Parameterised Finite Element Modelling of RC Beam - Simulia

2006 ABAQUS Users’ Conference 95

Parameterised Finite Element Modelling of RC Beam Shear Failure

V. Birtel, P. Mark

Ruhr-University Bochum, Universitätsstr. 150, 44780 Bochum, Germany Institute for Reinforced and Prestressed Concrete Structures

Abstract: A spatial finite element model of reinforced concrete (RC) beams with rectangular cross sections, typical side aligned stirrups and distributed or edge concentrated longitudinal reinforcement is presented. It is parameterised in its properties of geometry, material, discretisation and loads in biaxial directions. The concrete volume is discretisised into 8 or 20-node solid elements. Truss elements discretely model each single reinforcement bar. They are coupled to the concrete elements using the ″embedded modelling″ technique. The ″concrete damage plasticity″ model of ABAQUS is used to describe the nonlinear material behaviour of concrete. Suitable material functions and material parameters are derived and verified to experimental data of (cyclic) uniaxial, biaxial or triaxial stress tests. Energy criteria and internal length parameters ensure almost mesh independent results of the simulations. An elasto-plastic material model with a gradually rising plastic branch is adopted for the reinforcing steel. The parametric model is verified to experimental data of uniaxial shear tests taken from the literature. Afterwards, it is used to establish a data base of biaxial shear resistances to check developed biaxial shear design formulas that base on simple strut and tie models. More than 100 simulations guarantee an extended and reliable verification that experiments – almost none of them are available in the literature – are not able to give. Moreover, the arrangement of the stirrups is optimised in dependence upon the distribution of the longitudinal reinforcement to minimise reinforcement amounts and increase bearing capacities.

Keywords: Reinforced concrete, shear failure, parameterisation, embedded modelling

1. Introduction

The shear design of reinforced concrete beams bases on strut and tie models, first developed by (Ritter, 1899) and (Mörsch, 1927), respectively. They realised that the complex inner states of stresses can be idealised by tensile and compressive struts. Today, almost every design code – e. g. Eurocode 2 – uses this basic principle. However, loads are assumed to act along one principle axis of the cross section. There are no design rules for biaxial loadings with inclined resultant shear forces that do not comply with vertical or horizontal directions.

Formulas for the resistances of the tensile and the compressive shear struts of beams under biaxial shear forces have been developed by the coauthor (Mark, 2004; Mark 2005). They are valid for beams with rectangular cross sections, typical side aligned stirrups, normal strength concrete and arbitrary distributions of the longitudinal reinforcement. Moreover, there is no limitation for an

96 2006 ABAQUS Users’ Conference

inclination of the shear force. Thus, the verification of the formulas has to cover extended parameter variations. It is performed numerically with a special finite element model.

2. Parametric finite element model

It is obvious that almost every variation in geometry, material properties as well as load or reinforcement arrangement requires a complete revise of a finite element model. To avoid such efforts, it is convenient to generate different similar models from just one parametric input file. Then, the user has to specify only a few variable parameters (Table 1). Even the mesh generation is included in the parameterisation.

Table 1. Model parameter.

The spatial finite element model idealises three-point-bending tests of RC beams under biaxial loadings (Figure 2). Its boundary conditions and force applications are predefined and the symmetries of geometry and load are utilised to halve the model structure and thus save computing times.

Figure 1. Notation, generalised treatment of biaxial shear forces.


The two shear force components Vy and Vz are related to the aspect ratio of the cross section to obtain a shear force inclination αV that is free of dimensions (Figure 1).

bh

VV

z

yV =α

(1)

Vertical forces Vres = Vz a referred to by αV = 0, while a diagonally directed shear force Vres yields αV = 1. It is indispensable to hold 0 ≤ αV ≤ 1, otherwise the notations of Vy, Vz, h and b have to be exchanged.

Figure 2. RC structure and FE model with parameters.

