ON THE DISTORTIONAL BUCKLING, POST-BUCKLING AND STRENGTH OF

COLD-FORMED STEEL LIPPED CHANNEL COLUMNS UNDER FIRE CONDITIONS

Alexandre Landesmann*a

& Dinar Camotimb

aCivil Eng. Program, Federal University of Rio de Janeiro (COPPE/UFRJ), Brazil

bDepartment of Civil Engineering and Architecture, ICIST/IST, Technical University of Lisbon, Portugal

*Corresponding author − E-mail: alandes@coc.ufrj.br

Abstract: This paper reports the results of an ongoing shell finite element investigation on the distortional

post-buckling behaviour, ultimate strength and design of cold-formed steel lipped channel columns (centrally

compressed members) subjected to high temperatures typically caused by fire conditions. Two column collapse

situations are dealt with, corresponding to different loading strategies: (i) application of an increasing

compressive load to columns subjected to a constant (uniform) temperature distribution, in order to obtain

failure loads, and (ii) application of a progressive temperature raise to axially compressed column, in order to

obtain failure temperatures – the latter approach provides a more realistic simulation of fire conditions. The

steel material behaviour at high temperatures is described by the constitutive model prescribed in Eurocode 3

for cold-formed steel. After validating the numerical model adopted, through the comparison with results of

simulations reported in the literature and based on experimentally obtained stress-strain laws, the paper

presents numerical results concerning lipped channel columns made of various steel grades under fire

conditions – they consist of (i) non-linear equilibrium paths, yielded by steady state and transient column

analyses, and (ii) the corresponding failure loads/stresses and temperatures.

Keywords: Fire design, cold-formed steel, lipped channel columns, ultimate strength, distortional buckling,

distortional post-buckling, distortional failure.

1. INTRODUCTION

It is well-known that the use of cold-formed steel profiles in the construction of the structural frames

for either industrial (predominantly) or residential buildings has become increasingly popular in the

last few years, mainly due to their high structural efficiency (high strength-to-weight ratio), remarkable

fabrication versatility and increasingly low production and erection costs.

The knowledge about the structural behaviour of cold-formed steel members at room temperature has

advanced considerably in the last few years and, moreover, such advances have been incorporated in

design specifications at a fairly rapid rate. However, the same does not hold true for cold-formed steel

members subjected to elevated temperatures, namely those caused by fire. Indeed, it is fair to say that

the research activity devoted to cold-formed steel members under fire conditions was only initiated in

this century and is still rather scarce, as attested by the quite small number of available publications

(most of the existing information concerns hot-rolled members). Among them, the works of Outinen et

al. [1], Kaitila [2], Feng et al. [3,4], Lee et al. [5], Feng & Wang [6], Chen & Young [7,8,9,10] and

Ranawaka & Mahendran [11,12] deserve to be specially mentioned.

In addition, only the last two of the above studies address members affected by distortional buckling, an

instability phenomenon that often governs the behaviour and strength of lipped members. Therefore, it is

quite important to assess the structural response of such members under elevated temperatures. The aim of this

paper is to report the results of an ongoing shell finite element investigation on the distortional post-buckling

behaviour, ultimate strength and design of cold-formed steel lipped channel section columns (centrally

compressed members) subjected to elevated temperatures typically caused by fire conditions. Two column

collapse situations are considered, corresponding to two different loading strategies: (i) application of an

increasing compressive load to columns subjected to a constant (uniform) temperature distribution, in order to

obtain failure loads) and (ii) application of a gradual temperature raise to axially compressed columns, in order

to obtain failure temperatures − the latter approach provides a more realistic simulation of actual fire conditions.

The steel material behaviour at elevated temperatures is simulated by the model prescribed in

Eurocode 3 (part 1.2) (EC3-1.2 [13]) for cold-formed steel. Other stress-strain laws, based on

experimental investigations reported by Chen & Young [8,9,10] and Ranawaka & Mahendran [12],

are used exclusively in the context of validations studies.

The column post-buckling behaviour and ultimate strength are determined through geometrically and

materially non-linear shell finite element analyses carried out in the code ANSYS [14], incorporating critical-

mode (distortional) initial geometrical imperfections with small amplitudes. After validating the numerical

model adopted, through comparisons with shell finite element results reported by Chen & Young [8] and

Ranawaka [15], the paper presents a numerical investigation concerning the distortional post-buckling

behaviour and ultimate strength of lipped channel columns made of various steel grades and subjected to

fire conditions, involving temperatures ranging from 20°C (“room temperature”) to 600°C. The numerical

results presented and discussed consist of non-linear equilibrium paths yielded by steady state and transient

column analyses, which make it possible to obtain failure loads/stresses and failure temperatures,

respectively. Moreover, the relation existing between the data and solutions of the steady state and transient

analyses is addressed.

2 NUMERICAL ANALYSIS

2.1 Shell Finite Element Modelling

Column Discretization. The columns were discretized into fine meshes of SHELL181 elements

(ANSYS nomenclature – 4-node thin-shell elements with six d.o.f. per node and full integration, and

accounting for shear deformation). Convergence studies showed that 6.25 mm × 6.25mm (length ×

width) element meshes provide accurate results with a moderate computational effort.

End Support Conditions. Columns with two end support conditions are analyzed: (i) locally and globally

pinned end sections that can warp freely and (ii) locally and globally fixed end sections with warping

prevented − the latter were considered only in order to replicate the numerical results reported by Chen &

Young [8] and Ranawaka [15]. In the pinned case, the transverse displacements of all end cross-section

nodes were prevented, while keeping the axial (warping) displacements and all the rotations free. In the

fixed case, rigid plates were attached to the end cross-sections, thus precluding the occurrence of local

displacements/rotations and warping. Then, all rigid plate translations and rotations were prevented, with the

exception of the axial translations (associated with the load application).

Loading. The column axial compression is applied by means of either (i) sets of concentrated forces acting on

both end section nodes (pinned columns) or (ii) two concentrated forces applied on the rigid plates points

corresponding to the end section centroids. The above forces are applied in small increments, by means of the

ANSYS automatic load stepping procedure1.

Initial Geometrical Imperfections. The initial geometrical imperfections incorporated in the columns freshly

analyzed in this work exhibit the critical (distortional) buckling mode shape and an amplitude equal to 10% of

the wall thickness t, a value also adopted by other researchers (e.g., Dinis & Camotim [16]). In the validation

studies, involving column analyses by Chen & Young [8] and Ranawaka [15], the initial geometrical

imperfections reported by those researchers were considered: again the critical buckling mode shape, but now

an amplitude equal to either the wall thickness t or measured values.

