A simulation-based large deflection and inelastic

Journal of Constructional Steel Research 60 (2004) 1495–1524 www.elsevier.com/locate/jcsr

A simulation-based large deflection and inelastic analysis of steel frames under fire Chi Kin Iu, Siu Lai Chan Department of Civil and Structural Engineering , The Hong Kong Polytechnic University, Kowloon, Hong Kong Received 15 July 2003; received in revised form 20 February 2004; accepted 12 March 2004

Abstract This paper presents an accurate and robust geometric and material nonlinear formulation to predict structural behaviour of unprotected steel members at elevated temperatures. A fire analysis including large displacement effects for frame structures is presented. This finite element formulation of beam–column elements is based on the plastic hinge approach to model the elasto-plastic strain-hardening material behaviour. The Newton–Raphson method allowing for the thermal-time dependent effect was employed for the solution of the nonlinear governing equations for large deflection in thermal history. A combined incremental and total formulation for determining member resistance is employed in this nonlinear solution procedure for the efficient modeling of nonlinear effects. Degradation of material strength with increasing temperature is simulated by a set of temperature–stress–strain curves according to both ECCS and BS 5950 Part 8, which implicitly allows for creep deformation. The effects of uniform or non-uniform temperature distribution over the section of the structural steel member are also considered. Several numerical and experimental verifications are presented. # 2004 Elsevier Ltd. All rights reserved. Keywords: Fire engineering; Large deflection analysis; Inelastic analysis

1. Introduction Fire analysis and design comprise thermal expansion and material degradation. The latter mostly follows British Standards [1], ECCS [2] and Eurocode 3 [3]. From

Corresponding author. Tel.: +852-2766-6047; fax: +852-2334-6389. E-mail address: [email protected] (S.L. Chan).

0143-974X/$ - see front matter # 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2004.03.002

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C.K. Iu, S.L. Chan / Journal of Constructional Steel Research 60 (2004) 1495–1524

the British Steel test results tabulated from Wainman and Kirby [4] and Kirby and Preston [5], temperature–stress–strain relationship data are provided for Grade v 43A and 50B steels with yield stresses for temperature at 800 C and strains up to 2%. Sharples [6] constructed a set of simple empirical formulae for elastic modulus and yield stress to approximate the tabulated test results in BS5950 Part 8. In the European steel structure design code for fire resistance, Anderberg [7] and Talamona et al. [8], amongst others, also based their studies on steel behaviour under fire on this temperature–stress–strain curves which are described by a bilinear–elliptic equation given in Eurocode 3. When a structural steel frame is under fire, the strength and stiffness of the steel v structure significantly deteriorates after 200 C. It is, therefore, a common practice for design specifications to require some degrees of insulation or protection to an individual steel member so that the integrity of the structure can be preserved for a sufficient period. Generally speaking, the fire resistance of individual elements is classified on the basis of the minimum time that the elements can withstand a standard fire to ISO 834 [9]. It inevitably and conservatively involves a high construction cost for structural fire design when the overall structural behaviour of steel frames under fire cannot be predicted. An alternative direction of fire safety concern is to grasp an understanding of the structural behaviour of a building under fire. Nonlinear structural analysis has lagged significantly behind investigation of the response of steel structures to provide integrity design for fire safety of a building. It gives adequate information to an engineer to adopt a more reliable and cost effective performance based design solution. When a steel frame is heated, the steel member deteriorates rapidly and exhibits geometric nonlinearity due to changes in geometry, resulting in a necessity to utilize such nonlinear solution techniques to obtain instability structural behaviour under fire. Most fire analyses utilize the Newton–Raphson method to predict large deflection effects. In the early 1970s, Cheng and Mak [10] developed a general theory and algorithm to investigate the thermo-creep deformation and buckling behaviour of steel structures at elevated temperatures. In the mid-1980s, Jeanes [11] investigated the numerical solution of the time-history structural behaviour of two-dimensional floor systems subjected to fire, which was based on the previous work [12]. In this study, members in the frame are modeled by the beam–column elements and the non-conforming triangular plate bending element is used to model the floor slab. In continental Europe, researchers including Dotreppe et al. [13] and the ARBED Research Center [14,15] studied the structural behaviour of beams, columns and frames under fire by the finite element method. Based on previous work, Schleich et al. [16] simulated the effect of thermal-time dependence using the incremental-iterative Newton–Raphson solution procedure. Franssen [17] presented a fire analysis which considers non-uniform temperature distribution, material yielding and geometric nonlinearity. Franssen [18] has further extended his previous work to modify the constitutive law including unloading response and has


