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Practical method for calculating the buckling temperature of the web-post in a cellular steel beam in fire
Thin-Walled Structures 85 (2014) 441–455
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Practical method for calculating the buckling temperature of the web-post in a cellular steel beam in fire Peijun Wang n, Xudong Wang, Mei Liu Civil Engineering Department, Shandong University, Jinan 250061, Shandong Province, China
art ic l e i nf o
a b s t r a c t
Article history: Received 5 August 2014 Received in revised form 23 September 2014 Accepted 23 September 2014 Available online 17 October 2014
Buckling behaviors of web-posts in a cellular steel beam at elevated temperatures in a fire were studied using the Finite Element Method (FEM) analysis and available analytical models. The buckling temperatures obtained by the analytical models differed greatly to those obtained from the FEM simulation. Among these analytical models, the buckling temperature obtained through the strut model based on BS5950-1 agreed with the FEM result the best. It is more reasonable to take the width of the compression stress band in the web-post as the effective width of the strut. Numerical parametric studies showed that the width of the compression stress band varied with the opening diameter, the opening distance and the web thickness. A simplified method was proposed to calculate the effective width of the strut. The accuracy of the strut model integrating the new effective width was validated against the FEM simulations. The obtained buckling temperature of the web-post using the modified strut model agreed well with the FEM simulation result. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Cellular steel beam Web-post buckling Strut model Effective width Fire Buckling temperature
1. Introduction The cellular steel beam (CSB) can be made through cutting an H-section steel beam in a zigzag pattern along the web and then re-welding the web-post together or through cutting circular holes in the web directly. The obtained CSB has a higher strength to weight ratio and allows service integration to be installed within the beam depth, as shown in Fig. 1. For the discontinuous in the web, local failures modes, such as the web-post buckling and the Vierendeel bending failure at the perforated section, may happen in a CSB [1]. In the fire situation, the design of a CSB is getting more complex for the degrading of steel at high temperatures and the non-uniform thermal strain across the section [2–4]. For a CSB with a high web height to thickness ratio, the web-post buckling failure was the main concern. Web-post buckling behaviors in CSBs had been investigated experimentally and numerically recently. At ambient temperature, the buckling of the web-post is defined as when the out-of-plan displacement in a web-post suddenly increases [5,6]. The web opening shape may also affect the buckling strength of the web-post. Wang et al. [7] had studied the web-post buckling strength of a CSB with fillet corner hexagonal web openings. Analytical models have also been developed to simply estimate the buckling load of the web-post in a CSB [8–13]. For the Young’s modulus of steel degrades faster than the yield strength with the elevating temperature, the CSB is more vulnerable to
the web-post buckling failure in the fire situation. A CSB was failed by Vierendeel mechanism failure at ambient could be failed by web-post buckling in the fire situation. Though modifying the degrading function of the yield strength of steel at high temperatures, Bihina et al. [10] calculated the critical temperature of a composite cellular beam in a fire. Bitar et al. [11] proposed an empirical model to calculate the buckling strength of a composite cellular beam based on the method presented in SCI publication 100 [12]. Lawson et al. [13] proposed a method to determine the maximum compressive stress in the webpost. Instead of checking the critical section in the web-post, the strut model treated the web-post as a compression strut and checked its stability based on the column buckling curves in BS5950-1 [14]. In this paper, three analytical models for assessing the webpost buckling behaviors at ambient temperature were used to find the buckling strength and buckling temperature of the web-post in the fire condition. A simplified method through modifying the current strut model was proposed to calculate the buckling temperature of a web-post in the CSB. The new effective width in the modified strut model took the width of the compression band in the web-post which was obtained through FEM analysis. And a simplified method was presented to calculate the new effective width. The buckling temperature of the web-post predicted by the proposed strut model was validated by FEM analysis.
2. Finite element model n
Corresponding author. Tel./fax: þ 86 531 88392843. E-mail address: [email protected] (P. Wang).
http://dx.doi.org/10.1016/j.tws.2014.09.019 0263-8231/& 2014 Elsevier Ltd. All rights reserved.
The finite element software ABAQUS was used to simulate buckling behaviors of the web-post in a CSB at the ambient
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Cutting line Height of the original beam Cutting line Web-post Height of the cellular steel beam Cutting part Service intergration Welding position
Opening distance
Width of the web-post
Fig. 1. Cellular steel beam.
temperature and in the fire situation. The numerical simulation included three steps: (1) Obtaining the load bearing capacity of a CSB under the concentrated load at middle span at ambient temperature through utilizing the Riks method in ABAQUS; (2) Obtaining the temperature fields in the unprotected CSB or the unprotected composite cellular beam exposed to standard fire using the thermal analysis; (3) Obtaining the buckling temperature of the web-post in CSB at a given load ratio in the fire situation. The load ratio is the ratio of the applied load to the load bearing capacity at ambient temperature obtained in step (1). The load ratio takes 0.5 here for illustration.
2.1. Mechanical model at ambient temperature The studied CSB was made through cutting circular holes in the web of a steel beam with section of UB457 152 52. In order to prevent the flexural torsional buckling failure of the CSB, the lateral displacement of the flange at the middle span was fixed. Web stiffeners were placed at the supports and the mid-span where the concentrated force was applied to prevent the buckling of the web under the local concentrated force. Initial out-of-plane imperfection of the web-post took the first buckling mode obtained from the buckling analysis of ABAQUS. The magnitude of the initial-out-of-plane imperfection was 1 mm. The CSB was meshed by the S4R element in ABAQUS, a 4-node reduced integration shell element, as shown in Fig. 2. The steel was grade S355 steel with Young’s modulus of 200 GPa and the Poisson’s ratio of 0.3. Geometrical and mechanical properties of the studied beams were listed in Table 1 [6]. The FEM model was calibrated against experimental results reported by Tsavdaridis and D’Mello [6]. Mesh convergence studies showed that the S4R shell element with the mesh size of 10 mm could give accurate results, as shown in Fig. 3. Comparison of load-deflection curves and failure modes of the web-post obtained from the proposed FEM model and the test were shown in Fig. 4(a) and (b). The ultimate load obtained from the FEM simulation was greater than that from the test [6], which might be caused by the assumption in the magnitude of initial out-of-plan imperfection of the web-post. In the test carried out by Tsavdaridis and D’mello [6], no measured initial imperfection was reported. Redwood and Demirdjian [5] recorded a measured maximum imperfection of the web in their tests which was about 1 mm (26.3 times of tw/200). Undoubtedly, the web-post with a greater initial geometric imperfection has a smaller ultimate load. However, at present there were no researches that presented a larger initial imperfection than 1 mm. In the following parametric study, the initial imperfection was taken as 1 mm. FEM simulation results reported by Tsavdaridis and D’Mello [6] were also presented in Fig. 4(a). The simulated ultimate load presented by Tsavdaridis and D’mello [6] was greater than that obtained from the FEM model
presented here, for a much smaller imperfection of tw/200 (0.038 mm) was introduced in their model. In the FEM simulation, the CSB was failed by the web-post buckling, which agreed well with the test result, as shown in Fig. 4(b). The outof-plane displacements in the upper and lower part of the web-post were in the opposite direction and the web-post buckled in an S-shape mode along the section height. 2.2. Thermal-mechanical coupled model in the fire situation 2.2.1. Thermal-mechanical coupled model Heat transfer analysis was carried out first to obtain the temperature field in a CSB and a composite cellular beam exposed to the ISO834 standard fire. Points where the temperatures were reported for verification were shown in Figs. 2 and 5. In the thermal analysis model, the CSB was meshed using the 4-node heat transfer quadrilateral shell element DS4. The concrete slab in the composite cellular beam was meshed using the 8-node linear heat transfer brick element DC3D8. Both the radiation and convection heat transfer between the fire and the structural component were included in the model, as shown in Fig. 6. The CSB exposed to fire from four sides and the composite cellular steel beam exposed to fire from three sides. The emissivity of the surfaces and the convective heat transfer coefficient were taken as 0.8 and 25 W/m2 K 1, respectively. The convective heat transfer coefficient was taken as 4 W/m2 K 1 for the unexposed side of the concrete slab in the composite cellular beam. Thermal properties of steel and concrete varied with temperatures and took those described in ENV1993-1-2 [15] and ENV1994-1-2 [16]. In the mechanical model, the DS4 element to mesh the CSB was replaced by S4R structural shell element and the DC3D8 element to mesh the concrete slab was replaced by C3D8R structural brick element. The temperature field in the CSB and the concrete slab were defined by reading the result file generated during the thermal analysis. Mechanical properties of steel and concrete at elevated temperatures followed the descriptions in EN 1993-1-2 [15] and ENV1994-1-2 [16]. The ABAQUS/Explicit was used in the simulation to avoid convergence problems which were often encountered when doing the geometric and material non-linear numerical analysis. The buckling temperature was defined at which the lateral displacement of the web-post suddenly increased. 2.2.2. Model verification 2.2.2.1. Verification of the thermal model. EN1993-1-2 [15] provided equations to calculate the temperature elevation of an unprotected steel structural component exposed to fire. Δθa;t ¼ ksh
Am =V _ hnet Δt cα ρα
ð1Þ
where ksh was the correction factor for the shadow effect; Am =V was the section factor for unprotected steel members [1/m]; Am was the surface area of the member per unit length [m2/m]; V was the volume of the member per unit length [m3/m]; ca was the
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Fig. 2. Finite element model of a CSB.
Table 1 Geometrical and mechanical properties of the beams.
Upper member dimension Section Depth Lower member dimension Section Depth Openings characteristic Number Number of circular openings Number of elongated openings Cell diameter Cell pitch Concrete slab dimension Thickness Width Upper member strength Flange yield strength Web yield strength Lower member strength Flange yield strength Web yield strength Concrete strength Compressive strength
Cellular steel beam
Composite cellular steel beam
Beam A1 [6]
Beam 1 [10]
Beam 3 [10]
UB457 152 52 224.9 mm
IPE360
IPE360
255 mm
255 mm
UB457 152 52 224.9 mm
IPE450
IPE450
300 mm
300 mm
4 4
13 12
13 12
0
1
1
315 mm 409.5 mm
375 mm 600 mm
357 mm 600 mm
– –
120 mm 2200 mm
120 mm 2200 mm
360 MPa 375 MPa
460 MPa 480 MPa
451 MPa 478 MPa
360 MPa 375 MPa
438 MPa 475 MPa
438 MPa 475 MPa
–
31.0 MPa
33.5 MPa
Fig. 3. Mesh convergence study.
specific heat of steel [J/kg K]; h_ net was the design value of the net heat flux per unit area [W/m2]; Δt was the time interval [s] and ρa was the unit mass of steel [kg/m3]. Temperature elevation of the CSB obtained by Eq. (1) and the FEM simulation were shown in Fig. 6. The flange temperatures obtained from Eq. (1) were a little lower than those from the FEM simulation. However, the web temperatures obtained by Eq. (1) and the FEM simulation agreed very well, which showed the validation of the thermal model. The temperatures at the solid web and at the perforated web were nearly the same, as shown in Figs. 7 and 8, which showed that the openings had very little influences on the temperature distribution of the web in a CSB. Usually the temperature in the web was a little higher than those in the flanges. Non-uniform temperature distribution across the section will cause additional compression thermal stress to the hotter part. For the studied CSB, the maximum temperature difference is about 40 1C, as shown in Fig. 6. Hence, the compression thermal stress in the web is approximately σ th ¼ α ΔT ET ¼ 1:4 10 5 40 1:2 10 5 ¼ 67:2 MPa
ð2Þ
where α is the thermal expansion coefficient; ΔT is the temperature difference; ET is the Young’s modulus of steel at temperature T, and T takes the value of 600 1C here. Undoubtedly, the additional thermal stress will greatly influence behaviors of the beam in fire. In the thermal-mechanical coupled model, the non-uniform temperature distribution in the structural member was included by reading the thermal analysis result file. The thermal model was also verified by test results on the temperature elevation of unprotected cellular steel beams carried out by Bailey [17]. The dimension of the beam and locations of thermocouples are shown in Fig. 9(a) and (b), respectively. Fig. 9(c) showed the comparison of the temperature at the centre of the web-post (position C7) of the cellular beam and that at the corresponding position on the solid beam (position S5) measured in the test and obtained through FEM simulation. Fig. 9(d) showed the comparison of temperatures at the bottom flange on the solid beam (position S7 and S8) and at the bottom flange of the cellular beam below the webpost (position C16 and C17) obtained from fire test and FEM simulation. It can be seen that temperatures obtained through the proposed FEM model agreed very well with test results. We can have confidence in the simulation results of the thermal model. 2.2.2.2. Verification of the thermal-mechanical coupled model. Test results on composite cellular steel beams, Beam 1 and Beam 3, reported by Bihina et al. [10] were used to verify the thermal-mechanical coupled
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Fig. 4. Comparison of the load-deflection curves and the failure modes. (a) Comparison of load-displacement curves. (b) Comparison of the buckling modes.
