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Beams with corrugated web at elevated temperature, analytical and numerical models for heat transfer
Fire Safety Journal 86 (2016) 83–94
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Beams with corrugated web at elevated temperature, analytical and numerical models for heat transfer
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Jaroslav Vácha, Petr Kyzlík, Ioan Both , František Wald Czech Technical University in Prague, Department of Steel and Timber Structures, Thakurova 7, CZ 166 29 Praha, Czech Republi
A R T I C L E I N F O
A BS T RAC T
Keywords: Corrugated web beam Elevated temperature Heat transfer Aspect ratio View factor
In the case of corrugated web beams, based on experimental results, it was noticed that temperature development along the height of the web has a distinct pattern compared to the flat web beams. The particularity consists of a temperature decrease in the web, in the vicinity of the flange-to-web junction, assigned to the heat transfer by conduction and radiation. On this assumption, starting from the simple relations given in the fire design codes an analytical model is developed considering the heat conduction from the web to the flanges and the radiation heat transfer between web and flanges. The paper presents the results of the proposed analytical method against the experimental results and the numerical simulation. Both analytical and numerical models are compared to the experimental results obtained from the isolated element fire test and the real building fire test. A parametric study on the influence of the geometric dimensions is performed regarding also the time interval for the temperature development. For short term fire exposure, the web exhibits non-uniform temperature along the web height, while for long term fire exposure the entire cross-section tends to a homogeneous temperature.
1. Introduction
with time using the previous mentioned relations depends on the heat transfer by convection and radiation, while conduction is not considered. Owing its small massivity, the temperature in the thin steel part i.e. the beam web, increases at a higher rate than thick parts i.e. the flanges, therefore conduction plays an important role, especially at the junction area between flange and web. In fire design computation, a great importance is attributed to the temperature distribution which can be regarded as a critical stage of fire design since most mechanical models predict similar deformation/ time characteristics according to [3]. As mentioned above, in the case of high conductivity materials i.e. steel, the heat conduction has a great importance for significant temperature difference between two parts of steel cross-section. For the commonly used steel profiles with flat web, the temperature distribution within the parts of cross-section has minor effect along the part thickness and function of the exposure temperatures, it may be regarded either as a uniform temperature or as a thermal gradient through the cross-section [4,5]. Direct exposure to fire of cold-formed steel members exhibit no distinguishable temperature difference due to conduction. In the case of thin-walled steel panels, where the steel profiles are protected, a greater temperature difference is developing between the flange closer to the fire exposed face of the wall and the flange fixed to the unexposed face of the wall leading to conditions proper for observing heat conduction effect [6].
Mostly used in bridge engineering, the beams with corrugated webs have increased their use in the industrial structural systems providing not only material savings but also an attractive aspect through the trapezoidal or sinusoidal corrugation of the web. The design of industrial buildings involves the fire design situations therefore the response of these beams at elevated temperature should be evaluated. Due to the shape of the web, the height-to-width aspect ratio and the greater flange-to-web thickness ratio of corrugated web beams in comparison to the flat web beams, the thermal response of such beams implies particularities of temperature development over the steel crosssection. In a preceding paper [1], the authors presented experimental results for both isolated elements and real-scale building tests and this paper is a continuation of the study of elevated temperature influence on beams with corrugated web from the point of view of thermal response. The aspect ratio of the web of such beams would categorize the cross-section into a thin-walled profile but the thicker flanges makes these beams behave distinctly for both thermal and structural actions. The temperature development in commonly used flat web steel beam profiles can accurately be determined using the standardized relations recommended in EN 1993-1-2 [2]. The temperature increase
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Corresponding author present address at: Politehnica University of Timisoara, Department of Steel Structures and Structural Mechanics, Ioan Curea 1, 300224, Timisoara, Romania. E-mail addresses: [email protected] (J. Vácha), [email protected] (P. Kyzlík), [email protected] (I. Both), [email protected] (F. Wald).
http://dx.doi.org/10.1016/j.firesaf.2016.09.001 Received 19 May 2015; Received in revised form 9 September 2016; Accepted 22 September 2016 Available online 04 November 2016 0379-7112/ © 2016 Elsevier Ltd. All rights reserved.
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Nomenclature
Φ ΦA − B, i
area of top face of top flange on which heat is radiated [m2] Area of the ith element considered as an emitter Ai Am / V section factor of steel member [m−1] section factor of flange [m−1] Am, f / Vf Am, w / Vw section factor of web [m−1] C correction factor K emission coefficient L mean beam length [m] Li length of an element I [m] ΔQ1, ΔQ2 heat fluxes from the web to the top and bottom flanges [J] ΔQ1, n , ΔQ2, n intermediate heat fluxes from web to top and bottom flange [J] ΔQi heat flux from element i-1 to element i [J] V volume of the gas between flange and steel deck [m3] WTA, (B ), (C ) symbol for web thicknesses of 2, 2.5 and 3 mm bf 1, bf 2 width of top and bottom flange [m] specific heat of steel [J/kg K] ca h1w , h2w partial web height for upper part of web and lower part of web [m] h1w, n +1 intermediate values for partial web height [m] h1w,0 , h2w,0 initial partial web heights [m] ̇ net heat flux per unit area [W/m2] hnet ̇ ,c net heat flux to unit surface area due to convection [W/ hnet m2] ̇ ,r net heat flux to unit surface area due to radiation [W/m2] hnet hnet , r , A, i radiative heat transfer for element i of the web [W/m2] hnet , r , B radiative heat flux for the flange [W/m2] h wL width of the web strip with min 5% lower temperature than the maximum temperature point on web [m] hsi vertical distance from flange to element i [m] ksh correction factor for the shadow effect m f 1, m f 2 masses of the top flange and bottom flange [kg] mass of element i [kg] mi mw mass of the web [kg] s corrugated web length of the web [m] distance from half flange centroid to element i [m] si t f 1, t f 2 thickness of top and bottom flange [m] ti thickness of element i [m] ti −1, ti +1 thicknesses of the preceding adjacent element and next adjacent element [m] tw web thickness [m] Δt time interval [s] 2w wave length of the sinusoidal web [m]
A
αc β δ1, δ 2 δ1, n, δ 2, n
δi εA, εB εf εm ϕA, ϕB κTR λ θA, i θB
θi θf 1, θf 2 θg θm θr θw Δθ1, Δθ2
Δθa, t Δθa, t , A, i Δθa, t , B
Δθi, i −1 Δθi, i +1
Δθi Δθw ρa σ
configuration factor configuration factor for radiative heat transfer from web to flange coefficient of heat transfer by convection [W/m2 K] angle of incidence of heat radiation [°] mean distance of conductive heat transfer from web to top and bottom flange [m] intermediate values for mean distances of conductive heat transfer [m] distance from element i-1 to element i [m] emissivity of the web and the flange emissivity of flames, of the fire surface emissivity of the member surfaces relative orientation angles [°] covering factor thermal conductivity [W/mK] temperature of ith element on the web in radiative heat transfer computation [°C] flange temperature for radiative heat transfer computation [°C] temperature of element i [°C] temperature of the top flange, bottom flange [°C] gas temperature in the fire compartment [°C] temperature of the member surface [°C] effective radiation temperature of the fire environment [°C] web temperature [°C] temperature variation of top flange, bottom flange and web [°C] the increase of temperature in an unprotected steel member during a time interval Δt [°C] web temperature variation due to radiation of element i [°C] flange temperature variation due to radiation from web [°C] temperature variation of element i induced by the heat transfer from element i-1 to element i [°C] temperature variation of element i induced by the heat transfer from element i to element i+1 [°C] total temperature variation of element i caused by the heat transfer from element i-1 to i and from i to i+1 [°C] temperature variation of the web [°C] unit mass of steel [kg/m3] Stephan Boltzmann constant (=5,67×10−8 [W/m2 K4])
heated part of the cross-section, i.e. the web, to colder parts of crosssection, i.e. the flanges. The procedure starts by establishing first the point of highest temperature in the web and then, by dividing the web and the flanges into finite elements, the temperature variation along the web height is determined. The comparison of results is performed with respect to finite element heat transfer analysis and experimental results obtained from the tests performed at Czech Technical University in Prague, presented by the authors in [1].
The measurements performed by Feng et al. [7] revealed a higher temperature in the flange close to the unexposed face of the wall for the specimens that had continuous web between flanges than the specimens where the flange had a service hole. Another situation which exhibits steel conduction effect was presented by Wang [8] for partially protected beams, where, for a web protected on one quarter of its height, the temperature in the flange had a higher value than for the case of a fully protected cross-section. The heat transfer principles for furnace tests are clearly described in [3] for both convection and radiation interaction between furnace and elements, while in [9] it is presented the influence of steel type on thermal interaction properties. Moreover, due to high temperature difference at early stages of heating it might be considered that a radiation interaction exists between parts of steel beam. The paper presents an analytical model which assesses the temperature distribution in a corrugated web steel beam considering the heat conduction from the web to the flanges and the radiation of more
2. Heat transfer in steel beams The fire resistance of structural elements during fire is highly influenced by the temperature, thus, the accuracy of predicting thermal field in the cross-section is proportional to the precision of the thermomechanical analysis results. The beams with corrugated web may be considered a non-conventional steel profile and, as presented in the following, has a distinct distribution of temperatures than in the case of 84
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point for the case of distinct temperatures between the bottom and the top flanges [11]. For this, the heat fluxes from the web to the top flange, ΔQ1, and to the bottom flange, ΔQ2 , are calculated function of the mean distance of thermal transfer and of the temperature difference between web and top flange, bottom flange, respectively (4), (5).
Table 1 Temperature of IPE400 according to EN 1993-1-2. IPE400
Temperatures [°C]
Time [min]
Full cross-section kshAm/V=123.2 m−1
5 10 15 20
209.9 446.6 612.3 706.8
Flange Am/ V=155.7 m−1 248.4 502.7 653.3 725.7
Web Am/ V=232.6 m−1 325.2 582.5 695.8 737.1
2.1. Prescriptive incremental procedure In section 4.2.5 of EN 1993-1-2 [2] simple relations for temperature increase in a time interval less than 5 s are given for both unprotected and protected steelwork. The current study regards unprotected steel sections for which relation (1) is applicable. Regardless of the material properties of steel, the increase of temperature in a steel section is influenced by the profile geometry and the net heat flux per unit are (1).