Each critical buckling mode shape was determined by means of a linear stability analysis, also

performed in ANSYS and adopting exactly the same shell finite element mesh employed to carry out

the subsequent non-linear (post-buckling) analysis − this approach makes it very easy to “transform”

the output of the linear stability analysis into an input of the non-linear one.

1 Obviously, this loading description concerns only the steady state analyses (columns subjected to an increasing axial

compression). The application of the rising temperature required to perform the transient analyses is addressed ahead in

the paper − sub-section “Types of Non-Linear Analysis”.

2.2 Steel Material Behaviour

Some researchers have proposed experimentally-based analytical expressions to model the variation of the

cold-formed steel material behaviour with the temperature, namely Outinen et al. [1], Chen & Young

[7,8,9,10] and Ranawaka & Mahendran [12]. The temperature dependence is invariably taken into account by

means of reduction factors applicable to the steel Young’s modulus (ke), proportional limit stress (kp) and

yield stress (ky) − Figure 1 illustrates the qualitative differences between the steel constitutive laws at room and

elevated temperatures. Note that the stress-strain curve ceases to be bi-linear for temperatures above 100ºC −

indeed, it then exhibits a considerable amount of non-linear strain-hardening, which makes it indispensable

to distinguish between the yield and proportional limit stresses (the latter is almost always taken as the 0.2%

proof stress). Moreover, it is now widely recognized that that the reduction factors applicable to hot-rolled steel

members do not remain valid for cold-formed steel ones − this explains why distinct values are prescribed by

the current design codes (e.g., EC3-1.2).

Figures 2(a)-(b) make it possible to compare the cold-formed steel ky and ke values prescribed by EC3-1.2 [13]

with those proposed by Outinen et al. [1], Chen & Young [8] and Ranawaka & Mahendran [11]. One readily

observes that there are significant discrepancies between the various values − this fact indicates that the

reduction factors depend heavily on the particular cold-formed steel grade and suggests that further research is

required in order to obtain additional information to clarify this matter adequately.

The numerical results presented in this work are determined by adopting the constitutive model

prescribed by EC3-1.2, defined by the expressions

( ) ( )

.

0.52

2

. . . .

.

for

for

T p T

T p T y T y p T y T

y T

E

c b a a

ε ε ε

σ σ ε ε ε ε ε

σ

⋅ ≤

= − + − − < <

. . for y T u Tε ε ε

≤ ≤

(1)

( )( )2

. . . .y T p T y T p T Ta c Eε ε ε ε= − − +,

( )2 2

. .y T p T Tb c E cε ε= − +

( )( ) ( )

2

. .

. . . .2

y T p T

y T p T T y T p T

cE

σ σ

ε ε σ σ

−=

− − −

Where the (i) proportionality limit stress σp.T, (ii) yield stress σy.T, (iii) Young’s modulus ET, (iv) proportionality

limit strain εp.T=σp.T/ET are functions of the temperature T (given in tabular form). Moreover, the strains associated

with the yield (proof) and ultimate stresses (σy.T, σu.T) are always εy.T=0.02 and εu.T=0.15.

Since the experimentally-based constitutive models developed by Chen & Young [8,9] and Ranawaka &

Mahendran [11], were also employed in this work (for validation), they are briefly presented next. The model

of Chen & Young, based on a Ramberg-Osgood [17] type equation, is expressed as

.

.

. .

. . .

. . .

0.002 for

for

T

T

n

T TT y T

T y T

T m

T y T T y T

u T y T T y T

y T u T y T

E

E

σ σσ σ

σε

σ σ σ σε ε σ σ

σ σ

+ ≤

= − −

+ + > − (2)

( ).

.

,1 0.002

Ty T

T T y T

EE

n E σ=

+ ⋅ 20 0.6 , 1 350T Tn T m T= − = +

where (i) there is an analytical expression providing Ey.T (elastic modulus at temperature T), (ii) one still has

εu.T=0.15, (iii) the ultimate stress is given by σu.T=σy.T ku, where ku is the reduction factor of the EC3-1.2 model

and, (iv) σT is the applied stress at temperature T. Note that this equation was also adopted by Mirambell &

Real [18] and Rasmussen [19] to describe the stainless steel stress-strain curve at room temperature. Finally,

Ranawaka & Mahendran [11,12] also employed adopted a constitutive model based on an Ramberg-Osgood

type equation − however, εT is now obtained from the expression

.

T

y TT TT

T T TE E E

ησσ σ

ε β

= + (3)

where β=0.86 (value adopted by Outinen [20]) and the variation of ηT with the temperature is given by

7 3 3

.550 3.05 10 0.0005 0.215 62.653 20 800T T T T Tη −= − ⋅ + ⋅ − ⋅ + ≤ ≤ (4)

3

.250 0.000138 0.085468 19.212 350 800T T T Tη = ⋅ − ⋅ + ≤ ≤ (5)

for high and low strength steels. In the latter case, the stress-strain curves at T<350ºC were shown to exhibit a

yield plateau, thus meaning that ηT cannot be predicted by eq. (5), developed only for a gradually changing

stress-strain laws (Ranawaka [15]). In this work, it is assumed that eq. (4) holds approximately.

The code ANSYS [14] allows for the consideration of the multi-linear stress-strain curve adopted in this work.

Its first branch models the elastic range, up to the proportional limit stress, and exhibits the measured Young’s

modulus. The subsequent branches stand for the non-linear inelastic range, which accounts for (kinematic)

strain-hardening. It is assumed herein that Poisson’s ratio does not depend on the temperature and is equal to

0.3. Finally, note that, because the distortional post-buckling analyses carried out involve large inelastic strains,

the nominal (engineering) static stress-strain curve is replaced by a relation between the true stress and the

logarithmic plastic strain.

2.3 Temperature Evolution

In order to obtain the evolution of the column temperature as a function of the elapsed fire duration,

the 2D non-linear heat transfer finite model incorporated in the code SAFIR (Franssen et al. [21]) was

employed. The temperature evolution within the column cross-section, assumed longitudinally

uniform, is calculated in accordance with the standard fire design curve prescribed in Part 1.2 of

Eurocode 1 (EC1-1.2 [22]). Since the columns simulated in this paper are deemed to be completely

engulfed in flames, it was concluded that all their cross-sections and the surrounding air share the same

temperature (Landesmann et al. [23]) − moreover, the above standard temperature-time curve is applicable to

all the cross-section walls. Finally, it is worth noting that the thermal actions, including heat transfer

through convection and radiation, have also been included in the analyses at constant load − since the

fire duration is deemed small (less than 1 hour), creep effects were neglected.