Fig. 5. Thermal-mechanical coupled model of the composite cellular beam.
Radiation Convection
Air 4-side fire exposed
Fig.6. Thermal boundary conditions of the CSB and the composite cellular beam.
model, as shown in Fig. 8. Geometry and material properties of the test specimen were listed in Table 1. Comparison of temperatures, loaddeflection curves and failure modes between FEM and test results were shown in Figs. 10–12, respectively.
Temperatures in the upper flange of Beam 1 obtained from the FEM simulation were lower than those measured from the test [10]. However, the temperature difference was only about 50 1C. Temperatures in the lower flange and in the web obtained from
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the FEM simulations agreed very well with test results, as shown in Fig. 10, which showed the validation of the thermal analysis model. In the composite cellular beam, temperatures in the upper flange were much lower than those in the lower flange and the web. The great thermal compression stress must be included in the fire safety design for a composite steel beam. In the following structural analysis, the temperature field in the composite cellular beam was defined by reading the result file generated during the thermal analysis. The non-uniform temperature distribution in the composite beam can be precisely considered. Load-deflection curves of Beam 1 and Beam 3 obtained from the FEM simulation and the test agreed very well, as shown in Fig. 11, which showed the validity of the thermal-mechanical coupled model. The deformation of the web-post of Beam 1 and Beam 3 predicated by the FEM simulation agreed well with the test results, as shown in Fig. 12(a) and (b). They agreed well with each other. The web-post buckled in an S-shape in the fire, which is the same to the buckling mode of the web-post at the ambient temperature.
3. Parameter study Effects of the web-post geometry on the buckling behaviors and buckling temperatures of the web-post in a CSB in fire were studied using the verified thermal-mechanical coupled model. The studied CSB had the section of UB457 152 52. The applied concentrated load at the middle span was 50% of the ultimate load that was obtained from FEM simulation at ambient temperature. The opening diameter was 315 mm and the corresponding opening to depth ratio, d0/h, was 0.7, as shown in Fig. 13. d0 was the diameter of the circular opening; h was the section height of the cellular steel beam. The changing parameters were the web thickness, tw, and the opening distance, S. Values of the web thickness and the opening distance studied were:
Fig. 7. Temperature elevation of the CSB obtained by Eq. (1) and the FEM simulations.
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the web thickness: 3.9 mm, 5.0 mm, 6.0 mm, 7.6 mm, and 10.5 mm. The corresponding opening diameter to thickness ratio, d0/tw, were: 80.8, 63, 52.5, 41.4, and 30. the opening distance: 378.0 mm, 409.5 mm, 441.0 mm, 472.5 mm, 504.0 mm, 535.5 mm, and 567.0 mm. The corresponding opening distance to diameter ratio, S/d0, were: 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, and 1.8.
3.1. Buckling of the web-post at ambient temperature The CSB might be failed by the buckling in the web-post or Vierendeel mechanism plastic failure at the perforated section at the ambient temperature. Which failure mode will occur depends on the d0/tw and S/d0 of the web-post. The shear strength of the web-post with different d0/tw and S/d0, obtained through FEM simulations, were shown in Fig. 14. The points covered by a solid circle in Fig. 14 represented that the web-post was failed by Vierendeel mechanism plastic failure at the perforated section. That was, for the CSB with a stocker web-post, the plastic failure at the perforated section was the reason that caused the failure of the CSB [18]. When the CSB was failed by the web-post buckling, the stress in the web-post might not reach the yield stress of steel both at the web-post and the perforated section, as shown in Fig. 15. There was only one plastic hinge at the perforated section, which means that the Vierendeel mechanism had not been fully developed when the web-post buckled. On the other hand, when the CSB was failed by the Vierendeel mechanism plastic failure, steel in the flanges and the web at the perforated section were all yield. Four plastic hinges were fully developed at the perforated section. The stress in the web-post might be lower than the yield stress of steel, as shown in Fig. 16. 3.2. Buckling temperature of the web-post in a fire The buckling temperature of the web-post was defined as at which its lateral displacement of the web-post suddenly increased. For the web-posts with different d0/tw, the development of the lateral displacement with the temperature elevation was shown in Fig. 17. The definition of the buckling temperature was also shown in Fig. 17. Buckling temperatures of the web-post obtained through FEM simulation, Tcr,FEM, with different were d0/tw and S/d0 were listed in Table 2. With the increase in S/d0 or d0/tw, the web-post was getting stockier and the buckling temperature increased. For example, for the CSB with d0/tw of 80.8, the buckling temperature increased from 285 1C to 345 1C when the S/d0 increased from 1.2 to 1.8. For the web-post with d0/tw of 30, the CSB was failed by Vierendeel mechanism failure when S/d0 increased to the range of 1.6 to 1.8 (Fig. 14). The changing of failure modes with the increase in S/d0 was shown in Fig. 18. The failure modes of the same CSB may be different in the fire and at ambient temperature. In the fire situation, only the CSB with d0/tw of 30 and S/d0 in the range of 1.6 to 1.8 were failed by Vierendeel mechanism failure, as shown in Fig. 18 and listed in Table 2. However, at ambient temperature,
Fig. 8. Temperature distributions in the CSB at different heating time.
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Fig. 10. Comparison of the temperatures of Beam 1 obtained by tests and FEM simulations.
Fig. 11. Comparison of the vertical defection obtained by tests and FEM simulations.
1.5 to 1.8 were failed by Vierendeel mechanism failure when d0/tw was 41.4, as shown in Fig.16. The failure modes of the CSB under fire situation should be re-assessed. It will lead to unsafe design if we simply adopted the design method for ambient temperature to the fire safety design of a web-post.
4. Practical method for calculating the buckling temperature of a web-post in fire The buckling temperature of the web-post in a CSB can be determined through σ w;f i r σ w;f i;Rd
ð3Þ
where σw,fi is the maximum compression stress in the web-posts; and σw,fi,Rd is buckling resistance of the web-posts in the fire situation. Fig. 9. Verification of the thermal model by Bailey’s test results. (a) Dimension of the solid web beam and positions where temperatures were measured. (b) Dimension of the cellular steel beam and positions where temperatures were measured. (c) Comparison of temperatures at the web. (d) Comparison of temperatures at the flange.
4.1. Models for calculating the compression stress in the web-post
the CSB with d0/tw of 30 and S/d0 in the range of 1.3 to 1.8 were all failed by Vierendeel mechanism failure. In the fire situation, no CSBs were failed by Vierendeel mechanism failure when d0/tw was 41.4. At ambient temperature, the CSBs with S/d0 in the range of
Three analytical models for calculating the compression stress in the web-post,σ w;f i , were studied, which were the Bitar’s model [11], the Lawson’s model [13] and the strut model [6].
P. Wang et al. / Thin-Walled Structures 85 (2014) 441–455
4.1.1. The Bitar’s model Bitar et al. [11] assumed the horizontal section across the web-post as the critical section, as shown in Fig. 19. The position of the critical section, where the compression stress reached its maximum value, was determined by vq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 2 2 u S=d0 þ 8 S=d0 2 S=d0 d0 t dw ¼ ð4Þ 2 2
447
where dw was the distance from the middle of the web-post to the critical section. The width of the critical section, lw, was obtained by
0 lw ¼ d0 @
S d0
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 2dw A 1 d0
ð5Þ
Fig. 12. Comparison of the failure mode obtained by tests and FEM simulations. (a) Deformation shape of Beam 1 after test. (b) Deformation shape of Beam 3 after test.
Fig. 13. Dimension of the studied CSB.
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Table 2 Buckling temperatures of the web-posts obtained through FEM simulation for CSBs with load ratio of 0.5 (in 1C). S/d0
1.2 1.3 1.4 1.5 1.6 1.7 1.8
d0/tw 80.8
63
52.5
41.4
30
285 348 369 371 374 375 345
453 473 491 508 524 538 508
526 540 540 553 565 576 540
551 578 578 588 597 606 607
568 588 598 615 – – –
–: Represents the CSB failed by Vierendeel mechanism plastic bending failure.
Fig. 14. Ultimate strength of the CSBs by FEM simulations.
Fig. 17. Lateral displacement of the web-posts.
The principal compressive stress, σw,fi,Ed, in the critical section in the fire situation was calculated by σ w;f i;Ed ¼ Fig. 15. Stress distribution and deformation of the CSB with S/d0 ¼1.3, d0/tw ¼41.4. (a) Stress distribution in the web-post. (b) Deformation of the web-post.
6M w;f i;Ed 2 2 lw t w 1 4 dw =d0
ð6Þ
where Mw,fi,Ed, was the bending moment in the critical section in a fire situation and was obtained through M w;f i;Ed ¼ V h;f i;Ed dw þ M h;f i;Ed
ð7Þ
Mh,fi,Ed and Vh,fi,Ed were the bending moment and horizontal shear force at the middle of the web-post, respectively. 4.1.2. The Lawson’s model Lawson et al. [13] assumed the section that passed through center of the circular opening as the critical section, as shown in Fig. 20. The stresses varied along the section and with the changing of the incline angle. The critical angle when the maximum compressive stress occurred in the critical section depended on the width of the web-post. Horizontal equilibrium due to tension or compression stresses, σ, and shear stress, τ, at an angle θ in the web-post was obtained as S V h ¼ ðσ sin θ þ τ cos θÞ tw ð8Þ cos θ d0
Fig. 16. Stress distribution and deformation of the CSB with S/d0 ¼1.7, d0/tw ¼41.4. (a) Stress distribution in the web-post. (b) Deformation in the web-post.
where Vh was the horizontal shear force at the middle of the webpost. Moment equilibrium was obtained by taking moments around the point A, which was the intersection of the two planes at an angle θ through the centers of the adjacent openings. For an approximately uniform stress in the critical section, equilibrium
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449
Fig. 18. Failure modes of the CSBs in a fire.
was obtained from: V h y ¼ σt w x2
ð9Þ
where y ¼ 0:5 tan θ and x ¼ 0:5 S= cos θ d0 . The angle at which the maximum compression stress occurred in the section was established by d0 cos θ ¼ þ 4S Fig. 19. Web-post buckling model developed by Bitar et al. [11].
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 d0 þ 0:5 4S
ð10Þ
The maximum compression stress, relative to the horizontal shear stress at mid-height of the web-post, τh, was given by: σ max ¼ τh
ðS d0 Þ cos θ ðS d0 cos θÞ sin θ
ð11Þ
The shear stress could be obtained by the relationship between Vh and Vv [19], was given by: Vh S ¼ V v dG τh ¼
Fig. 20. Web-post buckling model developed by Lawson et al. [13].
Vh S0 t w
ð12Þ
ð13Þ
S0 was the width of the web-post. Vh and Vv were the horizontal and vertical shear force on the web-post, respectively.