Am / V ̇ hnet ⋅Δt ca⋅ρa
̇ , r = Φ⋅εm⋅εf ⋅σ⋅[(θr + 273)4 − (θm + 273)4 ] hnet
(3)
ΔQ2 =
λ (θw − θf 2 )⋅1⋅tw⋅Δt δ2
(5)
Δθ1 =
ΔQ1 λ 1 = (θw − θf 1)⋅tw⋅Δt⋅ m f 1⋅ca δ1 t f 1⋅bf 1⋅ρa ⋅ca
(6)
Δθ2 =
ΔQ1 λ 1 = (θw − θf 2 )⋅tw⋅Δt⋅ m f 2⋅ca δ2 t f 2⋅bf 2⋅ρa ⋅ca
(7)
Δθw =
⎤ ΔQ1 + ΔQ2 ⎡ λ λ 1 = ⎢ (θf 1 − θw ) + (θf 2 − θw ) ⎥⋅tw⋅Δt⋅ ⎦ ⎣ δ1 m w⋅ca δ2 tw⋅bw⋅ρa ⋅ca
The resulting temperature increase/decrease is dependent on the mean distances of heat transfer δ1and δ 2 determined according to relations (9) and (10) as the distance from the centroid of upper/lower part of the web to the midpoint on flange center line between flange end and web-to-flange junction, as presented in Fig. 1.
(1)
(2)
(4)
(8)
The net heat flux per unit area is the result of adding the net convective heat flux (2) and the net radiative heat flux (3), according to EN 1991-1-2 [10].
̇ , c = αc⋅(θg − θm ) hnet
λ (θw − θf 1)⋅1⋅tw⋅Δt δ1
The temperature variation due to heat transfer from web to flanges in time Δt is determined as:
temperatures calculated according to EN 1993-1-2 [2] and EN 1991-12 [10].
Δθa, t = ksh
ΔQ1 =
Currently adopted values for the coefficient of heat transfer by convection, αc , and the steel surface emissivity, εm , are 25 W/m2 K and 0.7, respectively [2]. The recommended value in Section 3 of EN19911-2 [10] for both the configuration factor, Φ , and flame emissivity, εf , is 1. It is to be mentioned that the coefficient of heat transfer by convection may change its value if the temperature-time curve is different from the standard fire curve, ISO 834. Either full cross-section or part of section may be considered for establishing the section factor Am/V, whereas more accurate results are provided by the latter approach. The case of full cross-section tends to lead to temperatures less than the web temperatures and higher temperatures when compared to flange temperatures. Table 1 shows the temperatures calculated in this manner for an IPE400 beam subjected to 20 min of the standard ISO-834 fire. This sections are influenced by the shadow effect coefficient ksh, leading to temperatures of entries cross-section less than temperatures of each part taken separately. 2.2. Analytical approach Two additional heat fluxes are proposed for the case of beams with corrugated web, namely the heat transfer by conduction and the radiation from the web to the flanges. 2.2.1. Heat transfer by conduction Since the heating of corrugated web is significantly faster than the heating of flanges and there is contact between the two parts, the necessary conditions for heat transfer by conduction are achieved. The basic starting assumption is considering a uniform temperature for every part of the cross-section computed using the simple relations given by EN 1993-1-2 [2]. A first step is to establish the position of the highest temperature
Fig. 1. Conduction heat transfer from web to flanges.
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δ1 =
tf1 bf 1 h1w + + 2 2 4
(9)
δ2 =
tf 2 bf 2 h2w + + 2 2 4
(10)
. Starting from the point where the heat transfers from web to the top flange, the partial web height h1w, is equal to the ratio of the heat flux towards top flange to the heat fluxes to both flanges, (11). Similarly, it is judged for the bottom flange, (12).
h1w =
ΔQ1 ⋅h w ΔQ1 + ΔQ2
(11)
h2w =
ΔQ2 ⋅h w ΔQ1 + ΔQ2
(12)
Due to iterative procedure and the inverse proportionality between ΔQi and δi , the highest temperature point tends to oscillate around the correct position. Hence there is a need for a sub-step iterative procedure to obtain a steady position. For this, the partial height related to the top flange may be computed according to relation (13).
ΔQ1, n ⋅h w = h1w, n +1 = ΔQ1, n + ΔQ2, n
1 ⋅(θw δ1, n 1 δ1, n
⋅(θw − θf 1) +
− θf 1) 1 ⋅(θw δ 2, n
− θf 2 )
⋅h w (13)
Our experience and monitoring of calculation results suggests that ten steps are sufficient for a converging solution. It must also be said that the initial values of h1w and h2w , in the heat conduction temperature increase calculation, are estimated according to (14).
h1w,0 = h2w,0 =
hw 2
(14)
By this procedure two uncommon scenarios may arise:
• •
If one of the flanges has the same temperature as the web, it results h1w = h w . The heat from the entire web transfers to the other flange with different temperature. If one of the flanges has a higher temperature than the web and the other flange is colder than the web, the length h1w comes out longer than the real web length h w . Heat fluxes (incoming and outgoing) partly eliminate each other in “fictive” length h1w − h w . The “released” flux represents heat flux which only goes through the web, one flange heats up the other and it has no influence on the web temperature. Remaining heat flux heats/cools the web and it is equal to the difference(Qw − Qf 1) − (Qf 2 − Qw ). If this difference is equal to zero, thus, the web temperature is an average of both flange temperatures, the length h1w is infinity. Incoming and outgoing heat fluxes are equal to each other, the heat goes only through the web and the web temperature does not change.
Because the temperatures of the top flange, bottom flange and web are developing at different rates, the ratio of heat fluxes and the mean heat transfer distances are varying continuously. The calculation procedure should be performed to obtain the quantities in the following order: temperature differences, mean heat transfer distances; heat flux into the top and bottom flange, change of temperature of flanges and web. The influence of temperature increase/decrease due to heat conduction is incorporated back into the step-by-step method. In every step, the increase/decrease due to heat conduction is added to the increment due to heat transfer from fire to steel member by convection and radiation. After computing the position of the highest temperature point, the section may be considered to be divided in two parts i.e heat transfer part from web to top flange and heat transfer part from web to bottom flange. Each part is then divided in finite length elements along the web
Fig. 2. Division of steel profile in elements. (a) Finite elements on the cross-section. (b) Numbering of finite elements.
and flange respectively, as principled presented in Fig. 2. The size of the finite elements is considered by the user such that accurate results are obtained. Either equal elements or variable length elements can be defined. The results in this paper are obtained by dividing the web into elements of 20 mm. The basic premise is that the heat is passing through an element and is consumed for heating it. By adapting the basic equations for heat conduction to the current situation, the following relations result for the analytical model. 86
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a part of the incremental method of calculating the temperatures in the cross-section presented in the following.
2.2.2. Heat transfer by radiation Radiation between component parts of cross-section is usually neglected either due to the small temperature difference within crosssection or due to the view factor developed between separate parts which leads to minor changes of temperatures. In the current study, the configuration factor in relation (3) is computed for the radiation from the hot surface of the web which acts as an emitter to the flange which is considered to be the absorber. Depending on the amount of emitted and absorbed energy the temperature tends to decrease in the web and increase in the flange [12]. In relation (3), the configuration factor or view factor is usually considered to have a unitary value. For the proposed methodology this value, measuring the fraction of radiative heat leaving a surface and arriving at the receiving surface, is computed according to Annex G of EN 1991-1-2 [10]. The computation is based on the same partitioning of the web as for the conductive heat transfer in Section 2.2.1 while the flange is divided in only two parts (Fig. 3). Each element of the web is considered to be an emitter while each side of the flange is considered to be an absorber. To solve the radiative heat transfer between the individual parts of a cross-section, only half of the cross-section is considered. The configuration factor is calculated for each element of the web according to the geometric dimensions in (20), (21) and (22) and Fig. 3. Considering the web to be the emitter and the flange to be the absorber, the configuration factor ΦA − B, i (23) can be computed similarly to the relation given in Annex G of EN 1991-1-2 [10] for the orientation presented in Fig. 3.
Fig. 3. Radiation heat transfer from web to flanges.
ΔQi =
λ ⋅(θi − θi −1)⋅(ti −1⋅1)⋅Δt δi
ΔQi +1 =
λ ⋅(θi +1 − θi )⋅(ti +1⋅1)⋅Δt δi +1
(15) (16)
The temperature increase/decrease in each element i (17) is computed function of the temperature difference from adjacent elements (18), (19).
Δθi = Δθi, i −1 + Δθi, i +1
(17)
Δθi, i −1 =
ΔQi λ Δt = ⋅(θi − θi −1)⋅(ti −1⋅1)⋅ mi⋅c δi ti⋅L i⋅1⋅ρ⋅c
(18)
Δθi, i +1 =
ΔQi +1 λ Δt = ⋅(θi +1 − θi )⋅(ti +1⋅1)⋅ mi⋅c δi +1 ti⋅L i⋅1⋅ρ⋅c
(19)
Δθi, i +1 =
ΔQi +1 λ Δt = ⋅(θi +1 − θi )⋅(ti +1⋅1.15)⋅ mi⋅c δi +1 ti⋅L i⋅1⋅ρ⋅c
(19a)
Having the temperatures computed by relations given in EN 19931-2, the temperature difference at the junction between web and flange will stand as an input condition for the heat flux. Application of relations (15)–(19) presented above for the cross-sections which are frequently used in practice i.e. WTA500, WTB750 or WTC1000, shows that the web affected areas have a length from 30 mm up to maximum 100 mm from the flange-to-web junction. The rest of the web is experiencing negligible temperature decrease. In the case of small height beams the web thickness is small and the influenced length along the web is also smaller, close to 30 mm. In order to compensate the increased length due to corrugation, relation (19) is modified by a coefficient of 1.15, relation (19a), for the web element which is in contact with the flange. This value represents the ratio between the wavelength of the web and the longitudinal projected length. Only the shape of the corrugation may increase or decrease the value of 1.15 but not the thickness of the web. The described model of the heat conduction initiated by the temperature difference between the flange and the web is designed as
Fig. 4. Flow-chart of the analytical model.
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Fig. 5. Radiation on corrugated web.
