A method for calculating the curvature-resistance moment curve of a composite beam cross section of steel and uhpc

Abstract

The application provides a calculation method for a cross-section curvature-resistance bending moment curve of a combined structure beam of steel and UHPC, which is obtained by iteratively calculating the resistance bending moment M r The obtained sequence and M r The sequence is taken as the horizontal coordinate and the vertical coordinate respectively, and the cross-section curvature-resistance bending moment curve of the combined structure beam is obtained. The calculation method solves the problem of lacking a method for obtaining the cross-section curvature-resistance bending moment curve through design calculation in the prior art, facilitates engineers to evaluate the performance of materials in the elastic and plastic stages, and helps to optimize the structural design of the combined structure beam.

Description

Technical Field

[0001] This invention relates to the field of bridge engineering technology, specifically to a method for calculating the cross-sectional curvature-resistance moment curve of a composite structural beam made of steel and UHPC. Background Technology

[0002] Traditional prefabricated bridges are characterized by their large quantity and wide application, with the main structural forms being prestressed hollow slab bridges, prestressed T-beam bridges, and prestressed small box girder bridges. The above-mentioned bridges generally have the following characteristics: (1) They use ordinary concrete (NC) as the main structural material, which is heavy and prone to cracking; (2) The steel bars and prestressing tendons are arranged in the concrete body, and their distribution and material usage are limited by the concrete structure size and pouring quality requirements, resulting in the internal force arm not being greater than the concrete section height when under bending, which is inefficient.

[0003] With the increasing popularity of ultra-high performance concrete (UHPC), existing bridge technologies have recently attempted to use UHPC to replace non-concrete composites (NC) as the main building material to reduce self-weight and improve performance, such as steel-UHPC bridge deck composite beams. However, this approach has drawbacks, including insufficient utilization of material properties and high steel consumption and welding workload. To fully utilize the superior properties of steel and UHPC materials and further improve the economy and applicability of steel-UHPC composite beams, some novel composite beam structures have been proposed. Among the currently published technologies, Chinese patents with publication numbers CN112391932A and CN112342889A both propose a UHPC-T-shaped beam using structural steel as external reinforcement. The underlying concept is to utilize UHPC to bear the high-pressure stress in the upper flange and the low-tensile stress in the composite web, while using structural steel to bear the high-tensile stress in the lower part, thus fully utilizing the material properties.

[0004] Curvature-moment resistance curves provide information on the behavior of composite structural beams under different loading conditions, revealing key properties of materials such as yield point, ultimate strength, and ductility. They demonstrate how the beam responds to applied bending moments, which is crucial for understanding the overall performance of the structure. This allows engineers to assess the material's performance in both elastic and plastic stages, aiding in optimizing structural design. Engineers can adjust beam dimensions, shape, or materials based on this data to achieve optimal performance and economy. Currently, curvature-moment resistance curves are primarily obtained through model testing, which significantly limits efficiency, economy, and convenience. Published techniques do not provide a method for obtaining cross-sectional curvature-moment resistance curves through design calculations.

[0005] In summary, there is an urgent need for a method to calculate the curvature-resistance moment curve of the cross-section of a composite structural beam made of steel and UHPC in order to solve the problems existing in the prior art. Summary of the Invention

[0006] The purpose of this invention is to provide a method for calculating the cross-sectional curvature-bending moment resistance curve of a composite structural beam made of steel and UHPC, aiming to solve the problem that existing technologies lack methods for obtaining the cross-sectional curvature-bending moment resistance curve through design calculations. The specific technical solution is as follows:

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

[0008] S1. Set constant parameters, including calculating the error convergence value. Incremental curvature iteration calculation Incremental calculation of neutral axis position ;

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

[0010] S3. Divide the cross-section of the composite structure 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-beam outside the UHPC concrete are divided into unit layers respectively. At the same time, the cross-section of all longitudinal steel bars in the uppermost layer of the composite structural beam and the cross-section of all longitudinal steel bars in the bottommost layer of the composite structural beam are each taken as a unit layer.

[0012] S4, Take N The initial value is 1;

[0013] S5. Calculate the cross-sectional curvature of the composite structure beam. ;

[0014] S6. Assume the neutral axis position of the composite structure beam. ;

[0015] S7. Calculate the strain of each element layer, and calculate the stress of each element layer based on the strain of each element layer and the corresponding nonlinear constitutive model of the element layer.

[0016] S8. Calculate the resultant force in the cross-section of the composite structural beam based on the stress and area of ​​each unit layer. F ;

[0017] S9, if satisfied Not greater than Then proceed to step S10; if not satisfied. Not greater than Then correct Then repeat steps S7-S9;

[0018] S10. Determine if there is compressive strain at the upper edge of the UHPC concrete. Less than its compressive ultimate strain Or, the tensile strain at the lower edge of the T-beam. Greater than its ultimate tensile strain; if it exists, then let Then repeat steps S6-S10; if it does not exist, proceed to step S11.

