Non-discretisation formulation for the non-linear analysis of semi-rigid steel frames at elevated temperatures

Computers and Structures 88 (2010) 207–222

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Non-discretisation formulation for the non-linear analysis of semi-rigid steel frames at elevated temperatures A. Heidarpour *, M.A. Bradford Centre for Infrastructure Engineering and Safety, School of Civil and Environmental Engineering, The University of New South Wales, UNSW Sydney, NSW 2052, Australia

a r t i c l e

i n f o

Article history: Received 2 February 2009 Accepted 25 October 2009 Available online 24 November 2009 Keywords: Fire loading Flexibility approach Geometric non-linear Inelastic Semi-analytical Semi-rigid

a b s t r a c t This paper proposes a flexibility-based method for the analysis of steel frame structures with semi-rigid joints subjected to fire loading. The semi-analytical non-discretisation formulation is able to capture material and geometric non-linearities. The modelling of the cross-sections as comprising of elastic and inelastic domains allows for the spread of yielding to be modelled, while the geometric non-linearity included in the method allows for catenary action to develop in the heated compartment within the frame. The accuracy of the technique is verified against an independent analysis using ABAQUS as well as by analysing a rigid steel frame with 104 members. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Under increases of temperature caused by fire, the component materials in buildings are susceptible to progressive deterioration, and in particular the yield strength and elastic modulus of steel deteriorate relatively quickly. Because of this, numerical modelling of steel framed buildings subjected to fire necessarily requires both material and geometric non-linear effects, and their interactions, to be considered. When assessing the realistic performance of building frames subjected to a typical compartment fire, the effects of the restraints of the heated flexural members at their ends and of the cooler portions of the frame outside of the compartment must be included. Obtaining this information from full-scale testing is both difficult and very expensive, and so numerical techniques are essential for the design and evaluation of building frames subjected to elevated temperatures caused by fire. This paper is concerned with the development of such a numerical technique, which is both accurate and efficient. The behaviour of steel frames at elevated temperatures has been researched extensively over recent years, with many significant contributions to their numerical modelling having been presented. Li and Jiang [1] proposed a technique of analysis for the non-linear behaviour of steel frames subjected to fire, while a formulation for the non-linear analysis of two-dimensional steel frames under fire conditions using the finite element method was presented by Saab and Nethercot [2]. This was based on the * Corresponding author. Tel.: +61 2 9385 5029; fax: +61 2 9385 9747. E-mail address: [email protected] (A. Heidarpour). 0045-7949/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2009.10.004

formulation of El-Zanaty and Murray [3] at ambient temperatures, which was modified for the deterioration of the material strength with increasing temperature using a set of non-linear stress–strain temperature relationships based on a Ramberg–Osgood formulation in which thermal creep effects were included implicitly. Subsequent studies undertaken by Burgess, Plank and co-workers [4– 6] have significantly refined and extended this approach. Including the joints is important, in order to establish that joint robustness [5,7] will not deteriorate in such a way as to lead to partial or progressive structural collapse. A methodology for a numerical technique for studying the large-displacement inelastic behaviour of building frames subjected to a localised fire, which can consider the simultaneous effects of axial force, bending moments and thermal expansion, was proposed by Liew et al. [8]. By following the change of the axial force–moment interactions at regular time intervals s during the fire event, they were able to predict the mode and critical time scr for structural failure, as well as the important intermediate events such as failure of an individual member and redistribution of the load to the surrounding cooler parts of the structure. Bailey [9] reported the development of computer software used for simulating the behaviour of steel framed buildings in fire; in particular the basic formulation of one-dimensional elements used to model three-dimensional frame behaviour was discussed. This methodology was able to consider the effects of semi-rigid joints, lateral-torsional buckling, continuous floor slabs and strain reversal. The results of four full-scale tests, which were carried out on a modern multi-storey composite steel building, were used to O’Connor and Martin [10] to generate a numerical model for steel

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framed buildings subjected to fire loading. Their results indicated that unprotected steelwork behaves significantly better in frames than when acting as isolated members, thereby indicating a need for robust computational techniques for evaluating the performance of entire structures in deference to single isolated members. Based on a finite element method, a direct iteration technique capable of predicting the non-linear behaviour of steel frames at elevated temperature was proposed by Zhao [11]. In this, the second-order effects of large deflections, the progressive softening of the steel, the gradual penetration of the inelastic zone both through the section and along the member, and non-uniform distributions of temperature within the steel members were all taken into consideration. An extension of the Rankine formula for frames under fire conditions, which was based on the principle of virtual work, was presented by Tan and Tan [12]. The formulation included a simple expression for the buckling coefficient of frames subjected to fire. An approach which was based on elastic and plastic methods for fire resistance analysis was reported by Wong [13]. The elastic method included the effect of interaction between the thermal loading and static loading in which the load ratio in each member was calculated, resulting in a formulation whereby the limiting temperature of each member could be obtained. However, the plastic method in Wong’s paper [13] was based on the plastichinge concept for the calculation of the critical temperature of frames at collapse, using the fact that the collapse load factor of a frame is a linear function of the collapse temperature for the same collapse mode. A second-order elastic–plastic-hinge method and a finite element model for the analysis of plane frames in fire were presented by Toh et al. [14], with methods being derived from the same two-noded co-rotational beam element and being based on engineering plasticity theorems. Within a discretised structural framework, an updated Lagrangian approach was used by Vimonsatit et al. [15] to incorporate nonlinear geometric effects on the equilibrium and constitutive conditions. An iterative predictor–corrector scheme coupled with mathematical programming was used in the numerical procedures. A finite element formulation of beam–column elements which was based on the plastic-hinge approach to model elasto-plastic strain hardening steel material subjected to fire was developed by Iu and Chan [16]. The Newton–Raphson method allowing for the thermaltime-dependent effect was used in the solution strategy for the governing non-linear equations for large deflection in the time domain. Additional refinements to allow for the gradual spread of yielding on the member were considered by Junior and Creus [17], while a numerical approach for the inelastic transient analysis of steel frame structures subjected to explosion followed by fire has been developed by Liew and Chen [18]. The approach adopted the use of beam–column elements and fibre elements to enable a realistic modelling of the overall framework subjected to a localised explosion followed by a fire. The influence of blast loads on the fire resistance of a multi-storey steel frame was also studied in this work [18]. It is based on a mixed element approach, in which critical members that are subjected to the direct action of explosion and fire were modelled using single shell elements, while the other members were modelled using beam elements [19]. Izzuddin and co-workers [20–24] developed a number of numerical approaches to handle the analysis of steel frames subjected to fire and blast loading. In these, an adaptive procedure was formulated, which similarly to the work of Chen and Liew [18,19] proposed an adaptive finite element technique to refine the meshing near the location of the blast and fire. Ma and Liew [25] presented a beam–column element for simulating the inelastic behaviour of three-dimensional steel frames, in which yielding of the steel section was modelling by two nested stress-resultant surfaces and the degradation of the strength of

