A Distributed Plasticity Model For

ELSEVIER PII: S0141-0296(97)00107-7

Engineering Structures, Vol. 20, No. 8, pp. 663-676, 1998 © 1998 Elsevier Science Ltd

All rights reserved. Printed in Great Britain 0141-0296/98 $19.00+ 0.00

A distributed plasticity model for concrete-filled steel tube beam- columns with interlayer slip Jerome F. Hajjar

Department of Civil Engineering, 500 Pillsbury Drive SE, University of Minnesota, Minneapolis, MN 55455-0220, USA

Paul H. Schiller

Barr Engineering Company, 8300 Norman Center Drive, Minneapolis, MN 55437-1026, USA

Aleksandr Molodan

Department of Civil Engineering, University of Minnesota, Minneapolis, MN 55455-0220, USA

(Received February 1997; revised version accepted April 1997)

A fiber-based distributed plasticity finite element formulation is presented to perform three-dimensional monotonic analysis of square or rectangular concrete-filled steel tube (CFT) beam-columns. This stiffness-based beam-column element fomulation accounts comprehensively for all significant geometric nonlinearity exhibited by CFT beam-columns as part of composite frame structures, and the steel and concrete constitutive models account for the significant inelastic phenomena which are seen in CFT experiments. In addition, the finite element formulation accounts for slip between the steel and concrete components of the CFT by incorporation of a nonlinear slip interface. This formulation is able to capture behavior ranging from perfect bond to immediate slip. The calibration and verification of the slip formulation are presented, and the finite element model is verified against experiments of CFT beam-columns subjected to monotonic loading. Results from a preliminary study are presented on the effect of slip on CFT beam- column and composite frame behavior. A related paper extends this formulation to cyclic analysis of composite CFT frames, and provides details of the steel and concrete constitutive models. © 1998 Elsevier Science Ltd.

Keywords: concrete-filled steel tube, distributed plasticity finite element, bond, slip

I. Introduction

One- and two-way low- to moderate-rise unbraced composite structures, consisting of steel I-girders framing rigidly into concrete-filled steel tube (CFT) beam-columns, exploit the many advantages that a CFT has to offer. This paper presents an overview of a three-dimensional, geometrically and materially nonlinear distributed plasticity finite element model for square or rectangular CFT beam- columns used in these types of composite frames. In con- trast to much of the computational research conducted on

CFTs to date, which has focused mainly on computing the ultimate axial and flexural capacity of individual members t, the current formulation provides an efficient method to study both the significant stress-strain behavior of a CFT member and the load-deformation behavior of CFTs as part of complete composite frames.

A stiffness-based beam-column finite element forms the basis of the CFT model. This formulation utilizes the fiber element approach for modeling distributed plasticity. The model discretizes the CFT cross-sections at the beam ends into a grid of fibers (as seen in Figure 1), and the stress-

663

664 A distributed plasticity model for CFT beam-columns with slip: J. F. Hajjar et al.

. . . o ....

End J

Superscr ipt c: concrete Superscr ip t s: steel

Concre te core . J slips relative to

steel tube

p* • ..-.,--..~ . . . . . , . . . . .~

\pc \

Origin , O , is in plane of~steel cross sect ion

Figure I Distributed plasticity concrete-filled tube finite element model

frame behavior. Hajjar et al. 22 extend this model to cyclic analysis of composite CFT frames, provide details of the constitutive models and their calibration to material tests and CFT experiments which yield moment-curvature results, and compare the results to cyclic experiments of CFT beam-columns and subassemblages of steel wide- flange girders framing into CFTs. This fiber analysis formulation is geared for conducting monotonic or cyclic static behavioral studies of composite CFT subassemblages or complete composite CFT structures. It is also suited for conducting comprehensive parametric studies which, in conjunction with experiments, may provide the data required to improve the accuracy and scope of current non- seismic and seismic design specification provisions for CFTs.

strain behavior is tracked explicitly at each fiber. Numerical integration is used through the cross-section to yield a stiffness formulation based upon stress-resultants, such that the element degress-of-freedom (DOFs) are all located at the centroidal axis of the member at each of its ends. The use of these types of finite elements has been well documented for decades by many researchers and has been shown to provide an accurate representation of the behavior of beam- column elements in structural frames. A wide variety of fiber (or, in two-dimensions, ' layered') elements have been used to model steel wide-flange beam-columns 2-7, reinforced concrete beam-columns s t2, and steel reinforced concrete beam-columns j3-~5, to name only a few. Very few models of this type have been developed for CFT beam- columns as part of frames. Bode 16, Bridge ~7, Tomii and Sakino zS, Shakir-Khalil and Zeghiche ~9, Kawaguchi et al. 2°, and Tsuji et al. 2~ have presented similar formulations, pre- dominantly for monotonic analysis of individual CFTs in two dimensions. Fiber element formulations require less computing time than three-dimensional continuum finite elements, yet they permit direct modeling of the variation of material properties across the element cross-section, and they may account explicitly for such effects as residual stresses or initial plastic strains.

The uniaxial steel and concrete constitutive models in this work are based upon comprehensive multi-axial plasticity formulations which account for the significant inelastic phenomena exhibited in CFT experiments and which greatly affect the behavior of CFTs. In addition, this CFT finite element formulation accounts for slip between the steel and concrete components of the CFT by incorporation of a nonlinear slip interface. This interface allows axial movement of the concrete core with respect to the steel tube. It is able to capture behavior ranging from perfect bond to immediate slip, and it permits modeling of the gradual transfer of stress between the steel tube and concrete core in the connection region of CFT beam-columns. The current formulation extends existing composite beam computational models which include slip to elements which incorporate both geometric and material nonlinearity. Cali- bration and verification of the slip formulation is presented, as is verification of the complete beam-column model vs monotonic beam-column experiments for CFTs having a wide range of material strengths and cross-section dimensions to verify the accuracy and robustness of the formulation. A comparable steel wide-flange fiber element is also implemented, based on the model of White 5, and a final example of a multistory composite CFT frame is presented, including an assessment of the effect of slip on the overall

1.1. Scope of the concrete-filled steel tube fiber model

The work presented here is part of an ongoing research program to develop analysis formulations capable of modeling the geometrically and materially nonlinear behavior of three-dimensional CFT beam-columns and their connections as part of composite frame systems. The fiber element formulation complements a concentrated plasticity CFT finite element model developed by Gourley and Hajjar, which is suitable for conducting static, transient dynamic, and eigenvalue buckling analysis of composite CFT frames23 26. While less compact than the concentrated plasticity element, this distributed plasticity formulation provides more detailed information which is critical for assessing the effect of slip on CFT member and frame behavior, conducting parametric studies of CFT subassemblages to generate a comprehensive suite of axial-flexural interaction diagrams, and assessing member and frame ductility.

The current research is limited to frame structures consisting of steel I-beams and/or square or rectangular CFTs. The steel constitutive model calibration has been conducted for ASTM A500 Grade B steel for CFTs, and ASTM A36 or A572 Grade 50 steel for wide-flange members. The concrete constitutive model calibration has been conducted for concrete strengths (f ' ) up to 50 MPa (see References 22 and 27 for calibration and verification of the constitutive models). The CFTs are assumed to be completely filled with concrete and to have no reinforcement or shear connectors. Confinement of the concrete core of the CFT by the steel tube is accounted for through calibration of the concrete constitutive model. Longitudinal residual stresses produced in cold-formed, welded steel tubes vary through the thickness 28 and are accounted for indirectly in the steel constitutive model used in the fiber analysis through calibration to stress-strain curves obtained from tests of hollow structural section tension coupon specimens, which retain these residual stresses 22'29. Effects which are not modeled directly in the current formulation include local buckling, nonlinearity due to shear or torsion (since shear and torsional forces are expected to be small in CFT frame members), shear deformations due to flexure (i.e. Euler- Bernoulli beam theory is assumed), time dependent effects on the materials, and post-collapse behavior.

