TECHNICAL UNIVERSITY OF CLUJ-NAPOCA

FACULTY OF CIVIL ENGINEERING

2016

Ph.D. Thesis – Extended Abstract

ADVANCED ANALYSIS OF STEEL–CONCRETE COMPOSITE STRUCTURES

by

Ștefan-Marius BURU

Supervisor:

Prof. Cosmin-Gruia CHIOREAN, Ph.D.

Table of contents

1 Introduction. Objective and layout of the thesis ......................................................... 1

2 Literature review .......................................................................................................... 6

2.1 Experimental studies ............................................................................................... 6

2.2 Analytical studies .................................................................................................. 10

2.3 Numerical studies ................................................................................................. ̀ 15

2.3.1 Numerical studies via finite element commercial packages ............................. 15

2.3.2 Numerical studies via one dimensional special purpose finite elements ........... 19

2.4 Conclusions ........................................................................................................... 27

3 Provisions of the main design standards on composite steel-concrete structural members ..................................................................................................................... 29

3.1 Equilibrium. Degree of shear connection. .............................................................. 30

3.2 Compatibility. Degree of interaction. ..................................................................... 33

3.3 Provisions related to composite beams ................................................................... 35

3.3.1 Ultimate limit state ......................................................................................... 35

3.3.2 Serviceability limit state ................................................................................. 40

3.4 Provisions related to composite columns ............................................................... 43

3.4.1 Members subjected to axial compression ........................................................ 43

3.5 Conclusions ........................................................................................................... 45

4 Elastic differential equilibrium equation of composite beams with incomplete interaction ................................................................................................................... 47

4.1 Introduction ........................................................................................................... 47

4.2 Newmark model for elastic analysis of composite beams with incomplete interaction ............................................................................................................. 47

4.2.1 Different forms of differential equation........................................................... 51

4.2.2 Solving differential equation for various loading patterns ............................... 54

4.2.3 Evaluation of integration constants ................................................................. 58

4.2.4 General solution of the differential equation ................................................... 62

4.3 Conclusions ........................................................................................................... 70

5 Mathematical formulation of the proposed analysis method ................................... 71

5.1 Introduction ........................................................................................................... 71

5.2 Basic assumptions and constitutive material and shear connection models ............. 73

5.3 Elasto-plastic cross-section analysis....................................................................... 78

5.3.1 Elasto-plastic flexural rigidity of cross-section under full composite action..... 79

5.3.2 Elasto-plastic flexural rigidity of cross-section under partial composite action .............................................................................................................. 91

5.3.3 Evaluation of the axial force in the concrete slab. Member effective degree of composite action ....................................................................................... 100

5.3.4 Evaluation of tangent stiffness and stress resultant ........................................ 106

5.3.5 Derivation of equivalent transverse shear stiffness for composite beam......... 108

5.4 Inelastic member analysis .................................................................................... 113

5.4.1 Elasto-plastic tangent stiffness matrix and equivalent nodal loads ................. 113

5.4.2 The second order effects on element tangent stiffness matrix ........................ 122

5.4.3 The effects of finite joint size and joint flexibility ......................................... 123

5.4.4 Nonlinear geometrical global effects ............................................................. 127

5.5 Numerical solution techniques ............................................................................. 129

5.6 Conclusions ......................................................................................................... 132

6 Nonlinear analysis of composite steel-concrete structural systems using Abaqus software ....................................................................................................... 134

6.1 Determining factors in numerical simulation ....................................................... 134

6.1.1 Material constitutive models ......................................................................... 135

6.1.2 Finite element type ....................................................................................... 138

6.1.3 Modeling of residual stresses and initial geometrical imperfections .............. 142

6.1.4 Load application and analysis procedure ....................................................... 143

6.2 Verification example ........................................................................................... 144

6.3 Conclusions.......................................................................................................... 148

7 Validation of the proposed analysis model .............................................................. 149

7.1 Numerical examples on composite beams ............................................................ 149

7.1.1 Simply supported composite beams tested by Chapman & Balakrishnan....... 149

7.1.2 Simply supported composite beams tested by Nie & Cai ............................... 171

7.1.3 Simply supported composite beam tested by de Fabbrocino & Pecce ............ 173

7.1.4 Simply supported composite beam tested by Aribert et.al ............................. 175

7.1.5 Two-span continuous composite beams tested by Ansourian......................... 178

7.2 Numerical examples on composite columns......................................................... 185

7.2.1 Composite columns proposed by Chiorean ................................................... 185

7.3 Numerical examples on composite frames ........................................................... 190

7.3.1 Portal frame .................................................................................................. 190

7.3.2 Six-story composite plane frame ................................................................... 192

7.3.3 Six-story composite space frame ................................................................... 196

7.4 Conclusions ......................................................................................................... 200

8 Final remarks ........................................................................................................... 204

8.1 Conclusions ......................................................................................................... 204

8.2 Contributions of the present study........................................................................ 210

8.3 Future research .................................................................................................... 212

References *The table of contents is related to the full version of the thesis

1

Introduction. Objectives of the study.

