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 beoordeling van betonelementen worden normaal gesproken uitgevoerd op doorsnede- (1D-element) of puntniveau (2D-element). Deze procedure wordt beschreven in alle normen voor constructief ontwerp, bijvoorbeeld in (EN 1992-1-1 of ACI 318-19), en wordt gebruikt in de dagelijkse constructieve praktijk. Het is echter niet altijd bekend of wordt niet altijd gerespecteerd dat deze procedure alleen toelaatbaar is in gebieden waar de Bernoulli-Navier-hypothese van vlakke rekverdeling van toepassing is (aangeduid als B-gebieden). De plaatsen waar deze hypothese niet van toepassing is, worden discontinuïteits- of verstoorde gebieden genoemd (D-gebieden). Voorbeelden van B- en D-gebieden van 1D-elementen zijn weergegeven in (Fig. 1). Dit zijn bijvoorbeeld opleggingen, delen waar geconcentreerde belastingen worden aangebracht, locaties waar een abrupte verandering van de doorsnede optreedt, 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 Discontinuïteitsgebieden (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 ontwerpnormen en tegenwoordig vaak door ontwerpers worden gebruikt. Deze modellen zijn mechanisch consistente en krachtige hulpmiddelen. Merk op 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 essentie nog steeds als handberekening toegepast. De toepassing ervan op praktijkconstructies is omslachtig en tijdrovend, omdat iteraties nodig zijn en meerdere belastingsgevallen in beschouwing moeten worden genomen. Bovendien is deze methode niet geschikt voor het toetsen van bruikbaarheidscriteria (vervormingen, scheurwijdten, enz.).</p>\n<p>De interesse van constructeurs in een betrouwbaar en snel hulpmiddel om D-gebieden te ontwerpen, leidde tot de beslissing om de nieuwe Compatible Stress Field Method te ontwikkelen, een methode voor computerondersteund ontwerp met spanningsvelden die het automatisch ontwerpen en toetsen van constructieve betonelementen onderworpen aan belasting in het vlak mogelijk maakt.</p>\n<p>De Compatible Stress Field Method (CSFM) is een continue, op EEM gebaseerde spanningsveldanalysemethode waarin klassieke spanningsveldoplossingen worden aangevuld met kinematische overwegingen, d.w.z. de rektoestand wordt door de gehele constructie heen geëvalueerd. Hierdoor kan de effectieve druksterkte van beton automatisch worden berekend op basis van de toestand van dwarsrek, op een vergelijkbare manier als bij drukveldanalyses die rekening houden met compression softening (Vecchio and Collins 1986; Kaufmann and Marti 1998) en de EPSF-methode (Fernández Ruiz and Muttoni 2007). Bovendien houdt de CSFM rekening met tension stiffening, wat realistische stijfheden aan de elementen geeft, en dekt de methode alle voorschriften van de ontwerpnormen (inclusief aspecten van bruikbaarheid en vervormingscapaciteit) die door eerdere benaderingen niet consequent werden behandeld. De CSFM gebruikt de gangbare eenassige constitutieve wetten die door ontwerpnormen voor beton en wapening worden voorgeschreven. Deze zijn bekend in de ontwerpfase, wat het mogelijk maakt de partiële veiligheidsfactormethode toe te passen. Hierdoor hoeven ontwerpers geen aanvullende, vaak willekeurige materiaaleigenschappen op te geven, zoals doorgaans vereist is bij niet-lineaire EEM-analyses, waardoor de methode uitstekend geschikt is voor de ingenieurspraktijk.</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 die gezamenlijk is 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>Workflow en doelstellingen</h3>\n<p>Het doel van de wapeningsontwerptools in de <a data-item-id=\"42ce7f6b-6491-4224-a01e-c4c0072ed1cd\" href=\"\">CSFM</a> is om ontwerpers te helpen de locatie en de vereiste hoeveelheid wapeningsstaven efficiënt te bepalen. De volgende tools zijn beschikbaar om de gebruiker in dit proces te helpen/begeleiden: lineaire berekening en <a data-item-id=\"decdf07d-a46b-5894-9a22-793436e318c7\" href=\"\">topologie-optimalisatie</a>.</p>\n<p>Wapeningsontwerptools houden rekening met eenvoudigere constitutieve modellen dan de modellen die worden gebruikt voor de uiteindelijke verificatie van de constructie. Daarom moet de definitie van de wapening in deze stap worden beschouwd als een voorontwerp dat tijdens de laatste verificatiestap bevestigd/verfijnd moet worden. Het gebruik van de verschillende wapeningsontwerptools wordt geïllustreerd aan de hand van het model in Fig. 3, dat bestaat uit één uiteinde van een enkelvoudig ondersteunde ligger met variabele hoogte, belast met 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 gebruikt om het gebruik van de wapeningsontwerptools te illustreren.}}}\\]</em></p>\n<h3>Lineaire analyse</h3>\n<p>De lineaire analyse gaat uit van lineair-elastische materiaaleigenschappen en houdt geen rekening met wapening in het betonnen gebied. Het is daarom een zeer snelle berekening die een eerste inzicht geeft in de locaties van trek- en drukgebieden. 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 Resultaten van de lineaire analysetool voor het bepalen van de wapeningsindeling}}}\\]</em></p>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{(rood: gebieden in druk, blauw: gebieden in trek).}}}\\]</em></p>\n<h3>Topologie-optimalisatie</h3>\n<p>Topologie-optimalisatie is een methode die tot doel heeft de optimale verdeling van materiaal in een bepaald volume te vinden voor een bepaalde belastingconfiguratie. De topologie-optimalisatie geïmplementeerd in <em>Idea StatiCa Detail</em> maakt gebruik van een lineair eindige-elementenmodel. Elk eindig element kan een relatieve dichtheid hebben van 0 tot 100%, wat de relatieve hoeveelheid gebruikt materiaal weergeeft. Deze elementdichtheden zijn de optimalisatieparameters in het optimalisatieprobleem. De resulterende materiaalverdeling wordt als optimaal beschouwd voor de gegeven belastingenset als deze de totale rekenergie van het systeem minimaliseert. Per definitie is de optimale verdeling ook de geometrie met de grootst mogelijke stijfheid voor de gegeven belastingen.</p>\n<p>Het iteratieve optimalisatieproces begint met een homogene dichtheidsverdeling.<em> </em>De berekening wordt uitgevoerd voor meerdere totale volumefracties (20%, 40%, 60% en 80%), waardoor de gebruiker het meest praktische resultaat kan selecteren. 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 Resultaten van de topologie-optimalisatie ontwerptool met 20\\% en 40\\% effectief volume}}}\\]</em></p>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{(rood: gebieden in druk, blauw: gebieden in trek).}}}\\]</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 voorstellen (1D eindige elementen). Daarom wordt de wapening niet diffuus in de 2D eindige elementen van het beton ingebed, maar expliciet gemodelleerd en ermee verbonden. In het rekenmodel wordt een vlakke spanningstoestand beschouwd.</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 volledige <a data-item-id=\"a11adc2d-9c84-4667-8061-600660e1ad87\" href=\"\">wanden</a> en liggers, als details (delen) van liggers (geïsoleerd discontinuïteitsgebied, ook wel afgesneden uiteinde genoemd), kunnen worden gemodelleerd. Bij wanden en volledige liggers moeten de opleggingen zodanig worden gedefinieerd dat een (extern) isostatische (statisch bepaalde) of hyperstatische (statisch onbepaalde) constructie ontstaat. De belastingsoverdracht bij de afgesneden uiteinden van liggers wordt geïntroduceerd door middel van een speciale Saint-Venant-overgangszone, die zorgt voor een realistische spanningsverdeling in het geanalyseerde detailgebied.</p>"
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"value": "<p>Beoordeling van de constructie met behulp van het CSFM gebeurt aan de hand van twee verschillende analyses: één voor de bruikbaarheidsgrenstoestand en één voor belastingscombinaties in de uiterste grenstoestand. Bij de analyse van de bruikbaarheidsgrenstoestand wordt ervan uitgegaan dat het gedrag van het element bij de uiterste grenstoestand voldoende is en dat de vloeicondities van het materiaal niet worden bereikt bij belastingsniveaus in de bruikbaarheidsgrenstoestand. Deze aanpak maakt het mogelijk om vereenvoudigde constitutieve modellen (met een lineair traject van het spanning-rekdiagram van beton) te gebruiken voor de bruikbaarheidsanalyse, om de numerieke stabiliteit en de rekensnelheid te verbeteren. Het wordt daarom aanbevolen om de hieronder beschreven werkwijze te volgen, waarbij de analyse van de uiterste grenstoestand als eerste stap wordt uitgevoerd.</p>\n<h3>Analyse van de uiterste grenstoestand</h3>\n<p>De verschillende verificaties die door specifieke ontwerpnormen worden vereist, worden beoordeeld op basis van de directe resultaten die het model oplevert. UGT-verificaties worden uitgevoerd voor de betonsterkte, de wapeningssterkte en de verankering (aanhechtingsschuifspanningen).</p>\n<p>Om ervoor te zorgen dat een constructief element een efficiënt ontwerp heeft, wordt sterk aanbevolen om eerst een voorlopige analyse uit te voeren waarbij rekening wordt gehouden met de volgende stappen:</p>\n<ul>\n <li>Kies een selectie van de meest kritische belastingscombinaties.</li>\n <li>Bereken alleen belastingscombinaties voor de uiterste grenstoestand (UGT).</li>\n <li>Gebruik een grove 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 wordt zeer snel berekend, waardoor ontwerpers de detaillering van het constructieve element efficiënt kunnen beoordelen en de analyse opnieuw kunnen uitvoeren totdat aan alle verificatie-eisen is voldaan voor de meest kritische belastingscombinaties. Zodra aan alle verificatie-eisen van deze voorlopige analyse is voldaan, wordt aanbevolen om de volledige set uiterste belastingscombinaties op te nemen en een fijne meshgrootte te gebruiken (de door het programma aanbevolen meshgrootte). De gebruiker kan de meshgrootte wijzigen via de vermenigvuldigingsfactor, die waarden kan aannemen van 0,5 tot 5 (Fig. 19).</p>\n<p>De basisresultaten en verificaties (spanning, rek en benuttingsgraad (d.w.z. de berekende waarde/grenswaarde volgens de norm), evenals de richting van de hoofdspanningen bij betonelementen) worden weergegeven met behulp van verschillende plots, waarbij druk over het algemeen in rood en trek in blauw wordt weergegeven. Globale minimum- en maximumwaarden voor de gehele constructie kunnen worden gemarkeerd, evenals minimum- en maximumwaarden voor elk door de gebruiker gedefinieerd onderdeel. In een apart tabblad van het programma kunnen geavanceerde resultaten worden weergegeven, zoals tensorwaarden, vervormingen van de constructie en wapeningsverhoudingen (effectief en geometrisch) die worden gebruikt voor het berekenen van de tension stiffening van wapeningsstaven. Bovendien kunnen belastingen en reacties voor geselecteerde combinaties of belastingsgevallen worden weergegeven.</p>\n<h3>Analyse van de bruikbaarheidsgrenstoestand</h3>\n<p>BGT-beoordelingen worden uitgevoerd voor spanningsbeperking, scheurwijdte en doorbuigingslimieten. Spanningen in beton- en wapeningselementen worden gecontroleerd volgens de toepasselijke norm, op een vergelijkbare manier als beschreven voor de UGT.</p>\n<p>De bruikbaarheidsanalyse bevat bepaalde vereenvoudigingen van de constitutieve modellen die worden gebruikt voor de analyse van de uiterste grenstoestand. 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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 vervolgens met elkaar verbonden met behulp van multi-point constraints (MPC-elementen). Hierdoor kan de wapening een willekeurige, relatieve positie ten opzichte van het beton innemen. Als de verankeringslengte moet worden geverifieerd, worden aanhechtings- en verankeringseinde-veerelementen ingevoegd tussen de wapening en de MPC-elementen.</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 behulp van vierhoekige en driehoekige schaalelementen, CQUAD4 en CTRIA3. Deze kunnen respectievelijk worden gedefinieerd door vier of drie knopen. In deze elementen wordt uitsluitend vlakke spanning verondersteld, d.w.z. spanningen of rekken in de z-richting worden niet in beschouwing genomen.</p>\n<p>Elk element heeft vier of drie integratiepunten die op ongeveer 1/4 van de elementgrootte zijn geplaatst. Op elk integratiepunt in 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 stijfheden <em>E</em><sub>1</sub>, <em>E</em><sub>2</sub> geëvalueerd volgens het opgegeven spanning-rekdiagram van beton, zoals weergegeven in Fig. 2. Opgemerkt moet worden dat het effect van compression softening het gedrag in de belangrijkste drukrichting koppelt aan de actuele toestand van de andere hoofdrichting.</p>\n<h3>Wapening</h3>\n<p>Wapeningsstaven worden gemodelleerd met tweeknoops 1D “staaf”-elementen (CROD), die alleen axiale stijfheid hebben. Deze elementen zijn verbonden met speciale “bond”-elementen die zijn ontwikkeld om het slipgedrag tussen een wapeningsstaaf en het omringende beton te modelleren. Deze bond-elementen worden vervolgens door middel van MPC-elementen (multi-point constraint) verbonden met de mesh die het beton weergeeft. Deze aanpak maakt onafhankelijke mesh-vorming van wapening en beton mogelijk, terwijl hun onderlinge verbinding later wordt gewaarborgd.</p>\n<h3>Aanhechtingselementen</h3>\n<p>De verankeringslengte wordt geverifieerd door de aanhechtingsschuifspanningen tussen betonelementen (2D) en wapeningsstaafelementen (1D) in het eindige-elementenmodel te implementeren. Hiertoe is een “bond”-eindig-elementtype ontwikkeld.</p>\n<p>De definitie van het bond-element is vergelijkbaar met die van een schaalelement (CQUAD4). Het wordt ook gedefinieerd door 4 knopen, maar in tegenstelling tot een schaal heeft het alleen een niet-nul stijfheid in afschuiving tussen de twee bovenste en twee onderste knopen. In het model zijn de bovenste knopen verbonden met de elementen die de wapening voorstellen en de onderste knopen met die welke het beton voorstellen. Het gedrag van dit element wordt beschreven door de aanhechtingsspanning, τ<em><sub>b</sub></em>, als een 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 bond-slip-relatie, <em>G</em><em><sub>b</sub></em>, is als volgt gedefinieerd:</p>\n<p>\\[G_b = k_g \\cdot \\frac{E_c}{Ø}\\]</p>\n<p>waarbij:</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> in geval van 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>, opgegeven in de betreffende geselecteerde ontwerpnormen EN 1992-1-1 of ACI 318-19, worden gebruikt om de verankeringslengte te verifiëren. 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 ontwerpnormen, maakt het mogelijk de basisverankeringslengte van de staven (<em>l</em><em><sub>b,net</sub></em>) te verminderen met een bepaalde factor β (hieronder aangeduid als de ‘verankeringscoëfficiënt’). De rekenwaarde van de verankeringslengte (<em>l</em><em><sub>b</sub></em>) wordt vervolgens 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> komt overeen met de activering van de wapeningsstaaf aan het uiteinde ervan bij een percentage van zijn maximale capaciteit dat wordt 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 door middel van een veerelement aan het uiteinde van de staaf (Fig. 15), dat wordt gedefinieerd door het constitutieve model weergegeven in Fig. 15b. De maximale kracht die door deze veer wordt overgebracht (<em>F</em><em><sub>au</sub></em>) is:</p>\n<p>\\[F_{au} = \\beta \\cdot A_s \\cdot f_{yd}\\]</p>\n<p>waarbij:</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 beton bij druk (σ</strong><em><strong><sub>c</sub></strong></em><strong><sub>2</sub></strong><em><strong><sub>r</sub></strong></em><strong>) en wapeningsspanningen (σ</strong><em><strong><sub>sr</sub></strong></em><strong>) ter plaatse van de scheuren, waarbij de treksterkte van het beton wordt verwaarloosd (σ</strong><em><strong><sub>c</sub></strong></em><strong><sub>1</sub></strong><em><strong><sub>r</sub></strong></em><strong> = 0), behalve voor het stiffening effect op de wapening.</strong> Door rekening te houden met tension stiffening kunnen de gemiddelde rekken van de wapening (ε<em><sub>m</sub></em>) worden gesimuleerd. Er wordt uitgegaan van fictieve, roterende, spanningsvrije scheuren die zonder slip openen (Fig. 2a), en er wordt ook rekening gehouden met het evenwicht ter plaatse van de scheuren samen met de gemiddelde rekken van de wapening. </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 aannames nauwkeurige voorspellingen opleveren voor gewapende staven onderworpen aan belasting in het vlak (Kaufmann 1998; Kaufmann en Marti 1998), mits de aanwezige 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 consistent met de principes van moderne ontwerpnormen, die grotendeels gebaseerd zijn op de plasticiteitstheorie.</p>\n<p>Echter, <strong>de CSFM is niet geschikt voor slanke elementen</strong> zonder dwarswapening, aangezien relevante mechanismen voor dergelijke elementen, zoals korrelinterlock, resterende trekspanningen aan de scheurtip en deuvelwerking – die allemaal direct of indirect afhankelijk zijn van de treksterkte van het beton – buiten beschouwing worden gelaten. Hoewel sommige ontwerpnormen het ontwerp van dergelijke elementen op basis van semi-empirische bepalingen toestaan, is de CSFM niet bedoeld voor dit type potentieel bros construeren.