″Compression only″ springs assure a numerically stable changeover of forces from the solid beam to the steel supports. They also exclude unrealistic tensile bearing reactions. The concrete body is discretised in nb x nh elements in the section plane and nl +8 (4) elements in the longitudinal direction. They represent the ″host elements″ in the applied concept of ″embedded elements″. 8-node C3D8 or 20-node C3D20 solid elements with linear or quadratic interpolation functions are chosen from the ABAQUS element library. T3D2 truss elements – lying embedded in the concrete


volume –model stirrups and longitudinal bars. Their coordinates are calculated in the global reference system regarding concrete cover, rebar diameters as well as stirrup and bar arrangements. No bond slip is assumed between reinforcement and concrete. The elongation lb of the concrete body behind the supports just operates as an anchorage zone for the longitudinal bars similar to comparable anchorage zones in experiments.

The nonlinear set of equations is solved with the modified ″Static Riks″ arc-length method (ABAQUS, 2003). Here, the method often achieved its best effectiveness with a limitation of the arc-length increment to ∆lmin = 10-10 and no upper threshold value for ∆lmax.

3. Material models

The complex, nonlinear material behaviour of concrete is described by the elasto-plastic damage model ″concrete damaged plasticity″ (ABAQUS, 2003) that was developed by (Lubliner et al., 1989) and elaborated by (Lee & Fenves,1998). It uses a yield surface F in the space of effective stresses σ of combined Drucker-Prager and Rankine type and assumes isotropic damage d as well as non-associated flow. Its basic equations read:

{ }

σσε

εεσhε

σεσσεσσεεDσ

∂∂

=

=

−=≤∈−=

)(ˆ)~,(~

))~,(1(,0)~,()(0

G

dF

pl

plplpl

plplpl

λ&&

&&

(2)

A bilinear relation models the uniaxial stress-strain behaviour of the reinforcing steel. It is characterised by the modulus of elasticity Es, the yield strength fy and the gradual rise of the second branch (Es1).

4. Material functions and parameters

The ″concrete damaged plasticity″ model requires the following material functions:

stress-strain relations for the uniaxial behaviours under compressive as well as tensile loadings including cyclic un- and reloading,

functions for the evolutions of the damage variables dc and dt under compressive and tensile loadings, respectively.

4.1 Uniaxial loading conditions

The stress-strain behaviour under sustained compressive loading is modelled in three phases (Figure 3).

ccc E εσ =)1( (3)


cm

c

c

cm

cci

cccm

cci

c f

fE

fE

1

1

21

)2(

)2(1

)/(

εεε

εεε

σ−+

−=

1

1

21

)3( 222

−

⎟⎟⎠

⎞⎜⎜⎝

⎛+−

+=

c

cccc

cm

ccmcc f

fεεγεγεγσ

(4)

(5)

The first two sections describe the ascending branch up to the peak load fcm at εc1. Their formulations are similar to the recommendations of the Model Code (CEB-FIB, 1993). The third and descending branch takes account for its dependency on the specimen geometry (Vonk,1993; Van Mier, 1984) to ensure almost mesh independent simulation results. Thus, σc(3) incorporates within the descent function γc the constant crushing energy Gcl (Krätzig & Pölling, 2004) as a material property in addition to an internal length parameter lc derived from the grid structure of the element mesh.

fcm

0,4 fcm

σc

εc

Ec

1−ccEσin

cε

plcε

elcε

)1( cc dE −incc

plc b εε =

εc1

(1) (2) (3)

loading path

0 42 6 80

10

20

[MPa]σc

εc [‰]

Sinha, Gerstle& Tulin (1964)

modelbc = 0,7

bc = 0,3

Figure 3. Stress-strain relation for (cyclic) compressive loading, experiments

acc. (Sinha et al., 1964).

The evolution of the compressive damage component dc is linked to the corresponding plastic strain εc

pl which is determined proportional to the inelastic strain εcin = εc - σcEc

-1 using a constant factor bc with 0 < bc ≤ 1.

1

1

)1/1(1 −

−

+−−=

cccpl

c

ccc Eb

Edσε

σ

(6)

A value bc = 0,7 fits well with experimental data of cyclic tests (Figure 3, right). So, most of the inelastic compressive strain maintains after unloading. Generally, unloading and subsequent reloading up to the monotonic path occur linearly with no hysteretic loops.