[0019] S11. Calculate the bending moment resisting the composite structural beam based on the acting moments of each unit layer. ;

[0020] S12. Determine whether the upper edge compressive strain of the UHPC concrete is satisfied. Subtract its compressive ultimate strain Less than or equal to Alternatively, the ultimate tensile strain of the T-beam minus the tensile strain at its lower edge. Less than or equal to If satisfied, proceed to step S13; otherwise, record the current ( , Let ) be a coordinate point on the cross-sectional curvature-resistance bending moment curve. N = N After +1, repeat steps S5-S12;

[0021] S13, based on all records ( , Data output: cross-sectional curvature-resistance bending moment curve.

[0022] In the preferred embodiment of the above technical solutions, the nonlinear constitutive models of each material in step S2 are specifically as follows:

[0023] The nonlinear constitutive model for UHPC concrete is as follows:

[0024] (1),

[0025] in, For the strain of UHPC concrete. For the stress of UHPC concrete, The elastic modulus of UHPC concrete; This represents the ultimate compressive strain of the UHPC concrete. The tensile elastic limit strain of UHPC concrete; The tensile limit strain of UHPC concrete; The fracture strain of UHPC concrete; The tensile elastic limit stress of UHPC concrete; This represents the ultimate tensile stress of UHPC concrete.

[0026] The nonlinear constitutive models for the T-beams and reinforcing bars in the composite structural beam are as follows:

[0027] (2),

[0028] in, For the strain of the T-shaped steel beam or reinforcing bar. For the stress of T-shaped steel beams or reinforcing bars, The elastic modulus of the T-shaped steel beam or reinforcing bar; This refers to the ultimate compressive strain of the T-shaped steel beam or reinforcing bar. This refers to the tensile yield strain of the T-shaped steel beam or reinforcing bar. The strain hardening initiation strain of the T-shaped steel beam or reinforcing bar; This represents the ultimate tensile strain of the T-shaped steel beam or reinforcing bar; if there is no yield plateau, then... ; The yield stress of the T-shaped steel beam or reinforcing bar; This refers to the ultimate tensile stress of the T-shaped steel beam or reinforcing bar.

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

[0030] UHPC concrete i strain of each unit layer Positive values ​​represent tensile strain, and negative values ​​represent compressive strain; in the T-shaped steel beam... k strain of each unit layer Positive values ​​represent tensile strain, and negative values ​​represent compressive strain; the strain of all longitudinal reinforcement bars at the bottom layer of the composite structural beam corresponds to the strain of the unit layer. The strain of all longitudinal reinforcement bars in the uppermost layer of the composite structural beam corresponds to the strain of the unit layer. ;

[0031] in, For UHPC concrete, the first i The vertical distance from the center of each unit layer to the upper surface of the composite structural beam. The first in the T-shaped steel beam k The vertical distance from the center of each unit layer to the upper surface of the composite structural beam. This refers to the vertical distance from the center of each unit layer corresponding to all the longitudinal reinforcement bars at the bottom layer of the composite structural beam to the upper surface of the composite structural beam. This refers to the vertical distance from the center of each unit layer corresponding to all longitudinal reinforcement bars in the uppermost layer of the composite structural beam to the upper surface of the composite structural beam.

[0032] In the preferred embodiment of the above technical solution, step S7 involves calculating the stress of each element layer based on the strain of each element layer and the corresponding nonlinear constitutive model. Specifically:

[0033] The stress of a single layer in UHPC concrete is obtained according to formula (1). Positive values ​​are tensile stress, and negative values ​​are compressive stress; the stress of a unit layer in a T-shaped steel beam is obtained according to formula (2). Positive values ​​are tensile stress, and negative values ​​are compressive stress; according to formula (2), the stress of all longitudinal reinforcements at the bottom layer of the composite beam is obtained. According to formula (2), the stress of all longitudinal reinforcements in the uppermost layer of the composite structural beam corresponding to the unit layer is obtained. .

[0034] In the preferred embodiment of the above technical solutions, the resultant force within the cross-section of the composite structural beam in step S8... F for:

[0035] (4),

[0036] in, For UHPC concrete, the first i The area of ​​each unit layer The first in the T-shaped steel beam k The area of ​​each unit layer This refers to the area of ​​the unit layer corresponding to all the longitudinal reinforcement bars at the bottom layer of the composite structural beam. This represents the area of ​​the unit layer corresponding to all the longitudinal reinforcement bars in the top layer of the composite structural beam.

[0037] In the preferred embodiment of the above technical solutions, the resisting bending moment of the composite structural beam in step S11 is... for:

[0038] (5).

[0039] In the preferred embodiment of the above technical solutions, the minimum compressive strain selected from each unit layer of the UHPC concrete is the upper edge compressive strain of the UHPC concrete. The maximum tensile strain selected from each unit layer of the T-shaped steel beam is the lower edge tensile strain of the T-shaped steel beam. .