In the last decades, composite steel-concrete members have seen widespread use as parts of the structural system of multistory buildings and bridges. Composite members are comprised of both steel (such as hot rolled I section profiles) and concrete components (such as reinforced concrete slabs). In order that these two components act as one, the connection between them must be ensured either by natural bond (as is the case of composite columns having steel sections fully or partially encased in concrete volumes) or by using mechanical shear connectors arranged at steel-concrete interface (as is the case of common composite beams in which a steel joist support a concrete slab). The main reason for the use of composite structural members lies in the benefits generated by coupling steel and concrete materials. The concrete material exhibits high rigidity and significant strength when is subjected to compression, but it has precarious behavior when is subjected to tension and is prone to creep and shrink with time. On the other hand structural steel is characterized by high load capacity and significant ductility but is prone to local and lateral buckling and moreover, steel members have weak fire resistance. Consequently, the benefits achieved with the optimal and rational use of steel and concrete are obvious. For example, adopting composite columns having steel sections fully encased in concrete volumes provides high fire protection and also the steel sections are less prone to buckle. Another type of composite columns is represented by concrete filled steel tubes. In this case, the steel section takes place of permanent formwork for concrete volumes. Another advantage of this type of composite member lies in increasing the concrete compressive strength because it is triaxially restrained and furthermore, the steel component is less likely to locally buckle. On the other side, a common form of composite beams is represented by a concrete slab supported by a cold formed steel profile interconnected with stud shear connectors. In this case, best use of concrete and steel materials is being made, as concrete is efficient in compression while the steel component is effective in tension. There are many situations in which the upper flange of the steel component is on the compression side of the composite section, so it is prone to buckle locally, but the concrete slab generally prevents this possibility. Moreover, using suchlike composite beams leads to reduced span to depth ratios compared to the classical forms of steel or concrete beams. The afore-mentioned benefits come together with complex behaviour aspects that are of current concern in the specialized literature although significant research has been reported in the past decades.

The design philosophy of composite members adopted by the main design standards (European EN 1994-1-1 [1], Australian AS 2327.1 [2], American ANSI/AISC 360-10 [3]) is that of limit states design associated with various types of global analyses: first order elastic, plastic and nonlinear. This semi-probabilistic approach supersedes the classical deterministic ones such as allowable stress design and plastic design within which the generic design condition implies a single parameter, labeled as safety factor, and thus the random nature of the variables that defines the structure safety (applied loads, mechanical properties of the materials, geometry of the structure components, etc.) is taken into account by means of a single parameter. Two types of analyses are associated with the deterministic design methods mentioned before: first order elastic and plastic analysis. The first one, still applied in current design practice, does not include directly the material and geometric nonlinearities, the later

2

ones are taken into account indirectly in the design process through interaction equations [4]. Although in the framework of the plastic design method, the material nonlinearity effects are approximately taken into account, usually by adopting the plastic hinge approach, other effects which have considerable influence on the strength and stability of the structure members (mechanical initial imperfections such as residual stresses, gradual yielding throughout the cross-sections, geometric local and global imperfections, geometric nonlinearity) are not directly considered in the global analysis and consequently are included indirectly through design interaction equations [4]. By adopting the limit states method as current design approach involves that, the main variables that describe the stochastic nature of structures behaviour are affected by distinct coefficients, thus eliminates the use of a single safety factor which is associated with deterministic methods of design. Usually, in limit states design approach the global structural analysis is conducted with either of first order elastic, rigid plastic or nonlinear analysis. Regardless of the chosen method of analysis, it should take into account, directly or indirectly, the effects of the main aspects that influence the strength, rigidity and stability of structural members such as material nonlinearity, local and global geometric nonlinearity, initial mechanical imperfections as residual stresses, initial geometric imperfections such as out of plumbness and out of straightness. By employing the first order elastic and rigid plastic analysis, the aforementioned effects are considered in an indirect manner, through provisions and interaction equations provided in design standards. Moreover, taking into account a combination of these effects is generally prohibitive. Furthermore, it is worth noting that each structural member of the structure needs to be individually checked [5]. Under these circumstances, the development and implementation in current design practice of higher order nonlinear static analyses becomes necessary. The main requirement of such analyses is that of being able to take into account all the variables that have major influence on the structural behaviour of members and framed structures, while separate member capacity checks are avoided. Obviously, in this category may be included the nonlinear analyses conducted via displacement based finite element commercial packages which have the capacity to capture in a rigorous manner the gradual development of plastic zones and the effect of geometric nonlinearities, but in many situations the problem size becomes considerable, implying a large number of finite elements and consequently the analysis times are also large even if the current technology provides very powerful hardware systems. Within the context, of the necessity of an analysis approach that eliminates the disadvantages of the existing methods of global analyses, from one point of view, and of continuous development of hardware and software technology, from other point of view, in the last three decades, considerable research have been reported in order to develop a new concept of analysis in conjunction with limit states design, which allows the simultaneous consideration of the factors that have direct impact on the structures behaviour and which is designated in the specialized literature as advanced analysis. In this view, advanced analysis denotes any method of global analysis that is able to capture the strength, rigidity and stability of a structural system in such a way that individual member checks are not necessary, thus providing a more realistic prediction of structural performance due to exterior actions [6–8]. Significant research on advanced analysis of steel structures have been reported among others by Chen & Toma [6], Kim & Chen [7] and Chiorean [8,9], while various approaches of advanced analysis of composite framed structures have been reported by Liew et al. [10],

3

Chiorean [11], Iu et al. [12] among others. Compared to the advanced analysis methods developed to study the behaviour of steel structures, those that are suitable for composite structural systems must treat in addition the connection between steel and concrete and therefore they are more complex than those suitable to steel structures. Generally, in composite columns the full interaction assumption is reasonable (especially when steel sections are fully encased in concrete volumes situation in which a large contact area between steel and concrete is provided and significant adhesion between them is developed), but in composite beams the interaction between steel and concrete is incomplete and must be taken into account on the strength and rigidity of such structural members. In some studies published in specialized literature (Iu [12,13], Liew et al. [10]), the effects of partial shear connection (and implicitly of partial interaction) are taken into account approximately at cross sectional level by evaluating the flexural stiffness using relationships proposed in design standards. On the other hand, modeling each component of the composite beam (steel profile and concrete slab) with one dimensional special purpose finite elements (Dall’Asta & Zona [14–16], Ranzi et al. [17–19], Zona et al. [20,21] among others) allows to rigorously include the partial interaction effects, but in this way the required number of finite elements may become large, and moreover, the use of only one finite element per physical member is prohibited.

