As discussed in Ref. (Chen, 2022), five beam-slab specimens (4900 mm × 2400 mm × 70 mm) with different reinforcement ratios and layouts were tested. Three specimens (named Specimens S1 to S3) were subjected to the same ISO834 fire scenarios (3 h). For Specimens S1 to S3, each interior beam L7 was directly heated during each test, and then the burners were shut-off. The residual strength tests (named Slabs S1-PF to S3-PF) were then conducted. In addition, two unheated specimens (Specimens S4 and S5) were the reference specimens. The details of each specimen are presented in Figure 1A–D. Other details of the fire tests are available in Ref. (Chen, 2022).

After the fire tests, the residual load-carrying capacities of Specimens S1-PF to S3-PF, as well as two reference Specimens S4 and S5, were investigated. The vertical mid-span deflection of each panel (Points V-A and V-B) was measured, as shown in Figure 2A,B. The material properties and the experienced maximum temperatures of the specimens, as shown in Table 1.

Figure 3A–E show the load-mid-span deflection curve of each panel in five specimens (Chen, 2022). On one hand, the limit loads of Specimens S1-PF, S2-PF, S3-PF, S4 and S5 were 130, 120, 70, 140 and 140 kN, respectively. In addition, for the fire-damaged specimens (the reference specimens), the limit deflection of these panels ranged from 70.9 mm (108.7) to 105.4 mm (160.5), with the average value of 94.3 mm (131.5). On the other hand, for the panels, the flexural failure (concrete crushing at the corners) and local punching shear failure appeared, and for the beams, the flexural-shear cracks appeared near to the connection as well as the concrete crushing at the column end. In all, for the fire-damaged specimens, they still had higher limit carrying capacities and ductility, and the specimens with higher reinforcement ratio have larger residual carrying capacities.

In this paper, the basic assumptions are as follows:

1) According to the observed cracking patterns of the test specimens (Chen, 2022), the yield line failure model was assumed (Shen et al., 1993) in this paper, as shown in Figure 4A. The position parameters ξ、ξ′ and η do not change with the vertical deformation of the specimens.

As shown in Figure 7A–D, the boundary conditions of one beam include several cases, including two ends fixed, one end fixed and one end simply supported and two sides simply supported. In addition, the load type included two cases, including the trapezoidal load type and the triangular load type, and the details can be found in Ref. (Chen, 2022).

According to Ref. (Chinese National Standard, 2015), the short-term stiffness B _s of reinforced concrete members is B s = E s A s h 0 2 1.15 ψ + 0.2 + 6 a E ρ 1 + 3.5 r ′ f (10) where E _s is the elastic modulus of reinforcement; E _c is elastic modulus of concrete; A _s is the cross-section area of longitudinal reinforcement in tension zone; h ₀ is the effective height of the beam; ψ is the parameter. a _E = E _s/E _c; ρ is the reinforcement ratio; r _f ′ is the ratio.

The stiffness reduction coefficient of the beam is calculated as (Shen, 1993): α 0 = { 1 M 0 ≤ 0.3 M u 0.4 + 0.875 ( 1 − M 0 / M u ) 0.3 M u ≤ M 0 ≤ M u (11) where M _u is the ultimate bending moment; M ₀ is the bending moment.

According the stiffness reduction coefficient α ₀, the short-term stiffness B _s´is: B s ′ = α 0 B s (12)

For different boundary conditions, the mid-span deflection of the beam is:

(1) Two sides simply supported

As shown in Figure 7A, under the trapezoidal load, the mid-span deformation of the beam is: w max ( a ) = q 1 B s ( − 1 48 n 2 L 1 4 + 5 384 L 1 4 + 1 120 n 4 L 1 4 ) ,   q 1 = q l / 2 ,   n = 1 2 − l 2 L 1 (13) where w _max(a) is the maximum displacement of the mid-span, q is the ultimate load of the slab; q ₁ is the load transferred from the slab to the long side of the beam (Figure 7A). n is the parameter.

As shown in Figure 7B, under the triangular load, the mid-span deformation of the beam is: w max ( b ) = q 2 l 1 4 120 B s ,   q 2 = q n L (14) where w _max(b) is the maximum displacement of the mid-span, q is the ultimate load of the slab; q ₂ is the load transferred from the slab to the short side of the beam (Figure 7B).