[0040] In the preferred embodiment of the above technical solutions, step S9 employs the influence coefficient method for iterative correction. Specifically:

[0041] S9.1 Assume a new neutral axis position for the composite structural beam. ,in ;

[0042] S9.2, Repeat steps S7-S8 to calculate... The resultant force within the cross section of the corresponding composite structural beam ;

[0043] S9.3 Calculate the change in resultant force within the cross-section of the composite structural beam. ,in ;

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

[0045] In the preferred embodiment of the above technical solution, step S3 further includes: obtaining the vertical distance from the center of each unit layer to the upper surface of the composite structural beam and the area of ​​each unit layer.

[0046] In the preferred embodiment of the above technical solutions, step S13 specifically includes:

[0047] by The value is the X coordinate. Establish a Cartesian coordinate system with the Y coordinate as the value, and record all ( , Data by Arrange the values ​​in ascending order, and use the point plotting method to record all ( , Data according to The order of the values ​​is plotted as a curvature-resistance moment curve in a Cartesian coordinate system.

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

[0049] This invention provides a method for calculating the cross-sectional curvature-resistance moment curve from a design calculation perspective. The cross-sectional curvature-resistance moment curve provides behavioral information of composite structural beams under different load conditions, revealing key properties of the material such as yield point, ultimate strength, and ductility. It demonstrates how the beam responds to the applied bending moment, facilitating engineers to evaluate the material's performance in the elastic and plastic stages. This helps optimize structural design, and engineers can adjust the beam's size, shape, or material based on this data to achieve optimal performance and economy.

[0050] In the calculation method of this invention, different constitutive models are adopted according to the performance characteristics of UHPC, T-shaped steel beam and steel bar, which fully considers the contribution of material strain hardening performance to resist bending moment. Compared with the ideal elastic-plastic model, it can obtain the true performance of the structure and reduce waste.

[0051] In the calculation method of this 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 manually setting the iteration increment.

[0052] In the calculation method of this invention, the condition of the top strain of UHPC concrete reaching the UHPC compressive ultimate strain or the bottom strain of T-beam reaching the steel ultimate strain is used as a judgment condition, which can simultaneously take into account the two failure modes of UHPC crushing and steel reaching the tensile ultimate strength.

[0053] In addition to the objectives, features, and advantages described above, the present invention has other objectives, features, and advantages. The invention will now be described in further detail with reference to the figures. Attached Figure Description

[0054] The accompanying drawings, which form part of this application, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an undue limitation of the invention. In the drawings:

[0055] Figure 1 It is an axonometric drawing of a composite beam structure made of steel and UHPC;

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

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

[0058] Figure 4 This is a flowchart of the method for calculating the curvature-resistance bending moment curve of the cross section of the composite structure beam according to the present invention;

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

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

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

[0062] Figure 8 This is a schematic diagram of strain distribution within the cross-section of a composite beam.

[0063] Figure 9 This is a schematic diagram of the stress distribution within the cross-section of a composite beam.

[0064] Figure 10 This is a schematic diagram of the curvature-resistance moment curve output in the application case;

[0065] Among them, 100 is the composite structural beam, 110 is the UHPC bridge deck, 111 is the transverse reinforcement of the bridge deck, 112 is the longitudinal reinforcement of the bridge deck, 120 is the UHPC rib, 121 is the bottom open stirrup, 122 is the top open stirrup, 123 is the bottom longitudinal reinforcement, 124 is the second bottom longitudinal reinforcement, 125 is the longitudinal distribution reinforcement, 126 is the transverse short reinforcement, 130 is the T-shaped steel beam, 131 is the steel bottom plate, 132 is the steel web, 133 is the steel tenon, and 134 is the mortise. Detailed Implementation

[0066] To facilitate understanding of the present invention, a more complete description is provided below, along with preferred embodiments. However, the present invention can be implemented in many different forms and is not limited to the embodiments described herein. Rather, these embodiments are provided to provide a thorough and complete understanding of the disclosure of the present invention.

[0067] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains. The terminology used herein in the description of the invention is for the purpose of describing particular embodiments only and is not intended to be limiting of the invention.

[0068] Example:

[0069] Figures 1-3 The diagram illustrates a composite structural beam of steel and UHPC. The composite structural beam 100 includes a UHPC bridge deck 110 and a composite rib plate (not shown) located on the bottom surface of the UHPC bridge deck 110. The composite rib plate includes a UHPC rib plate 120 and a T-shaped steel beam 130. The UHPC rib plate 120 is located between the UHPC bridge deck 110 and the T-shaped steel beam 130, that is, the UHPC rib plate 120 and the UHPC bridge deck 110 are cast together to form a T-shaped structure. The upper edge of the T-shaped steel beam 130 is embedded in the lower end of the UHPC rib plate 120.