2.4 Types of Non-Linear Analysis

Two different non-linear column analyses are performed, corresponding to distinct loading strategies: (i)

application of an increasing compressive load on a column exhibiting a constant temperature distribution

(steady case) or (ii) application of an initial compressive force, which remains constant, followed by a

progressive temperature increase (transient case). In the first case, it is assumed that the initial (imperfect)

column geometry already includes the thermal expansion associated with heating it to the pre-defined

temperature − then, the analyses of columns with two distinct temperatures only differ in the steel material

behaviour adopted. The aim of the steady analyses is to determine the variation of the column failure load with

the temperature. In the transient analyses, which correspond to a more realistic simulation of natural fire

conditions, one begins by applying an axial load with a pre-defined value to an initially imperfect column at

“room temperature” T=20ºC (obviously, the applied load must be lower than the associated ultimate value)

and by recording the ensuing column deflected equilibrium state. Then, incremental temperature increases are

sequentially prescribed to the column, leading it to neighbouring (more deflected) equilibrium states − in order

to obtain a new equilibrium state, the shell finite element analysis must take into account (i) the variation of the

steel stress-strain curve, (ii) the thermal strains caused by the temperature rise and (iii) the geometrically non-

linear effects due to the continuous presence of the axial load. The aim of these analyses is to assess the

variation of the column failure temperature with the applied load − in other words, to determine for which

temperature the applied load ceases to correspond to a column equilibrium state. Both analyses are carried by

means of an incremental-iterative technique employing Newton-Raphson’s method and an arc-length control

strategy. The relation between the results yielded by the steady state and transient analyses (failure loads and

temperatures) is addressed further ahead this paper.

At this stage, it should be noted that, in the transient analyses, the temperature values are prescribed at all the

column nodes and incrementally raised by means of the ANSYS automatic load stepping. The thermal

strains stemming from these temperature increments are imposed2 and a column equilibrium state is

sought taking into account both (i) the stress-strain law variation with T and (ii) the geometrically non-

linear effects associated with the compressive force existing in the column before the temperature rises.

3 VALIDATION STUDIES

3.1 Introduction

In order to validate the use of the ANSYS shell finite element model, one begins by reproducing numerical

simulations recently reported in the literature. These simulations concern cold-formed steel lipped channel

fixed columns that were experimentally and numerically analyzed by (i) Chen & Young [8] (columns C1 and

C2− high strength steel at room temperature) and by (ii) Ranawaka [15] (columns C3 − low and high

strength steels at room and elevated temperatures). Table 1 shows these column dimensions.

3.2 Room Temperature

The C1 and C2 columns, analyzed by Chen & Young [8], (i) were made of high strength zinc-coated steel

characterized by E=210 GPa, ν=0.3 and σy,20=450 MPa, and (ii) were analyzed in the code ABAQUS [24],

adopting discretizations into 10 mm×10 mm meshes of S4R5 shell elements (ABAQUS nomenclature − 4-node

thin-shell elements with five d.o.f. per node and reduced integration, and accounting for shear deformation).

The analyses (i) included initial geometrical imperfections with a critical (local or distortional) buckling mode

shapes and an amplitude equal to the wall thickness t, and (ii) did not take into account residual stress effects.

As for the C3 columns, analyzed by Ranawaka [15] at both room and elevated temperatures, they (i) were

made of both low and high strength steel grades G250 and G550, characterized by E=210 GPa, ν=0.3 and

σy,20=250 or σy,20=550 MPa, and (iii) were also analyzed in ABAQUS [24], but adopting discretizations into 5

mm×5 mm meshes of S4 elements (4-node thin-shell elements with six d.o.f. per node and full integration, and

accounting for shear deformation). The columns contained initial geometrical imperfections and flexural

residual stresses with configurations and amplitudes corresponding to the average values measured in the

tested specimens3.

The numerical analyses carried out in this work included initial geometrical imperfections with critical

(local or distortional) buckling mode shapes and amplitude of either 10% of the wall thickness (C1, C2

columns − 0.15 mm) or the reported measured average values (C3 columns − 0.65 mm and 0.4 mm for

2 The thermal strains are imposed under the assumption that the steel linear thermal expansion coefficient does not depend on the

temperature T − its value is always taken as 1.2x10-5 per ºC.

3 More details about the initial geometrical imperfection and residual stress measurements can be found in the work of

Ranawaka [15].

the G250 and G550 steel grades). No residual stresses were included in the analyses4. The table in Figure

3(a) provides the column ultimate loads obtained with simulations performed by the authors (Pu.obt) and

by the other researchers (Pu.rpt) − the percentage differences are also given.

The plot in Figure 3(b) shows a comparison between the obtained and reported equilibrium paths concerning

the C2 column − applied load vs. axial shortening. At this stage, it is worth mentioning that (i) all C1 and

C2 columns failed locally and (ii) both C3 columns failed distortionally. Since there is excellent agreement

between the obtained and reported (i) ultimate loads values (the differences never exceed 3.8% and

most of them fall below 1%) and (ii) equilibrium path configurations, one may consider that the shell

finite element model employed in this work as properly validated for room temperature analysis.

3.3 Elevated Temperature

The table in Figure 4(a) indicates the C3 columns analyzed by Ranawaka [15] at elevated temperatures

(ranging from 200 to 800ºC) that were also simulated in this work − they all have length L=20 cm and failed

distortionally. Their material behaviour dependence on the temperature follows the model described in eqs.

(3)-(5) and the column (i) room temperature material properties and (ii) initial geometrical imperfections

adopted are those indicated in the previous sub-section (the residual stresses are again neglected).

The above table provides the column Pu.obt and Pu.rpt values, together with the associated percentage

differences. The plot in Figure 4(b) shows a comparison between the equilibrium paths concerning the column

made of G550 steel at 800ºC − applied load vs. axial shortening. Although still quite acceptable, the agreement

between the obtained and reported ultimate loads values is not as good as at room temperature (the differences

now reach about 8% and several of them exceed 5%), most likely due some (virtually inevitable)

discrepancies in the numerical modelling of the steel material behaviour under elevated temperatures.

Nevertheless, there is a very good correlation between the equilibrium paths plotted in Figure 4(b). Therefore,

it seems fair to assume that the shell finite element model employed in this work is also adequately

validated for elevated temperature analysis.

