Method for calculating curvature-resistance bending moment curve of cross section of steel and UHPC (Ultra High Performance Concrete) composite structural beam

Abstract

The invention provides a method for calculating a cross section curvature-resistance bending moment curve of a steel and UHPC (Ultra High Performance Concrete) combined structural beam, which comprises the following steps of: iteratively calculating resistance bending moments Mr corresponding to different cross section curvatures # imgabs0 #, and respectively taking an obtained # imgabs1 # sequence and an obtained Mr sequence as a horizontal coordinate and a vertical coordinate, so as to obtain the cross section curvature-resistance bending moment curve of the combined structural beam. According to the calculation method, the problem that a method for obtaining a cross section curvature-resistance bending moment curve through design calculation is lacked in the prior art is solved, engineers can conveniently evaluate the performance of the material in the elastic and plastic stages, and optimization of the structural design of the composite structural beam is facilitated.

Description

Technical Field

[0001] The invention relates to the technical field of bridge engineering, and in particular to a method for calculating a cross-section curvature-resistance bending moment curve of a composite structural beam of steel and UHPC. Background Art

[0002] Traditional prefabricated bridges are characterized by large quantity and wide application, and their main structural forms are prestressed hollow slab bridges, prestressed T-beam bridges, and prestressed small box girder bridges. The above bridges generally have the following characteristics: (1) Ordinary concrete (NC) is used as the main structural material, which is heavy and easy to crack; (2) The steel bars and prestressed bars are arranged in the concrete body, and their distribution and material consumption are limited by the concrete structure size and casting quality requirements, resulting in the internal force arm being less than the concrete section height when subjected to bending, and low efficiency.

[0003] With the popularity of ultra-high performance concrete (UHPC), existing bridge technologies in recent years have attempted to use UHPC to replace NC as the main building material to achieve the requirements of reducing deadweight and improving performance, such as steel beam-UHPC bridge deck composite beams, but there are also defects such as the material properties cannot be fully utilized, the amount of steel used and the amount of welding work are large. In order to make full use of the excellent performance of steel and UHPC materials and further improve the economy and applicability of steel-UHPC composite beams, some new composite beam structures have been proposed. Among the currently disclosed technologies, Chinese patents with publication numbers CN112391932A and CN112342889A both propose a UHPC-T-beam using steel sections as external reinforcement. The idea is to use UHPC to bear the high pressure stress of the upper flange and the low tensile stress of the composite web, and use steel sections to bear the high tensile stress of the lower part, making full use of the material properties.

[0004] The curvature-resistance bending moment curve provides information about the behavior of composite structural beams under different load conditions, reveals key characteristics such as the yield point, ultimate strength and ductility of the material, and shows how the beam responds to the applied bending moment. It is essential to understand the overall performance of the structure, and allows engineers to evaluate the performance of the material in the elastic and plastic stages, which helps optimize the structural design. Engineers can adjust the size, shape or material of the beam based on this data to achieve the best performance and economy. At present, the curvature-resistance bending moment curve is mainly obtained by model testing, which is greatly limited in efficiency, economy and convenience. The published technology does not provide a method for obtaining the cross-sectional curvature-resistance bending moment curve through design calculation.

[0005] In summary, a method for calculating the cross-sectional curvature-resistance bending moment curve of a composite structural beam of steel and UHPC is urgently needed to solve the problems existing in the prior art. Summary of the invention

[0006] The present invention aims to provide a method for calculating the cross-sectional curvature-resistance bending moment curve of a composite structural beam of steel and UHPC, aiming to solve the problem in the prior art that there is a lack of a method for obtaining the cross-sectional curvature-resistance bending moment curve by design calculation. The specific technical solution is as follows:

[0007] A method for calculating the cross-sectional curvature-resistance bending moment curve of a composite structural beam of steel and UHPC comprises the following steps:

[0008] S1. Set constant parameters, including the calculation error convergence value θ and the curvature iteration calculation increment and the neutral axis position iteratively calculate the increment Δx n ;

[0009] S2. Input the cross-sectional dimensions of the composite structural beam, the properties of each material in the composite structural beam and the nonlinear constitutive model;

[0010] S3. Divide the cross section of the composite structural beam into unit layers, specifically:

[0011] The cross section of the UHPC concrete of the composite structural beam and the cross section of the T-shaped steel beam outside the UHPC concrete are divided into unit layers respectively. At the same time, the cross section of all the longitudinal steel bars in the uppermost layer of the composite structural beam and the cross section of all the longitudinal steel bars in the lowermost layer of the composite structural beam are respectively regarded as a unit layer;

[0012] S4, take the initial value of N equal to 1;

[0013] S5. Calculate the curvature of the cross section of the composite structural beam

[0014] S6. Assume that the neutral axis position x of the composite structural beam n ;

[0015] S7, calculating and obtaining the strain of each unit layer, and calculating and obtaining the stress of each unit layer according to the strain of each unit layer and the nonlinear constitutive model corresponding to the unit layer;

[0016] S8. Calculate the resultant force F in the cross section of the composite structure beam according to the stress and area of ​​each unit layer;

[0017] S9, if |F| is not greater than θ, then proceed to step S10; if |F| is not greater than θ, then correct x n Then repeat steps S7 to S9;

[0018] S10. Determine whether there is an upper edge compressive strain ε of UHPC concrete top Less than its compressive limit strain ε cu , or, the tensile strain ε at the lower edge of the T-beam bottom is greater than its ultimate tensile strain; if it exists, let Δxn =0.1Δx n Then repeat steps S6 to S10, if it does not exist, go to step S11;

[0019] S11. Calculate the resisting bending moment M of the composite structure beam according to the acting moment of each unit layer r ;

[0020] S12. Determine whether the upper edge compressive strain ε of UHPC concrete is satisfied top Subtract its compressive limit strain ε cu Less than or equal to 0.1·θ, or, the ultimate tensile strain of the T-beam minus the tensile strain at its lower edge ε bottom Less than or equal to 0.1·θ; if satisfied, proceed to step S13; if not satisfied, record the current (φ, M r ) as a coordinate point of the cross-sectional curvature-resistance bending moment curve, set N=N+1 and then repeat steps S5 to S12;

[0021] S13, according to all records (φ, M r )Data output: Cross-sectional curvature-resistance bending moment curve.

