Simplified calculation model for load-bearing cold-formed steel composite walls under fire conditions

Abstract

The conventional simplified model only restricts the bending buckling around the minor axis and overall torsional buckling, which is not suitable for external sandwiched cold-formed steel composite walls. Moreover, a solution to stud–track connections must be achieved in establishing the overall structure model. In this article, a simplified calculation model is proposed to accurately and efficiently reveal the fire performance of cold-formed steel composite walls. A tension spring is adopted to simulate the boundary condition that limits the axial thermal expansion of the studs at elevated temperature. Meanwhile, the simplified applications of the panel constraints and stud–track connections are also given in details. Finite element analysis using the developed simplified calculation model is conducted to simulate five full-scale cold-formed steel composite walls with different configurations. Comparisons between the finite element analysis and fire test results show an overall agreement on the failure modes, cold flange temperatures and lateral deflections at mid-height of the studs. These results demonstrate that the developed simplified calculation model is able to simulate the fire performance and predict the lateral deflection of the external sandwiched cold-formed steel composite walls accurately. Finally, the key factors affecting the lateral deflection of the studs are analysed.

Keywords

fire resistance lateral deformation load-bearing cold-formed steel walls numerical simulation simplified calculation model

Introduction

As the primary vertical load-bearing and space-dividing components, cold-formed steel (CFS) composite walls are composed of steel studs with panel linings on both sides. The performance of CFS composite walls under fire conditions is important for the structural fire-resistant designs. There were three forms of CFS wall configuration. Assemblies lined with single or double layers panel with or without cavity insulation were widely adopted in conventional CFS walls. A large number of experimental research on the conventional CFS walls have been carried out. However, such walls have poor fire resistance. A new CFS wall system based on a composite panel in which the insulation was sandwiched between two panels instead of cavity insulation was proposed in Kolarkar’s research (Kolarkar, 2011). Chen and Ye (2013a: 145–157) presented a detailed experimental investigation on six full-scale external sandwiched CFS walls. The research object of this article is the external sandwiched walls.

Fire tests and numerical simulation analyses are two main methods to investigate the fire resistance of CFS composite walls. The numerical simulation analysis of load-bearing CFS composite walls under fire conditions is a transient process involving thermal and structural behaviours and including the usage of finite element and simplification of calculation models. Many finite element software programmes such as ABAQUS, ANSYS, FLUENT, SAFIR and TASEF (Alfawakhiri, 2002; Feng et al., 2003a: 679–707, 2003b: 365–394; Gerlich et al., 1996; Gunalan, 2011; Keerthan and Mahendran, 2013; Thomas, 2010; Zhao, 2000; Zhao et al., 2005) can simulate heat transfer processes. The accurate calculation of the heat transfer process with commercial finite element software provides a foundation for thermal–mechanical coupling simulation.

The proposal of CFS simplified model is the key point of numerical calculation due to the complicated connections of a composite wall between multiple studs and wall panels. Alfawakhiri (2002) simplified the problem of CFS walls exposed to fire as a single-stud buckling in the case of non-uniform temperature rise and then compiled corresponding calculation analysis programme. Their numerical analysis results agreed well with three fire resistance tests. Feng et al. (2003b: 365–394) discussed the failure modes of CFS walls under two non-uniform temperature distributions along the cross-section of the studs and derived an axial bearing capacity calculation formula for the walls at high temperatures based on Eurocode3 Part 1.3. Their model was established in ABAQUS for steady-state numerical simulations, and the selection of parameters, such as basic parameters, meshing, boundary conditions and initial defects, was discussed. Based on ABAQUS, Gunalan and Mahendran (2013) simulated seven CFS walls of different configurations under transient and steady conditions, and the results indicated that the failure modes were similar between finite element analysis (FEA) and fire tests.

The above simplified models, which adopted a single stud instead of the wall, contained the following problems. (1) The simplified model, which only restricted the bending buckling around minor axis and overall torsional buckling, was not suitable for the external sandwiched walls, especially because of the neglect of the axial constraint. (2) The non-uniform temperature distribution of the stud cross-section measured by the test was taken as the temperature load. It was difficult to realize the temperature transfer of the stud cross-section over time using the above temperature load application method in FEA. (3) Axial restraints provided by the floor system and non-fired components were not considered. (4) Meanwhile, the single-stud model without providing a solution for the stud–track connection could not realize the process simulation and damage tracking of the whole structure. Thus, it was necessary to establish a simplified model for CFS composite walls to consider the axial constraint and solve the complex contact connection of the stud–track. Then, the precise thermal–mechanical coupling analysis of the CFS composite walls and further fire analysis of whole structure could be realized.