The current study can be integrated in the current preoccupations related to development and improvement of advanced analysis methods of steel concrete composite structural systems and aims to develop numerical methods that are able to take into account the main factors which directly influence the structural behaviour of such composite structures. Special attention will be paid to the effect of partial shear connection (and partial interaction) between the steel joists and concrete slabs of composite beams.

1 Layout of the thesis

This thesis is structured into eight chapters as follows: Chapter 1 – Introduction. Objective and outline of the thesis – gives a brief description of

steel concrete composite structural members and the main advantages of them are highlighted. Then the global analysis and design requirements, in accordance with limit states design procedure, are described and the necessity of more rational nonlinear advanced analyses that fulfill the design standards provisions is emphasized, thus becoming analyses tools that may be applied in current design process.

Chapter 2 – Literature review– presents a review of current literature relevant to experimental, analytical and numerical studies focused on elastic and inelastic behaviour of composite steel-concrete structural systems, particular attention being paid to researches that deal with composite beams under full and partial shear connection, because throughout this work the attention is directed mainly on the analysis of these structural members. The studies outlined in this chapter highlights that the topic of this thesis is of current interest.

Chapter 3 – Provisions of the main design standards on composite steel-concrete structural members – provides a review of the main provision related to strength and

4

rigidity of composite beams and columns, specified in current design standards (European EN 1994-1-1 [1], Australian AS 2327.1 [2], American ANSI/AISC 360-10 [3], Chinese GB 50017 [22]). Once again, special attention is paid on identifying the design philosophy associated to evaluation of strength and stiffness of composite beams with partial shear connection. This particular investigation points out the inconsistency between the provisions suggested in the aforementioned standards regarding the evaluation of strength and especially the stiffness of composite beams with partial shear connection. In this context, Fig. 1.1 shows the degree of shear connection versus mid-span deflection comparative curves for simply supported composite beam E1 tested experimentally by Chapman and Balakrishnan [23], drawn in accordance with the above mentioned design standards.

Fig. 1.1 Mid-span deflection vs. Degree of shear connection curves

In Fig. 1.1 the dotted lines mark the minimum value of shear connection degree, imposed by the mentioned design codes. In the first part of this chapter, some fundamental theoretical aspects related to composite beams are presented, including the concepts of degree of shear connection and degree of interaction.

Chapter 4 – Elastic differential equilibrium equation of composite beams with incomplete interaction – presents the general framework of first order elastic analysis of composite beam with incomplete (partial) interaction. Based on the analytical model proposed by Newmark [24], various forms of differential equilibrium equation related to composite beams are presented. The dissimilarity between them consists in the variable chosen as primary unknown: the transverse deflection, the curvature, the slip between steel and concrete components and the axial force in the concrete slab. Then, for the latter form (with axial force in concrete slab as unknown), the complete solutions for simply supported and fix-ended composite beams subjected to various loading schemes (one-point, two-point and uniform loads) are exposed. This study emphasize that the internal axial force in concrete slab associated to composite beams with partial interaction can be expressed as a fraction of the axial force in concrete slab under full interaction.

0

10

20

30

0 0.2 0.4 0.6 0.8 1

Mid

-spa

n de

flect

ion

[mm

]

Degree of shear connection

AISC/ANSI 360-10AS 2327.1EN 1994-1-1 - proppedEN 1994-1-1 - unproppedGB 50017

5

As an example, for simply supported composite beam E1 tested experimentally by Chapman & Balakrishnan [23], Fig. 1.2 shows the variation, along the beams length, of the function f(αL) that alter the internal axial force in concrete slab under full interaction between steel and concrete components in order to obtain the value of internal axial force in concrete slab under partial interaction.

Fig. 1.2 Variation of f(αL) function along beams length for various load patterns.

The study conducted in this chapter, within the elastic range of materials can give also accurate results relevant to elastic deflections of composite beams under partial shear connection (and implicitly under partial interaction). For the same example, Fig. 1.3 shows the mid-span deflection versus degree of shear connection curves, calculated according to Newmark’s [24] analytical model and compared to those obtained using EN 1994-1-1[1] provisions. As it can be seen, the deflections of composite beams with partial shear connection are evaluated only approximately by EN 1994-1-1[1] standard.

Fig. 1.3 Mid-span deflection vs. Degree of shear connection curves

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

f(αL)

Relative distance from left support

One-point load

Two-point load

Uniform load

0

10

20

30

0 0.2 0.4 0.6 0.8 1

Mid

-spa

n de

flect

ion

[mm

]

Degree of shear connection

EN 1994-1-1 propped

EN 1994-1-1 unpropped

Newmark model

6

Chapter 5 – Mathematical formulation of the proposed analysis method – describes the main fundamental aspects related to the proposed analysis method of composite steel-concrete structural systems, detailing the peculiarities associated with inclusion of spread of plasticity in sections and along members length , local and global geometrical effects, mechanical and geometrical initial imperfections, semi-rigid connections and finite size joints. Essentially, nonlinear inelastic analysis employed herein uses the accuracy of the fiber elements approach for inelastic member analysis and address its efficiency and modelling shortcomings both to element level, through the use of only one 2-noded element to model each physical member, and to cross-sectional level through the use of path integral approach to numerical integration of the cross-sectional nonlinear characteristics. Within the framework of flexibility-based formulation, a 2-noded 3D frame element with 12 DOF (Fig. 1.4) able to take into account the distributed plasticity and partial composite action (relevant to composite beams) is developed.