[0022] Preferably, in the above technical solution, the nonlinear constitutive model of each material in step S2 is specifically:

[0023] The nonlinear constitutive model of UHPC concrete is:

[0024]

[0025] Among them, ε u is the strain of UHPC concrete, σ u is the stress of UHPC concrete, E u is the elastic modulus of UHPC concrete; ε cu is the ultimate compressive strain of UHPC concrete; ε te is the tensile elastic limit strain of UHPC concrete; ε tu is the ultimate tensile strain of UHPC concrete; ε tr is the fracture strain of UHPC concrete; f te is the tensile elastic limit stress of UHPC concrete; f tu is the ultimate tensile stress of UHPC concrete;

[0026] The nonlinear constitutive models of T-beam and steel bars in the composite structural beam are:

[0027]

[0028] Among them, ε s is the strain of the T-beam or steel bar, σs is the stress of the T-beam or steel bar, E s is the elastic modulus of the T-beam or steel bar; ε' y is the ultimate compressive strain of the T-beam or steel bar; ε y is the tensile yield strain of the T-beam or steel bar; ε st is the strain hardening starting strain of the T-beam or steel bar; ε' u is the ultimate tensile strain of the T-beam or steel bar; if there is no yield platform, then ε y =ε st ;f y is the yield stress of the T-beam or steel bar; f u It is the ultimate tensile stress of T-beam or steel bar.

[0029] In the above technical solution, preferably, the strain of each unit layer calculated in step S7 is:

[0030] Strain of the i-th unit layer in UHPC concrete Positive values ​​are tensile strains, negative values ​​are compressive strains; the strain of the kth unit layer in the T-beam is Positive values ​​are tensile strains, negative values ​​are compressive strains; the strains of all longitudinal reinforcements in the bottom layer of the composite structural beam corresponding to the unit layer The strain of all longitudinal reinforcements in the uppermost layer of the composite structural beam corresponding to the unit layer

[0031] Among them, y i is the vertical distance from the center of the i-th unit layer in UHPC concrete to the upper surface of the composite structural beam, y k is the vertical distance from the center of the kth unit layer in the T-shaped steel beam to the upper surface of the composite structure beam, y' s ' is the vertical distance from the center of the unit layer corresponding to all longitudinal reinforcements at the bottom layer of the composite structural beam to the upper surface of the composite structural beam, y' s It is the vertical distance from the center of the unit layer corresponding to all the longitudinal reinforcements in the uppermost layer of the composite structural beam to the upper surface of the composite structural beam.

[0032] Preferably, in the above technical solution, in step S7, the stress of each unit layer is calculated according to the strain of each unit layer and the nonlinear constitutive model corresponding to the unit layer, specifically:

[0033] According to formula (1), the stress σ of the unit layer in UHPC concrete is obtained u,i =σ u (ε u,i ), positive value is tensile stress, negative value is compressive stress; according to formula (2), the stress σ of the unit layer in the T-beam is obtained h,k =σ s (εh,k ), positive value is tensile stress, negative value is compressive stress; according to formula (2), the stress σ" of all longitudinal reinforcements in the bottom layer of the composite structure beam is obtained. s =σ s (ε” s ); According to formula (2), the stress σ' of all longitudinal reinforcements corresponding to the unit layer of the top layer of the composite structure beam is obtained s =σ s (ε' s ).

[0034] Preferably, in the above technical solution, the resultant force F in the cross section of the composite structural beam in step S8 is:

[0035] F=∑A i σ u,i +∑A k σ h,k +A″ s σ″ s +A′ s σ′ s (4)

[0036] Among them, A i is the area of ​​the i-th unit layer in UHPC concrete, A k is the area of ​​the kth unit layer in the T-beam, A s " is the area of ​​the unit layer corresponding to all longitudinal reinforcements at the bottom layer of the composite structural beam, A s ' is the area of ​​the unit layer corresponding to all the longitudinal reinforcements in the uppermost layer of the composite structural beam.

[0037] In the above technical solution, preferably, the resistance bending moment M of the composite structural beam in step S11 is r for:

[0038]

[0039] In the above technical solution, the minimum compressive strain is selected from each unit layer of UHPC concrete, which is the upper edge compressive strain ε of UHPC concrete. top , the maximum tensile strain is selected from each unit layer of the T-beam, which is the tensile strain of the lower edge of the T-beam ε bottom .

[0040] In the above technical solution, preferably, in step S9, the influence coefficient method is used to iteratively correct x n , specifically:

[0041] S9.1. Assume that the new neutral axis position x' of the composite structural beam n , where x' n =x n +Δx n ;

[0042] S9.2, repeat steps S7-S8 to calculate x' n The resultant force F' in the cross section of the corresponding composite structural beam;

[0043] S9.3. Calculate the change in the resultant force in the cross section of the composite structural beam, ΔF, where ΔF = F'-F;

[0044] S9.4. Correction of the neutral axis position x of the composite structural beam n ,in

[0045] Preferably, in the above technical solution, step S3 further comprises: obtaining the vertical distance between the center of each unit layer and the upper surface of the composite structure beam and the area of ​​each unit layer.

