Numerical modeling of composite beam to reinforced concrete wall joints

Engineering Structures 52 (2013) 747–761

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Numerical modeling of composite beam to reinforced concrete wall joints Part I: Calibration of joint components José Henriques a,⇑, Luís Simões da Silva a, Isabel B. Valente b a b

ISISE, Civil Engineering Department, University of Coimbra, Pinhal de Marrocos, 3030 Coimbra, Portugal ISISE, Civil Engineering Department, University of Minho, Campus de Azurém, 4800-058 Guimarães, Portugal

a r t i c l e

i n f o

Article history: Received 24 August 2012 Revised 24 January 2013 Accepted 5 March 2013 Available online 29 April 2013 Keywords: Longitudinal reinforcement bars Solid elements Reinforcement–concrete interaction Composite behavior Plasticity Contact

a b s t r a c t This paper presents the first part of a numerical study to simulate composite beam to reinforced concrete wall joints using the finite element software ABAQUS. In detail, several benchmark examples dealing with different components of the joint are simulated in the validation and calibration process of the finite element package. Moreover, these simulations consider the analysis of: (i) type of finite element, 3D solid first and second order elements; (ii) material constitutive law for steel and concrete; (iii) interactions, reinforcement–concrete bond, composite behavior and mechanical contact. The validation of the simulated benchmark examples is accomplished by means of convergence studies and comparison with experimental tests. The accuracy obtained within these benchmark examples puts in evidence the appropriate simulation of the different phenomena to be dealt within the analysis of composite beam to reinforced concrete wall joints in a companion paper. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction 1.1. Objectives A common structural solution for office and car park buildings combines reinforced concrete walls and composite members. In such structural systems, the joints between the structural elements are a challenge, as the lack of knowledge on the joint behavior is reflected in the absence of general solutions and design approaches. Historically, the study of structural joints has been focused on steel and composite joints [1–4] and less attention has been paid to joints between members of different nature, e.g. composite beam to reinforced concrete wall, where column bases are the exception [5,6]. To address this issue, the recent RFCS project entitled ‘‘InFaSo’’ [7] envisaged the development of simple and efficient steel-to-concrete structural joints. Different types of joints were studied and amongst the proposed solutions is the composite beam to reinforced concrete wall joint depicted in Fig. 1. During the ‘‘InFaSo’’ project, the behavior of this moment resisting joint was essentially studied by means of experimental tests which indicated an adequate and efficient structural performance [7]. A limited number of tests were performed and therefore, a numerical study was required to obtain a more complete description of the joint behavior. ⇑ Corresponding author. Tel.: +351 919125472. E-mail address: [email protected] (J. Henriques). 0141-0296/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engstruct.2013.03.041

Developing a numerical model is not a simple task, and its difficulty increases with the complexity of the problem to be reproduced. For example, modeling a simple steel beam may be considered simpler than modeling a joint (three-dimensional problem). The numerical tool is the first obstacle to overcome. The complexity is then further increased when the modeling includes material and geometrical non-linearity, reinforced concrete, contact and composite interaction. For all these reasons, the development and the validation of a numerical model is a laborious task to perform. This is further evidenced by the reduced numerical work previously performed on composite joints. In [8,9] some initial work was undertaken in this field but not in the most consensual way. Later, more generally-acceptable approaches were implemented in [10,11]. However, these studies were focused on the presentation and validation of their corresponding models, and a detailed investigation of the joint behavior was not accomplished. Moreover, when dealing with composite beam to reinforced concrete wall joints, the absence of numerical approaches is notorious. The general aim of this paper is to present the calibration of a finite element model needed to reproduce the complex phenomena involved in the joint configuration illustrated in Fig. 1. Accordingly, the following issues are calibrated through benchmarking:  type of 3D solid finite element, first order elements and second order elements;  constitutive models for steel and concrete;

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a

f g

d

a –Longitudinal reinforcement bars b –Steel bracket c –Anchor plate d –Headed anchors e –Steel plate welded to steel bracket f –Steel beam end plate g –Steel contact plate

b e c

is welded on the external surface of the steel bracket, creating a ‘‘nose’’ that prevents the beam to slip out of the steel bracket. An end plate (f) welded on the steel beam is used to achieve mechanical interlock with the steel bracket. Finally, the joint is completed with a steel contact plate (g) between the steel beam end-plate and the anchor plate. 2.2. Identification of the joint components numerically simulated The joint was only tested under hogging bending moment. The main components that influence the joint response are identified in Fig. 2 and consist in the following:

Fig. 1. Joint configuration proposed in [7].

 reinforcement–concrete interaction, perfect bond and approximation to the real bond behavior;  composite behavior, steel beam to concrete slab interaction;  mechanical contact. The numerical calibration and successful validation of the above phenomena is then exploited for the simulation of the complete composite beam to reinforced concrete wall joint. This second part of the work is presented in a companion paper [12]. 1.2. Adopted methodology The numerical tool used is the non-linear finite element package ABAQUS (v6.11) [13]. ABAQUS provides a complete and flexible solution for a large range of problems. The number and the variety of finite elements available in the ABAQUS library are quite large. Geometric and material non-linearity are included in the analysis. Static-Riks FEA solver is used for the solution of non-linear problems. The Static-Riks solver in Abaqus is the ‘‘modified Riks method’’ to obtain nonlinear static equilibrium solutions for unstable problems. In this method, the basic algorithm remains the Newton method. However, the solution is viewed as the discovery of a single equilibrium path in space defined by the nodal variables and the loading parameter. The development of the solution requires that this path is traversed as far as required. At any time, a finite radius of convergence is found. Regarding the mesh, the software offers both automatic and manual meshing techniques. These techniques have application on a wide range of engineering problems, and therefore, their complete description is outside the scope of the paper and more detailed information about the software is available in [13]. The numerical models developed consist of three-dimensional solid elements where geometrical and material non-linearity are taken into account. The calibration and validation of the models is performed through convergence of results and comparison with the experimental results.

