0-847a-4d2d-8c6c-5430e5e9c43d\" data-image-id=\"ceefbba0-847a-4d2d-8c6c-5430e5e9c43d\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 74\\qquad Serviceability material models}}}\\]</em></p>\n<h3>Doorbuiging</h3>\n<p>Directe doorbuigingen en totale doorbuigingen worden bepaald voor elke combinatie waarvoor de doorbuigingsbeoordeling is ingeschakeld. </p>\n<ul>\n <li>Voor directe doorbuigingen wordt de elasticiteitsmodulus <em>E</em><em><sub>c</sub></em> volgens AASHTO LRFD (2024) artikel 5.4.2.4 gebruikt. </li>\n <li>Voor totale doorbuigingen wordt de effectieve elasticiteitsmodulus <em>E</em><em><sub>c,eff</sub></em> volgens AASHTO LRFD (2024) artikel C5.12.5.3.6 gebruikt. </li>\n</ul>\n<p>Zie het hoofdstuk '<em>Materiaalmodellen (AASHTO) - Beton – Bruikbaarheid</em>' in dit document.</p>\n<p>De doorbuigingstoetsing zelf wordt ingeschakeld in het bovenste lint. De gebruiker stelt de grenswaarden voor de doorbuiging in volgens AASHTO LRFD (2024) artikel 2.5.2.6.2, afhankelijk van het type element dat wordt geanalyseerd.</p>\n<figure data-asset-id=\"ddf1f284-82ac-44ca-a815-6e64b4471afe\" data-image-id=\"ddf1f284-82ac-44ca-a815-6e64b4471afe\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/b95113f5-c76d-4a13-8962-99fd504ce2f2/Deflection%20check%20AASHTO.png\" data-asset-id=\"ddf1f284-82ac-44ca-a815-6e64b4471afe\" data-image-id=\"ddf1f284-82ac-44ca-a815-6e64b4471afe\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 75\\qquad Maximum allowable deflection value}}}\\]</em></p>\n<p>Doorbuigingen bij ingekorte uiteinden kunnen niet worden getoetst.</p>\n<h3>Scheurwijdte</h3>\n<p>Scheurwijdten en scheurrichtingen worden alleen berekend voor langetermijneffecten (met <em>E</em><em><sub>c,eff</sub></em> volgens AASHTO LRFD (2024) artikel C5.12.5.3.6) voor combinaties waarvoor de scheurwijdtebeoordeling is ingeschakeld. De toetsingen op basis van door de gebruiker opgegeven grenswaarden zijn als volgt:</p>\n<p>\\[\\frac{w}{w_{lim}}\\]</p>\n<p>waarin:</p>\n<p><em>w</em> scheurwijdte berekend met de EEM-analyse,</p>\n<p><em>w</em><em><sub>lim</sub></em> grenswaarde van de scheurwijdte, gedefinieerd door de gebruiker.</p>\n<p>De grenswaarde <em>w</em><em><sub>lim</sub></em> moet worden bepaald op basis van het elementtype en de milieuklasse volgens AASHTO LRFD (2024) artikel 5.6.7 en de bijbehorende toelichting. </p>\n<p>Er zijn twee manieren om scheurwijdten te berekenen (gestabiliseerde en niet-gestabiliseerde scheurvorming). In het algemene geval (gestabiliseerde scheurvorming) wordt de scheurwijdte berekend door de rekken te integreren over 1D-elementen van wapeningsstaven. De scheurrichting wordt vervolgens berekend uit de drie dichtstbijzijnde integratiepunten (gerekend vanaf het midden van het betreffende 1D-eindige element van de wapening) van 2D-betonelementen. Deze benadering voor het berekenen van de scheurrichtingen komt weliswaar niet overeen met de werkelijke positie van de scheuren, maar levert toch representatieve waarden op. Deze leiden tot scheurwijdteresultaten die kunnen worden vergeleken met de door de norm vereiste scheurwijdten ter plaatse van de wapeningsstaaf.</p>"
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"value": "<h3>Beton - Sterkte</h3>\n<p>Het betonmodel dat in CSFM voor sterkteberekeningen is geïmplementeerd, is gebaseerd op de parabolisch-plastische spanning-rekcurve. De treksterkte wordt verwaarloosd, zoals gebruikelijk is bij klassiek gewapend betonontwerp.</p>\n<figure data-asset-id=\"1ce5c049-0015-4d84-8bd2-9bacc8e4b5b4\" data-image-id=\"1ce5c049-0015-4d84-8bd2-9bacc8e4b5b4\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/dc47139c-3c53-4397-bfa6-71fe09d5c24b/Concrete%20stress-strain%20diagram%20CSFM%20-%20AUS.png\" data-asset-id=\"1ce5c049-0015-4d84-8bd2-9bacc8e4b5b4\" data-image-id=\"1ce5c049-0015-4d84-8bd2-9bacc8e4b5b4\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 76\\qquad The stress-strain diagram of concrete for Strength analysis}}}\\]</em></p>\n<p>De implementatie van CSFM in <em>IDEA StatiCa Detail</em> hanteert voor beton onder druk geen expliciet bezwijkcriterium in termen van rek (na het bereiken van de piekspanning wordt een plastische tak aangehouden met ε<em><sub>c</sub></em><sub>0</sub> met een maximale waarde van 5%, terwijl AS 3600 Cl. 8.3.1 uitgaat van een uiterste rek van minder dan 0,3%). Door deze vereenvoudiging kan het vervormingsvermogen van constructies die door druk bezwijken niet worden geverifieerd. De sterkte wordt echter correct voorspeld wanneer, naast de factor voor gescheurd beton (<em>k</em><em><sub>c</sub></em><sub>2</sub>, gedefinieerd in (Fig. 77)), ook de toename van de brosheid van beton bij toenemende sterkte wordt meegenomen door middel van de reductiefactor <em>\\(\\eta_{fc}\\)</em> zoals gedefinieerd in <em>fib</em> Model Code 2010:</p>\n<p>\\[f'_{c,lim}=\\alpha_{2}\\cdot\\phi_{s}\\cdot \\beta \\cdot \\eta_{fc}\\cdot f'_{c}\\]</p>\n<p>\\[{\\eta _{fc}} = {\\left( {\\frac{{30}}{{{f'_{c}}}}} \\right)^{\\frac{1}{3}}} \\le 1\\]</p>\n<p>waarin:</p>\n<p><em>α</em><sub>2</sub> de reductiefactor is voor de druksterkte van beton, gedefinieerd in AS 3600 Cl. 8.3.1<br>Bij gebruik van een parabool-rechthoek spanning-rekdiagram moet de maximale drukspanning met deze factor worden gereduceerd. Hierdoor wordt de spanningsverdeling in de drukzone zodanig gemiddeld dat de resulterende druksterkte kleiner is dan of gelijk is aan de druksterkte berekend met een spanning-rekdiagram met een dalende plastische tak<em>. </em>Een analoge benadering is gedefinieerd voor het rechthoekige spanningsblok in Hoofdstuk 8.1.3.</p>\n<p><em>Φ</em><em><sub>s </sub></em>de spanningsreductiefactor voor beton is. De standaardwaarde is ingesteld volgens AS 3600 Table 2.2.3.</p>\n<p><em>β</em> de reductiefactor is vanwege de aanwezigheid van dwarsscheuren (in deze tekst ook aangeduid als <em>k</em><em><sub>c</sub></em><sub>2</sub>)</p>\n<p><em>f'</em><em><sub>c</sub></em> de cilinderdruksterkte van beton is (in MPa voor de definitie van <em>\\( \\eta_{fc} \\)</em>).</p>\n<figure data-asset-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\" data-image-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/085222c7-055a-4870-9bcb-8f18bd65620f/Compression%20softening%20CSFM.PNG\" data-asset-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\" data-image-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 77\\qquad The compression softening law.}}}\\]</em></p>\n<p><em>β</em> is een reductiefactor die op dezelfde principes is gebaseerd als de factor voor de effectieve druksterkte, gedefinieerd in Hoofdstuk 2.2.3. De literatuur waarop deze factor is gebaseerd, is (inclusief de context van de norm AS3600) te vinden in AS3600:2018 Sup 1:2022 CL. C2.2.3.</p>\n<h3>Beton – Bruikbaarheid</h3>\n<p>De bruikbaarheidsanalyse bevat enkele vereenvoudigingen van de constitutieve modellen die voor de sterkteanalyse worden gebruikt. De plastische tak van de spanning-rekcurve van beton onder druk wordt buiten beschouwing gelaten, terwijl de elastische tak lineair en oneindig is. Het compression softening-model wordt niet toegepast. Deze vereenvoudigingen verbeteren de numerieke stabiliteit en de rekensnelheid en verminderen de algemeenheid van de oplossing niet, zolang de resulterende materiaalspanningen in de bruikbaarheidsgrenstoestand duidelijk onder hun vloeigrens blijven (zoals vereist door AS3600). De vereenvoudigde modellen voor bruikbaarheid zijn daarom alleen geldig als aan alle toetsingseisen wordt voldaan.</p>\n<figure data-asset-id=\"1a187098-8984-42f2-b203-d261cab0f727\" data-image-id=\"1a187098-8984-42f2-b203-d261cab0f727\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/5b3dc17b-2a5b-4258-8495-b5d436e4885b/Concrete%20stress-strain%20for%20serviceability%20-%20AUS.png\" data-asset-id=\"1a187098-8984-42f2-b203-d261cab0f727\" data-image-id=\"1a187098-8984-42f2-b203-d261cab0f727\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 78\\qquad Concrete stress-strain diagrams implemented for serviceability analysis: short- and long-term verifications.}}}\\]</em></p>\n<p><br></p>\n<p><strong>Langetermijneffecten</strong></p>\n<p>In de bruikbaarheidsanalyse worden de langetermijneffecten van beton meegenomen met de rekenwaarde van de kruipcoëfficiënt volgens AS 3600 CL 3.1.8 (<em>φ</em><em><sub>cc</sub></em>, standaard een waarde van 2,5), die de secant-elasticiteitsmodulus van beton (<em>E</em><em><sub>c</sub></em>) als volgt aanpast:</p>\n<p>\\[E_{c,eff} = \\frac{E_{c}}{1+\\varphi_{cc}}\\]</p>\n<p>Belastingstappen worden achtereenvolgens berekend in de volgorde: voorspanning - permanent - veranderlijk, waarbij voor elke stap de passende effectieve elasticiteitsmodulus wordt gebruikt, zoals weergegeven in Fig. 78. Kruipfactoren worden door de gebruiker gedefinieerd in de materiaaleigenschappen en moeten worden berekend volgens AS 3600 CL 3.1.8.3</p>\n<figure data-asset-id=\"7c1e2af1-4d0f-46da-8cf0-d5bee4931cf3\" data-image-id=\"7c1e2af1-4d0f-46da-8cf0-d5bee4931cf3\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/f9c75c70-4a16-4077-963e-7ccbed22202a/Desgn%20creep%20factor%20-%20AUS.png\" data-asset-id=\"7c1e2af1-4d0f-46da-8cf0-d5bee4931cf3\" data-image-id=\"7c1e2af1-4d0f-46da-8cf0-d5bee4931cf3\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 79\\qquad Definition of the design creep factor}}}\\]</em></p>\n<p><strong>Korte-termijneffecten</strong></p>\n<p>Voor de kortetermijntoetsingen wordt een aparte berekening uitgevoerd waarin alle belastingen zonder de tijdsafhankelijke factor voor langdurige belastingen worden berekend. Beide berekeningen, voor lange- en korte-termijntoetsingen, zijn weergegeven in Fig. 78.</p>\n<h3>Wapening</h3>\n<p>Voor de niet-voorgespannen wapening wordt een perfect elastisch-plastisch spanning-rekdiagram met een gedefinieerd vloeipunt aangehouden, zie AS 3600 Section 3.2. Voor de definitie van dit diagram zijn alleen de basiseigenschappen van de wapening nodig – de sterkte en de elasticiteitsmodulus.</p>\n<p>Het spanning-rekdiagram van de wapening kan ook door de gebruiker worden gedefinieerd, maar in dat geval is het niet mogelijk om het tension stiffening-effect mee te nemen (de scheurwijdte kan dan niet worden berekend). </p>\n<figure data-asset-id=\"b5b99d46-a4ed-4625-853e-cdc4c4ede122\" data-image-id=\"b5b99d46-a4ed-4625-853e-cdc4c4ede122\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/4e33b934-9d0f-4ba7-9764-4f31801c752b/Steel%20stress-strain%20diagram%20CSFM%20-%20AUS.png\" data-asset-id=\"b5b99d46-a4ed-4625-853e-cdc4c4ede122\" data-image-id=\"b5b99d46-a4ed-4625-853e-cdc4c4ede122\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 80 \\qquad Stress-strain diagram of reinforcement}}}\\]</em></p>\n<p>waarin:</p>\n<p><em>Φ</em><em><sub>s </sub></em>de sterktereductiefactor voor wapening is. De standaardwaarde is ingesteld volgens AS 3600 Table 2.2.3.</p>\n<p><em>f</em><em><sub>y</sub></em> de vloeigrens van de wapening is</p>\n<p><em>E</em><em><sub>s</sub></em> de elasticiteitsmodulus van de wapening is</p>\n<p>Tension stiffening (Fig. 81) wordt automatisch meegenomen door de ingevoerde spanning-rekrelatie van het blanke wapeningsstaal aan te passen, zodat de gemiddelde stijfheid van de in het beton ingebedde staven wordt weergegeven (ε<em><sub>m</sub></em>).</p>\n<figure data-asset-id=\"c9465d3e-05e3-4514-a218-3a96876ed503\" data-image-id=\"c9465d3e-05e3-4514-a218-3a96876ed503\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/b27b5ab6-24ea-410b-901a-fccbd7e4005f/Tension%20stiffening%20CSFM%20-%20AUS.png\" data-asset-id=\"c9465d3e-05e3-4514-a218-3a96876ed503\" data-image-id=\"c9465d3e-05e3-4514-a218-3a96876ed503\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 81\\qquad Scheme of tension stiffening.}}}\\]</em></p>"
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"value": "<p>De Compatible Stress Field Method voldoet aan moderne ontwerpnormen. Aangezien de rekenmodellen uitsluitend standaard materiaaleigenschappen gebruiken, kan het in de ontwerpnormen voorgeschreven formaat met partiële veiligheidsfactoren zonder aanpassing worden toegepast. Op deze manier worden de invoerbelastingen met factoren vermenigvuldigd en de karakteristieke materiaaleigenschappen verlaagd met de betreffende spanningsreductiefactoren, precies zoals bij conventionele betonberekeningen.</p>\n<p>De waarden van de <strong>spanningsreductiefactoren</strong> zijn voorgeschreven in AUS 3600 Cl. 2.2.3. De standaardwaarden voor beton en wapening zijn ingesteld volgens Tabel 2.2.3</p>\n<figure data-asset-id=\"61735d28-361b-4275-b2d7-9ca00e01ebcf\" data-image-id=\"61735d28-361b-4275-b2d7-9ca00e01ebcf\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/1d32796c-ae70-42fb-a3d3-4542e785f5b1/Stress%20reduction%20factors_AUS.png\" data-asset-id=\"61735d28-361b-4275-b2d7-9ca00e01ebcf\" data-image-id=\"61735d28-361b-4275-b2d7-9ca00e01ebcf\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 82\\qquad The setting of stress reduction factors in IDEA StatiCa Detail.}}}\\]</em></p>\n<p><br></p>\n<p><strong>Belastingsfactoren</strong> voor sterktecombinaties worden gedefinieerd volgens AS 3600 Cl. 4.2.2. Belastingsfactoren voor bruikbaarheidscombinaties worden bepaald volgens Tabel 4.1. Voor alle templates zijn de belastingsfactoren al vooraf gedefinieerd.</p>\n<figure data-asset-id=\"c986c0fc-2e9a-42e1-95b4-1055d3ae76e2\" data-image-id=\"c986c0fc-2e9a-42e1-95b4-1055d3ae76e2\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/887ee546-c598-41fd-b494-c43ccbc55194/Load%20factors%20AUS.png\" data-asset-id=\"c986c0fc-2e9a-42e1-95b4-1055d3ae76e2\" data-image-id=\"c986c0fc-2e9a-42e1-95b4-1055d3ae76e2\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 83\\qquad The setting of load factors in Idea StatiCa Detail.}}}\\]</em></p>"
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"value": "<p>De verschillende controles die AS 3600 voorschrijft, worden beoordeeld op basis van de directe resultaten van het model. De controles worden uitgevoerd voor de betonsterkte, de sterkte van de wapening en de verankering (afschuifspanningen in de aanhechting).</p>\n<p>De <strong>betonsterkte</strong> op druk wordt bepaald als de verhouding tussen de maximale hoofddrukspanning <em>f</em><em><sub>c</sub></em> (ook σ<sub>2</sub> in Aanvullende resultaten) uit de FE-analyse en de grenswaarde <em>f'</em><em><sub>c,lim</sub></em>.</p>\n<p>De <strong>sterkte van de wapening</strong> wordt zowel op trek als op druk bepaald als de verhouding tussen de spanning in de wapening ter plaatse van de scheuren <em>f</em><em><sub>s</sub></em> en de opgegeven grenswaarde <em>f</em><em><sub>sy,lim</sub></em>.</p>\n<p>De <strong>afschuifspanning in de aanhechting</strong> wordt onafhankelijk bepaald als de verhouding tussen de aanhechtingsspanning τ<em><sub>b</sub></em>, berekend met de FE-analyse, en de rekenwaarde van de uiterste aanhechtingsspanning <em>f</em><em><sub>bu</sub></em>.</p>\n<p>Voor het bepalen van de rekenwaarde van de uiterste aanhechtingsspanning <em>f</em><em><sub>bu</sub></em> wordt in de applicatie formule C13.1.2.2 uit AS3600:2018 Sup 1:2022 gehanteerd.</p>\n<p>\\[f_{bu}=\\frac{k_{2}}{k_{1} \\cdot k_{3}} \\cdot (0.5 \\cdot \\sqrt{f'_{c}})\\]</p>\n<p>Hierbij geldt <em>f'</em><em><sub>c</sub></em><em> ≤ 65 MPa</em> (in de formule in MPa) en worden de <em>k</em>-factoren als volgt bepaald volgens AS 3600 Cl. 13.1.2.2:</p>\n<p><em>k</em><em><sub>3</sub></em><em> = 0.7</em> (conservatieve waarde voor alle wapening)<br><em>k</em><em><sub>2</sub></em><em> = (132 - d</em><em><sub>b</sub></em><em>) / 100</em> (<em>d</em><em><sub>b</sub></em> is de diameter van het wapeningsstaal in millimeters)<br>= 1.3 voor een horizontale staaf met meer dan 300 mm beton gestort onder de staaf, anders 1.0</p>\n<p><em>k</em><em><sub>1</sub></em> wordt automatisch afgeleid uit de positie van de wapening in het model en uit de stortrichting, die in de applicatie voor elk projectitem als volgt kan worden ingesteld.</p>\n<figure data-asset-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\" data-image-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/e00845bc-3d60-4315-a8b3-67d4a52666a4/Direction%20of%20concreting.png\" data-asset-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\" data-image-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 84\\qquad Direction of concreting}}}\\]</em></p>\n<p>De basisverankeringslengte <em>L</em><em><sub>sy,tb</sub></em> wordt berekend volgens formule 13.1.2.2 in AS 3600 als volgt:</p>\n<p>\\[L_{sy,tb}=\\frac{0.5\\cdot k_{1}\\cdot k_{3}\\cdot f_{sy}\\cdot d_{b}}{k_{2}\\cdot \\sqrt{f'_{c}}}\\ge 29 \\cdot k_{1}\\cdot d_{b}\\]</p>\n<p>Zoals uit de formule blijkt, is de basisverankeringslengte <em>L</em><em><sub>sy,tb</sub></em> aan de onderzijde begrensd. Daarom moet de rekenwaarde van de uiterste aanhechtingsspanning <em>f</em><em><sub>bu</sub></em> in de applicatie op dezelfde wijze worden begrensd, zodat het volgende geldt:</p>\n<p>\\[f_{bu}\\le \\frac{f_{sy}}{116 \\cdot k_{1}} \\]</p>\n<p>Hierbij is <em>f</em><em><sub>sy</sub></em> in MPa.</p>\n<p>De afleiding van de begrenzing van <em>f</em><em><sub>bu</sub></em> is als volgt:</p>\n<p>\\[f_{bu}= \\frac{f_{sy}\\cdot A_{s}}{ \\pi \\cdot d_{b} \\cdot L_{sy,tb}}=\\frac{f_{sy}\\cdot \\pi \\cdot d_{b}^{2}}{4 \\cdot \\pi \\cdot d_{b} \\cdot 29 \\cdot k{1} \\cdot d_{b}} =\\frac{f_{sy}}{116 \\cdot k_{1}} \\]</p>\n<p>Het is ook mogelijk om <strong>gladde wapeningsstaven</strong> te modelleren. Meer informatie vindt u hier: <a data-item-id=\"182f8ba8-899b-44fc-a1c7-59d562ef8c6c\" href=\"\">Gladde wapeningsstaven in Detail</a></p>\n<p><br></p>\n<p><strong>Totale kracht </strong><em><strong>F</strong></em><em><strong><sub>tot</sub></strong></em><strong> en grenskracht </strong><em><strong>F</strong></em><em><strong><sub>lim</sub></strong></em></p>\n<p>De totale kracht <em><strong>F</strong></em><em><strong><sub>tot</sub></strong></em> is een resultaat van de eindige-elementenanalyse en kan op twee manieren worden bepaald.