the material at elevated temperatures was formulated according to an effective yield strength concept. A finite element based non-linear transient heat transfer analysis, following the main guidelines proposed by the Eurocode for steel structures under fire conditions, was performed by Landesmann et al. [26], while a new sub-frame model and an isolated member model to ascertain the fire resistance of beams and columns subjected to a compartment fire was presented by Huang and Tan [27] in which the boundary restraints were represented as a combination of linear and rotational springs, where the spring stiffnesses were derived based on the assumption of semi-rigid beam-to-column connections. Based on a series of stress–strain curves obtained experimentally for various temperature levels, an artificial neutral network was employed by Hozjan et al. [28] in the material modelling of the steel. Santiago et al. [29] performed a numerical parametric study of a structural system consisting of an exposed steel beam restrained between a pair of fire-protected steel columns. The numerical model accounted for initial geometric imperfections, non-linear temperature gradient over the cross-section, geometrical and material non-linearity and temperature-dependent material properties. While it can be seen that there have been a number of innovative numerical formulations, the computational modelling of large frame structures subjected to fire is difficult, and despite advances in high-speed processing, accurate solutions for the response which can be used to assess potential partial and progressive collapse in these large structures is problematic. Software such as ABAQUS [30] is used widely by computational structural-fire researchers for member modelling, but it is very time-consuming when used to model full-scale structures and its scope is limited. Recognising that accurate determination of the thermal regime is difficult and often not justified, approximate solutions based on the plastic-hinge concept model have been argued, for which inelasticity is developed in zero-length hinges at the member ends with the remainder of the member being elastic [16,31]. This formulation can lead to efficient and robust solutions, but they are known to be approximate. On the other hand, plastic-zone models which allow for gradual yield propagation through the cross-section and along the length are accurate, but possess too many structural freedoms and Gauss–Legendre integration points to render then applicable for large frames, and so adaptive techniques such as those described previously [20,21] have evolved. It is widely recognised that efficacious numerical algorithms for modelling large structures subjected to fire loading are much-needed to elucidate the full-range structural response. This paper presents a formulation intended to circumvent some of the numerical difficulties encountered in the modelling steel frame structures under fire loading, and its focus is on the numerical scheme rather than on structural applications. It proposes a semi-analytical flexibility-based technique, which captures both geometric and material non-linearity without the need for discretisation of the member domain as is needed in conventional finite element treatments. The model uses only one element per member, which is able to handle numerically the non-linearities formulated elsewhere for a single member in analytic format [31]. The modelling of a generic steel cross-section as being composed of elastic and inelastic regions allows for the effect and spread of yielding, while the geometric non-linearity included in it allows for catenary action to develop. The advantage of the technique is that it overcomes the large discretisation schemes needed using plastic-zone methods, which have been established elsewhere to be problematic both in terms of computational efficiency and accuracy. It is also advantageous in providing an accurate alternative to plastic-hinge methods, whose accuracy is compromised because of the lumping of inelasticity at the ends of the member. The proposed formulation is used to study both a rigid and a semi-rigid

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frame, whose results are shown to agree with the computationally intensive ABAQUS modelling. 2. Flexibility formulation The general methodology for the two-dimensional non-linear analysis of a steel frame at elevated temperatures and its modelling is developed in this section. The model is based on achieving member equilibrium in each individual member which is followed by enforcing compatibility of the displacements between the beams as columns, as well as equilibrium of the forces at the joints. 2.1. Member equilibrium A general isolated portion of a frame compartment subjected to fire loading is depicted in Fig. 1a. Equilibrium of forces for each individual member is achieved under the following assumptions:  The steel section obeys Euler–Bernoulli beam theory, such that plane sections before the deformation remain plane after the deformation.  A beam element is denoted as ‘positive-oriented’ when it is oriented in the positive x-direction (from left to right: Fig. 1b).  A column is denoted as ‘positive-oriented’ when it is oriented in the positive y-direction (from bottom to top: Fig. 1c).

209

 Each cross-section is considered to be subjected to a thermal representation or regime represented symbolically by TðsÞ, where s is the time from the commencement of the fire at s ¼ s0 , so that a temperature T can be assigned to each point Pðx; y; z; sÞ 2 R4 in a space–time continuum in which the timedependent thermal regime can be determined from the relevant fire curve, with the time s being the independent variable.  The temperature is constant through the thickness of each crosssection, but it varies through the depth of the cross-section and along its length. The thermal profile for each element can be determined using a mechanical thermal analysis, which is outside of the scope of the present paper.  The analysis is considered to be a transient isothermal one, with the problem being simplified such that d( )/ds = 0. A positive-oriented element of length L is assumed to consist of a generic cross-section denoted X 2 R2 (Fig. 2a), with the element being geometrically prismatic in the domain K ¼ ½0; L 2 R1 , but under external and thermal loading it may deform in such a manner that part(s) of the cross-section X behave elastically and the other part(s) behave inelastically. Generically, the elastic and plastic portions are denoted, respectively, as Xe ; Xp 2 R2 so that Xe [ Xp ¼ X and Xe \ Xp ¼ 0 (Fig. 2b). It is noted that the domains Xe and Xp are not prismatic within K and that their variation in this domain is usually unknown a priori.

Fig. 1. Frame idealisation.

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Fig. 2. Cross-section: (a) before loading; (b) after loading; (c) total strain diagram; (d) thermal profile.

Each steel beam element i (Fig. 1b) is assumed to be attached to longitudinal elastic restraints of stiffnesses k0i and kLi , as well as counterpart rotational springs of elastic stiffnesses r 0i and rLi which are located at the geometric centroid of the steel cross-section at the ends x ¼ 0 and x ¼ L of the local member axes. These are used to model the semi-rigid joints in a steel framed structure and are taken for simplicity as being elastic, but non-linear spring models can be incorporated relatively easily into the analysis. Each individual steel element is loaded with a distributed load of intensity qi ðxÞ which is sustained throughout the thermal regime TðsÞ, as shown in Fig. 1b and c. For a positive-oriented element, the total strain over the crosssection can be represented by (Fig. 2c)

in which N e is the axial force acting on the elastic part of the crosssection ðXe Þ and Npa is the elastic force in each inelastic part of the cross-section ðXp Þ, being given by

etot ¼ ðy  yn Þj;

r¼

ð1Þ

in which yn is the location of the neutral fibre of the steel section below the local reference level, which is assumed to be coincident with the top fibre of the steel cross-section. For the analysis herein, the appropriate membrane strain is taken from the Green–Lagrange strain tensor as [32,33]

em

! 2 du 1 d v ; ¼ þ dx 2 dx2

ð2Þ

where u and v are the displacements in the x and y directions, respectively. The bending strain vanishes at the elastic centroid of the steel section, and therefore it can be shown that Eq. (1) has the alternate form

etot ¼ em þ ðy  yÞj;

ð3Þ

 is the location of the elastic centroid of the steel section in which y below the reference level. The variation of the thermal (nonmechanical) strain over a cross-section X distant x from the left (bottom) side of the beam (column) can be represented by

eh ¼ HTðx; y; sÞ;

ð4Þ

where H is the coefficient of thermal expansion (assumed as being constant herein without loss of generality) and Tðx; y; sÞ is the temperature at any time s at the point Pðx; y; z; sÞ in the member, derived from the fire regime T, which represents the variation of temperature along the length of the element x as well as through the cross-sectional depth y. By invoking the Duhamel–Neumann equation and dropping the s term for ease of formulation, the mechanical strain can be written as

ee ¼ etot  eh ¼ ðy  yn Þj  HTðx; yÞ:

ð5Þ

For a given cross-section, the total magnitude of the axial force in the steel section can be written as

N ¼ Ne þ

X a¼1;2

N pa ;

ð6Þ

Ne ¼

Z

rdAe and Npa ¼

Ae

Z

rdApa ða ¼ 1; 2Þ;

