Effect of transient strain on strength of concrete and CFT columns in fire–Part 1: Elevated-temperature analysis on a Shanley-like column model

Engineering Structures 44 (2012) 379–388

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Engineering Structures

journal homepage: www.elsevier .com/ locate /engstruct

Effect of transient strain on strength of concrete and CFT columns in fire – Part 1:Elevated-temperature analysis on a Shanley-like column model

Shan-Shan Huang ⇑, Ian W. BurgessDepartment of Civil and Structural Engineering, The University of Sheffield, Sir Frederick Mappin Building, Mappin Street, Sheffield S1 3JD, UK

a r t i c l e i n f o

Article history:Received 29 August 2011Revised 10 May 2012Accepted 23 May 2012Available online 9 July 2012

Keywords:Transient strainConcrete-filled tubular columnStructural fire engineeringShanley-like model

0141-0296/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.engstruct.2012.05.032

⇑ Corresponding author. Tel.: +44 114 222 5727; faE-mail address: [email protected] (S.-S. Huang).

a b s t r a c t

This paper presents elevated-temperature analysis on a Shanley-like column model, as part of a study onthe effect of transient strain on the strength of concrete and concrete-filled tubular columns in fire. Threehigh-temperature concrete material models are applied and the structural behaviours of the Shanley-likemodel using these three material models are compared. The effects of transient strain of concrete havebeen investigated by comparing the results of the analyses with and without considering this property,under the assumption that the temperature distribution within the model is uniform. The model has beenevaluated against the tangent-modulus and reduced-modulus critical buckling loads at elevated temper-atures. Numerical analyses have been carried out under both steady-state and transient heating scenar-ios, in order to investigate the influence of each on high-temperature structural analysis of the typedescribed in this paper. It is seen that considering transient strain causes a considerable reduction ofthe buckling resistance, irrespective of the concrete material models and loading–heating schemes used.

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1. Introduction

Concrete is now recognised to have a unique material propertyat high temperatures, defined either as transient strain (TS) or asload-induced thermal strain (LITS) [1–7]. Both of these definitionsdescribe a phenomenon in which pre-compressed concrete, withits preload being maintained during heating, experiences a muchlarger compressive strain after heating than when it is loaded tothe same level after being pre-heated to the same temperature.This additional strain is high in magnitude and is not recoverable,and therefore any structural analysis of a concrete structure sub-ject to fire which neglects the transient strain property will notbe able to represent the real behaviour of the structure, and will re-sult in an incorrect prediction. Unfortunately this phenomenon isneither clearly acknowledged in the Eurocodes [8]; nor is it consid-ered in the majority of analyses of concrete structures in fire.

Since concrete columns subjected to accidental fires are nearlyalways pre-compressed when they are heated, they are clearly vul-nerable to the effects of transient strain. The nature of transientstrain determines its significance. In stocky columns it causes con-siderable extra contraction across the whole cross-section, and thishas been observed in many tests on short cylinders. However, asthe slenderness of the columns increases, such tests become lessrelevant as the failure mode switches from material crushing tolateral buckling. The way in which transient strain affects buckling

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in such cases forms a gap in current knowledge which needs to beinvestigated.

A ‘‘Shanley-like’’ [9] column model (Fig. 1) has been establishedpreviously [10] to attempt to shed light on the mechanics of the ef-fect of TS on buckling of heated concrete columns. Its characteris-tics were programmed for numerical analysis. The model wasevaluated against the tangent-modulus and reduced-modulus crit-ical buckling loads, and was found to be effective in representingthe progressive change in the regions of loading and unloadingduring inelastic buckling. In this paper this model is used to studythe effects of transient strain of concrete on inelastic buckling athigh temperatures.

There are different macroscopic mathematical models of con-crete straining at high temperatures, of which the best-establishedare due to Anderberg and Thelandersson [1], Khoury [4,5] and Ter-ro [11,12] and Schneider et al. [13]. The Anderberg & Thelanders-son and Schneider models were developed for ordinaryconcretes, and there is no evidence that they are applicable tohigh-strength concrete, whereas the Khoury model covers nucle-ar-reactor-type high-strength concretes. The main distinction be-tween these three models is their classification of the total straininto components. In this study these models, all taking into ac-count transient strain, are applied to the Shanley-like model, to as-sess their differences in predicting critical loads for the simplifiedcolumn. In order to isolate this effect from that of temperature dis-tributions across the section due to different heating rates, it is as-sumed that the temperature distribution within the column modelis uniform.