2. CFT element formulation with interlayer slip

The occurrence of slip between the steel and concrete components of rectangular CFTs has been noted by a number of researchers. Much of the past work has specifically

A distributed plasticity model for CFT beam-columns with slip: J. F. Hajjar et al. 665

studied whether a strong bond between the concrete and steel is required for adequate CFT behavior, and whether shear connectors, ribs in the tube, or some other explicit means of achieving bond are required 3°. The experiments which have been conducted to determine slip and bond characteristics in rectangular and circular CFTs have included push-out tests 3] 35, tests of CFTs in pure flexure 36, and tests steel I-girders framing into CFTs using simple shear-tab connections 37-39. In addition, several building codes for composite construction recommend using mechanical shear connectors if ~:he calculated interface stress in a CFT column exceeds a limiting value 4°'4~. However, the effect of slip on the overall performance of CFTs remains a subject of research, and i! is probable that for frame structures, the effect of loss of bond is relatively small, especially with respect to the global behavior of the frame. To permit comprehensive modeling of this phenomenon, axial slip is permitted between the steel and concrete materials in this fiber model, and behavior ranging from perfect bond to immediate slip may be simulated.

The current formulation is intended to model the behaviour of CFT beam-columns in which the two materials are allowed to slip with respect to each other along the member's centroidal axis. Similar formulations have been applied to composite beams composed of a steel girder and concrete deck connected by a line of shear connectors 42 44 For CFTs, the centroidal a~es of the steel tube and the concrete core coincide in the initial state. Slip is, therefore, not induced due to flexure until the section becomes unsym- metric due to plastification or cracking of the materials. Slip may also be induced whea shear force is transferred from girders which frame into CFT columns at fully-restrained and, especially, simple connections.

Amadio and Fragiacomo 43 presented a two-dimensional layered composite beam element to model the slip between an elastic steel beam and a concrete slab modeled as a vis- coelastic material to capture creep and shrinkage. This formulation provides the basis for the current work, which extends Amadio's model to CFTs which include three- dimensional behavior, geometric nonlinearity, material nonlinearity of both the slJtp interface and of the component steel and concrete materials, and, in a related paper, cyclic excitation =. Schiller and Hajjar 4~ provide details of the geometrically nonlinear stiffness formulation, the force recovery procedure, the incremental/iterative Newton- Raphson nonlinear solution strategy, and results of CFT analyses used to verify the accuracy of the geometrically nonlinear element formulation. Features of this formulation which relate directly to the inclusion of slip in the model are outlined below.

2.1. Virtual work equation of equilibrium with interlayer slip For this work, the kinematic relationships for a CFT beam- column with interlayer slip are based on the assumption that the steel and concrete are separated by a layer of springs which determine the load transfer between the two materials based on a nonlinear spring stiffness. Thus, to track the differential movement between these materials for a three-dimensional geometrically nonlinear CFT arbitrarily oriented in space, and to allow for automated assembly of CFT elements into the global stiffness matrix of a composite frame during geometrically nonlinear analysis, three additional translational DOFs are added to each end of the conventional 12 DOF fiber beam element (i.e. nine DOFs

are modeled at each joint - - see Figure 1). These additional DOFs allow the steel and concrete to have independent axial deformations at the element level, while allowing for elements with different orientations of their longitudinal axes to frame into a joint (these differing orientations may be due either to the original topology of the frame, or to geometric nonlinearity of CFT elements originally aligned in a column stack). As will be described later, penalty functions are used in the global stiffness matrix to impose shear constraints, such that the transverse displacements of the steel and concrete at the ends of each element are con- strained to be the same.

To establish this element formulation, the virtual work equation of equilibrium 46 is modified to separate the steel and concrete contributions to the internal strain energy, and to include the strain energy associated with the deformation of the layer of springs:

f 2.UO c 2dV~. + / 2d}'~ f

2v ~ 2~eo :2v' 2,,,g ~j

"{"f 21 27r 28to Zdl = f 2s ~ 2~ ~uc 2dSc P

+ | zt~ 8u~ 2dgs J 2sS

(1)

Left superscripts denote the configuration in which a quan- tity is measured, and left subscripts denote a reference state. In this work, the current (unknown) state is referred to as configuration C2, the most recent converged state is referred to as configuration C 1, and the initial undeformed state is referred to as configuration CO. In equation ( 1 ), "r is the Cauchy stress tensor, 6e is the engineering strain tensor corresponding to the virtual displacements, 8u is the virtual displacement imposed in configuration C2, t is the surface traction vector (body forces are neglected in this work), 7r is the force per unit area transferred between the steel and concrete through the slip interface, tO is the deformation of the slip interface 43, V is the volume of the element, S is the surface of the element, and I is the steel-concrete interface. A right superscript c denotes concrete and a right superscript s denotes steel.

An updated Lagrangian (UL) incremental formulation, coupled with a corotational coordinate system, is used in this work to account for all significant geometric nonlinear behavior and path-dependent material behavior. This formulation follows the work of White s, Morales 47, and Yang and Kuo 48. Using a corotational approach, the element stiffness matrix is formed in a corotational coordinate system which 'corotates' with the rigid body motion of the element. In this system, only deformations which cause element straining are considered. This corotational stiffness matrix is then transformed to incorporate the rigid body modes, resulting in the final 18 DOF local element tangent stiffness matrix 45.

2.2. Slip kinematics and load-slip relation The slip formulation is illustrated in Figure 1. Since the concrete is encased within the steel, the two materials are assumed to have the same transverse displacements and rotations; slip stress is only caused by differential axial strain between the steel and concrete layers. This differential axial strain can be computed directly by considering the interpolation functions for axial displacement of the two


materials. Since the geometric centers of the two materials coincide, the same local axes can be used for both materials. The incremental deformation in the spring layer is, thus, given by 43

A~O = Au~, - Au~ (2)

where Auc represents the incremental axial deformation between the element ends along the centroidal axis. Equ- ation (3) then relates the incremental force transferred between the materials per area of interface, ATr, and the incremental deformation of the interface, A~O:

2~_~_ I,jT.]_ A~.j. - I,jT .t_ ]~ml// (3)

where ~: is the tangent stiffness of the interface (units of force/length3). The incremental linearization of the nonlinear load-slip relationship allows the slip interface stiffness, ~:, to be updated at the beginning of each global load increment or iteration.

2.3. Fiber element approach for modeling spread of plasticity To approximate the volume integrals which appear in the virtual work equation of equilibrium using the fiber element method, each integral of equation (1) is first decomposed into area and length integrals. The area integrals are then approximated by numerical integration over the fibers which comprise each cross-section. By evaluating the area integrals during incremental analysis using the current tangent modulus of each fiber in the cross-section, spread of plasticity through the cross-section is accounted for. The tangent modulus of each individual fiber is computed based on the appropriate material constitutive model and the fiber strain, which in turn is computed from the local element displacements 45.

For the integral of the virtual work term incorporating the increment in slip deformation, this decomposition may be represented by:

La,/,.a, q,.p'dx (4) /~.A~O.3, q/dl = f ,L

where p is the perimeter of the steel tube, L is the element length, and dx is a differential element along the length of the CFT.

In this work, only the torque and the rigidity term associated with torque are not calculated through numerical integration, since this element formulation does not account directly for shear strains. The torsional rigidity at a cross- section is assumed to remain constant over the entire load history. This approximation is believed to be sufficient for modeling the behavior of rectangular CFTs as members of structural frames, as these members exhibit high torsional rigidity 49.

2.4. Finite element interpolation functions Once numerical integration is used to compute the element forces, moments, and rigidities at the end nodes, interpolation functions are used to approximate the change in these quantities along the element length. In conventional beam finite element formulations, the axial force is assumed to be constant in the element. This assumption remains true in the current formulation for the total force at any point

along the element length, but the steel and concrete axial forces, taken separately, may vary linearly along the length of the member as they transfer force through the slip interface. The torque is assumed to remain constant along the length, and a linear interpolation along the length is used for the bending moments and higher-order force resultant terms 4~. Linear interpolation functions are also chosen to represent the change in axial, flexural and torsional rigidities along the element length.

Conventional cubic Hermitian shape functions are used to describe the transverse deformations and rotations which are caused by flexure, and a standard linear shape function is used to describe the torsional rotation of the element. A quadratic shape function is used to describe the axial deformation, which requires the addition of midpoint axial DOF for both the steel and concrete. The current formulation assumes that external loads are applied only at the element ends. Thus, the midpoint DOFs are removed from the element stiffness matrix through static condensation.

