Numerical Analysis of Punching Shear Failure ofReinforced Concrete Slabs Karsten Winkler and Friedhelm Stangenberg Ruhr-University Bochum, Universitätsstr. 150, 44780 Bochum, Germany Institute for Reinforced and Prestressed Concrete Structures Abstract: Nearly no load bearing behaviour of reinforced concrete members allows such varied interpretations and complex discussions as the shear behaviour. Especially the three-dimensional problem of the punching shear failure of reinforced concrete members is internationally discussed. Nevertheless up to now, there is no unified design approach or even an overall accepted design model. Especially for large structural members, as they are commonly used in industrial structures and high-rise structures, the experimental background is missing. Because of extensive costs and very high test loads numerical solutions and FE-simulations are indispensable. By using Abaqus, finite-element simulations are currently performed at the institute for reinforced and prestressed concrete structures at the Ruhr-University Bochum. For the main investigation of the punching shear problem, the general conditions as well as the applicability of the constitutive law and the discretization have to be investigated, proved and verified first. The main goals of these preliminary investigations are the determination of influences of different element types, the discretization of the reinforcement and the size effects of the used material model. The concrete is modelled by 8-nodes or 20-nodes solid continuum elements. Regarding the concrete material behaviour, a non-linear user-defined material model based on the concrete damaged plasticity model is used. For the parametric study, small and medium-sized slabs are simulated for comparison and verification with experimental data to ensure realistic results of the large-sized structures. The article will report the latest results of these simulations and the special problems of simulating the punching shear failure of reinforced concrete slabs. Keywords: Concrete, Constitutive Model, Damage, Experimental Verification, Failure, Fracture. 1. Introduction The actual German and European code provisions for the design and construction of reinforced concrete slabs against punching shear failure base on semi-empirically and statistically developed formulations. Although there are different models and approaches available that try to describe and explain the punching shear behaviour and failure, even nowadays formulations and design parameters for punching have to be found by experimental analysis. In this context the empirical background is generally derived from small-scale tests. Usually, experimental data for punching shear failure can be found for slab thicknesses smaller than around 30.0 cm (CEB-FIP, 1993). 2008 Abaqus Users’ Conference 1 as they can be found in shear and punching shear failure. This means.Especially industrial structures and high-rise buildings require extraordinary large concrete members. On the other hand. Up to now. finite-element simulations are currently performed with Abaqus. the discretization of the reinforcement and the size effects reflected by the used material model. are influenced by a pronounced size effect. The main goals of these investigations are the determination of influences of different element types. The validation of the punching shear failure loads of thick slabs by testing down-scaled experimental slabs is according to the disregard of the size effect not admissible. The currently existing tests satisfy neither geometrical similarity nor similarity to the material characteristics. former research of size effects in punching shear failure by (Bazant. the applicability as well as the general conditions of the constitutive law and the fineness of the mesh have to be investigated. For the main investigation of the punching shear problem. These large reinforced concrete members are mostly needed as footings for heavily-loaded single-columns or for transferring enormous big single loads. wide-spanned loading and test fields. 2007). the requirements and conditions for testing large slabs with high depths result in very high test loads. Especially brittle failure modes. 1987) resulted in a formulated size-effect law. that with an increasing effective depth of the concrete member. no generally admitted conclusions or predictions can be derived from the existing experimental results. 