Mechanical response of a partially restrained column exposed to localised fires

Fire Safety Journal 67 (2014) 82–95

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Mechanical response of a partially restrained column exposed to localised fires David Lange n, Johan Sjö strö m SP Technical Research Institute of Sweden, Department of Fire Research, Box 857, Borås 501 15-SE, Sweden

art ic l e i nf o

a b s t r a c t

Article history: Received 25 November 2013 Received in revised form 22 April 2014 Accepted 11 May 2014

Recent trends in structural fire engineering research have focussed on the response of buildings with large open plan spaces to so-called travelling fires. These fires travel horizontally across the floor plate of a building and result in time and spatially varying thermal exposure and response of the structure to the fire. What has received little attention, however, is the effect that non-uniform thermal exposure has on columns. Recent tests conducted at SP demonstrated the effect of a small non-uniformity of thermal exposure, resulting in a thermal gradient of around 1 1C/mm, on a column exposed to a pool fire. The curvature resulting from a non-uniform thermal exposure where the column is pinned, or in cases where the column is partially restrained, will result in an eccentricity in the column’s loading and large second order effects. This paper describes the effect of thermal exposure varying in both the horizontal and vertical axes to columns by means of including this thermal boundary in a solution of classical Euler beam theory. The resulting solution allows for a variation in the stiffness of the rotational restraint at both ends of the column and a non-uniform temperature exposure through the column’s section and along its height. The resulting equations help to understand better the impact of the assumptions of ‘lumped capacitance’ on steel columns, suggesting a challenge to this assumption in some instances, as well as to enhance understanding of the impact of non-uniform fires on steel columns. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Analytical method Partially restrained column Localised fires Non-uniform heating of column

1. Introduction Loss of strength and stiffness in steel as a result of high temperatures means that exposure of steel columns to fires may reduce their load bearing capacity in fire. This may lead to potential instability in a structure where appropriate alternate load paths or mechanisms are not available. Design procedures for columns in fire, such as those described in Eurocode 3 [1], are typically based on the assumption that the temperature distribution in columns is uniform. The objective of design is therefore to prevent or limit the effects of increased temperature such that the column is able to maintain its strength and stiffness for a specified period of time to ensure the safe evacuation of all building occupants as well as stability of the structure for a period following evacuation. Alternative methods for structural fire engineering now focus on the optimisation or removal of passive protection, allowing components to heat up so long as the structure can be shown through calculation or otherwise to remain stable for a given period of time.

n

Corresponding author. Tel.: þ 46 10 516 5000. E-mail address: [email protected] (D. Lange).

http://dx.doi.org/10.1016/j.firesaf.2014.05.013 0379-7112/& 2014 Elsevier Ltd. All rights reserved.

Past research into the behaviour of steel columns in fire has focused on evaluating the response of columns heated while considering the interaction with the surrounding structure. For example, some authors have studied the response of steel columns with different restraint conditions experimentally. In Ref. [2], the authors report on a series of tests which were carried out to evaluate the response of steel columns with axial restraint. They state that an increase in axial restraint stiffness is normally coincident with an increase in rotational restraint stiffness. The authors conclude that the negative effects of increasing axial restraint stiffness are offset by the positive effects of increasing the rotational restraint stiffness. Other research has been carried out to evaluate the effect of rotational restraint on steel columns in fire [3], where a number of medium scale tests were performed on partially restrained steel columns subject to fire. Using varying levels of restraint it was found that increasing the restraint increased the failure temperature without having a considerable effect on the restraint forces. Other research has focussed on developing analytical solutions to describe the mechanical response of heated columns in fire considering the thermal and mechanical boundary conditions. In Ref. [4], the authors study the effect of axial restraint on ‘þ’ shaped steel columns in fire. Using a specially formulated finite element code, the authors apply uniform temperatures. They

D. Lange, J. Sjö strö m / Fire Safety Journal 67 (2014) 82–95

Nomenclature A cp E Eref F, F i f fr

area heat capacity modulus of elasticity reference modulus of elasticity force in section and slice, respectively slope of the thermal gradient with height slope of the variation (with height) in residual curvature under partial restraint fP slope of the variation (with height) in residual curvature, under partial restraint after application of axial load I tot total second moment of area of the sections I zz second p ffiffiffiffiffiffiffiffiffiffi moment of area about the z axis k P=EI kRA, kRB rotational spring stiffness at connection a and b, respectively L length of column M resultant moment in column section M rPðaÞ ; M rPðbÞ restraining moment from the partial restraint to the applied load P at a and b, respectively M rthðaÞ , M rthðbÞ restraining moment from the partial restraint to thermal expansion at a and b, respectively

conclude in their study that axial restraint to the column can significantly increase the compressive load in a heated column and that rotational restraint appears to generally increase the fire resistance time of a column. This is in line with the findings described above and in Ref. [2]. Quiel and Garlock have presented a closed form solution for determining the demand on both a beam and a column in an assembly as part of a perimeter framing system of a structure [4]. Dwaikat and Kodur have also presented a simplified version of this methodology [5]. The method accounts for a thermal gradient through the beam section, resulting in a displacement to the P–M interaction diagram. The solution includes consideration of both temperature dependent material properties as well as the interaction between the two members, including push-out forces acting on a column caused by expansion of the beam. In Ref. [5], the authors also demonstrate that a thermal gradient through the weak axis has little effect on the P–M diagram, suggesting that a thermal gradient in this axis may be neglected. All of this work assumes uniform temperature along a columns height throughout the analysis. However, as has been discussed elsewhere [6,7], fires are rarely uniform and are known to travel around a compartment. Alternatively, localised fires may lead to non-uniform temperature distributions within a compartment and throughout an element of structure. Non-uniform heating along a column’s height as well as the impact of localised fires on steel beams has been considered elsewhere [8]. Tan and Yuan have studied the response of pin-ended steel columns exposed to a two zone fire, with the upper portion of the column heated to a higher extent than the lower part of the column, [9]. However they assume lumped capacitance (or uniform heating) in the two areas of the column. Elsewhere, researchers have considered non-uniform thermal exposure as a result of removal of passive fire protection [10] and derive an expression for the behaviour of a steel column when exposed to fire and with portions of a protected covering removed. The model accounts for partial axial restraint and is shown to have good performance when compared with a finite element model.