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developed a user-friendly computer program SAFIR using a fine grid of elements over each cross-section [19]. In the United Kingdom, Terro [20] studied the structural behaviour of general three-dimensional building structures under fire. Material and geometric nonlinearities were also taken into account. At the same time, Saab and Nethercot [21] presented the nonlinear structural behaviour of a two-dimensional frame under fire using the finite element method including the thermal-time dependence effect. Their basic formulation of the nonlinear solution at normal temperature was mainly based on the research work from EI-Zanaty and Murray [22] on which Najjar and Burgess [23] further based their work to incorporate three-dimensional behaviour of unprotected steel members, including the warping effect. Furthermore, Bailey [24] used this work to develop the computer program 3DFIRE. This basic nonlinear formulation allows modeling of semi-rigid connections, lateral-torsional buckling, continuous floor slabs and strain reversal. Wang and Moore [25] developed an analytical finite element method to predict the response of structures under fire. Second-order geometrical nonlinearity, residual stress and initial deflection were included. The material nonlinear effect is modeled by the plastic zone method used by numerous previous researchers. Liew et al. [26] presented a fire analysis using one element per member and thereby obtained a realistic representation of material and geometrical nonlinear behaviour of the overall framework. Material behaviour is considered by allowing for the formation of plastic hinges formed at the element ends and at the mid-span. The member is then divided into two new elements, when yielding occurs at the mid-span. The nonlinear solution procedure is based on thermal-time independence for different temperature–stress–strain curves. Using the plastic hinge method, the present fire analysis is based on the nonlinear inelastic finite element program, GMNAF [27], which incorporates the thermal effects for steel structures. The present fire analysis includes both uniform and non-uniform temperature distribution over the member section. An incrementaliterative solution procedure based on the Newton–Raphson method is employed to model the effect of thermal-time dependence. The proposed formulation is derived in an updated Lagrangian local system, which allows large deformation effects by using relative small strain–displacement relationships. Material nonlinearity is simulated by inserting a spring in both the tangent and secant stiffness formulation in which secant stiffness can be based on either the incremental or the total formulation in this fire analysis. The material properties at elevated temperatures are according to both BS5950 Part 8 and Eurocode 3 for different verifications. For verification of the present fire analysis, several theoretical examples are solved for verifying both the effects of material degradation and thermal expansion. The fire test results from beams, column and small-scale frames are also compared. The proposed method has the uniqueness of reliable prediction of structural behaviour under fire without numerical divergence.

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2. Assumptions in the formulation The following assumptions are made for the kinematic formulation in this proposed fire analysis: 1. A transverse plane is assumed to remain plane and normal to the beam axis throughout deformation, i.e. the Bernoulli–Euler hypothesis. 2. The cross-sectional area of the member is assumed undistorted due to applied load and elevated temperature. 3. Shear deformation is ignored. 4. Warping deformation is neglected, because the complexity lies in the assessment of the mechanism of transferring warping effect from one member to another, leading to difficulty in generalization of the warping transfer relation between members. 5. The beam–column element is assumed doubly symmetric in its cross-section so that the shear center and the centroid coincide. 6. The strain is small, but arbitrarily large deformation and rotation are allowed, 7. Effects from higher than second-order terms are neglected in the formulation. 8. Only independent or conservative loading is considered where the applied loading is independent of the displacement.

3. Thermal strain and curvature in fire analysis Based on experimental observations, the total strain of steel at elevated temperatures can be subdivided into thermal strain, instantaneous stress related strain and creep strain as et ¼ eth þ er

ð1Þ

in which et is total deformation strain, er is instantaneous stress related strain and eth is thermal strain which consists of both thermal axial strain and curvature. The thermal axial strain eth,a is given by eth;a ¼ aDT ¼ aðtþ1 T t TÞ

ð2Þ

in which a is the coefficient of thermal expansion. It is taken as a constant of 14 v 106 per C in the present fire analysis. t+1T and tT are the temperature at next (t þ 1)th and current tth thermal configuration, respectively. The thermal bowing effect is simulated by thermal curvature /th given by /th ¼

et eb abtþ1 T ¼ D D

ð3Þ

in which et and eb are the thermal strains at the extreme top and bottom fibers, respectively, D is the depth of the member section and b is the ratio of the tem-


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perature difference between the top and bottom surface of the member. It should be noted that this expression is according to independently free axial elongation at the top and bottom most fibers with linear relation that assumed no shear effect between fibers over the member section.

4. Basic formulation of the proposed fire analysis The incremental internal strain energy dU is based on Castigliano’s first theorem in terms of stress and strain components as ð4Þ

dU ¼ rdet

in which r and et are the internal stress and total strain, respectively. Only the thermal strains due to axial expansion and non-uniform temperature distribution across the major axis is considered. Thus, we have eth ¼ eth;a y/th ¼ aDT y

abT D

ð5Þ

In a nonlinear analysis using the finite element formulation, it is customary to use Green’s strain tensor to express the strain–displacement relation as et ¼

@u 1 @v 2 1 @w 2 @2v @2w þ þ y 2 z 2 @x 2 @x 2 @x @x @x

ð6Þ

The stress-related strain er is expressed from the total strain extracted from the thermal strain as compressive strain. It is assumed that the material obeys Hooke’s law as follows: r ¼ Eer ¼ E ðet eth Þ

ð7Þ

This implies that the stress–strain relation Eq. (7) is based on a particular temperature–stress–strain relationship. Substituting Eq. (7) into Eq. (4), the incremental internal strain energy is obtained. The incremental internal strain energy function can be accumulated by integration over the range of strain and the internal strain energy can be written as U¼

ð

dU ¼ E e

ðð ð

ðe eth Þ de dA dx

ð8Þ

l A e

The range of strain e varies from 0 to et at a particular temperature–stress–strain relationship for the updated Lagrangian formulation in which the undeformed configuration is regarded as zero strain.

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By back substitution, the internal strain energy function can be rewritten as follows: et 1 2 e eth e dA dx l A 0 l A 2 0 ðð abT du 1 dv 2 1 dw 2 þ E aDT y þ D dx 2 dx 2 dx l A d2 v d2 w y 2 z 2 dA dx dx dx

U ¼E

ð ð ð et

ðe eth Þ de dA dx ¼ E

ðð

ð 2 ð 2 du 2 E du dv E du dw dx þ dx þ dx 2 l dx dx 2 l dx dx l dx ð 2 ð 2 2 ð 2 2 E d v E d v d w 2 d w y2 z dx þ E yz þ dx þ dx 2 2 l dx2 2 l dx2 dx dx2 l ð 2 2 ð 4 ð 4 EA dv dw EA dv EA dw dx þ dx þ dx þ 4 l dx dx 8 l dx 8 l dx ð 2 ð 2 ð du EA dv EA dw EAaDT dx aDT dx dx aDT dx 2 dx 2 l l l dx ð 2 ð 2 abTE abTE dw 2 d v y yz dx dx 2 D dx D dx2 l l