[0046] In the above technical solution, preferably, step S13 is specifically:

[0047] by The value is the X coordinate, M r The value is the Cartesian coordinate system for the Y coordinate, and all the recorded (φ, M r )Data by The values ​​are arranged in ascending order, and the point plotting method is used to plot all the recorded (φ, M r )Data according to The order of the values ​​is plotted in a Cartesian coordinate system as a curvature-resistance moment curve.

[0048] The application of the technical solution of the present invention has the following beneficial effects:

[0049] The present invention provides a calculation method for obtaining a cross-sectional curvature-resistance bending moment curve from a design calculation perspective. The cross-sectional curvature-resistance bending moment curve provides behavior information of a composite structural beam under different load conditions, reveals key properties such as the yield point, ultimate strength and ductility of the material, and shows how the beam responds to an applied bending moment, making it easy for engineers to evaluate the performance of the material in the elastic and plastic stages, and helps to optimize the structural design. Engineers can adjust the size, shape or material of the beam based on these data to achieve optimal performance and economy.

[0050] In the calculation method of the present invention, different constitutive models are used according to the performance characteristics of UHPC, T-shaped steel beams and steel bars, and the contribution of material strain hardening performance to resistance to bending moment is fully considered. Compared with the ideal elastic-plastic model, the real performance of the structure can be obtained and waste can be reduced.

[0051] In the calculation method of the present invention, the influence coefficient method is used to search for the neutral axis position, which helps to speed up the search and solves the problem of non-convergence caused by artificially setting the iteration increment.

[0052] In the calculation method of the present invention, the top strain of the UHPC concrete reaches the UHPC compressive limit strain or the bottom strain of the T-shaped steel beam reaches the steel limit strain as a judgment condition, which can take into account both the UHPC crushing and the steel reaching the tensile limit strength failure mode.

[0053] In addition to the above-described purposes, features and advantages, the present invention has other purposes, features and advantages. The present invention will be further described in detail with reference to the accompanying drawings. BRIEF DESCRIPTION OF THE DRAWINGS

[0054] The drawings constituting a part of this application are used to provide a further understanding of the present invention. The exemplary embodiments of the present invention and their descriptions are used to explain the present invention and do not constitute an improper limitation of the present invention. In the drawings:

[0055] Figure 1 It is an axonometric view of the composite structural beam of steel and UHPC;

[0056] Figure 2 yes Figure 1 Structural diagram of the middle vertical stirrup group;

[0057] Figure 3 yes Figure 1 Schematic diagram of the structure of the steel tenon and mortise;

[0058] Figure 4 It is a flow chart of a method for calculating a curvature-resistance bending moment curve of a cross section of a composite structural beam of the present invention;

[0059] Figure 5 It is a schematic diagram of the nonlinear constitutive model of UHPC concrete;

[0060] Figure 6 It is a schematic diagram of the nonlinear constitutive model of a T-beam or steel bar;

[0061] Figure 7 It is a schematic diagram of the cross section of a composite structural beam;

[0062] Figure 8 It is a schematic diagram of strain distribution in the cross section of the composite structure beam;

[0063] Fig. 9 It is a schematic diagram of stress distribution in the cross section of the composite structure beam;

[0064] Fig.10 It is a schematic diagram of the curvature-resistance bending moment curve output in the application case;

[0065] Among them, 100, composite structure beam, 110, UHPC bridge deck, 111, bridge deck transverse reinforcement, 112, bridge deck longitudinal reinforcement, 120, UHPC rib, 121, lower open stirrups, 122, upper open stirrups, 123, the bottom layer of longitudinal reinforcement, 124, the second bottom layer of longitudinal reinforcement, 125, longitudinal distribution reinforcement, 126, transverse short reinforcement, 130, T-shaped steel beam, 131, steel bottom plate, 132, steel web, 133, steel tenon, 134, mortise. DETAILED DESCRIPTION

[0066] In order to facilitate the understanding of the present invention, the present invention will be described more fully below, and preferred embodiments of the present invention are provided. However, the present invention can be implemented in many different forms and is not limited to the embodiments described herein. On the contrary, the purpose of providing these embodiments is to make the understanding of the disclosure of the present invention more thorough and comprehensive.

[0067] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as those commonly understood by those skilled in the art of the present invention. The terms used in the specification of the present invention herein are only for the purpose of describing specific embodiments and are not intended to limit the present invention.

[0068] Example:

[0069] Figure 1-Figure 3 A composite structural beam of steel and UHPC is illustrated, the composite structural beam 100 comprises a UHPC bridge deck 110 and a composite rib (not shown) located on the bottom surface of the UHPC bridge deck 110, the composite rib comprises a UHPC rib 120 and a T-shaped steel beam 130, the UHPC rib 120 is located between the UHPC bridge deck 110 and the T-shaped steel beam 130, that is, the UHPC rib 120 and the UHPC bridge deck 110 are cast together to form a T-shaped structure, and the upper edge of the T-shaped steel beam 130 is buried inside the lower end of the UHPC rib 120.

[0070] Furthermore, the T-shaped steel beam 130 includes a steel bottom plate 131 and a steel web 132 disposed on the steel bottom plate 131, the two forming a T-shaped structure, and the upper edge of the steel web 132 (i.e., the side close to the UHPC rib 120) is alternately provided with steel tenons 133 and mortises 134 along the longitudinal bridge direction. Figure 3 As shown, Figure 3 The gray area in the figure is the mortise 134, and the distance between two adjacent tenons 133 is e. x The height of the steel tenon (i.e. the depth of the mortise) is h D The steel tenon 133 and the mortise 134 are buried in the concrete at the lower end of the UHPC rib 120, and the distance between the lower edge of the UHPC rib 120 and the bottom edge of the mortise 134 is CD .

