Sep 25, 2015 - Welded studs and tie bars ensure shear transfer between steel and .... studs were modelled as B31 beams, a 2-node linear Timoshenko ... free, and the load was applied as displacement along the vertical axis. ... modelling techniques for SC structures [7][11][12][13]. .... co, March 21-24 1993, Vol.2, ASME.
Nordic Steel Construction Conference 2015 Tampere, Finland 23-25 September 2015
NON-LINEAR FINITE ELEMENT MODELLING OF STEELCONCRETE-STEEL MEMBERS IN BENDING AND SHEAR Marc Donnadieua, Ludovic Fülöpb a
Institut Français de Mécanique Avancée, IFMA b VTT Technical Research Centre of Finland
Abstract: Steel-Concrete-Steel (SC) construction comprises two steel plates with concrete infill. Welded studs and tie bars ensure shear transfer between steel and concrete. SC has structural advantages and leads to faster construction compared to classical reinforced concrete solutions [1][2][3][4], and advantage especially important in the industrial sectors. In this study we developed a general purpose modelling tool for evaluating the bending and shear strength of SC members. We show that most numerical modelling techniques found in the technical literature cannot predict the general behaviour of SC members. This work aimed to develop a single finite element model, which is able to predict all failure modes relevant to SC members.
1 Introduction This work aims to develop a finite element model which is able to predict the following failure modes in SC structures: steel plate yielding, steel plate buckling, de-bonding of tensile sheet driven by stud shear failure, concrete crushing and vertical or horizontal shear failure. Methodologically, we started from a simple model and improved it gradually as the shortcomings of the simpler models were revealed. During this process, it has been shown that some modelling solutions are inconsistent, while others models are strictly calibrated for particular failure modes observed during the experiments confirming the model results. This makes it impossible to apply the modeling methods for beams experiencing other modes of failure, for instance in a parametric study. The use of superfluous model parameters, hard to calibrate against physical quantities has also been avoided. The outcome of the study is a finite element modelling methodology calibrated against a broader range of experimental results [5][6][7][8], which permits accurate prediction of the behaviour of SC members in bending and shear. After further calibration, we plan to use the methodology for parametric modeling of SC structures.
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2 Methodology The finite element models (FEM) were mainly calibrated against experimental results described by Oduyemi and Wright [5] and Varma et al [7]. The procedure to elaborate of the models comprises the following steps: 1. Developing the model, computationally as cheap and simple as possible. 2. Checking the behaviour prediction against experimental results. 3. Improving the finite element model until reaching a satisfactory degree of accuracy.
3 Initial modeling complexity This section presents the initial FEM, designed to be as simple as possible. 3.1 Material properties All known materials properties were taken from the papers reporting the experiments in order to be able to compare the FEM predictions with test results [5][7]. Supplementary properties, not reported in the initial papers were estimated based on codes or technical literature. In order to describe accurately the non-linear behaviour of concrete the damaged plasticity (DP) model was used based on literature [9][10]. The DP model is a continuum damage model based on concrete plasticity. It requires the definition of the two uni-axial failure mechanisms, tensile cracking and compressive crushing of the concrete. The evolution of the failure surface is controlled by two hardening variables, both equivalent plastic strain, in tension and compression, provided in tabular format in ABAQUS. For each experiment, cube strength, cylinder splitting strength, and Young´s modulus are reported together with the test results on the SC member [5][7]. However, cylinder strength and strain at the peak stress were not reported and these properties were approximated as the value from the closest concrete grade defined in Eurocode 4 [10]. The density was taken equal to 2400 kg/m³ and the Poisson’s ratio as 0.2. The uni-axial compression behaviour is defined according to the Eurocode 2 formulation [10]: 2 k ; with 0 f c cm c cu1 1 (k 2) c
(1)
c1
k 1.05
E cm
c1
f cm Concrete elastic modulus at the origin (E c) needs to be defined in ABAQUS, but it is not directly used in the formulation above. The secant modulus Ecm is defined as in Fig. 1, while the elastic modulus (Ec) at the onset of the curve can be approximated as Ec=1.05×Ecm [10]. However, this approximate is not sufficiently precise and creates discontinuities between the elastic and the plastic part of the curve. Hence it was not used in the model. Instead, the concrete behaviour was considered elastic only between the origin and the first increment corresponding to a very small compressive strain c=0.00005. The remaining part of the curve is following equation 1.