2 2007 ABAQUS Users’ Conference . analysed and validated for large concrete members. to cause a pure and undisturbed punching shear failure. Therefore these experiments require tremendous. such as foundations and individual footings with very large effective depths up to 7. proved and verified. what makes numerical solutions and FE-simulations indispensable. Regarding this type of reinforced concrete members. the actual code provisions and parameters should be proved. In this context. In this regard. Recent preliminary studies discover and analyse proper finite element models and parameters for the simulation of punching shear failure. these experimental investigations result in extensive costs. if and to what extent realistic and admissible declarations and results for the punching shear behaviour of (thick) reinforced concrete members can be achieved by finite element simulations. At the institute for reinforced and prestressed concrete structures at the Ruhr-University Bochum. because the used material properties as well as geometrical dimensions are differing in a wide range. the question arises. the main attention is to represent as much as essential parameters close to reality. the experimental basis and background is completely missing. Concerning this matter. the related shear failure load is decreasing. Regarding the fact that the simulation results of thick slabs cannot be verified with full-scale experiments. Particularly. the size-effect law for punching shear failure could not be verified and validated doubtless caused by a lack of missing comparable experimental results.0 m (Klöker. Nevertheless. this is up to two reasons: On the one hand. Figure 1. The failure state is reached when an inclined crack forms around the column. Without shear reinforcement the punching shear failure performs in a brittle manner within the discontinuity region of the highly stressed slab at the column. Cracking pattern and cylindrical cone of the punching shear failure.5 ⋅ d . Accordingly. 2001). In this regard. starting at the edge of the load application zone. Design concept for punching shear failure according to DIN 1045-1 (DIN 1045-1. the design value of the punching shear resistance without shear reinforcement for normal strength concrete is formulated by: 2008 Abaqus Users’ Conference 3 . c rcrit d θ θ h Figure 2. Punching shear failure of reinforced concrete slabs Punching shear failure of reinforced concrete slabs occurs when concentrated loads are initiated. At first.and axial stresses.7° . With a typical cylindrical punching failure cone (Figure 1) the column separates from the slab. which corresponds to an assumed crack inclination of θ = 33. beginning with the approach of different shear crack angles as well as the consideration of different shear transferring mechanisms and resulting in varying values of maximum shear failure loads. With respect to the German code provisions of DIN 1045-1 (DIN 1045-1.2. 2001) the punching shear loads have to be proved in a control diameter rcrit (Figure 2). different design concepts and design models are obvious. causing very high shear. Increasing the load causes tangential cracks around the load application zone. the combined stress performance leads to radial cracks. Comparison of international code provisions leads to different approaches of calculating the punching shear failure. For non-shear reinforced concrete slabs this control diameter calculates to rcrit = 1. this assimilation of the size factor changes as well as any formulation for the assimilation of the size effect itself changes according to the number of statistical analysed test results. or even to an inefficient design of slabs with large effective depths. Referring to the German Code DIN 1045-1. Also international discussions on shear failure and shear behaviour of reinforced concrete members in general and especially on punching shear failure demonstrate the need for further research in these topics. max depends on the concrete shear capacity: ν Rd . Finite Element model for punching Essential for a finite element simulation of the punching shear problem of large structural concrete members is the detailed verification of discretization and material model using experimental test results. there will be new modifications and formulations of the design concept against punching shear failure. Summarizing all approximate assumptions.