EA U¼ 2

ð

ð 2 ð 2 ð 2 2 du 2 P dv P dw EIz d v dx þ dx þ dx þ dx 2 dx 2 dx 2 dx dx 2 l l l l ð 2 2 ð 2 2 ð EIy d w d v d w du dx þ EJ þ dx dx EAaDT 2 2 2 dx 2 l dx l dx l dx ð 2 ð 2 ð 2 EA dv EA dw abTEIz d v aDT dx aDT dx dx ð9Þ 2 2 l dx 2 l dx dx D l

EA U¼ 2

ð

The external work done, V, is equal to the applied force multiplied by the corresponding displacement as follows: V ¼ f dk g T f f k g

ð10Þ

in which fdk g and ffk g are the vectors of the corresponding displacement and external applied force at kth degrees-of-freedom, respectively, and fdk gT ¼ hu1 ; u2 ; v1 ; v2 ; w1 ; w2 ; hx1 ; hx2 ; hz1 ; hz2 ; hy1 ; hy2 i is the displacement vector.


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The total potential energy for fire analysis is equal to the sum of Eqs. (9) and (10) as P¼U V Y

¼

ð 2 ð 2 ð 2 2 du 2 P dv P dw EIz d v dx þ dx þ dx þ dx 2 l dx 2 l dx 2 l dx2 l dx ð 2 2 ð 2 2 ð EIy d w d v d w du dx þ EJ þ dx dx EAaDT 2 2 2 dx 2 l dx l dx l dx ð 2 ð 2 ð 2 EA dv EA dw abTEIz dv dx aDT dx aDT dx 2 2 l dx 2 l dx dx D l EA 2

ð

fdk gT ffk g

ð11Þ

The nodal displacement functions of the element at elevated temperatures are assumed to be the same as those at room temperature. The nodal displacement functions based on the total nodal displacement in co-rotational coordinate are given by x x u ¼ 1 u1 þ u2 l l 2x2 x3 x2 x3 þ 2 hz 1 þ þ 2 hz 2 v¼ x ð12Þ l 2 l 3 l 2 l 3 2x x x x þ 2 hy1 þ 2 hy2 w¼ x l l l l The derivatives of the nodal displacement functions can be expressed as

du u1 u2 0 T u1 ¼ þ ¼ Bu u2 dx l l

0 T hz 1 dv 4x 3x2 2x 3x2 ¼ 1 þ 2 hz 1 þ þ hz2 ¼ Bv h dx l l l2 l

z2 2 00 T hz1 d v 4 6x 2 6x ¼ þ 2 hz1 þ þ 2 hz2 ¼ Bv ð13Þ hz 2 dx2 l l l l

T hy 1 dw 4x 3x2 2x 3x2 ¼ 1 þ 2 hy1 þ 2 hy2 ¼ B0w h dx l l l l

y2 00 T hy1 d2 w 4 6x 2 6x ¼ þ 2 hy1 þ 2 hy2 ¼ Bw hy2 dx2 l l l l in which B0u , B0v and B0w are the first derivatives of the nodal displacement functions of virtual displacement in axial and transverse y- and z-directions, respectively. B00v and B00w are the second derivatives of the nodal displacement functions of virtual displacement at the transverse y- and z-directions, respectively. The formulations of secant stiffness and tangent stiffness formulation are then derived from the total potential energy function in Eq. (11). The secant stiffness

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½Ks is the first variation of the internal strain energy equation given by

0 0 T u1 dx f u1 u2 g Bu Bu u2 l

ð T hz 1 P @ dx þ f hz1 hz2 g B0v B0v 2 @dk l hz 2

ð T hy 1 P @ hy1 hy2 B0w B0w þ dx 2 @dk l hy 2

ð 00 00 T hz1 EIz @ þ dx f hz1 hz2 g Bv Bv 2 @dk l hz 2

ð T hy 1 EIy @ hy1 hy2 B00w B00w þ dx 2 @dk l hy 2

ð T hz 1 @ hy1 hy2 B00w B00v þ EJ dx @dk l hz 2

ð 0 T u1 @ Bu dx EAaDT @dk l u2

ð 0 0 T hz 1 EA @ dx aDT f hz1 hz2 g Bv Bv 2 @dk l hz 2

ð T hy1 EA @ hy1 hy2 B0w B0w aDT dx 2 @dk l hy2

ð 00 T hz1 abTEIz @ Bv dx D @dk l hz 2

@U EA @ ¼ @dk 2 @dk

ð

ð14Þ

Integrating the shape function over the whole length of the element, the secant stiffness equation is expressed as 9 9 8u u 9 8 8 4l l 4l l 1 2 > > > > > > = = = h h h h < < < z z y y 1 2 1 2 @U l 30 30 30 30 þP þP ¼ EA u þ u 2> > > > @dk ; ; ; : 1 : l h þ 4l h > : l h þ 4l h > z1 z2 y y l 30 30 30 1 30 2 9 9 8 8 4 2 4 2 > >

= = < hz 1 þ hz 2 > < hy 1 þ hy 2 > 1 l l l l þ EIz þ EIy EAaDT > > 1 ; ; :2h þ4h > :2h þ4h > z z y y l 1 l 2 l 1 l 2 9 9 8 8 > > > 4l hz l hz = > 4l hy l hy = < < 1 2 1 2 30 30 EAaDT 30 EAaDT 30 > > ; ; : l h þ 4l h > : l h þ 4l h > z z y y 30 1 30 2 30 1 30 3

abT 1 EIz ð15Þ D 1


1503

The tangent stiffness matrix can be obtained by taking a second variation of the total potential energy in Eq. (11) as @2P ¼ ½KT ¼ ½KL þ ½KG þ ½Kth @dj @dk

