This is the author’s version of a work that was submitted/accepted for pub-lication in the following source:

Iu, C.K., Bardford, M.A., & Chen, W.F.(2009)Second-order inelastic analysis of composite framed structures based onthe refined plastic hinge method.Engineering Structures, 31(3), pp. 799-813.

This file was downloaded from: https://eprints.qut.edu.au/66964/

c© Copyright 2009 Elsevier

Notice: Changes introduced as a result of publishing processes such ascopy-editing and formatting may not be reflected in this document. For adefinitive version of this work, please refer to the published source:

https://doi.org/10.1016/j.engstruct.2008.12.007

1

Second‐Order Inelastic Analysis of Composite Framed Structures Based on 

the Refined Plastic Hinge Method 

C.K. Iu1* , M.A. Bradford2 and W.F. Chen3 

1,2Centre for Infrastructure Engineering and Safety 

School of Civil and Environmental Engineering 

The University of New South Wales 

UNSW Sydney, NSW 2052, Australia  

3Department of Civil and Environmental Engineering 

University of Hawaii at Manoa 

2540 Dole Street, Holmes 383, Honolulu, Hawaii, HI 96822, USA 

_____________ 

1 Research Associate:  

Email: jerryiu@unsw.edu.au;   Phone: +61 2 9385 5029;   Fax: +61 2 9385 9747 

 

2 Scientia Professor and Australian Government Federation Fellow: 

Email:  m.bradford@unsw.edu.au;   Phone: +61 2 9385 5014;   Fax: +61 2 9385 9747 

 

3Professor 

Email: chenwf@eng.hawaii.edu;   Phone: +808 956 7727 

 

* Corresponding author 

 

 

2

Abstract

Composite steel-concrete structures experience non-linear effects which arise from both

instability-related geometric non-linearity and from material non-linearity in all of their

component members. Because of this, conventional design procedures cannot capture the true

behaviour of a composite frame throughout its full loading range, and so a procedure to

account for those non-linearities is much needed. This paper therefore presents a numerical

procedure capable of addressing geometric and material non-linearities at the strength limit

state based on the refined plastic hinge method. Different material non-linearity for different

composite structural components such as T-beams, concrete-filled tubular (CFT) and steel-

encased reinforced concrete (SRC) sections can be treated using a routine numerical

procedure for their section properties in this plastic hinge approach. Simple and conservative

initial and full yield surfaces for general composite sections are proposed in this paper. The

refined plastic hinge approach models springs at the ends of the element which are activated

when the surface defining the interaction of bending and axial force at first yield is reached; a

transition from the first yield interaction surface to the fully plastic interaction surface is

postulated based on a proposed refined spring stiffness, which formulates the load-

displacement relation for material non-linearity under the interaction of bending and axial

actions. This produces a benign method for a beam-column composite element under general

loading cases. Another main feature of this paper is that, for members containing a point of

contraflexure, its location is determined with a simple application of the method herein and a

node is then located at this position to reproduce the real flexural behaviour and associated

material non-linearity of the member. Recourse is made to an updated Lagrangian

formulation to consider geometric non-linear behaviour and to develop a non-linear solution

strategy. The formulation with the refined plastic hinge approach is efficacious and robust,

and so a full frame analysis incorporating geometric and material non-linearity is tractable.

By way of contrast, the plastic zone approach possesses the drawback of strain-based

procedures which rely on determining plastic zones within a cross-section and which require

lengthwise integration. Following development of the theory, its application is illustrated

with a number of varied examples.

Key Words:

3

Beam-columns; composite members; geometric non-linearity; material non-linearity; plastic

hinge method; initial and full yield surfaces.

4

1. Introduction

Composite structural members and frames which contain steel and concrete components are

designed to best exploit the structural advantages of these two component materials. This

well-known symbiotic structural relationship is utilised widely in contemporary design and

construction solutions worldwide, resulting in stronger and stiffer frames than those afforded

by the counterpart bare steel frame structure. Despite this widespread use of steel-concrete

composite systems, accurate numerical procedures for their analysis and design have not

evolved in the same way as for steel frames, either with or without semi-rigid joints, and as a

consequence it is recognised widely that prescriptive codes of practice possess many

shortcomings when applied to the safe and economical design of composite structures. This

paper is therefore concerned with the development of a robust finite element-based numerical

procedure to model the material and geometric non-linearity in a composite building frame,

both accurately and efficiently.

Many cases of loading on a composite framed building, such as that caused by its self-weight,

imposed live loading, wind loading as well as extreme loads such as fire and earthquake, will

lead to situations for which the concrete will experience compressive stress and to potential

crushing, tensile stress and therefore to cracking, as well as situations for which the steel may

yield, or may buckle in compression. Because a multi-storey composite frame is often

subjected to significant gravity loading as well as lateral loading, it behaves as a sway frame

and so both geometric non-linearity and flexural buckling are of significance. Consequently,

any numerical procedure which claims to replicate the structural behaviour of a real

composite framed building at ultimate loading must account for material non-linearity of the

composite component as a whole, as well as geometric non-linearity of the frame.

It is both simple and convenient to investigate the strength of a composite cross-section under

both axial and bending actions using interaction yield surfaces. Ultimate strength surfaces for

a short reinforced concrete member subjected to biaxial bending moments at a given level of

axial loading were developed in early work by curve fitting of experimental data (e.g. Bresler

[1], Furlong [2], Meek [3], Bradford and Gilbert [4]). It was found that the interaction surface

between axial load and bending at ultimate was complicated and non-linear. The most useful

failure interaction surface in the form of biaxial bending, in which the presence of axial force

5

is implicitly considered, has been proposed for the concrete columns (e.g. Bresler [1] and

Chen and Shoraka [5]). Matsui et al. [6] concluded that this interaction equation in the form

of biaxial bending is also suitable for a steel-reinforced concrete column (SRC), in which the

steel is encased with concrete. Material strength curves of this type may be used to include

material non-linearity (Oehlers and Bradford [7]), and they underpin design procedures such

as in the Eurocode 4 [8]. Recent work on these failure surfaces of reinforced concrete has

been presented by Cedolin et al. [9].

The effect of confinement in a concrete-filled tubular (CFT) composite column is known to

enhance its cross-sectional strength significantly (Oehlers and Bradford [7]), because the axial

strength capacity of the concrete is improved by the triaxial loading regime produced in it by

the confining steel tube. Knowles and Park [10] tested circular and square concrete-filled

steel hollow sections. Their results showed that the circular steel tube provided confinement

of the concrete core, and this confinement enhanced the ultimate loads of the circular columns.

Enhanced ultimate loads were observed only for short columns, but no benign confinement

effects were observed for square or rectangular CFT columns. The section shape has a

profound effect on the confining action; square or rectangular CFT columns confine the

concrete by plate bending whereas circular tubes provide restraint through hoop membrane

stresses. The effects of the section shape on the ultimate load were investigated by Tomii et

al. [11], while Schneider [12] investigated CFT columns with different slenderness ratios

experimentally. His study showed that circular steel tubes have significant post-yield

ductility because of the membrane hoop tension developed in the steel tube. Semi-empirical

design procedures for CFT columns were developed by Bradford and Nguyen [13], and others.