The formation of this local element tangent stiffness matrix permits calculation of incremental deformations based upon an increment in applied load. After the incremental global deformations are computed, forces and moments are recovered at the element ends. Strains at each fiber are first computed, followed by assessment of the stress at each fiber based on the constitutive formulation. Stress-resultants are then computed using numerical integration. Schiller and Hajjar 45 present the details of the force recovery procedure adopted for the current formulation, including a discussion of the technique for including the high-order geometrically nonlinear terms of the virual work equation of equilibrium which are nonlinear in the displacements, recovery of element centroidal axis strain and curvature from the element end forces, and the technique used for materially nonlinear force recovery. The terms retained in the tangent stiffness matrix and force recovery procedure account sufficiently for all significant P -A and P-6 effects within the CFT beam-column, and the formulation is accurate for problems involving moderate rigid body rotations and small incremental strains.

2.5. Constraint of concrete and steel transverse displacements If 18 DOF CFT beam-column elements are used as members in a structural frame model, shear deformation compatibility constraints are required between the steel and concrete translational DOFs at each joint. For most practical structures, the CFT beam-columns will be aligned in vertical column stacks, requiring two global compatibility conditions (perpendicular to the axis of the column stack) at each joint. This may not always be the case, however, as other structural systems which may have arbitrary orientations of members entering each joint are possible. For this reason, a general formulation for shear compatibility constraints is used in this work.

Penalty functions imposed in the global coordinate system are used to enforce compatibility conditions on the CFT steel and concrete translational DOFs 46. In this approach, a large, but finite, stiffness is assigned between corresponding steel and concrete transverse displacements, thus constraining their values to be nearly identical. These constraint equations are determined at the beginning of the analysis, and the constraints are oriented in the directions of the local element transverse displacements, perpendicular to the centroidal axis of each CFT at each joint.


When conducting geometrically nonlinear analysis using constraint equations, the directions of shear constraint are assumed to rotate with the structural node at which they act to reflect the changing geometry of the structure. This approach assumes that all CFT members are free of internal hinges or member releases, an assumption which is consist- ent with the scope of the current formulation.

In addition to the required shear constraints, there are practical analysis situations when the steel and concrete should not be allowed to slip in the axial direction at a structural joint. These situations arise when mechanical shear connectors are provided at the joint, when there is a change in CFT cross-seclion size at a joint, when CFT elements frame into a joint from different directions, or when a beam-to-column connection includes diaphragms or through-bolts such that they enforce axial strain compatibility of the steel and concrete at a joint. In order to prevent slip at a joint in the computational model, an additional penalty function constraint equation may be specified in the direction of the element centroidal axis.

3. Constitutive modeling

The constitutive models for the steel and concrete simulate the significant aspects of inelastic behavior found in CFT members. These formulations and their calibration for the current model are discussed in detail in References 22 and 27, and are summarized only briefly here. Key effects which influence monotonic behavior of CFTs and which are modeled in this work include a gradually changing modulus of both steel and concrete with increased monotonic loading; retention of the elastic modulus upon unloading for the steel; modulus degradation of the concrete upon load reversal, tensile behavior of the concrete; and multiaxial stress effects such as confinement of the concrete core. In addition, the constitutive formulations have been adopted for cyclic loading of CFTs, in which additional key behavioral phenomena include strength degradation; stiffness deterioration; a vanishing zone of linear behavior with cyclic loading; and the Baushinger effect22.27,

The steel formulation, adapted from Shen et al. ~°, models the rounded shape of the stress-strain curve found in cold- formed tube steel, a decreasing elastic zone with increased plastic straining, and the different stress-strain behavior exhibited in the corners and flanges of cold-worked steel tubes. The concrete formulation, adapted from Chen and Buyukozturk 5~, models strength and stiffness degradation by means of a cumulative., damage parameter, and the post- peak behavior of the c,~ncrete is calibrated to account implicitly for the effects of confinement of the concrete core by the steel tube. The concrete formulation also works well for fibers which cycle into tension and then back into compression. Both of the material constitutive models are formulated in multiaxial stress space, allowing for future extension of the current tormulation to account directly for confinement. In the present work, only the uniaxial components of these models are currently activated.

4. Calibration of the load-slip relation

The focus of experimental research related to slip in CFTs has generally been to improve the recommendations which account for bond in various building c o d e s 3 j -34 ' 39 A majority of the published bond experiments present the

results of push-out studies, and focus exclusively on bond strength, rather than the full load-slip behavior. However, tests performed by Shakir-Khali133'34 have provided insight into the entire load-slip behavior of rectangular CFT sections. Graphs of the load-slip relationship for push-out tests of CFTs both with and without shear connectors suggest a simple bilinear relation. The spgcimens tend to lose their initial slip stiffness suddenly, at a point which corresponds to the bond strength of the interface. A very low stiffness is observed after the interface bond breaks down; a value near zero is assumed in this work. Consequently, only bond strength and initial slip stiffness must be calibrated. The calibration focuses on CFTs having no internal shear connectors or ribs. However, through recalibration or extension of the slip constitutive relation, this formulation is equally applicable to CFTs having explicit means of retaining bond.

Connection studies representing steel I-girders framing into CFTs with simple shear tabs or tee-stubs have been conducted by Dunberry et al . 37 and Shakir-Khali139 to determine the effect of slip on CFT columns in which only a portion of the column axial force enters at the connection, with the remaining axial force introduced at the column end. These studies provide detailed data about the load transfer and slip around CFT connections, and are believed to provide a more accurate representation of the load transfer at these connections than traditional push-out studies. These tests often exhibit bond strength and initial slip stiffness which are higher than those observed in push-out tests on similar CFT sections, due primarily to the added fric- tional resistance from pinching which occurs as the tube starts to deform or buckle in the connection region. Conse- quently, rather than use the results of push-out studies for calibration, the results of connection studies are utilized.

There are only a limited number of these types of connection studies documented thoroughly in the literature for use in the calibration of the initial value of ~: from equation (4). Thus, in this work, its value varies as an inverse function of the perimeter of the tube, such that ~: = k~v/p, where k~, o is a calibrated constant value of slip stiffness per unit length of interface, having units of force/length 2. Use of this relation effectively makes the initial value of the term k.p from equation (4) equal to a constant. The value of k~, o is thus calibrated directly.

4.1. Calibration of initial slip stiffness For calibration of initial slip stiffness, kslip, data is desired in the materially elastic range in order to eliminate the effects of the material model calibration on the slip calibration. Such data is available from Shakir-Khali139 from strain measurements taken during connection testing. In this publication, the strain profile along a CFT specimen is presented for the entire loading, providing detailed data about the straining effects near a CFT connection which may result from slip. The test specimens were constructed from square structural tubing with tee-cleat connections. Loading and test parameters for specimen E6 are shown in Figure 2. In the computational model, a node is assigned at each of the strain gage positions shown in Figure 2. The load applied at the connection is split between the three nodes in the connection region (gages 5, 6 and 7) in the ratio of 25-50-25% to simulate the transfer of load along the connection plate.

Figure 3 shows the measured strains in the steel at a level of applied load, P, equal to 600 kN, vs the height along the specimen for the computational model and the experiment.


Figure 2

P Tube: 150x150x5 e = 1511

~- D= 150m~ k~ " " ' | 7 7 5 t : 5 .0mm

-- [ f~. = 386.9 MPa 150 f , = 465.0 MPa

" 150

' ~ p 180 E~= 3 . 5GP ~-y 0------, too.

i , ~ ~ 5 ; 5 ~ d i s t a n c e between gages in mm

1, "~'gage number 925

Experimental setup of Shakir-KhaliP ~, specimen E6

4

o

o

0

]

i • Experiment (Shakir-Khalil, 1994, Specimen E6)

!--Analysis: kstip = 106 MPa -~.-Analysis: k,,e = 104 MPa

Analysis: k,, e = 5x103 MPa I .... Analysis: katie = 103 MPa .

connection l re~ion t

-0.0002 £s

i

i

i

i f

0

• i ~ t z : ' /

L

i

i. t

i t : i i I I

-0.0004 -0.0006

Figure3 Measured vs computed strain in the steel tube at a load of 600 kN

The results for several different values of initial slip stiffness are shown. The introduction of the load to the steel tube in the connection region is clearly seen, along with localized strain concentration below the connection.