ν Rd . the assimilation factor for the size effect κ reads by: κ = 1+ 200 ! ≤ 2. According to impractically verification of simulation results with the structural behaviour of full-scale members. max = 1. ct (2) The semi-empirical assimilation factor for the size effect κ was found by statistical analyses of shear tests of rectangular and profiled beams.0 d [mm] (3) Comparing international code provisions. Although there is no evidence that size effects of uniaxial shear for beams and punching shear are comparable or even the same.14 ⋅ κ ⋅ (100 ⋅ ρ l ⋅ f ck )1 / 3 ⋅ d (1) The formulation considers the following parameters: • The characteristic value of the concrete cylinder strength: f ck • The flexural reinforcement ratio: ρ l • The effective depth of the slab: d • The assimilation factor for the size effect of the effective depth: κ The maximum punching shear capacity ν Rd . the German code assimilates the empirical evaluated size effect law of punching shear and uniaxial shear of beams by the same size factor. ct = 0. if the calculated punching shear failure load according to code provisions leads to unsafe. 3. For the conversion to the new European code provisions. the question arises. the resulting finite element model has to be sufficient to reasonably 4 2007 ABAQUS Users’ Conference .5 ⋅ν Rd . suitable and admissible results for the simulation of the three-dimensional state of stress corresponding to the punching shear failure can only be derived from the elastoplastic damage model “concrete damaged plasticity” (Abaqus. According to these requirements. the first section represents the linear-elastic branch. Pölling. proved and analysed. Mark. 2003). 2000. In this regard. which represent the evolution of the damage variables under compressive loadings d c and tensile loadings d t are shown in Equation 4 and Equation 5. the stress-strain relation behaviour of concrete under uniaxial compressive loading can be divided into three domains. the modality of discretization of the reinforcement and the discretization of support and load application area. For the implementation of the concrete model. two types of material functions have to be defined. 1989) and elaborated by (Lee. suitable formulations for the stress-strain relations of concrete are given in (CEB-FIP. functions and parameters for concrete Although there are different possibilities to describe the complex. 3. which also includes cyclic un. the material model assumes a nonassociated flow rate as well as isotropic damage. by preliminary studies the different possibilities of discretization and approximation of structural as well as material behaviour are researched.1 Material model.and reloading. Functions. the type. respectively. nonlinear material behaviour of concrete in Abaqus. Developed by (Lubliner. Among other things. stress-strain relations represent the uniaxial material behaviour under compressive and tensile loadings. The simulations are verified and evaluated by different types of experiments. 2006).simulate the punching shear behaviour of small-scale slabs as well as slabs with large effective depths. Parametric studies figure out constraints and limits for material and discretization parameters. Some of the main goals are the influence of the used material model. In this regard. which can be formulated as a linear-elastic function of the secant modulus of elasticity Ec : σ c(1) = Ec ⋅ ε c 2008 Abaqus Users’ Conference (6) 5 . dc = 1 − dt = 1 − σ c ⋅ E c−1 ε cpl ⋅ (1 bc − 1) + σ c ⋅ E c−1 ε tpl σ t ⋅ Ec−1 ⋅ (1 bt − 1) + σ t ⋅ Ec−1 (4) (5) According to (CEB-FIP. geometrical shape and size of elements. the finite element model is not finally completed and is modified and improved continuously. 1998). As shown in Figure 3. 1993. 1993). sc sc fcm gc = 0 fcm e cpl = bc × e cin gcu gcl 0. This allows the advantage of using the secant modulus as a material parameter that complies with the standards. 1993. 1993) the linear branch ends at σ c = 0.33 ⋅ f cm . for verification of experiments with partial given material parameters. Mark. 2 ⎛ε ⎞ Eci ⋅ − ⎜⎜ c ⎟⎟ f cm ⎝ ε c1 ⎠ = ⋅ f cm ⎛ ⎞ ε ε 1 + ⎜⎜ E ci ⋅ c1 − 2 ⎟⎟ ⋅ c f cm ⎝ ⎠ ε c1 εc σ c( 2) (7) According to this. While according to (CEB-FIP.4×fcm 0. the modulus of elasticity can be taken from code approximations. Equation 7 describes the ascending branch of the uniaxial stress-strain relation for a compression loading up to the peak load f cm at the corresponding strain level ε c1 . 