Numerical analysis of aluminum alloy reticulated shells with gusset joints under fire conditions

Abstract

In this study, a numerical analysis was conducted on aluminum alloy reticulated shells (AARSs) with gusset joints under fire conditions. First, a thermal-structural coupled analysis model of AARSs considering joint semi-rigidity was proposed and validated against room-temperature and fire tests. The proposed model can also be adopted to analyze the fire response of other reticulated structures with semi-rigid joints. Second, a parametric analysis was conducted based on the numerical model to explore the buckling behavior of K6 AARS with gusset joints under fire conditions. The results indicated that the span, height-to-span ratio, height of the supporting structure, and fire power influence the reduction factor of the buckling capacity of AARSs under fire conditions. In contrast, the reduction factor is independent of the number of element divisions, number of rings, span-to-thickness ratio, and support condition. Subsequently, practical design formulae for predicting the reduction factor of the buckling capacity of K6 AARSs were derived based on numerical analysis results and machine learning techniques to provide a rapid evaluation method. Finally, further numerical analyses were conducted to propose practical design suggestions, including the conditions of ignoring the ultimate bearing capacity analysis of K6 AARS and ignoring the radiative heat flux.

Graphical abstract

Keywords

aluminum alloy reticulated shell / gusset joint / numerical analysis / fire resistance

1 Introduction

Aluminum alloys are increasingly being applied in large-space spatial structures owing to their advantages of corrosion resistance, high strength-to-weight ratio, and favorable appearance. As one of the most popular types of aluminum alloy structures, single-layer reticulated shells can have multiple and even complex structural forms to adapt to aesthetic requirements, for example, spherical shells, cylindrical shells, and free-form shells. Hence, the traditional design concerning the fire resistance of structural components may not be economical [1]. To resolve this problem, the concept of performance-based fire resistance design is proposed, which calls for the evaluation of the structural performance objectives, for example, the resistance or residual deformation of a specific structure or its structural components, under the designed fire scenarios.

Large-space fires differ from compartment fires because the volume of the fire is significantly smaller than that of the interior space, and the air temperature cannot be regarded as being uniformly distributed. Hence, a large-space fire scenario first generates a nonuniformly distributed air temperature field. Next, the structural components receive the heat flux from the air and fire through heat convection and radiation. Finally, thermal expansion and changes in the material properties cause the responses of the structure. Therefore, performance-based fire resistance design for large-space fire scenarios requires data, including the air temperature field, temperature development of structural components, and structural response.

Conducting fire tests is the most direct method to obtain the data mentioned above. In addition to fire tests on different types of steel structures [2−8], fire tests on aluminum alloy structures have also been conducted in recent years. Guo et al. [9] conducted fire tests on a single-layer spherical aluminum alloy reticulated shell (AARS), in which eight large-space fire scenarios were investigated. No damage to the shell specimen was observed after the fire tests, and the structural displacement was elastic. Zhu et al. [10] proceeded with two destructive tests on the same shell specimen. The failure mode of the shell specimen under the designed fire scenario was collapse, and the structural components failed by melting, rupture, and flexural-torsional buckling. As the aforementioned two studies focused on spherical reticulated shells, Yin et al. [11] designed a full-scale fire test on a cylindrical AARS specimen to evaluate the air temperature field. The test results revealed that the critical air temperature when the specimen collapsed was 330 °C, indicating that the existing suggestion of the critical temperature limit, i.e., 150 °C, has a large safety margin.

Although fire tests can produce immediate data, they consume a considerable amount of time and resources. As the fire tests conducted in Refs. [9−11] recorded the thermal or structural responses of the specimen, it was possible to simulate the fire process using numerical analysis, including the air temperature field, temperature development of the structural components, and global structural response. Regarding the simulation of the air temperature field, Refs. [9,11] used the fire dynamic simulation (FDS) software and the empirical formula proposed by Du and Li [12]. The comparison indicated that the field simulation can produce a relatively accurate prediction of the air temperature field; however, the parameters in the empirical formula should be reasonably adjusted. To simulate the temperature development of aluminum alloy structural components, Zhu et al. [13] proposed an iterative calculation method and highlighted that the radiative heat flux produced by the fire in large-space fire scenarios cannot be ignored. Nonetheless, a simulation method for the structural response of aluminum alloy structures, that is, the thermal-structural coupled analysis method, has not been proposed, which dramatically hinders the performance-based fire resistance design process and the promotion of aluminum alloy structures. Notably, typical joint systems of aluminum alloy structures, such as the gusset joint system, are typically semi-rigid [14−17], and their semi-rigidity has been proven to influence the structural performance at room-temperature [18]. However, the numerical model established for AARS at room-temperature [19,20] cannot be directly applied to fire analysis because a temperature-dependent variation in the joint rigidity is included [21].