M tot PðaÞ mi P r T

ΔT

T 0x T 0xa ; T 0xb vth vp vrth z zi

α εT ε∅ λ ρ smax sY

83

total moment at a after application of axial load, P the modular ratio of slice i axial load on column radial distance to the extreme fibre temperature (varying along the column and through the section) equivalent temperature increase of a section equivalent thermal gradient across a section T 0x ðy ¼ 0Þ and T 0x ðLÞ, equivalent thermal gradient at lower and upper end of the column thermal deflection deflection as a result of applied load deflection of a partially restrained column under thermal effects only arbitrary reference position distance in section to neutral axis at slice i from the arbitrary reference position coefficient of thermal expansion thermal strain curvature strain thermal conductivity density stress in the extreme fibre yield stress

Usmani et al. [11] describes fundamental behaviour of members subject to various restraints by providing analytical solutions to e.g. deflections and stresses of beams subject to different thermal loading, including a thermal gradient across a beam. Tsalikis et al. studied the effect of a thermal gradient on the response of a pinned steel column [12]. Using a thermal gradient which varies through the section, the authors included thermal bowing in a simple application of beam theory to an unrestrained column. This paper presents an overview of an experiment which is published elsewhere but which indicates the presence and magnitude of the likely thermal gradients through a steel section exposed to fire. We then review the means of calculating the thermal gradient and average temperature increase based on a reference value of the stiffness. This allows us to account for a temperature gradient through the column’s cross section which also varies along the column’s length. This type of thermal exposure is demonstrated to occur as a result of a localised fire around a column’s base, and it is expected that the temperature gradients would be worse in the event of a fire located adjacent to the column as opposed to directly underneath the column as in the experiment which is discussed. This is achieved through analysis of the response of a column of length L under a load, P, with a thermal gradient which varies along the height of the column. Two separate cases are studied, one where the column is simply supported, Fig. 1a, and one where the column is partially restrained, Fig. 1b, with ends restrained by springs of rotational stiffness k RA and k RB . An extension of Euler beam theory to evaluate the response of the steel column which is exposed to a localised fire is presented. The solution accounts for partial restraint of the column’s ends, and allows for rotational restraints of different stiffness to be prescribed to the ends of the column.

2. Non-uniform temperature exposure In order to facilitate the discussion of the temperature exposure of unprotected steel columns subjected to localised fires, reference

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is made to a series of tests which were carried out at SP and which are reported in Refs. [13,14]. These tests illustrate the potential for a localised fire to lead to a non-uniform temperature distribution both through the section of a steel column and along the height of the column. The test procedure as well as some of the results are summarised here, however for more detail the relevant references should be consulted. An unloaded, 6 m high steel column with a circular section (internal radius 90 mm, external radius of 100 mm) was suspended 200 mm above a series of pool fires. Steel temperatures were measured using thermocouples embedded about 1 mm into the steel, at every metre along the column’s height. At 2 and 4 m height temperatures were measured also around the section. The column was subject to several different fire exposures but only one is presented here, a 1.9 m diameter diesel pool fire, to demonstrate the effect non-uniformity of flames has on the temperatures observed in the steel element. The same behaviour was, however, seen in all the tests. During the fire tests, the plume was seen to be tilted to one side of the steel column as a result of unintentional natural ventilation conditions in the burn hall, Fig. 2. While these results do illustrate the case, a fire which is located adjacent to the column may result in greater temperature nonuniformities both across the section and along the column’s height as opposed to one which is positioned around the column. As a result of non-uniform heating of the column during the tests, significant thermal deflections were observed although no measurements were made of their magnitude.

Fig. 1. Column deflections (a) simply supported column, and (b) a partially restrained column.

The recorded steel temperatures for the 1.9 m diameter diesel fire are shown in Fig. 3 (the fuel was ignited at 2 min). The hottest region is close to the fire source and the steel reaches almost 800 1C at 1 m height after about 15 min of fire. The temperature decreases with height. The temperatures in Fig. 3a are labelled according to their height and the position on the column’s section, with positions 1 and 3 being opposite to one another. At 2 m the difference between position 1 and 3 is clearly very large and after 8 to 10 min of fire the difference is ca. 250 1C suggesting a thermal gradient of the order of 1.2 1C/mm. At 4 m, although the difference is not as high there is still a clear thermal gradient through the section. It may therefore be seen that the thermal gradient reduces along the height of the column which may be expected in this case. Fig. 3b shows the measured temperatures after 10 min along the height of the column, in the test only two data points were available on one side of the column, however it can be seen from the other side where more data points were available that a linear distribution gives a not unreasonable approximation to the measured temperatures in this case.

3. Calculation of temperature gradient through the column section In order to evaluate the response of a column to a gradient in temperature through the column’s section, and which varies along the length of the column it is necessary to determine the temperature distribution at various points in the column’s section. In structural fire engineering it is typical to idealise the temperature distribution through a composite section as a mean temperature increase and an equivalent uniform thermal gradient. Carrying out a similar analysis of a steel section, this means that where the section is subject to heating from one side the temperature distribution may be approximated to be linear. Because the thermal gradient varies with the height of the column in this analysis and because of temperature dependency of the stiffness, the equivalent thermal gradient and equivalent temperature increase can be estimated based on a single reference value of the stiffness. This will allow any calculations of the thermal deflection and any resulting thermo-mechanical loads to take advantage of this reference value while varying only the equivalent thermal gradient. The thermal gradient is evaluated using a method detailed in Ref. [15], which was developed for the analysis of composite sections. The method is based on a similar method for determining the effects of temperature on composite bridge decks and is

Fig. 2. The flame from diesel Ø ¼1.1 m. The flames are tilted towards position 1 [14].

