Residual Shear Capacity of Post-Fire RC Beams under Indirect Loading

Abstract

Building fire is one of the most frequent disasters. The mechanical properties of concrete and steel will deteriorate to different degrees under fire exposure, thus weakening the bearing capacity of reinforced concrete (RC) members. Therefore, it is of great theoretical and practical significance to carry out research on the bearing capacity of RC members subjected to fire. To study the residual shear capacity of post-fire RC beams under indirect loading, static load tests were carried out on six full-scale RC beams to investigate the development of diagonal cracks and failure modes under an indirect load. Numerical models were established with ABAQUS to study the effect of the shear span-to-depth ratio and stirrup ratio on the residual shear capacity of the tested beams after fire exposure, and the shear capacity calculation method of indirectly loaded beams after high temperature is proposed. The results indicate that although the fire had no obvious affect on the failure mode of the beam under indirect loading, the fire aggravated the failure degree of the beams without additional transverse reinforcement. Moreover, the ultimate bending capacity of beams with additional transverse reinforcement decreased obviously after fire. The residual shear capacity of indirectly-loaded beams subjected to fire decreased compared with the directly-loaded beams due to the variation in the shear span-to-depth ratio and stirrup ratio. The shear capacity theoretical calculation results of the indirectly-loaded beam after high temperature were in good agreement with the tested results, and the average absolute error was 3.88%. Therefore, this calculation method for an indirectly loaded beam with a shear span-to-depth ratio less than 3.0 after fire as proposed in this paper was effective. The findings of this study are expected to provide a reference for the improved fire-resistant design of RC structures under indirect loading.

1. Introduction

For the calculation of the shear capacity of reinforced concrete (RC) beams, the compilation of the concrete structure design code [1] is mainly based on the test results of ordinary RC beams under direct loading. However, in general cast-in-place concrete structures, the load is not directly loaded on the main beam, but indirectly loaded through the secondary beam or slab, so the loading mode in the actual structure is different from the direct loading mode commonly used in the test, as shown in Figure 1. Therefore, the shear failure mechanism of indirectly-loaded RC beams via secondary beams is obviously different from that of RC direct-loaded beams, causing the shear capacity of the primary beam to be considerably lower [2,3,4,5,6,7,8,9,10,11]. In order to solve this problem of the shear bearing capacity, the structural design of RC beams under indirect loading usually requires additional transverse reinforcement to strengthen the connection between the primary and secondary beams. However, the mechanical properties of concrete and steel will deteriorate to different degrees under fire exposure, thus weakening the bearing capacity of reinforced concrete (RC) members. Therefore, the occurrence of fire will endanger the safety of reinforced concrete (RC) buildings [12,13,14,15,16,17].

At present, the research on the shear capacity of fire-damaged RC beams with indirect loading is insufficient, and is mainly focused on beams under indirect loading at ambient temperatures. A group of different loading positions for reactive powder concrete (RPC) beams was tested by Cao et al. [9], and the influences of different loading on cracking, mid-span deflection, shear capacity, and failure patterns were analyzed. The results show that the development trend of the cracks in the RPC beams with upper cross beam was similar to the trend with equal height cross beam, and the tensile damage occurred in main RPC beams with the lower cross beams, which decreased the ultimate bearing capacity by about 25% in comparison with the capacity under direct loading. Zhou et al. [10] tested three RC beams with 500-MPa high-strength stirrups to analyze their structural behavior and failure modes under direct and indirect loading. It was found that the shear performance of high-strength concrete beams under indirect loading was lower than that of the beams under direct loading. At the same time, the calculation formula of the shear capacity in the Chinese design code [1] does not consider the difference between the shear capacity of directly and indirectly loaded beams. For indirectly loaded beams, only the provision of additional transverse reinforcement at the concentrated force is made. Therefore, it is necessary to study the failure mechanism and residual shear capacity of indirectly loaded RC beams after fire.