ð7Þ

Apa

where Ae and ðAp1 þ Ap2 Þ are the areas of the domains Xe and Xp , respectively. A bilinear constitutive model, with a small transition between the linear portions proposed by Lie [34], is used to represent the stress–strain curve at elevated temperatures. This curve consists of two separate equations as

(

Es;T  ee ðc1 ee þ c2 Þry;T  c3 r2y;T =Es;T

ee 6 eP ; ee > eP ;

ð8Þ

in which

eP ¼

c2 ry;T  c3 r2y;T =Es;T ; Es;T  c1 ry;T

ð9Þ

and where ry;T is the yield stress at the elevated temperature T. The first part of Eq. (8) describes the linear elastic portion and the second the inelastic portion of the steel material, and the coefficient are taken as c1 ¼ 12:5; c2 ¼ 0:975 and c3 ¼ 12:5 [34]. The temperature-dependent elastic modulus and yield strength are obtained empirically from the Australian AS4100 [35], which have the convenient representation in terms of retention functions gE and gy as

gE ðTÞ ¼

Es;T ¼ Es;0

gy ðTÞ ¼

ry;T ry;0

(

T 0  C < T 6 600  C; 1 þ 2000 lnðT=1100Þ 690ð1T=1000Þ T53:5

600  C < T 6 1000  C;  1 0  C < T 6 215  C; ¼ ð905  TÞ=690 215  C < T 6 905  C;

ð10Þ ð11Þ

in which Es;0 and ry;0 are the elastic modulus and yield strength of the steel at ambient temperature, respectively. The magnitudes of the axial forces acting on the elastic and inelastic domains Xe and Xp can be found by substituting Eqs. (5) and (8) into Eq. (7), so that

Ne ¼ ðEBe  yn EAe Þj  N eT ;

ð12Þ

Npa ¼ c1 ½ðSBpa  yn SApa Þj  NpTa  ða ¼ 1; 2Þ;

ð13Þ

where the axial forces NeT and NpTa which depend on the thermal regime T and the associated elevated-temperature mechanical properties of the steel are given by

NeT ¼ H Z NpTa ¼

Z

Es ðTðyÞÞdAe ;   ðc  c r =E Þ ry ðTðyÞÞ  HTðyÞ  2 3 y;T s;T dApa c1 Ap a

ð14Þ

Ae

ða ¼ 1; 2Þ; ð15Þ

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and the thermal section properties are defined as

Z

EAe ¼

Es ðyÞdAe ;

ZAe

SApa ¼

EBe ¼

Z

yEs ðyÞdAe ; Z ¼ yry ðyÞdApa

ð16Þ

Ae

ry ðyÞdApa ; SBpa

Apa

ða ¼ 1; 2Þ:

in which the flexibility coefficients b1 and b2 are P EAe þ c1 a¼1;2 SApa b1 ¼     2 ; P P P EIe þ c1 a¼1;2 SIpa EAe þ c1 a¼1;2 SApa  EBe þ c1 a¼1;2 SBpa

ð17Þ

ð30Þ

Apa

A relationship between the axial force acting on the whole section X and the thermal section properties can be obtained by substituting Eqs. (12)–(17) into Eq. (6), so that

" N ¼ yn EAe þ EBe þ c1

X

ðSBpa  yn SApa Þj  NT ;

NpTa :

ð19Þ

a¼1;2

The location of the neutral axis can be obtained from Eq. (18), producing

EBe þ c1

P

a¼1;2 SBpa

P

a¼1;2 SIpa

P EBe þ c1 a¼1;2 SBpa   2 : P P EAe þ c1 a¼1;2 SApa  EBe þ c1 a¼1;2 SBpa ð31Þ

ð18Þ

where

X

EIe þ c1



#

a¼1;2

NT ¼ N eT þ c1

b2 ¼ 

 ðN þ N T Þ=j

The magnitude of the internal bending moment Mint at each crosssection of the steel element can be expressed in terms of the bending moment, shear force and axial force at the left side of the beam element (denoted M 0 ; R0 and N 0 ) or at the bottom side of the column element (also denoted M0 ; R0 and N 0 ) using static equilibrium of the element in its deformed configuration as

  v0 þ vÞ  Mint ¼ M 0 þ R0 ðx  u0  uÞ þ N0 ðy

Z

xu0 u

qðx  u  fÞdf; u0

ð20Þ

ð32Þ

In a similar way, the internal bending moment on the whole crosssection about the reference axis is a summation of the elastic and inelastic moments obtained from

in which u0 and v 0 are the axial and vertical deformations at the left side of the beam or column element and u and v are the axial and transverse displacement fields along the element, respectively. In addition, axial force equilibrium leads to

yn ¼

EAe þ c1

X

M ¼ Me þ

P

a¼1;2 SApa

:

M pa ;

ð21Þ

a¼1;2

N cos h ¼ N 0 þ V sin h in the horizontal direction, and to

where

Me ¼

Z

yrdAe

and M pa ¼

Z

Ae

yrdApa

ða ¼ 1; 2Þ:

N sin h ¼ R0  V cos h 

Apa

2

 2 3 P EBe þ c1 a¼1;2 SBpa 7 X 6 M ¼ 4EIe þ c1 SIpa  5j P EAe þ c1 a¼1;2 SApa a¼1;2   P EBe þ c1 a¼1;2 SBpa ðN þ N T Þ þ  MT ; P EAe þ c1 a¼1;2 SApa

SIe ¼

ð23Þ

y2 ry ðyÞdApa ;

ð25Þ

Apa

M pTa ;

yEs ðTðyÞÞdAe ;   ðc2  c3 ry;T =Es;T Þ dApa yry ðTðyÞÞ  HTðyÞ  c1 Ap a

xu0 u

 qdf  R0 sin h

 Z xu0 u  qðx  fÞdf  b2 ½NT þ N0 cos h u Z0 xu0 u   qdf  R0 sin h : 

ð35Þ

ð36Þ

u0

Integrating Eq. (36) numerically over the domain K leads to the relationship between the slopes at the two ends of the element as

ð26Þ

in which

Z

Z

j ¼ b1 ½MT þ M0 þ R0 ðx  u0  uÞ þ N0 ðy  v 0 þ v Þ

hL  h0 ¼

a¼1;2

M eT ¼ H Z M pTa ¼

in the vertical direction, in which V is the shear force and h the slope of a cross-section located at a distance x from the origin. From Eqs. (33) and (34),

G 1 X  j jðxj Þ; w L 2 j¼1

n

and the moment MT is obtained from

M T ¼ M eT þ c1

ð34Þ

and the relationship between the curvature and internal actions given by Eq. (29) can then be rearranged using Eqs. (32) and (35) as

ð24Þ

X

qdf;

u0

y2 Es ðyÞdAe ;

Z Ae

xu0 u

u0

N ¼ N0 cos h 

where the thermal section properties are defined as

Z

Z

ð22Þ

The internal moment given by Eq. (21) can be written as a function of the internal axial force and the thermal section properties by substituting Eqs. (5) and (8) into Eqs. (22) and (21) as

EIe ¼

ð33Þ

ð27Þ

Ae

where xj 2 K is the designation evaluation point in the Gauss– Legendre integration technique, nG the number of integration points  j the associated weighting factor. In a similar fashion, the difand w ference in the deflections at the ends of the element can be related to the slopes in Eq. (37) using

