RC frame structure elastoplasticity analysis method based on deflection curve method and section reaction increment correlation equation

Abstract

The invention provides an RC frame structure elastic-plastic analysis method based on a deflection curve method and a section reaction increment correlation equation. According to the method, an improved deflection curve method is provided in the calculation of the elastic-plastic internal force of the reinforced concrete continuous beam, a basic unknown quantity is expanded from bearing reaction to section reaction, and the improved method is applied to a reinforced concrete plane frame structure. And an RC frame structure elastic-plastic analysis method based on a deflection curve method and a section reaction increment correlation equation is established.

Description

Technical Field

[0001] The present invention relates to the technical field of building construction, and in particular to an elastic-plastic analysis method for RC frame structures based on a deflection curve method and a section response increment correlation equation. Background Art

[0002] Reinforced concrete frame structures, known for their simple construction, ease of construction, and excellent mechanical properties, are still widely used in existing buildings and future multi-story structures. As existing frame structures age, load-bearing components such as beams and columns, as well as the concrete and steel materials, inevitably age, affecting their performance, creating safety hazards and even leading to accidents such as collapse, posing a serious threat to public health and property. Therefore, efficiently and accurately analyzing the mechanical properties of existing frame structures is a challenge that structural engineers must address.

[0003] Traditional force and displacement methods are typically used to solve the elastic response of structures under load. However, for structures like reinforced concrete, which experience large differences in tension and compression and exhibit cracks, the resulting solutions are inaccurate. The finite element method (FEM) employs a stepwise linear approximation approach to address material and geometric nonlinearities. Each incremental step requires the calculation of a new system stiffness matrix. As accuracy requirements increase, the computational effort increases significantly. Furthermore, each approximation step is based on the results of the previous step, leading to significant error accumulation and distortion, slow convergence, and increased computation time. The extra-large incremental algorithm separates the linear equilibrium and coordination equations from the nonlinear constitutive relations, avoiding the repeated computation of the changing global matrix and the nonlinear system of equations, improving solution efficiency. However, it treats all generalized internal forces of all elements as fundamental unknowns, resulting in a large number of fundamental unknowns and a complex computational process. This makes it difficult for currently commonly used structural finite element analysis methods, including concentrated plasticity analysis methods typified by plastic hinges, to effectively and efficiently address problems such as concrete frame structures.

[0004] Therefore, it is of great significance to develop an elastic-plastic analysis method for RC frame structures based on the deflection curve method and the section response increment correlation equation. Summary of the Invention

[0005] The purpose of the present invention is to provide an elastic-plastic analysis method for RC frame structures based on the deflection curve method and the section response increment correlation equation to solve the problems existing in the prior art.

[0006] The technical solution adopted to achieve the purpose of the present invention is as follows: an elastic-plastic analysis method for RC frame structures based on the deflection curve method and the cross-sectional response increment correlation equation, comprising the following steps:

[0007] 1) According to the form of the plane frame structure to be analyzed, select the basic unknowns and target displacements, and divide the rod into micro units.

[0008] 2) Under the conditions of basic unknowns and known external loads, the internal forces of the structure are determined according to the nodal force balance equations along a regular path. The axial forces, bending moments, and shear forces at the endpoints and intermediate micro-segments of all beam and column elements in the structure are then solved. A linear superposition analysis method for internal forces is employed at the element and structural levels. Nonlinear analysis methods are used to account for second-order effects under the influence of axial force and deflection in the calculation of beam and column cross-section internal forces.

[0009] 3) Calculate the component deflection curve and target point displacement using the deflection curve method. The curvature φ of any section is calculated using the NM-φ relationship. When calculating member deformation, consider the additional bending moment generated by the axial force during component deflection. The sum of this additional moment and the bending moment solved by the force balance equation is used to calculate the curvature φ.

[0010] 4) Establish the incremental correlation equation between the increments of the basic unknowns and the target displacement increments, and calculate the increment matrix. In the incremental mode, the target displacement increments caused by small increments of different basic unknowns are independent of each other and can be linearly superimposed.

[0011] 5) Correct the basic unknowns based on the difference between the target displacement and the boundary conditions. Iterate the calculation using the corrected basic unknowns as the new initial values ​​and recalculate the target displacement until the calculated target displacement meets the boundary conditions. The iterative calculation uses the Newton-Raphson method.