4 No residual stresses were included in the analyses carried out by Chen & Young [8] and Ranawaka [15] explicitly

mentioned that the influence of the residual stresses on the C3 column (distortional) failure loads is negligible − about 0.2%.

4 DISTORTIONAL POST-BUCKLING BEHAVIOUR AND FAILURE

The ANSYS shell finite element model validated in the previous section is now employed to perform a

parametric study aimed at assessing the post-buckling and ultimate strength behaviour of simply

supported lipped channel columns that both buckle and fail in distortional modes. Previous work by

the authors (Landesmann et al. [23]) showed that, regardless of the temperature, columns with the

cross-section dimensions given in Figure 5(a) and length L=60 cm exhibit distortional critical

buckling modes − Figure 5(b) provides an overall view of this buckling mode type, which is

characterized by the simultaneous occurrence of cross-section wall transverse bending and rigid-body

motions of the flange-lip assemblies. The various curves depicted in Figure 5(c) provide the variation

of the critical elastic buckling stress σcr with the column length L (logarithmic scale) and temperature T −

it should be pointed out that (i) T varies between 20/100ºC (“room temperature” material behaviour)

and 800ºC, and (ii) σcr is normalized with respect to σcr.D.20=154.9 MPa (minimum critical stress

associated with a single half-wave distortional buckling mode at T=20ºC)5. The observation of these

curves shows that (i) all of them may be obtained from the top one by means of a “vertical translation” with a

value that depends exclusively on the Young’s modulus erosion caused by the rising temperature, and

that, as mentioned earlier (ii) L=60 cm always corresponds to σcr.D.T (minimum single half-wave

critical distortional stress at temperature T). Moreover, note that σcr.D.T drops by about 90% when the

temperature rises from 20/100ºC to 800ºC (Young’s modulus decreases from 205 to 18.5 GPa).

The columns whose post-buckling behaviour and ultimate strength are analyzed in this work are built from

four European standard steel grades: S150, S235, S355 and S460 (E=205 GPa, ν=0.3 and σy,20=150, 235, 355,

460 MPa) − as mentioned earlier, the variation of the steel constitutive law with the temperature T is assumed

to follow the EC3-1.2 [13] model for light-gauge steels. All the columns are concentrically compressed and

contain critical-mode (distortional) initial geometrical imperfections with an amplitude equal to 10% of the

wall thickness and involving outward flange-lip motions (see Fig. 5(b))6. Next, one addresses separately the

determination of (i) failure loads (steady state analyses, in which columns subjected to a constant uniform

temperature are acted by an increasing axial compression) and (ii) failure temperatures (transient analyses, in

which columns under a constant axial compression are subjected to an increasing temperature) − some

attention is also devoted to the relation between these failure loads and temperatures.

5 These curves were obtained through buckling analyses carried out in using the code GBTUL (Bebiano et al. [25]), based

on a recent Generalised Beam Theory (GBT) formulation (Bebiano et al. [26]), and adopting the variation of Young’s

modulus with T prescribed in the EC3-1.2. 6 As shown by Prola & Camotim [27] and Silvestre & Camotim [28], the column distortional post-buckling behaviour exhibits a non-

negligible asymmetry with respect to the flange-lip motion sense (outward or inward). These authors showed that the

ultimate load associated with outward distortional failure is always lower than its inward counterpart (in lipped channel columns).

4.1 Column Failure Loads at Constant Temperature

Figure 6 shows the non-linear (post-buckling) equilibrium paths of initial imperfect columns made of

four steel grades (S150, S235, S355, S460) and subjected to with six uniform temperatures: 20/100ºC

(20ºC corresponds to “room temperature” and the steel material properties are deemed unchanged up to

100ºC), 200ºC, 300ºC, 400ºC, 500ºC, 600ºC. Each equilibrium path relates (i) the applied stress σ,

normalized with respect to σcr.D.20=154.9 MPa (critical distortional buckling stress at room temperature), with

(ii) the (outward) vertical displacement of the mid-span lips. The triangles indicate the various column

ultimate stress ratios (σu /σcr.D.20).

The observation of the column post-buckling equilibrium paths and ultimate stresses displayed in

Figures 6(a)-(f) prompts the following remarks:

(i) Obviously, the various column ultimate stresses decrease as T increases.

(ii) At room temperature, the columns always collapse immediately after the onset of yielding (i.e.,

there is no elastic-plastic strength reserve). In the S150 and S235 columns practically no ductility

precedes failure.

(iii) As T increases, the S150 and S235 columns exhibit some ductility prior to failure that is associated with

a visible (but very small) amount of elastic-plastic strength reserve. On the other hand, the S355 and

S460 columns failure always follows the onset of yielding, regardless of the temperature.

(v) In order to quantify the column ultimate stress erosion stemming from the rising temperature, Figure 7

shows the variation of the ultimate stress ratio σu.D.T /σu.D.20 with the temperature T for the four steel

grades considered − dashed lines. Also given in these figures is the variation of the critical stress ratio

σcr.D.T /σcr.D.20 with T − solid line. On the other hand, Table 2 indicates the σu.D.T /σcr.D.T values, which

provide the amount of post-critical strength reserve associated with each simulated column. The analysis

of these results leads to the following conclusions:

(v.1) Obviously, the variation of σcr.D.T /σcr.D.20 with T is the same for all steel grades − recall that σcr.D.T

depends exclusively on the reduction of the Young’s modulus with the temperature (see Fig. 5(c)).

(v.2) Since the variation of the ultimate stress is also affected by the (v1) proportionality limit and yield

stresses, and (v2) the shape of the stress-strain curve, there is a clear dependence on T.

(v.3) In the S460 columns the critical and ultimate stress ratio values practically coincide, which

means that, at least for this particular column geometry, the critical (distortional) and

ultimate stresses are equally affected by the temperature. Indeed, the σu.D.T /σcr.D.T values

are all very similar, even if they slightly decrease with T − they vary from 1.042 to 1.003, thus

indicating a very small post-critical strength.

(v.4) As the yield stress decreases, the column critical stress ratio values progressively fall below

their ultimate stress counterparts, which means that, again for this particular column geometry, the

σu.D.T /σcr.D.T values decrease more significantly with T − Table 2 shows that this ratio falls by

about 4% (S460), 7% (S355), 15% (S235) and 20% (S150).

(v.5) The σu.D.T /σcr.D.T increase with the yield stress (steel grade) grows with T − it varies between

about 20% (100ºC) and 37% (600ºC).