D. Lange, J. Sjö strö m / Fire Safety Journal 67 (2014) 82–95

85

Fig. 3. Measured steel temperatures for the larger diesel fire (Ø ¼1.9 m) (a) plotted against time (solid and dashed lines represent position 1 and 3, respectively [14]); and (b) measured temperature in the steel column after 10 min.

summarised here to aid discussion. Fig. 4 shows the division of an arbitrary section, subject to one dimensional heating, into n slices along with an estimation of the temperature of each slice and the temperature dependent material properties of each slice. Each slice, i, has temperature dependent material properties, and a distance, zi, to its neutral axis measured from an arbitrary reference position, z, on the major axis. The resultant force and moment in a fully restrained section arising from the temperature increase in each slice is determined from n

F ¼ ∑ Ei Ai αi T i i¼1

n

M ¼ ðF  zÞ  ∑ F i zi i¼1

ð1Þ

ð2Þ

where Fi is the thermal force in slice i, given by F i ¼ Ei Ai αi T i . As described by Usmani and Lamont, the second moment of area of the section is based on the parallel axis theorem with the contribution of each of the areas corrected by a suitable modular ratio to a given reference modulus. This gives a second moment of area of the heated section about the major axis (for consistency with the global axis definitions used throughout this paper, the major axis for bending in the section is assumed to be about the z-axis) n

I zz ¼ ∑ ðI zzi =mi Þ þ ðAi =mi Þðzi  zÞ i¼1

ð3Þ

where I zzi and Ai are the second moment of area and the area of the slice i respectively, and mi is the modular ratio of the slice. A similar procedure may be used to find the effective area of the section n

A ¼ ∑ ðAi =mi Þ i¼1

ð4Þ

Because each of the slices has a different temperature and the coefficient of thermal expansion is dependent upon the temperature, the net coefficient of thermal expansion of the section must also be modified appropriately based upon the contribution of each of the slices. This is done by simply averaging the expansion over the total area, accounting for the modular ratio

α¼

1 n ∑ α ðA =m Þ Ai¼1 i i i

ð5Þ

Finally, the total thermal strain εT , curvature strain ε∅ and the equivalent temperature increase ΔT and equivalent thermal gradient, T 0x on the section may be determined from the following equation based on any arbitrary reference stiffness (again, for consistency with the global axis definitions used throughout this paper, the temperature gradient through the section is through the

Fig. 4. The division of the steel section into slices, adapted from [14].

x-axis of the section)

εT ¼ ðF=Eref AÞ; and ΔT ¼ εT =α

ð6Þ

ε∅ ¼ ðM=Eref Itot Þ; and T 0x ¼ ε∅ =α

ð7Þ

When determining any distribution of thermal gradient along a column’s height it is necessary to take at least two sections of the column for analysis. In this case, the same reference stiffness must be used for both sectional analyses. This allows any thermal deflection to be calculated without having to account for a temperature dependent stiffness along the column. More complex however is the consideration of differences in thermal expansion in the sectional analyses. In order to account for this, it is proposed to use the average value of thermal expansion from the sectional analyses when determining the thermal deflections. This is deemed reasonable since it will be approximately equal to the coefficient of thermal expansion at the mid height where the thermal deflections are largest.

4. Column behaviour under non-uniform thermal gradient This section details the mechanical response of columns assuming the different mechanical boundary conditions shown in Fig. 1. The solution proposed is a 2-dimensional model which does not take account for axial deformation or 3D effects such as torsion. 4.1. Case 1—Simply supported 4.1.1. Thermal deflections Consider the behaviour of the simply supported column shown in Fig. 1a. The column is of height (or length) L, is pinned at the bottom and is free to translate along the y-axis at the top. The

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column is exposed to a thermal gradient through the section and which varies along the height of the column, T 0x ðyÞ, and an average temperature increase which also varies along the height of the column, ΔTðyÞ. When determining the deflected shape of the column it is assumed that the net thermal expansion may be ignored and that only the geometric effect of the thermal gradient is considered. In this analysis, the geometric change associated with the temperature increase is considered first, the deflected shape resulting from the temperature change is denoted vth . Throughout this paper all derivatives are taken with respect to the height of the column, along the y-axis, except the thermal gradient which is always expressed as a gradient in the x-axis through the section. To simplify the nomenclature used, derivatives in the y-axis are not explicit in the subscripts. For a simply supported element subject to a thermal gradient only, the thermal curvature (v″th , the second derivative of the thermally deflected shape) may be determined from the thermal gradient

illustrates three possible variations in thermal gradient, including an example where the thermal gradient is constant along the height of the column and a case where the thermal gradient changes sign along the column’s height (representing e.g. a situation of an adjacent fire tilted towards the column). All of these examples have the same average value of thermal gradient along the height of the column. Fig. 5b shows the resulting deflected shapes for these cases. The constant thermal gradient results in a symmetrical displacement of the column. With increasing variation in the gradient, with the same value at midheight; the maximum deflection is larger and the location of the maximum deflection moves towards the end of the column with the highest absolute value of thermal gradient. For interest, in the simply supported case shown, the maximum thermal deflection for linearly varying thermal gradient is to be found at

v″th ¼  α

y¼ 

T 0x ðyÞ

ð8Þ

Assuming that the thermal gradient varies linearly along the column height, i.e. T 0x ðyÞ ¼ T 0xa þ f y (where T 0xa is the thermal gradient at position a on the column ðy ¼ 0Þ, f is the slope of the thermal gradient with height between position a and position b where the thermal gradient is denoted T 0xb ), and assuming that any displacement of the top of the column as a result of thermal effects is small and may be neglected, Eq. (8) can be integrated twice (accounting for the boundary conditions vth ð0Þ ¼ vth ðLÞ ¼ 0).

