A method for calculating residual bending stiffness of a t-shaped composite beam after fire

Abstract

The application discloses a kind of T-shaped composite beam post-fire residual bending stiffness calculation method, belongs to the technical field of fire safety risk assessment, comprising the following steps: (1) fire test design is carried out, numerical model is established to T-shaped composite beam under fire test using Abaqus software, numerical simulation is carried out by model;(2) on the basis of verified finite element model, the working condition is expanded, the influence of different parameters on the residual bending stiffness of T-shaped composite beam after fire is simulated, the parameters include the superposition parameter gamma of T-shaped composite beam, the friction coefficient λ of superposition surface and fire time t;(3) according to the result obtained by simulation, the fitting of the residual bending stiffness of T-shaped composite beam after fire under different working conditions is carried out using SPSS data analysis software, finally the calculated value is compared with the test value or simulation value to verify the accuracy of the calculation method.The application can effectively improve the reliability of T-shaped composite beam bending stiffness evaluation after fire by accurate calculation.

Description

Technical Field

[0001] This invention belongs to the field of fire safety risk assessment technology, specifically relating to a method for calculating the residual bending stiffness of a T-shaped composite beam after a fire. Background Technology

[0002] It is known that European standards specify a method for calculating the flexural stiffness of simply supported reinforced concrete beams after a fire. This method is based on the stiffness of the beam at room temperature, with a reduction in its area coefficient: (EI) z =[k c (θ M )] 2 E c I z This method has three drawbacks: First, it fails to consider the reduction in steel reinforcement strength under fire conditions. The stress performance of steel reinforcement decreases with increasing temperature, altering the neutral axis position of the cross-section and ultimately changing the moment of inertia. Therefore, the degradation of steel reinforcement strength reduces the stiffness of the T-shaped composite beam. Second, it fails to consider the influence of the precast slab thickness. When the precast slab thickness decreases, the overall integrity and stability of the composite beam decline, its strength deteriorates more severely at high temperatures, its fire resistance decreases, and it increases the risk of structural collapse under fire. Third, the roughness of the composite surface also affects crack propagation and the distribution of the cross-sectional temperature field under fire conditions, further impacting the beam's stiffness degradation. In actual engineering, composite beams use steel reinforcement of varying strengths, precast slab thicknesses, and surface roughness. The residual bending stiffness of T-shaped composite beams after a fire varies under different working conditions. Therefore, a calculation method applicable to the residual bending stiffness of T-shaped composite beams after a fire should be proposed. Summary of the Invention

[0003] To overcome the shortcomings of the prior art, this invention discloses a method for calculating the residual bending stiffness of a T-shaped composite beam after a fire. The purpose is to determine the influence of different parameters on the residual bending stiffness of the T-shaped composite beam after a fire. The parameters include the composite parameters of the T-shaped composite beam, the friction coefficient of the composite surface, and the fire exposure time. The residual bending stiffness of the specimen after the fire is determined by comprehensively considering all parameter factors.

[0004] To achieve the above objectives, the technical solution of the present invention is as follows:

[0005] A method for calculating the residual bending stiffness of a T-shaped composite beam after a fire includes the following steps:

[0006] (1) Conduct fire test design, use Abaqus software to establish a numerical model of the T-shaped composite beam under fire test, and perform numerical simulation through the model;

[0007] (2) Based on the verified finite element model, the working conditions are expanded to simulate the influence of different parameters on the residual bending stiffness of the T-shaped composite beam after the fire. The parameters include the composite parameter γ of the T-shaped composite beam, the friction coefficient λ of the composite surface and the fire exposure time t.

[0008] (3) Based on the simulation results, SPSS data analysis software was used to fit the residual bending stiffness of T-shaped composite beams after fire under different working conditions. Finally, the calculated values ​​were compared with the experimental or simulated values ​​to verify the accuracy of the calculation method.

[0009] Preferably, in step (1), when designing the fire test, considering the size of the fire furnace and the requirements of the current concrete structure design code, 21 T-shaped cross-section beams are designed and manufactured, including 14 precast T-shaped composite beams and 7 cast-in-place beams. The length of both the T-shaped composite beams and the cast-in-place beams is 3000mm, the total height of the cross-section is 300mm, the flange height is 100mm, the flange width is 450mm, and the beam rib width is 150mm. Unlike the cast-in-place beams, the flange of the T-shaped composite beams is composed of a cast-in-place part and a precast part. The height h of the precast part has two types: 40mm and 60mm, and the ratio of the precast part to the total height of the flange part is 0.4 and 0.6, respectively. According to the requirements of the concrete structure design code for the thickness of the protective layer, the thickness of the protective layer of both the T-shaped composite beams and the cast-in-place beams is 30mm. A uniformly distributed load is applied to the beams by load blocks, and the load ratio is 44%.