The stress-strain relation σt(εt) for tensile loading consists of a linear part up the strength fct and a nonlinearly descending part that depends on the specimen geometry (Figure 4). The latter function is derived from the stress-crack opening relation (Hordijk,1992)

[ ] 22

)1()/(1)( 31

31

c

c

wwc

cct

t ecwwewwc

fw

c −−

+−+=σ c1 = 3, c2 = 6,93

(7)

using the principle of the ″Fictitious Crack Model″ (Hillerborg, 1983). Thus, a product of the inelastic strain and an internal length parameter lt replaces the crack opening w to yield σt = σt(w = ltεt

in = lt(εt - σtEc-1)) and w is smeared over the average element length lt = Ve

⅓. As intended, σt(εt) then encloses the ratio of fracture energy GF and lt (Bazant & Oh,1983).

σt(w) acc. Hordijk

MC 90

w [µm]0 40 80 120

0

½

1 ctt f/σ

(1992)

experimental dataReinhardt, Cornelissen

(1984)

wc = 180µmdmax = 16mm≈ C30/37

0 4,82,4

fctm = 2,56 MPa, lt = 25 mmwc = 180 µm, dmax = 16 mm

2

1

0

[MPa]σt

εt [‰]

Reinhardt, Cornelissen (1984)

σt(εt) bt = 0,1

Figure 4. Stress-crack opening and stress-strain relations for (cyclic) tensile loading, experiments acc. (Reinhardt & Cornelissen, 1984).

Similar to (6) the damage dt depends on εtpl and an experimentally determined parameter bt = 0,1

(Figure 4, right). So, unloading is assumed to return almost back to the origin and to leave only a small residual strain.

1

1

)1/1(1 −

−

+−−=

cttpl

t

ctt Eb

Edσε

σ

(8)

4.2 Crack closure and stiffness recovery

Damage d isotropically reduces the initial elastic stiffness parameters in D0 to gain D = (1-d)D0. It arises from damage partitions associated with tensile and compressive loadings (dt and dc), respectively.

)1)(1(1 cttc dsdsd −−=−

)ˆ( , )ˆ(½1 σσ rsrs ct =−=

(9)


Stiffness recovery or stiffness loss effects due to crack closure or crack reopening are derived from uniaxial loading scenarios and the corresponding crack pattern (Figure 5). They are introduced via the functions st and sc in the following way:

If the load changes from tension to compression, cracks nucleated in tension close and stiffness fully recovers (Figure 5, left). Thus, dt is not transferred to the compressive side, as sc = 0 if σ < 0.

On the contrary, load changes from compression to tension close cracks perpendicular to the load and keep parallel cracks open (Figure 5, right). Consequently, half of dc is transferred to the tensile side, because st = 1 - ½ = ½ if σ > 0.

σt

σc

µ→ 0

tensile test compressive test

steel bars fordeformationcontrolled testing

almost no lateral restraint at the

loading platens Figure 5. Typical crack pattern (Van Mier, 1992) of uniaxial tensile and

compressive tests.

0

1

2

3

4

-0,5 0 0,5 1 1,5 2 2,5 3

(2,9 | 4,1)

(4,7 | 5,5)

(5,8 | 10,2)

experimentsimulationq/fc

p/fc

experimental data acc.

Kupfer (1973), Linse & Aschl (1976), van Mier (1984) Schickert & Winkler (1977), Hampel & Curbach (2000) Mills & Zimmermann (1970), Scholz et al. (1992)

Figure 6. Comparisons of bi- and triaxial strength results to experimental data.


4.3 Multiaxial loading conditions

Calculated uniaxial, biaxial and triaxial strength results of normal strength concrete agree well with experimental data taken from the literature (Kupfer & Gerstle,1973; Linse & Aschl,1976; Van Mier,1984) (Figure 6). This holds true for tensile, compressive as well as combinations of tensile and compressive loadings with low confining pressures. If high confinements occur, resistances are often overestimated by the material model (cp. (ABAQUS, 2003)). The three examples in Figure 6 indicate that these overestimations can have pronounced extents. But the ratios of hydrostatic pressures p and the von Mises equivalent stresses q still fit well to the experimental ones.

