Numerical analysis and theoretical calculation of tensile membrane action region of RC simply supported slabs under fire

Tensile membrane action (TMA) could significantly enhance the ultimate bearing capacity of reinforced concrete (RC) simply supported slabs under fire conditions. To address this, this paper proposes an analytical method for determining the ultimate bearing capacity of RC slabs at elevated temperatures by explicitly accounting for the influence of TMA. In this method, the TMA region is reasonably defined via equations for an elliptical boundary, and the coefficient of increase in ultimate load-carrying capacity is derived. Through a combination of fire testing, numerical analysis, and theoretical calculations, the study investigated the TMA region and the increase in the ultimate bearing capacity, revealing that temperature changes have little effect on the geometry and size of the TMA region. Furthermore, upon comparing three failure criteria to validate the accuracy of calculations, it was found that a deflection limit of w = l/10 provided good agreement, thereby facilitating further analysis of the ultimate load-carrying capacity of simply supported slabs.

1 Introduction

Currently, worldwide research primarily focuses on the mechanical properties of simply supported slabs, continuous slabs, and composite floor systems under normal temperature conditions. Studies on the mechanical properties of simply supported slabs under fire conditions have mainly been limited to experimental research on fire resistance performance, with relatively few theoretical analysis and numerical simulations of the ultimate bearing capacity under corresponding fire conditions. Therefore, it is necessary to conduct numerical simulations and theoretical analysis of the mechanical properties of simply supported slabs under fire conditions.

In recent years, many experimental studies have been carried out on structural members, such as beams, slabs, columns, simply supported slabs, continuous slabs, and composite floor systems. In 2009, Omer et al. (Omer et al., 2010a; Omer et al., 2010b) proposed a method for calculating the ultimate bearing capacity of simply supported concrete slabs under fire. This method considered the bond-slip characteristics between the reinforcement and the concrete, and it assumed that reinforcement damage was the only type of damage. In 2015, Nguyen and Tan (2015) proposed a method for calculating the ultimate bearing capacity of the floor at elevated temperature. The theory considered the effect of the deformation of beams on the ultimate bearing capacity of the floor at elevated temperature. In 2016, Herraiz and Vogel (2016) proposed that the load-displacement relationship curve of concrete slab could be divided into three stages for derivation based on Bailey’s theory. This approach considered the bond-slip force between the reinforcement and the concrete while accounting for the damage criterion of reinforced concrete. It also established a theoretical model of the load-carrying capacity of the concrete slab. In 2017, Burgess (2017) identified two types of yield line damage modes based on the classical yield line theory, combined with the strength ratio of reinforcing bars per unit width in both directions. Burgess also proposed a gradual damage criterion for the reinforcing bars along the inclined yield line. The criterion obtained the load-deflection relationship curves for the ascending and descending phases. In 2019, Kang et al. (2020) proposed an analytical model for reinforced concrete beam-column sub-assemblages under compressive arch action and catenary action.

In addition to experiments on the load-bearing capacity of simply supported slabs at elevated temperature, numerous scholars have conducted extensive theoretical analysis on the ultimate load-bearing capacity of concrete two-way simply supported slabs under fire conditions. These primarily include the classical yield line theory (Shen et al., 1993), Bailey’s theory (Lennon et al., 1999; Bailey et al., 2000; Bailey and Moore, 2000a; Bailey and Moore, 2000b; Bailey, 2001; Bailey, 2003; Bailey, 2004; Bailey and Toh, 2007a; Bailey and Toh, 2007b; Bailey and Toh, 2008), Li’s theory (Li et al., 2017), and Dong’s theory (Dong, 2010; Dong and Fang, 2010), etc. Besides, (Wang et al., 2021a; Wang et al., 2021b; Wan et al., 2018) proposed the concrete bi-directional slab damage mode and an elliptic equation based on the plastic hinge line theory. They deduced the equilibrium equations of the post slab’s internal forces and established a method for calculating the post-disaster slab’s residual bearing capacity. The results showed that for the post-disaster slabs, the bending and punching theories should be used to analyze the residual bearing capacity respectively, and the minimum value of both can be used as the residual ultimate bearing capacity.