In this paper, aiming to consider the common fire safety scientific problem of indirectly loaded beams, a research method combining experimental research, theoretical analysis, and numerical analysis was adopted to study the shear performance of post-fire beams with indirect loads. Firstly, the experimental research of six full-scale beams with indirect loading was carried out to study the shear performance with the help of a large fire simulation test system. The additional transverse reinforcement and load ratio were selected as critical parameters to investigate the failure characteristics and residual shear capacity of the indirectly loaded beams. Secondly, based on Abaqus 6.5 finite element analysis software [18], the numerical analysis method of indirectly loaded beams after fire was established, and the accuracy of this method was verified according to the experimental results, considering the influence of the shear span-to-depth ratio λ and stirrup ratio ρ_sv on the residual bearing capacity. Based on the shear capacity calculation method of beams at room temperature, the strength reduction coefficient of the material was introduced according to the experimental results and numerical analysis results, and the residual shear capacity calculation methods of the indirect methods after fire were proposed. This solution could provide a basis for establishing resistance design standards for relevant structures and engineering practice.

2. Experimental Program

2.1. Test Specimens

Taking load ratio and additional transverse reinforcement as the tested parameters, six specimens under indirect loading were designed by Song [11], following the Chinese design code [1], in order to investigate the shear performance of indirectly loaded primary beams after experiencing high temperature. Beams CWJJL1, JJLH1, and JJLH2 were designed according to the design principle of “strong bending and weak shear” to ensure shear failure, whereas beams CWJJL2, JJLH3, and JJLH4 were designed according to the principle of “strong shear and weak bending” to study the effect of additional transverse reinforcement on the shear performance of beams under indirect loading. The main parameters, dimensions, and reinforcement details of the beams are shown in

Table 1. Main parameters of the beams under indirect loading.

Specimen	Cross-Section/mm	ρsv/%	ρ/%	λ	Additional Reinforcement	Preload/kN	t/min
CWJJL1	250 × 400	0.15	0.82	2.1	/	0	0
JJLH1	250 × 400	0.15	0.82	2.1	/	0	90
JJLH2	250 × 400	0.15	0.82	2.1	/	0.4 Pu	90
CWJJL2	250 × 400	0.15	0.82	2.1	6C6 + 2C12	0	0
JJLH3	250 × 400	0.15	0.82	2.1	6C6 + 2C12	0	90
JJLH4	250 × 400	0.15	0.82	2.1	6C6 + 2C12	0.4 Pu	90

and Figure 2. In the names of the specimens, the letters “CWJJL” indicate that the specimens were not exposed to fire, while those with “JJLH” were exposed to fire. P_u is the ultimate capacity of the reference beam at ambient temperature; P is twice the load at the loading point of the single secondary beams, that is, the load at the loading point of the single secondary beams is P/2; and ρ_sv, ρ, λ, and t are the stirrup ratio, longitudinal reinforcement ratio, shear span-to-depth ratio, and fire exposure time, respectively.

2.2. Test Set-Up and Instrumentation

2.2.1. Fire Test Set-Up and Instrumentation

The fire tests were conducted in a furnace chamber, as shown in Figure 3. Three sides of four beams, JJLH1−JJLH4, were exposed to fire temperatures. Because the indirectly loaded beams were subjected to the fire tests together with other beams, four indirectly loaded beams were divided into two batches. After reaching the set fire time, the supply to the furnace was switched off, and the beams were cooled down naturally to room temperature. After natural cooling to room temperature at the end of the heating action, the after-fire static load bearing capacity tests of the specimens were carried out.

In the fire tests, the furnace temperature, temperature distribution in cross-section, applied load, and deflection for each beam were measured, as shown in Figure 4. The heating system of the furnace chamber adopted the international standard ISO834 fire curve [19] to control the heating, while the furnace temperature was measured using a thermocouple. The LVDTs layout of the beam during thermomechanical coupling is shown in Figure 4a, while the LVDTs layout of the beam without load during the fire test is shown in Figure 3f. The thermocouple arrangements on the cross-section of the beams are shown in Figure 4b.

2.2.2. Static Load Test Set-Up and Instrumentation

The static load test device of the normal indirect loading specimen was the same as that of the indirect loading beam after fire exposure, as shown in Figure 5. The arrangement of the reinforcement strain measuring points is shown in Figure 6. The vertical deformation of the test beam was measured by LVDTs, and the arrangement of LVDTs was consistent with that of the indirectly loaded beam under thermomechanical coupling, as shown in Figure 4a.