Fig. 1.4 Flexibility based 2-noded 12 DOF beam and column elements

7

The nonlinear effects of partial composite action between the concrete and steel components of composite beams are taken into account innovatively by introducing, in an approximately manner, the internal axial force of the concrete slab as function of the member effective degree of composite action. The internal axial force in the concrete slab, under partial composite action ( cN int ), is assumed to be a fraction of the axial force in the concrete slab under full composite action ( cfN ), as shown in Fig. 1.5. This formulation is based on the assumption that the

distribution of the axial force in the concrete slab at all cross-sections along the beam is known and can be estimated as a function of the degree of composite action of the member, hence, the inelastic response, at the cross-sectional level, can be formulated by means of three equilibrium equations.

Fig. 1.5 Cross section of composite beam under partial composite action

In this way, gradual yielding throughout the cross-section subjected to the combined action of axial force and bending moment is described through basic equilibrium, compatibility, material and shear connection nonlinear constitutive equations, the states of strain, stress and yield stress are monitored explicitly during each step of the analysis. Tangent flexural and axial rigidity of the cross-section are derived and then using the flexibility approach the elasto-plastic tangent stiffness matrix and equivalent nodal load vector of the beam-column element including the shear deformability of the composite beam is developed. In this respect, an equivalent transverse shear stiffness for partially connected composite beam has been introduced and used in the general force-displacement relationships. In this way the elements of the stiffness matrix and equivalent nodal loads can be obtained analytically and readily evaluated by computing the correction coefficients that affect the elastic flexibility coefficients and equivalent loads and numerical integrations are required only to evaluate these correction coefficients. Using an updated Lagrangian formulation, the global geometrical effects are considered by updating at each load increment the geometry of the structure. The proposed numerical procedure for composite members has been implemented in the general static purpose computer program, NEFCAD [5,9], extending the capabilities of this software application.

8

Chapter 6 – Nonlinear analysis of composite steel-concrete structural systems using Abaqus software – describes an alternative numerical model for nonlinear analysis of steel-concrete composite structures, generated within Abaqus [25] software, which is able to take into account the main factors that dominate the structural behaviour of such systems. Among other features, modeling details related to the type and dimensionality of selected finite elements, material constitutive models, shear studs modeling, analysis procedures, are presented. For composite beams, three numerical models (A, B, C in Tab. 1.1) are proposed and verified, the differences between them consisting in the finite element types selected to simulate each component of the composite system, as can be seen in Tab. 1.1. The advantages and disadvantages of each numerical model are then discussed.

Tab. 1.1 Finite element families used in the proposed numerical models for composite beams

Model A B C

Concrete slab solid C3D8I solid C3D8I solid C3D8I Steel profile solid C3D8I shell S4R beam B31

Reinforcements truss T3D2 truss T3D2 truss T3D2 Shear connectors solid C3D8I connector CONN3D2 connector CONN3D2

In the following, the proposed numerical models are validated by comparison against Chapman and Balakrishnan [23] experimental tests on simply supported composite beam E1. Fig. 1.6 presents the comparative load versus mid-span deflection curves obtained by the proposed finite element numerical models in conjunction with experimental results retrieved from [23].

Fig. 1.6 Load vs. Mid-span deflection for simply supported composite beam E1

As can be seen, model A predicts with high accuracy the initial stiffness of the composite beam, but overestimates the ultimate load capacity of the system, most likely due to numerical problems exhibited by solid finite elements. Running on a computer with two 2.80 GHz clock speed CPUs (having eight cores) and 48 GB RAM, the present analysis is performed in approximately 7 hours. On the other

0

150

300

450

600

0 20 40 60 80

Load

(kN

)

Mid-span deflection (mm)

E1

ExperimentAbaqus_Model AAbaqus_Model BAbaqus_Model C

9

hand, the curve predicted with model B is in very close agreement with the experimental one and it’s generated in 10 minutes on the same computer. Finally, it can be observed, that model C slightly underestimates the initial stiffness and the ultimate load capacity while the analysis time is slightly lower than that needed by model B. Based on these details, model B is selected as alternative numerical model for composite beams and will be often used in next chapter.

Chapter 7 – Validation of the proposed analysis model – summarizes the main numerical experiments carried out to validate the proposed analysis procedure and also to reveal certain characteristics and peculiarities related to the behaviour of composite steel-concrete structural systems. The computational examples were primarily focused on composite beams whose structural response (strength and stiffness) is significantly influenced by the level of connection between steel and concrete components, aiming thus to highlight the capacity and efficiency of the proposed approach to capture the partial interaction (and partial shear connection) effects. The predicted results are compared against experimental ones, as well as against those obtained with Abaqus software (based on the numerical model described in Chapter 6) and those retrieved from specialized literature. The performance of the proposed analysis method is also verified on composite columns and on composite frames, whose nonlinear behaviour is dominated by both material and geometric nonlinearities. Hereinafter, some of the main results are presented.

a. Elastoplastic cross-sectional analysis

Fig. 1.7 Bending moment vs. Curvature curves for A3, A5, A6 beams cross sections

0

150

300

450

600

0 20 40 60

Bend

ing

mom

ent (

kNm

)

