UHPC composite beam section stress distribution calculation and drawing method and system and storage medium

Abstract

The invention relates to the technical field of bridge engineering, in particular to a UHPC composite beam section stress distribution calculation and drawing method and system and a storage medium, and the method comprises the steps: obtaining geometric parameters and material parameters of a UHPC composite beam, and building a nonlinear constitutive model of each material; a curvature value phi is given, and unit layer division is carried out on the cross section of the UHPC composite beam; obtaining an approximate solution of a neutral axis by using a dichotomy, and then obtaining an accurate solution of the neutral axis by using a Newton method; calculating strain and stress of each unit layer based on plane section assumption; and generating a continuous stress distribution curve according to the stress of each unit layer. According to the method, stress distribution curves of the UHPC composite beam under different curvatures can be obtained, and structural stress calculation and section arrangement optimization of the UHPC composite beam are guided.

Description

Technical Field

[0001] The present invention relates to the technical field of bridge engineering, and in particular to a method, system and storage medium for calculating and drawing stress distribution of a UHPC composite beam cross section. Background Art

[0002] With the development and application of ultra-high performance concrete (UHPC), UHPC-steel composite beams have been widely used in bridge engineering due to their excellent mechanical properties. Accurate calculation of the stress distribution of composite beam sections is of great significance for structural design and performance evaluation.

[0003] At present, among the disclosed technologies, the Chinese patent application with application number CN202311748675.X discloses a steel-UHPC composite beam and its design method using a mixed connection of multiple PBL connectors, wherein a composite beam design method based on the bending limit state is proposed, but the strain hardening characteristics of UHPC are not fully considered; the Chinese patent application with application number CN202411139448.1 discloses a method for determining the bending bearing capacity of a high-strength steel-UHPC composite beam, wherein a calculation method considering material nonlinearity is proposed, but the calculation process is complicated, and it is impossible to clarify the cross-sectional stress distribution of the UHPC-steel composite beam under the whole process of stress.

[0004] Although the existing technology provides a variety of different methods for the stress analysis of UHPC-steel composite beams, the following problems are common: 1) Due to the significant nonlinear characteristics of UHPC materials, especially the strain hardening behavior in the tension zone, the traditional linear elastic calculation method is difficult to accurately reflect its true stress state; 2) When solving the neutral axis position, the conventional iterative algorithm is prone to oscillation divergence and poor convergence; 3) It is impossible to clearly define the cross-sectional stress distribution of the UHPC-steel composite beam during the entire stress process.

[0005] By understanding the stress distribution and calculation of the UHPC-steel composite beam section, we can obtain the stress characteristic points such as steel yielding, steel bar yielding, UHPC cracking, and UHPC crushing during the stress process, and clarify the order in which the composite section reaches these characteristic points during the stress process, thereby guiding the structural stress calculation and section layout optimization of the UHPC-steel composite beam. Therefore, a method, system, and storage medium for calculating and drawing the stress distribution of the UHPC composite beam section are urgently needed to solve the problems existing in the existing technology. Summary of the invention

[0006] The purpose of the present invention is to provide a method for calculating and drawing the cross-sectional stress distribution of a UHPC composite beam, aiming to solve the problem that the prior art cannot clearly define the cross-sectional stress distribution of a UHPC-steel composite beam during the stress process. The specific technical solution is as follows: A method for calculating and drawing stress distribution of a UHPC composite beam section, comprising: S1. Obtain the geometric parameters and material parameters of the UHPC composite beam and establish the nonlinear constitutive model of each material; S2, given curvature value φ , and divide the cross section of the UHPC composite beam into unit layers; S3, using the bisection method to obtain an approximate solution of the neutral axis, and using Newton's method based on the approximate solution of the neutral axis to obtain an accurate solution of the neutral axis; S4. Calculate the strain and stress of each unit layer based on the plane section assumption; S5. Generate a continuous stress distribution curve according to the stress of each unit layer.

[0007] Preferably, the bisection method is used to obtain an approximate solution of the neutral axis, specifically: A1. Set the initial search interval to [ a , b ] = [0, h ],in h is the total height of the cross section of the UHPC composite beam; A2. Calculate the neutral axis a Total axial force of the cross section at and located in b Total axial force of the cross section at ; A3. If It means there is no solution in the current search interval, and the calculation ends; if Then proceed to step A4; A4. Calculate the midpoint of the current search interval , and calculate the neutral axis at c Total axial force of the cross section at ; A5. If satisfied or , then output the midpoint of the current search interval c is the approximate solution for the neutral axis , otherwise, update the search interval and repeat steps A2-A5; in, is the convergence tolerance set in the bisection method, is the interval length convergence threshold; the specific method of updating the search interval in step A5 is: if , then let To update the search interval; if , then let to update the search interval.

