Guidelines for Nonlinear Finite Element Analysis of Concrete Structures

Transcription

1 P rd /P exp [%] Rijkswaterstaat Technical Document (RTD) Guidelines or Nonlinear Finite Element Analysis o Concrete Structures Doc.nr.: RTD :2017 Version: 2.1 Status: Final Date: 15 June Water. Wegen. Werken. Rijkswaterstaat

2 Page: 2 o 69 Guidelines or Issued: 15 June 2017 Nonlinear Finite Element Analysis o Version: 2.1 Concrete Structures Status: Final RTD: :2017 Please reer to this report as: M.A.N. Hendriks, A. de Boer, B. Belletti, Guidelines or Nonlinear Finite Element Analysis o Concrete Structures, Rijkswaterstaat Centre or Inrastructure, Report RTD:1016-1:2017, This document is one rom a series o documents. At the time o writing, the ollowing documents have been drated: RTD : RTD : RTD A: RTD B: RTD C: Guidelines or Nonlinear Finite Element Analysis o Concrete Structures Validation o the Guidelines or Nonlinear Finite Element Analysis o Concrete Structures - Part: Overview o results Validation o the Guidelines or Nonlinear Finite Element Analysis o Concrete Structures - Part: Reinorced beams Validation o the Guidelines or Nonlinear Finite Element Analysis o Concrete Structures - Part: Prestressed beams Validation o the Guidelines or Nonlinear Finite Element Analysis o Concrete Structures - Part: Slabs

3 Guidelines or Page: 3 o 69 RTD: :2017 Status: Final PREFACE The Dutch Ministry o Inrastructure and the Environment is concerned with the saety o existing inrastructure and expected re-analysis o a large number o bridges and viaducts. Nonlinear inite element analysis can provide a tool to assess saety using realistic descriptions o the material behavior based on actual material properties. In this way, a realistic estimation o the existing saety can be obtained utilizing hidden capacities (de Boer et al., 2015). Nonlinear inite element analyses have intrinsic model and user actors that inluence the results o the analysis. This document provides guidelines to reduce these actors and to improve the robustness o nonlinear inite element analyses. The guidelines are developed based on scientiic research, general consensus among peers, and a long-term experience with nonlinear analysis o concrete structures by the contributors. This version o the guidelines can be used or the inite element analysis o basic concrete structural elements like beams, girders and slabs, reinorced or prestressed. The guidelines can also be applied to structures, like box-girder structures, culverts and bridge decks with prestressed girders in composite structures. The guidelines are restricted to be used or existing structures. The guidelines have been developed with a two-old purpose. First, to advice analysts on nonlinear inite element analysis o reinorced and pre-stressed concrete structures. Second, to explain the choices made and to educate analysts, because ultimately the analysts stays responsible or the analysis and the results. An inormed user is better capable to make educated guesses; something that everybody perorming nonlinear inite element analyses is well aware o. Max Hendriks, Ane de Boer, Beatrice Belletti June 2017

4 Page: 4 o 69 Guidelines or Issued: 15 June 2017 Nonlinear Finite Element Analysis o Version: 2.1 Concrete Structures Status: Final RTD: :2017 CONTRIBUTORS This is version 2.1 o the guidelines. The authors o this revision are: Pro.dr.ir. M.A.N. Hendriks, Delt University o Technology, Netherlands, and Norwegian University o Science and Technology, Norway Dr.ir. A. de Boer, Ministry o Inrastructure and the Environment, Netherlands Assoc. pro.dr. B. Belletti, University o Parma, Italy and ormerly Delt University o Technology, Netherlands Apart rom the names listed above, the main contributors o version 1.0 were: Ir. J.A. den Uijl, ormerly Delt University o Technology, Netherlands Dr.ir. P.H. Feenstra, ormerly Delt University o Technology, Netherlands, and presently Exponent, United States Dr. C. Damoni, ormerly Delt University o Technology, Netherlands and University o Parma, Italy Inormation: rbk-ino@rws.nl

5 Guidelines or Page: 5 o 69 RTD: :2017 Status: Final CONTENTS 1 INTRODUCTION Format Applicability Deviations Case studies Disclaimer MODELING General Units Material Properties Concrete Reinorcement Constitutive Models Model or Concrete Model or Reinorcement Model or Concrete-Reinorcement Interaction Finite Element Discretization Finite Elements or Concrete Finite Elements or Reinorcement Meshing Algorithm Minimum Element Size Maximum Element Size Prestressing Existing Cracks Loads Boundary Conditions Support and load plates Symmetry ANALYSIS Loading Sequence Load Incrementation Equilibrium Iteration Convergence Criteria LIMIT STATE VERIFICATIONS Serviceability Limit State (SLS) Ultimate Limit State (ULS) Global Resistance Factor Method (GRF) Partial Saety Factor Method (PF) Estimation o Coeicient O Variation o Resistance Method (ECOV) 57 5 REPORTING OF RESULTS Finite element analysis input check list Finite element results check list Finite element model checks... 63

6 Page: 6 o 69 Guidelines or Issued: 15 June 2017 Nonlinear Finite Element Analysis o Version: 2.1 Concrete Structures Status: Final RTD: : REFERENCES Annex A: Material input values or Saety Formats... 67

7 Guidelines or Page: 7 o 69 1 INTRODUCTION This document provides guidelines or nonlinear inite element analyses o existing concrete structures and inrastructures, like bridges and viaducts. The guidelines could be applied to beams, slabs, box girders, tunnels, culverts, etc.. The members can contain prestressing as well as normal reinorcement. 1.1 Format The ormat is similar to the ib documents: On the right-hand side, the guideline as brie as possible. On the let-hand side, the comments and explanations o the guidelines and, where appropriate, reerences to literature. 1.2 Applicability The guidelines in this document are intended to be applied to nonlinear inite element analysis or the saety veriication o reinorced and prestressed concrete structures under quasi-static, monotonic loading. These guidelines cannot be applied to any other kind o analysis. For instance, these guidelines are not intended or modeling cyclic and dynamic loading, such as earthquake or wind loads, and are not intended to model transient eects, such as creep and shrinkage.