[0070] Furthermore, the T-shaped steel beam 130 includes a steel base plate 131 and a steel web plate 132 disposed on the steel base plate 131, which together form a T-shaped structure. The upper edge of the steel web plate 132 (i.e., the side closest to the UHPC rib plate 120) is alternately provided with steel tenons 133 and mortises 134 along the longitudinal direction of the bridge. For example... Figure 3 As shown, Figure 3 The gray area in the image is mortise 134, and the spacing between two adjacent tenons 133 is... The height of the steel tenon (i.e., the depth of the mortise) is The steel tenon 133 and mortise 134 are embedded 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 [missing information]. .

[0071] Furthermore, the UHPC rib 120 includes a rib concrete layer and a rib reinforcement cage located within the rib concrete layer. The rib reinforcement cage includes a bottom longitudinal reinforcement 123, a second bottom longitudinal reinforcement 124, longitudinal distribution reinforcement 125, transverse short reinforcement 126, and a vertical stirrup group (not shown in the figure). The vertical stirrup group includes alternating lower open stirrups 121 and upper open stirrups 122 arranged along the longitudinal direction of the bridge, such as... Figure 1 and Figure 2 As shown, the lower opening stirrup 121 and the steel tenon 133 are configured in a one-to-one correspondence, and the upper opening stirrup 122 and the mortise 134 are configured in a one-to-one correspondence. The steel tenon 133 is inserted into the opening at the lower end of the lower opening stirrup 121, and the lower end of the upper opening stirrup 122 is placed in the mortise 134. Both sides of the lower end of the upper opening stirrup 122 are provided with transverse short steel bars 126, which are spaced apart from the lower end of the upper opening stirrup 122 (i.e., they do not overlap). The lower end of the lower opening stirrup 121 is positioned lower than the upper opening stirrup 122. At the lower end, the bottommost longitudinal steel bar 123 is used to connect the lower ends of each lower open stirrup 121 longitudinally, and the next bottommost longitudinal steel bar 124 is used to connect the lower ends of each upper open stirrup 122 longitudinally. Specifically, in this embodiment, there are two of each bottommost longitudinal steel bar 123 and the next bottommost longitudinal steel bar 124, which are symmetrically arranged on both sides of the steel web 132. The interior of the vertical stirrup group has multiple longitudinally distributed steel bars 125 arranged in the vertical direction. Specifically, each of the longitudinally distributed steel bars 125 passes through the alternating upper open stirrups 122 and lower open stirrups 121 longitudinally.

[0072] Furthermore, the UHPC bridge deck 110 includes a bridge deck concrete layer and a bridge deck reinforcement skeleton located in the bridge deck concrete layer. The bridge deck reinforcement skeleton includes an upper transverse reinforcement group, a lower transverse reinforcement group, and bridge deck longitudinal reinforcement 112. Multiple bridge deck longitudinal reinforcement 112 are arranged at intervals along the transverse direction of the bridge. The upper transverse reinforcement group is located above the bridge deck longitudinal reinforcement 112. The upper transverse reinforcement group includes multiple bridge deck transverse reinforcement 111 arranged at intervals along the longitudinal direction of the bridge. The bridge deck longitudinal reinforcement 112 overlaps with the bridge deck transverse reinforcement 111 in the upper transverse reinforcement group. The lower transverse reinforcement group is located below the bridge deck longitudinal reinforcement 112. The lower transverse reinforcement group includes multiple bridge deck transverse reinforcement 111 arranged at intervals along the longitudinal direction of the bridge. The bridge deck transverse reinforcement 111 in the lower transverse reinforcement group does not overlap with the bridge deck longitudinal reinforcement 112.

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

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

[0075] To facilitate engineers' understanding of the behavior of composite structural beams under different load conditions and to reveal key material properties such as yield point, ultimate strength, and ductility, thereby enabling composite structural beams to achieve optimal performance and economy, this embodiment provides a method for calculating the cross-sectional curvature-moment resistance curve of a composite structural beam made of steel and UHPC. Figure 4 As shown, the calculation method includes the following steps:

[0076] A1. Set constant parameters, including calculating the error convergence value. Incremental curvature iteration calculation Incremental calculation of neutral axis position ;

[0077] Preferred, The value of is set according to the required accuracy of the cross-sectional curvature-resistance bending moment curve, and is generally equal to ( ); Calculate the convergence value of the error The value of is generally Incremental calculation of neutral axis position iteration The value is generally 0.1 mm;

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

[0079] Unlike ordinary concrete, UHPC concrete exhibits strain hardening characteristics, and its strength after cracking still contributes to flexural bearing capacity (strain softening). To fully consider its influence, a tri-segmented model is used for UHPC tensile strength, and a linear elastic model is used for UHPC compressive strength. For the T-beams and reinforcing bars in the composite structural beams, a tri-segmented model considering yield plateau and strain hardening is used. Because the project focuses on maximum bearing capacity, the material constitutive models, except for those for UHPC tensile strength, do not consider the descending segment.