Therefore, this article proposed a wall simplified wall calculation model, in which a tension spring was used to solve the axial limitation boundary condition of the stud at elevated temperatures. Taking a simplified model with three studs as an example, the simplifications of wall panel restraint, stud–track connection and temperature load application were given. Based on the standard fire tests of five full-scale CFS composite walls, the failure mode, cold flange temperature and out-of-plane deformation were compared between fire test and FEA to verify the correctness and accuracy of the proposed simplified wall calculation model mentioned in this article. Finally, their influences of temperature rise rate and vertical load ratio on the lateral deflections were analysed, which could provide a reference for effectively predicting the lateral deflection.

Simplified calculation model for CFS composite walls

Axial constraint

In the actual structure, the axial constraints of the inner steel skeleton were provided by floor system and lining panels (Figure 1, the panels were plasterboard (PB)). The constraints of the floor system existed before the composite walls and floor were detached. The panels were no longer provided constraints when they lost thermal insulation or connection to the studs. There are no experimental measurements and analytical values for the constraints of the studs provided by wall panels and other components. Moreover, the contributions of different elements in the axial constraint were difficult to distinguish. Therefore, this article analysed the consequences of axial restraint. In the fire tests, the thermal axial expansion value approximates 0 (Chen and Ye, 2013a). The main effect of axial restraint is to constrain the thermal axial expansion deformation of the studs at elevated temperature. This article considered the common constraint without distinction.

Figure 1.

Axial constraint of a CFS composite wall in an actual structure.

Solution of the axial constraint

In the actual structure, the CFS composite wall subjected to fire on one side which produced thermal gradient along the studs section (Figure 2(a)). The thermal gradient caused the expansion deformation of the stud hot and cold flanges to be different (Figure 2(b)). ΔL₂ is the difference expansion deformation between the hot and cold flanges. Therefore, a bowing deformation to the fire side was generated. When the axial constraint was not limited at top of the stud (Figure 2(b), ΔL₁ is the axial expansion with free axial boundary), the stud did not have temperature stress. When the top of the stud was axial constrained (the axial expansion deformation is smaller than ΔL₁), temperature stress was generated and finally, the total stress was superposition of temperature stress and stress generated by the axial force. Then the stud exhibited local or overall buckling under vertical loading and temperature stresses. In the actual structure, the axial expansion of stud was constrained by the floor system through the upper track (Figure 1). The upper end cross-section of the stud was kept horizontal under the track constraints even if the stud exhibited overall or local buckling. Therefore, the constraint provided by the floor system was constant. The decreases in the panel properties at elevated temperature, the breakage of panel parts and the disengagement of screws diminishing the axial constraint provided by the cladding sheet. However, the random influencing factors were difficult to consider. The axial constraint was the largest at the initial stage of the fire. Therefore, this article analysed the mechanical state of the stud before the overall bending occurred to calculate the axial constraint, which was applied during entire fire process. The temperature transfer of the stud section and the external heat exchange were present throughout the whole process.

Figure 2.

Deformation analysis of the studs: (a) thermal gradient of stud section, (b) thermal bowing deformation of stud and (c) pre-buckling deformation of the stud with constraint.

The boundary conditions in numerical calculation analysis including the usage of the completely constrained or common spring which both restricted the axial shorten deformation of the stud prevented the occurrence of instability damage (Models 2 and 3 in section ‘Comparison of different axial constraints’). This condition did not match the actual situation. Therefore, a tension spring which limited the axial expansion and eliminated shorten deformation was set on the coupling control point between the top (bottom) tracks and studs. The value of axial expansion stiffness was calculated in the following, while the axial shorten stiffness was set to 0.

The pre-buckling deformation of the stud without an external load is shown in Figure 2(c), where L is the height of the stud and ΔL is the expansion value of the stud at T (°C) under free boundary condition. The expansion force ΔP, the tension spring stiffness k_s and the stud stiffness k_c,T coincided with the following relationship

{\begin{matrix} Δ P = k_{s} Δ L_{2} \\ Δ P = k_{c, T} Δ L_{1} \end{matrix}

(1)

From formula (1), the following formula was available

k_{s} = \frac{Δ L_{1}}{Δ L_{2}} k_{c, T}

(2)

By defining $μ = Δ L_{1} / Δ L_{2}$ , the following expression is yielded

k_{s} = μ k_{c, T}

(3)

where $k_{c, T} = E_{T} A / L$ , wherein E_T is the steel elastic modulus of steel at T (°C), A is the cross-sectional area of stud, ΔL₁ is the retraction value of the stud due to the limitation boundary conditions and ΔL₂ is the axial deformation of the tension spring.