8 Guidelines or Page: 8 o Deviations The analyst is ultimately responsible or the model, the analysis, and the interpretation o results. The analyst has thereore the right to deviate rom these guidelines. In the case the guidelines are not ollowed, the analysis report should explicitly mention this and the analyst should show suicient proo that the alternative method or model will result in accurate and reliable results using benchmarks agreed on by both principal and analyst. Additional inormation on the numerical analyses can be ound in various other publications (Belletti et al., 2011, 2013, 2014), (de Boer et al. 2014). 1.4 Case studies As part o the development o these guidelines a number o case studies have been perormed. An overview o the numerical analyses is provided in a separate validation report, which is released alongside with the guidelines (Belletti et al., 2017). 1.5 Disclaimer Although the editors have done their utmost best to ensure that any inormation given is accurate, no liability o any kind, including liability or negligence, can be accepted in this respect by the organization involved, its employees, or the Authors o this document.

9 Guidelines or Page: 9 o 69 2 MODELING Modeling a structure consists o a number o sequential steps which should be taken deliberately to ensure the quality o the overall analysis. A inite element model consists o a number o entities. First, the unit system or the analysis should be decided. Next, the material and sectional properties are deined or all the parts o the structure. Then, the inite element discretization is created and boundary conditions and loads are applied to the model. Since these guidelines are written or assessing the reliability o a structure, in general a ull model o the structure is necessary with permanent loads and those variable loads or which the load-carrying capacity is to be ound. 2.1 General A inite element model o a structure is an abstraction o the physical structure with a number o assumptions, generalizations, and idealizations. The abstraction process has two distinct steps: irst, the abstraction rom the structure to the mechanical model, and then the abstraction rom the mechanical model to the inite element model. In the irst step, assumptions and simpliications have to be made regarding to which extent and to which detail the structure has to be modeled, how the boundaries o the model are described, which loads on the structure are signiicant and how they are described, et cetera. The second step is to discretize the mechanical model into a inite element model, and attach the necessary attributes such as material models, boundary conditions, and loading to the inite element model. It is important to use a consistent set o units when generating input or a inite element program. A units check should be used to ensure that the set o units lead to results in the required units. The Finite Element Method has no inherent notion o units; it deals only with numbers. Finite element programs, however, sometimes require certain input in predeined units. 2.2 Units A consistent set o units should be used and the input o the inite element program should always be checked with a units check. The preerred system o units is listed in the table below.

10 Guidelines or Page: 10 o 69 The program will take care that the unit system is consistent. Note that the preerred length unit is in meters. A length unit o millimeters is oten used but special care should be taken with in particular dead weight and the gravity constant, g = 10 m/s 2 = 10 N/kg, and the interpretation o output such as eigenrequencies and the units o the stress plots, or instance. Entity Unit Alternative unit Length Meter m Millimeter mm Mass Kilogram kg Time Second s Temperature Celsius C 2.3 Material Properties Material properties should relect the current physical state o the structure. From these properties the model parameters are derived, dependent on the particular model used in the inite element analysis. For the guidelines, material properties or concrete and reinorcing steel are discussed only. The number o the concrete grade indicates the characteristic cylinder compressive strength ck. Following the Model Code 2010, the most important material properties o concrete can be related to this property and are listed in the table below Concrete The concrete properties should be derived rom the provisions o the ib Model Code 2010 (ib, 2013). For Serviceability Limit State analysis characteristic values o the material properties should be used (see section 4.1). For ailure Ultimate Limit State

11 Guidelines or Page: 11 o 69 Parameter (Model Code 2010) Characteristic cylinder compressive strength Mean compressive strength Design compressive strength ck cm cd ck Minimum reduction actor o min min 0.4; compressive strength due to lateral (40% o the strength remains) cracking Lower-bound characteristic tensile ctk, min 0. 7 ctm strength Mean tensile strength (or C50) 2/ 3.3 Design tensile strength ctm ctd cc ck c 0 ck ctk,min Fracture energy G F c 73 cm Compressive racture energy, (Nakamura and Higai, 2001) GC 250 G F Young s modulus ater 28 days 1/ 3 cm 0 10 E ci Ec (Initial) Poisson ratio ν = 0.15 Density plain concrete ρ = 2400 kg/m 3 Density reinorced concrete ρ = 2500 kg/m 3 Concrete saety coeicient c =1.5 Long term eect coeicient cc <1 analyses either characteristic, design or mean values o the material properties should be used, in accordance with the saety ormat (see section 4.2 and Table 1 and Table 2 in Annex A).