In addition, it is notable that in performance-based fire resistance design, field simulation and thermal-structural coupled analysis are computationally expensive and are not responsive to the needs of practical engineers at the initial/concept design stage. Specifically, repeated thermal-structural coupled analyses are required if the cross-sections of the members are adjusted, and additional field simulations are included if the fire scenario is adjusted owing to changes in the architectural composition. The above problems call for rapid evaluation methods and practical design suggestions to provide valuable information to designers at the initial/concept design stage.

In this study we conducted a numerical analysis of AARS with semi-rigid gusset joints under fire conditions. First, the numerical model of the AARS at room-temperature and its restrictions were briefly reviewed. Second, a numerical model for fire analysis was proposed and validated against fire test data. Next, a parametric analysis of the ultimate bearing capacity of K6 AARS under fire conditions was conducted based on the validated thermal-structural coupled analysis model, and the corresponding mechanisms were analyzed. A practical design formula for predicting the ultimate bearing capacity of K6 AARS was proposed based on further numerical analysis and machine learning techniques. Finally, practical design suggestions, including the conditions of ignoring the ultimate bearing capacity analysis of K6 AARS and ignoring the radiative heat flux, were proposed based on further numerical analysis.

2 Numerical model

2.1 Model at room-temperature and its restrictions

Based on a room-temperature static experiment on a K6 AARS specimen with semi-rigid gusset joints shown in Fig.1 [22], Xiong et al. [19] established and verified a numerical model to simulate the stability behavior of the specimen using the general finite element software ANSYS [23]. In the numerical model, BEAM188 was used to simulate the member and the joint zones. Owing to the existence of a joint plate, the out-of-plane bending stiffness of the joint zone was significantly larger than that of the member. Thus, the elastic modulus of the element at the joint zone was set to 100E, where E is the elastic modulus of the member. Note that a value of 100 was determined based on trial and error [19]. The two-node nonlinear spring element COMBIN39 was used to simulate the out-of-plane bending stiffness of the gusset joint, where the nonlinear stiffness parameters were calculated using the four-line model proposed by Guo et al. [24]. To reduce the computational cost, the other degrees-of-freedom (DOF), including the three translational DOFs, in-plane and torsional rotational DOFs, and warping DOF, are coupled between the member and the joint zone.

Through experimental and numerical investigations, Guo et al. [21] highlighted that the rigidity of a gusset joint is temperature dependent. However, the real constants of the COMBIN39 element, which is, the out-of-plane bending stiffness of the joint, cannot be varied directly or indirectly using the birth and death element method. Therefore, the numerical model at room-temperature cannot be directly applied to fire analysis.

2.2 Numerical model for fire analysis

To consider the temperature-dependent joint rigidity, the MPC184 element, which is a multi-point constraint element based on the Lagrange multiplier method, was used to replace the COMBIN39 element in the room-temperature numerical model. Specifically, the pin sub element shown in Fig.2 was adopted. The pin sub element is a two-node single-DOF element that can rotate around axis 1 of the local coordinate system of nodes i and j. In Fig.2, e_m,i and e_m,j are the unit vectors of nodes i and j in the m direction (m = 1,2,3) of their local coordinate systems. We denote u_i and u_j as the resultant displacement vectors of nodes i and j, respectively, and the constraint conditions of the pin element can be described as

(1)\{\begin{array}{c}{\mathit{u}}_i={\mathit{u}}_j\\ {\mathit{e}}_{1,i}\cdot {\mathit{e}}_{2,j}=0\\ {\mathit{e}}_{1,i}\cdot {\mathit{e}}_{3,j}=0\end{array}.

The nonlinear stiffness, damping properties, and Coulomb friction of the pin element at different temperatures were simulated by defining the material properties. This element also supports geometric nonlinear and linear perturbation analyses.

When using the pin sub element of the MPC184 element to simulate the nonlinear bending stiffness of the gusset joint, the following points should be noted.

1) The two nodes of the element must have the same spatial coordinates.

2) After defining an element, the SECDATA command must be used to define a shared local coordinate system for both nodes.

3) The JOIN option in the TB command is used to define the nonlinear moment-rotation curve of the joint and input the points on the curve using the TBDATA command.

4) Use the TBTEMP command to shift different temperatures for the moment−rotation curves.

Therefore, the simplified numerical model for a member in an AARS is established as shown in Fig.3, where the elements selected for the member and the joint zone are the same as those described in Section 2.1. Note that we do not consider the stiffness reduction of the joint zone owing to the elevated temperature, as no joints fail before the members in the fire test conducted as reported in Ref. [10].