Under fire conditions, fully utilizing TMA can substantially increase the ultimate bearing capacity of simply supported slabs. Based on this research, this paper proposes an analytical method to calculate the ultimate bearing capacity of simply supported reinforced concrete slabs at elevated temperature, considering the effect of TMA. This analytical method reasonably determines the defining equations of the elliptic boundary in the TMA region and deduces the TMA increase coefficient through a series of force and moment balancing. The theoretical calculations are mutually verified through the experimental phenomena and simulation results, which are in a good agreement. The three cases of w = w_test, w = l/10 and w = l/20 are compared and analyzed respectively. The experimental study of ultimate bearing capacity of simply supported slab under fire in the literature (Bailey and Toh, 2007b; Lim and Wade, 2002; Wan et al., 2010) is analyzed obtain the influence laws of load levels on the mechanical properties of simply supported slab at elevated temperature. The properties include cracks, in-plane and out-of-plane displacements, region of the tensile membrane effect, ultimate loads and damage modes of simply supported slab.

2 Theoretical analysis

2.1 Basic assumptions

In this paper, a method for analyzing the ultimate bearing capacity of simply supported slabs at elevated temperature, which considers the influence of tensile membrane effects, is proposed. This method is used to solve for the region of tensile membrane effects under specific assumptions:

1. The end supports of the slabs are simply supported.

2. The aspect ratio of rectangular panels shall not exceed 3.

3. The large deformation state of the slab is approximated by an ellipsoidal surface.

2.2 Model parameters

The internal forces distribution of the simply supported slabs at elevated temperature are shown in Figure 1. Based on the classical plastic hinge theory (Shen et al., 1993) and equilibrium Equation 22, the location parameters (n, α, K and b) of the plastic hinge line and the internal forces can be obtained.

Figure 1

Figure 1. (a) Plate division diagram (b). Plates and internal forces in the concrete slab.

In Fig.1(b), L(l) is the length (width) of the slab; α is the angle defining the yield line pattern of slab panel; n is the factor defining the yield-line pattern; b is the membrane force parameter; K is the ratio of yield force per unit width of reinforcement bar in y direction to yield force per unit width of reinforcement bar in x direction; x_c is the width of the compressive membrane force at, EG, which is defined as (l/2 - imaginary axis).

The key internal forces are defined as follows based on temperature. C_1,θ is the resultant force in the compressed zone of plastic hinge at elevated temperature. C_2,θ is the resultant force in the compressed zone of concrete in the uncracked region. T_1,θ and T_2,θ is the resultant tensile forces of reinforcement in different segments at elevated temperature. S_θ is the in-plane shear force between slabs at elevated temperature, while T_0,θ is the yield force per unit width of reinforcement in the x-direction at temperature θ. The forces should be coupled with the results of the temperature field analysis for calculation.

Based on the internal force distribution at the inter-panel connections, shown in Figure 1b, the resulting distribution characteristics of tensile membrane actions in simply supported slabs under fire conditions reveals the existence of distinct points (Point I₁, Point I₂, Point I₃, Point I₄), shown in Figure 2, along the plastic hinge lines where the membrane force equals zero. These points satisfy the equilibrium conditions T_{θ =} C_θ, which the coordinates of Point I₁, designated as (x₀, y₀). And the red elliptic curve is established to be the demarcation line of the tension and compression membrane effects, with the points (Point I₁, Point I₂, Point I₃, Point I₄) being passed through. The region of the reinforcement mesh inside the curve is subjected to the tensile membrane effect, while the concrete at the edge of the slab outside the curve is subjected to the compression membrane effect.

Figure 2

Figure 2. Ellipse diagram of membrane action.

As shown in Figure 2, the center of the rectangular plate (Point F) is assumed to be the origin and the intersection points (Points B and C) of the yield line in the middle region are the two foci of the elliptic equation. Thus, the elliptical equation is established as shown in Equation 1:

$\sqrt{{\left(x_0-\left(\frac{L}{2}-nL\right)\right)}^2+{y_0}^2}+\sqrt{{\left(x_0+\left(\frac{L}{2}-nL\right)\right)}^2+{y_0}^2}=2\phi (1)$

Where: 2φ is the length of the real axis.