Nomenclature

A areaB model widthCr damping coefficient of the rotational damper (Nmms)Cv damping coefficient of the vertical damper (Ns/mm)E Young’s moduluse strainFs,ij reaction force on spring j (j = 1,n from column centre to

edge) on either convex side (i = 1) or concave side (i = 2)Ff

s spring force recorded at the end of the previous loadstep

Fpfs spring force recorded one load step before

Fu ultimate compressive strengthFu0 ambient-temperature compressive strength

Ftu tensile strength

FUL spring force when unloading startsF� compressive force at x�

k stiffness at linear-elastic stagek0 initial stiffness of the compressive curve at ambient

temperaturek� slope of the linear descending branch of the compres-

sive curveL model lengthMcr reaction moment on the rotational damperP applied loadPe Euler buckling loadPt tangent-modulus critical buckling loadPr reduced-modulus critical buckling load

Pp proportional limitr stressru0 ultimate compressive stress at ambient-temperatureT temperaturet timeh rotation_h velocity of hh0 initial imperfectionu vertical movement_u velocity of uxij deformation of spring j (j = 1, n from column centre to

edge) on either convex side (i = 1) or concave side (i = 2)_xij velocity of xij

xth thermal displacementxr instantaneous stress-related displacement_xr velocity of xr

xfr xr recorded at the end of the previous load step

xpfr xr recorded one load step before

xcr creep displacementxtr transient displacementxel elastic displacementxtr,cr transient creep displacementxm mechanical displacementxu xr at the ultimate compressive strengthxUL spring deformation when unloading startsx� displacement at the transition between the parabolic

branch and the linear descending branch in the com-pression quadrant

380 S.-S. Huang, I.W. Burgess / Engineering Structures 44 (2012) 379–388

2. Steady-state heating analysis on the Shanley-like model

The Shanley-like column model is firstly analysed under steady-state heating scenario, with the mechanical loading applied at con-stant temperature. The aim is to find a failure load, at which overallbuckling of the model occurs. In incorporating transient strain, pre-loading prior to heating is assumed, and so the analysis simulates apre-loaded column which is heated under load to a stabilised uni-

Fig. 1. Shanley-like column model.

form temperature; it is then unloaded and reloaded from zero atthis temperature until failure occurs.

2.1. Mathematical model and calculation procedure

The geometry of the Shanley-like model, the fundamental for-mulations and solution procedure have been described in detailin a previous paper [10], and so only a brief summary of somekey points will be given here. Eqs. (1) and (2) are derived fromthe equations of motion, and define the relationship between theinternal and external forces and the rates of the two degrees offreedom.

_u ¼ P �X2

i¼1

Xn

j¼1

Fs;ij

!,Cv ð1Þ

_h ¼ ðPðh0 þ hÞL� B2n

Xn

j¼1

j � ðFs;2j � Fs;1jÞÞ=Cr ð2Þ

The displacements of the springs are related to the two DoFsthrough Eqs. (3) and (4), on the assumption of a linear strain-gradient:

u ¼ x1jþx2j

2

h ¼ x2j�x1jjBn

9=;) x1j ¼ u� jB

2n h

x2j ¼ uþ jB2n h

(ð3Þ

_u ¼ _x1jþ _x2j

2

_h ¼ _x2j� _x1jjBn

9=; )

_x1j ¼ _u� jB2n

_h

_x2j ¼ _uþ jB2n

_h

(ð4Þ

Depending on the material model used, the force–displacementrelationships of the springs differ, and this affects the calculationprocedure, so the applications of the three material models are de-scribed separately.

S.-S. Huang, I.W. Burgess / Engineering Structures 44 (2012) 379–388 381

2.1.1. Loading schemeThe applied force is increased step by step, each load step taking

a finite number of time steps. These time steps are necessary to al-low the internal forces to balance the external forces so that a newstatic equilibrium is reached (when the velocities _u and _h vanishand the dampers have zero force eventually); this equilibrium po-sition is used as the initial condition for the next load step. This isrepeated until the rotation of the model is seen to diverge, indicat-ing failure by runaway buckling. In this way, the buckling load ofthe model at any given temperature is assessed. The whole processcan be repeated at different temperatures, giving a relationship be-tween the critical buckling loads of the model and temperature.

2.1.2. Application of the Anderberg & Thelandersson concrete modelThe high-temperature concrete model of Anderberg and

Thelandersson [1], which was derived on the basis of tests onquartzite-aggregate concrete, is applied first. For convenience inanalysing the multi-spring Shanley-like model, the stress–strainrelationships for equivalent solid bars are converted to force–displacement relationships by substituting Eq. (5) into the originalformulations.

r ¼ Fs=A; e ¼ x=L; E ¼ LA

k ð5Þ

A square bare-concrete (Grade C30) column is simulated, so theequivalent area of each spring forming part of the cross-section isA ¼ B2

2n. The decomposition of the total strain is converted to the fol-lowing form:

x ¼ xth þ xr þ xcr þ xtr ð6Þ

The thermal strain eth(xth = eth � L) is a simple function of tem-perature, which is directly given by the mean of the measuredthermal strain curves in the original model. The EC2 [8] equationfor thermal strain of siliceous-aggregate concrete compares rea-sonably well with this curve and is used in this analysis. The creepdisplacement xcr is neglected due to its insignificance for accidentalfire applications. The relationship between the spring force Fs andthe instantaneous stress-related displacement xr is illustrated inFig. 2.

