Yield design-based analysis of high rise concrete walls subjected to fireloading conditions

Duc Toan Pham a,⇑, Patrick de Buhan b, Céline Florence a, Jean-Vivien Heck a, Hong Hai Nguyen a

a Université Paris-Est, Centre Scientifique et Technique du Bâtiment (CSTB), 84 avenue Jean Jaurès, Champs-sur-Marne, 77447 Marne-la-Vallée Cedex 2, Franceb Université Paris-Est, Laboratoire Navier (Ecole des Ponts ParisTech, IFSTTAR, CNRS UMR 8205), 6-8 avenue Blaise Pascal, Cité Descartes, Champs-sur-Marne, 77455Marne-la-Vallée Cedex 2, France

a r t i c l e i n f o

Article history:Received 6 June 2014Revised 13 January 2015Accepted 14 January 2015Available online 11 February 2015

Keywords:High rise wallsReinforced concreteFire loadingYield design approach

a b s t r a c t

Relying on a simplified one dimensional beam-like schematization of the problem, a yield design-basedapproach is developed for analyzing the potential failure of high rise walls (that are larger than thedimensions of experimental test furnaces) under fire conditions. The implementation of the method com-bines two original features: first, the preliminary determination of interaction diagrams reflecting thelocal decrease in strength of the wall due to thermal loading; second, the thermal-induced geometrychanges which are explicitly accounted for in the overall failure design of the wall. Application of theapproach is illustrated in either evaluating the fire resistance of a wall of given height or predictingthe maximum height that the wall could reach for a prescribed fire exposure time. First results of thisanalysis point to the conclusion that wall failure due to fire loading is highly sensitive to its height.

! 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Assessing the ultimate load bearing performance of reinforcedconcrete members subjected to fire exposure, and devisingappropriate design methods, have been the subject of an increasingnumber of contributions in the last decades (see among manyothers Lie and Celikkol [1], Lie and Irwin [2], Dotreppe et al. [3],Franssen and Dotreppe [4] or El Fitiany and Youssef [5]). Quiterecently, attention has been more specifically focused on the deter-mination of axial force-bending moment interaction diagrams of areinforced concrete section subjected to a fire induced temperaturegradient (Caldas et al. [6], Law and Gillie [7]). The yield designapproach in particular and its related lower and upper boundmethods (Chen [9], Salençon [10]) have proved to be a suitableframework for determining such interaction diagrams in a rigorousway, either under ambient temperature (Averbuch [11], Koechlinand Potapov [12]), or when subjected to a temperature gradient(Pham et al. [13]).

Increasingly involved in the construction of tall industrial build-ings, high rise concrete walls are large size reinforced concretestructures for which the evaluation of the fire resistance requiresa more sophisticated approach than for conventional, i.e. smaller

size structures. Indeed, the sole degradation of the stiffness andstrength properties of reinforced concrete due to severetemperature increase, cannot explain as such the collapse of thesestructures. Due to the thermal-induced deformations, such slenderstructures exhibit important out-of-plane (horizontal)displacements, which in turn lead to an eccentricity of the gravityload (self-weight) with respect to the initial undeformedconfiguration. As a consequence, bending moments are generatedin the wall in addition to the pre-existing compressive axial forcedistribution, which is usually known as a second order (or P-delta)effect (see for instance the classical textbook by Bazant and Cedolin[14]). As the eccentricity increases, the moment due to self-weighteccentricity also increases, thus subjecting the wall to higherbending moments and associated curvature deformations. At thesame time, but independently, elevated temperature leads to adegradation of constituent materials. Consequently, it is theconjunction of fire-induced material strength degradation withdeveloping bending effects which may trigger the overall failureof the structure, even before the occurrence of any bucklingphenomenon.

The purpose of the present contribution is to extend the range ofapplication of the yield design approach (Salençon [10]) in order toanalyze the global stability of high rise walls, taking the geometrychanges induced by the thermal loading into account. Thiscontribution will demonstrate how it is possible to combine theglobal deformed configuration analysis with that of the localcross-sectional strength degradation. Unlike most of the classical

http://dx.doi.org/10.1016/j.engstruct.2015.01.0220141-0296/! 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +33 1 61 44 81 89, +33 6 09 26 01 38; fax: +33 1 6468 85 23.

E-mail addresses: phamductoanvn@yahoo.com, ductoan.pham@cstb.fr(D.T. Pham).

Engineering Structures 87 (2015) 153–161

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approaches which are based on conventional limitations of thestrains undergone by the concrete material and reinforcing steel,the yield design approach only requires that stress (and not strain)limitations be assigned to the constituent materials in the form of astrength or failure criterion, with no consideration for other mate-rial properties, such as for instance the elastic stiffness characteris-tics. From a fundamental viewpoint, the yield design (or limitanalysis) reasoning is based on verifying the compatibility betweenstatic equilibrium of a structure subjected to given loading condi-tions in a defined geometric configuration, and strength conditionsexpressed through the above mentioned materials failure criteria,thus avoiding to perform computational time consuming incremen-tal calculations.