Fig. 6. Finite element model of the heat transfer analysis.. 0.5 ⎡ ⎛ bf ⎞ 2 ⎤ si = ⎢hs2, i + ⎜ ⎟ ⎥ ⎢⎣ ⎝ 4 ⎠ ⎥⎦
cos(ϕA, i ) =
cos(ϕB, i ) =
ΦA − B, i =
Fig. 7. (a) Specimens before test, (b) Specimens during test.
the section factors for web and flange are computed as: (20)
hs, i si
Am, w / Vw =
(21)
(22)
cos(ϕA, i )⋅ cos(ϕB, i ) π⋅si2
⋅Ai =
hs, i⋅bf 4⋅si4⋅π
⋅(L i⋅1)
(23)
The radiative heat flux is determined according to relations (24) and (26) and used to calculate the temperature decrease (25) and increase (27) of the web and flange, respectively. Due to the heat transfer from both sides of the web, the temperature decrease is considered with a factor of 2 in relation (25),
hnet , r , A, i = ΦA − B, i⋅εA⋅εB⋅σ⋅[(θB + 273)4 − (θA, i + 273)4 ] Δθa, t , A, i
Am, w / Vw = 2⋅ ⋅hnet , r , A, i⋅Δt ρa ⋅ca
hnet , r , B =
∑ {ΦA−B,i⋅εA⋅εB⋅σ⋅[(θA,i + 273)4 − (θB + 273)4]} i =1
Δθa, t , B = 2⋅
Am, f / Vf ρa ⋅ca
⋅hnet , r , B⋅Δt
(28)
The values for the temperature change of the web elements and the flange elements are arranged as part of the incremental method for calculating cross-section temperature. For each step the temperature computation is performed according to the incremental procedure for the specified time period for each individual element, thus each element will have various temperatures in subsequent steps, together with the heat conduction from web to flange. For the situation of a fire with cooling phase, the flange may have a higher temperature than the web. The sign of radiative fluxes change and the temperature variation reverse their sign for both web and flange. It must be mentioned that a condition for the temperature difference is added and the view factor is replaced by:
bf 4⋅si
1 1 Am, f / Vf = tw tf
(24)
ΦB − A, i =
hs, i⋅bf 4⋅si4⋅π
⋅(
bf − t f 2
⋅1)
(29)
The evaluation of the hotter part of the cross-section must be performed and inserted in the flowchart of the proposed analytical approach as presented in Fig. 4. For the specimens in contact with the trapezoidal steel sheet, a covering factor κtr is considered to account the smaller influence of heat transfer to the top flange. The value of the covering factor is introduced in the section factor computation of the top flange, (30). According to the geometric configuration of the trapezoidal steel sheet TR150/280/ 0.75 the ratio of the gaps volume to the total volume of the steel deck is 0.7.
(25) (26)
(27)
In relations (24) and (26), the emissivities εA and εB are considered both with a value of 0.7. Being an exchange of heat, only between one side of each element 88
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Am / V =
(κTR⋅bf + bf + 2⋅t f ) bf ⋅t f
(30)
A configuration factor due to web corrugation may be considered by the simple relation (31) where the flat web and the sinusoidal pattern are evaluated according to Fig. 5. By a simple reasoning, we can conclude that a perpendicular projection of the corrugated web could be considered as an area exposed to fire while the configuration factor remains 1. Angle β is the angle of incidence of heat radiation in the given point, measured with respect to the web normal.
Φ=
β dA ∫ cos π⋅s 2
(31)
. 3. Finite element model Either for plane or spatial domain, several papers regarding validation of heat transfer analysis using finite element codes may be found in the literature [7,13]. For the present study a 3-dimensional heat transfer analysis using ABAQUS [14] was performed. Beside the influence of conduction within cross-section, the model was created so that the influence of corrugation of the web over the temperature in the flange regarding the position with respect to the flange end may be evaluated, thus, the length of the model is chosen equal to the wavelength of the web corrugation as presented in Fig. 6. All the corrugated web beams monitored in the experiments performed in [1] are modeled according to the considerations presented in the following. The system of measurement is considered to be kg, m, s and °C. Fig. 8. Temperature of specimen 1 WTC 500-300×20: (a) top flange, (b) bottom flange.
3.1. Geometry The model is created by defining one part, i.e. the web, and
Fig. 9. Web temperature of specimen 1 WTC 500-300×20: (a) mid-height, (b) lower part. Fig. 10. (a) temperature monitoring positions (b) web temperature of specimen 9, WTB 750-250×12, level D.
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Fig. 11. Web temperature of specimen 9, WTB 750-250×12: (a) level A, (b) level B, (c) level C, (d) level E, (e) level F, (f) level G.
extruding its both ends to form the flanges. This technique eliminates the use of interaction definition between individual parts and allows the conduction between web and flanges. The geometric dimensions are taken from the experiments presented in [1]. In order to simulate the configuration as close as possible to the reality, two sections are defined and assigned to the flanges and one section for the web part. For both top and bottom flanges, a shell offset definition is set to the bottom surface and top surface, respectively.
3.3. Interactions The heat transfer from the environment is considering convection and radiation. For convective interaction between environment and steel, a convection coefficient is defined with a value of 25 W/m2 K. The radiation between furnace and steel profile requires several settings in order to use correctly the unit system with Celsius degrees. First a value of −273.15 is defined for the absolute zero temperature and the value of Stefan-Boltzmann constant is set to 5.67e-8. Radiative heat transfer is defined in Abaqus using an emissivity coefficient and an ambient temperature. The emissivity is considered as recommended in EN1993-1-2 [2] with a value of 0.7 for all the surfaces exposed to the gas temperatures except the top flange where, function of geometric configuration and position of steel decking, an emissivity factor is computed as presented in [15,16]. The emissivity of flames for the fire exposed surfaces is considered to be 1.0. For the top flange the total emissivity of hot gas εf depends on the emission coefficient K and thickness or the mean beam length L [15,17], as presented in relation (32).
3.2. Material properties A great advantage of the general purpose finite element code, ABAQUS [14], is the possibility to use temperature dependent mechanical and thermal properties. The necessary properties needed for a heat transfer analysis are conductivity, specific heat and density. The first two are temperature dependent and their pattern is defined according to EN 1993-1-2 [2] while the latter is considered to have a constant value of 7850 kg/m3.
εf = 1 − e−K ⋅ L 90
(32)
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for the temperature increase. Specimen 9 from [1] was positioned such that the trapezoidal steel sheet was in contact with the top flange. For this case both analytical and numerical model provide consideration of top covering as described in Sections 2.2 and 3.4. The temperatures measured during the test and computed by the analytical and numerical models are depicted in Figs. 10 and 11. The temperature measurement levels are resumed from [1] in Fig. 10(a). As a comparison to the recommended method provided by the design code [2], the temperatures obtained for level D accommodate the reference temperature increase of the web. Fig. 11(a)–(f) present the temperature distribution along the web height of Specimen 9. At the levels closer to the web-to-flange junction, i.e A and G, the temperatures show a great difference from the reference values at the level D, especially in the first 15 min of the test. The period exhibiting temperature difference is smaller as the web points are situated further from the flange. The analytical results follow in a good accordance the experimentally determined temperatures of the web while the numerical model has the tendency to predict the temperature in a conservative sense.
The tests performed in [1] involved diesel burners for the isolated specimens and wood cribs for the real-scale building test. These two cases imply the use of emissivity coefficients of 0.43 and 0.8, respectively as recommended in [15,18,19]. Since the emissivity coefficients were empirically determined and their accuracy is questionable [15] an approximate value for the mean beam length may be expressed according to relation (33).
L = C⋅
4⋅V A
(33)
where: V is the total volume of the gas between flange and steel deck, A is the area on which heat is radiated and C is a correction factor of 0.9. By the previously mentioned procedure an emissivity is computed for each of the following situations: 0.27 for the specimens 500 mm under the steel deck, 0.1 for the specimens with trapezoidal steel sheet on top and 0.073 for the specimens in the real building test, in Mokrsko [1]. Convection and radiation sink/ambient temperature are defined by tabulated amplitude with the values recorded in the corresponding test as presented in [1]. The radiation between web and flange is not considered in the finite element model. Depending on the test a constant temperature predefined field is considered as follows: 15 °C and 20 °C for isolated element test stage 1 and stage 2, respectively and 15 °C for the real building test, [1].
4.2. Results for building fire test Real fire tests imply complex phenomena of fluid dynamics and interactions between enclosed elements, liable for a greater error range of results. A great deviation from the experimental results was obtained by the analytical and numerical computation in the case of the top flange of the corrugated web beam CS2 used in the fire test in Mokrsko, Fig. 12. The test recordings for the bottom flange are closely followed by the computational models results, Fig. 11b. This implies either that the interaction between the concrete slab and the steel flange is not disclosed by the computational models as well as it may be a subject of fluid dynamics. The recordings are considered as presented in [1] and summarized in Fig. 13(b).
3.4. Element type A smaller range of elements and element controls are available in heat transfer analysis. The most commonly used type is the first order 4-node heat transfer quadrilateral shell, DS4, also selected for this study. Each element edge was set to have an approximately 10 mm length. 3.5. Analysis control A general step of heat transfer analysis is selected for the numerical analysis with a transient response for results. Although the maximum number of increments needed to be increased, it depends mostly on the time period of the analysis. The characteristic setting of the analysis step which influences the results is the maximum increment size set to 5 s for both fire tests. The total analysis time for the isolated elements tests is 1800 s while for the building fire tests is 4400 s. 4. Validation of the proposed model The following presents the comparison of the results obtained by: tests measurements, proposed analytical methodology and numerical analysis. It is to be mentioned that the results obtained analytically do not include the radiative heat transfer. The influence of radiative heat transfer between flange and web is presented in Section 4.3. 4.1. Results for the isolated elements test The results obtained for the isolated elements test have a similar trend for all specimens and for the three computational options; therefore there are presented the results for one specimen suspended 300 mm below the ceiling, Specimen 1, and for one specimen in contact with the trapezoidal steel deck, Specimen 9, [1] (Fig. 7). Figs. 8 and 9 present the temperature development in the flanges and the web parts according to the positions specified in [1]. The results show similar evolutions of temperature. For both top and bottom flange the value presenting deviation from the experimental results is represented by 735 °C. This temperature represents the value which defines the peak value of the specific heat of steel and due to the slow increase of the gas temperature at that point and the massivity of the flanges, the analytical and numerical models require a longer time
Fig. 12. Temperature of the beam CS2, WTB 500-220×15: (a) top flange, (b) bottom flange.
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Fig. 14. Temperature of beam CS2, WTB 500-220×15: (a) web upper part, (b) web lower part.
hotter part of the beam to the cooler parts. By enabling the radiation part of analytical model, the temperature field suffers the following changes compared to the results obtained without considering the radiation:
Fig. 13. (a) Midheight web temperature, (b) Measurement position.
The temperature measurements at the web mid-height and the computational results presented in Fig. 13(a) are in close agreement, the predicted values being conservative temperatures. For the upper and bottom part of the web the analytical model results are closer to the experimental results than the numerical predicted temperatures, Fig. 14(a) and (b). An aspect common especially for Fig. 12(a) and (b) and Fig. 13(a) is observed on the cooling phase of fire when the slope of the experimental curve is descending at a lower rate than the analytical or numerical curves. Although of less importance, the phenomenon might influence the behavior of structures subjected to parametric curves. A possible cause for the temperature divergence on the cooling phase might be due to the heat that was retained by the ceiling in the experiment. The ceiling of the test setup is made of a trapezoidal steel sheet covered with insulating material. During the increasing phase of the fire, it accumulates heat which is afterward radiated toward the steel beam, leading to a lower decrease rate of the temperature especially on the top flange. Since the plate-thermometers were placed lower, at the level of the web midheight, the radiation of the ceiling was not recorded properly.