i. Longitudinal reinforcement: according to the tests, this is the governing component. Therefore, the accuracy of the numerical model heavily dependents on this component. The post-elastic behavior of the steel reinforcement bar is therefore crucial. ii. Reinforcement–concrete interaction: given the role of the longitudinal reinforcement, its interaction with concrete in the highest stressed region, located in the slab–wall interface, is non-negligible. iii. Slab–wall interface: only the longitudinal reinforcement performs the connection between these two members. iv. Composite behavior: the level of interaction influences the load capacity of the composite beam and the slip between the steel beam and the concrete slab contributes to the joint rotation. v. Beam bottom flange in compression: the yielding of the bottom flange of the steel beam may limit the resistance of the joint. vi. Anchor plate in compression: concrete crushing or yielding of the steel components can introduce limitations to the load carrying capacity of the joint. 3. Description of the non-linear finite element model 3.1. Type of finite elements The joints proposed for finite element modeling embody situations of material discontinuity, yielding, stress concentration, contact and composite behavior. These complex 3D phenomena are reproduced by adopting hexahedra solid elements [13], which

iii.

Load i., ii.

2. Composite beam to reinforced concrete wall joint

iv.

2.1. Joint configuration

v. Two zones may be distinguished in the joint configuration illustrated in Fig. 1. The upper zone consists of a reinforced concrete slab where the longitudinal reinforcement (a) is extended and anchored to a reinforced concrete wall. The wall and the slab are concreted in different stages and therefore no continuity is assumed between these concrete members. In the bottom zone, the steel beam sits on a steel bracket (b) welded to a plate (c) that is anchored to the reinforced concrete wall. The anchorage of the plate to the wall is performed with headed anchors (d). A steel plate (e)

vi.

Fig. 2. Identification of the main components to be analyzed.

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are used to model the majority of the parts of the specimens. Exception are the ordinary reinforcement in the concrete wall and concrete slab, where truss elements are used, and the shear studs in the composite beam modeled with spring elements (axial springs). In ABAQUS, the finite elements C3D8 (8 nodes) (Fig. 3a) and C3D20 (20 nodes) (Fig. 3b) are continuum stress/displacement 3D solid finite elements of first and second order, respectively, either with reduced or full integration. The 8-node element with reduced integration is chosen for the general numerical simulations. A sensitivity analysis is performed with respect to the element type in order to check the required mesh density for application of this type of elements. For this purpose the second order element (C3D20R) with reduced integration is used to obtain the ‘‘correct’’ solution. This issue is dealt later within several of the benchmark studies. The truss element (T3D2) used to model the ordinary reinforcement is a 2-node linear 3D truss element that can only transmit axial forces.

3.2. Interactions 3.2.1. Reinforcement–concrete interaction In ABAQUS, the 3D modeling of reinforced concrete may be performed using the following options: rebar layer embedded in the concrete; truss element (2-noded elements) embedded in concrete; and solid (continuum) elements, embedded or not in the concrete. Due to the importance of the longitudinal reinforcement in the composite beam (single tension component in the joint), it is modeled with 3D solid elements. For this type of element, two strategies may be defined to model the interaction with the con-

8

7

8

7

15

16 6

5

14 6

13

5

19

20

17 4

18

3

4

3

11

12 1

2

1

10 9

(a)

2

(b)

Fig. 3. a) Linear order solid element with 8 nodes (C3D8); b) Quadratic order solid element with 20 nodes (C-3D20).

crete: (i) embedded, which corresponds to perfect bond behavior (rigid link between reinforcement and concrete nodes) or (ii) bond behavior, whereby the bond between concrete and reinforcement is modeled by an approximation of the bond–slip response. The first technique consists of physically superposing the two parts. It is based on master and slave regions, where the nodes of the embedded region (slave, the reinforcement) displace by the same amount as the closest node of the host region (master, the concrete). Such type of modeling enforces a perfect bond between master and slave. However, it is only valid when stress transfer is medium–low. For highly stressed regions, e. g. near cracks, there are different strains in the concrete and in the reinforcement, as slip occurs due to the loss of bond. Therefore, modeling the interaction with perfect bond leads to excessive stresses in concrete and stiffer response of the reinforcement. Two strategies may be considered to model the bond behavior in the reinforcement–concrete interface: (i) cohesive elements; and (ii) contact with cohesive behavior. Both techniques imply that the reinforcement and the concrete elements are not superposed, contrarily to the embedded technique. Consequently, their implementation is time consuming. In order to reduce the modeling time, the ‘‘real’’ bond behavior is only applied in regions where the expected transfer of stresses is higher, which is around the wall–slab interface. The perfect bond model is applied in all the other parts of these reinforcement bars and for all ordinary reinforcement. 3.2.2. Steel beam–reinforced concrete slab interaction (composite behavior) In ABAQUS [13], the composite behavior between the steel beam and the concrete slab may be modeled by using the following options: (i) modeling the shear studs physically and establishing contact between concrete and studs; (ii) using special elements, either connectors’ elements or springs elements. The first option is the most realistic, but it leads to increased calculation time and convergence difficulties due to contact problems. The geometry of the shear studs also adds significant difficulties for the mesh generation. For the second option there are two alternatives, the connector and the spring elements. The main difference between these alternatives is that connectors are much more generic, allowing the combination of different degrees of freedom in a single connector element. In the case of springs, properties have to be defined for each degree of freedom; however, its definition is simpler. Thus, the use of springs is preferred because of their satisfactory accuracy, simplicity of definition, less time consumption and better convergence. This option was also adopted by Gil and Bayo [11] where the composite behavior was successfully modeled in

iii i

vi

iv

ii v

ix

viii

Fig. 4. Localization of the different parts in contact.

vii

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ABAQUS [13] using this type of elements. Each spring only connects one degree of freedom between two nodes. Thus, to model the shear connection, two springs are needed per shear connector, one for longitudinal and one for normal (vertical) interaction. 3.2.3. Mechanical contacts For the joint configuration presented in Fig. 1, besides the interactions described above, reinforcement–concrete and concrete slab–steel beam, several other interactions have to be treated. The following interactions are considered (Fig. 4): i. ii. iii. iv. v. vi. vii. viii. ix.