An analysis of the results indicates that the computational model matches the experimental strain measurements fairly well at this load level, especially outside the connection region 45. The model does not predict the strains within the connection region as accurately, due predomi- nantly to the assumed pattern of load distribution in the connection region (i.e. 25-50-25%), but the shape of the strain distribution exhibits a similar rate of load transfer as the experimental results. The strain measurements of gages 1, 2 and 3 above the connection show that the majority of load is transferred to the concrete in the connection region (since the strain differs little between these gages), a trend which is only matched well by the highest of the k~p values. All values of k~, v produce approximately the same average percent error, between 6 and 7%, but the highest value of slip stiffness produces the lowest standard deviation, indicating a better match to the shape of the strain distribution. The value for ks,p of 103 MPa (which is representative of the results from push-out studies) produces a strain diagram that over-predicts the length of the load- transfer region. Note that computational parametric studies

conducted by Schiller and Hajjar 45 in which the concrete core is pushed through the steel tube shows that values of k,r~p above approximately 104 MPa provide nearly perfect bond.

Similar correlation is achieved at a load, P, of 800 kN, and from a second specimen, E8, which has the same dimensions and strain gage positions as specimen E6, but a slightly higher concrete compressive strength. These comparisons thus show evidence that the value of k,,p from the push-out studies (approximately 103 MPa) consistently overpredicts the length of the load transfer region around the connections. The experimental results from this set of test, as well as the observations of other researchers 37, suggests that there is little or no slip outside the load transfer region. This condition is only predicted accurately by the highest value of k~,p. The value of initial slip stiffness of kslip of lO 4 MPa consistently provides the best balance of mean error and standard deviation in the results, and this value of initial slip stiffness is thus chosen for the current computational model.

4.2. Calibration of bond strength Push-out studies may not be representative of the bond strength at simple beam-to-column connections due to the pinching action caused by connection rotation. Thus, in order to study bond strength, connection specimens from Dunberry et al . 37 are utilized. A much higher percentage of the total load is applied at the connections for this series of tests compared to Shakir-Khali139. This creates more slip and bond loss and provides a better indication of the bond strength at these connections. Dunberry et al . 37 present graphs of the experimentally measured slip, strain, and total load in the steel and concrete at a load level near the limit point of the column. Note that material nonlinearity in the steel and concrete contribute significantly to the straining of these specimens near their ultimate loads.

Dunberry et al . 37 present extensive data for specimen D1. This test used the same nominal tube size and had approximately the same material properties as the Shakir-KhaliP 9 specimens E6 and E8, which were used for calibration of initial slip stiffness. Figure 4 shows the experimental setup and test parameters. A detailed description of the strain gage locations is not provided by the authors, but the speci-

t

• j

_ _ t , i

4 b

i

T,

Tube: 152.4x 152.4x4.83 ! P = 1622 kN

D = 152.4 nun t = 4.83 mm

590mm f~ = 443MPa f~ = 588 MPa

180mm fc'= 29.6 MPa 180ram Ec= 19.3GPa

1050mm

Figure 4 Experiment test setup of Dunberry et a/37, specimen D1

A distr ibuted plasticity model for CFT beam-columns with slip: J. F. Hajjar et al. 669

mens were instrumented to measure steel strain with strain gages, and concrete strain and slip with demic gages.

The computational model uses 11 elements, with the loading applied to the top, middle and bottom of the connection region in a ratio of 2 0 - 2 0 - 6 0 % to simulate the gradual transfer of load in this region. This loading scheme is different than the one used in the previous section, since it better reflects the load transfer seen in this specific experiment (Schiller and Hajjar 45 present the loading scheme used for the initial slip stiffness studies to demonstrate the effect that the loading pattern has on the accuracy of the computational results for this type of connection). The initial elastic modulus of the concrete was not reported, and a value of 19.3 GPa was selected for the analysis so as to match the experimentally observed concrete rigidity 45.

Figure 5 shows the comparison of the computational model to the published experimental results for various values of bond strength, J'bo,d" Slip between the concrete core and the steel tube is shown at points along the length of the steel tube.

The results show that the highest values offbo,a provide the most accurate comparison to the experimental slip profile. However, the value of fbo,a of 0.6 MPa provides the best comparison with the maximum slip, with an error of 2.5% 45. The slip profile in Figure 5 presents evidence that the computational model predicts the shape and magnitude of the slip along the columa with better accuracy below the connection than above it. This phenomenon can again be attributed to connection rotation and the pinching mech- anism which results.

The distribution of load between the steel and concrete portions of specimen D1 along the length of the column is presented by Dunberry el al. 37 for a load level slightly below the ultmate load of the column. t Figure 6 shows the

177Y'A

4 'k' , . .

[ : 1 " o=on .::.-..~m;'~:~'.'5. 2-'-- . . . .

~ , ~ " ~ Experiment (Dunberry et al., 1987, * ~ Specimen D1)

~ - Analysis, fbo,~ = 0.4 MPa .... Analysis,fbo,,~ = 0.5 MPa

6 ...... Analysis,fbo,~ = 0.6 MPa ¢, ..... Analysis, fbo,~ = 0.7 MPa

* . . . . Analysis, fbo,a = 0.8 MPa i ~ i I i i i i I i ~ p t I

0 0.05 0.1 0.15 Slip (mm)

Figure 5 Measured vs computed slip between the steel tube and concrete core

¢The total load in this specimen, and the proportion applied at the connection, may be calculated by adding the forces in the steel and concrete at various points along the length. The published experimental graph indicates that the total applied loads are 680 kN at the top of the specimen and 880 kN at the connection. This data contradicts the tabulated values of Dunberry et a l? 7, who state that 50% of the total load is applied at the connection, and that the total load graphed is 1622 kN. In order to compare to the published results, however, the applied loads in the analysis are

E

I°

a !:

9

#

o

! • Concrete, experimental (Dunberry et al., 1987, • Steel, experimental Specimen D 1)

- - Analysis,f0o.a = 0.4 MPa "-- Analysis,fbo.a = 0.6 MPa .... Analysis, fbo.d---'- 1.0 MPa

~.. • connection ~ - ~ load on i' [ transfer

~ region

C o n c r e t e " " ~"Steel

Load (kN)

Figure 6 Measured vs computed axial force in the steel tube and concrete core

comparison between the published results and the prediction of the computational model for various values of bond strength. A statistical analysis of the results suggests that a bond strength of 1.0 MPa produces the best results when compared to the experimental data (e.g. 6.86% error for the concrete load) 45. However, the other values presented show similar percent error (e.g. 7.29% error for fbon~ = 0.6 MPa for the concrete load). A bond strength of 0.6 MPa gives acceptable results and matches the maximum slip much better than the value of 1.0 MPa. Both of the higher values of bond strength in the computational model match the overall shape of the load diagram more accurately than the lowest value (note that a value of bond strength of 0.4 MPa corresponds to that specified in Reference 40, while a value of approximately 0.1 MPa corresponds to that specified in Reference 41 ). This trend is especially apparent in the concrete load in the region above the connection. The experimental data points suggest that all the load transfer from the steel to the concrete occurs in the region indicated on the graph. Only the highest values of bond strength result in a load transfer within this region, with the lowest bond strength value showing transfer over a much larger region.

Based on the examples provided in this section, a bond strength of 0.6 MPa provides the best correlation between experimental and computational results. This value provides the best prediction of total slip, loss of bond around the connection region, and transfer of load from the steel tube to the concrete core after loss of bond occurs. The calibration has also shown that the computational model is more accurate for the region at the compression end of a connection than the tension end because of localized behavior in the experiments caused by connection rotation.

4.3. Verification o f slip formula t ion

As verification of these calibrated slip parameters, two further experiments are investigated, holding all calibrated parameters fixed. The series of tests conducted by Dunberry et al. 37 included a specimen, C1, which was loaded only at the connections, and had no cap or other means of pre-

assumed to be those which are computed by summing the steel and concrete loads in the published experimental graph.