Rather than recording the plastic strain as in the original model,to determine the unloading route, the concurrent spring force FUL

and instantaneous stress-related displacement xUL at the transitionfrom loading to unloading are recorded. Linear unloading with theinitial stiffness k of the compressive curve starts at Point 1, whosecoordinate is (xUL, FUL), and ends when the tensile strength Ft

u isreached. A simplification is introduced to describe the behaviourin tension. The arrows in Fig. 2 indicate the allowed directions ofloading and unloading. This simplification is justified in many

Fig. 2. The relationship between the spring force and the instantaneous stress-related displacement based on the Anderberg & Thelandersson model.

cases by the fact that the tensile stresses in concrete are insignifi-cant to the structural behaviour. In addition, the fact that the tran-sient strain is absent in tension makes the tensile behaviour ofconcrete insignificant in this analysis.

The model of transient displacement for temperatures above500 �C (Eq. (8)) cannot rationally be applied in a steady-state heat-ing analysis, and so it is not used; Eq. (7) is adopted for the fulltemperature range.

For 20 �C 6 T 6 500 �C xtr ¼ �ktrFs

Fu0xth ð7Þ

For T > 500 �C Dxtr ¼ L � 0:1� 10�3DTFs

Fu0

� �ð8Þ

where ktr is assumed to be equal to 2, which is an intermediate va-lue taken from the range 1.8–2.35 given in the original model.

Another very important characteristic of transient strain, whichneeds to be accounted for, is its irrecoverability. This is done bypreventing the value of the transient displacement xtr fromdecreasing between adjacent load steps. At constant temperaturethe variation of xtr depends solely on the spring force level, but onlythe spring forces at static equilibrium positions are used to accountfor the irrecoverability of xtr. Therefore, the original model is com-plemented by assuming that, if the previous two load steps show apotential decrease of spring force, the value of xtr is not allowed toalter from its previous value.

After all, the force–displacement relationship of the springs isdefined as Eq. (9). On the basis that the displacement of each springis the sum of three components, two of which are load-dependent,all three parts (9a, 9b and 9c) of this equation set must simulta-neously be satisfied:

xtr;t þ xr;t ¼ xt � xth ð9aÞ

If Ffs P 0 and Ff

s P Fpfs then

xtr;t ¼ �ktrFs;tFu0

xth

Otherwisextr;t ¼ xtr;t�Dt

8>>>><>>>>:

ð9bÞ

Fs;t ¼ Fuxr;t

xu2� xr;t

xu

� �Loading Stages ðN1Þ and ðN2Þ ð9c1Þ

Fs;t ¼ k�xr;t þ F� Loading Stage ðDÞ ð9c2ÞFs;t ¼ kxr;t þ FUL;t � kxUL;t Loading Stage ðUL=RLÞ; Fs;t�Dt P Ft

u ð9c3ÞFs;t ¼ Ft

u Loading Stage ðUL=RLÞ; Fs;t�Dt < Ftu ð9c4Þ

where x� = xu(1 � k�/k), F� = Fu(1 � k�/k)2 and k� ¼ �880 AL

ðunit is MPaÞ.If xr;t�Dt P 0; _xr;t�2Dt P 0; _xr;t�Dt < 0 and xr;t�Dt > xUL;t�Dt , then

the transition from loading to unloading is detected:

FUL;t ¼ ðFs;t�2Dt þ Fs;t�DtÞ=2xUL;t ¼ ðxr;t�2Dt þ xr;t�DtÞ=2

ð10Þ

The calculation procedure within each time step is illustrated inFig. 3. Relaxation with explicit time integration is used for thenumerical algorithm. The two DoFs ut and ht at time t are calculatedfrom their values ut�Dt and ht�Dt and their velocities _ut�Dt and _ht�Dt

at the end of the previous time step, as u and h are assumed toincrease linearly with time within each time increment. Thedisplacement of each spring xt at time t, and its velocity _xt�Dt atthe end of the previous time step are then calculated fromut ; ht ; _ut�Dt and _ht�Dt using the linear strain-gradient assumption(Eqs. (3) and (4)).

The loading stage (N1), (N2), (D) or (UL/RL) of a spring on itsforce–displacement curve is detected as shown in Fig. 4. This deter-mines which of the four formulae should be used as Eq. (9c).

Fig. 3. Calculation procedure within each time step of the steady-state heatinganalysis using the Anderberg & Thelandersson concrete model.

Fig. 4. Detection of loading stage when Anderberg & Thelandersson concrete modeland steady-state heating scenario are applied.

382 S.-S. Huang, I.W. Burgess / Engineering Structures 44 (2012) 379–388

Unloading (Stage (UL)) is detected when the velocity _xr;t�Dt of theinstantaneous stress-related displacement at the end of the previ-ous time step is negative, which indicates a decrease of the instan-taneous stress-related displacement xr. This applies to all the timesteps within one load step, except the initial one in which all thevelocities are zero. In the initial time step, unloading is detectedwhen the instantaneous stress-related displacement xf

r, recordedat the end of the previous load step, decreases from its value xpf

r

one load step before. When unloading (UL) is detected, either Eq.(9 c3) or Eq. (9 c4) is used.