The main original feature of the present approach lies in the factthat the geometric configuration of the high rise wall on which theyield design approach is to be performed, is not known a priori, butshould be determined from a preliminary thermo-elastic calcula-tion accounting for geometrically non-linear second order effects.The paper is thus organised as follows. Section 2 describes the sim-plified one dimensional beam-column model adopted for the highrise wall, the corresponding problem statement and the three-stepprocedure employed for solving the latter. Section 3 is devoted to adetailed presentation of step n"1 of the procedure, consisting inpredicting the thermal-induced deflected configuration of thebeam-wall, by means of a non-linear thermo-elastic analysis.Section 4 essentially relies on the results of a previous paper (Phamet al. [13]) making it possible to evaluate the local decrease of thebearing capacity of reinforced concrete sections in the form oftemperature-dependent interaction curves (step n"2). Section 5 isconcerned with the implementation of the last step, aggregatingthe results of the two previous steps, resulting in a final illustrativeapplication of the whole procedure presented in Section 5.

2. Problem statement and outline of solution procedure

The problem under consideration is the potential instability orfailure of a high rise reinforced concrete wall subjected to itsown weight on the one hand and to fire exposure on the otherhand. In the following, the boundary conditions on the lateral sidesof the wall are prescribed in such a way that it can be simply mod-elled as a one dimensional vertical beam as sketched in Fig. 1.

The beamlike wall is uniformly exposed to fire on one side aswell as to its self-weight. Following the standard curve of temper-ature versus time advocated by design codes [15] for modelling theaction of a fire on a structure, a heat transfer analysis may be firstlycarried out on the wall. In the case of a simple wall member such asthat considered here, one-dimensional heat propagation across thewall thickness suggests that the field of temperature increase

resulting from such a thermal loading will depend on the thick-ness-coordinate only.

Fig. 1 provides a first insight into the basic mechanism whichmay explain why failure of a high rise wall under fire loadingmay occur. In its initial configuration, that is prior to fire loading,the wall is a straight vertical beam subjected to its own verticalweight, resulting in a linearly increasing distribution of axial com-pressive force N along the wall (Fig. 1(a)). The wall is generallydesigned so as to avoid any buckling phenomenon, while the max-imum compressive force at its base remains far below the com-pressive strength of the reinforced concrete section. As it will beexplained later on in more details, the transverse gradient of tem-perature due to fire exposure on one side of the wall will induce auniform thermal curvature of the beam and, as a direct conse-quence, out of plane transversal displacements will appear, leadingto a deformed configuration of the wall (Fig. 1(b)). Under such con-ditions, simple equilibrium considerations imply that any wallcross section is subjected to a significant bending moment M inaddition to the already existing axial compressive force N.

Apart from this first decisive phenomenon which could beattributed to an overall structural change of geometry (secondorder effect), experimental evidence clearly shows that the severetemperature increases associated with fire exposure, lead to animportant degradation of the stiffness as well as strength proper-ties of the reinforced concrete materials, namely plain concreteand steel reinforcements. It is the combined effect of these twophenomena (change of geometry on the one hand, decay of thematerial properties on the other hand) which may trigger the over-all failure of the high rise wall.

The calculation and design procedure proposed and developedin this paper is derived from the implementation of the above con-siderations and their formulations in a rigorous and mechanicallyconsistent framework. The analysis is performed in three main suc-cessive steps.

! Step n" 1. Determination of the wall deformed configuration andgeneralized stress distribution. This step consists first in evaluat-ing the equilibrium configuration of the wall under the com-bined action of thermal gradient and self-weight, then incalculating the resulting local solicitations (axial force andbending moment) in each section.! Step n" 2. Determination of temperature dependent interaction

diagrams. The objective of this phase, which is completelyindependent from the first one, is to determine the axial force-bending moment yield strength capacities of any wall cross-section as a function of the prescribed temperature gradient.! Step n" 3. Yield analysis and design of the wall in its deformed

configuration determined in step n"1, on account of its reducedstrength properties evaluated in step n"2.

The whole calculation procedure is sketched in Fig. 2.

3. Step n" 1. Thermal-induced equilibrium configuration

As mentioned earlier, the whole procedure is performed on asimplified one dimensional model of the wall, schematized as aninitially straight vertical beam of height H, articulated at both endsas shown in Fig. 3, the bottom end being kept fixed, while the topend is free to translate vertically. The wall is subjected to its self-weight characterized by a constant linear density w and a uniformtemperature gradient along its height, resulting in a preliminarydeformed shape of equation uh(x), where u denotes the transversaldisplacement. This thermal-induced change of geometry impliesan out-of-plane eccentricity of the self-weight and then additionalelastic bending deformations, resulting in a new equilibriumdeformed shape of equation u(x).