[0071] Furthermore, the UHPC rib 120 includes a rib concrete layer and a rib reinforcement skeleton located in the rib concrete layer, wherein the rib reinforcement skeleton includes a bottom layer of longitudinal reinforcement 123, a second bottom layer of longitudinal reinforcement 124, a longitudinal distribution reinforcement 125, a transverse short reinforcement 126 and a vertical stirrup group (not shown in the figure); the vertical stirrup group includes lower open stirrups 121 and upper open stirrups 122 alternately arranged along the longitudinal bridge direction, such as Figure 1 and Figure 2 As shown, the lower open stirrup 121 and the steel tenon 133 are arranged in a one-to-one correspondence, the upper open stirrup 122 and the mortise 134 are arranged in a one-to-one correspondence, the steel tenon 133 is inserted into the opening at the lower end of the lower open stirrup 121, the lower end of the upper open stirrup 122 is placed in the mortise 134, and transverse short steel bars 126 are provided on both sides of the lower end of the upper open stirrup 122, and the transverse short steel bars 126 are spaced from the lower end of the upper open stirrup 122 (that is, the two are not overlapped); the position of the lower end of the lower open stirrup 121 is lower than that of the upper open stirrup 122 At the lower end, the bottom layer of longitudinal steel bars 123 is used to connect the lower ends of the lower open stirrups 121 along the longitudinal direction, and the second bottom layer of longitudinal steel bars 124 is used to connect the lower ends of the upper open stirrups 122 along the longitudinal direction. Specifically, in this embodiment, the number of the bottom layer of longitudinal steel bars 123 and the second bottom layer of longitudinal steel bars 124 are both two, and they are symmetrically arranged on both sides of the steel web 132; a plurality of longitudinal distribution steel bars 125 are arranged inside the vertical stirrup group along the vertical direction, and specifically, a single longitudinal distribution steel bar 125 passes through the alternately arranged upper open stirrups 122 and lower open stirrups 121 along the longitudinal direction.

[0072] Furthermore, the UHPC bridge deck 110 includes a bridge deck concrete layer and a bridge deck steel bar skeleton located in the bridge deck concrete layer, the bridge deck steel bar skeleton includes an upper transverse steel bar group, a lower transverse steel bar group and a bridge deck longitudinal steel bar 112; a plurality of bridge deck longitudinal steel bars 112 are arranged at intervals along the transverse direction of the bridge, the upper transverse steel bar group is located on the upper side of the bridge deck longitudinal steel bars 112, the upper transverse steel bar group includes a plurality of bridge deck transverse steel bars 111 arranged at intervals along the longitudinal direction of the bridge, the bridge deck longitudinal steel bars 112 are overlapped with the bridge deck transverse steel bars 111 in the upper transverse steel bar group; the lower transverse steel bar group is located on the lower side of the bridge deck longitudinal steel bars 112, the lower transverse steel bar group includes a plurality of bridge deck transverse steel bars 111 arranged at intervals along the longitudinal direction of the bridge, and the bridge deck transverse steel bars 111 in the lower transverse steel bar group are not overlapped with the bridge deck longitudinal steel bars 112.

[0073] Furthermore, the upper end of the lower open stirrup 121 is connected to the bridge deck reinforcement skeleton, specifically, overlapped with the bridge deck longitudinal reinforcement 112 .

[0074] Preferably, the bridge deck concrete layer and the rib concrete layer are both cast using UHPC concrete, and the UHPC concrete is an ultra-high performance concrete having ultra-high durability and ultra-high mechanical properties.

[0075] In order to facilitate engineers to understand the behavior information of composite structural beams under different load conditions, reveal the key characteristics of materials such as yield point, ultimate strength and ductility, so that composite structural beams can achieve the best performance and economy. This embodiment provides a method for calculating the cross-sectional curvature-resistance bending moment curve of composite structural beams of steel and UHPC, such as Figure 4 As shown, the calculation method includes the following steps:

[0076] A1. Set constant parameters, including the calculation error convergence value θ and the curvature iteration calculation increment and the neutral axis position iteratively calculate the increment Δx n ;

[0077] Preferably, The value of is set according to the accuracy of the cross-sectional curvature-resistance bending moment curve, generally equal to 10 -4 ~10 -3 (m -1 );The value of the calculation error convergence value θ is generally 10 -5 ; Iterative calculation increment Δx of neutral axis position n The value is generally 0.1mm;

[0078] A2. Input the cross-sectional dimensions of the composite structural beam, the properties of each material and the nonlinear constitutive model;

[0079] Different from ordinary concrete, UHPC concrete has the characteristics of strain hardening, and its strength after cracking still contributes to the bending bearing capacity (strain softening). In order to fully consider its influence, UHPC tensile resistance adopts a three-fold line model, UHPC compression resistance adopts a linear elastic model, and T-shaped steel beams and steel bars in the composite structure beams are tensile. The three-fold line model considering the yield platform and strain hardening is used. Because the project focuses on the maximum bearing capacity, except for UHPC tensile resistance, the material constitutive model does not consider the descending section.

[0080] Preferably, Figure 5 As shown in Figure 2, the nonlinear constitutive model of UHPC concrete is:

[0081]

[0082] Among them, ε uis the strain of UHPC concrete, σ u is the stress of UHPC concrete, E u is the elastic modulus of UHPC concrete; ε cu is the ultimate compressive strain of UHPC concrete; ε te is the tensile elastic limit strain of UHPC concrete; ε tu is the ultimate tensile strain of UHPC concrete; ε tr is the fracture strain of UHPC concrete; when ε u =ε cu When u =f cu , reaches the ultimate compressive stress, f cu is the compressive ultimate stress of UHPC concrete; f te is the tensile elastic limit stress of UHPC concrete, f te =E u ·ε te ;f tu is the tensile ultimate stress of UHPC concrete, and the corresponding strain is ε tu .