2.3 Internal force equations

2.3.1 Internal forces and bending moments at elevated temperature

Take plate ③ (half of plate ①), shown in Figure 1b, for force analysis, assuming that the compressive membrane force is triangularly distributed on slab edges (EG) and its maximum value is N.

According to the distribution of internal forces at each cross-section of plate ③ and the balance of internal forces, the equilibrium Equation 2 can be obtained as:

$C_{2,\mathrm{\theta }}=\frac{N_{\theta }{x}_c}{2}=K{T}_{0,\mathrm{\theta }}\left(\frac{l}{2}-x_c\right)+C_{1,\mathrm{\theta }}\cos \theta -T_{2,\mathrm{\theta }}\cos \theta -S_{\mathrm{\theta }}\mathit{sin}\theta (2)$

Where N is the membrane force at point E position.; In addition, the moments are obtained by solving for the point E in the rigid plate ③ as shown in Equation 3.

$T_{2,\mathrm{\theta }}\left[\left(\frac{\mathit{cos}\theta \times L}{2}-\frac{\frac{L}{2}-\alpha L}{\mathit{cos}\theta }\right)\frac{1}{\mathit{tan}\theta }-\frac{\sqrt{{\left(\alpha L\right)}^2+\frac{l^2}{4}}}{3\left(1+k\right)}\right]-K{T}_{0,\mathrm{\theta }}\left(\frac{l}{2}-x_c\right)\times \left(\frac{l}{4}+\frac{x_{\mathrm{c}}}{2}\right)+\frac{1}{3}{C}_{2,\mathrm{\theta }}{x}_{\mathrm{c}}-\frac{T_{1,\mathrm{\theta }}}{4}\left(\frac{L}{2}-\alpha L\right)+C_{1,\mathrm{\theta }}\left[\frac{\mathit{sin}\theta L}{2}-\frac{k\sqrt{{\left(\alpha L\right)}^2+\frac{l^2}{4}}}{3\left(l+k\right)}\right]+S_{\mathrm{\theta }}\frac{L}{2}\mathit{cos}\theta =0(3)$

2.3.2 Load-carrying capacity increase coefficient

At elevated temperature, the increase in the ultimate bearing capacity of two-way simply supported slabs is mainly reflected in two aspects: the increase coefficients due to membrane effect (e_1m,θ and e_2m,θ), as well as the increase coefficients resulting from the axial force action at the plastic hinge lines (e_{1b, θ} and e_{2b, θ}).

The concrete slab cross-section bending moment resistance moments M₀₁ and M₀₂, without considering the influence of tensile membrane effect, are respectively shown in Equations 4–6.

$M_{01,\mathrm{\theta }}=T_{0,\mathrm{\theta }}{d}_1\left(\frac{3+g_{1,\mathrm{\theta }}}{4}\right)(4)$

$M_{02,\mathrm{\theta }}=K{T}_{0,\mathrm{\theta }}{d}_2\left(\frac{3+g_{2,\mathrm{\theta }}}{4}\right)(5)$

$g_{1,\mathrm{\theta }}=\left[\frac{d_1}{2}-\frac{T_{0,\mathrm{\theta }}}{f_{\text{cu},\mathrm{\theta }}}\right]/\frac{d_1}{2}{g}_{2,\theta }=\left[\frac{d_1}{2}-\frac{T_{0,\mathrm{\theta }}}{f_{\text{cu},\mathrm{\theta }}}\right]/\frac{d_2}{2}(6)$

Where d₁ and d₂ are the effective heights of the double-layer reinforcement at the bottom of the slab, respectively; f_cu,θ is the concrete cube compressive strength following elevated temperature reduction.

As shown in Figure 3, the concrete slab is assumed to have a deflection of w in the span of the slab at the limit state:

Figure 3

Figure 3. Internal forces on the plate. (A) Plate ①. (B) Plate ②.