12 Guidelines or Page: 12 o 69 The parameters in the equations above are listed in the table below. 4 8MPa E c MPa Typical values or concrete C40 are listed in the ollowing table. Parameter Value Unit ck 35.5 N/mm 2 cc 1 - cd N/mm 2 ctm 3.24 N/mm 2 ctk,min 2.27 N/mm 2 ctd 1.51 N/mm 2 E ci N/mm 2 G F Nmm/mm 2 ρ kg/mm 3

13 Guidelines or Page: 13 o Reinorcement Parameter Characteristic yielding strength Characteristic ultimate strength Mean yielding strength 10 Design yielding strength Design ultimate strength Class A: ( t / y ) k 1.05 Class B: ( t / y ) k 1.08 Class C: 1.15 ( t / y ) k 1.35 Class D: 1.25 ( t / y ) k 1.45 Design ultimate strain yk tk ym yd td yk tk yk uk 2.5% uk 5% uk 7% uk 8% ud 0. 9 uk Poisson ratio ν = 0.3 Density steel ρ = 7850 kg/m 3 Steel saety coeicient s =1.15 s s Steel or bars The material properties or the bars should be determined rom data sheets provided by the manuacturer, or rom original speciications. I material properties are determined on test bars, the in-situ values can be used. In other cases the properties should be derived rom the provisions o the ib Model Code Hardening can be approximated by a bilinear diagram.

14 Guidelines or Page: 14 o 69 Typical values or steel B450C are listed in the ollowing table. Parameter Value Unit Value Unit yk N/m N/mm 2 tk N/m N/mm 2 E s N/m N/mm 2 ρ 7850 kg/m kg/mm 3 Parameter Characteristic 0.1% proo stress Characteristic ultimate tensile strength Characteristic percentage total elongation at maximum orce Design 0.1% proo stress Design ultimate tensile strength p0. 1k ptk puk p0. 1d p0. 1k Poisson ratio ν = 0.3 Density steel ρ = 7850 kg/m 3 Steel saety coeicient s =1.1 ptd ptk s s Steel or prestressing tendons The material properties or the prestressing steel should be determined rom data sheets provided by the manuacturer, or rom original speciications. I material properties are determined on test bars, the in-situ values can be used. In other cases the properties should be derived rom the ib Model Code Hardening can be approximated by a bilinear diagram.

15 Rijkswaterstaat Centre or Inrastructure Guidelines or Page: 15 o 69 Typical values or seven wire-low relax strands are listed in the ollowing table. Parameter Value Unit Value Unit p0.1k N/m N/mm 2 ptk N/m N/mm 2 E s N/m N/mm 2 ρ 7850 kg/m kg/mm Constitutive Models Constitutive models, also known as material models, used in a inite element context speciy the constitutive behavior (the stress-strain relationship) that is assumed or the materials in the structure. The material models are oten simpliied abstractions o the true material behavior. Compared to the ixed model, the rotating model usually results in a lowerlimit ailure load because it does not suer as much rom spurious stresslocking. The stress-locking phenomena is present in the ixed crack model where stresses rotate signiicantly ater crack ormation resulting in a considerable overestimation o the ailure load (Rots 1988). I a ixed crack model is used, an adequate shear retention model should be used (see ). For beams and slabs without stirrups the adequacy o the shear retention model should be proved explicitly. Alternatively the rotating crack model Model or Concrete For concrete, a total strain-based rotating crack or ixed crack model is preerred.

16 Guidelines or Page: 16 o 69 should be used. The linear-elastic material properties are the Young s modulus and the Poisson ratio. The latter is assumed equal to 0.15, irrespective o the concrete strength. I the applied cracking model does not include a decrease o the Poisson eect during progressive cracking an additional analysis with a Poisson ratio equal to 0.0 should be considered. A reduced Young s modulus should be used with a reduction actor equal to 0.85 to account or initial cracking due to creep, shrinkage, and such. The initial Young s modulus can be determined according to the Model Code 2010 provisions given in Section 2.3.1, or, alternatively, the simpliied relationship according to the Eurocode-2 can be used. cm Ec 0.85Eci 0.85Ec 0 cm0 with E co = MPa and cm0 = 10 MPa. The uniaxial stress-strain diagram or tension is shown in the igure below. The exponential-type sotening diagrams such as the Hordijk relationship or the exponential sotening diagram is preerred since this diagram will result in more localized cracks and consequently will avoid large areas o diuse cracking. The area under the stress-strain curve should be equal to the racture energy divided by the equivalent length. Ater complete sotening, i.e. when virtually no stresses are transmitted, the crack is said to be ully open. In case o a multi-linear stress-strain diagram, a predeined Linear-elastic properties An isotropic linear-elastic material model based on the Young s modulus and Poisson ratio should be used Tensile Behavior An exponential sotening diagram is preerred. The parameters are the tensile strength, t, the racture energy, G F, and the equivalent length, h eq. For the description o h eq, reerence to section is made. A multi-linear approximation o the exponential uniaxial stress-strain diagram can be used i exponential sotening is not available. The apparent Poisson ratio should be reduced ater cracking ater crack initiation.