[0083] like Figure 5 As shown in the figure, the second section of the first quadrant in the nonlinear constitutive model of UHPC considers the contribution of UHPC tensile strain hardening to the bending bearing capacity, and the third section considers the contribution of UHPC to the bending bearing capacity after cracking.

[0084] Preferably, Figure 6 As shown in the figure, the nonlinear constitutive models of the T-beam and steel bars in the composite structural beam are:

[0085]

[0086] Among them, ε s is the strain of the T-beam or steel bar, σ s is the stress of the T-beam or steel bar, E s is the elastic modulus of the T-beam or steel bar; ε' y is the ultimate compressive strain of the T-beam or steel bar; ε y is the tensile yield strain of the T-beam or steel bar; ε st is the strain hardening starting strain of the T-beam or steel bar; ε' u is the ultimate tensile strain of the T-beam or steel bar; for the case of no yield platform, such as some specifications of steel bars, ε y =ε st ;

[0087] When ε s =ε' y When s =f'y , reaching the ultimate compressive stress, f' y is the compressive limit stress of the T-beam or steel bar; f y is the yield stress of the T-beam or steel bar, f y =E s ·ε y ;f u is the ultimate tensile stress of the T-beam or steel bar, and the corresponding strain is ε u .

[0088] like Figure 6 As shown in the figure, the second section of the first quadrant in the nonlinear constitutive model is the yield platform considering the stress-strain relationship of the T-shaped steel beam or steel bar, and the third section is the contribution of the strain hardening of the T-shaped steel beam or steel bar to the bending bearing capacity.

[0089] A3. Figure 7 As shown in the figure, the cross section of the UHPC concrete of the composite structural beam is divided into unit layers, and the vertical distance y from the center of each unit layer to the upper surface of the composite structural beam is obtained. i And the area A of each unit layer i ; Divide the cross section of the T-shaped steel beam outside the UHPC concrete into unit layers, and obtain the vertical distance y between the center of each unit layer and the upper surface of the composite structure beam k And the area A of each unit layer k ; Take the cross section of all longitudinal reinforcements at the bottom layer of the composite structural beam as a unit layer and obtain its area A s " and the vertical distance y' from the upper surface of the composite structural beam s '; Take the cross section of all longitudinal reinforcements in the top layer of the composite structural beam as a unit layer and record its area A s ' and the vertical distance y from the upper surface of the composite structural beam s ;

[0090] Preferably, i represents the i-th unit layer in the UHPC concrete, and k represents the k-th unit layer in the T-shaped steel beam.

[0091] Specifically, the upper surface of the composite structure beam is the upper surface of the UHPC bridge deck, that is, the bridge deck; the bottom layer of longitudinal steel bars of the composite structure beam refers to the bottom layer of longitudinal steel bars 123 in the UHPC rib plate, and the top layer of longitudinal steel bars of the composite structure beam refers to the bridge deck longitudinal steel bars 112 in the UHPC bridge deck.

[0092] Preferably, all unit layers should be on the same cross section of the composite structural beam. When dividing the cross section of the UHPC concrete and the cross section of the T-shaped steel beam outside the UHPC concrete into unit layers, layered forced dividing lines are arranged at all locations where the cross-sectional width and material change, so as to facilitate the calculation of the area of ​​each unit layer and subsequent internal forces. According to the above unit layer division rules, it can be seen that except for the longitudinal steel bars at the bottom and top layers of the composite structural beam, the rest of the steel structure embedded in the UHPC concrete (referring to the steel bars embedded in the UHPC concrete and part of the steel web) is not considered for calculation, which can facilitate the calculation.

[0093] Preferably, the area of ​​a single unit layer in UHPC concrete and T-shaped steel beam is calculated as follows:

[0094] A=b·h' (3),

[0095] Where: b is the width of the unit layer, which is determined by the cross-sectional shape; h' is the height of the unit layer. To ensure the accuracy of the calculation and analysis, the height h' of each unit layer shall not exceed 1 / 50h; h is the total height of the cross section.

[0096] A4. Set the initial value of N to 1;

[0097] A5. Calculate the curvature of the cross section of the composite structural beam

[0098] A6. Assume that the neutral axis position of the composite structural beam is x n , where x n =x n,0 ; Preferably, x n,0 The general value is h / 2.

[0099] A7. Calculate the strain ε of the unit layer in UHPC concrete based on the plane section assumption u,i , the strain of the unit layer in the T-beam ε h,k , the strain of all longitudinal reinforcements in the bottom layer of the composite structure beam corresponding to the unit layer ε" s and the strain ε' of all longitudinal reinforcements corresponding to the unit layer in the top layer of the composite structure beam s ,like Figure 8 As shown;

[0100] Specifically, the strain of the i-th unit layer in UHPC concrete is Positive values ​​are tensile strains, negative values ​​are compressive strains; the strain of the kth unit layer in the T-beam is Positive values ​​are tensile strains, negative values ​​are compressive strains; the strains of all longitudinal reinforcements in the bottom layer of the composite structural beam corresponding to the unit layer The strain of all longitudinal reinforcements in the uppermost layer of the composite structural beam corresponding to the unit layer