ða ¼ 1; 2Þ: ð28Þ

G 1 X  j hðxj Þ; L w 2 j¼1

n

vL  v0 ¼

Eq. (23) can then be rearranged so that the curvature j at a crosssection is written as a function of the internal actions in the form

in which

j ¼ b1 ðM þ MT Þ  b2 ðN þ NT Þ;

nG X 1  k jðxk Þ ðj ¼ 1; . . . ; nG Þ hðxj Þ ¼ hðxj1 Þ þ ðxj  xj1 Þ w 2 k¼1

ð29Þ

ð37Þ

ð38Þ

ð39Þ

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A. Heidarpour, M.A. Bradford / Computers and Structures 88 (2010) 207–222

is the slope at x ¼ xj . The axial deformation of the locus of the elastic  can be found by using Eqs. (1) and (2) as centroid in K at y ¼ y

u ¼ u0 þ

x

Z 0



1 Þj  ðdv =dxÞ2 dx; ðy  y 2

 nG 1 X 1   yn ðxj Þjðxj Þ  ½hðxj Þ2  ;  j ½y L w 2 j¼1 2

ð41Þ

in which yn ðxj Þ is the location of the neutral axis at x ¼ xj given by Eq. (20). Eqs. (37), (38) and (41) form the basis for a typical positive-oriented element, and can be written in matrix format as

 ¼ Kf þ b; d

T

f ¼ fM 0 ; R0 ; N0 g:

0

d

7 1 5;

0

ð51Þ

0 1 0 2 1=r0i 0 6 ¼4 0 0

3 2 3 1=r Li 0 0 0 7 6 7 i 0 5 and TrkL ¼ 4 0 0 0 5; ð52Þ 0 1=k0i 0 0 1=kLi

0

in which k0i and r 0i are the spring stiffness at the left side of element i and kLi and r Li the spring stiffnesses as the right side. Although the connection matrices Tirk0 and TirkL are based on elastic stiffnesses of the end restraints, the proposed formulation is quite general and can be modified to include the effects of an inelastic semi-rigid joint. Substituting Eq. (47) into Eq. (43) leads to a relationship between the displacement field and the internal end forces of each beam element, which can then be rearranged using Eq. (42) as

ð43Þ

ð45Þ

f ;d

3

Using the equations of equilibrium for each beam element i produces a relationship between the internal end forces of the element as

In Eq. (42), K is a flexibility matrix and b a constant vector, which can be determined from

b¼b þb þb ;

Tirk0

0

    i i i i i Td dtbcL  dtbc0 þ TirkL f L ¼ Ki þ Tirk0 f 0 þ b :

ð44Þ

K ¼ K0 þ Kd þ Kf ;d ;

0

ð42Þ

where

 T ¼ fðh  h Þ; ðv  v Þ; ðu  u Þg; d L 0 L 0 L 0

1

6 Td ¼ 4 0

ð40Þ

and therefore the relationship between the axial deformations at the two ends of the element can be obtained from

wL  w0 ¼

2

ð46Þ

i

i

i

f L ¼ Tie f 0 þ be ;

ð53Þ

ð54Þ

in which

2

3i 1 ðL  u0  uL Þ ðv 0 þ v L Þ 6 7 Tie ¼ 4 0  sin hL 5 ; cos hL 0 sin hL cos hL

ð55Þ

d

in which Kd and b are the flexibility matrix and constant vector which depend on the displacement along the element due to the eff ;d fects of large deflections, Kf ;d and b are the flexibility matrix and constant vector which depend both on the displacements and force 0 vector f, and K0 and b are the flexibility matrix and constant vector which depend only on the geometric properties of the element at elevated temperatures. Full details of these matrices are given in Appendix A. 2.2. Compatibility equations The displacements at the ends of member i, represented in Eq. (43), must be compatible with the end displacements of the other members which are connected to member i. Without loss of generality, it is assumed that the connection between each individual beam and column is semi-rigid, where the semi-rigid connection is simulated by elastic and translational springs (Fig. 1a), and so the relationship between the rotation and deflection at each end of the beam and column is given by i

i

i

db0 ¼ Td dtbc0 þ Tirk0 f 0 i

i

i

i

and dbL ¼ Td dtbcL þ TirkL f L ;

ð47Þ

i

in which db0 and dbL are the left and end displacement vectors of i i beam i, respectively, and dtbc0 and dtbcL are the top end displacement vectors of the left and right bottom column which is connected to the left or right side of beam i, respectively. These displacement vectors can be obtained from iT

v 0 u0 ii and diTbL ¼ h hL v L uL ii ; iT iT dtbc0 ¼ h ht0 v t0 ut0 ii and dtbcL ¼ h htL v tL utL ii ;

db0 ¼ h h0

i f0

¼ h M0

R0

N0 i

i

and

Eq. (53) can be manipulated by using Eq. (55) for each beam element i and Eq. (42) for each column at each side of the beam as

  i i i i i i Td dbbcL þ KibcL f bcL þ bbcL  dbbc0  Kibc0 f bc0  bbc0   i i i ¼ Ki þ Tirk0  TirkL Tie f 0 þ b  TirkL be ; i dbbcL

in which and are the bottom end displacement vector of those bottom columns which are connected to the right and left i i side of the beam element i, respectively, KibcL ; Kibc0 and bbcL ; bbc0 are flexibility matrices and constant vectors for these bottom columns and are given by Eqs. (45) and (46). Compatibility of displacements between the upper and lower columns at each beam-to-column joint indicates that

dbtc ¼ dtbc ;

ð58Þ

and using Eq. (42) leads to

dbtc ¼ dbbc þ Kbc f bc þ bc ;

ð59Þ

ð49Þ

dbccL ¼ dbbc;bsL þ

i

i

X

ðKbck f bck þ bbck Þ;

ð60Þ

nR1

and similarly i

¼ h ML

ð57Þ

i dbbc0

ð48Þ

i fL

iT fL

ð56Þ

in which dbtc and dtbc are the displacement vectors at the bottom of the upper column and the top of the lower column, respectively. i i The displacement vectors dbbcL and dbbc0 in Eq. (57) can be rewritten as a function of the displacement vectors of all the other columns which are connected to the bottom of this column using compatibility of the displacements given by Eqs. (58) and (59) as

while the force vectors and represent the magnitudes of the internal forces at each end of the beam as iT f0

D R ET R Lu u R Lu u i Lu u be ¼  u 0 L qðL  fÞdf;  cos hL u 0 L qdf;  sin hL u 0 L qdf; : 0 0 0

RL

i

NL i :

ð50Þ

The transformation matrix Td and the connection matrices Tirk0 and TirkL in Eq. (47) can be expressed as

i

dbcc0 ¼ dbbc;bs0 þ

X

ðKbck f bck þ bbck Þ;

ð61Þ

nL1 i

i

in which dbbc;bsL and dbbc;bs0 are the displacement vectors of the base column at the right side and left of the beam element i, respectively,

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P

and nR1 ð Þ; nL1 ð Þ is a summation on all columns which are located at the right and left side of beam element i, respectively. Substituting Eqs. (60) and (61) into Eq. (57) then leads to a general formulation for beam element i as i

i

Td dbbc;bsL  dbbc;bc0 þ

X

bbck 

nR

X

! bbck

i

i

 b þ TirkL be

nL

nR

ð62Þ

nL

Whilst Eq. (62), which can be expanded for each beam element i, satisfies member equilibrium at each beam and column and well as compatibility of displacements, equilibrium of forces at each beam-to-column joints has not yet been considered. For a typical internal joint (Fig. 3), the equations of equilibrium can be written as

¼ NbcL þ Ntc0 ;

R0i  RLj N0i  NLj ¼ RbcL  Rtc0 :

ð63Þ

Eq. (63) can be represented in matrix form using Eq. (54) for the lower column and beam element j, so that

h  i i j j j : f bbc ¼ T1 e bbc be bbc  f btc  Td f 0  Te f 0  be

ð64Þ

Eq. (64) indicates that the magnitude of the internal forces at the bottom side of each column can be written as a function of the internal forces of those beams which are connected to the top side of that column.