[0012] Furthermore, step 1) specifically includes the following sub-steps:

[0013] 1.1) Construct a planar frame. The planar frame structure is uniquely defined by the following parameters: the number of frame layers (CS), the total number of frame column columns (ZS), and the information for each layer (KJ). Each layer's information (KJ) includes the total number of column columns in the current layer, the position of the initial column within the total column columns, the position of the missing column within the total column columns, the position of the span containing the missing beam within the overall layout, the spans (SPAN) of each frame span, and the floor heights (SH) of each frame layer. The information for each layer (KJ) is summarized into a matrix, with the spans (SPAN) being a matrix and the floor heights (SH) being a matrix. Nodes, column elements, and beam elements are numbered layer by layer, following a bottom-up, left-to-right numbering convention. With node 1 as the zero point, the right as the positive x direction, and the upward as the positive y direction, a plane rectangular coordinate system is established. The coordinates of each node are calculated based on the input frame form parameters. In addition, the following data are obtained: the number of nodes in the frame structure, the number of beam units, the number of column units, the node number and its corresponding coordinates, the column unit number and the nodes at both ends, the beam unit number and the nodes at both ends, the unit number connected around the node, the number of nodes in each layer and each column and their number.

[0014] 1.2) Each unit is divided into micro-segments. All cross-sectional information within each micro-segment is the same. The information of each unit micro-segment includes cross-sectional information and material information. The cross-sectional information includes unit type, unit number, cross-sectional size and concrete cover thickness. The material information includes concrete strength, steel bar strength, beam bottom reinforcement, top reinforcement and distribution information of different reinforcement along the length of the beam unit. The calculable loads on the frame structure include node horizontal loads, node vertical loads, beam unit full span uniformly distributed transverse loads and beam unit span transverse concentrated loads. For node loads, record the node number, node vertical load and node horizontal load. For loads on the beam, record the beam unit number, uniformly distributed dead load, uniformly distributed live load, number of concentrated loads, the size of each concentrated load and its distance from the left end point of the beam unit.

[0015] 1.3) Determine the basic unknowns and target displacements. The basic unknowns include the initial deformation unknowns and the section response unknowns. The initial deformation unknowns are the initial deformations used in calculating displacements. The initial deformation unknowns are located at the starting point of numerical integration. The section response unknowns are the section responses at a certain location inside the structure used in calculating displacements. The section responses are generalized internal forces and generalized displacements related to internal forces. Determine the selection rules for the basic unknowns, using the curvature of the beam unit end and the shear force at the top of the column unit at a specific position as the section response unknowns. The total number of the initial deformation unknowns and section response unknowns is equal to the number of target displacements.

[0016] Furthermore, in step 1.2), each unit is divided into several micro-segments according to the calculation accuracy requirements, and all information of each unit micro-segment is compiled into a TXT file database.

[0017] Furthermore, in step 2), the axial force of the beam unit is positive in tension and negative in pressure. The shear force of the beam unit is positive when the cut micro-segment rotates in the clockwise direction, and negative when it does not. The bending moment of the beam unit is positive when the lower edge of the beam is in tension and the upper edge is in compression, and negative when it does not. The axial force of the column unit is positive in tension and negative in pressure. The shear force of the column unit is positive when the cut micro-segment rotates in the clockwise direction, and negative when it does not. The bending moment of the column unit is positive when the left edge of the column is in tension and the right edge is in compression, and negative when it does not. The vertical load is positive when it is downward and negative when it is upward. The horizontal load is positive to the right and negative to the left.

[0018] Furthermore, in step 2), the nodes with basic unknown curvature at the ends of the beam units connected by nodes are marked as regular nodes. The beam units with curvature at both ends that are not all as basic unknowns are marked as special nodes. In the process of calculating the internal forces at the ends of the units connected by regular nodes, under the condition that the relationship between the axial force N-curvature φ-bending moment M of the rod section is known, the node or unit is taken as a detached body and the result is obtained through the force balance condition. In the process of calculating the internal forces at the ends of the units connected by special nodes, the combination of the beam unit and the column unit is taken as a detached body, and according to the basic unknowns in the combination and the external loads and the internal forces transmitted from the adjacent units, the unknown bending moment and shear force at the beam end are obtained by establishing a set of equations through the force balance condition of the combination.