[0010] Preferably, in step (1), the numerical simulation includes the following specific steps:

[0011] (11) Fire test simulation: Abaqus software was used to simulate the mechanical properties of T-shaped composite beams under uniformly distributed load at high temperature using sequential thermo-mechanical coupling: that is, firstly, a temperature field model was established, and then the obtained temperature field was used as a known condition to perform thermal stress analysis to obtain the stress and strain field at high temperature. Finally, a numerical model was established to simulate the static loading test after high temperature.

[0012] In the T-beam model established for analyzing heat transfer problems, the concrete uses an eight-node linear heat transfer hexahedral solid element, and the steel reinforcement uses a two-node heat transfer element. The steel reinforcement and concrete are bound together with a tie to ensure the heat transfer between the steel reinforcement and concrete. The temperature is increased according to the experimentally measured furnace temperature. Except for the top surface, the other surfaces of the beam are exposed to the fire and are uniformly exposed to the fire.

[0013] The temperature field in the temperature field model is exported and preset as a predefined field into the stress analysis model. The concrete in this model uses an eight-node hexahedral element. The element is reduced and integrated using hourglass control. The steel reinforcement uses a two-node linear three-dimensional truss element. The steel reinforcement and concrete are connected by an embedded method to ensure that the two parts are subjected to the same force. The connection method at the overlapping surface of the T-shaped composite beam is to use Cartesian elements to create a dimensionless three-dimensional spring to simulate the interaction at the overlapping surface.

[0014] (12) Conduct static loading test simulation after fire: After the fire, apply progressively increasing concentrated loads at two three-point positions on the beam until the beam is completely destroyed and reaches the ultimate bearing capacity value. In this model, the concrete adopts an eight-node hexahedral element, the steel reinforcement adopts a two-node linear three-dimensional truss element, and the steel reinforcement and concrete adopt an embedded connection method to ensure that the two parts are subjected to the same force.

[0015] (13) Finite element simulation verification: Plot the experimental and simulated values ​​of the temperature field at different measuring points at the mid-span section of the T-shaped composite beam specimen under fire, plot the deflection at the mid-span position under fire and after fire, and compare and verify the experimental and simulated values.

[0016] Preferably, step (2) includes the following specific steps:

[0017] (21) Considering the residual bending stiffness B of the T-shaped composite beam after high temperature Pt The residual bending B of a cast-in-place beam under the same working conditions Rt Based on the reduction, the relationship between the two is as follows:

[0018] B Pt =(1-β)B Rt (1)

[0019] In equation (1), β is the reduction factor;

[0020] The residual bending stiffness of the cast-in-place beam after high temperature is calculated by the method of mesh superposition, without considering the tensile strength of the concrete in the tension zone, and meshing is performed on the concrete in the compression zone.

[0021] Based on the plane section assumption, the strain at any location in the compression zone of the concrete after high temperature is expressed as:

[0022]

[0023] In equation (2), ε ct Let x be the concrete strain in the compression zone, h0 be the effective height of the cross section, and x be the concrete strain in the compression zone. c ε is the height of the neutral axis. yt This represents the yield strain of the reinforcing steel.

[0024] The stress expression for each mesh is obtained based on the stress-strain relationship after high temperature, where equation (2) is related to the neutral axis height x. c The functional expression; when the sum of the compressive forces of the concrete equals the sum of the tensile forces provided by the tensile steel, the neutral axis x can be obtained by solving the equation. c The value is expressed as:

[0025]

[0026] In equation (3), σ ct A represents the compressive strength of the concrete in the unit after high temperature. ij f is the area of ​​each unit. yt Here, k represents the tensile strength of the steel reinforcement after high temperature, and A represents the number of longitudinal tensile bars. s This represents the cross-sectional area of ​​a single tension longitudinal reinforcement bar;

[0027] Based on the existing relationship between the elastic modulus of concrete and the elastic modulus of steel bars after high temperature and temperature, the corresponding elastic modulus values ​​of concrete and steel bars after high temperature are obtained for each grid. The contributions of all concrete and steel elements to the flexural stiffness are summed, and the T-shaped cast-in-place beam after high temperature B... Rt The residual bending stiffness after that is:

[0028]

[0029] In equation (4), B Rt E represents the residual flexural stiffness of the T-shaped cast-in-place beam after high temperature. ct E represents the elastic modulus of concrete after high temperature. st I represents the elastic modulus of the steel reinforcement after high temperature. ij Let I be the moment of inertia of element (i,j) about the neutral axis. st Let be the moment of inertia of the tensioned steel bar about its neutral axis;

[0030] (22) Parameter analysis:

[0031] In formula (1), the reduction factor β comprehensively considers the influence of fire exposure time t, superposition parameter γ, and superposition surface friction coefficient λ on the residual bending stiffness of the T-shaped composite beam; superposition parameter γ refers to the ratio of the height of the precast slab to the flange height of the T-shaped composite beam; based on the different values ​​of fire exposure time t, superposition parameter γ, and superposition surface friction coefficient λ, a table of residual bending stiffness of the T-shaped composite beam under different working conditions is drawn; based on the table of residual bending stiffness, the relationship between the residual bending stiffness of the T-shaped composite beam and fire exposure time t, superposition parameter γ, and superposition surface friction coefficient λ is analyzed.