Taking the membrane forces in Plate ① and ②, the moments for the deflections at the supports of each slab are obtained, which in turn give M_1m,θ and M_2m,θ. Based on the bending resistance moment without the effect of the membrane action M_01,θ and M_02,θ, the increase coefficients e_1m,θ and e_2m,θ can be obtained as shown in Equations 7, 8:

$e_{1\mathrm{m},\theta }=\frac{M_{1\mathrm{m},\theta }}{M_{01,\theta }L}=\frac{4Kb}{3+g_{1,\theta }}\left(\frac{\omega }{d_1}\right)\left(1-2\alpha +\frac{\alpha \left(2-k\right)}{3}-\frac{l^2\left(1-2\alpha \right)}{8\left[{\left(\alpha L\right)}^2+{\left(\frac{l}{2}\right)}^2\right]}\right)(7)$

$e_{2\mathrm{m},\theta }=\frac{M_{2\mathrm{m},\theta }}{M_{02,\theta }l}=\frac{4b}{3+g_{2,\theta }}\left(\frac{\omega }{d_2}\right)\left(\frac{2-k}{6}+\frac{\alpha L^2\left(1-2\alpha \right)}{4\left[{\left(\alpha L\right)}^2+{\left(\frac{l}{2}\right)}^2\right]}\right)(8)$

The bearing capacity of the plastic hinge line of a concrete slab under the action of an axial force is calculated by the following formula (Equations 9, 10)

$\frac{M}{M_0}=1+{\alpha }_0\left(\frac{N}{T_0}\right)-{\beta }_0{\left(\frac{N}{T_0}\right)}^2(9)$

${\alpha }_0=\frac{2\times g_0}{3+g_0},{\beta }_0=\frac{1-g_0}{3+g_0}(10)$

Where: g₀ is the proportion of concrete compressive stress region.

For plate ①, on the AB side, located at a point projected on the x-axis at a distance x from point B, then N_x,θ is obtained as shown in Equation 11:

$N_{\mathrm{x},\theta }=bK{T}_{0,\theta }\left(\frac{x\left(k+1\right)}{\alpha L}-1\right)(11)$

Substituting into Equation 9 and integrating yields Equation 12:

$2{\displaystyle \underset{0}{\overset{\alpha L}{\int }}}\frac{M}{M_0}dx=2\alpha L\left[1+\frac{{\alpha }_{1,\theta }b}{2}\left(k-1\right)-\frac{1}{3}{\beta }_{1,\theta }{b}^2\left(k^2-k+1\right)\right]=Z_{1,\theta }(12)$

For plate ① BC section, N_θ = -bKT_0,θ, which Equation 13 can be obtained:

${\displaystyle \underset{0}{\overset{L-2\alpha L}{\int }}}\frac{M}{M_0}dx=\left(L-2\alpha L\right)\left(1-{\alpha }_{1,\theta }b-{\beta }_{1,\theta }{b}^2\right)=Z_{2,\theta }(13)$

For the plate ① GF section, assume that the concrete of GF section cracks and all the reinforcement of GF section yields. According to N_θ = -KT_0,θ, it Equation 14 can be obtained:

$2{\displaystyle \underset{0}{\overset{\frac{l}{2}-x_c}{\int }}}\frac{M}{M_0}dy=2\left(\frac{l}{2}-x_c\right)\left(1-K{\alpha }_{2,\theta }-{\beta }_{2,\theta }{K}^2\right)=Z_{3,\theta }(14)$

Ultimately, according to Equations 12–15 can be shown:

$e_{1\mathrm{b},\theta }=\frac{Z_{1,\theta }}{L}+\frac{Z_{2,\theta }}{L}+\frac{Z_{3,\theta }}{l}(15)$

For plate ②, on the AB side, located at a point projected on the y-axis at a distance y from point B, then N_y,θ is obtained as shown in Equation 16:

$N_{y,\theta }=bK{T}_{0,\theta }\left(\frac{2y\left(k+1\right)}{l}-1\right)(16)$

Substituting into Equation 9 and integrating yields Equation 17:

$2{\displaystyle {\int }_0^{\frac{l}{2}}}\frac{M}{M_0}dy=l\left[1+\frac{{\alpha }_{2,\theta }bK}{2}\left(k-1\right)-\frac{{\beta }_{2,\theta }{b}^2{K}^2}{3}\left(k^2-k+1\right)\right]=Y_{\theta }(17)$