</p>\n<h4>Beton</h4>\n<p>Het betonmodel dat in de CSFM is geïmplementeerd, is gebaseerd op de uniaxiale drukconstitutieve wetten die door ontwerpnormen worden voorgeschreven voor het ontwerp van dwarsdoorsneden, 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 meer vereenvoudigde elastisch-ideaal-plastische relatie. Bij toetsing volgens de ACI-norm is het alleen mogelijk om het parabool-rechthoek spanning-rek diagram te gebruiken. Zoals eerder vermeld, wordt de treksterkte verwaarloosd, zoals ook gebruikelijk is bij klassiek gewapend betonontwerp.</p>\n<p>De effectieve drukvastheid wordt automatisch bepaald voor gescheurd beton op basis van de hoofdtrekrek (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. De geïmplementeerde reductierelatie (Fig. 2e) is een generalisatie van het voorstel van de <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, wat 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> houdt geen rekening met een expliciet bezwijkcriterium in termen van rek voor beton bij druk (d.w.z. er wordt uitgegaan van een oneindig plastische tak nadat de piekspanning is bereikt). Deze vereenvoudiging maakt het niet mogelijk om het vervormingsvermogen van constructies die op druk bezwijken te verifiëren. 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>) gedefinieerd in (Fig. 2e), ook rekening wordt gehouden met de toename van de brosheid van beton naarmate de sterkte toeneemt, door middel van de reductiefactor <em>\\( \\eta_{fc} \\)</em> gedefinieerd in de <em>fib</em> Model Code 2010 als volgt:</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>waarbij:</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 als gevolg van de aanwezigheid van dwarsscheurvorming</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 is ook een reductie van de factor <em>k</em><em><sub>c</sub></em><sub>2</sub> vanwege de stabiliteit van de berekening. Deze reductie heeft geen invloed op de totale sterkte van staven. Uitgaande van de waarde <em>f</em><em><sub>cd</sub></em> als de gefactoreerde sterkte van beton (rekenwaarde), wordt de waarde van <em>k</em><em><sub>c</sub></em><sub>2</sub> verminderd volgens de volgende 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 afgelezen uit de<br> grafiek weergegeven 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 rechtstreeks afgelezen uit de grafiek van Fig. 2f</p>\n<h4>Wapening</h4>\n<p>Het geïdealiseerde bilineaire spanning-rek diagram voor kale wapeningsstaven, zoals doorgaans gedefinieerd door ontwerpnormen (Fig. 2d), wordt gehanteerd. 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 gedefinieerde spanning-rek relatie worden opgegeven.</p>\n<p>Er wordt rekening gehouden met tension stiffening door de ingevoerde spanning-rek relatie van de kale wapeningsstaaf aan te passen, om zo de gemiddelde stijfheid van de in het beton ingebedde staven (ε<em><sub>m</sub></em>) vast te leggen.</p>\n<h4>Aanhechtingsmodel</h4>\n<p>Bond-slip tussen wapening en beton wordt in het eindige-elementenmodel geïntroduceerd door de vereenvoudigde, star-perfect-plastische constitutieve relatie in Fig. 2f te hanteren, waarbij <em>f</em><em><sub>bd</sub></em> de rekenwaarde (gefactoreerde waarde) is van de uiterste aanhechtingsspanning zoals gespecificeerd door de ontwerpnorm voor de specifieke aanhechtingscondities.</p>\n<p>Dit is een vereenvoudigd model met als enig doel het toetsen van aanhechtingsvoorschriften volgens ontwerpnormen (d.w.z. verankering van wapening). De reductie van de verankeringslengte bij gebruik van haken, lussen en vergelijkbare staafvormen kan in rekening worden gebracht door een bepaalde capaciteit 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 opleggingen (afb. 7) en componenten voor belastingoverdracht (afb. 8) beschikbaar.</p>\n<h3>Opleggingen</h3>\n<p>Een puntoplegging kan op verschillende manieren worden gemodelleerd om ervoor te zorgen dat spanningen niet in één punt geconcentreerd zijn, maar over een groter gebied worden verdeeld. De eerste optie is een verdeelde puntoplegging (afb. 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-oplegging (afb. 7d) kan daarentegen alleen binnen een betonvolume met een gedefinieerde effectieve straal worden geplaatst. Deze wordt vervolgens door middel van stijve elementen verbonden met de knopen van het wapeningsnet binnen deze straal. Daarom is het vereist om een wapeningskooi rondom de patch-oplegging te definiëren.</p>\n<p>Voor een nauwkeuriger modellering van bepaalde realistische scenario's zijn er nog twee andere opties voor puntoplegging. Ten eerste is er een puntoplegging met een oplegplaat van gedefinieerde breedte en dikte (afb. 7b). Het materiaal van de oplegplaat kan worden gespecificeerd, en de gehele oplegplaat wordt onafhankelijk gemesht. Ten tweede is er een hangende oplegging beschikbaar (afb. 7e), die kan worden gebruikt voor het modelleren van hijsankers of hijsdeuvels.</p>\n<p>Een lijnoplegging (afb. 7c) kan worden gedefinieerd op een rand (door de lengte ervan op te geven) of binnen een element (door middel van een polylijn). Het is ook mogelijk om de stijfheid en/of het niet-lineaire gedrag ervan te specificeren (oplegging in druk/trek of alleen in druk).</p>\n<ul>\n <li>Lees gedetailleerde beschrijvingen in<strong> </strong><a data-item-id=\"5a121972-f384-4f14-8788-9da298e1aae1\" href=\"\"><strong>Types opleggingen in IDEA StatiCa Detail</strong></a></li>\n</ul>\n<h3>Belastingoverdragende componenten</h3>\n<p>Het inleiden van belastingen in de constructie kan eveneens op verschillende manieren worden gemodelleerd. Voor puntlasten kan een oplegplaat (afb. 8a) worden gebruikt, op dezelfde manier als bij puntoplegging, waarbij de geconcentreerde belasting dankzij een stalen plaat met gedefinieerde breedte en dikte over een groter gebied wordt verdeeld. </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 ofwel rechtstreeks op het oppervlak van de constructie worden aangebracht met een gedefinieerde werkingsstraal (de belasting wordt op de betonelementen aangebracht), ofwel via een speciaal overdrachtsmiddel genaamd patch-belasting (afb. 8b en afb. 9). Patch-belasting maakt het mogelijk om de belasting rechtstreeks over te dragen naar de gedefinieerde wapening binnen het gebied van de effectieve straal. Om de correcte werking van de patch-belasting te waarborgen, is het noodzakelijk om een groep wapeningsstaven te definiëren die met de belasting worden verbonden (in de wapeningseigenschappen). Wanneer de te verbinden wapening niet is gedefinieerd, is het belastingoverdrachtsmechanisme hetzelfde als bij een puntlast die op een staafoppervlak is geplaatst, en wordt de belasting via de randvoorwaarden overgedragen naar de betonelementen, niet rechtstreeks 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 hangende belasting (afb. 8c). De gebruiker kan een partieel belast gebied (afb. 8d) gebruiken, waarmee de drukcapaciteit van beton kan worden verhoogd volgens Eurocode (het is niet mogelijk om dit type belastingoverdragende component te gebruiken wanneer ACI is ingesteld). De constructie kan ook worden belast met lijnlasten op de randen, via een algemene polylijn, of met vlaklasten. De Detail applicatie is in staat om automatisch rekening te houden met het eigen gewicht in de analyse.</p>\n<p><br></p>"
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"value": "<p>In veel gevallen hoeven we slechts een bepaald detail (deel) van een constructief staaf te modelleren, zoals de oplegging van een ligger, een opening in het midden van de ligger, enz. Deze aanpak kan leiden tot oplegconfiguraties die instabiel maar toelaatbaar zijn in <em>IDEA StatiCa Detail</em> (inclusief het geval zonder opleggingen). In dergelijke gevallen is het echter ook noodzakelijk om de doorsnede te modelleren die de verbinding met het aangrenzende B-gebied vertegenwoordigt, inclusief de inwendige krachten op deze doorsnede die aan het evenwicht voldoen. In bepaalde gevallen (bijvoorbeeld bij het modelleren van een liggeroplegging) kunnen deze inwendige krachten automatisch door het programma worden bepaald.</p>\n<p>Tussen het B-gebied en het geanalyseerde discontinuïteitsgebied wordt automatisch een Saint-Venant-overgangszone aangemaakt 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. Aangezien het enige doel van de Saint-Venant-zone is om een juiste spanningsverdeling in de rest van het model te bereiken, worden er geen resultaten uit dit gebied weergegeven bij de verificatie en worden hier geen stopcriteria in aanmerking genomen.</p>\n<p>De rand van de Saint-Venant-zone die het verkorte uiteinde van de ligger vertegenwoordigt, wordt gemodelleerd als star, d.w.z. deze mag roteren maar moet vlak blijven. Dit wordt bereikt door alle FEM-knopen van de rand te verbinden met een afzonderlijke knoop in het traagheidsmiddelpunt van de doorsnede met behulp van een star lichaamselement<em> </em>(RBE2). De inwendige krachten 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 Overdracht van inwendige krachten bij een verkort uiteinde.}}}\\]</em></p>"
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"value": "<p>Er wordt een standaard volledig Newton-Raphson (NR) algoritme gebruikt om de oplossing van een niet-lineair EEM-probleem te vinden. </p>\n<p>Over het algemeen convergeert het NR-algoritme meestal niet wanneer de volledige belasting in één stap wordt aangebracht. Een gebruikelijke aanpak, die ook hier wordt toegepast, is om de belasting sequentieel in meerdere increments aan te brengen en het resultaat van het vorige belastingsincrement te gebruiken om de Newton-oplossing van een volgend increment te starten. Hiertoe is een belastingscontrole-algoritme geïmplementeerd bovenop de Newton-Raphson-methode. 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 belastingscontrole-algoritme is het vinden van de kritieke belasting, die overeenkomt met bepaalde \"stopcriteria\" – specifiek de maximale rek in beton, de maximale slip in aanhechtingselementen, de maximale verplaatsing in verankeringselementen en de maximale rek in wapeningsstaven. De kritieke belasting wordt gevonden met behulp van 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 van de helft van de grootte van het vorige. Dit proces wordt herhaald totdat de kritieke belasting is gevonden binnen een bepaalde foutmarge.</p>\n<p>Voor beton werd het stopcriterium ingesteld op een rek van 5% bij druk (d.w.z. ongeveer een orde van grootte groter dan de werkelijke bezwijkrek van beton) en 7% bij trek op de integratiepunten van schaalelementen. Bij trek werd de waarde zo ingesteld dat de grensrek in de wapening, die gewoonlijk ongeveer 5% bedraagt zonder rekening te houden met tension stiffening, eerst wordt bereikt. Bij druk werd de waarde gekozen uit verschillende 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 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 Constitutieve relatie van aanhechtings- en verankeringselementen gebruikt voor de verificatie van de verankeringslengte:}}}\\]</em></p>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{(a) aanhechting-afschuifspanning-slipresponsie van een aanhechtingselement; (b) kracht-verplaatsingsresponsie van een verankeringselement.}}}\\]</em></p>\n<p>Voor wapening wordt het stopcriterium gedefinieerd in termen van spanningen. Aangezien de spanningen ter plaatse van de scheur worden gemodelleerd, komt het criterium bij trek overeen met de treksterkte van de wapening, rekening houdend met de veiligheidscoëfficiënt. Dezelfde waarde wordt gebruikt voor het criterium bij 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 wordt gebruikt bij normtoetsingen en α = 10.</p>"
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"value": "<p>De resultaten worden afzonderlijk gepresenteerd voor beton en voor wapeningselementen. De spannings- en rekwaarden in het beton worden berekend op de integratiepunten van schaalelementen. Aangezien het echter niet praktisch is om de gegevens op deze manier weer te geven, worden de resultaten standaard gepresenteerd in knopen, zoals de maximale waarde van de drukspanning van aangrenzende Gauss-integratiepunten in verbonden elementen (Fig. 18). Opgemerkt dient te worden dat deze weergave de resultaten aan gedrukte randen van staven lokaal kan onderschatten in het geval dat de afmeting van het eindige element vergelijkbaar is met de hoogte van de drukzone.</p>\n<figure data-asset-id=\"5633d094-25c8-46e3-a481-843b6082214b\" data-image-id=\"5633d094-25c8-46e3-a481-843b6082214b\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/9dac87f5-fd94-41db-bcb2-c56897b22a45/Result%20presentation.PNG\" data-asset-id=\"5633d094-25c8-46e3-a481-843b6082214b\" data-image-id=\"5633d094-25c8-46e3-a481-843b6082214b\" alt=\"\"></figure>\n<p><em>Fig. 18 - Eindig element van beton met integratiepunten en knopen: presentatie van de resultaten voor beton in knopen en in eindige elementen.</em></p>\n<p>De resultaten voor de eindige elementen van de wapening zijn ofwel constant voor elk element (één waarde – bijv. voor staalspanningen) of lineair (twee waarden – voor aanhechtingsresultaten). Voor hulpelementen, zoals elementen van oplegplaten, worden alleen vervormingen gepresenteerd.</p>"
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"value": "<h3>Beton - UGT</h3>\n<p>Het betonmodel dat in de CSFM is geïmplementeerd, is gebaseerd op de constitutieve wetten voor eenassige druk zoals voorgeschreven door EN 1992-1-1 voor het ontwerp van doorsneden, die alleen afhankelijk zijn van de druksterkte. Het parabool-rechthoekdiagram gespecificeerd in EN 1992-1-1 Cl. 3.1.7 (1) (Fig. 24a) wordt standaard gebruikt in de CSFM, maar ontwerpers kunnen ook kiezen voor een meer vereenvoudigde elastisch-ideaalplastische relatie volgens EN 1992-1-1 Cl. 3.1.7 (2) (Fig. 24b). De treksterkte wordt verwaarloosd, zoals gebruikelijk is bij klassiek gewapend betonontwerp.</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 De spanning-rekdiagrammen van beton voor UGT: a) parabool-rechthoekdiagram; b) bilineair diagram.}}}\\]</em></p>\n<p>De implementatie van de CSFM in <em>IDEA StatiCa Detail</em> houdt geen expliciet bezwijkcriterium in termen van rek aan voor beton onder druk (d.w.z. na het bereiken van de piekspanning wordt een plastische tak beschouwd 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 een uiterste rek van minder dan 0,35% aanneemt). Deze vereenvoudiging maakt het niet mogelijk om het vervormingsvermogen van constructies die op druk bezwijken te verifiëren. De uiteindelijke 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)), de toenemende 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 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>waarbij:</p>\n<p>α<em><sub>cc</sub></em> is de coëfficiënt die rekening houdt met langetermijneffecten op de druksterkte en met ongunstige effecten die voortvloeien uit de wijze waarop de belasting wordt aangebracht. Deze is volgens EN 1992-1-1 Cl. 3.1.6 (1). De standaardwaarde is 1,0.</p>\n<p><em>k</em><em><sub>c </sub></em>is de algemene reductiefactor van de druksterkte</p>\n<p><em>k</em><em><sub>c</sub></em><sub>2</sub> is de reductiefactor als gevolg van de aanwezigheid van dwarsscheurvorming</p>\n<p><em>f</em><em><sub>ck</sub></em> is de karakteristieke cilindersterkte van beton (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 De compression softening-wet.}}}\\]</em></p>\n<h3>Beton - BGT</h3>\n<p>De bruikbaarheidsanalyse bevat bepaalde vereenvoudigingen van de constitutieve modellen die worden gebruikt voor de analyse van de uiterste grenstoestand. De plastische tak van het spanning-rekdiagram van beton onder druk wordt buiten beschouwing gelaten, terwijl de elastische tak lineair en oneindig is. Compression softening wordt niet in rekening gebracht. Deze vereenvoudigingen verbeteren de numerieke stabiliteit en rekensnelheid en beperken de algemeenheid van de oplossing niet, zolang de resulterende materiaalspanningslimieten bij bruikbaarheid duidelijk onder hun vloeipunten liggen (zoals vereist door Eurocode). Daarom zijn de vereenvoudigde modellen die voor bruikbaarheid worden gebruikt alleen geldig als aan alle verificatievereisten 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 Spanning-rekdiagrammen van beton geïmplementeerd voor bruikbaarheidsanalyse: korte- en langetermijnverificaties.}}}\\]</em></p>\n<p><br></p>\n<p><strong>Langetermijneffecten</strong></p>\n<p>Bij de bruikbaarheidsanalyse worden de langetermijneffecten van beton in rekening gebracht met behulp van een effectieve oneindige kruipcoëfficiënt (\\(\\varphi\\), standaard aangenomen als 2,5), die de secans-elasticiteitsmodulus van beton (<em>E</em><em><sub>cm</sub></em>) aanpast volgens EN 1992-1-1, sectie 3.1.4 (3) resp. 7.4.3 (5) als volgt:</p>\n<p>\\[E_{c,eff} = \\frac{E_{cm}}{1+\\varphi}\\]</p>\n<p>Bij het in rekening brengen van langetermijneffecten wordt eerst een belastingstap met alle permanente belastingen berekend rekening houdend met de kruipcoëfficiënt (d.w.z. met gebruik van de effectieve elasticiteitsmodulus van beton, <em>E</em><em><sub>c,eff</sub></em>), en vervolgens worden de bijkomende belastingen berekend zonder de kruipcoëfficiënt (d.w.z. met gebruik van <em>E</em><em><sub>cm</sub></em>). Daarnaast wordt, om kortetermijnverificaties uit te voeren, een andere berekening uitgevoerd waarbij alle belastingen worden berekend zonder de kruipcoëfficiënt. Beide berekeningen voor lange- en kortetermijnverificaties zijn weergegeven in Fig. 26.</p>\n<p>Kruipfactoren worden door de gebruiker gedefinieerd in de materiaaleigenschappen en dienen te 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 de blote wapeningsstaven gedefinieerd in EN 1992-1-1, sectie 3.2.7 (Fig. 27) beschouwd. De definitie van dit diagram vereist alleen dat de basiseigenschappen van de wapening bekend zijn tijdens de ontwerpfase (sterkte- en ductiliteitsklasse). Wanneer bekend, kan de werkelijke spanning-rekrelatie van de wapening (warmgewalst, koudvervormd, gehard en zelf-ontlaten, …) in rekening worden gebracht. Het spanning-rekdiagram van de wapening kan door de gebruiker worden gedefinieerd, maar in dit geval is het onmogelijk om het tension stiffening-effect aan te nemen (het is onmogelijk om de scheurbreedte te berekenen). Het gebruik van het spanning-rekdiagram met een horizontale bovenste tak maakt de verificatie van de constructieve duurzaamheid onmogelijk. Daarom is handmatige verificatie van 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 Spanning-rekdiagram van wapening: a) bilineair diagram met een schuine bovenste tak; b) bilineair diagram}}}\\) \\( \\textsf{\\textit{\\footnotesize{met een horizontale bovenste tak.}}}\\)</em></p>\n<p><br></p>\n<p>Tension stiffening (Fig. 28) wordt automatisch in rekening gebracht door de invoerspanning-rekrelatie van de blote wapeningsstaaf aan te passen om de gemiddelde stijfheid van de in het beton ingebedde staven vast te leggen (ε<em><sub>m</sub></em>).</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 Schema van tension stiffening.}}}\\]</em></p>"