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Fig. 1: Concrete stress-strain curve in uni-axial compression
The tension behaviour of concrete was defined according to Wang and Hsu [9]: Ec if t t t cra 0.4
f cm
t
cra
if
(2) t
cra
t
Finally, material properties were converted in true values, and concrete damage dc was defined to range from zero for undamaged material to one for the total loss of load-bearing capacity:
dc
1
tk
if
tk
tk 1
; k
c,t (tension, compression)
(3)
tk max
Steel was modelled by a tri-linear stress–strain curve as shown in Fig. 2. The material behaviour is initially elastic, followed by strain softening and then perfect plasticity.
Fig. 2: Steel stress-strain curve
3.2 Geometry and element types For simplicity studs and tie-bars were modeled without weld or head only the shank being considered, permitting the modelling of studs and ties with beam elements. Steel plates were
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modelled with shell elements to allow compression buckling. Stud diameter and plate thicknesses were therefore defined in the section module of ABAQUS. The concrete infill has been modelled with 3D solid elements. One of these models is illustrated in Fig. 3.
Fig. 3: First model sophistication level
The steel plates were modelled with S4R shell elements, a 4-node, quadrilateral, stress/displacement shell element with reduced integration and a large-strain formulation. The studs were modelled as B31 beams, a 2-node linear Timoshenko beam which allows transverse shear deformation. ABAQUS assumes that the transverse shear behaviour of Timoshenko beams is linear elastic with a fixed modulus and, thus, independent of the response of the beam section to axial stretch and bending. Finally, concrete was modelled with C3D8R hexahedron elements, an 8-node linear brick, reduced integration with hourglass control. 3.3 Constraints and interactions Constraints and interactions needed to be defined between steel and concrete. For simplicity, studs were modelled as wires embedded in concrete material. This means that the translational degrees of freedom of the embedded node are constrained to the interpolated values of the corresponding degrees of freedom of the host element. This technique is usually used to define contact between rebar and concrete in classical reinforced concrete structures, and has its limitations when used to model dowel action as in this case. Regarding interactions between steel plates and concrete, no particular constraint was applied besides hard contact in perpendicular direction. This choice allows the development of potential plate buckling failure modes. In reality, the two materials are also linked by steelconcrete chemical bounds, which are considered negligible compare to the capacity of studs. 3.4 Loads, supports, and boundary conditions In order to be as realistic as possible, both the loading and the supporting elements were modelled as rigid surfaces. A reference node was associated to each surface and used to apply the boundary conditions. At the support only the translational degree along the vertical axis and the rotational degree around the longitudinal axis were blocked. A tie constraint was applied between the support and the bottom plate in order to avoid relative displacement. Regarding the loading elements, only the rotational degree around the transversal axis was left free, and the load was applied as displacement along the vertical axis. Loading element and support dimensions were taken from drawings of the experiments. In order to make the FE model computationally cheaper, we always useed the symmetry of the problem within length and width when possible.