Calculation of the tension spring parameters

The simplified calculation model (Figure 3) was calculated in FEA with different μ (formula (3)). The temperature load was applied according to the ISO-834 curve. The geometric dimensions are shown in Figure 3(b). Q345 steel was used in the simplified model in which the yield strength was 339.2 MPa at ambient temperature (the Poisson ratio was 0.3 and the density was 7680 kg/m³). The material properties were obtained by the transient testing (Ye and Chen, 2012) (Figure 4(a) and (c)). Material tests under transient state conditions gave nominal stress–strain curves whereas ABAQUS required stress–strain relationship in terms of true stress and logarithmic plastic strain. Therefore, engineering stress–strain data obtained from material tests (Ye and Chen, 2012: 947–957) were converted to true stress and logarithmic plastic strain values (Figure 4(b)) using equation (4)

\begin{matrix} σ_{true} = σ_{eng} (1 + ε_{eng}) \\ ε_{true} = \ln (1 + ε_{eng}) - \frac{σ_{true}}{E} \end{matrix}

(4)

where σ_eng is the engineering stress, ε_eng is the engineering strain, σ_true is the true stress, ε_true is the logarithmic plastic strain and E is the elastic modulus.

Figure 3.

Simplified calculation model: (a) simplified calculation model in the FEA and (b) the geometric dimensions of simulation model (thickness was 0.9 mm).

Figure 4.

Mechanical and thermal parameters at high temperatures: (a) elastic modulus, (b) stress–strain curves and (c) thermal parameters.

The axial and lateral deformations of the stud showed (Figure 5) that a larger value of μ meant a smaller axial deformation at the end of the stud and a greater lateral deflection in the middle of the stud. When μ ≥ 5, the axial deformation was approximately 0 and the lateral deflection curve was almost coincident, which suggested that the stud expansion was completely limited by the boundary conditions at elevated temperatures.

Figure 5.

Deformation of the simplified wall calculation model with different values of μ: (a) axial deformation and (b) lateral deflection.

Comparison of different axial constraints

To examine the influence of the axial constraint on the failure modes of CFS composite walls subjected to fire on one side, four individual models were established referring to fire tests of composite walls in the literature (Chen et al., 2013a: 145–157) (the wall S1 in Figure 11(g)). Except for the boundary constraint, the four models shared the same geometric dimensions (Figure 3(b)), mechanical performance parameters and thermo-physical parameters (Figure 4). The temperature load (temperature load of S1 in Figure 13) was applied to the hot flange in Figure 3(a), and the vertical load was 20 kN (the temperature load and the vertical load were the same as those of the wall S1 in Figure 11(g)). The initial defects of the four models were the eigen value buckling shown in Figure 10(a). The initial defect coefficient was 0.33. The determination of the initial defect coefficient is discussed in detail in section ‘Initial imperfections’.

All the four models were hinged at the bottom and the top boundary constraints were different (Figure 6). Model 1 was axially free and Model 2 was completely fixed. The linear spring was set to constrain the axial deformation in Model 3. A non-linear tension spring was used in Model 4. The vertical expansion stiffness (positive Z-axis direction) was calculated according to formula (3). E_T is the steel elastic modulus of steel at T (°C) according to Figure 4(a). A is the cross-sectional area of stud which section side is shown in Figure 6(a). L is 3 m and μ takes 5 according to the analysis of Figure 5. The vertical compression stiffness was 0 (negative Z-axis direction) which means the limitation of the axial expansion deformation and freedom of axial compression deformation. The spring stiffness value of Model 3 was the same as the axial expansion stiffness of Model 4.

Figure 6.

Boundary conditions: (a) Model 1, (b) Model 2, (c) Model 3 and (d) Model 4.

A calculation model for a single stud was established, and thermal–mechanical coupling calculation was performed in FEA. Figure 7 shows the overall deformation and local buckling of studs under four different boundary conditions. Except for Model 2 (the axial-fixed constraint boundary condition), the other models all exhibited overall bend deformation around the Y-axis and local buckling in the middle of the cold flange. Model 2 appeared local buckling at top of the studs due to the over constraints in Z axial. This phenomenon because thermal bowing to the fire side was the main deformation of the studs due to thermal gradient in the cross-section and consequent differential degradation of the mechanical properties of the steel, which was the same as the fire test.

Figure 7.

(a) Failure modes and (b) axial deformations.

The comparisons (Figure 7(b)) of the axial deformation among four calculation models and the fire test wall (the wall S1 in Figure 11(g)) showed that the axial deformation of Model 4 was the most similar to the test wall. The stud exhibited axial expansion deformation (positive direction of the Z-axis in Figure 7(a)) induced by the initial elevated temperature, and the deformation converted to the opposite direction due to the thermal bowing. There was an inflection point in the axial deformation curve in Figure 7(b) (the positive value in Figure 7(b) represents the deformation of the positive direction of Z-axis in Figure 7(a)). A comparison between Models 1 and 4 showed that axial constraint accelerated the overall bending deformation of the stud. Model 2 with a rigid boundary constraint exhibited lower calculation convergence using implicit static step. Model 2 was calculated using dynamic explicit step. When the thermal and mechanical coupling simulation was performed by the dynamic explicit calculation method, a large amount of calculation time was required. When the finite element software used a dynamic explicit step, the duration of the fire experience was usually shortened to save calculation costs (1 h concentrated to 12–20 s). For Model 2, the temperature rise time of 70 min was shortened to 14 s when the thermo-mechanical coupling calculation adopted dynamic explicit step. The failure mode of Model 2 was different from the fire test. The boundary conditions of Model 2 did not suit for the actual situation. There was no inflection point in the axial deformation curve of Model 3 due to the axial compression stiffness of the tension spring which inhibited the overall bending of the stud. In summary, the constraint boundary conditions of Model 4 were the most suitable for composite walls.