ð16Þ

in which ½KL and ½KG are the linear and geometric stiffness given by Chan [28] and ½Kth is the thermal geometric stiffness which simulates the change of geometry due to rising temperature given by 3 2 0 6 7 60 Sym: 7 6 5l 7 6 7 6 6 7 6 0 0 7 6 5l ð17Þ Kth;11 ¼ EAaDT 6 7 0 0 7 60 0 7 6 1 2l 7 60 0 0 7 6 10 15 4 1 2l 5 0 0 0 0 10 15

5. Nonlinear thermal incremental-iterative procedure In structural fire analysis, the steel structure exhibits geometric nonlinearity resulting in a necessity to utilize nonlinear solution procedures to obtain the structural response near collapse. An incremental-iterative solution procedure based on the Newton–Raphson scheme is extended and shown in Fig. 1. The flowchart showing the overall present fire analysis is illustrated in Fig. 2. The Newton–Raphson scheme based on the combined incremental and total formulation automatically allows for the thermal-time dependence effect. The incremental displacement can be determined by the tangent stiffness in Eq. (18), and the total displacement is obtained by accumulation of increment displacements in Eq. (19): n t fDugiþ1 n t fugiþ1

n ¼t ½KT ðfRgni Þ1 t fwgi

ð18Þ

¼t fugni þt fDugniþ1

n t ½KT ðfRgi Þ

ð19Þ fRgni . t fwgni

is the incremental tangent stiffness at is the unbain which lanced force between applied forces and internal member resistances. The incremental displacement in global coordinates can be transformed to the member deformation t fDue gniþ1 using the transformation matrix ½LT for mapping of global to local coordinates. t fDRe gniþ1 can then be evaluated in local coordinates as n t fDue giþ1

¼ ½LTt fDugniþ1 ð20Þ n n n ð21Þ t fDRe giþ1 ¼ Ks ðt fRgi Þ t fDue giþ1 n n Ks t fRgi is the incremental secant stiffness at t fRgi . After determination of the

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Fig. 1. Incremental-iterative Newton–Raphson scheme for fire structural analysis.

member resistant forces, total member resistances in global coordinates are accumulated by the incremental member resistance as follows: n t fRgiþ1

¼t fRgni þ ½T½Lt fDRe gniþ1

ð22Þ

In which ½T is the transformation matrix from member coordinate to nodal coordinate. Therefore, the unbalanced forces t fwgniþ1 are obtained as n t fwgiþ1

¼t ff gn t fRgniþ1

ð23Þ

The above incremental procedure is repeated for a specified temperature level until force equilibrium is achieved. Before the load incremental procedure is completed to specified load level, the applied load is kept constant throughout the heating sequence. The next thermal cycle starts with a similar procedure of total formulation. The member resistances determined from the total secant stiffness using the total displacements save computational time. This is termed as the thermal-time dependence effect. The total displacement at the last thermal cycle satisfying equilibrium for new thermal cycle is given by 1 tþ1 fugi

¼t fugnþ1 iþm

1 tþ1 fue gi

ð24Þ

¼ ½LTtþ1 fug1i ð25Þ h i 1 n 1 fue g1i ð26Þ tþ1 fRe gi ¼tþ1 Ks tþ1 fugi ;t fRgiþm tþ1 h i in which tþ1 Ks tþ1 fug1i ;t fRgniþm is the total secant stiffness relation based on the new temperature–stress–strain relationship at (t þ 1)th thermal cycle. (i þ m)th is the number of iterations required to achieve equilibrium at the last thermal cycle.


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Fig. 2. Flowchart for present nonlinear fire analysis basing on combined incremental and total formulation.

It is commonly recognized that the incremental formulation is efficient in modeling the nonlinear effects or to trace the nonlinear equilibrium path. However, in order to include the thermal-time dependent effect, it is necessary to use total formulation to calculate the member resistance using an updated temperature–stress–strain relationship. Since tracing equilibrium path by incremental procedure is more important to nonlinearity than the determination of member resistance by total secant stiffness, nonlinear effects are also incorporated effectively in this fire analysis. Member resistance at the new thermal cycle is weakened due to material deterioration at elevated temperatures. An unbalanced force is, therefore, generated at the new temperature–stress–strain curve given by Eq. (28), after the member resistance

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is transformed from member coordinate to nodal coordinate in Eq. (27). For the sake of eliminating this unbalanced force, the similar incremental procedures of the constant load Newton–Raphson method is utilized as Eqs. (18)–(23) and the resistance in global coordinate is computed as 1 tþ1 fRgi ¼ 1 tþ1 fwg1

½T½Ltþ1 fRe g1i

¼tþ1 ff g

ð27Þ

tþ1 fRg11

ð28Þ

The basic principle of this combined incremental and total thermal procedure is to minimize the difference between external applied forces and internal member resistance, defined as fwg. The process is repeated throughout the heating sequence until divergence is detected. The convergence check for equilibrium is carried out using the following criterion: ½DuT ½Du < 0:001 ½uT ½u;