In the initial time step, when xfr P xpf

r , or in the rest of the timesteps when _xr;t�Dt P 0, loading which follows the original

compressive curve (N1), (N2) or (D) or the linear reloading path(RL) is detected. When the instantaneous stress-related displace-ment xr,t�Dt at the end of the previous time step is greater thanor equal to xUL,t, the original compressive curve is applied and,depending on the relationship between xr,t�Dt, xu and x�, eitherEq. (9 c1) or Eq. (9 c2) is used. Otherwise, the linear reloading pathof Eq. (9 c3) or Eq. (9 c4) is applied.

Solving Eq. (9), its three variables: (the transient displacementxtr,t, the instantaneous stress-related displacement xr, t and theconcurrent spring force Fs,t) can be found. The same procedure isrepeated for each of the springs. Finally, the velocities of the twoDoFs _ut and _ht are calculated from Eqs. (1) and (2) for use in thenext time step.

2.1.3. Application of the Khoury & Terro concrete modelThe high-temperature concrete model given by Khoury [4] and

Terro [11,12] is also applied to the Shanley-like model. This con-crete model was suggested for Portland-cement-based concretesin general, irrespective of the type of aggregate or cement blendused. An update on the strain decomposition of this concrete mod-el has been given by Khoury [5], but no further development of themathematical models of the strain components is introduced inthis new version. Conversion of the stress–strain relationships toforce–displacement relationships is again carried out, using Eq.(5). The decomposition of the total strain is converted to the fol-lowing form:

x ¼ xth þ xel þ xtr;cr ð11Þ

The same model of the thermal displacement xth as used for theAnderberg & Thelandersson model is applied in this section forconvenience in comparing, in particular, the load-related proper-ties of the three concrete models. Taking a typical value Va = 65%of the aggregate content of concrete by volume, the transient creepdisplacement xtr,cr is:

xtr;cr ¼ L � f ðTÞ � 0:032þ 3:226Fs

Fu0

� �ð12Þ

where f(T) = �(A0 + A1T + A2T2 + A3T3 + A4T4) � 10�6; A0 = 43, A1 =�2.725, A2 = �6.248 � 10�2, A3 = 2.193 � 10�4 and A4 = �2.769 �10�7;Fu0 = ru0 � A; C30 concrete is used, so ru0 = 30 MPa.

The irrecoverability of xtr,cr is treated in similar fashion to theirrecoverability of xtr as formulated in Eq. (9b). Due to the simpli-fications introduced by Khoury & Terro, a relationship betweenthe spring force Fs and the total spring displacement x, as formu-lated in Eq. (13), can be achieved.

For loading stage (L), if Ffs P 0 and Ff

s P Fpfs :

Fs;t ¼ ðxt � xth � 0:032L � f ðTÞÞ 3:226L � f ðTÞFu0

þ 1k0

� ��ð13aÞ

Otherwise at loading stage (L):

Fs;t ¼ k0ðxt � xtr;cr;t�Dt � xthÞ ð13bÞ

For loading stage ðUL=RLÞ; if Fs;t�Dt P Ftu:

Fs;t ¼ kxt þ FUL;t � kxUL;t ð13cÞ

For loading stage ðUL=RLÞ; if Fs;t�Dt < Ftu:

Fs;t ¼ Ftu ð13dÞ

In terms of unloading and possible reloading (UL/RL), the refer-ence point (xUL,FUL) should be taken from the full force–displace-ment (Fs � x) curve, and unloading should be detected on thebasis of the total spring displacement x and its velocity _x, ratherthan the instantaneous stress-related displacement xr and its

S.-S. Huang, I.W. Burgess / Engineering Structures 44 (2012) 379–388 383

velocity _xr, as in the previous section. Therefore, the transitionfrom loading to unloading is detected if xt�Dt P xth þ 0:032L � f ðTÞ;_xt�2Dt P 0; _xt�Dt < 0 and xt�Dt > xUL;t�Dt .

The corresponding spring force FUL and displacement xUL arecalculated as:

FUL;t ¼ ðFs;t�2Dt þ Fs;t�DtÞ=2xUL;t ¼ ðxt�2Dt þ xt�DtÞ=2

ð14Þ

The initial value of the spring displacement x0 and the initial va-lue of the displacement xUL,0 at this transition are both equal toxth + 0.032L � f(T), given by Eq. (13a) when Fs = 0.

Another important issue is that, although the full force–dis-placement (Fs–x) curve is used, the stiffness of the linear unloadingpath should still be the initial stiffness k of the curve of forceagainst instantaneous stress-related displacement (Fs–xr), becausenone of the spring displacement components except the elastic dis-placement xel is recoverable. Using the initial stiffness of the Fs–xcurve would lead to an erroneous prediction of the spring displace-ment, especially for its irrecoverable components. The relationshipbetween r and er was not measured in the experiments carried outby Khoury [2,3], although a model was given by Khoury [4] on thebasis of Schneider’s [6] test results. Therefore, for simplification,the initial stiffness k of the Fs–xr curve is derived from the Young’smodulus E of the Schneider model, which is related to the load-his-tory factor a ¼ rhistory

ru0. The literature lacks a clear definition of rhistory,

and so a is simplified and converted to a ¼ Ffs

Fu0. The original model

only gives the formulation of E for a = 0, a = 0.1 and a P 0.3, and soit is assumed that linear interpolation determines E for intermedi-ate values of a.