2.3 Validation of the model for fire analysis

Because the proposed numerical model for fire analysis is adaptive to the bending stiffness of the joint under different temperatures, this section verifies the model using room-temperature test data [22] and fire test data [10].

a) Room temperature-test

Fig.4 shows a comparison of load−displacement curves of the top joint of the shell specimen obtained from the experimental data and the proposed numerical model, where P is the load and δ is the vertical displacement. As the curve obtained using the proposed model is in good agreement with the experimental curve, it can be concluded that the proposed numerical model is also reliable for simulating the buckling behavior of reticulated shells with semi-rigid joints at room-temperature.

b) Fire test

Fig.5(a) plots the material constitutive model of the 6063-T5 aluminum alloy used for the AARS specimen in Ref. [10]. Notably, the elastic modulus and nominal yield strength of the material at 600 °C are set at 1% of those under room-temperature to simulate the melting of the material. Based on the out-of-plane bending stiffness model of aluminum alloy gusset joints at elevated temperatures [21], the out-of-plane moment−rotation curves of the joints in the AARS specimen [10] at different temperatures are calculated and shown in Fig.5(b). Note that the elevated-temperature constitutive model shown in Fig.5(b) is determined by the room-temperature tensile test described by Zhu et al. [10], the elevated-temperature material property reduction factors, and the constitutive models recommended in the Eurocode 9 [25].

Destructive fire tests D-1 and D-2 described by Zhu et al. in Ref. [10] were used to validate the established numerical model because the deformation of the specimen was large. By including the measured member temperature data with that of the numerical model, a comparison of the experimental and numerical displacement−time curves of typical joints in the AARS specimen in fire tests D-1 and D-2 [10] are shown in Fig.6 and Fig.7, respectively, where the upward displacement was positive. Di is the symbol of the ith displacement transducer, the detailed location of which is given in Ref. [10]. In addition, only the displacement curves before the fire became very faint in test D-1 (shown in Fig.6), that is, the time range was 0–800 s. A comparison of the numerical and experimental deformations of the specimens is shown in Fig.8. It can be concluded from the good agreement of the curves and the identical deformation patterns that the proposed numerical model can simulate the thermal expansion, stiffness degradation, and fire-induced collapse of AARS with semi-rigid gusset joints. Notably, the numerical simulation of test D-2 was terminated at the end of the test, which was at 614 s, whereas the photo shows the ultimate deformation when the fire was finally extinguished. Although there were slight differences, it can still be concluded that the sinking deformation at the center of the specimen within the fire duration was well simulated.

The proposed model can also be used to simulate the buckling behavior of other types of reticulated structures with semi-rigid joints under fire conditions when different bending stiffness models are used for the MPC184 element.

Because the readings of the displacement sensors were reset to zero before tests D-1 and D-2, the displacement induced by the vertical load applied before the tests did not influence the comparisons shown in Fig.6 and Fig.7.

3 Parametric study

Zhu et al. [10] analyzed the mechanism of fire-induced collapse of K6 AARSs. They concluded that the ferrule effect was the main cause of the thermal compressive forces in the ring members. The failure of the structure was the outcome of the degradation of the material properties, ferrule effect, and catenary action.

As the K6 reticulated shell is one of the most commonly used spatial structural types, this section further investigates the ultimate bearing capacity of K6 AARS with gusset joints under fire conditions, based on the proposed numerical model. The main motivation is that the ultimate bearing capacity of the structure is also a concern in performance-based fire resistance design. To illustrate the variation in the ultimate bearing capacity during the fire process, we define k_Λ(t) as the reduction factor of the ultimate bearing capacity at time t of the fire process, as follows:

(2)k_{\mathrm{\Lambda }}\left(t\right)=\frac{\mathrm{\Lambda }\left(t\right)}{\mathrm{\Lambda }\left(0\right)},

where Λ(0) and Λ(t) are the elastoplastic buckling capacity of AARS with gusset joints at times 0 and t of the fire process, respectively. Note that Λ(0), the room-temperature buckling capacity, can be calculated based on the formulae proposed in Ref. [19] or the room-temperature numerical model introduced in Section 2.1, whereas Λ(t) should be determined based on the numerical model proposed in Section 2.2.

3.1 Analysis scheme

To explore the calculation method of k_Λ(t) (abbreviated as k_Λ hereinafter) under common fire conditions, numerical models were established based on the following analysis scheme:

1) span L: 25, 30, 40 m;

2) height-to-span ratio f/L: 1/3, 1/4, 1/5;

3) number of rings: 10, 12, 14;

4) cross-section: H250 × 200 × 8 × 10, H300 × 200 × 10 × 14, and H400 × 250 × 10 × 16;

5) support conditions: pinned support and fixed support at the periphery;

6) fire power Q: 2, 8, 25 MW;

7) height of the supporting structure H: 0, 5, 10 m;

8) fire location: at the center (location 1), and corner (location 2).