17 Guidelines or Page: 17 o 69 equivalent length has to be taken into account that should be based on the element size as much as possible. Figure 1 Exponential sotening Figure 2 Hordijk sotening

18 Guidelines or Page: 18 o 69 Figure 3 Multi-linear sotening The exponential sotening relationship is given by cr t exp u The sotening curve according to Hordijk (Hordijk 1991) is given by 3 cr cr cr 3 cr t 1 c1 exp c2 1 c1 expc2 0 u u u u cr 0 u The usual parameters are c 1 =3.0 and c 2 =6.93. For both curves, the maximum stress is given by the tensile strength t and shape o the sotening diagram is governed by the ultimate strain parameter ε u. For exponential sotening the ultimate strain parameter is given by

19 Guidelines or Page: 19 o 69 GF u h The ultimate strain parameter in case o Hordijk sotening is given by GF u h eq t eq t The selection o a shear retention model is only relevant or ixed crack models. In a conservative variable shear retention model the secant shear stiness degrades at the same rate as the secant tensile stiness due to cracking. Alternatively, or beams, a variable shear retention model can be used in which de shear stiness gradually reduces to zero or a crack width o hal the average aggregate size. Constant shear retention models are not advisable, or should at least be accompanied with thorough post-analysis checks o spurious principal tensile stresses. The compressive behavior o concrete is rather complicated; especially the post-peak behavior is complex and depends to some extend on the boundary conditions o the experimental setup. The experimental behavior under uniaxial compression shows a sotening relationship ater the peak strength. Under increasing levels o lateral coninement, concrete in compression shows an increasing strength and increasing ductility (see ). On the other hand, the compressive strength should be reduced, Shear Behavior For ixed crack models a variable shear retention model is strongly recommended. For beams and slabs without stirrups the adequacy o the variable shear retention model should be veriied explicitly Compressive Behavior The compressive behavior should be modeled such that the maximum compressive stress is limited. Parabolic stress strain diagram with sotening branch is recommended. The sotening branch should be based on the compressive racture energy value (see 2.3.1) in order to reduce mesh size sensitivity during compressive strain localization.

20 Guidelines or Page: 20 o 69 speciically in case o lateral cracking in plane stress models (see ). The preerred model is based on a compressive racture energy, G c, (Feenstra 1993, Cervenka and Cervenka 2010), regularized with a crushingband width (see ). The (automatic) determination o the crushingband width o h eq ollows the same lines as or tension sotening and the cracking-band width, but should now be based on the principal compression strain direction. The compressive sotening is a unction o the compressive racture energy, based on the tensile racture energy value (see 2.3.1). The parabolic diagram can be used to model this, see Figure below. Alternatively a model with a parabolic ascending branch ollowed by a linear sotening can be used. Figure 4 Parabolic compression diagram The above parabolic curve is deined as:

21 Guidelines or Page: 21 o 69 u j c j u c u c j c / c j c / c c / c j / c c / c j c j / c / c j c i i i i j denotes the (negative) compressive strain or the case o progressive compression. The unction is partitioned by: E c c / 3 5 / c c c c c u h G 2 3 Models which only limit the compressive strength, like the simple elastoplastic diagram shown below, are not advisable. Analyses with such models should always be accompanied with a post-analysis check o the compressive strains.

22 Guidelines or Page: 22 o 69 Figure 5 Elasto-plastic compression diagram with e c Ec. Ec i e 0 c i e This holds also or the parabola-rectangular diagram used or the design o cross-sections rom the Eurocode-2: Figure 6 Parabola-rectangular compression diagram

23 Guidelines or Page: 23 o 69 n 1 1 i c2 0 c i c2 c c2 The parameters o the curve are n=2, ε c2 = -2.0, and ε cu2 = -3.5 or compressive strengths lower than 50 MPa. The initial slope o the curve should be equal to the linear-elastic Young s modulus. However, the initial slope is ully determined by the parameters o the curve resulting in E c n c2 c Neither o the relationships given above model the strength degradation ater the peak strength. In the post-analysis check or these non-sotening models compressive ailure o the structure is identiied as reaching o an ultimate compressive strain (-3.5 ) somewhere in the structure. The area over which the compressive strains are averaged should be motivated. The compressive stress-strain diagram o Thoreneldt, see below, is not advisable.

24 Guidelines or Page: 24 o 69 Figure 7 Thoreneldt compression diagram The Thoreneldt curve is deined as n c nk c2 n 1 c2 where c n and 1 i c2 0 k c 0.67 i c2 63 The strain at the maximum stress is deined as

25 Guidelines or Page: 25 o 69 c2 n n 1 E c c Note that the parameters o the Thoreneldt curve are not unit-ree and that the compressive strength needs to be deined in MPa. Also, the curve shows a sotening behavior and inite element results are consequently mesh-dependent since they are not regularized with a crushing-band width h eq. Although tension-compression interaction is an important eature o the constitutive behavior o concrete, the behavior is rather complicated and or existing models the parameters are sometimes diicult to interpret. Attention should be given to the inite element results since ignoring tension-compression interaction is a non-conservative assumption. A reduction o the compressive strength resulting rom lateral cracking should be taken into account. Dierent models that take into account the tension-compression interaction are available in literature (Vecchio & Collins 1993, Hsu 2010). Some o these models only reduce the compressive strength, leading to a reduction o the Young s modulus or low values o compressive strain. Some other more reined models reduce both the compressive strength and the peak compressive strain so that the initial stiness o the structure is not reduced, see Figure below Tension-Compression Interaction Tension-compression interaction needs to be addressed and taken into account in the modeling o concrete structures subjected to multi-axial stress state.