3. Experimental Results and Discussion

3.1. Fire Test Application and Analysis

Because of the limitations of furnace equipment on fire tests, it is impossible to provide strict temperature control according to the ISO834 temperature curve [19], as shown in Figure 7. The measured furnace temperatures were basically lower than the temperature specified by the ISO834 temperature curve, but generally matched the geometry of the ISO834 in other aspects. These temperature differences with the other batch could be caused by the reuse of the furnace system or the compactness of the cover slabs.

The measured temperatures in the cross-section of the beams during the fire tests are shown in Figure 8. As the experimental results of the indirectly loaded beams have been detailed in the reference [12], this paper only provides a brief introduction of representative beams JJLH1−JJLH2. The following could be observed:

(1)

The maximum temperature experienced by the temperature measuring point of the beam mostly occurred in the cooling section.

(2)

The temperatures near the bottom faces of the preloaded beams were higher than those of the unloaded beams, and their temperature peaks were relatively high.

3.2. Static Load Test Results and Discussion

3.2.1. Shear Capacity and Failure Mode of Beams

The failure modes of the room beams and post-fire beams in the static load tests are illustrated in Figure 9. In the static load test, the indirectly loaded beam was loaded at the overhanging end of the secondary beam, and the jack was used to control the load. The static load tests were controlled by the load, and each load level was 5 kN. Therefore, the total ultimate load F applied by the jack was twice that of the failure load P_u (two load point), and their ultimate capacities are shown in

Table 2. Ultimate load of beams.

Beams	Total Ultimate Load F (kN)	Ultimate Load Pu (kN)	Failure Mode
CWJJL1	318	159	shear
CWJJL2	408	204	bending
JJLH1	335	168	shear
JJLH2	315	158	shear
JJLH3	350	175	bending
JJLH4	330	165	bending

. The following conclusions can be obtained:

(1)

The fire-damaged beams JJLH1, JJLH2, JJLH3, and JJLH4 failed with more flexural and diagonal cracks than the beams CWJJL1 and CWJJL2 at ambient temperature, and the cracks were wider, as shown in Figure 9.

(2)

Beams JJLH1 and JJLH2, without additional reinforcement, occurred shear failure caused by the diagonal cracks at the junction of the secondary and main beam. However, the projection length of the damaged diagonal crack of the beam after fire exposure was longer than that of the comparison beam at ambient temperature.

(3)

Beams JJLH3 and JJLH4 with additional reinforcement exhibited bending failure, but the ultimate bearing capacity of the post-fire beam was significantly lower than that of the comparison beam at room temperature. Compared with the beam at ambient temperature, the ultimate bearing capacity of beam JJLH3 and JJLH4 decreased by about 14.2% and 19.1%, respectively, with an average decrease of 16.7%.

(4)

The load holding of beams under fire aggravated the reduction in the bearing capacity of the beam after fire exposure. The shear capacity of the JJLH2 beam after thermomechanical coupling was about 5.9% lower than that of beam JJLH1 only subjected to fire.

3.2.2. Vertical Displacement of Specimen

The load P−vertical displacement f curves of the beams under indirect loading are shown in Figure 10. The following conclusions can be obtained:

(1)

The vertical deflections (f = 20.77 mm) of specimen CWJJL1 were small and shear failure occurred at ambient temperature, while the vertical deflections (f = 48.27 mm) of specimen CWJJL2 were large and flexural failure occurred.

(2)

The high temperatures action had little effect on the vertical displacement of beams under indirect loading failing in the shear, but had a great influence on that of the beams failing in bending.

4. Numerical Research

In order to clarify the effect of the shear span-to-depth ratio and stirrup ratio on the shear performance of indirectly loaded beams subjected to fire exposure, ABAQUS finite element [18] was used to simulate the experimental beams. The thermal and mechanical performance parameters of concrete and reinforcement were reasonably selected, and the thermomechanical coupling model of beam under indirect loading subjected to fire was established and verified by the experimental results. The finite element (FE) analysis of the post-fire RC beams was carried out using the method of temperature field analysis first and then thermal mechanical coupling analysis, which is mainly divided into temperature filed analysis and force field analysis.