Eqs. (62) and (64) form a basis for which to develop a non-discretisation semi-analytical formulation for the non-linear analysis of steel frames at elevated temperatures. For the frame shown in Fig. 1, the equation for the internal forces at the left side of the beams of at the bottom side of the columns (with regard to positively-oriented elements) can be represented by

Cb Cc

 ¼

Kbb

Kbc

Kcb

Kcc



0 0 .. .

0 0 .. .    Kbb mm



0

 .. .

0 0 .. .

0

   Kbb nb nb

0

7 7 7 7 7; 7 7 7 5

ð66Þ

in which the matrix Kbb mm is determined from

ð67Þ

where Ki is given by Eq. (45). The elements of Kbc in Eq. (65), which is an nb  nc matrix (where nb and nc are the number of all beams and columns, respectively), can be determined as follows:  If column j is at the bottom and at the right side of beam i, then ½Kbc ij ¼ Td ½Kc j .  If column j is at the bottom and at the left side of beam i, then ½Kbc ij ¼ Td ½Kc j .  If column j is at the top of beam i, then ½Kbc ij ¼ 0: It is noted that the terms ½Kc j can be obtained from Eq. (45). In a similar manner, the elements of the nc  nb matrix Kcb can be determined as follows:  If the left side of beam j is connected to the top of column i, then ½Kcb ij ¼ Ti1 ec Td .  If the right side of beam j is connected to the top of column i, j then ½Kcb ij ¼ Ti1 ec Td Teb .  If beam j is connected to the bottom of column i, then ½Kcb ij ¼ 0. In Eq. (66), Kcc is the nc  nc matrix which can be determined as follows:

3. Solution procedure



Kbb 11

3

Kbb mm ¼ Ki þ Tirk0  TirkL Tit ;

2.3. Force equilibrium at joints

M 0i  MLj ¼ MbcL  M tc0 ;

Kbb

6 6 6 6 ¼6 6 6 6 4

0 !

  X X i ¼ Ki þ Tirk0  TirkL Tie f 0  Td Kbck f bck  Kbck f bck :



2

P

fb ; fc

ð65Þ

in which f b is the vector of the internal forces at the left side of all beams, f c is the vector of the internal forces at the bottom side of all columns, Kbb is the diagonal square matrix

 If the top side of column i is connected to the bottom side of colh i . umn j, then ½Kcc ij ¼ T1 ec i

 All diagonal elements are unity, viz. ½Kcc mm ¼ 1.  All other terms are zero. In Eq. (66), the constant vectors Cb and Cc can be determined from

CTb ¼ h Cb1

Cb2

   Cbnb i and CTc ¼ h Cc1

Cc2

. . . Ccnc i; ð68Þ

in which Cbm and Ccn are given by

Fig. 3. Boundary actions acting on typical internal joint at elevated temperature.

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Cbm ¼ Td

m dbbc;bsL



m dbbc;bs0

þ

X nR

m bbck



X

! m bbck

m

m

 b þ Tm rkL be

nL

ðm ¼ 1; 2; . . . ; nb Þ;   n j ðn ¼ 1; 2; . . . ; nc Þ: be þ Td be Ccn ¼ T1n e

ð69Þ ð70Þ

j

In Eq. (70), be is taken into account for the beam whose right side is connected to the top of column element n, and is given by Eq. (56). For the purposes of this paper, Eq. (65) forms a system of ðnb  nc Þ equations which must be solved simultaneously to find the internal forces to the left side of the beams f b or at the lower ends of the columns f c , the displacement vectors for the base of the columns dbbc;bsL and dbb;bs0 . Computationally, non-linear equations of this order are less than is found in conventional discretisation models for frames, requiring significantly less computational effort to extract the solution at elevated temperatures. Because the deflections v and propagation of yielding along each individual member are not known a priori, an iterative solution is used which takes the following steps: Step 1. The initial values of f b and f c are determined using an elastic analysis, in which the yielded domain Xp at each Gauss–Legendre station x ¼ xj for each element is taken as zero, and the deflection is initially set to zero. Step 2. Updated values of yp1 and yp2 (Fig. 2) at each cross-section xj 2 K of element i are determined by invoking von Mises’ yield criterion at elevated temperature given by

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2x þ 3r2xz ¼ ry;T

The first example is the one-bay two-storey symmetric rigid frame shown in Fig. 4a whose beams and columns are all positively-oriented, while the second example is the two bay one-storey unsymmetrical semi-rigid frame shown in Fig. 4b. The beams are subjected to uniformly distributed loads q (inclusive of self weight and live loading) which are sustained throughout the fire event T; q = 185 and 110 kN/m for the beams located in the first and second floor of the symmetric rigid frame and q = 35 and 55 kN/m for the beams of the unsymmetrical semi-rigid frame. It is assumed that the beams are subjected to a thermal profile determined from the thermal regime TðsÞ at a specific time s resulting from a fire in the ground floor of the symmetric frame and in the left hand bay of the unsymmetrical frame, which is linear over the depth of the cross-section. This produces temperatures T b and bT b at the bottom and top fibres of the steel beams (Fig. 4c) and they are constant along the length of the beams, whereas it is assumed that the columns are fire-protected and therefore the effects of temperatures above the ambient one in the columns are negligible. Tables 1 and 2 list the magnitudes of T b and b, as well as the cross-sectional geometry, and it is assumed that Es;0 ¼ 200 kN=mm2 , ry;0 ¼ 300 N=mm2 and H ¼ 11  106  C1 . It is further assumed that the beams are restrained at their ends by translational and rotational elastic restraints such that

ð71Þ

in which rx is the normal stress produced by the axial force and bending moment, rxz is the shear stress that is evaluated based on the average shear stress (shear force divided by cross-sectional area of the web) and ry;T is the elevated-temperature yield strength in Eq. (11). Step 3. Using Eqs. (38) and (40), the magnitudes of the slope, transverse and axial deformations at each cross-section xj 2 K of element i are calculated. Step 4. The mechanical properties of each cross-section of element i at elevated temperature at the location xj 2 K are found from Eqs. (14)–(17) and Eqs. (24)–(28). Step 5. Substituting the updated values of yp1 and yp2 , as well as the updated values of the displacement field and mechanical properties at the cross-section xj 2 K are determined using Steps 2–4 and Eq. (65), leading to a new system of equations which are solved to find updated values of f b and f c . Step 6.Steps 2–5 are repeated until the convergence criterion

  kpk k  kpk1 k abs < etol ; kpk k

ð72Þ

is satisfied, in which k is the iteration number, k k the Euclidean norm, etol are pre-determined convergence tolerance and

pT ¼ h f b

f c i;

ð73Þ

is the load vector.