670 A distributed plasticity model for CFT beam-columns with slip: J. F. Hajjar e t a l .

uncapped

I j

J

b

Figure 7 men C1

220 mm

180 mm 1'80 mm

920 mm

T

Tube: 101.8x101.8x4.82 ] P = 608 kN l D = 101.8nun t = 4 .82mm 0 . 8

fy = 374 MPa

L= i

24.2 MPa I

Ec= 17.9 GPa 0 . 6

M/Mo

0.4

Experimental parameters of Dunberry et aL 37, speci-

venting slip at the top of the specimen. Figure 7 gives the experimental parameters for this test. A plot of the experimentally observed load in the concrete over the height of the column is presented for a given load of P = 608 kN. Figure 8 shows the comparison of the experimental data points to the computational model using the calibrated parameters. The shape and magnitude of the load distribution is well predicted by the computational model, with the load gradually entering the concrete core starting above the connection and continuing below it.

Lu and Kennedy 36 present the results of flexural testing of a rectangular CFT, bent about its major axis, in which the experimentally observed slip was reported for specimen CB33. The CFT consisted of a 254 × 152 × 6.4 rectangular steel tube, with a yield stress of 377 MPa, and a concrete strength of 45.2 MPa. The specimen was simply supported, 3040 mm long, and had equal concentrated loads applied 766 mm from each end. The beam had no end plates or other means of preventing slip along the specimen. Figure 9 shows the comparison between the experimentally observed slip and the slip predicted by the computational model using the calibrated slip parameters (M,, is the ultimate moment strength of the CFT). This graph provides evidence that the calibration based on connection tests is accurate for other types of loading situations as well. The

¢1

• Experiment (Dunberry et al., 1987, " ~ Specimen C1)

[ I • I

0 50 100 150 200 Load (kN)

Figure8 Measured vs computed concrete axial force in uncapped column specimen

0.2

m m

ental u and Kennedy, 1994, Specimen CB33)

- - Analysis

0 t I [

0 0.05 0.1 0.15

Slip (mm)

Figure9 Measured vs computed slip in uncapped flexural specimen

slip and the point where loss of bond occurs are predicted well by the computational model.

5. Verification of the concrete-filled steel tube fiber model

Verification of the CFT fiber element is performed by com- paring the finite element results to over thirty different experimental studies of CFT beams, eccentrically loaded beam-columns, nonproportionally loaded beam-columns, and composite frames composed of steel 1-girders rigidly framing into CFT columns, holding all calibration parameters fixed 45. Experimental verification studies were selected to provide data for CFTs having a wide range of parameters, such as material strengths, D/t ratio, L/D ratio, and the method of applied loading j. Figures lOa-h show the comparison between the fiber model and experimental monotonic load-deflection curves for several of the verification studies (details of each test are provided in Table 1 and Reference 45). Each test is referenced by the nomencla- ture of the experimentalist. Figures lOa and b are beams loaded in pure flexure by point loads, including both major and minor axis bending. Figures lOc-f are eccentrically loaded beam-columns~ including comparisons to tests from three different experimentalists. These four examples include both stocky and slender beam-columns subjected to uniaxial or biaxial bending, for a range of material strengths. Figures 10g and h n8 are annealed specimens which are loaded nonproportionally, with varying levels of constant axial force being applied in the different tests (in the table, Po is the ultimate axial strength of the member), followed by application of bending moment.

The percent error in load was computed for each of the verification problems at the deformation level corresponding to the experimental maximum load, or at the level corresponding to the end of the analysis, if this occurs first, and then again at half of that deformation level. The peak strengths for all monotonic comparisons were underpred- icted with an average error of 1.8%, with a standard deviation of 7.0%. At half of the deformation value at peak strength, the average error in the strengths for all monotonic


25O

2OO

Moment 150

(kN-m) 100

50

l,Oi f " " /. Moment /

(kN-m) 50

• Experimental

- Computational

i L 0 0.00000 0 . 0 0 0 1 0 0.00020

Curvature (rad/mm) a) Lu and Kennedy (1994), Specimen CB33

Load 1500

(kN) 1000 I " ~ t ~ .Experimental

500 ~ - ~ - Computational V

O f I I I 0 5 10 15

Deflection (ram) c) Bridge 1'1976), Specimen SHC-1

2500 ~axial Bendi~

2000 ~ ~ " - ~ ~ '

Load 1500

(kN) 1000 / " Experimental

500 / / - Computational 0 I I i I

0 5 10 15 20 Deflection (mm)

e) Bridge (1976), Specimen SHC-5 15 . . . .

10 Moment (~'q-m) /

5 _ _ °

Figure 10

0

800

Load 600

(~,4) 4oo

200

I I

Load (kN)

10

8 Moment (l~/-m) 6

4

2

0

• Experimental

- Computational

I 0.00000 0.00010

Curvature (rad/mm) b) Lu and Kennedy (1994), Specimen CB53

lOOO ~

- Computational

0 I I I 0 10 20 30

Deflection (mm) d) Cederwall et al (1991), Specimen 10

250 • "

200

150 100 / - Computational

/ 50

0 I I I 0 10 20 30

Deflection (mm) f) Shakir-Khalil and Zeghiche (1989), Spec. 6

12 E~---~t~

" ~ ~ ~ t a l / - Computation "al

0.00020

0 t t I I t J I 0 0.005 0.01 0.015 0 0 .005 0 . 0 1 0 .015 0.02

Rotation (rad) Rotation (rad) g) Tomii and Sakino (1979), Specimen 111-3 h) Tomii and Sakino (1979), Specimen III-6 Comparison of experimental and computational monotonic Ioadin 9 results


Table 1 Concrete-f i l led steel tube monoton ica l ly loaded ver i f icat ion exper iments

Test (specimen) Actual tube D/t major L/D major f" (MPa) fy (MPa) Other data d imensions (mm) (minor) (minor)

Lu and Kennedy 36 253.4 × 152.0 x 39.7 (23.75) 12 (20) 45.2 377 (CB33) 6.17

Lu and Kennedy 36 253.4 × 152.0 × 39.7 (23.75) 8.8 (14.7) 42.1 377 (CB53) 6.17

Bridge 17 (SHC-1) 203.7 x 203.9 x 20 10.5 29.9 291 9.96

Cederwal l et al. 52 120 x 120 x 8 15 25 39 397 e = 20 mm (10) Bridge 17 (SHC-5) 202.6 x 203.2 x 20 15 44.3 319 e = 38 mm

10.0 c~ = 30 °2 Shakir-Khali l and 120 x 80 x 4.47 24 (16) 23 (34.5) 45 343.3 er~.jor = 24 mm Zegiche 19 (6) eminor = 16 mm Tomi i and 100 x 100 x 2.99 33 3 20.6 289 P/Po = 0.30 Sakino TM (111-3) Annealed tube Tomi i and 100 x 100 x 2.99 33 3 20.6 289 P/Po = 0.60 Sakino TM (111-6) Annealed tube

Ma jo r axis bending; P=0 ; load appl ied 766 mm f rom each end of 3040 mm beam Mino r axis bending; P= 0; load appl ied 463 mm f rom each end of 2231 mm beam e = 38 mm 1

1Eccentricity o f appl ied axial load f rom centroidal axis of member 2Angle of appl ied axial load with respect to major principal axis o f cross-sect ion (i.e. a = 0 ° induces minor axis bending)

comparisons was 0.15%, with a standard deviation of 6.7% 45 .

For the verification studies, 10 fibers were used in each flange of the steel tube (with one fiber through the tube thickness), and the concrete core was meshed with a 10 by 10 grid of fibers. Four elements were used along the length of each member.

6. Effect of slip on concrete-filled tube beam- column behavior

6.1. Effect of bond and slip on flexural behavior The purpose of including slip in the formulation of the CFT element is primarily to determine whether or not loss of bond occurs in CFT members used as part of structural frames. It has already been seen in the verification and calibration of the slip parameters that slip may affect the region near a connection consisting, for example, of a steel girder and/or brace element framing into a CFT with a shear tab, as may be found in braced frame structures. This section, as well as the final example of a four-story frame, provide the results of a preliminary study to determine the effect of slip on behavior of CFT beam-columns, as may be found in unbraced frames having fully-restrained connections.