4.1. Finite-Element Model

Figure 10 shows the configuration of the beam’s finite element (FE) model, and its dimensions and reinforcement were consistent with the previous experimental beams, as shown in Figure 2. In this FE model, the concrete and reinforcement frames were connected using the surface-based Tie constraint provided in ABAQUS, and the three surfaces of the rectangular beam were subjected to fire, as shown in Figure 11.

For the initial conditions, the initial temperature could be defined through the predefined field, and the whole model was given room temperature of 20 °C. For the boundary conditions, the bottom and two sides of the beam were the fire surfaces. Because of the existence of a non-fire surface under the three-side fire condition, the fire surface and non-fire surface needed to be defined separately. The preset temperature rise curve was used for the fire surface, and the room temperature was 20 °C for the non-fire surface.

For the beam’s surfaces subjected to fire, the convective heat transfer coefficient was assumed to be a constant of 25 W/(m·K) and the thermal emissivity was recommended to be 0.7. For the no-fire surface, the convective heat transfer coefficient was assumed to be a constant of 9 W/(m·K). For the whole model, the Stefan−Boltzmann constant was assumed to be 5.67 × 10⁻⁸ W·m⁻²·°C⁻⁴ [20]. Dense grids were used for the entire model to improve the calculation accuracy. The concrete unit size was 20 mm and the reinforcement unit size was 35 mm.

4.2. Thermal Properties

The thermal properties of the concrete and steel, including the density, specific heat capacity, and heat conductivity, have been thoroughly studied by other researchers [21,22,23,24]. The density and specific heat capacity of concrete and steel can be determined according to Eurocode 2 [21], and their heat conductivity is defined according to Lie [22]. In order to make the simulation results more accurate, when verifying the temperature field, it is necessary to take the measured furnace temperature for comparison. Take the first batch tested beam JJLH1 as a representative, the temperature rise curve used in the simulation was the measured temperature rise curve of the first batch fire test, as shown in Figure 7. Figure 12 and Figure 13 show the comparison of the simulated and measured temperature at the cross-section of beam, where the letter T means the measured temperature of the test beams in the fire test and FE means the simulated temperature.

4.3. Force Field Analysis

In the force field analysis, the constitutive models of concrete and steel were the main contents [25,26,27,28], and the concrete constitutive [25,26] used the concrete damaged plasticity (CDP) model in ABAQUS. The concrete adopted a solid eight node linear hexahedron element (C3D8R) and the reinforcement skeleton adopted a two node linear truss element (T3D2). The contact constraint was set in the whole model and the friction coefficient of penalty was 0.3. During the finite element mechanical analysis, the mesh division of the RC specimen was consistent with that of the temperature field analysis, so as to read the temperature of each element in the calculation result of the temperature field. In the mechanical analysis, the concrete adopted continuous solid elements, which were not suitable for the cracking and complex mechanical analysis. Therefore, ABAQUS/Explicit could be used for explicit dynamic analysis, and the concrete cracking and crushing could be achieved by deleting the element.

In the force field analysis, the highest temperature obtained for the temperature field analysis was imported into the force field as the predefined field, for which the analysis step was set to initial. Because the numerical research and theoretical analysis mainly studied the shear capacity of the beams, while beams JJLH3 and JJLH4 were subjected to bending failure in the static load test, only two beams, JJLH1 and JJLH2, were considered in the force field analysis section. At this time, the analysis step setting was mainly to load the model, use the load control, and then calculate the residual shear bearing capacity after fire exposure. Figure 14 and Figure 15 show the numerical results and experimental results of the beams under indirect loading after fire, such as the load−vertical displacement curves and failure form of beams. The FE simulation results were basically consistent with the experimental results, as shown in Figure 14 and Figure 15. This shows that the FE model established in this paper could simulate the actual loading situation well and was suitable for the shear performance analysis of post-fire beams under indirect loading.

Combined with the failure mode and mechanism analysis of the beam under indirect loading (as shown in Figure 9 and Figure 15), it can be concluded that the indirectly loaded beam had shear failure at the junction 1-1 of the primary and secondary beams at room temperature, while beams JJLH1 and JJLH2 after experiencing high temperature had a small shear bearing capacity at the support, which also led to the extension of the failure trace of the indirectly loaded beam to the support after high temperature and shear failure in the shear bending section. Then, the bearing capacity of the indirectly loaded beam after experiencing high temperature was taken as the calculated bearing capacity.