α 1 αL vth ðyÞ ¼  f y3  αT 0xa y2 þ ð3T 0xa þf LÞy 2 6 6

ð9Þ

The same procedure may be followed for other variations in thermal gradient with height so long as it may be expressed using a continuous function. The resulting deflected shape for a range of different thermal gradient variations is shown in Fig. 5. The column chosen is the same as the column which was used in the experiments which are described above with simple material properties chosen for illustration (L ¼6 m; I ¼5.4e7 mm4; E¼ Eref ¼ 200,000 MPa; α ¼1.4e 5). This column will be used throughout this section to illustrate the different stages in the analysis. Fig. 5a

T 0xa  f

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðT 0xa =f Þ2 þ T 0xa L=f þ L2 =3

ð10Þ

4.1.2. Mechanical deflections Introducing the load, P, as shown in Fig. 1a on the column with initial deflection corresponding to the thermally deflected shape induces a P  δ moment and a corresponding increase in the deflection as a result of the application of the load, vp. From beam theory, the moment over the height of a column with load P and initial deflection vth is given by: M ¼  Pðvp þ vth Þ ¼ EIv″p ð11Þ pffiffiffiffiffiffiffiffiffiffi Writing k ¼ P=EI this gives us the following differential equation 2

2

v″p þ k vp ¼  k vth

ð12Þ

with boundary conditions vp ð0Þ ¼ vp ðLÞ ¼ 0. The solution to Eq. (12) is given by: vp ðyÞ ¼ C 1 sin ky þ C 2 cos ky  vth þv″th =k

2

ð13Þ

Fig. 5. (a) Linear variations of effective thermal gradients along a 6 m long column, (b) the corresponding thermal deflections assuming no net expansion (only thermal bowing).

D. Lange, J. Sjö strö m / Fire Safety Journal 67 (2014) 82–95

with

α

1

½v″thðaÞ cos kL v″thðbÞ  ¼ 2 ½T 0xb  T 0xa cos kL 2 k sin kL k sin kL 1 α C 2 ¼  2 v″thðaÞ ¼ 2 T 0xa k k

C1 ¼

Thus, the total deflection, v, of the column is given by vðyÞ ¼ C 1 sin ky þ C 2 cos ky þ

v″th k

ð14Þ

2

For the linearly varying cases in Fig. 5 Eq. (14) is written vðyÞ ¼ C 1 sin ky þ C 2 cos ky  αT 0x ðyÞ=k

2

ð15Þ

The total deflection for the column in Fig. 1a subject to different loads is shown in Fig. 6. The three cases of thermal boundary conditions given in Fig. 5a are shown. 4.1.3. Failure criteria The largest stress in the column section will occur in an extreme fibre, and is not constant along the column’s height

smax ðyÞ ¼ 

MðyÞr P þ I A

ð16Þ

where r is the radial distance to the extreme fibre. Since MðyÞ ¼  Pðvth ðyÞ þ vp ðyÞÞ ¼  PðvðyÞÞ, this can be written as:

smax ðyÞ ¼ 

PðvðyÞÞr P þ I A

ð17Þ

Which, for the linearly varying case can be written as:   P sin ky þT 0xa cos ky T 0x ðyÞ smax ðyÞ ¼ þ rEα ðT 0xb  T 0xa cos kLÞ A sin kL ð18Þ Fig. 7 shows the stresses along the height of the column discussed above subject to loads of 200, 400 and 800 kN for different thermal gradients. Again, the same three distributions of thermal gradient are considered. Increasing thermal deflections result in larger P  δ moments and mechanical deflections as well as correspondingly larger stresses in the extreme fibre. For a thermal gradient which varies along the length of the column the maximum stress in an extreme fibre will not occur at mid-height. In order to determine the failure load of a column under these conditions the stress in the extreme fibre along the column’s length must be compared with the yield stress in the same fibre along the column’s length. For a given thermal

0

0

87

exposure the yield stress will vary with the temperature in the fibre along the y-axis, sY ðT f Þ. Values for the temperature dependent yield stress of steel are taken from EN 1993-1-2 [1]. This should be based not only on the average increase in column temperature, ΔT, but also the thermal gradient. Fig. 7 shows the stress in the extreme fibre of a steel column exposed to the same thermal gradients as in the example used above to illustrate thermal and total deflections. Plotted on the same figure is the temperature dependent yield stress in the column, assuming a linear temperature distribution in the extreme fibre from 550 1C at y¼0 to 150 1C at y¼ L. Failure is assumed to occur when the stress exceeds the yield stress, smax ðyÞ 4 sY ðyÞ. Increasing the thermal gradient increases the thermal deflection and correspondingly increases the stress in the extreme fibre. Increasing the thermal gradient at one of the ends of the column while maintain the same gradient at the mid height will result in higher stresses forming towards the end with the larger gradient. 4.2. Case 2—Partial rotation restraint 4.2.1. Thermal deflection Including in the above model the effects of partial rotational restraint, as shown in Fig. 1b, requires the inclusion of the resisting moment from rotational springs and the corresponding reduction in the rotations at the supports. In this representation the rotational restraints can have two different values of stiffness, kRA and kRB, as a result of either construction details or the effect of an increase in temperature of the construction which reduces the stiffness of the rotational restraint. Everything else in this example remains the same as the previous, simply supported, case. Considering only the thermally induced deflections of the column, the partial restraint reduces the rotation at the top and bottom of the column through the application of a restraining moment. Exemplifying once more using a linear variation of T 0x along the length of the column, the free rotation at a and b as a result of the thermal gradient are given by the values of the first derivative of vth at y ¼a and at y¼b, denoted hereafter v0thðaÞ and v0thðbÞ v0thðaÞ ¼

αL 6

ð3T 0xa þ f LÞ

ð19Þ

α αL v0thðbÞ ¼  f L2  αT 0xa L þ ð3T 0xa þ f LÞ 6 2

0

0

0

ð20Þ

0

Fig. 6. Total deflection of a column under different loads; (a) Txa ¼Txb ¼ 3 1C/mm, (b) Txa ¼ 6 1C/mm Txb ¼ 0 1C/mm, and (c) Txa ¼ 9 1C/mm Txb ¼  3 1C/mm.

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0

0

0

0

Fig. 7. Stress in the extreme fibre of the column with varying levels of load, (a) Txa ¼ Txb ¼ 3 1C/mm, (b) Txa ¼9 1C/mm Txb ¼  3 1C/mm.