Concrete slab to concrete wall. Top flange of the steel beam to concrete slab. Steel beam to beam end-plate. Beam end-plate to steel contact plate. Beam end-plate to steel bracket. Steel contact plate to anchor plate. Anchor plate to steel bracket. Anchor plate to concrete wall. Headed anchors on anchor plate to concrete wall.

These interactions are included in the analysis as contact problems. Accordingly, two types of contacts are considered: (a) hard contact with and/or without friction, where contact is modeled based on contact-pairs, master–slave surface interaction; (b) rigid link (tie option), which is also based on master and slave interaction, but no sliding is allowed and the parts are rigidly connected. The types of contact used for the above list of interactions are identified in Table 1 and also illustrated in Fig. 4. Note that welds are not modeled and therefore the tie option is used to connect welded steel parts. 3.3. Constitutive laws and mechanical properties The constitutive laws used to model the mechanical behavior of the different materials and interactions considered include five types: Concrete Damage Plasticity; Isotropic Material; Bond (stress–slip); Uniaxial Spring (force–displacement) and Hard Contact (with and without friction). The behavior of concrete may be modeled in ABAQUS [13] using three types of constitutive laws: Concrete Smeared Cracking; Concrete Damage Plasticity; Drucker–Prager. Based on previous work [11], the Concrete Damage Plasticity was chosen because it is simpler to model and more stable for the numeric calculation. The Concrete Damage Plasticity constitutive model is defined by a uniaxial compression and tension response (Fig. 5), where five constitutive parameters are needed to identify the shape of the flow potential surface and the yield surface [14]. In this constitutive model, a non-linear stress–strain relation is assumed for compression. As the entire stress–strain curve is not available from the test reports and only the usual parameters are reported (compression strength and Young’s modulus), the concrete uniaxial compressive

Table 1 Identification of the type of contacts considered in the model.

Fig. 5. Concrete uniaxial stress–strain response in compression and tension.

behavior is obtained by applying Eq. (1), as specified in [15] for non-linear analysis.

rc =fcm ¼ ðkg  g2 Þ=ð1 þ ðk  2ÞgÞ

ð1Þ

In (1), rc is the concrete stress, fcm is the mean concrete cylinder compressive strength, k and g are two factors determined according to Eqs. (2) and (3), Ecm is the secant modulus of elasticity of concrete, ec is the concrete strain, ec1 is the compressive strain at the peak stress fcm and ecu1 is the ultimate compressive strain in the concrete.

g ¼ ec =ec1

ð2Þ

k ¼ 1; 05Ecm jec1 j=fcm

ð3Þ

In tension, the behavior is assumed elastic up to the onset of cracking and then followed by tension softening. Tension softening may be introduced in the calculations by means of a Stress-Cracking Strain curve, a Stress-Displacement curve or considering Fracture Energy. In the absence of experimental information to characterize the strain softening response of the concrete in tension, the concept of facture energy was used in the models [13]. The fracture energy is defined as the energy required to form a single crack that according to [16] and expressed in (4) can be estimated as a function of the mean compressive strength of the concrete. Finally, for the five constitutive parameters (w – dilatation angle; e – flow potential eccentricity; fb0/fc0 – ratio of initial equibiaxial compressive yield stress to initial compressive yield stress; k – ratio of second stress invariant on the tensile meridian; l – viscosity parameter) required to complete the definition of the constitutive model, no information was available from the experimental tests and therefore default values [13] were used (see Table 2).

Gf ¼ 73ðfcm Þ0;18

ð4Þ

friction and with friction (both options were tested)

To model the behavior of structural steel and reinforcement, the classical isotropic material law that implements the von Mises plasticity model (isotropic yielding) is used. For the generality of the steel parts, an elasto plastic behavior with hardening is

friction friction

Table 2 Additional parameters to define CDP constitutive model.

Interaction

Type of contact

i ii iii iv v vi vii viii ix

(a) Without (a) Without (b) (b) (a) Without (a) Without (b) (a) Without (a) Without

friction friction

w

e

fb0/fc0

K

M

38°

0.1

1.16

0.67

0

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assumed. For the longitudinal reinforcement, the governing component of the joint, the true stress–strain (rtrue-etrue) material curve available from the experimental tests is considered. The latter properties are calculated as expressed in (5) and (6) [17] using the nominal properties (e, r) obtained in the tests.