Curvature (x10-6 mm-1)

A3

Experiment

Proposed approach0

150

300

450

600

0 20 40 60 80

Bend

ing

mom

ent (

kNm

)

Curvature (x10-6 mm-1)

A5

Experiment

Ban&Bradford

Proposed approach

0

150

300

450

600

0 20 40 60 80

Bend

ing

mom

ent(

kNm

)

Curvature (x10-6 mm-1)

A6

ExperimentBan&BradfordProposed approach

10

Due to the fact that in the proposed approach the key elements of the elasto-plastic formulation are established at the cross-sectional level, the first set of numerical experiments is conducted at the cross-section level. In order to prove the reliability, efficiency and numerical stability of the proposed approach several moment-curvature analyses have been performed considering the material and shear connection properties of beam cross-sections (A3, A5 and A6) used in the experimental tests by Chapman and Balakrishnan [23]. The material properties of the tested specimens can be found in [23] and also in the full version of the thesis. The results of the proposed approach for the moment-curvature response are compared with test results in Fig. 1.7. For specimens A5 and A6 the results are also compared with an analytical model developed by Ban & Bradford [26] for elasto-plastic cross-sectional behavior of composite beams with partial shear-connection. In [26] the partial shear connection is defined at cross-sectional level assuming the slip strain at the steel-concrete interface to be a function of the degree of shear connection. As can be seen, good agreement exists between the proposed approach, test results [23] and other analytical results [26]. In the following, some results of a comprehensive parametric and sensitivity analysis of cross-sectional behavior are presented. Fig. 1.8 shows moment-curvature response of A5 and A6 cross-sections, by means of adopting different values for concrete strength. As it can be seen, varying the experimental concrete compressive strength with ± 10% and ± 20% generates changes in the ultimate load capacity value of at most 5%.

Fig. 1.8 Bending moment vs. Curvature curves for different concrete compressive strengths

Fig. 1.9 Bending moment vs. Curvature curves for different flange yield stresses

0

150

300

450

600

0 20 40 60

Bend

ing

mom

ent (

kNm

)

Curvature (x10-6 mm-1)

A5

fc=29.7 MPafc=27.23 MPafc=24.75 MPa - experimentfc=22.28 MPafc=19.8 MPa

0

150

300

450

0 20 40 60

Bend

ing

mom

ent (

kNm

)

Curvature (x10-6 mm-1)

A6

fc=28.46 MPafc=26.09 MPafc=23.72 MPa - experimentfc=21.35 MPafc=18.98 MPa

0

150

300

450

600

0 20 40 60

Bend

ing

mom

ent (

kNm

)

Curvature (x10-6 mm-1)

A5

fy_flanges=282.6 MPafy_flanges=259 MPafy_flanges=235.5 MPa - experimentfy_flanges=212 MPafy_flanges=188.4 MPa

0

150

300

450

0 20 40 60

Bend

ing

mom

ent (

kNm

)

Curvature (x10-6 mm-1)

A6

fy_flanges=288.2 MPafy_flanges=264.2 MPafy_flanges=240.2 MPa - experimentfy_flanges=216.2 MPafy_flanges=192.2 MPa

11

Fig. 1.10 Bending moment vs. Curvature curves for different web yield stresses

Fig. 1.9 and Fig. 1.10 shows the cross-sectional response for specimens A5 and A6 respectively for a fixed slab concrete strength and adopting different values for the flanges and the web yield stresses. As it can be seen, for the composite cross-sections analysed, the strength properties of the flanges, have the greatest influence on the overall response of the composite cross-sections.

Fig. 1.11 Effect of degree of composite action: (a) moment-curvature response; (b) moment-tangent

flexural stiffness

0

150

300

450

600

0 20 40 60

Bend

ing

mom

ent (

kNm

)

Curvature (x10-6 mm-1)

A5

fy_web=354 MPafy_web=324.5 MPafy_web=295 MPa -experimentfy_web=265.5 MPafy_web=236 MPa

0

150

300

450

0 20 40 60

Bend

ing

mom

ent (

kNm

)

Curvature (x10-6 mm-1)

A6

fy_web=321.2 MPafy_web=295.5 MPafy_web=267.7 MPa - experimentfy_web=240.9 MPafy_web=214.2 MPa

0

150

300

450

600

0 20 40 60 80

Bend

ing

mom

ent(

kNm

)

Curvature (x10-6 mm-1)

A6f(γ)=1f(γ)=0.942f(γ)=0.888f(γ)=0.841f(γ)=0.811f(γ)=0.784f(γ)=0.748f(γ)=0.698f(γ)=0.623f(γ)=0.497f(γ)=0.248f(γ)=0.142f(γ)=0

(a)

0

150

300

450

600

0 2 4 6 8 10 12

Ben

ding

mom

ent (

kNm

)

Flexural stiffness (x104 kN·cm2)

A6 f(γ)=1f(γ)=0.942f(γ)=0.888f(γ)=0.841f(γ)=0.811f(γ)=0.784f(γ)=0.748f(γ)=0.698f(γ)=0.623f(γ)=0.497f(γ)=0.248f(γ)=0.142f(γ)=0

(b)

12

In the following the proposed moment-curvature analysis is used to assess the influence of the effects of partial shear connection. Fig. 1.11 shows the comparative bending moment-curvature and bending moment-tangent flexural rigidity diagrams of A6 cross-section, considering different values for the degree of composite action by means of adopting different values for the function f ranging from 1f (i.e. full composite action) and 0f (i.e. no composite action). It can be observed that, as expected, on decreasing the

level of shear connection (i.e. decreasing the values of f ) the strength and stiffness of the cross-section is reduced. However, the decrease in the ultimate strength capacity did not seem to be as significant as the decrease in the tangent flexural rigidity.

b. Nonlinear analysis of composite beams

This section presents the global analysis of E1 composite beam, tested experimentally by Chapman & Balakrishnan [23]. In the proposed approach, one half of the beam is modelled by a single inelastic flexibility- based element with 7 Gauss-Lobatto integration points along the element length. The necessity to divide the beam into two elements is due only to have a node at mid-span of the beam where load-deflection curve is explicitly monitored. The mid-span deflection vs. applied load comparative curves is plotted in Fig. 1.12.