The total rotation at a and b also has a component resulting from the restraining moment at a and b. Denoting the restraining moments at a and b from the partial restraint to thermal expansion as M rthðaÞ and M rthðbÞ respectively, as well as the deflection of a partially restrained column under thermal effects as vrth , and therefore the rotation at a from the restraining moment at a as r0 vr0 aa , and the rotation at a from the restraining moment at b as vab , the restraining moments at a and b may be related to the rotations at a as: r vr0 aa ¼  M thðaÞ L=3E ref I

ð21Þ

r vr0 ab ¼  M thðbÞ L=6E ref I

ð22Þ

Similar equations may then be written for the components of the rotations at b arising from the contribution of the restraining r0 moments at a and b, vr0 bb and vba . Summing the rotations at a and b gives the total rotation at the ends including the result of partial restraint to the thermally induced curvature r0 v0thðaÞ þ vr0 aa þ vab

ð23Þ

0 r0 r0 vr0 thðbÞ ¼ vthðbÞ þ vbb þ vba

ð24Þ

vr0 thðaÞ

¼

Multiplying these expressions by the rotational stiffness at a and b respectively gives the restraining moment at a and b to the thermal curvature r0 M rthðaÞ ¼ kRA ðv0thðaÞ þvr0 aa þ vab Þ

ð25Þ

r0 M rthðbÞ ¼ kRB ðv0thðbÞ þvr0 bb þ vba Þ

ð26Þ

Eqs. (25) and (26) may be solved simultaneously to obtain the moments arising from the partial restraint to thermal bowing at both ends. These moments effectively reduce the rotations arising from the thermal gradient, opposing a part of the thermal curvature strain as an elastic thermo-mechanical strain. Equating the unrestrained thermal curvature to a notional moment, the remaining curvature at a and b in the partially restrained column can then be determined from (note the use of E and Eref) vr″ thðaÞ ¼

Eref I αT 0xa  M rthðaÞ EI

ð27Þ

vr″ thðbÞ ¼

Eref I αðT 0xb Þ  M rthðbÞ EI

ð28Þ

Since the thermal gradient varies linearly with height, the residual thermal curvature also has a linear variation. Denoting the slope of the variation in residual curvature (following application of the restraining moment from the springs to reduce the thermal effect) with height f r r″ f r ¼ ðvr″ thðbÞ  vthðaÞ Þ=L

ð29Þ

The deflected shape can now be calculated in the same way as for a simply supported column, by taking the second integral of the curvature, Eq. (30), and applying the boundary conditions vrth ð0Þ ¼ vrth ðLÞ ¼ 0 once more we obtain the equation of the deflected shape, Eq. (31). r″ vr″ th ðyÞ ¼ vthðaÞ þ f r y

ð30Þ

1 1 L vrth ðyÞ ¼  f r y3  vr″ y2 þ ð3vr″ þf r LÞy 6 2 thðaÞ 6 thðaÞ

ð31Þ

Fig. 8 shows some examples of the deflected shape of the column described above given different distributions of thermal gradient along the length of the column and different stiffness of end restraint. Fig. 8a to c show examples of the thermal deflections with the same stiffness at each end of the column, ranging from 1  108 N/mm to 1  1010 N/mm, and with the same distributions of gradient as shown in Fig. 5. The impact of increasing the rotational stiffness is clear, with significantly smaller deflections in the column with increasing rotational stiffness of the restraints. Fig. 8d shows the effect of different rotational restraint stiffness on the column, with a much smaller rotational stiffness at the top of the column the gradient of the deflected shape is much larger at the upper end of the column. 4.2.2. Mechanical deflection In order to determine the effect of the application of the vertical load on the deflected shape where partial restraint is provided, the following steps must be taken: starting with the an initial shape which corresponds to the deflection of the column from thermal effects including the effects of the partial end restraint, release the end restraints and calculate the expression

D. Lange, J. Sjö strö m / Fire Safety Journal 67 (2014) 82–95

89

Fig. 8. Column thermally deflected shape given different distributions of thermal gradient and different combinations of partial restraint at the ends (a) kRA ¼ kRB ¼1  108 N/mm; (b) kRA ¼kRB ¼ 1  109 N/mm; (c) kRA ¼kRB ¼ 1  1010 N/mm; and (d) kRA ¼ 1  1010 N/mm and kRB ¼1  108 mm.

for the deflection of the now pinned column under a compressive load P with an initial shape from the previous step. Then, using the same procedure as was described above to determine the applied moment from the end restraint to the thermal curvature determine the restraining moments to the mechanical load which are imposed by the rotational restraints. At this stage the moment which is distributed through the column as a result of the partial end restraint to thermal effects is not included, and will be added in later. Returning to Eq. (12), we now write the moment in the column with an initial shape equal to vrth as: 2

2

v″p þ k vp ¼ k vrth

ð32Þ

The solution to which is, similar to Eq. (13), and referring to Eqs. (30) and (31) vp ðyÞ ¼ C 1 sin ky þ C 2 cos

2 ky  vrth þ vr″ th =k

ð33Þ

With constants C 1 ¼  C 2 cos kL C2 ¼

M rthðaÞ

vr″ th k

2

simultaneously rP0 M rPðaÞ ¼ kRA ðv0PðaÞ þ vrP0 aa þ vab Þ

ð36Þ

rP0 M rPðbÞ ¼ kRB ðv0PðbÞ þvrP0 bb þ vba Þ

ð37Þ

Finally, by superposition, we can combine the effects of thermal expansion, partial restraint to thermal expansion, and the partially restrained increase in curvature under an applied load P (which must be opposite in sign to curvature arising from the moment which restrains thermal expansion) vr″ PðaÞ ¼ vr″ PðbÞ ¼

Eref I αT 0xa  M rthðaÞ M rPðaÞ EI Eref I αðT 0xa þ f LÞ M rthðbÞ  M rPðbÞ EI

Eq. (33) gives the deflected shape of a pinned column with initial shape equal to vrth and subject to axial load P applied at the top. We now apply partial end restraints and calculate once more the restraining moment which is applied by the springs of stiffness kRA and kRB. Following the same procedure as above, we denote the restraining moments at a and b from the partial restraint to the applied load P as M rPðaÞ and M rPðbÞ respectively, and the rotation at a from the restraining moment at a as vrP0 aa , and the rotation at a from ' , The restraining moments at a the restraining moment at b as vrP ab and b may be related to the rotations at a as (again, similar expressions may be written to describe the moments as a result of rotation at b) r vrP0 aa ¼  M PðaÞ L=3EI