2006) and can be calculated from: Eci = 2 3 ⋅ Ec 2 ⎛ f ⎞ 4 f 5 ⋅ ⎜⎜ cm ⎟⎟ − ⋅ cm + ⋅ Ec 3 ε c1 3 ⎝ ε c1 ⎠ (8) Section three in Figure 3 represents the post-peak branch and is described by Equation 9: σ c(3) 6 ⎛ 2 + γ c ⋅ f cm ⋅ ε c1 γ ⋅ε 2 ⎞ =⎜ − γ c ⋅εc + c c ⎟ ⎜ 2 ⋅ f cm 2 ⋅ ε c1 ⎟⎠ ⎝ −1 (9) 2007 ABAQUS Users’ Conference .4 ⋅ f cm . the modified parameter Eci corresponds to the modulus of elasticity in Equation 7 (CEB-FIP.4×fcm gc ® ¥ Ec gc > 0 ec1 1 Ec×(1-dc) ec 2 s c × Ec. the stress-strain relations are negligible modified and the first section is expanded up to σ c = 0. Stress-strain relation for uniaxial compressive loading.1 e cin e cpl 3 ec e cel Figure 3. Anymore. an uniaxial stress-strain relation for the used concrete including an utilizable part of the descending branch has to be given. 1993.2 h = 100mm h = 200mm e1[‰] 0 0 2 4 6 8 10 Figure 4.4 0.The post-peak behaviour depends on the descent function γ c : γc = π 2 ⋅ f cm ⋅ ε c1 ! ⎡ f ⎛ b ⋅ f ⎞⎤ 2 ⋅ ⎢ g cl − cm ⎜⎜ ε c1 ⋅ (1 − bc ) + c cm ⎟⎟⎥ 2 Ec ⎠⎦⎥ ⎝ ⎣⎢ 2 >0 (10) Basing on the assumption that the constant crushing energy Gcl (Pölling. 2008 Abaqus Users’ Conference 7 . This procedure is also used for the simulation of punching shear experiments.8 0.6 h = 50mm 0. 1984).0 0. However. 2000) is a material property. s1 / s1 p experiment simulation 1. In the literature a wide differing range of values for the crushing energy Gcl can be found. 1984) to almost eliminate mesh dependencies of the simulation results: g cl = Gcl lc (11) Herein lc represents the characteristic length of the simulated or tested specimen. Equation 11 considers its dependency on the geometry of the tested or simulated specimen (Vonk. Comparison between simulation results and experimental data according to (Van Mier. simulations of path controlled uniaxial compression tests are used to validate an admissible parameter for Gcl . 1984). Van Mier.2 1.0 kN / m . Figure 4 shows the simulation results of three concrete specimens with different heights and the same quadratic base area tested by (Van Mier. The best approximation was found using a crushing energy of Gcl = 19. for most publications of experimental data these information is unfortunately missing. Concerning this matter. In this regard. The simulations with Abaqus and the described material functions provide very good agreement with the given experimental values. σ t ( w) f ct ⎛ ⎛ w = ⎜1 + ⎜⎜ c1 ⋅ ⎜ wc ⎝ ⎝ ⎞ ⎟⎟ ⎠ 3⎞ ⎟⋅e ⎟ ⎠ − c2 ⋅ w wc − ( ) w ⋅ 1 + c13 ⋅ e − c 2 wc (13) The free parameters could be experimentally determined to c1 = 3 and c 2 = 6. For multiaxial loadings. the material model was tested by simulating uniaxial. 1973). This is due to the fact that the sector of mixed compression-tension-stress is approximated by a linear function. 1983). Stress-strain relation for uniaxial tension loading.dt) et s t × E c-1 e tin pl et e tel Figure 5. Up to the maximum concrete tension strength. basing on the fictitious crack model of (Hillerborg. the mixed-strength sectors seem to deviate from experimental relations and the published function definitions in (Abaqus. 1992).The description of the stress-strain relation for tensile loading is divided into two sections. 2003). the linear part is calculated from: σ t(1) = Ec ⋅ ε t (12) st 1 2 fctm pl e t = bt × e tin Ec Ec × (1 . The function d t (Equation 5) regulates the tension damage and includes the experimentally determined parameter bt . While the implemented “concrete damaged plasticity” material model reflects very successful the only-compression and only-tension ranges. 1992). The descent branch of the stress-strain relation of concrete loaded in uniaxial tension can be derived from a stress-crack opening relation (Equation 13) according to (Hordijk. The simulation results are in great accordance with the experimental data.93 (Hordijk. biaxial and triaxial strength experiments for normal strength concrete. Figure 6 shows the comparison of the obtained simulation results for the described material functions and experimental results of biaxial loadings on concrete specimen by (Kupfer. 8 2007 ABAQUS Users’ Conference . 8 -0.2 biaxial tension s11 fcm 0. 1973) simulation . rectangular slabs with rectangular load application zones and longitudinal supports have been considered.0.s22 fcm uniaxial 0 2 tension uniaxial compression -1. 