CFT columns are susceptible to local buckling of the steel tube, but this uniaxial buckling

mode is attained at larger stresses than if the tube was hollow because of the restraint

provided by the stiff concrete core. This effect must be included if the CFT is thin-walled,

and semi-empirical bases for design of circular tubes have been proposed by Bradford et al.

[14] and for rectangular box columns by Uy [15]. In summary, then, it is possible to make

recourse to these findings to provide a semi-empirical strength interaction curve for a

composite column cross-section which may be a SRC member or a CFT member with

empirical formulations for the effects of confinement and local buckling.

6

Numerical techniques for simulating the structural response of composite columns have

grown in prominence over the last decade or so. Mirza and Skrabek [16] presented a

numerical model to determine the nominal strength of SRC columns using their moment-

curvature-thrust relationship and moment equilibrium, assuming a deflected shape for the

beam-column element. A similar procedure to incorporate concrete material non-linearity

was used by Bradford and Nguyen [13] and Bradford [17], and for CFT columns by Mursi

and Uy [18]. Hajjar and Gourley [19] presented a detailed plastic zone analysis of short

square CFT columns, validated with experiments, to obtain an accurate representation of the

cross-section strength of a CFT subjected to a combination of biaxial bending and axial

compression.

For investigating the stability of a composite frame, the description of kinematic deformations

for use in a total and an updated Lagrangian formulation is a necessity. Initial studies using a

total Lagrangian formulation (Mallet and Marcal [20], Zienkiewicz and Taylor [21] and Wen

and Rahimzadeh [22]) were made in the development of the numerical approaches for

modelling the geometric non-linearity. To extend his work to model geometric non-linearity

for a beam-column element, Hajjar et al. [23] developed a stiffness-based finite element

approach for CFT composite members using an updated Lagrangian formulation, which was

able to capture the P- and P- geometric non-linearity in a restrained CFT column, as well as

material non-linearity in the form of distributed plasticity and inter-layer slip between the

steel and concrete. Using a displacement-based finite element formulation, Pi et al. [24] [25]

presented a second order inelastic analysis which included the shear force versus slip

relationship derived from empirical data. Material non-linearity was included with a plastic

zone formulation, with a total Lagrangian formulation being adopted in tackling the geometric

non-linear response. This method was shown to be very versatile and accurate for isolated

members, but its plastic zone formulation in a total Lagrangian representation is

comparatively complicated for use in frame structures with many members, and a reason for

the alternative approach adopted in the present paper.

The objective of this study is to present a non-linear inelastic analysis of composite framed

structures so that the complicated non-linear response of the structure can be obtained in an

efficacious fashion. This stiffness-based finite element formulation resorts to a plastic hinge

approach (Iu and Chan [26]) which has previously been implemented for steel frames; the

7

refined plastic hinge has been modified from an inelastic analysis (Iu [27]) of composite

beams, and can be adapted to different arbitrary composite sections in the beam-column

element. Initial and full yield surfaces for both axial and bending effects which act on

composite sections are proposed, which are conservative for the design purposes, and they are

consistent with plastic hinge formulations that appear in design codes (such as Eurocode 4

[8]). This proposed plastic hinge is able to capture the gradual transition from initial yield to

full plasticity under the interaction of axial and bending effects, which describes relationship

between the load and displacement due to material non-linearity. This refined plastic hinge

approach is versatile and adaptive to different kinds of material by using the appropriate

failure surface. It allows for faster and better-controlled convergence than the plastic zone

method. The geometric non-linearity is included with respect to an updated Lagrangian

formulation, for which the geometry of the structure is updated continuously at each iteration.

Following the development of the proposed plastic hinge model for the composite frames, its

accuracy and scope are established with a number of comparative examples. In particular, a

large-scale composite space frame is studied to demonstrate the efficacy of the present

numerical procedures, which is the one of the prime criteria of engineering design.

2. Basic Stiffness Formulation

Second order analysis in a line finite element representation is usually implemented from

Green’s strain tensor t, with reference to the centroidal axis of the element. The element is

assumed to be prismatic with principal axes y and z and a longitudinal axis x; xyz forming a

right handed Cartesian axis system. This longitudinal strain t consists of a linear term

(u/x), non-linear terms (½(v/x)2 and ½(w/x)2) due to member bowing, as well as

bending terms (y2v/x2 and z2w/x2), so that

wzvywvut 2212

21 , (1)

in which u, v and w are the displacements in the x, y and z directions respectively and ( )

( )/x.

In the elastic range within the element in the plastic hinge modelling, the normal stress is

defined in terms of the normal strain t as

8

tE , (2)

and invoking Castigliano’s first theorem of strain energy for the internal strain energy U

produces

tEU δδ , (3)

which may be integrated over the volume (vol) of the element to produce

L

AvolxAEvolEU

tt

00

221

0

dd)d(d

. (4)

By noting that the axes are centroidal, Eqn (4) leads to

LL

yL

zLL

zwEI

zvEI

xwP

xvP

xuEA

U0

2

0

2

0

2

0

2

0

2 d2

d2

d2

d2

d2

, (5)

in which P is the axial force, Iy and Iz are the second moments of area about respective axis.

The plastic hinge approach herein isolates inelastic effects to the member ends, so that it is

elastic with elastic modulus E for x (0, L). The plastic hinge contribution is discussed

subsequently.

The external work done V equals the product of the applied forces and the corresponding

displacements as

fdTV , (6)

in which d and f are column vectors of the displacements and corresponding external applied

forces. The applied load f is assumed to be independent of the displacement d and so is

conservative; this assumption is valid for general structural engineering problems where the

deformations are small enough to not cause large changes of the points of application of the

loads. The external work V is a linear function of the displacement vector d, so that this

component disappears in the derivation of the tangent stiffness (second derivative) for the

structure.

9

The total potential energy for second order elastic analysis is = U – V, so that using Eqs.

(5) and (6) for an element with incremental displacements u, v and w gives

12

1

T

0

2

0

2

0

2

0

2

0

2 d2

d2

d2

d2

d2 k

kk

LLy

Lz

LL

zwEI

zvEI

xwP

xvP

xuEA

fd . (7)

Representing the displacement vector by

T

zzyyxxuu 21212121 d (8)

and integrating the conventional finite element interpolation functions given by

211 uuu ; 22

1221 zz xxv ;

22

1221 yy xxw (9)

where = x/L, the secant stiffness equation may be obtained from

.42

24

42

24

4

4

304

4

30

21

21

21

21

21

21

21

21

21

21

yy

uyz

zz

zzz

yy

yy

zz

zz

k

L

EI

L

EI

PLPLuu

uu

L

EAU

d

(10)

The tangent stiffness matrix may be obtained from

GLtkj

KKKdd

2

, (11)

in which KL and KG are the 6 6 linear and geometric stiffness matrices of an element. The

structural tangent stiffness matrix can then be assembled using

elements

TT

elements

T LNTKTLLLKK teT , (12)

10

in which T is the transformation matrix relating the member forces to the element forces in

the local coordinate system, L is the transformation matrix from local to global coordinates

and N is a matrix introduced to allow for the work done by rigid body movement, which is

neglected in the stiffness formulation based on an updated Lagrangian formulation.