rtrue ¼ rð1 þ eÞ

ð5Þ

etrue ¼ lnð1 þ eÞ

ð6Þ

tion, kss and ktt, are obtained by approximation of the model described in Fig. 6a, as expressed in (9). Regarding knn, according to Gan [19], the stiffness of the normal traction is higher than in the shear direction and is expressed in (10):

kss ¼ ktt ¼ smax =S1

ð9Þ

knn ¼ 100kss ¼ 100ktt The theoretical basis for the modeling of the reinforcement– concrete interaction behavior is the stress–slip relation proposed by Eligehausen et al. [18]. It constitutes the basis of the model prescribed in [16] and is represented in Fig. 6a. It is characterized by an increasing non-linear branch, up to a maximum stress (smax). Then, depending on the concrete confinement, a plateau may be assumed followed by a linearly decreasing branch that achieves a second plateau at the ultimate frictional bond resistance (sf). The slip at maximum stress (S1 and S2) and at ultimate frictional bond resistance (S3) depends on the confinement and bond conditions. Given the absence of experimental data, the stress–slip relationship was computed from the concrete resistance, as prescribed in [16]. In ABAQUS, this type of behavior can only be approximated by the traction–separation law (Fig. 6b). For the increasing branch only a linear response is allowed. Subsequently, the beginning and evolution of damage can be reproduced. This type of behavior is valid for both cohesive elements and contact elements with cohesive behavior. The elastic definition is written in terms of nominal tensile stresses and nominal strains. In the case of cohesive elements, defining a unit length, separations (displacements) may be used directly instead of strains. According to [13], for contact with cohesive behavior, deformations are considered directly as separations (displacements). The constitutive relation can be uncoupled or coupled, as expressed in (7) and (8), respectively.

8 9 2 knn > < tn > = 6 T ¼ ts ¼ 4 0 > : > ; tt 0 8 9 2 knn > < tn > = 6 T ¼ t s ¼ 4 kns > : > ; tt knt

0 kss 0 kns kss kst

38 9 > < dn > = 7 0 5 ds ¼ Kd > : > ; dt ktt

ð7Þ

38 9 knt > < dn > = 7 kst 5 ds ¼ Kd > : > ; dt ktt

ð8Þ

0

In Eqs. (7) and (8), tn represents the nominal tractions in the normal direction, ts and tt represent the nominal stresses in two local shear directions, respectively; dn, ds and dt are the displacements related to corresponding nominal strains (en, es, et); and kij are the stiffness coefficients. There is not much information on how to determine the stiffness coefficients (kij) and consequently it was decided to use the uncoupled behavior. The coefficients related to the shear deforma-

(a)

751

ð10Þ

The mechanical behavior of the springs representing the shear connectors depend on the direction of the spring. For the connection in the normal direction, this spring is assumed to be fully rigid and therefore an infinite stiffness is considered. For the connection in the longitudinal direction, the response of the spring is tested in three different forms: (i) elastic; (ii) elastic-perfectly plastic and (iii) full non-linear. For the two first options, the properties are obtained from EN 1994-1-1 [20] and these rely on: initial stiffness [ksc]; resistance [PRk] and ultimate slip [du]. The full non-linear response is based on the Ollgaard et al. [21] model. The application of these three models is analyzed later. Finally, regarding the other interactions, the contact and interface stresses are treated using the ‘‘hard’’ contact model with frictional behavior. In the ‘‘hard’’ contact: (i) pressure is transmitted when nodes of slave surface contact the master surface; (ii) no penetration is allowed; (iii) no limit to the pressure is assumed when the surfaces are in contact. The frictional behavior is guaranteed by the stiffness (penalty) method and the sliding conditions between bodies are reproduced with the classical isotropic Coulomb friction model. For more detailed information reference is given to [13]. 4. Calibration of the finite element modeling: benchmark examples 4.1. Longitudinal steel reinforcement bar: discretization and stress– strain curve Two benchmark examples are carried out to accurately model the governing component of the joint. In the first case, as solid finite elements of hexahedral shape are used, the bar cross-section has to be discretized so that the resulting polygon circumscribed by the circular cross-section (Fig. 7a) minimizes the error of a non-circular shape. In addition, the required discretization to use C3D8R type of elements is calibrated performing a convergence analysis using the second order elements with reduced integration (C3D20R). The example illustrated in Fig. 7b is very simple and considers the simulation of a steel bar with 16 mm diameter (size of longitudinal reinforcement bar in the reference specimen of the joint configuration under study) subject to a tensile load. The load is applied at one edge of the bar, imposing displacements, and at the opposite

(b)

Fig. 6. (a) Bond–slip model proposed by Eligehausen et al. [18] and prescribed by the Model Code [16]; (b) typical traction–separation response available in ABAQUS [13].

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Supported section

Load application section

(a) Bar cross-section discretization

(b) Reinforcement bar in tension

Fig. 7. Benchmark example to calibrate the reinforcement bar discretization.

edge of the bar all nodes are fully fixed in all degrees of freedom. The material properties assigned to the steel bar were obtained in tests reported in [7]. The yield and the ultimate strength of the steel are 450 N/mm2 and 745 N/mm2, respectively. The required discretization of the cross-section using the C3D8R elements is obtained through convergence of results. For this purpose, the second order elements (C3D20R) are used to obtain the correct solution. Table 3 summarizes the performed simulations. The comparison of results is performed by means of force–deformation in the bar axial direction. The force represents the applied load and the deformation regards to the displacement at the edge of the applied load. Fig. 8 shows the performance of the different models and includes an analytical calculation of the problem. The curves of the models using the second order elements and of the analytical calculation are barely undistinguishable, indicating the correct solution (convergence obtained). The application of the first order elements demonstrate that the edge of the cross-section should be discretized in at least 20 nodes to obtain a good approximation. The main deviations are observed in the plastic range where the maximum load capacity is underestimated in the models with less cross-section edge nodes, e.g. using only 8 nodes the maximum load is 11% lower. The second benchmark example for the reinforcement bar analysis regards the two following aspects: (i) the type of stress–strain curve for the material behavior, nominal or true; (ii) a failure criterion. A typical uniaxial tension test (Coupon Test) of a steel reinforcement bar is used as benchmark. This test was part of the experimental program presented in [22]. The numerical simulation considers the same loading strategy as illustrated in Fig. 8, but the diameter of the bar is now 12 mm. The nominal yield and ultimate strength are 494 N/mm2 and 705 N/mm2, respectively. Two numerical simulations were performed to compare the obtained response when considering the following material properties: (i) nominal stress–strain curve; (ii) true stress – logarithmic strain curve. The results of the numerical simulation are presented in Fig. 9. The force–elongation (Dl) curve measured in the tested