[0008] Preferably, Newton's method is used to obtain an accurate solution, and the specific steps are as follows: B1. Take ,make ; B2. Calculate the neutral axis Total axial force of the cross section ; B3. If satisfied , then output the current is the accurate solution for the neutral axis position; if it is not satisfied, then the current Is it less than N If it is less than, go to step B4, if it is not less than, end the calculation; N is the maximum number of iterations, is the convergence tolerance set in Newton's method, ; B4. Calculate and obtain the total axial force of the current section The derivative of ; B5. Update the neutral axis position to , and let Then repeat step B2; wherein, For the The relaxation factor for the iteration.

[0009] Preferably, the relaxation factor is dynamically adjusted according to the iterative axial force error, and is expressed as: (3), Among them, the axial force error It is expressed as: (4), in: For the The total axial force of the cross section obtained by the iteration; Indicates The relaxation factor at the iteration; is the adjustment coefficient. If the axial force errors of two consecutive times show a decreasing trend, If the axial force errors between two adjacent times tend to increase, Less than 1, if the two adjacent axial force errors are equal, then is equal to 1.

[0010] Preferably, the calculation method of the total axial force of the cross section is as follows: According to the curvature and the set neutral axis position, the strain of each unit layer in the cross section is obtained based on the plane section assumption; According to the nonlinear constitutive model of the material and the strain of each unit layer, the stress of each unit layer is calculated; The total axial force in the cross section of the UHPC composite beam is calculated based on the stress and area of ​​each unit layer.

[0011] Preferably, the unit layer division of the cross section of the UHPC composite beam is specifically: The cross section of the UHPC concrete of the UHPC composite 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 top layer of the UHPC composite beam and the cross section of all the longitudinal steel bars in the bottom layer of the UHPC composite beam are respectively regarded as a unit layer.

[0012] Preferably, the nonlinear constitutive model of UHPC concrete is: (1), in, is the strain of UHPC concrete, is the stress of UHPC concrete, is the elastic modulus of UHPC concrete; is the ultimate compressive strain of UHPC concrete; is the tensile elastic limit strain of UHPC concrete; is the ultimate tensile strain of UHPC concrete; is the fracture strain of UHPC concrete; is the tensile elastic limit stress of UHPC concrete; is the ultimate tensile stress of UHPC concrete.

[0013] Preferably, the nonlinear constitutive models of the T-beam and the steel bars in the UHPC composite beam are: (2), in, is the strain of the T-beam or reinforcement, is the stress of the T-beam or steel bar, is the elastic modulus of the T-beam or steel bar; is the ultimate compressive strain of the T-beam or steel bar; is the tensile yield strain of the T-beam or reinforcement; is the strain hardening starting strain of the T-beam or steel bar; is the ultimate tensile strain of the T-beam or steel bar; if there is no yield plateau, then ; is the yield stress of the T-beam or steel bar; It is the ultimate tensile stress of T-beam or steel bar.

[0014] The present invention also provides a UHPC composite beam cross-section stress distribution calculation and drawing system, the system adopts the above-mentioned UHPC composite beam cross-section stress distribution calculation and drawing method, the system comprises: Parameter input module, used to input geometric parameters, material parameters and calculation control parameters; the calculation control parameters include convergence tolerance and , Interval length convergence threshold , maximum number of iterations N , initial relaxation factor and initial adjustment coefficient ; Meshing module, used to divide the cross section of UHPC composite beam into unit layers; Iterative calculation module, using the bisection method to obtain the approximate solution of the neutral axis, and using Newton's method based on the approximate solution of the neutral axis to obtain the exact solution of the neutral axis; Stress calculation module, used to calculate the strain and stress of each unit layer; The result processing module is used to generate continuous stress distribution curves.

[0015] The present invention also provides a storage medium, in which a computer program is stored. When the computer program is run, the method for calculating and drawing the stress distribution of the cross section of the UHPC composite beam is executed.

[0016] The application of the technical solution of the present invention has the following beneficial effects: The method of the present invention adopts a nonlinear constitutive model of UHPC concrete, takes into account the strain hardening characteristics of UHPC, can accurately reflect its real stress state, and provide an accurate analysis basis for subsequent calculations; the method of the present invention can obtain the stress distribution curve of the UHPC composite beam under different curvatures (by giving different curvature values), that is, the method of the present invention can obtain the stress distribution of the UHPC composite beam under various typical stress conditions, obtain the stress characteristic points such as steel yielding, steel yielding, UHPC cracking, UHPC crushing, etc. during the stress process, and clarify the order in which the composite section reaches these characteristic points during the stress process, thereby guiding the structural stress calculation and section layout optimization of the UHPC composite beam, and ensuring the safe design of the UHPC composite beam.