5. Parametric Study

5.1. Shear Capacity Analysis

Combined with the above experimental results and numerical analysis, the shear bearing capacity of the indirectly loaded beams after fire exposure was analyzed. To clarify the influence of the shear span-to-depth ratio λ and stirrup ratio ρ_sv on the shear performance of indirectly loaded beams after fire, simulation calculations of the direct loading and indirect loading beam after fire were carried out, and the reduction degree of the indirectly loaded beams after experiencing high temperature was calculated. The comparison results of the numerical simulation analysis of the directly loaded beams are mentioned in [29], thus the comparison model of the directly loaded beam was not included in this manuscript. The beams in

Table 3. Comparison of the failure load between the direct and indirectly loaded beam after fire exposure.

No.	λ	s/mm	ρsv/%	t/min	Pu1/kN	Pu2/kN	W/%
1	1.50	120	0.19	90	229.45	263.03	12.76
2	1.50	150	0.15	90	217.60	251.08	13.33
3	1.50	200	0.11	90	204.77	238.90	14.28
4	1.75	120	0.19	90	199.77	217.60	7.22
5	1.75	150	0.15	90	188.42	204.77	7.98
6	1.75	200	0.11	90	183.56	197.84	8.19
7	2.10	120	0.19	90	179.29	192.23	6.73
8	2.10	150	0.15	90	167.50	184.42	9.17
9	2.10	200	0.11	90	152.97	167.38	8.61
10	2.50	120	0.19	90	173.38	180.12	3.74
11	2.50	150	0.15	90	161.36	172.75	6.59
12	2.50	200	0.11	90	150.64	159.02	5.26
13	2.84	120	0.19	90	152.44	159.01	4.13
14	2.84	150	0.15	90	150.73	154.38	2.36
15	2.84	200	0.11	90	143.43	150.64	4.78

were designed following the “strong bending and weak shear” design principle to ensure shear failure, and all of the specimens in Table 3 showed obvious shear failure in the shear span area, meeting the design expectations. The simulation results are shown in Table 3, where P_u1 is the failure load of the indirectly loaded beam after experiencing high temperature, P_u2 is the failure load of the directly loaded beam after experiencing high temperature, W is the reduction coefficient of the bearing capacity of the indirectly loaded beam after experiencing high temperature compared with the directly loaded beam after high temperature (W = (P_u2 − P_u1)/P_u2), and s is stirrup spacing.

It can be seen from Table 3 that under the same shear span-to-depth ratio, the shear capacity of the indirectly loaded beams after fire increased with the increase in stirrup ratio. With the increase in the shear span ratio, the reduction in the shear capacity of the indirectly loaded beams relative to the direct loading decreased after experiencing a high temperature. When the shear span-to-depth ratio was greater than 2.5, the shear capacity of the indirectly loaded beams after fire was basically the same as that of the directly loaded beams.

Referring to the formula for reducing the shear performance of the indirectly loaded beams at ambient temperature and fitting the data in Table 3 through the nonlinear curve fitting method, it can be concluded that the formula for reducing the shear bearing capacity of indirectly loaded beams relative to directly loaded beams after fire is expressed by Equation (1), as shown in Figure 16:

$$W=\frac{44.13}{{\left(100{\rho }_{\mathrm{sv}}\right)}^{0.18}{\lambda }^{1.83}}\%$$

(1)

Equation (1) is applicable to high-temperature beams with a shear span-to-depth ratio less than 3.0. The reduction degree of the indirectly loaded RC beam in Table 3 was calculated using Equation (1), and the comparison between the calculated value and the simulated value is shown in Figure 17. It can be seen from Figure 17 that the simulated value of the indirectly loaded beam after experiencing a high temperature was in good agreement with the calculated value of Equation (1).