</p>\n<p>\\[F_{tot}=A_{s} \\cdot f_{s}\\]</p>\n<p>waarbij <em>A</em><em><sub>s</sub></em> de oppervlakte van de wapeningsstaaf is en <em>f</em><em><sub>s</sub></em> de spanning in de staaf.</p>\n<p>Of als de som van de verankeringskracht <em>F</em><em><sub>a </sub></em>en de aanhechtingskracht <em>F</em><em><sub>bond</sub></em><em>.</em></p>\n<p>\\[F_{tot}=F_{a}+F_{bond}\\]</p>\n<p>waarbij <em>F</em><em><sub>a</sub></em> de werkelijke kracht in de verankeringsveer is en <em>F</em><em><sub>bond</sub></em> de aanhechtingskracht, die kan worden verkregen door de aanhechtingsspanning <em>τ</em><em><sub>b</sub></em> te integreren over de lengte van de wapeningsstaaf <em>l.</em></p>\n<p>\\[F_{bond}=C_{s} \\cdot \\int_{0}^{l}\\tau_{b}\\left( x \\right)dx\\]</p>\n<p>C<sub>s</sub> is de omtrek van de wapeningsstaaf.</p>\n<p>De grenskracht <em><strong>F</strong></em><em><strong><sub>lim</sub></strong></em> is de maximale kracht in het element van de wapeningsstaaf, rekening houdend met de <strong>sterkte</strong> van de wapening en ook met de <strong>verankeringsomstandigheden </strong>(aanhechting tussen beton en wapening en verankeringshaken, lussen, enz.).</p>\n<p>\\[F_{lim}=min\\left( F_{lim,bond}+F_{au},F_{u} \\right)\\]</p>\n<p>\\[F_{u}=f_{y,lim}\\cdot A_{s}\\]</p>\n<p>\\[F_{au}=\\beta\\cdot f_{y,lim}\\cdot A_{s}\\]</p>\n<p>\\[F_{lim,bond}=C_{s}\\cdot l \\cdot f_{bu}\\]</p>\n<p>waarbij C<sub>s</sub> de omtrek van de wapeningsstaaf is en <em>l</em> de lengte vanaf het begin van de wapeningsstaaf tot het beschouwde punt.</p>\n<figure data-asset-id=\"d3675eaf-0adb-4512-9366-58e4bdf171b1\" data-image-id=\"d3675eaf-0adb-4512-9366-58e4bdf171b1\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/1a6bbdca-e56b-47e1-a85f-00d4317689a8/Flim.png\" data-asset-id=\"d3675eaf-0adb-4512-9366-58e4bdf171b1\" data-image-id=\"d3675eaf-0adb-4512-9366-58e4bdf171b1\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 85\\qquad Definition of the limit force Flim}}}\\]</em></p>\n<p><br></p>\n<p>\\[F_{lim,2}=F_{lim,1}+F_{lim,add}\\]</p>\n<p>waarbij <em>F</em><em><sub>lim,add</sub></em> de extra kracht is, berekend uit de grootte van de hoek tussen aangrenzende elementen. <em>F</em><em><sub>lim,2</sub></em> moet altijd kleiner zijn dan <em>F</em><em><sub>u</sub></em>.</p>\n<p><br></p>\n<p>De beschikbare <strong>verankeringstypen</strong> in CSFM omvatten een rechte staaf (d.w.z. zonder reductie voor het verankeringseinde), standaard cog, standaard haak, perfecte aanhechting en doorlopende staaf. Al deze typen worden, samen met de bijbehorende verankeringscoëfficiënten β, voor langswapening weergegeven in Fig. 86. De waarden van de toegepaste verankeringscoëfficiënten zijn afgeleid uit AS 3600 Cl. 13.1.2. Opgemerkt moet worden dat CSFM drie typen verankeringseinden onderscheidt: (i) geen reductie van de verankeringslengte, (ii) een reductie van 50% van de verankeringslengte bij een genormaliseerde verankering en (iii) perfecte aanhechting.</p>\n<figure data-asset-id=\"ea687a47-41cc-487f-b7b9-2ed97bfb2932\" data-image-id=\"ea687a47-41cc-487f-b7b9-2ed97bfb2932\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/021688e6-24c8-441b-8210-9f0bb4377e75/Available%20anchorage%20types%20for%20longitudinal%20rebars_AUS.png\" data-asset-id=\"ea687a47-41cc-487f-b7b9-2ed97bfb2932\" data-image-id=\"ea687a47-41cc-487f-b7b9-2ed97bfb2932\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 86\\qquad Available anchorage types and respective anchorage coefficients for longitudinal reinforcing bars in CSFM:}}}\\]</em></p>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{(a) straight bar; (b) Standard cog; (c) Standard hook; (d) perfect bond; (e) continuous bar}}}\\]</em></p>\n<p>De verankeringscoëfficiënt voor beugels is altijd β = 1.0.</p>\n<p>Om aan AS 3600 te voldoen, moet de verankeringsveer in de berekening worden gebruikt. 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"value": "<h3>Beton - Sterkte</h3>\n<p>Het betonmodel dat in CSFM is geïmplementeerd voor sterkteberekeningen is gebaseerd op de parabolisch-plastische spanning-rekkromme. De treksterkte wordt verwaarloosd, zoals gebruikelijk is bij het klassieke ontwerp van gewapend beton. Voor de ontwerpnorm CSA wordt alleen het parabool-rechthoekdiagram voor beton ondersteund.</p>\n<figure data-asset-id=\"75c1ac5f-22d1-4eee-8e6b-136e3d35e377\" data-image-id=\"75c1ac5f-22d1-4eee-8e6b-136e3d35e377\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/0e30004f-06ce-46e3-a200-775a2e5e0e5a/fig_8-01_concrete_stress_strain_strength.png\" data-asset-id=\"75c1ac5f-22d1-4eee-8e6b-136e3d35e377\" data-image-id=\"75c1ac5f-22d1-4eee-8e6b-136e3d35e377\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 90\\qquad The stress-strain diagram of concrete for Strength analysis}}}\\]</em></p>\n<p>De implementatie van CSFM in IDEA StatiCa Detail houdt geen rekening met een expliciet bezwijkcriterium in termen van rekken voor beton onder druk (na het bereiken van de piekspanning wordt een plastische tak aangehouden met ε<sub>c</sub> met een maximumwaarde van 5%, terwijl CSA A23.3:24 artikel 10.1.3 uitgaat van een maximale betondrukrek van 0,0035). Door deze vereenvoudiging kan het vervormingsvermogen van constructies die onder druk bezwijken niet worden geverifieerd. De sterkte wordt echter correct voorspeld wanneer, naast de factor voor gescheurd beton (k<sub>c2</sub> gedefinieerd in Fig. 90), ook de toename van de brosheid van beton bij toenemende sterkte wordt meegenomen via de reductiefactor η<sub>fc</sub> zoals gedefinieerd in fib Model Code 2010:</p>\n<p>\\[f'_{c,lim}=\\alpha\\cdot\\phi_{c}\\cdot k_{c}\\cdot f'_{c}\\]</p>\n<p>\\[k_{c}=\\eta_{fc}\\cdot k_{c2}\\]</p>\n<p>\\[{\\eta _{fc}} = {\\left( {\\frac{{30}}{{{f'_{c}}}}} \\right)^{\\frac{1}{3}}} \\le 1\\]</p>\n<p>waarbij:</p>\n<p><em>α</em> de factor is die rekening houdt met de verschillen tussen de sterkte in het werk en de sterkte van de standaardcilinder volgens CSA A23.3:24 artikel 10.1.6. De opmerking bij dit artikel vereist dat spanningsblokken gebaseerd zijn op spanning-rekkrommen met een piekspanning van ten hoogste 0,9 f'<sub>c</sub>. De factor sluit ook aan bij de eis om de drukspanning in drukdiagonalen te beperken volgens artikel 11.4.2.3. De standaardwaarde is 0,85 en kan worden aangepast in de Projectinstellingen.<br>Opmerking: α is één door de gebruiker gedefinieerde waarde die wordt toegepast op het parabool-rechthoek spanning-rekdiagram. Verwar deze niet met de factor α<sub>1</sub> van de equivalente rechthoekige betonspanningsverdeling, die in CSA A23.3:24 artikel 10.1.7 als sterkteafhankelijk wordt gedefinieerd (α<sub>1</sub> = 0,85 − 0,0015 f'<sub>c</sub> ≥ 0,67, vergelijking 10.1).</p>\n<p><em>Φ</em><em><sub>c</sub></em> is de weerstandsfactor voor beton volgens CSA A23.3:24 artikel 8.4.2. </p>\n<p><em>k</em><em><sub>c2</sub></em> is de reductiefactor als gevolg van de aanwezigheid van dwarsscheuren (compression softening).</p>\n<p><em>f'</em><em><sub>c</sub></em> is de voorgeschreven druksterkte van beton (in MPa voor de definitie van <em>η</em><em><sub>fc</sub></em>).</p>\n<p>Bij gebruik van een parabool-rechthoek spanning-rekdiagram is het noodzakelijk om de maximale drukspanning te reduceren met de factor α. Hierdoor wordt de spanningsverdeling in de drukzone zodanig gemiddeld dat de resulterende druksterkte kleiner is dan of gelijk is aan de druksterkte die wordt berekend met een spanning-rekdiagram met een dalende plastische tak. Een vergelijkbare benadering is gedefinieerd voor de equivalente rechthoekige betonspanningsverdeling in CSA A23.3:24 artikel 10.1.7, waarbij de gefactoreerde betonspanning <em>α</em><em><sub>1</sub></em><em> Φ</em><em><sub>c</sub></em><em> f'</em><em><sub>c</sub></em> wordt verondersteld gelijkmatig verdeeld te zijn over de equivalente drukzone.</p>\n<figure data-asset-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\" data-image-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/085222c7-055a-4870-9bcb-8f18bd65620f/Compression%20softening%20CSFM.PNG\" data-asset-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\" data-image-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 91\\qquad The compression softening law.}}}\\]</em></p>\n<p><em>k</em><em><sub>c2</sub></em> is een reductiefactor die gebaseerd is op dezelfde aannames als de beperking van de drukspanning in drukdiagonalen in CSA A23.3:24 artikel 11.4.2.3, met dit verschil dat in CSFM de aanwezigheid van een hoofdrek (trek) loodrecht op de hoofdspanning (druk) voor elk eindig element wordt getoetst (en niet alleen voor de drukdiagonalen en knoopgebieden van het staafwerkmodel).</p>\n<h3>Beton – Bruikbaarheid</h3>\n<p>De bruikbaarheidsberekening bevat enkele vereenvoudigingen van de constitutieve modellen die voor de sterkteberekening worden gebruikt. De plastische tak van de spanning-rekkromme van beton onder druk wordt buiten beschouwing gelaten, terwijl de elastische tak lineair en oneindig is. De compression softening wordt niet meegenomen. Deze vereenvoudigingen verbeteren de numerieke stabiliteit en de rekensnelheid en verminderen de algemeenheid van de oplossing niet, zolang de resulterende materiaalspanningen in de bruikbaarheidsgrenstoestand duidelijk onder hun vloeigrens blijven (conform de bruikbaarheidseisen van CSA A23.3:24 artikel 9.8 en artikel 18.3 voor voorgespannen staven). De vereenvoudigde modellen die voor bruikbaarheid worden gebruikt, zijn daarom alleen geldig als aan alle toetsingseisen is voldaan.</p>\n<p><strong>Langetermijneffecten</strong></p>\n<p>De langetermijn constitutieve wet wordt gebruikt voor de berekening van de scheurwijdte, de totale doorbuiging en de spanningsbeperking van voorgespannen staven wanneer het langetermijneffect is geselecteerd in het bovenste lint. </p>\n<p>In de IDEA StatiCa Detail applicatie wordt de elasticiteitsmodulus <em>E</em><em><sub>c</sub></em> aangepast om het langetermijngedrag van de constructie te bepalen via de tijdsafhankelijke factor <em>s</em>. De aangepaste elasticiteitsmodulus wordt aangeduid als <em>E</em><em><sub>c,eff</sub></em> – zie Figuur 92.</p>\n<p>Als we aannemen dat de vervorming van het element wordt uitgedrukt als rek, kan worden geschreven:</p>\n<p>\\[\\epsilon_{tot} = \\epsilon_{0} + \\epsilon_{creep} = \\epsilon_{0} \\cdot (1+s)\\]</p>\n<p>waarbij:</p>\n<p><em>ε</em><em><sub>0</sub></em> de kortetermijnrek is (zonder invloed van kruip) en <em>ε</em><em><sub>creep</sub></em> de rek is die door kruip wordt veroorzaakt.</p>\n<p>Met de wet van Hooke kunnen we schrijven:</p>\n<p>\\[E_{c,eff} = \\frac{f_{c}}{\\epsilon_{tot}}\\]</p>\n<p>Door \\(\\epsilon_{tot} = \\epsilon_{0} \\cdot (1+\\xi)\\) en \\(\\epsilon_{0} = f_{c} / E_{c}\\) te substitueren, krijgen we:</p>\n<p>\\[E_{c,eff} = \\frac{E_{c}}{1+s}\\]</p>\n<p>waarbij:<br><em>E</em><em><sub>c</sub></em> de elasticiteitsmodulus van beton is zoals gedefinieerd in CSA A23.3:24 artikel 8.6.2.2<br><em>s</em> de tijdsafhankelijke factor is. Drie factoren kunnen door de gebruiker worden gedefinieerd in het tabblad Materialen en moeten worden bepaald in overeenstemming met de factor voor doorbuigingen onder langdurige belastingen ζ<sub>s</sub> uit CSA A23.3:24 artikel 9.8.2.5 (artikel 9.8.4.4 voor voorgespannen staven):</p>\n<figure data-asset-id=\"a8a59d43-d248-4fda-8c1c-139fe673e512\" data-image-id=\"a8a59d43-d248-4fda-8c1c-139fe673e512\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/9385417d-7ed8-49b6-8fdc-8bb1eab30e5e/Concrete%20stress-strain%20for%20serviceability%20-%20CSA.png\" data-asset-id=\"a8a59d43-d248-4fda-8c1c-139fe673e512\" data-image-id=\"a8a59d43-d248-4fda-8c1c-139fe673e512\" alt=\"\"></figure>\n<p><br><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 92\\qquad Concrete stress-strain diagrams implemented for serviceability analysis: short- and long-term verifications.}}}\\]</em></p>\n<p><strong>Kortetermijneffecten</strong></p>\n<p>Om de kortetermijntoetsingen uit te voeren, wordt een tweede berekening uitgevoerd waarin alle belastingen zonder de tijdsafhankelijke factor worden berekend. Beide berekeningen voor de lange- en kortetermijntoetsingen zijn weergegeven in Fig. 92.</p>\n<h3>Wapening</h3>\n<p>Voor de niet-voorgespannen wapening wordt een perfect elastisch-plastisch spanning-rekdiagram met een gedefinieerde vloeigrens aangehouden. De wapeningsmaterialen zijn gedefinieerd volgens CSA G30.18:21. Voor de definitie van dit diagram zijn alleen de basiseigenschappen van de wapening nodig – de sterkte en de elasticiteitsmodulus.</p>\n<p>Het spanning-rekdiagram van de wapening kan ook door de gebruiker worden gedefinieerd, maar in dat geval is het niet mogelijk het tension stiffening effect mee te nemen (het is niet mogelijk de scheurwijdte te berekenen). </p>\n<figure data-asset-id=\"2d9c6401-28af-4bfe-bc92-1d6f830f7c93\" data-image-id=\"2d9c6401-28af-4bfe-bc92-1d6f830f7c93\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/77dadff9-85d4-402e-94e5-a3725f908933/Steel%20stress-strain%20diagram%20CSFM%20-%20ACI.png\" data-asset-id=\"2d9c6401-28af-4bfe-bc92-1d6f830f7c93\" data-image-id=\"2d9c6401-28af-4bfe-bc92-1d6f830f7c93\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 93 \\qquad Stress-strain diagram of reinforcement}}}\\]</em></p>\n<p>waarbij:</p>\n<p><em>Φ</em><em><sub>s</sub></em><em> </em>de weerstandsfactor voor wapening is volgens CSA A23.3:24 artikel 8.4.3.</p>\n<p><em>f</em><em><sub>sy</sub></em> de vloeigrens van de wapening is</p>\n<p><em>E</em><em><sub>s</sub></em> de elasticiteitsmodulus van de wapening is</p>\n<p>10% is gekozen als de grensrek waarbij de berekening wordt gestopt. Dit wordt als veilig beschouwd op basis van ASTM A955/A955M-20c artikel 7.</p>\n<p>Tension stiffening (Fig. 94) wordt automatisch meegenomen door de invoer van de spanning-rekrelatie van het blote wapeningsstaal aan te passen, zodat de gemiddelde stijfheid van de in het beton ingebedde staven wordt vastgelegd (ε<em><sub>m</sub></em>).</p>\n<figure data-asset-id=\"c9add949-2ad5-4922-8e6c-0d75fb47cb70\" data-image-id=\"c9add949-2ad5-4922-8e6c-0d75fb47cb70\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/c045fcb6-32c6-4a92-aa15-24530fb11484/Tension%20stiffening%20CSFM%20-%20ACI.png\" data-asset-id=\"c9add949-2ad5-4922-8e6c-0d75fb47cb70\" data-image-id=\"c9add949-2ad5-4922-8e6c-0d75fb47cb70\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 94\\qquad Scheme of tension stiffening.}}}\\]</em></p>"
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"value": "<p>De Compatible Stress Field Method voldoet aan moderne ontwerpnormen. Omdat de rekenmodellen uitsluitend standaard materiaaleigenschappen gebruiken, kan het in de ontwerpnormen voorgeschreven formaat met partiële veiligheidsfactoren zonder aanpassing worden toegepast. Op deze manier worden de invoerbelastingen gefactoreerd en de opgegeven materiaaleigenschappen gereduceerd met de bijbehorende weerstandsfactoren, precies zoals bij conventionele betonberekeningen.</p>\n<p>De waarden van de materiaalweerstandsfactoren zijn voorgeschreven in CSA A23.3:24, artikel 8.4. De gefactoreerde betonsterkte is gedefinieerd in artikel 8.4.2 en de gefactoreerde kracht in wapening en spanelementen in artikel 8.4.3. De standaardwaarden in IDEA StatiCa Detail komen overeen met de waarden uit deze artikelen voor constructies die zijn ontworpen volgens artikel 8. Ze zijn dus niet aangepast aan de gereduceerde materiaalfactoren die zijn toegestaan voor prefabbetoncentrales die gecertificeerd zijn volgens artikel 16.1.3. Wordt een dergelijk geval beoordeeld, dan kan de gebruiker de waarden van de weerstandsfactoren wijzigen in de Projectinstellingen.</p>\n<figure data-asset-id=\"27f1f3f8-4d11-4585-bd0b-70180ea21803\" data-image-id=\"27f1f3f8-4d11-4585-bd0b-70180ea21803\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/ddf3d73b-8052-4900-87b7-38242fba4936/Rezistance%20factors%20CSA.png\" data-asset-id=\"27f1f3f8-4d11-4585-bd0b-70180ea21803\" data-image-id=\"27f1f3f8-4d11-4585-bd0b-70180ea21803\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 95\\qquad The setting of resistance factors in IDEA StatiCa Detail.}}}\\]</em></p>\n<p>waarbij:</p>\n<ul>\n <li>Φ<sub>c</sub> de weerstandsfactor voor beton is volgens CSA A23.3:24, artikel 8.4.2.</li>\n <li>Φ<sub>s</sub> de weerstandsfactor is voor wapeningsstaal en ingestorte stalen ankers volgens CSA A23.3:24, artikel 8.4.3 a).</li>\n <li>Φ<sub>p</sub> de weerstandsfactor is voor spanelementen voor voorspanning volgens CSA A23.3:24, artikel 8.4.3 b). </li>\n</ul>\n<p>Naast de materiaalweerstandsfactoren wordt in de Projectinstellingen de factor α gedefinieerd, die rekening houdt met de verschillen tussen de sterkte in het werk en de sterkte van de standaardcilinder. Deze factor wordt beschreven in hoofdstuk 8.1. De standaardwaarde van 0,85 weerspiegelt zowel de eis van CSA A23.3:24, artikel 10.1.6, als de limiet voor de drukspanning in drukdiagonalen uit artikel 11.4.2.3.</p>\n<figure data-asset-id=\"6a37fbcf-6032-4f79-b377-4103af3bbe77\" data-image-id=\"6a37fbcf-6032-4f79-b377-4103af3bbe77\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/62863d25-8bd8-4cd1-b62c-2e29a7fd3382/Clause%2010%20factor%20alfa%20CSA.png\" data-asset-id=\"6a37fbcf-6032-4f79-b377-4103af3bbe77\" data-image-id=\"6a37fbcf-6032-4f79-b377-4103af3bbe77\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 96\\qquad The setting of the factor α in IDEA StatiCa Detail}}}\\]</em></p>\n<p>Belastingfactoren en belastingcombinaties worden bepaald volgens de geldende bouwvoorschriften, zoals vereist in CSA A23.3:24, artikel 8.2.1 en artikel 8.3.1. CSA A23.3:24 zelf specificeert eisen in overeenstemming met de National Building Code of Canada (artikel 1.1). Een aangepaste versie van de bepalingen voor belastingcombinaties is opgenomen in CSA A23.3:24, bijlage C (informatief). De belastingcombinaties voor de uiterste grenstoestanden staan in tabel C.1 a) voor belastinggevallen zonder kraanbelastingen en in tabel C.1 b) voor belastinggevallen met kraanbelastingen. Maatgevend is de combinatie die het meest kritieke effect geeft. Voor de belastingfactoren van de horizontale gronddruk H, de beperkte vervormingseffecten uit voorspanning P en de opgelegde vervorming T gelden volgens artikel C.1.2.4 respectievelijk de waarden 1,5, 1,0 en 1,25.