½DwT ½Dw < 0:001 ½wT ½w

ð29Þ

6. Plastic hinge in stiffness formulation Material nonlinear effect in the proposed fire analysis is simulated through the plastic hinge approach. This approach is used due to its high convergence rate and simplicity in stiffness formulation. It is assumed that the beam element remains elastic and material yielding is lumped to a zero-length spring at the end nodes. In the incremental formulation, this plastic hinge method is based on the previous works by Yau and Chan [29] and Chan and Chui [30] in which the value of this spring for partial yielding varies from 1 to 0. Since the member resistance at the new temperature–stress–strain curve is based on the total secant stiffness approach, their spring stiffness formulation is modified and incorporated in Eq. (26) for total formulation. This spring stiffness models effectively the load redistribution effect after yielding. Considering the force equilibrium and displacement compatibility conditions, the total moment–rotation relationship can be written as Eq. (30). All material yielding is simulated by the end spring stiffness S1 and S2: 9 2 8 S1 Mc1 > > > > = 6 < S1 Mb1 ¼6 4 0 M > > b2 > > ; : Mc2 0

S1 S1 þ K11 K21 0

0 K12 K22 þ S2 S2

9 38 hc1 > 0 > > > = < 0 7 7 hb1 S2 5> h > > ; : b2 > S2 hc2

ð30Þ

in which Mc1 and Mc2 are the connection moments at the near and far node of the member, respectively. hc1 and hc2 are the nodal rotations of the connection at the near and far end, respectively. Similarly, hb1 and hb2 are nodal rotations of the beam at the corresponding nodes. Fig. 3 shows the locations of hc1, hb1, hb2 and hc2 graphically.


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Fig. 3. Locations of connection and beam rotations about their corresponding spring stiffness.

Using Eq. (30), the quantities and after condensation of the rotations of beam and substitution into the elastic beam–column formulation, we have

1 K11 K12 S1 ðK22 þ S2 Þ S2 K21 hc1 Mc1 ¼ M K K S K S ð K þ S Þ hc2 K j j c2 21 22 1 12 2 11 1

2 1 S1 K11 ðK22 þ S2 Þ S2 K12 Mc1 S2 K12 ðK11 þ S1 Þ S2 K11 K21 hc1 ¼ 2 Mc2 hc2 jK j S1 K21 ðK22 þ S2 Þ S1 K22 K12 S2 K22 ðK11 þ S1 Þ S2 K21

ð31Þ in which ! ! ! K11 þ S1 K12 ! ! ! ¼ ðK11 þ S1 ÞðK22 þ S2 Þ K12 K21 : jK j ¼ ! K21 K22 þ S2 ! 2Pl 2EI Pl and K11 ¼ K22 ¼ 4EI l þ 15 ; K12 ¼ K21 ¼ l 30 . S1 and S2 are the spring stiffness at the first and second node at the end of the element, respectively, which combine the effects of semi-rigid connection and material yielding (see [30]). When the moment Mc1 at the near node of an element exceeds the plastic moment capacity Mp of the section and the member is elastic at the second node, the spring stiffness at this yielded node is then reduced as Eq. (32) so that the additional moment over the plastic capacity can be redistributed to the elastic spring of another node possessing this yielded spring stiffness. This yielded spring stiffness is also incorporated in the tangent stiffness formulation. For this case,

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Mc1 ¼ Mp and S2!1, the spring stiffness about the first yielded node is given as ðK11 K22 K12 K21 þ S2 K11 ÞMp 2 h þ S K h ðS þ K ÞM K11 ðK22 þ S2 Þ K12 1 2 12 2 2 22 p ððK11 K22 K12 K21 Þ=S2 þ K11 ÞMp S1 ¼ lim 2 =S h þ K h ð1 þ K =S ÞM S2!1 K11 ðK22 =S2 þ 1Þ K12 2 1 12 2 22 2 p K11 Mp S1 ¼ K11 h1 þ K12 h2 Mp S1 ¼

ð32Þ

Similarly, the spring stiffness of the second node for the first yielded moment can be expressed in Eq. (33) with elastic moment at the near node. In the present case, the similar conditions are Mc2 ¼ Mp and S1!1 as S2 ¼

K22 Mp K21 h1 þ K22 h2 Mp

ð33Þ

It should be noted that spring stiffness S1 and S2 must be positive, because elastic moments at end nodes must be greater than the plastic moment Mp. Otherwise, the spring stiffness will remain elastic. When two nodes of the element are also yielded, the additional moments over the plastic moment capacity are redistributed to other members through the equilibrium equation of the whole structure. The moments about the yielded nodes of the element are assigned to the plastic moment capacity, Mp. On the other hand, the spring stiffness for incremental formulation including the strain-hardening effect developed from Chan and Chui [30] is shown in Eq. (34). The strain-hardening effect is essential to the structural behaviour at fire condition, because this effect dominates the plastic limit at a higher temperature ! ! ! 6EI !! Mpr M ð34Þ þ l!! S¼ ! l M Mer in which Mer and Mpr are the reduced first yielded and plastic moment, respectMpr M , must be greater than 0, when ively. The non-dimensional parameter, MM er Mer < M < Mpr . Another non-dimensional parameter, l, is to model strain-hardening behaviour and ranges from 0 to 10. This value is not sensitive and a small value of spring stiffness for gradual yielding greater than the stiffness of the strainhardening effect is adequate. Under incremental formulation, the spring stiffness in Eq. (34) can be incorporated in both the incremental tangent stiffness in Eq. (18) and the incremental secant stiffness in Eq. (21) to trace the nonlinear equilibrium path. Alternatively, when the total formulation is used, the total spring stiffness in Eq. (32) or Eq. (33) determines the total member resistance at the start of each temperature level on the basis of total formulation. However, the incremental spring stiffness can model gradual yielding and strain-hardening material behaviour, whereas the total spring stiffness cannot simulate such effects. It causes an inconsistency in material representation under incremental and total formulations. This inconsistency of determination of total resistance at new temperature levels only causes large out-of-