4.3. Influence of radiation between cross-section parts In order to assess the effect of all three ways of heat transfer, the following presents the effect of the heat transfer by radiation from the
Fig. 15. Numerical model of parametric study.
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higher rate compared to heat redistribution by conduction. The numerical study is based on a simple 3D model as presented in Fig. 15 according to the specifications mentioned in Section 3. The variable parameters are the web thickness, the flange thickness and the flange width, while the web height and length are considered to have constant values of 500 mm and 155 mm, respectively. The gas temperature increase is defined according to the Standard Fire Curve.
Table 2 Web strip width with min 5% lower temperature than the maximum temperature point on web. hwL [mm] tw=2 mm
tw=2.5 mm
tw=3 mm
tf Time
10
15
20
25
30
10
15
20
25
30
10
15
20
25
30
5 min 10 min 15 min 20 min 25 min 30 min
40 20 10 0 0 0
40 20 20 0 0 0
40 30 20 10 0 0
40 30 20 20 10 0
50 30 20 20 20 10
30 10 10 0 0 0
40 20 10 0 0 0
40 30 20 10 10 0
50 30 20 10 10 10
50 40 30 20 20 10
30 10 10 0 0 0
40 20 10 0 0 0
40 30 10 10 0 0
50 30 20 10 10 10
50 40 30 20 10 10
5.1. Influence of the flange-to-web thickness ratio
Fig. 16. Web temperatures for flange-to-web ratio of: (a) 3.33, (b) 15.
The web influence over the flange and vice-versa depends on the massivity of each part. A thinner web heats faster but it does not have enough “power” to transmit the heat, while a thicker web heats slower but has more heat conducting capacity. A series of numerical simulations were performed using 2, 2.5 and 3 mm web thicknesses and 10, 15, 20, 25, 30 mm flange thicknesses. The extreme flange-to-web thickness ratios resulted from the previous mentioned dimensions are 3.33 and 15. To avoid the influence of flange the study was conducted for a width of 400 mm, which proves to be a correct assumption from the second parametric study.
– For the specimen WTC 500-200×20, situated 300 mm below the ceiling, at time t=15 min, in elements next to the flange, the temperature decreases with maximum 22 °C, which represents only 3%. This temperature decrease is reduced to 0.4% at web midheight. The flange temperature is increased by 1.3%. For 30 min, the temperature decreases with less than 1.5% at the web-to-flange junction, and this decrease is smaller as the web element is further from the flange, to a reduction by 0.2% at web midheight. The flange temperature is increased by 0.8%. – For the specimen in contact with the ceiling, WTB 750-250×12, after a time interval of 15 min the temperature in the web decreases by 1.8–0.2% as the web element is further away from the flange. The flange temperature increases by 0.8%. For 30 min, the web temperatures decrease by 0.5–0.1% varying along the height of the web. For this time interval the flange temperature increases by 0.2%. The temperature changes are only presented for 15 and 30 min in relation with the practice of assessing the fire resistance. As the temperature in the steel beams were obtained by averaging test recordings from multiple positions [1] and the temperature deviation is greater than the temperature difference in the analytical model, caused by the radiation between the web and the flange, it is considered that this effect is of less importance. 5. Parametric study The heat exchange between the web and the flange by conduction indicates a non-uniform temperature along the web and possibly the flange. To investigate the temperature affected area a parametric study was performed considering the dimensions range available in the product catalogue of Zeman & Co GmbH [20]. The fabricated beams available have a web thickness, flange thickness and a flange width of 2–3 mm, 10–30 mm and 200– 430 mm, respectively. Although several web heights are available, from 500 mm, the thinness of this part suggests that there is marginal influence of this parameter over the temperature distribution on the web, since the heat transfer by convection and radiation will have a
Fig. 17. Flange temperature distribution for flange-to-web ratio of: (a) 3.33, (b) 15.
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the temperatures of the protected steel elements. Function of the protection solution, the conduction may or may not be influencing the temperature field. Engineering judgement, advanced models and experimental studies might reveal if temperature difference between flange and web appears so that the present analytical model, considering conduction heat transfer, is suitable to be used. The influence of the non-uniform temperature distribution on the structural behavior of beams with corrugated web is an ongoing study performed at the Czech Technical University in Prague and will be presented in a forthcoming article.
Fig. 16 shows that a significant decrease of temperature may extend to 60 mm along the web height but only in initial phases of fire. For relevant time of fire exposure i.e. 15 min or 30 min, the area with a 5% decreased temperatures is reduced significantly as shown in Table 2. 5.2. Influence of the flange width The width of a flange may increase in some cases the massivity of the flange e.g. for small widths and high thicknesses, thus a smaller temperature increase rate. Having a greater difference of temperature the condition for heat conduction would be even more favorable. The considered flange widths were 200 mm, 300 mm and 400 mm. Fig. 17 presents two extreme cases: 1) web thickness of 3 mm and a flange of 10 mm thick and 2) a 2 mm thick web and a flange thickness of 30 mm. The influence over the flange is visible only for higher section factors i.e. small flange thickness, and thicker webs. Also, the temperature influence is visible only for the first phase of temperature increase, up to 10 min. The temperature increase in the flange, which has a non-conservative effect, extends only 40 mm from the flange-toweb junction the same response exhibited also for wider flanges. After 15 min the temperatures developed in the flange do not exceed the temperatures predicted according to EN 1993-1-2 [2], represented with a red dashed line for every 5 min interval, in Fig. 17.
Acknowledgement This publication was supported by the European social fund within the framework of realizing the project “Support of inter-sectorial mobility and quality enhancement of research teams at Czech Technical University in Prague”, CZ.1.07/2.3.00/30.0034. Period of the project's realization 1.12.2012–30.6.2015. References [1] Jaroslav Vácha, Petr Kyzlík, Ioan Both, František Wald, Beams with corrugated web at elevated temperature,experimental results, Thin-Walled Struct. (2015). http:// dx.doi.org/10.1016/j.tws.2015.02.026. [2] C.E.N, EN 1993-1-2, Eurocode 3: Design of steel structures – Part 1-2: General rules – Structural fire design. Brussels, 2005. [3] T.R. Kay, B.R. Kirby, R.R. Preston, Calculation of the heating rate of an unprotected steel member in a standard fire resistance test, Fire Saf. J. 26 (4) (1996) 327–350. [4] Martin Gillie, Analysis of heated structures: nature and modelling benchmarks, Fire Saf. J. 44 (5) (2009) 673–680. [5] Bin Zhao, Mohsen Roosefid, Experimental and numerical investigations of steel and concrete composite floors subjected to ISO fire condition, in: Proceedings of the Sixth International Conference on Structures in fire, 2010 East Lansing, pp. 407– 416. [6] M. Feng, Y.C. Wang, J.M. Davies, Axial strength of cold-formed thin-walled steel channels under non-uniform temperatures in fire, Fire Saf. J. 38 (2003) 679–707. [7] M. Feng, Y.C. Wang, J.M. Davies, Thermal performance of cold-formed thin-walled steel panel systems in fire, Fire Saf. J. 38 (4) (2003) 365–394. [8] Y.C. Wang, Composite beams with partial fire protection, Fire Saf. J. 30 (4) (1998) 315–332. [9] L. Gardner, K.T. Ng, Temperature development in structural stainless steel sections exposed to fire, Fire Saf. J. 41 (3) (2006) 185–203. [10] C.E.N, EN 1991-1-2, Eurocode 1: Actions on structures – Part 1-2: General actions – Actions on structures exposed to fire. Brussels, 2002. [11] K.Petr, Fire resistance of Beams with Corrugated Web (Ph.D Thesis). Czech Technical University in Prague. Prague(in Czech), 2012. [12] Vacha Jaroslav, Influence of nonhomogeneous temperature on fire resistance of beams with corrugated web (Ph.D Thesis), Czech Techical University in Prague, Prague, 2015 (in Czech). [13] E.OzyurtY.C.WangK.H.Tan, Elevated temperature resistance of welded tubular joints under axial load in the brace member, Eng. Struct. vol. 59, pp. 574–586, February 2, 0014 [14] ABAQUS, ABAQUS Documentation 6.11, Dassault Systèmes, Providence, RI, USA, 2011. [15] U.Wickström, Heat Transfer in Fire Technology. Luleå tekniska universitet, Luleå. 2012. [16] T.JánaY.C.WangF.WaldK.Horová, Temperatures and thermal boundary conditions in reverse channel connections to concrete filled steel sections during standard and natural fire tests, Fire Saf. J., 2015. [17] Dougal Drysdale, An Introduction to Fire Dynamics, John Wiley & Sons, Ltd, 2011. [18] B. Hagglund, L.E. Persson, The heat radiation from petroleum fires, Stockh. FOA Rep. C20126-D6 (A3) (1976). [19] K. Sato, T. Kunimoto, Memoirs of the Faculty of Engineering, Kyoto University 31 (47) (1969). [20] Zenam Beteiligungsgesellschaft mbH. [Online]. 〈www.zeman-stahl.com/en/ products/sin-profiles〉
6. Conclusions The paper presents a study on the temperature development in steel beams with sinusoidal corrugated web. Starting from the simple relations provided by fire design codes, an analytical model which can consider the conduction and the radiation between web and flange, is proposed for calculating the temperature distribution in a steel crosssection. Also, numerical heat transfer analyses were performed considering conduction of steel. Both analytical and finite element models are able to compute the non-uniform temperature distribution over the corrugated web height and predict the temperature variation in the cross-section in an acceptable conservative range. The study reveals the non-uniform temperature variation on the web height and in the flange, at early stages of fire, while for longer fire exposure, the temperature in the cross-section tends toward homogeneous web and flange temperatures. Based on this fact it is considered to predict a better response in the early stages of a thermo-mechanical analysis. For the real scale building test the gas temperatures included also the cooling phase of fire. The descending branch shows a different cooling rate of elements. The experimental results show a lower rate for cooling while the analytical and numerical results show rapidly descending temperatures. This phenomenon questions the validity of the convective coefficients on the cooling phase of elements. The parametric study reveals that the geometric dimensions have a higher influence on web temperature development for cross-section with greater flange-to-web thickness ratio. The increased massivity, given by the width of the flange, has no influence on the temperature development in the cross-section. Since the analytical model described in the paper is based on the relations given in EN1993-1-2 for the unprotected steel elements, the model may be adapted, in particular cases, to the protected steel elements considering the provisions of the design codes. Basically, radiation and convection are not directly present in the computation of
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Beams with corrugated web at elevated temperature, analytical and numerical models for heat transfer
crossmark
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Jaroslav Vácha, Petr Kyzlík, Ioan Both , František Wald Czech Technical University in Prague, Department of Steel and Timber Structures, Thakurova 7, CZ 166 29 Praha, Czech Republi
A R T I C L E I N F O
A BS T RAC T
Keywords: Corrugated web beam Elevated temperature Heat transfer Aspect ratio View factor
In the case of corrugated web beams, based on experimental results, it was noticed that temperature development along the height of the web has a distinct pattern compared to the flat web beams. The particularity consists of a temperature decrease in the web, in the vicinity of the flange-to-web junction, assigned to the heat transfer by conduction and radiation. On this assumption, starting from the simple relations given in the fire design codes an analytical model is developed considering the heat conduction from the web to the flanges and the radiation heat transfer between web and flanges. The paper presents the results of the proposed analytical method against the experimental results and the numerical simulation. Both analytical and numerical models are compared to the experimental results obtained from the isolated element fire test and the real building fire test. A parametric study on the influence of the geometric dimensions is performed regarding also the time interval for the temperature development. For short term fire exposure, the web exhibits non-uniform temperature along the web height, while for long term fire exposure the entire cross-section tends to a homogeneous temperature.