[0032] Preferably, in step (22), the fire exposure time t is 30 min, 60 min, 90 min, 120 min, 150 min, and 180 min, respectively; the superposition parameter γ is 0.4, 0.5, and 0.6, respectively; and the friction coefficient λ of the superposition surface is 0.4, 0.6, and 0.8, respectively.

[0033] Preferably, in step (22), the relationship between the residual bending stiffness of the T-shaped composite beam and the fire exposure time t, the composite parameter γ, and the friction coefficient λ of the composite surface is analyzed using the following method:

[0034] (221) Analysis of the relationship between the residual bending stiffness of the T-shaped composite beam after high temperature and the fire exposure time t: Under the condition that the composite parameter γ and the friction coefficient λ of the composite surface are the same, the change of the reduction coefficient between the cast-in-place beam and the T-shaped composite beam is plotted in the figure, and the relationship between the fire exposure time and the reduction coefficient of the T-shaped composite beam is analyzed based on the plotted figure.

[0035] (222) Analysis of the relationship between the residual bending stiffness of the T-shaped composite beam after high temperature and the composite parameter γ: When the fire exposure time t is the same as the friction coefficient λ of the composite surface, the change of the reduction coefficient between the cast-in-place beam and the T-shaped composite beam is plotted. Based on the plotted graph, the relationship between the composite parameter γ and the reduction coefficient of the T-shaped composite beam is analyzed.

[0036] (223) Analysis of the relationship between the residual bending stiffness of the T-shaped composite beam after high temperature and the friction coefficient λ of the composite surface: Under the condition that the fire exposure time t and the composite parameter γ are the same, the change of the reduction coefficient of the residual bending stiffness of the T-shaped composite beam is plotted. Based on the plotted graph, the relationship between the friction coefficient λ of the composite surface of the T-shaped composite beam and the reduction coefficient is analyzed.

[0037] Preferably, step (3) includes the following specific steps:

[0038] Using SPSS statistical analysis software, the reduction coefficient relationship between T-shaped cast-in-place beams and T-shaped composite beams is given as follows:

[0039]

[0040] In equation (5), t∈[30min,180min], λ∈[0.4,0.8], γ∈[0.4,0.6], l∈[0,44%], and the correlation coefficient of equation (5) is R. 2 =0.979.

[0041] Preferably, step (3) further includes: to verify the effectiveness of the proposed calculation method for the residual bending stiffness of the T-shaped composite beam after high temperature, an error analysis is performed between the calculated value and the experimental and simulated values ​​of the residual bending stiffness of the T-shaped composite beam after high temperature.

[0042] The beneficial effects of this invention's method for calculating the residual bending stiffness of T-shaped composite beams after a fire are as follows: This invention provides a method for calculating the residual bending stiffness of T-shaped composite beams after high-temperature exposure, considering the influence of fire exposure time t, the composite parameter γ (the ratio of the precast slab height to the flange height of the T-shaped composite beam), and the friction coefficient λ of the composite surface on the residual bending stiffness of the T-shaped composite beams. This calculation method focuses on the fact that in actual engineering, as the fire exposure time changes, the composite parameter and the friction coefficient of the composite surface vary in different building structures, which in turn affects the beam's residual bending stiffness. If the residual bending stiffness of T-shaped composite beams after high-temperature exposure cannot be accurately calculated, the safety and applicability of the structure after a fire cannot be accurately assessed. In severe cases, this increases the risk of structural collapse after a fire, threatening people's lives and property, and hindering firefighting, post-disaster reinforcement and repair, and reuse. Therefore, it is necessary to accurately calculate the residual bending stiffness of T-shaped composite beams after high-temperature exposure. This invention, through accurate calculation, can effectively improve the reliability of the bending stiffness assessment of T-shaped composite beams after a fire. Attached Figure Description

[0043] Figure 1 Specimen numbering rule diagram;

[0044] Figure 2 Dimensions, reinforcement details, and thermocouple arrangement of the T-shaped composite beam;

[0045] Figure 3 Temperature field cloud map of T-shaped composite beam;

[0046] Figure 4 A three-dimensional schematic diagram of a T-shaped composite beam with springs;