The general calculation procedure is the same as that for theAnderberg & Thelandersson model. The only differences are thatEq. (13) is used to replace Eq. (9) and that the spring force Fs is di-rectly calculated from the concurrent spring displacement x. Fig. 5shows the process for detection of the loading stage, which is alsosimilar to that used in the previous section, except that it is doneon the basis of the total spring displacement x and velocity _x;and that there are only three loading stages.

2.1.4. Application of the Schneider concrete modelThe application of the concrete model given by Schneider et al.

[13] is described in this section. This model was developed forthree types (quartzite, limestone and lightweight) of structuralconcretes. Eq. (5) is again used to convert the stress–strain rela-tionships to force–displacement relationships. The decompositionof the total strain becomes:

x ¼ xth þ xm ð15Þ

Fig. 5. Detection of loading stage when Khoury & Terro concrete model is applied.

The same model of thermal displacement xth used in the previ-ous two sections is again applied in this section. The mechanicaldisplacement is given by:

xm ¼Fs

kð1þuÞ ð16Þ

where u is solely related to temperature for any given type of con-crete. Its value for quartzite concrete is used, for convenience incomparing the three models, since the Anderberg & Thelanderssonmodel is based solely on tests on quartzite-aggregate concrete. Asuggested value of moisture content w = 2% is adopted [7].

Although the recoverable elastic displacement xel is included inthe mechanical displacement xm, since the magnitude of xel is verysmall the irrecoverability of xm is accounted for in the same way asdescribed in the previous two sections. As in applying the Khoury &Terro model, this force–displacement relationship is also a full con-stitutive relationship between the spring force Fs and total springdisplacement x:

For loading stage ðLÞ; if Ffs P 0 and Ff

s P Fpfs :

Fs;t ¼kðxt � xthÞ

1þuð17aÞ

Otherwise at loading stage (L):

Fs;t ¼ Fs;t�Dt ð17bÞ

For loading stage ðUL=RLÞ; if Fs;t�Dt P Ftu:

Fs;t ¼ kxt þ FUL;t � kxUL;t ð17cÞ

For loading stage ðUL=RLÞ; if Fs;t�Dt < Ftu:

Fs;t ¼ Ftu ð17dÞ

Therefore, the loading stage is again detected on the basis of xand its velocity _x and hence the transition from loading to unload-ing is detected, if xt�Dt P xth; _xt�2Dt P 0; _xt�Dt < 0 and xt�Dt >

xUL;t�Dt . The concurrent spring force and displacement are calcu-lated as:

FUL;t ¼ ðFs;t�2Dt þ Fs;t�DtÞ=2xUL;t ¼ ðxt�2Dt þ xt�DtÞ=2

ð18Þ

The initial value of the spring displacement x0 and the initial va-lue of the displacement xUL,0 at this transition are both equal to xth.The stiffness of the linear unloading path should still be the initialstiffness k of the Fs � xr curve. The definition of k has been de-scribed in detail in the previous section. The general calculationprocedure is the same as that for the previous two models, exceptthat Eq. (17) is used to calculate the spring force Fs,t on the basis ofthe concurrent spring displacement xt. The loading stage, deter-mining which of the four formulae in Eq. (17) should be used, is de-tected in the same way as described in the previous section, asshown in Fig. 5.

2.2. Behaviour of the Shanley-like column model under steady-stateuniform heating

2.2.1. Results of analysis using the Anderberg & Thelandersson modelAn example model was created, whose dimensions were chosen

to ensure that the failure mode of the model at any given temper-ature is overall buckling in the inelastic range. The specification ofthe model is given in Table 1. A model with only four springs (twoon each side) is adopted for convenience in understanding the ana-lytical results at elevated temperatures. A convergence test wasdone on the size of the time increment Dt. Due to the nature ofthe calculation procedure itself this analysis is quite independentfrom the size of Dt as long as it is fine enough for the numericalprocess to proceed. When an oversized Dt is applied a numerical

Table 1Specification of the example model analysed.

L(mm)

B(mm)

n h0

(rad)Cv(Ns/mm)

Cr

(Nmms)Dt(s)

DP(N)

150 15 2 1E�12 2000 2000 1E�4 100

384 S.-S. Huang, I.W. Burgess / Engineering Structures 44 (2012) 379–388

instability ensues, but otherwise very little difference is made tothe results by varying Dt.

Figs. 6 and 7 plot selected structural responses of the model at500 �C. The results of the analysis are divided into two groups, oneincluding the transient straining of concrete and one without, inorder to investigate its influence on the mechanics of inelasticbuckling at high temperature. Fig. 6 shows the total spring dis-placement and its three components, for the two springs at theedges of the model, throughout the time steps of the successiveload steps. The force–displacement relationships Fs,ij � xr,ij of boththe outer and inner springs during the same period are plotted inFig. 7.