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"value": "<p>De Compatible Stress Field Method voldoet aan moderne ontwerpnormen. Omdat de rekenmodellen alleen gebruikmaken van standaard materiaaleigenschappen, kan het formaat van partiële veiligheidsfactoren zoals voorgeschreven in de ontwerpnormen zonder aanpassing worden toegepast. Op deze manier worden de invoerbelastingen gefactoreerd en worden de karakteristieke materiaaleigenschappen gereduceerd met de betreffende veiligheidscoëfficiënten die in de ontwerpnormen zijn voorgeschreven, precies zoals bij een conventionele betonanalyse. De waarden van de materiaalveiligheidsfactoren die zijn voorgeschreven in EN 1992-1-1 hoofdstuk 2.4.2.4 zijn standaard ingesteld, maar de gebruiker kan de veiligheidsfactoren wijzigen in de Code and calculation settings (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 De instelling van materiaalveiligheidsfactoren in Idea StatiCa Detail.}}}\\]</em></p>\n<p><br></p>\n<p>Belastingveiligheidsfactoren moeten door de gebruiker worden gedefinieerd in Combination rules voor elke niet-lineaire combinatie van belastinggevallen (Fig. 30). Voor alle templates die zijn geïmplementeerd in <a data-item-id=\"b4790cf9-a605-45b3-b41b-e36909ad4291\" href=\"\">Idea StatiCa Detail</a>, zijn 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 De instelling van partiële belastingfactoren in Idea StatiCa Detail.}}}\\]</em></p>\n<p><br></p>\n<p>Door gebruik te maken van geschikte, door de gebruiker gedefinieerde combinaties van partiële veiligheidsfactoren, kunnen gebruikers ook rekenen met de CSFM volgens de globale resistance factor method (Navrátil, et al. 2017), maar deze aanpak wordt in de ontwerppraktijk vrijwel nooit gebruikt. Sommige richtlijnen bevelen het gebruik van de globale resistance factor method aan voor niet-lineaire analyses. Echter, bij vereenvoudigde niet-lineaire analyses (zoals de CSFM), waarbij alleen die materiaaleigenschappen nodig zijn die ook bij conventionele handberekeningen worden gebruikt, blijft het gebruik van het partiële veiligheidsformaat de voorkeur verdienen.</p>"
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"value": "<p>De verschillende verificaties die door EN 1992-1-1 vereist worden, worden beoordeeld op basis van de directe resultaten van het model. UGT-verificaties worden uitgevoerd voor betonsterkte, wapeningssterkte en verankering (aanhechtingsschuifspanningen).</p>\n<p>De <strong>betonsterkte</strong> in druk wordt geëvalueerd als de verhouding tussen de maximale hoofddrukspanning σ<em><sub>c </sub></em>= σ<em><sub>c</sub></em><sub>2</sub> verkregen 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 in trek als in druk geëvalueerd als de verhouding tussen de spanning in de wapening ter plaatse van 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{voor bilineair diagram met hellende bovenste tak}}\\)</p>\n<p>\\(σ_{s,lim} = \\frac{f_{yk}}{γ_s}\\qquad\\qquad\\,\\,\\,\\,\\textsf{\\small{voor bilineair diagram met horizontale bovenste tak}}\\)</p>\n<p>waarbij:</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 van de treksterkte <em>f</em><em><sub>tk</sub></em> tot de vloeispanning,<br> \\(k = \\frac{f_{tk}}{f_{yk}}\\)</p>\n<p><em>γ</em><em><sub>s </sub></em><sub> </sub>is de partiële veiligheidsfactor voor wapening</p>\n<p>De <strong>aanhechtingsschuifspanning</strong> wordt onafhankelijk geëvalueerd als de verhouding tussen de aanhechtingsspanning τ<em><sub>b</sub></em> berekend door 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>waarbij:</p>\n<p><em>f</em><em><sub>ctd</sub></em><sub> </sub> de rekenwaarde van de betontreksterkte volgens EN 1992-1-1 art. 3.1.6 (2) is. Vanwege de toenemende brosheid van beton met hogere sterkte is <em>f</em><em><sub>ctk,0.05</sub></em><sub> </sub>beperkt 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 betrekking heeft op de kwaliteit van de aanhechtingsconditie en de positie van de staaf tijdens het storten van het beton (Fig. 31).</p>\n<p>η<sub>1</sub> = 1,0 wanneer ‘goede’ omstandigheden worden verkregen en</p>\n<p>η<sub>1</sub> = 0,7 voor alle andere gevallen en voor staven in constructieve elementen gebouwd met glijbekisting, tenzij kan worden aangetoond dat er ‘goede’ aanhechtingsomstandigheden bestaan</p>\n<p>η<sub>2</sub> betrekking heeft op 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 Figuur 8.2 - Beschrijving van aanhechtingsomstandigheden.}}}\\]</em></p>\n<p>In IDEA StatiCa Detail wordt rekening gehouden met de aanhechtingsomstandigheden volgens Fig. 31 c) en d). De richting van het storten van het beton kan in de applicatie voor elk projectonderdeel 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 verificaties worden uitgevoerd met inachtneming van de betreffende grenswaarden voor de respectieve delen van de constructie (d.w.z. ondanks dat er slechts één kwaliteit is voor zowel het beton als de wapening, zullen de uiteindelijke spanning-rekdiagrammen in elk deel van de constructie verschillen 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 is hier te vinden: <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 \\sigma_{s}\\]</p>\n<p>waarbij <em>A</em><em><sub>s</sub></em> de oppervlakte van de wapeningsstaaf is en <em>σ</em><em><sub>s</sub></em> de spanning in de staaf.</p>\n<p>Of als een 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 is 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 staaf 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}=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>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 punt van interesse.</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 Definitie van de grenskracht 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 op basis van 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>verankeringstypes</strong> in de CSFM omvatten een rechte staaf (d.w.z. geen reductie aan het uiteinde van het anker), bocht, haak, lus, gelaste dwarsstaaf, perfecte aanhechting en doorlopende staaf. Al deze types, samen met de bijbehorende verankeringscoëfficiënten β, worden getoond in Fig. 32 voor langswapening en in Fig. 33 voor beugels. De waarden van de gehanteerde verankeringscoëfficiënten zijn in overeenstemming met EN 1992-1-1 sectie 8.4.4 Tab. 8.2. Er dient te worden opgemerkt dat ondanks de verschillende beschikbare opties, de CSFM drie soorten verankeringsuiteinden onderscheidt: (i) geen reductie van de verankeringslengte, (ii) een reductie van 30% van de verankeringslengte in het geval van 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 Beschikbare verankeringstypes en bijbehorende verankeringscoëfficiënten voor langswapeningsstaven in de CSFM:}}}\\]</em></p>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{(a) rechte staaf; (b) bocht; (c) haak; (d) lus; (e) gelaste dwarsstaaf; (f) perfecte aanhechting; (g) doorlopende staaf.}}}\\]</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 Beschikbare verankeringstypes en bijbehorende verankeringscoëfficiënten voor beugels.}}}\\]</em></p>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Gesloten beugels: (a) haak; (b) bocht; (c) overlap. Open beugels: (d) haak; (e) doorlopende staaf.}}}\\]</em></p>\n<p>Om te voldoen aan EN 1992-1-1 moet de verankeringsveer worden gebruikt in de berekening; de verankeringsveer wordt aangepast met de β-coëfficiënt, zodat de gebruiker een van de beschikbare verankeringstypes moet gebruiken bij het definiëren van de begin- en eindcondities van de wapening. </p>"
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"value": "<p>Bij het ontwerpen van betonconstructies komen we twee grote groepen gedeeltelijk belaste gebieden (PLA) tegen - de eerste groep omvat opleggingen, terwijl de andere bestaat uit verankeringsgebieden. Volgens de huidige geldige normen voor het ontwerp van gewapend betonconstructies EN 1992-1-1 hfst. 6.7 (<em>Fig. 34</em>), moet voor gedeeltelijk belaste gebieden rekening worden gehouden met lokaal verbrijzelen van beton en transversale trekkrachten. Voor een gelijkmatig verdeelde belasting op een gebied, <em>A</em><em><sub>c0</sub></em>, kan de drukcapaciteit van beton met maximaal een factor drie worden verhoogd, afhankelijk van het rekenkundige verdeelgebied <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 Gedeeltelijk belaste gebieden volgens EN 1992-1-1.}}}\\]</em></p>\n<p>Het gedeeltelijk belaste gebied moet voldoende worden gewapend met dwarswapening die is ontworpen om de splijtkrachten die in het gebied optreden, over te dragen. Voor het ontwerp van dwarswapening in gedeeltelijk belaste gebieden wordt volgens de Eurocode de Staafwerk-methode gebruikt. Zonder de vereiste dwarswapening is het niet mogelijk om een verhoging van de drukcapaciteit van het beton in aanmerking te nemen.</p>\n<p><br></p>\n<p><strong>Gedeeltelijk belaste gebieden 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 Fictieve drukdiagonalen met eindige-elementen-mesh van het beton.}}}\\]</em></p>\n<p>Met behulp van de CSFM is het mogelijk om gewapend betonconstructies te ontwerpen en te toetsen, waarbij rekening wordt gehouden met de invloed van de toenemende drukweerstand van beton in gedeeltelijk belaste gebieden. Omdat de CSFM een wand (2D)-model is en de gedeeltelijk belaste gebieden een ruimtelijke (3D) taak vormen, was het noodzakelijk om een oplossing te vinden die deze twee verschillende soorten taken combineert (<em>Fig. 35</em>). Als de functie \"gedeeltelijk belaste gebieden\" is geactiveerd, wordt de toelaatbare kegelgeometrie gemaakt volgens de Eurocode (<em>Fig. 34</em>). Alle geometrische conflicten worden volledig in 3D opgelost voor de opgegeven geometrie van het betonnen staaf en de afmetingen van elk PLA. Vervolgens wordt een rekenmodel van het gedeeltelijk belaste gebied 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. 36\\qquad Toelaatbare kegelgeometrieën.}}}\\]</em></p>\n<p>De aanpassing van het materiaalmodel bleek een ongeschikte aanpak te zijn, wat voornamelijk kwam doordat de toewijzing van eigenschappen aan de eindige-elementen-mesh problematisch is. Er werd vastgesteld dat een aanpak die onafhankelijk is van de eindige-elementen-mesh een geschiktere oplossing is. Voor de bekende drukkegelgeometrie worden volledig coherente fictieve drukdiagonalen gemaakt (<em>Fig. 35</em> <em>en Fig. 37</em>). Deze drukdiagonalen hebben identieke materiaaleigenschappen als het beton dat in het model wordt gebruikt, inclusief het spanning-rekdiagram. De vorm van de kegel bepaalt de richting van de drukdiagonalen, die de belasting geleidelijk over het PLA verdelen naar het rekenkundige verdeelgebied. De oppervlaktedichtheid van de fictieve drukdiagonalen is variabel bij elk deel van de kegel en voegt een fictief betonoppervlak toe in de belastingsrichting. Op het niveau van het belaste gebied (<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> een oppervlak van de oplegging is dat wordt aangenomen in het 2D-rekenmodel), en dit oppervlak neemt lineair af tot nul richting het rekenkundige verdeelgebied (<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 Fictieve drukdiagonalen in het rekenmodel}}}\\]</em></p>\n<p>De weerstand van het gedeeltelijk belaste gebied wordt verhoogd volgens de verhouding van het rekenkundige verdeelgebied en het belaste gebied, zoals vastgelegd in EN 1992-1-1 (6.7). Er moet in gedachten worden gehouden dat dit een rekenmodel is dat de spanningstoestand over een gedeeltelijk belast gebied, waarvan de werkelijke spanningsstroming veel gecompliceerder is, niet nauwkeurig kan beschrijven. Deze oplossing maakt echter een correcte verdeling van de belasting over het gehele model mogelijk, met inachtneming van de verhoogde belastingscapaciteit van het gedeeltelijk belaste gebied. Bovendien introduceert het correct transversale spanningen in dit gebied.</p>\n<p>Bij gebruik van de functie Gedeeltelijk belaste gebieden om de toename van de drukcapaciteit van beton te simuleren, is het noodzakelijk om de normtoetsing afzonderlijk uit te voeren volgens EN 1992-1-1, paragraaf 6.7 (2). De transversale 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 is geïmplementeerd voor sterkteberekeningen in CSFM is gebaseerd op de parabolisch-plastische spanning-rekkromme voor beton, gebaseerd op de parabolische spanning-rekkromme van de Portland Cement Association, zoals beschreven in PCA's Notes on ACI 318-99 Building Code Requirements for Structural Concrete, Figuur 6-8. De treksterkte wordt verwaarloosd, zoals gebruikelijk is bij klassiek gewapend betonontwerp.</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> houdt geen rekening met een expliciet bezwijkcriterium in termen van rek voor beton in druk (d.w.z. na het bereiken van de piekspanning wordt een plastische tak beschouwd met ε<em><sub>c</sub></em><sub>0</sub> met een maximale waarde van 5%, terwijl ACI 318-19 Cl. 22.2.2.1 uitgaat van een grensrek van minder dan 0,3%). Deze vereenvoudiging maakt het niet mogelijk om het vervormingsvermogen van constructies die bezwijken door druk te verifiëren. 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)), rekening wordt gehouden met de toenemende brosheid van beton naarmate de sterkte toeneemt, door middel van de reductiefactor <em>\\(\\eta_{fc}\\)</em> zoals gedefinieerd in 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 druksterkte van beton, gedefinieerd in ACI 318-19 Cl. 22.2.2.4.1. Bij gebruik van een parabool-rechthoek-spanning-rekdiagram is het noodzakelijk om de maximale drukspanning met deze factor te reduceren. Dit middelt de spanningsverdeling in de drukzone op een zodanige manier dat de resulterende druksterkte kleiner is dan of gelijk is aan de druksterkte berekend met een spanning-rekdiagram met een afnemende plastische tak<em>.</em></p>\n<p><em>Φ</em><em><sub>c </sub></em>is de sterktereductiefactor voor beton. De standaardwaarde is ingesteld volgens ACI 318-19 Tabel 24.2.1 (b)(f).</p>\n<p><em>k</em><em><sub>c</sub></em><sub>2</sub> is de reductiefactor als gevolg van de aanwezigheid van dwarsscheurvorming.</p>\n<p><em>f'</em><em><sub>c</sub></em> is de cilindersterkte van beton (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 gebaseerd op dezelfde aannames als de knoopzonecoëfficiënt <em>β</em><em><sub>n</sub></em>, gegeven in ACI 318-19 Tabel 23.9.2, met dien verstande dat in CSFM voor elk eindig element wordt gecontroleerd of er een hoofdtrekspanning loodrecht op de hoofddrukspanning aanwezig is (niet alleen voor knopen van het Staafwerk-model).