[0101] A8. Calculate the stress σ of each unit layer in UHPC concrete according to the nonlinear constitutive model of the material. u,i , the stress σ of each unit layer in the T-beam h,k , the stress of all longitudinal reinforcements in the bottom layer of the composite structure beam corresponding to the unit layer σ” s and the stress σ' of the corresponding unit layer of all longitudinal reinforcements in the top layer of the composite structure beam s ,like Fig. 9 As shown;

[0102] Specifically, according to formula (1), the stress σ of the unit layer in UHPC concrete is obtained: u,i =σ u (ε u,i ), positive value is tensile stress, negative value is compressive stress; according to formula (2), the stress σ of the unit layer in the T-beam is obtained h,k =σ s (ε h,k ), positive value is tensile stress, negative value is compressive stress; according to formula (2), the stress σ" of all longitudinal reinforcements in the bottom layer of the composite structure beam is obtained. s =σ s (ε” s ); According to formula (2), the stress σ' of all longitudinal reinforcements corresponding to the unit layer of the top layer of the composite structure beam is obtained s =σ s (ε' s ).

[0103] A9. Calculate the resultant force F in the cross section of the composite structure beam based on the stress and area of ​​each unit layer;

[0104] Preferably, the resultant force F in the cross section of the composite structural beam is:

[0105] F=∑A i σ u,i +∑A k σ h,k +A″ s σ″ s +A′ s σ′ s (4).

[0106] A10, if |F| is not greater than (i.e., less than or equal to) θ, then proceed to step A11; if |F| is not greater than θ, then use the influence coefficient method to iteratively correct x n Then repeat steps A7-A10;

[0107] Preferably, the influence coefficient method is used to iteratively correct x n Specifically:

[0108] A10.1. Assume that the new neutral axis position of the composite structural beam is x' n , where x' n =x n +Δx n ;

[0109] A10.2. Repeat steps A7-A9 to calculate x' n The resultant force F' in the cross section of the corresponding composite structural beam;

[0110] A10.3. Calculate the change in the resultant force in the cross section of the composite structural beam, ΔF, where ΔF = F'-F;

[0111] A10.4. Correction of the neutral axis position x of the composite structural beam n ,in

[0112] Follow steps A10.1-A10.4 to complete x n The iterative correction and influence coefficient method can speed up the search and solve the problem of non-convergence caused by artificially set iterative increments.

[0113] A11. Determine whether there is an upper edge compressive strain ε of UHPC concrete top Less than its compressive limit strain ε cu , or, the tensile strain ε at the lower edge of the T-beam bottom is greater than its ultimate tensile strain; if it exists, let Δx n =0.1Δx n Then repeat steps A6 to A11, if it does not exist, go to step A12;

[0114] Specifically, if there is a compressive strain ε at the upper edge of UHPC concrete top Less than its compressive limit strain ε cu , or, the tensile strain ε at the lower edge of the T-beam bottom Greater than its ultimate tensile strain ε' u1 , it means that during the calculation process, the compressive strain on the upper edge of the UHPC exceeds its compressive limit strain, or the tensile strain on the lower edge of the T-shaped steel beam exceeds its ultimate tensile strain, indicating that the step size taken in the calculation is too large.

[0115] Preferably, the upper edge compressive strain ε of UHPC concrete is top and the tensile strain ε at the lower edge of the T-beam bottom All are obtained from step A7. The minimum compressive strain is selected from each unit layer of UHPC concrete as ε top , the maximum tensile strain is selected from each unit layer of the T-beam as ε bottom .

[0116] A12. Calculate the resisting bending moment M of the composite structure beam according to the moment of each unit layer in the UHPC concrete, the moment of each unit layer in the T-shaped steel beam, the moment of all longitudinal steel bars corresponding to the unit layer in the bottom layer of the composite structure beam, and the moment of all longitudinal steel bars corresponding to the unit layer in the top layer of the composite structure beam. r ;

[0117] Preferably, the composite structural beam resists the bending moment M r for:

[0118]

[0119] A13. Determine whether the upper edge compressive strain ε of UHPC concrete is satisfied top Subtract its compressive limit strain ε cu Less than or equal to 0.1·θ, or, the ultimate tensile strain of the T-beam minus the tensile strain at its lower edge ε bottom Less than or equal to 0.1·θ; if satisfied, proceed to step A14; if not satisfied, record the current (φ, M r ) as a coordinate point of the cross-sectional curvature-resistance bending moment curve, set N=N+1 and repeat steps A5-A13;

[0120] Search for the maximum resisting moment M r The corresponding cross-sectional curvature When the strain at the top of the UHPC concrete reaches the UHPC compressive limit strain or the strain at the bottom of the T-shaped steel beam reaches the steel limit strain as the judgment condition, the two failure modes of UHPC crushing and steel reaching the tensile limit strength can be taken into account at the same time. In order to take into account both the calculation efficiency and the reliability of convergence, the judgment limit is 0.1·θ, which can ensure that the stress error is within 0.1MPa, which is sufficient in engineering accuracy.

[0121] A14, according to all records (φ, M r )Data output: Cross-sectional curvature-resistance bending moment curve.

[0122] Specifically, The value is the X coordinate, M r The value is the Cartesian coordinate system for the Y coordinate, and all the recorded (φ, M r )Data by The values ​​are arranged in ascending order, and the point plotting method is used to plot all the recorded (φ, M r )Data according to The order of the values ​​is plotted in a Cartesian coordinate system as a curvature-resistance moment curve.

[0123] Application examples:

[0124] In this case, based on the specific structural dimensions of a composite structural beam, the above calculation method is used to obtain the cross-sectional curvature-resistance bending moment curve of the composite structural beam, as follows:

[0125] Step 01, set constant parameters: calculate error convergence value θ = 10 -5 , curvature iteration calculation increment Δφ=10 -3 (m -1 ), iterative calculation increment Δx of the neutral axis position n =0.1mm.