Some of the springs on the convex side start to unload when theapplied force and its consequent rotation are sufficiently large. Thisresults in differential instantaneous stress-related displacementxr,ij and differential transient displacement xtr,ij among the springs,because these are both load-related. This naturally causes differen-tial total spring displacement xij among the springs, as illustratedin Fig. 6. At failure, the difference in displacement between theconcave- and convex-side springs diverges.

The inclusion of transient strain causes the buckling load of themodel to decrease significantly compared with the case when thetransient displacement is set to zero. This corresponds to the muchlower spring-load levels and much smaller xr,ij shown in the ‘TS’parts of Figs. 6 and 7 than those shown in the ‘no TS’ parts. Fig. 7shows that without considering TS the force–displacement rela-tionships of the springs show evident nonlinearity, long before fail-ure occurs, whilst with the involvement of TS the spring forces areso low that the spring force–displacement curves still appear to belinear. Although, when TS is included, the displacements xr,ij aremuch smaller than when there is no TS, the total spring displace-mentsxij and the two DoFs u and h have magnitudes very similarto those resulting when TS is excluded, because they are comple-mented by the transient displacement xtr,ij. This may explain themuch lower buckling load when xtr is taken into account.

Plotting, at various temperatures, the ultimate buckling loads ofthe model, with and without considering TS, together with theidealised critical buckling loads Pe, Pr and Pt and the proportional

Fig. 6. Development of the total spring displacement and its three componen

limit Pp, gives Fig. 8. Without including TS, the buckling load ofthe model at an arbitrary temperature lies between Pt and Pr atthe same temperature. This is consistent with the statement inShanley’s theory [9] that the tangent-and reduced-modulus criticalbuckling loads are the lower and upper bounds of the inelasticbuckling loads, if Pt and Pr are calculated on the basis of theforce–displacement (Fs–xr) relationship given by Fig. 2, which doesnot include TS. The critical buckling loads are closer to Pt than to Pr,which results from the decreasing tangent modulus of the nonlin-ear force–displacement curve. A similar phenomenon is also de-scribed by Shanley for structural materials with reducing tangentmoduli.

The indication from the Shanley-like column model is that theinvolvement of TS (at least for the Anderberg & Thelandersson con-crete model) may cause a large reduction of the buckling resistanceof an intermediate-length concrete column at elevated tempera-tures. This result is revealing, because the effects of TS on the buck-ling of concrete columns are usually ignored in structural firemodelling, which may lead to unsafe prediction of the capacitiesof such elements. This is less surprising if the stress-dependenceof TS is considered. The study of the mechanics of inelastic buck-ling, an example of which (at 500 �C) is described above, indicatesthat, when a column starts to bend, the bending causes differentialstresses between its concave and convex faces. At elevated temper-atures this difference between the compressive stresses will causedifferential TS, which will cause a further difference of the strainsbetween the two sides, leading to further bending. Due to therather large magnitude of TS this iterative effect will be significant,and actually causes the column to lose stability at a much lowerload than when TS is absent.

2.2.2. Model comparisonsThe high-temperature concrete models given by Khoury & Terro

and Schneider were also implemented in the same four-springmodel used with the Anderberg & Thelandersson concrete model.Since the decomposition of the total strain is different in each ofthese models, it is not possible to compare each strain componentindividually, and so the comparisons shown in this section are allbased on total strain.

Fig. 9 shows the buckling loads of the multi-spring model over arange of temperatures, applying the three high-temperature con-crete models. It should be noted that the buckling loads have beennominalised against the critical buckling load at room temperaturewhen using the Anderberg & Thelandersson (A&T) model. Thisfigure also plots the high-temperature strength reduction factors

ts over time, with and without the consideration of transient straining.

Fig. 7. Compressive force–displacement curves of the springs under the steady-state heating scenario.

0

2000

4000

6000

8000

10000

0 200 400 600 800

Pcr

(N)

T (ºC)

PePrPtPpTSNo TS

Fig. 8. Buckling loads of the model with and without TS, compared with theoreticalbuckling loads and proportional limits at elevated temperatures.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 200 400 600 800

Pcr

/P20

,A&

T

T (ºC)

A&T model_ TSK&T modelSchneider modelA&T model_ No TS

Fig. 9. Comparison of various high-temperature concrete material models involv-ing TS on the buckling resistance of the multi-spring model.

S.-S. Huang, I.W. Burgess / Engineering Structures 44 (2012) 379–388 385

for normal-weight concrete given by Eurocode 2 [8], which tracksthe results given by adopting the A&T model without TS in thehigh-temperature and low-temperature regions. It is shown thatthe two curves resulting from the A&T and Khoury & Terro (K&T)models are very close to each other over the whole temperaturerange, whilst the curve from the Schneider model is quite separate

from them, except that it almost converges with them at temper-atures above 600 �C. Below this range, the A&T and K&T modelslead to lower predictions of buckling resistance than Schneider.The A&T model, which has the most refined strain decompositionamong the three models, proved suitable for use in the rest of thisresearch.