Simplification method

Panel constraint simplification

The lateral constraint provided by the panels on both sides limited the torsion buckling (around the Z-axis) and overall bending buckling (around the X-axis) of the stud from the axial compression test of CFS walls at ambient temperature (Liu, 2013). Therefore, panel constraints were simplified to Y-direction restriction (Figure 8(a)) (Alfawakhiri, 2002; Feng et al., 2003b: 365–394; Gerlich et al., 1996; Gunalan and Mahendran, 2013; Kaitila, 2002) in the mid-line of the flange, and the interval of the constraint was the same as the actual interval of the tapping screws. Axial constraint (Z-direction) of the panels was applied by the tension spring. The constraint on the X-direction was not considered since out-of-plane stiffnesses of the panels (PB, glass magnesium board (GMB), etc.) were small.

Figure 8.

Simplified wall calculation model: (a) simplification of the wall panel restraint and (b) simplification of the wall panel restraint.

Simplification of the stud–track connection

In the actual construction, studs and tracks were assembled by tapping screws. Therefore, the internal force was transmitted by the contact between the studs and tracks under vertical loading. The connections set as actual structures easily led to convergence problems including element penetration due to a larger number of complicated contacts such as surface-to-line contact (between stud end cross-section and track flange) and the surface-to-surface contact (between the stud flange and track flange). Therefore, the stud–track connection was simplified as follows:

A Cartesian element (Figure 8(c)) with stiffness degeneration was used to simulate the connection of tapping screws, and the detailed parameter values were obtained from Ye et al. (2016).

Motion coupling interaction was provided between the end section of the stud and top (bottom) track. The coupling control points were on the tracks with coordinates that were the same as the geometric centroid of the stud cross-section. Moreover, ‘123456’ degrees of freedom were completely coupled (Figure 8(d)). This setting method represented the contact force of the track webs and stud cross-sections in the rail structure (Figure 8(d)). The setting method with motion coupling interaction could effectively transmit the vertical force from the track to the stud, which reduces the contact to solve the element penetration problem and improve the calculation convergence.

Applied method of the temperature load

Under fire conditions, the CFS walls suffered from elevated temperatures on one side. The heat transfer process is shown in Figure 9. The non-uniform temperature distribution of the stud cross-section was caused by the heat exchange between the hot air in the cavity and the steel stud and the heat transfer of the stud cross-section.

Figure 9.

Heat transfer process of the CFS walls.

For the sake of simplifying the calculation, temperature rise curve at the hot flange of the stud was directly used as the temperature load with considering the thermal insulation of the lining panels. The main reason for temperature rise of the studs is the thermal conduction of the steel and the thermal exchange with the hot air. Due to the fast heat transfer of the steel, the thermal gradient of the stud section is mainly caused by heat conduction. Therefore, only thermal conduction was considered in the simplified model. The good agreement of the cold flange temperature variation between numerical simulation and fire test value is shown in Figure 17 in section ‘Temperature of the cold flange’ verified the accuracy of such heat transfer model. Thermal–mechanical coupling transient analysis was used to achieve the temperature transfer process from the hot flange through the web to the cold flange. Assuming that the hot flange of each stud had the same distance from the heat source, there was no temperature gradient along the height of the stud. Such assumption has been proven by the fire test specimens (Chen et al., 2013a) and adopted by other research (Gunalan and Mahendran, 2013).

Verification of the wall simplified calculation model

The boundary conditions of the wall (Figure 8(b)) and single-stud simplified model (Figure 3(a)) were both limited to ‘126’ degrees of freedom on the side of the applied load and ‘1236’ degrees of freedom on the fixed end. From the comparison of eigen modes (Figure 10) and load (buckling load of the first eigen mode for the simplified single-stud model was 38.45 kN and buckling load of Cases 1–3 eigen modes for the simplified wall calculation model were 38.45 kN (Case 1), 38.45 kN (Case 2) and 38.46 kN (Case 3), respectively) of the two simplified models, it can be seen that the buckling modes and loads were the same, which showed that the stud–track connection method could effectively transmit the internal force.

Figure 10.

Eigen modes: (a) first buckling eigen mode for a simplified single-stud model and (b) buckling eigen modes for the simplified wall calculation model.