Clearly if sufficient axial force is applied to only the steel or concrete of a CFT, slip may be induced. The effect of slip on flexural behavior is more difficult to quantify than it is for axial behavior, because material nonlinearity is required before any interface stress develops in flexure. While complexities such as contact between the concrete core and the steel tube are not modeled directly in this formulation, the calibrated slip model provides strong correlation with experimental results of CFTs subjected to flexure. Several comparisons to results from Lu and Kennedy 36 are presented in Schiller and Hajjar 45. The number of elements along the length and fiber mesh densities at each cross-section were similar to those used for the CFT verification studies. For each test, the parameters of the computational slip model were varied to study the effect of bond

strength on the moment-curvature behavior of the members. It was found that reducing the initial slip layer stiffness by two orders of magnitude reduced the initial flexural stiffness of the members by less than 1%, and it did not noticeably change the ultimate moment strength of these members at the point corresponding to the maximum curvature obtained in each experiment, due largely to the fact that the bond stress of 0.6 MPa was not breached in these studies. Similarly, reducing the bond strength to one-third of the calibrated value (0.2 MPa) decreased the maximum strength attained in the specimens by an average of only 2%. In general, the magnitude of strength, stiffness and deformation of the members changed very little for the full duration of loading in all of these parametric studies, and the computational curves all match the experimental results equally well.

6.2. Composite frame example The CFT fiber element formulation was used to analyze a four-story composite CFT frame structure subjected to a combination of gravity and wind loading. Figure 11 illustrates the members, geometry, structural parameters and unfactored loading. The frame is proportioned so as to achi- eve all factored load combinations (specifically, a load factor of 1.0/0.9 = 1.1 1 times the factored loads to account for the AISC LRFD 53 resistance factor in the analysis) without collapsing due to combined geometric and material nonlinear behavior. In the analysis for the AISC LRFD 53 wind load combination, which controls the design, the gravity and lateral loads are increased proportionally, so that at a load factor of 1.0, the applied loads equal the factored loads for the load combination. The frame shown actually reaches its structural limit point at a load factor of approximately 1.4, thus yielding a safe and serviceable design. The computational model utilizes the 18 DOF CFT beam-column fiber element for the columns, and a comparable 12 DOF steel beam-column fiber element for the girders 5,47. Twelve elements are used to model each girder and column, and the fiber mesh densities at each cross-section are similar to those used for the CFT verification studies.


V12

29,2 kN ~ / 4 Stories v / 2 1 v ~ v ~v g~v ~ v ~ v ~v®v/2 [email protected]

27,3 ® v / 2 j v ® ¢ ~ v ~ ~ v ~ v ~ v ® ~ _ ~ =15 '84m

25.0 kN ~ ' ®

/-/7

V = 186.8, kN • All loads shown are unfactored. Gravity

4 Bays @ 6.10 m = 24.40 m load is divided equally Girders: between dead and live.

W410x60; fy= 248.2 MPa /~.nalysis Cases:

CFT Sections: Case 1: No Slip Permitted ~TS 177.8x177.8x6.35 Case II: Slip Restrained at Bottom of @TS 152.4x152.4x4.76 Column Stacks; ~TS 254.0x203.2x6.35 Calibrated Parameters ~)TS 203.2x 152.4x6.35 ®TS 203.2x203.2x6.35 Case HI: Case II, kslip = 102 MPa

~TS 152.4xl 52.4x6.35 Case IV: Case II, fOond = 0.1 MPa

fy= 317.2 MPa, fc" = 34.5 MPa

Figure 11 Four-story f rame computat ion model

The AISC LRFD interaction equation values for the individual members were also computed at a load factor of 1.0. Even at this load level, almost all of the lower story columns have exceeded the allowable LRFD interaction equation value. The distributed plasticity analysis not only indicates that the frame has a higher ultimate capacity, but also that the members are largely elastic at a load factor of 1.0. This result illustrates the generally conservative approach that design codes have adopted currently for CFT members.

Four cases were investigated to determine the effect of slip parameters on the behavior of the frame. As seen in Figure 1 I, in Case I the steel girder was assumed to engage the concrete directly through the connection, and no slip was permitted at the joints. In the other cases, the steel girder was assumed to engage only the tube, load was transferred to the concrete through the slip interface, and the concrete was free to protrude from the top of the column stack. Case II used the calibrated parameters for fhond and k~,p. In Case III, the value of k~, v was decreased to 10 2 MPa. In Case IV, the value of fbo.d was decreased to 0.1 MPa. Each analysis was carried out to the limit point of the structure, There was no perceivable difference between the entire load-deformation response of the analyses, thus indicating that slip has a negligible effect on the overall load-deformation behavior of this frame.

However, the potential effects of slip on local behavior of CFTs are more evident. Figure 12 shows the slip and bond stress along the length of each CFT column stack for Cases II, III, and IV. The slip plotted as positive to the right of each column stack indicates that the concrete core is moving vertically relative to the steel tube (i.e. it tends to protrude through the top of the column stack). Figure 13 shows the axial force in tae concrete core and in the steel tube along the length of each CFT column stack for the same cases. All results are shown at a load factor of 1.0 for the wind load combination. The bond stress is not breached in Case II at this load level, although bond stresses exceed 0.6 MPa at the top of each column stack at a load level just over 1.1. The concentration of slip and bond stress at each connection is evident (Figure 12), as is the gradual transfer of load from the steel tube to the concrete core both above and below each connection. The effects of tension just above many of the connections are evident as

i •

(

Scale in mm i J i

0 1

,°° f °"

l.I .

Case II

- - Case IH

....... Case IV

(

t a) Slip in CFTs at a factored load level of 1.0

E> °°

°,

>

>

C a s e II Scale in M P a

', ', I ~ ¢ I ~ ~ C a s e I I I 0 0 . 3 0 . 6

. . . . . . . C a s e I V

Figure 12 Slip and bond stress in the composite frame at the factored design load

well, with the compressive axial force in the steel tube decreasing rapidly over a short distance as a portion of the gravity load at the floor level is carried by the tube above the connection as well as below (Figure 13b). However, the magnitude of the slip is relatively small in this case, as may be expected for unbraced framing action, and it does not change significantly even up to the failure load.

For Case III, the slip is an order of magnitude larger than for Case II because of the reduced slip stiffness (Figure 12a). Correspondingly, the transfer of load from the connection regions occurs over nearly the entire length of each CFT column. As compared to that of Case II, the concrete load for Case III is seen in Figure 13 to increase consistently along the member length, while the steel tube continuously sheds load to the concrete along the tube length. In addition, because of this more gradual transfer of load, the maximum interface stress values in the CFTs of Case III are seen to be well below those of Case II (Figure 12b).

For Case IV, the bond stress of 0.1 MPa is breached at a load factor of 0.19. Since the slip stiffness is small at locations which have breached fbo,d, the bond stress is effectively capped at this value, as is seen by the dotted line in Figure 12b. The magnitude of slip is more than dou- ble that of Case II at a factored level of 1.0, and the transfer of loads from the connections takes place over nearly the

674 A distributed plasticity model for CFT beam-columns with slip: d. F. Hajjar et al.

t t l Case II

S c a l e in kN P I ) I - - Case In

0 1500 . . . . . . . Case IV

Compression

a) A x i a l fo rce in C F T c o n c r e t e c o r e s at a f a c t o r e d load l eve l o f 1.0

1 P

0

I I

Scale in kin I I I

1500

C o m p r e s s i o n

I f

Case II

- - Case HI

. . . . . . . Case IV

b) A x i a l f o r ce in C F T steel tubes at a f ac to r ed load l eve l o f 1.0

Figure 13 Axial force in the concrete core and steel tube in the composite frame at the factored design load

entire length of the CFT, as in Case Ill (Figure 13). The bond stress and slip profile at the failure load of the structure (not shown) indicate substantially more slip and bre- aching of the bond strength occurring along nearly the entire length of the three central CFT column stacks, indicating that continued loading substantially increases the bond loss in these members.