ð34Þ

r vrP0 ab ¼  M PðbÞ L=6EI

ð35Þ

Summing the rotations at both ends, we can write similar equations to (23) and (24), before multiplying by the stiffness of the partial restraint to obtain the moments resulting from the partial restraint for the case in hand which may again be solved

ð39Þ

Denoting slope of the variation in residual curvature (following application of the restraining moment from the springs to reduce the thermal effect and the curvature from the applied load) with height f P r″ f P ¼ ðvr″ PðbÞ vPðaÞ Þ=L

P

ð38Þ

ð40Þ

The deflected shape is once more calculated by taking the second integral of the curvature, Eq. (41), and applying the boundary conditions vrP ð0Þ ¼ vrP ðLÞ ¼ 0 we obtain the equation of the deflected shape, Eq. (42). r″ vr″ P ðyÞ ¼ vPðaÞ þ f P y

ð41Þ

1 1 L y2 þ ð3vr″ þ f P LÞy vrP ðyÞ ¼  f P y3  vr″ 6 2 P 6 PðaÞ

ð42Þ

Fig. 9 shows some examples of the resulting total deflected shapes calculated from Eq. (42). The four cases shown are as follows: (a) kRA ¼kRB ¼2  108 N/mm, thermal gradient is constant along the height of the column at 3 1C/mm; (b) kRA ¼kRB ¼ 2  109 N/mm, thermal gradient varies along the height of the column from 5 1C/mm at the base to  1 1C/mm at the top; (c) kRA ¼kRB ¼2  1010, thermal gradient varies along the height of the column from 6 1C/mm at the base to 0 1C/mm at the top; (d) kRA ¼1  1010 N/mm and kRB ¼1  108 N/mm, thermal gradient varies along the height of the column from 9 1C/mm at the base to  3 1C/mm at the top. On each panel is plotted the thermal deflection based on the mechanical boundary conditions and the

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Fig. 9. Total deflections of a column subject to different non-uniform thermal gradient along its length and different loads.

thermal gradient varying along the length of the column, as well as the final deflected shape based on the calculation method proposed and, in each panel, 3 different applied loads.

The distribution of moment along the length of the beam is still linear and therefore the moment as a function of y is given by: MðyÞ ¼  Pðvrth ðyÞ þ vrP ðyÞÞ þ

4.2.3. Failure criteria As with the pinned case, the failure criteria for the partially restrained column are where the maximum stress in an extreme fibre of the column exceeds the temperature dependent yield stress of the column. Referring once more to Eq. (17), the maximum stress as a function of the height in the column can be determined by substitution of the appropriate equations of deflection from the partially restrained case. However there is also a moment arising from the partial restraint at both ends which must be added to the total moment in this instance. 0 r r M tot PðaÞ ¼ E ref I αT xð0Þ  M thðaÞ  M PðaÞ

ð43Þ

0 r r M tot PðbÞ ¼ E ref I αðT xð0Þ þf LÞ  M thðbÞ  M PðbÞ

ð44Þ

tot ðM tot PðbÞ  M PðaÞ Þ y þ M tot PðaÞ L

ð45Þ

Substitution of this into Eq. (17) yields the following expression for the maximum stress in an extreme fibre

smax ðyÞ ¼ 

tot tot ½  Pðvrth ðyÞ þ vrP ðyÞÞ þ ðM tot PðbÞ  M PðaÞ =LÞy þ M PðaÞ r

I

þ

P A ð46Þ

Fig. 10 shows a number of examples of the maximum stress in a partially restrained column. The four cases shown are as follows: (a) kRA ¼ kRB ¼2  108 N/mm, thermal gradient is constant along the height of the column at 3 1C/mm; (b) kRA ¼kRB ¼2  109 N/mm, thermal gradient varies along the height of the column from 5 1C/mm at the base to  1 1C/mm at the top; (c) kRA ¼kRB ¼ 2  1010, thermal gradient varies along the height of the column from 6 1C/mm at the base to 0 1C/mm at the top; (d) kRA ¼1  1010 N/mm and kRB ¼1  108 N/mm, thermal gradient varies along the height of the column from 9 1C/mm at the base to  3 1C/mm at the top. Where the stress in the column under a load P crosses the dashed line which represents the yield stress as a function of temperature

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Fig. 10. Stress in an extreme fibre of a column subject to different non-uniform thermal gradient along its length and different loads.

then the column may be assumed to fail. A higher rigidity of the rotational restraint results in a smaller thermal deflection and a lower stress in the extreme fibre, when compared to a column with lower rotational restraint stiffness at the ends. The importance of the non-uniform thermal gradient is clear since the highest stresses in all of the column’s, except for the case where the gradient is uniform, clearly occur in the half of the column which is subject to the higher magnitude of thermal gradient. The y-ordinate at which the stress in the column crosses the yield stress is important since it may give an indication of the failure mode. Where the stress crosses the yield stress at the bottom of the column, then bending stress as a result of the P  δ moment is small and failure may be assumed to be dominated by squashing— in such instances the thermal deflection makes little difference. Whereas if the stress crosses the yield stress closer to mid height of the column then the P  δ moment is larger as a result of larger thermal deflections and bending may be assumed to be the dominant failure mechanism.

5. Comparison with numerical analyses In order to demonstrate the capabilities of the proposed model and to validate the assumptions made the deflections calculated

using the proposed model are compared in this section with a number of finite element models of columns with different degrees of mechanical restraint and exposed to different thermal boundary conditions. Since the column without partial restraint at the ends is equivalent to the sprung column with no rotational spring stiffness, the comparison is restricted to a comparison of the column with partial restraint, varying the degree of restraint. The finite element model which is used for comparison is comprised of linear formulation Timoshenko beam elements. Partial restraint is provided by additional beam elements which are placed at the top and bottom of the beam, above and below the ‘length’ which defines the column and outwith the restraints for the column which represent simply supported end conditions. These additional elements are then restrained in order to provide additional rotational restraint to the column ends. Rotational stiffness is adjusted by changing the stiffness of these elements. In all cases the top of the column is permitted to translate vertically. The analysis comprises two steps, a thermal step where the thermal gradient and any temperature increase are applied so that the thermal deflection may be evaluated; and then a subsequent step where the mechanical load is applied. In order to account for P-delta effects non-linear geometric effects are accounted for in the analysis. The analysis is performed in the Abaqus FEA [16].