Introduction to the fire test

To explore the fire performance and fire-resistant time of CFS walls, 11 full-scale CFS wall specimens were examined in Chen et al.’s (2012: 242–254, 2013a: 145–157) research. Five walls (Figure 11(g)) were chosen for analysis in this article. The steel frames used in the full-scale load-bearing CFS wall models were built to a height of 3380 mm and a width of 3000 mm, as shown in Figure 11(a). The studs and tracks used in the test frames were fabricated from Q345 galvanized steel sheets with a nominal base metal thickness of 0.9 mm; the measured yield strength and elastic modulus of this Q345 steel were 339.2 and 203,600 MPa, respectively, at ambient temperature. The frames consisted of five vertical studs with dimensions of 89 mm × 50 mm × 13 mm × 0.9 mm (Figure 11(a) for S1–S4) or 140 mm × 70 mm × 13 mm × 0.9 mm lipped channel sections (Figure 11(c) for S5), spaced at 600 mm centres. The frames were made by attaching the studs to the top and bottom tracks made of 89 mm × 50 mm × 0.9 mm (Figure 11(b) for S1–S4) or 140 mm × 70 mm × 0.9 mm (Figure 11(d) for S5) unlipped channel sections. The locations of the displacement transducers and thermocouples are shown in Figure 11(e). The connections and assemblies were described in detail in the literature (Chen et al., 2012: 242–254, 2013a: 145–157).

Figure 11.

Details of construction and locations of displacement transducers and thermocouple (Chen et al., 2012: 242–254). (a) Steel frame, (b) S1, (c) S2–S4, (d) S5, (e) displacement transducers and thermocouple locations, (f) cross-section sizes of the studs and the tracks (mm) and (g) details of CFS wall specimens in the fire test.

Numerical simulation analysis of CFS composite walls

The numerical simulations of the five walls were performed based on the simplified wall calculation model mentioned in section ‘Introduction’. The geometric parameters are as follows: the height of the wall is 3 m; the spacing between studs is 0.6 m; the studs and tracks of S1–S4 are made of 0.9-mm-thick C89 steel and U89 steel and the studs and tracks of S5 are made of 1.2-mm-thick C140 steel and U140 steel (Figure 11(f)).

Material properties

Mechanical parameters

The model used cold-formed thin-walled Q345 steel (Ye and Chen, 2012), and the material parameters were exactly the same as the single-stud model of section ‘Calculation of the tension spring parameters’.

Thermal parameters

The specific values of the heat transfer coefficient, specific heat and thermal expansion coefficient are given in Figure 4(c).

Simplified calculation model for CFS walls

The constraints of the wall panel were set according to the simplified method described in section ‘Panel constraint simplification’ (Figure 12(b)). The restraint interval was 150 mm, which was the same as the real interval of the tapping screws in the test.

Figure 12.

Simplified calculation model: (a) coupled connection of stud–track, (b) Y-direction constraint and (c) Cartesian element.

The connections between the studs and tracks were applied as described in section ‘Simplification of the stud–track connection’. Figure 12(a) shows the coupled connections between the studs and tracks, and Figure 12(c) shows the Cartesian element. The studs were connected with Cartesian elements in double C-type studs according to the actual positions of the tapping screws.

The stiffness of the tension spring was determined by the material properties listed in section ‘Material properties’ and formula (3). The values of μ more than five were all reasonable. Considering the simplicity of the calculation, μ takes 10.

Temperature load

The average temperature at hot flanges of studs 3 and 4 measured by the thermocouple at section B in Figure 11 was used as temperature load. The temperature load was applied to the hot flange of the studs shown in Figure 12. The specific temperature rise curve is shown in Figure 13.

Figure 13.

Temperature loads. (a) S1 and S2; (b) S3, S4 and S5.

Load and boundary conditions

A concentrated vertical load was applied at the coupled control point of the stud–top track (Figure 12). The load for S1–S3 was 20 kN, while the load for S4 was 40 kN and the load for S5 was 35 kN, which was 65% of the ultimate bearing capacity under ambient temperature conditions.

The webs of the tracks (top and bottom) were restrained (Figure 12), where the top track was restrained in ‘126’ degrees of freedom and the bottom track was restrained in ‘1236’ degrees of freedom.