</p>\n<h3>Beton – Bruikbaarheid</h3>\n<p>De analyse van de bruikbaarheidsgrenstoestand bevat bepaalde vereenvoudigingen van de constitutieve modellen die worden gebruikt voor de sterkteanalyse. De plastische tak van de spanning-rekkromme van beton in druk wordt buiten beschouwing gelaten, terwijl de elastische tak lineair en oneindig is. Compression softening wordt niet in rekening gebracht. Deze vereenvoudigingen verbeteren de numerieke stabiliteit en berekeningssnelheid en verminderen de algemeenheid van de oplossing niet, zolang de resulterende materiaalspanningslimieten bij bruikbaarheid duidelijk onder hun vloeigrens liggen (zoals vereist door ACI). Daarom zijn de vereenvoudigde modellen die worden gebruikt voor bruikbaarheid alleen geldig als aan alle verificatievereisten wordt 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>Lange-termijneffecten</strong></p>\n<p>Het langetermijngedrag van de constructie, zoals langetermijndoorbuigingen of de berekening van scheurbreedtes veroorzaakt door blijvende belastingen, wordt beïnvloed door kruip van beton. ACI 318-19 in paragraaf 24.2.4.1.3 definieert de tijdsafhankelijke factor voor blijvende belastingen – ξ, die het kruipeffect voor een bepaalde duur van de blijvende belasting weergeeft.</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 via de factor ξ. De aangepaste elasticiteitsmodulus wordt aangeduid als <em>E</em><em><sub>c,eff</sub></em> – zie Figuur 40.</p>\n<p>Uitgaande van de aanname dat de vervorming van het element wordt uitgedrukt door rek, kan worden geschreven dat:</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> een korte-termijnrek is (zonder de invloed van kruip) en <em>ε</em><em><sub>creep</sub></em> een rek is veroorzaakt door kruip.</p>\n<p>Met behulp van 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}\\) in te vullen, krijgen we:</p>\n<p>\\[E_{c,eff} = \\frac{E_{c}}{1+\\xi}\\]</p>\n<p>De duur van de blijvende belasting voor het bepalen van de factor ξ kan afzonderlijk worden ingesteld voor elke lange-termijn gebruikslastcombinatie.</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 scheurbreedtes worden vervolgens berekend met een aangepast materiaalmodel, waarbij het effect van drukverfijning automatisch in rekening wordt gebracht door de aard van de EEM-analyse. Het is daarom niet nodig deze verder te vermenigvuldigen met de factor gedefinieerd in 24.2.4.1.1.</p>\n<p><strong>Korte-termijneffecten</strong></p>\n<p>Om korte-termijnverificaties uit te voeren, wordt een andere berekening uitgevoerd waarbij alle belastingen worden berekend zonder de tijdsafhankelijke factor voor blijvende belastingen. Beide berekeningen voor lange- en korte-termijnverificaties zijn weergegeven in Fig. 40.</p>\n<h3>Wapening</h3>\n<p>Voor de niet-voorgespannen wapening wordt een perfect elastisch-plastisch spanning-rekdiagram met een gedefinieerd vloeipunt beschouwd, zie ACI 319-19 Cl. 20.2.1. 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 rekening te houden met het tension stiffening-effect (het is niet mogelijk om de scheurbreedte 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 Tabel 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>10% wordt 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. 43) wordt automatisch in rekening gebracht door het invoerspanning-rekverband van de blote wapeningsstaaf aan te passen, om zo de gemiddelde stijfheid van de in het beton ingebedde staven vast te leggen (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. 43\\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 alleen gebruikmaken van standaard materiaaleigenschappen, kan het formaat van partiële veiligheidsfactoren zoals voorgeschreven in de ontwerpnormen zonder aanpassing worden toegepast. Op deze manier worden de invoerbelastingen gefactoreerd en worden de karakteristieke materiaaleigenschappen gereduceerd met behulp van de betreffende sterktereductiefactoren, precies zoals bij conventionele betonanalyse.</p>\n<p>Waarden van <strong>sterktereductiefactoren</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 dat in de applicatie wordt opgelost, afschuiving-gestuurd is (op basis van Tabel 21.2.1 (b), (f), (g)). Het is echter mogelijk om elk type element te modelleren. Als daarom een op druk of trek gestuurd element wordt beoordeeld, heeft de gebruiker de mogelijkheid om de waarde van de sterktereductiefactor te 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 De instelling van sterktereductiefactoren in IDEA StatiCa Detail.}}}\\]</em></p>\n<p><br></p>\n<p><strong>Belastingsfactoren</strong> voor sterktecombinaties dienen te worden gedefinieerd volgens ACI 318-19 Tabel 5.3.1.</p>\n<p>Behalve zoals vermeld in Hoofdstuk 34, worden belastingscombinaties op gebruiksniveau niet gedefinieerd in ACI 318-19. Het wordt aanbevolen om combinatieregels te gebruiken op basis van Appendix C van ASCE/SEI 7-16. Voor alle templates zijn de belastingsfactoren al vooraf gedefinieerd.</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 De instelling van belastingsfactoren in Idea StatiCa Detail.}}}\\]</em></p>"
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"value": "<p>De verschillende verificaties die door ACI 318-19 vereist zijn, worden beoordeeld op basis van de directe resultaten die het model oplevert. Verificaties worden uitgevoerd voor betonsterkte, wapeningssterkte en verankering (aanhechtingsschuifspanningen).</p>\n<p>De <strong>betonsterkte</strong> in druk wordt geëvalueerd als de verhouding tussen de maximale hoofddrukspanning <em>f</em><em><sub>c</sub></em> (ook σ<sub>2</sub> in Aanvullende resultaten) verkregen 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 in trek als in druk geëvalueerd 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>aanhechtingsschuifspanning</strong> wordt onafhankelijk geëvalueerd als de verhouding tussen de aanhechtingsspanning τ<em><sub>b</sub></em> berekend door de EEM-analyse en de aanhechtingssterkte <em>f</em><em><sub>bu</sub></em>.</p>\n<p>De ACI-norm behandelt de aanhechtingssterkte echter niet expliciet, maar werkt met de berekening van de zogenaamde ontwikkelingslengte, die beschreven wordt in Sectie 25.4.2. Aangezien de aanhechtingssterkte een fundamentele invoerparameter is voor het bepalen van de ontwikkelingslengte, zie R25.4.1.1 en ACI Committee 408 1966, kan de aanhechtingssterkte als volgt berekend worden:</p>\n<p>Laten we aannemen dat als we de wapeningsstaaf verankeren in een betonblok tot de ontwikkelingslengte <em>l</em><em><sub>d</sub></em> of groter, het uittrekken van de wapening zal leiden tot breuk van de wapening en niet tot het uittrekken van het beton. Dit kan met de volgende formule geschreven worden.</p>\n<p>\\[\\pi\\cdot d_{b} \\cdot l_{d} \\cdot f_{bu}=f_{y}\\cdot A_{s}\\]</p>\n<p>waarbij:</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 ontwikkelingslengte is, <em>f</em><em><sub>bu</sub></em> de aanhechtingssterkte is, <em>f</em><em><sub>y</sub></em> de vloeigrens van de wapening is, en <em>A</em><em><sub>s</sub></em> de oppervlakte van de wapeningsstaaf is.</p>\n<p>Uit het voorgaande kan de formule voor het berekenen van de aanhechtingssterkte eenvoudig worden afgeleid:</p>\n<p>\\[f_{bu}=\\frac{f_{y}\\cdot A_{s}}{\\pi\\cdot d_{b} \\cdot l_{d} }\\]</p>\n<p>De ontwikkelingslengte <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>waarbij:</p>\n<p><em>C = 25</em> (2,1 voor metrisch) voor staven nr. 6 en kleiner en geribde draden, <em>C = 20</em> (1,7 voor metrisch) voor staven nr. 7 en groter, λ = 1,0 voor normaal gewicht beton, <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 ongecoate of verzinkte (gegalvaniseerde) 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 wapeningsklasse, en <em>ψ</em><em><sub>t</sub></em> wordt automatisch afgeleid van de positie van de wapening in het model en van de stortrichting die voor elk projectonderdeel als volgt in de applicatie 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 met betrekking tot de geschikte grenswaarden voor de respectievelijke delen van de constructie (dat wil zeggen, ondanks dat er een enkele klasse is voor zowel beton als wapeningsmateriaal, zullen de uiteindelijke spanning-rek-diagrammen in elk deel van de constructie verschillen door tension stiffening- en compression softening-effecten).</p>\n<p>Er is ook een optie 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><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 gedefinieerd worden.</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 is.</p>\n<p>Of als een 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 is die verkregen kan worden door integratie van de aanhechtingsspanning <em>τ</em><em><sub>b</sub></em> 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 de <strong>verankeringsvoorwaarden</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 is vanaf het begin van de wapeningsstaaf tot het punt van interesse.</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>waarbij <em>F</em><em><sub>lim,add</sub></em> de extra kracht is die berekend wordt 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 aan het verankeringseinde), een haak van 90 graden, een haak van 180 graden, perfecte aanhechting en een doorlopende staaf. Al deze typen, samen met de bijbehorende verankeringscoëfficiënten β, worden weergegeven in Fig. 48 voor langswapening. De waarden van de gehanteerde verankeringscoëfficiënten zijn afgeleid uit de vergelijking van de vergelijking uit sectie ACI 318-19 25.4.3.1 en vergelijkingen ontleend aan sectie ACI 318-19 25.4.2.3. Er dient opgemerkt te worden dat, ondanks de verschillende beschikbare opties, CSFM 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) 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 te voldoen aan ACI moet de verankeringsveer in de berekening worden gebruikt; de verankeringsveer wordt aangepast met de β-coëfficiënt, dus de gebruiker moet een van de beschikbare verankeringstypen gebruiken bij het definiëren van de begin- en eindvoorwaarden van de wapening. </p>"
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"value": "<p>Bij het ontwerpen van betonconstructies onderscheiden we twee grote groepen gedeeltelijk belaste vlakken (PLA) – de eerste hiervan omvat <strong>opleggingen</strong>, terwijl de andere bestaat uit <strong>verankeringszones</strong>. </p>\n<p>Volgens de momenteel geldende normen voor het ontwerp van gewapend betonconstructies ACI 318-19 hfst. 22.8, moet bij <strong>opleggingen</strong> rekening worden gehouden met lokale verbrijzeling van beton en dwarse trekkrachten. Voor een gelijkmatig verdeelde belasting op een oppervlak, <em>A</em><em><sub>c1</sub></em>, mag de drukcapaciteit van beton met maximaal een factor twee worden verhoogd, afhankelijk van het rekenkundige verdeelvlak <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 Gedeeltelijk belaste vlakken voor opleggingen volgens ACI 318-19}}}\\]</em></p>\n<p>Voor <strong>verankeringszones</strong> met voorspanning met nagerekt staal moet ACI 318-19 hfst. 25.9 worden gevolgd.</p>\n<p>Het gedeeltelijk belaste vlak moet voldoende worden gewapend met dwarswapening die is ontworpen om de splijtkrachten die in het vlak optreden, over te dragen. Zonder de vereiste dwarswapening is het niet mogelijk om een verhoging van de drukcapaciteit van het beton in aanmerking te nemen.</p>\n<p><br></p>\n<p><strong>Gedeeltelijk belaste vlakken 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 Fictieve drukdiagonalen met eindige-elementenmesh van beton.}}}\\]</em></p>\n<p>Met CSFM is het mogelijk om gewapende betonconstructies te ontwerpen en te toetsen, waarbij rekening wordt gehouden met de invloed van de toenemende drukweerstand van beton in gedeeltelijk belaste vlakken. Omdat CSFM een wandmodel (2D) is en gedeeltelijk belaste vlakken een ruimtelijke (3D) taak vormen, moest een oplossing worden gevonden die deze twee verschillende soorten taken combineert (<em>Fig. 50</em>). Als de functie \"gedeeltelijk belaste vlakken\" is geactiveerd, wordt de toelaatbare kegelgeometrie gecreëerd volgens de ACI (<em>Fig. 49</em>). Alle geometrische conflicten worden volledig in 3D opgelost voor de opgegeven geometrie van het betonelement en de afmetingen van elk PLA. Vervolgens wordt een rekenmodel van het gedeeltelijk belaste vlak 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. 51\\qquad Toelaatbare kegelgeometrieën.}}}\\]</em></p>\n<p>De aanpassing van het materiaalmodel bleek een ongeschikte aanpak te zijn, voornamelijk omdat het toewijzen van eigenschappen aan de eindige-elementenmesh problematisch is. Er werd vastgesteld dat een aanpak die onafhankelijk is van de eindige-elementenmesh een geschiktere oplossing is. Voor de bekende drukkegelgeometrie worden volledig coherente fictieve drukdiagonalen gecreëerd (<em>Fig. 51</em> <em>en Fig. 52</em>). Deze drukdiagonalen hebben identieke materiaaleigenschappen als het beton dat in het model wordt gebruikt, inclusief het spanning-rekdiagram. De vorm van de kegel bepaalt de richting van de drukdiagonalen, die de belasting geleidelijk over het PLA naar het rekenkundige verdeelvlak verdelen. De oppervlaktedichtheid van de fictieve drukdiagonalen is variabel bij elk deel van de kegel, en er wordt een fictief betonoppervlak toegevoegd in de belastingsrichting. Op het niveau van het belaste vlak (<em>A</em><em><sub>c1</sub></em>) wordt een fictief betonoppervlak toegevoegd volgens de verhouding \\(\\sqrt{A_{c1} \\cdot A_{c2}} - A_{real}\\) (waarbij <em>A</em><em><sub>real</sub></em> een oppervlak van de oplegging is dat in het 2D-rekenmodel wordt aangenomen), en dit oppervlak neemt lineair af tot nul richting het rekenkundige verdeelvlak (<em>A</em><em><sub>c2</sub></em>). Deze oplossing zorgt ervoor dat de drukspanning in het beton constant is over het gehele 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_{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 Fictieve drukdiagonalen in het rekenmodel}}}\\]</em></p>\n<p>De weerstand van het gedeeltelijk belaste vlak wordt verhoogd volgens de verhouding van het rekenkundige verdeelvlak en het belaste vlak, zoals vastgelegd in ACI 318-19 hfst. 22.8. Er moet worden opgemerkt dat dit een rekenmodel is dat de spanningstoestand over een gedeeltelijk belast vlak, waarvan de werkelijke verdeling veel complexer is, niet nauwkeurig kan beschrijven. Deze oplossing maakt echter een correcte verdeling van de belasting over het gehele model mogelijk, met inachtneming van de verhoogde belastingscapaciteit van het gedeeltelijk belaste vlak. Bovendien worden hierbij correct dwarsspanningen in dit vlak geïntroduceerd om de wapening voor splijtkrachten correct te ontwerpen.</p>\n<p>De toelaatbare <strong>oplegspanning</strong> van <em>0.85f</em><em><sub>c</sub></em><em>'</em> staat vermeld in Tabel 22.8.3.2. De dichtheid is beperkt zodat de maximale dubbele capaciteit, gegeven in de formule in Tabel 22.8.3.2(b), niet wordt overschreden. </p>\n<p>Voor de <strong>verankeringszones</strong> wordt PLA op dezelfde manier gebruikt als voor opleggingen in de applicatie. Daarom moeten de lokale zones die zijn gedefinieerd in ACI 318-19 hoofdstuk 25.9 handmatig worden getoetst volgens ACI 318-19 25.9.3. Het PLA wordt daarom alleen gebruikt om te voorkomen dat het rekcriterium in de lokale zone wordt overschreden en zo de berekening voortijdig wordt gestopt. Aan de andere kant kan, volgens ACI 318-19, art. 25.9.4.3.1 (b), de wapening die de bersting en afschilfering van de spanningen in het vlak weerstaat, direct en voordelig worden geverifieerd in de applicatie.</p>"
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"value": "<p>Controles op de bruikbaarheidsgrenstoestand worden uitgevoerd voor spanningsbeperking, scheurwijdte en doorbuigingsgrenzen. Spanningen worden gecontroleerd in beton- en wapeningselementen volgens ACI 318-19, op een vergelijkbare manier als gespecificeerd voor de Sterkte.</p>\n<h3>Spanningsbeperking</h3>\n<p>Toelaatbare drukspanningen in beton bij gebruikslast moeten worden gecontroleerd voor voorgespannen staven van Klasse U en T. Op basis van Tabel R24.5.2.1 is er geen controle op spanningsbeperking vereist voor beton dat als gescheurd wordt beschouwd. 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 zijn onderworpen, wordt in ACI 318-19 24.5.4.1 gespecificeerd als <em>0.6f</em><em><sub>c</sub></em><em>'. </em>De drukspanningsgrens van <em>0.45f</em><em><sub>c</sub></em><em>'</em> is vastgesteld om de kans op bezwijken van voorgespannen betonnen staven als gevolg van herhaalde belastingen te verkleinen. Deze grens leek ook redelijk om overmatige kruipvervorming te voorkomen. Bij hogere spanningswaarden nemen kruiprekken doorgaans sneller toe naarmate de aangebrachte spanning toeneemt.</p>\n<p>De drukspanning in het beton wordt geëvalueerd 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>verkregen uit de EEM-analyse voor bruikbaarheid en de grenswaarde, 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>Op basis van het geselecteerde combinatietype (langetermijn of kortetermijn) wordt ofwel de langetermijn- ofwel de kortetermijndoorbuiging geëvalueerd. De maximaal toelaatbare doorbuigingswaarde moet door de gebruiker worden bepaald en moet in overeenstemming zijn met 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 is het mogelijk om de doorbuigingen door eigen gewicht <em>Δ</em><em><sub>DL</sub></em> en veranderlijke belasting <em>Δ</em><em><sub>LL</sub></em> afzonderlijk weer te geven, evenals de totale doorbuiging <em>Δ</em><em><sub>Tot</sub></em><sub> </sub>(blijvend+veranderlijk), terwijl tegelijkertijd de vervormde vorm wordt weergegeven.</p>\n<p>Doorbuigingen bij afgesneden uiteinden kunnen niet worden gecontroleerd.</p>\n<h3>Scheurwijdte</h3>\n<p><br></p>\n<p>Scheurwijdtes en scheurrichtingen worden berekend voor kortetermijn- of langetermijncombinaties voor de bruikbaarheidsgrenstoestand. Aangezien ACI geen grenswaarden voor scheurwijdtes rechtstreeks voorschrijft, moet de gebruiker een grenswaarde voor de scheurwijdte <em>w</em><em><sub>lim</sub></em> opgeven.</p>\n<p>De controles worden als volgt weergegeven:</p>\n<p>\\[\\frac{w}{w_{lim}}\\]</p>\n<p>waarbij:</p>\n<p><em>w</em> kortetermijn- of langetermijnscheurwijdte berekend door 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 methode voor het berekenen van scheurwijdtes die in de applicatie wordt gebruikt, en die in dit document ook in meer detail wordt beschreven, is in overeenstemming met ACI 224R-01. Het is daarom mogelijk om ACI 224R-01 Tabel 4.1 te gebruiken om de grenswaarde van scheurwijdtes te bepalen.</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 scheurwijdtes te berekenen (gestabiliseerde en niet-gestabiliseerde scheurvorming). In het algemene geval (gestabiliseerde scheurvorming) wordt de scheurwijdte berekend door de rekken op 1D-elementen van wapeningsstaven te integreren. De scheurrichting wordt vervolgens berekend vanuit de drie dichtstbijzijnde (vanaf het middelpunt van het gegeven 1D eindige-elementensegment van de wapening) integratiepunten van 2D-betonelementen. Hoewel deze benadering voor het berekenen van de scheurrichtingen niet overeenkomt met de werkelijke positie van de scheuren, levert deze nog steeds representatieve waarden op die leiden tot scheurwijdteresultaten die vergeleken kunnen worden met de door de norm vereiste scheurwijdtewaarden op de positie van de wapeningsstaaf.</p>"