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4.1.3. The strut model The strut model considered compressive and tensile forces acting across the web-post on opposite diagonals. A diagonal strut under compression was designed to calculate the compressive stresses of the web-post, as shown in Fig. 21. For simplification, the vertical shear force in the upper T section was used to calculate the compressive stress acting on the strut. The effective width of the strut, be, was taken as a half of the total width of the web-post: be ¼
S0 2
ð14Þ
The compressive stress in the strut was: σ¼
V v =2 Vv ¼ be t w S0 t w
The length of the strut was given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 le ¼ 0:5 S20 þd0
ð15Þ
ð16Þ
The yield stress of the steel in the web was 375 MPa at ambient temperature and the applied load ratio was 0.5. However, for the CSB with S/d0 of 1.2 and d0/tw of 80.8, the predicated compression stresses in the web-post were 418 MPa and 276 MPa by the Bitar’s model and the Lawson’s model, respectively. Theoretically, they should be around 0.5 375 MPa that was 187.5 MPa, where the 0.5 was the load ratio and the 375 was the yield strength of steel. The Bitar’s model and the Lawson’s model gave a too great compression stress in the web-post. The compression stress predicated by the strut model was 138 MPa, which was a reasonable value although it was a little lower. The strut model will be used to calculate the compression stress in the web-post at ambient temperature and in a fire. Table 3 Compressive stresses in the strut calculated by the Bitar’s Model for CSBs with load ratio of 0.5 (in MPa). S/d0
4.2. Compression of the stresses obtained by different analytical models Compressive stresses in the web-post calculated by the Bitar’s model [11], the Lawson’s model [13] and the strut model [6] were listed in Tables 3 and 5, respectively. The applied load ratio was 0.5. The opening distance will greatly affect the compression stresses in the web-post. With the increase in S/d0, the compression stress in the web-post decreases. However, the compression stresses predicated by different models differed greatly. For example, for the web-posts had d0/tw of 80.8, when S/d0 increased from 1.2 to 1.8, the compression stresses calculated by the Bitar’s model decreased from 418 MPa to 133 MPa, as listed in Table 3. The stresses calculated by the Lawson’s model and strut model were much smaller. They decreased from 276 MPa to 79 MPa calculated by the Lawson’s model and from 138 MPa to 42 MPa calculated by the strut model, as listed in Tables 4 and 5. The compression stress in the web-post decreased with the increase in the web thickness if the Bitar’s model was employed, as listed in Table 3. For the web-post with S/d0 of 1.2, when d0/tw decreased from 80.8 to 41.4, the compression stress decreased from 418 MPa to 272 MPa. However, if the Lawson’s model or the strut model were employed, the compression stress in the webpost seemed not to be affected by the changing of web thickness, as listed in Tables 4 and 5. For the same web-post with S/d0 of 1.2, when d0/tw decreased from 80.8 to 41.4, the compression stress changed from 276 MPa to 298 MPa if they were calculated by the Lawson’s model and from 138 MPa to 147 MPa by the strut model.
d0/tw 80.8
1.2 1.3 1.4 1.5 1.6 1.7 1.8
63.0
52.5
41.4
30.0
σcal
σFEM
σcal
σFEM
σcal
σFEM
σcal
σFEM
σcal
σFEM
418 277 215 181 159 144 133
82.0 86.0 79.3 70.5 67.5 60.3 56.1
409 313 264 231 206 184 163
83.0 71.4 70.4 69.8 65.2 62.7 57.0
343 262 221 193 172 154 136
111 82.9 70.7 72.1 70.0 66.1 61.5
272 208 175 154 137 122 108
112 96.1 83.8 80.5 79.8 71.5 66.9
199 152 128 112 100 89 79
137 111.9 92.4 91.4 – – –
–: Represents the CSB failed by the Vierendeel mechanism plastic bending failure.
Table 4 Compressive stresses in the strut calculated by the Lawson’s model for CSBs with load ratio of 0.5 (in MPa). S/d0
d0/tw ¼ 80.8
1.2 1.3 1.4 1.5 1.6 1.7 1.8
63.0
52.5
41.4
30.0
σcal
σFEM
σcal
σFEM
σcal
σFEM
σcal
σFEM
σcal
σFEM
276 181 138 114 99 88 79
82.0 86.0 79.3 70.5 67.5 60.3 56.1
270 204 170 146 128 112 97
83.0 71.4 70.4 69.8 65.2 62.7 57.0
275 211 178 156 138 123 109
111 82.9 70.7 72.1 70.0 66.1 61.5
298 230 193 168 148 131 114
112 96.1 83.8 80.5 79.8 71.5 66.9
340 240 191 160 137 117 100
137 111.9 92.4 91.4 – – –
–: Represents the CSB failed by the Vierendeel mechanism plastic bending failure.
Fig. 21. Strut model for predicating the compression stress in the web-post. (a) Strut in wide web-post (b) strut in middle width web-post (c) strut in narrow web-post.
P. Wang et al. / Thin-Walled Structures 85 (2014) 441–455
4.3. Buckling resistance of the web-post in a fire
4.4. Buckling resistance of the web-post calculated based on BS5950-1
According to the strut model, the width and the length of the compression strut were calculated by Eqs. (14) and (16). The buckling resistance of the strut in a fire, σw,fi,Rd, was calculated based on BS5950-1 [14] considering the degrading of steel at high temperatures. A Perry formula was used for calculating the buckling resistance of the strut: σ¼
f E;θ f y;θ
φθ þ φ2θ f E;θ f y;θ
ð17Þ
0:5
and φθ ¼
f y;θ þ ðη þ 1Þf E;θ 2
ð18Þ
where fE,θ was the Euler bucking stress of the strut. f E;θ ¼
451
π 2 Eθ λ2
ð19Þ
λ was the slenderness of the web-post pffiffiffiffiffiffi le 12 λ¼ tw
The buckling resistance of the strut based on BS5950-1 [14] considering the strength and Young’s modulus reduction of steel in a fire was listed in Table 6. Temperatures at which the buckling resistance in Table 6 were calculated were listed in Table 2. With the increase of the temperature, the buckling resistance of the web-post reduces. The buckling temperature is the temperature at which the buckling resistance equals to the compression stress in the web-post, as defined by Eq. (3). Through comparing the compression stresses predicated by the strut model that were listed in Table 5 and the buckling resistance that were listed in Table 6, we can find that, if the buckling temperatures took those listed in Table 2, the buckling resistance predicated by BS5950-1 were a little smaller than the compression stress. The temperatures given in Table 2 were the buckling temperature obtained through the FEM simulation. That was, BS5950-1 might underestimate the buckling temperature of the web-post in a fire.
ð20Þ
4.5. Critical temperatures of the web-post obtained from analytical models
le was the length of the strut and calculated by Eq. (16). The Perry factor η was given by:
The critical temperature of the web-post can be obtained by the following steps:
η ¼ 0:001αðλ λ0 Þ
ð21Þ
where π 2 Eθ λ0 ¼ 0:2 f y;θ
!0:5 ð22Þ
α was the Robertson factor that was determined considering the section type and taken as 8.0 for a Class 4 cross-section. Table 5 Compressive stresses in the strut calculated by the strut model for CSBs with load ratio of 0.5 (in MPa). S/d0
d0/tw 80.8
1.2 1.3 1.4 1.5 1.6 1.7 1.8
63.0
52.5
41.4
30.0
σcal
σFEM
σcal
σFEM
σcal
σFEM
σcal
σFEM
σcal
σFEM
138 98 77 64 55 48 42
82.0 86.0 79.3 70.5 67.5 60.3 56.1
134 110 94 81 71 61 52
83.0 71.4 70.4 69.8 65.2 62.7 57.0
136 113 98 86 76 66 58
111 82.9 70.7 72.1 70.0 66.1 61.5
147 123 106 93 81 70 60
112 96.1 83.8 80.5 79.8 71.5 66.9
166 127 104 87 74 62 52
137.0 111.9 92.4 91.4 – – –
–: Represents the CSB failed by the Vierendeel mechanism plastic bending failure.
Table 6 Buckling resistance predicated by BS 5950-1 at the temperature given in Table 2 (in MPa). S/d0
1.2 1.3 1.4 1.5 1.6 1.7 1.8
(1) Using the strut model to determine the compressive stresses in the web-post under a given load, as listed in Table 5 when the load ratio is 0.5; (2) Making an assumption on the buckling temperature of webpost; (3) Following the description in BS5950-1 and considering the degrading of steel at high temperatures to determine the buckling resistance at the assumed temperature; (4) If the buckling resistance calculated in step (3) equals to the compression stress calculated in step (1), the temperature assumed in step (2) is the buckling temperature of the webpost. Otherwise, make a new assumption on the buckling temperature and re-calculate the buckling resistance of the web-post;
d0/tw 80.8
63
52.5
41.4
30
63 56 53 48 44 42 39
78 73 67 58 50 43 46
82 75 71 64 55 47 54
104 83 80 70 61 55 51
131 105 102 90 – – –
–: Represents the CSB failed by the Vierendeel mechanism plastic bending failure.
When the load ratio was 0.5, the buckling temperatures calculated following the above procedures were listed in Table 7. δ in Table 7 was the error between the buckling temperatures calculated through the analytical model and that from FEM simulation. δ¼
T cr;str T cr;FEM 100% T cr;FEM
ð23Þ
where Tcr,str was the buckling temperature calculated using the strut model; Tcr,FEM was the temperature obtained through FEM simulation. The strut model will underestimate the buckling temperature of the web-post, as listed in Table 7. For the thin web-post, the temperature difference was more than 90%. The strut model has to be modified to improve its accuracy in predicating the buckling temperature of a web-post in a fire. 4.6. Modified strut model for calculating the buckling temperature of a web-post in fire Three parameters might affect the bucking resistance of the strut at ambient temperature and in a fire situation: (1) the length of strut, l; (2) the buckling length factor, μ; (3) the effective width, be. The buckling length of the strut was calculated by Eq. (16). The length of strut took the diagonal length of the web-post, as shown in Fig. 22. The buckling factor, μ, took the value of 0.5 for there were
452
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Table 7 Buckling temperature of the web-post calculated using the strut model for CSBs with load ratio of 0.5 (in 1C). S/d0
d0/tw 80.8
63.0 δ (%)
Tcr,str (1C) a
1.2 1.3 0.4 0.5 1.6 1.7 1.8
Tcr,str (1C)
92.98 94.25 94.58 94.61 52.67 36.80 20.29
20 20a 20a 20a 177 237 275
52.5
a
20 20a 194 293 365 420 467
41.4
30.0
δ (%)
Tcr,str (1C)
δ (%)
Tcr,str (1C)
δ (%)
Tcr,str (1C)
δ (%)
95.58 95.77 60.49 42.32 30.34 21.93 8.07
177 379 442 479 504 516 528
66.35 29.81 18.15 13.38 10.80 10.42 2.22
452 511 532 548 561 573 584
17.97 11.59 7.96 6.8 6.03 5.45 3.79
526 573 597 620 – – –
7.39 2.55 0.17 0.81
–: Represents the CSB failed by the Vierendeel mechanism plastic bending failure. a
Represents the web-post buckles at ambient temperature.
Fig. 22. Direction and width of the compression principal stresses band in the web-post at buckling at ambient temperature.
Table 8 Measured effective width of the strut form FEM analysis and comparison with be for CSBs with load ratio of 0.5 (in mm). S/d0
S0 (mm)
be (mm)
d0/tw 80.8
1.2 1.3 1.4 1.5 1.6 1.7 1.8
63 94.5 126 157.5 189 220.5 252
31.5 47.3 63.0 78.8 94.5 110.3 126.0
63.0
52.5
41.4
30.0
be,FEM (mm)
be,m/be
be,FEM (mm)
be,m/be
be,FEM (mm)
be,m/be
be,FEM (mm)
be,m/be
be,FEM (mm)
be,m/be
60.16 80.79 88.83 103.16 116.23 124.58 142.38
1.91 1.71 1.41 1.31 1.23 1.13 1.13
59.22 75.12 79.38 102.37 128.52 147.73 128.52
1.88 1.59 1.26 1.30 1.36 1.34 1.02
47.56 72.76 85.68 105.52 122.85 142.22 147.42
1.51 1.54 1.36 1.34 1.30 1.29 1.17
51.97 82.21 93.24 116.55 138.92 148.84 142.38
1.65 1.74 1.48 1.48 1.47 1.35 1.13
39.69 66.62 85.05 96.86 – – –
1.26 1.41 1.35 1.23
–: Represents the CSB failed by the Vierendeel mechanism plastic bending failure.
two half wave lengths along the strut. For the restraints to the strut provided by the flange and the effect of the tension membrane action provide by the tension stress were ignored, the buckling stress of the web-post was underestimated by the strut model, especially for a thin web-post [21]. Traditionally, the effective width of the compression strut took a half of the width of the web-post [6,20], which was not coincidence with the FEM simulation results. A new effective width of strut is proposed in this paper based on the width of the compression stress band in the web-post.