Element type and size

Mesh size has a great impact on the results of simulation. The stimulated mesh sizes are 1 cm × 1 cm, 2 cm × 2 cm, 3 cm × 3 cm, 4 cm × 4 cm, 5 cm × 5 cm and 8 cm × 8 cm. The ISO834 curve was selected as the temperature load. The failure modes of studs with different mesh sizes were compared (Figure 14(a) to (f)). Meanwhile, the comparison of the simulation time is shown in Figure 14(g). All the models used the same computer to calculate. The central processing unit (CPU) of the computer is i7-5820 and the calculation frequency is 3.3 GHz. The models with element sizes of 1–8 cm appeared local buckling at hot flange of the studs. Meanwhile, in order to determine the position of the self-tapping screw, the model has been cut to small part according to the distance between the self-tapping screws which is 15 cm (Figure 12(b)). Therefore, when the mesh size was a factor of 15, the mesh could be divided evenly in the model. Therefore, combining the calculation efficiency and the uniformity of the mesh division, a 5 cm mesh size was chosen for calculation.

Figure 14.

Failure mode of the models with different mesh sizes: (a) 1 cm × 1 cm, (b) 2 cm × 2 cm, (c) 3 cm × 3 cm, (d) 4 cm × 4 cm, (e) 5 cm × 5 cm, (f) 8 cm × 8 cm and (g) calculation time of the model with different mesh sizes.

The S4R shell element with a mesh size of 5 cm × 5 cm was selected in the calculation model. The S4R element is a four-node, quadrilateral element with reduced integral integration, hourglass mode control and limited thin strain. It provided accurate results as same as the S4 element but it required less memory spaces and times. This type element ensured sufficient degrees of freedom for buckling deformations of the CFS studs.

Residual stress and cold bending effect

Residual stress causes premature yielding and decreasing stiffness, but it had a small effect on the ultimate stress. The residual stress proposed in the literature (Young and Rasmussen, 1998) showed that the bending residual stress of the press-formed lipped channel could be neglected when it was less than 7% of the yield strength. Moreover, the cold bending effect gradually disappeared with increasing temperature. Consequently, the effect of the residual stress and the cold bending were not considered in the simplified wall calculation model.

Initial imperfections

Both the buckling modes and loads of the simplified wall calculation model (Figure 11(b)) were similar to the three-stud model (coefficients are 1.9678, 1.9774, 1.98, 1.9827 and 1.997). Hence, the first five buckling modes were used as the initial defects of the structure.

Considering the influence of different initial defect coefficients ξ, the numerical simulation results showed that when ξ was 0.3 or 0.33, the failure mode of the simplified wall calculation model was the same as that of the fire test (Figure 15(a)). When ξ was 0.89, 0.9 or 1, the failure modes differed greatly from the test results, and the calculation was not convergent (Figure 15(b)). Therefore, this article adopted 0.33 as the initial defect coefficient.

Figure 15.

Initial geometric imperfection coefficients ξ and strain diagrams with different ξ: (a) ξ = 0.33 and (b) ξ = 0.9.

Analysis of the simulation results

Comparison of the failure time

The comparisons of test and numerical simulation results of five wall pieces (Table 1) showed that the fire resistance time (ISO834-1) was similar in the two methods and the error was small. A shorter fire resistance time in the numerical simulation suggested that the simplified wall calculation model tended to be conservative, which is favourable for the fire-resistant design of CFS composite walls.

Table 1.

Failure times.

Test number	FEA time (min)	Test time (min)	Error (%)
S1	70.3	71.0	1.0
S2	127.6	137.0	6.9
S3	158.3	159.0	0.4
S4	154.9	165.0	6.1
S5	151.0	153.0	1.3

FEA: finite element analysis.

Comparison of the failure modes

There were two failure modes in the fire tests. Figure 16 lists the detailed comparisons between the tests and simulations of S1 and S5. The failure modes were identical for both tests and simulations. During the fire tests, lateral deflections (Figure 16(a) and (d)) of the overall frame towards the fire side and local buckling at the fixed end of the hot flange (Figure 16(b) and (e)) and the mid-height region of the cold flange (Figure 16(c) and (f)) could be observed. The same deformation and buckling were observed in the FEA. For S5, compression damage of the entire section at the end of the studs was observed (Figure 16(g) to (j)) in both the simulations and tests. However, without considering the limitation of the out-of-plane stiffness of the autoclaved lightweight concrete (ALC) board in the simplified wall model, the lateral deflection towards the fire side in the simulation differed from the fire test results. The failure modes of S2–S4 were the same as that of S1.

Figure 16.

Comparison between the fire test and simulation analysis: (a) failure mode of S1 in fire test, (b) end of the stud, (c) mid-height of the stud, (d) failure mode of S1 in the simulation analysis, (e) end of the stud, (f) mid-height of the stud, (g) failure mode of S5 in fire test, (h) end of the stud, (i) failure mode of the S5 in the simulation analysis and (j) end of the stud.

Temperature of the cold flange

The accurate simulation of the heat transfer process guaranteed the thermal–mechanical coupling calculation for CFS composite walls. Therefore, it was necessary to analyse the distinction of the temperature in the cold flange for five wall specimens between the tests and simulations (Figure 17). The temperature of the cold flange in the test was taken from the thermocouple in section B (Figure 11).

Figure 17.