</p>\n<p>De belastingcombinaties voor de bruikbaarheidsgrenstoestanden worden genomen als de combinatie die het ongunstigste effect geeft voor de beschouwde grenstoestand, zoals vereist in artikel 8.3.3.</p>\n<p>Omdat IDEA StatiCa Detail ook wordt gebruikt voor het ontwerp van brugdetails – zoals landhoofdbalken, dwarsdragers en landhoofden – moet worden opgemerkt dat de belastingcombinaties en belastingfactoren van de National Building Code of Canada uit bijlage C gelden voor gebouwen. Voor snelwegbruggen in Canada is het maatgevende document de Canadian Highway Bridge Design Code CSA S6, die eigen belastingen, belastingcombinaties en belastingfactoren definieert. In dergelijke gevallen worden de belastingfactoren en combinaties in plaats daarvan bepaald volgens CSA S6. Het is de verantwoordelijkheid van de gebruiker om de belastingcombinaties te definiëren die relevant zijn voor de te beoordelen constructie en voor de editie van de norm die door de bevoegde autoriteit wordt afgedwongen.</p>\n<figure data-asset-id=\"075cd332-27e0-42ce-9042-ab405cd4453a\" data-image-id=\"075cd332-27e0-42ce-9042-ab405cd4453a\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/2b56741b-83ac-4d27-bc28-cf744cf86d23/Load%20factors%20CSA.png\" data-asset-id=\"075cd332-27e0-42ce-9042-ab405cd4453a\" data-image-id=\"075cd332-27e0-42ce-9042-ab405cd4453a\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 97\\qquad The setting of load factors in Idea StatiCa Detail.}}}\\]</em></p>\n<p>De volgende typen belastinggevallen zijn beschikbaar in de applicatie:</p>\n<ul>\n <li>Voorspanning</li>\n <li>Permanent</li>\n <li>Veranderlijk</li>\n</ul>\n<p>Aan elke combinatie is een grenstoestand toegekend. Deze bepaalt zowel de materiaalmodellen die in de analyse worden gebruikt als de normtoetsingen die worden uitgevoerd:</p>\n<ul>\n <li><strong>UGT</strong> – de sterktetoetsingen die in hoofdstuk 8.3 worden beschreven, worden uitgevoerd met het parabool-rechthoekmodel voor beton.</li>\n <li><strong>BGT</strong> – de bruikbaarheidstoetsingen die in hoofdstuk 8.5 worden beschreven, worden uitgevoerd met de lineaire materiaalmodellen voor beton. Er worden twee subtypen van de bruikbaarheidscombinatie onderscheiden:\n <ul>\n <li><strong>Blijvende belasting</strong> – voorspanning plus permanente belasting, gebruikt voor de toetsing van de spanningsbegrenzing van voorgespannen staven onder blijvende belasting.</li>\n <li><strong>Totale belasting</strong> – voorspanning, permanente en tijdelijke belasting, gebruikt voor de toetsing van de spanningsbegrenzing van voorgespannen staven onder totale belasting.</li>\n </ul>\n </li>\n</ul>\n<p>Voor elke bruikbaarheidscombinatie bepaalt de gebruiker bovendien of de combinatie actief is voor de toetsing van de scheurwijdte en voor de toetsing van de doorbuiging.</p>\n<figure data-asset-id=\"2adb3c90-7b6d-473e-a14b-42f116fff503\" data-image-id=\"2adb3c90-7b6d-473e-a14b-42f116fff503\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/87f20241-0920-4ebc-be8a-8ce069ea59a4/Limit%20state%20definition%20CSA.png\" data-asset-id=\"2adb3c90-7b6d-473e-a14b-42f116fff503\" data-image-id=\"2adb3c90-7b6d-473e-a14b-42f116fff503\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 98\\qquad The setting of load cases and combinations in IDEA StatiCa Detail}}}\\]</em></p>\n<p>Voor elk template bevat het programma vooraf gedefinieerde basiscombinaties die moeten worden aangevuld, afhankelijk van het element dat wordt verwerkt.</p>"
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"value": "<p>De verschillende toetsingen die CSA A23.3:24 vereist, worden beoordeeld op basis van de directe resultaten van het model. De toetsingen worden uitgevoerd voor de betonsterkte, de wapeningssterkte en de verankering (aanhechtingsschuifspanningen).</p>\n<p>De <strong>betonsterkte</strong> op druk wordt bepaald als de verhouding tussen de maximale hoofddrukspanning <em>f</em><em><sub>c</sub></em> (ook <em>σ</em><em><sub>c2</sub></em><em> </em>in de aanvullende resultaten) uit de FE-analyse en de grenswaarde <em>f'</em><em><sub>c,lim</sub></em>.</p>\n<p>De <strong>sterkte van de wapening</strong> wordt zowel op trek als op druk bepaald als de verhouding tussen de spanning in de wapening ter plaatse van de scheuren <em>f</em><em><sub>s</sub></em> en de gespecificeerde grenswaarde <em>f</em><em><sub>sy,lim</sub></em>.</p>\n<p>De <strong>aanhechtingsschuifspanning</strong> wordt afzonderlijk bepaald als de verhouding tussen de aanhechtingsspanning τ<sub>b</sub> uit de FE-analyse en de rekenwaarde van de aanhechtingssterkte <em>f</em><em><sub>bu</sub></em>.</p>\n<p>CSA A23.3:24 definieert de aanhechtingssterkte niet expliciet. Het is echter de grootheid waarvan de bepalingen voor de verankeringslengte in artikel 12 zijn afgeleid, zodat deze eruit kan worden herleid. De afleiding verloopt in enkele stappen.</p>\n<p>De verankeringslengte <em>l</em><em><sub>d</sub></em> is per definitie de inbeddingslengte waarover de aanhechting alleen de volledige vloeikracht van de staaf in het beton overdraagt. Als een staaf over <em>l</em><em><sub>d</sub></em> of meer is ingebed, leidt uittrekken tot vloeien van de staaf in plaats van bezwijken van de aanhechting. Door de kracht die wordt overgedragen door een uniforme aanhechtingsspanning <em>f</em><em><sub>bu</sub></em> op de omtrek van de staaf gelijk te stellen aan de vloeikracht van de staafdoorsnede, volgt:</p>\n<p>\\[\\pi\\cdot d_{b} \\cdot l_{d} \\cdot f_{bu}=f_{yg}\\cdot A_{b}\\]</p>\n<p>waarin:</p>\n<ul>\n <li><em>d</em><em><sub>b</sub></em> de nominale staafdiameter is en <em>π d</em><em><sub>b</sub></em><em> l</em><em><sub>d</sub></em> het aanhechtingsoppervlak;</li>\n <li><em>A</em><em><sub>b</sub></em><em> = π d</em><em><sub>b</sub></em><em><sup>2</sup></em><em> / 4</em> de doorsnede van de staaf is;</li>\n <li><em>f</em><em><sub>yg</sub></em> de gespecificeerde vloeigrens van de wapening is.</li>\n</ul>\n<p>Dit kunnen we schrijven als:</p>\n<p>\\[f_{bu}=\\frac{f_{yg} \\cdot A_{b}}{\\pi \\cdot d_{b} \\cdot l_{d}}=\\frac{f_{yg} \\cdot \\frac{\\pi \\cdot d_{b}^{2}}{4}}{\\pi \\cdot d_{b} \\cdot l_{d}}=\\frac{f_{y} \\cdot d_{b}}{4 \\cdot l_{d}}\\]</p>\n<p>De vereenvoudigde vergelijkingen voor de verankeringslengte op trek van geribbelde staven staan in CSA A23.3:24 Tabel 12.1 (artikel 12.2.3). De applicatie hanteert het tweede, conservatievere geval met de aanduiding <em>Overige gevallen</em>. </p>\n<p>In de applicatie zijn de modificatiefactoren uit CSA A23.3:24 Tabel 12.2 (artikel 12.2.4) k<sub>c</sub> = 1,0, omdat alleen ongecoate wapening wordt ondersteund, en k<sub>a</sub> = 1,0, omdat alleen beton met normale dichtheid wordt ondersteund. De toegepaste uitdrukking is daarom</p>\n<p>\\[l_{d}=0{,}6 \\cdot k_{1} k_{s} k_{g} \\cdot \\frac{f_{y}}{\\sqrt{f'_{c}}} \\cdot d_{b}\\]</p>\n<p>Door deze uitdrukking in de evenwichtsvoorwaarde te substitueren, vervalt de verankeringslengte:</p>\n<p>\\[f_{bu}=\\frac{f_{yg} \\cdot d_{b}}{4 \\cdot 0{,}6 \\cdot k_{1} k_{s} k_{g} \\cdot \\frac{f_{yg}}{\\sqrt{f'_{c}}} \\cdot d_{b}}\\]</p>\n<p>Zowel <em>f</em><em><sub>yg</sub></em> als <em>d</em><em><sub>b</sub></em> komen voor in de teller en in de noemer en vallen daarom weg:</p>\n<p>\\[f_{bu}=\\frac{\\sqrt{f'_{c}}}{4 \\cdot 0{,}6 \\cdot k_{1} k_{s} k_{g}}=\\frac{1}{2{,}4} \\cdot \\frac{\\sqrt{f'_{c}}}{k_{1} k_{s} k_{g}}\\]</p>\n<p>Uitwerking van de constante, 1 / 2,4 = 0,4167, afgerond op 0,42, levert de in de applicatie toegepaste rekenwaarde van de aanhechtingssterkte op:</p>\n<p>\\[f_{bu}=0{,}42 \\cdot \\frac{\\sqrt{f'_{c}}}{k_{1} \\cdot k_{s} \\cdot k_{g}}\\]</p>\n<p>De modificatiefactoren worden bepaald volgens CSA A23.3:24 Tabel 12.2 (artikel 12.2.4):</p>\n<figure data-asset-id=\"f34c70d7-218f-4c0b-b198-8ecc1d1dc177\" data-image-id=\"f34c70d7-218f-4c0b-b198-8ecc1d1dc177\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/5caad6cb-d8bc-4fe6-a92a-35869c377ef5/bond%20modification%20factors%20CSA.png\" data-asset-id=\"f34c70d7-218f-4c0b-b198-8ecc1d1dc177\" data-image-id=\"f34c70d7-218f-4c0b-b198-8ecc1d1dc177\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 99\\qquad Modification factors}}}\\]</em></p>\n<p>Conform CSA A23.3:24 artikel 12.1.2 bedraagt de maximaal toelaatbare waarde van <em>√f'</em><em><sub>c</sub></em> bij de berekening van verankeringslengten 8 MPa.</p>\n<p>De factor <em>k</em><em><sub>1</sub></em> wordt automatisch afgeleid uit de positie van de wapening in het model volgens de onderstaande figuur: </p>\n<figure data-asset-id=\"f0e69ed2-6640-446f-94b0-f06ed48a87d8\" data-image-id=\"f0e69ed2-6640-446f-94b0-f06ed48a87d8\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/bc7cd4e7-291d-4204-83f3-17e9d9cc7b3b/Bond%20conditions%20-%20CSA.png\" data-asset-id=\"f0e69ed2-6640-446f-94b0-f06ed48a87d8\" data-image-id=\"f0e69ed2-6640-446f-94b0-f06ed48a87d8\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 100\\qquad Description of bond conditions; a) b) ‘good’ bond conditions for all bars; c) d) unhatched zone – ‘good’ bond conditions, hatched zone – ‘poor’ bond conditions}}}\\]</em></p>\n<p>Bij slechte aanhechtingsomstandigheden is <em>k</em><em><sub>1</sub></em><em>=1,3</em></p>\n<p>De stortrichting kan in de applicatie worden ingesteld.</p>\n<figure data-asset-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\" data-image-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/e00845bc-3d60-4315-a8b3-67d4a52666a4/Direction%20of%20concreting.png\" data-asset-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\" data-image-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 101\\qquad Direction of concreting}}}\\]</em></p>\n<p>De aanhechtingsschuifspanning en de aanhechtingssterkte van staven op druk worden analoog aan die van staven op trek berekend, maar daarbij worden de vergelijkingen voor de verankeringslengte op druk uit CSA A23.3:24 artikel 12.3.2 gebruikt.</p>\n<p>Het is ook mogelijk om <strong>gladde wapeningsstaven</strong> te modelleren. Meer informatie is hier te vinden: <a data-item-id=\"182f8ba8-899b-44fc-a1c7-59d562ef8c6c\" href=\"\">Gladde wapeningsstaven in Detail</a></p>\n<p><br></p>\n<p><strong>Totale kracht </strong><em><strong>F</strong></em><em><strong><sub>tot</sub></strong></em><strong> en grenskracht </strong><em><strong>F</strong></em><em><strong><sub>lim</sub></strong></em></p>\n<p>De totale kracht <em><strong>F</strong></em><em><strong><sub>tot</sub></strong></em> is een resultaat van de eindige-elementenanalyse en kan op twee manieren worden bepaald.</p>\n<p>\\[F_{tot}=A_{s} \\cdot f_{s}\\]</p>\n<p>waarin <em>A</em><em><sub>s</sub></em> de doorsnede van de wapeningsstaaf is en <em>f</em><em><sub>s</sub></em> de spanning in de staaf.</p>\n<p>Of als de som van de verankeringskracht <em>F</em><em><sub>a </sub></em>en de aanhechtingskracht <em>F</em><em><sub>bond</sub></em><em>.</em></p>\n<p>\\[F_{tot}=F_{a}+F_{bond}\\]</p>\n<p>waarin <em>F</em><em><sub>a</sub></em> de werkelijke kracht in de verankeringsveer is en <em>F</em><em><sub>bond</sub></em> de aanhechtingskracht, die kan worden verkregen door de aanhechtingsspanning <em>τ</em><em><sub>b</sub></em> te integreren over de lengte van de wapeningsstaaf <em>l.</em></p>\n<p>\\[F_{bond}=C_{s} \\cdot \\int_{0}^{l}\\tau_{b}\\left( x \\right)dx\\]</p>\n<p>C<sub>s</sub> is de omtrek van de wapeningsstaaf.</p>\n<p>De grenskracht <em><strong>F</strong></em><em><strong><sub>lim</sub></strong></em> is de maximale kracht in het element van de wapeningsstaaf, rekening houdend met de <strong>sterkte</strong> van de staaf en ook met de <strong>verankeringsomstandigheden </strong>(aanhechting tussen beton en wapening en verankeringshaken, lussen, enzovoort).</p>\n<p>\\[F_{lim}=min\\left( F_{lim,bond}+F_{au},F_{u} \\right)\\]</p>\n<p>\\[F_{u}=f_{y,lim}\\cdot A_{s}\\]</p>\n<p>\\[F_{au}=\\beta\\cdot f_{y,lim}\\cdot A_{s}\\]</p>\n<p>\\[F_{lim,bond}=C_{s}\\cdot l \\cdot f_{bu}\\]</p>\n<p>waarin C<sub>s</sub> de omtrek van de wapeningsstaaf is en <em>l</em> de lengte vanaf het begin van de wapeningsstaaf tot het beschouwde punt.</p>\n<figure data-asset-id=\"d3675eaf-0adb-4512-9366-58e4bdf171b1\" data-image-id=\"d3675eaf-0adb-4512-9366-58e4bdf171b1\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/1a6bbdca-e56b-47e1-a85f-00d4317689a8/Flim.png\" data-asset-id=\"d3675eaf-0adb-4512-9366-58e4bdf171b1\" data-image-id=\"d3675eaf-0adb-4512-9366-58e4bdf171b1\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 102\\qquad Definition of the limit force Flim}}}\\]</em></p>\n<p><br></p>\n<p>\\[F_{lim,2}=F_{lim,1}+F_{lim,add}\\]</p>\n<p>waarin <em>F</em><em><sub>lim,add</sub></em> de extra kracht is die wordt berekend uit de grootte van de hoek tussen naburige elementen. <em>F</em><em><sub>lim,2</sub></em> moet altijd kleiner zijn dan <em>F</em><em><sub>u</sub></em>.</p>\n<p><br></p>\n<p>De beschikbare <strong>verankeringstypen</strong> voor langs- en dwarswapening volgen de haken en buigingen die zijn gedefinieerd in CSA A23.1:19 artikel 6.6.2. Een standaardhaak is in artikel 6.6.2.2 gedefinieerd als een halfronde buiging (180 graden) of een buiging van 90 graden met de voorgeschreven verlenging aan het vrije uiteinde van de staaf. Voor de verankering van beugels en bindbeugels is een buiging van 90 graden of van 135 graden gedefinieerd. De in CSFM beschikbare verankeringstypen voor langswapening zijn een rechte staaf (dus zonder reductie voor het verankeringseinde), een buiging van 90 graden, een buiging van 180 graden, volledige aanhechting en een doorlopende staaf. Al deze typen zijn, samen met de bijbehorende verankeringscoëfficiënten β, weergegeven in Fig. 103.</p>\n<figure data-asset-id=\"85c164c0-d864-4723-8c34-a84a426100b2\" data-image-id=\"85c164c0-d864-4723-8c34-a84a426100b2\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/b76bc446-995d-4d16-8ef9-4aa26671edda/Available%20anchorage%20types%20for%20longitudinal%20rebars.png\" data-asset-id=\"85c164c0-d864-4723-8c34-a84a426100b2\" data-image-id=\"85c164c0-d864-4723-8c34-a84a426100b2\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 103\\qquad Available anchorage types and respective anchorage coefficients for longitudinal reinforcing bars in CSFM:}}}\\]</em></p>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{(a) straight bar; (b) 90-degree hook; (c) 180-degree hook; (d) perfect bond; (e) continuous bar}}}\\]</em></p>\n<p>Opgemerkt moet worden dat CSFM, ondanks de verschillende beschikbare opties, drie typen verankeringseinden onderscheidt: (i) geen reductie van de verankeringslengte, (ii) een reductie van 30% van de verankeringslengte in het geval van een genormaliseerde verankering en (iii) volledige aanhechting. De toegepaste verankeringscoëfficiënten zijn als volgt:</p>\n<p>De verankeringscoëfficiënt voor beugels (beschikbaar voor het Beam-element) is altijd β = 1,0.</p>\n<p>Om aan CSA A23.3 te voldoen, moet in de berekening de verankeringsveer worden gebruikt. De verankeringsveer wordt gewijzigd met de coëfficiënt β, dus de gebruiker moet bij het definiëren van de begin- en eindvoorwaarden van de wapening een van de beschikbare verankeringstypen gebruiken. </p>"
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"value": "<p>Bij het ontwerpen van betonconstructies komen we twee grote groepen gedeeltelijk belaste oppervlakken (PLA) tegen. De eerste groep omvat <strong>opleggingen</strong>, de tweede bestaat uit <strong>verankeringsgebieden</strong>.</p>\n<p>Volgens de huidige normen voor het ontwerp van gewapende betonconstructies moeten bij <strong>opleggingen</strong> het plaatselijk verbrijzelen van het beton en dwarstrekkrachten worden beschouwd. Voor een gelijkmatig verdeelde belasting op een oppervlak A<sub>1</sub> mag de drukcapaciteit van het beton, afhankelijk van het rekenverdelingsoppervlak A<sub>2</sub>, met maximaal een factor twee worden verhoogd. Zie CSA A23.3:24 artikel 10.8.1.</p>\n<p>\\[B_{r}=0{,}85 \\cdot \\Phi_{c} \\cdot f'_{c} \\cdot A_{1}\\]</p>\n<p>behalve wanneer het draagvlak aan alle zijden breder is dan het belaste oppervlak. In dat geval mag de opleggingsweerstand op het belaste oppervlak worden vermenigvuldigd met</p>\n<p>\\[\\sqrt{\\frac{A_{2}}{A_{1}}} \\leq 2\\]</p>\n<p>waarin:</p>\n<ul>\n <li>A<sub>1</sub> het belaste oppervlak is;</li>\n <li>A<sub>2</sub> het rekenverdelingsoppervlak is, gemeten in het vlak dat wordt bepaald door de toelaatbare spreiding van de belasting;</li>\n <li>Φ<sub>c</sub> de weerstandsfactor voor beton is volgens artikel 8.4.2, met een standaardwaarde van 0,65 (zie hoofdstuk 8.2).</li>\n</ul>\n<p>Voor verankeringszones van voorspanning met nagerekt staal moet CSA A23.3:24 artikel 18.13 worden gevolgd. Artikel 18.13.1 vereist dat verankeringszones van voorspanning met nagerekt staal worden ontworpen om de gespecificeerde treksterkte van de spanelementen te weerstaan. Artikel 18.13.2 staat toe dat het ontwerp met een van de volgende methoden wordt uitgevoerd: evenwicht op basis van Staafwerk-modellen volgens artikel 11.4; elastische spanningsanalyse met eindige elementen methoden of een gelijkwaardige methode; methoden op basis van proeven; of vereenvoudigde vergelijkingen waar van toepassing. Artikel 18.13.4 vereist bovendien dat gebieden met spanningsconcentraties door abrupte doorsnedeveranderingen voldoende worden gewapend.</p>\n<p>De tweede van deze methoden is degene die de CSFM vertegenwoordigt. Hierdoor is Detail direct toepasbaar voor het ontwerp van verankeringszones van spanelementen volgens CSA A23.3:24.</p>\n<p>Het gedeeltelijk belaste oppervlak moet voldoende worden voorzien van dwarswapening, ontworpen om de splijtkrachten in dit gebied over te dragen, zoals vereist door artikel 10.8.2. Dit artikel schrijft wapening in oplegzones voor om barst-, splijt- en afschilferkrachten te weerstaan. Zonder de vereiste dwarswapening mag geen verhoging van de drukcapaciteit van het beton worden aangehouden.