Fig. 8. Force–deformation curves of the convergence study for reinforcement bar discretization using first order elements (C3D8R).

bar is used for comparison. If the nominal stress–strain curve is used instead of the true stress–logarithmic strain curve, a worse approximation is obtained when plastic deformation appears. This result is expected and demonstrates that ABAQUS [13] requires the true stress–logarithmic strain if the problem enters in the large deformation range. Common to both numerical simulations is a descending branch appearing earlier than in the experimental test. This deviation is justified by the fact that the correction of the nominal curve to real properties using expressions (5) and (6) is only valid up to the phenomenon of necking occurs in the bar. After this stage, the data

Table 3 Performed models to calibrate reinforcement modeling. Model ID

Description

R-C3D20R-8 R-C3D20R-16 R-C3D8R-8 R-C3D8R-16 R-C3D8R-20 R-C3D8R-32

Element Element Element Element Element Element

type: type: type: type: type: type:

C3D20R cross-section edge nodes: 8 C3D20R cross-section edge nodes: 16 C3D8R cross-section edge nodes: 8 C3D8R cross-section edge nodes: 16 C3D8R cross-section edge nodes: 20 C3D8R cross-section edge nodes: 32

Fig. 9. Force–elongation curves comparing the type of nominal and true stress– strain material properties.

J. Henriques et al. / Engineering Structures 52 (2013) 747–761

obtained in the Coupon Test cannot reproduce with sufficient approximation the properties of the material. The considerable reduction of the bar cross-section, in the necking region, leads to a decrease of the external applied load; however, the material strength is still increasing (hardening). The decrease of strength is only observed when damage of the material takes place. Though, the nominal properties are computed in function of the initial cross-section and therefore, a reduction of the material strength is reported before it really occurs. Therefore, the true properties determined with (5) and (6) are affected by this phenomenon. Without sufficient information to correct the material curve, subsequently to necking (identified by the maximum strength of the material obtained from the Coupon Test) a plateau was considered (Fig. 10a). As result, in the numerical simulations when the reduction of the cross-section begins to be significant and the stress on the material has achieved its maximum, a decrease of the applied load is observed. In what regards to a failure criterion, it was decided to introduce a limitation on the material strain. Thus, in the material properties curve, when a maximum strain of 7.5% is achieved, in accordance with the tests report [22], the stress of the material is decreased to zero, as illustrated in Fig. 10a). Fig. 10b) shows the force–deformation curve that compares the experimental results with the numerical simulations. The model without failure criterion is also included. As expected, the simulation with failure criterion limits the response before the test results. However, taking into account the simplicity of the considered failure criterion, the approximation is acceptable. Furthermore, the ultimate strain assumed is also affected by the local phenomenon described above, as the value considered is obtained from nominal results.

753

4.2. Reinforcement–concrete interaction The numerical simulation of the reinforcement–concrete interaction was first evaluated using experimental tests on reinforcement stirrups performed at the University of Stuttgart within the InFaSo project [7]. These tests consisted of pull-out tests of reinforcement bars with a hook shape embedded in concrete blocks. For detailed information on the tests reference is given to [23]. Fig. 11 shows the numerical model developed for this benchmark example. Profiting from the symmetry of the problem, only half of the specimen is considered. Solid (continuum) elements type C3D8R are used in both parts; reinforcement and concrete. Only in the straight part of the reinforcement bar embedded in the concrete, the bond–slip behavior is considered. In this region, the concrete block elements do not superpose with the reinforcement bar elements (hole created in the concrete block). In this way, the contact interface between concrete and reinforcement was handled numerically as contact problem with bond behavior. This technique was described in the prior section. For the hook, the embedded technique, also described above, is used. The force–deformation curves presented in Fig. 12 compare numerical and experimental results. The force represents the pull-out load applied to the reinforcement bars and the deformation corresponds to the displacements measured in the reinforcement bar at the concrete surface. The numerical curve is close to a linear relation which may be justified by the traction–separation law approximating a linear stress–slip response, as described in Section 3.3. Besides this deviation, there is a good approximation of the numerical model. The plateau observed in the numerical result is due to the yielding of the reinforcement. The latter occurs

Fig. 10. Application of a failure criterion to the numerical simulations.

Fig. 11. Benchmark example to calibrate the numerical modelling of the reinforcement–concrete interaction.

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beam and the deformation corresponds to the mid span deflection. The experimental curve is included. The presence of reinforcement is clearly noticed, as the simulation with reinforcement almost achieves the experimental resistance. The model without reinforcement presents difficulty to converge already at a low load level, as the concrete quickly achieves its tension capacity and there is no reinforcement to transfer the stresses at the bottom part of the beam. Fig. 15 shows the von Mises stresses developed in the reinforcement for an applied load of approximately 22 kN. For this load level, the longitudinal reinforcement achieves its yield capacity, at the mid-span of the beam.