Note that items 1), 2), 3), and 5) are determined according to engineering experience, while the cross-section of the members, that is, item 4), is determined based on the principle of avoiding the in-plane buckling of the member [26] and local buckling of the cross-section [27]. Identical cross sections are assigned to all members because the gusset joint requires the height of the H-shaped members to be the same. The illustration of these parameters is shown in Fig.9.

According to the heat release rate−time curves of ordinary large-space fire combustibles provided in the NFPA handbook [28], the fire duration does not exceed 2000 s. To study the variation trend of the elasto-plastic ultimate bearing capacity of AARS during the entire fire process, the fire duration t_max was selected as 2400 s, and the interval of the evaluation of the ultimate bearing capacity was 240 s, considering both the accuracy and the computational cost.

The empirical formula proposed by Du and Li [12] without a drop in temperature was used to calculate the air temperature field, as suggested in Ref. [9]. The temperature development of the aluminum alloy structural components was calculated according to the point assumption-based method proposed in Ref. [13].

6061-T6 aluminum alloy was selected as the material for the numerical model. The material properties at room and elevated temperatures were determined according to the Eurocode [25].

3.2 Influence of number of element divisions

Member buckling directly influences the global stability of reticulated shells [29] and it can be simulated by simulating a member with multiple elements [30]. However, increasing the number of element divisions significantly affects the total number of nodes and elements in the numerical model, resulting in a higher computational cost. Therefore, in this section we investigate the influence of the number of element divisions on the reduction factor, k_Λ.

Fig.10 shows the k_Λ–t curves of the two fire locations when different numbers of segments are used to simulate the member. The span of the numerical model was 40 m, the height-to-span ratio was 1/3, the cross-section was H400 × 250 × 10 × 16, the number of rings was 12, the support condition was pinned support, the height of the supporting structure was 0 m, and the fire power was 8 MW. The k_Λ–t curves coincide when the number of element divisions increases from one to four, indicating that k_Λ is independent of the number of element divisions. Note that the parameters that are not specifically described in the following sections are the same as those mentioned above.

This is because the cross-sections are designed to prevent member buckling and local buckling, as discussed in Section 3.1. Therefore, there was no interaction between member buckling and global buckling in the numerical examples. To reduce the computational cost, only one element was used to simulate a member in the following numerical analysis. The shape function of the element adopts a cubic polynomial to ensure accuracy.

3.3 Influence of span

When the height-to-span ratio and span-to-thickness ratio are constant, the span influences the distribution of the air temperature field. Fig.11 shows the k_Λ–t curves of two fire locations with different spans, which are 25, 30, and 40 m.

The following conclusions can be obtained from Fig.11.

1) When the span is 25 m, k_Λ increases at the initial stage of the fire process. When the span is 30 or 40 m, the value of k_Λ monotonically decreases with time. This is because the air temperature field and thermal expansion were more uniformly distributed when the span was smaller. As the maximum temperature of the structural components was small at the initial stage of the fire process, the reduction in the material properties was not significant. At this time, the thermal expansion can be regarded as a minor variation in the structural shape, which can result in an increase in the ultimate bearing capacity [31].

2) When the fire location moved from the center to the corner, k_Λ significantly decreased at the late stage of the fire process, whereas the reduction in k_Λ became less significant with an increase in the span. This is because the asymmetric air temperature field induced by the fire at the corner is more disadvantageous than the symmetric air temperature field induced by the fire at the center, and the extra compressive forces owing to the ferrule effect become more severe.

3.4 Influence of height-to-span ratio

The shape of spherical AARSs is determined by the height-to-span ratio, which also affects global stability [19] and the air temperature field. Fig.12 shows the k_Λ–t curves of the two fire locations under various height-to-span ratios, which are 1/3, 1/4, and 1/5.

The following conclusions can be derived from Fig.12.

1) With a decrease in the height-to-span ratio, k_Λ gradually decreases during the fire process. Therefore, the reduction in the ultimate bearing capacity of AARS with small height-to-span ratios was more severe under fire conditions. This is because the compressive forces induced by the ferrule effect are more unfavorable for AARS with small height-to-span ratios. Specifically, AARS with small height-to-span ratios bear more external load through their bending stiffness rather than through the membrane stiffness, so that they are more sensitive to the extra compressive forces, which results in a reduction in the bending stiffness [26]. In addition, as the top joint is closer to the fire source, the average temperature of the structural components is higher, and the degradation in material properties is more dramatic.

2) When the fire source moved from the center to the corner, the k_Λ of the AARS with various height-to-span ratios decreased, and the maximum decline occurred when the height-to-span ratio was 1/4. This is because when the fire source is located at the center, the favorable effect of symmetric thermal expansion [9] can neutralize the reduction in the ultimate bearing capacity.