(v.6) All the σu.D.T /σcr.D.T values concerning the S235 and S150 columns fall below 1.0, which means

that their collapses occur before the critical applied stress level is reached. Moreover, those values

decrease with both the temperature and the yield stress, thus implying that plasticity effects play

an increasing role in column failure.

Figure 8 shows the von Mises stress distributions occurring at the distortional collapse (σ=σu.D.T) of columns

made of S460 steel and subjected to four different temperatures (T=20-400-600-800ºC) − note that the “stress

scales” are different for each distribution. One observes that, since the thermal action effects are negligible

(uniform temperature and free-to-deform columns), the distortional failure modes are virtually identical in the

four columns, i.e., they do not depend on the temperature. Moreover, the four von Mises stress distributions are

also qualitatively rather similar − the higher stresses always occur in the vicinity of the lip free ends. It is worth

noting that the collapse is always triggered by the yielding of the web-flange edge regions in the vicinity of the

column mid-span. Quantitatively speaking, the stress values obviously decrease as the temperature rises and

continuously erodes the steel material behaviour.

Figure 9(a) illustrates the distortional post-buckling equilibrium path obtained − it concerns a S460 column

subjected to 20/100ºC (i.e., room temperature condition). As for Figure 9(b), it provides information

about the evolution of column deformed configuration and plastic strain distribution − 4 plastic strain

diagrams are presented, corresponding to distinct equilibrium states located along the equilibrium path

(their locations are indicated in Figure 9(a)). It is worth noting that (i) the deformed configuration

corresponding to point 1 is amplified 5 times, and that (ii) point 3 corresponds to the occurrence of the column

collapse − i.e., the associated deformed configuration provides the shape of the column failure mode.

4.2 Column Failure Temperatures at Constant Load

The curves shown in Figures 10(a)-(d) concern the behaviour of initially imperfect columns that (i) are first

subjected to 20, 40, 60 and 80% of their distortional critical buckling load at room temperature (Pcr.D.20) and

(ii) then subjected to small uniform temperature increments (starting at T=20ºC). After each increment, one

seeks a column equilibrium configuration that (i) accommodates the change in the steel constitutive (stress-

strain) relationship, (ii) includes the geometry change due to the thermal strains stemming from the

temperature increase and (iii) incorporates the corresponding geometrically non-linear effects associated with

the (constant) applied load. The temperature is incrementally raised until no equilibrium configuration can be

reached (for that applied load), which means that the column collapses for that specific temperature value –

it constitutes the column failure temperature (Tu).

The observation of the column equilibrium paths and failure temperatures presented in Figures 10(a)-

(d) leads the following comments:

(i) Obviously, the failure temperature of a given column drops as the initially applied load level P increases.

(ii) The ductility prior to failure generally increases with the steel grade. No clear correlation can be

established with the column applied load level.

(iii) As expected, the non-linearity of the equilibrium path ascending branch grows with the applied load

level − the geometrically non-linear effects due to the axial compression (and felt throughout the whole

heating process) are more pronounced.

(iv) Table 3 provides the Tu values obtained, making it possible to assess how the yield stress and initial

applied load level influence the column temperature failure. One observes that the Tu percentage

drop, as P/Pcr.D.20 varies from 0.20 to 0.80, decreases as the yield stress increases: the drops are of

77% (S150), 60% (S235), 56.5% (S355) and 55.6% (S460). Note, however, that the percentage drop is

practically the same for the S355 and S460 columns − indeed, the Tu values are almost identical

for these two sets of columns.

4.3 Relation between Failure Loads/Stresses and Temperatures

One question naturally arises concerning the relation existing between the data and solutions of the steady

state (failure loads/stresses at constant temperature) and transient (failure temperatures at constant

load/stress) analyses. In order to investigate this issue, Figures 11(a)-(b) plot curves σu /σcr.D.20 (T) and

Tu (σ /σcr.D.20) − while (i) the former are dashed with small triangles identifying the constant temperatures

considered (see Table 2), (ii) the latter are dotted with small squares identifying the constant initial

stresses/loads dealt with (see Table 3). One readily observes that, regardless of the steel grade, the two curves

(obtained by joining the small triangles and squares) practically coincide, which means that (i) the failure

stress/load at temperature T0 (σu.T0) is virtually identical to (ii) the applied stress (σ) that leads to a failure

temperature Tu=T0. This can be confirmed by looking at the steady state analysis the S460 column at

temperature T=500 ºC, which led to σu /σcr.D.20=0.602 − Table 3 shows that a transient analysis of this

same column initially compressed at σu /σcr.D.20=0.60 yields the failure temperature Tu=500ºC. In this

context, it is worth noting that the experimental results recently reported by Ranawaka & Mahendran [11]

point out to this same conclusion, as the differences between the applied/failure temperature and load sets

are negligible. Thus, it was shown that the result of a steady state analysis (much easier to perform) matches

that of the “corresponding” transient analysis.

4.4 Column Failure Times at Constant Load

On the basis of the well known member temperature evolution, expressed by the time-temperature relation

shown in Figure 12 (e.g., EC1-1.2 [22]), it is possible to determine the time required for the columns to fail

as function of the applied load value − these “failure times” are also plotted in Figure 12 versus the

applied load ratio P /Pcr.D.20 for the different steel grades. The observation of these plots leads to the

following remarks (note that, obviously, they are the logical consequence of the earlier remarks concerning the

failure temperature results):

(i) Naturally, the failure time drops as (i1) the applied load ratio P /Pcr.D.20 increases and (i2) the steel

yield stress decreases − nevertheless, there is very little difference between the failure times of the S460

and S355 columns, which suggests that there is a yield stress threshold after which it does not influence

the failure time7.

(ii) For each steel grade, it seems fair to say that the failure time is approximately inversely proportional

to the applied load level.

(iii) In the columns analysed, the failure times varied between 1.04 min (S150 column under P /Pcr.D.20=0.80)

and 10.11 min (S460 column under P /Pcr.D.20= 0.20). Since these values are very small, the application

of fire protection insulation to increase the column failure time becomes indispensable.