1. Introduction
with time using the previous mentioned relations depends on the heat transfer by convection and radiation, while conduction is not considered. Owing its small massivity, the temperature in the thin steel part i.e. the beam web, increases at a higher rate than thick parts i.e. the flanges, therefore conduction plays an important role, especially at the junction area between flange and web. In fire design computation, a great importance is attributed to the temperature distribution which can be regarded as a critical stage of fire design since most mechanical models predict similar deformation/ time characteristics according to [3]. As mentioned above, in the case of high conductivity materials i.e. steel, the heat conduction has a great importance for significant temperature difference between two parts of steel cross-section. For the commonly used steel profiles with flat web, the temperature distribution within the parts of cross-section has minor effect along the part thickness and function of the exposure temperatures, it may be regarded either as a uniform temperature or as a thermal gradient through the cross-section [4,5]. Direct exposure to fire of cold-formed steel members exhibit no distinguishable temperature difference due to conduction. In the case of thin-walled steel panels, where the steel profiles are protected, a greater temperature difference is developing between the flange closer to the fire exposed face of the wall and the flange fixed to the unexposed face of the wall leading to conditions proper for observing heat conduction effect [6].
Mostly used in bridge engineering, the beams with corrugated webs have increased their use in the industrial structural systems providing not only material savings but also an attractive aspect through the trapezoidal or sinusoidal corrugation of the web. The design of industrial buildings involves the fire design situations therefore the response of these beams at elevated temperature should be evaluated. Due to the shape of the web, the height-to-width aspect ratio and the greater flange-to-web thickness ratio of corrugated web beams in comparison to the flat web beams, the thermal response of such beams implies particularities of temperature development over the steel crosssection. In a preceding paper [1], the authors presented experimental results for both isolated elements and real-scale building tests and this paper is a continuation of the study of elevated temperature influence on beams with corrugated web from the point of view of thermal response. The aspect ratio of the web of such beams would categorize the cross-section into a thin-walled profile but the thicker flanges makes these beams behave distinctly for both thermal and structural actions. The temperature development in commonly used flat web steel beam profiles can accurately be determined using the standardized relations recommended in EN 1993-1-2 [2]. The temperature increase
⁎
Corresponding author present address at: Politehnica University of Timisoara, Department of Steel Structures and Structural Mechanics, Ioan Curea 1, 300224, Timisoara, Romania. E-mail addresses: [email protected] (J. Vácha), [email protected] (P. Kyzlík), [email protected] (I. Both), [email protected] (F. Wald).
http://dx.doi.org/10.1016/j.firesaf.2016.09.001 Received 19 May 2015; Received in revised form 9 September 2016; Accepted 22 September 2016 Available online 04 November 2016 0379-7112/ © 2016 Elsevier Ltd. All rights reserved.
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Nomenclature
Φ ΦA − B, i
area of top face of top flange on which heat is radiated [m2] Area of the ith element considered as an emitter Ai Am / V section factor of steel member [m−1] section factor of flange [m−1] Am, f / Vf Am, w / Vw section factor of web [m−1] C correction factor K emission coefficient L mean beam length [m] Li length of an element I [m] ΔQ1, ΔQ2 heat fluxes from the web to the top and bottom flanges [J] ΔQ1, n , ΔQ2, n intermediate heat fluxes from web to top and bottom flange [J] ΔQi heat flux from element i-1 to element i [J] V volume of the gas between flange and steel deck [m3] WTA, (B ), (C ) symbol for web thicknesses of 2, 2.5 and 3 mm bf 1, bf 2 width of top and bottom flange [m] specific heat of steel [J/kg K] ca h1w , h2w partial web height for upper part of web and lower part of web [m] h1w, n +1 intermediate values for partial web height [m] h1w,0 , h2w,0 initial partial web heights [m] ̇ net heat flux per unit area [W/m2] hnet ̇ ,c net heat flux to unit surface area due to convection [W/ hnet m2] ̇ ,r net heat flux to unit surface area due to radiation [W/m2] hnet hnet , r , A, i radiative heat transfer for element i of the web [W/m2] hnet , r , B radiative heat flux for the flange [W/m2] h wL width of the web strip with min 5% lower temperature than the maximum temperature point on web [m] hsi vertical distance from flange to element i [m] ksh correction factor for the shadow effect m f 1, m f 2 masses of the top flange and bottom flange [kg] mass of element i [kg] mi mw mass of the web [kg] s corrugated web length of the web [m] distance from half flange centroid to element i [m] si t f 1, t f 2 thickness of top and bottom flange [m] ti thickness of element i [m] ti −1, ti +1 thicknesses of the preceding adjacent element and next adjacent element [m] tw web thickness [m] Δt time interval [s] 2w wave length of the sinusoidal web [m]
A
αc β δ1, δ 2 δ1, n, δ 2, n
δi εA, εB εf εm ϕA, ϕB κTR λ θA, i θB
θi θf 1, θf 2 θg θm θr θw Δθ1, Δθ2
Δθa, t Δθa, t , A, i Δθa, t , B
Δθi, i −1 Δθi, i +1
Δθi Δθw ρa σ
configuration factor configuration factor for radiative heat transfer from web to flange coefficient of heat transfer by convection [W/m2 K] angle of incidence of heat radiation [°] mean distance of conductive heat transfer from web to top and bottom flange [m] intermediate values for mean distances of conductive heat transfer [m] distance from element i-1 to element i [m] emissivity of the web and the flange emissivity of flames, of the fire surface emissivity of the member surfaces relative orientation angles [°] covering factor thermal conductivity [W/mK] temperature of ith element on the web in radiative heat transfer computation [°C] flange temperature for radiative heat transfer computation [°C] temperature of element i [°C] temperature of the top flange, bottom flange [°C] gas temperature in the fire compartment [°C] temperature of the member surface [°C] effective radiation temperature of the fire environment [°C] web temperature [°C] temperature variation of top flange, bottom flange and web [°C] the increase of temperature in an unprotected steel member during a time interval Δt [°C] web temperature variation due to radiation of element i [°C] flange temperature variation due to radiation from web [°C] temperature variation of element i induced by the heat transfer from element i-1 to element i [°C] temperature variation of element i induced by the heat transfer from element i to element i+1 [°C] total temperature variation of element i caused by the heat transfer from element i-1 to i and from i to i+1 [°C] temperature variation of the web [°C] unit mass of steel [kg/m3] Stephan Boltzmann constant (=5,67×10−8 [W/m2 K4])
heated part of the cross-section, i.e. the web, to colder parts of crosssection, i.e. the flanges. The procedure starts by establishing first the point of highest temperature in the web and then, by dividing the web and the flanges into finite elements, the temperature variation along the web height is determined. The comparison of results is performed with respect to finite element heat transfer analysis and experimental results obtained from the tests performed at Czech Technical University in Prague, presented by the authors in [1].
The measurements performed by Feng et al. [7] revealed a higher temperature in the flange close to the unexposed face of the wall for the specimens that had continuous web between flanges than the specimens where the flange had a service hole. Another situation which exhibits steel conduction effect was presented by Wang [8] for partially protected beams, where, for a web protected on one quarter of its height, the temperature in the flange had a higher value than for the case of a fully protected cross-section. The heat transfer principles for furnace tests are clearly described in [3] for both convection and radiation interaction between furnace and elements, while in [9] it is presented the influence of steel type on thermal interaction properties. Moreover, due to high temperature difference at early stages of heating it might be considered that a radiation interaction exists between parts of steel beam. The paper presents an analytical model which assesses the temperature distribution in a corrugated web steel beam considering the heat conduction from the web to the flanges and the radiation of more
2. Heat transfer in steel beams The fire resistance of structural elements during fire is highly influenced by the temperature, thus, the accuracy of predicting thermal field in the cross-section is proportional to the precision of the thermomechanical analysis results. The beams with corrugated web may be considered a non-conventional steel profile and, as presented in the following, has a distinct distribution of temperatures than in the case of 84
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point for the case of distinct temperatures between the bottom and the top flanges [11]. For this, the heat fluxes from the web to the top flange, ΔQ1, and to the bottom flange, ΔQ2 , are calculated function of the mean distance of thermal transfer and of the temperature difference between web and top flange, bottom flange, respectively (4), (5).
Table 1 Temperature of IPE400 according to EN 1993-1-2. IPE400
Temperatures [°C]
Time [min]
Full cross-section kshAm/V=123.2 m−1
5 10 15 20
209.9 446.6 612.3 706.8
Flange Am/ V=155.7 m−1 248.4 502.7 653.3 725.7
Web Am/ V=232.6 m−1 325.2 582.5 695.8 737.1
2.1. Prescriptive incremental procedure In section 4.2.5 of EN 1993-1-2 [2] simple relations for temperature increase in a time interval less than 5 s are given for both unprotected and protected steelwork. The current study regards unprotected steel sections for which relation (1) is applicable. Regardless of the material properties of steel, the increase of temperature in a steel section is influenced by the profile geometry and the net heat flux per unit are (1).