Fig. 1.12 Load vs. Mid-span deflection for simply supported composite beam E1

As it can be seen the behaviour of the composite beam predicted by the proposed analysis procedure and the advanced three-dimensional finite element model implemented in Abaqus, described in Chapter 6, is in close agreement with that of experimental test. It is worth noting that, the results predicted by the proposed (Nefcad) procedure include the effect of shear deformability by means of an equivalent shear stiffness which takes into account de concrete contribution. Fig. 1.12 indicates that the proposed model successfully predicts the elastic part stiffness and the ultimate load capacity of the composite beam, while a slightly stiffer behaviour is observed on the first part of the post-elastic portion of the curve. The results are also consistent with those published by Queiroz et al. [27] and El-Lobody & Lam [28], in which the nonlinear behaviour of all components of the cross-section is explicitly modelled by using advanced three-dimensional finite element approaches.

0

150

300

450

0 20 40 60 80

App

lied

load

(kN

)

Mid-span deflection (mm)

E1

Experiment

Queiroz s.a.

El-Lobody&Lam

Proposed (Nefcad)

Proposed (Abaqus)

13

The effectiveness of the proposed procedure is further assessed by varying the degree of shear connection (by means of varying the number of shear connectors-partial shear connection) and comparing the predicted curves with those obtained with more complex finite element analysis, as shown in Fig. 1.13 and Fig. 1.14.

Fig. 1.13 Concentrated load vs. Mid-span deflection curves for different degrees of shear connection:

(a) global behaviour; (b) initial stiffness

0

50

100

150

200

250

300

0 3 6 9 12 15(a)0

150

300

450

600

0 20 40 60 80 100A

pplie

d lo

ad(k

N)

Mid-span deflection (mm)

E1-concentrated load

136% - Proposed (Nefcad)136% - Proposed (Abaqus)136% - Queiroz et.al. (Ansys)100% - Proposed (Nefcad)100% - Proposed (Abaqus)100% - Queiroz et.al. (Ansys)71% - Proposed (Nefcad)71% - Proposed (Abaqus)71% - Queiroz et.al. (Ansys)

(b)

0

50

100

150

200

250

300

0 3 6 9 12 15(a)0

150

300

450

0 20 40 60 80 100

App

lied

load

(kN

)

Mid-span deflection (mm)

E1-concentrated load

118% - Proposed (Nefcad)118% - Proposed (Abaqus)118% -Queiroz et.al. (Ansys)89% - Proposed (Nefcad)89% - Proposed (Abaqus)89% -Queiroz et.al. (Ansys)47% - Proposed (Nefcad)47% - Proposed (Abaqus)47% -Queiroz et.al. (Ansys)

(b)

0

20

40

60

80

100

0 3 6 9 12 15(a)0

50

100

150

200

0 30 60 90 120

App

lied

load

(kN

/m)

Mid-span deflection (mm)

E1-uniform load

136% - Proposed (Nefcad)136% - Proposed (Abaqus)136% - Queiroz et.al (Ansys)100% - Proposed (Nefcad)100% - Proposed (Abaqus)100% - Queiroz et.al (Ansys)71% - Proposed (Nefcad)71% - Proposed (Abaqus)71% - Queiroz et.al (Ansys)

(b)

14

Fig. 1.14 Uniform load vs. Mid-span deflection curves for different degrees of shear connection:

(a) global behaviour; (b) initial stiffness

Two load cases are considered: mid-span point load and uniformly distributed load [27]. The levels of shear connection, ranging from 47% to 136%, are determined using the material and shear connector properties retrieved from experimental procedure. For mid-span point load case, the full load-mid-span deflection curves for all cases having different level of shear connection are shown in Fig. 1.13.b. while the initial branch of these curves are represented in Fig. 1.13.a. As expected on decreasing the level of shear connection the beam became more flexible, with reduced strength and stiffness. For uniformly distributed load case, the full load vs. mid-span deflection curves are shown in Fig. 1.14.b whereas the initial branches of the curves are depicted in Fig. 1.14.a. By comparing the load-deflection curves depicted in Fig. 1.13 and Fig. 1.14, it can be observed, that the proposed method (Nefcad) predicts fairly well the nonlinear behaviour and ultimate load capacity of the system when lower levels of shear connection are considered.

c. Nonlinear analysis of a six-story composite frame

The accuracy of the proposed procedure is verified on Vogel’s six-story two-bay frame with proportionally applied distributed gravity loads and concentrated lateral loads. The Vogel’s pure steel frame, studied previously by other researchers, has been modified and included in this verification study. The geometry, member and material properties, and the applied loads are reported in the full version of the thesis and can be also found in [29]. In order to evaluate the effects of the finite size of the joints and to make possible comparisons with more advanced nonlinear FEM solutions (Abaqus), in the proposed approach (Nefcad) the frame has been modelled considering member end offsets and assuming effective rigid joint size of one-half the true size of the joint. Fig. 1.15 shows the applied load factor versus lateral deflection curves predicted by the proposed approach for three different structural systems: pure steel frame, composite beams and steel columns and composite beams and columns. Full interaction is here assumed. It can be observed that the predicted curves are in close agreement with those obtained with Abaqus software, the proposed procedure being able to accurately capture both strength and stiffness of analyzed structures.