Consequently, the incremental displacements () evaluated from the tangent stiffness include

the rigid body movement, and the geometry of the structure is then updated by accumulation

of both neutral member deformations and rigid body movements.

3. Non-Linear Solution Procedure

A non-linear incremental-iterative procedure is useful for tracing the non-linear equilibrium

path resulting from geometric and material non-linear effects. The incremental-iterative

solution procedure resorts to the constant load method in a Newton-Raphson scheme, as

shown in Fig. 1.

Using a Newton-Raphson formulation, the total potential energy i+1 at the (i+1)-th iteration

can be linearised using the first term in a Taylor expansion. Hence at a solution point,

0ddddd

jj

i

kk

i

k

i 1 , (13)

so that

0ddddd

jj

i

kk

i

k

i UVU. (14)

In Eq. (14),

sik

iUK

d

(15)

is the secant stiffness matrix at the i-th iteration and

Tij

i

kj

i

k

UK

dddd

(16)

11

is the tangent stiffness matrix at this iteration. Equation (14) may therefore be written as

uKuKf Ts , (17)

in which Ksu = i k

kiU d represents the internal member resistances and f is the vector

of the applied loading. The incremental formulation of Eq. (17) is

uKf T , (18)

in which f is a prescriptive load vector or unbalanced force and u is the vector of

incremental displacements due to the load increment and unbalanced force. For a constant

load method, such as the Newton-Raphson technique, f is the same for each load increment

throughout the non-linear solution procedure.

Thus the incremental displacement can then be determined using the tangent stiffness matrix

after this load increment is known, so that

niT

ni fKu 1 , (19)

with the total displacement vector being accumulated by

ni

nni uuu 1 . (20)

The incremental displacements in global coordinates can be transformed to the member

deformations neiu by the transformation matrix LT for mapping global coordinates to local

coordinates. The incremental member resistance in member coordinates neiR can then be

evaluated in local coordinates as

ni

nei uLu T (21)

and

neis

nei uKR , (22)

12

in which Ks is the incremental secant stiffness matrix at neiu . After determination of the

member resistances, the incremental member resistance is accumulated into the total member

resistance vector in global coordinates as

nei

nni RTLRR 1 , (23)

in which T is the transformation matrix from member to global coordinates. The unbalanced

force ni 1f at the second iteration is obtained as

ni

ni

ni Rff 1 , (24)

where nif and n

iR are the total applied load and member resistance vectors at the n-th load

cycle. This incremental procedure is repeated until an equilibrium solution is achieved, or

until divergence is detected.

4. Partial Shear Connection Effects

In common composite building structures, the beam and slab components may not be

connected rigidly, so that relative slip occurs at their interface, leading to a decrease in

strength (partial shear connection effect) and in stiffness (partial shear interaction effect) of

the composite beam relative to the sum of the individual components when they have full

interaction. For composite columns, the axial force usually dominates which leads to

insignificant shear slip at the interface between steel section and concrete. The effect of

partial shear interaction on composite columns is therefore ignored. In the presence of partial

shear interaction between the steel beam and concrete slab, the effective second moment of

area of the composite beam Ieff is using adopted from the the AISC [28] as

sff

fseff II

C

NII , (25)

in which Nf and Cf are respectively the total shear capacity contributed from the shear studs

and total shear capacity in the longitudinal direction from the steel beam or concrete slab,

whichever is less, so the ratio of Nf/Cf can be treated as degree of shear connection, Is and If

13

are the second moment of inertia of steel section and the composite section with full

interaction, respectively.

Similarly, the effective plastic section modulus Seff including partial shear connection can be

expressed as

sff

fseff SS

C

NSS , (26)

in which Ss and Sf are the plastic section modulus of the steel section and the composite

section with full composite action, respectively. The elastic section modulus of steel section

can be expressed as

c

effeff y

IZ or

c

effeff yD

IZ

, (27)

whichever is smaller on the safer side. An elastic stress distribution across the composite

section is therefore ensured. When a composite beam is subjected to both sagging and

hogging moment, the flexural properties of the composite beam are different either side of the

point of contraflexure. Therefore, Eqs. (25) to (27) should be only applied to the composite

beam in either its sagging or hogging moment region. It is also beneficial to assign the

constant section properties to an element in accordance with its moment distributions. The

reduction in the plastic and elastic moment capacities due to the partial shear connection is

accounted for by Eqs. (25) and (27) respectively, which reduce the yield surfaces as discussed

in Section 5; the deterioration of stiffness due to partial shear interaction can be accounted by

Eq. (25) as addressed in Section 6.

5. Implementation of Material Non-Linearity

5.1 Constitutive laws

Normal structural steel behaviour is characterised by a stress-strain representation as in Fig.

2(a), comprising of an elastic range, gradual and full-yielding, and then by a strain hardening

range until eventual fracture. It is commonly assumed that this behaviour is the same for

tension and compression. Herein, the steel is assumed to be isotropic elastic and fully ductile

14

in the elastic range in the direction of longitudinal normal stress. On the other hand, the

stress-strain representation for concrete in compression is usually characterised by an elastic

range, gradual yielding to its ultimate strength and then by material softening until failure, as

shown in Fig. 2(b). In tension, concrete is usually accepted to be elastic-brittle; its tensile

strength is usually neglected in strength design and it is also neglected in the plastic hinge

model of this paper. In the proposed plastic hinge approach, the material behaviour of the

composite section is integrally considered as a whole.

5.2 Cross-section properties

Cross-sectional properties are useful in describing the material behaviour at a section. For

example, the second moments of area Iy, Iz are related to the elastic stiffness of the member,

whilst the elastic section modulus Ze and plastic section modulus Zp are useful for measuring

the bending strength of the member.

The locations of the elastic and plastic neutral axes are needed for determining section

properties related to bending behaviour at the force-displacement relationship and cross-

section strength capacity, respectively. Unfortunately, the determination of their location is

not trivial and obvious, because of the interdependence between the composite section

properties and these neutral axes when concrete section being in tension is neglected. Further,

because of the variety of composite sections encountered in practice, the material non-linear

analysis of a composite cross-section is best tackled numerically. The location of the elastic

and plastic neutral axes and section properties for Iy, Iz and Ze, Zp, respectively, can be

determined iteratively using this well-recognised procedure by invoking the appropriate

stress-strain material curve depending on load state (Sun et al. [29]).

Another approach, which is conceptually different for locating the elastic and plastic neutral

axes, is proposed in this paper. The location of the elastic neutral axis can scrutinize the

minimal condition that yEdA = 0 and the plastic neutral axis from dA = 0; both these

minimal conditions refer to elastic bending and plastic axial force equilibrium conditions,

which depend on constant material properties, such as E and only in lieu of searching the

stress-strain material curve. Figure 3 shows a layering or segmentation of the cross-section

for T-beam, rectangular CFT, SRC and circular CFT sections. Depending on the fineness of

the layering or segmentation, once the neutral axes have been located, the neutral axes remain

15

unchanged under the assumption of the plastic hinge approach that the elastic beam-column

element is elastic through the entire incremental-iterative process. In other words, this

procedure is an optimization process involving no iteration, which is implemented n times

with respect to each layer defined for the composite section. It therefore leads to reliable and

efficient solutions for these locations.

