those of the Sampoong Department Store occurred in 1995 [2] or Bullocks department in 1994 [3] indicated that two way shear strength of slab-column ..... N.J. Gardner, J. Huh, L. Chung, âLessons from Sampoong Department Store. Collapse.
Nonlinear Finite Element Analysis of Reinforced Concrete Slab-Column Connections under Gravity Loading
S. Derogar1, P. Mandal2, C. Ince1, T. Michellitsch3, 1
Faculty of Engineering and Architecture, Yeditepe University, Istanbul, Turkey 2 School of Mechanical, Aerospace and Civil Engineering, The University of Manchester, UK 3 Institut Jean le Rond d'Alembert, CNRS UMR 7190, Université Pierre et Marie Curie, Paris 6, France
Abstract This paper addresses an implementation of a three-dimensional nonlinear finite element model to assess the behaviour of slab-column connections under gravity loading. The commercial finite element package, ABAQUS, was used to efficiently capture the load deformation behaviour of slab-column connections. The geometric and material model adopted for the simulations are investigated and the model parameters are determined. Experimental data on slab-column connection reported by Li [1] is used to validate and calibrate the proposed FEM models. Keywords: Nonlinear Finite Model, Slab-Column Connections, ABAQUS.
1 Introduction One of the most common concrete floor systems is the flat plate. A flat plate structural system consists of a slab with uniform thickness supported directly on columns without any beams. Flat plate structural systems offer several advantages when compared with other concrete floor systems. Because of the absence of beams, flat plates provide architectural flexibility, more clear space, less total building height, easier formwork, and consequently shorter construction time. Flat plates can suffer from a serious problem that can limit performance of such systems since brittle punching failure can occur due to transfer of shear force between slabs and columns. Failure of flat-plate structures initiated by punching failure, including those of the Sampoong Department Store occurred in 1995 [2] or Bullocks department in 1994 [3] indicated that two way shear strength of slab-column connections and the mechanics of punching shear failure have not been well understood. Therefore, the need for experimental research to provide further inside into the design equations is vital. Experimental tests are expensive and also time consuming. Therefore, more interest has been given to numerical simulations. This paper investigates the numerical modelling on punching shear behaviour of slab
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column connections. The finite element analysis is carried out using ABAQUS, a general purpose FE package. The material properties and nonlinear solutions are implemented in the FE model and are therefore discussed here. Slab PSSA from Li [1] is numerically investigated and therefore the experimental result will be briefly explained here.
2 Slab PSSA Slab PSSA shown in Figure 1, tested by Li [1] was selected for the numerical analysis. The slab was 175 mm thick and 2,000 mm square with a 200×200×200 mm column stub located centrally below the plate. The slab was reinforced with bottom and top flexural reinforcement mats, as shown in Figure 1. For determining the flexural capacity assumptions regarding the possible yield line failures need to be made. Failure is likely to occur by 4 diagonal yield lines. For the 4 diagonal yields line mechanism, by using virtual work [4] the force that produces flexural failure is Vf
8ml s l1 c
(1)
where m is the bending moment per unit slab width at yielding of the top flexural reinforcement, ls is dimension of the slab specimen (2000 mm), l1 is the span between the supports ( loading points, 1700 mm), and c is the column dimension (200 mm).
2000 mm Ø16@ 200 mm
200
200
4 Ø16
Ø6 @50 mm
Ø16@ 200 mm
Slab
Bottom Bars
Column stub
Column Ø12@ 200 mm
Schematic representation of experimental set-up
Slab and reinforcement details
Figure 1: Test setup by Li [1]
2
Ø16@ 200 mm
Column
2000 mm
5 97 7,
Top Bars
Reacting ring frames
Jacks
m m
Loading plate Loading ring frames
The flexural capacity of the slab PSSA using Equation 1 was calculated and is 490 kN. The detail of slab PSSA is shown in Table 1. Vf is the flexural capacity of the slab. Table 1: experimental test by Li [1] Reinf. Yield Compressive Effective Column ratio, strength, strength, f’c depth, d dimension, fy (Mpa) (MPa) (mm) c (mm) (%) PSSA 0.72 500 32.3 140 200
Vtest (kN)
Vf (kN)
454
490
3 Material properties 3.1 “Concrete smeared cracking” vs. “Concrete damage plasticity” ABAQUS [5] offers three material models for modelling the nonlinear behaviour of concrete. These are namely: concrete smeared cracking, brittle cracking and concrete damage plasticity. Brittle cracking can only be used in dynamic analysis in ABAQUS/Explicit therefore is not addressed in this paper. Concrete smeared cracking and concrete damage plasticity, were evaluated to verify their capabilities in modelling the slabs tested herein. For this purpose, Slab PSSA was used to perform the evaluation. Concrete smeared cracking uses the associated flow assumption which overestimates the volumetric plastic strain due to simplification of the compressive behaviour. The model uses a fixed angle crack model to detect the cracks results in shear stress locking problem. This can lead to convergence problem and results the analysis to be stopped in early loading due to numerical instability.