0

20

40

60

80

100

0 3 6 9 12 15(a)0

50

100

150

0 30 60 90 120 150

App

lied

load

(kN

/m)

Mid-span deflection (mm)

E1-uniform load

118% - Proposed (Nefcad)118% - Proposed (Abaqus)118% - Queiroz et.al (Ansys)89% - Proposed (Nefcad)89% - Proposed (Abaqus)89% - Queiroz et.al (Ansys)47% - Proposed (Nefcad)47% - Proposed (Abaqus)47% - Queiroz et.al (Ansys)

(b)

15

Fig. 1.15 Load factor vs. top story lateral displacement. Full interaction case.

The effectiveness of the proposed analysis method is further assessed on Vogel’s frame with composite beams and columns by considering two levels of shear connection (η=200% and η=40%) between the components of composite beams while full interaction is assumed for composite columns. The predicted curves are very similar with those obtained with Abaqus software, as can be seen in Fig. 1.16. Through this numerical example, the differences between full interaction and full shear connection are clearly highlighted, as the curves corresponding to full interaction are different from those associated to η=200%, which pertains to full shear connection range.

Fig. 1.16 Load factor vs. top story lateral displacement. Partial shear connection effect.

0.0

0.3

0.6

0.9

1.2

1.5

1.8

0 50 100 150 200 250 300 350 400

App

lied

load

fact

or

Lateral displacement (mm)

Vogel's frame

Nefcad_composite beams and columnsAbaqus_composite beams and columnsNefcad_composite beams and steel columnsAbaqus_composite beams and steel columnsNefcad_pure steel frameAbaqus_pure steel frame

0.0

0.3

0.6

0.9

1.2

1.5

1.8

0 50 100 150 200 250 300 350 400

App

lied

load

fact

or

Lateral displacement (mm)

Vogel's frame

Nefcad_full interaction

Abaqus_full interaction

Nefcad_η=200%

Abaqus_η=200%

Nefcad_η=40%

Abaqus_η=40%

16

3. Final remarks

The studies undertaken in this thesis fit in the actual scientifically research concerns related to the analysis and design of composite steel-concrete structural systems. The design philosophy of these types of structures adopted by the main design codes is that of limit states design in conjunction with various types (first order elastic, plastic and nonlinear) of global analyses. The analysis and design of composite members has to include, directly or indirectly, the influence of the prime factors which overlooks their behaviour. Within these circumstances arises a major concern in the development of advanced analysis procedures in accordance with limit states design, able of including in a rigorous manner the factors influencing strength, stiffness and stability of structures, thus individual member check being avoided. Therefore, the main objective of this study relies on development and improvement of advanced analysis methods for composite structures, which generates a number of secondary aims, briefly described in the previous sections.

Based on the numerical studies conducted in the present thesis, the following conclusions regarding the performances and capabilities of the proposed procedure as well as the structural behaviour of composite members and frames can be drawn:

a. Concerning the accuracy and computational efficiency of the proposed analysis method:

The described procedure is able to take into account the distributed plasticity and partial composite action (partial interaction and partial shear connection) between the concrete slab and the steel beam. Gradual yielding throughout the cross-section subjected to the combined action of axial force and bending moment is described through basic equilibrium, compatibility, material and shear connection nonlinear constitutive equations. Tangent flexural and axial rigidity of the cross-section are derived and then, using the flexibility approach, the elasto-plastic tangent stiffness matrix and equivalent nodal loads vector of the beam-column element including the shear deformability of the composite beam has been developed;

The proposed numerical approach for cross-sectional analysis allows the accomplishment of comprehensive studies capable of underlining certain behaviour particularities and pointing out the main factors that dominate the inelastic response of composite cross-sections under partial composite action. The parametric and sensitivity studies followed in this study have underlined the influence of peculiar effects such as: partial shear connection by considering different values for the degree of shear connection, shear studs adopted behaviour, material constitutive relations, material properties (yield stresses, strain values corresponding to strain hardening onset), residual stresses effect;

The model presents an additional ability to include the effects of shear deformations in the partially connected composite beams by means of an equivalent transverse shear stiffness, derived by using energetic principles;

Numerical studies undertaken on simply supported as well as continuous composite beams with partial shear connection certifies the ability and accuracy of the model to trace the elastic and inelastic behaviour by usually using only one element per member. The global behavior of the composite beams with partial composite action, predicted by

17

the proposed approach, correlate reasonable well with the experimental results and advanced finite element models, but with computational efficiency. It has been demonstrated that this model is applicable to composite beams using solid slabs with different material properties, various geometry ratios for slabs, varied cross section parameters, different beam lengths and various loading scenarios;

Studies conducted on composite columns, composite portal frames as well as on 2D and 3D composite frames, having high degree of static indeterminacy, indicates that the combined effects of material and geometric nonlinearities are accurately taken into account by using only one 2-noded element for each structural member, proving once more the computational efficiency of the model

b. Concerning the global behaviour of composite structural systems

Studies undertaken both with Nefcad and Abaqus software on composite beams, indicate that partial interaction phenomenon is exhibited regardless on the level of shear connection, thus relative slip between steel and concrete components will be detected even in the case of full shear connection. However, for degrees of shear connection greater than 100%, increasing the number of shear studs does not produce a significant increase neither on stiffness nor strength capacity of the beam. On the other hand, the decrease of stiffness and ultimate load capacity is more pronounced in partial shear connection range;