2008 Abaqus Users’ Conference 9 . The corresponding parameterized discretization is illustrated in Figure 8.2 Figure 6.2 -0.4 . sizes and quantity.4 -0. Therefore. a parameterized discretization of the mesh has been created which allows not only the variation of geometrical parameters.0 biaxial compression -1.0.3 Mesh generation and discretization of reinforced concrete slabs For the preliminary studies. Biaxial strength of concrete: Comparison of experiments of (Kupfer.2 Material model for reinforcing steel The uniaxial material behaviour of the reinforcing steel is modelled by a bilinear stress-strain relation. a wide range of discretization variants has to be considered. 1973) and simulation. A typical test setup for rectangular concrete slabs according to punching failure is shown in Figure 7. the modulus of elasticity E s as well as the ultimate strength f u and the corresponding ultimate strain εu .6 -0.8 -1.6 ax is of sy m m et ry experiments (Kupfer. the reinforcing steel material model only requires the yield strength f y .0 -0. For first verifications. 3.2 -0.2 -1. Therefore. 3. but also enables the modification of element types. Because of the very complex three-dimensional stress state within the punching area of the slab. and for application of high-order elements or finer discretizations. To satisfy the constraints of the used material model. the concrete mesh has to be almost cubic. the symmetry will be utilized to shorten the calculation processes and to reduce data interpretation. The reinforcement is discretized as three-dimensional 8-node truss elements (T3D8). Preliminary studies show that for the discretization of bending reinforcement. the whole system is discretized. 10 2007 ABAQUS Users’ Conference . Figure 8. unintentional discretization and mesh influences on the crushing and fracture energy can be avoided (Pölling. Finite element discretization of test setups for punching. 2000). (B) = control positions for instrumentation (A) (B) flexible bearing pads lc a d load application zone l Figure 7. For preliminary analyses of the global bearing behaviour of the slabs. of course. Horizontal projection and sectional drawing of typical test setups for punching. also unsymmetrical test setups and excentric load or support cases can be analysed. there’s no remarkable difference by using three-dimensional truss or beam elements. Therefore. For a more detailed analysis of inner stress states and crack propagation. Thus.(A). three-dimensional solid (continuum) elements are used for the discretization of concrete elements. 4. The geometrical and material parameters of the chosen slab and corresponding simulation parameters are shown in Table 1 and Table 2. Therefore. even linear or plane bedded support as well as simplified models for ground. regarding experimental studies and verification results. neoprene bearing pads according to Figure 7 are used to reduce the reaction of constraints influenced by the support. Therefore. To manage the nonlinear simulation. 2008 Abaqus Users’ Conference 11 . 3. Regarding these first studies. the material properties for the discretized column can be modified independently from the concrete’s material of the slab. as it affects the punching behaviour of footings. limitations of the arc-length increments as well as modifications in the time incrementation parameters often improve the convergence behaviour of the simulations. The discretization as spring elements enables the simulation of different types of support.4 Support and load The conditions and requirements of support and load application area are affecting the experimental as well as the simulation results. To analyse the influence of stiffness of the load application member. respectively.To model the composite between concrete and reinforcing steel. only rectangular columns or application zones are discretized. respectively. Verification of the model As a first verification and validation of the described and discretized finite element model. In experimental setups. Rectangular shaped concrete slabs of different heights. the reinforcement is implemented as embedded elements in the concrete’s host elements. Concerning this matter. distributed element loads or single node loads. for example. Within the simulation the vertical support of the reinforced concrete slabs is realised by “compression only” springs. the assumed ideal bonding behaviour between concrete and reinforcing steel has to be taken into account. For the static simulation the “modified Riks method” is used to solve the nonlinear set of equations. The load application is realized either by surface loads. the method uses the arc-length to measure the progress of the solution. frequently long-winded variations of the solution and control parameters have to be performed. supported by longitudinal neoprene bearing pads have been tested until failure occurred. The column is discretized similar to the concrete mesh of the slab. Concerning this matter. punching shear tests on reinforced concrete slabs done by (Li. while loads and displacements are solved simultaneously. also three-dimensional 8 node solid continuum elements are taken. 