By doing this, the procedure represents the material behaviour integrally with the plastic

hinge formulation instead of sampling each fibre through a section as is needed in the

techniques reliant on the plastic zone formulations. In other words, this numerical procedure

is equivalent to numerical integration in the plastic zone method. The peculiar feature of this

methodology is that the section properties are evaluated merely for either sagging or hogging

moment distribution, and remain unchanged during the iterative-incremental procedure under

assumption of the plastic hinge approach. Hence the member is discretised into two elements

at the point of contraflexure. The detailed methodology and implication of this methodology

is presented in Iu [27].

It is of interest to note that, for a symmetric composite column section, the elastic and plastic

behaviour under sagging and hogging are the same. Hence, the aforementioned numerical

procedure (element discretisation process) is implemented for the composite columns

irrespective of the moment distribution, with no element discretisation being needed at the

point of contraflexure for a composite column

5.3 Material yield surface

The cross-section properties of a composite beam-column section are evaluated with respect

to biaxial bending and axial force resultants separately, but these resultants interact which are

considered by the yield surfaces as depicted in Fig. 4. To allow for the gradual yielding on

the general composite section, the initial yield criterion is required, as it is of significance that

the full plasticity on the composite section generally occurs long after its initial yield.

Unfortunately, less literature has reported the initial yield condition of the composite sections.

A proposed initial yield for the composite section is therefore directly modified from the

initial yield surface of a steel section (King et al. [30]) by simply replacing the composite

section properties, which are characterised linearly and represented empirically by the factor

i(f) given by

16

pyy

yy

pxx

xx

yai M

Mf

M

Mf

P

P

f , (28)

in which fx and fy is shape factor of the composite cross section for the major and minor axis,

respectively, and a, x and y are factor to take residual stress into account for axial effects

and bending effects about major and minor axes, respectively, f=(P, Mx, My) is the vector of

stress resultants.

The full yield failure of a composite section has attracted research activity, as noted

previously. Nevertheless, most of these full yield surface models are complicated to use, and

the axial force effect is considered implicitly through a complex procedure, such as in

Eurocode 4 [8]. To formulate the plastic hinge stiffness conveniently, a simple and

conservative fully yielded surface is postulated for general composite sections, in which the

axial force effect is considered linearly and explicitly as represented by the factor f (f) in

yx n

py

y

n

px

x

yf M

M

M

M

P

P

f . (29)

Initial yielding is defined by the stress resultant set at a cross-section f = (P, Mx, My) such that

i(f) = 1 and full yielding by f(f) = 1; solutions for f such that f(f) > 1 are not admissible in

the procedure unless strain-hardening behaviour is included. The exponents nx and ny depend

on the column dimensions, amount and distribution of longitudinal steel reinforcement and

lateral ties, the stress-strain characteristics of steel and concrete, and the like.

6. Plastic Hinge Formulation

Figure 5 illustrates the bending and axial plastic hinges employed in the current approach.

When the factor i(f) in Eq. (30) at a node exceeds unity, the bending and axial refined plastic

hinge is activated; the stiffness of the axial spring being

a

i

fa L

EAS

1

1

f

f, (30)

and the stiffness of the flexural spring being

17

b

i

fb L

EIS

1

1

f

f, (31)

in which EA/L and EI/L are the elastic axial and flexural stiffnesses of the composite member,

and a and b are strain hardening parameters which can be selected as positive values (Iu

[27]). Both Sa [0, ) and Sb [0, ). These spring stiffnesses are able to represent the

material non-linear response in the load-deformation relationship for the composite section,

including the elastic domain, gradual yielding and the strain hardening response as depicted in

Fig. 6. Hence this refined plastic hinge formulates the elastic-gradual-plastic material model

with the strain-hardening effect in the force state (load-deformation relationship).

It is interesting to remark that the interaction between the bending and axial actions is

incorporated into this refined plastic hinge stiffness Sa and Sb (elastic-gradual-plastic model)

by twofold considerations. First, in the material model, the interaction effects are taken into

account using the initial i(f) and full yield f(f) criteria in Eqs. (30) and (31). This governs

the occurrence of gradual or full material yielding. Second, in the spring stiffness formulation,

the interaction effect by both axial and bending actions can simultaneously influence the

spring stiffness Sa or Sb so as to experience the gradual yielding, which controls the load-

deformation relationship by the gradual stiffness degradation. Eventually, the gradual

yielding behaviour resulting from both compression and bending is accounted for, which is of

significance for composite beam-columns.

The incremental stiffness relationship for the beam-column element in Fig. 5 is

.2

2

1

1

22

223332

231221

11

11

2

2

1

1

e

s

e

s

e

a

bb

bb

bb

bb

a

e

s

e

s

e

a

u

u

SS

SSKK

KSKS

SS

K

S

M

M

M

M

P

P

(32)

When the axial and bending effects are uncoupled, the moment and axial force equilibrium in

Eq. (32) can be considered separately.

18

Because of the insertion of plastic hinges at the ends of the element, additional internal

rotational degrees of freedom (s1 and s2) are introduced, as shown in Fig. 5.

Condensation of an internal freedom in the member, such as Me1 = Me2 = 0, allows Eq. (32)

to be decomposed, producing

2

1

2

1

2

1

2

1

2

1

e

e

b

b

s

s

b

b

s

s

S

S

S

S

M

M

(33)

and

2

1

22221

12111

2

1

2

1

0

0

e

e

b

b

s

s

b

b

SKK

KSK

S

S

. (34)

After matrix condensation as in Eqs. (33) and (34), the incremental rotational deformations of

the beam-column element e1 and e2 can be evaluated from the incremental joint rotations

s1 and s2 from the equation

2

1

2

1

1

22221

12111

2

1

s

s

b

b

b

b

e

e

S

S

SKK

KSK

2

1

1112211

1222221

2

1 1

s

s

bbb

bbb

e

e

SKSKS

KSSKS

K

, (35)

where 2112222111 KKSKSKK ss . Once the incremental rotations at element e are

known from Eq. (35), which includes the material non-linear effect, the incremental moment

resistance of beam-column element can be evaluated Me from elastic element stiffness as

2

1

2221

1211

2

1

e

e

e

e

KK

KK

M

M

2

1

1112211

1222221

2221

1211

2

1 1

s

s

bbb

bbb

e

e

SKSKS

KSSKS

KK

KK

KM

M

. (36)

Equation (36) therefore can be used to determine the member resistance from the rotational

deformations of the joints, and this is equivalent to the incremental secant stiffness

19

formulation of Eq. (10) by expressing the plastic hinge stiffness in local coordinates in the

non-linear solution procedure expressed in Eq. (22).

For axial forces, the incremental stiffness relationship is given by

e

aa

e

aa

e

a

u

u

LEA

S

u

u

K

S

P

P

11

, (37)

where the incremental axial force in the linear axial spring Pa is equal to the axial force in

the beam-column element Pe. The total incremental axial deformation u is equal to the

sum of the incremental axial deformation in the spring ua and in the element ue, and

consequently the axial resistance of the element is

LEA

P

S

Puuu

aba

uLEAS

L

EAS

Pa

a

, (38)

which describes the axial resistance of the element in terms of the axial displacement and so is

equivalent to the secant stiffness for a plastic axial spring.