4. Numerical verification The semi-analytical mechanical-based formulation developed in the previous section has been verified by considering three examples, whose results are compared with those of the ABAQUS software [30].

Fig. 4. Steel frames under fire loading.

A. Heidarpour, M.A. Bradford / Computers and Structures 88 (2010) 207–222

Figs. 5–15 illustrate the accuracy and efficiency of the proposed model, by comparing the solutions obtained from it with those of ABAQUS. It can be seen that the non-discretisation model using ‘‘one element” with 8 Gauss–Legendre stations in each agrees well with the finite element solutions obtained from ABAQUS, which used 120 elements for the rigid frame and 105 elements for the semi-rigid frame. Figs. 5 and 11 show the variation of the magnitude of the moment at the bottom side of column C1 of the rigid frame and column C2 of the semi-rigid frame. For both cases there is no significant change in this moment with the value of b, while the magnitude of the moment in column C1 increases at elevated temperatures and decreases at elevated temperatures in C2. A similar scenario to that of the base moment of column C1 occurs for the horizontal reaction, as shown in Fig. 6. The magnitude of the compressive axial force in column C2 of the semi-rigid frame is depicted in Fig. 12. As expected, the presence of the thermal gradient in the beams leads to a reduction in the compressive axial force induced in the column (because of ‘‘thermal bowing” [32]), while this force is fairly insensitive to the magnitude of the temperature T b in the bottom fibre of the steel beam cross-section. Unlike the axial force in column C2, the magnitude of the axial force in each beam of the rigid frame at elevated temperatures (Figs. 7 and 8) depends significantly on both the temperature at the bottom fibre of the steel beam and the thermal gradient through the depth of the steel beam. A larger compressive force is developed in beam B1 with an increase in temperature T b , while a reverse scenario exists for beam B2 for which catenary action develops when reference temperature is more than 400 °C and b1 ¼ b2 ¼ 1. In addition, although the magnitude of the compressive axial force in beam B2 in the semi-rigid frame increases with an increase of temperature, it is not sensitive to the variation of b, as shown in Fig. 13.

Table 1 Cross-section and thermal profile properties of rigid frame. Element

Cross-section properties d (mm)

bf (mm)

tf (mm)

tw (mm)

T b (°C)

b (Fig. 4c)

B1a B2

612 612

229 229

19.6 19.6

11.9 11.9

100–700 50–350

0.8 or 1.0 0.6 or 1.0

C1–C2 C3–C4

327 260

311 256

25.0 17.3

15.7 10.5

0 0

0 0

Thermal profile

a Temperature at bottom fibre of this beam is chosen as the reference temperature.

Table 2 Cross-section and thermal profile properties of semi-rigid frame. Element

Cross-section properties bf (mm)

tw (mm)

T b (°C)

b (Fig. 4c)

612 612

229 229

19.6 19.6

11.9 11.9

100–700 70–490

0.6 or 1.0 0.6 or 1.0

327 260

311 256

25.0 17.3

15.7 10.5

0 0

0 0

B1 B2 C1 C2

a

Thermal profile

tf (mm)

d (mm)

a Temperature at bottom fibre of this beam is chosen as the reference temperature.

i

i

kl ¼ kr ¼ b c

    Es;0 Ai Es;0 Ii and ril ¼ r ir ¼ bc ; Li Li

ð74Þ

where bc is an arbitrary coefficient, taken herein as bc ¼ 1 for the rigid frame and bc ¼ 10 for the semi-rigid frame. The values of Ai ; Ii and Li for each element i are given in Tables 1 and 2, and in Fig. 4.

6.00E+02

4.00E+02

Proposed Model (β1 =β2 =1.0)

3.00E+02

ABAQUS (β1 =β2 =1.0)

2.00E+02

Proposed Model (β1 =0.8, β 2 =0.6) ABAQUS (β1 =0.8, β 2 =0.6)

1.00E+02 0.00E+00

100

200

300

400

500

600

700

Reference temperature (°C) Fig. 5. Moment at bottom of column C1 of rigid frame.

0 0

Horizontal reaction (kN)

Moment (kN.m)

5.00E+02

0

-50

100

200

300

215

400

500

600

700

Reference temperature (°C)

-100

Proposed Model (β1 =β2 =1.0)

-150

Proposed Model (β1 =0.8, β 2 =0.6)

ABAQUS (β1 =β2 =1.0) ABAQUS (β1 =0. 8, β 2 =0.6)

-200 -250 -300 Fig. 6. Horizontal reaction at bottom of column C1 of rigid frame.

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50

Axial force (kN)

0 -50

0

100

200

300

400

500

600

700

Reference temperature (°C)

-100 -150

Proposed Model (β1 =β 2 =1.0) ABAQUS (β1 =β2 =1.0)

-200

Proposed Model (β 1 =0.8, β2 =0.6)

-250

ABAQUS (β1 =0.8, β2 =0.6)

-300

Axial force (kN)

Fig. 7. Axial force in beam B1 of rigid frame.

60

Proposed Model (β1 =β2 =1.0)

40

ABAQUS (β 1 =β2 =1.0) Proposed Model (β1 =0.8, β 2 =0.6) ABAQUS (β1 =0.8, β2 =0.6)

20

Reference temperature (°C)

0

0

100

200

300

400

500

600

700

-20 -40 -60 -80 Fig. 8. Axial force in beam B2 of rigid frame.

Mid-span deflection (mm)

0.00 -2.00

0

100

200

300

400

500

600

700

Reference temperature (°C)

-4.00 -6.00 -8.00

Proposed Model (β1 =β 2 =1.0)

-10.00

ABAQUS (β1 =β 2 =1.0)

-12.00

Proposed Model (β1 =0.8, β 2 =0.6) ABAQUS (β1 =0.8, β 2 =0.6)

-14.00

Fig. 9. Deflection at mid-span of beam B2 of rigid frame.

1.00E-03

Rotation (radian)

5.00E-04

Reference temperature (°C)

0.00E+00

0

100

200

300

400

500

600

700

-5.00E-04 -1.00E-03 -1.50E-03

Proposed Model (β1 =β2 =1.0)

-2.00E-03

ABAQUS (β 1 =β2 =1.0)

-2.50E-03

Proposed Model (β1 =0.8, β2 =0.6) ABAQUS (β1 =0.8, β2 =0.6) Fig. 10. Rotation at left of beam B1 of rigid frame.

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0 -50 0

100

200

300

400

500

600

-100

Moment (kN.m)

700

Reference temperature (°C)

-150

Proposed Model (β =1.0)

-200

ABAQUS (β =1.0)

-250

Proposed Model (β =0.6)

-300

ABAQUS (β =0.6)

-350 -400 -450 Fig. 11. Moment at bottom of column C2 of semi-rigid frame.

0

Axail force (kN)

0

100

200

300

400

500

600

700

-50

Reference temperature (°C)

-100

Proposed Model (β =1.0) ABAQUS (β =1.0)

Proposed Model (β =0.6)

-150

ABAQUS (β =0.6)

-200 -250 -300 Fig. 12. Axial force in column C2 of semi-rigid frame.

0

100

200

300

400

500

600

700

-50

Reference temperature (°C)

-100

Proposed Model (β =1.0) ABAQUS (β =1.0) Proposed Model (β =0.6)

-150

ABAQUS (β =0.6) -200

-250 Fig. 13. Axial force in beam B2 of semi-rigid frame.