Fig. 1. Schematic diagram of a high rise wall subjected to fire loading: (a) initial and(b) deformed configurations.

154 D.T. Pham et al. / Engineering Structures 87 (2015) 153–161

Note that the high rise wall is not subjected to any other pre-scribed load than its self-weight, which means in particular that noload is applied on its top. Consequently, assuming that the rotationat any point of the deformed beam remains small enough, that is:

0 " x " H : du=dxj j ¼ ju0ðxÞj& 1 ð1Þ

the axial compressive force N at any point x is simply equal to theweight of the part of the beam of height H–x located above thispoint. Hence the resulting linear distribution:

NðxÞ ¼ wðx' HÞ ð2Þ

which remains constant whatever the subsequent horizontaldeflections u experienced by the wall, provided that condition (1)is satisfied.

3.1. Thermal deformed shape

Upon heating, the side of the wall exposed to fire and thus totemperature increase tends to expand much more than its unex-posed side, resulting in a global curvature of the wall which willbe first calculated as a function of the temperature increase acrossthe wall thickness h(y), 'h 6 y6+h, and the associated thermalstrain distribution eth(y) (see red1 dashed line in Fig. 4) equal to:

'h " y " þh : ethðyÞ ¼ aðyÞhðyÞ ð3Þ

where a(y) is the coefficient of thermal expansion of the material(concrete or reinforcing steel) located at level y in the wallthickness.

The kinematics of each plane cross-section of the wall located atpoint x, will be characterized by its mid-plane (y = 0) axial straineh(x) on the one hand, its curvature vh(x) on the other hand (seeFig. 4), so that the total strain linear distribution is:

eðx; yÞ ¼ ehðxÞ ' vhðxÞy ð4Þ

The thermo-elastic constitutive behavior of concrete subjectedto temperature increase must take the so-called Load InducedThermal Strain (LITS: see for example Law and Gillie [8]) intoaccount. This is achieved here by considering that the concreteYoung’s modulus is a decreasing function of the temperatureincrease, that is by applying a reduction factor to the modulus atambient temperature. Such a reduction factor, provided by Euro-code 2-Part 1-2 [18], is shown in Fig. 5, which also represent thereduction factor applicable to steel reinforcement.

Thus, a linear thermo-elastic behavior of the concrete materialcan be expressed by the following constitutive relationship:

ðrc ' r0c Þðx; yÞ ¼ EcðhðyÞÞ ehðxÞ ' vhðxÞy' eth

c ðyÞ! "

ð5Þ

where r0c ðx; yÞ denotes the initial (that is prior to thermal loading)

stress distribution and Ec(h(y)) is the concrete Young’s moduluswhich, as mentioned above, is dependent of the temperatureincrease h(y). Similarly, the axial force in the steel reinforcementn"k (k = 1 to K) located in the plane y = nk, may be expressed as:

ðns ' n0s Þðx; nkÞ ¼ AkEsðhðnkÞÞðehðxÞ ' vhðxÞnk ' eth

s ðnkÞÞ ð6Þ

in which n0s ðx; nkÞ is the initial axial force, Ak the reinforcing bar

cross sectional area and Es(h(nk)) the temperature dependent steelYoung’s modulus.

Now the stress and axial force increments appearing in the lefthand members of the constitutive relationships (5) and (6) areself-equilibrated, that is in equilibrium with zero resultant axialforce and bending moment:Z h

'hðrc ' r0

c ÞðyÞdyþX

k

ðns ' n0s ÞðnkÞ ¼ 0;

'Z h

'hyðrc ' r0

c ÞðyÞdy'X

k

nkðns ' n0s ÞðnkÞ ¼ 0 ð7Þ

Introducing Eqs. (5) and (6) into Eqs. (7) leads to the followingsolution:

eh ¼CD' BEAC ' B2 and vh ¼

BD' AEAC ' B2 ð8Þ

where A, B, C, D, and E are constants calculated as:

A ¼Z h

'hEcðhðyÞÞdyþ

X

k

EsðhðnkÞÞAk;

B ¼Z h

'hyEcðhðyÞÞdyþ

X

k

nkEsðhðnkÞÞAk;

C ¼Z h

'hy2EcðhðyÞÞdyþ

X

k

n2k EsðhðnkÞÞAk;

D ¼Z h

'hEcðhðyÞÞeth

c ðyÞdyþX

k

EsðhðnkÞÞeths ðnkÞAk;

E ¼Z h

'hyEcðhðyÞÞeth

c ðyÞdyþX

k

nkEsðhðnkÞÞeths ðnkÞAk ð9Þ

Therefore, the resulting axial strain and curvature are constantalong the wall height, and it can be easily verified that D = E = 0 andthen eh = vh = 0 when eth(y) = h(y) = 0.