5. CONCLUDING REMARKS

This paper reported the available results of an ongoing shell finite element investigation on the distortional

buckling, post-buckling and ultimate strength behaviour of simply supported cold-formed steel lipped

channel columns subjected to (i) concentric compression and (ii) elevated temperatures caused by fire

conditions. The variation of the steel material behaviour with the temperature was assumed to follow the

model prescribed in EC3-1.2, for cold-formed steel, and the geometrically and materially non-linear

7 Only indirectly, through the value of σcr.D.20, which obviously increases with the yield stress.

response of the lipped channel columns was determined by means of shell finite element analyses

performed in ANSYS. The column collapse was computed for two loading strategies: (i) application of an

increasing compressive load on a column subjected to a constant temperature distribution (obtaining failure

loads) or (ii) application of an increasing temperature to an initially compressed column (obtaining failure

temperatures) − using the well known time-temperature curve prescribed in EC1-1.2, the failure temperatures

were also “transformed” into failure times. Among the various conclusions already drawn from this ongoing

investigation, the following ones deserve to be specially mentioned:

(i) It was shown that the differences between the applied/failure temperature and load sets are

negligible, which means that the results of the steady state analyses match those of the

“corresponding” transient analyses. Therefore, the (distortional) failure of columns under fire conditions

can be fully investigated by resorting only to failure loads under elevated temperatures − steady

state analyses are easier to perform than transient ones. Nevertheless, the transient analyses can

also yield failure times, which provide very useful information concerning the stringency of the

need to have fire protection insulation.

(ii) Since the column failure load drop with the temperature stems from (ii1) the reduction of the steel

Young’s modulus, proportionality limit stress and yield stress, and (ii2) the variation of the

stress-strain curve shape, it cannot be adequately “measured” by the associated critical load

decrease, exclusively due to the Young’s modulus reduction. Nevertheless, it was found that, for

a high enough steel grade (S460 in this case), the column critical and ultimate loads are equally

affected by the temperature, probably because failure is mostly governed by instability effects −

further research is needed to clarify this issue, which may have far-reaching implications on the

design of high-strength steel columns under fire conditions.

Finally, it is worth mentioning that further developments of this work (presently under way) include analysing

the buckling, post-buckling and ultimate strength behaviour, under fire conditions, of cold-formed steel

columns with other end support conditions (fixed columns) and/or other cross-section shapes (rack-

section columns) − the results obtained will be reported in the near future. Moreover, since there are no

specific rules to predict the ultimate load of cold-formed steel columns exhibiting distortional collapses at high

temperatures, the failure loads gathered during this ongoing research effort should be very helpful in

establishing guidelines for the design of such members under fire conditions. The authors have already

done some preliminary work towards achieving this goal (Landesmann & Camotim [29]), which

consisted of comparing numerical and experimental lipped channel column ultimate loads with the predictions

yielded by the application of the (slightly modified) provisions of the Direct Strength Method (e.g., Schafer

[30,31]) for distortional failure at room temperature8 − note that the Direct Strength Method was recently

included in Appendix 1 of the North American Specification for cold-formed steel structures (AISI [32]).

REFERENCES

1. Outinen, J., Kaitila, O., Makelainen, P., A study for the development of the design of steel structures in fire

conditions, Proceedings of First International Workshop on Structures in Fire (Copenhagen), 2000, 267-281.

2. Kaitila, O., Imperfection sensitivity analysis of lipped channel columns at high temperatures,

Journal of Constructional Steel Research, 2002, 58(3), 333-51.

3. Feng, M., Wang, Y.C. and Davies, J.M., Structural behaviour of cold-formed thin-walled short steel channel

columns at elevated temperatures − Part 1: experiments, Thin-Walled Structures, 2003a, 41(6), 543-70.

4. Feng, M., Wang, Y.C. and Davies, J.M., Structural behaviour of cold-formed thin-walled short steel

channel columns at elevated temperatures − Part 2: design calculations and numerical analysis, Thin-Walled

Structures, 2003b, 41(6), 571-94.

5. Lee, J.H., Mahendran, M. and Makelainen, P., Prediction of mechanical properties of light gauge steels at

elevated temperatures, Journal of Constructional Steel Research, 2003, 59(12), 1517-32.

6. Feng, M. and Wang, Y.C., An analysis of the structural behaviour of axially loaded full-scale cold-formed

thin-walled steel structural panels tested under fire conditions, Thin-Walled Structures, 2005, 43(2), 291-332.

7. Chen, J. and Young, B., Corner properties of cold-formed steel sections at elevated temperatures, Thin-

Walled Structures, 2006, 44(2), 216-23.

8. Chen J. and Young B., Cold-formed steel lipped channel columns at elevated temperatures,

Engineering Structures, 2007a, 29(10), 2445-56.

9. Chen, J. and Young, B., Experimental investigation of cold-formed steel material at elevated

temperatures. Thin-Walled Structures, 2007b, 45(1), 96-110.

10. Chen, J. and Young, B., Design of high strength steel columns at elevated temperatures, Journal of

Constructional Steel Research, 2008, 64(6), 689-703.

11. Ranawaka, T. and Mahendran, M., Distortional buckling tests of cold-formed steel compression

members at elevated temperatures, Journal of Constructional Steel Research, 2009, 65(2), 249-59,.

8 The authors followed an approach already (partially) explored by other researchers, namely Chen & Young [8,10] and Ranawaka &

Mahendran [11].

12. Ranawaka, T. and Mahendran, M., Numerical modelling of light gauge cold-formed steel

compression members subjected to distortional buckling at elevated temperatures, Thin-Walled

Structures, 2010, 48(4-5), 334-344,.

13. Comité Européen de Normalisation (CEN). Eurocode 3: Design of Steel Structures – Part 1-2:

General Rules – Structural Fire Design, Brussels, 2005.

14. Swanson Analysis Systems (SAS). ANSYS Reference Manual (vs. 8.1), 2004.

15. Ranawaka, T., Distortional Buckling Behaviour of Cold-Formed Steel Compression Members at

Elevated Temperatures. Ph.D. Thesis, Queensland University of Technology, Australia, 2006.

16. Dinis, P.B. and Camotim, D., On the use of shell finite element analysis to assess the local

buckling and post-buckling behaviour of cold-formed steel thin-walled members, in C.A.M. Soares et al.

(eds.): Book of Abstracts of III European Conference on Computational Mechanics: Solids, Structures and

Coupled Problems in Engineering (Lisbon, 5-9/6), Springer (Dordrecht), 689, 2006. (full paper in CD-

ROM Proceedings)

17. Ramberg, W. and Osgood, W.R., Description of stress–strain curves by three parameters, NACA

Technical Note 902, 1943.

18. Mirambell, E. and Real, E., On the calculation of deflections in structural stainless steel beams: an

experimental and numerical investigation. Journal of Constructional Steel Research, 2000, 54(1), 109-33.