0.00 -5.00 0

Mid-span deflection (mm)

Axail force (kN)

0

100

200

300

400

500

600

700

Reference temperature (°C)

-10.00 -15.00 -20.00 -25.00 -30.00

Proposed Model (β =1.0)

-35.00

ABAQUS (β =1.0) Proposed Model (β =0.6)

-40.00

ABAQUS (β =0.6)

-45.00 Fig. 14. Deflection at mid-span of beam B2 of semi-rigid frame.

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0.00E+00

0

100

200

300

400

500

600

Rotation (radian)

-2.00E-03

700

Reference temperature (°C)

-4.00E-03 -6.00E-03 -8.00E-03 -1.00E-02

Proposed Model (β =1.0) ABAQUS (β =1.0) Proposed Model (β =0.6) ABAQUS (β =0.6)

-1.20E-02 Fig. 15. Rotation at left of beam B2 of semi-rigid frame.

Figs. 9 and 14 show that the magnitude of the deflection at midspan of beam B2 in both the rigid and semi-rigid frames increases with an increase of the temperature T b and with an increase in the gradient parameter b. Similar behaviour can be seen for the rotation at the right side of beam B1 in the rigid frame and B2 in the semi-rigid frame (Figs. 10 and 15), in which a larger rotations develop with an increase of the thermal gradient through the steel beam cross-section. In order to evaluate the accuracy and efficiency of the proposed model for analysing large steel frames subjected to fire loading, the rigid steel frame comprising of 48 beams and 56 columns shown in Fig. 16 was analysed and the results compared with those obtained using ABAQUS [30]. All of the beams were subjected to sustained uniformly distributed loads and the fire compartment was taken as being in the fourth bay of the ground floor; the geometric properties of the members and the thermal regime are given in Table 3 and the mechanical properties of the steel are the same as for the previous examples. Figs. 17 and 18 show the variation of the axial force and midspan deflection, respectively, in five elements chosen randomly from the 48 beams in the frame. Although there is a slight discrepancy in the mid-span deflection between the results, that

Table 3 Cross-section and thermal profile properties for large steel frame. Element

Cross-section properties

Thermal profile

d (mm)

bf (mm)

tf (mm)

tw (mm)

Tb (°C)

b (Fig. 4c)

B1 & B2 & B6 B3 & B5 B4a B7–B18 B19–B24 B25–B42 B43–B48

612 612 612 612 533 533 460

229 229 229 229 209 209 191

19.6 19.6 19.6 19.6 15.6 15.6 16.0

11.9 11.9 11.9 11.9 10.2 10.2 9.9

300 350 400 250 250 200 200

0.8 0.8 0.8 0.6 0.6 0.4 0.4

C1-C28 C29-C56

327 260

311 256

25.0 17.3

15.7 10.5

0 0

0 0

a Temperature at bottom fibre of this beam is chosen as the reference temperature.

is due to small numerical instabilities occur at large deformations beyond yielding, it can be seen that the proposed model is, overall, quite accurate in comparison with ABAQUS. The same conclusion can be made for the 8 columns selected, whose variation of axial force and shear force is shown in Figs. 19 and 20, respectively.

Fig. 16. Large scale steel frame with rigid joints under fire loading.

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1000

Axial force (kN)

500 0 -500

0

100

200

300

400

500

600 700 Reference temperature (°C)

-1000 -1500 -2000

B4(Proposed Model) B9(Proposed Model) B20(Proposed Model) B25(Proposed Model) B47(Proposed Model)

B4(ABAQUS) B9(ABAQUS) B20(ABAQUS) B25(ABAQUS) B47(ABAQUS)

Fig. 17. Axial force in selected beams in large scale steel frame.

Reference temperature (°C) Mid-span deflection(mm)

0 -2 0 -4

100

200

300

400

500

600

700

-6 -8 -10 -12 -14 -16 -18 -20

B4(Proposed Model) B9(Proposed Model) B20(Proposed Model) B25(Proposed Model) B47(Proposed Model)

B4(ABAQUS) B9(ABAQUS) B20(ABAQUS) B25(ABAQUS) B47(ABAQUS)

Fig. 18. Mid-span deflection at selected beams in large scale steel frame.

Reference temperature (°C) 0 -250 0

100

200

300

400

500

600

700

Axial force (kN)

-500 -750 -1000 -1250 -1500 -1750 -2000 C4(Proposed Model)

C4(ABAQUS)

C10(Proposed Model)

C10(ABAQUS)

C16(Proposed Model)

C16(ABAQUS)

C22(Proposed Model)

C22(ABAQUS)

C33(Proposed Model)

C33(ABAQUS)

C41(Proposed Model)

C41(ABAQUS)

C49(Proposed Model)

C49(ABAQUS)

C55(Proposed Model)

C55(ABAQUS)

Fig. 19. Axial force in selected columns in large scale steel frame.

5. Concluding remarks This paper has formulated a flexibility method for undertaking the analysis of large steel frames subjected to fire loading. Contemporary methods of analysis for these framed structures are either plastic-zone or plastic-hinge based; the former are widely known to be problematic from a computational standpoint because of the fine discretisation needed to represent accurately the spread of plasticity while the latter are known to produce only

approximate results because of the lumping of inelastic effects at the member ends when one element per member is used. The technique is a non-adaptive formulation which possesses the accuracy merits of the plastic-zone approach, as well as the computational advantages of the plastic-hinge approach by requiring only one element per member. By expressing the Green–Lagrange strain tensor in a form which retains only non-linear membrane strains but which linearises the curvature, expressions for the curvature, slope and deflection along

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A. Heidarpour, M.A. Bradford / Computers and Structures 88 (2010) 207–222

300 250

Shear force (kN)

200 150 100 50 0 -50 0

100

200

300

400

500

600

700

Reference temperature (°C)

-100 -150 -200 C4(Proposed Model) C16(Proposed Model) C33(Proposed Model) C49(Proposed Model)

C4(ABAQUS) C16(ABAQUS) C33(ABAQUS) C49(ABAQUS)

C10(Proposed Model) C22(Proposed Model) C41(Proposed Model) C55(Proposed Model)

C10(ABAQUS) C22(ABAQUS) C41(ABAQUS) C55(ABAQUS)

Fig. 20. Shear force in selected columns of large scale steel frame.

the member may be established readily in this semi-analytical approach, leading to the flexibility formulation in which the flexibility matrix is made up of separate terms which are associated with displacements along the member due to: (i) large deflections, (ii) displacements and forces and (iii) geometric properties at elevated temperatures. The proposed method was used to model three frames; one small frame with rigid joints, one small frame with flexible joints and one large rigid-jointed frame with 104 members, and the results were compared with those of ABAQUS. In all cases, the agreement between the proposal here and of the computationally intensive ABAQUS software was shown to be very good, with the flexibility approach using only one element per member. It provides a platform for undertaking rapid and accurate analyses of steel frames with a large number of members throughout the thermal event that is an attractive alternative to plastic-zone and plastic-hinge techniques.