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Fig. 11. Comparison between the proposed methodology (lines) and the results of finite element analysis (symbols).

Evaluation of the analytical method is shown in Fig. 11. In all cases the symbols are used to represent the numerical analyses whereas the lines represent the analytical method described. In all instances shown the column is of 3 m height. Fig. 11a shows the column with a rotational stiffness of 1.5  109 N/mm at the top and the bottom. The modulus of elasticity and the coefficient of thermal expansion are constant (E ¼Eref ¼ 200,000 MPa; α ¼1.4e  5), thermal gradient is 4 1C/mm along the entire height of the column and the applied load is 1  106 N. Excellent agreement can be seen between the model proposed and the finite element calculation. Fig. 11b shows the same case with a nonuniform thermal gradient in the column, 4 1C/mm at the base of the column decreasing linearly to 0 1C/mm at the top of the column, P ¼5  105 N. In this case the deflections predicted by the proposed method are slightly higher than the finite element model; nevertheless good agreement is seen. The difference in the total deflections in this instance can be attributed to a small difference in the predicted thermal displacement. The same effect can be seen in Fig. 11c where the same column under the same loading is subject to a thermal gradient decreasing from 1.5 1C/mm at the base to 0 1C/mm at the top. Changing the rotational restraint in the model has the effect shown in Fig. 11d from 1.5  109 N/mm at the top to 1.5  107 mm, with a thermal gradient reducing linearly from 4 1C/mm at the bottom to  2 1C/mm at the top.

Once more excellent agreement is shown between the proposed methodology and the finite element results. In the examples shown here and in the examples shown earlier in the article, the thermal gradient is relatively small. With increasing thermal gradient and subsequent increasing thermal deflection the error in the proposed method is likely to increase since the analytical model proposed is based on small displacements. Nevertheless, as demonstrated by the experiments discussed in Section 2, thermal gradients in steel sections exposed to localised fires are likely to be small, of the order of 1 to 1.5 K/mm. The limitations of the proposed method may therefore be deemed acceptable.

6. Proposed analysis methodology With increasing temperature there is a corresponding decrease in the strength and stiffness of a steel section. Although the temperature dependent strength is accounted for above the temperature dependent stiffness is not accounted for directly in the analysis method. In order to make use of the method which is described above, the temperature dependent material properties, including the coefficient of thermal expansion, must be reconciled somehow; while it would be possible to include in the above

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derivations the temperature dependent stiffness and thermal expansion this would add an additional level of complexity to the above formulae. Therefore, the following methodology is proposed for analysis, accounting for the temperature dependent material properties. 1. Calculate the temperature distribution of the section based on some suitable fire exposure, for example a travelling fire, a plume adjacent to the column, or external flames impinging on an external column (do not assume lumped capacitance). 2. Using the method which is described in Section 3, determine the mechanical properties as well as equivalent areas and section moduli based on the temperature and a reference value of the stiffness for the section split into slices and then determine the equivalent thermal gradient and temperature increase in the section. The thermal deflection can be based upon the reference stiffness which is used when calculating the thermal gradient and the average temperature increase in the section; however the mechanical deflection should be calculated based upon the mean stiffness accounting for the average equivalent temperature increase from the two sections. In order to better reconcile the thermal and the mechanical deflections, the reference stiffness used for the calculation of thermal deflections should be the same stiffness that is used for the mechanical calculation. 3. Using the reference value of the stiffness and the average temperature dependent coefficient of thermal expansion from the sectional analyses determine the thermal deflection of the section, accounting for the mechanical boundary conditions. 4. Determine the mechanical deflection of the column as a result of the applied load and P  δ moments arising from the eccentricity, assuming stiffness equal to the average stiffness in the column. 5. Calculate the stress in the extreme fibre and compare this with the yield stress in the extreme fibre along the full length of the column. Where the stress in the column exceeds the yield stress at a particular height then the column may be assumed to have failed. In order to evaluate the effect of non-uniform stiffness on the column, and to justify the use of the average stiffness in Step 4, a comparison is made between the deflected shapes assuming temperature dependent stiffness in a finite element model and the proposed methodology using the stiffness at mid-height to calculate the total deflection. The model and the boundary conditions, including thermal gradient are the same as the model which is shown in Fig. 11d. In this case however the column is exposed to a temperature which varies linearly between 550 1C at the base to 83 1C at the top of the column. The mean stiffness of the column which is used for the calculation of the displacement in the analytical model is 164 kN/mm2, corresponding to the stiffness of steel at a temperature of 315 1C (the temperature at mid height). The steel temperature in the numerical analysis is assumed to vary according to the temperature dependent properties in Eurocode 3. The comparison between the two methods is shown in Fig. 12. It can be seen that the proposed method over predicts the total displacement as a result of differences in the mechanical response and the thermal displacement. The application of mechanical load to the finite element model translates the position of maximum deflection upwards towards the mid-height of the column. While this happens the deflection below the mid-height reduces slightly, which means that the mechanical portion of the deflection is slightly negative. This effect is not captured in the proposed analytical methodology which results in slightly larger predictions

Fig. 12. Comparison between the proposed method assuming an average stiffness and finite element analysis accounting for temperature dependent stiffness.