Step

n°1 Step

n° 2

Thermal gradient

Thermal curvature

Equilibriumdeformed

configura!on

(N, M)distribu!ons

Reducedinterac!on

diagrams

Materialstrength

degrada!on

Step n°3Yield design assessment

Heat transfer analysis

Fig. 2. A three-step evaluation procedure for the yield design of fire loaded high risevertical walls.

Fig. 3. Schematic diagram of thermal and equilibrium deformed shapes of the wallunder fire loading.

1 For interpretation of color in Figs. 4, 12 and 13 the reader is referred to the webversion of this article.

D.T. Pham et al. / Engineering Structures 87 (2015) 153–161 155

It follows that the thermal deformed shape of the wall is char-acterized by the out-of-plane displacement uh(x) such that:

d2uh

dx2 ¼ vh ð10Þ

Integrating twice the above differential equation immediatelyyields, on account of the boundary conditions uh(x = 0)=uh(x = H) = 0:

uhðxÞ ¼vh

2xðx' HÞ ¼ 1

2qh

xðx' HÞ ð11Þ

where qh represents the constant radius of curvature of the wall.The condition (1) of small rotation at any point of the beam issatisfied as far as qh) H.

3.2. Total deformed shape as a result of second order effect

The thermal-induced change of geometry characterized by theparabolic Eq. (11), entails an eccentricity of the self-weight w withrespect to the initial vertical plane and thus a bending moment dis-tribution Mh(x). Owing to the fact that the wall, modelled as acurved beam, is statically determinate, the bending moment distri-bution can be straightforwardly calculated as follows:

MhðxÞ ¼ wx' H

H

Z H

0uhðsÞdsþ

Z H

xuhðsÞ ' uhðxÞ½ +ds

# $ð12Þ

or on account of (11):

MhðxÞ ¼w

12qh

xðH ' xÞð5H ' 4xÞ ð13Þ

In order to calculate the additional bending deformations asso-ciated with this moment distribution, the same kind of reasoningas that previously used for evaluating the thermal-induced defor-mations of the wall will be performed. Denoting by DN(x) andDM(x) the increments of solicitations applied to any section ofthe wall located at point x, and by De(x) and Dv(x) the correspond-ing elastic response in terms of mid-plane axial strain and curva-ture increments at this same section, we can write the followingconstitutive relations similar to (5) and (6):

Drcðx; yÞ ¼ EcðhðyÞÞ DeðxÞ ' DvðxÞyð ÞDnsðx; nkÞ ¼ AkEsðhðnkÞÞðDeðxÞ ' DvðxÞnkÞ

ð14Þ

which may be incorporated into the equilibrium conditions:Z h

'hDrcðyÞdyþ

X

k

DnsðnkÞ ¼ DN;

'Z h

'hyDrcðyÞdy'

X

k

nkDnsðnkÞ ¼ DM ð15Þ

leading to the following linear relations:

DNDM

% &¼

A 'B'B C

% &DeDv

% &ð16Þ

where the expressions of coefficients A, B, and C are given by (9).Since in the present situation DN(x) = 0 and DM(x) = Mh(x), wefinally obtain:

DvðxÞ ¼ d2

dx2 ½DuðxÞ+ ¼ MhðxÞðEIÞh

ð17Þ

where

ðEIÞh ¼ C ' B2=A ð18Þ

the wall section flexural stiffness under pure bending. Relations (9)show that in the particular case of a homogeneous concrete sectionreinforced by two symmetrically placed steel bars (y = ± n) at ambi-ent temperature, B = 0 and the expression of this coefficient simpli-fies to:

ðEIÞh¼20,C ¼ C ¼ 2h3

3Ec þ n2AsEs

!ð19Þ

3.3. A simple iterative procedure

The integration of Eq. (17), on account of (13) along with theboundary conditions, gives:

Fig. 4. Strain and stress distributions in a reinforced concrete wall section due to the application of a fire-induced temperature gradient.

0

0.2

0.4

0.6

0.8

1

0 300 600 900 1200

Dec

reas

e of

You

ng's

mod

ulus

Temperature θ (°C)

Concrete

Steel

Fig. 5. Reduction factors of concrete and steel reinforcement Young’s modulus asfunctions of temperature increase (Eurocode 2-Part 1-2).