2000) have been simulated. Parameter Denotation l = 145. However. Especially close to the failure load. ν = 0.1 damage parameters according to (Pölling. 12 2007 ABAQUS Users’ Conference .16 Kc = 2 3 Dilatation angle according to (Lee. bt = 0. Especially non-proportional high strength up to the occurrence of first cracks and smaller displacements compared to the experimental values can be observed.3 ‰ Yield strain of tension reinforcement Table 2.195 ⋅ wc ⋅ fctm Crushing and fracture energies bc = 0. variation and modification of the control parameters for the “Static Riks”-solution as well as the time incrementation becomes more and more important. Concerning this matter.Table 1. 2000). the simulations run more and more instable.7 . Geometrical and material parameters of the experiment (Li. Simulation parameters. 2000) wc = 180μm Maximum crack opening according to (Hordijk.0 cm. 1998) Ratio of biaxial to uniaxial compressive strength according to (Kupfer. d = 20. 1973) Second stress invariant ratio In Figure 9 the load-displacement curvatures of simulation and experiment are given.5 cm Length of shearing zone ρl = 0.83 % Ratio of tension reinforcement fcm = 39. Comparing the maximum failure loads. Relative to the mean displacement of the column.4 MPa Average concrete cylinder strength fctm = 3. 2000). Parameter Denotation Gcl = 19 kN/m.2 ψ = 30° Poisson’s ratio α f = 1. G f = 0.0 cm Side length of the quadratic column a = 52.13 ‰ Strain at fcm f y = 465 MPa Yield strength of tension reinforcement ε y = 2. the diagrams show the comparative displacements of the control positions (A) and (B) according to Figure 7 and corresponding to the described instrumentation of the experiments by (Li. 1992).27 MPa Average concrete tension strength Ecm = 29425 MPa Secant modulus of elasticity ε c1 = -2. the simulation leads to a quite small aberration of about 2 % to the experimental value.0 cm Side length and effective depth of the quadratic slab lc = 20. the general agreement with the experimental results is not satisfying. Therefore. Therefore. However. The comparison of simulation of the concrete’s material behaviour with test results shows very good approximation results. Load displacement curvature of experiment and simulation. Comparing the crack pattern corresponding to the fineness of the discretized mesh.0 u (mm) control position (B) Figure 9. but the typical punching shear cone can be identified. 5. The article reports about the preparation and verification of material model and mesh generation for simulating reinforced concrete slabs. intensive analyses are actually performed.0 control position (A) 1. A closer look into the slab shows.0 2. an intense dependence can be observed.F (kN) F (kN) 900 900 600 600 300 0 0 1 2 3 4 5 6 experiment simulation 300 experiment simulation u (mm) 0 0. 2008 Abaqus Users’ Conference 13 . the typical inclined cracks – represented by the plastic strain results – agree quite well with the crack pattern of punching shear experiments (Figure 1). the inclination of the simulated cracks is to steep. Illustrated in Figure 10. Crack pattern of the simulation of punching shear failure. Figure 10. The verification of the presented finite element model is shown by simulation results of punching shear failure experiments. Conclusions and further development A finite element model to simulate the punching shear failure of rectangular reinforced concrete slabs is presented.0 3. Comparatively coarse discretizations cause numerical problems and do not lead to suitable results. that a general punching shear behaviour can be recognized. a closer look at the comparison of theoretical assumptions. Bazant.und Spannbetontragwerken. Furthermore. Vol. D. 2005. 6. H. 84. 9. “Analysis of one single crack. Berlin. 1983. 6. J. RWTH Aachen. pp. 2003. 233-243. Summarizing it must be considered. Hillerborg.“ Heron 37 (1). Ziegler. “Durchstanzen schubbewehrter Flachdecken im Bereich von Innenstützen. 7.“ PHDthesis. Ulke. 223-249. 2001.experimental test results and simulation results shows that there are still questions left yet. Z.. CEB-FIP.“ Beuth Verlag. P. Elsevier. Hegger. 