[0047] Figure 5 Comparison of simulated and experimental temperature field values;

[0048] Figure 6 Comparison of simulated and experimental deflection values;

[0049] Figure 7 A schematic diagram of the mesh generation method calculation;

[0050] Figure 8 The variation of residual bending stiffness of T-shaped composite beams with fire exposure time;

[0051] Figure 9 The variation of residual bending stiffness of T-shaped composite beams with composite parameters;

[0052] Figure 10 The variation of residual bending stiffness of T-shaped composite beam with friction coefficient of composite surface. Detailed Implementation

[0053] The following description is merely a preferred embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

[0054] In the initial embodiment, the present invention provides a method for calculating the residual bending stiffness of a T-shaped composite beam after a fire, comprising the following steps:

[0055] (1) Conduct fire test design, use Abaqus software to establish a numerical model of the T-shaped composite beam under fire test, and perform numerical simulation through the model;

[0056] (2) Based on the verified finite element model, the working conditions (i.e., different parameter conditions represent different working conditions) are expanded to simulate the influence of different parameters on the residual bending stiffness of the T-shaped composite beam after the fire. The parameters include the composite parameter γ of the T-shaped composite beam, the friction coefficient λ of the composite surface and the fire exposure time t.

[0057] (3) Based on the simulation results, SPSS data analysis software was used to fit the residual bending stiffness of T-shaped composite beams after fire under different working conditions. Finally, the calculated values ​​were compared with the experimental or simulated values ​​to verify the accuracy of the calculation method.

[0058] In a further embodiment, in step (1), when designing the fire test, considering the size of the fire furnace and the requirements of the current concrete structure design code, 21 T-shaped cross-section beams are designed and manufactured, including 14 precast T-shaped composite beams and 7 cast-in-place beams. The length of both the T-shaped composite beams and the cast-in-place beams is 3000mm, the total height of the cross-section is 300mm, the flange height is 100mm, the flange width is 450mm, and the beam rib width is 150mm. Unlike the cast-in-place beams, the flange of the T-shaped composite beams is composed of a cast-in-place part and a precast part. The height h of the precast part has two types: 40mm and 60mm, and the ratio of the precast part to the total height of the flange part is 0.4 and 0.6, respectively. According to the requirements of the concrete structure design code for the thickness of the protective layer, the thickness of the protective layer of both the T-shaped composite beams and the cast-in-place beams is 30mm. A uniformly distributed load is applied to the beams by load blocks, and the load ratio is 44%.

[0059] The test groups are shown in Table 1. The specimen numbering rules in Table 1 are as follows: Figure 1 As shown. The reinforcement diagram for the T-beam is as follows. Figure 2 As shown.

[0060]

[0061] In a further embodiment, step (1) of the numerical simulation includes the following specific steps:

[0062] (11) Fire test simulation: Abaqus software was used to simulate the mechanical properties of T-shaped composite beams under uniformly distributed load at high temperature using sequential thermo-mechanical coupling: that is, firstly, a temperature field model was established, and then the obtained temperature field was used as a known condition to perform thermal stress analysis to obtain the stress and strain field at high temperature. Finally, a numerical model was established to simulate the static loading test after high temperature.

[0063] In the T-beam model established for analyzing heat transfer, the concrete was modeled using eight-node linear heat transfer hexahedral solid elements (C3D8), and the reinforcing steel was modeled using two-node heat transfer elements (DC1D2). Ties were used to bind the reinforcing steel and concrete to ensure heat transfer between them. The temperature was increased according to the experimentally measured furnace temperature. Except for the top surface, all other surfaces of the beam were exposed to the fire and uniformly heated. The temperature field results for the cast-in-place beam and the T-shaped composite beam are as follows: Figure 3 As shown;

[0064] The temperature field from the temperature field model is exported and preset as a predefined field into the stress analysis model. The concrete in this model uses eight-node hexahedral elements (C3D8R), which are then reduced through integration and controlled by an hourglass. The reinforcing steel uses two-node linear three-dimensional truss elements (T3D2). The reinforcing steel and concrete are connected via an embedded method to ensure that both parts share the load. The connection at the overlapping surface of the T-shaped composite beam uses Cartesian elements to create a dimensionless three-dimensional spring to simulate the interaction at the overlapping surface. Figure 4 As shown;

[0065] (12) Conduct static loading test simulation after fire: After the fire, apply progressively increasing concentrated loads at two three-point positions on the beam until the beam is completely destroyed and reaches the ultimate bearing capacity value. In this model, the concrete adopts an eight-node hexahedral element (C3D8R), the steel reinforcement adopts a two-node linear three-dimensional truss element (T3D2), and the steel reinforcement and concrete adopt an embedded connection method to ensure that the two parts are subjected to the same force.