3.5 Influence of number of rings

The ultimate bearing capacity of K6 AARS improves with an increase in the number of rings at room-temperature [19]. Fig.13 shows the k_Λ–t curves of the two fire locations for various numbers of rings, which are 10, 12, and 14. Note that the cross-section considered in this section is H300 × 200 × 10 × 14. As shown in Fig.13, the effect of the number of rings on k_Λ was not significant.

To explore the mechanism, Fig.14 shows the ultimate states of the AARS with different numbers of rings at t = 2400 s when the fire was located at the center. It can be observed that the shells share the same failure mechanism, which is, excessive bulge deformation at the outermost ring. Notably, this failure mechanism is in accordance with the analysis by Zhu et al. [10], in which the compressive forces of members at the outermost ring are the highest among all members owing to the ferrule effect. Specifically, although the ultimate bearing capacity of AARS at room-temperature can be increased by increasing the number of rings [19], the relative stiffness of the outermost ring remains almost unchanged, resulting in identical values of k_Λ. As a result, the number of rings is taken as a constant value when deriving the formula for k_Λ in Section 4.

3.6 Influence of span-to-thickness ratio

When the span was constant, the ultimate bearing capacity of a K6 AARS at room-temperature increased with a decrease in the span-to-thickness ratio [19]. Fig.15 shows the k_Λ–t curves of the two fire locations under different span-to-thickness ratios, which are 160, 400/3, and 100. As the span of the numerical model was 40 m, the corresponding cross-sections of the members were H250 × 200 × 8 × 10, H300 × 200 × 10 × 14, and H400 × 250 × 10 × 16.

It can be observed from Fig.15 that the k_Λ–t curves are almost coincident. Fig.16 shows the ultimate states of the AARS with different span-to-thickness ratios at t = 2400 s when the fire was located at the center. Notably, the ultimate state of the AARS with a span-to-thickness ratio of 400/3 is the same as that in Fig.14(b). The mechanism is almost the same as that described in Section 3.5. Because member buckling has already been avoided, although enlarging the cross-section of the member leads to an increase in the global stiffness, the relative stiffness of the outermost ring is not changed. Therefore, the value of k_Λ remained unchanged. Consequently, the span-to-thickness ratio is taken as a constant value when deriving the formula for k_Λ in Section 4.

3.7 Influence of support condition

The support condition influences the room-temperature elasto-plastic buckling capacity of AARS, as there is an approximate 10% reduction in the buckling capacity when the support condition varies from fixed to pinned [19]. Fig.17 shows the k_Λ–t curves of the two fire locations under different support conditions, which are pinned and fixed. Note that the cross-section considered in this section is H250 × 200 × 8 × 10. It can be observed from Fig.17 that the influence of the support conditions on k_Λ was not significant.

Fig.18 shows the ultimate states of the AARS with different support conditions at t = 2400 s when the fire was located at the center. It can be observed that the failure mode remains the same when the support conditions are different, that is, the relative stiffness of the outermost ring is not changed by the support condition. Therefore, in the subsequent numerical analysis, only the pinned-supported AARSs were analyzed to reduce the computational cost.

3.8 Influence of height of supporting structure

The height of the supporting structure directly affects the air temperature field. Fig.19 shows the k_Λ–t curves of the two fire locations under different heights of the supporting structure, which are 0, 5, and 10 m. Note that the cross-section considered in this section is H300 × 200 × 10 × 14.

The following conclusions can be derived from Fig.19.

1) With an increase in the height of the supporting structure, k_Λ increases during the fire process. This is because the height of the supporting structure can also be regarded as the minimum distance between the fire source and structural components. Hence, the air temperature near the structural components and the radiative heat flux from the fire source decrease with an increase in the height of the supporting structure [13]. Consequently, the degradation in material properties, as well as the thermal compressive force induced by the ferrule effect, becomes less severe.

2) With an increase in the height of the supporting structure, the rate of increase in k_Λ is reduced. Therefore, the economic benefit of significantly adjusting the structural layout to ensure fire safety is low for large-space structures.

3.9 Influence of fire power

Fire power directly affects the maximum temperature of the air temperature field [9]. Fig.20 shows the k_Λ–t curves of the two fire locations under different fire powers, which are 2, 8, and 25 MW. Note that the cross-section considered in this section is H300 × 200 × 10 × 14.

It can be concluded from Fig.20 that with an increase in the fire power, k_Λ decreased during the fire process. This is owing to the increase in the temperature at each height of the fire centerline and the radiative heat flux with an increase in the fire power. Therefore, the reduction in material properties and extra compressive forces caused by the high temperature became more significant, resulting in a more severe reduction in the ultimate bearing capacity.

4 Practical design formula

In the initial/concept design stage, it is necessary to predict the ultimate bearing capacity of the AARSs under fire conditions. In this section, a practical design formula for the reduction coefficient k_Λ is derived based on further parametric analysis.

4.1 Form of formula

The following rules can be obtained by observing the k_Λ–t curves in Sections 3.2 to 3.9.