[0080] Preferred, such as Figure 5 As shown, the nonlinear constitutive model of UHPC concrete is:

[0081] (1),

[0082] in, For the strain of UHPC concrete. For the stress of UHPC concrete, The elastic modulus of UHPC concrete; This represents the ultimate compressive strain of the UHPC concrete. The tensile elastic limit strain of UHPC concrete; The tensile limit strain of UHPC concrete; For the fracture strain of UHPC concrete; when hour, Reaching the ultimate compressive stress, This represents the ultimate compressive stress of UHPC concrete. The tensile elastic limit stress of UHPC concrete. = ; The tensile ultimate stress of UHPC concrete is given by the strain at this stress. .

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

[0084] Preferred, such as Figure 6 As shown, the nonlinear constitutive models for the T-beams and reinforcing bars in the composite structural beam are:

[0085] (2),

[0086] in, For the strain of the T-shaped steel beam or reinforcing bar. For the stress of T-shaped steel beams or reinforcing bars, The elastic modulus of the T-shaped steel beam or reinforcing bar; This refers to the ultimate compressive strain of the T-shaped steel beam or reinforcing bar. This refers to the tensile yield strain of the T-shaped steel beam or reinforcing bar. The strain hardening initiation strain of the T-shaped steel beam or reinforcing bar; This represents the ultimate tensile strain of the T-shaped steel beam or reinforcing bar; for cases without a yield plateau, such as certain sizes of reinforcing bars, then... ;

[0087] when hour, The compressive stress reaches the ultimate compressive stress. This refers to the ultimate compressive stress of the T-shaped steel beam or reinforcing bar. The yield stress of the T-shaped steel beam or reinforcing bar. = ; This represents the ultimate tensile stress of the T-shaped steel beam or reinforcing bar, at which point the corresponding strain is... .

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

[0089] A3, such as Figure 7 As shown, the cross-section of the UHPC concrete in the composite structural beam is divided into unit layers, and the vertical distance from the center of each unit layer to the upper surface of the composite structural beam is obtained. and the area of ​​each unit layer The cross-section of the T-shaped steel beam outside the UHPC concrete is divided into unit layers, and the vertical distance from the center of each unit layer to the upper surface of the composite structural beam is obtained. and the area of ​​each unit layer The cross-section of all longitudinal reinforcement bars at the bottom layer of the composite structural beam is taken as a single unit layer, and its area is obtained. and the vertical distance from the upper surface of the composite structural beam The cross-sections of all longitudinal reinforcement bars in the top layer of the composite structural beam are treated as a single unit layer, and their areas are recorded. and the vertical distance from the upper surface of the composite structural beam ;

[0090] Preferred, i Indicates the first in UHPC concrete i Each unit layer, k Indicates the first in the T-shaped steel beam k Each unit layer.

[0091] Specifically, the upper surface of the composite structural beam is the upper surface of the UHPC bridge deck, i.e., the bridge deck; the bottom longitudinal reinforcement of the composite structural beam refers to the bottom longitudinal reinforcement 123 in the UHPC rib plate, and the top longitudinal reinforcement of the composite structural beam refers to the bridge deck longitudinal reinforcement 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-beam located outside the UHPC concrete into unit layers, forced dividing lines are arranged at all locations where the cross-sectional width and material change, 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 reinforcement of the bottom and top layers of the composite structural beam, the remaining steel structure embedded in the UHPC concrete (referring to the reinforcement embedded in the UHPC concrete and part of the steel web) is not considered in the calculation, which facilitates the calculation.

[0093] Preferably, the area calculation method for a single unit layer in UHPC concrete and T-shaped steel beams is as follows:

[0094] (3),

[0095] in: The width of the unit cell is determined by the cross-sectional shape. The height of each unit layer is [height]. To ensure the accuracy of the calculation and analysis, the height of each unit layer is [height]. No more than 1 / 50 h , h This represents the total height of the cross-section.

[0096] A4, take N The initial value is 1;

[0097] A5. Calculate the cross-sectional curvature of composite structural beams. ;

[0098] A6. Assume the neutral axis position of the composite structure beam. ,in Preferred, Generally, the possible values ​​are: h / 2.

[0099] A7. Calculate the strain of a single layer in UHPC concrete based on the plane section assumption. Strain of unit layer in T-shaped steel beam The strain of all longitudinal reinforcement bars at the bottom layer of the composite structural beam corresponding to the unit layer The strain of the unit layer corresponding to all longitudinal reinforcement bars in the top layer of the composite structural beam ,like Figure 8 As shown;

[0100] Specifically, in UHPC concrete, the first i strain of each unit layer Positive values ​​represent tensile strain, and negative values ​​represent compressive strain; in the T-shaped steel beam... k strain of each unit layer Positive values ​​represent tensile strain, and negative values ​​represent compressive strain; the strain of all longitudinal reinforcement bars at the bottom layer of the composite structural beam corresponds to the strain of the unit layer. The strain of all longitudinal reinforcement bars in the uppermost layer of the composite structural beam corresponds to the strain of the unit layer. .