26 Guidelines or Page: 26 o 69 Figure 8 Compression sotening models As an example the reduction o the compressive strength trend or Model B is shown below. Figure 9 Reduction o the compressive strength

27 Guidelines or Page: 27 o 69 The ormulation o the reduction coeicient reported below. c,red 1 c 1 K c where lat K c lat is the tensile strain and 0 is the compressive peak strain. However the reduction o the compressive strength should be limited in order to avoid excessive reduction that leads to a non-realistic response o the structure (see 2.3.1, ). min Compression-compression interaction is an important eature to model coninement eects. Although modeling this eect is necessary to ully understand the nonlinear behavior o concrete, ignoring coninement eects is a conservative assumption and thereore permitted. The equivalent length, related to the dimensions o the inite element, is crucial to reduce mesh size dependency (Bazant and Oh 1983; Crisield 1984; Rots 1988). User-assigned values or this parameter are usually inaccurate and increase the user and model actors o the simulation. A irst method is to assign a value based on the area or volume o the element (Rots 1988; Feenstra 1993), but this method will not be accurate in case o Compression-Compression Interaction Compression-compression interaction does not need to be modeled Equivalent Length The equivalent length, also known as the crack-band width, is an essential parameter in constitutive models that describe a sotening stress-strain relationship. An automatic procedure or determining the equivalent length, or crack-band width, should be used. The preerred method is a method based on the initial direction o the crack and the element dimensions. Alternatively, a method based on the area or volume o the

28 Guidelines or Page: 28 o 69 distorted elements and elements with a high aspect-ratio. An improved method has been proposed by Oliver (Oliver 1989) with improvements suggested by Govindjee et al. (Govindjee, Kay et al. 1995). inite element can be used. The equivalent length should be based on the element dimensions and the crack directions with respect to the element alignment (Oliver, 1989). It is advised to supplement this procedure with an additional orientation actor (Cervenka, 1995, Cervenka and Cervenka, 2010). For quadratic quadrilateral elements with a square shape (dimensions h x h) and with a crack direction along one o the diagonals this would lead to an estimated crack-band width o h eq = 2 h. For the same square elements with a crack direction along one o its edges this would simply lead to h eq = h. h a b h h eq = a h eq = b h eq = 2 h Figure 10 Examples o equivalent length based on element dimensions and crack direction For other shapes and or other crack directions other results will apply. It is

29 Guidelines or Page: 29 o 69 advised to make use o an automatic determination o h eq by the inite element program. I the inite element program does not have an option or a variable crack-band width determination depending on the crack orientation, the user should either choose or a conservative (i.e. large) estimation o h eq or check the used crack-band width a posteriori based on the obtained crack orientations and element alignment. For rectangular elements (dimensions a x b) with a crack direction along edge a this would lead to h eq = b. Note, that in smeared cracking, the ratio G /h eq determines the actual sotening. For obtaining conservative results, instead o increasing the h eq, reduction o racture energy G can be applied Model or Reinorcement Reinorcing steel exhibits an elasto-plastic behavior where the elastic limit is equal to the yield strength o the steel. The post-yield behavior is known as hardening that should be modeled according to the speciications o the reinorcing and pre-stressing steel. I no hardening speciications are available, a nominal hardening modulus, or instance E har = 0.02 E s, can be used. This will improve the stability o the analysis Model or steel bars An elasto-plastic material model with hardening should be used.

30 Guidelines or Page: 30 o 69 Figure 11 Stress-strain diagram or steel The stress-strain relationship is characterized by the deinition o the 0.1% proo stress, by the ultimate tensile strength and by the percentage total elongation at maximum orce, see Figure below Model or prestressing steel An elasto-plastic material model with hardening should be used to approximate the stress-strain relationship.

31 Guidelines or Page: 31 o 69 Figure 12 Stress-strain diagram or prestressing steel Model or Concrete-Reinorcement Interaction Concrete-reinorcement interaction is the main mechanism or stress redistribution ater cracking in concrete structures with bonded reinorcement. Although the mechanisms are governed at the micro- and meso-scale with rather complex inter-dependencies, which can only be properly modeled using dense inite element discretizations with dedicated constitutive models, the models at the macro-level can be simpliied signiicantly. Redistribution o stresses rom concrete to reinorcement ater cracking occurs is an essential load-carrying mechanism in reinorced and prestressed concrete. The behavior o a reinorced bar in tension is Tension-stiening The interaction eect o distributed cracking and stress-redistribution to the reinorcement need to be taken into account.

32 Guidelines or Page: 32 o 69 governed by the number o cracks that are present ater a stabilized crack patterns has developed. The number o cracks that can develop is dependent on dierent structural and material properties such as reinorcement ratio, reinorcement diameter, tensile strength, and such. Even ater a stabilized crack pattern has developed, the stiness o the reinorced tensile member is higher than the stiness o the reinorcement alone. This eect is oten reerred to as tension-stiening. A conservative assumption is to ignore the tension-stiening component and only account or the energy dissipated in the cracks that develop during the loading process. I the element size is smaller than the estimated average crack spacing, the tension-sotening model can be used, provided that the analysis leads to an realistic crack spacing. Otherwise, the amount o energy that can be dissipated within a inite element should be related to the average crack spacing and the size o the element. I the crack spacing is equal to s r,max and the equivalent length equal to h eq, then the amount o released energy is given by G n G RC F cr F where the number o cracks, n cr, is given by n cr h eq max 1, s r,max