of the deflection. Nevertheless the analytical model shows good correlation with the finite element analysis. 6.1. Example In the discussion and derivations presented above, the examples which are given to illustrate the effects of the thermal gradient assume a very large gradient through a column section. In reality, and as evidenced in the tests reported in Section 2, the thermal gradient across a steel column is unlikely to be more than 1 to 1.5 1C/mm at the most. In this section, an example of the methodology applied to a 4 m long column exposed on one side to a pool fire is presented. The column is a slender column, UK Section 203  203  46. Three mechanical boundary conditions are studied: a simply supported column, a column with kRA ¼kRB ¼ 1  1010 N/mm and a column with kRA ¼1  1010 N/mm and kRB ¼1  108 N/mm. The fire exposure, chosen arbitrarily, comes from a pool fire with a heat release rate of 2 MW. The plume temperature is determined from the simple model presented in Appendix C of EN 1991-1-2 [17]. A simple finite element heat transfer model was used to determine the temperature distribution in the steel section (although alternative methods could be employed). The objective was to have a demonstrative temperature distribution without labouring on this aspect of the analysis. Material properties were not varied with temperature and were: cp ¼ 475 J/kg K, λ ¼ 44.5 W/m K and ρ ¼7850 kg/m3. Emissivity of the steel was assumed to be 0.8, and of the flame 1. The radiation temperature was assumed equal to the plume temperature and the ambient air temperature to be 20 1C, with a convective heat transfer coefficient of 10 W/m2 K. Radiation was applied only to one side of one of the flanges. Using the results of the heat transfer calculation to determine the thermal gradient and mean temperature increase in the section as described in Section 3, we obtain the gradients at a and b, at various times, as summarised in Table 1 along with the mean temperature in both of the flanges and in the web of the beam at both locations. 20 slices were taken of the section to determine the gradient, although there is no need to take as many as this. The reference modulus of elasticity in the calculation was

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200,000 N/mm2. This was also taken in this analysis as the mean of modulus of elasticity in both of the sections at the top and bottom of the column which was used to determine the mechanical deflections since the temperature of the web and the cold flange was never over 200 1C in the analysis. The coefficient of thermal expansion, α, used was 1.4  10  5. Defining a margin of safety as the difference between the yield stress in the extreme fibre and the actual stress in the extreme fibre ðM ¼ sy ðyÞ  smax ðyÞÞ, the results for the pinned case are presented in Fig. 13. The applied load on the column is 1000 kN. Results are shown for times of 2, 4, 10 and 20 min. The column fails between 4 and 10 min as a result of the effects of thermal bowing and the resulting P  δ moment from the applied load. Also shown

Table 1 Thermal gradients determined in the example analysis. 0

Time (s)

Posn

T1 (1C)

T2 (1C)

T3 (1C)

T x (1C/mm)

120 240 600 1200 120 240 600 1200

a

220 352 530 550 32.5 42.5 65 83

30 50 140 180 20 22 30 40

20 20 40 50 20 20 22.5 23

0.516 1.08 0.73 0.7 0.04 0.122 0.22 0.325

b

is the margin of safety of the column at a time of 4 min when exposed to the critical load at that time (P¼ 1164 kN). Initial yield under this load and at this time occurs at a height of 1250 mm. Fig. 14a and b show the same figures for the partially restrained columns described. Fig. 13a shows the case where kRA ¼kRB ¼ 1  1010 N/mm. The effect of the partial restraint is clear, and the location of the first yield is lower, approximately 200 mm from the base of the column. In this case, the failure is a combination of the axial load as well as the mechanical moment arising from the partial restraint. Fig. 13b shows the case where kRA ¼1  1010 N/mm and kRB ¼1  108 N/mm. In this instance, the stiffer restraint ‘attracts’ a larger moment, resulting in a failure at the base of the column. A comparison between these three examples gives a clear indication of the effect of the moment arising from the partial restraint on the margin of safety in the column—provision of partial restraint reduces the thermal deflection and therefore the magnitude of the resulting P  δ moment. However it also attracts moment to the ends of the column, increasing the stress in the extreme fibre in this example at the column’s base. Practically, this suggests that a simply supported column subject to non-uniform thermal gradients and non-uniform heating will fail in bending somewhere near to the mid height of the column since this is where the largest moments exist. However in the example presented, with higher temperatures at the base of the column along with higher gradients and where the column is suitably short or stocky, or the thermal gradients small, the column will start to fail at the base as a result of squashing. Where the distribution of temperature is reversed, i.e. higher temperatures at the top of the column then this trend will also be reversed, with squashing occurring at the top of the column. Where the column is partially restrained, some of the thermal curvature and the curvature induced by the P  δ moment will be reduced resulting in a mechanical moment at the supports which is opposite in direction to the P  δ moment. This however means that the stress in the extreme fibre at either the base of the column or the top of the column, depending upon the relative level of restraint, may exceed the temperature dependent yield stress in that fibre. The exact location of the point of first yield depends upon the temperature distribution in the column as well as the relative stiffness of the restraints.

7. Discussion and conclusions

Fig. 13. Example of the margin of safety of a pinned column under non-uniform thermal gradients across the section.

Travelling and localised fires for design are becoming increasingly common in structural fire engineering, yet the application of such complex temperature fields to structural engineering has typically been limited to finite element analysis. In this paper we

Fig. 14. A example of the margin of safety if a partially restrained column under non-uniform thermal gradients (a) kRA ¼ kRB ¼ 1  1010 N/mm; and (b) kRA ¼1  1010 N/mm; kRB ¼ 1  108 N/mm.

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have presented a study of the effects of non-uniform heating along the height of a column on said column’s behaviour. From fundamental principles of structural behaviour we have included the effects of a temperature gradient through a column’s section which varies along the column’s length. Accounting for varying levels of partial rotational restraint we have derived the thermally deflected shape as well as the deflected shape following application of a mechanical load and as a result of P  δ moments. It has been shown through example that the failure of the column exposed to localised fire is strongly dependent upon the relative level of restraint provided as well as the thermal deflection arising from the thermal gradient, an aspect which would not be captured if lumped capacitance was assumed. The method discussed here could be easily adapted to represent composite beams and 1-way spanning slabs. With bending added as a load case the effects of travelling fires on horizontal construction could easily be determined from the analytical expressions developed here. It therefore represents not only an analytical solution for a column under non-uniform temperature effects, it is also a first step towards developing an analytical model which describes structural behaviour under travelling fire conditions. Extending the method towards this aim and evaluating the limits of its application is the subject of ongoing work by the authors. In the context of modern building design, the tall slender column which was the subject of the fire tests which are summarised in this paper and upon which the analytical method is described is an unusual feature. It may however feature as part of, e.g. a façade system or a light weight structure such as might be found in an airport. The method is however also shown through example to be suitable for use when studying other columns, with slightly shorter length and different section size. Nevertheless, the effects of differential heating on columns with larger section size are likely to be limited and the research presented herein is targeted specifically at slender columns. Acknowledgements This research was funded by The Swedish Civil Contingencies Agency, MSB, as part of the Safe Multibygg project.

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