[0101] A8. Calculate the stress in each unit layer of UHPC concrete based on the nonlinear constitutive model of the material. Stress in each unit layer of T-shaped steel beam The stress of all longitudinal reinforcement bars at the bottom layer of the composite structural beam corresponding to the unit layer. The stress of all longitudinal reinforcement bars in the uppermost layer of the composite structural beam corresponding to the unit layer ,like Figure 9 As shown;

[0102] Specifically, the stress of a single layer in UHPC concrete is obtained according to formula (1). Positive values ​​are tensile stress, and negative values ​​are compressive stress; the stress of a unit layer in a T-shaped steel beam is obtained according to formula (2). Positive values ​​are tensile stress, and negative values ​​are compressive stress; according to formula (2), the stress of all longitudinal reinforcements at the bottom layer of the composite beam is obtained. According to formula (2), the stress of all longitudinal reinforcements in the uppermost layer of the composite structural beam corresponding to the unit layer is obtained. .

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

[0104] Preferably, the resultant force within the cross-section of the composite structural beam... F for:

[0105] (4).

[0106] A10, if satisfied Not greater than (i.e., less than or equal to) Then proceed to step A11; if not satisfied. Not greater than Then the influence coefficient method is used for iterative correction. Then repeat steps A7-A10;

[0107] Preferably, the influence coefficient method is used for iterative correction. Specifically:

[0108] A10.1 Assume the new neutral axis position of the composite structural beam. ,in ;

[0109] A10.2, Repeat steps A7-A9 to calculate... The resultant force within the cross section of the corresponding composite structural beam ;

[0110] A10.3 Calculate the change in resultant force within the cross-section of the composite structural beam. ,in ;

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

[0112] This can be completed by following steps A10.1-A10.4. The iterative correction and influence coefficient method can speed up the search and solve the problem of non-convergence caused by manually setting the iteration increment.

[0113] A11. Determine if there is compressive strain at the upper edge of the UHPC concrete. Less than its compressive ultimate strain Or, the tensile strain at the lower edge of the T-beam. Greater than its ultimate tensile strain; if it exists, then let Then repeat steps A6-A11; if it does not exist, proceed to step A12.

[0114] Specifically, if there is compressive strain at the upper edge of the UHPC concrete... Less than its compressive ultimate strain Or, the tensile strain at the lower edge of the T-shaped steel beam. Greater than its ultimate tensile strain If the calculation results in the upper edge compressive strain of the UHPC exceeding its ultimate compressive strain, or the lower edge tensile strain of the T-beam exceeding its ultimate tensile strain, it indicates that the step size used in the calculation is too large.

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

[0116] A12. Calculate the bending moment resisted by the composite structural beam based on the action moments of each unit layer in the UHPC concrete, the action moments of each unit layer in the T-beam, the action moments of the corresponding unit layers of all longitudinal reinforcements at the bottom layer of the composite structural beam, and the action moments of the corresponding unit layers of all longitudinal reinforcements at the top layer of the composite structural beam. ;

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

[0118] (5).

[0119] A13. Determine whether the upper edge compressive strain of UHPC concrete is satisfied. Subtract its compressive ultimate strain Less than or equal to Alternatively, the ultimate tensile strain of the T-beam minus the tensile strain at its lower edge. Less than or equal to If satisfied, proceed to step A14; otherwise, record the current ( , Let ) be a coordinate point on the cross-sectional curvature-resistance bending moment curve. N = N After +1, repeat steps A5-A13;

[0120] Search for the maximum resisting bending moment Corresponding cross-sectional curvature Simultaneously using the condition that the top strain of the UHPC concrete reaches the UHPC compressive ultimate strain or the bottom strain of the T-beam reaches the steel's ultimate strain as the criterion can simultaneously account for both UHPC crushing and steel reaching its tensile ultimate strength failure modes. To balance computational efficiency and convergence reliability, the criterion limit is set to [value missing]. This ensures that the stress error is within 0.1 MPa, which is sufficient for engineering applications.

[0121] A14. Based on all records ( , Data output: cross-sectional curvature-resistance bending moment curve.

[0122] Specifically, with The value is the X coordinate. Establish a Cartesian coordinate system with the Y coordinate as the value, and record all ( , Data by Arrange the values ​​in ascending order, and use the point plotting method to record all ( , Data according to The order of the values ​​is plotted as a curvature-resistance moment curve in a Cartesian coordinate system.

[0123] Application examples:

[0124] In this case, based on the specific structural dimensions of a composite beam, the cross-sectional curvature-resistance moment curve of the composite beam was obtained using the above calculation method, as follows:

[0125] Step 01: Set constant parameters: Calculate the error convergence value Curvature iteration calculation increment Iterative calculation of the increment of the neutral axis position 0.1 mm .