4.4 Material parameters

Table 2 summarises the material parameters of the concrete model. It distinguishes between parameters related to the uniaxial and to the multiaxial behaviour. Strength and stiffness parameters are taken from European Standards. But of course, this choice is variable.

Table 2. Material parameters, * uniaxial loading, ** multiaxial loading. Parameter Denotation

Ec = Ecm, fcm, fctm bc = 0.7, bt = 0.1 GF = 0,195wc fctm , Gcl = 15 kN/m wc = 180 µm, εc1 = -2,2‰

acc. Eurocode 2 (ENV 1992-1-1, 1992) damage parameters, (0 < bc, bt ≤ 1) (Krätzig & Pölling, 2004) fracture and crushing energies (Vonk, 1993) max. crack opening (Hordijk, 1992), strain at fcm (CEB-FIB, 1993)

*

ν = 0,2 ψ = 30° αf = 1,16 (→ α = 0,12) ε = 0,1 Kc = ⅔ (→ γ = 3)

Poisson’s ratio dilation angle (Lee & Fenves, 1998) ratio biaxial to uniaxial compressive strength (Kupfer & Gerstle, 1973) parameter of the flow potential G second stress invariant ratio

**

The following parameters are adopted for the reinforcing steel of stirrups and longitudinal bars: Es = 200.000, Es1 = 1111, fy = 500 (all values given in [MPa]).

5. Verification of the FE model

The parametric model is verified to the data of two well documented uniaxial shear experiments (αV = 0), both currently carried out with a three-point-bending test set-up, rectangular cross sections and normal strength concretes. The first was accomplished by the coauthor (Mark, 2004), the second by (Toongoenthong & Maekawa, 2005).

Figure 7 shows specimen details and reinforcement arrangements of the first test by (Mark, 2004). All dimensions are given in [mm]. The girder spans over leff = 3000 mm. For the sake of simplicity, standard strength values according to Eurocode 2 were adopted for concrete and steel during the numerical simulations, although the actual test values slightly differed: compressive concrete strength 38 MPa (test 35,8 MPa), steel yield strength 500 MPa (test 570 MPa).


Figure 7. Load- displacement response and specimen details, experimental data

acc. (Mark, 2004).

The global load-displacement relation of the test is well met by the simulations, as the element grids are sufficiently fine. This applies to both isoparametric solid element types with linear or quadratic interpolation functions. Thus, the linear element is selected for further investigations to limit computational efforts that rapidly increase, when using the 20-node solid. Initially, the stiffness of the girder is overestimated. However, the deviations of about 5% in the peak loads are mainly due to the assumed reduced yield strength of stirrups and longitudinal bars. They almost vanish, if fy is corrected to its actual experimental value.

Figure 8 shows similar simulation results for the recalculation of the test data of (Toongoenthong & Maekawa, 2005), where leff = 2000 mm. Peak load and even the descending branch acceptably agree with the experimental ones, if the element mesh does not get too course. The overestimation of initial stiffness properties remains.

Figure 8. Load-displacement response and specimen details, experimental data

acc. (Toongoenthong & Maekawa, 2005).

Figure 9 proofs that the numerical model is also capable to describe local test results like cracking, the formation of shear struts or the redistribution of forces onto the vertical stirrup legs, at least in


an indirect way. Hence, experimentally derived crack pattern at the lateral surfaces of the concrete body are displayed together with the distributions of concrete and stirrup stresses, both calculated for the peak load. Compressive concrete struts nucleate arch-like or strait inclined with an average angle of about 40° to the girder axis (white areas of the stress distributions, broken lines in the truss model), as concrete cracks in tension. They lie almost parallel to the surface cracks. Stirrups take over vertical tensile stresses and pronounced deflections as well as redistributions of inner forces take place. As expected, stirrups yield localised, just where cracks cross.

Figure 9. Experimental crack pattern and calc. distribution of concrete stresses (top), stirrup stresses and experimental crack pattern, truss mechanism (bottom).