[0126] Step 02: Obtain the cross-sectional dimensions of the composite structural beam, the properties of each material, and the nonlinear constitutive model;

[0127] The nonlinear constitutive model of UHPC is shown in formula (1), and the nonlinear constitutive model of T-beam and steel bar is shown in formula (2).

[0128] Elastic modulus E of UHPC concrete u =45.7Gpa, compressive limit strain ε cu =0.0029, ultimate compressive stress f cu =145Mpa; tensile elastic limit stress f of UHPC concrete te =10.87MPa, ultimate tensile stress f tu =11.66MPa, tensile elastic limit strain ε te =0.0002174, ultimate tensile strain ε tu =0.00293, fracture strain ε tr =0.01.

[0129] Elastic modulus E of steel bar (HRB400) s =200GPa, compressive limit strain ε′ y =0.002, ultimate compressive stress f′ y =400MPa; yield stress of steel bar f y =400Mpa, ultimate stress f u =565MPa, yield strain ε y = 0.002, strain hardening starting strain ε without yield platform st =ε y , ultimate tensile strain ε' u =0.016.

[0130] Elastic modulus E of T-beam (Q355) s =206GPa, yield stress f y =355Mpa, ultimate tensile stress f u =512MPa, yield strain ε y=0.001723, strain hardening starting strain ε st =0.0198, ultimate tensile strain ε' u =0.035;

[0131] Step 03, dividing the cross section of the composite structural beam into unit layers according to step A3;

[0132] Among them, the total height of the cross section h = 1200mm, the distance y″ between the unit layer corresponding to all longitudinal reinforcements at the bottom layer of the composite structural beam and the upper edge of the cross section s =728mm, the sum of the cross-sectional areas of all longitudinal reinforcements at the bottom layer of the composite structural beam (i.e. the area of ​​the corresponding unit layer) is A″ s =402.1mm 2 , the distance y′ between the unit layer corresponding to all longitudinal reinforcements in the top layer of the composite structure beam and the upper edge of the section s =39mm, the sum of the cross-sectional areas of all longitudinal reinforcements in the top layer of the composite structural beam (i.e. the area of ​​the corresponding unit layer) is A′ s =3217.6mm 2 .

[0133] Step 04: Execute steps A4-A14 to obtain the cross-sectional curvature-resistance bending moment curve as follows: Fig.10 As shown, from Fig.10 It can be seen that the maximum bending moment of the composite structural beam in this case is 5338.3KN.m, and the corresponding cross-sectional curvature is is 0.103;

[0134] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. For those skilled in the art, the present invention may have various modifications and variations. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. A method for calculating the cross-sectional curvature-resistance bending moment curve of a composite structural beam of steel and UHPC, characterized in that: The following steps are involved: S1. Set constant parameters, including calculation error convergence value Curvature Iteration Increment and the neutral axis position iteratively calculate the increment Δx n ; S2. Input the cross-sectional dimensions of the composite structural beam, the properties of each material in the composite structural beam and the nonlinear constitutive model; S3. Divide the cross section of the composite structural beam into unit layers, specifically: The cross section of the UHPC concrete of the composite structural beam and the cross section of the T-shaped steel beam outside the UHPC concrete are divided into unit layers respectively. At the same time, the cross section of all the longitudinal steel bars in the uppermost layer of the composite structural beam and the cross section of all the longitudinal steel bars in the lowermost layer of the composite structural beam are respectively regarded as a unit layer; S4, take the initial value of N equal to 1; S5. Calculate the curvature of the cross section of the composite structural beam S6. Assume that the neutral axis position x of the composite structural beam n ; S7, calculating and obtaining the strain of each unit layer, and calculating and obtaining the stress of each unit layer according to the strain of each unit layer and the nonlinear constitutive model corresponding to the unit layer; S8. Calculate the resultant force F in the cross section of the composite structure beam according to the stress and area of ​​each unit layer; S9, if |F| is not greater than Then go to step S10; if |F| is not greater than Then correct x n Then repeat steps S7 to S9; S10. Determine whether there is an upper edge compressive strain ε of UHPC concrete top Less than its compressive limit strain ε cu , or, the tensile strain ε at the lower edge of the T-beam bottom is greater than its ultimate tensile strain; if it exists, let Δx n =0.1Δx n Then repeat steps S6 to S10, if it does not exist, go to step S11; S11. Calculate the resisting bending moment M of the composite structure beam according to the acting moment of each unit layer r ; S12. Determine whether the upper edge compressive strain ε of UHPC concrete is satisfied top Subtract its compressive limit strain ε cu Less than or equal to Or, the ultimate tensile strain of the T-beam minus the tensile strain at its lower edge ε bottom Less than or equal to If it is satisfied, go to step S13; if it is not satisfied, record the current (φ, M r ) as a coordinate point of the cross-sectional curvature-resistance bending moment curve, set N=N+1 and then repeat steps S5 to S12; S13, according to all records (φ, M r )Data output: Cross-sectional curvature-resistance bending moment curve.