[0066] (13) Finite element simulation verification: Plot the experimental and simulated values ​​of the temperature field at different measuring points on the mid-span section of the T-shaped composite beam specimen under fire conditions (e.g., ...). Figure 5 As shown), plot the deflection at the mid-span position before and after the fire (e.g. Figure 6 As shown in the figure, the experimental values ​​are compared and verified with the simulated values.

[0067] In a further embodiment, step (2) includes the following specific steps:

[0068] (21) The current Chinese code for concrete structure design stipulates that the mechanical properties of T-shaped composite beams at normal temperature can be calculated using the same method as cast-in-place beams of the same dimensions, concrete strength, and reinforcement. However, the aforementioned fire tests and static loading tests have demonstrated that due to the presence of the composite surface, the mechanical properties of T-shaped composite beams after high temperatures cannot be replaced by cast-in-place beams under the same conditions. Therefore, considering the residual bending stiffness B of the T-shaped composite beam after high temperatures... Pt The residual bending B of a cast-in-place beam under the same working conditions Rt Based on the reduction, the relationship between the two is as follows:

[0069] B Pt =(1-β)B Rt (1)

[0070] In equation (1), β is the reduction factor;

[0071] The residual flexural stiffness of cast-in-place beams after high temperature is calculated using a mesh stacking method, neglecting the tensile strength of the concrete in the tension zone, and meshing the concrete in the compression zone; for example Figure 7 As shown.

[0072] like Figure 7 As shown, based on the plane section assumption, the strain at any location in the compression zone of the concrete after high temperature is expressed as:

[0073]

[0074] In equation (2), ε ct Let x be the concrete strain in the compression zone, h0 be the effective height of the cross section, and x be the concrete strain in the compression zone. c ε is the height of the neutral axis. yt This represents the yield strain of the reinforcing steel.

[0075] The stress expression for each mesh is obtained based on the stress-strain relationship after high temperature, where equation (2) is related to the neutral axis height x. c The functional expression; when the sum of the compressive forces of the concrete equals the sum of the tensile forces provided by the tensile steel, the neutral axis x can be obtained by solving the equation. c The value is expressed as:

[0076]

[0077] In equation (3), σ ct A represents the compressive strength of the concrete in the unit after high temperature. ij f is the area of ​​each unit. yt Here, k represents the tensile strength of the steel reinforcement after high temperature, and A represents the number of longitudinal tensile bars. s This represents the cross-sectional area of ​​a single tension longitudinal reinforcement bar;

[0078] Based on the existing relationship between the elastic modulus of concrete and the elastic modulus of steel bars after high temperature and temperature, the corresponding elastic modulus values ​​of concrete and steel bars after high temperature are obtained for each grid. The contributions of all concrete and steel elements to the flexural stiffness are summed, and the T-shaped cast-in-place beam after high temperature B... Rt The residual bending stiffness after that is:

[0079]

[0080] In equation (4), B Rt E represents the residual flexural stiffness of the T-shaped cast-in-place beam after high temperature. ct E represents the elastic modulus of concrete after high temperature. st I represents the elastic modulus of the steel reinforcement after high temperature. ij Let I be the moment of inertia of element (i,j) about the neutral axis. st Let be the moment of inertia of the tensioned steel bar about its neutral axis;

[0081] The calculated values ​​of the residual bending stiffness of the cast-in-place beam after high temperature obtained using the above calculation method are listed in Table 2, and the comparison with the experimental values ​​is also listed in Table 2.

[0082]

[0083] (22) Parameter analysis:

[0084] In formula (1), the reduction factor β comprehensively considers the influence of fire exposure time t, superposition parameter γ, and superposition surface friction coefficient λ on the residual bending stiffness of the T-shaped composite beam; superposition parameter γ refers to the ratio of the height of the precast slab to the flange height of the T-shaped composite beam; based on the different values ​​of fire exposure time t, superposition parameter γ, and superposition surface friction coefficient λ, a table of residual bending stiffness of the T-shaped composite beam under different working conditions is drawn; based on the table of residual bending stiffness, the relationship between the residual bending stiffness of the T-shaped composite beam and fire exposure time t, superposition parameter γ, and superposition surface friction coefficient λ is analyzed.

[0085] In a further embodiment, in step (22), the fire exposure time t is 30 min, 60 min, 90 min, 120 min, 150 min, and 180 min, respectively; the superposition parameter γ is 0.4, 0.5, and 0.6, respectively; and the friction coefficient λ of the superposition surface is 0.4, 0.6, and 0.8, respectively. The residual bending stiffness of the T-shaped superposition beam under different working conditions is shown in Table 3.