156 D.T. Pham et al. / Engineering Structures 87 (2015) 153–161

u1ðxÞ ¼ uhðxÞ þ DuðxÞ ð20Þ

with:

DuðxÞ ¼ w720qhðEIÞh

xðx' HÞ 12x3 ' 33Hx2 þ 17H2ðx' HÞh i

ð21Þ

corresponding to the following updated expression for the bendingmoment distribution calculated through Eq. (12) where uh has to bereplaced by u1:

M1ðxÞ¼MhðxÞþw2

1440qhðEIÞhxðx'HÞ½20x4'76Hx3þ89H2x2'11H3x'28H4+

ð22Þ

The same iterative procedure can be applied once again by solv-ing the second order differential equation:

M1ðxÞ ¼ ðEIÞhd2

dx2 ½u2ðxÞ ' uhðxÞ+ ð23Þ

resulting in a new polynomial expression u2, of degree 8, for thewall deformed shape. This procedure is carried out until conver-gence of the sequence of computed transversal displacements isobserved:uðxÞ ¼ lim

i!þ1uiðxÞ and MðxÞ ¼ lim

i!þ1MiðxÞ ð24Þ

The corresponding distributions of axial force NðxÞ ¼ wðx' HÞand bending moment M(x) may be represented in a form of a curvedrawn in the (N, M) plane as shown in Fig. 3.

Such an iterative procedure, which could appear time consum-ing, actually remains quite simple, since only easily tractable poly-nomial forms are involved for both the total deformed shape andbending moment distribution. As seen for example in Fig. 6, thedeformed shapes corresponding to the second and the third itera-tions are almost coincident with the converged solution:u2ðxÞ ffi u3ðxÞ ffi uðxÞ ¼ lim

n!þ1unðxÞ ð25Þ

Furthermore, as it can be clearly seen from Fig. 7, the differencebetween the initial thermal-induced deformation uh and the fullyequilibrated configuration u is becoming significant for high risewalls only (namely H > 8 m, for a wall thickness of 0.15 m). Thedeformed shape and corresponding out-of-plane displacementsof the wall due to thermal loading are strongly increasing withthe height. While for the 6 m-high wall, the additional horizontaldisplacement due to further elastic bending associated to theself-weight eccentricity, remains negligible, the response is quitedifferent for the 10-m and 12-m high walls. Fig. 7 for instanceshows that the maximum deflection of the 12-m high wall is morethan five times larger than that of the 6-m wall (1.10 m instead of0.20 m).

3.4. Comparison with results of analytical model and numericalsimulations

As an alternative to the above described iterative procedure, a(semi)analytical closed-form solution to the above problem canbe worked out directly. Indeed, expressing the equilibrium onthe (unknown) final deformed configuration, one gets the follow-ing integral equation:

MðxÞ ¼ wx' H

H

Z H

0uðsÞdsþ

Z H

x½uðsÞ ' uðxÞ+ds

# $ð26Þ

which, in conjunction with the thermo-elastic constitutiverelationship:

MðxÞ ¼ ðEIÞhd2

dx2 ½uðxÞ ' uhðxÞ+ ð27Þ

and after derivation with respect to x, leads to the following thirdorder differential equation:

d3

dx3 uðxÞ þ wðEIÞh

ðH ' xÞ ddx

uðxÞ ¼ c ð28Þ

where c is a constant to be determined.The exact solution to the latter equation can be found making

use of hypergeometric functions as proposed for instance by Sam-paio and Hundhausen [16] for analyzing the buckling of a beam-column subjected to its own weight.

In addition, a numerical simulation of the same problem hasbeen carried out for comparison purposes, with the help of a finiteelement software [17], able to take large displacements, and hencegeometry changes, into account in the analysis of a structure suchas a high rise wall under fire loading. Fig. 8 clearly demonstratesthat the three calculation methods (iterative procedure, analyticaland numerical models) produce results which are in excellentagreement with each other.

0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1 1.2

Height(m)

Displacement (m)

)( )(32 xu xu≅

)(1 xu)(xuθ

Fig. 6. Sequence of wall deformed configurations obtained from the iterativeprocedure (note that the abscissa scale is strongly dilated).

0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1 1.2

Height(m)

Displacement (m)

)(xu)(xuθ

Fig. 7. Initial thermal-induced and final equilibrated deformed configurations of awall as a function of height.

D.T. Pham et al. / Engineering Structures 87 (2015) 153–161 157

4. Step n" 2: determination of temperature dependentinteraction diagrams (Pham et al. [13])

As mentioned earlier, independently of the change of geometryand related development of bending moment distributions alongthe wall, the fire-induced temperature increase has a quite signif-icant impact in terms of material properties degradation. Basedupon experimental evidence, empirical relationships have thusbeen established linking the reduction of material stiffness andstrength properties to the temperature increase [18]. This will haveobvious consequences on the wall bending stiffness, as alreadypointed out in the previous section; but still more importantlyon its resistance usually expressed in terms of axial force-bendingmoment interaction diagrams.