[0019] Furthermore, before step 2), there is also a step of considering all elastic-plastic stress states of the reinforced concrete rectangular section from the beginning of stress to destruction, and establishing an analytical expression for the relationship between axial force N, bending moment M and curvature φ.

[0020] Furthermore, based on five basic calculation models for different stress states, analytical relationships were established for reinforced concrete rectangular sections within the elastic-plastic range. The five basic calculation models for different stress states are labeled Model 1, Model 2, Model 3, Model 4, and Model 5. In Model 1, the entire cross-section is in compression, and the compressive strain at the concrete edge is greater than the peak strain. In Model 2, the entire cross-section is in compression, and the compressive strain at the concrete edge is less than the peak strain. In Model 3, the cross-section is partially in tension, and the compressive strain at the concrete edge is less than the peak strain. In Model 4, the cross-section is partially in tension, and the compressive strain at the concrete edge is greater than the peak strain. In Model 5, the entire cross-section is in tension. In the N-φ-M relationship calculation, the limiting curvature and limiting axial force of the cross-section under the characteristic stress state are first solved. Then, the ranges of the curvature φ and the axial force N are determined in sequence, and then the stress state of the cross-section is determined. The corresponding basic model is substituted into the solution to solve for the bending moment M. In the NM-φ relationship calculation, the range enclosed by the NM full failure curve of the section is divided into regions. The regional node coordinates and boundary curve equations are solved according to the characteristic stress state. Each region corresponds to a specific stress state and calculation model. The cross-section stress state is judged by the region where the coordinate point (M, N) is located, and the curvature φ is solved by substituting it into the corresponding basic model.

[0021] Furthermore, in step 4), the elastic-plastic deflection curves of the frame beams and columns are calculated along a regular path from left to right and from top to bottom. The displacements of each node along each calculation path are extracted, and the difference from the target displacement is obtained. Under this load state, the increments of each basic unknown are traversed to obtain the contribution to the target displacement difference, forming an increment matrix and increment-related equations under the current basic unknown conditions.

[0022] In step 5), the basic unknowns are corrected by the difference between the target displacement calculation value and the boundary conditions and the incremental matrix. The corrected basic unknowns are used as new initial values ​​for iterative calculation, and the target displacement is recalculated until the target displacement calculation value meets the boundary conditions, thereby obtaining the true solution of the structural internal force and deformation.

[0023] The present invention also discloses a computer-readable storage medium storing a computer program for implementing the above-mentioned elastic-plastic analysis method of RC frame structure based on the deflection curve method and the section response increment correlation equation.

[0024] The present invention also discloses an RC frame structure analysis device based on the deflection curve method and incremental cross-sectional response equations. The device comprises a processor, an input unit, an output unit, and a storage unit. The input unit is capable of receiving basic frame parameters from an external source. The output unit is capable of presenting external performance information in the form of a matrix. The storage unit is configured to store a sequence of instructions executed by the processor for implementing the aforementioned elastic-plastic analysis method for RC frame structures based on the deflection curve method and incremental cross-sectional response equations, as well as data during operation.

[0025] The technical effect of the present invention is unquestionable: an improved deflection curve method is proposed in the calculation of the elastic-plastic internal forces of reinforced concrete continuous beams, the basic unknown quantity is expanded from the support reaction to the cross-sectional reaction, and the improved method is applied to reinforced concrete plane frame structures, establishing an elastic-plastic analysis method for RC frame structures based on the deflection curve method and the cross-sectional reaction increment correlation equation. BRIEF DESCRIPTION OF THE DRAWINGS

[0026] Figure 1 It is a schematic diagram of a special-shaped frame;

[0027] Figure 2 Schematic diagram of node type;

[0028] Figure 3 The horizontal force balance of the node is separated from the body;

[0029] Figure 4 The lateral force and bending moment balance of the column element is separated from the body;

[0030] Figure 5 The lateral force and bending moment balance of the beam element is separated from the body;

[0031] Figure 6 The node is out of body moment balance;

[0032] Figure 7 The vertical force balance of the node detached body;

[0033] Figure 8 To break away from the internal force calculation process;

[0034] Figure 9 This is a simplified diagram for calculating the internal forces of assembly 1;