</p>\n<p><br></p>\n<p><strong>Gedeeltelijk belaste oppervlakken in CSFM</strong></p>\n<figure data-asset-id=\"77fdebe4-afac-4ee7-aee5-716984d4e6d3\" data-image-id=\"77fdebe4-afac-4ee7-aee5-716984d4e6d3\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/3dcea2b1-7700-46f3-a938-4c08204d52e8/Fictitious%20struts.PNG\" data-asset-id=\"77fdebe4-afac-4ee7-aee5-716984d4e6d3\" data-image-id=\"77fdebe4-afac-4ee7-aee5-716984d4e6d3\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 104\\qquad Fictitious struts with concrete finite element mesh.}}}\\]</em></p>\n<p>Met CSFM is het mogelijk om gewapende betonconstructies te ontwerpen en te toetsen, waarbij de invloed van de toenemende drukweerstand van beton in gedeeltelijk belaste oppervlakken wordt meegenomen. Omdat CSFM een wandmodel (2D) is en gedeeltelijk belaste oppervlakken een ruimtelijk (3D) vraagstuk vormen, was het nodig een oplossing te vinden die deze twee verschillende typen vraagstukken combineert (<em>Fig. 105</em>). Als de functie “gedeeltelijk belaste oppervlakken” is geactiveerd, wordt de toelaatbare kegelgeometrie gecreëerd volgens de CSA (<em>Fig. 104</em>). Alle geometrische botsingen worden volledig in 3D opgelost voor de opgegeven geometrie van de betonnen staaf en de afmetingen van elk PLA. Vervolgens wordt een rekenmodel van het gedeeltelijk belaste oppervlak gemaakt.</p>\n<figure data-asset-id=\"05c2e193-bc14-42b5-bc07-da8610febda8\" data-image-id=\"05c2e193-bc14-42b5-bc07-da8610febda8\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/6ae87bd2-682b-4b92-ab1f-4b12e9d3a0df/Cone%20geometry.png\" data-asset-id=\"05c2e193-bc14-42b5-bc07-da8610febda8\" data-image-id=\"05c2e193-bc14-42b5-bc07-da8610febda8\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 105\\qquad Allowable cone geometries.}}}\\]</em></p>\n<p>Het aanpassen van het materiaalmodel bleek een ongeschikte aanpak, vooral omdat het toewijzen van eigenschappen aan de eindige elementen mesh problematisch is. Een aanpak die onafhankelijk is van de eindige elementen mesh bleek een geschiktere oplossing. Voor de bekende geometrie van de drukkegel worden volledig coherente fictieve drukdiagonalen gecreëerd (<em>Fig. 105</em> <em>en Fig. 106</em>). Deze drukdiagonalen hebben dezelfde materiaaleigenschappen als het beton dat in het model wordt gebruikt, inclusief het spanning-rek diagram. De vorm van de kegel bepaalt de richting van de drukdiagonalen, die de belasting geleidelijk over het PLA naar het rekenverdelingsoppervlak verdelen. De oppervlaktedichtheid van de fictieve drukdiagonalen varieert in elk deel van de kegel en voegt een fictief betonoppervlak toe in de belastingsrichting. Ter hoogte van het belaste oppervlak (<em>A</em><em><sub>1</sub></em>) wordt een fictief betonoppervlak toegevoegd volgens de verhouding \\(\\sqrt{A_{1} \\cdot A_{2}} - A_{real}\\) (waarbij <em>A</em><em><sub>real</sub></em> het oppervlak van het steunpunt is dat in het 2D-rekenmodel wordt aangenomen). Dit oppervlak neemt lineair af tot nul in de richting van het rekenverdelingsoppervlak (<em>A</em><em><sub>2</sub></em>). Deze oplossing zorgt ervoor dat de drukspanning in het beton constant is over het volledige kegelvolume.</p>\n<figure data-asset-id=\"aff079fa-74f7-4575-a46b-8e589950238a\" data-image-id=\"aff079fa-74f7-4575-a46b-8e589950238a\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/1dae350c-2f3a-445d-930f-f383e991dcca/Partially%20loaded%20areas%20-%20ACI.png\" data-asset-id=\"aff079fa-74f7-4575-a46b-8e589950238a\" data-image-id=\"aff079fa-74f7-4575-a46b-8e589950238a\" alt=\"\"></figure>\n<p>\\[\\rho \\left( {\\beta ,z} \\right) = \\left( {\\sqrt {\\frac{A_{2}}{A_{1}}} - \\frac{A_{real}}{A_{1}}} \\right)\\,\\cdot\\,\\left( {1 - \\frac{z}{h}} \\right)\\,\\cdot\\,\\frac{1}{{\\cos \\beta }}\\]</p>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 106\\qquad Fictitious struts in the computational model}}}\\]</em></p>\n<p>De weerstand van het gedeeltelijk belaste oppervlak wordt verhoogd volgens de verhouding tussen het rekenverdelingsoppervlak en het belaste oppervlak, zoals vastgelegd in CSA A23.3:24 artikel 10.8.1. Houd er rekening mee dat dit een rekenmodel is dat de spanningstoestand over een gedeeltelijk belast oppervlak, waarvan de werkelijke krachtswerking veel complexer is, niet exact kan beschrijven. Deze oplossing maakt echter een correcte verdeling van de belasting over het hele model mogelijk, met inachtneming van de verhoogde draagcapaciteit van het gedeeltelijk belaste oppervlak. Bovendien worden dwarsspanningen in dit gebied correct geïntroduceerd, zodat de wapening voor splijtkrachten correct kan worden ontworpen.</p>\n<p>De dichtheid wordt zo beperkt dat de maximale verdubbeling van de capaciteit, gegeven door de limiet √(A<sub>2</sub>/A<sub>1</sub>) ≤ 2 uit artikel 10.8.1, niet wordt overschreden.</p>\n<p>Voor de verankeringszones wordt PLA in de applicatie op dezelfde manier gebruikt als voor opleggingen. De drukspanning ter plaatse van het verankeringsmiddel zelf moet daarom handmatig worden getoetst aan de eisen van artikel 18.13. In dit geval wordt PLA gebruikt om te voorkomen dat het rekcriterium in de lokale zone wordt overschreden en de berekening daardoor voortijdig stopt. De wapening die de barst-, splijt- en afschilferkrachten weerstaat, vereist door artikel 10.8.2, en de wapening van de gebieden met spanningsconcentraties, vereist door artikel 18.13.4, kunnen daarentegen direct en met voordeel in de applicatie worden geverifieerd.</p>"
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"value": "<p>De bruikbaarheidsgrenstoestanden van CSA A23.3:24 zijn gedefinieerd in artikel 8.1.4: doorbuigingen volgens artikel 8.1.4.1, lokale schade en scheurvorming volgens artikel 8.1.4.2, en trillingen volgens artikel 8.1.4.3. Bruikbaarheidstoetsingen in Detail worden uitgevoerd voor spanningsbegrenzing, scheurwijdte en doorbuigingslimieten. Trillingen worden niet beoordeeld.</p>\n<p>Alle bruikbaarheidstoetsingen worden uitgevoerd met de lineaire materiaalmodellen die in hoofdstuk 8.1 zijn beschreven, voor de bruikbaarheidscombinaties die in hoofdstuk 8.2 zijn beschreven. Er zijn zowel een kortetermijn- als een langetermijnberekening beschikbaar, waarbij respectievelijk de gemiddelde elasticiteitsmodulus <em>E</em><em><sub>c</sub></em> en de effectieve modulus <em>E</em><em><sub>c,eff</sub></em><sub>,</sub> worden gebruikt.</p>\n<h3>Spanningsbegrenzing</h3>\n<p>De toetsing van de spanningsbegrenzing wordt alleen uitgevoerd voor voorgespannen staven, d.w.z. alleen als het belastinggeval Voorspanning in het model aanwezig is, conform artikel 18.3.</p>\n<p>CSA A23.3:24 vereist dat zowel de spanningen in het beton als in de spanelementen worden getoetst. De toelaatbare spanningen in spanelementen uit artikel 18.4 gelden voor de vijzelkracht en voor de toestand direct na het overdragen van de voorspanning, d.w.z. ze moeten tijdens het spannen worden geverifieerd; dit valt buiten de reikwijdte van het model en wordt daarom niet beoordeeld in Detail.</p>\n<p>De drukspanning in het beton wordt bepaald als de verhouding tussen de maximale hoofddrukspanning <em>f</em><em><sub>c</sub></em><em> = σ</em><em><sub>c2</sub></em> uit de FE-analyse voor bruikbaarheid en de grenswaarde. De limieten zijn die van artikel 18.3.2, dat van toepassing is op spanningen onder gespecificeerde belastingen en voorspanning na aftrek van alle voorspanverliezen:</p>\n<figure data-asset-id=\"9b3ba577-076c-4f6b-a629-a45d9894c473\" data-image-id=\"9b3ba577-076c-4f6b-a629-a45d9894c473\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/83423ed4-9e4e-4819-973b-fadb4306a55e/Compressive%20Stress%20Limits%20in%20Prestressed%20Concrete%20at%20Service%20Limit%20State%20-%20CSA.png\" data-asset-id=\"9b3ba577-076c-4f6b-a629-a45d9894c473\" data-image-id=\"9b3ba577-076c-4f6b-a629-a45d9894c473\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 107\\qquad Concrete compressive stress limits at service loads}}}\\]</em></p>\n<p>In de applicatie wordt <em>voorspanning plus permanente belasting</em> behandeld als een langdurige belasting (Sustain load) en <em>voorspanning, permanente en tijdelijke belasting</em> als een totale belasting (Total load).</p>\n<figure data-asset-id=\"ac528856-0620-4e95-9877-ea4415ba38b5\" data-image-id=\"ac528856-0620-4e95-9877-ea4415ba38b5\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/bf637c0b-a972-4836-a119-45f59df9ed58/Combination%20types%20-%20AASHTO.png\" data-asset-id=\"ac528856-0620-4e95-9877-ea4415ba38b5\" data-image-id=\"ac528856-0620-4e95-9877-ea4415ba38b5\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 108\\qquad Serviceability combination types}}}\\]</em></p>\n<p>De twee limieten komen overeen met de twee subtypen van de bruikbaarheidscombinatie die in hoofdstuk 8.2 zijn beschreven: de combinatie met langdurige belasting wordt getoetst aan 0,45 f'<sub>c</sub> en de combinatie met totale belasting aan 0,60 f'<sub>c</sub>. Beide waarden kunnen worden aangepast in de projectinstellingen en elk van beide toetsingen kan afzonderlijk worden uitgeschakeld.</p>\n<figure data-asset-id=\"fbc3aacf-e2f7-4cc0-b101-2f0fa684380e\" data-image-id=\"fbc3aacf-e2f7-4cc0-b101-2f0fa684380e\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/2c6c69d5-b379-4463-96e4-7552c71aea93/Stress%20limitation%20settings_CSA.png\" data-asset-id=\"fbc3aacf-e2f7-4cc0-b101-2f0fa684380e\" data-image-id=\"fbc3aacf-e2f7-4cc0-b101-2f0fa684380e\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 109\\qquad The setting of the stress limitation parameters}}}\\]</em></p>\n<p>De spanning in het beton wordt berekend met lineaire spanning-rek-diagrammen. Voor elke bruikbaarheidscombinatie worden twee berekeningen uitgevoerd:</p>\n<ul>\n <li><strong>korte termijn</strong> – de gemiddelde elasticiteitsmodulus van het beton E<sub>c</sub> wordt gehanteerd;</li>\n <li><strong>lange termijn</strong> – de effectieve elasticiteitsmodulus van het beton E<sub>c,eff</sub>, inclusief kruip, wordt gehanteerd voor alle rekenstappen, d.w.z. voor het aanbrengen van de voorspanning, de permanente belastingen en de tijdelijke belastingen.</li>\n</ul>\n<p>De kruipcoëfficiënten worden gedefinieerd als materiaaleigenschappen, zoals beschreven in hoofdstuk 8.1. Als de resultaten voor de werkelijke betonspanning f<sub>c</sub> worden weergegeven, is de functionaliteit voor het aanpassen van de paletlimiet beschikbaar.</p>\n<figure data-asset-id=\"b9ef304f-a055-4045-9ba8-6b74c494bddb\" data-image-id=\"b9ef304f-a055-4045-9ba8-6b74c494bddb\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/de69aea9-f032-4643-9b9b-dcd15607b9f5/Results%20of%20the%20stress%20limitation%20check_CSA.png\" data-asset-id=\"b9ef304f-a055-4045-9ba8-6b74c494bddb\" data-image-id=\"b9ef304f-a055-4045-9ba8-6b74c494bddb\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 110\\qquad Results of the stress limitation check}}}\\]</em></p>\n<h3>Decompressie</h3>\n<p>Detail biedt ook een decompressietoetsing voor de voorspanwapening als een spanelement in het model is gedefinieerd. De toetsing verifieert dat het beton ter hoogte van het spanelement onder druk blijft, wat de voorwaarde is die ten grondslag ligt aan de trekspanningslimiet van artikel 18.3.2 c) voor de voorgedrukte getrokken zone.</p>\n<h3>Doorbuiging</h3>\n<p>CSA A23.3:24 vereist de berekening van drie verschillende typen doorbuiging, conform artikel 9.8.2 voor niet-voorgespannen staven en artikel 9.8.4 voor voorgespannen staven:</p>\n<ul>\n <li><strong>Onmiddellijke</strong> – de doorbuiging veroorzaakt door de gedefinieerde belastingcombinatie, berekend met de gemiddelde elasticiteitsmodulus van het beton E<sub>c</sub>, conform artikel 9.8.2.2;</li>\n <li><strong>Langetermijn</strong> – de aanvullende doorbuiging veroorzaakt door de effecten van kruip en krimp, conform artikel 9.8.2.5;</li>\n <li><strong>Totale</strong> – de doorbuiging veroorzaakt door de gedefinieerde belastingcombinatie, berekend met de effectieve elasticiteitsmodulus E<sub>c,eff</sub>. De effectieve modulus wordt voor elke rekenstap – voorspanning, eigen gewicht en veranderlijke belasting – afzonderlijk berekend, met de kruipcoëfficiënten die in de betoneigenschappen zijn gedefinieerd, zoals beschreven in hoofdstuk 8.1.</li>\n</ul>\n<p>De langetermijndoorbuiging is het verschil tussen de totale en de onmiddellijke doorbuiging.</p>\n<p>De berekening wordt uitgevoerd voor alle combinaties die actief zijn voor de doorbuigingstoetsing, waarbij lineaire spanning-rek-diagrammen voor beton met de relevante elasticiteitsmodulus worden gehanteerd.</p>\n<p>De resultaten worden weergegeven in de doorbuigingstoetsing. De gebruiker activeert of deactiveert de normtoetsing voor de totale en de onmiddellijke doorbuiging en definieert voor beide de grenswaarde.</p>\n<figure data-asset-id=\"94e2d55d-6acc-4677-9eba-71eb8230fd37\" data-image-id=\"94e2d55d-6acc-4677-9eba-71eb8230fd37\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/8a3f43f8-bb03-4f34-a2a6-a40da697f9af/Deflection%20check%20CSA.png\" data-asset-id=\"94e2d55d-6acc-4677-9eba-71eb8230fd37\" data-image-id=\"94e2d55d-6acc-4677-9eba-71eb8230fd37\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 111\\qquad Maximum allowable deflection value}}}\\]</em></p>\n<p>De gegevensweergave toont een tabel met de resultaten van de onmiddellijke, de langetermijn- en de totale doorbuiging voor de geselecteerde belastingcombinatie.</p>\n<p>De maximaal toelaatbare berekende doorbuigingen zijn gegeven in CSA A23.3:24 tabel 9.3, die onderscheid maakt naar het type staaf en de te beschouwen doorbuiging.</p>\n<p>De grenswaarden moeten door de gebruiker worden bepaald op basis van het type element dat wordt geanalyseerd. Houd er rekening mee dat de limieten van tabel 9.3 voor het derde en vierde type staaf verwijzen naar het deel van de totale doorbuiging dat optreedt na het aanbrengen van de niet-constructieve elementen, terwijl Detail de totale doorbuiging vanaf het begin van de belastingsgeschiedenis weergeeft. De opmerkingen bij tabel 9.3 staan toe dat de langetermijndoorbuiging wordt verminderd met de hoeveelheid waarvan wordt berekend dat deze vóór dat aanbrengen optreedt; deze reductie is de verantwoordelijkheid van de gebruiker.</p>\n<h3>Scheurwijdte</h3>\n<p>CSA A23.3:24 beheerst scheurvorming indirect in artikel 10.6.1 door middel van de scheurbeheersingsparameter z. De opmerking bij artikel 8.1.4.2 stelt expliciet dat \"this Standard does not specifically limit crack widths\", en de opmerking bij artikel 10.6.1 voegt toe dat de eisen van dat artikel mogelijk onvoldoende zijn voor constructies die worden blootgesteld aan zeer agressieve omstandigheden of die waterdicht moeten worden ontworpen.</p>\n<p>De parameter z vertegenwoordigt een potentiële mate van scheurvorming in plaats van een fysieke scheurwijdte. Het onderliggende verband houdt geen rekening met de scheurafstand, die niet betrouwbaar kan worden voorspeld, zodat een waarde van z niet kan worden omgerekend naar een scheurwijdte en daarmee kan worden vergeleken.</p>\n<p>Aangezien CSA A23.3:24 geen procedure geeft voor de berekening van een scheurwijdte, bepaalt Detail de scheurwijdte met de algemene CSFM-procedure, die onafhankelijk is van de ontwerpnorm: de gemiddelde rekken die langs de wapeningsstaven worden berekend, worden geïntegreerd over de scheurafstand die volgt uit het Tension Chord Model. De scheurwijdtetoetsing wordt uitgevoerd voor alle bruikbaarheidscombinaties waarvoor de eigenschap \"Actief voor de scheurwijdtetoetsing\" is geactiveerd.</p>\n<p>De berekeningen worden uitgevoerd met lineaire spanning-rek-diagrammen voor beton. Voor elke bruikbaarheidscombinatie die aan de scheurwijdtetoetsing is toegewezen, worden de hierboven beschreven kortetermijn- en langetermijnberekeningen uitgevoerd.</p>\n<p>De toetsing wordt als volgt weergegeven:</p>\n<p>\\[\\frac{w}{w_{lim}}\\]</p>\n<p>waarin:</p>\n<ul>\n <li>w de kortetermijn- of langetermijnscheurwijdte is, berekend met de FE-analyse;</li>\n <li>w<sub>lim</sub> de door de gebruiker gedefinieerde grenswaarde van de scheurwijdte is.</li>\n</ul>\n<p>Aangezien CSA A23.3 geen grenswaarden voor scheurwijdten voorschrijft, moet de gebruiker w<sub>lim</sub> specificeren op basis van de blootstellingsomstandigheden en de eisen van het project.</p>\n<p>Richtlijnen zijn te vinden in CSA S6:14, Canadian Highway Bridge Design Code, artikel 8.12.3.1 en tabel 8.6, die de maximale scheurwijdten in bruikbaarheidsgrenstoestanden specificeert voor het toepasselijke type constructief onderdeel en de toepasselijke blootstelling:</p>\n<figure data-asset-id=\"6314246f-310c-4b09-97b2-62afb0553116\" data-image-id=\"6314246f-310c-4b09-97b2-62afb0553116\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/29ed7393-c339-4488-9451-9a3c3299a02c/Wlim%20recomendation%20CSA.png\" data-asset-id=\"6314246f-310c-4b09-97b2-62afb0553116\" data-image-id=\"6314246f-310c-4b09-97b2-62afb0553116\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 112\\qquad Reasonable crack widths for reinforced concrete under service load}}}\\]</em></p>\n<p>Deze waarden zijn vastgesteld voor verkeersbruggen. Voor gebouwen die zijn ontworpen volgens CSA A23.3 vormen ze geen verplichte eis, maar ze vertegenwoordigen de limieten die in de Canadese ontwerppraktijk worden gehanteerd voor de overeenkomstige blootstellingsomstandigheden.</p>\n<p>Ze komen ook overeen met de grootheid die de applicatie bepaalt. CSA S6:14 artikel 8.12.3.2 berekent de scheurwijdte als w = k<sub>b</sub> β<sub>c</sub> s<sub>rm</sub> ε<sub>sm</sub></p>\n<p>d.w.z. als het product van een scheurafstand <em>s</em><em><sub>rm</sub></em> en een gemiddelde wapeningsrek <em>ε</em><em><sub>sm</sub></em>, wat dezelfde grootheid is die de CSFM verkrijgt door de berekende rekken langs de wapeningsstaven te integreren over de scheurafstand. De limieten van tabel 8.6 zijn daarom gedefinieerd voor een scheurwijdte van dezelfde aard als die welke door de applicatie wordt weergegeven, in tegenstelling tot een limiet die is afgeleid van een parameter voor de verdeling van de staven, zoals <em>z</em>.</p>"