Fig. 12. Force–deformation curve evaluating the application of a bond model to simulate reinforcement–concrete interaction.

early in the numerical calculations because nominal properties (EC2 values) of the steel are used instead of the material properties obtained in experimental tests which were not available. In the same figure, the result of a similar model with perfect bond between reinforcement and concrete in the entire interface is included. It can be observed that for low load levels, the behavior of the latter is close to the experimental observations. For higher loads, a stiffer response is obtained. Fig. 13a) shows the von Mises stresses along the reinforcement bar for an applied load of about 100 kN. Note the degradation of stresses in the reinforcement bar starting at the concrete surface and following within the embedded part. The bond stresses in the contact surface are shown in Fig. 13b). Degradation of these stresses is also observed with the embedment depth. The second benchmark example to evaluate the reinforcement– concrete interaction consists in a reinforced concrete beam simply supported and loaded at the mid-span with a concentrated load. The description of these experimental tests is outside the scope of this paper and detailed information may be found in [24]. The evaluation was mainly focused on the effect of the longitudinal reinforcement on the beam response, verifying its activation. In this way, two models were performed: one with and one without reinforcement. The obtained force–deformation curves are compared in Fig. 14, where the force is the total load applied at the

4.3. Connection of concrete members using only longitudinal reinforcement In the absence of specific experimental data, an example of the connection between composite beam and a reinforced concrete wall, as illustrated in Fig. 16, was idealized to perform the benchmark of two concrete members only connected by steel reinforcement bars. The main objective of this example is to verify the adequacy of the model to simulate the connection at the tension zone of the joint when using only the longitudinal reinforcement bar and subsequently to assess the influence of

Fig. 14. Force–deformation curve identifying the participation of the reinforcement modelled with the embedded technique.

Fig. 13. Reinforcement and bond stresses in the numerical model implementing the contact with bond behavior.

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embedded technique

Load application Symmetry plane

Fig. 15. Von Mises stresses distribution in the reinforcement in the benchmark example to evaluate the reinforcement–concrete interaction in a reinforced concrete beam.

Wall edge fixed

Wall-slab interface

Beam bottom flange fixed

Load

Fig. 16. Benchmark example for connection of concrete parts using only longitudinal reinforcement.

the reinforcement–concrete interaction model on this connection. Thus, only a portion of the composite beam and wall is modeled. The concrete slab, wall and the steel beam are considered elastic (Ec = 32,000 N/mm2 and Es = 210,000 N/mm2) and full interaction is assumed. For the steel reinforcement bars elastic–plastic behavior is considered with a yield and ultimate strengths of 450 N/mm2 and 600 N/mm2, respectively. The simulated portion of the wall, at the edge opposite to the wall–slab interface, and the steel beam bottom flange, are fully fixed. Two simulations are performed in

order to evaluate the application of ‘‘real’’ (Traction–Separation) bond or perfect (Embedded technique) bond model. The wall–slab interface is modeled with ‘‘Hard’’ contact model allowing normal pressure when in contact and free separation. Fig. 17 shows the obtained results in terms of applied load versus beam deflection. It is clear that the embedded option provides a stiffer response than the previous case. Furthermore, because of the perfect bond assumption, the model response is linear, as the yield of the longitudinal reinforcement is achieved for much higher

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considering the bond model to approximate the real interaction between the concrete and the reinforcement bars in this critical zone of the joint. 4.4. Composite behavior The modeling of the composite behavior in a steel–concrete composite beam is validated using as benchmark example the test of a simply supported composite beam reported in [25]. This example is also used for validation of the software VULCAN in [26]. The latter numerical results are included here for general comparison. The geometric dimensions and the material properties considered for this benchmark example are given in Fig. 19 and in Table 4. The beam is loaded with a concentrated load at the mid-span. More detailed information is available in the given references. In the numerical model, the connection between the concrete slab and the steel beam is performed with spring elements, as described in the previous section. The quality of the developed model is shown in Fig. 20a) comparing the obtained load to beam midspan deflection curve with the experimental result [25] and the numerical calculation performed in VULCAN [26]. In the latter case, the result of a model with zero interaction is included to verify the effect of the concrete slab–steel beam interaction. A good agreement is observed between the numerical model, experimental test and numerical simulation performed in VULCAN [26]. The simulation with zero interaction (only contact between slab and steel beam) demonstrates that the degree of composite action is relevant because it influences the maximum load to be attained. The end-slip measured experimentally and predicted in the numerical models are compared in Fig. 20b). Here, a good agreement is also observed, confirming the accuracy of the model developed in ABAQUS. In the previous model, the behavior of the shear connection in the composite beam considers a non-linear shear–slip relation according to Ollgaard et al. [21]. As the shear connection was not taken to its ultimate deformation, like in the tests performed in [7], two additional simulations were performed to assess the influence of considering a linear relation between shear and slip, instead of the non-linear model. These two models considered the following: (i) linear elastic relation (CB-LIN); (ii) elastic–plastic relation (CB-EP). The three constitutive models assigned to the springs that model the shear connection are shown in Fig. 21a). The maximum force on the shear connectors is 48% of their capacity. Fig. 21b) shows the global force–deflection curves for the com-

Fig. 17. Load–deformation curve comparing the models of connection between concrete parts using only reinforcement bars.

Fig. 18. Load–separation curves comparing the models with approximation to ‘‘real’’ bond behavior and with perfect bond.

loads. Fig. 18 shows the load–separation relation between wall and slab for both models. There is almost no separation in the case of the model with perfect bond because of the rigid connection between reinforcement and concrete nodes that conditions the deformation in the interface. Furthermore, in the case of the perfect bond, the reinforcement remains in the elastic domain for much higher loads than in the case of the model with approximation to ‘‘real’’ bond behavior. In the latter, the maximum is limited by the steel strength of the reinforcement which is attained for a load of approximately 160 kN. These results emphasize the need of

P

279.4

5500

279.4

1219

152.4 Shear studs 8Φ12

20Φ12

tw = 6.7

40Φ12 307.1

tf = 11.8 Fig. 19. Geometry of the benchmark example for the composite behavior (dimensions in mm) [25].

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J. Henriques et al. / Engineering Structures 52 (2013) 747–761 Table 4 Material properties, stud number and dimension of the benchmark example for the composite behavior [25]. fy,sb (N/mm2)

fc (N/mm2)

fy,r (N/mm2)

No. studs

d (mm)

fu (N/mm2)

302

27

600

68

19

600

Fig. 21. Evaluation of the influence of the shear connection behavior.