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4 Results and explored model improvements With this level of modeling sophistication, the behaviour of SC beams is predicted with good accuracy when the failure is governed by plate buckling, plate yielding, concrete cracking and concrete crushing. However, when the failure is due to studs shear failure or interactions between studs and concrete, results are not accurate. This means that the concrete formulation is adequate, and shell elements are properly predicting the behaviour of plates. However, the model cannot be used to estimate failure involving studs. The observed problems may result from the way studs are modelled, but also from either the stud material properties or the interaction definition between studs and concrete. The following, section presents ways to improve the model tested by the authors. 4.1 Stud/concrete interactions Studs defined with beam elements exert concentrated pressure on the concrete because of their geometry. Moreover they cannot take correctly into account the localized shear deformation and failure of the steel. After defining studs with the 3D geometry using brick elements results were still inaccurate; the ultimate load levels being too high compared to experiments. We concluded that the problem was generated by the embedded condition. In fact, the embedded part does not replace the host entity, leading to an increase of mass and stiffness corresponding to the addition of the two materials. Replacing embedded condition by holes in concrete could be a solution, with the price of requiring more refined meshing of the concrete. 4.2 Stud/plates interactions One way to improve the model was to define a stud-plate interaction based on the shear resistance of studs. Several ways are possible in ABAQUS by placing fasteners, connectors, cohesive elements, surface constraints, etc. between the stud-wire and the plate-shell. After trial-modelling with these solutions, the correct level of ultimate loading was found with cohesive elements. This is also a very practical mesh-independent method, so the plate can be meshed disregarding the stud positions. But, properties for the cohesive element need to be defined, representing all mechanical properties of the stud-concrete interaction. Deriving such properties is simple based on push-out test [11] or analytical formulas, but the model becomes dependent on pre-calibrated parameters. During deformation, the de-bounding of studs is also not captured, the model exhibiting excessive stiffness in early stages of deformation. 4.3 Damage formulation for studs So far steel stress-strain curve was been defined tri-linear, the material following a perfect plasticity state after the ultimate point. One possible model improvement is to introduce damage formulation for the studs according to Fig. 4. Progressive ductile and shear damage models in ABAQUS were used to take in account failure modes. Damage evolution laws were inputted in ABAQUS in tabular form as damage variables as function of equivalent plastic displacement. However, in the analysed models damage criterion in the studs was not reached, because failure modes related to concrete crushing in the stud-concrete interaction was controlling the ultimate resultant forces.
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Fig. 4: Theoretical damage formulation for studs
4.4 Improvements summary Several ways of model improvement have been explored. They were collected from suggested modelling techniques for SC structures [7][11][12][13]. For each technique, the validity of the complex interaction parameters or material properties was confined to the particular cases examined. But, when applying these methods on structures with varying failure modes, it appears that it is impossible to predict two different failure modes without changing material or interaction properties, which is highlighting the incoherence of the techniques.
5 Proposed modeling complexity The proposed FE model is based on the initial model described in Chaper 4 with the improvements highlighted below. 5.1 Material properties For the final modelling proposal steel properties for stud have been refined. Since Nelson studs are standardized products, the stress strain curve reported by Pavlovic et al [13] has been used (Fig. 5). This curve was introduced in ABAQUS in true values, represented by the yielding point, the ultimate point and the fracture point. The stud material damage formulation from Fig. 4 was also preserved in the model, even if it is never activated in the SC elements tested by us. Other material properties were left unchanged.
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Fig. 5: Experimental stress-strain curve [13]
5.2 Geometry and element types Improvements on geometry concern only studs and concrete. Studs and tie bars were now defined with the entire geometry, holes being created in concrete to host studs and ties. One of these models is illustrated in Fig. 6. As it can be observed, the complexity of the geometry is greatly increased and partitioning is necessary to achieve a more orderly meshing.