19. Rasmussen, K.J.R., Full-range stress–strain curves for stainless steel alloys, Journal of

Constructional Steel Research, 2003, 59(1), 47-61.

20. Outinen J., Mechanical Properties of Structural Steels at Elevated Temperatures, Licentiate Thesis,

Helsinki University of Technology, Finland, 1999.

21. Franssen, J.M., Kodur, V. and Mason, J., User’s Manual for SAFIR 2004 (computer program to analyse

structures subjected to fire), University of Liège, 2004.

22. Comité Européen de Normalisation (CEN), Eurocode 1: Actions on Structures – Part 1-2: General

Actions – Actions on Structures Exposed to Fire, Brussels, 2002.

23. Landesmann, A., Camotim, D. and Batista, E.M., On the distortional buckling, post-buckling and strength

of cold-formed steel lipped channel columns subjected to elevated temperatures, In F. Wald, P. Kallerová, J.

Chlouba (eds.): Proceedings of International Conference on Applications of Structural Fire Engineering

(Prague), 2009, A8-A13.

24. Hibbit, Karlsson & Sorensen Inc. (HKS). ABAQUS Standard (vrs. 6.5), 2004.

25. Bebiano, R., Pina, P., Silvestre, N. and Camotim, D., GBTUL 1.0β – Buckling and Vibration

Analysis of Thin-Walled Members, DECivil/IST, Technical University of Lisbon, 2008.

(http://www.civil.ist.utl.pt/gbt)

26. Bebiano, R., Silvestre, N. and Camotim, D., GBTUL - A code for the buckling analysis of cold-

formed steel members, In R. LaBoube, W.-W. Yu (eds.): Proceedings of 19th International Specialty

Conference on Recent Research and Developments in Cold-Formed Steel Design and Construction (St.

Louis, 14-15/10), 2008b, 61-79.

27. Prola, L.C. and Camotim, D., On the distortional post-buckling behaviour of cold-formed lipped channel

steel columns, Proceedings of Structural Stability Research Council Annual Stability Conference, (Seattle 24-

26/4), 2002, 571-590.

28. Silvestre, N. and Camotim, D., Local-plate and distortional post-buckling behaviour of cold-formed

steel lipped channel columns with intermediate stiffeners, Journal of Structural Engineering (ASCE), 2006,

132(4), 529-540.

29. Landesmann, A. and Camotim, D., Distortional failure and design of cold-formed steel lipped channel

columns under fire conditions, Proceedings of Structural Stability Research Council Annual Stability

Conference (Orlando, 12-15/5), 2010. (in press)

30. Schafer, B.W., Direct Strength Method Design Guide, American Iron & Steel Institute (AISI),

Washington DC, 2005.

31. Schafer B.W., Review: the Direct Strength Method of cold-formed steel member design, Journal

of Constructional Steel Research, 2008, 64(7-8), 766-88.

32. American Iron and Steel Institute (AISI). Appendix I of the North American Specification (NAS)

for the Design of Cold-formed Steel Structural Members: Design of Cold-Formed Steel Structural

Members with the Direct Strength Method, Washington DC, 2007.

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0

Strain (%)

No

rma

lize

d s

tres

s600º

200º20ºC

Ranawaka & Mahendran

Chen & Young

EC3-1.2

ε

()

.20

Ty

σσ

Figure 1. Cold-formed steel constitutive laws at room and high temperatures (ε ≤ 2%).

0.0

0.2

0.4

0.6

0.8

1.0

0 100 200 300 400 500 600 700 800

Yie

ld s

tres

s re

du

ctio

n f

act

or

(ky)

Temperature T (ºC)

Chen & Young

Ranawaka

& Mahendran

EC3-1.2

Outinen

0.0

0.2

0.4

0.6

0.8

1.0

0 100 200 300 400 500 600 700 800

Yo

un

g’s

mod

ulu

s re

du

ctio

n f

act

or

(ke)

Temperature T (ºC)

Chen & Young

Ranawaka

& Mahendran

EC3-1.2

Outinen

Figure 2. Variation of (a) ky and (b) ke with the temperature (T≤800°C).

(a) (b)

Column C2

0

20

40

60

80

100

120

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Axial shortening (mm)

Ap

pli

ed

lo

ad

(k

N)

Reported by Chen & Young (2007a)

Obtained

Figure 3. (a) Column ultimate loads obtained (Pu.obt) and reported (Pu.rpt), and (b) comparison between

the equilibrium paths concerning the C2 column.

C L

(cm)

Pu.obt

(kN)

Pu.rpt

(kN)

u.obt u.rpt

u.rpt

P -P

P

C1 28 99.9 100.2 0.29%

C1 100 94.5 91.0 -3.80%

C1 150 83.9 84.8 1.11%

C2 100 104.5 103.5 -0.99%

C3G250 20 12.5 12.4 -0.40%

C3G550 20 19.75 20.14 1.94%

(a) (b)

Column C3 G550

800ºC

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.0 0.2 0.4 0.6 0.8 1.0

Axial shortening (mm)

Ap

pli

ed

lo

ad

(k

N)

Reported by Ranawaka (2006)

Obtained

Figure 4. (a) Column Pu.obt and Pu.rpt values, and (b) comparison between the equilibrium paths

concerning the C3 column made of G550 steel at 800ºC.

C T

(ºC)

Pu.obt

(kN)

Pu.rpt

(kN)

u.obt u.rpt

u.rpt

P -P

P

C3G250 200 9.80 9.45 3.70%

C3G250 500 3.80 3.61 5.26%

C3G250 800 0.84 0.79 6.33%

C3G550 350 15.78 14.60 8.08%

C3G550 500 2.90 2.75 5.45%

C3G550 650 0.73 0.76 -3.95%

C3G550 800 7.62 7.51 1.41%

(a) (b)

Figure 5. Lipped channel column (a) cross-section dimensions, (b) distortional buckling mode shape

and (c) σcr/σcr.D.20 vs. L elastic buckling curves.

0

0.4

0.8

1.2

1.6

1 10 100 1000

Column length

σσ σσc

r/σσ σσ

cr.