A companion paper (Pham et al. [13]) has been specifically ded-icated to the determination of such axial force-bending momentinteraction diagrams of a reinforced concrete section whensubjected to a fire-induced temperature increase, through astraightforward computational procedure based upon the yielddesign (or limit analysis) theory. The key input data of the analysiswas the introduction of the above referred experimentally-basedrelationships between the degradation of local material strengthproperties and the fire-induced temperature increase. The interac-tion diagram associated with a prescribed temperature profileacross the wall thickness has been thus exactly determined fromimplementing both the lower bound static and upper bound kine-matic methods. The key points and main results of the analysisare recalled hereafter.

4.1. Solving an auxiliary yield design problem

With reference to an orthonormal coordinate system Oxyz(Fig. 9), the interaction diagrams are determined from the solutionto a yield design problem defined on a reinforced concrete wall sec-tion modelled as a square parallelepiped of thickness 2h along theOy-axis, and lengths conventionally taken equal to unity along theOx and Oz-axes (see Pham et al. [13], for more details). It consists ofhomogeneous concrete reinforced by longitudinal steel bars placedalong the Ox-direction. Such a reinforced concrete beam is sub-jected to mechanical loading conditions defined as follows.

! body forces (weight) are taken equal to zero;! the left hand section (x = 0) is in smooth contact with a fixed

and rigid vertical punch, while the right hand section (x = 1) isin smooth contact with a rigid punch in horizontal translationof velocity at mid-plane level (y = 0) and rotation of angularvelocity about the z-axis;! the four remaining sides of the wall section are stress free.

Considering any kinematically admissible (K.A.) velocity field U,defined as complying with the velocity boundary conditionsdepending on _d and _a, the work of the external forces in any suchfield, may be put in the following form:

WeðUÞ ¼ N _dþM _a ð29Þ

where N and M may be interpreted as the axial force along Ox andbending moment about Oz exerted on the right hand side of the wallsection, which can easily be calculated from the stress distributionacross the wall thickness. The representative wall section is thussubjected to a two-parameter loading mode.

According to the yield design reasoning (see among others thequite recent textbook by Salençon [19]), the so-called domain Kof potentially safe loads (N, M), is defined as the set of loads whichcan be equilibrated by a stress distribution in the beam (stress ten-sor fields in the concrete, tensile force distributions along the rein-forcements), verifying the respective strength limitations ofconcrete and steel at any point of the wall. The boundary of thisdomain, locus of the extreme or failure loads, is called the interac-tion diagram of the reinforced concrete wall section subjected tocombined axial and bending loadings.

The strength properties of the constituent materials are speci-fied as follows:

(a) The plain concrete material obeys a truncated Mohr-Coulombstrength criterion characterized by its uniaxial tensile andcompressive resistances denoted by ft and fc, respectively.According to experimentally established relationships, suchcharacteristics take the form:

f tðyÞ ¼ ktðhðyÞÞf t ; f cðyÞ ¼ kcðhðyÞÞf c ð30Þ

where kt and kc are non-dimensional coefficients decreasingwith temperature h, and thus depending on the y-coordinate,conventionally equal to one for h = 20 "C (ambienttemperature).

(b) Likewise, any longitudinal reinforcing bar n"k of cross-section Ak, located at y = nk, is made of a steel of uniaxialtensile-compressive yield strength equal to fy, so that theoverall strength of the bar under tensile or compressive axialforce may be expressed as:

n0ðnkÞ ¼ ksðhðnkÞÞf yAk ð31Þ

where ks is a non-dimensional decreasing function of temper-ature increase.

0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1 1.2

Height(m)

Displacement (m)

Fig. 8. Comparison of predictions from iterative, analytical and numerical methodson the problem of the stability of a high rise wall under combined self-weight andthermal loading.

Fig. 9. Auxiliary yield design problem: reinforced concrete wall section subjected tothe combined action of axial force and bending moment.

158 D.T. Pham et al. / Engineering Structures 87 (2015) 153–161

4.2. Interaction diagram as a two-parameter yield surface

Given any value e, comprised between 'h and +h, the twofollowing uniaxial stress fields in concrete, along with axial forcesin the reinforcements are considered (see Fig. 10). It simply meansthat in configuration (a) both concrete and reinforcing bars reachtheir positive tensile (resp. negative compressive) strengths whenlocated below (resp. above) the plane of equation y = e. Theopposite applies in configuration (b).

It can be immediately observed that the above stress distribu-tions, depending on the eccentricity parameter e, automaticallysatisfy the respective strength conditions of the concrete materialand steel reinforcing bars. Besides, they satisfy the equilibriumequations in the absence of body forces, while the discontinuityof stress when crossing the y = e plane remains admissible. The cor-responding values of the loading parameters in equilibrium withsuch stress distributions may be easily calculated, leading to thedetermination of a yield surface in the (N, M)-plane, which repre-sents a lower bound to the interaction diagram. As explained inmuch more details in (Pham et al. [13]) this lower bound solutioncould be recovered from implementing the upper bound kinematicapproach of yield design. It therefore describes the exact failurecurve (interaction diagram) in the (N, M)-plane. It should beemphasized that the solution thus obtained has also been favour-ably compared with Eurocode-based predictions in the case ofambient temperature, as well as with available experimentalresults (Pham et al. [13]).