Generally, the FE model is appropriate to describe the shear failure mechanism of RC beams close to reality. This includes the typical stirrup yielding, the branched cracking and the redistributions of tensile stresses to stirrups and longitudinal bars. Furthermore, peak loads (or shear resistances) only slightly differ from experimental ones. So, those numerically determined resistances are well suited to reliably verify practical design formulas for shear resistances of RC beams, even in cases of biaxial loadings.

6. Biaxial shear

For the verification of the biaxial shear design formulas more than 100 single simulations are performed with the parametric model varying the basic parameters in ranges that typically occur in practical applications. The variations cover concrete properties (24 ≤ fcm ≤ 58 MPa, normal strength concrete), the aspect ratio (1 ≤ h/b ≤ 2,5), the shear load inclination (0 ≤ αV ≤ 1), the mechanical reinforcement ratios of stirrups and longitudinal bars as well as different typical edge concentrated or side aligned distributions of the longitudinal reinforcement. Of course, usual detailing rules (ENV 1992-1-1, 1992) like minimum distances of stirrups and bars, minimum


concrete covers or anchorage lengths are fulfilled. High strength concrete is not considered by now.

Generally, failure arises from stirrup yielding, so it is the tensile strut that fails, just as it is expected from uniaxial shear failures in cases of rectangular cross sections. The shear failure is checked for all single calculations to reject bending failures that of course are not requested here. Even if the preliminary dimensioning of the longitudinal reinforcement should prevent such unfavourable failure modes, they sometimes occur, especially if the reinforcement amounts of the stirrups are small.

The shear resistances Vsim are extracted from the maximum forces Fmax = 2Vsim of the simulations and compared to the ultimate shear resistance VR3 (design formula) of the tensile strut (Figure 10). Experimental ratios Vexp/VR3 of several uniaxial and two biaxial shear tests (Mark, 2004) are added. Effects of the load inclination are removed by the mechanical reinforcement ratio ωw2 that adjusts the ratios Vsim/VR3 and Vexp/VR3 on the level of uniaxial shear and thus allows comparisons of uniaxial and biaxial results.

Figure 10. Comparisons of biaxial shear resistances to numerical and experimental data.

Numerically and experimentally determined ratios fall into the same scatter range. So the design formulas describe uniaxial and biaxial shear resistances with almost the same accuracy and the well known decreasing relationship (Reineck, 2001) between Vsim/exp/VR3 and ωw – governed by the ″concrete partition″ Vc for small ωw – emerges. Furthermore, most of the ratios – especially the ones derived from biaxial loadings – exceed unity, so they conservatively underestimate actual resistances.


Biaxial shear exhibits its own specific characteristics. Typical features are the formation of complex, three dimensional distributions of concrete stresses with stiffening effects in the stirrup’s corners or a more localised and side concentrated yielding of stirrup legs. An additional one is illustrated in Figure 11: As compressive shear struts rest on the longitudinal bars under tension, they bend aside those bars and thus the stirrup legs, if the bars are located at the section sides and not at the corners of the stirrups. Additional, horizontally oriented stirrups help to improve the situation. They reduce lateral deformations and increase bearing capacities.

Figure 11. Effect of additional stirrups in cases of distributed longitudinal bars.

7. Conclusions

Spatial finite element models with concrete solids and embedded truss elements – modelling stirrups and longitudinal bars – are very suitable to numerically simulate the load bearing behaviour of RC beams. On the one hand, they reliably evaluate global parameters like ultimate loads or deformations close to reality. This even holds true for complex loading conditions like biaxial ones. On the other hand, they allow extended variations of basic parameters that experiments – due to their demand on time and costs – are not able to give. Parametric input files reduce the user’s effort for such variations to a minimum.

Moreover, simulations open the view to the inside of RC girders. Here, the discrete modelling of stirrups and longitudinal bars is especially appropriate, as then their results can be separately


displayed and monitored. Stresses and strains become visual at all times of the simulation and at all positions in the beams to properly understand the inner, nonlinear load bearing mechanisms governed by cracking and pronounced load redistributions.