33 Guidelines or Page: 33 o 69 The crack spacing is related to the (equivalent) reinorcement ratio and the (equivalent) diameter o the reinorcing bars. For instance, the Eurocode-2 provides guidelines or calculating the crack spacing or stabilized cracking, s, eq sr,max k3c k1k2k 4 s, e with c the cover o the main reinorcement, s, eq the (equivalent) diameter o the reinorcing bars, and ρ s,e the eective reinorcement ratio, ρ s,e = A s / A c,e. The parameters k 1 to k 4 are given in the table below. k 1 k 2 k 3 k or high-bond bars 1.6 or plain bars 0.5 or pure bending 1.0 or pure tension 3.4 (recommended value) (recommended value) The eective area o concrete in tension can be estimated using the provision in the Model Code 1990 (see Fig o the Model Code 1990, CEB-FIP, 1993).

34 Guidelines or Page: 34 o 69 (a) beam Figure 13 Eective area (b) slab For a beam, the eective concrete area is determined by A h b c, e c, e with b the width o the beam and h c,e the eective height, h c, e h x/ 3;2. h d min 5 The parameter x in this equation is the depth o the neutral axis. For a slab structure, the eective concrete area is calculated per unit width, with the eective height given by h c e h x/ 3;2.5c 2 min, The underlying assumption o the calculation o the crack spacing is that the crack direction and the reinorcement are approximately orthogonal. In case the cracks will develop under a signiicant angle with the reinorcement, or i an orthogonal reinorcement grid is used, the crack spacing should be calculated using the directional average

35 Guidelines or Page: 35 o 69 1 s cos sin s r s,max, y r,max, z where is the angle between the reinorcement along y direction and the principal tensile stress direction and s r,max, y, sr,max, z are the crack spacing calculated according to the Eurocode 2. For inite elements with dimensions much larger than the crack spacing, it is practical to assign an ultimate strain in the tension-sotening diagram that is equal to the yield strain o the reinorcement. Note that this can only be applied in an area equal to the eective concrete area around the main reinorcement. For other parts o the structure, a regular, racture energybased tension-sotening model should be used. Taking into account slip between reinorcement and concrete will result in more accurate results. The Model Code 2010 provides bond-slip relations. However, robust and easy-to-use models are not commonly available in commercial inite element codes. Instead, a perect-bond assumption is considered suicient. In that case special care should be taken when calculating the crack opening in the Serviceability Limit State veriication (see 4.1). Although taking into account dowel action will result in more accurate results, robust and easy-to-use models are not commonly available in commercial inite element codes Slip Slip between reinorcement and concrete can be modeled i an appropriate model is available Dowel Action Dowel action o reinorcement can be modeled i an appropriate model is available.

36 Guidelines or Page: 36 o Finite Element Discretization When using the Finite Element Method to perorm a numerical simulation o the behavior o a structure, the mechanical model o the structure needs to be divided in a number o elements. Various aspects are inluencing the quality o the results o the analysis and the most important aspects are the shape o the elements used; the degree o interpolation o the displacement ield; and the numerical integration scheme or the internal state since we tacitly assumed that the internal state is deined as a stressstrain relationship and not based on generalized orces and deormations Finite Elements or Concrete Linear elements will show locking behavior in certain cases. In most inite element programs these linear elements have been improved but quadratic elements are still better suited because they can described more deormation modes and are better capable o describing more complex ailure modes such as shear ailure. For analyzing beams the preerred element is an 8-node quadrilateral element or 2D simulations and a 20-node hexahedral element or 3D simulations. For analyzing slabs the preerred element is a 20-node hexahedral element. I necessary, quadratic triangular and quadratic Shape and Interpolation Elements with quadratic interpolation o the displacement ield should be used. Preerably a quadrilateral shape or a hexahedral shape should be used in 2D and 3D, respectively.

37 Guidelines or Page: 37 o 69 tetrahedral elements can be used in 2D and 3D, respectively. Quadratic triangle Quadratic quadrilateral Quadratic tetrahedral Quadratic hexahedron Figure 14 Preerred continuum elements

38 Guidelines or Page: 38 o 69 For large slab structures, modeling with continuum-based inite elements is not practical because o the large amounts o inite elements needed to accurately describe the stresses in the structure. Structural elements such as beam elements and (lat) shell elements can be used to model large-scale structures where it is not easible anymore to model with continuum elements. However, these types o structural elements are not capable to model shear ailure and additional post-analysis checks should be carried out to ascertain that a shear ailure mode is not overlooked. The preerred elements are also quadratic elements, such as 3-node beams in 2D and 3D, and 6-node triangular and 8-node quadrilateral shell elements or 2.5D analysis. Also, models with a combination o structural elements and continuum elements can be considered.

39 Guidelines or Page: 39 o 69 Quadratic 2D beam Quadratic 3D beam Quadratic triangular shell Quadratic quadrilateral shell Figure 15 Preerred structural elements Reduced-order integration or quadratic elements can lead to spurious modes when the stiness o the element becomes small due to extensive Numerical Integration Full integration should be used.