The frame was also analyzed for the AISC LRFD 53 gravity load combination using the calibrated parameters to determine if loss of bond is detected under the increased vertical loading. For this loading combination, the interface stress of 0.6 MPa was breached at 0.68 times the factored load, suggesting that the increased vertical load that must be transferred at connections due to factored gravity loading may cause bond loss in CFT members before the design load level of the frame is reached.

For this specific frame, the top of each column stack consistently has the highest magnitudes of slip and bond stress for all loading combinations. Capping the CFTs at the top of the column stacks alleviates the slippage at that location, but localized slip in the connection regions at each floor level is still seen when using the calibrated parameters. Global behavior, however, is unaffected by slip in these analyses for all load combinations. Of course, engag- ing the concrete directly in each connection, and thus restraining slip, facilitates immediate transfer of the load to the connection region.

7. Conclusions

This paper summarizes the development and use of a fiber- based distributed plasticity finite element formulation for three-dimensional concrete-filled steel tube beam-columns. The formulation may be used to simulate the behavior of composite frames consisting of steel I-girders framing into CFTs, subjected to monotonic static loading; a companion paper 2~ outlines the formulation for cyclic loading. The fiber element approach discretizes the CFT element end cross-sections into a grid of fibers, and the steel and concrete stress-strain behavior is tracked explicitly at each fiber. In addition, this formulation permits axial slip between the concrete core and steel shell of the CFT, so as to permit study of the effect of slip on CFT beam-column behavior as part of braced or unbraced frame structures. Details of the calibration and verification of the slip parameters are presented, based on comparison to tests of steel I-girders framing into CFTs with shear connections. The calibrated parameters suggest that little slip is experienced in a CFT member before the bond strength of the slip interface is breached. In addition, the calibrated value of bond strength used for analysis is higher than the value rec- ommended by design codes, suggesting that the rec- ommended design values may be conservative. The beam- column element is verified against experiments of individual CFTs having a wide range of cross-section properties, material properties and lengths, subjected to combined axial force and uniaxial or biaxial bending moment, and loaded either proportionally or nonproportionally.

A preliminary investigation is included which indicates that bond strength may be breached in unbraced frames, possibly before the design load level is reached, if slip is not restrained at the joints (e.g. if the concrete core is not engaged directly at the connection), due either to gravity load or wind load combinations. However, in these studies, even for the more extreme conditions, slip is seen to have little effect on the global behavior of a composite CFT frame, or on the strength achieved by a CFT member subjected to flexure. Nevertheless, understanding the effect of slip more fully on the behavior of CFTs in composite frame structures warrants further comprehensive parametric studies; experimental and analytical research in this area is ongoing 54. Documenting the effect of varying CFT per- imeters or member lengths on the slip calibration requires additional experimental tests of steel I-beams framing into CFTs for comparison, and establishing the effect of slip on a wider variety of frame configurations and boundary conditions is needed.

This stress-resultant based distributed plasticity formulation provides detailed information on CFT behavior, yet remains efficient and suitable for studying complete composite CFT frames. The formulation is especially amenable to conducting static, nonlinear seismic 'push-over' analysis of composite CFT frames, or for conducting advanced analysis directly for static design of frames z6. Cyclic behavioral studies may also be undertaken; transient dynamic analysis is pending for this CFT formulation.

Acknowledgements

The authors would like to thank Professor H. Shakir-Khalil, University of Manchester, Professor T. Usami, Nagoya University, and Professor O. Buyukozturk, Massachusetts Institute of Technology, for their generous sharing of infer-


mation relevant to this research, and Katherine A. Fetterer for assisting with the analysis of the four-story frame. Funding for this research was provided by the National Science Foundation (Grant no. CMS-9410473) under Dr Shih-Chi Liu and Dr M. P. Singh, and by the University of Minnesota Department of Civil Engineering through a Sommerfeld Fellowship for the third author and through additional research funding. The authors gratefully acknowledge this support.

References 1 Gourley, B. C., Hajjar, J. F. and Schiller, P. H. 'A synopsis of studies

of the monotonic and cyclic behavior of concrete-filled steel tube beam-columns', Report no. ST-93-5.2, Department of Civil Engineer- ing, University of MinneapoLis, Minneapolis, MN, 1995

2 Wright, E. W. and Gaylord, E. H. 'Analysis of unbraced multistory steel rigid frames', J. Struct. Div., ASCE 1968, 94(ST5), 1143-1163

3 Alvarez, R. J. and Birnstiel, C. qnelastic analysis of multistory multi- bay frames', J. Struct. Div., ASCE 1969, 95(STll) , 2477-2503

4 Kanchanalai, T. 'The design and behavior of beam-columns in unbraced steel frames', AISI Proj. no. 189, Civil Engineering/Structural Research Laboratory (CESRL) Rep. no. 77- 2, Department of Civil Engineering, Structural Research Laboratory, University of Texas, Austin, TX, 1977

5 White, D. W. 'Material and geometric nonlinear analysis of local planar behavior in steel frames using interactive computer graphics', M.S. thesis, Cornell University, Ithaca, NY, 1985

6 Izzuddin, B. A. and Elnashai, A. S. 'Adaptive space frame analysis. Part II: a distributed plastici:y approach', Proc. Inst. of Civil Engng 1993, 99, 317-326

7 Challa, V. R. M. and Hall, J. F. 'Earthquake collapse analysis of steel frames', Earthq. Engng Struct. Dyn. 1994, 23( I 1 ), 1199-1218

8 Krishnamoorthy, C. S. 'Finite element models for inelastic analysis of reinforced concrete frames and application to limit state design', Computational plasticity: models, software and applications, Owen, D. R. J., Hinton, E. and Onate, E. (eds), Pineridge Press, Swansea, 1987

9 Taucer, F., Spacone, E. ancl Filippou, F. C. 'A fiber beam-column element for seismic response analysis of reinforced concrete structures', Report no. UCB/EERC-91/17, Earthquake Engineering Research Center, University of California, Berkeley, CA, 1991

10 Chang, G. A. and Mander, J. B. 'Seismic energy based fatigue damage analysis of bridge columns: Part I - - evaluation of seismic capacity', Rep. no. NCEER-94-0006, National Center for Earthquake Engineering Research, State University of New York, Buffalo, NY, 1994

11 lzzuddin, B. A., Karayanni,;, C. G. and Elnashai, A. S. 'Advanced nonlinear formulation tor reinforced concrete beam-columns', J. Struct. Engng, ASCE 1994, 120(10), 2913-2934

12 Ahmad, S. H. and Weerakoon, S. L. 'Model for behavior of slender reinforced concrete column:~ under biaxial bending', ACI Struct. J. 1995, 92(2), 188-198

13 Roik, K. and Bergmann, R. 'Composite columns', Const. Steel Des.: An lnt Guide, Dowling, P. J , Harding, J. E. and Bjorhovde, R. (eds), Elsevier Applied Science, London, 1992, pp. 443-469

14 Elnashai, A. S. and Elghazouli, A. Y. 'Performance of composite steel concrete members under earthquake loading: 1. Analytical model', Eorthq. Engng Struct. Dyn. 1993, 22(4), 315-344

15 EI-Tawil, S. and Deierlein, G. G. 'Inelastic dynamic analysis of mixed steel-concrete space frames', Structural Engineering Rep. no. 96-5, School of Civil and F, nvironmental Engineering, Coruell Uni- versity, Ithaca, NY, 1996

16 Bode, H. 'Columns of steel tubular sections filled with concrete design and applications', Acier Stahl 1976, 11(12), 388-393

17 Bridge, R. Q. 'Concrete filled steel tubular columns', Rep. R283, School of Civil and Mining Engineering, University of Sydney, Syd- ney, Australia, 1976

18 Tomii, M. and Sakino, K. 'Experimental studies on the ultimate moment of concrete-filled square tubular beam-columns', Trans., Arch. Inst. Japan 1979, 27S, 55-63

19 Shakir-Khalil, H. and Zeghiche, Z. 'Experimental behavior of concrete-filled rolled rectangular hollow section columns', The Struct. Engineer 1989, 67(19), 345-353 Kawagucbi, J., Morino, S., Atsumi, H. and Yamamoto, S. 'Strength deterioration behavior of concrete-filled steel tubular beam-columns', Proc. Third Int. Conf on S,'eel-Concrete Comp. Struct., Wakabaya-