Temperature in the cold flange of stud: (a) S1 and S2, (b) S3 and S4 and (c) S5.

The obtained results indicated close agreement in the variation trend and temperature of the cold flange for all wall pieces between the fire test and FEA (Figure 17). Thus, the simplification method was consistent with the actual condition.

The temperature variation rates for S1 and S2 were similar in the early and last stages (Figure 17(a)). However, consistent with S1 and S2 appeared in the temperature range (2500 s < time < 5000 s) because the aluminium silicate wool between the two PBs on the fire side delayed temperature transmission.

Due to the composite panels on the fire side being filled with aluminium silicate wool, the temperature curve of the cold flange for S3–S5 could be divided into three phases: the early temperature rise phase (rises quickly to approximately 100°C), the medium temperature platform phase (maintained at 100°C) and the last temperature rise phase (temperature increases again).

Lateral deformation

U_x was the average lateral deflections at the mid-height of studs 3 and 4 in Figure 16(a). The deformation of the test wall took the average of displacement transducers (3) and (4) in Figure 11(e). The lateral deflections were compared as follows (Figure 18):

The failure modes of five walls all bended to the furnace side (the positive value of U_x meant that the stud bent to the fire side).

The appearance times of local buckling at the mid-height of the stud were similar for S1–S3 (Figure 18(a) to (c)). However, for S4 (Figure 18(d)), the occurrence of local buckling was earlier in the FEA than that in the fire test. The reason is explained in the following. The S4 studs are double C-shaped section. In the actual test, the double C-shaped section studs are connected by self-tapping screws and the connection is non-linear diminution with the temperature variation. Limited to existing test equipment and conditions, there is no steel–steel connection data at elevated temperature. The numerical simulation model used Cartesian elements which adopted force–displacement curve at ambient temperature to simulated the self-tapping screws connection. Such method led to stronger stud–stud connection in the simulation than fire rest and neglected the influence of contact frictional force. Therefore, the occurrence of local buckling in S4 was earlier in the FEA than that in the fire test.

Due to the stronger connection of the stud–track in simplified model, there were differences in the maximum lateral deflections of S1–S4. However, this result had no influence on the failure mode.

Compared with the other four walls, the failure mode of S5 was local damage at the ends of the studs (Figure 16(h)). The lateral deflection in the fire test and numerical simulation was small (Figure 18(e)). Although there was a difference between fire test and numerical simulation, the fire resistance time and failure mode were not affected. The differences between numerical simulations and experiments are explained in the following. S5 is a typical external CFS walls. The ALC board is used as the lining board at outdoor side. Due to the large out-of-plane stiffness of ALC board, the out-of-plane constraint of the studs needs to be considered. In order to solve the above problems, detailed research (Chen and Ye, 2019) is carried out to simplify the external wall. Therefore, the simplified model of this article applies to the calculation of interior walls.

Figure 18.

Lateral deflections (X-direction): (a) S1, (b) S2, (c) S3, (d) S4 and (e) S5.

Influencing factors of the lateral deformation

Despite the same geometrical parameters, material properties, boundary conditions and vertical loads, there were still differences in the lateral deflections of S1–S3 caused by the different temperature loads due to the different types of composite wall panels. Moreover, the vertical load was another factor affecting the lateral deformation (Gunalan, 2013).

Temperature rise rate

The assembly forms of the CFS composite walls, which consist of panels and chemical fibres in the cavity between panels, delayed the temperature transfer to meet the requirements of fire resistance leading to a multi-line temperature rise curve (Figure 19(a)). The ISO834 standard curve could not be directly used as a temperature load for the simplified calculation model in this article.

Figure 19.

Influencing factors–temperatures rise rate α: (a) temperature rise curves of S1 and S2, (b) temperature rise curves under different α values, (c) lateral deformations under different α values and (d) fitted curve of α and T_c.

From the temperature rise curves (Figure 19(a)) at the hot flange of studs for S1 and S2, it can be seen that when the temperature was less than 135°C, the curves were similar for the two walls. In addition, when the temperature was greater than 135°C, the maximum temperatures of the two walls were similar (S1 is 336°C and S2 is 364°C) but the temperature rise rates α in the two walls were significantly different. Because the parameters, including geometrical dimensions, panel constraints and vertical loads, were the same, the different values of the temperature rise rate α above 135°C were the key factor for the different lateral deformations of S1 and S2.

To investigate the effect of α on the lateral deformation, temperature rise curves with different rise rates were obtained (Figure 19(b)) based on the temperature curves of the four CFS composite walls (S1–S4) whose B1 layer was PB (Figure 11(g)). When the temperature was less than 135°C, α was the average temperature rise rate (0.031°C/s) of the four walls. When the temperature was greater than 135°C, α was in the range of [0.01, 0.5].