A final, but important, comment should be made regarding thepossible role played by thermal stresses in the determination ofthe interaction diagram. Apart from deteriorating the materialproperties of the reinforced concrete components, fire loading con-ditions imply quite significant changes of the initial state of thestructure due to the generation of thermo-mechanical stresses. Itis clear however, that such stress fields are self-equilibrated (thatis in equilibrium with N = M = 0), having as a consequence noinfluence on the values of the limit loads and thus of the interactiondiagram. Indeed, it is a well-known result of associated elastoplas-ticity and related limit analysis theorems (see among many otherreferences Salençon [20] or de Buhan [21]), that such limit or failureloads are independent of the elastic characteristics, loading pathfollowed up to them or, still more decisively, initial state of stressof the structure, due for instance to a previously applied thermalloading. As a consequence, the sole influence to be expected fromfire loading on the interaction diagram of the reinforced concretewall section must be attributed to the degradation of the concreteand steel strength properties.

5. Step n" 3: stability analysis of the wall in its deformedconfiguration

5.1. Principle of the method

On account of the knowledge of the local solicitations (N,M)(x) along the wall determined from step n"1 on the one hand,and strength capacities (interaction diagrams) of the heatedcross-section calculated in step n"2 on the other hand, theprocess could now be completed. Assessing the stability of thestructure (in the sense of yield design) simply consists incomparing the combined bending moments-axial forces distribu-tions resulting from the equilibrium of the wall in its deformedconfiguration, with the interaction diagram modified by theapplication of thermal loading.

More precisely, for a given height and a given fire exposure, thestability of a wall is ensured as far as the curve representing thesolicitation distribution along the wall height remains entirelyinside the strength domain delimited by the interaction diagramin the (N, M)-plane. Collapse occurs at the section where the curveof solicitation distribution becomes tangent to the interaction dia-gram as shown in Fig. 11. It should be noted that the failure of onesection, as considered in this approach, implies the complete fail-ure of the entire wall. This is due to the fact that the consideredwall-beam structure is statically determinate, the first yield pointand ultimate limit load being coincident.

5.2. Implementing the method on an illustrative example

For illustrative purpose, the approach is now implemented onthe problem stated above. The following example will help clarifyand quantify the negative effect of high temperature increase onthe stability of such slender structures in two different ways: thedegradation of the wall strength capacities expressed through thereduced interaction diagram on the one hand, the thermal-inducedout-of-plane change of geometry of the wall which generatesbending moments in addition to compressive loads on the otherhand.

Assuming that the wall is exposed to an ISO 834 fire [15] on itsright hand face with different time durations (60, 90 and 120 min),the following set of quite representative data has been selected:

! Rectangular cross-section 0.15 . 1 m2.! Normal weight concrete with siliceous aggregates exhibiting

the following strength characteristics at ambient temperature(20 "C): fc = 32 MPa, ft = 2.5 MPa.

Fig. 10. Stress profiles in the wall cross-section used in the lower bound static approach of yield design.

D.T. Pham et al. / Engineering Structures 87 (2015) 153–161 159

! Two symmetrically placed layers of 10 hot rolled steel reinforc-ing bars of diameter 6 mm with 3 cm of concrete cover at topand bottom: fy = 500 MPa.! Material properties are considered to be temperature depen-

dent according to experimental curves provided by Eurocode2-Part 1-2 [18].! Constant vertical self-weight density of w = 3.5 kN/m.

A preliminary heat transfer analysis, aimed at evaluating thetemperature increase distribution across the wall thickness, shouldbe first conducted. Fig. 12 displays the temperature profiles acrossthe wall thickness obtained for instance by the SAFIR computer pro-gram [22], corresponding to 0, 60, 90 and 120 min fire durations.

Introducing these thermal gradients into the step n"2 calcula-tion procedure presented in Section 4, the corresponding interac-tion diagrams could be determined as shown in Fig. 13. It thusclearly appears from the latter figure, that temperature increaseaffects the strength properties of the reinforced concrete section,in the form of a quite significant reduction of the strength domain.The fire loading leads to an increase of temperature (Fig. 12),resulting in a decrease of material strength parameters associatedwith a ‘‘shrinkage’’ of the interaction diagram (Fig. 13), and thus toa much smaller global resistance of the reinforced concrete wall.

Focussing on a given fire exposure, say 120 min, for which theinteraction diagram (represented by the closed red curves ofFigs. 12 and 13) remains constant, our objective will be first todetermine from which height failure of the wall will occur.