Fig. 6: Geometry of the proposed model
5.3 Constraints and interactions General contact definition remained unchanged, but the chemical bound between studs/ties and concrete were included using tie constraints. 5.3 Loads, supports, and boundary conditions The general contact condition was not sufficient to avoid penetration of the loading element into the beam. Thus tie constraint was applied between the plate and both for loading and support elements. Displacement along the longitudinal axis was freed in order to maintain the same degrees of freedom. 5.4 Results Before starting to compare predictions with test results, it is important to mention that some parameters (material, geometry, supports etc.) have not been accurately reported with all ex-
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periments. This lack of accurate knowledge can lead to some approximation and inaccuracies. The selected results presented should be interpreted in light of these potential sources of inaccuracies. We settled for modeling results based on first approximation of the unknown parameters, without making further attempts to bring model results closer to test results. Two beams are reported, B2 tested by Oduyemi and Wright [5] and SP1-5 tested by Varma et al [7]. The geometries are presented in Fig. 7 for the half of the beams. In both cases the loading was 4 point bending type. But the two beams are very different, in scale, shear reinforcement type and expected mode of failure.
(a)
(b) Fig. 7: Geometries of (a) beam B2 and (b) SP1-5 as used in the models (not all geometric parameters could be recovered from the test reports, some are estimated)
5.4.1 Failure by bottom plate yielding and concrete crushing The beam B2 tested by Oduyemi and Wright [5] failed by yielding of the bottom steel plate and by crushing of the concrete on the upper side. Experimental results and FEM analysis results are presented in Fig. 8. This failure mode is correctly predict by the model. The first cracks occur at the section near the load point and start around studs. The force-displacement prediction is reasonably accurate. The bottom plate starts to yield at the same load and displacement, but the ultimate load is 10kN higher, an inaccuracy probably caused by material property estimates. 5.4.2 Failure by shearing of the concrete The beam SP1-5 tested by Varma et al [7] failed by vertical shear. Experimental results and FEM analysis resulsts are presented in Fig. 8. The FEM result is quite accurate in terms of load and displacement, and the failure mode is also correctly attributed to sudden and brittle shear failure of the concrete. However, a time increment of 0.0001 is not small enough to guaranty the stability of the ABAQUS Explicit model.
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6 Conclusion Based on the presented model results and further calibration modeling we can conclude that the updated FE model, as presented in Chapter 5 of the paper, predicts with reasonable accuracy a broad range of failure modes relevant to SC structures: yielding of the steel plate in tension, de-bonding of tensile sheet related to stud shear failure, yielding of the steel plate in compression, buckling of the steel plate in compression, concrete crushing, and shear failure in concrete. It is also an advantage that the model only needs to be supplied with material properties for steel and concrete, avoiding the need to calibrate superfluous parameters e.g cohesive elements mimicking stud-concrete interaction or steel-concrete bond. 100
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800
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F(kN)
F(kN)
60 50
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B2, Model 0
0 0
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Deflection (mm)
100
125
0
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Deflection (mm)
Fig. 8: Experiment vs. FE modeling result for beam B2 and SP1-5
The time increment in ABAQUS Explicit needs to be decreased in case of sudden failure modes, in order to improve stability of the response. The FE models have a run-time of about 10-12 hours on an average 4-5 core computer, making them potential candidates for a limited parametric study. At large deformations an adaptive Lagragian-Eularian meshing may solve problems of distorted element, which was often observed to be the limiting error for the response. The modeling technique described here were used for developing a general purpose PYTHON based plug-in in ABAQUS in order to allow quick generation of SC beam geometries, and permit work with different geometric configuration efficiently.
Acknowledgments The work here has been supported by the RFCS “SC for Industrial, Energy and Nuclear Construction Efficiency (SCIENCE)” project. An industrial placement from the French Institute of Advanced Mechanics (IFMA) generously supported the work of the lead author in Finland. The lead author also thanks member of the VTT’s Infrastructure Health group, for the constant support, patience, and the time spent to explain their work.
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Notation Ecm Ec fcm c t tc tt c t c1 cr cra cu1
Secant elastic modulus of concrete in compression Initial elastic modulus of concrete in compression Concrete cylinder compressive strength Compressive stress Tensile stress True compressive stress True tensile stress Compressive strain Tensile strain Strain at ultimate strength Concrete crushing strain Concrete cracking strain Stress at failure
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