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"value": "<h3>Beton - Sterkte</h3>\n<p>Het betonmodel dat is geïmplementeerd voor sterkteberekeningen in CSFM is gebaseerd op de AASHTO LRFD strength-design uitgangspunten van evenwicht en rekcompatibiliteit. In overeenstemming met 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> houdt geen rekening met een expliciet bezwijkcriterium in termen van rek voor beton onder druk (d.w.z. na het bereiken van de piekspanning wordt een plastische tak beschouwd 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 van minder dan 0,3%). Deze vereenvoudiging maakt het niet mogelijk om de vervormingscapaciteit van constructies die bezwijken op druk te verifiëren. 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)), rekening wordt gehouden met de toename van de brosheid van beton naarmate de sterkte toeneemt, door middel van de reductiefactor <em>\\(\\eta_{fc}\\)</em> zoals gedefinieerd in <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 van de betondrukvastheid zoals gedefinieerd in AASHTO LRFD (2024) Artikel 5.6.2.2. Bij gebruik van een parabool-rechthoek spanning-rek-diagram is het noodzakelijk om de maximale drukspanning met deze factor te reduceren. Dit middelt de spanningsverdeling in de drukzone zodanig dat de resulterende drukvastheid kleiner is dan of gelijk is aan de drukvastheid berekend met een spanning-rek-diagram met een aflopende plastische tak<em>.</em></p>\n<p><em>Φ</em><em><sub>c </sub></em>is de weerstandsfactor voor beton. De standaardwaarde is ingesteld volgens AASHTO LRFD (2024) Artikel 5.5.4.2.</p>\n<p><em>k</em><em><sub>c</sub></em><sub>2</sub> is de reductiefactor als gevolg van de aanwezigheid van dwarsscheurvorming.</p>\n<p><em>f'</em><em><sub>c</sub></em> is de cilindersterkte van het beton (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 gebaseerd op dezelfde uitgangspunten als de betonefficiëntiefactor <em>ν</em> gegeven in AASHTO LRFD (2024) 5.8.2.5.3a en Tabel 5.8.2.5.3a-1, met dat verschil dat in CSFM voor elk eindig element wordt gecontroleerd op de aanwezigheid van een hoofdtrekspanning loodrecht op de hoofddrukspanning (niet alleen voor knopen van het Staafwerk-model).</p>\n<h3>Beton – Bruikbaarheid</h3>\n<p>De bruikbaarheidsanalyse bevat bepaalde vereenvoudigingen van de constitutieve modellen die worden gebruikt voor de sterkteanalyse. 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 materiaalspanningslimieten bij bruikbaarheid duidelijk onder hun vloeipunten liggen (in overeenstemming met de AASHTO LRFD service limit state-benadering). De vereenvoudigde modellen die worden gebruikt voor bruikbaarheid zijn daarom alleen geldig als aan alle verificatievereisten is 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>Het langetermijn constitutieve model (de rode curve in Fig. 59) wordt gebruikt voor de berekening van de scheurwijdte, de totale doorbuiging en de spanningslimitering 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>waarbij:<br><em>E</em><em><sub>c</sub></em> de elasticiteitsmodulus is zoals gedefinieerd in AASHTO LRFD (2024) artikel 5.4.2.4<br><em>ψ</em> de kruipcoëfficiënt is zoals 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>Kortetermijneffecten</strong></p>\n<p>Om kortetermijnverificaties uit te voeren, wordt een aparte berekening uitgevoerd waarbij alle belastingen worden berekend zonder de kruipfactor. Beide berekeningen voor lange- en kortetermijnverificaties zijn weergegeven in Fig. 59.</p>\n<h3>Wapening</h3>\n<p>Voor de niet-voorgespannen wapening wordt een volledig elastisch-plastisch spanning-rek-diagram met een gedefinieerd vloeipunt beschouwd, zie AASHTO LRFD (2024) Artikel 5.4.3. Voor de definitie van dit diagram zijn alleen de basiseigenschappen van de wapening vereist – 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 om het tension stiffening effect te veronderstellen (het is niet mogelijk om 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>waarbij:</p>\n<p><em>Φ</em><em><sub>s </sub></em>is de weerstandsfactor voor wapening. De standaardwaarde is ingesteld volgens AASHTO LRFD (2024) Artikel 5.5.4.2.</p>\n<p><em>f</em><em><sub>y</sub></em> is de vloeigrens van de wapening</p>\n<p><em>E</em><em><sub>s</sub></em> is de elasticiteitsmodulus van de wapening</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. 61) wordt automatisch in rekening gebracht door de ingevoerde spanning-rek-relatie van de blote wapeningsstaaf aan te passen, om zo de gemiddelde stijfheid van de in het beton ingebedde staven vast te leggen (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 moderne ontwerpnormen. Aangezien de rekenmodellen alleen gebruikmaken van standaard materiaaleigenschappen, kan het formaat van partiële veiligheidsfactoren dat in de ontwerpnormen wordt voorgeschreven zonder enige aanpassing worden toegepast. Op deze manier worden de invoerbelastingen gefactoreerd en worden de karakteristieke materiaaleigenschappen gereduceerd met behulp van de betreffende sterktefactoren, precies zoals bij een conventionele betonanalyse.</p>\n<p>Waarden van <strong>sterktefactoren</strong> zijn voorgeschreven in AASHTO LRFD (2024) Artikel 5.5.4. De standaardwaarden voor beton en wapening zijn conservatief gekozen, uitgaande van de aanname dat het typische opgeloste voorbeeld een D-gebied is - een typisch geval voor de Staafwerkmethode. Het is echter mogelijk om elk type element te modelleren. Daarom heeft de gebruiker, indien een op druk of trek gecontroleerd element wordt beoordeeld, de mogelijkheid om de waarde van de sterktereductiefactor in de Voorkeuren te wijzigen.</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>Belastingsfactoren en belastingcombinaties moeten worden gedefinieerd volgens de AASHTO LRFD Bridge Design Specifications (2024), Artikel 3.4.1 en Tabellen 3.4.1-1 tot 3.4.1-6. AASHTO LRFD specificeert expliciet de belastingcombinaties voor de uiterste grenstoestand (Strength I tot Strength V), evenals de belastingcombinaties op gebruiksniveau (Service I tot Service IV), inclusief de bijbehorende belastingsfactoren 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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"value": "<p>De verschillende verificaties die door AASHTO worden vereist, worden beoordeeld op basis van de directe resultaten van het model. Verificaties worden uitgevoerd voor de betonsterkte, de wapeningssterkte en de verankering (aanhechtingsschuifspanningen).</p>\n<p>De <strong>betonsterkte</strong> in druk wordt geëvalueerd als de verhouding tussen de maximale hoofddrukspanning <em>f</em><em><sub>c</sub></em> (ook σ<sub>2</sub> in de aanvullende resultaten) verkregen 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 in trek als in druk geëvalueerd 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>aanhechtingsschuifspanning</strong> wordt onafhankelijk geëvalueerd als de verhouding tussen de aanhechtingsspanning τ<em><sub>b</sub></em> berekend door de EEM-analyse en de aanhechtingssterkte <em>f</em><em><sub>bu</sub></em>.</p>\n<p>Aangezien de aanhechtingssterkte echter niet expliciet is gedefinieerd in AASHTO, moet de waarde ervan worden bepaald met behulp van de vergelijkingen die de verankeringslengte definiëren. De aanhechtingssterkte is in feite de primaire invoer voor het bepalen van de verankeringslengte; zie bijvoorbeeld dit artikel AASHTO LRFD (2024) Article C5.10.8.2 of NCHRP Report 733, Attachment E pagina E-9.</p>\n<p>De berekening beschreven in AASHTO LRFD (2024) Article 5.10.8.2.1 en 5.10.8.2.2, die kennis vereist van de maximale hart-op-hart afstand van de dwarswapening binnen <em>l</em><em><sub>d</sub></em>, het aantal staven of draden dat wordt ontwikkeld langs het splijtvlak, de totale doorsnede van alle dwarswapening en andere geometrische grootheden die niet betrouwbaar kunnen worden bepaald in het model van de Detail applicatie voor algemene invoer, werd een aanpak overgenomen uit AASHTO LRFD (2014) Article 5.11.2.1.1 op de volgende manier:</p>\n<p>Laten we aannemen dat als we de wapeningsstaaf verankeren in een betonblok tot de verankeringslengte <em>l</em><em><sub>d</sub></em> of groter, het uittrekken van de wapening zal leiden tot breuk van de wapening en niet tot het uittrekken van het beton. Dit kan worden geschreven met de volgende formule.</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> het oppervlak van de wapeningsstaaf is</li>\n</ul>\n<p>Uit het voorgaande kan de formule voor het berekenen van de aanhechtingssterkte eenvoudig 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 basis-verankeringslengte voor trek <em>l</em><em><sub>db</sub></em> wordt bepaald in AASHTO LRFD (2014) Article 5.11.2.1.1 als volgt:</p>\n<p>Voor staaf 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> het oppervlak 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 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>Vervolgens wordt, door de basis-verankeringslengte <em>l</em><em><sub>db</sub></em> te vermenigvuldigen met factoren beschreven in AASHTO LRFD (2014) Article 5.11.2.1.2 en 5.11.2.1.3, de verankeringslengte <em>l</em><em><sub>d</sub></em> als invoer bepaald.</p>\n<p>Reductiefactoren die de verankeringslengte verlagen volgens 5.11.2.1.3 zijn in de applicatie altijd gelijk aan 1,0. De aanpassingsfactor voor bovenliggende horizontale of bijna horizontale wapening is gelijk aan 1,4 voor 'slechte' aanhechtingscondities, volgens de volgende afbeelding:</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 van het beton 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 andere factoren bepaald in 5.11.2.1.2 zijn gelijk aan 1,0, omdat alleen normaalgewicht beton wordt ondersteund en alleen ongecoate wapening wordt ondersteund.</p>\n<p>De aanhechtingsschuifspanning en aanhechtingssterkte van staven in druk worden analoog berekend aan staven in trek, maar hierbij worden vergelijkingen uit AASHTO LRFD (2014) Article 5.11.2.2 gebruikt.</p>\n<p>Er is ook een optie 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 gedefinieerd.</p>\n<p>\\[F_{tot}=A_{b} \\cdot f_{s}\\]</p>\n<p>waarbij <em>A</em><em><sub>b</sub></em> het oppervlak van de wapeningsstaaf is en <em>f</em><em><sub>s</sub></em> de spanning in de staaf is.</p>\n<p>Of als een 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 is 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 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_{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 is 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, 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>verankeringstypes</strong> in CSFM omvatten een rechte staaf (d.w.z. geen reductie van het staafuiteinde), een haak van 90 graden, een haak van 180 graden, perfecte aanhechting en een doorlopende staaf. Al deze types, samen met de bijbehorende verankeringscoëfficiënten β, worden weergegeven in Fig. 67 voor langswapening. De waarden van de gehanteerde verankeringscoëfficiënten zijn afgeleid uit de vergelijking van de vergelijking uit sectie AASHTO LRFD (2014) 5.11.2.1 en vergelijkingen uit sectie AASHTO LRFD (2014) 5.11.2.4.1. Er dient te worden opgemerkt dat, ondanks de verschillende beschikbare opties, CSFM drie types verankeringsuiteinden onderscheidt: (i) geen reductie van de verankeringslengte, (ii) een reductie van 30% van de verankeringslengte in het geval van 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 het balkelement) is altijd - β = 1,0.</p>\n<p>Om te voldoen aan AASHTO moet de verankeringsveer worden gebruikt in de berekening. De verankeringsveer wordt aangepast met de β-coëfficiënt, dus de gebruiker moet een van de beschikbare verankeringstypes gebruiken bij het definiëren van de begin- en eindcondities van de wapening. </p>"
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"value": "<p>Bij het ontwerpen van betonconstructies onderscheiden we twee grote groepen gedeeltelijk belaste vlakken (PLA – partially loaded areas) – de eerste bestaat uit <strong>opleggingen</strong>, terwijl de andere bestaat uit <strong>verankeringszones</strong>. </p>\n<p>Volgens de huidig geldende normen voor het ontwerp van gewapendbetonconstructies moeten voor <strong>opleggingen</strong> lokale verbrijzeling van beton en dwarse trekkrachten in beschouwing worden genomen. Bij een gelijkmatig verdeelde belasting over een vlak <em>A</em><em><sub>1</sub></em> mag de drukcapaciteit van het beton met een factor tot maximaal twee worden verhoogd, afhankelijk van het rekenkundig verdelingsvlak <em>A</em><em><sub>2</sub></em>. Zie AASHTO LRFD (2024) Artikel 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 Gedeeltelijk belaste vlakken voor opleggingen volgens AASHTO LRFD (2024) Artikel 5.6.5}}}\\]</em></p>\n<p>Voor voorgespannen (met nagerekt staal) <strong>verankeringszones</strong> moet AASHTO LRFD (2024) Artikel 5.8.4.4 worden aangehouden.</p>\n<p>Het gedeeltelijk belaste vlak moet voldoende worden gewapend met dwarswapening die is ontworpen om de splijtkrachten die in dit vlak optreden, over te dragen. Zonder de vereiste dwarswapening is het niet mogelijk om de drukcapaciteit van het beton te verhogen.</p>\n<p><br></p>\n<p><strong>Gedeeltelijk belaste vlakken 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 Fictieve diagonalen met eindige-elementenmesh van beton.}}}\\]</em></p>\n<p>Met CSFM is het mogelijk om gewapendbetonconstructies te ontwerpen en te beoordelen, waarbij de invloed van de toenemende drukweerstand van beton in gedeeltelijk belaste vlakken wordt meegenomen. Omdat CSFM een wand-(2D-)model is en gedeeltelijk belaste vlakken een ruimtelijk (3D-)vraagstuk vormen, was het noodzakelijk om een oplossing te vinden die deze twee verschillende soorten vraagstukken combineert (<em>Fig. 69</em>). Als de functie “gedeeltelijk belaste vlakken” is geactiveerd, wordt de toelaatbare kegelgeometrie aangemaakt volgens ACI (<em>Fig. 68</em>). Alle geometrische conflicten worden volledig in 3D opgelost voor de opgegeven geometrie van het betonelement en de afmetingen van elk PLA. Vervolgens wordt een rekenmodel van het gedeeltelijk belaste vlak 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 Toelaatbare kegelgeometrieën.}}}\\]</em></p>\n<p>De aanpassing van het materiaalmodel bleek een ongeschikte aanpak, vooral omdat de toewijzing van eigenschappen aan de eindige-elementenmesh problematisch is. Er werd vastgesteld dat een aanpak die onafhankelijk is van de eindige-elementenmesh een geschiktere oplossing biedt. Voor de bekende drukkegelgeometrie worden volledig coherente fictieve diagonalen aangemaakt (<em>Fig. 70</em> <em>en Fig. 71</em>). Deze diagonalen hebben dezelfde materiaaleigenschappen als het in het model gebruikte beton, inclusief het spanning-rekdiagram. De vorm van de kegel bepaalt de richting van de diagonalen, die de belasting geleidelijk over het PLA verdelen naar het rekenkundig verdelingsvlak. De oppervlaktedichtheid van de fictieve diagonalen varieert bij elk deel van de kegel en voegt een fictief betonoppervlak toe in de belastingsrichting. Op het niveau van het belaste vlak (<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> een oplegvlak is dat in het 2D-rekenmodel wordt aangenomen), en dit oppervlak neemt lineair af tot nul richting het rekenkundig verdelingsvlak (<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. 71\\qquad Fictieve diagonalen in het rekenmodel}}}\\]</em></p>\n<p>De weerstand van het gedeeltelijk belaste vlak wordt verhoogd volgens de verhouding tussen het rekenkundig verdelingsvlak en het belaste vlak, zoals vastgelegd in AASHTO LRFD (2024) Artikel 5.6.5. Er dient te worden opgemerkt dat dit een rekenmodel is dat de spanningstoestand over een gedeeltelijk belast vlak, waarvan het werkelijke verloop veel complexer is, niet nauwkeurig kan beschrijven. Deze oplossing maakt echter een correcte verdeling van de belasting over het gehele model mogelijk, met inachtneming van de verhoogde belastingscapaciteit van het gedeeltelijk belaste vlak. Bovendien worden hierdoor de dwarsspanningen in dit vlak correct geïntroduceerd, zodat de wapening tegen splijtkrachten juist kan worden ontworpen.</p>\n<p>De toelaatbare <strong>oplegspanning</strong> van <em>0.85f</em><em><sub>c</sub></em><em>'</em> staat vermeld in AASHTO LRFD (2024) Artikel 5.8.4.4. De dichtheid wordt begrensd zodat de maximale dubbele capaciteit uit de formule in 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, gedefinieerd in Artikel 5.8.4.4 en 5.8.4.5, handmatig worden gecontroleerd. Het PLA wordt daarom alleen gebruikt om te voorkomen dat het rekcriterium in de lokale zone wordt overschreden en de berekening daardoor voortijdig wordt gestopt. Daarentegen kan de wapening die weerstand biedt tegen bursting, oppervlakkig afspatten (spalling in-plane) en randtrekspanningen in algemene zones (gedefinieerd in Artikel 5.8.4.5) direct en op eenvoudige wijze in de applicatie worden geverifieerd.</p>"