Am / V ̇ hnet ⋅Δt ca⋅ρa
̇ , r = Φ⋅εm⋅εf ⋅σ⋅[(θr + 273)4 − (θm + 273)4 ] hnet
(3)
ΔQ2 =
λ (θw − θf 2 )⋅1⋅tw⋅Δt δ2
(5)
Δθ1 =
ΔQ1 λ 1 = (θw − θf 1)⋅tw⋅Δt⋅ m f 1⋅ca δ1 t f 1⋅bf 1⋅ρa ⋅ca
(6)
Δθ2 =
ΔQ1 λ 1 = (θw − θf 2 )⋅tw⋅Δt⋅ m f 2⋅ca δ2 t f 2⋅bf 2⋅ρa ⋅ca
(7)
Δθw =
⎤ ΔQ1 + ΔQ2 ⎡ λ λ 1 = ⎢ (θf 1 − θw ) + (θf 2 − θw ) ⎥⋅tw⋅Δt⋅ ⎦ ⎣ δ1 m w⋅ca δ2 tw⋅bw⋅ρa ⋅ca
The resulting temperature increase/decrease is dependent on the mean distances of heat transfer δ1and δ 2 determined according to relations (9) and (10) as the distance from the centroid of upper/lower part of the web to the midpoint on flange center line between flange end and web-to-flange junction, as presented in Fig. 1.
(1)
(2)
(4)
(8)
The net heat flux per unit area is the result of adding the net convective heat flux (2) and the net radiative heat flux (3), according to EN 1991-1-2 [10].
̇ , c = αc⋅(θg − θm ) hnet
λ (θw − θf 1)⋅1⋅tw⋅Δt δ1
The temperature variation due to heat transfer from web to flanges in time Δt is determined as:
temperatures calculated according to EN 1993-1-2 [2] and EN 1991-12 [10].
Δθa, t = ksh
ΔQ1 =
Currently adopted values for the coefficient of heat transfer by convection, αc , and the steel surface emissivity, εm , are 25 W/m2 K and 0.7, respectively [2]. The recommended value in Section 3 of EN19911-2 [10] for both the configuration factor, Φ , and flame emissivity, εf , is 1. It is to be mentioned that the coefficient of heat transfer by convection may change its value if the temperature-time curve is different from the standard fire curve, ISO 834. Either full cross-section or part of section may be considered for establishing the section factor Am/V, whereas more accurate results are provided by the latter approach. The case of full cross-section tends to lead to temperatures less than the web temperatures and higher temperatures when compared to flange temperatures. Table 1 shows the temperatures calculated in this manner for an IPE400 beam subjected to 20 min of the standard ISO-834 fire. This sections are influenced by the shadow effect coefficient ksh, leading to temperatures of entries cross-section less than temperatures of each part taken separately. 2.2. Analytical approach Two additional heat fluxes are proposed for the case of beams with corrugated web, namely the heat transfer by conduction and the radiation from the web to the flanges. 2.2.1. Heat transfer by conduction Since the heating of corrugated web is significantly faster than the heating of flanges and there is contact between the two parts, the necessary conditions for heat transfer by conduction are achieved. The basic starting assumption is considering a uniform temperature for every part of the cross-section computed using the simple relations given by EN 1993-1-2 [2]. A first step is to establish the position of the highest temperature
Fig. 1. Conduction heat transfer from web to flanges.
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δ1 =
tf1 bf 1 h1w + + 2 2 4
(9)
δ2 =
tf 2 bf 2 h2w + + 2 2 4
(10)
. Starting from the point where the heat transfers from web to the top flange, the partial web height h1w, is equal to the ratio of the heat flux towards top flange to the heat fluxes to both flanges, (11). Similarly, it is judged for the bottom flange, (12).
h1w =
ΔQ1 ⋅h w ΔQ1 + ΔQ2
(11)
h2w =
ΔQ2 ⋅h w ΔQ1 + ΔQ2
(12)
Due to iterative procedure and the inverse proportionality between ΔQi and δi , the highest temperature point tends to oscillate around the correct position. Hence there is a need for a sub-step iterative procedure to obtain a steady position. For this, the partial height related to the top flange may be computed according to relation (13).
ΔQ1, n ⋅h w = h1w, n +1 = ΔQ1, n + ΔQ2, n
1 ⋅(θw δ1, n 1 δ1, n
⋅(θw − θf 1) +
− θf 1) 1 ⋅(θw δ 2, n
− θf 2 )
⋅h w (13)
Our experience and monitoring of calculation results suggests that ten steps are sufficient for a converging solution. It must also be said that the initial values of h1w and h2w , in the heat conduction temperature increase calculation, are estimated according to (14).
h1w,0 = h2w,0 =
hw 2
(14)
By this procedure two uncommon scenarios may arise:
• •
If one of the flanges has the same temperature as the web, it results h1w = h w . The heat from the entire web transfers to the other flange with different temperature. If one of the flanges has a higher temperature than the web and the other flange is colder than the web, the length h1w comes out longer than the real web length h w . Heat fluxes (incoming and outgoing) partly eliminate each other in “fictive” length h1w − h w . The “released” flux represents heat flux which only goes through the web, one flange heats up the other and it has no influence on the web temperature. Remaining heat flux heats/cools the web and it is equal to the difference(Qw − Qf 1) − (Qf 2 − Qw ). If this difference is equal to zero, thus, the web temperature is an average of both flange temperatures, the length h1w is infinity. Incoming and outgoing heat fluxes are equal to each other, the heat goes only through the web and the web temperature does not change.
Because the temperatures of the top flange, bottom flange and web are developing at different rates, the ratio of heat fluxes and the mean heat transfer distances are varying continuously. The calculation procedure should be performed to obtain the quantities in the following order: temperature differences, mean heat transfer distances; heat flux into the top and bottom flange, change of temperature of flanges and web. The influence of temperature increase/decrease due to heat conduction is incorporated back into the step-by-step method. In every step, the increase/decrease due to heat conduction is added to the increment due to heat transfer from fire to steel member by convection and radiation. After computing the position of the highest temperature point, the section may be considered to be divided in two parts i.e heat transfer part from web to top flange and heat transfer part from web to bottom flange. Each part is then divided in finite length elements along the web
Fig. 2. Division of steel profile in elements. (a) Finite elements on the cross-section. (b) Numbering of finite elements.
and flange respectively, as principled presented in Fig. 2. The size of the finite elements is considered by the user such that accurate results are obtained. Either equal elements or variable length elements can be defined. The results in this paper are obtained by dividing the web into elements of 20 mm. The basic premise is that the heat is passing through an element and is consumed for heating it. By adapting the basic equations for heat conduction to the current situation, the following relations result for the analytical model. 86
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a part of the incremental method of calculating the temperatures in the cross-section presented in the following.
2.2.2. Heat transfer by radiation Radiation between component parts of cross-section is usually neglected either due to the small temperature difference within crosssection or due to the view factor developed between separate parts which leads to minor changes of temperatures. In the current study, the configuration factor in relation (3) is computed for the radiation from the hot surface of the web which acts as an emitter to the flange which is considered to be the absorber. Depending on the amount of emitted and absorbed energy the temperature tends to decrease in the web and increase in the flange [12]. In relation (3), the configuration factor or view factor is usually considered to have a unitary value. For the proposed methodology this value, measuring the fraction of radiative heat leaving a surface and arriving at the receiving surface, is computed according to Annex G of EN 1991-1-2 [10]. The computation is based on the same partitioning of the web as for the conductive heat transfer in Section 2.2.1 while the flange is divided in only two parts (Fig. 3). Each element of the web is considered to be an emitter while each side of the flange is considered to be an absorber. To solve the radiative heat transfer between the individual parts of a cross-section, only half of the cross-section is considered. The configuration factor is calculated for each element of the web according to the geometric dimensions in (20), (21) and (22) and Fig. 3. Considering the web to be the emitter and the flange to be the absorber, the configuration factor ΦA − B, i (23) can be computed similarly to the relation given in Annex G of EN 1991-1-2 [10] for the orientation presented in Fig. 3.
Fig. 3. Radiation heat transfer from web to flanges.
ΔQi =
λ ⋅(θi − θi −1)⋅(ti −1⋅1)⋅Δt δi
ΔQi +1 =
λ ⋅(θi +1 − θi )⋅(ti +1⋅1)⋅Δt δi +1
(15) (16)
The temperature increase/decrease in each element i (17) is computed function of the temperature difference from adjacent elements (18), (19).
Δθi = Δθi, i −1 + Δθi, i +1
(17)
Δθi, i −1 =
ΔQi λ Δt = ⋅(θi − θi −1)⋅(ti −1⋅1)⋅ mi⋅c δi ti⋅L i⋅1⋅ρ⋅c
(18)
Δθi, i +1 =
ΔQi +1 λ Δt = ⋅(θi +1 − θi )⋅(ti +1⋅1)⋅ mi⋅c δi +1 ti⋅L i⋅1⋅ρ⋅c
(19)
Δθi, i +1 =
ΔQi +1 λ Δt = ⋅(θi +1 − θi )⋅(ti +1⋅1.15)⋅ mi⋅c δi +1 ti⋅L i⋅1⋅ρ⋅c
(19a)
Having the temperatures computed by relations given in EN 19931-2, the temperature difference at the junction between web and flange will stand as an input condition for the heat flux. Application of relations (15)–(19) presented above for the cross-sections which are frequently used in practice i.e. WTA500, WTB750 or WTC1000, shows that the web affected areas have a length from 30 mm up to maximum 100 mm from the flange-to-web junction. The rest of the web is experiencing negligible temperature decrease. In the case of small height beams the web thickness is small and the influenced length along the web is also smaller, close to 30 mm. In order to compensate the increased length due to corrugation, relation (19) is modified by a coefficient of 1.15, relation (19a), for the web element which is in contact with the flange. This value represents the ratio between the wavelength of the web and the longitudinal projected length. Only the shape of the corrugation may increase or decrease the value of 1.15 but not the thickness of the web. The described model of the heat conduction initiated by the temperature difference between the flange and the web is designed as
Fig. 4. Flow-chart of the analytical model.
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Fig. 5. Radiation on corrugated web.