By back substituting Eq. (35) into Eq. (33), the incremental moment-rotation relationship is

reformulated with respect to the local or nodal coordinate system according to

2

1

3332

2322

11

2

1

s

s

s

s

u

KK

KK

K

M

M

P

, (39)

in which the axial spring stiffness has been superimposed into the formulation, as there is

uncoupling between the axial and bending effects in the element stiffness. In Eq. (39),

a

a

SK

KSK

11

1111 , (40)

20

KSKSSK bbb 22221122 , (41)

KKSSKK bb 12213223 , (42)

KSKSSK bbb 11122233 . (43)

Equation (39) is equivalent to the tangent stiffness formulation of Eq. (12), which is

incorporated into the solution procedure according to Eq. (19). The material non-linearity in a

composite bean-column element therefore manifest themselves in the structural equilibrium

path non-proportionally on the basis of the plastic hinge approach, through implementation in

the non-linear solution technique of this paper.

7. Verification Examples

This non-linear inelastic analysis is an extension work from Iu [27] to allow for both material

and geometric non-linearity in the composite beam-column element. The initial and full yield

surfaces postulated in this refined plastic hinge approach are verified with those from

different design codes for an SRC section with high and low steel ratios. Axially loaded

columns under uniaxial and biaxial bending are validated with experimental results. The

confinement effect on a circular CFT column is also emulated using this proposed approach.

To study the non-linear behaviour of the composite frames as well as to examine the efficacy

of proposed composite analysis, a simple composite frame and 6-storey composite space

frame are also included.

7.1 Initial and fully yield criterion by comparing with different design

codes

El-Tawil et al. [31] developed a procedure to derive the failure surfaces for a composite

column for use in design, and performed a comparison study of SRC columns for high and

low reinforcement ratios in which their method was compared with different design codes.

Figure 7 shows the two sections used in their study; Section C16-2 has a high steel ratio and

Section C4-2 a low steel ratio.

In this study, the failure surfaces from different design codes and their approach for uniaxial

and biaxial bending effects, as well as buckling, were evaluated. For uniaxial bending, the

21

comparison of failure surfaces for SRC columns using the approach of El-Tawil et al. [31],

those of different design codes and the present composite analysis are shown in Fig. 8 (for

high steel ratio) and Fig. 9 (for low steel ratio). It can be seen that the proposed initial and

full yield surfaces in the present composite analysis are conservative among those from

different codes. It is suggested that the term for the axial force should not be linear as given

in Eq. (30), especially when the axial force level is low.

For biaxial bending, the comparison of the failure surfaces of a SRC column determined from

different approaches is shown in Fig. 10 and 11 for high and low steel ratios respectively.

Similarly to uniaxial bending, it can also be seen that the proposed initial and full yield

surfaces in the present composite analysis are conservative to predict the material failure for

biaxial bending.

It should be noted that the present non-linear analysis of composite structures allows for

buckling (geometric non-linearity) directly through the numerical procedures, as mentioned in

Sections 2 and 3. When the load increment just reaches the initial and full yield surfaces, the

stress resultants (axial force and bending moments) were measured and plotted in Figs. 12 and

13 for high and low steel ratios respectively. Hence the load increment used in this

comparison study should be as fine as possible in order to ensure the smooth reproduction of

the yield surfaces for stability effect. It is also found that the proposed initial and full yield

surfaces are conservative, especially for the case of the low steel ratio. In summary, the initial

and full yield surfaces for stability effect from the present composite analysis are plotted

under different loading cases, so the contour lines of yield surfaces are discontinuous.

In conclusion, the proposed initial and full failure surfaces are simple and convenient to

formulate the present plastic hinge approach, and also involve no onerous implementation,

such as in design codes (Eurocode 4 [8]).

7.2 Experimental results of square CFT composite column under uniaxial

bending

Bridge [32] reported experimental results for eight slender square CFT composite columns

subjected to uniaxial bending. His columns SHC-1 and SHC-2 under concentric and eccentric

end compressive loading, respectively, were analysed by the numerical technique of this

22

paper, and their geometry and material properties are also shown in Fig. 14. Figure 15 shows

the load versus deflection plots for the two columns, in which it can be seen that the plastic

hinge-based numerical procedure replicates the experimental response well, especially for the

column which displays flexural buckling behaviour, and so is able to capture both geometric

and material non-linearity when subjected to the uniaxial bending.

7.3 Experimental results of steel encased concrete section under biaxial

bending

Virdi and Dowling [33] presented experimental results for nine SRC columns subjected to

biaxial bending and compression. The cross-section used is shown in Fig. 16 and the nine

specimens were divided into three batches of constant length; each batch of constant length

had different eccentric compression producing different biaxial bending moments. One batch

of length 732 mm (columns G, H and I) was analysed using the numerical procedure herein,

with the material properties including the yield strength of the steel section and cube strength

of the concrete being given in Virdi and Dowling [33]. The elastic modulus of the steel was

taken as 200 kN/mm2.

The load versus mid-height deflection of columns G, H and I are shown in Fig. 17, which

compares the present numerical solutions with the tests of Virdi and Dowling [33]. This

paper does not report the elastic modulus of the concrete (which is quite variable in nature)

and a modular ratio of 6.8 was used for the numerical results of column G only to correlate

the elastic modulus of the concrete, whereas the other columns H and I use modular ratio of

15.

The values of different concrete elasticity are chosen without loss of accuracy, since three

column specimens were of the same sections and member lengths but subjected to an axial

compression load with linearly proportional eccentricities. As a natural result, the elastic

flexural behaviour of three specimens should be also linear proportion, if and only if the

elastic modulus of these columns was same, which governs the behaviour in elastic range.

However, from the observation of their experimental results, the mid-height load-deflection

curves of the column G did not vary linearly from the other columns H and I, but those of the

columns H and I varied linearly accordingly. Thus, it is strongly reasonable that the elastic

modulus of column G was different from other two specimens. It can be seen from Fig. 17

23

that the agreement between test and theory is generally good. Therefore, the present

composite analysis can capture the geometric and material non-linearity of the composite

columns when subjected to the biaxial bending.

7.4 Experimental and numerical results of circular CFT section under

axial load

It is well-known that when concrete is confined, as in a circular CFT composite column, its

overall compressive strength increases because of the effect of the confinement. Schneider

[12] conducted tests on 14 CFT specimens which were all loaded axially in compression;

three were of circular, five of square and six of rectangular tubular steel section. The results

from one of the tests (C1) are shown in Fig. 18, with the material and section properties used

in the numerical studies being obtained from Schneider [12].