Then the increase coefficients e_2b,θ is obtained as shown in Equation 18:

$e_{2\mathrm{b},\theta }=\frac{Y_{\theta }}{l}=1+\frac{{\alpha }_{2,\theta }bK}{2}\left(k-1\right)-\frac{{\beta }_{2,\theta }{b}^2{K}^2}{3}\left(k^2-k+1\right)(18)$

2.3.3 Limit load capacity

According to the classical plastic hinge line theory (Huang et al., 2004), the yield load of a simply supported concrete slab at high temperatures is found to be as shown in Equation 19:

$P_{y,\theta }=\frac{24\mu M_0}{l^2}{\left[\sqrt{3+\frac{1}{\mu r^2}}-\frac{1}{\sqrt{\mu }r}\right]}^{-2}(19)$

The coefficients of increase in carrying capacity are e_1,θ and e_2,θ for plate ① and plate ②, respectively as shown in Equations 20, 21:

$e_{1,\theta }=e_{1\mathrm{m},\theta }+e_{1\mathrm{b},\theta }(20)$

$e_{2,\theta }=e_{2\mathrm{m},\theta }+e_{2\mathrm{b},\theta }(21)$

The ultimate bearing capacity P_limit,θ at high temperature can be obtained as shown in Equation 22:

$P_{\text{limit},\theta }=e_{\theta }\times P_{y,\theta }(22)$

3 Test validation

3.1 Introduction to the experiment

As shown in Table 1, the experimental data of 27 reinforced concrete slabs subjected to high-temperature loading tests, as documented in references (Bailey and Toh, 2007b; Lim and Wade, 2002; Wan et al., 2010), thereby providing a basis for theoretical validation. The slabs were square or rectangular and restrained on all four sides. The slabs’ thickness increased from 0.018 to 0.12 m. The compressive strength of the concrete is about 40 MPa, while the reinforcement bars with different diameters, yield strengths, ultimate strengths and spacing were involved. The tests were performed in a controlled furnace environment (Parametric Fire and ISO 834), which is shown below in Figure 4.

Table 1

Table 1. Details of RC slab tests at elevated temperature.

Figure 4

Figure 4. The fire curves for different tests.

As the test results show, damage to the specimens is usually caused by breaking the reinforcement bars, which forms full-depth cracks on shorter spans of rectangular plates and on one span of square plates. Using reinforcement breakage as the damage mode, the temperature range of the reinforcement at the time of specimen damage was 557 °C–898 °C. The material parameters of the slabs are determined by the weighted average temperature field method. For the compression strength of concrete, the influence of temperature distribution at both the top and bottom of the slab is considered through the analysis of the temperature gradient across the slab section. The strength of the reinforcement is determined based on the recorded temperature of the lower-layer steel bars.

3.2 Numerical analysis

3.2.1 Numerical model

Vulcan is a finite element analysis (FEA) program, which is developed specifically for the analysis of building performance in fire conditions (Huang et al., 2004; Huang et al., 2009). The program was used to simulate the temperature field and analyze the deformation behavior and mechanism of the bi-directional test slab.

The slab unit is a nine-node Gaussian unit that considers geometric non-linearities. The total number of model units is 96, as shown in Figure 5. The equivalent strength of the post-disaster concrete is calculated using the equivalent layering method from the literature (Wang and He, 2016), in conjunction with the maximum temperature experienced by the concrete. The strength under fire was used for the concrete and reinforcement under the assumption that they were well bonded.

Figure 5

Figure 5. Numerical analysis model of specimen.

For the temperature field analysis, the slab is divided into 10 layers. The values of the concrete surface radiation coefficient, flame radiation coefficient, and surface absorption coefficient are 0.9, 0.75, and 1.0, respectively.