8. References

1. ABAQUS Theory Manual, “Version 6.4,” ABAQUS Inc., USA, 2003. 2. Bazant, Z. P., and B. H. Oh, “Crack band theory for fracture of concrete,” Matériaux et

Constructions 16 (93), pp. 155-177, 1983. 3. CEB-FIP, “Model Code 1990,” Thomas Telford, London, 1993. 4. ENV 1992-1-1,Eurocode 2, “Design of concrete structures – part1 general rules and rules for

buildings,” 1992. 5. Hillerborg, A., “Analysis of one single crack,” Fracture mechanics of concrete, ed. by

Wittmann, and F. H. Elsevier, pp. 223-249, Amsterdam, 1983. 6. Hordijk, D. A., “Tensile and tensile fatigue behaviour of concrete; experiments, modeling and

analyses,” Heron 37(1), pp. 3-79, 1992. 7. Krätzig, W. B., and R. Pölling, ”An elasto-plastic damage model for reinforced concrete with

minimum number of material parameters,” Computer and Structures 82, pp. 1201-1215, 2004. 8. Kupfer, H. B., and K. H. Gerstle, “Behaviour of concrete under biaxial stresses,” Journal of

Engineering Mechanics Division 99 (EM4), pp. 853-866, 1973. 9. Lee, J., and G. L. Fenves, “Plastic-damage model for cyclic loading of concrete structures,”

J.Eng.Mechanics 124(8), pp. 892-900, 1998. 10. Linse, D., and H. Aschl,“ Versuche zum Verhalten von Beton unter mehrachsiger

Beanspruchung,“ test report, TU München, 1976 11. Lubliner, J., J. Oliver, S. Oller, and E. Onate, “A plastic-damage model for concrete,”

Int.J.Solids Structures 25(3), pp. 299-326, 1989. 12. Mark, P., “Design of reinforced concrete beams with rectangular cross sections against biaxial

shear forces” Beton- und Stahlbetonbau 100 (5), pp. 370-375, 2005. 13. Mark, P., “Reinforced Concrete Beams subject to biaxial Shear Forces: strut-and-tie models,

experiment and design approach,” Beton- und Stahlbetonbau 99 (9), pp. 744-753, 2004. 14. Mörsch, E., “Die Schubsicherung der Eisenbetonbalken,” Beton und Eisen 26 (2), pp. 27-35,

1927. 15. Reineck K.-H., “Hintergründe zur Querkraftbemessung in DIN 1045-1 für Bauteile aus

Konstruktionsbeton mit Querkraftbewehrung,” Bauingenieur 76, pp. 168-179, 2001. 16. Reinhardt, H. W., and H. A. W. Cornelissen, “Post-peak cyclic behaviour of concrete in

uniaxial tensile and alternating tensile and compressive loading,” Cement and Concrete Research 14 (2), pp. 263-270, 1984.

17. Ritter, W, “ Die Bauweise der Hennebique,” Schweizerische Bauzeitung 17, pp. 41-43, 59-61, 1899.


18. Sinha, B. P., K. H. Gerstle, and L.G. Tulin, “Stress-strain relations for concrete under cyclic loading,” Journal of the ACI 61 (2), pp. 195-211, 1964.

19. Toongoenthong, K., and K. Maekawa, “Computational Performance Assessment of Damaged RC Members with Fractured Stirrups,” Journal of Advanced Concrete Technology 3 (1), pp. 123-136, 2005.

20. Van Mier, J. G. M., “Strain-softening of concrete under multiaxial loading conditions,” PhD-thesis, Techn. Univ. Eindhoven, 1984.

21. Van Mier, J. G. M., “Assesment of strain softening curves for concrete,” Lecture Notes, TU Delft, 1992.

22. Vonk, R. A., “A micromechanical investigation of softening of concrete loaded in compression,” Heron 38 (3), pp. 3-94, 1993.

9. Acknowledgment

The authors thank the German Research Foundation (Deutsche Forschungs-Gemeinschaft DFG, http://www.dfg.de) for the financial support of the project "Experimental and numerical investigations of reinforced concrete girders under biaxial shear forces".

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Parameterised Finite Element Modelling of RC Beam - Simulia