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

[0127] The nonlinear constitutive model of UHPC is shown in Equation (1), and the nonlinear constitutive models of T-beams and reinforcing bars are shown in Equation (2).

[0128] Elastic modulus of UHPC concrete ultimate compressive strain Ultimate compressive stress The tensile elastic limit stress of UHPC concrete Tensile ultimate stress =11.66MPa, tensile elastic limit strain 174. Ultimate tensile strain Fracture strain .

[0129] Elastic modulus of steel reinforcement (HRB400) =200GPa, ultimate compressive strain 2. Ultimate compressive stress =400MPa; Yield stress of steel reinforcement =400 MPa, ultimate stress MPa, yield strain =0.002, strain hardening initiation strain without yield plateau = Ultimate tensile strain .

[0130] The elastic modulus of T-shaped steel beams (Q355) Yield stress =355 MPa, ultimate tensile stress MPa, yield strain =0.001723, strain hardening initiation strain =0.0198, ultimate tensile strain ;

[0131] Step 03: Divide the cross-section of the composite structure beam into unit layers according to step A3;

[0132] Among them, the total height of the cross section The distance from the top edge of the cross-section to the unit layer corresponding to all the longitudinal reinforcement bars at the bottom layer of the composite structural beam. The sum of the cross-sectional areas of all longitudinal reinforcement bars at the bottom layer of the composite structural beam (i.e., the area of ​​the corresponding unit layer) is: The distance from the top edge of the cross-section to the unit layer corresponding to all longitudinal reinforcement bars in the top layer of the composite structural beam. The sum of the cross-sectional areas of all longitudinal reinforcement bars in the top layer of the composite structural beam (i.e., the area of ​​the corresponding unit layer) is: .

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

[0134] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for calculating the cross-sectional curvature-moment resistance curve of a composite steel and UHPC beam, characterized in that, Includes the following steps: S1. Set constant parameters, including calculating the error convergence value. Incremental curvature iteration calculation Incremental calculation of neutral axis position ; S2. Input the cross-sectional dimensions of the composite beam, the properties of each material in the composite beam, and the nonlinear constitutive model; S3. Divide the cross-section of the composite structure beam into unit layers, specifically: The cross-section of the UHPC concrete of the composite structural beam and the cross-section of the T-beam outside the UHPC concrete are divided into unit layers. At the same time, the cross-sections of all longitudinal steel bars in the uppermost layer of the composite structural beam and the cross-sections of all longitudinal steel bars in the bottommost layer of the composite structural beam are each taken as a unit layer. S4, Take N The initial value is 1; S5. Calculate the cross-sectional curvature of the composite structure beam. ; S6. Assume the neutral axis position of the composite structure beam. ; S7. Calculate the strain of each element layer, and calculate the stress of each element layer based on the strain of each element layer and the corresponding nonlinear constitutive model of the element layer. S8. Calculate the resultant force in the cross-section of the composite structural beam based on the stress and area of ​​each unit layer. F ; S9, if satisfied Not greater than Then proceed to step S10; if not satisfied... Not greater than Then correct Then repeat steps S7-S9; S10. Determine if there is compressive strain at the upper edge of the UHPC concrete. Less than its compressive ultimate strain Or, the tensile strain at the lower edge of the T-beam. Greater than its ultimate tensile strain; if it exists, then let Then repeat steps S6-S10; if it does not exist, proceed to step S11. S11. Calculate the bending moment resisting the composite structural beam based on the acting moments of each unit layer. ; S12. Determine whether the upper edge compressive strain of the UHPC concrete is satisfied. Subtract its compressive ultimate strain Less than or equal to Alternatively, the ultimate tensile strain of the T-beam minus the tensile strain at its lower edge. Less than or equal to If satisfied, proceed to step S13; otherwise, record the current ( , Let ) be a coordinate point on the cross-sectional curvature-resistance bending moment curve. N = N After +1, repeat steps S5-S12; S13, based on all records ( , Data output: cross-sectional curvature-resistance bending moment curve.