Although the strain definitions and formulations of these threemodels are completely different, they all show a reduction of buck-ling resistance due to the involvement of transient strain, at leastfor the majority of the temperature range, compared with the casein which A&T is applied without the TS component.

3. Transient heating scenario

When structures are subject to accidental fires they are usuallyalready carrying in-service loading. In multi-storey frames the col-umns in particular continue to carry these pre-loads throughoutthe fire, although the forces in the elements of flooring systemsmay change radically due to thermo-mechanical effects. Therefore,especially for columns, it is more rational to seek a failure temper-ature at constant load than to seek a failure load at constant tem-perature. At least these two approaches should be compared, inorder to decide whether it is necessary to involve the complexityinherent in transient heating. Therefore, the transient heating sce-nario is also modelled with the Shanley-like model.

3.1. Mathematical model and calculation procedure

3.1.1. Loading schemeThe loading–heating process is more complex than that used in

the steady-state heating approach. The imposed load is firstly ap-plied in steps at ambient temperature. Then, maintaining the loadconstant, the transient heating is applied step-by-step, similar tothe application of step loading. In this way, the buckling tempera-ture of the model at any given load level is assessed. Repeating thisprocedure for various load levels, a relationship between the buck-ling temperature of the model and the applied load level is found.It should also be noted that the assumption that the temperaturedistribution within the multi-spring model is uniform is still ap-plied in this analysis, although it would be unrealistic in a solidcross-section.

386 S.-S. Huang, I.W. Burgess / Engineering Structures 44 (2012) 379–388

3.1.2. Application of the Anderberg & Thelandersson concrete modelWhen applying the step loading at ambient temperature, the

calculation procedure within each load step is exactly the sameas in the steady-state heating approach. For further application ofthe heating steps, the same mathematical model is still adopted,except that the full formula (both Eqs. (7) and (8)) of the transientstrain etr may be used under this transient heating condition, andso Eq. (9b) is replaced by:

If Ffs P 0 and Ff

s P Fpfs then

xtr;t ¼ �ktrFs;tFu0

xth ð20 �C 6 T 6 500 �CÞ

xtr;t ¼ xtr;t�Dt þ L � 0:1� 10�3DT Fs;tFu0

� �ðT > 500 �CÞ

Otherwise

xtr;t ¼ xtr;t�Dt

8>>>>>>>>>>>><>>>>>>>>>>>>:

ð19Þ

Fig. 10 shows the process for discrimination between loadingand unloading. While applying load at ambient temperature, it isthe same as in the steady-state heating analysis. For further heat-ing steps, it remains the same, except during the initial time step ateach temperature. In this time step, unloading is assumed not totake place, and the change of the force–displacement relationshipwith temperature is accounted for by the spring displacement xrremaining the same as at the end of the previous temperature step.The spring force Fs at this unchanged xr is then re-calculatedaccording to the updated force–displacement relationship.

Fig. 10. Detection of the loading stage when the Anderberg & Thelanderssonconcrete model and transient heating scenario are applied.

3.2. Effects of transient heating on the behaviour of the Shanley-likecolumn model

The four-spring model used in the steady-state heating analysishas been re-evaluated for the transient heating scenario. The struc-tural response of the model subject first to step loading and then tostep heating is shown below, up to the critical temperature beyondwhich overall buckling of the model occurs. A typical load ratioLR = 0.6 (imposed force = 60% of the buckling load of the model atroom temperature), is applied. The results of the analysis are againdivided into two groups, depending on whether transient strain ofconcrete is taken into account. In the figures, the labels ‘TS’ and ‘NoTS’ are again used to distinguish these two groups of results. Fig. 11shows the displacements of the two outer springs throughout thetime steps during which loading and then temperature increase oc-cur. The force–displacement relationship Ff

s;ij—xfr;ij of each spring at

the end of each load/temperature step during the same period isplotted in Fig. 12.

For all the load/temperature steps except the last one, thespring displacements are always initiated in the first time step ateach load or temperature, and then re-stabilise after a certain per-iod of time, as shown in Fig. 11. Without considering transientstrain, a sudden increase of T and sudden decreases of xth,ij and xij

in the initial range (at temperatures from 20 �C to 100 �C) are ob-served. This results from the rapid re-stabilisation of the motionin each temperature step, since the force–displacement (Fs–xr)curves of the springs do not alter within this temperature range.Alteration of the thermal displacement xth,ij with temperature willnot cause any additional Fs,ij because xth,ij is not load-related. Thisphenomenon does not take place when TS is included, becausethe alteration of the transient displacement xtr,ij with temperaturerise causes an additional Fs,ij due to the load-dependence of xtr,ij.

Fig. 11. Development of the total spring displacement and its three componentsunder step loading followed by step heating.

Fig. 12. Compressive force–displacement curves of the springs under the transient heating scenario.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 200 400 600 800

P/P

20,A

&T

T (ºC)

Transient heating_ A&T model_ TSTransient heating_ A&T model_ No TSStatic-state heating_ A&T model_ TSStatic-state heating_ K&T modelStatic-state heating_ Schneider modelStatic-state heating_ A&T model_ No TS

Fig. 13. Comparison between the steady-state and transient heating scenarios onthe buckling resistance of the multi-spring model.