40 Guidelines or Page: 40 o 69 cracking (De Borst and Rots 1989). Continuum elements should be integrated with the integration rules given in the igure below. Quadratic triangle: 7-point Hammer Quadratic quadrilateral: 3x3-point Gauss Quadratic tetrahedral: 4-point Hammer Quadratic hexahedron: 3x3x3- point Gauss Figure 16 Sampling points or continuum elements Other integration rules that result in ull integration are also available but Gaussian integration rules or quadrilaterals and hexahedrons and Hammer integration rules or triangles and tetrahedral are most commonly used.

41 Guidelines or Page: 41 o 69 For structural elements integration schemes are used in case the elements are numerically integrated. The integration scheme is a combination o an integration rule along the axis o the beam or in the plane o the slab, and through the thickness. Quadratic 2D beam: 3-point Gauss along the axis and 7-point Simpson through depth Quadratic 3D beam: 3-point Gauss along the axis and 7-point Simpson through depth and thickness Quadratic triangular shell: 7-point Hammer in-plane and 7-point Simpson through depth Quadratic quadrilateral shell: 3x3- point Gauss in-plane and 7-point Simpson through depth Figure 17 Sampling points or structural elements

42 Guidelines or Page: 42 o 69 The integration rule along the beam axis or in the plane o the slab should result in ull integration, or instance 3-point Gauss or a quadratic beam element. The through-depth integration rule should be capable o capturing a gradual stiness reduction due to cracking and crushing. In general, a 7- point Simpson rule is mostly suicient but an 11-point Simpson rule is necessary in certain cases and recommended in case o doubt. Embedded reinorcement has the advantage over explicitly modeling reinorcement with truss elements o overlay elements that the connectivity o the concrete elements does not have to be altered to model the reinorcement layout. Using overlay elements to model grid reinorcement has the disadvantage that shear stiness will be present while this term is usually ignored in embedded grid reinorcement. In most commercial inite element codes the use o embedded reinorcements entails that slip between reinorcement and concrete is ignored (see ). In modern embedded bond-slip models the advantages o embedded reinorcements and interace models are combined, such that slip can be modeled explicitly. The interpolation o the displacement degree o reedom o the reinorcement should be compatible with the element in which the reinorcement is embedded Finite Elements or Reinorcement Embedded reinorcement elements are preerred; both embedded bars and grids can be used Shape and Interpolation The same order o interpolation as the concrete elements should be used.

43 Guidelines or Page: 43 o 69 The reinorcement can be integrated with a reduced integration scheme since the reinorcement will not exhibit spurious modes since these are inhibited by the embedding element Numerical Integration Full or reduced integration can be used. The inite element discretization has a proound eect on the accuracy o a nonlinear inite element simulation. The shape o the generated inite elements can usually be checked by the program using various metrics such as aspect ratio, skewness, area over perimeter ratio, and such. These metrics should be used as much as possible to create a inite element discretization that has a limited number o distorted elements. Comparisons o results with dierent discretizations might provide additional conidence Meshing Algorithm The inite element mesh has to be generated using an algorithm that produces regular meshes with less than 5% o distorted elements. The minimum element size is usually determined by practical considerations. The computational time increases approximately quadratic with the number o elements and the number o elements should be limited in order to reduce the elapsed time or inishing the simulation Minimum Element Size There is no minimum element size requirement.

44 Guidelines or Page: 44 o 69 For sotening materials, the post-peak response can show a snap-back behavior when the equivalent length is too large. Since the equivalent length is related to the element size, the maximum element size is given by the initial slope o the post-peak stress-strain relationship. For exponential sotening, the initial post-peak slope is given by Maximum Element Size The element size is limited to ensure that the constitutive model does not exhibit a ''snap-back'' in the stress-strain relationship. cr t t exp cr cr 0 u u cr 0 u which should be larger than the Young's modulus, E. With u GF heq t, the equivalent length should be smaller than h eq EG F 2 t The maximum element edge length should be approximately hal o the maximum equivalent length. The maximum element size is also limited by the inherent inaccuracy o the inite element method. I the inite element discretization is too coarse, the stress ield will show considerable jumps rom one element to another since the stress ield is not continuous. As a guideline, or reinorced concrete members with standard reinorcement layouts, the element size should be less than the values in the table below. The maximum element size in the model should be chosen such that relatively smooth stress ields can be calculated.

45 Guidelines or Page: 45 o 69 Beam Structure Maximum element size 2D modeling l h min, D modeling l h b min,, Slab Structure Maximum element size 2D Modeling l b min, D Modeling l b h min,, where h the depth, l the span, and b the width, see Figure 13 on page 34. In other words: or 2D modeling o beams at least 6 elements over the height should be used. For beams or slabs with openings or other discontinuities more elements should be considered. Short-term prestress losses due to wobble, riction, and anchor retraction have to be taking into account. Long-term prestressing levels also change due to relaxation, shrinkage, and creep o the structure. The actual level o prestressing should be assessed as accurately as possible. I no data is available, the design prestressing level should be reduced to 70% or SLS and ULS simulations. For simulation o construction stages, the prestressing 2.6 Prestressing Prestressing should be applied taking into account prestress losses.