The axial response from the numerical model is reasonably consistent with that of the

experimental results, as well as the numerical results of Schneider [12]. However, the

ultimate load of the circular CFT column from the present analysis without considering

confinement effect is less than the results of Schneider [12]. To account for the confinement

effect, the section properties as mentioned in Section 4 can be proportionally enhanced with

respect to the concrete section. Therefore, the axial deformation is also plotted using the

present model with a 50% enhancement for the concrete as result of the confinement; this

produces an upper bound on the axial load versus deflection plot. It should be noted that the

effects of confinement of the concrete are dependent on the complex triaxial state of stress

that develops; this enhancement can reduce with increasing load and disappears altogether

with the onset of local buckling of the tube. However, the proposed approach simplifies the

post-yield response of confinement effect by the constant strain-hardening parameter in the

plastic hinge spring stiffness, which is independent of the stress state, so the enhancement of

the cross-section strength of a circular specimen increases proportionally reliant on the

present composite analysis.

7.5 Numerical results for a simple composite frame

24

The previous validations were for cross-sections and for isolated members, and so the validity

of the numerical procedure has been demonstrated here for a portal frame structure with steel

columns and a composite beam, whose behaviour has been reported by Liew et al. [34] using

another numerical scheme. In this study, the behaviour of a bare steel frame with or without a

composite beam and an entire composite frame with SRC columns are further compared. The

geometry of the frame, including the composite cross-sections, is given in Fig. 19. As shown

in this figure, the frame was subjected to both vertical and lateral loading, its bases were

pinned and the beam-to-column connections were taken as rigid. The composite columns and

beam were modelled by one respective type of beam-column element for simulating the

material and geometric non-linearity of the portal frame.

For the proposed composite analysis, the section properties of the composite beam element

depend merely on the sign of the bending moment and element discretisation at the point of

contraflexure is carried out, whilst the properties of the symmetric composite column element

is evaluated irrespective of the moment distribution and hence necessitates no element

discretisation process as discussed in Section 5.2. Therefore, a fictitious incremental-iterative

procedure (such as n = 0) commences for an arbitrary small load increment in order to

determine the moment distribution in the structure. Once the sign of the bending moment in

each member is known, the composite beam is split into different segments according to the

corresponding bending moment region. In other words, the composite beam element being

redefined from the present composite analysis experiences single curvature bending only.

After the section properties of the corresponding segment or new composite beam-column

element have been evaluated, the normal incremental-iterative Newton-Raphson procedure

(such as n = 1) begins from scratch as discussed in Section 2 and 3. Figure 20 shows the

moment distribution on the frame due to an arbitrarily small load increment. Since positive

and negative moments exist in the composite beam, a new node is inserted at the point of

contraflexure and the composite beam divided into two beam-column element as shown in

Fig. 20, which also displays the new node and element number. Despite a point of

contraflexure on a column, no element discretisation process is implemented for the

symmetric composite column section, because the flexural behavioiur under different moment

regions is identical.

25

According to Liew et al. [34], the flexural stiffness of the composite beam should be

approximated by using Ic. On the other hand, no approximate flexural stiffness of the

composite beam-column element is necessary in the present analysis to capture overall

composite beam behaviour, since all section properties are evaluated in accordance with the

rigorous procedure as mentioned in Section 5.2. The centroidal locus of the two columns and

composite beam is plotted in Fig. 21 according to the sign of moment region.

In this verification, no strain hardening is included. A mechanism then develops when both

fully-yielded hinges are formed at beam-column joints at which no further member resistance

is possible for resisting additional loading. The gradual yielding, which is of significance on

the composite member in general, is included in the material model of this study.

The behaviour of the horizontal displacement under the vertical and lateral load P was

studied in this numerical example, and is plotted in Fig. 22. The ultimate load factor of a bare

frame from Liew et al. [34] is 0.7775. In the present analysis, the ultimate loads factor of the

corresponding frame under elastic-plastic and elastic-gradual-plastic material model are

respectively 0.76 and 0.74 as indicated in Fig. 22. Further, the lateral displacement of a steel

frame with composite beam predicted from the present analysis is in good agreement with the

numerical results of Liew et al. [34] generally. The first initial yielded plastic hinge appears

at node 3 (steel beam to steel column connection) at about load factor of 0.75. Subsequently,

both fully yielded plastic hinge springs at both beam-to-column joints are activated till the

occurrence of numerical divergence at load factor of 1.02, which is coincident with the

ultimate load from Liew et al. [34]. In summary, the ultimate strength of this composite

frame is enhanced by 38% above that of the bare frame.

For the frame with both a composite beam and columns, its ultimate strength against sway is

identical at 1.02, but the lateral stiffness of the composite frame is larger than that of steel

frame with a composite beam. This implies that the composite columns contribute little

lateral load capacity but the stiffness of lateral stability, because the composite column section

cannot considerably enhance the strength capacity of the beam-to-column joint, particularly in

the hogging moment region. This is a crucial structural component to the lateral load capacity

of the frame.

26

Before further discussion of differences between the present composite analysis and

traditional composite design, it is of interest to briefly review the practical composite design

procedure in general for which the composite sections are evaluated and applied for the whole

composite member; linear elastic analysis is executed to determine loading distribution on the

composite structures. According to the linear elastic frame analysis, the ultimate strength of

the composite frame is obtained complying with codified composite design criteria for both

geometric and material non-linearity.

Without element discretisation at the point of contraflexure on a composite beam, the ultimate

loads of a composite frame with steel or composite columns are also higher in the consistent

manner as given in Fig. 22, which overstates both the ultimate strength and, the lateral

stiffness. Unfortunately, this process is normal design practice for composite member or

frame, and consequently yields an unsafe ultimate strength design of the composite structure

without onerous design criteria. It is therefore concluded that the proposed composite

analysis can robustly replicate the real behaviour at ultimate of a composite frame accurately,

even without the approximate flexural stiffness used in Liew et al. [34]. In this study, all

significant geometric and material non-linearities composite structures are directly and

numerically tackled through the proposed numerical procedure, which leads to a safe

composite design for the ultimate strength of the composite frames in the efficacious manner.

7.6 Numerical results for a six-storey composite space frame

In addition to a simple composite plane frame and isolated members, a rigid-jointed 6-storey

asymmetrical steel space frame has been modified and included in this verification as given in

Fig. 23, which was originally analysed by Orbison et al. [35] in 1982 and later studied by

Liew et al. [36] and Jiang et al. [37]. More recently, this large-scale steel frame has been

analysed to validate a higher-order element formulation associated with refined plastic hinge

approach by Iu and Bradford [38] [39]. This example converts the frame to a composite open

frame with the intent to demonstrate the efficacy of the present numerical procedures and to

verify the behaviour of a large-scale composite space frame using the present composite

analysis.

The steel beams in the frame rigidly support the concrete floor slab, and the steel columns are

encased partially in the concrete for fire protection, whose sections are also given in Fig. 23.

27

The yield strength and elastic modulus of steel are 250103kN/m2 and 2.07108kN/m2,

respectively, whilst the elastic modulus and compressive yield stress of concrete are

respectively 16103kN/m2 and 1.38107kN/m2; the composite frame is subjected to both

uniform gravity loads of 9.6kN/m2 on every floor and lateral loads of 53.376kN, which are

adopted as same as the previous studies for comparison; the lateral loads are applied at every

beam-column joints in transverse z direction; the member sections and geometric

configuration together with the structural plan and three-dimensional view of this 6-storey

composite frame are depicted in Fig. 23.