20

shi, M. (ed.), Fukuoka, Japan, 1991, Association of International Cooperation and Research in Steel-Concrete Composite Structures, 1991, pp. 119-124

21 Tsuji, B., Nakashima, M. and Morita, S. 'Axial compression behavior of concrete filled circular steel tubes', Proc. Third Int. Conf. Steel- Concrete Composite Struct., Wakabayashi, M. (ed.), Association International Cooperation and Research in Steel-Concrete Composite Structures, 1991, pp. 19-24

22 Hajjar, J. F., Molodan, A. and Schiller, P. H. 'A distributed plasticity model for cyclic analysis of concrete-filled steel tube beam-columns and composite frames', Engng Struct. 20(4-6), 398412

23 Gourley, B. C. and Hajjar, J. F. 'Cyclic nonlinear analysis of three- dimensional concrete-filled steel tube beam-columns and composite frames', Report no. ST-94-3, Department of Civil Engineering, Uni- versity of Minneapolis, Minneapolis, MN, 1994

24 Hajjar, J. F. and Gourley, B. C. 'Representation of concrete-filled steel tube cross-section strength', J. Struct. Engng, ASCE 1996, 122( I 1 ), 1327-1336

25 Hajjar, J. F., Gourley, B. C. and Olson, M. C. (for Part II) 'A cyclic nonlinear model for concrete-filled tubes. I: formulation. II: verification', J. Struct. Engng, ASCE 1997, 123(6), 736-754

26 Hajjar, J. F., Gourley, B. C., Schiller, P. H., Molodan, A. and Stillwell, K. A. "Seismic analysis of concrete-filled steel tube beam- columns and three-dimensional composite frames', Comp. Const. HI, Buckner, C. D. and Shahrooz, B. M. (eds), Engineering Foundation, ASCE, NY 1997, pp. 75-88

27 Molodan, A. and Hajjar, J. F. 'A cyclic distributed plasticity model for three-dimensional concrete-filled steel tube beam-columns and composite frames', Rep. no. ST-96-6, Department of Civil Engineer- ing, University of Minneapolis, Minneapolis, MN, 1997

28 Sherman, D. R. 'Tubular members', Const. Steel Des.: An Int. Guide, Dowling, P. J., Harding, J. E. and Bjorhovde, R. (eds), Elsevier Applied Science, London, 1992, pp. 91-104

29 Sully, R. M. and Hancock, G. J. 'Behavior of cold-formed SHS beam- columns', J. Struct. Engng, ASCE 1996, 122(3), 326-336

30 Goel, S. C. and Yamanouchi, H. (eds) Proc. 1992 US-Japan Work- shop on Composite and Hybrid Structures, Berkeley, CA, 1992, Research Report, Department of Civil and Environmental Engineer- ing, University of Michigan, Ann Arbor, MI, 1993

31 Virdi, K. S. and Dowling, P. K. 'Bond strength in concrete-filled steel tubes', IABSE Periodica 3/1980, IABSE Proc. P-33/80, 1980, pp. 125-139

32 Tomii, M. 'Bond check for concrete filled steel tubular columns', Comp. and Mixed Constr., Proc. of the US-Japan Joint Sere. on Comp. and Mixed Constr., Roeder, C. W. (ed.), University of Wash- ington, Seattle, WA, ASCE, NY, 1985, pp. 195-204

33 Shakir-Khalil, H. 'Pushout strength of concrete-filled steel hollow sections', The Struct. Engineer 1993, 71(13), 230-233

34 Shakir-Khalil, H. 'Resistance of concrete-filled steel tubes to pushout forces', The Struct. Engineer 1993, 71(13), 234-243

35 Shakir-Khalil, H. and Hassan, N. K. A. 'Push-out resistance of concrete-filled tubes', Tubular structures VI, Grundy, P., Holgate, A. and Wong, W. (eds), Melbourne, Australia, A. A. Balkema, Rotterdam, The Netherlands, 1994, pp. 285-291

36 Lu, Y. Q. and Kennedy, D. J. L. 'The flexural behaviour of concrete- filled hollow structural sections', Can. J. Civil Engng 1994, 21(1), I 1 1 - 1 3 0

37 Dunberry, E., LeBlanc, D. and Redwood, R. G. 'Cross-section strength of concrete-filled HSS columns at simple beam connections', Can. J. Civil Engng 1987, 14, 408-417

38 Shakir-Khalil, H. 'Connection of steel beams to concrete-filled tubes', Proc. 5th Int. Symp. Tubular Struct., Nottingham, UK, 1993, pp. 195-203

39 Shakir-Khalil, H. 'Beam connection to concrete-filled tubes', Tubular structures VI, Grundy, P., Holgate, A. and Wong, W. (eds), Mel- bourne, Australia, A. A. Balkema, Rotterdam, The Netherlands, 1994, pp. 357-364

40 British Standards Institute (BSI) BS 5400: Steel, concrete and composite bridges. Part 5: Code of practice for design of composite bridges, London, British Standards Institution, 1979

41 Architectural Institute of Japan (AIJ) AIJ Standard for structural calculation of mixed tubular steel-concrete composite structures, AIJ, Tokyo, Japan, 1980

42 Bradford, M. A. and Gilbert, R. I. 'Composite beams with partial interaction under sustained loads', J. Struct. Engng, ASCE 1992, 118(7), 1871-1883

43 Amadio, C. and Fragiacomo, M, 'A finite element model for the study of creep and shrinkage effects in composite beams with deformable shear connections', Construzioni Metalliche, 1993, 4, 213-228


44 Aribert, J.-M. 'Influence of slip on composite joint behaviour', Conns. in Stl Structs. I11: Beh., Strength and Des. R. Bjorhovde, A. Colson and R. Zandonini (eds), Pergamon, Amsterdam, 1995, pp. 11-22

45 Schiller, P. H. and Hajjar, J. F. 'A distributed plasticity formulation for three-dimensional rectangular concrete-filled steel tube beam columns and composite frames', Report no. ST-96-5, Department of Civil Engineering, University of Minneapolis, Minneapolis, MN, 1996

46 Bathe, K.-J. Finite element procedures in en~,ineering analysis, Pren- tice-Hall, Englewood Cliffs, NJ, 1982

47 Morales, L. E. 'Object-oriented software for advanced analysis of steel frames', M.S. thesis, Department of Civil Engineering, Purdue University, West Lafayette, IN, 1994

48 Yang, Y. B. and Kuo, S. R. Theory and analysis o f nonlinear framed structures, Prentice-Hall, Englewood Cliffs, NJ, 1994

49 Kitada, T. 'Ductility and ultimate strength of concrete-filled steel members', Stability and ductility o f steel structures under cyclic loading, Fukumoto, Y. and Lee, G. C, (eds), CRC Press, Boca Raton, FL, 1992, pp. 139-148

50 Shen, C., Mamaghani, I. H. P., Mizuno, E. and Usami, T. 'Cyclic behavior of structural steels. II. Theory', J. Engng Mech., ASCE 1995, 121(11), 1165-1172

51 Chert, E. S. and Buyukozturk, O. "Constitutive model for concrete in cyclic compression', J. Engng Mech., ASCE 1985, I l l , 797-813

52 Cederwall, K., Engstrom, B. and Grauers, M. 'High strength concrete used in composite columns', 2nd h~t. Syrup. on Utilization t~f High- Strength Conc., American Concrete Institute, 1991, pp. 195-214

53 American Institute of Steel Construction, Load and resistance.tactor design specification for structural steel buildings, Second edn, AISC, Chicago, IL, 1993

54 Roeder, C. W. 'Design requirements for shear connectors in encased steel (SRC) and concrete filled tube (CFT) construction'. Presen- tation made at the Third JTCC, U.S./Japan Cooperative Research Program on Composite and Hybrid Structures, Hong Kong, 1996, Department of Civil Engineering, University of Washington, Seattle, WA, 1996

A Distributed Plasticity Model For

Documents

Transcript of A Distributed Plasticity Model For