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"value": "<p>Beoordelingen van de bruikbaarheidsgrenstoestand worden uitgevoerd voor spanningsbeperking, scheurwijdte en doorbuigingslimieten. Spanningen worden in beton- en wapeningselementen gecontroleerd volgens AASHTO LRFD op een vergelijkbare manier als gespecificeerd voor Sterkte.</p>\n<h3>Spanningsbeperking</h3>\n<p>De betondrukspanning wordt alleen geëvalueerd voor voorgespannen staven (wanneer de belastingsgeval Voorspanning 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>verkregen uit FE-analyse voor bruikbaarheid en de limietwaarden, die zijn vastgesteld op basis van AASHTO LRFD Table 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>Prestress plus permanent load</em> behandeld als een blijvende belasting (Sustain load), en <em>Prestress, permanent, and transient load</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. 73\\qquad Serviceability combination types}}}\\]</em></p>\n<p>Daarnaast is het altijd mogelijk om een analyse uit te voeren voor zowel kortetermijn- als langetermijneffecten, met materiaalmodellen die al dan niet rekening houden met de kruipfactor — zie het gedeelte \"Material models (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>Onmiddellijke doorbuigingen en totale doorbuigingen worden geëvalueerd voor elke combinatie waarin de doorbuigingsevaluatie is ingeschakeld. </p>\n<ul>\n <li>Voor onmiddellijke 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>Material models (AASHTO) - Concrete – Serviceability</em>' in dit document.</p>\n<p>De doorbuigingscontrole zelf wordt ingeschakeld in het bovenste lint. De gebruiker stelt de doorbuigingslimietwaarden 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 afgeknotte uiteinden kunnen niet worden gecontroleerd.</p>\n<h3>Scheurwijdte</h3>\n<p>Scheurwijdtes en -oriëntaties worden alleen berekend voor langetermijneffecten (met gebruik van <em>E</em><em><sub>c,eff</sub></em> volgens AASHTO LRFD (2024) artikel C5.12.5.3.6) voor combinaties waarin de scheurwijdte-evaluatie is ingeschakeld. Verificaties op basis van door de gebruiker opgegeven limietwaarden zijn als volgt:</p>\n<p>\\[\\frac{w}{w_{lim}}\\]</p>\n<p>waarbij:</p>\n<p><em>w</em> scheurwijdte berekend door FE-analyse,</p>\n<p><em>w</em><em><sub>lim</sub></em> limietwaarde van de scheurwijdte gedefinieerd door de gebruiker.</p>\n<p>De limietwaarde <em>w</em><em><sub>lim</sub></em> moet worden bepaald op basis van het elementtype en de milieuklasse in overeenstemming met AASHTO LRFD (2024) artikel 5.6.7 en de bijbehorende toelichting. </p>\n<p>Er zijn twee manieren om scheurwijdtes te berekenen (gestabiliseerde en niet-gestabiliseerde scheurvorming). In het algemene geval (gestabiliseerde scheurvorming) wordt de scheurwijdte berekend door integratie van de rekken op 1D-elementen van wapeningsstaven. De scheurrichting wordt vervolgens berekend uit de drie dichtstbijzijnde (vanaf het middelpunt van het betreffende 1D eindige element van de wapening) integratiepunten van 2D-betonelementen. Hoewel deze benadering voor het berekenen van de scheurrichtingen niet overeenkomt met de werkelijke positie van de scheuren, levert deze nog steeds representatieve waarden op die leiden tot scheurwijdteresultaten die vergeleken kunnen worden met de door de norm vereiste scheurwijdtewaarden ter plaatse van de wapeningsstaaf.</p>"
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"value": "<h3>Beton - Sterkte</h3>\n<p>Het betonmodel dat is geïmplementeerd voor sterkteberekeningen in CSFM is gebaseerd op de parabolisch-plastische spanning-rekkromme. 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> houdt geen rekening met een expliciet bezwijkcriterium in termen van rek voor beton onder druk (d.w.z. na het bereiken van de piekspanning wordt een plastische tak beschouwd 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 grensrek van minder dan 0,3%). Deze vereenvoudiging maakt het niet mogelijk om de vervormingscapaciteit van constructies die bezwijken onder druk te verifiëren. 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)), rekening wordt gehouden met de toename van de brosheid van beton naarmate de sterkte toeneemt, door middel van de <em>\\(\\eta_{fc}\\)</em> reductiefactor gedefinieerd in <em>fib</em> Model Code 2010 als volgt:</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>waarbij:</p>\n<p><em>α</em><sub>2</sub> is de reductiefactor van de betondruksterkte, gedefinieerd in AS 3600 Cl. 8.3.1<br>Bij gebruik van een parabool-rechthoek spanning-rekdiagram is het noodzakelijk om de maximale drukspanning met deze factor te verminderen. Dit middelt de spanningsverdeling in de drukzone zodanig dat de resulterende druksterkte kleiner is dan of gelijk is aan de druksterkte berekend met een spanning-rekdiagram met een aflopende 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>is de spanningsreductiefactor voor beton. De standaardwaarde is ingesteld volgens AS 3600 Tabel 2.2.3.</p>\n<p><em>β</em> is de reductiefactor als gevolg van de aanwezigheid van dwarsscheurvorming (in deze tekst ook wel <em>k</em><em><sub>c</sub></em><sub>2</sub> genoemd)</p>\n<p><em>f'</em><em><sub>c</sub></em> is de betoncilindersterkte (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 is gebaseerd op dezelfde principes als een effectieve druksterktefactor, gedefinieerd in Hoofdstuk 2.2.3. De literatuur waartegen deze factor is bepaald, is te vinden (inclusief in de context van de AS3600-norm) in AS3600:2018 Sup 1:2022 Cl. C2.2.3.</p>\n<h3>Beton – Bruikbaarheid</h3>\n<p>De bruikbaarheidsanalyse bevat bepaalde vereenvoudigingen van de constitutieve modellen die worden gebruikt voor de sterkteanalyse. De plastische tak van de spanning-rekkromme van beton onder druk wordt buiten beschouwing gelaten, terwijl de elastische tak lineair en oneindig is. Er wordt geen rekening gehouden met de compression softening law. Deze vereenvoudigingen verbeteren de numerieke stabiliteit en rekensnelheid en verminderen de algemeenheid van de oplossing niet, zolang de resulterende materiaalspanningslimieten bij bruikbaarheid duidelijk onder hun vloeigrenzen liggen (zoals vereist door AS3600). Daarom zijn de vereenvoudigde modellen die worden gebruikt voor bruikbaarheid alleen geldig als aan alle verificatievereisten 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>Bij de bruikbaarheidsanalyse wordt rekening gehouden met de langetermijneffecten van beton door gebruik te maken van de rekenwaarde van de kruipcoëfficiënt volgens AS 3600 Cl 3.1.8 (<em>φ</em><em><sub>cc</sub></em>, standaard genomen als een waarde van 2,5), die de secantelasticiteitsmodulus 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>Belastingsincrementen worden achtereenvolgens berekend in de volgorde: Voorspanning - Permanent - Veranderlijk, waarbij voor elk increment de bijbehorende effectieve elasticiteitsmodulus wordt gebruikt, zoals weergegeven in Fig. 78. Kruipfactoren worden door de gebruiker gedefinieerd in de materiaaleigenschappen en dienen te 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>Kortetermijneffecten</strong></p>\n<p>Om kortetermijnverificaties uit te voeren, wordt een aanvullende berekening uitgevoerd waarbij alle belastingen worden berekend zonder de tijdsafhankelijke factor voor blijvende belastingen. Beide berekeningen voor lange- en kortetermijnverificaties zijn weergegeven in Fig. 78.</p>\n<h3>Wapening</h3>\n<p>Voor niet-voorgespannen wapening wordt een volledig elastisch-plastisch spanning-rekdiagram met een gedefinieerd vloeipunt beschouwd, zie AS 3600 Sectie 3.2. 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"value": "<p>De Compatible Stress Field Method voldoet aan moderne ontwerpnormen. Aangezien de rekenmodellen alleen standaard materiaaleigenschappen gebruiken, kan het formaat van partiële veiligheidsfactoren zoals voorgeschreven in de ontwerpnormen zonder enige aanpassing worden toegepast. Op deze manier worden de invoerbelastingen gefactoreerd en worden de karakteristieke materiaaleigenschappen gereduceerd met behulp van de respectievelijke spanningsreductiefactoren, precies zoals bij conventionele betonanalyse.</p>\n<p>Waarden van <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 dienen te worden vastgesteld volgens AS 3600 Cl. 4.2.2. Belastingsfactoren voor bruikbaarheidscombinaties dienen te 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 verificaties die vereist zijn door AS 3600 worden beoordeeld op basis van de directe resultaten van het model. Verificaties worden uitgevoerd voor betonsterkte, wapeningssterkte en verankering (aanhechting schuifspanningen).</p>\n<p>De <strong>betonsterkte</strong> bij druk wordt beoordeeld als de verhouding tussen de maximale hoofddrukspanning <em>f</em><em><sub>c</sub></em> (ook σ<sub>2</sub> in Hulpresultaten) verkregen uit de EE-analyse en de grenswaarde <em>f'</em><em><sub>c,lim</sub></em>.</p>\n<p>De <strong>sterkte van de wapening</strong> wordt beoordeeld bij zowel trek als druk 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>aanhechting schuifspanning</strong> wordt afzonderlijk beoordeeld als de verhouding tussen de aanhechtingsspanning τ<em><sub>b</sub></em> berekend door de EE-analyse en de rekenwaarde van de maximale aanhechtingsspanning <em>f</em><em><sub>bu</sub></em>.</p>\n<p>Voor de bepaling van de rekenwaarde van de maximale aanhechtingsspanning <em>f</em><em><sub>bu</sub></em> wordt formule C13.1.2.2 uit AS3600:2018 Sup 1:2022 toegepast in de applicatie.</p>\n<p>\\[f_{bu}=\\frac{k_{2}}{k_{1} \\cdot k_{3}} \\cdot (0.5 \\cdot \\sqrt{f'_{c}})\\]</p>\n<p>Waarbij <em>f'</em><em><sub>c</sub></em><em> ≤ 65 MPa</em> (in de formule in MPa), en de <em>k</em>-factoren worden bepaald uit AS 3600 Cl. 13.1.2.2 als volgt:</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 de staaf in millimeters)<br>= 1.3 voor een horizontale staaf met meer dan 300 mm beton gestort onder de staaf, of 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 van het beton, die in de applicatie per projectonderdeel kan worden ingesteld als volgt.</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 te zien is in de formule, is de basisverankeringslengte <em>L</em><em><sub>sy,tb</sub></em> naar beneden begrensd, en daarom moet de rekenwaarde van de maximale aanhechtingsspanning <em>f</em><em><sub>bu</sub></em> op dezelfde wijze worden begrensd in de applicatie, zodat het volgende van toepassing is:</p>\n<p>\\[f_{bu}\\le \\frac{f_{sy}}{116 \\cdot k_{1}} \\]</p>\n<p>Waarbij <em>f</em><em><sub>sy</sub></em> in MPa is.</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>Er is ook een optie om <strong>gladde staven</strong> te modelleren. Meer informatie is hier te vinden: <a data-item-id=\"182f8ba8-899b-44fc-a1c7-59d562ef8c6c\" href=\"\">Gladde staven 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 gedefinieerd.</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 verkregen kan worden 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 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 is vanaf het begin van de staaf 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 aanvullende 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. geen reductie van het ankereinde), standaard haak (cog), standaard haak (hook), perfecte aanhechting en doorgaande staaf. Al deze typen, samen met de bijbehorende verankeringscoëfficiënten β, zijn weergegeven in Fig. 86 voor langsliggende wapening. De waarden van de gehanteerde verankeringscoëfficiënten zijn afgeleid uit AS 3600 Cl. 13.1.2. Opgemerkt dient te worden dat CSFM drie typen ankereinden 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 te voldoen aan AS 3600 dient de verankeringsveer te worden gebruikt in de berekening; de verankeringsveer wordt aangepast door de β-coëfficiënt, zodat de gebruiker een van de beschikbare verankeringstypen moet gebruiken bij het definiëren van de begin- en eindcondities van de wapening. </p>"
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"value": "<p>De Compatible Stress Field Method (CSFM) is een rekenmethode gebaseerd op 2D-vlakspanningen, waarbij beton wordt gemodelleerd met 2D-eindige elementen waaraan 1D-wapeningselementen worden gekoppeld door middel van randvoorwaarden. Er kunnen ook speciale typen 1D-elementen die verankerde voorspanwapening voorstellen aan het model worden toegevoegd, die gemodelleerd kunnen worden als voorgerekt of nagerekt.</p>\n<p>Voorspanwapening wordt op dezelfde manier gemodelleerd als conventionele wapening, met lineaire elementen die de normaalkracht overdragen. Elk afzonderlijk voorspanwapeningselement wordt gekenmerkt door zijn oppervlakte en materiaaleigenschappen. Deze eigenschappen worden gegeven door de karakteristieke materiaalkromme volgens de gebruikte norm (EN 1992-1-1, ACI 318-19, enz.)</p>\n<p><strong>EUROCODE</strong></p>\n<p>Spanning-rek-diagram van voorspanwapening: a) Spanning-rek-diagram 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-rek-diagram van voorspanwapening: a) Spanning-rek-diagram; 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 op dezelfde manier via een aanhechtingsmodel verbonden met de 2D-elementen van het betonmodel 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 aanhechtingsmodel-elementen laten de relatieve vervorming van de voorspanwapening en het beton toe met geschikte niet-lineaire eigenschappen. Dit modelleert correct de samenhang van de wapening met het beton en ook het verankeringsmodel van de voorgerekte wapening. De eindaanpassingen 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, en de eindvoorspankracht wordt toegepast als een vlaklast op het betonmodel over een oppervlak ter grootte van de ankerplaat. Het model kan de lokale triaxiale spanning in het subverankeringsgebied niet correct beschrijven, en dit gebied moet afzonderlijk worden beschouwd. </p>\n<p>De tension stiffening van de wapening als gevolg van interacties met het beton wordt niet in rekening gebracht voor de voorspanwapening, 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 rechtlijnig aangebracht, waardoor er geen wrijvingsverliezen bij voorspanning optreden. Zodra de vereiste betonsterkte is bereikt, wordt de wapening losgemaakt van de ankerblokken, waardoor de voorspanwapening wordt geactiveerd en de krachten van de wapening naar het beton worden overgedragen. Dit effect is fysiek equivalent aan het onderkoelen van de wapening en wordt gemodelleerd door een initiële rek, vergelijkbaar met die van thermische belasting. Dit geeft een spanning-rek-diagram van voorgespannen wapening zoals weergegeven in bovenstaande afbeelding onder b). Het rekenmodel berekent automatisch de vervormingsrespons van de constructie op de toegepaste voorspanning, en bepaalt daardoor rechtstreeks de voorspanverliezen door elastische rek van het element.