Fig. 6. Finite element model of the heat transfer analysis.. 0.5 ⎡ ⎛ bf ⎞ 2 ⎤ si = ⎢hs2, i + ⎜ ⎟ ⎥ ⎢⎣ ⎝ 4 ⎠ ⎥⎦
cos(ϕA, i ) =
cos(ϕB, i ) =
ΦA − B, i =
Fig. 7. (a) Specimens before test, (b) Specimens during test.
the section factors for web and flange are computed as: (20)
hs, i si
Am, w / Vw =
(21)
(22)
cos(ϕA, i )⋅ cos(ϕB, i ) π⋅si2
⋅Ai =
hs, i⋅bf 4⋅si4⋅π
⋅(L i⋅1)
(23)
The radiative heat flux is determined according to relations (24) and (26) and used to calculate the temperature decrease (25) and increase (27) of the web and flange, respectively. Due to the heat transfer from both sides of the web, the temperature decrease is considered with a factor of 2 in relation (25),
hnet , r , A, i = ΦA − B, i⋅εA⋅εB⋅σ⋅[(θB + 273)4 − (θA, i + 273)4 ] Δθa, t , A, i
Am, w / Vw = 2⋅ ⋅hnet , r , A, i⋅Δt ρa ⋅ca
hnet , r , B =
∑ {ΦA−B,i⋅εA⋅εB⋅σ⋅[(θA,i + 273)4 − (θB + 273)4]} i =1
Δθa, t , B = 2⋅
Am, f / Vf ρa ⋅ca
⋅hnet , r , B⋅Δt
(28)
The values for the temperature change of the web elements and the flange elements are arranged as part of the incremental method for calculating cross-section temperature. For each step the temperature computation is performed according to the incremental procedure for the specified time period for each individual element, thus each element will have various temperatures in subsequent steps, together with the heat conduction from web to flange. For the situation of a fire with cooling phase, the flange may have a higher temperature than the web. The sign of radiative fluxes change and the temperature variation reverse their sign for both web and flange. It must be mentioned that a condition for the temperature difference is added and the view factor is replaced by:
bf 4⋅si
1 1 Am, f / Vf = tw tf
(24)
ΦB − A, i =
hs, i⋅bf 4⋅si4⋅π
⋅(
bf − t f 2
⋅1)
(29)
The evaluation of the hotter part of the cross-section must be performed and inserted in the flowchart of the proposed analytical approach as presented in Fig. 4. For the specimens in contact with the trapezoidal steel sheet, a covering factor κtr is considered to account the smaller influence of heat transfer to the top flange. The value of the covering factor is introduced in the section factor computation of the top flange, (30). According to the geometric configuration of the trapezoidal steel sheet TR150/280/ 0.75 the ratio of the gaps volume to the total volume of the steel deck is 0.7.
(25) (26)
(27)
In relations (24) and (26), the emissivities εA and εB are considered both with a value of 0.7. Being an exchange of heat, only between one side of each element 88
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Am / V =
(κTR⋅bf + bf + 2⋅t f ) bf ⋅t f
(30)
A configuration factor due to web corrugation may be considered by the simple relation (31) where the flat web and the sinusoidal pattern are evaluated according to Fig. 5. By a simple reasoning, we can conclude that a perpendicular projection of the corrugated web could be considered as an area exposed to fire while the configuration factor remains 1. Angle β is the angle of incidence of heat radiation in the given point, measured with respect to the web normal.
Φ=
β dA ∫ cos π⋅s 2
(31)
. 3. Finite element model Either for plane or spatial domain, several papers regarding validation of heat transfer analysis using finite element codes may be found in the literature [7,13]. For the present study a 3-dimensional heat transfer analysis using ABAQUS [14] was performed. Beside the influence of conduction within cross-section, the model was created so that the influence of corrugation of the web over the temperature in the flange regarding the position with respect to the flange end may be evaluated, thus, the length of the model is chosen equal to the wavelength of the web corrugation as presented in Fig. 6. All the corrugated web beams monitored in the experiments performed in [1] are modeled according to the considerations presented in the following. The system of measurement is considered to be kg, m, s and °C. Fig. 8. Temperature of specimen 1 WTC 500-300×20: (a) top flange, (b) bottom flange.
3.1. Geometry The model is created by defining one part, i.e. the web, and
Fig. 9. Web temperature of specimen 1 WTC 500-300×20: (a) mid-height, (b) lower part. Fig. 10. (a) temperature monitoring positions (b) web temperature of specimen 9, WTB 750-250×12, level D.
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Fig. 11. Web temperature of specimen 9, WTB 750-250×12: (a) level A, (b) level B, (c) level C, (d) level E, (e) level F, (f) level G.
extruding its both ends to form the flanges. This technique eliminates the use of interaction definition between individual parts and allows the conduction between web and flanges. The geometric dimensions are taken from the experiments presented in [1]. In order to simulate the configuration as close as possible to the reality, two sections are defined and assigned to the flanges and one section for the web part. For both top and bottom flanges, a shell offset definition is set to the bottom surface and top surface, respectively.
3.3. Interactions The heat transfer from the environment is considering convection and radiation. For convective interaction between environment and steel, a convection coefficient is defined with a value of 25 W/m2 K. The radiation between furnace and steel profile requires several settings in order to use correctly the unit system with Celsius degrees. First a value of −273.15 is defined for the absolute zero temperature and the value of Stefan-Boltzmann constant is set to 5.67e-8. Radiative heat transfer is defined in Abaqus using an emissivity coefficient and an ambient temperature. The emissivity is considered as recommended in EN1993-1-2 [2] with a value of 0.7 for all the surfaces exposed to the gas temperatures except the top flange where, function of geometric configuration and position of steel decking, an emissivity factor is computed as presented in [15,16]. The emissivity of flames for the fire exposed surfaces is considered to be 1.0. For the top flange the total emissivity of hot gas εf depends on the emission coefficient K and thickness or the mean beam length L [15,17], as presented in relation (32).
3.2. Material properties A great advantage of the general purpose finite element code, ABAQUS [14], is the possibility to use temperature dependent mechanical and thermal properties. The necessary properties needed for a heat transfer analysis are conductivity, specific heat and density. The first two are temperature dependent and their pattern is defined according to EN 1993-1-2 [2] while the latter is considered to have a constant value of 7850 kg/m3.
εf = 1 − e−K ⋅ L 90
(32)
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for the temperature increase. Specimen 9 from [1] was positioned such that the trapezoidal steel sheet was in contact with the top flange. For this case both analytical and numerical model provide consideration of top covering as described in Sections 2.2 and 3.4. The temperatures measured during the test and computed by the analytical and numerical models are depicted in Figs. 10 and 11. The temperature measurement levels are resumed from [1] in Fig. 10(a). As a comparison to the recommended method provided by the design code [2], the temperatures obtained for level D accommodate the reference temperature increase of the web. Fig. 11(a)–(f) present the temperature distribution along the web height of Specimen 9. At the levels closer to the web-to-flange junction, i.e A and G, the temperatures show a great difference from the reference values at the level D, especially in the first 15 min of the test. The period exhibiting temperature difference is smaller as the web points are situated further from the flange. The analytical results follow in a good accordance the experimentally determined temperatures of the web while the numerical model has the tendency to predict the temperature in a conservative sense.
The tests performed in [1] involved diesel burners for the isolated specimens and wood cribs for the real-scale building test. These two cases imply the use of emissivity coefficients of 0.43 and 0.8, respectively as recommended in [15,18,19]. Since the emissivity coefficients were empirically determined and their accuracy is questionable [15] an approximate value for the mean beam length may be expressed according to relation (33).
L = C⋅
4⋅V A
(33)
where: V is the total volume of the gas between flange and steel deck, A is the area on which heat is radiated and C is a correction factor of 0.9. By the previously mentioned procedure an emissivity is computed for each of the following situations: 0.27 for the specimens 500 mm under the steel deck, 0.1 for the specimens with trapezoidal steel sheet on top and 0.073 for the specimens in the real building test, in Mokrsko [1]. Convection and radiation sink/ambient temperature are defined by tabulated amplitude with the values recorded in the corresponding test as presented in [1]. The radiation between web and flange is not considered in the finite element model. Depending on the test a constant temperature predefined field is considered as follows: 15 °C and 20 °C for isolated element test stage 1 and stage 2, respectively and 15 °C for the real building test, [1].
4.2. Results for building fire test Real fire tests imply complex phenomena of fluid dynamics and interactions between enclosed elements, liable for a greater error range of results. A great deviation from the experimental results was obtained by the analytical and numerical computation in the case of the top flange of the corrugated web beam CS2 used in the fire test in Mokrsko, Fig. 12. The test recordings for the bottom flange are closely followed by the computational models results, Fig. 11b. This implies either that the interaction between the concrete slab and the steel flange is not disclosed by the computational models as well as it may be a subject of fluid dynamics. The recordings are considered as presented in [1] and summarized in Fig. 13(b).
3.4. Element type A smaller range of elements and element controls are available in heat transfer analysis. The most commonly used type is the first order 4-node heat transfer quadrilateral shell, DS4, also selected for this study. Each element edge was set to have an approximately 10 mm length. 3.5. Analysis control A general step of heat transfer analysis is selected for the numerical analysis with a transient response for results. Although the maximum number of increments needed to be increased, it depends mostly on the time period of the analysis. The characteristic setting of the analysis step which influences the results is the maximum increment size set to 5 s for both fire tests. The total analysis time for the isolated elements tests is 1800 s while for the building fire tests is 4400 s. 4. Validation of the proposed model The following presents the comparison of the results obtained by: tests measurements, proposed analytical methodology and numerical analysis. It is to be mentioned that the results obtained analytically do not include the radiative heat transfer. The influence of radiative heat transfer between flange and web is presented in Section 4.3. 4.1. Results for the isolated elements test The results obtained for the isolated elements test have a similar trend for all specimens and for the three computational options; therefore there are presented the results for one specimen suspended 300 mm below the ceiling, Specimen 1, and for one specimen in contact with the trapezoidal steel deck, Specimen 9, [1] (Fig. 7). Figs. 8 and 9 present the temperature development in the flanges and the web parts according to the positions specified in [1]. The results show similar evolutions of temperature. For both top and bottom flange the value presenting deviation from the experimental results is represented by 735 °C. This temperature represents the value which defines the peak value of the specific heat of steel and due to the slow increase of the gas temperature at that point and the massivity of the flanges, the analytical and numerical models require a longer time
Fig. 12. Temperature of the beam CS2, WTB 500-220×15: (a) top flange, (b) bottom flange.
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Fig. 14. Temperature of beam CS2, WTB 500-220×15: (a) web upper part, (b) web lower part.
hotter part of the beam to the cooler parts. By enabling the radiation part of analytical model, the temperature field suffers the following changes compared to the results obtained without considering the radiation:
Fig. 13. (a) Midheight web temperature, (b) Measurement position.
The temperature measurements at the web mid-height and the computational results presented in Fig. 13(a) are in close agreement, the predicted values being conservative temperatures. For the upper and bottom part of the web the analytical model results are closer to the experimental results than the numerical predicted temperatures, Fig. 14(a) and (b). An aspect common especially for Fig. 12(a) and (b) and Fig. 13(a) is observed on the cooling phase of fire when the slope of the experimental curve is descending at a lower rate than the analytical or numerical curves. Although of less importance, the phenomenon might influence the behavior of structures subjected to parametric curves. A possible cause for the temperature divergence on the cooling phase might be due to the heat that was retained by the ceiling in the experiment. The ceiling of the test setup is made of a trapezoidal steel sheet covered with insulating material. During the increasing phase of the fire, it accumulates heat which is afterward radiated toward the steel beam, leading to a lower decrease rate of the temperature especially on the top flange. Since the plate-thermometers were placed lower, at the level of the web midheight, the radiation of the ceiling was not recorded properly.