[0086]

[0087]

[0088]

[0089]

[0090] In a further embodiment, the relationship between the residual bending stiffness of the T-shaped composite beam and the fire exposure time t, the composite parameter γ, and the friction coefficient λ of the composite surface in step (22) is analyzed as follows:

[0091] (221) Analysis of the relationship between the residual bending stiffness of the T-shaped composite beam after high temperature and the fire exposure time t: Under the condition that the composite parameter γ and the friction coefficient λ of the composite surface are the same, the change of the reduction coefficient between the cast-in-place beam and the T-shaped composite beam is plotted in the figure (e.g. Figure 8 As shown in the figure, analyze the relationship between the fire exposure time and the reduction factor of the T-shaped composite beam; from the... Figure 8 It is evident that as the fire exposure time of the T-shaped composite beam increases from 30 minutes to 180 minutes, the reduction factor gradually increases. The longer the fire exposure time of the T-shaped composite beam, the more severe the fire crack development, the greater the deformation, the smaller the residual bending stiffness, and consequently the greater the reduction factor.

[0092] (222) Analysis of the relationship between the residual bending stiffness of the T-shaped composite beam after high temperature and the composite parameter γ: When the fire exposure time t is the same as the friction coefficient λ of the composite surface, the change of the reduction coefficient between the cast-in-place beam and the T-shaped composite beam is plotted as follows (e.g. Figure 9 As shown in the figure, analyze the relationship between the composite parameter γ and the reduction factor of the T-shaped composite beam based on the drawn diagram; Figure 9 It can be seen that the larger the composite parameter of the T-shaped composite beam, the smaller the reduction factor for the same fire exposure time, meaning the greater the residual bending stiffness. This is because a larger composite parameter results in a higher precast slab height in the flange portion, making its structure closer to that of a cast-in-place beam, leading to higher overall strength, smaller deflection under fire, and greater residual bending stiffness after the fire. Furthermore, the rate of decrease in the reduction factor gradually decreases with increasing fire exposure time, indicating that the stiffness of the T-shaped composite beam decreases rapidly before 90 minutes of fire exposure and then decreases slowly after 90 minutes.

[0093] (223) Analysis of the relationship between the residual bending stiffness of the T-shaped composite beam after high temperature and the friction coefficient λ of the composite surface: Under the condition that the fire exposure time t and the composite parameter γ are the same, the change of the reduction coefficient of the residual bending stiffness of the T-shaped composite beam is plotted as follows (e.g. Figure 10 As shown in the figure, the relationship between the friction coefficient λ and the reduction coefficient of the T-shaped composite beam is analyzed based on the drawn figure.

[0094] Depend on Figure 10It can be seen that the friction coefficient λ of the mating surface of the T-shaped composite beam has an overall effect on the residual bending stiffness of the T-shaped composite beam after high temperature: the larger the friction coefficient, the greater the residual bending stiffness. A higher friction coefficient at the mating interface results in greater bonding force between the precast and cast-in-place slabs on the T-shaped composite beam, leading to better overall load-bearing performance. At high temperatures, heat is less likely to diffuse through the gap between the two, thus preventing damage to the overall load-bearing performance of the T-shaped composite beam. Throughout the fire exposure process, the reduction coefficient decreases uniformly with the increase of the friction coefficient, meaning that a higher friction coefficient at the mating surface results in greater residual bending stiffness of the T-shaped composite beam after high temperature.

[0095] Preferably, step (3) includes the following specific steps:

[0096] Considering the combined effects of fire exposure time t, superposition parameter γ, and friction coefficient λ of the superposition surface on the residual bending stiffness of the T-shaped composite beam, the above parameter analysis shows that all three factors affect the reduction coefficient β to a considerable degree and none can be ignored. The data in Tables 2 and 3 are organized and analyzed, and the relationship between the reduction coefficients of the T-shaped cast-in-place beam and the T-shaped composite beam is given by regression analysis using SPSS software:

[0097]

[0098] To more reasonably calculate the residual bending stiffness of the T-shaped composite beam after high temperature, the above formula needs to be used within a certain range: In formula (5), t∈[30min,180min], λ∈[0.4,0.8], γ∈[0.4,0.6], l∈[0,44%], and the correlation coefficient of formula (5) is R. 2 =0.979.

[0099] Preferably, step (3) further includes: to verify the effectiveness of the proposed calculation method for the residual bending stiffness of the T-shaped composite beam after high temperature, an error analysis is performed between the calculated value and the experimental and simulated values ​​of the residual bending stiffness of the T-shaped composite beam after high temperature.

[0100] As shown in Table 4.