2. The method for calculating the cross-sectional curvature-resistance bending moment curve of a composite structural beam of steel and UHPC according to claim 1 is characterized in that: The nonlinear constitutive model of each material in step S2 is specifically: The nonlinear constitutive model of UHPC concrete is: Among them, ε u is the strain of UHPC concrete, σ u is the stress of UHPC concrete, E u is the elastic modulus of UHPC concrete; ε cu is the ultimate compressive strain of UHPC concrete; ε te is the tensile elastic limit strain of UHPC concrete; ε tu is the ultimate tensile strain of UHPC concrete; ε tr is the fracture strain of UHPC concrete; f te is the tensile elastic limit stress of UHPC concrete; f tu is the ultimate tensile stress of UHPC concrete; The nonlinear constitutive models of T-beam and steel bars in the composite structural beam are: Among them, ε s is the strain of the T-beam or steel bar, σ s is the stress of the T-beam or steel bar, E s is the elastic modulus of the T-beam or steel bar; is the ultimate compressive strain of the T-beam or steel bar; ε y is the tensile yield strain of the T-beam or steel bar; ε st is the strain hardening starting strain of the T-beam or steel bar; ε' u is the ultimate tensile strain of the T-beam or steel bar; if there is no yield platform, then ε y =ε st ;f y is the yield stress of the T-beam or steel bar; f u It is the ultimate tensile stress of T-beam or steel bar.

3. The method for calculating the cross-sectional curvature-resistance bending moment curve of a composite structural beam of steel and UHPC according to claim 2, characterized in that: The strain of each unit layer calculated in step S7 is specifically: Strain of the i-th unit layer in UHPC concrete Positive values ​​are tensile strains, negative values ​​are compressive strains; the strain of the kth unit layer in the T-beam is Positive values ​​are tensile strains, negative values ​​are compressive strains; the strains of all longitudinal reinforcements in the bottom layer of the composite structural beam corresponding to the unit layer The strain of all longitudinal reinforcements in the uppermost layer of the composite structural beam corresponding to the unit layer Among them, y i is the vertical distance from the center of the i-th unit layer in UHPC concrete to the upper surface of the composite structural beam, y k is the vertical distance from the center of the kth unit layer in the T-shaped steel beam to the upper surface of the composite structure beam, y' s ' is the vertical distance from the center of the unit layer corresponding to all longitudinal reinforcements at the bottom layer of the composite structural beam to the upper surface of the composite structural beam, y' s It is the vertical distance from the center of the unit layer corresponding to all the longitudinal reinforcements in the uppermost layer of the composite structural beam to the upper surface of the composite structural beam.

4. The method for calculating the cross-sectional curvature-resistance bending moment curve of a composite structural beam of steel and UHPC according to claim 3, characterized in that: In step S7, the stress of each unit layer is calculated based on the strain of each unit layer and the nonlinear constitutive model corresponding to the unit layer, specifically: According to formula (1), the stress σ of the unit layer in UHPC concrete is obtained u,i =σ u (ε u,i ), positive value is tensile stress, negative value is compressive stress; according to formula (2), the stress σ of the unit layer in the T-beam is obtained h,k =σ s (ε h,k ), positive value is tensile stress, negative value is compressive stress; according to formula (2), the stress σ″ of all longitudinal reinforcements in the bottom layer of the composite structure beam is obtained: s =σ s (ε″ s ); According to formula (2), the stress σ' of all longitudinal reinforcements in the uppermost layer of the composite structure beam is obtained. s =σ s (ε' s ).

5. The method for calculating the cross-sectional curvature-resistance bending moment curve of a composite structural beam of steel and UHPC according to claim 4, characterized in that: The resultant force F in the cross section of the composite structural beam in step S8 is: F=∑A i s u,i +∑A k s h,k +A″ s in s +A′ s in s (4), Among them, A i is the area of ​​the i-th unit layer in UHPC concrete, A k is the area of ​​the kth unit layer in the T-beam, A s " is the area of ​​the unit layer corresponding to all longitudinal reinforcements at the bottom layer of the composite structural beam, A s ' is the area of ​​the unit layer corresponding to all the longitudinal reinforcements in the uppermost layer of the composite structural beam.

6. The method for calculating the cross-sectional curvature-resistance bending moment curve of a composite structural beam of steel and UHPC according to claim 5, characterized in that: The resistance bending moment M of the composite structural beam in step S11 r for:

7. The method for calculating the cross-sectional curvature-resistance bending moment curve of a composite structural beam of steel and UHPC according to any one of claims 1 to 6, characterized in that: The minimum compressive strain selected from each unit layer of UHPC concrete is the upper edge compressive strain ε of UHPC concrete. top , the maximum tensile strain is selected from each unit layer of the T-beam, which is the tensile strain of the lower edge of the T-beam ε bottom .

8. The method for calculating the cross-sectional curvature-resistance bending moment curve of a composite structural beam of steel and UHPC according to any one of claims 1 to 6, characterized in that: In step S9, the influence coefficient method is used to iteratively correct x n , specifically: S9.

1. Assuming the new neutral axis position of the composite beam in S9.2, repeat steps S7-S8, and calculate The resultant force F' in the cross section of the corresponding composite structural beam; S9.

3. Calculate the change in the resultant force in the cross section of the composite structural beam, ΔF, where ΔF = F'-F; S9.

4. Correction of the neutral axis position x of the composite structural beam n ,in 9. The method for calculating the cross-sectional curvature-resistance bending moment curve of a composite structural beam of steel and UHPC according to any one of claims 1 to 6, characterized in that: Step S3 also includes: obtaining the vertical distance between the center of each unit layer and the upper surface of the composite structure beam and the area of ​​each unit layer.

10. The method for calculating the cross-sectional curvature-resistance bending moment curve of a composite structural beam of steel and UHPC according to any one of claims 1 to 6, characterized in that: Step S13 specifically includes: by The value is the X coordinate, M r The value is the Y coordinate to establish a Cartesian coordinate system, and all the recorded (φ, M r )Data by The values ​​are arranged in ascending order, and the point plotting method is used to plot all the recorded (φ, M r )Data according to The order of the values ​​is plotted in a Cartesian coordinate system as a curvature-resistance moment curve.

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