4.3. Influence of radiation between cross-section parts In order to assess the effect of all three ways of heat transfer, the following presents the effect of the heat transfer by radiation from the
Fig. 15. Numerical model of parametric study.
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higher rate compared to heat redistribution by conduction. The numerical study is based on a simple 3D model as presented in Fig. 15 according to the specifications mentioned in Section 3. The variable parameters are the web thickness, the flange thickness and the flange width, while the web height and length are considered to have constant values of 500 mm and 155 mm, respectively. The gas temperature increase is defined according to the Standard Fire Curve.
Table 2 Web strip width with min 5% lower temperature than the maximum temperature point on web. hwL [mm] tw=2 mm
tw=2.5 mm
tw=3 mm
tf Time
10
15
20
25
30
10
15
20
25
30
10
15
20
25
30
5 min 10 min 15 min 20 min 25 min 30 min
40 20 10 0 0 0
40 20 20 0 0 0
40 30 20 10 0 0
40 30 20 20 10 0
50 30 20 20 20 10
30 10 10 0 0 0
40 20 10 0 0 0
40 30 20 10 10 0
50 30 20 10 10 10
50 40 30 20 20 10
30 10 10 0 0 0
40 20 10 0 0 0
40 30 10 10 0 0
50 30 20 10 10 10
50 40 30 20 10 10
5.1. Influence of the flange-to-web thickness ratio
Fig. 16. Web temperatures for flange-to-web ratio of: (a) 3.33, (b) 15.
The web influence over the flange and vice-versa depends on the massivity of each part. A thinner web heats faster but it does not have enough “power” to transmit the heat, while a thicker web heats slower but has more heat conducting capacity. A series of numerical simulations were performed using 2, 2.5 and 3 mm web thicknesses and 10, 15, 20, 25, 30 mm flange thicknesses. The extreme flange-to-web thickness ratios resulted from the previous mentioned dimensions are 3.33 and 15. To avoid the influence of flange the study was conducted for a width of 400 mm, which proves to be a correct assumption from the second parametric study.
– For the specimen WTC 500-200×20, situated 300 mm below the ceiling, at time t=15 min, in elements next to the flange, the temperature decreases with maximum 22 °C, which represents only 3%. This temperature decrease is reduced to 0.4% at web midheight. The flange temperature is increased by 1.3%. For 30 min, the temperature decreases with less than 1.5% at the web-to-flange junction, and this decrease is smaller as the web element is further from the flange, to a reduction by 0.2% at web midheight. The flange temperature is increased by 0.8%. – For the specimen in contact with the ceiling, WTB 750-250×12, after a time interval of 15 min the temperature in the web decreases by 1.8–0.2% as the web element is further away from the flange. The flange temperature increases by 0.8%. For 30 min, the web temperatures decrease by 0.5–0.1% varying along the height of the web. For this time interval the flange temperature increases by 0.2%. The temperature changes are only presented for 15 and 30 min in relation with the practice of assessing the fire resistance. As the temperature in the steel beams were obtained by averaging test recordings from multiple positions [1] and the temperature deviation is greater than the temperature difference in the analytical model, caused by the radiation between the web and the flange, it is considered that this effect is of less importance. 5. Parametric study The heat exchange between the web and the flange by conduction indicates a non-uniform temperature along the web and possibly the flange. To investigate the temperature affected area a parametric study was performed considering the dimensions range available in the product catalogue of Zeman & Co GmbH [20]. The fabricated beams available have a web thickness, flange thickness and a flange width of 2–3 mm, 10–30 mm and 200– 430 mm, respectively. Although several web heights are available, from 500 mm, the thinness of this part suggests that there is marginal influence of this parameter over the temperature distribution on the web, since the heat transfer by convection and radiation will have a
Fig. 17. Flange temperature distribution for flange-to-web ratio of: (a) 3.33, (b) 15.
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the temperatures of the protected steel elements. Function of the protection solution, the conduction may or may not be influencing the temperature field. Engineering judgement, advanced models and experimental studies might reveal if temperature difference between flange and web appears so that the present analytical model, considering conduction heat transfer, is suitable to be used. The influence of the non-uniform temperature distribution on the structural behavior of beams with corrugated web is an ongoing study performed at the Czech Technical University in Prague and will be presented in a forthcoming article.
Fig. 16 shows that a significant decrease of temperature may extend to 60 mm along the web height but only in initial phases of fire. For relevant time of fire exposure i.e. 15 min or 30 min, the area with a 5% decreased temperatures is reduced significantly as shown in Table 2. 5.2. Influence of the flange width The width of a flange may increase in some cases the massivity of the flange e.g. for small widths and high thicknesses, thus a smaller temperature increase rate. Having a greater difference of temperature the condition for heat conduction would be even more favorable. The considered flange widths were 200 mm, 300 mm and 400 mm. Fig. 17 presents two extreme cases: 1) web thickness of 3 mm and a flange of 10 mm thick and 2) a 2 mm thick web and a flange thickness of 30 mm. The influence over the flange is visible only for higher section factors i.e. small flange thickness, and thicker webs. Also, the temperature influence is visible only for the first phase of temperature increase, up to 10 min. The temperature increase in the flange, which has a non-conservative effect, extends only 40 mm from the flange-toweb junction the same response exhibited also for wider flanges. After 15 min the temperatures developed in the flange do not exceed the temperatures predicted according to EN 1993-1-2 [2], represented with a red dashed line for every 5 min interval, in Fig. 17.
Acknowledgement This publication was supported by the European social fund within the framework of realizing the project “Support of inter-sectorial mobility and quality enhancement of research teams at Czech Technical University in Prague”, CZ.1.07/2.3.00/30.0034. Period of the project's realization 1.12.2012–30.6.2015. References [1] Jaroslav Vácha, Petr Kyzlík, Ioan Both, František Wald, Beams with corrugated web at elevated temperature,experimental results, Thin-Walled Struct. (2015). http:// dx.doi.org/10.1016/j.tws.2015.02.026. [2] C.E.N, EN 1993-1-2, Eurocode 3: Design of steel structures – Part 1-2: General rules – Structural fire design. Brussels, 2005. [3] T.R. Kay, B.R. Kirby, R.R. Preston, Calculation of the heating rate of an unprotected steel member in a standard fire resistance test, Fire Saf. J. 26 (4) (1996) 327–350. [4] Martin Gillie, Analysis of heated structures: nature and modelling benchmarks, Fire Saf. J. 44 (5) (2009) 673–680. [5] Bin Zhao, Mohsen Roosefid, Experimental and numerical investigations of steel and concrete composite floors subjected to ISO fire condition, in: Proceedings of the Sixth International Conference on Structures in fire, 2010 East Lansing, pp. 407– 416. [6] M. Feng, Y.C. Wang, J.M. Davies, Axial strength of cold-formed thin-walled steel channels under non-uniform temperatures in fire, Fire Saf. J. 38 (2003) 679–707. [7] M. Feng, Y.C. Wang, J.M. Davies, Thermal performance of cold-formed thin-walled steel panel systems in fire, Fire Saf. J. 38 (4) (2003) 365–394. [8] Y.C. Wang, Composite beams with partial fire protection, Fire Saf. J. 30 (4) (1998) 315–332. [9] L. Gardner, K.T. Ng, Temperature development in structural stainless steel sections exposed to fire, Fire Saf. J. 41 (3) (2006) 185–203. [10] C.E.N, EN 1991-1-2, Eurocode 1: Actions on structures – Part 1-2: General actions – Actions on structures exposed to fire. Brussels, 2002. [11] K.Petr, Fire resistance of Beams with Corrugated Web (Ph.D Thesis). Czech Technical University in Prague. Prague(in Czech), 2012. [12] Vacha Jaroslav, Influence of nonhomogeneous temperature on fire resistance of beams with corrugated web (Ph.D Thesis), Czech Techical University in Prague, Prague, 2015 (in Czech). [13] E.OzyurtY.C.WangK.H.Tan, Elevated temperature resistance of welded tubular joints under axial load in the brace member, Eng. Struct. vol. 59, pp. 574–586, February 2, 0014 [14] ABAQUS, ABAQUS Documentation 6.11, Dassault Systèmes, Providence, RI, USA, 2011. [15] U.Wickström, Heat Transfer in Fire Technology. Luleå tekniska universitet, Luleå. 2012. [16] T.JánaY.C.WangF.WaldK.Horová, Temperatures and thermal boundary conditions in reverse channel connections to concrete filled steel sections during standard and natural fire tests, Fire Saf. J., 2015. [17] Dougal Drysdale, An Introduction to Fire Dynamics, John Wiley & Sons, Ltd, 2011. [18] B. Hagglund, L.E. Persson, The heat radiation from petroleum fires, Stockh. FOA Rep. C20126-D6 (A3) (1976). [19] K. Sato, T. Kunimoto, Memoirs of the Faculty of Engineering, Kyoto University 31 (47) (1969). [20] Zenam Beteiligungsgesellschaft mbH. [Online]. 〈www.zeman-stahl.com/en/ products/sin-profiles〉
6. Conclusions The paper presents a study on the temperature development in steel beams with sinusoidal corrugated web. Starting from the simple relations provided by fire design codes, an analytical model which can consider the conduction and the radiation between web and flange, is proposed for calculating the temperature distribution in a steel crosssection. Also, numerical heat transfer analyses were performed considering conduction of steel. Both analytical and finite element models are able to compute the non-uniform temperature distribution over the corrugated web height and predict the temperature variation in the cross-section in an acceptable conservative range. The study reveals the non-uniform temperature variation on the web height and in the flange, at early stages of fire, while for longer fire exposure, the temperature in the cross-section tends toward homogeneous web and flange temperatures. Based on this fact it is considered to predict a better response in the early stages of a thermo-mechanical analysis. For the real scale building test the gas temperatures included also the cooling phase of fire. The descending branch shows a different cooling rate of elements. The experimental results show a lower rate for cooling while the analytical and numerical results show rapidly descending temperatures. This phenomenon questions the validity of the convective coefficients on the cooling phase of elements. The parametric study reveals that the geometric dimensions have a higher influence on web temperature development for cross-section with greater flange-to-web thickness ratio. The increased massivity, given by the width of the flange, has no influence on the temperature development in the cross-section. Since the analytical model described in the paper is based on the relations given in EN1993-1-2 for the unprotected steel elements, the model may be adapted, in particular cases, to the protected steel elements considering the provisions of the design codes. Basically, radiation and convection are not directly present in the computation of
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