46 Guidelines or Page: 46 o 69 levels should be increased to 110%. 2.7 Existing Cracks Existing cracks basically reduce the stiness in a local region o the structure. This can be modeled using a reduced tensile strength, reduced Young's modulus and reduced racture energy. Since the amount o reduction is diicult to assess, the existing crack pattern should be recreated using multiple load cases that lead to the observed pattern. Alternatively the cause o existing cracks is modeled explicitly. Possible causes include restrained volume changes or dierential support settlement. Dead weight and permanent loads should be modeled as a separate, initial load case. Including dead weight loading leads to a non-uniorm stress ield in general, which is beneicial in nonlinear analysis because constant-stress zones exhibit multiple localizations which are mostly spurious since only a small number o cracks will localize. The traic load is modeled using a predeined wheel coniguration that is applied to the structure. The wheel coniguration has to be in the most unavorable position, considering all relevant ailure modes o each structural part. Existing cracks in the structure should be taken into account whenever detailed inormation about location and crack widths is available. 2.8 Loads Loads on new structures should be applied according to the speciications in the Eurocode, the National Appendices or the RWS ROK-1.3 (Richtlijnen Ontwerpen Kunstwerken). For existing structures the Eurocode, the National 8700 serie or the RWS RBK 1.1 (Richtlijnen Beoordeling Kunstwerken) should be applied. Loads that should be taken into account, but are not limited to: 1. Dead weight and prestressing. 2. Permanent loads, such as asphalt, barriers and railings. 3. Traic loads, both distributed and combinations o axle loads (per lane). 4. Temperature loads

47 Guidelines or Page: 47 o 69 Temperature loads need to be applied in combination with all other load cases to ind the most conservative case. In general, a temperature gradient over the depth o the structure has to be modeled to account or daily temperature dierences, as well as a constant temperature dierence to account or annual temperature dierences. In certain cases, a concentrated load can be replaced by an equivalent displacement. This method is oten reerred to as displacement control and is oten more stable than load control where the orce is applied. However, displacement control restricts the displacement o a point to a prescribed value and is oten not suitable or structures with a multiple o loads and/or distributed loads such as dead weight loading. Displacement controlled analysis, albeit more stable than orce control, should be considered more research-oriented. 2.9 Boundary Conditions Boundary conditions are considered in this document the restraints on the displacements at certain points o the structure. They e.g. can represent the supports o a structure, or the load plate in an experiment. In case o structural symmetry and a symmetrical loading pattern, the inite element model can be reduced. Loads and supports are usually applied using load and support plates. These Support and load plates Unless the objective o the analysis is to study the detailed behavior o the

48 Guidelines or Page: 48 o 69 structural components can be included in the inite element model but special attention is needed since spurious high stress concentrations can occur due to the inite element discretization. These high stress concentrations can result in premature, numerical ailure that is not present in the real structure. To avoid stress concentrations due to loading, the load can be replaced by a distributed load over the area o the load plate. This approach assumes that the load plate is highly compliant; or instance a rubber block. Alternatively, a no-tension/no-riction interace could be used between the plate and the concrete, thus reducing local stress concentrations. loading and support points, the support and load plates should be modeled such that local stress concentrations are reduced. I the objective o the analysis is to study the behavior o the loading and/or support in detail, then the relevant part o the structure should be modeled and analyzed in detail. In case o a symmetrical structure with symmetrical loading, it could be decided to model only hal or a quarter o total the structure by applying the proper symmetry boundary conditions. Although this can reduce the computational costs, applying symmetry inherently assumes that the ailure mode is symmetric which is not correct in most cases Symmetry Using the symmetry o the structure and the loading should be used with care.

49 Guidelines or Page: 49 o 69 3 ANALYSIS A clear loading sequence plan should be motivated. This plan could include several loading sequences to be applied on the same inite element model. The loading sequence plan should ollow or instance the Eurocode 2, that considers dierent loading combinations or the Ultimate Limit State and the Serviceability Limit State veriications. An example o such an loading sequence plan, including one load combination o actions o a bridge in the Netherlands, looks like: 3.1 Loading Sequence The loading sequence should always contain initial load steps where dead weight, permanent loads, and, i appropriate, prestressing are applied to the structure. The loading sequence will depend on the limit state and on combinations o actions to be considered. Load Load Factor Remark step Dead weight & prestress 2 Permanent 1.0 Each load step can be divided in substeps, load increments, Concentrated according to the adopted variable Q k coeicients or the Distributed combination, requent, quasipermanent value o variable variable q ik 4 Permanent 0.15 action. Concentrated The occurrence o cracking and variable Q k convergence issues might also Distributed inluence the increments. variable q ik 7 Concentrated 0.1

50 Guidelines or Page: 50 o variable Q k Distributed variable q ik Concentrated variable Q k Distributed variable q ik Notes: The load actors up to step 8 are based on the Dutch code. The total load actor ater step 9 and 10 or the variable loads is which is the product o 1.25 and For actor 1.27 see section The load increment that would lead to the irst crack can easily be determined with a linear-static analysis. Subsequent load increments should be determined using an automated procedure such as the method based on the number o iterations o the previous step(s), the method based on external work, or any other method that takes into account the changing stiness in the structure. 3.2 Load Incrementation The load or which the ailure mechanism is studied should be applied incrementally with increments that are approximately 0.5 times the load increment that would lead to the irst crack. The load incrementation can be done manually but the preerred method is to apply a load incrementation method based on a measure o nonlinearity. A nonlinear analysis will, in general, result in an unbalance orce between the internal or restoring orces and the external orces (loads). Using an iterative procedure, the unbalance orce will be cancelled out and the 3.3 Equilibrium Iteration Equilibrium between internal and external orces should be achieved iteratively using a Newton-Raphson method with an arc-length procedure.