4.6.1. Effective width of the compression strut The width and direction of the compression strut in the webpost can be defined by those of the compression principal stress band. For example, for the CSB with S/d0 ¼1.5 and d0/tw ¼41.4, the distribution of the compressive principal stresses in the web-post at buckling was shown in Fig. 22. The band width of the
Table 9 Values of the coefficients. a0 ¼0.983227 b0 ¼ 0.623962 c0 ¼ 0.072041
a1 ¼ 00562 b1 ¼ 0.487153 c1 ¼ 0.07283
c2 ¼ 0.016533
compression principal stress was affected by the opening diameter, the opening distance and the web thickness, as listed in Table 8. be,FEM was the measured effective width of the compression strut from the FEM simulation. The effective strut width nearly equaled to the web-post width for the CSB with a thin web and narrow web-post, as listed in Table 10. For the web-post with S/d0 of 1.2 and d0/tw of 80.0, the actual web-post width was 63 mm. The effective width calculated by Eq. (14) was only 31.5 mm; however, the measured effective width in the FEM
P. Wang et al. / Thin-Walled Structures 85 (2014) 441–455
simulation was 60.16 mm. With the increase in the web thickness or the opening distance, the measured effective width was close to the effective width calculated by Eq. (14) which was a half of the web-post width. The effective width of the strut can be expressed by a function of S0, d0 and tw as: be;m ¼ κ
S0 2
κ ¼ a0 þ a1
ð24Þ d0 tw
a0 ¼ b0 þ b1
d0 tw
a1 ¼ c0 þc1
453
2 S S þ c2 d0 d0
ð27Þ
The coefficients in Eqs. (25)–(27) can be obtained through curve fitting of the FEM analysis results, as listed in Table 9. The comparison of the effective width calculated by Eq. (24) and the FEM simulation were listed in Table 10. δ in Table 10 was the error of the calculated effective width to that obtained through FEM simulation, which was calculated by be;m be;FEM 100% be;FEM
ð25Þ
δ¼
ð26Þ
For most cases, δ was less than 5%, which showed the correctness of the proposed equations.
ð28Þ
Table 10 Effective width calculate by Eq. (14) and comparison with FEM results for CSBs with load ratio of 0.5 (in mm). S/d0
S0 (mm)
be (mm)
d0/tw 80.8
1.2 1.3 1.4 1.5 1.6 1.7 1.8
63.0 94.5 126.0 157.5 189.0 220.5 252.0
31.5 47.3 63.0 78.8 94.5 110.3 126.0
63.0
52.5
41.4
30.0
be,cal (mm)
δ (%)
be,cal (mm)
δ (%)
be,cal (mm)
δ (%)
be,cal (mm)
δ (%)
be,cal (mm)
δ (%)
60.16 77.57 93.87 107.16 118.12 129.05 138.60
0.00 3.99 5.67 3.88 1.63 3.59 2.65
53.86 74.73 91.98 107.16 120.01 132.36 144.90
9.04 0.52 15.87 4.68 6.62 10.41 12.75
51.97 72.84 90.72 106.38 120.96 135.66 148.68
9.27 0.11 5.88 0.81 1.54 4.61 0.85
49.77 70.47 89.46 106.38 121.90 137.87 153.72
4.24 14.28 4.05 8.73 12.24 7.37 7.96
47.88 68.58 88.20 106.38 – – –
20.63 2.94 3.70 9.83
–: Represents the CSB failed by the Vierendeel mechanism plastic bending failure.
Table 11 Compressive stresses in the web-post obtained through the modified strut model for CSBs with load ratio of 0.5 (in MPa). S/d0
d0/tw 80.8
1.2 1.3 1.4 1.5 1.6 1.7 1.8
63.0
52.5
41.4
30.0
σcal
σFEM
σcal
σFEM
σcal
σFEM
σcal
σFEM
σcal
σFEM
72.2 57.3 54.6 48.8 44.7 42.4 37.1
82.0 86.0 79.3 70.5 67.5 60.3 56.1
71.2 69.1 74.6 62.3 52.2 45.5 50.9
83.0 71.4 70.4 69.8 65.2 62.7 57.0
90.0 73.3 72.0 64.1 58.4 51.1 49.5
111 82.9 70.7 72.1 70.0 66.1 61.5
89.0 70.6 71.6 62.8 55.1 51.8 53.0
112 96.1 83.8 80.5 79.8 71.5 66.9
131.7 90.0 77.0 70.7 – – –
137 111.9 92.4 91.4 – – –
–: Represents the CSB failed by the Vierendeel mechanism plastic bending failure.
Table 12 Buckling temperatures obtained from the modified strut mode and differences to FEM results for CSBs with load ratio of 0.5 (in 1C). S/d0
d0/tw 80.8
1.2 1.3 1.4 1.5 1.6 1.7 1.8
63.0
52.5
41.4
30.0
Tcr,A (1C)
δ (%)
Tcr,A (1C)
δ (%)
Tcr,A (1C)
δ (%)
Tcr,A (1C)
δ (%)
Tcr,A (1C)
δ (%)
111 297 382 405 402 384 364
61.05 14.66 3.52 9.16 7.49 2.40 5.51
449 493 503 505 506 509 516
0.88 4.23 2.44 0.59 3.44 5.39 1.57
529 544 550 552 553 566 558
0.57 0.74 1.85 0.18 2.12 1.74 3.33
569 582 589 593 597 602 611
3.27 0.69 1.90 0.85 0.00 0.66 0.66
568 588 598 615 – – –
5.11 7.48 8.70 7.64
–: Represents the CSB failed by the Vierendeel mechanism plastic bending failure.
454
P. Wang et al. / Thin-Walled Structures 85 (2014) 441–455
Table 13 Buckling temperatures of CSBs with load ratio of 0.4 obtained from the FEM simulation, the strut model and the modified strut mode (in 1C). d0/tw
80.8
63.0
52.5
41.4
30.0
Buckling temperature
Tcr,FEM (1C) Tcr,str (1C) (Tcr,str Tcr,FEM)/Tcr,FEM Tcr,A (1C) (Tcr,A Tcr,FEM)/Tcr,FEM Tcr,FEM (1C) Tcr,str (1C) (Tcr,str Tcr,FEM)/Tcr,FEM Tcr,A (1C) (Tcr,A Tcr,FEM)/Tcr,FEM Tcr,FEM (1C) Tcr,str (1C) (Tcr,str Tcr,FEM)/Tcr,FEM Tcr,A (1C) (Tcr,A Tcr,FEM)/Tcr,FEM Tcr,FEM (1C) Tcr,str (1C) (Tcr,str Tcr,FEM)/Tcr,FEM Tcr,A (1C) (Tcr,A Tcr,FEM)/Tcr,FEM Tcr,FEM (1C) Tcr,str (1C) (Tcr,str Tcr,FEM)/Tcr,FEM Tcr,A (1C) (Tcr,A Tcr,FEM)/Tcr,FEM
S/d0 1.2
1.3
1.4
1.5
1.6
1.7
1.8
377 20 94.69% 365 3.18% 524 121 76.91% 525 0.19% 553 431 22.06% 569 2.89% 577 525 9.01% 599 3.81% 615 568 7.64% 635 3.25%
473 20 95.77% 467 1.27% 551 334 39.38% 541 1.81% 565 504 10.80% 580 2.65% 587 555 5.45% 614 4.60% 615 606 1.46% 662 7.64%
473 201 57.51% 506 6.98% 564 423 25.00% 546 3.19% 576 523 9.20% 583 1.22% 588 571 2.89% 623 5.95% 623 635 1.93% 678 8.83%
492 329 33.13% 512 4.07% 575 468 18.61% 549 4.52% 587 535 8.86% 584 0.51% 598 583 2.51% 629 5.18% 638 653 2.35% 688 7.84%
525 400 23.81% 511 2.67% 589 502 14.77% 549 6.79% 615 549 10.73% 585 4.88% 607 593 2.31% 633 4.28% – –
525 432 17.71% 508 3.24% 586 516 11.95% 551 5.97% 606 558 7.92% 587 3.14% 624 602 3.53% 639 2.40% – –
509 453 11.00% 503 1.18% 551 531 3.63% 557 1.09% 565 569 0.71% 589 4.25% 586 615 4.95% 646 10.24% – –
–
–
–
–: Represents the CSB failed by the Vierendeel mechanism plastic bending failure. Table 14 Buckling temperatures of CSBs with load ratio of 0.7 obtained from the FEM simulation, the strut model and the modified strut mode (in 1C). d0/tw
80.8
63.0
52.5
41.4
30.0
Buckling temperature
Tcr,FEM (1C) Tcr,str (1C) (Tcr,str Tcr,FEM)/Tcr,FEM Tcr,A (1C) (Tcr,A Tcr,FEM)/Tcr,FEM Tcr,FEM (1C) Tcr,str (1C) (Tcr,str Tcr,FEM)/Tcr,FEM Tcr,A (1C) (Tcr,A Tcr,FEM)/Tcr,FEM Tcr,FEM (1C) Tcr,str (1C) (Tcr,str Tcr,FEM)/Tcr,FEM Tcr,A (1C) (Tcr,A Tcr,FEM)/Tcr,FEM Tcr,FEM (1C) Tcr,str (1C) (Tcr,str Tcr,FEM)/Tcr,FEM Tcr,A (1C) (Tcr,A Tcr,FEM)/Tcr,FEM Tcr,FEM (1C) Tcr,str (1C) (Tcr,str Tcr,FEM)/Tcr,FEM Tcr,A (1C) (Tcr,A Tcr,FEM)/Tcr,FEM
S/d0 1.2
1.3
1.4
1.5
1.6
1.7
1.8
20 20 0.00% 20 0.00% 215 20 90.70% 234 8.84% 419 20 95.23% 457 9.07% 485 225 53.61% 530 9.28% 522 457 12.45% 565 8.24%
20 20 0.00% 20 0.00% 304 20 93.42% 325 6.91% 440 20 95.45% 499 13.41% 485 419 13.61% 548 12.99% 534 533 0.19% 596 11.61%
148 20 86.49% 109 26.35% 332 20 93.98% 347 4.52% 440 221 49.77% 505 14.77% 501 472 5.79% 557 11.18% 546 566 3.66% 617 13.00%
148 20 86.49% 143 3.38% 332 20 93.98% 352 6.02% 460 301 34.57% 505 9.78% 515 505 1.94% 562 9.13% – –
183 20 89.07% 136 25.68% 332 20 93.98% 350 5.42% 460 360 21.74% 506 10.00% 529 520 1.70% 566 6.99% – –
148 20 86.49% 119 19.59% 359 175 51.25% 360 0.28% 478 405 15.27% 509 6.49% 543 534 1.66% 572 5.34% – –
20 20 0.00% 20 0.00% 359 261 27.30% 389 8.36% 478 440 7.95% 512 7.11% 516 550 6.59% 580 12.40% – –
–
–
–
–
–: Represents the CSB failed by the Vierendeel mechanism plastic bending failure.
4.6.2. Critical temperatures of the web-posts predicted by the modified strut model The buckling temperatures of the web-posts predicted by the modified strut model can be obtained by the following steps: (1) Using the modified strut model to determine the compressive stresses in the web-posts under a given load, as listed in Table 11;
(2) Making an assumption on the buckling temperature of webpost; (3) Following the description in BS5950-1 and considering the degrading of steel at high temperature to determine the buckling resistance at the assumed temperature; (4) If the buckling resistance calculated in step (3) equals to the compression stress calculated in step (1), the temperature assumed in step (2) is the buckling temperature of the web-
P. Wang et al. / Thin-Walled Structures 85 (2014) 441–455
post. Otherwise, make a new assumption and re-calculate the buckling resistance of the web-post; The comparison of the buckling temperature obtained from the modified strut model and the FEM simulation were listed in Table 12. δ is the error of the calculated buckling temperature to that obtained through FEM simulation δ¼
T cr;A T cr;FEM 100% T cr;FEM
ð29Þ
where Tcr,A is the temperature obtained through the modified strut model. For most cases, δ is less than 5%, as listed in Table 12, which shows the applicability of the strut model in predicating the buckling temperature of the web-post in a cellular steel beam in a fire. To show the applicability of the modified strut model, buckling temperatures of the web-post in CSBs with load ratio of 0.4 and 0.7 were calculated using the modified strut model. Buckling temperatures obtained by the FEM simulation, the traditional strut model and the modified strut model were all listed in Tables 13 and 14. The buckling temperature calculated by the modified strut model agreed with the FEM results much better than that obtained by the traditional strut model, as listed in Tables 13 and 14. 5. Conclusion Buckling behaviors of the web-post in a CSB in a fire were investigated through using a thermal-mechanical coupled finite element model. The Bitar’s model, the Lawson’s model and the strut model were used to calculate the compression stresses in the web-post under a given load. The Bitar’s model and the Lawson’s model gave an extraordinary high compression stress when the CSB had a narrow web-post and thin web thickness. The compression stress given by the strut model was at a reasonable value but its correctness was still need to be improved. The buckling temperature of the web-post will be underestimated if the traditional strut model was used to predication the buckling temperature of the web-post. A simplified method to calculate the effective width of the strut was proposed. The width of the strut was obtained through measuring the width of the compression principle stress band.
455
The buckling temperature obtained through the modified strut model agreed well with the FEM analysis.