[0017] The method of the present invention first obtains an approximate solution to the neutral axis position through the bisection method, and then obtains an accurate solution to the neutral axis position through the Newton method. The accurate neutral axis position can be quickly calculated to provide an accurate data basis for subsequent stress and strain calculations, and solves the problems of oscillation divergence and poor convergence in conventional iterative algorithms.

[0018] 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

[0019] 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: Figure 1 is an axonometric view of a composite structural beam of steel and UHPC (i.e., a UHPC composite beam) in Example 1; Figure 2 yes Figure 1 Structural diagram of the middle vertical stirrup group; Figure 3 yes Figure 1 Schematic diagram of the structure of the steel tenon and mortise; Figure 4 is a flow chart of the stress distribution calculation and drawing method in Example 1; Figure 5 is a schematic diagram of the nonlinear constitutive model of UHPC concrete in Example 1; Figure 6 is a schematic diagram of a nonlinear constitutive model of a T-shaped steel beam or steel bar in Example 1; Figure 7 is a schematic cross-sectional view of the composite structural beam (i.e., UHPC composite beam) in Example 1; Figure 8 is a curve of the relationship between the bending moment / neutral axis position and the curvature of the composite structural beam in Example 1; Fig. 9 yes Figure 8 Schematic diagram of the cross-sectional stress distribution curve results at point I (i.e. the lower edge of the steel beam reaches the tensile design strength); Fig.10 yes Figure 8 Schematic diagram of the cross-sectional stress distribution curve results at point A (i.e., the upper edge of the UHPC bridge deck reaches the ultimate compressive strain); 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

[0020] 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.

[0021] 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.

[0022] Embodiment 1: Figure 1-Figure 3 A composite structural beam of steel and UHPC (i.e., a UHPC composite beam) is illustrated. The composite structural beam 100 includes 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 includes 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 is cast together with the UHPC bridge deck 110 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.

[0023] 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 , the height of the steel tenon (i.e. the depth of the mortise) is 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 .

[0024] 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 2As 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.

[0025] 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.

[0026] 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 .

[0027] Preferably, the bridge deck concrete layer and the rib concrete layer are both cast using UHPC concrete, and the UHPC concrete is ultra-high performance concrete having ultra-high durability and ultra-high mechanical properties.

[0028] Furthermore, the lower end of the UHPC rib 120 is thickened at the buried steel tenon 133 and the mortise 134 (i.e., a reinforced horseshoe is provided) to ensure that the transverse distance from the steel tenon to the edge of the UHPC rib 120 is not less than 5 times the height of the steel tenon, thereby improving the tolerance of the centering error between the steel web 132 and the UHPC rib 120, and at the same time providing bending space for the upper open stirrups and the lower open stirrups and reserving sufficient protective layer thickness for the stirrups, thereby improving the bonding ability between the steel web and the UHPC rib 120.

[0029] In order to ensure the safe application of UHPC and steel composite structural beams, the stress distribution in the cross section of the composite structural beam under different curvatures (i.e., different typical stress conditions) is clarified to guide the structural force calculation and cross-sectional layout optimization of the composite structural beam. This embodiment provides a method for calculating and drawing the stress distribution in the cross section of a UHPC composite beam, such as Figure 4 As shown, the following steps are included: S1. Obtain the geometric parameters and material parameters of the UHPC composite beam and establish the nonlinear constitutive model of each material; Different from ordinary concrete, UHPC concrete has the characteristics of strain hardening, and its strength after cracking still contributes to the bending bearing capacity. 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 the T-shaped steel beam and steel bar tensile resistance in the UHPC composite beam adopt a three-fold line model that considers the yield platform and strain hardening. Because the project focuses on the maximum bearing capacity, except for UHPC tensile resistance, the material constitutive model does not consider the descending section.

[0030] Preferably, Figure 5 As shown in Figure 2, the nonlinear constitutive model of UHPC concrete is: (1), in, is the strain of UHPC concrete, is the stress of UHPC concrete, is the elastic modulus of UHPC concrete; is the ultimate compressive strain of UHPC concrete; is the tensile elastic limit strain of UHPC concrete; is the ultimate tensile strain of UHPC concrete; is the fracture strain of UHPC concrete; hour, , reaching the ultimate compressive stress, is the compressive ultimate stress of UHPC concrete; is the tensile elastic limit stress of UHPC concrete, = ; is the tensile ultimate stress of UHPC concrete, and the corresponding strain is .

[0031] 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.