1) The value of k_Λ is 1.0 at 0 s, because the fire has not affected the structure, which is, k_Λ(0) = 1.0.

2) The value of k_Λ decreases monotonically with time and reaches the minimum value of k_Λ,min at the end of the fire process, which is, k_Λ(2400) = k_Λ,min.

3) The value of k_Λ varies slowly in the early stage of the fire process, rapidly in the middle stage, and slowly again in the later stage. Therefore, the k_Λ−t curve should include two inflection points.

Thus, the calculation formula of k_Λ(t) can be constructed as

(3)k_{\mathrm{\Lambda }}\left(t\right)=\frac{k_{\mathrm{\Lambda },min}}{1+\left(k_{\mathrm{\Lambda },min}-1\right){\text{e}}^{-{10}^{-6}{t}^2}},

where t is the time (s), 0 ≤ t ≤ 2400. It can be observed from Eq. (3) that the formula for k_Λ(t) contains only one undetermined parameter k_Λ,min. Taking the curves of the fire power series in Section 3.9 as an example, Fig.21 shows a comparison between the numerical curve and the curve calculated using Eq. (3) when k_Λ,min is taken as the accurate result of the numerical curve. In Fig.21, the solid red line represents the curve obtained using Eq. (3). It can be observed that Eq. (3) can accurately predict the variation in the reduction factor of the ultimate bearing capacity during the entire fire process. Thus, the equation for k_Λ(t) can be determined based on two thermal-structural coupled elasto-plastic analyses.

4.2 Parametric analysis scheme

To further reduce the computational cost in practical engineering applications, a practical calculation method for k_Λ,min under common fire scenarios is proposed.

According to the parametric analysis in Sections 3.2 to 3.9, the value of k_Λ should only be related to the span, height-to-span ratio, height of the supporting structure, fire power, and fire location. As other parameters are kept constant, that is, the number of rings is 12, the span-to-thickness ratio is 100, and the support condition is a pinned support, the numerical models of 162 reticulated shells are established considering the following parameters:

1) span L: 25, 30, 40 m;

2) height-to-span ratio f/L:1/3, 1/4, 1/5;

3) fire power Q: 2, 8, 25 MW;

4) height of supporting structure H: 0, 5, 10 m;

5) fire location: at the center (location 1) and corner (location 2).

4.3 Regression analysis based on machine learning

As indicated in Section 4.1, Λ(0) and Λ(2400) were calculated to determine the value of k_Λ,min for each numerical model as follows:

(4)k_{\mathrm{\Lambda },min}=\frac{\mathrm{\Lambda }(2400)}{\mathrm{\Lambda }(0)}.

On this basis, using the Statistics and Machine Learning Toolbox in MATLAB R2020a [32], a support vector machine with a linear kernel function was used to fit the numerical results, and quadratic terms were introduced to improve the accuracy (see Ref. [30] for specific principles). Therefore, k_Λ,min can be calculated as follows:

(5)\begin{array}{rl}{k}_{\mathrm{\Lambda },min}=& p_{1}L+p_2\left(f/L\right)+p_{3}Q+p_{4}H+p_{1,1}{L}^2\\ & +p_{1,2}L\cdot \left(f/L\right)+p_{1,3}L\cdot Q+p_{1,4}L\cdot H\\ & +p_{2,2}{\left(f/L\right)}^2+p_{2,3}\left(f/L\right)\cdot Q+p_{2,4}\left(f/L\right)\cdot H\\ & +p_{3,3}{Q}^2+p_{3,4}Q\cdot H+p_{4,4}{H}^2+b,\end{array}

where p_i is the coefficient of the ith (1 ≤ i ≤ 4) linear term; p_i,j is the coefficient of the ijth (i ≤ j ≤ 4) quadratic term; and b is the undetermined bias. For the two fire locations, the fitted values of each parameter in Eq. (5) are listed in Tab.1. We consider the results calculated using Eq. (5) as the fitting value, and the results calculated by the thermal-structural coupled analysis as the actual value. Fig.22 shows a comparison of the fitting and actual values. It can be observed that the error is small, and therefore it is reasonable to use Eqs. (3) and (5), together with Tab.1, to rapidly predict the ultimate bearing capacity at the initial stage of the AARS design. Thus, repetitive thermal-structural coupled analyses under fire conditions can be avoided at the initial/concept design stage. However, it is worth noting that Eqs. (3) and (5) are only applicable to K6 AARS when the span is 25–40 m, height-to-span ratio is 1/5–1/3, height of the supporting structure is 0–10 m, and fire power is 2–8 MW.