For the structural analysis, the plate is divided into layers of non-uniform meshes along its thickness. For example, the node coordinates from the bottom to the top of the plate for MF1 are 0, 2.5, 5.09, 5.18, 12.35, 14.85, 17.35, and 19.7 mm. The plate is discretized using 96 nine-node Gaussian elements arranged in an 8 × 12 grid. Unless otherwise specified, the thermal and mechanical properties of high-temperature materials are modeled using the EC2 model (EN, 1992-1-1, 2004).

The main calculation steps in the simulation process are as follows:

1. The concrete is divided into multiple regions along the thickness direction.

2. The strength of the concrete in each region is considered constant.

3. According to the maximum temperature experienced by the concrete in each region, the reduction coefficient of each region is weighted to obtain the total reduction coefficient of the concrete’s strength in the entire cross-section.

4. The bond-slip effect between the reinforcement and the concrete is ignored.

3.2.2 Numerical results

Figure 6 shows the comparison between the tensile membrane effect regions obtained from numerical simulation of simply supported slabs (MF1∼MF6) under fire in literature (Bailey and Toh, 2007b) and the experimental results. The red ellipse indicates the calculated TMA region. The blue line represents the compressive tensile membrane effect, and the red line represents the tensile membrane effect. It indicates that the calculated elliptical boundary and crack distribution patterns are in good agreement with the experimental results.

Figure 6

Figure 6. Comparison of numerical simulation and experimental results under the limit state of simply supported plates MF1∼MF6. (A) MF1. (B) MF2. (C) MF3. (D) MF4. (E) MF5. (F) MF6.

For slabs 661, D147, and 4ES-2, the simulation results at 120 min are shown in Figure 7. The tensile membrane effect (red region) is most pronounced at the center of the plate. Tensile strength decreases from the center towards edges. The compressive membrane effect (blue region) exhibits a ring-shaped distribution along the edges of the plate, with maximum compressive stress occurring near endpoints of the short axis. The elliptical boundary derived from the theoretical calculations shows a high-level agreement with the simulation results, which validating the applicability of the elliptical equations.

Figure 7

Figure 7. Theoretical and Simulation results for slabs 661 (Left), D147 (Middle) and 4ES-2 (Right).

3.3 Comparative analysis of extreme loads

The classical yield line theory and the analysis method in this paper are used to calculate the ultimate bearing capacity of simply supported slabs at elevated temperature. The calculation results are shown in Table 2. The material properties for concrete and steel reinforcement at high temperatures are based on Eurocode (EN, 1992-1-1, 2004), while the properties for stainless steel are based on literature (Wu et al., 2022).

Table 2

Table 2. Measured and calculated ultimate loads of concrete slabs.

Different sources of evidence have been integrated in determining the failure criteria. The experimental criterion directly adopted the maximum deflection recorded in the tests (w = w_test). The fire condition criterion (w = l/10) was derived from studies by Bailey and Toh (Bailey and Toh, 2007b) and Jiang et al. (Jiang et al., 2022), capturing the deformation behavior of slabs under realistic fire conditions. The code-based criterion (w = l/10) adhered to the Code for Design of Concrete Structures (GB50010-2010) (GB50010-2010, 2015), reflecting conservative considerations in engineering design. Based on the experiments from previous research, the three damage criteria are analyzed respectively.

As shown in Table 2, using classical yield line theory, the P_limit,θ/P_test ratio ranges from 0.23 to 0.90 with a mean of 0.42, standard deviation of 0.165, and coefficient of variation (COV) of 39.08%, indicating considerable underestimation and high variability. For the mid-span displacement test value (w = w_test), the ratio ranges from 0.53 to 1.71 with a mean of 0.93, standard deviation of 0.302. and COV of 32.49, indicating improved accuracy and lower dispersion. For the criterion w = l/10, the ratio ranges from 0.51 to 1.99 with a mean of 0.96, standard deviation of 0.359, and COV of 37.34%, indicating the closest mean value but moderate scatter. For the criterion w = l/20, the ratio ranges from 0.40 to 1.54 with a mean of 0.78, standard deviation of 0.261, and COV of 33.50%, indicating a relatively conservative and stable option.