Combined with the simulation results in Table 3, the relationship between the shear bearing capacity of the directly loaded beam after fire exposure for 90 min and parameters such as shear span-to-depth ratio λ and stirrup spacing s are shown in Figure 18. The equation is expressed as follows:

$$P_{\mathrm{uT}2}=\frac{1.62}{\lambda +0.01}{f}_{\mathrm{t}}b{h}_0+0.81f_{yv}\frac{A_{sv}}{s}{h}_0$$

(2)

Therefore, the shear bearing capacity of the indirectly loaded beam after fire exposure is calculated using Equation (3):

$$\begin{array}{l}{P}_{\mathrm{uT}1}=(1-W)(\frac{1.62}{\lambda +0.01}{f}_{\mathrm{t}}b{h}_0+0.81f_{yv}\frac{A_{sv}}{s}{h}_0)\\ =(1-\frac{44.13}{{(100{\rho }_{\mathrm{sv}})}^{0.18}{\lambda }^{1.83}}\%)(\frac{1.62}{\lambda +0.01}{f}_{\mathrm{t}}b{h}_0+0.81f_{yv}\frac{A_{sv}}{s}{h}_0).\end{array}$$

(3)

5.2. Verification of Calculation Method

Combined with the experimental results and FE analysis, Equation (3) is used to calculate the shear capacity of the indirectly loaded beams. The theoretical calculation results V_cal of the indirectly loaded beam after fire were compared with the test results V_test, as shown in

Table 4. Comparison of the failure load between V_test and V_cal.

Data Sources	Specimens	λ	s/mm	ρsv/%	Vtest/kN	Vcal/kN	(Vtest- Vcal)/Vtest/%	Coefficient of Variation
Table 3	1	1.50	120	0.19	229.45	224.81	2.02	0.67
	2	1.50	150	0.15	217.60	213.69	1.79
	3	1.50	200	0.11	204.77	202.21	1.25
	4	1.75	120	0.19	199.77	206.77	−3.51
	5	1.75	150	0.15	188.42	195.75	−3.89
	6	1.75	200	0.11	183.56	184.51	−0.51
	7	2.10	120	0.19	179.29	186.16	−3.83
	8	2.10	150	0.15	167.50	175.18	−4.59
	9	2.10	200	0.11	152.97	164.07	−7.25
	10	2.50	120	0.19	173.38	168.04	3.08
	11	2.50	150	0.15	161.36	157.05	2.67
	12	2.50	200	0.11	150.64	145.98	3.09
	13	2.84	120	0.19	152.44	155.97	−2.32
	14	2.84	150	0.15	150.73	144.97	3.82
	15	2.84	200	0.11	143.43	133.92	6.63
Table 2	JJLH1	2.10	150	0.15	167.50	175.18	−4.58
	JJLH2	2.10	150	0.15	157.50	175.18	−11.22

and Figure 19.

Table 4 and Figure 19 show that the theoretical calculation results of the RC beam after experiencing a high temperature was basically consistent with the experimental results, and the average absolute error was 3.88 %. Therefore, the shear capacity calculation method of the indirectly loaded beam after experiencing high temperature, as proposed in this paper, was effective.

6. Conclusions

Combined with the experimental results, numerical simulation, and theoretical research, the shear capacity calculation method of post-fire beams under indirect loading was established. The main conclusions of this research are as follows:

(1)

The fire damage was not observed to affect the failure mode of the indirectly loaded primary beams, but the failure degree of the indirectly loaded beams without additional transverse reinforcement after fire was severe.

(2)

All beams with additional reinforcement exhibited bending failure, but the ultimate bearing capacity of the post-fire beam was significantly lower than that of the comparison beam at room temperature; the ultimate bearing capacity of the post-fire beam under indirect loading decreased by about 16.7% on average.

(3)

Under the same shear span-to-depth ratio, the residual shear capacity of the indirectly loaded beams increased with the increase in stirrup ratio, and the shear capacity reduction coefficient of the indirectly loaded beams decreased gradually after fire exposure.

(4)

The shear capacity calculation results of the indirectly loaded beam after fire were basically consistent with the experimental results, and the average absolute error was 3.88%. Therefore, the shear capacity calculation method of the indirectly loaded beam after fire exposure, as proposed in this paper, was effective.

The findings of this study are expected to be useful to researchers and designers looking to improve the performance of post-fire beams under indirect loading. In the future, the proposed equation may be combined with different fire conditions to predict the shear strength of indirectly loaded beams after fire exposure.