S.-S. Huang, I.W. Burgess / Engineering Structures 44 (2012) 379–388 387

The differential displacements across the springs, illustrated inFig. 11, result from the unloading of some of the convex-sidesprings, shown in Fig. 12. At failure, the difference of the spring dis-placements between the concave and convex sides diverges, whichcorresponds to a divergence of the model’s rotation. It should benoted that the spring displacements and forces, after significantdivergence of the rotation of the model has occurred, are excludedin the ‘TS’ parts of Figs. 11 and 12, in order to show their develop-ment before failure more clearly. The transient displacements xtr,ij

of the unloading springs do not decrease when both their xr,ij andxij decrease as the spring forces decrease. This is because the de-crease of xtr,ij between adjacent load/temperature steps is pre-vented in the numerical modelling, in order to account for theirrecoverability of TS.

Including transient strain in the analysis causes the critical tem-perature, at which buckling of the model occurs, to decrease signif-icantly from that when TS is not taken into account. This results inmuch lower absolute values of thermal displacements xth,ij in the‘TS’ part of Fig. 11 than those in the ‘no TS’ part of this figure. InFig. 12, the initial simultaneous gradients of the force–displace-ment curves are approximately linear, and are identical in boththe ‘No TS’ and ‘TS’ cases. They result from the step loading at roomtemperature, which does not induce significant rotation of themodel. After a heating step is imposed, without TS, the degradationof the material with temperature rise causes xr,ij to increase underalmost unchanged Fs,ij, until buckling occurs. However, the involve-ment of TS causes a considerable amount of rotation of the model,and evident differences of the spring forces and rotations happenat much lower temperatures than when TS is excluded. The springforce–displacement curves are still approximately linear just be-fore failure occurs, as seen in the ‘TS’ part of Fig. 12.

Plotting, for a range of load levels (nominalised against thecritical buckling load at 20 �C when using the A&T model), theultimate buckling temperatures of the models with and withoutTS, on the curves of Fig. 9, gives Fig. 13. In this figure, the circlesand triangles illustrate the results of the analysis in which a tran-sient heating scenario is applied. Comparing these with the resultsof the steady-state heating analysis shows that there is very little

difference between these two heating scenarios, suggesting thatthe steady-state heating approach is adequate when a uniformtemperature distribution within the model is assumed. However,the influence of the heating scenario is expected to be more signif-icant if a thermal gradient exists across the springs. In particular,the magnitude of the additional spring forces and displacementscaused by the thermal gradients could be significantly affectedby the loading and heating histories. Therefore, in further analysesin which non-uniform temperature distributions through columncross-sections are represented, the transient heating approachshould be applied for a better prediction of the spring forces anddisplacements on the basis of a more realistic loading–heatinghistory. It should also be noted that the use of the complementaryformula for xtr (Eq. (8)) using Anderberg & Thelanderssons’ con-crete model for temperatures above 500 �C leads to a lower predic-tion of the buckling resistance, and so the predicted buckling

388 S.-S. Huang, I.W. Burgess / Engineering Structures 44 (2012) 379–388

resistances based on the three material models are approximatelythe same beyond 600 �C.

4. Conclusions

Irrespective of the concrete material models and loading–heating schemes used, considering transient strain causes a con-siderable reduction of the buckling resistance of the model. Thissuggests that ignoring the effects of TS on the buckling of concretecolumns in structural modelling may lead to unsafe predictions ofthe capacities of such elements. The reduction of the bucklingresistance due to the inclusion of TS is induced by an interactiveprocess in which the bending of the column causes differentialstresses between its concave and convex sides, and in turn causesdifferential TS. This causes a further difference of strains betweenthe two sides, leading to further bending. This interactive effectis very significant, due to the large magnitude and irrecoverabilityof TS. It actually causes the column to lose stability at much lowerload/temperature levels in both the steady-state and transientheating analyses, than when TS is absent.

The comparison of the three concrete models in the steady-state heating analysis indicates that the A&T and K&T materialmodels give very close predictions of the failure loads of the modelat constant temperature, whilst the Schneider material model re-sults in a considerably higher prediction of critical load than theother two, at temperatures below 600 �C. Compared with the casein which the Anderberg & Thelandersson model is applied withoutTS, all three models show a reduction of the buckling resistancedue to the involvement of transient strain.

There is very little difference between the steady-state andtransient heating scenarios when a uniform temperature distribu-tion is assumed. However, when non-uniform temperature distri-butions through column cross-sections are simulated, thetransient heating approach is suggested for a more precise evalua-tion of the spring forces and displacements, on the basis of a morerealistic loading–heating history.

Acknowledgments

The principal author is grateful for the support of Corus GroupLtd. and the Engineering and Physical Sciences Research Councilof the United Kingdom, under a Dorothy Hodgkin PostgraduateAward.

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