[0032] Preferably, Figure 6 As shown in the figure, the nonlinear constitutive models of T-beam and steel bars in UHPC composite beams are: (2), in, is the strain of the T-beam or reinforcement, is the stress of the T-beam or steel bar, is the elastic modulus of the T-beam or steel bar; is the ultimate compressive strain of the T-beam or steel bar; is the tensile yield strain of the T-beam or reinforcement; is the strain hardening starting strain of the T-beam or steel bar; 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, ; when hour, , reaching the compressive limit stress, is the compressive ultimate stress of the T-beam or steel bar; is the yield stress of the T-beam or steel bar, = ; is the ultimate tensile stress of the T-beam or steel bar, and the corresponding strain is .

[0033] 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.

[0034] S2, given curvature value φ , and divide the cross section of the UHPC composite beam into unit layers; Preferably, the unit layer division of the cross section of the UHPC composite beam is specifically: The cross section of the UHPC concrete of the UHPC composite beam and the cross section of the T-shaped steel beam outside the UHPC concrete are divided into unit layers. At the same time, the cross section of all longitudinal steel bars in the top layer of the UHPC composite beam and the cross section of all longitudinal steel bars in the bottom layer of the UHPC composite beam are taken as a unit layer, such as Figure 7 As shown; Preferably, all unit layers should be on the same cross section of the UHPC composite 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 mandatory dividing lines are arranged at all locations where the cross-sectional width changes and the material changes, 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 UHPC composite beam, the rest of the steel structures embedded in the UHPC concrete (referring to the steel bars embedded in the UHPC concrete and part of the steel web) are not considered for calculation, which can simplify the calculation.

[0035] S3. Use the bisection method to obtain an approximate solution of the neutral axis, and use the Newton method (i.e., Newton iteration method) based on the approximate solution of the neutral axis to obtain the exact solution of the neutral axis; In this embodiment, the dichotomy method is used to obtain the approximate solution of the neutral axis, specifically: A1. Set the initial search interval to [ a , b ] = [0, h ],in h is the total height of the cross section of the UHPC composite beam; A2. Calculate the neutral axis a Total axial force of the cross section at and located in b Total axial force of the cross section at ; A3. If It means there is no solution in the current search interval, and the calculation ends; if Then proceed to step A4; A4. Calculate the midpoint of the current search interval , and calculate the neutral axis at c Total axial force of the cross section at ; A5. If satisfied or , then output the midpoint of the current search interval c is the approximate solution for the neutral axis , otherwise, the search interval is updated and steps A2-A5 are repeated; wherein, is the convergence tolerance set in the bisection method, is the interval length convergence threshold, The value can be 0.1 h ; Furthermore, the method of updating the search interval in step A5 is as follows: , then let To update the search interval; if , then let To update the search interval; The approximate solution of the neutral axis position can be solved through steps A1-A5. The approximate solution of the neutral axis position calculated at this time can generally be used for the subsequent stress and strain calculation of each unit layer. However, in order to obtain a more accurate neutral axis position in this embodiment, the Newton method is used on the basis of the approximate solution to further obtain an accurate solution. The specific steps are as follows: B1. Take ,make ; B2. Calculate the neutral axis Total axial force of the cross section ; B3. If satisfied , then output the current is the accurate solution for the neutral axis position; if it is not satisfied, then the current Is it less than N If it is less than, go to step B4, if it is not less than, end the calculation; N is the maximum number of iterations, is the convergence tolerance set in Newton's method, ; B4. Calculate and obtain the total axial force of the current section The derivative of ; B5. Update the neutral axis position to , and let Then repeat step B2; wherein, For the The relaxation factor for the iteration.

[0036] Preferably, the relaxation factor is dynamically adjusted according to the iterative axial force error (usually limited to 0.1≤ ≤0.5), to achieve a balance between computational efficiency and stability, the relaxation factor adjustment method is: (3), in, Indicates The relaxation factor at the iteration; is the adjustment coefficient. If the axial force errors of two consecutive times show a decreasing trend, Greater than 1 (e.g. ), to appropriately increase the relaxation factor to speed up the convergence speed; if the two adjacent axial force errors tend to increase, take Less than 1 (such as ), in order to reduce the relaxation factor and ensure the stability of calculation; if the errors of two adjacent axial forces are equal, then Equal to 1, that is, no adjustment is made to the relaxation factor.

[0037] Furthermore, the axial force error It is expressed as: (4), in, For the The total axial force of the cross section obtained by iteration.