3-79. Oh. A. R. 14 2007 ABAQUS Users’ Conference . Beutel. R. DIN EN 1992-1-1.“ Simulia. Y. J. especially for simulations of very brittle failure modes. CEB-FIP. USA. “Size effect in punching shear failure of slabs. Z. 2007. By parametric studies the influence of discretization parameters will be investigated und analysed in the future. Vol. 1983. 155-177. References 1. This includes the verification with other test results and test setups of reinforced concrete slabs with smaller depths. that further development and research is needed.. “Tensile and tensile fatigue behaviour of concrete.. ed. Karakas. A. “Model Code 1990. the preliminary studies will continue by determining problematic material and discretization parameters. 4. In this regard own experimental studies will complete the identification and analysis of needed simulation parameters. Berlin. yielding of bending reinforcement or the differing displacement results. before extrapolation to structural systems with bigger dimensions are possible. 1987.. 3. and M.4. modeling and analysis. Cao. pp. “Eurocode 2: Bemessung und Konstruktion von Stahlbeton.“ Beton.” Thomas Telford. Häusler. pp.und Stahlbetonbau. H. and B. the most problematic aspects are the lack of missing material data and setup facts. Bazant. 44-53. DIN 1045-1. “Punching of structural concrete slabs.. “Version 6. “Tragwerke aus Beton. and F.. 10. 1992. “Crack Band Theory for fracture of concrete. technical report. and F. 2003.” Fracture Mechanics of Concrete. 8. like the mismatch of crack inclination. Wittmann. Hordijk. A. 2001. Ricker. Currently.. and Z. pp.” ACI Structural Journal. 11. 2. 12. Abaqus Theory Manual.” Matériaux et Constructions. Beutel. 2006. pp.” bulletin 12. 16 (93). P. stabilization problems still occur during the simulation. M. Stahlbeton und Spannbeton. According to that. experiments. Vol. London 1993. Missing test results and incomplete descriptions of test and material parameters cause the need of new experimental programs to validate and verify the described and used finite element model. 101 (4).“ DIBT.“ Beuth Verlag. 5. “Ersatz oder Teilersatz von Querkraftversuchen durch Finite-Elemente-Berechnungen – Abschlussbericht. Hegger. Amsterdam. B. “Untersuchungen zum Durchstanzverhalten von Stahlbetonfundamenten. “A plastic-damage model for concrete.. “Ergebnisse dreiachsiger verformungsgesteuerter Belastungsversuche an Beton. “Strain-softening of concrete under multiaxial loading conditions. J. 101-112.” test report. 2000. B. “Influence of Size on punching shear strength of concrete slabs. “Maßstabseinfluss bei Schubversuchen im Licht der Bruchmechanik. 25 (3). Vol. 195-211. S. K. 100 (2). 892-900. D. pp. “Stress-strain relations for concrete under cyclic loading.“ Beton. and L. and G. Fenves. 16.. Eindhoven. G.. W. H.. H. K. 99. Canada. G. McGill University. 22.. J. Vol. Vol. 240-249.” Festschrift Professor Stangenberg "25 Jahre in Forschung.H. pp. Lehre und Praxis".. pp.“ Bauingenieur. “Eine praxisnahe.” master thesis. pp. 2008 Abaqus Users’ Conference 15 . H.. R.-W. A. 1989. K.“ Dissertation. M.. H. Kupfer. 26. Van Mier. J. “ Journal of the ACI. K. 23. 2006. 24. D. pp. H. 353-361. Vonk. 2007. 1973. Aschl. 1993. Tulin.. Montreal.” Journal of Engineering Mechanics. Van der Vlugt. Marx. Gerstle. Kuchma. 38 (3).” International Journal for Solids and Structures.. 1976. Li. Techn. 76 (1). 18. Gerstle. and S. G. pp. “Plastic-damage model for cyclic loading of concrete structures. and H. “Zweiachsig durch Biegung und Querkräfte beanspruchte Stahlbetonträger.“ ACI Structural Journal. “Behaviour of Concrete under Biaxial Stresses. 25. T. Kim.. Sinha. Univ. 21. Vol. Pölling. 20. Oliver. “A micromechanical investigation of softening of concrete loaded in compression.“ Habilitationsschrift. 62. 19-21. 3-94. 2000. schädigungsorientierte Materialbeschreibung für Stahlbeton. 1981. 1964. Bochum. L. and K.. Vol. and E. “Durchstanzen massiger Fundamentplatten aus Stahlbeton. Ruhr-Universität Bochum. Van Mier. L. 1984.” PHDthesis. P. J. pp. Reineck. 853-866. 124 (8). pp. S. R. M.13. 1987. Lee.” Heron Vol. Winkler. Vol. Linse. A. 14.. Lubliner. and B. Klöker. P. Reinhardt. ”Shear Database for Reincorced Concrete Members Without Shear Reinforcement. Ruhr-Universität Bochum. Mark. 17. 1998. “Versuche zum Verhalten von Beton unter mehrachsiger Beanspruchung. Onate.” Journal of the Engineering Mechanics Division. pp. K. 19. W. TU München. 61 (2). Oller.und Stahlbeton Vol. 299-326. J. B. 15. Reinhardt. and K. 2003.