1509

balanced forces and a larger number of iterations are required to restore the equilibrium. It does not, therefore, influence the accuracy of the nonlinear solutions. The accuracy depends only on the mathematical model in the secant stiffness. In a fire analysis, the total spring stiffness can only approximate the material behaviour as follows: . During the elastic range, the spring stiffness for incremental and total formulations are the same and infinite. . At onset of gradual yielding, the total spring stiffness is assumed to remain elastic. . At strain-hardening after the plastic limit, total spring stiffness is taken as fully yielded. 7. Illustration of solutions under fire The adapted nonlinear solution procedure allowing for the effects of thermal expansion and thermal bowing is illustrated in Figs. 4 and 5 for both unrestrained and restrained cases. They include simply supported and fixed end beams. The restraining effect is important in fire analysis, while the additional thermal forces can be developed in the restrained case so as to cause the geometric nonlinear effects at elevated temperature. In the case of a simply supported beam with roller supports, there is no axial and rotational restraint at the supports. When the beam is heated uniformly, ther-

Fig. 4. Thermal effects in a simply supported beam in the proposed fire analysis.

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Fig. 5. Thermal effects in a fixed ended beam in the proposed fire analysis.

mal axial strains and thermal curvature are added as compressive mechanical strains. The unbalanced forces at unrestrained support degrees-of-freedom are obtained due to thermal expansion and rotation. These unbalanced forces at the support degrees-of-freedom cause the corresponding incremental displacements according to the equilibrium equation (22). This relationship is summarized in Fig. 4. In the second case of a fixed ended beam, the axial expansion and rotation are not permitted at the restrained supports. The member reactions at the restrained degrees-of-freedom are similarly induced from the compressive mechanical strain due to thermal axial force and moment, when the equilibrium equation at the corresponding degree-of-freedoms for member reactions is not balanced. Therefore, these member reactions are converted to member forces. These additional member forces weaken the tangent stiffness of the beam and cause member bowing. Consequently, larger deformations are obtained. This case is shown in Fig. 5.

8. Numerical verification For verification of the present fire analysis, the nonlinear finite element program, GMNAF [27], is modified to incorporate the thermal effect for different structures. Both uniform and non-uniform temperature distribution over the member section are considered in this fire analysis. The verification of different cases is carried out in this section.


1511

8.1. Simply supported beams under various load ratios A 1.14 m long simply supported beam under various load ratios in different cases was investigated by Rubert and Schaumann [31] through the fire test, which is modeled by 10 cubic elements for different cases. The beam of IPE 80 I-section is subjected to a concentrated load at the mid-span and heated uniformly along the entire length. The beam is under various load ratios of F =Fu , in which F and Fu are, respectively, actual and ultimate point loads. Load ratios against yield stress are plotted in Fig. 6. The elastic modulus is assumed as 210 kN/mm2. A more detailed description of material properties and external load is given in Rubert and Schaumann [32]. This example is a simple bending problem by the applied point load, so there are no buckling or geometric nonlinear effects. On the other hand, material yielding and strain-hardening effect can also be included when the applied load exceeds the material strength at elevated temperature. Hence this case demonstrates the combined material nonlinearity and strain-hardening effect of a fire analysis. The strain-hardening parameter for this problem is selected to be 2 and is not very

Fig. 6. Deflections at mid-span of the simply supported beams against temperature.

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sensitive for the general problems. Greater values represent a larger stiffness after full yield, and vice versa. The mid-span deflections under various load ratios are also plotted in Fig. 6 which shows that the deflections are generally close to the test results for the full range of temperature. When a plastic hinge forms on the simply supported beam at lower temperature level, the runaway deflection only contributes to the strain-hardening effect. The results of analysis therefore show the adequacy of the analysis in efficient simulation of strain-hardening effect at elevated temperatures and under different load levels. The deflection curve under load ratios 0.85, 0.7 and 0.5 are according to the temperature–stress–strain relationship of ECCS Pt. 1.1 2. Since elastic modulus of temv perature over 600 C is undefined for ECCS Pt. 1.1, only the material relationship of BS5950 1 and Eurocode 3 Pt. 1.2 3 are instead compared for the case of load ratio 0.2. It is observed that the material behaviour according to Eurocode 3 is softer than that based on BS5950, as the material yielding of the beam is formed at lower temperature level. According to the different temperature–stress–strain curves, the deflection curve is therefore different from other curves of load ratios 0.85, 0.7 and 0.5. 8.2. Axially unrestrained steel column tests A number of steel columns were tested by Aasen [33]. The same reference numbers for the columns as those used by Aasen [33] are also adopted in this paper and all construction detail should refer to Ref. [33]. However, only axially unrestrained steel columns are considered for verification of second-order effects in this fire analysis. The numerical results of these column tests conducted by Poh and Bennetts [34] are also included in this verification. All the steel columns are discretized into 20 elements in this fire study. All columns are made from European rolled I-section IPE 160. The yield stress of these steel columns at room temperature is 448 MPa. The modulus of elasticity of the steel at room temperature is assumed to be 210 GPa. The temperature– stress–strain relationships for steel material recommended by Eurocode 3 were used for this numerical analysis, which are identical in tension and in compression. The examples of unrestrained column under compression can demonstrate the geometrically nonlinear effect due to thermal expansion. In these examples, the yield stress is chosen to remain elastic over the range of applied loads. Therefore, geometric nonlinearity of this proposed fire analysis can be verified through the experimental results of column tests. Lateral deflections and axial deformations of the column tests and results from Ref. [34] are plotted in Figs. 7–16 In the unrestrained column tests, the axial deformations of all unrestrained columns are also accurately modeled. This fire analysis shows the accuracy of modeling for thermal expansion. In the case of slender columns, lateral deflections for columns, numbered as 2, 5, 6, 8–10, are predicted accurately by the present numerical fire analysis. The lateral deflections at mid-