[0035] Figure 10 This is a simplified diagram for calculating the internal forces of assembly 2;

[0036] Figure 11 This is a simplified diagram for calculating the internal forces of a micro-segment of a beam element;

[0037] Figure 12 is the node displacement state;

[0038] Figure 13 is a schematic diagram of displacement transfer;

[0039] Figure 14 Schematic diagram of numerical integration of beam-column components;

[0040] Figure 15 are the node and unit numbers in Example 5;

[0041] Figure 16 are the basic unknowns and the displacement coordination conditions in Example 5;

[0042] Figure 17 Schematic diagram of boundary conditions in Example 5;

[0043] Figure 18 The displacement calculation path in Example 5;

[0044] Figure 19 Calculate the program flow chart for the framework form;

[0045] Figure 20 Flowchart of the program for calculating micro-segment information;

[0046] Figure 21 The flowchart of the program for basic unknown quantities and target displacement information;

[0047] Figure 22 This is the flow chart of the internal force calculation program;

[0048] Figure 23 This is the flow chart of the displacement calculation program;

[0049] Figure 24 Main program flow chart. DETAILED DESCRIPTION

[0050] The present invention will be further described below with reference to the following examples, but it should not be understood that the scope of the present invention is limited to the following examples. Without departing from the above technical ideas of the present invention, various substitutions and modifications can be made according to common technical knowledge and customary means in the art, and all should be included in the scope of protection of the present invention.

[0051] Example 1:

[0052] This embodiment provides an elastic-plastic analysis method for RC frame structures based on a deflection curve method and a cross-sectional response increment correlation equation, comprising the following steps:

[0053] 1) Select the basic unknowns and target displacements based on the form of the plane frame structure to be analyzed. Step 1) specifically includes the following sub-steps:

[0054] 1.1) Build a plane frame. Figure 1 , special-shaped frames, including multi-layer unequal spans, beams and columns at each floor, setbacks, cantilevers, etc. Figure 1 The following are examples of various special-shaped frames: 1a represents a column drawout, 1b represents a beam drawout, 1c represents a cantilever, and 1d represents a setback. Assuming that the number of drawout beams and columns per layer of a planar frame structure does not exceed one, the planar frame structure is uniquely determined by the following parameters: the number of frame layers CS, the total number of frame column columns ZS, and the per-layer information KJ. Each layer's information KJ includes the total number of column columns in the current layer, the position of the initial column within the total column columns, the position of the missing column within the total column columns, the position of the span containing the missing beam within the overall layout, the spans (SPAN) of each frame span, and the floor heights (SH) of each frame layer. The per-layer information KJ is summarized into a matrix, with the span (SPAN) being a matrix and the floor heights (SH) being a matrix. Nodes, column elements, and beam elements are numbered layer by layer, following a bottom-up, left-to-right numbering rule. With node 1 as the zero point, the right as the positive x direction, and the upward as the positive y direction, a plane rectangular coordinate system is established. The coordinates of each node are calculated based on the input frame form parameters. In addition, the following data are obtained: the number of nodes in the frame structure, the number of beam units, the number of column units, the node number and its corresponding coordinates, the column unit number and the nodes at both ends, the beam unit number and the nodes at both ends, the unit number connected around the node, the number of nodes in each layer and each column and their number.

[0055] 1.2) Each unit is divided into micro-segments. All cross-sectional information within each micro-segment is the same. The information of each unit micro-segment includes cross-sectional information and material information. The cross-sectional information includes unit type, unit number, cross-sectional size and concrete cover thickness. The material information includes concrete strength, steel bar strength, beam bottom reinforcement, top reinforcement and distribution information of different reinforcement along the length of the beam unit. The calculable loads on the frame structure include node horizontal loads, node vertical loads, beam unit full span uniformly distributed transverse loads and beam unit span transverse concentrated loads. For node loads, record the node number, node vertical load and node horizontal load. For loads on the beam, record the beam unit number, uniformly distributed dead load, uniformly distributed live load, number of concentrated loads, the size of each concentrated load and its distance from the left end point of the beam unit.