Fig. 20. Composite beam benchmark example comparison of results for evaluation of the model developed in ABAQUS.

posite beam. As the level of load in each stud is in the elastic range, where all the shear connection behavior models are similar, it can be observed that the model considered for the shear connection has negligible influence on the beam global behavior. Consequently, considering only a linear elastic relation is sufficiently accurate.

4.5. Steel beam bottom flange in compression To analyze the steel beam bottom flange in compression, an academic example was conceived to calibrate the mesh refinement required for the use of the C3D8R finite elements. This consists in a simple case of a cantilever beam loaded by a concentrated load at the free edge, as illustrated in Fig. 22. The geometric and the material properties of the problem are similar to those of steel beam used in the tests in [7]. The steel constitutive behavior is assumed elastic-perfectly-plastic and the yield strength considered the nominal value for a S355 steel (fy = 355 N/mm2). As for the reinforcement bar, several models were executed which are summarized in Table 5. First, the correct solution was obtained through a mesh convergence study of the models, using the second order elements (C3D20R). Then, the first order elements (C3D8R) are implemented and the mesh validated through comparison with results of the previous models. This analysis is mainly focused on the number of elements to use through the web thickness, and specially, through the compression flange thickness.

Subsequently, an optimization of the mesh regarding the size of the model is performed. Finally, the consideration of the fillet radius in the cross-section is evaluated. In the latter models, first tetrahedral elements are used in order to model the fillet radius with acceptable element aspect ratios. According to the ABAQUS manual [13], the first order elements of this type should be avoided so, the second order elements (C3D10) are used. Finally, a model calibrates the use of elements type C3D8R in the fillet radius. In all the previous models, the fillet radius is neglected. The evaluation of the mesh quality is made by means of moment–rotation curves calculated at the support section, as shown in Fig. 23. The bending moment (M) and rotation (U) are calculated using the applied load (F) and the deflection of the beam at the free edge, respectively. In terms of initial stiffness, all models provide the same results, showing convergence. However, when the plastic deformation appears, the use of C3D8R with only one element through the flange thickness, diverges from the other models. For the model CB-C3D8R-T1, the load decreases approximately 14%. With two elements through the thickness (CB-C3D8R-T2) a good approximation is now obtained (Fig. 23a). In the first models, the length of the elements is approximately 10 mm. Initially, higher lengths were not considered as this would result in elements with

F

1500 mm

IPE 300

Fig. 22. Cantilever beam example for calibration of the mesh required to use first order elements (C3D8R) in the steel beam.

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Table 5 Summary of the models performed to calibrate discretization of the steel beam. Model ID

Description

CB-C3D20R-T1

Element type: C3D20R Number of elements through Fillet radius not modeled Element type: C3D20R Number of elements through Fillet radius not modeled Element type: C3D8R Number of elements through Fillet radius not modeled Element type: C3D8R Number of elements through Fillet radius not modeled Element type: C3D8R Number of elements through Longer elements Fillet radius not modeled Element type: C3D10 Number of elements through Fillet radius modeled Element type: C3D8R Number of elements through Fillet radius modeled

CB-C3D20R-T2

CB-C3D8R-T1

CB-C3D8R-T2

CB-C3D8R-T2LE

CB-C3D10-T2FR

CB-C3D8R-T2FR

thickness: 1

thickness: 2

thickness: 1

thickness: 2

thickness: 2

thickness: 2

thickness: 2

high aspect ratios. Then, in order to reduce the model size, the length of the elements was increased from 10 mm to 20 mm (closer to the supported section) and 30 mm (near the free edge). This fact only affects the results for high plastic deformations, above 30 mrad, however, the deviations observed are acceptable. In what concerns the fillet radius of the steel profile, Fig. 24a) shows the effect of neglecting this part of the beam by comparing the previous models with model CB-C3D10-T2FR. The increase on the load capacity is approximately 4.5%. No variation is noticed in the initial stiffness. Experimentally, in [7], the load on the beam bottom flange was estimated to be approximately 780 kN.

According to [27], the yielding of the first fibers on IPE 300 subjected to pure bending occurs for a load on the beam bottom flange of approximately 684 kN. Although, the real properties of the material are expected to be higher than the nominal ones, these values indicate that yielding on the beam bottom flange may occur. Consequently, modeling the fillet radius should be considered. Thus, an additional numerical simulation (CB-C3D8R-T2RF) is performed to validate the use of C3D8R in the fillet radius. In order to implement this type of elements, the fillet radius is considered with a triangular shape inscribed in the real shape. In a conservative way, less material is considered than in reality. This option is taken in order to have a good aspect ratio of the elements so that convergence problems are avoided due to bad element shape. In Fig. 24b), it is verified that the two models considering the fillet radius converge.

4.6. Anchor plate in compression and yielding of the steel plates For the anchor plate in compression at the bottom part of the joint (Fig. 1), no specific experimental tests were available. The validation of the numerical model, namely type of elements and constitutive models, is here envisaged using experimental data available in the literature that considers similar loading conditions. For this purpose, experimental tests on T-stub in compression and T-stub in tension are used for validation purposes. In the first case, the concrete modeling is assessed. In the latter, the plastic model for the steel is evaluated. Hence, for the anchor plate in compression, the experimental tests on T-stub in compression performed in Prague [28] are used. In these tests, none of the specimens tested in compression was taken up to failure. Therefore, only the initial stiffness could be compared. In all tests the same size of concrete block (550  550  550 mm3) is used. Two types of tests are performed which consider the variation on the dimensions of the T-stub flange: test

Fig. 23. Moment–rotation curves comparing the models for calibration of the mesh discretization of the steel cantilever beam.