D.2

0

L = 60 cm

20-100 ºC

600 ºC

700 ºC

800 ºC

500 ºC

(c) (b)

100

130

12

.5

x

y

2

(a)

Dimensions in mm

0.0

0.2

0.4

0.6

0.8

1.0

0.0 2.0 4.0 6.0 8.0 10.0

Outward lip displacement (mm)

S460

S355

S235

S150

20σ σ . .u cr D

20/100 °C

20

σσ

..

cr

D

Elastic

0.0

0.2

0.4

0.6

0.8

1.0

0.0 2.0 4.0 6.0 8.0 10.0

Outward lip displacement (mm)

S460

S355

S235

S150

20σ σ . .u cr D

200 °C

20

σσ

..

cr

D

Elastic

0.0

0.2

0.4

0.6

0.8

1.0

0.0 2.0 4.0 6.0 8.0 10.0

Outward lip displacement (mm)

S460

S355

S235

S150

20σ σ . .u cr D

300 °C

20

σσ

..

cr

D

Elastic

0.0

0.2

0.4

0.6

0.8

1.0

0.0 2.0 4.0 6.0 8.0 10.0

Outward lip displacement (mm)

S460

S355

S235

S150

20σ σ . .u cr D

400 °C

20

σσ

..

cr

D

Elastic

0.0

0.2

0.4

0.6

0.8

1.0

0.0 2.0 4.0 6.0 8.0 10.0

Outward lip displacement (mm)

S460

S355

S235

S150

20σ σ . .u cr D500 °C

20

σσ

..

cr

D

Elastic

0.0

0.2

0.4

0.6

0.8

1.0

0.0 2.0 4.0 6.0 8.0 10.0

Outward lip displacement (mm)

S460

S355

S235

S150

20σ σ . .u cr D600 °C

20

σσ

..

cr

D

Elastic

Figure 6. Distortional post-buckling equilibrium paths concerning columns made of 4 steel grades

(S150, S235, S355, S460) and subjected to temperatures (a) 20/100 ºC, (b) 200 ºC, (c) 300 ºC, (d) 400 ºC,

(e) 500 ºC and (f) 600 ºC.

(c)

(a) (b )

(f)

(d)

(e)

0

0.2

0.4

0.6

0.8

1

0 200 400 600

Temperature T (ºC)

σu.

D.T

/σu.

D.2

0 o

r σ

cr.D

.T /σ

cr.D

.20

S355

S460

. . . .20cr D T cr Dσ σ

S150

σu.D.T /σu.D.20

S235

Figure 7. Variation of σu.D.T /σu.D.20 (dashed curves) and σcr.D.T /σcr.D.20 with T (common solid curve) for the

steel grades S150, S235, S355 and S460.

20ºC

400ºC

600ºC

800ºC

Figure 8. Von Mises stress distributions occurring at the distortional collapse of columns made of S460 steel

and subjected to four different temperatures.

165

137.5

110

82.5

55.0

27.5

0

MPa

460

383

307

230

153

76.7

0

MPa

50

41.7

33.3

25.0

16.7

8.3

0

MPa

460

383

307

230

153

76.7

0

MPa

0

0.2

0.4

0.6

0.8

1

0 4 8 12 16

Outward lip displacement (mm)

20

σσ

..

cr

D

1

4

32

Figure 9. (a) Distortional post-buckling equilibrium paths and (b) deformed configurations and plastic

strain locations at the four equilibrium points indicated (S460 column subjected to 20/100ºC).

1 3

2 4

(a)

(b)

0

100

200

300

400

500

600

700

0.0 1.0 2.0 3.0 4.0

Outward lip displacement (mm)

P/Pcr.D.20 = 0.20

P /Pcr.D.20 = 0.40

P /Pcr.D.20 = 0.60

P /Pcr.D.20 = 0.80

Failure temperature

oT

em

pe

ratu

re

(C

)T

S150

0

100

200

300

400

500

600

700

0.0 1.0 2.0 3.0 4.0

Outward lip displacement (mm)

P/Pcr.D.20 = 0.20

P /Pcr.D.20 = 0.40

P /Pcr.D.20 = 0.60

P /Pcr.D.20 = 0.80

Failure temperature

S235

0

100

200

300

400

500

600

700

0.0 1.0 2.0 3.0 4.0

Outward lip displacement (mm)

P/Pcr.D.20 = 0.20

P /Pcr.D.20 = 0.40

P /Pcr.D.20 = 0.60

P /Pcr.D.20 = 0.80

Failure temperature

oT

em

pe

ratu

re

(C

)T

S355

0

100

200

300

400

500

600

700

0.0 1.0 2.0 3.0 4.0

Outward lip displacement (mm)

P/Pcr.D.20 = 0.20

P /Pcr.D.20 = 0.40

P /Pcr.D.20 = 0.60

P /Pcr.D.20 = 0.80

Failure temperature

S460

Figure 10. Variation with P/Pcr.D.20 of the column response to a temperature increase, for columns

made of (a) S150, (b) S235, (c) S355 and (d) S460 steel.

(a) (b)

(c) (d)

Te

mp

era

ture

T (

°° °° C)

Te

mp

era

ture

T (

°° °° C)

0.0

0.2

0.4

0.6

0.8

1.0

0 200 400 600

Temperature T (ºC)

Failure temperature

Failure stress/load ratio

S150

20

σσ

..

cr

D

S355

0.0

0.2

0.4

0.6

0.8

1.0

0 200 400 600

Temperature T (ºC)

Failure temperature

Failure stress/load ratio

20

σσ

..

cr

D

S460S235

Figure 11. Plots σ /σcr.D.20 vs. T relating the data and solutions of the steady state and transient

analyses for (a) S150 + S355 and (b) S235 + S460 steel columns.

(a) (b)

0.0

0.2

0.4

0.6

0.8

1.0

0 4 8 12

Failure Time t u (min)

Temperature-time curve:

T = 20 + 345 log 10 (8t +1)

..2

0cr

DP

P

S150

S235

S460

S355

Figure 12. Variation of the column failure time (tu) with P /Pcr.D.20.

Table 1. Dimensions of the columns considered in the validation studies

Column bw

(mm)

bf

(mm)

bs

(mm)

t

(mm)

L

(cm)

C1 96 36 12 1.5 28-150

C2 96 48 12 1.5 100

C3 30 30 5 0.6 20

t bw

bf bs

Table 2. σu.D.T /σcr.D.T values concerning the columns analysed in this work

T (°°°°C) S150 S235 S355 S460

100 0.836 0.976 1.026 1.042

200 0.836 0.976 1.026 1.042

300 0.787 0.952 1.018 1.036

400 0.722 0.910 1.000 1.027

500 0.636 0.827 0.961 1.004

600 0.632 0.823 0.959 1.003

Table 3. Tu values concerning the columns analyzed in this work

P/Pcr.D.20 S150 S235 S355 S460

0.2 595 640 655 665

0.4 475 540 565 575

0.6 280 385 480 500

0.8 135 255 285 295