[0038] The calculation method of the total axial force of the cross section in this embodiment will be described in detail below, specifically: See also Figure 7 According to the curvature and the set neutral axis position, the strain of each unit layer in the cross section is obtained based on the plane section assumption: UHPC concrete i The strain of each unit layer , positive value is tensile strain, negative value is compressive strain; k The strain of each unit layer , positive value is tensile strain, negative value is compressive strain; the strain of all longitudinal reinforcements in the bottom layer of UHPC composite beam corresponding to the unit layer ; The strain of all longitudinal reinforcements in the uppermost layer of the UHPC composite beam corresponding to the unit layer .

[0039] in, is the position of the neutral axis. The positions of the neutral axis assumed by the bisection method and Newton's method are different when solving the position of the neutral axis. The value of is also different; It is the first i The vertical distance from the center of each unit layer to the upper surface of the UHPC composite beam, For the T-beam k The vertical distance from the center of each unit layer to the upper surface of the UHPC composite beam, is the vertical distance from the center of the unit layer corresponding to all the longitudinal reinforcements in the bottom layer of the UHPC composite beam to the upper surface of the UHPC composite beam, It is the vertical distance from the center of the unit layer corresponding to all the longitudinal reinforcements in the uppermost layer of the UHPC composite beam to the upper surface of the UHPC composite beam.

[0040] Calculate the stress of each unit layer in UHPC concrete according to the nonlinear constitutive model of the material , the stress of each unit layer in the T-beam , the stress of all longitudinal reinforcements in the bottom layer of UHPC composite beams corresponding to the unit layer The stress of all longitudinal reinforcements in the uppermost layer of the UHPC composite beam corresponds to the stress of the unit layer , Specifically, according to formula (1), the stress of the unit layer in UHPC concrete is obtained as , positive value is tensile stress, negative value is compressive stress; according to formula (2), the stress of the unit layer in the T-beam is obtained ), positive values ​​are tensile stress, and negative values ​​are compressive stress; According to formula (2), the stress of all longitudinal reinforcements in the bottom layer of the UHPC composite beam corresponding to the unit layer is obtained: According to formula (2), the stress of all longitudinal reinforcements in the uppermost layer of the UHPC composite beam corresponding to the unit layer is obtained: .

[0041] Calculate the total axial force in the cross section of the UHPC composite beam based on the stress and area of ​​each unit layer ; Preferably, the total axial force in the cross section of the UHPC composite beam is for: (5).

[0042] S4. Calculate the strain and stress of each unit layer based on the plane section assumption; After the position of the neutral axis is obtained in step S3, the strain and stress of each unit layer can be accurately calculated. The calculation method is consistent with the strain and stress calculation method when calculating the total axial force of the cross section in step S3, so it will not be repeated.

[0043] S5. Generate a continuous stress distribution curve according to the stress of each unit layer.

[0044] Preferably, in order to obtain a continuous and smooth stress distribution curve, in this embodiment, based on the stress value of each unit layer and the coordinates of each unit layer in the longitudinal direction (i.e., the height direction of the UHPC composite beam), the stress distribution curve is generated by interpolation. In this embodiment, the cubic spline interpolation method is used to process discrete stress points; the interval between adjacent calculation points is Inside, the stress values ​​are described by a cubic polynomial: (6), in: is the cubic coefficient, is the coefficient of the quadratic term, is the coefficient of the first-order term, is a constant term, and represents the coordinates of two adjacent discrete points, represents the coordinates of the calculated position, represents the stress distribution polynomial.

[0045] Furthermore, the coefficient , , and The determination of needs to meet the following continuity conditions: 1). The interpolation function must pass through all known stress points; 2). The first-order derivative is required to be continuous at the internal nodes to ensure the smoothness of stress changes; 3). The second-order derivative must also remain continuous to avoid sudden stress changes; 4). Using natural boundary conditions at the boundary, that is, the second-order derivative is zero, makes the overall stress distribution more reasonable.

[0046] It should be noted that the method for generating a continuous stress distribution curve is not limited to the cubic spline interpolation method in this embodiment. Those skilled in the art may also use other methods to generate a stress distribution curve. This embodiment will not list all feasible methods one by one.

[0047] Calculation accuracy verification: In order to verify the accuracy and applicability of the method in this embodiment, a systematic calculation and analysis of UHPC composite beams at different curvature levels was carried out. Taking a certain UHPC composite beam as an example, the cross-sectional parameters are as follows: UHPC bridge deck: width 1200mm, height 120mm; UHPC rib plate: width 120mm, height 480mm; Steel web: width 16mm, height 575mm; Steel bottom plate: width 400mm, height 25mm; Further set the initial value of the relaxation factor to 0.3 and the convergence tolerance Set to 1kN, convergence tolerance Set to 0.1kN; interval length convergence threshold 120mm; When the curvature is small ( φ =5×10 -6 / m), the neutral axis is located at 448.6mm from the top of the section. At this time, the upper edge of the bridge deck produces a compressive strain of 0.0000225, corresponding to a compressive stress of 1.03MPa, while the lower edge of the T-steel beam produces a tensile strain of 0.0000376, corresponding to a stress of 7.74MPa; this indicates that under a smaller curvature, all materials are in an elastic working state.