</p>\n<p>Aangezien de voorspankracht bekend is, en dus ook de voorspanspanning <em>σ</em><em><sub>pmo</sub></em>, wordt het materiaaldiagram van de wapening gebruikt voor de afhankelijkheid van spanning van vervorming en kan het geschreven worden als:</p>\n<p><em>\\[{{σ}_{p}}=~{{f}}({{ε}}-{{ε}_{0}})\\]</em></p>\n<p>Ervan uitgaande dat de voorspanning in de wapening lager is dan de vloeigrens (d.w.z. dat aan de voorwaarden gedefinieerd in EN 1992-1-1, hoofdstuk 5.10.3, wordt voldaan), kan de initiële vervorming ook berekend worden 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 net vóór het lossen<br><em>E</em><em><sub>p</sub></em> - elasticiteitsmodulus voor voorspanwapening</p>\n<p>Voorgerekte wapening is bijzonder doordat de verankering van de uiteinden tot stand komt via verschillende mechanismen - adhesie van de wapening en het beton op moleculair niveau, de wrijving die ontstaat aan het grensvlak tussen het oppervlak van de wapening en het beton, mechanisch indrukken van de spiraalwapening in het beton, en een toename van de diameter van de voorspanwapening, bekend als het wig-mechanisme of Hoyer-effect. De bovengenoemde effecten worden in het CSFM-rekenmodel meegenomen door de eigenschappen van het verankeringsmodel in het eindgebied van de voorgerekte wapening aan te passen.</p>\n<p>Interactie van voorgerekte wapening en beton: a) spiraalwapening die in het beton drukt; 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 wordt rechtstreeks op de constructie ondersteund, waardoor de verliezen door elastische rek van de constructie door voorspanning worden geëlimineerd. Zodra de gewenste voorspankracht is bereikt, wordt de wapening verankerd, en vervolgens worden de kabelkanalen geïnjecteerd, waardoor een aanhechting van de wapening met de constructie tot stand komt. Bij het modelleren van nagerekte wapening wordt de berekening daarom onderverdeeld in meerdere belastingsstappen - voorspannen, aanbrengen van overige permanente belastingen en aanbrengen van variabele belastingen.</p>\n<p>Eindige-elementen betonmesh 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>Belastingsstap \"voorspannen\"</h4>\n<p>Bij het voorspannen van de wapening wordt de stijfheid van de wapening niet meegenomen in de stijfheid van de constructie. In deze belastingsstap wordt de stijfheid van het lineaire element niet in het model beschouwd; de wapeningselementen worden vervangen door een vervangende belasting die overeenkomt met de voorspanspanning en het wapeningsoppervlak, zoals weergegeven in bovenstaande afbeelding. Nadat de volledige belasting door de voorspanning is bereikt en deze belastingsstap 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 voorspanspanning kan handmatig worden gedefinieerd over de lengte van de wapening of automatisch worden berekend op basis van de geometrie van de wapening. Als de automatische berekening van verliezen wordt gekozen, worden wrijvingsverlies (volgens EN 1992-1-1, 5.10.5.2, of ACI 318-19, 20.3.2) en wapeningsslip (indrukken van ankerwiggen) tijdens het verankeren in rekening gebracht. Aangezien alle voorspanwapening in één stap wordt aangebracht, wordt verlies door opeenvolgend voorspannen niet in rekening gebracht.</p>\n<h4>Volgende belastingsstappen met geactiveerde voorspanwapening</h4>\n<p>In de volgende belastingsstappen (aanbrengen van overige permanente en variabele belastingen) wordt dezelfde procedure gevolgd als voor voorgerekte wapening. De volledige stijfheid van de voorspanwapening wordt in rekening gebracht, de aanhechting tussen de wapening en het omringende beton wordt in rekening gebracht, en het spanning-rek-diagram 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 werd verkregen uit de voorgaande belastingsstap \"voorspannen\". Dankzij de aanhechting van de wapening en het beton wordt de verandering van voorspanning door de elastische vervorming van de constructie ten gevolge van de externe belasting correct in het model in rekening gebracht.</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 betonnen discontinuïteiten in IDEA StatiCa Detail</h2>\n<h3>1 Inleiding tot de CSFM-methode</h3>\n<p><a href=\"#general-introduction\">1.1 Algemene inleiding voor het constructief ontwerp van betondetails</a><br><a href=\"#main-assumptions-and-limitations\">1.2 Belangrijkste uitgangspunten en beperkingen</a><br><a href=\"#design-tools-for-reinforcement\">1.3 Ontwerptools voor wapening</a></p>\n<h3>2 Analysemodel van IDEA StatiCa Detail</h3>\n<p><a href=\"#introduction-to-finite-element-implementation\">2.1 Inleiding tot 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 Belastingoverdracht bij afgesneden liggeruiteinden</a><br><a href=\"#geometric-modification-of-cross-sections\">2.4 Geometrische aanpassing van dwarsdoorsneden</a><br><a href=\"#finite-element-types\">2.5 Typen eindige elementen</a><br><a href=\"#meshing\">2.6 Mesh-generatie</a><br><a href=\"#solution-method-and-load-control-algorithm\">2.7 Oplossingsmethode en belastingregelalgoritme</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 verificaties 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 uiterste grenstoestand</a><br><a href=\"#partially-loaded-areas\">4.4 Partieel belaste vlakken (PLA)<br></a><a href=\"#serviceability-limit-state-analysis\">4.5 Analyse bruikbaarheidsgrenstoestand</a></p>\n<h3>5 Constructieve verificaties 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 Sterkteverificaties</a><br><a href=\"#bearing-and-anchorage-zones-partially-loaded-areas\">5.4 Oplegvlakken en verankeringszones - Partieel belaste vlakken<br></a><a href=\"#serviceability-verifications\">5.5 Bruikbaarheidsverificaties</a></p>\n<h3>6 Constructieve verificaties 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 Sterktegrenstoestand</a><br><a href=\"#bearing-and-anchorage-zones-resistance-partially-loaded-areas\">6.4 Weerstand van oplegvlakken en verankeringszones – Partieel belaste vlakken</a><br><a href=\"#service-limit-state\">6.5 Bruikbaarheidsgrenstoestand</a></p>\n<h3>7 Constructieve verificaties 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 verankeringsverificaties</a><a href=\"#bearing-and-anchorage-zones-partially-loaded-areas\"><br></a><a href=\"#serviceability-checks\">7.4 Bruikbaarheidscontroles</a></p>\n<h3><a href=\"#prestressing-in-detail-model-description\">8 Voorspanning in Detail - Modelbeschrijving</a></h3>\n<p><br></p>\n<p><br></p>\n<h2>1 Inleiding tot 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 Analysemodel 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 verificaties volgens Eurocode</h2>\n<p>De beoordeling van de constructie met CSFM wordt uitgevoerd door middel van twee verschillende analyses: één voor bruikbaarheid en één voor belastingcombinaties in de uiterste grenstoestand. De bruikbaarheidsanalyse gaat ervan uit dat het uiterste gedrag van het element voldoende is, en dat de vloeicondities van het materiaal niet worden bereikt bij bruikbaarheidsbelastingniveaus. Deze aanpak maakt het mogelijk om vereenvoudigde constitutieve modellen te gebruiken (met een lineaire tak van het spanning-rek-diagram van beton) voor de bruikbaarheidsanalyse, om de numerieke stabiliteit en rekensnelheid te verbeteren.</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 verificaties volgens ACI 318-19</h2>\n<p>De beoordeling van de constructie met de CSFM wordt uitgevoerd door middel van twee verschillende analyses: één voor bruikbaarheid en één voor belastingcombinaties op sterkte. De bruikbaarheidsanalyse gaat ervan uit dat het gedrag onder gefactoreerde belastingen voldoende is, en dat de vloeicondities van het materiaal niet worden bereikt bij bruikbaarheidsbelastingniveaus. Deze aanpak maakt het mogelijk om vereenvoudigde constitutieve modellen te gebruiken (met een lineaire tak van het spanning-rek-diagram van beton) voor de bruikbaarheidsanalyse, om de numerieke stabiliteit en rekensnelheid te verbeteren.</p>\n<p>CSFM is in overeenstemming met ACI 318-19, hoofdstuk 6.8.1.1. Om ervoor te zorgen dat de CSFM voldoet aan de eisen van ACI 318-19, paragraaf 6.8.1.2, is er uitgebreid verificatieonderzoek uitgevoerd aan diverse universiteiten. Afzonderlijke artikelen met een samenvatting van de resultaten van verificatie en validatie 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 verificaties 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 verificaties volgens de Australische norm AS 3600 (2018)</h2>\n<p>De beoordeling van de constructie met de CSFM wordt uitgevoerd door middel van twee verschillende analyses: één voor bruikbaarheid en één voor belastingcombinaties op sterkte. De bruikbaarheidsanalyse gaat ervan uit dat het gedrag onder gefactoreerde belastingen voldoende is, en dat de vloeicondities van het materiaal niet worden bereikt bij bruikbaarheidsbelastingniveaus. Deze aanpak maakt het mogelijk om vereenvoudigde constitutieve modellen te gebruiken (met een lineaire tak van het spanning-rek-diagram van beton) voor de bruikbaarheidsanalyse, om de numerieke stabiliteit en rekensnelheid te verbeteren.</p>\n<p>De CSFM is een methode voor constructieve analyse 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 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 de paragrafen 6.6.4 en 6.6.5 - meer informatie is te vinden in AS3600:2018 Sup 1:2022 paragraaf C6.6 - zijn verificatie en validatie van de methode uitgevoerd aan diverse universiteiten. Afzonderlijke artikelen met een samenvatting van de resultaten van verificatie en validatie 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 drukcilindersterkte 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<h1><br></h1>\n<h2>8 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>ACI Committee 318. 2019. <em>Building Code Requirements for Structural Concrete (ACI 318-19) and Commentary</em>. Farmington Hills, MI: American Concrete Institute.</p>\n<p><br></p>\n<p>Alvarez, Manuel. 1998. <em>Einfluss des Verbundverhaltens auf das Verformungsvermögen von Stahlbeton</em>. 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.” <em>The Structural Engineer</em> 57A (1): 9–17.</p>\n<p><br></p>\n<p>Broms, Bengt B. 1965. “Crack Width and Crack Spacing In Reinforced Concrete Members.” <em>ACI Journal Proceedings</em> 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. <em>Non-Linear Finite Element Analysis of Solids and Structures</em>. Wiley.</p>\n<p><br></p>\n<p>European Committee for Standardization (CEN). 2015. <em>1 Eurocode 2: Design of concrete structures - Part 1-1: General rules and rules for buildings</em>. 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.” <em>ACI Structural Journal</em> 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. “<em>Compatible Stress Field Design Of Structural Concrete</em>. 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.” <em>Journal of Structural Engineering</em> 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. <em>Use of Topology Optimization in Concrete Reinforcement Design</em>. 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.” <em>Concrete International</em> 7 (12): 66–73.</p>\n<p><br></p>\n<p>Marti, P. 2013. <em>Theory of Structures: Fundamentals, Framed Structures, Plates and Shells</em>. 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.” <em>Structural Engineering International</em> 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. <em>A Solution for Walls and Details of Concrete Structures</em>. 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.” <em>PCI Journal</em> 32 (3): 74–150.</p>\n<p><br></p>\n<p>Standards Australia. 2018. <em>Concrete Structures (AS 3600:2018)</em>. Sydney, NSW: Standards Australia.</p>\n<p><br></p>\n<p>Standards Australia. 2022. <em>Concrete Structures – Commentary (Supplement 1 to AS 3600:2018)</em>. 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.” <em>ACI Journal</em> 83 (2): 219–31. </p>"
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"value": "<p>In de praktijk worden de <strong>Staafwerkmethode (Strut-and-Tie, S&T)</strong> en <strong>Stress Fields methoden</strong> standaard gebruikt om discontinuïteitsgebieden in gewapende en voorgespannen betonconstructies te ontwerpen. De <strong>Compatible Stress Field Method (CSFM)</strong> is ontwikkeld door deze klassieke theorieën uit te breiden, waardoor een hoge mate van automatisering mogelijk is en de methode consistent is met de ontwerpnorm. Ondanks de eenvoud biedt de methode een zeer realistische beschrijving van het gedrag van een betonconstructie, zowel in de uiterste grenstoestand (UGT) als 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) Schuifwand c) Ligger met tandopleggingen en openingen d) Brugpijler e) Brugdiafragma </em></p>\n<p>Standaardprocedures voor het ontwerpen van doorsneden van betonconstructies zijn toepasbaar in delen waar de Bernoulli-Navier-hypothese van vlakke rekverdeling geldt (B-gebied). De plaatsen waar deze hypothese niet geldt, worden <strong>discontinuïteitsgebieden (D-gebieden)</strong> genoemd. Dit omvat delen van constructies waar geconcentreerde belastingen optreden of waar sprake is van een plotselinge verandering van doorsnede, zoals tandopleggingen (Fig. 1c), gedrongen liggers, wanden met openingen (Fig. 1a, 1b), of consoles en paalkoppen. In de bruggenbouw zijn dit bijvoorbeeld opleggingen van pijlers (Fig. 1d), diafragma's (Fig. 1e), deviatoren, enz.</p>\n<h2>1. Staafwerkmethode</h2>\n<p>De basisaanname bij het definiëren van een staafwerkmodel is dat de treksterkte van beton wordt verwaarloosd. Een eenvoudig vakwerkmodel bestaat uit elementen die op druk en trek werken en het gedrag in de UGT weergeven. In het algemeen is dit geen complex probleem, en het definiëren van een basis staafwerkmodel (Fig. 2a) zou voor een ervaren ingenieur geen probleem moeten zijn. Echter, zelfs voor deze basistaak kan de correcte beoordeling van het model in overeenstemming met de ontwerpnorm 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) Staafwerkmodel optie 1 b) Staafwerkmodel optie 2 c) Staafwerkmodel optie </em></p>\n<p>Trekstaven, knoopvlakken en dwarse trekrek in drukdiagonalen moeten worden beoordeeld. Als het model niet aan de toetsing voldoet, moet de geometrie van het staafwerk worden aangepast, of moet een ander staafwerkmodel worden gekozen (Fig. 2b, 2c). Dit leidt er vaak toe dat de constructeur de geometrie van het staafwerkmodel slechts één keer kiest en alleen de wapening beoordeelt. Dit kan tot een aanzienlijke fout leiden. De keuze van het model is altijd een kwestie van ervaring. Voor complexere constructieve details is het kiezen van een staafwerk dat voldoende overeenkomt met het werkelijke gedrag van de constructie mogelijk niet zo eenvoudig als in het bovenstaande geval. Bovendien is de staafwerkmethode alleen een methode voor het ontwerp van uiterste grenstoestanden. Het maakt het ontwerp van <strong>bruikbaarheidsgrenstoestanden (vervorming, scheurvorming)</strong> niet mogelijk, wat kritische criteria zijn, vooral bij constructies van aanzienlijk belang, aangezien deze 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 vlakke spanningstoestand, d.w.z. een 2D-model. Toch is de methode nog steeds gebaseerd op een fundamentele en veilige aanname van de normen: <strong>beton werkt niet op trek</strong>, en alle trek moet door wapening worden overgedragen. De Compatible Stress Field Method (CSFM) is een verdere ontwikkeling van de staafwerk- en stress fields-methoden, waarbij de hierboven genoemde belangrijkste nadelen worden weggenomen: onzekerheden bij de modelkeuze, moeilijkheden bij automatisering en het onvermogen om bruikbaarheidsgrenstoestanden te beoordelen.</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 uitgelegd aan de hand van de <strong>vlakke spanningstoestand</strong> van het basisvlakelement van een gewapende betonconstructie. Fig. 3a toont het basis 2D-element in vlakke spanning zoals we dat kennen uit alle leerboeken over elasticiteit en sterkteleer. Dit is de spanning op één punt in de constructie, verkregen bijvoorbeeld door lineair-elastische analyse met de eindige-elementenmethode (FEM). Het element wordt onderworpen aan 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 zogenaamde <strong>hoofdspanningen</strong> en hun richting, gedefinieerd door de hoek θ, worden bepaald (Fig. 3b). Het element wordt vervolgens onderworpen aan de hoofdtrekspanning σ<sub>1</sub> en de hoofddrukspanning σ<sub>2</sub>.</p>\n<p>Hoe ziet de rek van hetzelfde element, geanalyseerd met CSFM, eruit? De rek wordt getoond in Figuur 3c. Het gedrukte beton verschijnt in de richting van de hoofddrukspanning σ<sub>2</sub>. En er ontstaat een spanningsveld met spanning σ<sub>c2</sub>. Zoals hierboven vermeld, is de basisaanname dat het beton niet op trek werkt. Daarom wordt de dwarse hoofdtrekspanning σ<sub>1</sub> niet door het beton overgedragen, en ontstaat er een scheur loodrecht op de richting. De spanning σ<sub>c1r</sub> moet daarom nul zijn. Om het bezwijken van ons 2D-element te voorkomen, moet alle trekspanning door de wapening worden overgedragen (in blauw aangegeven in Fig. 3c), die deel moet uitmaken van het rekenmodel. </p>\n<p>Als deze spanningsanalyse met CSFM <strong>continu over het gehele op te lossen 2D-gebied</strong> wordt uitgevoerd, 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 beton en wapening geeft de figuur ook de variërende 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 Totaalresultaten van IDEA StatiCa Detail </em></p>\n<p>De analyse van een detail of constructie met <strong>CSFM is gebaseerd op de eindige-elementenmethode</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 betreft. Belastingen worden tijdens de berekening incrementeel opgelegd, en de oplossing van het niet-lineaire stelsel van vergelijkingen wordt gevonden met de <strong>Newton-Raphson-methode</strong>. </p>\n<p>De fictieve gespreide scheuren (ε<sub>1</sub> is de gemiddelde waarde) worden \"gevormd\" loodrecht op de richting van de hoofdspanningen, die tijdens de niet-lineaire berekening kunnen veranderen naarmate het element bij elke belastingstoename \"geleidelijk scheurt\". Samengevat wordt een fictieve, spanningsvrije roterende scheur beschouwd. </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 de rektoestand, die continu zijn over het gehele op te lossen 2D-domein. Dit is een groot voordeel ten opzichte van klassieke staafwerkbenaderingen en maakt het mogelijk om het rekenmodel te automatiseren en te verfijnen, zoals beschreven in de volgende paragrafen.