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Practical method for calculating the buckling temperature of the web-post in a cellular steel beam in fire Peijun Wang n, Xudong Wang, Mei Liu Civil Engineering Department, Shandong University, Jinan 250061, Shandong Province, China
art ic l e i nf o
a b s t r a c t
Article history: Received 5 August 2014 Received in revised form 23 September 2014 Accepted 23 September 2014 Available online 17 October 2014
Buckling behaviors of web-posts in a cellular steel beam at elevated temperatures in a fire were studied using the Finite Element Method (FEM) analysis and available analytical models. The buckling temperatures obtained by the analytical models differed greatly to those obtained from the FEM simulation. Among these analytical models, the buckling temperature obtained through the strut model based on BS5950-1 agreed with the FEM result the best. It is more reasonable to take the width of the compression stress band in the web-post as the effective width of the strut. Numerical parametric studies showed that the width of the compression stress band varied with the opening diameter, the opening distance and the web thickness. A simplified method was proposed to calculate the effective width of the strut. The accuracy of the strut model integrating the new effective width was validated against the FEM simulations. The obtained buckling temperature of the web-post using the modified strut model agreed well with the FEM simulation result. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Cellular steel beam Web-post buckling Strut model Effective width Fire Buckling temperature
1. Introduction The cellular steel beam (CSB) can be made through cutting an H-section steel beam in a zigzag pattern along the web and then re-welding the web-post together or through cutting circular holes in the web directly. The obtained CSB has a higher strength to weight ratio and allows service integration to be installed within the beam depth, as shown in Fig. 1. For the discontinuous in the web, local failures modes, such as the web-post buckling and the Vierendeel bending failure at the perforated section, may happen in a CSB [1]. In the fire situation, the design of a CSB is getting more complex for the degrading of steel at high temperatures and the non-uniform thermal strain across the section [2–4]. For a CSB with a high web height to thickness ratio, the web-post buckling failure was the main concern. Web-post buckling behaviors in CSBs had been investigated experimentally and numerically recently. At ambient temperature, the buckling of the web-post is defined as when the out-of-plan displacement in a web-post suddenly increases [5,6]. The web opening shape may also affect the buckling strength of the web-post. Wang et al. [7] had studied the web-post buckling strength of a CSB with fillet corner hexagonal web openings. Analytical models have also been developed to simply estimate the buckling load of the web-post in a CSB [8–13]. For the Young’s modulus of steel degrades faster than the yield strength with the elevating temperature, the CSB is more vulnerable to
the web-post buckling failure in the fire situation. A CSB was failed by Vierendeel mechanism failure at ambient could be failed by web-post buckling in the fire situation. Though modifying the degrading function of the yield strength of steel at high temperatures, Bihina et al. [10] calculated the critical temperature of a composite cellular beam in a fire. Bitar et al. [11] proposed an empirical model to calculate the buckling strength of a composite cellular beam based on the method presented in SCI publication 100 [12]. Lawson et al. [13] proposed a method to determine the maximum compressive stress in the webpost. Instead of checking the critical section in the web-post, the strut model treated the web-post as a compression strut and checked its stability based on the column buckling curves in BS5950-1 [14]. In this paper, three analytical models for assessing the webpost buckling behaviors at ambient temperature were used to find the buckling strength and buckling temperature of the web-post in the fire condition. A simplified method through modifying the current strut model was proposed to calculate the buckling temperature of a web-post in the CSB. The new effective width in the modified strut model took the width of the compression band in the web-post which was obtained through FEM analysis. And a simplified method was presented to calculate the new effective width. The buckling temperature of the web-post predicted by the proposed strut model was validated by FEM analysis.
2. Finite element model n
Corresponding author. Tel./fax: þ 86 531 88392843. E-mail address: [email protected] (P. Wang).
http://dx.doi.org/10.1016/j.tws.2014.09.019 0263-8231/& 2014 Elsevier Ltd. All rights reserved.
The finite element software ABAQUS was used to simulate buckling behaviors of the web-post in a CSB at the ambient
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Cutting line Height of the original beam Cutting line Web-post Height of the cellular steel beam Cutting part Service intergration Welding position
Opening distance
Width of the web-post
Fig. 1. Cellular steel beam.
temperature and in the fire situation. The numerical simulation included three steps: (1) Obtaining the load bearing capacity of a CSB under the concentrated load at middle span at ambient temperature through utilizing the Riks method in ABAQUS; (2) Obtaining the temperature fields in the unprotected CSB or the unprotected composite cellular beam exposed to standard fire using the thermal analysis; (3) Obtaining the buckling temperature of the web-post in CSB at a given load ratio in the fire situation. The load ratio is the ratio of the applied load to the load bearing capacity at ambient temperature obtained in step (1). The load ratio takes 0.5 here for illustration.
2.1. Mechanical model at ambient temperature The studied CSB was made through cutting circular holes in the web of a steel beam with section of UB457 152 52. In order to prevent the flexural torsional buckling failure of the CSB, the lateral displacement of the flange at the middle span was fixed. Web stiffeners were placed at the supports and the mid-span where the concentrated force was applied to prevent the buckling of the web under the local concentrated force. Initial out-of-plane imperfection of the web-post took the first buckling mode obtained from the buckling analysis of ABAQUS. The magnitude of the initial-out-of-plane imperfection was 1 mm. The CSB was meshed by the S4R element in ABAQUS, a 4-node reduced integration shell element, as shown in Fig. 2. The steel was grade S355 steel with Young’s modulus of 200 GPa and the Poisson’s ratio of 0.3. Geometrical and mechanical properties of the studied beams were listed in Table 1 [6]. The FEM model was calibrated against experimental results reported by Tsavdaridis and D’Mello [6]. Mesh convergence studies showed that the S4R shell element with the mesh size of 10 mm could give accurate results, as shown in Fig. 3. Comparison of load-deflection curves and failure modes of the web-post obtained from the proposed FEM model and the test were shown in Fig. 4(a) and (b). The ultimate load obtained from the FEM simulation was greater than that from the test [6], which might be caused by the assumption in the magnitude of initial out-of-plan imperfection of the web-post. In the test carried out by Tsavdaridis and D’mello [6], no measured initial imperfection was reported. Redwood and Demirdjian [5] recorded a measured maximum imperfection of the web in their tests which was about 1 mm (26.3 times of tw/200). Undoubtedly, the web-post with a greater initial geometric imperfection has a smaller ultimate load. However, at present there were no researches that presented a larger initial imperfection than 1 mm. In the following parametric study, the initial imperfection was taken as 1 mm. FEM simulation results reported by Tsavdaridis and D’Mello [6] were also presented in Fig. 4(a). The simulated ultimate load presented by Tsavdaridis and D’mello [6] was greater than that obtained from the FEM model
presented here, for a much smaller imperfection of tw/200 (0.038 mm) was introduced in their model. In the FEM simulation, the CSB was failed by the web-post buckling, which agreed well with the test result, as shown in Fig. 4(b). The outof-plane displacements in the upper and lower part of the web-post were in the opposite direction and the web-post buckled in an S-shape mode along the section height. 2.2. Thermal-mechanical coupled model in the fire situation 2.2.1. Thermal-mechanical coupled model Heat transfer analysis was carried out first to obtain the temperature field in a CSB and a composite cellular beam exposed to the ISO834 standard fire. Points where the temperatures were reported for verification were shown in Figs. 2 and 5. In the thermal analysis model, the CSB was meshed using the 4-node heat transfer quadrilateral shell element DS4. The concrete slab in the composite cellular beam was meshed using the 8-node linear heat transfer brick element DC3D8. Both the radiation and convection heat transfer between the fire and the structural component were included in the model, as shown in Fig. 6. The CSB exposed to fire from four sides and the composite cellular steel beam exposed to fire from three sides. The emissivity of the surfaces and the convective heat transfer coefficient were taken as 0.8 and 25 W/m2 K 1, respectively. The convective heat transfer coefficient was taken as 4 W/m2 K 1 for the unexposed side of the concrete slab in the composite cellular beam. Thermal properties of steel and concrete varied with temperatures and took those described in ENV1993-1-2 [15] and ENV1994-1-2 [16]. In the mechanical model, the DS4 element to mesh the CSB was replaced by S4R structural shell element and the DC3D8 element to mesh the concrete slab was replaced by C3D8R structural brick element. The temperature field in the CSB and the concrete slab were defined by reading the result file generated during the thermal analysis. Mechanical properties of steel and concrete at elevated temperatures followed the descriptions in EN 1993-1-2 [15] and ENV1994-1-2 [16]. The ABAQUS/Explicit was used in the simulation to avoid convergence problems which were often encountered when doing the geometric and material non-linear numerical analysis. The buckling temperature was defined at which the lateral displacement of the web-post suddenly increased. 2.2.2. Model verification 2.2.2.1. Verification of the thermal model. EN1993-1-2 [15] provided equations to calculate the temperature elevation of an unprotected steel structural component exposed to fire. Δθa;t ¼ ksh
Am =V _ hnet Δt cα ρα
ð1Þ
where ksh was the correction factor for the shadow effect; Am =V was the section factor for unprotected steel members [1/m]; Am was the surface area of the member per unit length [m2/m]; V was the volume of the member per unit length [m3/m]; ca was the
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Fig. 2. Finite element model of a CSB.
Table 1 Geometrical and mechanical properties of the beams.
Upper member dimension Section Depth Lower member dimension Section Depth Openings characteristic Number Number of circular openings Number of elongated openings Cell diameter Cell pitch Concrete slab dimension Thickness Width Upper member strength Flange yield strength Web yield strength Lower member strength Flange yield strength Web yield strength Concrete strength Compressive strength
Cellular steel beam
Composite cellular steel beam
Beam A1 [6]
Beam 1 [10]
Beam 3 [10]
UB457 152 52 224.9 mm
IPE360
IPE360
255 mm
255 mm
UB457 152 52 224.9 mm
IPE450
IPE450
300 mm
300 mm
4 4
13 12
13 12
0
1
1
315 mm 409.5 mm
375 mm 600 mm
357 mm 600 mm
– –
120 mm 2200 mm
120 mm 2200 mm
360 MPa 375 MPa
460 MPa 480 MPa
451 MPa 478 MPa
360 MPa 375 MPa
438 MPa 475 MPa
438 MPa 475 MPa
–
31.0 MPa
33.5 MPa
Fig. 3. Mesh convergence study.
specific heat of steel [J/kg K]; h_ net was the design value of the net heat flux per unit area [W/m2]; Δt was the time interval [s] and ρa was the unit mass of steel [kg/m3]. Temperature elevation of the CSB obtained by Eq. (1) and the FEM simulation were shown in Fig. 6. The flange temperatures obtained from Eq. (1) were a little lower than those from the FEM simulation. However, the web temperatures obtained by Eq. (1) and the FEM simulation agreed very well, which showed the validation of the thermal model. The temperatures at the solid web and at the perforated web were nearly the same, as shown in Figs. 7 and 8, which showed that the openings had very little influences on the temperature distribution of the web in a CSB. Usually the temperature in the web was a little higher than those in the flanges. Non-uniform temperature distribution across the section will cause additional compression thermal stress to the hotter part. For the studied CSB, the maximum temperature difference is about 40 1C, as shown in Fig. 6. Hence, the compression thermal stress in the web is approximately σ th ¼ α ΔT ET ¼ 1:4 10 5 40 1:2 10 5 ¼ 67:2 MPa
ð2Þ
where α is the thermal expansion coefficient; ΔT is the temperature difference; ET is the Young’s modulus of steel at temperature T, and T takes the value of 600 1C here. Undoubtedly, the additional thermal stress will greatly influence behaviors of the beam in fire. In the thermal-mechanical coupled model, the non-uniform temperature distribution in the structural member was included by reading the thermal analysis result file. The thermal model was also verified by test results on the temperature elevation of unprotected cellular steel beams carried out by Bailey [17]. The dimension of the beam and locations of thermocouples are shown in Fig. 9(a) and (b), respectively. Fig. 9(c) showed the comparison of the temperature at the centre of the web-post (position C7) of the cellular beam and that at the corresponding position on the solid beam (position S5) measured in the test and obtained through FEM simulation. Fig. 9(d) showed the comparison of temperatures at the bottom flange on the solid beam (position S7 and S8) and at the bottom flange of the cellular beam below the webpost (position C16 and C17) obtained from fire test and FEM simulation. It can be seen that temperatures obtained through the proposed FEM model agreed very well with test results. We can have confidence in the simulation results of the thermal model. 2.2.2.2. Verification of the thermal-mechanical coupled model. Test results on composite cellular steel beams, Beam 1 and Beam 3, reported by Bihina et al. [10] were used to verify the thermal-mechanical coupled
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Fig. 4. Comparison of the load-deflection curves and the failure modes. (a) Comparison of load-displacement curves. (b) Comparison of the buckling modes.