It should be noted that machine learning techniques were used because the reduction factors of the ultimate bearing capacity of K6 AARS are highly nonlinear with respect to the parameters specified in Eq. (5). Thus, traditional curve-fitting techniques, although applicable, provide complex fitting formulae. An example of this can be found in Ref. [19]. In addition, we aimed to provide an explicit formula instead of a black box to better serve practical engineering applications. Therefore, we chose a support vector machine with a linear kernel function rather than a Gaussian kernel function or neural network to establish the machine learning model, although they may have a better regression performance.

However, there were still a few outliers, as shown in Fig.22. Although this indicates that overfitting does not exist in our trained model, we still need to emphasize that the thermal-structural coupled analysis should be conducted after determining the structural design scheme in the initial/concept design stage if the requirements proposed in Section 5.1 (for ignoring the ultimate bearing capacity analysis) are not satisfied.

5 Practical design suggestions

In addition to the formula for the ultimate bearing capacity, practical design suggestions, including conditions for ignoring the ultimate bearing capacity analysis of K6 AARS and conditions for ignoring the radiative heat flux, are proposed in this section. Our aim is to reduce the computational cost of the field simulation and the thermal-structural coupled analysis.

5.1 Conditions of ignoring the ultimate bearing capacity analysis of K6 AARSs

The analysis results in Sections 3.8 and 3.9 show that the reduction in the ultimate bearing capacity under fire becomes less significant with a decrease in Q and an increase in H. Therefore, there should be a critical combination of Q and H that makes the reduction in the ultimate bearing capacity negligible. To explore the critical combination to simplify the calculation process, in this section we conducted a further numerical analysis with respect to the influential parameters. Unvaried parameters included the span (40 m), number of rings (12), cross-section (H300 × 200 × 10 × 14), and support condition (pinned support). The influential parameters were varied according to the following scheme:

1) height-to-span ratio f/L: 1/3, 1/4, 1/5;

2) fire power Q: 2, 8, 15, 25, 35 MW;

3) height of supporting structure H: 20, 30, 40 m;

4) fire location: at the center (location 1) and corner (location 2).

Note that the fire duration is also conservatively taken as 2400 s.

We designate k_i (i = 1,2) as the minimum reduction factor of the ultimate bearing capacity of the entire fire process at the ith fire source position:

(6)k_i=1-\frac{{\mathrm{\Lambda }}_i(2400)}{\mathrm{\Lambda }(0)}.

The k_i values from the numerical analysis are summarized in Fig.23, and the following design suggestions are drawn.

1) If the designer can accept a reduction in the ultimate bearing capacity within 5%, then when f/L equals 1/5, Q is less than or equal to 8 MW, and the minimum distance between the fire source and the structure is greater than 20 m, the ultimate bearing capacity analysis under fire can be ignored.

2) If the designer can accept a reduction in ultimate bearing capacity within 10%, the ultimate bearing capacity analysis under fire can be ignored when the f/L equals 1/3 or 1/5, the fire power is less than or equal to 15 MW, and the minimum distance between the fire source and the structure is greater than 20 m; when f/L equals 1/4, the maximum Q should be limited to 8 MW.

3) When Q is greater than or equal to 25 MW, the ultimate bearing capacity of the structure under fire must be determined, because the reduction in the ultimate bearing capacity is significant.

5.2 Conditions of ignoring the radiative heat flux

Zhu et al. [13] highlighted that the radiative heat flux plays a more important role in large-space fires than compartment fires, yet the contribution of the radiative heat flux to the temperature development of structural components decays with an increase in the distance between the structural component and the fire source. Therefore, this section explores the critical condition that the radiative heat flux can be ignored and provides suggestions for architectural composition and simplification of the calculations. The basic model is the same as that described in Section 5.1, where f/L and H were fixed at 1/4 and 0, respectively. The influential parameters were varied based on the following scheme:

1) fire power Q: 2, 8, 25 MW;

2) radiation heat flux: considered, ignored;

3) fire source-structural component distance d_f: 1, 3, 5, 7 m;

4) fire location: below the roof, near the support.

The fire source-structural component distance d_f refers to the distance between the centroid of the fire source and structural components. The definitions of the fire location and d_f are shown in Fig.24.

Because the fire source is close to the structure, the empirical formula in Ref. [12] cannot be used to calculate the air temperature field. Hence, FDS models were established using the method proposed by Zhu et al. [10] to calculate the air temperature field. In addition, instead of the ultimate bearing capacity, the ultimate fire resistance time of the structure was used to evaluate the structural capacity of the structure under fire, and a uniformly distributed surface load of 1 kN/m² was considered. Here, we note that the fire-resistance time of the structure is defined as the moment when the stiffness matrix of the structure is singular, and no further load can be applied. The fire duration was set as 2400 s.

a) Fire under the roof

The numerical analysis results when the fire was under the roof are listed in Tab.2. In Tab.2, t_r and t_nr are the ultimate fire resistance times when the radiative heat flux is considered and ignored, respectively; e_t and e_T are the relative errors of the ultimate fire resistance time and maximum member temperature difference, respectively, defined as