In the present numerical modelling, one element is used for each member to reproduce the

overall load-displacement behaviour of the 6-storey composite space frame, which is same as

the numerical modelling in [36]. The main difference in these analyses is the inelastic method.

A plastic hinge model using a plastic interaction function was employed in Orbison et al. [35]

and Liew et al. [36], whereas a gradual element yielding by the plastic zone approach was

included in Jiang et al. [37]. According to the present composite analysis, the refined plastic

hinge spring has been formulated to allow for gradual yielding at the hinge under both axial

and bending actions with or without the strain-hardening effect.

The composite continuous beams in a framed structure are usually subjected to both sagging

and hogging bending moment, so one respective type of composite beam-column element is

assigned to the beam in either the sagging or the hogging moment region. The element

discretisation process for the composite space frame is redrawn in Fig. 24, where bold and

light line indicates the composite and steel beam section respectively; the point of

contraflexure is approximately located at mid-span of every beam due to lateral loading effect.

Thus, 96 composite beam-column elements are used in the final numerical modelling for this

composite frame.

Figure 25 shows the lateral displacements u and v versus load factor plots at the point A,

which is indicated in Fig. 23. It can be seen that the ultimate load of the composite frame is

enhanced from the bare steel frame by 13.6% for both elastic-plastic and elastic-gradual-

plastic material models; the lateral stiffness against both transverse directions of the

composite frame is also increased consistently compared to the bare frame. The lateral

displacement of u and v versus load factor plots for both material models are consistent. The

28

first yield of the frame occurs at the load factor of 0.77 and 0.97 for the elastic-gradual-plastic

and elastic-plastic models, respectively. In line with the previous results of [39], the twisting

behaviour of the composite structure is observed due to the asymmetric structural plan, which

leads to the asymmetric lateral loads.

The present composite analysis accounting for both geometric and material non-linearity is

efficacious, as it takes 10 seconds to complete the numerical analysis of this composite space

frame for over 100 load cycles on a modest personal computer.

8. Concluding Remarks

This paper has proposed a composite beam-column element which can successfully

incorporate both material and geometric non-linearity into its formulation. The inelastic

element is based on a refined plastic hinge spring at its ends, which can model the gradual

yielding from initial to full yield under the axial and bending interaction effect. Thus, the

elastic-gradual-plastic with strain-hardening material model in terms of a load-displacement

relation is included in the present analysis for composite structures. On the contrary, for the

alternate plastic zone method, the yield surfaces for the interaction of bending and axial force

are formulated in terms of stress resultants in deference to stresses which produce

computational inefficiencies and which render such stress-based approaches difficult for the

modelling of full structural frames. The refined plastic hinge model is versatile and adaptive

to different kinds of material, when the corresponding initial and full yield surfaces are used

in the spring stiffness formulation. As a result, the present approach is useful in its

incorporation of maternal non-linearity in composite beam-columns, because it captures

inelastic buckling caused by the interaction of geometric and material non-linearity in the

proposed refined plastic hinge approach. The method also allows for easy incorporation of

bending about both axes as well as axial force, so that three dimensional frame analyses can

be undertaken.

The approach herein implements a procedure for evaluating the section properties to represent

the member cross-section strength capacity in line with numerical integration used in the

plastic zone method. This procedure can be carried out robustly for any arbitrary cross-

section in general, including composite T-beams, SRC and CFT columns with or without

conventional reinforcement. Since members may, in general, be subjected through their

29

length to positive and negative bending so that they experience different material behaviour as

well as flexural behaviour, an adaptive-type element is used in which an additional node is

located at the point of contraflexure of the bending composite beams. The inelastic element

with end plastic hinges enables replication of the corresponding material non-linearity of

either positive or negative bending; a fictitious incremental-iterative procedure is therefore

carried out to determine the moment distribution under an arbitrarily small load increment.

This simple discretisation process leads the present composite analysis to a precise estimation

of the flexural behaviour of a composite beam. Ultimate strength analyses of composite

frames are therefore achieved through both the element discretisation process and the refined

plastic hinge approach. Finally, the present analysis was validated with the models of single

members, frames and large-scale composite building structures, in which shows the present

approach was shown to be accurate, effective and reliable.

In conclusion, the proposed composite analysis is not only reliant on the refined plastic hinge

approach, which is capable of evaluating the cross-section strength capacity of a whole

structure, but also it is based on the non-linear solution procedures as mentioned in Sections 2

and 3, which directly and numerically solve the geometric non-linearity of general composite

frames. It provides a platform for robust and efficacious design applications of general

frames with arbitrary composite sections at the strength limit state.

Acknowledgements

The work in this paper was supported by the University of Hawaii and by the Australian

Research Council through its Federation Fellowship Scheme.

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plastic hinge analysis of space frame structures”, Engineering Structures, 22, 1324-1338.

[37]. Jiang, X.M., Chen, H. and Liew, J.Y.R. (2002), “Spread of plasticity analysis of three-

dimensional steel frames”, Journal of Constructional Steel Research, 58, 193-212.

[38]. Iu, C.K. and Bradford, M.A. (2008), “Higher-order nonlinear analysis of steel structures

Part I: Elastic second-order formulation”, (to be submitted).

[39]. Iu, C.K. and Bradford, M.A. (2008), “Higher-order nonlinear analysis of steel structures

Part II: Refined plastic hinge method”, (to be submitted).

34

Figure 1. Incremental-iterative scheme of Newton-Raphson method

nf

1nf 1 nf

nf

1nf

Displacement

Loa

d

ni 1u n

iu

1nu

12 n

iu11 n

iu1 niu

nu

niR

1nR

12 n

iR11 n

iR1 niR

nR

Displacement

Loa

d

35

Figure 2. Stress-strain representation of material behaviour

cuf

cuf

(a) Steel material (b) Concrete material

plastic plateau strain hardening

elastic limit

Elasticlimit strain strain

stress stress

plasticlimit

EcE s

36

Figure 3. Segmentation of composite cross-sections

x1 x2 x3

y1

y2

y3

x1 x2 x3 x4 x5

Nc Nf Nw Nf Nc

Nc

Nf

Nw

Nf

Nc

Nf

Nw

Nf

CFT column Concrete core

Steel tube

Nw

Nw

x1

x2 x3 x4 x5 x6 x7

y1 y2 y3 y4

y5 y7 Reinforcement

Nc

Nt

Nw

Nb

y1 y2 y3 y4 y5

37

Figure 4. Yield surfaces of composite cross-section in terms of axial and biaxial bending

stress resultants

1

yn

py

yxn

px

x

yf M

M

M

M

P

P

1pyy

yy

pxx

xx

yai M

Mf

M

Mf

P

P

P

Mx

My

38

Figure 5. Composite beam-column element with axial and rotational springs

ai

f

L

EA

1

1

bi

f

L

EI

1

11e1sau

eu

2e 2s

bi

f

L

EI

1

1

2sM

1sM2eM

1eMeP

aP S b1

S b2

LS a

39

Deformation

Loa

d

Gradualyielding

Elastic

Es

Strain-hardening effect

Figure 6. Material behaviour in plastic hinge formulation in terms of load-deformation axes

40

660.4

660.4

50140.1

140.1

140.1

140.150

889

889

50

197.25

197.25

197.25

197.25

50W14 176 (16 No. 8)

W14 370 (16 No. 6)

(High steel ratio) (Low steel ratio)

Figure 7. Section, dimension and reinforcement pattern of SRC columns

41

-5000

0

5000

10000

15000

20000

25000

30000

0 500 1000 1500 2000 2500 3000 3500 4000 4500

Moment M (kNm)

Axi

al f

orce

P (

kN)

COSBIAN

ACI-Ult.