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Fig. 7. Lateral deflection and axial deformation against time for column no. 2.

height of short columns, such as column numbers 12–14 and 16, show the linear proportionality to the applied load. 8.3. Experimental results of thermal bowing beam and columns from Cooke [35] Cooke [35] tested one small-scale steel beam heated along a flange which results in a non-uniform temperature distribution across the member section. The beam section is welded from two flange plates of 50 mm 5 mm and one web plate of 70 mm 3 mm. Detailed information is then in Ref. [35]. The results show the


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thermal bowing effect against the material nonlinearity. The elastic modulus and yield stress are 194.55 kN/mm2 and 325 N/mm2, respectively. In addition, the numerical results of this example from Wang and Lennon [36] are compared. From Fig. 17, the mid-span deflection between numerical simulation from Wang and Lennon [36] and present fire analysis are in close agreement for the whole temperature ranges. Cooke [35] also provided experimental data of three unprotected steel columns under axial loads. All three columns are the same size except for their width, which are 40, 50 and 60 mm. The depths of all three columns are 60 mm whilst the flange

Fig. 10. Lateral deformation and axial deformation against time for column no. 8.


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and web thickness are 6 and 4 mm. The elastic modulus and yield stress for these three columns are 192.8 kN/mm2 and 346 N/mm2. The columns were heated on one flange. The temperature gradient between two flanges is detailed in Ref. [35]. Related properties are further given in Figs. 18–23. The tested lateral deflections at mid-height and axial deformation are compared with numerical results from Wang and Lennon [36] and those from present fire analysis plotted in Figs. 18–23. From the lateral deflection, the results of the steel column of 40 mm width are consistent with Cooke’s test results [35]. However, the results from the present fire analysis in the other two examples show an under-


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estimation to the tested results. The axial deformations from this fire analysis are predicted well with all the tested columns from Wang and Lennon [36]. The trend of the axial extension before the specimen approaches failure is correctly predicted. 8.4. Small-scale steel frame tests under uniform temperature distribution The small-scale steel frames with dimensions shown in Fig. 24 are heated uniformly in a test by Rubert and Schaumann [32]. There are three series of geometry for these steel frames, which are the inverted L-shaped frame (EHR), the singlebay portal frame (EGR) and the double-bay portal frame (ZSR). These frames are composed of the same IPE80 I-section. Their yield stress and other material properties are also shown in Fig. 24 and their elastic modulus is taken as 210 kN/



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mm2. The frames are also adequately restrained against out-of-plane deflection. The measured and computed deformations studied are plotted in Fig. 24. Fig. 25 shows the deformations of the EHR frame. The deformations from the present fire analysis are consistent with the results from the fire tests in general. v Two plastic hinges are formed at 400 C and the EHR frame can sustain not more v than one hinge before reaching a failure mechanism. After 400 C, the runaway deflections are due to the strain-hardening effect, which can only provide residual strength of the steel frame after yielding. The corresponding displacements of the frames EGR and ZSR versus the temperature rise are also plotted in Figs. 26 and 27. These figures show that the displacements between the analysis and the test are consistent until divergence and show material nonlinear effects have been correctly simulated by present fire analy-


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Fig. 17. Mid-span deflection for the simply supported beam by Cooke.

sis. The Euler buckling load of the compressive members in these frames at failure temperature is much higher than the applied compressive load. The geometric nonlinear effect of these frames should be, therefore, excluded in the test and this verification.

Fig. 18. Mid-height lateral deflection against time for column 40 60.


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Fig. 19. Axial deformation against time for column 40 60.

9. Conclusions The present fire analysis based on the second-order inelastic analysis at ambient conditions has been extended to study the structural behaviour at elevated temperature. The incremental-iterative Newton–Raphson method is utilized to solve the geometric nonlinearity using a combined incremental and total formulation. The thermal-time dependence is fully considered and the material yielding is mod-


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eled by the plastic hinges at both ends of an element with gradual yielding modeled by elasto-plastic spring stiffness. The strain-hardening effect is also included in the spring stiffness formulation of this proposed fire analysis. From the numerical examples, the proposed fire analysis is verified to include thermal effects, such as material deterioration and thermal expansion or bowing, under fire conditions of both uniform and non-uniform temperature distribution. From the experimental examples, the material and geometric nonlinear effects have




Fig. 24. Configurations of three small-scale steel frames.

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Fig. 25. Deformation of the frame EHR with temperature change.

been incorporated in the present fire analysis consistent with results. This fire analysis, therefore, shows that both geometric and material nonlinear behaviour at fire condition can be modeled reliably, as well as the different types of thermal effects. The proposed nonlinear fire analysis has a good convergent rate and requires less computational effort than fire analysis based on the plastic zone approach. An incremental-iterative solution procedure is required for tracing the equilibrium paths at different temperature levels. In addition, the thermal-time dependent effect is included and it greatly reduces the computational time when compared with the thermal-time independence analysis.

Fig. 26. Deformation of the frame EGR with temperature change.


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Fig. 27. Deformations of the frame ZSR with temperature change.