[0056] 1.3) Select basic unknowns and target displacements. Basic unknowns and target displacements are the control variables in this algorithm. The selection of basic unknowns needs to meet two conditions: (1) All internal forces of the entire structure can be obtained based on the basic unknowns; (2) The solution process is as simple and efficient as possible. To achieve the above goals, the basic unknowns and target displacements are determined regularly. According to the position relationship between beams and columns, the nodes that may exist in complex plane frames are classified into 10 types, such as Figure 2 a~2j. The basic unknowns include the initial deformation unknowns and the section reaction unknowns. The initial deformation unknowns are the initial deformations used in calculating displacements. The initial deformation unknowns are located at the starting point of numerical integration. The section reaction unknowns are the section reactions at a certain location inside the structure used in calculating displacements. The section reactions are generalized internal forces and generalized displacements related to internal forces. At the same time, the curvatures of the beam elements at specific locations and the shear forces at the tops of the column elements are selected as the section reaction unknowns. The number of the section reaction unknowns is equal to the number of redundant constraints of the structure. In this embodiment, the specific rules for determining the basic unknowns of the section reaction are as follows: except for the two sides of the D-type nodes (no column elements below the nodes) and the two ends of the beam elements connected to the E-type and F-type nodes (cantilevers), the curvatures of the ends of the other beam elements are all taken as basic unknowns; except for the rightmost column elements of each layer frame and the column elements connected below the I-type nodes (no beam elements on the right), the shear forces at the tops of the other column elements are all taken as basic unknowns. Regarding the principle of selecting the above-mentioned basic unknowns and displacement coordination conditions, the inventors listed a large number of planar frames of various forms and were able to select an equal number of basic unknowns and boundary conditions under this principle, confirming the feasibility of this method, which will be directly applied in subsequent analysis and calculations.

[0057] 2) Considering all stress states of the reinforced concrete rectangular section from the beginning of stress to failure, an analytical expression of the NM-φ relationship is established. In this embodiment, based on the basic calculation models of 5 different stress states, an analytical expression of the reinforced concrete rectangular section within the elastic-plastic range is established. Among them, the basic calculation models of the 5 different stress states are marked as model 1, model 2, model 3, model 4 and model 5. In model 1, the entire cross-section is compressed, and the compressive strain at the concrete edge is greater than the peak strain. In model 2, the entire cross-section is compressed, and the compressive strain at the concrete edge is less than the peak strain. In model 3, the cross-section is partially tensile, and the compressive strain at the concrete edge is less than the peak strain. In model 4, the cross-section is partially tensile, and the compressive strain at the concrete edge is greater than the peak strain. In model 5, the entire cross-section is tensile. In the calculation of the N-φ-M relationship, the limit curvature and limit axial force of the cross-section under the characteristic stress state are first solved, and then the range of the curvature φ and the axial force N are determined in turn, and then the stress state of the cross-section is determined, and the corresponding basic model is substituted to solve the bending moment M. In the NM-φ relationship calculation, the range enclosed by the NM full failure curve of the section is divided into regions. The regional node coordinates and boundary curve equations are solved according to the characteristic stress state. Each region corresponds to a specific stress state and calculation model. The cross-section stress state is judged by the region where the coordinate point (M, N) is located, and the curvature φ is solved by substituting it into the corresponding basic model.

[0058] 3) Under the conditions of basic unknowns and known external loads, the structural internal forces are calculated using equilibrium equations. The axial forces, bending moments, and shear forces at the endpoints and intermediate micro-segments of all beam and column elements in the frame structure are solved. Geometric nonlinear analysis methods are incorporated into the calculation of internal forces at the element and structural levels. Material nonlinear analysis methods are also considered in the calculation of internal forces at the cross-sectional level.

[0059] 3A) Internal Force Calculation Instructions

[0060] The internal forces of the frame structure are calculated based on basic unknown quantities and external loads. The goal is to solve the axial forces, bending moments and shear forces of all beam and column unit endpoints and all intermediate micro-segments in the frame structure under known conditions.

[0061] (1) Basic assumptions

[0062] ① The internal forces can be superimposed under loads in different directions, and the axial forces of beams and columns under horizontal and vertical loads are the same along their own lengths.

[0063] ② All nodes of the frame are ideal rigid nodes. The internal forces and moments of the structure are not reduced when they are transmitted to adjacent units through the nodes. That is, the ends of the units connected by the nodes satisfy force balance and moment balance.