[0048] As the curvature increases to moderate levels ( φ =5.4×10-3 / m), the neutral axis obviously moved up to 330.6mm from the top of the section, the compressive strain at the upper edge of the bridge deck increased to 0.000179, and the compressive stress increased to 81.59MPa accordingly; at the same time, the tensile strain at the lower edge of the T-beam increased significantly to 0.000469, at which time the T-beam had entered the yield stage, and the stress reached 355.0MPa; the calculation results of this stage clearly reflect the nonlinear working characteristics of the material.

[0049] When the curvature increases further to φ =2×10 -2 / m, the neutral axis continues to move upward to 143.2mm from the top of the section, the compressive strain at the upper edge of the bridge deck reaches 0.00286, and the corresponding compressive stress increases to 130.85MPa, which still does not reach the ultimate compressive stress; while the tensile strain at the lower edge of the T-beam increases to 0.02114. At this time, the T-beam has entered the strengthening stage, and the stress rises to 368.8MPa. These results fully demonstrate that the method of this embodiment can accurately capture the nonlinear mechanical behavior of the material under large deformation state.

[0050] Furthermore, based on the above UHPC composite beam section parameters, by inputting different curvature values, the relationship between the bending moment / neutral axis position and the section curvature during the full load process of the UHPC composite beam can be analyzed. The analysis results are as follows: Figure 8 As shown ( Figure 8 The neutral axis position in the figure indicates the distance between the neutral axis and the upper edge of the UHPC bridge deck). The characteristic points in the figure are: Point A: The upper edge of the UHPC bridge deck reaches the ultimate compressive strain (i.e. the composite beam reaches the ultimate bearing capacity); Point B: The lower end of the UHPC rib plate reaches the tensile elastic limit point; Point C: The lower end of the UHPC rib plate reaches the ultimate tensile strength; Point E: The lower edge of the steel bottom plate of the T-beam reaches the yield strain; Point F: The lower edge of the steel bottom plate of the T-beam reaches the strengthening strain; Point H: The upper edge of the UHPC bridge deck reaches the compressive design strength; Point I: The lower edge of the steel bottom plate of the T-shaped steel beam reaches the tensile design strength (i.e. the design bearing capacity of the composite beam); Figure 8 The whole process of stress of UHPC composite beam is clarified, and the change of bending moment / neutral axis position of UHPC composite beam with section curvature is revealed. Figure 8 The characteristic point of interest is plotted in the coordinate system. x is the stress value, the vertical axis yHere, by selecting the characteristic point I (when the composite beam reaches the design bearing capacity) and the characteristic point A (when the composite beam reaches the ultimate bearing capacity), the stress distribution of the cross section can be obtained as follows: Fig. 9 and Fig.10 shown.

[0051] Fig. 9 The curvature of the UHPC composite beam is shown in Figure 2. φ =1.84×10 -3 / m, the stress distribution of the cross section when the lower edge of the steel bottom plate of the T-shaped steel beam reaches the tensile design strength (corresponding to the design bearing capacity of the composite beam). At this time, the neutral axis is located 448.5mm from the upper edge of the UHPC bridge deck, the tensile stress of all longitudinal steel bars in the bottom layer of the UHPC composite beam is 48.5Mpa, and the compressive stress of all longitudinal steel bars in the top layer of the UHPC composite beam is 158.1Mpa. Fig. 9 The stress distribution in the UHPC area, T-beam area and interface can be intuitively understood. The UHPC from the upper edge of the UHPC bridge deck to 580mm away is in an elastic stress state, and the UHPC from 580mm from the upper edge of the UHPC bridge deck to the interface is in the tensile strain hardening stage. At this time, the UHPC tensile area has just entered the strain hardening stage and has not yet entered the stress drop stage. The entire steel beam is in the elastic stage, and only the lower edge of the steel bottom plate of the T-beam has reached the tensile design strength.