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"value": "<p>De Compatible Stress Field Method (CSFM) is een rekenmethode op basis van 2D-vlakspanningen, waarbij beton wordt gemodelleerd met 2D-eindige elementen waaraan 1D-wapeningselementen zijn gekoppeld via randvoorwaarden. Daarnaast kunnen speciale typen 1D-elementen aan het model worden toegevoegd die voorspanwapening met aanhechting representeren. Deze kunnen worden gemodelleerd als voorgerekt en als nagerekt.</p>\n<p>Voorspanwapening wordt op dezelfde wijze gemodelleerd als gewone wapening, met lineaire elementen die de normaalkracht overdragen. Elk afzonderlijk voorspanwapeningselement wordt gekarakteriseerd door zijn oppervlakte en materiaaleigenschappen. Deze eigenschappen volgen uit de karakteristieke materiaalkromme volgens de toegepaste norm (EN 1992-1-1, ACI 318-19, enz.)</p>\n<p><strong>EUROCODE</strong></p>\n<p>Spanning-rekdiagram van voorspanwapening: a) spanning-rekdiagram zoals gedefinieerd in EN 1992-1-1; b) initiële rek voor voorgerekte wapening</p>\n<figure data-asset-id=\"7d9fac4b-fa97-49d3-a624-ddfab1bf7dee\" data-image-id=\"7d9fac4b-fa97-49d3-a624-ddfab1bf7dee\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/aa25e678-c691-4887-9f8f-b5ae0c4a4fb2/prestressing%20model_Detail_01.png\" data-asset-id=\"7d9fac4b-fa97-49d3-a624-ddfab1bf7dee\" data-image-id=\"7d9fac4b-fa97-49d3-a624-ddfab1bf7dee\" alt=\"\"></figure>\n<p><strong>ACI</strong></p>\n<p>Spanning-rekdiagram van voorspanwapening: a) spanning-rekdiagram; b) initiële rek voor voorgerekte wapening</p>\n<figure data-asset-id=\"7b26f280-9951-4255-98c4-90f558de030f\" data-image-id=\"7b26f280-9951-4255-98c4-90f558de030f\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/1c112ef0-c06a-4141-9d09-1e3cfa42d079/prestressing%20model_Detail__ACI.png\" data-asset-id=\"7b26f280-9951-4255-98c4-90f558de030f\" data-image-id=\"7b26f280-9951-4255-98c4-90f558de030f\" alt=\"\"></figure>\n<p><br></p>\n<p>De wapeningselementen worden via een aanhechtingsmodel verbonden met de 2D-elementen van het betonmodel, op dezelfde manier als de klassieke betonwapening. </p>\n<ul>\n <li>Lees <a data-item-id=\"85424e98-41cd-4bdd-a978-e4b540a10be5\" href=\"\">Typen eindige elementen</a></li>\n</ul>\n<p>De elementen van het aanhechtingsmodel laten een relatieve vervorming van de voorspanwapening en het beton toe met geschikte niet-lineaire eigenschappen. Hiermee worden de hechting van de wapening aan het beton en ook het verankeringsmodel van de voorgerekte wapening correct gemodelleerd. De uiteindelijke voorzieningen van de nagerekte wapening, bijvoorbeeld de ankerplaat, worden gemodelleerd door een element met een stijfheid die overeenkomt met het anker aan het uiteinde van de voorspanwapening. De voorspankracht aan het uiteinde wordt als vlaklast op het betonmodel aangebracht over een oppervlak ter grootte van de ankerplaat. Het model kan de lokale drieassige spanningstoestand in het gebied onder het anker niet correct beschrijven. Dit gebied moet afzonderlijk worden beschouwd. </p>\n<p>De tension stiffening van de wapening als gevolg van interactie met het beton wordt bij voorspanwapening niet in rekening gebracht, omdat wordt aangenomen dat het beton in de nabijheid van de voorspanwapening op druk wordt belast.</p>\n<h2>Voorgerekte wapening</h2>\n<p>De voorgerekte wapening wordt voorgespannen vóór het storten van het element. De voorspanwapening wordt bijna altijd recht aangebracht, waardoor er geen wrijvingsverliezen optreden. Zodra de vereiste betonsterkte is bereikt, wordt de wapening losgelaten van de ankerblokken. Daardoor wordt de voorspanwapening geactiveerd en worden de krachten van de wapening op het beton overgedragen. Dit effect is fysisch equivalent aan onderkoeling van de wapening en wordt gemodelleerd met een initiële rek, vergelijkbaar met een thermische belasting. Dit levert een spanning-rekdiagram van de voorspanwapening op zoals getoond in de bovenstaande figuur onder b). Het rekenmodel berekent automatisch de vervormingsrespons van de constructie op de aangebrachte voorspanning en bepaalt daarmee direct de voorspanverliezen door elastische rek van het element.</p>\n<p>Omdat de voorspankracht bekend is, en daarmee ook de voorspanning <em>σ</em><em><sub>pmo</sub></em>, wordt het materiaaldiagram van de wapening gebruikt voor de afhankelijkheid van de spanning van de vervorming. Dit kan worden geschreven als:</p>\n<p><em>\\[{{σ}_{p}}=~{{f}}({{ε}}-{{ε}_{0}})\\]</em></p>\n<p>Uitgaande van een voorspanning in de wapening die lager is dan de vloeigrens (d.w.z. de voorwaarden gedefinieerd in EN 1992-1-1, hoofdstuk 5.10.3 zijn vervuld), kan de initiële vervorming ook worden berekend als:</p>\n<p><em>\\[{{ε}_{0}}=\\frac{{{σ}_{pm0}}}{{{E}_{p}}}\\]</em></p>\n<p><em>ε</em><em><sub>0</sub></em> - initiële rek door voorspanning<br><em>σ</em><em><sub>pm0</sub></em> - spanning vlak vóór het loslaten<br><em>E</em><em><sub>p</sub></em> - elasticiteitsmodulus van de voorspanwapening</p>\n<p>Voorgerekte wapening is specifiek doordat de verankering van de uiteinden via meerdere mechanismen verloopt: adhesie van wapening en beton op moleculair niveau, de wrijving die ontstaat op het contactvlak tussen het oppervlak van de wapening en het beton, het mechanisch indrukken van de spiraalwapening in het beton en een toename van de diameter van de voorspanwapening, bekend als het wigmechanisme of het Hoyer-effect. De genoemde effecten worden in het CSFM-rekenmodel meegenomen door de eigenschappen van het verankeringsmodel in het uiteindegebied van de voorgerekte wapening aan te passen.</p>\n<p>Interactie van voorgerekte wapening en beton: a) spiraalwapening die in het beton wordt gedrukt; b) Hoyer-effect</p>\n<figure data-asset-id=\"cd6cee68-68e6-44b3-921a-4ccf8cd4df35\" data-image-id=\"cd6cee68-68e6-44b3-921a-4ccf8cd4df35\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/035bbeed-4b37-4477-b848-8ee98b174f72/prestressing%20model_Detail_02.png\" data-asset-id=\"cd6cee68-68e6-44b3-921a-4ccf8cd4df35\" data-image-id=\"cd6cee68-68e6-44b3-921a-4ccf8cd4df35\" alt=\"\"></figure>\n<h2>Nagerekte wapening</h2>\n<p>De nagerekte wapening wordt voorgespannen nadat de constructie is gestort. Het voorspanapparaat steunt rechtstreeks op de constructie, waardoor de verliezen door elastische rek van de constructie als gevolg van de voorspanning worden uitgesloten. Zodra de gewenste voorspankracht is bereikt, wordt de wapening verankerd en worden vervolgens de kabelkanalen geïnjecteerd, waardoor aanhechting van de wapening aan de constructie ontstaat. Bij het modelleren van nagerekte wapening wordt de berekening daarom opgedeeld in meerdere belastingstappen: voorspanning, aanbrengen van overige permanente belastingen en aanbrengen van veranderlijke belastingen.</p>\n<p>Eindige-elementenmesh van het beton met gekoppelde 1D-voorspanwapeningselementen:</p>\n<figure data-asset-id=\"3b267c80-ee0e-457f-af00-f74c91a48d7d\" data-image-id=\"3b267c80-ee0e-457f-af00-f74c91a48d7d\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/a028db63-b458-44e7-945b-bedabb1a6785/prestressing%20model_Detail_03.png\" data-asset-id=\"3b267c80-ee0e-457f-af00-f74c91a48d7d\" data-image-id=\"3b267c80-ee0e-457f-af00-f74c91a48d7d\" alt=\"\"></figure>\n<h4>Belastingstap \"voorspanning\"</h4>\n<p>Bij het voorspannen van de wapening wordt de stijfheid van de wapening niet opgenomen in de stijfheid van de constructie. In deze belastingstap wordt de stijfheid van het lineaire element niet in het model meegenomen. De wapeningselementen worden vervangen door een vervangende belasting die overeenkomt met de voorspanning en het wapeningsoppervlak, zoals getoond in de bovenstaande figuur. Nadat de volledige belasting uit de voorspanning is bereikt en deze belastingstap is geconvergeerd, wordt de vervorming van het specifieke lineaire element afgelezen. Op basis van de vervorming wordt de initiële rek <em>ε</em><em><sub>0</sub></em> van de afzonderlijke lineaire elementen van de voorspanwapening bepaald.</p>\n<p>De voorspanning kan handmatig langs de lengte van de wapening worden gedefinieerd of automatisch worden berekend op basis van de geometrie van de wapening. Als de automatische berekening van verliezen is gekozen, worden wrijvingsverlies (volgens EN 1992-1-1, 5.10.5.2, of ACI 318-19, 20.3.2) en slip van de wapening (indrukken van ankerwiggen) tijdens het verankeren in rekening gebracht. Omdat alle voorspanwapening in één stap wordt aangebracht, wordt verlies door opeenvolgend voorspannen niet beschouwd.</p>\n<h4>Volgende belastingstappen met geactiveerde voorspanwapening</h4>\n<p>In de volgende belastingstappen (aanbrengen van overige permanente en veranderlijke belastingen) wordt dezelfde procedure gevolgd als bij voorgerekte wapening. De volledige stijfheid van de voorspanwapening wordt in rekening gebracht, de aanhechting tussen de wapening en het omringende beton wordt meegenomen en het spanning-rekdiagram van de voorspanwapening wordt aangepast met de initiële rek <em>ε</em><em><sub>0</sub></em>. Deze rek is voor elk element verschillend en is verkregen uit de voorgaande belastingstap \"voorspanning\". Door de aanhechting van de wapening aan het beton wordt de verandering van de voorspanning door de elastische vervorming van de constructie onder de uitwendige belasting correct in het model meegenomen.</p>"
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"value": "<p><br></p>\n<p>De theoretische achtergrond is gebaseerd op COMPATIBLE STRESS FIELD DESIGN OF STRUCTURAL CONCRETE<br>(Kaufmann et al., 2020)</p>\n<h2>Constructief ontwerp van betondiscontinuïteiten in IDEA StatiCa Detail</h2>\n<h3>1 Inleiding in de CSFM-methode</h3>\n<p><a href=\"#general-introduction\">1.1 Algemene inleiding in het constructief ontwerp van betondetails</a><br><a href=\"#main-assumptions-and-limitations\">1.2 Belangrijkste aannames en beperkingen</a><br><a href=\"#design-tools-for-reinforcement\">1.3 Ontwerptools voor wapening</a></p>\n<h3>2 Rekenmodel van IDEA StatiCa Detail</h3>\n<p><a href=\"#introduction-to-finite-element-implementation\">2.1 Inleiding in de implementatie van eindige elementen</a><br><a href=\"#supports-and-load-transmitting-components\">2.2 Opleggingen en belastingoverdragende componenten</a><br><a href=\"#load-transfer-at-trimmed-ends-of-beams\">2.3 Krachtsoverdracht bij afgekapte uiteinden van liggers</a><br><a href=\"#geometric-modification-of-cross-sections\">2.4 Geometrische aanpassing van doorsneden</a><br><a href=\"#finite-element-types\">2.5 Typen eindige elementen</a><br><a href=\"#meshing\">2.6 Meshing</a><br><a href=\"#solution-method-and-load-control-algorithm\">2.7 Oplossingsmethode en belastingsgestuurd algoritme</a><br><a href=\"#presentation-of-results\">2.8 Weergave van resultaten</a></p>\n<h3>3 Modelverificatie</h3>\n<p><a href=\"#limit-states-and-crack-width-calculation\">3.1 Grenstoestanden, scheurwijdteberekening en tension stiffening</a></p>\n<h3>4 Constructieve toetsingen volgens EUROCODE</h3>\n<p><a href=\"#material-models-en\">4.1 Materiaalmodellen (EN)</a><br><a href=\"#safety-factors\">4.2 Veiligheidsfactoren</a><br><a href=\"#ultimate-limit-state-analysis\">4.3 Analyse van de uiterste grenstoestand</a><br><a href=\"#partially-loaded-areas\">4.4 Deels belaste gebieden (PLA)<br></a><a href=\"#serviceability-limit-state-analysis\">4.5 Analyse van de bruikbaarheidsgrenstoestand</a></p>\n<h3>5 Constructieve toetsingen volgens ACI 318-19</h3>\n<p><a href=\"#material-models-aci\">5.1 Materiaalmodellen (ACI)</a><br><a href=\"#strength-reduction-and-load-factors\">5.2 Sterktereductie- en belastingfactoren</a><br><a href=\"#strength-verifications\">5.3 Sterktetoetsingen</a><br><a href=\"#bearing-and-anchorage-zones-partially-loaded-areas\">5.4 Oplegging- en verankeringszones - Deels belaste gebieden<br></a><a href=\"#serviceability-verifications\">5.5 Bruikbaarheidstoetsingen</a></p>\n<h3>6 Constructieve toetsingen volgens AASHTO</h3>\n<p><a href=\"#material-models-aahsto\">6.1 Materiaalmodellen (AASHTO)</a><br><a href=\"#resistance-and-load-factors\">6.2 Weerstands- en belastingfactoren</a><br><a href=\"#strength-limit-state\">6.3 Grenstoestand voor sterkte</a><br><a href=\"#bearing-and-anchorage-zones-resistance-partially-loaded-areas\">6.4 Weerstand van oplegging- en verankeringszones – Deels belaste gebieden</a><br><a href=\"#service-limit-state\">6.5 Bruikbaarheidsgrenstoestand</a></p>\n<h3>7 Constructieve toetsingen volgens AS 3600</h3>\n<p><a href=\"#material-models-aus\">7.1 Materiaalmodellen (AUS)</a><br><a href=\"#stress-reduction-and-load-factors\">7.2 Spanningsreductie- en belastingfactoren</a><br><a href=\"#strength-and-anchorage-verifications\">7.3 Sterkte- en verankeringstoetsingen</a><a href=\"#bearing-and-anchorage-zones-partially-loaded-areas\"><br></a><a href=\"#serviceability-checks\">7.4 Bruikbaarheidstoetsingen</a></p>\n<h3>8 Constructieve toetsingen volgens CSA A23.3:24</h3>\n<p><a href=\"#material-models-csa\">8.1 Materiaalmodellen (CSA)</a><br><a href=\"#resistance-and-load-factors-csa\">8.2 Weerstands- en belastingfactoren</a><br><a href=\"#ultimate-limit-state-csa\">8.3 Uiterste grenstoestand</a><br><a href=\"#bearing-and-anchorage-zones-resistance-partially-loaded-areas-csa\">8.4 Weerstand van oplegging- en verankeringszones – Deels belaste gebieden</a><br><a href=\"#serviceability-limit-state-csa\">8.5 Bruikbaarheidsgrenstoestand</a></p>\n<p><br></p>\n<h3><a href=\"#prestressing-in-detail-model-description\">9 Voorspanning in Detail - Modelbeschrijving</a></h3>\n<p><br></p>\n<p><br></p>\n<h2>1 Inleiding in de CSFM-methode</h2>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"n27e5ceb5_fe99_012a_949d_acfef5af50ee\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___general\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"n3cd1a150_2eeb_0108_47fd_33270a047501\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___main_assumptions_a\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"n3936277a_8378_01c6_4f3c_ca5b41e3c90a\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___general___reinforc\"></object>\n<h1><br></h1>\n<h2>2 Rekenmodel van IDEA StatiCa Detail</h2>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"dab79f23_0f63_019f_ec30_fdf52018a612\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___general___finite_e\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"n2bf9e88e_9f55_0185_625e_97c549378963\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___supports_and_load_\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"n793d9504_1df5_01b5_5681_d50ba0d6000f\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___load_transfer_at_t\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"bdb219b1_e3d4_018b_2868_14d83f2ed5f4\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___geometric_modifica\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"e4043d87_e123_014e_0afe_165906f09559\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___finite_element_typ\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"n2bfb4459_7177_019d_11fd_630319d04c6d\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___meshing\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"n51b1d71a_6358_013e_b4b2_968ffd2cfa9b\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___solution_method_an\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"n58732c1e_26a2_010a_2d98_254eab294b3e\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___presentation_of_re\"></object>\n<h1><br></h1>\n<h2>3 Modelverificatie</h2>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"f6b2313c_260f_01c4_e283_bd9a657bedc8\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___general___verifica\"></object>\n<h1><br></h1>\n<h2>4 Constructieve toetsingen volgens Eurocode</h2>\n<p>De beoordeling van de constructie met CSFM wordt uitgevoerd met twee verschillende analyses: één voor de bruikbaarheid en één voor de belastingcombinaties in de uiterste grenstoestand. De bruikbaarheidsanalyse gaat ervan uit dat het uiterste gedrag van het element voldoende is en dat de vloeicriteria van het materiaal bij belastingen op bruikbaarheidsniveau niet worden bereikt. Deze aanpak maakt het mogelijk om vereenvoudigde constitutieve modellen (met een lineaire tak van het spanning-rekdiagram van beton) te gebruiken voor de bruikbaarheidsanalyse. Dit verbetert de numerieke stabiliteit en de rekensnelheid.</p>\n<p><br></p>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"becc4dc4_31c1_015a_24bd_6fd5781aa38d\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___material_models__e\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"n80e5e573_65e7_01e2_efa9_64968112c891\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___safety_factors\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"n0b398e51_fa89_01ca_8900_6782bf33a122\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___ultimate_limit_sta\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"n44edb482_456c_017b_9c79_d10fda6588f1\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___partially_loaded_a\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"n9dcc9e8d_671f_01e8_2b3d_7869f9da5d56\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___serviceability_lim\"></object>\n<h1><br></h1>\n<h2>5 Constructieve toetsingen volgens ACI 318-19</h2>\n<p>De beoordeling van de constructie met de CSFM wordt uitgevoerd met twee verschillende analyses: één voor de bruikbaarheid en één voor de sterktebelastingcombinaties. De bruikbaarheidsanalyse gaat ervan uit dat het gedrag onder gefactoreerde belastingen voldoende is en dat de vloeicriteria van het materiaal bij belastingen op bruikbaarheidsniveau niet worden bereikt. Deze aanpak maakt het mogelijk om vereenvoudigde constitutieve modellen (met een lineaire tak van het spanning-rekdiagram van beton) te gebruiken voor de bruikbaarheidsanalyse. Dit verbetert de numerieke stabiliteit en de rekensnelheid.