[0101]

Claims

1. A method for calculating the residual bending stiffness of a T-shaped composite beam after a fire, characterized by the following steps: (1) Conduct fire test design, use Abaqus software to establish a numerical model of the T-shaped composite beam under fire test, and conduct numerical simulation through the model; (2) On the basis of the verified finite element model, the working conditions are expanded, and the influence of different parameters on the residual bending stiffness of the T-shaped composite beam after the fire is simulated, the parameters including the composite parameters of the T-shaped composite beam γ , the friction coefficient of the composite surface λ , and the fire time t ; (3) Based on the simulation results, SPSS data analysis software was used to fit the residual bending stiffness of the T-shaped composite beam after fire under different working conditions. Finally, the calculated values ​​were compared with the experimental or simulated values ​​to verify the accuracy of the calculation method. Step (2) includes the following specific steps: (21) Consider the residual bending stiffness of T-shaped composite beam after high temperature B Pt , the residual bending of cast-in-place beam under the same working condition is reduced on the basis of B Rt , the relationship between the two is: （1）； In formula (1), β is the reduction factor; The residual bending stiffness of the cast-in-place beam after high temperature is calculated by the method of mesh superposition, without considering the tensile strength of the concrete in the tension zone, and meshing is performed on the concrete in the compression zone. Based on the plane section assumption, the strain at any location in the compression zone of the concrete after high temperature is expressed as: （2）； In formula (2), ɛ ct is the strain of the concrete in the compression zone, h 0 is the effective depth of the cross section, x c is the height of the neutral axis, ɛ yt is the yield strain of the reinforcement. The stress expression of each grid is obtained according to the stress-strain relationship after high temperature, wherein ， Formula (2) is a function expression about the neutral axis height x c ; when the sum of the compression force of the concrete and the sum of the tension force provided by the tensioned steel are equal, the value of the neutral axis x c is solved by solving the equation, and the expression is: （3）； In formula (3), σ ct fcu is the compressive strength of the concrete after high temperature, A ij A is the area of each unit, f yt ft is the tensile strength of the steel after high temperature, k N is the number of tensile longitudinal reinforcement, A s As is the cross-sectional area of a single tensile longitudinal reinforcement; According to the existing relationship between the elastic modulus of high-temperature concrete and the elastic modulus of steel and temperature, the corresponding elastic modulus values of high-temperature concrete and steel of each grid are obtained, the contributions of all concrete and steel elements to the bending stiffness are added, and the residual bending stiffness of the T-shaped cast-in-place beam after high temperature is: B Rt ​ （4）； In formula (4), B Rt is the residual flexural rigidity of the post-high-temperature T-shaped cast-in-place beam, E ct is the elastic modulus of the post-high-temperature concrete, E st is the elastic modulus of the post-high-temperature steel bar, I ij is the unit is the moment of inertia of the neutral axis, I st is the moment of inertia of the neutral axis of the tensile steel bar; (22) Parameter analysis: In formula (1), the reduction coefficient β The fire duration time t、 The superposition parameter γ The friction coefficient of the superposition surface λ The influence of the T-shaped superposed beam residual bending stiffness; the superposition parameter γ refers to the ratio of the T-shaped superposed beam prefabricated plate height to the flange height; the residual bending stiffness table of the T-shaped superposed beam under different working conditions is drawn according to different values of the fire duration time t, the superposition parameter γ and the friction coefficient λ of the superposition surface; the relationship between the residual bending stiffness of the T-shaped superposed beam and the fire duration time t, the superposition parameter γ and the friction coefficient λ of the superposition surface is analyzed based on the residual bending stiffness table.

2. The method for calculating the residual flexural rigidity of a T-shaped composite beam after fire according to claim 1, characterized in that: In step (1), when designing the fire test, considering the size of the fire furnace and the requirements of the current concrete structure design code, 21 T-shaped cross-section beams were designed and fabricated, including 14 precast T-shaped composite beams and 7 cast-in-place beams. The length of both the T-shaped composite beams and the cast-in-place beams is 3000 mm, the total height of the cross-section is 300 mm, the flange height is 100 mm, the flange width is 450 mm, and the beam rib width is 150 mm. Unlike the cast-in-place beams, the flange of the T-shaped composite beams consists of a cast-in-place part and a precast part. The height h of the precast part has two types: 40 mm and 60 mm, and the ratio of the precast part height to the total height of the flange part is 0.4 and 0.6, respectively. According to the requirements of the concrete structure design code for the thickness of the protective layer, the thickness of the protective layer of both the T-shaped composite beams and the cast-in-place beams is 30 mm. A uniformly distributed load is applied to the beams by load blocks, and the load ratio is 44%.