AISC-Ult.

EURO-Ult.

ACI-design

AISC-design

EURO-design

Present analysis

Figure 8. C16-2 specimen under uniaxial bending (high steel ratio)

42

-5000

0

5000

10000

15000

20000

25000

30000

0 500 1000 1500 2000 2500 3000 3500 4000 4500

Moment (kNm)

Axi

al f

orce

(kN

)

COSBIAN

ACI-Ult.

AISC-Ult.

EURO-Ult.

ACI-design

AISC-design

EURO-design

Present analysis

Figure 9. C4-2 specimen under uniaxial bending (low steel ratio)

43

-5000

0

5000

10000

15000

20000

25000

30000

0 500 1000 1500 2000 2500 3000

Moment M (kNm)

Axi

al f

orce

P (

kN)

COSBIAN

ACI-Ult.

AISC-Ult.

ACI-design

AISC-design

Present analysis

Figure 10. C16-2 specimen under biaxial bending (high steel ratio)

44

-5000

0

5000

10000

15000

20000

25000

30000

0 500 1000 1500 2000 2500 3000

Moment M (kNm)

Axi

al f

orce

P (

kN)

COSBIAN

ACI-Ult.

AISC-Ult.

ACI-design

AISC-design

Present analysis

Figure 11. C4-2 specimen under biaxial bending (low steel ratio)

45

0

5000

10000

15000

20000

25000

30000

0 500 1000 1500 2000 2500 3000 3500 4000 4500

Moment M (kNm)

Axi

al f

orce

P (

kN)

COSBIAN (L/r=0)

COSBIAN (L/r=40)

ACI (L/r=40)

AISC (L/r=40)

EURO (L/r=40)

Present analysis (initial yield surface)

Present analysis (full yield surface)

Figure 12. C16-2 specimen subjected to buckling effect (high steel ratio)

46

0

5000

10000

15000

20000

25000

30000

0 500 1000 1500 2000 2500 3000 3500 4000

Moment M (kNm)

Axi

al f

orce

P (

kN)

COSBIAN (L/r=0)

COSBIAN (L/r=40)

ACI (L/r=40)

AISC (L/r=40)

EURO (L/r=40)

Present analysis (initial yield surface)

Present analysis (full yield surface)

Figure 13. C4-2 specimen subjected to buckling effect (low steel ratio)

47

Figure 14. Section and dimension of test specimen of the square CFT column

Steel (kN/m2) Concrete (kN/m2)

Elastic modulus 2.05108 2.33107 Steel yield stress/

Concrete compressive strength 2.91105 3.02104 20

00

P

200mm

200mm

e

10mm

48

0

500

1000

1500

2000

2500

3000

3500

0 2 4 6 8 10 12 14

Deflection at mid-height (mm)

Loa

d Q

(kN

)

Bridge [32]: SHC-1

Bridge [32]: SHC-2

Present analysis: SHC-1

Present analysis: SHC-2

Figure 15. Square CFT columns under uniaxial bending compared with tests

49

152

50.8

50.8

152 50.850.8

25.4

UC 152 152 23

12

v

u

Figure 16. Section, dimension and axes of SRC column

50

0

100

200

300

400

500

600

700

800

0 20 40 60 80 100 120 140 160

Deflection (mm)

Loa

d Q

(kN

)

Virdi and Dowling [33] (Column G)

Virdi and Dowling [33] (Column H)

Virdi and Dowling [33] (Column I)

Present analysis (Column G)

Present analysis (Column G)

Present analysis (Column H)

Present analysis (Column H)

Present analysis (Column I)

Present analysis (Column I)

Figure 17. Experimental SRC columns G, H and I under biaxial bending

51

Figure 18. Axial load and deformation curve of specimen C1

0

100

200

300

400

500

600

700

800

900

1000

0 2 4 6 8 10 12 14 16 18 20

Axial deformation (mm)

Axi

al loa

d (k

N)

Schneider [12]: Experimental results

Schneider [12]: Numerical results (ABAQUS)

Present analysis

Present analysis (with 50% confinement effect)

0.14m

0.60544m

52

L=5m

L

P P

P

A

A 1219

102

304W 12 27

Section A-A

E=200N/mm

y=248.2N/mm2

2

c=16N/mm2

B B

Section B-B

W 12 50 400

400

Figure 19. Geometry of steel portal frame with composite beam

53

19.9kNm 19.9kNm

1

2 3

4

5

1 3

42

(2.5072 , 5)

Figure 20. Element discretisation process according to moment distribution under arbitrary small load increment

54

Figure 21. Compatibility condition of the portal frame

41b

32b

31b2

2b

21b

12b

32b

41b

32s

41s

41

32 ss where

55

Figure 22. Lateral displacement of portal frame with composite beam

76.074.077.0

02.1 02.1

29.1 27.1

02.1

0

20

40

60

80

100

0 20 40 60 80 100 120 140 160 180 200

Lateral displacement (mm)

Loa

d P (

kN)

Liew et al. [34]: Numerical results of bare steel frame

Liew et al. [34]: Numerical results of composite frame

Present analysis of bare steel frame (Elastic-plastic model)

Present analysis of bare steel frame (Elastic-gradual-plastic model)

Present analysis of steel frame with composite beam

Present analysis of composite frame (composite beam & columns)

Present analysis of steel frame with composite beam (without discretization by moment regions)

Present analysis of composite frame (without discretization by moment regions)

56

W 12 26 W 12 53 W 12 87

W 10 60W 12 120W 12 87

1000mm

125mm

1000mm 1000mm

7.315m 7.315m

7.31

5m

W12 26W12 26

W12 26W12 26

W12

53

W12

87

W12

53

6 @

3.6

85m

=21

.948

m

W12

87W

1060

W10

60

W10

60

W12

87

W12

120

W12

120

W10

60

W12

87

W12

87

Point A

y

xz

Wind loaddirection

u

v

Figure 23. Configuration and member sections of a 6-storey composite space frame

57

Figure 24. Element discretisation of 6-storey composite frame

58

0

0.2

0.4

0.6

0.8

1

1.2

1.4

‐320 ‐270 ‐220 ‐170 ‐120 ‐70 ‐20

Lateral displacement (mm)

Load factor

L iew et al. [36] (R efined plas tic  hinge method)

J iang  et al. [37] (P las tic  zone method)

Iu and Bradford [38] [39] (Higher‐order analys is  of s teel s tructures )

P resent analys is  (R efined plas tic  hinge method of compos ite s tructures ) [elas tic‐plas tic]

P resent analys is  (R efined plas tic  hinge method of compos ite s tructures ) [elas tic‐gradual‐plas tic]

Figure 25. Lateral drifts of points A and B in x- and z-directions vs load factor

17.1

03.1

16.1 17.1

03.1

16.1

uv