</p>\n<p>CSFM is in overeenstemming met ACI 318-19, hoofdstuk 6.8.1.1. Om te voldoen aan de eisen van ACI 318-19, sectie 6.8.1.2, zijn aan diverse universiteiten uitgebreide verificatietests uitgevoerd. Afzonderlijke artikelen waarin de resultaten van verificatie en validatie zijn samengevat, zijn te vinden via de volgende link.</p>\n<ul>\n <li><a href=\"https://www.ideastatica.com/support-center-verifications?label=detail\">Verificaties: Detail 2D</a></li>\n</ul>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"e9ab9075_b7f3_0153_7fc4_15fc78797b89\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___material_models__a\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"n4161570e_a5d2_01f8_dad0_fa76fd2980cd\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___factors___aci\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"n5909e1f2_f63c_01c2_7825_0ef451e607b8\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___strength_analysis_\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"aca373b1_2142_0114_6c22_85448fa93541\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___bearing_and_anchor\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"f33091cd_bad1_0142_78bd_f7436149143f\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___serviceability_ver\"></object>\n<h1><br></h1>\n<h2>6 Constructieve toetsingen volgens AASHTO</h2>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"fec9abc1_4496_01a8_532d_724d9a784657\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___material_models__a_0b99d24\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"eaa60bb0_481e_019f_988e_f52bef1a4d86\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___resistance_and_loa\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"a17b4261_8a06_01ff_9079_248b3cb95255\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___strength_limit_sta\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"n00e48069_a41c_0115_a827_176fc8675a9e\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___bearing_and_anchor_69cbe39\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"b54485e8_ca3f_01d6_a3dc_554a7cc0a2df\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___service_limit_stat\"></object>\n<h1><br></h1>\n<h2>7 Constructieve toetsingen volgens de Australische norm AS 3600 (2018)</h2>\n<p>De beoordeling van de constructie met de CSFM wordt uitgevoerd met twee verschillende analyses: één voor de bruikbaarheid en één voor de sterktebelastingcombinaties. De bruikbaarheidsanalyse gaat ervan uit dat het gedrag onder gefactoreerde belastingen voldoende is en dat de vloeicriteria van het materiaal bij belastingen op bruikbaarheidsniveau niet worden bereikt. Deze aanpak maakt het mogelijk om vereenvoudigde constitutieve modellen (met een lineaire tak van het spanning-rekdiagram van beton) te gebruiken voor de bruikbaarheidsanalyse. Dit verbetert de numerieke stabiliteit en de rekensnelheid.</p>\n<p>De CSFM is een constructieve analysemethode die voldoet aan de algemene regels in hoofdstuk 6.1.1 en 6.1.2 en wordt gedefinieerd als (f) niet-lineaire spanningsanalyse in hoofdstuk 6.1.3 - verder uitgewerkt in hoofdstuk 6.6. </p>\n<p>De analyse met de CSFM houdt rekening met alle relevante niet-lineaire en inelastische effecten (behalve krimp) zoals gedefinieerd in 6.6.3. </p>\n<p>Om te voldoen aan de eisen in sectie 6.6.4 en 6.6.5 - meer informatie is te vinden in AS3600:2018 Sup 1:2022 sectie C6.6 - zijn verificaties en validaties van de methode uitgevoerd aan diverse universiteiten. Afzonderlijke artikelen waarin de resultaten van verificatie en validatie zijn samengevat, zijn te vinden via de volgende link.</p>\n<ul>\n <li><a href=\"https://www.ideastatica.com/support-center-verifications?label=detail\">Verificaties: Detail 2D</a></li>\n</ul>\n<p>Omdat IDEA StatiCa Detail een praktisch ontwerpprogramma is, wordt voor de berekeningen de gefactoreerde karakteristieke cilinderdrukssterkte na 28 dagen <em>f'</em><em><sub>c</sub></em> gebruikt, zoals beschreven in het volgende hoofdstuk.</p>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"ae2cb6c5_baa5_018b_66a6_532df9c21787\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___material_models__a_b7035a6\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"n98f37a1f_02b6_0106_8bdf_a012d81b5585\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___stress_reduction_a\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"e774fa22_dcb7_0117_de10_68cb1245f5ff\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___strength_analysis__8bc3bfe\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"n609eb824_9b85_01f5_b149_9c9a79e95cd1\"></object>\n<h2><br></h2>\n<h2>8 Constructieve toetsingen volgens CSA A23.3:24</h2>\n<p>De beoordeling van de constructie met de CSFM wordt uitgevoerd met twee verschillende analyses: één voor de bruikbaarheid en één voor de sterktebelastingcombinaties. De bruikbaarheidsanalyse gaat ervan uit dat het gedrag onder gefactoreerde belastingen voldoende is en dat de vloeicriteria van het materiaal bij belastingen op bruikbaarheidsniveau niet worden bereikt. Deze aanpak maakt het mogelijk om vereenvoudigde constitutieve modellen (met een lineaire tak van het spanning-rekdiagram van beton) te gebruiken voor de bruikbaarheidsanalyse. Dit verbetert de numerieke stabiliteit en de rekensnelheid.</p>\n<p>De CSFM is een constructieve analysemethode die voldoet aan de eisen voor eindige-elementenanalyse in CSA A23.3:24 clausule 9.5. De effecten van scheurvorming worden direct in de materiaalmodellen meegenomen (clausule 9.5.1), de verankering van de wapening wordt expliciet beschouwd (clausule 9.5.3), en scheurbeheersing en doorbuigingen worden getoetst als onderdeel van de bruikbaarheidstoetsingen (clausule 9.5.5). Omdat de CSFM een continue spanningsveldmethode is, is zij ook consistent met de analyse met Staafwerkmodellen die clausule 9.4 toestaat. De begrenzing van de drukspanning in de drukdiagonalen volgens clausule 11.4.2.3 komt tot uiting in de begrensde druksterkte van het beton (zie hieronder de definitie van de factoren α en k<sub>c2</sub> ).</p>\n<p>Verificatie en validatie van de methode zijn uitgevoerd aan diverse universiteiten. Afzonderlijke artikelen waarin de resultaten van verificatie en validatie zijn samengevat, zijn te vinden via de volgende link.</p>\n<ul>\n <li><a href=\"https://www.ideastatica.com/support-center-verifications?label=detail\">Verificaties: Detail 2D</a></li>\n</ul>\n<p>Omdat IDEA StatiCa Detail een praktisch ontwerpprogramma is, wordt voor de berekeningen de gefactoreerde gespecificeerde druksterkte van het beton f'<sub>c</sub> gebruikt, zoals beschreven in het volgende hoofdstuk.</p>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"n24c1498c_1fe8_0113_0fd4_f4039bfe0cd2\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___material_models__c\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"n9f9fdee5_b6ef_019a_3d79_bfc0c13f9ec3\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___resistance_and_loa_3be42f4\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"n10d99c41_9943_0100_14d1_6af7f28c25e0\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___strength_limit_sta_fde100c\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"n86151269_3bee_01e7_e678_4729d136134e\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___bearing_and_anchor_bfd09ef\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"n7f7cfd5a_d80d_0135_3f15_8c98650b9931\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___service_limit_stat_00c850c\"></object>\n<h2>9 Voorspanning - modelbeschrijving</h2>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"n94d73a69_8436_011d_4af4_c34dfae8c63b\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"prestressing_in_detail___model_description__body_\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"n156a6cf8_77d7_017d_7a73_ffb3f3e93d30\"></object>\n<h1><br></h1>\n<h2>Referenties</h2>\n<p>AASHTO. 2014. AASHTO LRFD Bridge Design Specifications. 7th ed. Washington, DC: American Association of State Highway and Transportation Officials.</p>\n<p><br></p>\n<p>AASHTO. 2024. AASHTO LRFD Bridge Design Specifications. 10th ed. Washington, DC: American Association of State Highway and Transportation Officials.</p>\n<p><br></p>\n<p>ACI Committee 318. 2019. Building Code Requirements for Structural Concrete (ACI 318-19) and Commentary. Farmington Hills, MI: American Concrete Institute.</p>\n<p><br></p>\n<p>Alvarez, Manuel. 1998. Einfluss des Verbundverhaltens auf das Verformungsvermögen von Stahlbeton. IBK Bericht 236. Basel: Institut für Baustatik und Konstruktion, ETH Zurich, Birkhäuser Verlag.</p>\n<p><br></p>\n<p>Beeby, A. W. 1979. “The Prediction of Crack Widths in Hardened Concrete.” The Structural Engineer 57A (1): 9–17.</p>\n<p><br></p>\n<p>Broms, Bengt B. 1965. “Crack Width and Crack Spacing In Reinforced Concrete Members.” ACI Journal Proceedings 62 (10): 1237–56. https://doi.org/10.14359/7742.</p>\n<p><br></p>\n<p>Burns, C.. 2012. “Serviceability Analysis of Reinforced Concrete Members Based on the Tension Chord Model.” IBK Report Nr. 342, Zurich, Switzerland: ETH Zurich.</p>\n<p><br></p>\n<p>Crisfield, M. A. 1997. Non-Linear Finite Element Analysis of Solids and Structures. Wiley.</p>\n<p><br></p>\n<p>CSA Group. 2014. Canadian Highway Bridge Design Code (CSA S6-14). Toronto, ON: CSA Group.</p>\n<p><br></p>\n<p>CSA Group. 2024. Design of Concrete Structures (CSA A23.3:24). Toronto, ON: CSA Group.</p>\n<p><br></p>\n<p>European Committee for Standardization (CEN). 2015. 1 Eurocode 2: Design of concrete structures - Part 1-1: General rules and rules for buildings. Brussels: CEN, 2005.</p>\n<p><br></p>\n<p>Fernández Ruiz, M., and A. Muttoni. 2007. “On Development of Suitable Stress Fields for Structural Concrete.” ACI Structural Journal 104 (4): 495–502.</p>\n<p><br></p>\n<p>Kaufmann, W., J. Mata-Falcón, M. Weber, T. Galkovski, D. Thong Tran, J. Kabelac, M. Konecny, J. Navratil, M. Cihal, and P. Komarkova. 2020. “Compatible Stress Field Design Of Structural Concrete. Berlin, Germany.”AZ Druck und Datentechnik GmbH, ISBN 978-3-906916-95-8.</p>\n<p><br></p>\n<p>Kaufmann, W., and P. Marti. 1998. “Structural Concrete: Cracked Membrane Model.” Journal of Structural Engineering 124 (12): 1467–75. https://doi.org/10.1061/(ASCE)0733-9445(1998)124:12(1467).</p>\n<p><br></p>\n<p>Kaufmann, W.. 1998. “Strength and Deformations of Structural Concrete Subjected to In-Plane Shear and Normal Forces.” Doctoral dissertation, Basel: Institut für Baustatik und Konstruktion, ETH Zürich. https://doi.org/10.1007/978-3-0348-7612-4.</p>\n<p><br></p>\n<p>Konečný, M., J. Kabeláč, and J. Navrátil. 2017. Use of Topology Optimization in Concrete Reinforcement Design. 24. Czech Concrete Days (2017). ČBS ČSSI. https://resources.ideastatica.com/Content/06_Detail/Verification/Articles/Topology_optimization_US.pdf.</p>\n<p><br></p>\n<p>Marti, P. 1985. “Truss Models in Detailing.” Concrete International 7 (12): 66–73.</p>\n<p><br></p>\n<p>Marti, P. 2013. Theory of Structures: Fundamentals, Framed Structures, Plates and Shells. First edition. Berlin, Germany: Wiley Ernst & Sohn.</p>\n<p>http://sfx.ethz.ch/sfx_locater?sid=ALEPH:EBI01&genre=book&isbn=9783433029916.</p>\n<p><br></p>\n<p>Marti, P., M.Alvarez, W. Kaufmann, and V. Sigrist. 1998. “Tension Chord Model for Structural Concrete.” Structural Engineering International 8 (4): 287–298.</p>\n<p>https://doi.org/10.2749/101686698780488875.</p>\n<p><br></p>\n<p>Mata-Falcón, J. 2015. “Serviceability and Ultimate Behaviour of Dapped-End Beams (In Spanish: Estudio Del Comportamiento En Servicio y Rotura de Los Apoyos a Media Madera).” PhD thesis, Valencia: Universitat Politècnica de València.</p>\n<p><br></p>\n<p>Meier, H. 1983. “Berücksichtigung Des Wirklichkeitsnahen Werkstoffverhaltens Beim Standsicherheitsnachweis Turmartiger Stahlbetonbauwerke.” Institut für Massivbau, Universität Stuttgart.</p>\n<p><br></p>\n<p>Navrátil, J., P. Ševčík, L. Michalčík, P. Foltyn, and J. Kabeláč. 2017. A Solution for Walls and Details of Concrete Structures. 24. Czech Concrete Days.</p>\n<p><br></p>\n<p>Schlaich, J., K. Schäfer, and M. Jennewein. 1987a. “Toward a Consistent Design of Structural Concrete.” PCI Journal 32 (3): 74–150.</p>\n<p><br></p>\n<p>Standards Australia. 2018. Concrete Structures (AS 3600:2018). Sydney, NSW: Standards Australia.</p>\n<p><br></p>\n<p>Standards Australia. 2022. Concrete Structures – Commentary (Supplement 1 to AS 3600:2018). Sydney, NSW: Standards Australia.</p>\n<p><br></p>\n<p>Vecchio, F.J., and M.P. Collins. 1986. “The Modified Compression Field Theory for Reinforced Concrete Elements Subjected to Shear.” ACI Journal 83 (2): 219–31. </p>"
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"value": "<p>In de praktijk worden de <strong>Staafwerkmethode (S&T)</strong> en de <strong>spanningsveldmethoden</strong> standaard gebruikt voor het ontwerpen van discontinuïteitsgebieden in gewapende en voorgespannen betonconstructies. De <strong>Compatible Stress Field Method (CSFM)</strong> is ontwikkeld door deze klassieke theorieën uit te breiden. De methode maakt een hoge mate van automatisering mogelijk en is consistent met de norm. Ondanks de eenvoud geeft de methode een zeer realistische beschrijving van het gedrag van een betonconstructie, zowel in de uiterste grenstoestand (UGT) als in de bruikbaarheidsgrenstoestand (BGT). De CSFM is geïmplementeerd in <a data-item-id=\"b4790cf9-a605-45b3-b41b-e36909ad4291\" href=\"\">IDEA StatiCa Detail</a>. </p>\n<figure data-asset-id=\"a7b3dcf1-10ed-4b44-99e3-f59b4bd2a7fe\" data-image-id=\"a7b3dcf1-10ed-4b44-99e3-f59b4bd2a7fe\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/7fd8d041-20d1-40a8-9a71-eb9cdce27155/7.png\" data-asset-id=\"a7b3dcf1-10ed-4b44-99e3-f59b4bd2a7fe\" data-image-id=\"a7b3dcf1-10ed-4b44-99e3-f59b4bd2a7fe\" alt=\"\"></figure>\n<p><em>Fig. 1 a) Wand met openingen b) Wandschijf c) Ligger met tandopleggingen en openingen d) Brugpijler e) Brugdiafragma </em></p>\n<p>Standaardprocedures voor het ontwerpen van doorsneden van betonconstructies zijn van toepassing in gebieden waar de hypothese van Bernoulli-Navier over een vlakke rekverdeling geldt (B-gebied). De plaatsen waar deze hypothese niet geldt, worden <strong>discontinuïteitsgebieden (D-gebieden)</strong> genoemd. Hiertoe behoren delen van constructies waar geconcentreerde belastingen optreden of waar een plotselinge verandering van de doorsnede is, zoals tandopleggingen (Fig. 1c), gedrongen liggers, wanden met openingen (Fig. 1a, 1b), of consoles en paalkoppen. In de brugbouw zijn dit bijv. pijlerkoppen (Fig. 1d), diafragma's (Fig. 1e), deviatoren, enz.</p>\n<h2>1. Staafwerkmethode</h2>\n<p>Het uitgangspunt bij het definiëren van een S&T-model is dat de treksterkte van beton wordt verwaarloosd. Een eenvoudig vakwerkmodel bestaat uit elementen die op druk en trek werken en die het gedrag in de UGT weergeven. Over het algemeen is dit geen complex probleem en het definiëren van een basis S&T-model (Fig. 2a) zou voor een ervaren ingenieur geen probleem moeten zijn. Maar zelfs voor deze basistaak kan de correcte toetsing van het model conform de norm een tijdrovend, handmatig en iteratief proces zijn.</p>\n<figure data-asset-id=\"59f28d4a-b793-4501-a11a-6ae6245cab70\" data-image-id=\"59f28d4a-b793-4501-a11a-6ae6245cab70\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/7aecb55f-dac0-47db-a25c-48082e5e70c4/Untitled%20design.png\" data-asset-id=\"59f28d4a-b793-4501-a11a-6ae6245cab70\" data-image-id=\"59f28d4a-b793-4501-a11a-6ae6245cab70\" alt=\"\"></figure>\n<p><em>Fig. 2 a) S&T-model optie 1 b) S&T-model optie 2 c) S&T-model optie </em></p>\n<p>Trekstaven, knoopgebieden en de dwarse trekrek in drukdiagonalen moeten worden getoetst. Als het model de toetsing niet doorstaat, moet de S&T-geometrie worden aangepast of moet een ander S&T-model worden gekozen (Fig. 2b, 2c). Dit leidt er vaak toe dat de constructeur de geometrie van het S&T-model slechts één keer kiest en alleen de wapening toetst. Dit kan tot een aanzienlijke fout leiden. De keuze van het model is altijd een kwestie van ervaring. Bij complexere constructieve details is hetkiezen van een S&T-model dat voldoende overeenkomt met het werkelijke gedrag van de constructie mogelijk niet zo eenvoudig als in bovenstaand geval. Bovendien is de S&T een methode voor het ontwerpen van uitsluitend uiterste grenstoestanden. De methode maakt het niet mogelijk om <strong>bruikbaarheidsgrenstoestanden (vervorming, scheurvorming)</strong> te toetsen. Dit zijn kritische criteria, vooral bij constructies van groot belang, omdat ze rechtstreeks van invloed zijn op de levensduur van de constructie.</p>\n<h2>2. Compatible Stress Field Method - CSFM</h2>\n<p>CSFM is een moderne <strong>niet-lineaire methode</strong> voor de analyse van D-gebieden en elementen waarvan het gedrag kan worden vereenvoudigd tot een vlakspanningstoestand, d.w.z. een 2D-model. De methode is echter nog steeds gebaseerd op een fundamenteel en veilig uitgangspunt van de normen: <strong>beton neemt geen trek op</strong> en alle trek moet door wapening worden overgedragen. De Compatible Stress Field Method (CSFM) is een doorontwikkeling van de S&T- en spanningsveldmethoden. Zij neemt de hierboven genoemde belangrijkste nadelen weg: onzekerheden bij de modelkeuze, de moeilijke automatisering en het onvermogen om bruikbaarheidsgrenstoestanden te toetsen.</p>\n<figure data-asset-id=\"6552ad81-c0fa-4071-9b95-00d09eb9fea4\" data-image-id=\"6552ad81-c0fa-4071-9b95-00d09eb9fea4\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/9e760312-8401-44c8-aae1-020b514876fb/2.png\" data-asset-id=\"6552ad81-c0fa-4071-9b95-00d09eb9fea4\" data-image-id=\"6552ad81-c0fa-4071-9b95-00d09eb9fea4\" alt=\"\"></figure>\n<p><em>Fig. 3 a) Vlakke rek b) Hoofdspanning c) CSFM</em></p>\n<p>Het principe van CSFM kan worden toegelicht aan de hand van de <strong>vlakspanningstoestand</strong> van het basis vlakke element van een gewapende betonconstructie. Fig. 3a toont het basis 2D-element in vlakspanning, zoals we dat kennen uit alle leerboeken over elasticiteit en sterkteleer. Dit is de spanning in één punt van de constructie, bijvoorbeeld verkregen met een lineair elastische analyse met de Eindige Elementen Methode (FEM). Het element wordt belast door een horizontale normaalspanning σ<sub>x</sub>, een verticale normaalspanning σ<sub>z</sub> en een schuifspanning τ<sub>xz</sub>. Uit deze spanningen kunnen de zogenoemde <strong>hoofdspanningen</strong> en hun richting, gedefinieerd door de hoek θ, worden bepaald (Fig. 3b). Het element wordt vervolgens belast door de hoofdtrekspanning σ<sub>1</sub> en de hoofddrukspanning σ<sub>2</sub>.