3. The method for calculating the residual bending stiffness of a T-shaped composite beam after a fire as described in claim 1, characterized in that: in step (1), the numerical simulation includes the following specific steps: (11) Fire test simulation: Abaqus software was used to simulate the mechanical properties of T-shaped composite beams under uniformly distributed load at high temperature using sequential thermo-mechanical coupling: that is, firstly, a temperature field model was established, and then the obtained temperature field was used as a known condition to perform thermal stress analysis to obtain the stress and strain field at high temperature. Finally, a numerical model was established to simulate the static loading test after high temperature. In the T-beam model established for analyzing heat transfer problems, the concrete uses an eight-node linear heat transfer hexahedral solid element, and the steel reinforcement uses a two-node heat transfer element. The steel reinforcement and concrete are bound together with a tie to ensure the heat transfer between the steel reinforcement and concrete. The temperature is increased according to the experimentally measured furnace temperature. Except for the top surface, the other surfaces of the beam are exposed to the fire and are uniformly exposed to the fire. The temperature field in the temperature field model is exported and preset as a predefined field into the stress analysis model. The concrete in this model uses an eight-node hexahedral element. The element is reduced and integrated using hourglass control. The steel reinforcement uses a two-node linear three-dimensional truss element. The steel reinforcement and concrete are connected by an embedded method to ensure that the two parts are subjected to the same force. The connection method at the overlapping surface of the T-shaped composite beam is to use Cartesian elements to create a dimensionless three-dimensional spring to simulate the interaction at the overlapping surface. (12) Conduct static loading test simulation after fire: After the fire, apply a progressively increasing concentrated load at two third points on the beam until the beam is completely destroyed and reaches the ultimate bearing capacity. In this model, the concrete adopts an eight-node hexahedral element, the steel reinforcement adopts a two-node linear three-dimensional truss element, and the steel reinforcement and concrete adopt an embedded connection method to ensure that the two parts are subjected to the same force. (13) Finite element simulation verification: Plot the experimental and simulated values ​​of the temperature field at different measuring points at the mid-span section of the T-shaped composite beam specimen under fire, plot the deflection at the mid-span position under fire and after fire, and compare and verify the experimental and simulated values.

4. The method for calculating the residual bending stiffness of a T-shaped composite beam after a fire, as described in claim 1, is characterized by: The step (22) is the fire time t 30 min, 60 min, 90 min, 120 min, 150 min, 180 min respectively; overlapping parameters γ 0.4, 0.5, 0.6 respectively; friction coefficient of overlapping surface λ 0.4, 0.6, 0.8 respectively.

5. The method for calculating the residual bending stiffness of a T-shaped composite beam after a fire, as described in claim 4, is characterized by: In step (22), the relationship between the residual bending stiffness of the T-shaped composite beam and the fire exposure time t, the composite parameter γ, and the friction coefficient λ of the composite surface is analyzed as follows: (221) Analysis of the relationship between the residual bending stiffness of T-shaped composite beams after high temperature and the fire duration t : Under the same conditions of the composite parameters γ and the friction coefficient of the composite surface λ , the change of the reduction factor between the cast-in-place beam and the T-shaped composite beam is plotted into a graph, and the relationship between the fire duration of the T-shaped composite beam and the reduction factor is analyzed according to the graph drawn. (222) Relationship analysis between residual bending stiffness of T-shaped composite beam after high temperature and composite parameters γ : When the fire time t and the friction coefficient of the composite surface λ are the same, the change of the reduction factor between the cast-in-place beam and the T-shaped composite beam is plotted into a graph, and the relationship between the composite parameter γ of the T-shaped composite beam and the reduction factor is analyzed according to the plotted graph; (223) Analysis of the relationship between the residual bending stiffness of T-shaped composite beams after high temperature and the friction coefficient of the composite surface: ensuring the fire exposure time λ t and the composite parameters γ Under the same conditions, the change of the reduction coefficient of the residual bending stiffness of T-shaped composite beams is plotted into a graph. According to the graph drawn, the relationship between the friction coefficient λ of the composite surface of the T-shaped composite beam and the reduction coefficient is analyzed.​ 6. The method for calculating the residual bending stiffness of a T-shaped composite beam after a fire as described in claim 1, characterized in that step (3) includes the following specific steps: Using SPSS statistical analysis software, the reduction coefficient relationship between T-shaped cast-in-place beams and T-shaped composite beams is given as follows: （5）； in formula (5), t ∈ [30 min, 180 min], λ ∈ [0.4, 0.8], γ ∈ [0.4, 0.6], l ∈ [0, 44%], the correlation coefficient of formula (5) is R 2 = 0.

979.

7. The method of calculating the residual flexural stiffness of a T-shaped composite beam after fire according to claim 6, characterized in that: Step (3) further includes: to verify the effectiveness of the proposed calculation method for the residual bending stiffness of the T-shaped composite beam after high temperature, an error analysis is performed between the calculated value and the experimental and simulated values ​​of the residual bending stiffness of the T-shaped composite beam after high temperature.

Images