Fig. 5. Thermal-mechanical coupled model of the composite cellular beam.
Radiation Convection
Air 4-side fire exposed
Fig.6. Thermal boundary conditions of the CSB and the composite cellular beam.
model, as shown in Fig. 8. Geometry and material properties of the test specimen were listed in Table 1. Comparison of temperatures, loaddeflection curves and failure modes between FEM and test results were shown in Figs. 10–12, respectively.
Temperatures in the upper flange of Beam 1 obtained from the FEM simulation were lower than those measured from the test [10]. However, the temperature difference was only about 50 1C. Temperatures in the lower flange and in the web obtained from
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the FEM simulations agreed very well with test results, as shown in Fig. 10, which showed the validation of the thermal analysis model. In the composite cellular beam, temperatures in the upper flange were much lower than those in the lower flange and the web. The great thermal compression stress must be included in the fire safety design for a composite steel beam. In the following structural analysis, the temperature field in the composite cellular beam was defined by reading the result file generated during the thermal analysis. The non-uniform temperature distribution in the composite beam can be precisely considered. Load-deflection curves of Beam 1 and Beam 3 obtained from the FEM simulation and the test agreed very well, as shown in Fig. 11, which showed the validity of the thermal-mechanical coupled model. The deformation of the web-post of Beam 1 and Beam 3 predicated by the FEM simulation agreed well with the test results, as shown in Fig. 12(a) and (b). They agreed well with each other. The web-post buckled in an S-shape in the fire, which is the same to the buckling mode of the web-post at the ambient temperature.
3. Parameter study Effects of the web-post geometry on the buckling behaviors and buckling temperatures of the web-post in a CSB in fire were studied using the verified thermal-mechanical coupled model. The studied CSB had the section of UB457 152 52. The applied concentrated load at the middle span was 50% of the ultimate load that was obtained from FEM simulation at ambient temperature. The opening diameter was 315 mm and the corresponding opening to depth ratio, d0/h, was 0.7, as shown in Fig. 13. d0 was the diameter of the circular opening; h was the section height of the cellular steel beam. The changing parameters were the web thickness, tw, and the opening distance, S. Values of the web thickness and the opening distance studied were:
Fig. 7. Temperature elevation of the CSB obtained by Eq. (1) and the FEM simulations.
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the web thickness: 3.9 mm, 5.0 mm, 6.0 mm, 7.6 mm, and 10.5 mm. The corresponding opening diameter to thickness ratio, d0/tw, were: 80.8, 63, 52.5, 41.4, and 30. the opening distance: 378.0 mm, 409.5 mm, 441.0 mm, 472.5 mm, 504.0 mm, 535.5 mm, and 567.0 mm. The corresponding opening distance to diameter ratio, S/d0, were: 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, and 1.8.
3.1. Buckling of the web-post at ambient temperature The CSB might be failed by the buckling in the web-post or Vierendeel mechanism plastic failure at the perforated section at the ambient temperature. Which failure mode will occur depends on the d0/tw and S/d0 of the web-post. The shear strength of the web-post with different d0/tw and S/d0, obtained through FEM simulations, were shown in Fig. 14. The points covered by a solid circle in Fig. 14 represented that the web-post was failed by Vierendeel mechanism plastic failure at the perforated section. That was, for the CSB with a stocker web-post, the plastic failure at the perforated section was the reason that caused the failure of the CSB [18]. When the CSB was failed by the web-post buckling, the stress in the web-post might not reach the yield stress of steel both at the web-post and the perforated section, as shown in Fig. 15. There was only one plastic hinge at the perforated section, which means that the Vierendeel mechanism had not been fully developed when the web-post buckled. On the other hand, when the CSB was failed by the Vierendeel mechanism plastic failure, steel in the flanges and the web at the perforated section were all yield. Four plastic hinges were fully developed at the perforated section. The stress in the web-post might be lower than the yield stress of steel, as shown in Fig. 16. 3.2. Buckling temperature of the web-post in a fire The buckling temperature of the web-post was defined as at which its lateral displacement of the web-post suddenly increased. For the web-posts with different d0/tw, the development of the lateral displacement with the temperature elevation was shown in Fig. 17. The definition of the buckling temperature was also shown in Fig. 17. Buckling temperatures of the web-post obtained through FEM simulation, Tcr,FEM, with different were d0/tw and S/d0 were listed in Table 2. With the increase in S/d0 or d0/tw, the web-post was getting stockier and the buckling temperature increased. For example, for the CSB with d0/tw of 80.8, the buckling temperature increased from 285 1C to 345 1C when the S/d0 increased from 1.2 to 1.8. For the web-post with d0/tw of 30, the CSB was failed by Vierendeel mechanism failure when S/d0 increased to the range of 1.6 to 1.8 (Fig. 14). The changing of failure modes with the increase in S/d0 was shown in Fig. 18. The failure modes of the same CSB may be different in the fire and at ambient temperature. In the fire situation, only the CSB with d0/tw of 30 and S/d0 in the range of 1.6 to 1.8 were failed by Vierendeel mechanism failure, as shown in Fig. 18 and listed in Table 2. However, at ambient temperature,
Fig. 8. Temperature distributions in the CSB at different heating time.
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Fig. 10. Comparison of the temperatures of Beam 1 obtained by tests and FEM simulations.
Fig. 11. Comparison of the vertical defection obtained by tests and FEM simulations.
1.5 to 1.8 were failed by Vierendeel mechanism failure when d0/tw was 41.4, as shown in Fig.16. The failure modes of the CSB under fire situation should be re-assessed. It will lead to unsafe design if we simply adopted the design method for ambient temperature to the fire safety design of a web-post.
4. Practical method for calculating the buckling temperature of a web-post in fire The buckling temperature of the web-post in a CSB can be determined through σ w;f i r σ w;f i;Rd
ð3Þ
where σw,fi is the maximum compression stress in the web-posts; and σw,fi,Rd is buckling resistance of the web-posts in the fire situation. Fig. 9. Verification of the thermal model by Bailey’s test results. (a) Dimension of the solid web beam and positions where temperatures were measured. (b) Dimension of the cellular steel beam and positions where temperatures were measured. (c) Comparison of temperatures at the web. (d) Comparison of temperatures at the flange.
4.1. Models for calculating the compression stress in the web-post
the CSB with d0/tw of 30 and S/d0 in the range of 1.3 to 1.8 were all failed by Vierendeel mechanism failure. In the fire situation, no CSBs were failed by Vierendeel mechanism failure when d0/tw was 41.4. At ambient temperature, the CSBs with S/d0 in the range of
Three analytical models for calculating the compression stress in the web-post,σ w;f i , were studied, which were the Bitar’s model [11], the Lawson’s model [13] and the strut model [6].
P. Wang et al. / Thin-Walled Structures 85 (2014) 441–455
4.1.1. The Bitar’s model Bitar et al. [11] assumed the horizontal section across the web-post as the critical section, as shown in Fig. 19. The position of the critical section, where the compression stress reached its maximum value, was determined by vq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 2 2 u S=d0 þ 8 S=d0 2 S=d0 d0 t dw ¼ ð4Þ 2 2
447
where dw was the distance from the middle of the web-post to the critical section. The width of the critical section, lw, was obtained by
0 lw ¼ d0 @
S d0
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 2dw A 1 d0
ð5Þ
Fig. 12. Comparison of the failure mode obtained by tests and FEM simulations. (a) Deformation shape of Beam 1 after test. (b) Deformation shape of Beam 3 after test.
Fig. 13. Dimension of the studied CSB.
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Table 2 Buckling temperatures of the web-posts obtained through FEM simulation for CSBs with load ratio of 0.5 (in 1C). S/d0
1.2 1.3 1.4 1.5 1.6 1.7 1.8
d0/tw 80.8
63
52.5
41.4
30
285 348 369 371 374 375 345
453 473 491 508 524 538 508
526 540 540 553 565 576 540
551 578 578 588 597 606 607
568 588 598 615 – – –
–: Represents the CSB failed by Vierendeel mechanism plastic bending failure.
Fig. 14. Ultimate strength of the CSBs by FEM simulations.
Fig. 17. Lateral displacement of the web-posts.
The principal compressive stress, σw,fi,Ed, in the critical section in the fire situation was calculated by σ w;f i;Ed ¼ Fig. 15. Stress distribution and deformation of the CSB with S/d0 ¼1.3, d0/tw ¼41.4. (a) Stress distribution in the web-post. (b) Deformation of the web-post.
6M w;f i;Ed 2 2 lw t w 1 4 dw =d0
ð6Þ
where Mw,fi,Ed, was the bending moment in the critical section in a fire situation and was obtained through M w;f i;Ed ¼ V h;f i;Ed dw þ M h;f i;Ed
ð7Þ
Mh,fi,Ed and Vh,fi,Ed were the bending moment and horizontal shear force at the middle of the web-post, respectively. 4.1.2. The Lawson’s model Lawson et al. [13] assumed the section that passed through center of the circular opening as the critical section, as shown in Fig. 20. The stresses varied along the section and with the changing of the incline angle. The critical angle when the maximum compressive stress occurred in the critical section depended on the width of the web-post. Horizontal equilibrium due to tension or compression stresses, σ, and shear stress, τ, at an angle θ in the web-post was obtained as S V h ¼ ðσ sin θ þ τ cos θÞ tw ð8Þ cos θ d0
Fig. 16. Stress distribution and deformation of the CSB with S/d0 ¼1.7, d0/tw ¼41.4. (a) Stress distribution in the web-post. (b) Deformation in the web-post.
where Vh was the horizontal shear force at the middle of the webpost. Moment equilibrium was obtained by taking moments around the point A, which was the intersection of the two planes at an angle θ through the centers of the adjacent openings. For an approximately uniform stress in the critical section, equilibrium
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449
Fig. 18. Failure modes of the CSBs in a fire.
was obtained from: V h y ¼ σt w x2
ð9Þ
where y ¼ 0:5 tan θ and x ¼ 0:5 S= cos θ d0 . The angle at which the maximum compression stress occurred in the section was established by d0 cos θ ¼ þ 4S Fig. 19. Web-post buckling model developed by Bitar et al. [11].
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 d0 þ 0:5 4S
ð10Þ
The maximum compression stress, relative to the horizontal shear stress at mid-height of the web-post, τh, was given by: σ max ¼ τh
ðS d0 Þ cos θ ðS d0 cos θÞ sin θ
ð11Þ
The shear stress could be obtained by the relationship between Vh and Vv [19], was given by: Vh S ¼ V v dG τh ¼
Fig. 20. Web-post buckling model developed by Lawson et al. [13].
Vh S0 t w
ð12Þ
ð13Þ
S0 was the width of the web-post. Vh and Vv were the horizontal and vertical shear force on the web-post, respectively.