Through theoretical calculation and experimental validation, the ultimate load-bearing capacity of simply supported slabs at elevated temperature can be analyzed using the following three failure criteria, each demonstrating specific characteristics in terms of accuracy and reliability. The measure displacement criterion (w = w_test) is found to be stable but conservative predictions. The failure criterion (w = l/10) achieves the closest agreement with experimental data, representing an optimal balance between accuracy and practical applicability. In contrast, the normative reference criterion (w = l/20) is demonstrated to a better stability compared to the classical yield line theory but remains conservative, potentially underestimating the contribution of tensile membrane action to structural capacity as discussed in references (Cui et al., 2022; Burgess and Chan, 2020). The comparative performance of different analytical methods and failure criteria is shown in Figure 8.

Figure 8

Figure 8. Comparison between the test load and calculated values.

As shown in Figure 8, the yield line load-to-test load ratio is essentially below the rightmost dashed line. This indicates that the yield line theory greatly underestimates the ultimate load-carrying capacity of simply supported slabs at high temperatures. The failure criterion (w = l/10) provides the most rational approach for evaluating the ultimate load-bearing capacity of simply supported slabs under fire conditions, effectively integrating theoretical predictions with experimental observations while maintaining structural safety requirements.

3.4 Effect of maximum test temperature on ultimate load capacity

The experimental measurements at the failure point showed temperatures of 700 °C for the reinforcement, 770 °C at the top of the concrete slab, and 580 °C at the bottom. Consequently, the average temperature across the slab section was calculated to be 675 °C. Based on a measured temperature ratio of 0.96 between concrete and steel reinforcement, the degradation of material properties for ultimate load capacity calculation is determined.

Take the specimen details of MF1 as an example to analyze the corresponding values of ultimate load under different reinforcement temperatures, and the calculation results are shown in Figure 9. The temperature change has little effect on the geometry and size of the TMA region (Hqi et al., 2024), and the ultimate load capacity of the plate is not significantly weakened when the temperature of the steel reinforcement does not exceed 400 °C (Bailey and Toh, 2007b). After the temperature exceeds 400 °C, the ultimate load decreases rapidly, owing to the deterioration of the yield strength of the steel and compression strength of the concrete. Thus, the temperature increase exhibits a pronounced detrimental effect on the ultimate bearing capacity. A temperature of 400 °C is recommended to be adopted for fire protection design. Additional reinforcement should be provided in critical regions, such as fire-exposed surfaces and beam-column joints, to compensate for the strength loss at elevated temperature.

Figure 9

Figure 9. Temperature-ultimate load curve of steel.

4 Conclusion

This paper proposes a calculation method for determining the ultimate bearing capacity of simply supported slabs at elevated temperature, which accounting for the effects of TMA. The Vulcan simulation program is used to validate the method with the experiment data from the former research. The three different failure criteria have also been evaluated to improve computational accuracy, thereby enhancing the theoretical framework available for the fire protection design.

Calculation theory for the ultimate bearing capacity of simply supported slabs at elevated temperatures. The main conclusions are as follows:

1. An analytical method proposed in the article has been validated to have a good agreement with the representative examples from the former research.

2. The fire condition criterion (w = l/10) has been demonstrated to be the most reasonable, exhibiting the closet fit to test data and effectively balancing safety with practical capacity. This criterion is recommended for conventional fire resistance design.

3. A rapid decline in the ultimate load occurs once the reinforcement temperature exceeds 400 °C due to the degradation of steel yield strength and concrete compressive strength. It is recommended to configure additional reinforcement in critical areas to compensate for strength loss at elevated temperature.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

Author contributions

YJ: Writing – review and editing, Methodology, Conceptualization. QY: Formal Analysis, Writing – review and editing, Writing – original draft, Methodology, Investigation. SW: Investigation, Methodology, Writing – original draft. DY: Writing – original draft, Software, Investigation, Visualization. FT: Investigation, Writing – review and editing, Software. YY: Investigation, Software, Writing – review and editing. GG: Resources, Writing – review and editing, Investigation.

Funding

The author(s) declared that financial support was not received for this work and/or its publication.

Conflict of interest

Author QY was employed by China Construction Eighth Engineering Division Co., Ltd.

The remaining author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Generative AI statement

The author(s) declared that generative AI was not used in the creation of this manuscript.

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