Fig. 24. Moment–rotation curves evaluating the effect of modeling the fillet radius.

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Fig. 25. Numerical models of the T-stub in compression tests reported in [28].

Fig. 26. Comparison between experimental and numerical results for the T-stub in compression benchmark example.

IPE 300

d Bolts M12 8.8 L=55

20 40

460

20 30

90

30

(a) T-stub specimen (dimensions in mm) [29]

(b) Numerical model

Fig. 27. FE model to assess the plastic model for steel plates.

type I – 200  300  10 mm3; test type II – 335  100  12 mm3. In addition to differences in geometry, also the loading system varies. The two test types are illustrated in Fig. 25, where the cylinder represents the hydraulic jack. The numerical models developed in ABAQUS are 3D using the type of finite elements, constitutive laws and interactions described above. Two types of elements (TI/IIC3D8R and TI/II-C3D20R) were been tested and the results are shown in Fig. 26. The curves represent the force–deformation

response and compare experimental with numerical results. For the test type I, it can be observed that there is a good agreement between experimental and numerical results. According to the numerical model, the resistance is considerably above 600 kN; however, the test was stopped at approximately 600 kN without failure of the specimen. Consequently, the model accuracy cannot be verified beyond this load. In what concerns to the test type II, the quality of the approximation decreases. However, it should

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Fig. 28. Results of the T-stub in tension numerical calculations.

be noticed that for the same type of concrete, at about 200 kN, a considerable loss of stiffness occurs. The tests report did not present any justification for this observation. Numerically, at this approximate load level, the steel bar representing the web of the T-stub reaches its yield capacity. Consequently, the response is governed by this part of the T-stub. Finally, it should be mentioned that convergence of results is obtained with two numerical models with different type of elements. For validation of the plastic model of the steel parts, one of the tests on T-stub in tension, reported in [29], was selected. The Tstub is produced from an IPE 300 and the bolts are non-preloaded, as illustrated in Fig. 27a. Detailed information on the geometrical and material properties of the test specimen may be checked in [29]. Fig. 27b illustrates the numerical model developed. Profiting from the geometry of the problem, only one quarter of the specimen is considered. Two models are developed, one considering the use of first order elements (TS-C3D8R) and another that uses second order elements (TS-C3D20R). Experimentally, failure resulted from relevant flange yielding. However, inelastic deformations were also registered in the bolts. This indicates that the failure is between mode 1 and mode 2. The load–deformation curve that compares experimental and numerical results is shown in Fig. 28a. The deformation corresponds to the variation of the distance d (Dd). The results show a good agreement between tests and numerical models. A small deviation is observed in the plastic region which may be attributed to bolt tightening that is not considered in the numerical model. The two models with different type of elements, first and second order elements, demonstrate the convergence of the numerical results. Fig. 28-b) illustrates the deformation of the T-stub and the elements ‘‘actively yielding’’ for a load of approximately 170 kN. The formation of the plastic hinges in the flange is clearly identified.

5. Conclusions The first part of a study devoted to the analysis of a composite beam to reinforced concrete wall joint by means of finite elements calculations was presented. Giving the large variety of phenomena in this joint, such as plasticity, concrete, composite action, reinforcement–concrete interaction, and mechanical contact, the validation and calibration of the finite element model is performed using benchmark examples for each of the main joint components. The finite element software used is ABAQUS. This numerical tool was shown to be capable of reproducing the characteristic phenomena embodied in this type of joint. The comparison between experimental and numerical results, as well as the convergence studies highlight its degree of accuracy. Through the simulation of simpler examples, the findings of this numerical study and the calibrations performed are adopted in a second follow-up paper

dealing with the numerical simulation of the composite beam to reinforced concrete wall joint using the finite element software ABAQUS. The following aspects are highlighted:  Longitudinal reinforcement bars: solid elements type C3D8R may be used but required that at least 20 nodes through bar cross-section edge are considered; the von Mises plasticity model (isotropic yielding) can be used for the material constitutive law and the true stress-logarithmic strain curve should be given; failure criterion can be considered simply by limiting the material strain.  Reinforcement–concrete interaction: in the critical region of the joint (near joint face), the longitudinal reinforcement–concrete interaction should approximate the real bond–slip interaction; for all other remaining parts of this reinforcement and for all ordinary reinforcement, the perfect bond model may be implemented.  Composite beam: the steel beam to concrete slab interaction is realized using simple spring elements (uniaxial elements) which represent the real shear connectors; for each shear connector, two springs are used, one for the vertical direction and one for the longitudinal direction (slip direction); the behavior of the vertical springs is assumed to be infinitly rigid while for the longitudinal springs, a linear elastic response is sufficient; the properties of the later are obtained according to [20].  Steel beam: solid element type C3D8R may be used as long as at least 2 elements through flange and web thickness are considered; the fillet radius should be approximated as the steel beam may be in onset of yielding.  Anchor plate in compression: the concrete behavior is modeled using the Concrete Damage Plasticity model; solid elements type C3D8R may be used and the aspect ratio should be the lowest as possible.  Steel pieces at the bottom part of the joint: the material behavior may be reproduced using the von Mises Plasticity model (as for the longitudinal reinforcement bars); the use of solid elements type C3D8R requires at least 3 elements through the plate thickness. Acknowledgement Community’s Research Fund for Coal and Steel (RFCS) under grant agreement no. RFSR-CT- 2007-00051. References [1] Zoetemeijer P. A design method for the tension side of statically-loaded bolted beam-to-column joints. Heron 1974;20(1):1–59. [2] da Silva Simões, Simões L, Cruz PJ. Experimental behaviour of end-plate beamto-column composite joints under monotonical loading. Eng Struct 2001;23(11):1383–409.

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