</p>\n<p>Hoe ziet de rek van hetzelfde element eruit wanneer het met CSFM wordt geanalyseerd? De rek is weergegeven in Figuur 3c. Het beton onder druk treedt op in de richting van de hoofddrukspanning σ<sub>2</sub>. Hierdoor ontstaat een spanningsveld met spanning σ<sub>c2</sub>. Zoals hierboven vermeld, is het uitgangspunt dat beton geen trek opneemt. De dwarse hoofdtrekspanning σ<sub>1</sub> wordt dus niet door het beton overgedragen en loodrecht op die richting ontstaat een scheur. De spanning σ<sub>c1r</sub> moet daarom nul zijn. Om bezwijken van ons 2D-element te voorkomen, moet alle trekspanning door de wapening worden overgedragen (blauw weergegeven in Fig. 3c). Deze wapening moet deel uitmaken van het rekenmodel. </p>\n<p>Wordt deze spanningsanalyse met CSFM<strong> continu over het gehele 2D-gebied</strong> uitgevoerd, dan is het resultaat een continu drukveld in het beton plus trek- en drukspanningen in de wapening. Een vereenvoudigde grafische weergave van het CSFM-spanningsveld is te zien in Figuur 4. Naast de benuttingsgraden van het beton en de wapening geeft de figuur ook de wisselende richtingen van de berekende spanningen σ<sub>c2</sub> langs de gebieden weer.</p>\n<figure data-asset-id=\"9739b6d6-2cbc-4745-a590-4a85f7e1862f\" data-image-id=\"9739b6d6-2cbc-4745-a590-4a85f7e1862f\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/dc56c80f-67f3-481d-a2c8-26dacc258bb2/csfm%20explained%20%281%29.png\" data-asset-id=\"9739b6d6-2cbc-4745-a590-4a85f7e1862f\" data-image-id=\"9739b6d6-2cbc-4745-a590-4a85f7e1862f\" alt=\"\"></figure>\n<p><em>Fig. 4 Globale resultaten uit IDEA StatiCa Detail </em></p>\n<p>De analyse van een detail of constructie met <strong>CSFM is gebaseerd op de Eindige Elementen Methode</strong>. Het beton wordt gemodelleerd met 2D-wandelementen en de wapening met 1D-staafelementen (Fig. 7). De analyse wordt niet in één stap uitgevoerd, omdat het een niet-lineair probleem is. Tijdens de berekening worden de belastingen stapsgewijs aangebracht en de oplossing van het niet-lineaire stelsel vergelijkingen wordt gevonden met de <strong>Newton-Raphson-methode</strong>. </p>\n<p>De fictieve uitgesmeerde scheuren (ε<sub>1</sub> is de gemiddelde waarde) worden \"gevormd\" loodrecht op de richting van de hoofdspanningen. Deze richting kan tijdens de niet-lineaire berekening veranderen doordat het element bij elke belastingstap \"progressief scheurt\". Samengevat wordt uitgegaan van een fictieve spanningsvrije roterende scheur. </p>\n<p>Het resultaat van de FEM-oplossing met CSFM is een compatibel spanningsveld (d.w.z. het beton breekt in het model niet op in afzonderlijke, onafhankelijk werkende drukdiagonalen) en een rektoestand die continu zijn over het gehele op te lossen 2D-domein. Dit is een belangrijk voordeel ten opzichte van klassieke S&T-benaderingen en maakt het mogelijk het rekenmodel te automatiseren en te verfijnen, zoals beschreven in de volgende alinea's.</p>\n<figure data-asset-id=\"c5bf3113-2223-4ddc-bbab-2131db37ac0c\" data-image-id=\"c5bf3113-2223-4ddc-bbab-2131db37ac0c\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/40e41bf7-4a81-4245-bd1b-843963c104e8/3.png\" data-asset-id=\"c5bf3113-2223-4ddc-bbab-2131db37ac0c\" data-image-id=\"c5bf3113-2223-4ddc-bbab-2131db37ac0c\" alt=\"\"></figure>\n<p><em>Fig. 5 Principe van compression softening van beton</em></p>\n<p>De eenvoudige formulering van CSFM maakt het mogelijk het standaard uniaxiale parabool-rechthoekdiagram voor de spanning-rek van beton onder druk conform de norm te gebruiken. Zoals bekend neemt de druksterkte van beton af wanneer het beton door dwarse scheuren is aangetast (Fig. 5). Dit zogenoemde <strong>compression softening </strong>effect wordt in de methode verwerkt door automatisch rekening te houden met de effectieve druksterkte van het beton. </p>\n<p>Op basis van de grootte van de dwarse trekrek ε<sub>1</sub> wordt de reductiefactor k<sub>c</sub> bepaald en wordt het spanning-rekdiagram van het beton aangepast (Fig. 5). Aangezien het rekveld in de gehele constructie bekend is, kan de effectieve druksterkte van het beton automatisch worden berekend in afzonderlijke doorsneden, afhankelijk van de lokale grootte van de dwarse trekrek ε<sub>1</sub>.</p>\n<figure data-asset-id=\"6c73faf0-64d4-41ce-b816-520ccadff05a\" data-image-id=\"6c73faf0-64d4-41ce-b816-520ccadff05a\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/51628156-5014-4501-8f28-9f3b93783840/1.png\" data-asset-id=\"6c73faf0-64d4-41ce-b816-520ccadff05a\" data-image-id=\"6c73faf0-64d4-41ce-b816-520ccadff05a\" alt=\"\"></figure>\n<p><em>Fig. 6 Principe van tension stiffening</em></p>\n<p>Bovendien houdt de CSFM rekening met het <strong>stiffening effect</strong> van het op trek belaste beton tussen de scheuren op de wapening, de zogenoemde tension stiffening. In het rekenmodel wordt de gemiddelde wapeningsrek ε<sub>m</sub> gebruikt. Vervolgens wordt het spanning-rekdiagram van de wapening aangepast (Fig. 6). Hierdoor kan de stijfheid van een door scheuren aangetaste gewapende betonconstructie realistisch worden weergegeven. Het blijft echter zo dat de treksterkte van beton niet bijdraagt aan de uiterste draagkracht. De maximale spanning in de wapening σ<sub>sr</sub> in de scheuren is maatgevend voor het ontwerp (Fig. 6).</p>\n<p>CSFM gebruikt gangbare uniaxiale materiaalmodellen (spanning-rekdiagrammen) die in de <strong>normen</strong> zijn gedefinieerd. Voor de toetsing in de UGT wordt vervolgens de standaardaanpak met partiële factoren gebruikt. De eenvoud van de methode maakt haar geschikt voor de ingenieurspraktijk en is consistent met de normen. </p>\n<p>Hoewel het een niet-lineaire FEM-analyse is, hoeft de constructeur geen extra materiaaleigenschappen en betonkarakteristieken in de berekening in te voeren. Deze zijn in de ontwerpfase mogelijk niet eens beschikbaar en zijn bijvoorbeeld wel nodig voor niet-lineaire FEM-analyses op basis van breukmechanica. Zoals eerder aangegeven is een belangrijk voordeel van de CSFM-analyse, naast de uiterste grenstoestanden, de mogelijkheid om <strong>bruikbaarheidsgrenstoestanden te toetsen: doorbuigingen, spanningsbeperkingen en met name scheurwijdte</strong>.</p>\n<figure data-asset-id=\"6c090b06-f906-4e6e-9016-d73de172f321\" data-image-id=\"6c090b06-f906-4e6e-9016-d73de172f321\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/8fd3f66a-63b9-4aba-bb10-9e12889ed7d4/Finite%20element%20model.png\" data-asset-id=\"6c090b06-f906-4e6e-9016-d73de172f321\" data-image-id=\"6c090b06-f906-4e6e-9016-d73de172f321\" alt=\"\"></figure>\n<p><em>Fig. 7 Voorbeeld van een weergave van het eindige-elementenmodel in IDEA StatiCa Detail</em></p>\n<p>(Fig. 7) Het FEM-model in CSFM bestaat uit verschillende typen eindige elementen:</p>\n<ul>\n <li>1D-element met axiale stijfheid voor de wapening</li>\n <li>2D-isoparametrisch element voor het beton</li>\n <li>Eindveren voor het verankeringsmodel van de wapening met eindafwerking</li>\n <li>Speciaal 2D-element om de aanhechting tussen wapening en beton te modelleren</li>\n <li>Stijve en interpolerende randvoorwaarden (Multi-Point Constraints, MPC) tussen aanhechtingselementen en beton</li>\n</ul>\n<p>Als de ontworpen wapening bros bezwijken van het element voorkomt, geeft de CSFM aantoonbaar zeer goede voorspellingen van het gedrag en de uiterste draagkracht van de constructie, ondanks de eenvoud van de formulering. De methode is met andere woorden niet geschikt voor bijvoorbeeld het ontwerp van liggers zonder dwarskrachtwapening, die mogelijk bros gedrag vertonen. <a data-item-id=\"1e879886-9e36-49e1-acb1-e6001361531f\" href=\"\">Verificaties</a> van de methode, inclusief experimenten, zijn opgenomen in [1]. Een uitgebreidere beschrijving van de methode valt buiten het bestek van dit artikel en is ook te vinden in <a data-item-id=\"0000c94c-b603-48c4-8d31-bc56d7c95886\" href=\"\">Theoretical Background</a>.</p>\n<p>Het is duidelijk dat de principes van CSFM algemeen zijn en dat de toepassing dus niet beperkt is tot D-gebieden, maar kan worden gebruikt om hele staven te modelleren, bijv. prefab liggers, en waar het element kan worden vereenvoudigd tot een vlak 2D-model. De methode en de implementatie ervan in software (IDEA StatiCa Detail) zijn ook uitgebreid met de mogelijkheid om <strong>voorgespannen wapening en wapening met voorspanning met nagerekt staal</strong> op te geven.</p>\n<h2>3. Voorbeeld van het ontwerp van een pijlerkop</h2>\n<p>De praktische toepassing van CSFM wordt getoond aan de hand van het ontwerp van de brugpijlerkop in Figuur 8. Dit is de tweede pijler van een doorgaande brug met drie overspanningen van 30,0 m, 42,0 m en 30,0 m. De kop van de gewapende betonpijler is ontworpen in beton C40/50 en heeft een dikte (in de lengterichting van de brug) van 2,0 m.</p>\n<figure data-asset-id=\"9d541a10-b879-4d35-a6f8-4e85fa9843a6\" data-image-id=\"9d541a10-b879-4d35-a6f8-4e85fa9843a6\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/7bd9ebbf-5d30-448a-93e2-c89641d05677/4.png\" data-asset-id=\"9d541a10-b879-4d35-a6f8-4e85fa9843a6\" data-image-id=\"9d541a10-b879-4d35-a6f8-4e85fa9843a6\" alt=\"\"></figure>\n<p><em>Fig. 8 Pijlerkop: a) Samenvatting van het ontwerp; b) Drukspanning in het beton in de UGT; c) Trekspanning in de wapening in de UGT; d) Scheurwijdte in de BGT</em></p>\n<p>Aan de bovenzijde van de pijlerkop werd eerst een dwarsligger ontworpen met wapening B500 20xϕ28+20xϕ25 in de bovenste vier lagen. Figuur 8a toont een samenvattend ontwerp in de uiterste grenstoestand, met de drukspanningen in het beton, de richtingen van de drukspanningen en de spanningen in de wapening. Een gedetailleerdere spanningsverdeling in het beton en de wapening is weergegeven in Figuur 8b en 8c. De dwarswapening zit net onder de vloeigrens en ook de spanningen in het beton (en de bijbehorende rekken) zijn in de UGT bevredigend. Het resultaat van de scheurwijdteberekening (Fig. 8d) laat echter zien dat het ontwerp niet voldoet in de <strong>BGT</strong>: w<sub>max</sub> = 0,36 mm > w<sub>lim</sub> = 0,3 mm. <strong>Om aan de grenswaarde voor de scheurwijdte te voldoen, is het noodzakelijk de wapening</strong> van de dwarsligger te vergroten tot 20xϕ32+20xϕ28. Bij w<sub>lim</sub> = 0,2 mm (bijv. een pijler nabij een weg waar strooizout wordt verspreid, milieuklasse XF2) zou de wapening van de dwarsligger zelfs moeten worden vergroot tot 24xϕ32+24xϕ28.</p>\n<h2>Conclusie</h2>\n<p>CSFM past bij de ingenieurspraktijk, omdat de methode <strong>eenvoudige materiaalmodellen gebruikt die in een norm zijn gedefinieerd</strong>. Naast de uiterste grenstoestanden maakt zij ook het ontwerpen van bruikbaarheidsgrenstoestanden mogelijk. Een toetsing hiervan was met S&T-modellen tot nu toe moeilijk voor te stellen. Door de implementatie van de methode in <strong>IDEA StatiCa Detail</strong> kan de respons van de constructie realistisch worden vastgelegd en kunnen discontinuïteitsgebieden en grotere samenstellingen efficiënt en veilig worden ontworpen en getoetst.</p>\n<p>De CSFM is voornamelijk ontwikkeld door het werk van professor Walter Kaufmann, hoofd van de leerstoel Structural Engineering aan het Swiss Federal Institute of Technology (ETH) Zürich. Hij en zijn team hebben ook <a data-item-id=\"0dd36e25-63b2-4d63-a33e-6043644fda4f\" href=\"\">de methode en de softwareimplementatie ervan geverifieerd</a>.</p>\n<h2>Literatuur</h2>\n<p>[1] KAUFMANN, Walter, et al.: Compatible stress field design of structural concrete, ETH Zurich, 2020, ISBN 978-3-906916-95-8,</p>\n<p>[2] KAUFMANN, W., MARTI, P.: Structural Concrete: Cracked Membrane Model. Journal of Structural Engineering 124 (12): 1467-75, 1998 https://doi.org/10.1061/(ASCE)0733-9445(1998)124:12(1467)</p>\n<p>[3] KRAUS, M., M. WEBER, W. KAUFMANN, W, BOBEK, L.: Numerical analysis of experimentally tested frame corners with opening moments using the Compatible Stress Field Method (CSFM). In: Computational Modelling of Concrete and Concrete Structures, pp. 694-03. CRC Press, 2022 <a href=\"https://doi.org/10.1201/9781003316404\">https://doi.org/10.1201/9781003316404</a></p>\n<h2>Auteur</h2>\n<p>Ing. Pavel Kaláb, Ph.D.</p>\n<p>IDEA StatiCa s.r.o.</p>\n<p><br></p>"
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"value": "<figure data-asset-id=\"6abf376c-12d5-4198-8788-f091af429104\" data-image-id=\"6abf376c-12d5-4198-8788-f091af429104\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/65823418-0ec0-4658-bfa1-31bf83f189a2/Untitled.png\" data-asset-id=\"6abf376c-12d5-4198-8788-f091af429104\" data-image-id=\"6abf376c-12d5-4198-8788-f091af429104\" alt=\"\"></figure>\n<p>Traditionele methoden voor het ontwerpen van de wapening in <a data-item-id=\"1623b411-ff55-567e-974c-f9e7bbca3ea7\" href=\"\">framehoeken</a> zijn gebaseerd op een reeks detailleringsregels en equivalente Staafwerk-modellen. Voor uiteenlopende geometrieën en belastingsituaties zijn in de literatuur talrijke <a data-item-id=\"5d38b30f-eef7-5387-8017-d50064ddfb20\" href=\"\">Staafwerk-modellen</a> beschikbaar. De keuze van het juiste model hangt daarom sterk af van de ervaring van de ingenieur.<br>Een gedetailleerde toetsing van een Staafwerk-model, inclusief verankeringscontroles en het in rekening brengen van dwarstrekspanningen in drukdiagonalen, is tijdrovend en arbeidsintensief.</p>\n<figure data-asset-id=\"d134bec5-6945-4748-9abe-ed75602e4247\" data-image-id=\"d134bec5-6945-4748-9abe-ed75602e4247\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/bf899ce1-1209-4838-8203-84756669f79b/image%20%283%29.png\" data-asset-id=\"d134bec5-6945-4748-9abe-ed75602e4247\" data-image-id=\"d134bec5-6945-4748-9abe-ed75602e4247\" alt=\"\"></figure>\n<p>Het doel van deze webinar is om aan de hand van een concreet voorbeeld van een framehoek in de tunnelbekleding van een met grond aangevulde tunnel een ontwerpaanpak te presenteren die is gebaseerd op de <a data-item-id=\"eaab962d-ba44-4ee0-8fa7-45193c9f52b5\" href=\"\"><strong>CSFM (Compatible Stress Field Method)</strong></a>, zoals geïmplementeerd in <strong>IDEA StatiCa Detail</strong>.<br>Deze materieel niet-lineaire methode vervangt het Staafwerk-model volledig en biedt, naast de toetsing in de uiterste grenstoestand, ook een beoordeling van de <a data-item-id=\"9e7e995c-6e74-422f-af6e-88a8d7fe047f\" href=\"\"><strong>bruikbaarheidsgrenstoestand</strong></a> (spanningsbeperkingen en scheurwijdte).</p>\n<figure data-asset-id=\"10808b63-a505-40b4-98da-a96e9e2326e2\" data-image-id=\"10808b63-a505-40b4-98da-a96e9e2326e2\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/068eba30-25f2-4fe9-8195-b8e177e1cba4/Detail_results.png\" data-asset-id=\"10808b63-a505-40b4-98da-a96e9e2326e2\" data-image-id=\"10808b63-a505-40b4-98da-a96e9e2326e2\" alt=\"\"></figure>\n<p>Tijdens de webinar wordt de framehoek gemodelleerd als een uitgesneden segment van de constructie. Er wordt uitgelegd hoe de juiste <a data-item-id=\"5a121972-f384-4f14-8788-9da298e1aae1\" href=\"\">oplegcondities </a>en de <a data-item-id=\"38cbe005-0e1e-4d75-ae8a-2ef9dcee4c2b\" href=\"\">belasting van dit segment</a> in het model worden toegepast. <strong>De resultaten van de CSFM-analyse en de interpretatie ervan worden uitgebreid besproken.</strong><br>Tot slot laat een gedetailleerde demonstratie zien hoe u met de software wapening ontwerpt voor dwarstrekspanningen in de “hoofddiagonaal” van de framehoek en hoe u de wapeningsindeling optimaliseert.</p>\n<h3>Wilt u meer webinars bekijken?</h3>\n<p>Lees meer over het ontwerpen van discontinuïteitsgebieden in gewapend en voorgespannen beton met IDEA StatiCa:</p>\n<ul>\n <li><a data-item-id=\"8a052360-12ef-4eef-bbb9-7ea5dc3e604f\" href=\"\">Master Your Design - Precast and Infrastructure Beams</a></li>\n <li><a data-item-id=\"8584d7a0-afb0-469a-bcd3-f40167a0d3eb\" href=\"\">Master Your Design - Pier Caps</a></li>\n <li><a data-item-id=\"499c5309-0a5c-4d9f-b492-9dc02f354d36\" href=\"\">Master Your Design - Prestressed Beam with Openings</a></li>\n</ul>"