(2) One side fixed and one side simply supported

The cross-section bending moment is defined as: M c 1 = q 2 [ ( L − n L ) ( L 2 + x ) − 1 3 n L ( L 2 + x ) 3 − L 2 4 ( 1 − 2 n 2 + n 3 ) ( 1 2 − x L ) ] , − L 2 ≤ x ≤ − ( L 2 − n L ) (15)

The rotation angle and the deflection of the beam can be expressed as: d 2 w d x 2 = M B s (16) d w c 1 d x = ∫ M c 1 B s d x = q 2 B s [ 1 2 ( L − n L ) ( L 2 + x ) 2 − 1 12 n L ( L 2 + x ) 4 + L 3 8 ( 1 − 2 n 2 + n 3 ) ( 1 2 − x L ) 2 ] + C c 1 w c 1 = ∫ d w c 1 d x d x = q 2 B s [ 1 6 ( L − n L ) ( L 2 + x ) 3 − 1 60 n L ( L 2 + x ) 5 − L 4 24 ( 1 − 2 n 2 + n 3 ) ( 1 2 − x L ) 3 ] + C c 1 x + D c 1 (17)

The bending moment is defined as: M c 2 = q 2 [ ( L − n L ) ( L 2 − x ) − 1 3 n L ( L 2 − x ) 3 − L 2 4 ( 1 − 2 n 2 + n 3 ) ( 1 2 − x L ) ] ,   L 2 − n L ≤ x ≤ L 2 (18)

Similarly, the rotation angle and the deflection of the beam can be expressed as: d w c 2 d x = ∫ M c 2 B s d x = q 2 B s [ − 1 2 ( L − n L ) ( L 2 − x ) 2 + 1 12 n L ( L 2 − x ) 4 + L 3 8 ( 1 − 2 n 2 + n 3 ) ( 1 2 − x L ) 2 ] + C c 2 (19) w c 2 = ∫ d w c 2 d x d x = q 2 B s [ − 1 6 ( L − n L ) ( L 2 − x ) 3 + 1 60 n L ( L 2 − x ) 5 − L 4 24 ( 1 − 2 n 2 + n 3 ) ( 1 2 − x L ) 3 ] + C c 2 x + D c 2 (20)

The bending moment is defined as: M c 3 = q 2 [ − x 2 + L 4 4 − 1 3 n 2 L 2 − L 2 4 ( 1 − 2 n 2 + n 3 ) ( 1 2 − x L ) ] ,   − ( L 2 − n L ) ≤ x ≤ L 2 − n L (21)

Similarly, the rotation angle and the deflection can be expressed as: d w c 3 d x = ∫ M c 3 B s d x   = q 2 B s [ − 1 3 x 3 + ( L 4 4 − 1 3 n 2 L 2 ) x + L 3 8 ( 1 − 2 n 2 + n 3 ) ( 1 2 − x L ) 2 ] + C c 3 (22) w c 3 = ∫ d w c 3 d x d x = q 2 B s [ − 1 12 x 4 + 1 2 ( L 4 4 − 1 3 n 2 L 2 ) x 2 − L 4 24 ( 1 − 2 n 2 + n 3 ) ( 1 2 − x L ) 3 ] + C c 3 x + D c 3 (23)

The boundary conditions are: x = L 2 ,   w c 2 = 0 x = − L 2 ,   w c 1 = 0 x = − ( L 2 − n L ) ,   w c 1 = w c 3   d w c 1 d x = d w 3 d x x = L 2 − n L ,   w c 2 = w c 3   d w c 2 d x = d w 3 d x

According to the above boundary conditions, the mid-span deflection of the beam is defined as: w max ( c ) = q 1 B s ( 5 384 L 1 4 − 1 48 n 2 L 1 4 + 1 120 n 4 L 1 4 ) + x 1 c 16 B s L 1 2 ,   x 1 c = − q 1 L 1 2 8 ( 1 − 2 n 2 + n 3 ) (24)

As shown in Figure 7D, under the triangular load, the mid-span deflection of the beam is: w max ( d ) = 1 120 B s q 2 l 1 4 + 1 16 B s x 1 d l 1 2 ,   x 1 d = − 5 q 2 l 1 2 64 (25)