Load (kN)
400 350
Stoped due to numerical instability
300 250 200 150 100
Plasticity damage concrete model Concrete smeared cracking model
50 0 0
2
4 6 8 Displacement (mm)
10
12
Figure 2: Comparison of load-deflection behaviour of Slab PSSA based on smeared cracking and damage plasticity models.
3
Figure 2 show that the concrete damage plasticity model is able to continue the analysis up to failure while smeared cracking model stops in early stage of loading due to numerical stability problems.
3.2 Concrete Damage Plasticity Plasticity theory is a mathematical representation of the mechanical behaviour of solids. It can be used for translation of physical reality for ductile materials such as metals or a model that approximates the behaviour under certain circumstances for brittle materials such as concrete. In problems where the tension, with the crack development, plays a significant role, such as shear failure in reinforced concrete structures, the usual procedure is to apply plasticity theory in the compression zone and treat the zones in which at least one principal stress is tensile by one of several versions of fracture mechanics [6]. Since the critical surfaces are similar in the biaxial behaviour of concrete, a yield function is used in plasticity based models. The size of yield function is based on the material properties defined for the uniaxial behaviour of concrete. The yield surface is defined as the material no longer acts elastic and the failure surface is based on the ultimate strengths. This means that in the biaxial tensile meridian the yield surface is equal to the failure surface. In compression, the material is usually assumed to be initially elastic up to 30-60 % of the compressive strength [7]. There are several failure (or yield) criteria developed for concrete materials [6] such as Drucker-prager and Mohr-Coulomb criteria. For steel usually the Von Mises failure criteria is used (further details can be obtained in ABAQUS user’s manual, 2008). According to Lubliner [6], these criteria do not represent experimental results for concrete precisely otherwise they are suitably modified. For instance, one modification is to use a combination of the Mohr-Coulomb and Drucker-Prager yield functions, where the Drucker-Prager is used for biaxial compression and the Mohr-Coulomb is used otherwise. The biaxial yield function used in concrete damage plasticity was developed by Lubliner [6] which also includes the modifications that was proposed by Lee and Fenves [8]. It is vital to understand the constitutive parameters which describe the material properties in finite element modelling to obtain authentic results. A parametric study for dilation angle is carried out in order to determine the optimised value to be implemented in the numerical analysis. Dilation angle measures the inclination of the plastic potential which reaches for high confining pressure. Low values of the dilation angle produce brittle behaviour while higher values produce more ductile behaviour. A parametric study on dilation angle for Slab PSSA is presented in Figure 3. Figure 3 shows that the difference in the behaviour of the slab is fairly small when the dilation angle is between 30º to 38º. In this paper, a dilation angle of 38º was used. Malm [9] also made the same conclusion on dilation angle for the analysis of a reinforced concrete beam subjected to four-point bending test. He used a dilation angle of 35º and 38º in his analysis. Jankowiak and Lodygowski [10] presented a work on identification of dilation angle for the use in ABAQUS. They determined a dilation angle of 38º based on
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minimisation of the error of the biaxial failure envelope of Kupfer [11] and the yield surface in ABAQUS. 700 600 Slab PSSA Dilation 10 Dilation 15 Dilation 20 Dilation 25 Dilation 30 Dilation 38 Dilation 50
Load (kN)
500
400 300 200 100 0
0
5
10
15
20
25
30
35
Deflection (mm)
Figure 3: Parametric study on the effect of dilation angle on the response slabs
3.3
Compressive Behaviour
In this study the uniaxial tensile and compressive response of the concrete are used along with the concept of damage plasticity model. The uniaxial compressive concrete model from Eurocode 2 [12] was adopted in this paper. Also the concrete stress strain curve was adjusted to continue beyond the crushing point to avoid stopping the analysis.