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"value": "<h2>Inleiding </h2>\n<p>Dit artikel behandelt het gebruik van de <a data-item-id=\"eaab962d-ba44-4ee0-8fa7-45193c9f52b5\" href=\"\">CSFM</a> voor het analyseren van afschuifbezwijking in liggers met een kleine hoeveelheid beugels. Hiervoor is een selectie geanalyseerd van proeven op vrij opgelegde gewapend betonliggers door Huber (2016), Piyamahant (2002) en Vecchio en Shim (2004). Deze proeven omvatten een groot aantal parameters, waaronder verschillende afmetingen, afschuifslankheden en hoeveelheden afschuif- en langswapening. Dit gedeelte beschrijft de analyse van 17 experimenten uit deze proefseries met de CSFM. Daarbij wordt onderzocht in hoeverre de CSFM zeer uiteenlopende bezwijkvormen correct kan modelleren. Deze reiken van afschuifbezwijking met en zonder breuk van de beugels tot buigbezwijking en gemengd afschuif-buigbezwijking. </p>\n<p>Proefopstelling: Fig. 6.17 toont de geometrie, de proefopstellingen en de wapeningsindelingen van de geanalyseerde experimenten. Tabel 6.10 geeft informatie over de afschuifwapening (diameter (<em>Ø</em><em><sub>t</sub></em>) , hart-op-hartafstand (<em>s</em><em><sub>t</sub></em>) en geometrisch wapeningspercentage (<em>ρ</em><em><sub>t,geo</sub></em>)), de buigwapening (aantal (<em>n</em><em><sub>l</sub></em>) en diameter (<em>Ø</em><em><sub>l</sub></em>)) en de geometrie (effectieve hoogte (<em>d</em>), afschuifslankheid (<em>a/d</em>) en breedte van de liggers (<em>b</em>)). Bij de proeven R1000m60 en R500m351 van Huber (2016) werden eensnedige haken toegepast, terwijl alle andere proeven tweesnedige gesloten beugels gebruikten. Bij de geanalyseerde proeven van Piyamahant (2002) bleven de geometrie en de buigwapening constant, terwijl deze bij de andere twee onderzoeken werden gevarieerd. </p>\n<figure data-asset-id=\"7a127e09-323a-47c2-bb68-f6d7305df916\" data-image-id=\"7a127e09-323a-47c2-bb68-f6d7305df916\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/b04ab1b6-8164-42a4-8153-b0bca529109d/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20fig%206.17.png\" data-asset-id=\"7a127e09-323a-47c2-bb68-f6d7305df916\" data-image-id=\"7a127e09-323a-47c2-bb68-f6d7305df916\" alt=\"\"></figure>\n<figure data-asset-id=\"ae1b1eb6-ac3f-45d5-908f-58e08887a726\" data-image-id=\"ae1b1eb6-ac3f-45d5-908f-58e08887a726\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/f016576c-b58f-4934-8a91-92a8f02ada8b/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20table%2010.png\" data-asset-id=\"ae1b1eb6-ac3f-45d5-908f-58e08887a726\" data-image-id=\"ae1b1eb6-ac3f-45d5-908f-58e08887a726\" alt=\"\"></figure>\n<figure data-asset-id=\"460dd1dc-d307-4f49-b7d8-f46b081238f6\" data-image-id=\"460dd1dc-d307-4f49-b7d8-f46b081238f6\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/1c39493d-2611-4b20-8595-b15bd109181c/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20table%2010-2.png\" data-asset-id=\"460dd1dc-d307-4f49-b7d8-f46b081238f6\" data-image-id=\"460dd1dc-d307-4f49-b7d8-f46b081238f6\" alt=\"\"></figure>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"four_point_bending_tests_on_t_beams__definition_of\"></object>\n<h2>Materiaaleigenschappen</h2>\n<p>De materiaaleigenschappen van de afschuifwapening, de buigwapening en het beton die in de CSFM-analyse zijn gebruikt, staan samengevat in Tabel 6.11. De meeste materiaaleigenschappen die voor de <a data-item-id=\"2ebdaf9c-827f-4fd6-9f82-28bc96970a64\" href=\"\">CSFM</a>-analyse nodig waren, zijn te vinden in de bijbehorende proefverslagen. De waarden die moesten worden aangenomen, zijn aangegeven in Tabel 6.11. </p>\n<figure data-asset-id=\"8493aef1-7d61-4eca-8e25-aeda94512c10\" data-image-id=\"8493aef1-7d61-4eca-8e25-aeda94512c10\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/d40a91ae-9e7a-4cbe-9980-8bc295693359/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20table%206.11.png\" data-asset-id=\"8493aef1-7d61-4eca-8e25-aeda94512c10\" data-image-id=\"8493aef1-7d61-4eca-8e25-aeda94512c10\" alt=\"\"></figure>\n<figure data-asset-id=\"b38d80f8-07f0-4992-b707-36da923011d3\" data-image-id=\"b38d80f8-07f0-4992-b707-36da923011d3\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/037428f1-0d3f-4404-9321-db2a2a0f0c8c/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20table%206.11-2.png\" data-asset-id=\"b38d80f8-07f0-4992-b707-36da923011d3\" data-image-id=\"b38d80f8-07f0-4992-b707-36da923011d3\" alt=\"\"></figure>\n<figure data-asset-id=\"78bf381f-664d-41ad-a30e-cb1dc0204a8e\" data-image-id=\"78bf381f-664d-41ad-a30e-cb1dc0204a8e\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/6388402c-7243-4c35-9936-241e9bbc86e5/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20table%206.11-3.png\" data-asset-id=\"78bf381f-664d-41ad-a30e-cb1dc0204a8e\" data-image-id=\"78bf381f-664d-41ad-a30e-cb1dc0204a8e\" alt=\"\"></figure>\n<h2>Modelleren met de CSFM</h2>\n<p>De geometrie, de wapening en de <a data-item-id=\"50ed723b-9b87-4870-a69f-e05b5a8a8150\" href=\"\">oplegging en belasting </a>zijn in de CSFM gemodelleerd volgens de proefopstellingen. Fig. 6.18 toont als voorbeeld het modelleren van Proef A3 van Vecchio en Shim (2004).</p>\n<figure data-asset-id=\"be95afab-6efa-4ca2-af2d-187495214492\" data-image-id=\"be95afab-6efa-4ca2-af2d-187495214492\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/ee8f6a35-2573-4491-a59a-c75249bae2ec/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20fig%206.18.png\" data-asset-id=\"be95afab-6efa-4ca2-af2d-187495214492\" data-image-id=\"be95afab-6efa-4ca2-af2d-187495214492\" alt=\"\"></figure>\n<p>Voor elke proef zijn vier numerieke berekeningen uitgevoerd met de volgende parameters:</p>\n<ul>\n <li>De mesh-grootte, variërend van 5 (standaardwaarde voor deze voorbeelden) via 10 tot 20 eindige elementen over de liggerhoogte. Omdat de standaard mesh al zeer grof is, zijn in dit onderzoek alleen fijnere meshes geanalyseerd. De mesh met 10 elementen is gebruikt, behalve in M0. </li>\n <li> Het wel of niet in rekening brengen van het tension stiffening effect. Standaard wordt tension stiffening in de CSFM meegenomen. </li>\n <li>Het wel of niet in rekening brengen van mogelijk niet-gestabiliseerde scheurvorming in beugels. Indien meegenomen (standaard) definieert het Pull-Out Model (POM) de tension stiffening in beugels (de geometrische wapeningspercentages van alle liggers liggen onder (<em>ρ</em><em><sub>cr</sub></em>), zodat het Tension Chord Model nooit wordt gebruikt). Indien uitgeschakeld, houden de modellen rekening met tension stiffening door middel van het TCM.</li>\n</ul>\n<p><em>\\[ρ_{\\text{cr}} = \\frac{f_{\\text{ct}}}{f_{\\text{y}} - (n-1)f_{\\text{ct}}}\\]</em></p>\n<p>waarin<em>:</em></p>\n<ul>\n <li>\\(f_y\\)<em> - </em>vloeigrens van de wapening</li>\n <li>\\(f_{ct}\\) - treksterkte van het beton</li>\n <li>\\(n = \\frac{E_s}{E_c}\\) - elasticiteitsmodulusverhouding</li>\n</ul>\n<p>Tabel 6.12 toont de parameters die in elke numerieke berekening zijn gebruikt. M0 komt overeen met het model met de standaardinstellingen in de CSFM.</p>\n<figure data-asset-id=\"f066bc6c-98fe-48c4-b90d-1eb356184726\" data-image-id=\"f066bc6c-98fe-48c4-b90d-1eb356184726\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/654a5eaa-78a1-4ba4-aa24-c24a2bb360bd/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20table%206.12.png\" data-asset-id=\"f066bc6c-98fe-48c4-b90d-1eb356184726\" data-image-id=\"f066bc6c-98fe-48c4-b90d-1eb356184726\" alt=\"\"></figure>\n<h2>Vergelijking met experimentele resultaten</h2>\n<p>Dit gedeelte bevat vergelijkingen tussen de breukbelastingen en bezwijkvormen volgens de <a data-item-id=\"8b9399db-b927-491a-a50f-c66ad97560af\" href=\"\">CSFM</a> en de experimentele resultaten. Om de CSFM ook te verifiëren voor het gedrag in de bruikbaarheidsgrenstoestand en voor het vervormingsvermogen, zijn voor geselecteerde liggers de belasting-vervormingsresponsies van het model vergeleken met die uit de proeven.</p>\n<h4>Bezwijkvormen en breukbelastingen</h4>\n<p>Tabel 6.13 vat de in de proeven gemeten uiterste dwarskrachten (<em>V</em><em><sub>u,exp</sub></em>) samen, evenals de door de CSFM voorspelde uiterste dwarskrachten (<em>V</em><em><sub>u,calc</sub></em>) en de bijbehorende bezwijkvormen. Deze tabel geeft ook het gemiddelde en de variatiecoëfficiënt (CoV) van de verhoudingen tussen de gemeten en de berekende breukbelastingen voor elk numeriek model. In alle analyses (behalve M3, waarin tension stiffening is verwaarloosd) voorspelde de CSFM afschuifbezwijking in de beugels. Dit komt goed overeen met de bezwijkmechanismen die in de proeven van Huber (2016) en Piyamahant (2002) zijn waargenomen, maar niet met die in Vecchio en Shim (2004). Doordat de bezwijkvormen in dit geval niet goed werden getroffen, waren de schattingen van de breukbelasting enigszins conservatief. Over het geheel genomen geven de standaardparameters goede schattingen van de sterkte, maar iets aan de onveilige kant (gemiddeld 6%).</p>\n<p>De gevoeligheid van de sterktevoorspellingen van de CSFM voor de verschillende geanalyseerde numerieke parameters is weergegeven in Fig. 6.19 aan de hand van de verhouding tussen de experimentele en de berekende uiterste dwarskrachten (<em>V</em><em><sub>u,exp</sub></em><em>/V</em><em><sub>u,calc</sub></em>). De breukbelasting is duidelijk gevoelig voor de gekozen grootte van de eindige elementen (zie Fig. 6.19 a). Het maximale verschil tussen de grofste en de fijnste mesh (M0 en M2) bedraagt 36% (Proef 4 van Piyamahant (2002)), met een gemiddeld verschil van ongeveer 15%. De voorspellingen met de standaardparameters (5 eindige elementen over de hoogte van de ligger in model M0) overschatten de experimentele sterkte enigszins (ongeveer 5%). Bij het verfijnen van de mesh tot 10 of 20 eindige elementen over de hoogte van de ligger (respectievelijk modellen M1 en M2) worden uitstekende sterktevoorspellingen verkregen, die iets aan de veilige kant van de breukbelastingen liggen. Er zijn geen veranderingen in de bezwijkvormen waargenomen bij het variëren van de meshgrootte van de eindige elementen. Zelfs de resultaten met de standaard meshgrootte zijn zeer bevredigend, gezien het feit dat verschillende experimenten bros afschuifbezwijken vertoonden, wat met rekenmethoden moeilijk te voorspellen is.</p>\n<p>De manier waarop tension stiffening wordt meegenomen heeft een zeer relevante invloed op de sterktevoorspellingen, zoals blijkt uit Fig. 6.19 b-c. Het meenemen van tension stiffening in de beugels door middel van het POM (de standaardinstelling in de CSFM) leidt gemiddeld tot een uitstekende overeenstemming met de experimentele resultaten (zie Fig. 6.19 b). Het verwaarlozen van tension stiffening leidt echter tot een gemiddelde overschatting van de breukbelasting van ongeveer 22% (zie Tabel 6.12). Bij het verwaarlozen van tension stiffening verandert de bezwijkvorm in buigbezwijking (zie Tabel 6.12) en komen de waargenomen afschuifbezwijkvormen niet overeen. De resultaten zijn ook zeer gevoelig voor de gekozen compression softening-relatie. Zoals te zien is in Fig. 6.19 c, geeft het gebruik van het Tension Chord Model in de beugels (model M4) in plaats van het Pull-out Model (model M1) iets betere resultaten dan het verwaarlozen van tension stiffening (model M3), maar worden de breukbelastingen nog steeds sterk overschat, met ongeveer 15% (zie Tabel 6.12). Hieruit kan worden geconcludeerd dat het gebruik van het Pull-Out-Model in deze voorbeelden cruciaal is voor het correct modelleren van het draaggedrag. </p>\n<figure data-asset-id=\"31352232-bbe5-4fb4-b54e-b0809010c318\" data-image-id=\"31352232-bbe5-4fb4-b54e-b0809010c318\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/4c6d49ee-719b-4472-9a67-06b062b2df81/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20table%206.13.png\" data-asset-id=\"31352232-bbe5-4fb4-b54e-b0809010c318\" data-image-id=\"31352232-bbe5-4fb4-b54e-b0809010c318\" alt=\"\"></figure>\n<figure data-asset-id=\"2fcb4c3b-de08-410f-93c2-989b65091df0\" data-image-id=\"2fcb4c3b-de08-410f-93c2-989b65091df0\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/f3c49504-d203-4f82-a25e-0ab44da27aaa/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20table%206.13-2.png\" data-asset-id=\"2fcb4c3b-de08-410f-93c2-989b65091df0\" data-image-id=\"2fcb4c3b-de08-410f-93c2-989b65091df0\" alt=\"\"></figure>\n<figure data-asset-id=\"48a1d13e-083a-4eb5-927c-4812e5f4e37d\" data-image-id=\"48a1d13e-083a-4eb5-927c-4812e5f4e37d\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/6fcfdbf1-9666-447a-988f-40df2e112587/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20table%206.13-3.png\" data-asset-id=\"48a1d13e-083a-4eb5-927c-4812e5f4e37d\" data-image-id=\"48a1d13e-083a-4eb5-927c-4812e5f4e37d\" alt=\"\"></figure>\n<figure data-asset-id=\"82f061b1-5fb4-4ea7-8342-375c6db59907\" data-image-id=\"82f061b1-5fb4-4ea7-8342-375c6db59907\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/23242854-167d-40ac-8182-edd05e68f870/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20fig%206.19.png\" data-asset-id=\"82f061b1-5fb4-4ea7-8342-375c6db59907\" data-image-id=\"82f061b1-5fb4-4ea7-8342-375c6db59907\" alt=\"\"></figure>\n<figure data-asset-id=\"3ccbbd8a-d639-4133-be1c-6e4e9c0abade\" data-image-id=\"3ccbbd8a-d639-4133-be1c-6e4e9c0abade\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/a1037f6e-2499-4ade-bb10-681d8ea41e54/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20fig%206.19-2.png\" data-asset-id=\"3ccbbd8a-d639-4133-be1c-6e4e9c0abade\" data-image-id=\"3ccbbd8a-d639-4133-be1c-6e4e9c0abade\" alt=\"\"></figure>\n<figure data-asset-id=\"52e40f52-8d8f-4d36-a4c6-7885d25979c3\" data-image-id=\"52e40f52-8d8f-4d36-a4c6-7885d25979c3\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/2391dda1-eba7-4458-8142-a7ff352da592/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20fig%206.19-3.png\" data-asset-id=\"52e40f52-8d8f-4d36-a4c6-7885d25979c3\" data-image-id=\"52e40f52-8d8f-4d36-a4c6-7885d25979c3\" alt=\"\"></figure>\n<p>Fig. 6.20 toont de continue spanningsveldresultaten (hoofddrukspanningen (<em>σ</em><em><sub>c</sub></em>) en staalspanningen (<em>σ</em><em><sub>sr</sub></em>) bij de scheuren) voor de proefstukken A1 en A3 van Vecchio en Shim (2004), waarin de voorspelde afschuifbezwijkingen zijn gemarkeerd. Deze resultaten zijn berekend met de numerieke parameters van M1 (standaardparameters, behalve de meshgrootte, die de helft van de standaardwaarde bedraagt). Zoals uit de spanningsvelden blijkt, bevindt de drukspanning in de drukzone ten gevolge van buiging zich in het plastische bereik (99,5 %). Door de gehanteerde criteria voor het verbrijzelen van het beton treedt het breken van de beugels echter eerder op dan het verbrijzelen van het beton. </p>\n<figure data-asset-id=\"72d9b512-35a8-4c59-bf70-2fa1c8913dba\" data-image-id=\"72d9b512-35a8-4c59-bf70-2fa1c8913dba\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/86e2648e-f52e-4162-8ad8-a9615fc572c6/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20fig%206.20.png\" data-asset-id=\"72d9b512-35a8-4c59-bf70-2fa1c8913dba\" data-image-id=\"72d9b512-35a8-4c59-bf70-2fa1c8913dba\" alt=\"\"></figure>\n<figure data-asset-id=\"5faec16e-18c3-4336-aaaf-92b23a315aed\" data-image-id=\"5faec16e-18c3-4336-aaaf-92b23a315aed\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/65718f07-1206-42f2-990a-97e7e18d0547/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20fig%206.20-2.png\" data-asset-id=\"5faec16e-18c3-4336-aaaf-92b23a315aed\" data-image-id=\"5faec16e-18c3-4336-aaaf-92b23a315aed\" alt=\"\"></figure>\n<h2>Belasting-vervormingsgedrag</h2>\n<p>De berekende belasting-vervormingsresponsies, verkregen met de numerieke parameters van M1 (met het TCM voor de buigwapening en het POM voor de beugels) en M3 (waarbij alle tension stiffening effecten worden verwaarloosd), zijn in Fig. 6.21 vergeleken met de gemeten belasting-vervormingsresponsies voor de proeven R500m352, T1, A1 en A3. De belasting <em>V</em> komt overeen met de aangebrachte dwarskracht en <em>u</em> met de doorbuiging halverwege de overspanning (zie Fig. 6.20a).</p>\n<figure data-asset-id=\"4f5fd5df-0fbe-493e-aab6-f3613ea8a079\" data-image-id=\"4f5fd5df-0fbe-493e-aab6-f3613ea8a079\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/b2a3cfdf-9892-4c29-b6b0-a26d1f47bb22/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20fig%206.21.png\" data-asset-id=\"4f5fd5df-0fbe-493e-aab6-f3613ea8a079\" data-image-id=\"4f5fd5df-0fbe-493e-aab6-f3613ea8a079\" alt=\"\"></figure>\n<figure data-asset-id=\"43e6f9e6-0867-450f-86f0-447652d4cba9\" data-image-id=\"43e6f9e6-0867-450f-86f0-447652d4cba9\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/e90b12fb-ba6f-4ce8-97c3-ebca7c6212f6/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20fig%206.21-2.png\" data-asset-id=\"43e6f9e6-0867-450f-86f0-447652d4cba9\" data-image-id=\"43e6f9e6-0867-450f-86f0-447652d4cba9\" alt=\"\"></figure>\n<figure data-asset-id=\"005c3655-78cd-4a12-b72d-adb1628d298d\" data-image-id=\"005c3655-78cd-4a12-b72d-adb1628d298d\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/ba5f9efa-97c3-461f-b6fc-5e3fdd4f0ff9/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20fig%206.21-3.png\" data-asset-id=\"005c3655-78cd-4a12-b72d-adb1628d298d\" data-image-id=\"005c3655-78cd-4a12-b72d-adb1628d298d\" alt=\"\"></figure>\n<p>Door rekening te houden met tension stiffening effecten kunnen de experimentele doorbuigingen over de gehele belastinggeschiedenis redelijk goed worden voorspeld, hoewel de doorbuigingen bij maximale belasting enigszins worden onderschat. Met name in Proef A3 van Vecchio en Shim (2004) kan het in de experimenten waargenomen plateau als gevolg van het vloeien van de buigwapening niet goed worden gevat in de numerieke analyse, omdat eerst het breken van de beugels wordt voorspeld. Het verwaarlozen van tension stiffening effecten leidt tot een overschatting van de breukbelastingen en vervormingen. Deze uitspraken voor de analyses zonder tension stiffening gelden ook bij gebruik van de parameters van M4 (het TCM zowel in de beugels als in de buigwapening).</p>\n<h2>Conclusies</h2>\n<p>Ten aanzien van de vergelijking tussen de CSFM-resultaten en het waargenomen gedrag in de geanalyseerde proeven op vrij opgelegde liggers met een kleine hoeveelheid beugels kunnen de volgende conclusies worden getrokken: </p>\n<ul>\n <li>De CSFM geeft goede schattingen van de breukbelasting, die bij gebruik van de standaard numerieke parameters enigszins wordt overschat (gemiddeld met 5%). Gecombineerde bezwijkvormen door afschuiving en verbrijzelen van het beton op buiging zijn moeilijk te vatten. De CSFM voorspelt bezwijken door het breken van de beugels, wat leidt tot sterktevoorspellingen die aan de conservatieve kant liggen. </li>\n <li>De voorspellingen van de breukbelasting zijn enigszins gevoelig voor variaties in de meshgrootte van de eindige elementen. De beste voorspellingen worden verkregen wanneer de standaard eindige-elementenmesh wordt verfijnd. Daarom wordt aanbevolen om bij het uitvoeren van de definitieve toetsingen altijd de invloed van de elementgrootte op de resultaten te onderzoeken. </li>\n <li>Het verwaarlozen van tension stiffening leidt tot een zeer sterke overschatting van de breukbelasting en het vervormingsvermogen. Zelfs wanneer tension stiffening in de beugels met het Tension Chord Model wordt gemodelleerd, ligt de voorspelde breukbelasting duidelijk aan de onveilige kant. 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