S1 was simulated with different temperature rise curves, as shown in Figure 19(b). The lateral deformation results (Figure 19(c)) showed that different α values produced different lateral deformations with different first inflection points, whose physical meaning represents the buckling temperature T_c when the cold flanges exhibited local buckling at the mid-height of the stud. The comparisons between the curve fitted to different α and T_c values and the fire test results are shown (Figure 19(d)) as follows: the fitting curve was in good agreement with the fire test results of S2–S4; α and T_c were inversely proportional because a faster temperature rise increased the non-uniform temperature distribution between the cold and hot flanges, which resulted in a more rapid appearance of local buckling.

The temperature rise rate α was determined by the thermal conductivity and thickness of the panels and the insulation cavity between the panels in the CFS composite walls. To meet a higher fire resistance rating (FRR), panels with smaller thermal conductivity were the best choice because they could reduce the temperature rise rate, delay local buckling of the stud to limit lateral deformation and extend the fire resistance time of CFS composite walls.

Vertical load ratio

The analysis of the lateral deformations at the mid-height of the four walls with different vertical load ratios β (ratio of the applied load to the ultimate load) (Figure 20) showed that different values of β indicated different second inflection points of the lateral deformation curve, indicating that the temperature where the stud exhibited overall buckling was different.

Figure 20.

Lateral deformations of different β values: (a) S1, (b) S2, (c) S3 and (d) S4.

For S1, when β ≥ 0.8, axial compression failure rather than overall buckling bending to the fire side occurred, mainly due to a sudden increase in the temperature rise rate at the last stage (after the inflection point of temperature shown in Figure 13, which means that layer B1 lost heat insulation). The reason for the above phenomenon was that when the vertical load was greater than or equal to the ultimate load while the deformation of the stud was small, axial compression failure occurred (the early segment in Figure 20(a)). When 0.8 > β > 0.4, overall buckling of the stud occurred, and larger β values meant lower buckling temperatures. The reason for this phenomenon was that as the out-of-plane deformation increased under increasing temperature, overall buckling occurred before the stud reached the ultimate axial pressure state. In addition, a larger load corresponded to a lower ultimate temperature; when β ≤ 0.4, there was no overall buckling of the stud (Figure 20). The reason for the lack of buckling was that the temperature rise curve used in the numerical simulation was taken from the fire test. The ultimate refractory temperature of the simulation model was smaller than that of the fire test (the vertical load ratio of test wall S1 was 0.65, which was larger than 0.4) because overall buckling could not occur. When β > 0.6, S2–S4 all exhibited overall buckling, and the larger the β value was, the lower the corresponding instability temperature. When β ≤ 0.6, overall buckling was not observed in any of the four walls for the same reason as S1.

Moreover, compared with S1, there was a platform in the temperature rise curves for the other three walls due to aluminium silicate wool (Figure 2) filled in the two panels on the fire side (Figure 13). Therefore, even if the vertical load ratio was large, it was difficult for the stud to achieve the ultimate axial pressure state. However, with gradually increasing temperature, gradually increasing out-of-plane deformation eventually led to overall buckling instead of axial pressure damage.

Conclusion

In this article, a simplified calculation model for CFS composite walls is proposed to solve the axial constraint boundary condition of studs at elevated temperatures. The conclusions are described as follows:

A tension spring is set between the top and bottom tracks to limit stud expansion at high temperatures, and the calculation formula for the stiffness of the tension spring is derived. The panel constraint is simplified to limit the Y degree of freedom at the stud flange. The stud–track is connected with a Cartesian element and complete coupling. The solution and simplified method can be applied to thermal–mechanical coupling numerical simulations of CFS composite walls and structures.

From the discussion of the fire test and numerical simulation results, the simplified calculation model proposed in this article can effectively simulate fire failure modes and predict cold flange temperatures and lateral deformation of CFS composite walls that are exposed to fire on one side. In addition, the simplified model is applicable to different types of CFS composite walls with various panels (gypsum board, GMB and ALC board) and stud forms (C90, C140 and double C90).

The key factors affecting the lateral deformations of walls are the temperature rise rate α and vertical load ratio β. The temperature rise rate α is inversely proportional to the buckling temperature T_c at the cold flange of the stud. For the CFS composite walls whose B1 layer is PB, the local buckling temperature of the stud can be deduced according to the fitting formula of α and T_c. The vertical load ratio β is related to the overall buckling of the stud. When β is larger (β ∈ (0.4, 0.8) for a wall without a cavity, and β > 0.6 for a wall whose cavity is filled with aluminium silicate wool), the stud exhibits overall buckling.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship and/or publication of this article: The work described in this article was fully supported by the National Key Programme Foundation of China (51538002) and the Fundamental Research Funds for the Central Universities, Research and Innovation Project for College Graduates of Jiangsu Province (KYLX16_0252).

ORCID iD

Wenwen Chen

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