(7)e_{\text{t}}=\frac{t_{\text{nr}}-t_{\text{r}}}{t_{\text{r}}}\times 100\mathrm{\%},

(8)e_{\text{T}}=\underset{t}{max}\left[\frac{T_{\text{nr}}(t)-T_{\text{r}}(t)}{T_{\text{r}}(t)}\right]\times 100\mathrm{\%},

where T_r(t) and T_nr(t) are the member temperatures at time t, when the radiative heat flux is considered and ignored, respectively. ΔT_max is the maximum member temperature difference during the entire fire process and is defined as

(9)\mathrm{\Delta }{T}_{max}=\underset{t}{max}\left[T_{\text{nr}}(t)-T_{\text{r}}(t)\right].

Notably, e_t was negative only when Q was 8 MW and d_f was 5 m, indicating that the ultimate fire resistance time was increased by considering the radiative heat flux. Fig.25 and Fig.26 plot the member temperature contours at the ultimate state of the structure when Q is 8 MW, and d_f is 1 m and 5 m, respectively. As shown in Fig.25, the peak temperature of the structure increased significantly when radiative heat flux was considered. In contrast, the peak temperature increased only slightly when radiative heat flux was considered as shown in Fig.26 and the member temperature field also changes. As the thermal expansion becomes more uniform for the state shown in Fig.26(a), t_r becomes larger than t_nr. Therefore, to ensure the accuracy of the structural fire analysis results, the influence of the radiative heat flux should be considered when calculating the temperature development of structural components.

Nonetheless, the relationship t_r ≤ t_nr holds true for other situations. Based on the data in Tab.2, it can be concluded that if the designer can accept a maximum of 10% of e_t, the radiative heat flux (when the fire is under the roof) can be ignored when d_f is greater than the critical value d_f,min,r.

(10)

b) Fire near the support

The numerical analysis results when the fire is near the support are listed in Tab.3, where the relationship t_r ≤ t_nr holds for all situations. By comparing Tab.2 and Tab.3, it can be concluded that when the fire was located near the support, the effect of increasing d_f on improving the ultimate fire resistance time of the structure was more significant than that when the fire was under the roof.

Similarly, according to the results in Tab.3, it can be concluded that if the designer can accept a maximum of 10% of e_t, the radiative heat flux (when the fire is near the support) can be ignored when d_f is greater than the critical value d_f,min,s.

(11)

When d_f was small, the adverse effect of the radiative heat flux significantly reduced the ultimate fire resistance time of the structure. Therefore, Eqs. (10) and (11) are referred to in order to limit the minimum values of d_f when conducting architectural composition to determine the fire scenarios.

6 Conclusions

Based on existing research findings on the high-temperature mechanical performance of aluminum alloy materials and structural components, this study conducted numerical analyses of AARS under fire conditions considering joint semi-rigidity. The main contributions of this study are as follows.

1) A numerical model of AARS under fire conditions considering joint semi-rigidity was established and verified against room-temperature and fire tests. The proposed model can also be applied to the thermal-structural coupled analysis of other structures with semi-rigid joints.

2) The reduction factor of the ultimate bearing capacity of K6 AARS under common fire conditions is related to the span, height-to-span ratio f/L, height of the supporting structure, and fire power Q, whereas it is independent of the number of element divisions, number of rings, span-to-thickness ratio, and support condition. The main mechanism of this phenomenon is that the failure of AARS is highly associated with the extra compressive force at the outmost ring induced by the ferrule effect, and the influential parameters will affect the relative stiffness of the outmost ring under fire conditions.

3) The reduction in the ultimate bearing capacity of the K6 AARS is rapid at the stable combustion stage of the fire duration and is slow at the initial and decay stages.

4) Practical design formulae were derived by machine learning via 324 thermal-structural coupled analysis results to serve the initial/concept design of K6 AARS.

5) Conditions for ignoring the ultimate bearing capacity of K6 AARS were proposed. Specifically, if the designer can accept a reduction in the ultimate bearing capacity within 5%, the conditions are f/L = 1/5, Q ≤ 8 MW, and the minimum distance between the fire source and the structure is greater than 20 m. If the designer can accept a reduction in the ultimate bearing capacity within 10%, the conditions are f/L = 1/3 or 1/5, Q ≤ 15 MW, and the minimum distance between the fire source and the structure is greater than 20 m; when f/L = 1/4, the maximum Q should be limited to 8 MW.

6) The conditions for ignoring the radiative heat flux were proposed by limiting the minimum value of the distance between the centroid of the fire source and the structural component d_f. It is noted that the limit values of d_f should be referred to in order to limit the minimum values of d_f when conducting architectural composition to determine the fire scenarios. This conclusion can be applied when designing AARSs with any structural form.

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