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4.1.3. The strut model The strut model considered compressive and tensile forces acting across the web-post on opposite diagonals. A diagonal strut under compression was designed to calculate the compressive stresses of the web-post, as shown in Fig. 21. For simplification, the vertical shear force in the upper T section was used to calculate the compressive stress acting on the strut. The effective width of the strut, be, was taken as a half of the total width of the web-post: be ¼
S0 2
ð14Þ
The compressive stress in the strut was: σ¼
V v =2 Vv ¼ be t w S0 t w
The length of the strut was given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 le ¼ 0:5 S20 þd0
ð15Þ
ð16Þ
The yield stress of the steel in the web was 375 MPa at ambient temperature and the applied load ratio was 0.5. However, for the CSB with S/d0 of 1.2 and d0/tw of 80.8, the predicated compression stresses in the web-post were 418 MPa and 276 MPa by the Bitar’s model and the Lawson’s model, respectively. Theoretically, they should be around 0.5 375 MPa that was 187.5 MPa, where the 0.5 was the load ratio and the 375 was the yield strength of steel. The Bitar’s model and the Lawson’s model gave a too great compression stress in the web-post. The compression stress predicated by the strut model was 138 MPa, which was a reasonable value although it was a little lower. The strut model will be used to calculate the compression stress in the web-post at ambient temperature and in a fire. Table 3 Compressive stresses in the strut calculated by the Bitar’s Model for CSBs with load ratio of 0.5 (in MPa). S/d0
4.2. Compression of the stresses obtained by different analytical models Compressive stresses in the web-post calculated by the Bitar’s model [11], the Lawson’s model [13] and the strut model [6] were listed in Tables 3 and 5, respectively. The applied load ratio was 0.5. The opening distance will greatly affect the compression stresses in the web-post. With the increase in S/d0, the compression stress in the web-post decreases. However, the compression stresses predicated by different models differed greatly. For example, for the web-posts had d0/tw of 80.8, when S/d0 increased from 1.2 to 1.8, the compression stresses calculated by the Bitar’s model decreased from 418 MPa to 133 MPa, as listed in Table 3. The stresses calculated by the Lawson’s model and strut model were much smaller. They decreased from 276 MPa to 79 MPa calculated by the Lawson’s model and from 138 MPa to 42 MPa calculated by the strut model, as listed in Tables 4 and 5. The compression stress in the web-post decreased with the increase in the web thickness if the Bitar’s model was employed, as listed in Table 3. For the web-post with S/d0 of 1.2, when d0/tw decreased from 80.8 to 41.4, the compression stress decreased from 418 MPa to 272 MPa. However, if the Lawson’s model or the strut model were employed, the compression stress in the webpost seemed not to be affected by the changing of web thickness, as listed in Tables 4 and 5. For the same web-post with S/d0 of 1.2, when d0/tw decreased from 80.8 to 41.4, the compression stress changed from 276 MPa to 298 MPa if they were calculated by the Lawson’s model and from 138 MPa to 147 MPa by the strut model.
d0/tw 80.8
1.2 1.3 1.4 1.5 1.6 1.7 1.8
63.0
52.5
41.4
30.0
σcal
σFEM
σcal
σFEM
σcal
σFEM
σcal
σFEM
σcal
σFEM
418 277 215 181 159 144 133
82.0 86.0 79.3 70.5 67.5 60.3 56.1
409 313 264 231 206 184 163
83.0 71.4 70.4 69.8 65.2 62.7 57.0
343 262 221 193 172 154 136
111 82.9 70.7 72.1 70.0 66.1 61.5
272 208 175 154 137 122 108
112 96.1 83.8 80.5 79.8 71.5 66.9
199 152 128 112 100 89 79
137 111.9 92.4 91.4 – – –
–: Represents the CSB failed by the Vierendeel mechanism plastic bending failure.
Table 4 Compressive stresses in the strut calculated by the Lawson’s model for CSBs with load ratio of 0.5 (in MPa). S/d0
d0/tw ¼ 80.8
1.2 1.3 1.4 1.5 1.6 1.7 1.8
63.0
52.5
41.4
30.0
σcal
σFEM
σcal
σFEM
σcal
σFEM
σcal
σFEM
σcal
σFEM
276 181 138 114 99 88 79
82.0 86.0 79.3 70.5 67.5 60.3 56.1
270 204 170 146 128 112 97
83.0 71.4 70.4 69.8 65.2 62.7 57.0
275 211 178 156 138 123 109
111 82.9 70.7 72.1 70.0 66.1 61.5
298 230 193 168 148 131 114
112 96.1 83.8 80.5 79.8 71.5 66.9
340 240 191 160 137 117 100
137 111.9 92.4 91.4 – – –
–: Represents the CSB failed by the Vierendeel mechanism plastic bending failure.
Fig. 21. Strut model for predicating the compression stress in the web-post. (a) Strut in wide web-post (b) strut in middle width web-post (c) strut in narrow web-post.
P. Wang et al. / Thin-Walled Structures 85 (2014) 441–455
4.3. Buckling resistance of the web-post in a fire
4.4. Buckling resistance of the web-post calculated based on BS5950-1
According to the strut model, the width and the length of the compression strut were calculated by Eqs. (14) and (16). The buckling resistance of the strut in a fire, σw,fi,Rd, was calculated based on BS5950-1 [14] considering the degrading of steel at high temperatures. A Perry formula was used for calculating the buckling resistance of the strut: σ¼
f E;θ f y;θ
φθ þ φ2θ f E;θ f y;θ
ð17Þ
0:5
and φθ ¼
f y;θ þ ðη þ 1Þf E;θ 2
ð18Þ
where fE,θ was the Euler bucking stress of the strut. f E;θ ¼
451
π 2 Eθ λ2
ð19Þ
λ was the slenderness of the web-post pffiffiffiffiffiffi le 12 λ¼ tw
The buckling resistance of the strut based on BS5950-1 [14] considering the strength and Young’s modulus reduction of steel in a fire was listed in Table 6. Temperatures at which the buckling resistance in Table 6 were calculated were listed in Table 2. With the increase of the temperature, the buckling resistance of the web-post reduces. The buckling temperature is the temperature at which the buckling resistance equals to the compression stress in the web-post, as defined by Eq. (3). Through comparing the compression stresses predicated by the strut model that were listed in Table 5 and the buckling resistance that were listed in Table 6, we can find that, if the buckling temperatures took those listed in Table 2, the buckling resistance predicated by BS5950-1 were a little smaller than the compression stress. The temperatures given in Table 2 were the buckling temperature obtained through the FEM simulation. That was, BS5950-1 might underestimate the buckling temperature of the web-post in a fire.
ð20Þ
4.5. Critical temperatures of the web-post obtained from analytical models
le was the length of the strut and calculated by Eq. (16). The Perry factor η was given by:
The critical temperature of the web-post can be obtained by the following steps:
η ¼ 0:001αðλ λ0 Þ
ð21Þ
where π 2 Eθ λ0 ¼ 0:2 f y;θ
!0:5 ð22Þ
α was the Robertson factor that was determined considering the section type and taken as 8.0 for a Class 4 cross-section. Table 5 Compressive stresses in the strut calculated by the strut model for CSBs with load ratio of 0.5 (in MPa). S/d0
d0/tw 80.8
1.2 1.3 1.4 1.5 1.6 1.7 1.8
63.0
52.5
41.4
30.0
σcal
σFEM
σcal
σFEM
σcal
σFEM
σcal
σFEM
σcal
σFEM
138 98 77 64 55 48 42
82.0 86.0 79.3 70.5 67.5 60.3 56.1
134 110 94 81 71 61 52
83.0 71.4 70.4 69.8 65.2 62.7 57.0
136 113 98 86 76 66 58
111 82.9 70.7 72.1 70.0 66.1 61.5
147 123 106 93 81 70 60
112 96.1 83.8 80.5 79.8 71.5 66.9
166 127 104 87 74 62 52
137.0 111.9 92.4 91.4 – – –
–: Represents the CSB failed by the Vierendeel mechanism plastic bending failure.
Table 6 Buckling resistance predicated by BS 5950-1 at the temperature given in Table 2 (in MPa). S/d0
1.2 1.3 1.4 1.5 1.6 1.7 1.8
(1) Using the strut model to determine the compressive stresses in the web-post under a given load, as listed in Table 5 when the load ratio is 0.5; (2) Making an assumption on the buckling temperature of webpost; (3) Following the description in BS5950-1 and considering the degrading of steel at high temperatures to determine the buckling resistance at the assumed temperature; (4) If the buckling resistance calculated in step (3) equals to the compression stress calculated in step (1), the temperature assumed in step (2) is the buckling temperature of the webpost. Otherwise, make a new assumption on the buckling temperature and re-calculate the buckling resistance of the web-post;
d0/tw 80.8
63
52.5
41.4
30
63 56 53 48 44 42 39
78 73 67 58 50 43 46
82 75 71 64 55 47 54
104 83 80 70 61 55 51
131 105 102 90 – – –
–: Represents the CSB failed by the Vierendeel mechanism plastic bending failure.
When the load ratio was 0.5, the buckling temperatures calculated following the above procedures were listed in Table 7. δ in Table 7 was the error between the buckling temperatures calculated through the analytical model and that from FEM simulation. δ¼
T cr;str T cr;FEM 100% T cr;FEM
ð23Þ
where Tcr,str was the buckling temperature calculated using the strut model; Tcr,FEM was the temperature obtained through FEM simulation. The strut model will underestimate the buckling temperature of the web-post, as listed in Table 7. For the thin web-post, the temperature difference was more than 90%. The strut model has to be modified to improve its accuracy in predicating the buckling temperature of a web-post in a fire. 4.6. Modified strut model for calculating the buckling temperature of a web-post in fire Three parameters might affect the bucking resistance of the strut at ambient temperature and in a fire situation: (1) the length of strut, l; (2) the buckling length factor, μ; (3) the effective width, be. The buckling length of the strut was calculated by Eq. (16). The length of strut took the diagonal length of the web-post, as shown in Fig. 22. The buckling factor, μ, took the value of 0.5 for there were
452
P. Wang et al. / Thin-Walled Structures 85 (2014) 441–455
Table 7 Buckling temperature of the web-post calculated using the strut model for CSBs with load ratio of 0.5 (in 1C). S/d0
d0/tw 80.8
63.0 δ (%)
Tcr,str (1C) a
1.2 1.3 0.4 0.5 1.6 1.7 1.8
Tcr,str (1C)
92.98 94.25 94.58 94.61 52.67 36.80 20.29
20 20a 20a 20a 177 237 275
52.5
a
20 20a 194 293 365 420 467
41.4
30.0
δ (%)
Tcr,str (1C)
δ (%)
Tcr,str (1C)
δ (%)
Tcr,str (1C)
δ (%)
95.58 95.77 60.49 42.32 30.34 21.93 8.07
177 379 442 479 504 516 528
66.35 29.81 18.15 13.38 10.80 10.42 2.22
452 511 532 548 561 573 584
17.97 11.59 7.96 6.8 6.03 5.45 3.79
526 573 597 620 – – –
7.39 2.55 0.17 0.81
–: Represents the CSB failed by the Vierendeel mechanism plastic bending failure. a
Represents the web-post buckles at ambient temperature.
Fig. 22. Direction and width of the compression principal stresses band in the web-post at buckling at ambient temperature.
Table 8 Measured effective width of the strut form FEM analysis and comparison with be for CSBs with load ratio of 0.5 (in mm). S/d0
S0 (mm)
be (mm)
d0/tw 80.8
1.2 1.3 1.4 1.5 1.6 1.7 1.8
63 94.5 126 157.5 189 220.5 252
31.5 47.3 63.0 78.8 94.5 110.3 126.0
63.0
52.5
41.4
30.0
be,FEM (mm)
be,m/be
be,FEM (mm)
be,m/be
be,FEM (mm)
be,m/be
be,FEM (mm)
be,m/be
be,FEM (mm)
be,m/be
60.16 80.79 88.83 103.16 116.23 124.58 142.38
1.91 1.71 1.41 1.31 1.23 1.13 1.13
59.22 75.12 79.38 102.37 128.52 147.73 128.52
1.88 1.59 1.26 1.30 1.36 1.34 1.02
47.56 72.76 85.68 105.52 122.85 142.22 147.42
1.51 1.54 1.36 1.34 1.30 1.29 1.17
51.97 82.21 93.24 116.55 138.92 148.84 142.38
1.65 1.74 1.48 1.48 1.47 1.35 1.13
39.69 66.62 85.05 96.86 – – –
1.26 1.41 1.35 1.23
–: Represents the CSB failed by the Vierendeel mechanism plastic bending failure.
two half wave lengths along the strut. For the restraints to the strut provided by the flange and the effect of the tension membrane action provide by the tension stress were ignored, the buckling stress of the web-post was underestimated by the strut model, especially for a thin web-post [21]. Traditionally, the effective width of the compression strut took a half of the width of the web-post [6,20], which was not coincidence with the FEM simulation results. A new effective width of strut is proposed in this paper based on the width of the compression stress band in the web-post.