[0052] Fig.10 The curvature of the UHPC composite beam is shown in Figure 2. φ =2.08×10 -2 / m, stress distribution in the cross section when the upper edge of the UHPC bridge deck reaches the ultimate compressive strain. At this time, the neutral axis is located 140.7mm away from the upper edge of the UHPC bridge deck. The tensile stress of all longitudinal steel bars in the bottom layer of the UHPC composite beam is 484.0Mpa, and the compressive stress of all longitudinal steel bars in the top layer of the UHPC composite beam is 406.0Mpa. The UHPC from the upper edge of the UHPC bridge deck to 160mm away from it is in an elastic stress state, the UHPC from 160mm to 280mm from the upper edge of the UHPC bridge deck is in the tensile strain hardening stage, giving full play to its tensile resistance, the UHPC from 280mm from the upper edge of the UHPC bridge deck to the steel-UHPC interface is in the tensile stress drop stage, the steel beams from the interface to 1080mm from the upper edge of the UHPC bridge deck have reached the tensile yield stage, and from 1080mm from the upper edge of the UHPC bridge deck to the bottom of the steel beam, the steel beam has entered the tensile strain strengthening stage.

[0053] contrast Fig. 9 and Fig.10It can be seen that under the stress state of the design bearing capacity of the UHPC composite beam, the material strength of the steel beam and the UHPC is not fully utilized, that is, compared with the ultimate bearing capacity, the composite beam still has a large safety margin. It can be found that the cross-sectional stress calculation and drawing method of the UHPC composite beam provided in this embodiment can effectively guide the design and optimization of the UHPC composite beam.

[0054] The method of this embodiment adopts the nonlinear constitutive model of UHPC concrete, takes into account the strain hardening characteristics of UHPC, can accurately reflect its actual stress state, and provide an accurate analysis basis for subsequent calculations; the method of this embodiment can obtain the stress distribution curve of the UHPC composite beam under different curvatures (by giving different curvature values), that is, the method of this embodiment can obtain the stress distribution of the UHPC composite beam under various typical stress conditions, obtain the stress characteristic points such as steel yielding, steel yielding, UHPC cracking, and UHPC crushing during the stress process, and clarify the order in which the composite section reaches these characteristic points during the stress process, thereby guiding the structural stress calculation and section layout optimization of the UHPC composite beam, and ensuring the safe design of the UHPC composite beam.

[0055] The method of this embodiment first obtains an approximate solution to the neutral axis position through the bisection method, and then obtains the accurate solution to the neutral axis position through the Newton method. The accurate neutral axis position can be quickly calculated to provide an accurate data basis for subsequent stress and strain calculations, and solves the problem that conventional iterative algorithms are prone to oscillation divergence and poor convergence.

[0056] Embodiment 2: This embodiment provides a system for calculating and drawing stress distribution of a UHPC composite beam cross section. The system adopts the method for calculating and drawing stress distribution of a UHPC composite beam cross section in Embodiment 1, and specifically includes: Parameter input module, used to input geometric parameters, material parameters and calculation control parameters; the calculation control parameters include convergence tolerance and , Interval length convergence threshold , maximum number of iterations N , initial relaxation factor and initial adjustment coefficient ; Meshing module, used to divide the cross section of UHPC composite beam into unit layers; Iterative calculation module, using the bisection method to obtain the approximate solution of the neutral axis, and using Newton's method based on the approximate solution of the neutral axis to obtain the exact solution of the neutral axis; Stress calculation module, used to calculate the strain and stress of each unit layer; The result processing module is used to generate continuous stress distribution curves.

[0057] Embodiment 3: This embodiment provides a storage medium, in which a computer program is stored. When the computer program is run, the method for calculating and drawing the stress distribution of the UHPC composite beam cross section in Embodiment 1 is executed.

[0058] 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 and drawing stress distribution of a UHPC composite beam cross section, characterized in that: include: S1. Obtain the geometric parameters and material parameters of the UHPC composite beam and establish the nonlinear constitutive model of each material; S2, given curvature value φ , and divide the cross section of the UHPC composite beam into unit layers; S3, using the bisection method to obtain an approximate solution of the neutral axis, and using Newton's method based on the approximate solution of the neutral axis to obtain an accurate solution of the neutral axis; S4. Calculate the strain and stress of each unit layer based on the plane section assumption; S5. Generate a continuous stress distribution curve according to the stress of each unit layer.

2. The method for calculating and drawing stress distribution of a UHPC composite beam cross section according to claim 1, characterized in that: The bisection method is used to obtain the approximate solution of the neutral axis, specifically: A1. Set the initial search interval to [ a , b ] = [0, h ],in h is the total height of the cross section of the UHPC composite beam; A2. Calculate the neutral axis a Total axial force of the cross section at and located in b Total axial force of the cross section at ; A3. If It means there is no solution in the current search interval, and the calculation ends; if Then proceed to step A4; A4. Calculate the midpoint of the current search interval , and calculate the neutral axis at c Total axial force of the cross section at ; A5. If satisfied or , then output the midpoint of the current search interval c is the approximate solution for the neutral axis , otherwise, update the search interval and repeat steps A2-A5; in, is the convergence tolerance set in the bisection method, is the interval length convergence threshold; the specific method of updating the search interval in step A5 is: if , then let To update the search interval; if , then let to update the search interval.