As already observed for instance in Fig. 7, the higher the wall,the larger is the relative difference between the thermal deformedshape and the total deformed shape. As a practical consequence,such a second order effect has no significant influence on the globalfire stability of normal height walls (apart from the strength reduc-tion which remains the same), but becomes much more critical forhigh rise walls. This decisive point is further illustrated in Fig. 14where, for clarity purpose, the region of the (N, M)-plane close tothe origin has been magnified. It may be observed from this figurethat the solicitation curve intersects the interaction diagram assoon as the wall becomes higher than 12 m. More precisely, itmay be shown that collapse occurs for a wall of about 11.4 m-high(corresponding solicitation curve becoming tangent to the interac-tion diagram), whereas it still remains well below the interactiondiagram for a 10-m high wall (purple dashed line).

Conversely, for a 13-m high wall, the minimum fire duration forwhich collapse occurs can also be estimated from the abovedescribed procedure.

In such a case, as shown in Fig. 15, the increase of fire exposureduration results in a lowering of the interaction diagram, simulta-neously to an expansion of the solicitation curve, due to the change

Fig. 11. Principle of the stability analysis of the wall in its deformed configuration.

0

3

6

9

12

15

0 300 600 900 1200

Thickness(cm)

θ (°C)

min120

min90min60

min0

t

Fig. 12. Calculated temperature profiles across the wall thickness for different fireexposures.

-0.12

-0.06

0

0.06

0.12

-5.5 -3.5 -1.5 0.5

M(MNm/m)

N (MN/m)

t↑min60

min0

min90

min120

Fig. 13. Evolution of the interaction diagrams as a function of fire exposure.

-0.06

-0.03

0

0.03

0.06

0.09

0.12

-4 -3 -2 -1 0 1

M( MNm)

N (MN)

0

0.03

-0.05 0

1 m0

1 m2

m6

Fig. 14. Yield design analysis of walls of different heights exposed to a given fireexposure of 120 min.

160 D.T. Pham et al. / Engineering Structures 87 (2015) 153–161

of geometry. Failure occurs when the two corresponding curvescome into contact. In the present case Fig. 15 indicates that thewall of height 13 m will not resist more than 90 min in fire, thecritical section being located at about 5 m from the bottom ofthe wall.

6. Concluding remarks

Based on a simplified one-dimensional modelling of the prob-lem, this paper has proposed and developed a rigorous, compre-hensive and consistent approach for evaluating the fire resistanceof high rise concrete walls. The method is based on the applicationof the yield design theory at two stages: first as regards the deter-mination of the wall strength capacities expressed in the form ofan interaction diagram parameterized by the thermal loading; thenas concerns the analysis of the load-bearing capacity of the wall inits fire-induced deformed configuration. The entire procedure hasbeen implemented on an illustrative case study illustrating eachstep of the evaluation process, making it possible to highlight bothfire effects on the individual members’ strength properties and sec-ond order effects due to thermal-induced geometry changes.Among the first most important conclusions that can be drawnfrom the results of this case study, is the fact that the failure ofthe wall is extremely sensitive to an even relatively slight increaseof its height.

Due to its simplicity, mainly attributable to the (semi)analyticalformulations involved in the analysis, the model and relatedcalculation procedure are particularly well suited for a preliminaryengineering design of high rise wall in fire, providing useful guide-lines in the crucial matter of structural safety assessment. It allowsperforming parametric studies in a rather quick way, without itbeing necessary to resort to complex numerical simulations.Furthermore the procedure could easily be generalized to morecomplex thermal loading conditions, due for instance to the factthat the wall is not uniformly exposed to fire along its whole height.

Of course, the main limitation of the present approach lies in itsinability to still provide reliable predictions as soon as the simpli-fied one-dimensional beam model adopted in this contribution, isno more valid. This is for instance the case of rectangular rein-forced concrete panels, the lateral vertical sides of which are sim-ply supported, thus preventing out-of-plane displacements alongthe wall four edges. The structure is then to be adequately schema-tized as a two-dimensional plate (or slab), but the general principleof the procedure described in this paper remains exactly the same.

To that end, developments are currently under way concerningthe formulation of a strength criterion of the plate expressed interms of in-plane membrane forces and bending moments, therebyextending the usual concept of interaction diagrams. Furthermore,owing to the fact that, unlike for the simple beam model, the struc-ture is no longer statically determinate, the questions of thermal-induced change of geometry and yield design assessment in thedeformed configuration have to be considered in a somewhat morecomplex framework.

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0

0.01

0.02

0.03

0.04

-0.06 -0.04 -0.02 0

M(MNm)

N (MN)

09 nim

06 nim

021 nim

wH−

Fig. 15. Yield design-based analysis of a 13 m-high wall.

D.T. Pham et al. / Engineering Structures 87 (2015) 153–161 161