Acknowledgements This fire analysis as a part of the project of ‘‘Numerical simulation for structural steel member at elevated temperature’’ has been carried out with the support of the research grants from The Hong Kong Polytechnic University and valuable advice and supervision from Prof. S.L. Chan in Civil and Structural Engineering Department. References [1] BS5950 Part 8. Structural use of steelwork in building: code of practice for fire resistant design, 1990. [2] ECCS European recommendations for the fire safety of steel structures: technical committee 3—fire safety of steel structures, 1985. [3] Eurocode 3, Design of steel structures, Part. 1.2: Structural Fire Design (Draft) European Committee for standisation, 1993. [4] Wainman DE, Kirby BR. Compendium of UK standard fire test data, unprotected structural steel. vol. 1, Ref. No. RS/RSC/S10328/1/87/B and vol. 2, Ref. No. RS/R/S1199/88/B, 1986. [5] Kirby BR, Preston RR. High temperature properties of hot-rolled, structural steels for use fire engineering design studies. Fire Safety Journal 1988;13:27–37. [6] Sharpes R. The strength of partially exposed steel columns in fire. Mphil dissertation, Department of Civil and Structural Engineering, University of Sheffield, 1987. [7] Anderberg Y. Modeling steel behaviour. Fire Safety Journal 1988;13:17–26. [8] Talamona D, Kruppa J, Franssen JM, Recho N. Factors influencing the behaviours of steel columns exposed to fire. Journal of Fire Protection Engineering 1996;8(1):31–43. [9] ISO 834. Fire resistance tests: elements of building construction, international standard 834, 1975. [10] Cheng WC, Mak CK. Computer analysis of steel frames in fire. Journal of Structural Division, ASCE 1975;101(ST4):855–67. [11] Jeanes DC. Applications of the computer in modeling the endurance of structural steel floor systems. Fire Safety Journal 1985;9:119–35. [12] Jeanes DC. Predicting fire endurance of steel structures, Preprint 82-033, ASCE Convention, Nevada, April 26–30, (1982), American Society of Civil Engineers.

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[13] Dotreppe JC, Franssen JM, Schleich JB. Computer aided fire resistance for steel and composite structures. Review acier/Stahl/Steel No. 3, 1984. [14] ARBED—Research, Luxembourg/Department of Bridges And Structural Engineering, University of Liege, Belgium—REFAO/CAFIR. Computer assisted analysis of the fire resistance of steel and composite steel-concrete structures. CEC research 7210-SA/502, Technical reports 1 to 6, 1982/85. [15] ARBED— Research, Luxembourg, University of Liege, Belgium, Buckling curves in case of fire. CEC research 7210-SA/515/931/316/618, 1992/95. [16] Schleich JB, Dotreppe JC, Franssen JM. Numerical simulations of fire resistance tests on steel and composite structural elements or frames. Fire Safety Science: Proceedings of the First International Symposium. 1986, p. 311–23. [17] Franssen JM. Etude du comportement au feu des structures mixtes acier-beton (CEFICOSS). A study of the behaviour of composite steel-concrete structures in fire. These de Doctorat. Belgique: Universite de Liege; 1987. [18] Franssen JM. The unloading of building materials submitted to fire. Fire Safety Journal 1990;16:213–37. [19] Franssen JM, Kodur VKR, Mason J. User’s manual for SAFIR: a computer program for analysis of structures submitted to fire. Internal Report SPEC/2000_03, 2000. Belgium: University of Liege, Ponts et Charpentes; 2000. [20] Terro MJ. Numerical modeling of thermal and structural response of reinforced concrete structures in fire. PhD thesis, Department of Civil Engineering, Imperial college of Science, Technology And Medicine, January 1991. [21] Saab HA, Nethercot DA. Modelling steel frame behaviour under fire conditions. Engineering Structures 1991;13:371–82. [22] EI-Zanaty MH, Murray DW. Nonlinear finite element analysis of steel frames. Journal of Structural Division, ASCE 1983;109(ST2):353–68. [23] Najjar SR, Burgess IW. A nonlinear analysis for three-dimensional steel frames in fire conditions. Engineering Structures 1996;18(1):77–89. [24] Bailey CG. Development of computer software to simulate the structural behaviour of steel-framed building in fire. Computers and Structures 1998;67:421–38. [25] Wang YC, Moore DB. Steel frames in fire: analysis. Engineering Structures 1995;17(6):462–72. [26] Liew RJY, Tang LK, Holmaas T, Choo YS. Advanced analysis for the assessment of steel frames in fire. Journal of Constructional Steel Research 1998;47:19–45. [27] Chan SL. GMNAF, geometric and material nonlinear analysis of frames. Program manual, 1993. [28] Chan SL. Nonlinear analysis of structures composed of thin-walled beam–columns. Doctoral thesis, The Department Of Civil Engineering, University of Queensland, 1987. [29] Yau CY, Chan SL. Inelastic and stability analysis of flexibly connected steel frames by spring-inseries model. Journal of Structural Division, ASCE 1994;120(ST10):2803–19. [30] Chan SL, Chui PPT. A generalized design-based elastoplastic analysis of steel frames by section assemblage concept. Engineering Structures 1997;19(8):628–36. [31] Rubert A, Schaumann P. Structural steel and plane frame assemblies under fire action. Fire Safety Journal 1986;10:173–84. [32] Rubert A, Schaumann P. Temperaturabhangige werkstoffeigenschaften von baustahl bei brandbeanspruchung. Stahlbau 1985;3:81–6. [33] Aasen B. An experimental study on steel column behaviour at elevated temperatures. Research report, Division of Steel Structures, University of Trondheim, Norwegian Institute of Technology, Norway, 1985. [34] Poh KW, Bennetts ID. Behaviour of steel columns at elevated temperature. Journal of Structural Division, ASCE 1995;121(ST4):676–84. [35] Cooke GME. The structural response of steel I-section members subjected to elevated temperature gradients across the section. Doctoral thesis, Department of Civil Engineering, The City University, London, 1987. [36] Wang YC, Lennon T. A computer program for structural analysis at elevated temperatures. BRE client report CR77/92, Building research establishment, Garston, Watford, 1992.

A simulation-based large deflection and inelastic

A simulation-based large deflection and inelastic

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