3. The method for calculating and drawing stress distribution of a UHPC composite beam cross section according to claim 2 is characterized in that: Newton's method is used to obtain the exact solution. The specific steps are as follows: B1. Take ,make ; B2. Calculate the neutral axis Total axial force of the cross section ; B3. If satisfied , then output the current is the exact solution for the neutral axis position; If not satisfied, judge the current Is it less than N If it is less than, go to step B4, if it is not less than, end the calculation; N is the maximum number of iterations, is the convergence tolerance set in Newton's method, ; B4. Calculate and obtain the total axial force of the current section The derivative of ; B5. Update the neutral axis position to , and let Then repeat step B2; wherein, For the The relaxation factor for the iteration.

4. The method for calculating and drawing stress distribution of a UHPC composite beam cross section according to claim 3 is characterized in that: The relaxation factor is dynamically adjusted according to the iterative axial force error and is expressed as: （3）， Among them, the axial force error It is expressed as: （4）， in: For the The total axial force of the cross section obtained by the iteration; Indicates The relaxation factor at the iteration; is the adjustment coefficient. If the axial force errors of two consecutive times show a decreasing trend, If the axial force errors between two adjacent times tend to increase, Less than 1, if the two adjacent axial force errors are equal, then Equal to 1.

5. The method for calculating and drawing stress distribution of a UHPC composite beam cross section according to claim 1, characterized in that: The calculation method of the total axial force of the section is as follows: According to the curvature and the set neutral axis position, the strain of each unit layer in the cross section is obtained based on the plane section assumption; According to the nonlinear constitutive model of the material and the strain of each unit layer, the stress of each unit layer is calculated; The total axial force in the cross section of the UHPC composite beam is calculated based on the stress and area of ​​each unit layer.

6. The method for calculating and drawing stress distribution of a UHPC composite beam cross section according to claim 1, characterized in that: The specific unit layer division of the cross section of the UHPC composite beam is: The cross section of the UHPC concrete of the UHPC composite 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 top layer of the UHPC composite beam and the cross section of all the longitudinal steel bars in the bottom layer of the UHPC composite beam are respectively regarded as a unit layer.

7. The method for calculating and drawing stress distribution of a UHPC composite beam cross section according to claim 1, characterized in that: The nonlinear constitutive model of UHPC concrete is: （1）， in, is the strain of UHPC concrete, is the stress of UHPC concrete, is the elastic modulus of UHPC concrete; is the ultimate compressive strain of UHPC concrete; is the tensile elastic limit strain of UHPC concrete; is the ultimate tensile strain of UHPC concrete; is the fracture strain of UHPC concrete; is the tensile elastic limit stress of UHPC concrete; is the ultimate tensile stress of UHPC concrete.

8. The method for calculating and drawing stress distribution of a UHPC composite beam cross section according to claim 1, characterized in that: The nonlinear constitutive models of T-beam and steel bars in UHPC composite beams are: （2）， in, is the strain of the T-beam or reinforcement, is the stress of the T-beam or steel bar, is the elastic modulus of the T-beam or steel bar; is the ultimate compressive strain of the T-beam or steel bar; is the tensile yield strain of the T-beam or steel bar; is the strain hardening starting strain of the T-beam or steel bar; is the ultimate tensile strain of the T-beam or steel bar; if there is no yield plateau, then ; is the yield stress of the T-beam or steel bar; It is the ultimate tensile stress of T-beam or steel bar.

9. A UHPC composite beam cross-section stress distribution calculation and drawing system, characterized in that: The system adopts the method described in any one of claims 1 to 8, and the system comprises: Parameter input module, used to input geometric parameters, material parameters and calculation control parameters; the calculation control parameters include convergence tolerance and , Interval length convergence threshold , maximum number of iterations N , initial relaxation factor and initial adjustment coefficient ; Meshing module, used to divide the cross section of UHPC composite beam into unit layers; Iterative calculation module, using the bisection method to obtain the approximate solution of the neutral axis, and using Newton's method based on the approximate solution of the neutral axis to obtain the exact solution of the neutral axis; Stress calculation module, used to calculate the strain and stress of each unit layer; The result processing module is used to generate continuous stress distribution curves.

10. A storage medium, characterized in that: The storage medium stores a computer program, and when the computer program is run, the method for calculating and drawing the stress distribution of the cross section of a UHPC composite beam according to any one of claims 1 to 8 is executed.

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