4.6.1. Effective width of the compression strut The width and direction of the compression strut in the webpost can be defined by those of the compression principal stress band. For example, for the CSB with S/d0 ¼1.5 and d0/tw ¼41.4, the distribution of the compressive principal stresses in the web-post at buckling was shown in Fig. 22. The band width of the
Table 9 Values of the coefficients. a0 ¼0.983227 b0 ¼ 0.623962 c0 ¼ 0.072041
a1 ¼ 00562 b1 ¼ 0.487153 c1 ¼ 0.07283
c2 ¼ 0.016533
compression principal stress was affected by the opening diameter, the opening distance and the web thickness, as listed in Table 8. be,FEM was the measured effective width of the compression strut from the FEM simulation. The effective strut width nearly equaled to the web-post width for the CSB with a thin web and narrow web-post, as listed in Table 10. For the web-post with S/d0 of 1.2 and d0/tw of 80.0, the actual web-post width was 63 mm. The effective width calculated by Eq. (14) was only 31.5 mm; however, the measured effective width in the FEM
P. Wang et al. / Thin-Walled Structures 85 (2014) 441–455
simulation was 60.16 mm. With the increase in the web thickness or the opening distance, the measured effective width was close to the effective width calculated by Eq. (14) which was a half of the web-post width. The effective width of the strut can be expressed by a function of S0, d0 and tw as: be;m ¼ κ
S0 2
κ ¼ a0 þ a1
ð24Þ d0 tw
a0 ¼ b0 þ b1
d0 tw
a1 ¼ c0 þc1
453
2 S S þ c2 d0 d0
ð27Þ
The coefficients in Eqs. (25)–(27) can be obtained through curve fitting of the FEM analysis results, as listed in Table 9. The comparison of the effective width calculated by Eq. (24) and the FEM simulation were listed in Table 10. δ in Table 10 was the error of the calculated effective width to that obtained through FEM simulation, which was calculated by be;m be;FEM 100% be;FEM
ð25Þ
δ¼
ð26Þ
For most cases, δ was less than 5%, which showed the correctness of the proposed equations.
ð28Þ
Table 10 Effective width calculate by Eq. (14) and comparison with FEM results for CSBs with load ratio of 0.5 (in mm). S/d0
S0 (mm)
be (mm)
d0/tw 80.8
1.2 1.3 1.4 1.5 1.6 1.7 1.8
63.0 94.5 126.0 157.5 189.0 220.5 252.0
31.5 47.3 63.0 78.8 94.5 110.3 126.0
63.0
52.5
41.4
30.0
be,cal (mm)
δ (%)
be,cal (mm)
δ (%)
be,cal (mm)
δ (%)
be,cal (mm)
δ (%)
be,cal (mm)
δ (%)
60.16 77.57 93.87 107.16 118.12 129.05 138.60
0.00 3.99 5.67 3.88 1.63 3.59 2.65
53.86 74.73 91.98 107.16 120.01 132.36 144.90
9.04 0.52 15.87 4.68 6.62 10.41 12.75
51.97 72.84 90.72 106.38 120.96 135.66 148.68
9.27 0.11 5.88 0.81 1.54 4.61 0.85
49.77 70.47 89.46 106.38 121.90 137.87 153.72
4.24 14.28 4.05 8.73 12.24 7.37 7.96
47.88 68.58 88.20 106.38 – – –
20.63 2.94 3.70 9.83
–: Represents the CSB failed by the Vierendeel mechanism plastic bending failure.
Table 11 Compressive stresses in the web-post obtained through the modified strut model for CSBs with load ratio of 0.5 (in MPa). S/d0
d0/tw 80.8
1.2 1.3 1.4 1.5 1.6 1.7 1.8
63.0
52.5
41.4
30.0
σcal
σFEM
σcal
σFEM
σcal
σFEM
σcal
σFEM
σcal
σFEM
72.2 57.3 54.6 48.8 44.7 42.4 37.1
82.0 86.0 79.3 70.5 67.5 60.3 56.1
71.2 69.1 74.6 62.3 52.2 45.5 50.9
83.0 71.4 70.4 69.8 65.2 62.7 57.0
90.0 73.3 72.0 64.1 58.4 51.1 49.5
111 82.9 70.7 72.1 70.0 66.1 61.5
89.0 70.6 71.6 62.8 55.1 51.8 53.0
112 96.1 83.8 80.5 79.8 71.5 66.9
131.7 90.0 77.0 70.7 – – –
137 111.9 92.4 91.4 – – –
–: Represents the CSB failed by the Vierendeel mechanism plastic bending failure.
Table 12 Buckling temperatures obtained from the modified strut mode and differences to FEM results for CSBs with load ratio of 0.5 (in 1C). S/d0
d0/tw 80.8
1.2 1.3 1.4 1.5 1.6 1.7 1.8
63.0
52.5
41.4
30.0
Tcr,A (1C)
δ (%)
Tcr,A (1C)
δ (%)
Tcr,A (1C)
δ (%)
Tcr,A (1C)
δ (%)
Tcr,A (1C)
δ (%)
111 297 382 405 402 384 364
61.05 14.66 3.52 9.16 7.49 2.40 5.51
449 493 503 505 506 509 516
0.88 4.23 2.44 0.59 3.44 5.39 1.57
529 544 550 552 553 566 558
0.57 0.74 1.85 0.18 2.12 1.74 3.33
569 582 589 593 597 602 611
3.27 0.69 1.90 0.85 0.00 0.66 0.66
568 588 598 615 – – –
5.11 7.48 8.70 7.64
–: Represents the CSB failed by the Vierendeel mechanism plastic bending failure.
454
P. Wang et al. / Thin-Walled Structures 85 (2014) 441–455
Table 13 Buckling temperatures of CSBs with load ratio of 0.4 obtained from the FEM simulation, the strut model and the modified strut mode (in 1C). d0/tw
80.8
63.0
52.5
41.4
30.0
Buckling temperature
Tcr,FEM (1C) Tcr,str (1C) (Tcr,str Tcr,FEM)/Tcr,FEM Tcr,A (1C) (Tcr,A Tcr,FEM)/Tcr,FEM Tcr,FEM (1C) Tcr,str (1C) (Tcr,str Tcr,FEM)/Tcr,FEM Tcr,A (1C) (Tcr,A Tcr,FEM)/Tcr,FEM Tcr,FEM (1C) Tcr,str (1C) (Tcr,str Tcr,FEM)/Tcr,FEM Tcr,A (1C) (Tcr,A Tcr,FEM)/Tcr,FEM Tcr,FEM (1C) Tcr,str (1C) (Tcr,str Tcr,FEM)/Tcr,FEM Tcr,A (1C) (Tcr,A Tcr,FEM)/Tcr,FEM Tcr,FEM (1C) Tcr,str (1C) (Tcr,str Tcr,FEM)/Tcr,FEM Tcr,A (1C) (Tcr,A Tcr,FEM)/Tcr,FEM
S/d0 1.2
1.3
1.4
1.5
1.6
1.7
1.8
377 20 94.69% 365 3.18% 524 121 76.91% 525 0.19% 553 431 22.06% 569 2.89% 577 525 9.01% 599 3.81% 615 568 7.64% 635 3.25%
473 20 95.77% 467 1.27% 551 334 39.38% 541 1.81% 565 504 10.80% 580 2.65% 587 555 5.45% 614 4.60% 615 606 1.46% 662 7.64%
473 201 57.51% 506 6.98% 564 423 25.00% 546 3.19% 576 523 9.20% 583 1.22% 588 571 2.89% 623 5.95% 623 635 1.93% 678 8.83%
492 329 33.13% 512 4.07% 575 468 18.61% 549 4.52% 587 535 8.86% 584 0.51% 598 583 2.51% 629 5.18% 638 653 2.35% 688 7.84%
525 400 23.81% 511 2.67% 589 502 14.77% 549 6.79% 615 549 10.73% 585 4.88% 607 593 2.31% 633 4.28% – –
525 432 17.71% 508 3.24% 586 516 11.95% 551 5.97% 606 558 7.92% 587 3.14% 624 602 3.53% 639 2.40% – –
509 453 11.00% 503 1.18% 551 531 3.63% 557 1.09% 565 569 0.71% 589 4.25% 586 615 4.95% 646 10.24% – –
–
–
–
–: Represents the CSB failed by the Vierendeel mechanism plastic bending failure. Table 14 Buckling temperatures of CSBs with load ratio of 0.7 obtained from the FEM simulation, the strut model and the modified strut mode (in 1C). d0/tw
80.8
63.0
52.5
41.4
30.0
Buckling temperature
Tcr,FEM (1C) Tcr,str (1C) (Tcr,str Tcr,FEM)/Tcr,FEM Tcr,A (1C) (Tcr,A Tcr,FEM)/Tcr,FEM Tcr,FEM (1C) Tcr,str (1C) (Tcr,str Tcr,FEM)/Tcr,FEM Tcr,A (1C) (Tcr,A Tcr,FEM)/Tcr,FEM Tcr,FEM (1C) Tcr,str (1C) (Tcr,str Tcr,FEM)/Tcr,FEM Tcr,A (1C) (Tcr,A Tcr,FEM)/Tcr,FEM Tcr,FEM (1C) Tcr,str (1C) (Tcr,str Tcr,FEM)/Tcr,FEM Tcr,A (1C) (Tcr,A Tcr,FEM)/Tcr,FEM Tcr,FEM (1C) Tcr,str (1C) (Tcr,str Tcr,FEM)/Tcr,FEM Tcr,A (1C) (Tcr,A Tcr,FEM)/Tcr,FEM
S/d0 1.2
1.3
1.4
1.5
1.6
1.7
1.8
20 20 0.00% 20 0.00% 215 20 90.70% 234 8.84% 419 20 95.23% 457 9.07% 485 225 53.61% 530 9.28% 522 457 12.45% 565 8.24%
20 20 0.00% 20 0.00% 304 20 93.42% 325 6.91% 440 20 95.45% 499 13.41% 485 419 13.61% 548 12.99% 534 533 0.19% 596 11.61%
148 20 86.49% 109 26.35% 332 20 93.98% 347 4.52% 440 221 49.77% 505 14.77% 501 472 5.79% 557 11.18% 546 566 3.66% 617 13.00%
148 20 86.49% 143 3.38% 332 20 93.98% 352 6.02% 460 301 34.57% 505 9.78% 515 505 1.94% 562 9.13% – –
183 20 89.07% 136 25.68% 332 20 93.98% 350 5.42% 460 360 21.74% 506 10.00% 529 520 1.70% 566 6.99% – –
148 20 86.49% 119 19.59% 359 175 51.25% 360 0.28% 478 405 15.27% 509 6.49% 543 534 1.66% 572 5.34% – –
20 20 0.00% 20 0.00% 359 261 27.30% 389 8.36% 478 440 7.95% 512 7.11% 516 550 6.59% 580 12.40% – –
–
–
–
–
–: Represents the CSB failed by the Vierendeel mechanism plastic bending failure.
4.6.2. Critical temperatures of the web-posts predicted by the modified strut model The buckling temperatures of the web-posts predicted by the modified strut model can be obtained by the following steps: (1) Using the modified strut model to determine the compressive stresses in the web-posts under a given load, as listed in Table 11;
(2) Making an assumption on the buckling temperature of webpost; (3) Following the description in BS5950-1 and considering the degrading of steel at high temperature to determine the buckling resistance at the assumed temperature; (4) If the buckling resistance calculated in step (3) equals to the compression stress calculated in step (1), the temperature assumed in step (2) is the buckling temperature of the web-
P. Wang et al. / Thin-Walled Structures 85 (2014) 441–455
post. Otherwise, make a new assumption and re-calculate the buckling resistance of the web-post; The comparison of the buckling temperature obtained from the modified strut model and the FEM simulation were listed in Table 12. δ is the error of the calculated buckling temperature to that obtained through FEM simulation δ¼
T cr;A T cr;FEM 100% T cr;FEM
ð29Þ
where Tcr,A is the temperature obtained through the modified strut model. For most cases, δ is less than 5%, as listed in Table 12, which shows the applicability of the strut model in predicating the buckling temperature of the web-post in a cellular steel beam in a fire. To show the applicability of the modified strut model, buckling temperatures of the web-post in CSBs with load ratio of 0.4 and 0.7 were calculated using the modified strut model. Buckling temperatures obtained by the FEM simulation, the traditional strut model and the modified strut model were all listed in Tables 13 and 14. The buckling temperature calculated by the modified strut model agreed with the FEM results much better than that obtained by the traditional strut model, as listed in Tables 13 and 14. 5. Conclusion Buckling behaviors of the web-post in a CSB in a fire were investigated through using a thermal-mechanical coupled finite element model. The Bitar’s model, the Lawson’s model and the strut model were used to calculate the compression stresses in the web-post under a given load. The Bitar’s model and the Lawson’s model gave an extraordinary high compression stress when the CSB had a narrow web-post and thin web thickness. The compression stress given by the strut model was at a reasonable value but its correctness was still need to be improved. The buckling temperature of the web-post will be underestimated if the traditional strut model was used to predication the buckling temperature of the web-post. A simplified method to calculate the effective width of the strut was proposed. The width of the strut was obtained through measuring the width of the compression principle stress band.
455
The buckling temperature obtained through the modified strut model agreed well with the FEM analysis.
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