The parametric and sensitivity studies conducted at element level indicates that the global behaviour of composite beams is mostly affected by the yield stress of flanges, while other factors suchlike yield stress of web, concrete compressive strength, strain at onset of strain hardening and shear studs stiffness, have less pronounced effect on the global response of analyzed beams;

It is demonstrated that the residual stresses have a great influence on the stiffness and strength capacity of composite members (columns) subjected to important compressive axial forces;

The studies conducted on composite frames reveal the increase of stiffness and load capacity when the actual size of beam-column joints is taken into account. Further, these studies also indicates that the global behaviour of composite framed structures is less sensitive to partial shear connection effects as compared to composite beams.

18

Selected references [1] Eurocode 4: Design of Composite Steel and Concrete Structures - Part 1-1: General

Rules and Rules for Buildings 2004. [2] Standards Australia. AS 2327.1-2003 Composite structures Part 1 : Simply supported

beams. Sydney: 2003. [3] American Institute of Steel Construction. ANSI/AISC 360-05. Specification for

Structural Steel Buildings. Chicago: 2005. [4] Chiorean C. Calculul Neliniar al Structurilor Vol. 1 Structuri Plane. Cluj Napoca:

UTPRESS; 2009. [5] Chiorean CG. Aplicații software pentru analiza neliniară a structurilor în cadre.

U.T.PRES; 2006. [6] Chen W-F, Toma S. Advanced analysis of steel frames. Boca Raton: CRC Press; 1994. [7] Kim S-E, Chen W-F. Design guide for steel frames using advanced analysis program.

Eng Struct 1999;21:352–64. [8] Chiorean CG. Contribuții la analiza avansată a structurilor în cadre. Teză de doctorat.

Universitatea Tehnică din Cluj-Napoca, 2001. [9] Chiorean CG. A computer method for nonlinear inelastic analysis of 3D semi-rigid

steel frameworks. Eng Struct 2009;31:3016–33. doi:10.1016/j.engstruct.2009.08.003. [10] Liew JYR, Chen H, Shanmugam NE. Inelastic Analysis of Steel Frames with

Composite Beams. J Struct Eng 2001;127:194–202. [11] Chiorean CG. A computer method for nonlinear inelastic analysis of 3D composite

steel–concrete frame structures. Eng Struct 2013;57:125–52. [12] Iu CK, Bradford M a., Chen WF. Second-order inelastic analysis of composite framed

structures based on the refined plastic hinge method. Eng Struct 2009;31:799–813. [13] Iu CK. Nonlinear analysis for the pre- and post-yield behaviour of a composite

structure with the refined plastic hinge approach. J Constr Steel Res 2016;119:1–16. [14] Dall’Asta A, Zona A. Non-linear analysis of composite beams by a displacement

approach. Comput Struct 2002;80:2217–28. [15] Dall’Asta A, Zona A. Three-field mixed formulation for the non-linear analysis of

composite beams with deformable shear connection. Finite Elem Anal Des 2004;40:425–48.

[16] Dall’Asta A, Zona A. Comparison and validation of displacement and mixed elements for the non-linear analysis of continuous composite beams. Comput Struct 2004;82:2117–30.

[17] Ranzi G, Gara F, Ansourian P. General method of analysis for composite beams with longitudinal and transverse partial interaction. Comput Struct 2006;84:2373–84.

[18] Ranzi G, Gara F, Leoni G, Bradford M a. Analysis of composite beams with partial shear interaction using available modelling techniques: A comparative study. Comput Struct 2006;84:930–41.

[19] Ranzi G, Bradford M a. Nonlinear analysis of composite beams with partial shear interaction by means of the direct stiffness method. Steel Compos Struct 2009;9:131–58.

[20] Zona A, Barbato M, Conte JP. Nonlinear Seismic Response Analysis of Steel – Concrete Composite Frames. J Struct Eng 2008;134:986–97.

[21] Zona A, Ranzi G. Finite element models for nonlinear analysis of steel–concrete composite beams with partial interaction in combined bending and shear. Finite Elem Anal Des 2011;47:98–118.

[22] Ministry of Construction of the People’s Republic of China. GB 50017 - 2003 Code for design of steel structures. Beijing: 2003.

19

[23] Chapman JC, Balakrishnan S. Experiments on composite beams. Struct Eng 1964;42:369–83.

[24] Newmark NM, Siess CP, Viest IM. Tests and analysis of composite beams with incomplete interaction. Proc Soc Exp Stress Anal 1951;9.

[25] Abaqus - User’s Manual. Hibbit, Karlsson & Sorenson; 2011. [26] Ban H, Bradford MA. Elastoplastic Cross-Sectional Behavior of Composite Beams

with High-Strength Steel : Analytical Modeling. J Struct Eng 2014. [27] Queiroz FD, Vellasco PCGS, Nethercot D a. Finite element modelling of composite

beams with full and partial shear connection. J Constr Steel Res 2007;63:505–21. [28] El-Lobody E, Lam D. Finite Element Analysis of Steel-Concrete Composite Girders.

Adv Struct Eng 2003;6:267–81. [29] Chiorean CG, Buru SM. Nonlinear inelastic analysis of 3D frameworks with partial

shear connection. Proc. th Int. Colloquim Stab. Ductility Steel Struct., Wiley Ernst & Sohn; 2016, p. 877–85.