2. The method for calculating the cross-sectional curvature-resistance moment curve of a composite steel and UHPC beam according to claim 1, characterized in that, The nonlinear constitutive models for each material in step S2 are as follows: The nonlinear constitutive model for UHPC concrete is as follows: （1）， in, For the strain of UHPC concrete. For the stress of UHPC concrete, The elastic modulus of UHPC concrete; This represents the ultimate compressive strain of the UHPC concrete. The tensile elastic limit strain of UHPC concrete; The tensile limit strain of UHPC concrete; The fracture strain of UHPC concrete; The tensile elastic limit stress of UHPC concrete; This represents the ultimate tensile stress of UHPC concrete. The nonlinear constitutive models for the T-beams and reinforcing bars in the composite structural beam are as follows: （2）， in, For the strain of the T-shaped steel beam or reinforcing bar, For the stress of T-shaped steel beams or reinforcing bars, The elastic modulus of the T-shaped steel beam or reinforcing bar; This refers to the ultimate compressive strain of the T-shaped steel beam or reinforcing bar. This refers to the tensile yield strain of the T-shaped steel beam or reinforcing bar. The strain hardening initiation strain of the T-shaped steel beam or reinforcing bar; This represents the ultimate tensile strain of the T-shaped steel beam or reinforcing bar; if there is no yield plateau, then... ; The yield stress of the T-shaped steel beam or reinforcing bar; This refers to the ultimate tensile stress of the T-shaped steel beam or reinforcing bar.

3. The method for calculating the cross-sectional curvature-resistance moment curve of a composite steel and UHPC beam according to claim 2, characterized in that, The specific steps for calculating the strain of each unit layer in step S7 are as follows: UHPC concrete i strain of each unit layer Positive values ​​represent tensile strain, and negative values ​​represent compressive strain; in the T-shaped steel beam... k strain of each unit layer Positive values ​​represent tensile strain, and negative values ​​represent compressive strain; the strain of all longitudinal reinforcement bars at the bottom layer of the composite structural beam corresponds to the strain of the unit layer. ; The strain of all longitudinal reinforcement bars in the uppermost layer of the composite structural beam corresponds to the strain of the unit layer. ; in, For UHPC concrete, the first i The vertical distance from the center of each unit layer to the upper surface of the composite structural beam. The first in the T-shaped steel beam k The vertical distance from the center of each unit layer to the upper surface of the composite structural beam. This refers to the vertical distance from the center of each unit layer corresponding to all the longitudinal reinforcement bars at the bottom layer of the composite structural beam to the upper surface of the composite structural beam. This refers to the vertical distance from the center of each unit layer corresponding to all longitudinal reinforcement bars 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 moment curve of a composite steel and UHPC beam according to claim 3, characterized in that, In step S7, the stress of each element layer is calculated based on the strain of each element layer and the corresponding nonlinear constitutive model. Specifically: The stress of a single layer in UHPC concrete is obtained according to formula (1). Positive values ​​are tensile stress, and negative values ​​are compressive stress; the stress of a unit layer in a T-shaped steel beam is obtained according to formula (2). Positive values ​​are tensile stress, and negative values ​​are compressive stress; according to formula (2), the stress of all longitudinal reinforcements at the bottom layer of the composite beam is obtained. ; According to formula (2), the stress of all longitudinal reinforcements in the uppermost layer of the composite structural beam corresponding to the unit layer is obtained. .

5. The method for calculating the cross-sectional curvature-resistance moment curve of a composite steel and UHPC beam according to claim 4, characterized in that, The resultant force within the cross-section of the composite structural beam in step S8 F for: （4）， in, For UHPC concrete, the first i The area of ​​each unit layer The first in the T-shaped steel beam k The area of ​​each unit layer This refers to the area of ​​the unit layer corresponding to all the longitudinal reinforcement bars at the bottom layer of the composite structural beam. This represents the area of ​​the unit layer corresponding to all the longitudinal reinforcement bars in the top layer of the composite structural beam.

6. The method for calculating the cross-sectional curvature-resistance moment curve of a composite steel and UHPC beam according to claim 5, characterized in that, The resisting moment of the composite structural beam in step S11 for: （5）。 7. The method for calculating the cross-sectional curvature-resistance moment curve of a composite steel and UHPC beam according to any one of claims 1-6, characterized in that, The minimum compressive strain selected from each unit layer of the UHPC concrete is the upper edge compressive strain of the UHPC concrete. The maximum tensile strain selected from each unit layer of the T-shaped steel beam is the lower edge tensile strain of the T-shaped steel beam. .

8. The method for calculating the cross-sectional curvature-resistance moment curve of a composite steel and UHPC beam according to any one of claims 1-6, characterized in that, In step S9, the influence coefficient method is used for iterative correction. Specifically: S9.1 Assume a new neutral axis position for the composite structural beam. ,in ; S9.2, Repeat steps S7-S8 to calculate... The resultant force within the cross section of the corresponding composite structural beam ; S9.3 Calculate the change in resultant force within the cross-section of the composite structural beam. ,in ; S9.4 Correcting the neutral axis position of the composite structural beam ,in .

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

10. The method for calculating the cross-sectional curvature-moment resistance curve of a composite steel and UHPC beam according to any one of claims 1-6, characterized in that, Step S13 is as follows: by The value is the X coordinate. Establish a Cartesian coordinate system with the Y coordinate as the value, and record all ( , Data by Arrange the values ​​in ascending order, and use the point plotting method to record all ( , Data according to The order of the values ​​is plotted as a curvature-resistance moment curve in a Cartesian coordinate system.

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