3.4
Tensile Behaviour
ABAQUS provides two approaches to describe the softening behaviour of cracked concrete. The first method is based on strength criteria whereas the other approach adopts fracture energy cracking criteria. The approach based on stress-strain relationship will introduce mesh sensitivity into the results if significant regions of the model do not contain reinforcement [7]. It means that the analysis will not converge to a unique solution. The fracture energy criterion which was developed by Hillerborg [13] overcomes the mesh sensitivity issues for the former approach. The crack behaviour of concrete was described based on the fracture energy. The fracture energy, Gf, is a material parameter that describes the amount of energy required to open a unit area of a crack. The area under stress-displacement curve (Figure 4) corresponds to the fracture energy. Three different stress-displacement curves can be used to model tension softening. They are linear, bilinear and exponential which are based on analytical expressions derived from curve-fitting of experimental test data. The simplest way to introduce crack is to use linear approximation. The crack opening corresponding to a stress free crack with linear tension stiffening can be calculated as shown in the Equation 2.
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2Gf . (2) ft Linear and bilinear tension softening models are used extensively in modelling plane concrete however; Malm [9] reported that the exponential tension softening model is by far the best and most accurate model. Figure 4 illustrates the bilinear and exponential tension softening models.
c
(b) Figure 4: (a) Bilinear and (b) exponential tension softening model. The following equations are used to develop exponential tension softening model:
f ( ) f (c ) ft c
(3)
where f ( ) is a displacement function given by c 3 c f ( ) 1 1 exp 2 (4) c c where, ω is the crack opening displacement and ωc is the crack opening displacement at which stress can no longer be transferred (ωc=5.14 Gf /ft for normal weight concrete). c1 is a material constant and is 3.0 for normal weight concrete. c2 is a material constant and is 6.93 for normal weight concrete. The fracture energy of concrete, Gf, is calculated from Equation 5 suggested by MC90 [9] was used to estimate Gf in this chapter.
f Gf Gf0 cm f cm0
0.7
(5)
0.95 0.0053d max , dmax is the maximum aggregate size, fcm is the 8 mean concrete compressive strength, and fcm0=10MPa.
where Gf0 0.0024
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4
Mesh Size
Since punching shear failure is characterized by crack localisation in certain areas, mesh density can be a crucial problem. Therefore, a mesh sensitivity analysis was carried out in this study. Also the sensitivity to the mesh size is linked to the tension stiffening option. Hence, the sensitivity of the FE model to mesh size should be seen in view of the tension stiffening values used. Tension stiffening in ABAQUS can be introduced by the strain approach and the displacement approach (fracture energy approach). In strain approach, for the same tension stiffening, the response for the denser mesh becomes softer. For this purpose, Slab PSSA with different mesh sizes was considered here. The nominal mesh sizes were 25 mm, 45 mm, 85 mm, and 175 mm. For every case, the aspect ratio was kept one as far as possible. As it is shown in Figure 4, , the mesh is assumed to be converged when an increase in mesh density had negligible effect on the results obtained. For mesh sensitivity, a linear displacement-type, tension stiffening curve based on fracture energy which defines the ultimate displacement value u0 at the crack when the tensile strength drops to zero. A typical displacement value of u0=0.15 mm yielded different responses for the meshes examined here. Base on the results shown in Figure 5, the convergence study implies that the 45 mm mesh size converges to 25 mm mesh sizes. Mesh size of 85 mm and 175 mm appears to have unstable behaviour after first cracking. Therefore, a mesh size of 25 mm was adopted for the rest of the analysis. 500 450
Load (kN)
400 350 300
Slab PSSA
250 200
Mesh- 25 mm
150
Mesh- 45 mm
100
Mesh- 85 mm
50
Mesh-170 mm
0 0
5
10
15
20
25
Displacement (mm)
Figure 5: Mesh sensitivity analysis of Slab PSSA
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Numerical Instability
Concrete material model which develops strain softening and stiffness degradation, often experiences severe convergence problems. To avoid some of these convergence difficulties, viscoplastic regularisation of the constitutive equation could be used. Another way of reducing convergence difficulties is the use of artificial damping in combination with concrete damaged plasticity model. This may
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introduce large artificial damping forces on elements undergoing severe damage since ABAQUS calculates the damping stress based on the undamaged elastic stiffness (ABAQUS, User’s Manual 2008). Another technique is to change the convergence criteria and allowing a larger number of iterations before convergence is checked. The other option also suggested by Malm and Holmgren, (2008) is the possibility of converting a static problem to a quasi-static problem to use an explicit solution. Malm [15] also suggested reducing the tolerances to 10% of default value and increasing the number of iterations between to two and four times of default value, before the rate of convergence is checked. Analysis presented in this paper, are performed with an implicit solution scheme. Also the simulations performed in ABAQUS had a trend to interrupt due to convergence difficulties. The most effective method to continue beyond this in the analysis suggested by the author was to increase the number of iteration. This will lead in small time increments when reaching a point with convergence difficulties and thereby increasing the time. This would be done by addition of Volumeproportional damping to the model to stabilize the unstable static problem in ABAQUS/Standard (ABAQUS user’s manual, 2008). To obtain an optimum value for the damping factor requires performing post-analysis comparison of the energy dissipated by viscous damping (ALLSD) to the total strain energy (ALLIE) [5].