Kd1;1 ¼ 0;

K01;1 K02;1

n

K02;2 ¼

g L X  j ðxj  xj1 Þ  I k8 ; w  4 j¼1

n

g L X  j ðxj  xj1 Þ  I k2 ;  w 4 j¼1

ng X



g  X Ly  j ðxj  xj1 Þ  I k1 ;  w 4 j¼1

vb1 ðxk Þ  vqb2 ðxk Þ;

k¼1

 ng  ng   L X sin hðxj Þ LX j  j hðxj1 Þðxj  xj1 Þ I k2  I k7  Kd3;2 ¼   J j7  w w 2 j¼1 4 j¼1 xA ðxj Þ

Kd3;3 ¼

ng    LX  j ðxj  xj1 Þ2 I k2  I k7 I k3  I k5  I k4 þ I k6 ; w 8 j¼1

 ng  ng   L X cos hðxj Þ LX  j hðxj1 Þðxj  xj1 Þ y j I k1 þ I k8 w J j8 þ w   xA ðxj Þ 2 j¼1 4 j¼1

;d Kf3;1 ¼

n

K02;3 ¼

n

g g L X L X  j hðxj1 Þðxj  xj1 Þ  I k1    j ðxj  xj1 Þ2 Kd3;1 ¼   w w 4 j¼1 8 j¼1

ng    LX  j ðxj  xj1 Þ2 I k8 þ y I k1 I k3  I k4  I k5 þ I k6 : w 8 j¼1

;d Kf1;1 ¼ 0;

 L L Ly ¼ Ij1 ; K01;2 ¼ I j1 ; K 01;3 ¼ I j1 ; 2 2 2 ng L X  j ðxj  xj1 Þ  I k1 ; ¼  w 4 j¼1

Kd2;1 ¼ 0;

n

Kd2;3 ¼



Flexibility Matrix: K ¼ K0 þ Kd þ Kf ;d

L j I ; 2 8

n



Appendix A. Flexibility terms

Kd1;3 ¼

g L X  j ðxj  xj1 Þ  I k7 ; w Kd2;2 ¼   4 j¼1

Acknowledgement The work in this paper was supported by the Australian Research Council, through a Federation Fellowship awarded to the second author.

L Kd1;2 ¼   I j7 ; 2



;d Kf1;2 ¼ 0;

;d Kf1;3 ¼ 0;

;d Kf2;1 ¼ 0;

;d Kf2;2 ¼ 0;

;d Kf2;3 ¼ 0;

ng ng   LM 0 X LR0 X  j ðxj  xj1 Þ2 I k2  j ðxj  xj1 Þ2 I k1 I k2  I k7 w w  1  16 j¼1 8 j¼1 ng   LN 0 X  j ðxj  xj1 Þ2 I k1 y I k1 þ I k8 ; w 8 j¼1

g  2 LR0 X  j ðxj  xj1 Þ2 I k2  I k7 w  16 j¼1

n

K03;1 ¼ K03;2 ¼

L j L J   2 1 8 L j J ; 2 2

ng X

   j ðxj  xj1 Þ2  I k1  I k3  I k5  I k4 ; w

j¼1

K03;3 ¼

 j Ly J : 2 1

;d Kf3;2 ¼

þ

ng ng   X LN 0 X  j ðxj  xj1 Þ2 vwb1b2 ðxk Þ vb1 ðxk Þ þ vv b1b2 ðxk Þ ; w y 8 j¼1 k¼1

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A. Heidarpour, M.A. Bradford / Computers and Structures 88 (2010) 207–222

;d Kf3;3 ¼

ng  2 LN 0 X  j ðxj  xj1 Þ2 y I k1  I k8 : w 16 j¼1 0

d

Constant Vector: b ¼ b þ b þ b

J1 ¼

0

b2 ¼

0

b3 ¼

d

b1 ¼

L 4

f ;d

J3 ¼

J4 ¼ J5 ¼

ng  LX  j NT ðxj Þ w L j J 3  J j4  J j5 þ : 2 2 j¼1 xA ðxj Þ

d

ng X

L 2

ng X

 j hðxj1 Þ þ w

j¼1

L 4

ng X

 j ðxj  xj1 ÞI k6 ; w

j¼1

   j hðxj1 Þðxj  xj1 Þ I k3  I k4  I k5 þ I k6 : w

f ;d

f ;d b3

 j M T ðxÞb1 ðxÞkðxÞ; w

ng X

 j N T ðxÞb2 ðxÞkðxÞ; w

ng X

Z  j b1 ðxÞkðxÞ w

xu0 uðxÞ

 qðx  fÞdf ;

u0

Z  j b2 ðxÞkðxÞ sin hðxÞ J6 ¼ w

xu0 uðxÞ

 qdf ;

u0

J7 ¼

ng X

 j kðxÞ½ðu0 þ uðxÞÞb1 ðxÞ þ b2 ðxÞ sin hðxÞ; w

j¼1

J8 ¼

ng X

 j kðxÞ½ðv ðxÞ  v 0 Þb1 ðxÞ  b2 ðxÞ cos hðxÞ; w

j¼1

j¼1

b1 ¼ 0;

ng X

 j xb1 ðxÞkðxÞ; w

j¼1

j¼1

  ng ng LX 1 LX L d  j h2 ðxj1 Þ   w vqb2 ðxj Þ kðxj Þ  b3 ¼ 2 j¼1 b2 ðxj Þ  xA ðxj Þ 4 j¼1 4 

ng X

j¼1

j¼1

b2 ¼

J2 ¼

j¼1

   j ðxj  xj1 Þ I k3  I k4  I k5 ; w

L j I ; 2 6

 j b1 ðxÞkðxÞ; w

j¼1

 L j 0 b1 ¼ I  I j4  I j5 ; 2 3 ng X

ng X

f ;d

b2 ¼ 0;

ng  2 L X  j ðxj  xj1 Þ2 I k3  I k4 w ¼ 16 j¼1 ng h   i L X  j ðxj  xj1 Þ2 I k5 2I k4 þ I k5  2I k6  2I k6 I k4 w 16 j¼1   ng   2 L X  j ðxj  xj1 Þ2 I k6  2I k3 I k5  I k6 :  w 16 j¼1

I jn ¼ I n ðx ¼ xj Þ;

I kn ¼ I n ðx ¼ xk Þ;

J kn ¼ J n ðx ¼ xk Þ;

n ¼ 1; . . . ; 8

J jn ¼ J n ðx ¼ xj Þ and

x x xk ¼ xj1 þ j 2 j1 ð1 þ fm Þ; fm 2 ½1 1 is a designated Legendre evaluation point.

Gauss–



In these equations,

  x ðxÞ  B kðxÞ ¼ y ; xA ðxÞ

xA ðxÞ ¼ EAe þ c1

2 X

xB ðxÞ ¼ EBe þ c1

2 X

SBpa ;

a¼1

SApa ;

a¼1

I1 ¼

ng X

 j b1 ðxÞ; w

I2 ¼

ng X

j¼1

I4 ¼

ng X

 j xb1 ðxÞ; w

I3 ¼

ng X

j¼1

j¼1

 j N T ðxÞb2 ðxÞ; w

j¼1

I5 ¼

ng X j¼1

 j b1 ðxÞ w

Z

xu0 uðxÞ

 qðx  fÞdf ;

u0

 j b2 ðxÞ sin hðxÞ I6 ¼ w

Z

xu0 uðxÞ

 qdf ;

u0

I7 ¼

ng X

 j ½ðu0 þ uðxÞÞb1 ðxÞ þ b2 ðxÞ sin hðxÞ; w

j¼1

I8 ¼

ng X j¼1

 j ½ðv ðxÞ  v 0 Þb1 ðxÞ  b2 ðxÞ cos hðxÞ; w

 j M T ðxÞb1 ðxÞ; w

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