</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 de CSFM maakt het mogelijk om het standaard eenassige parabolisch-rechthoekige spanning-rekdiagram voor beton op druk volgens de ontwerpnorm te gebruiken. Zoals bekend, neemt de drukvastheid van beton af wanneer het beton beschadigd is door dwarsscheuren (Fig. 5). Dit zogenaamde <strong>compression softening</strong>-effect wordt in de methode meegenomen door automatisch rekening te houden met de effectieve drukvastheid van het beton. </p>\n<p>Op basis van het niveau van de dwarse hoofdtrekrek 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 door de hele constructie bekend is, kan de effectieve drukvastheid van het beton automatisch worden berekend in individuele doorsneden, afhankelijk van het lokale niveau van de dwarse hoofdtrekrek 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 getrokken beton tussen de scheuren op de wapening, het zogenaamde 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). Dit maakt een realistische weergave mogelijk van de stijfheid van een gewapende betonconstructie die is aangetast door scheuren. Het blijft echter waar dat de treksterkte van beton niet bijdraagt aan de uiteindelijke draagkracht. De maximale spanning in de wapening σ<sub>sr</sub> in de scheuren is bepalend voor het ontwerp (Fig. 6).</p>\n<p>CSFM gebruikt gangbare eenassige materiaalmodellen (spanning-rekdiagrammen) zoals gedefinieerd in <strong>ontwerpnormen</strong>. De standaardaanpak, de partiële veiligheidsfactormethode, wordt vervolgens gebruikt om de UGT te beoordelen. De eenvoud van de methode maakt haar geschikt voor de ingenieurspraktijk en is consistent met de ontwerpnormen. </p>\n<p>Hoewel het om een niet-lineaire FEA-analyse gaat, hoeft de constructeur geen aanvullende materiaaleigenschappen en betonkarakteristieken in de berekening in te voeren die in de ontwerpfase mogelijk nog niet eens beschikbaar zijn en die nodig zijn voor bijvoorbeeld niet-lineaire FEA-analyses op basis van breukmechanica. Zoals reeds aangegeven, is een groot voordeel van CSFM-analyse, naast de uiterste grenstoestanden, de mogelijkheid om <strong>bruikbaarheidsgrenstoestanden te beoordelen: 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 wapening</li>\n <li>2D-isoparametrisch element voor beton</li>\n <li>Eindveren voor het verankeringsmodel van de wapening met eindbehandeling</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, is aangetoond dat de CSFM zeer goede voorspellingen geeft van het gedrag en de uiteindelijke draagkracht van de constructie, ondanks de eenvoud van de formulering. Met andere woorden, de methode is bijvoorbeeld niet geschikt voor het ontwerp van liggers zonder dwarskrachtwapening, die potentieel bros gedrag vertonen. <a data-item-id=\"1e879886-9e36-49e1-acb1-e6001361531f\" href=\"\">Verificaties</a> van de methode, inclusief experimenten, zijn te vinden in [1]. Een meer gedetailleerde 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 ervan dus niet beperkt is tot D-gebieden, maar kan worden gebruikt om hele staven te modelleren, bijvoorbeeld 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 en nagerekt voorgespannen wapening</strong> te specificeren.</p>\n<h2>3. Voorbeeld van het ontwerp van een pijleroplegging</h2>\n<p>De praktische toepassing van CSFM wordt getoond in het ontwerp van de brugpijleroplegging 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 gewapend betonnen pijler is ontworpen in beton C40/50 en de dikte ervan (in de langsrichting van de brug) is 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 Pijleroplegging: a) Samenvatting ontwerp; b) Drukspanning in beton in UGT; c) Trekspanning in wapening in UGT; d) Scheurwijdte in BGT</em></p>\n<p>Bovenaan de pijleroplegging werd eerst een dwarsligger van B500-wapening 20xϕ28+20xϕ25 - de bovenste vier lagen - ontworpen. 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 meer gedetailleerde spanningsverdeling in het beton en de wapening wordt vervolgens gedocumenteerd in Figuur 8b en 8c. De dwarswapening zit net onder de vloeigrens; ook de spanningen in het beton (en de bijbehorende rekken) zijn bevredigend in UGT. Het resultaat van de scheurwijdteberekening (Fig. 8d) laat echter zien dat het ontwerp niet voldoet in <strong>BGT</strong>: w<sub>max</sub> = 0,36 mm > w<sub>lim</sub> = 0,3 mm. <strong>Om aan de grenswaarde van de scheurwijdte te voldoen, is het noodzakelijk de wapening</strong> van de dwarsligger te vergroten naar 20xϕ32+20xϕ28. In het geval van w<sub>lim</sub> = 0,2 mm (bijvoorbeeld een pijler nabij een weg met zoutsproei, milieuklasse XF2), zou de wapening van de dwarsligger zelfs moeten worden vergroot naar 24xϕ32+24xϕ28.</p>\n<h2>Conclusie</h2>\n<p>CSFM is geschikt voor de ingenieurspraktijk omdat het <strong>eenvoudige materiaalmodellen gebruikt die in een ontwerpnorm zijn gedefinieerd</strong>. Naast de uiterste grenstoestanden maakt het ook het ontwerp van bruikbaarheidsgrenstoestanden mogelijk. Iets waarvan de beoordeling voorheen moeilijk voor te stellen was bij gebruik van staafwerkmodellen. Door de methode te implementeren in <strong>IDEA StatiCa Detail</strong> is het vervolgens mogelijk om het gedrag van de constructie realistisch vast te leggen en discontinuïteitsgebieden en grotere samenstellingen efficiënt en veilig te ontwerpen en te beoordelen.</p>\n<p>De CSFM is voornamelijk ontwikkeld door het werk van professor Walter Kaufmann, hoofd van de leerstoel Constructietechniek aan de 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": "<h2>Inleiding </h2>\n<p>Dit artikel bespreekt het gebruik van de <a data-item-id=\"eaab962d-ba44-4ee0-8fa7-45193c9f52b5\" href=\"\">CSFM</a> om afschuivingsbezwijken te analyseren in liggers met geringe hoeveelheden beugels. Hiertoe wordt een selectie van proeven, uitgevoerd op eenvoudig opgelegde gewapend betonliggers door Huber (2016), Piyamahant (2002) en Vecchio en Shim (2004), geanalyseerd. Deze proeven omvatten een groot aantal parameters, waaronder verschillende afmetingen, afschuifslankheden en hoeveelheden afschuif- en langswapening. Dit hoofdstuk beschrijft de analyse van 17 experimenten uit deze onderzoeken met de CSFM, waarbij het vermogen van de CSFM wordt onderzocht om zeer uiteenlopende bezwijkvormen correct te modelleren, variërend van afschuivingsbezwijken met en zonder breuk van de beugels tot buigingsbezwijken en gemengde afschuiving-buigingsbezwijken. </p>\n<p>Proefopstelling Fig. 6.17 toont de geometrie, proefopstellingen en wapeningsconfiguraties van de geanalyseerde experimenten. Informatie over de afschuifwapening (diameter (<em>Ø</em><em><sub>t</sub></em>), hart-op-hart afstand (<em>s</em><em><sub>t</sub></em>) en geometrische wapeningsverhouding (<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 geometrie (effectieve hoogte (<em>d</em>), afschuifslankheid (<em>a/d</em>) en breedte van de liggers (<em>b</em>)) wordt weergegeven in Tabel 6.10. Proeven R1000m60 en R500m351, uitgevoerd door Huber (2016), hadden eenbenige haken, terwijl bij alle andere proeven tweebenige gesloten beugels werden gebruikt. In de geanalyseerde proeven van Piyamahant (2002) bleven de geometrie en de buigwapening constant, terwijl deze in de andere twee onderzoeken varieerden. </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 zijn gebruikt in de CSFM-analyse zijn samengevat in Tabel 6.11. De meeste materiaaleigenschappen die nodig waren voor de <a data-item-id=\"2ebdaf9c-827f-4fd6-9f82-28bc96970a64\" href=\"\">CSFM</a>-analyse waren beschikbaar in de bijbehorende proefrapporten. 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, wapening, <a data-item-id=\"50ed723b-9b87-4870-a69f-e05b5a8a8150\" href=\"\">oplegging en belastingscondities</a> werden in de CSFM gemodelleerd volgens de proefopstellingen. Fig. 6.18 toont het modelleren van Test A3 van Vecchio en Shim (2004) als voorbeeld.</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 werden vier numerieke berekeningen uitgevoerd met de volgende parameters:</p>\n<ul>\n <li>De mesh-grootte, die varieerde van 5 (standaardwaarde voor deze specifieke voorbeelden), via 10 tot 20 eindige elementen over de hoogte van de ligger. Aangezien de standaardmesh al zeer grof is, worden in dit onderzoek alleen fijnere meshes geanalyseerd, en de mesh met 10 elementen werd gebruikt behalve in M0. </li>\n <li> Het al dan niet in rekening brengen van het tension stiffening-effect. Standaard wordt tension stiffening in rekening gebracht in de CSFM. </li>\n <li>Het al dan niet in rekening brengen van mogelijk niet-gestabiliseerde scheurvorming in beugels. Wanneer dit in rekening wordt gebracht (standaard), definieert het Pull-Out Model (POM) tension stiffening in beugels (de geometrische wapeningsverhoudingen van alle liggers liggen onder (<em>ρ</em><em><sub>cr</sub></em>), dus het Tension Chord Model wordt nooit gebruikt). Wanneer dit is 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>waarbij<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}\\) - modulaire verhouding</li>\n</ul>\n<p>Tabel 6.12 toont de parameters die zijn gebruikt in elke numerieke berekening. 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 hoofdstuk bevat vergelijkingen tussen de uiterste belastingen en bezwijkvormen die worden geleverd door de <a data-item-id=\"8b9399db-b927-491a-a50f-c66ad97560af\" href=\"\">CSFM</a> en de experimentele resultaten. Om de CSFM ook te verifiëren voor bruikbaarheidsgedrag en vervormingscapaciteit, worden de belasting-vervormingsresponsen van het model vergeleken met die van de proeven voor geselecteerde liggers.</p>\n<h4>Bezwijkvormen en uiterste belastingen</h4>\n<p>Tabel 6.13 vat de uiterste afschuifkrachten samen die in de proeven zijn gemeten (<em>V</em><em><sub>u,exp</sub></em>), de uiterste afschuifkrachten voorspeld door de CSFM (<em>V</em><em><sub>u,calc</sub></em>), en de respectievelijke bezwijkvormen. Deze tabel geeft ook het gemiddelde en de variatiecoëfficiënt (CoV) van de verhoudingen tussen de gemeten en berekende uiterste belastingen voor elk numeriek model. In alle analyses (behalve M3, waarin tension stiffening werd verwaarloosd) werd een afschuivingsbezwijken in de beugels voorspeld door de CSFM. Dit komt goed overeen met de bezwijkmechanismen die zijn waargenomen in de proeven van Huber (2016) en Piyamahant (2002), maar komt niet overeen met die waargenomen in Vecchio en Shim (2004). Het niet correct vastleggen van de bezwijkvormen leidde in dit geval tot enigszins conservatieve schattingen van de uiterste belasting. Over het algemeen leveren de standaardparameters goede sterkteschattingen op, 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 wordt weergegeven in Fig. 6.19 door middel van de verhouding tussen experimentele en berekende uiterste afschuifkrachten (<em>V</em><em><sub>u,exp</sub></em><em>/V</em><em><sub>u,calc</sub></em>). De uiterste belasting is duidelijk gevoelig voor de gekozen grootte van de eindige elementen (zie Fig. 6.19 a). Het maximale verschil tussen de grofste en fijnste mesh (M0 en M2) bedraagt 36% (Test 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 licht (ongeveer 5%). Bij het verfijnen van de mesh naar 10 of 20 eindige elementen over de hoogte van de ligger (respectievelijk modellen M1 en M2), kunnen uitstekende sterktevoorspellingen worden bereikt die iets aan de veilige kant van de uiterste belastingen liggen. Er werden geen veranderingen in de bezwijkvormen waargenomen bij het variëren van de mesh-grootte van de eindige elementen. Zelfs de resultaten met de standaard mesh-grootte zijn zeer bevredigend, gezien het feit dat verschillende experimenten brosse afschuivingsbezwijken vertoonden, die moeilijk te voorspellen zijn met ontwerpmethoden.</p>\n<p>De manier waarop tension stiffening in rekening wordt gebracht, heeft een zeer relevante invloed op de sterktevoorspellingen, zoals te zien is in Fig. 6.19 b-c. Het in rekening brengen 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 uiterste belasting van ongeveer 22% (zie Tabel 6.12). Bij het verwaarlozen van tension stiffening verandert de bezwijkvorm in buigingsbezwijken (zie Tabel 6.12) en komen de waargenomen afschuivingsbezwijkvormen niet overeen. De resultaten zijn ook zeer gevoelig voor de beschouwde compression softening-relatie. Zoals te zien is in Fig. 6.19 c, levert het gebruik van het Tension Chord Model in de beugels (model M4) in plaats van het Pull-out Model (model M1) iets betere resultaten op dan bij het verwaarlozen van tension stiffening (model M3), maar overschat nog steeds de uiterste belastingen sterk met ongeveer 15% (zie Tabel 6.12). Daarom kan worden geconcludeerd dat het gebruik van het Pull-Out-Model in deze voorbeelden cruciaal is voor een correcte modellering van het draagvermogen. </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 resultaten van het continue spanningsveld (hoofddrukspanningen (<em>σ</em><em><sub>c</sub></em>) en staalspanningen (<em>σ</em><em><sub>sr</sub></em>) bij de scheuren) voor proefstukken A1 en A3 van Vecchio en Shim (2004), waarin de voorspelde afschuivingsbezwijken zijn gemarkeerd. Deze resultaten zijn berekend met de numerieke parameters M1 (standaardparameters, behalve de mesh-grootte, die de helft van de standaardwaarde bedraagt). Zoals te zien is aan de spanningsvelden, bevindt de drukspanning in de drukzone door buiging zich in de plastische tak (99,5%). Echter, door het gehanteerde criterium voor betonverbrijzeling, treedt de breuk van de beugels op vóórdat betonverbrijzeling plaatsvindt. </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-vervormingsrespons</h2>\n<p>De berekende belasting-vervormingsresponsen, verkregen met de numerieke parameters van M1 (waarbij het TCM voor de buigwapening en het POM voor de beugels wordt beschouwd) en M3 (waarbij eventuele tension stiffening-effecten worden verwaarloosd), worden vergeleken met de gemeten belasting-vervormingsresponsen in Fig. 6.21 voor de proeven R500m352, T1, A1 en A3. De belasting <em>V</em> komt overeen met de aangebrachte afschuifkracht en <em>u</em> komt overeen met de doorbuiging in het midden van 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 redelijk goed worden voorspeld over de gehele belastingsgeschiedenis, hoewel de doorbuigingen bij piekbelasting iets worden onderschat. Met name in Test A3 van Vecchio en Shim (2004) kan het waargenomen plateau in de experimenten, veroorzaakt door het vloeien van de buigwapening, niet goed worden vastgelegd in de numerieke analyse, aangezien eerst breuk van de beugels wordt voorspeld. Het verwaarlozen van tension stiffening-effecten leidt tot een overschatting van de uiterste belastingen en vervormingen. Deze uitspraken voor de analyses zonder tension stiffening gelden ook bij gebruik van de M4-parameters (waarbij het TCM zowel voor de beugels als voor de buigwapening wordt gebruikt).</p>\n<h2>Conclusies</h2>\n<p>De volgende conclusies kunnen worden getrokken met betrekking tot de vergelijking van de CSFM-resultaten en het waargenomen gedrag in de geanalyseerde proeven, uitgevoerd op eenvoudig opgelegde liggers met geringe hoeveelheden beugels: </p>\n<ul>\n <li>De CSFM levert goede schattingen van de uiterste belasting op, die licht wordt overschat (gemiddeld met 5%) bij gebruik van de standaard numerieke parameters. Het is moeilijk om gecombineerde bezwijkvormen door afschuiving en betonverbrijzeling bij buiging vast te leggen; de CSFM voorspelt bezwijken door breuk van de beugels, wat leidt tot sterktevoorspellingen die aan de conservatieve kant liggen. </li>\n <li>De voorspellingen van de uiterste belasting zijn enigszins gevoelig voor variaties in de mesh-grootte van de eindige elementen. De beste voorspellingen worden verkregen wanneer de standaard eindige-elementenmesh wordt verfijnd. Daarom wordt altijd aanbevolen om de invloed van de grootte van de eindige elementen op de resultaten te onderzoeken bij het uitvoeren van definitieve verificaties. </li>\n <li>Het verwaarlozen van tension stiffening leidt tot een zeer uitgesproken overschatting van de uiterste belasting en de vervormingscapaciteit. Zelfs bij het modelleren van tension stiffening in de beugels door middel van het Tension Chord Model, ligt de voorspelde uiterste belasting duidelijk aan de onveilige kant. 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