6 Load deformation behaviour Figure 6 shows the comparison of load deformation of Slab PSSA with the ABAQUS model. 500
Load (kN)
400 300 Test- PSSA ABAQUS
200 100 0 0
2
4 6 Displacement (mm)
8
10
12
Figure 6: Comparison between experimental and model prediction of slab PSSA It can be seen from the Figure 5 that both models are in good agreement with the experiments. ABAQUS models for both the slabs showed stiffer behaviour when the slabs start to have flexural cracks compared to the test results. Flexural cracks start
8
to happen at loads over 200 kN. This is expected since in the FE model, a full bond between the flexural reinforcement and concrete was assumed by embedding the flexural reinforcement to the concrete.
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Internal steel reinforcement
Strain gauges on the flexural reinforcement were installed for Slab PSSA as reported by Li [1]. It has been highlighted earlier in this chapter that FE analysis smears the effect of cracks and, in itself, does not differentiate between cracked and uncracked sections. The possible prediction of the experimental deflection is obtained by the level of tension stiffening in FE analysis. Therefore, prediction of the average rebar strain between cracks is likely to be obtained by FE analysis. This strain is usually less than that of an individual rebar and this is shown in Figure 7.
Figure 7: Comparison between strain gauge measurements of flexural rein. in Slab PSSA with ABAQUS predictions It is clear from Figure 6 that the predictions of reinforcement strains using FE model reasonably well. However the rebar strain predicted by the model is less than the actual reinforcement strain in Gauges S3 and S4. On the other hand, up to a load of about 400 kN, the strain predicted by the model compares very well with the measured strain in strain gauge S3 and S4. It has to be noted that predicted strain is
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underestimated in the model beyond this load. The amount of tension stiffening added to the concrete model could be the possible explanation for such discrepancy. As it is very well recognised that, the punching behaviour is characterised by localization of strains near to the loading area, in contrary to the flexural behaviour in which cracking is nearly distributed all over the member. So, the model may overestimate the strains in some regions while underestimating them in other regions. It is shown here that Slab 1 failed in pure punching failure mode while Slab PSSA failed by flexural and punching shear mode.
8 Conclusions In this paper, nonlinear finite element analyses were carried out to simulate the deflection behaviour of slab column connection. Slab PSSA from Li [1] was used in the simulations. The simulations also showed good agreements with the yield line analysis for the Slab PSSA. The nonlinear finite element analysis with ABAQUS can simulate slab-column connections behaviour in terms of load-deflection curves and stiffness in both the pre and the post-cracking stages. However, for such analysis, a proper evaluation of tension stiffening is vital. For slabs with low reinforcement ratio such as reported in Li [1], the ultimate load reaches while the flexural reinforcements are yielding therefore, an accurate estimation of the tension stiffening is essential.
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[2]
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J. Lee, and G. L. Fenves, “A plastic-damage concrete model for earthquake analysis of dams” Earthquake Engineering & Structural Dynamics, 1998, Vol. 27, No. 9, pp. 937–956. R. Malm, “Predicting shear type crack initiation and growth in concrete with non-linear finite element method.” TRITA-BKN. Bulletin 97, 2009, Royal Institute of Technology (KTH), Stockholm. T. Jankowiak, T. Lodygowski, “Identification of parameters of concrete damage plasticity constitutive model” Foundation of Civil and Environmental Engineering, 2005, No 6, ISSN 1642-9303, pp. 53-69. H. Kupfer, H. K. Hilsdorf, H. Rusch, “Behaviour of concrete under biaxial stresses.” ACI Structural Journal, 1969, Vol. 65, No. 8, pp. 656-666. EN 1992-1-1: 2004, Eurocode 2: Design of Cncrete Structures, Part 1-1: General rules and rules for buildings, December 2004. Hillerborg, A., Modeer, M., Petersson, P.E., 1976, “Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements” Vol. 6, No. 6, pp. 773-782. CEB-FIP Model Code 1990: Design Code, Comitee Euro-International De Beton, Bulletin D’Information No. 203–305, Lausanne, Switzerland. R. Malm, and J. Holgren, “Cracking in deep beams owing to shear loading. Part 2: Non-linear analysis.”Magazine of Concrete Research, 2008, Vol. 60, No. 5, pp. 381-388.
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