According to the principle of virtual work, the external work of the specimen is: W = q l 2 12 [ 6 λ − ( ξ + ξ ′ ) ] + 1 3 q ( h 1 S 1 + h 2 S 2 + h 3 S 3 + h 4 S 4 ) (26) λ = L / l ;   h 1 = x 1 η l η 2 l 2 + 4 ;   h 2 = x 2 ( 2 − η ) l ( 2 − η ) 2 l 2 + 4 ;   h 3 = y 1 ξ l ξ 2 l 2 + 4 ;   h 4 = y 2 ξ ′ l ξ ′ 2 l 2 + 4 S 1 = [ L − l 4 ( ξ + ξ ′ ) ] ( η l / 2 ) 2 + 1 ;           S 2 = [ L − l 4 ( ξ + ξ ′ ) ] [ ( 2 − η ) l / 2 ] 2 + 1 ;   S 3 = 1 2 l ( ξ l / 2 ) 2 + 1 ;   S 4 = 1 2 l ( ξ ′ l / 2 ) 2 + 1 where x ₁、x ₂、y ₁ and y ₂ are the ratio between the beam’s ultimate displacement w (w _max(a)∼w _max(f)) and the slab’s ultimate displacement (l/20); η, ξ and ξ′ are the position parameters.

The internal work done by each slab yield line is defined as D = [ ( 1 + β 1 ) ( 1 − x 1 ) 2 λ η + ( 1 + β 1 ′ ) ( 1 − x 2 ) 2 λ 2 − η + ( 1 + β 2 ) ( 1 − y 1 ) 2 k ξ + ( 1 + β 2 ′ ) ( 1 − y 2 ) 2 k ξ ′ ] M y ,   k = M x / M y ;   β 1 = α y M y ′ / M y ; β 1 ′ = α y M y ′ ′ / M y ; β 2 = α x M x ′ / M x ; β 2 ′ = α x M x ′ ′ / M x (27)

If D = W, we have q = 24 M y A 1 A 2 (28) A 1 = ( 1 + β 1 ) ( 1 − x 1 ) λ η + ( 1 + β 1 ′ ) ( 1 − x 2 ) λ 2 − η + ( 1 + β 2 ) ( 1 − y 1 ) k ξ + ( 1 + β 2 ′ ) ( 1 − y 2 ) k ξ ′ ,   A 2 = [ 6 λ − ( ξ + ξ ′ ) ] l 2 + 4 ( h 1 S 1 + h 2 S 2 + h 3 S 3 + h 4 S 4 )

According to the above equations, the calculation process of the model is as follows:

(1) Firstly, it is assumed that all beams are not damaged.

The vertical deformation of the beams is not considered, i.e., x ₁, x ₂, y ₁ and y ₂ are 0. Only the torsional stiffness effect of the edge beam is considered, and the torsional stiffness effect of the interior beam is not considered.

The torsional stiffness (k’ _x and k’ _y ) of the edge beam is calculated as well as the ratio (α _x and α _y) . Then, according to Eq. 27, the ratio (β ₁, β′ ₁, β ₂ and β′ ₂) is calculated. The ultimate load of the slab (q) can be obtained by Eq. 28.

(2) Secondly, it is assumed that all beams are not damaged.

The vertical deformation of all beams are not 0. In other words, the ratios of the mid-span ultimate displacement of the beam (w (w _max(a)∼w _max(d))) and l/20 (x ₁、x ₂、y ₁、y ₂), and repeat Step (1), the ultimate load of the slab (q ₁) is calculated by Eq. 28.

(3) Thirdly, it is assumed that all beams are damaged.

The torsional stiffness of the edge beam is considered, and the parameters (k’ _x and k’ _y ) are calculated as well as the bending moment ratios (α _x and α _y ). Next, according to Eq. 27, the ratio (β ₁、β′ ₁、β ₂、β′ ₂) is calculated. Based on the three methods discussed in the literature (Xing, 1993; Huang et al., 2013), the minimum value of the limit load (q´) is obtained.

(4) If q ₁ is less than q´, the beam will not fail, only the panel fails. According to Step (2), recalculate the ultimate load (q ₂) of the panel. Note that, q in step 2 is replaced with q ₁.

The flow chart of the present method is shown in Figure 8.

Table 3 shows the calculated and experimental values of the ultimate load of concrete slabs calculated by the three methods, including the yield-line method (four edges with simply supported), the yield-line method (four edges with fixed-end) and the present methods. On one hand, for two yield-line methods, the ratios (P _s /P _test and P _f /P _test ) are 0.835 and 1.629, respectively. The reason is that the effect of the boundary condition was underestimated or overestimated. On the other hand, for the present method, the ratio (P _sp /P _test ) is 1.029.