## 4 Calculation of Bearing Capacity Based on Yield Hinge Line Theory
### Verification  
Figures 27 and 28 show the relationship between the calculated carrying capacity of two-way slab specimens under fire and the temperature change of bottom reinforcement based on plate balance method and energy method, respectively. The traditional method in the figure is the yield hinge line theoretical formula, without considering the tensile membrane effect. Table 5 shows the comparison of limit carrying capacity of two-way slabs under fire. As can be seen from Figures 27 and 28, when the temperature of the bottom reinforcement exceeds 400 \({}^{\circ}\)C, the limit carrying capacity of the slab decreases rapidly. This is because the yield strength of the reinforcement will be rapidly reduced when temperature exceeds 400 \({}^{\circ}\)C [47]. According to the limit load-carrying capacity calculated by the formula in this paper, when the vertical out-of-plane mid-span deflection is lower than \(v_{0}\), the carrying capacity is constant. However, when the vertical out-of-plane mid-span deflection is higher than \(v_{0}\), the carrying capacity of the slab begins to increase because the membrane effect of the slab begins to appear. Figure 27: Curve of bearing capacity with temperature (plate balance method). (**a**) Slab CS1. (**b**) Slab CS2. According to the calculation based on the traditional plate balance method, the slabs CS1 and CS2 are damaged when the bottom reinforcement temperature is 572 \({}^{\circ}\)C and 603 \({}^{\circ}\)C; that is, the limit carrying capacity is lower than 5.00 kN/m\({}^{2}\), and the vertical out-of-plane mid-span deflections of each slab corresponding to the reinforcement temperature are span/28 and span/24, respectively, which has not exceeded the deflection limit of span/20 [48; 49; 50], and the slab still has a certain fire resistance. It is obvious that a traditional calculation method does not consider the improvement of the bearing capacity of the two-way slab membrane effect at high temperature, which leads to the conservative calculation results. According to the calculation formula proposed in this paper, the limit bearing capacity of the specimen when the fire turned off are 5.53 kN/m\({}^{2}\) and 5.31 kN/m\({}^{2}\), respectively, and the corresponding vertical out-of-plane mid-span deflections are span/22 and span/23, respectively, which is basically consistent with the test results. According to the calculation based on the traditional energy method, the slab CS1 and CS2 are damaged when the bottom reinforcement temperature reaches 657 \({}^{\circ}\)C and 656 \({}^{\circ}\)C and the vertical out-of-plane mid-span deflections of each slab corresponding to the reinforcement temperature is span/22 and span/24, respectively, which has not reached the failure criterion of the two-way slab (span/20). As can be seen, there are problems between the traditional energy method and the traditional plate balance method. The calculation results are still conservative, but the deviation degree is lower than that of the plate balance method. According to the calculation formula proposed in this paper, the limit load-carrying capacity of the slabs when the fire is turned off is 5.54 kN/m\({}^{2}\) and 5.40 kN/m\({}^{2}\), respectively, which is slightly larger than the calculated value of the plate balance method proposed in this paper, with a deviation of about 0.2-1.7%, which is basically consistent with the test results. Figure 29 shows the comparison of the calculation results of the carrying capacity of the two slabs obtained by Equation (15) when the vertical out-of-plane mid-span deflections is span/15, span/20, span/25, span/30, and span/35, respectively. The temperature of reinforcement at the bottom of the slab shall be taken as the test result. Calculation result of  
Figure 28: Curve of bearing capacity with temperature (energy method). (**a**) Slab CS1. (**b**) Slab CS2.