[0064] (2) Rules for the positive and negative signs of internal forces

[0065] In the calculation of structural internal forces, the sign of the internal force variable indicates the nature or direction of the internal force. The sign of the internal force will have a direct impact on the calculation results and is an important basis for judging the stress state of the structure. Therefore, clarifying the positive and negative signs of the internal forces is crucial in the process of establishing equilibrium equations and solving calculation results. In the calculations in this section, the positive and negative signs of the internal forces are stipulated as follows:

[0066] The axial force of the beam unit is positive when tension is applied and negative when pressure is applied; the shear force of the beam unit is positive when the cut micro-segment rotates clockwise, and negative when the opposite is true; the bending moment of the beam unit is positive when the lower edge of the beam is subjected to tension and the upper edge to compression, and negative when the opposite is true.

[0067] The axial force of the column unit is positive with tension and negative with pressure; the shear force of the column unit is positive when the cut micro-segment rotates clockwise, and negative when the opposite is true; the bending moment of the column unit is positive when the left edge of the column is tensile and the right edge is compressive, and negative when the opposite is true.

[0068] (3) Load sign requirements

[0069] Vertical loads are positive when pointing downward and negative when pointing upward; horizontal loads are positive when pointing to the right and negative when pointing to the left.

[0070] When calculating bending moments at beam ends, the N-φ-M relationship is used to directly determine the cross-sectional bending moments given known axial forces and curvature. While this process requires cross-sectional information and involves nonlinear problems, other internal force calculations can be solved using equilibrium equations simply by treating nodes, beams, and columns as separate entities. When solving equilibrium equations for separate entities, in addition to fundamental unknowns and external loads, internal forces transmitted from adjacent elements are also considered.

[0071] 3B) Calculation of internal forces at unit ends

[0072] The calculation of the internal forces at the ends of the frame structure unit starts from the A-type node, from the top to the bottom, from left to right, taking the node or beam and column units as detached bodies, and solving the equations through force balance and moment balance based on the basic unknown quantities of the cross-sectional reaction and the known external loads. Starting from the top of the frame, the horizontal force balance of the node detached body can be used to calculate the axial force of the beam unit based on the basic unknown quantity of the column top shear force and the horizontal external load of the node. The calculation diagram is as follows Figure 3 For a general node in the frame, that is, a node with units connected on all sides, V a represents the shear force at the bottom of the column element above the node, V b represents the shear force at the top of the column element below the node, N1 represents the axial force of the beam element on the left side of the node, N2 represents the axial force of the beam element on the right side of the node, and F x represents the lateral external load on the node. The following equilibrium equation can be obtained:

[0073] F x +N2+V a-N1-V b =0 (1)

[0074] Starting from the top node of the frame, the calculation is performed from left to right. The first node to be calculated is the type A node. The force condition of the node separated from the body is shown in Figure (b). At this time, V a =0, N1=0, F x is the known horizontal load of the node, V b Assuming that the column top shear force is the basic unknown quantity, the column axial force N2 on the left side of the node can be obtained; then calculate the type b node, the node is separated from the body and the force is shown in Figure (c). At this time, V a = 0, N1 is equal to the beam axial force calculated from the adjacent node on the left side of the node, V b 、F x As a known quantity, N2 can be obtained; for type c nodes, the stress condition of the node detached from the body is shown in Figure (d). The shear force V at the top of the column below this type of node is b It is not a basic unknown quantity, V a =0, N2=0, N1 is equal to the beam axial force calculated from the adjacent node on the left side of the node, and V can be obtained. b Through the above process, the axial force of the beam element connected to the top node of the frame and the shear force at the top of the column element below can be obtained. For general floor nodes, when solving the axial force of the beam around the node, the shear force V at the bottom of the column element above the node is also needed. a , is obtained by balancing the lateral forces and moments of the column element detached from the known column top shear force. Figure 4 , V a represents the shear force at the top of the column element, V b represents the shear force at the bottom of the column element, M a represents the bending moment at the top of the column element, M b represents the bending moment at the bottom of the column element. There is no lateral load in the middle of the column element, which results in the following equilibrium equation:

[0075] Shear force at column bottom:

[0076] V b =V a (2)

[0077] Bending moment at column base:

[0078] M b =V a L C +M a (3)

[0079] Shear force in the column at a distance x from the top:

[0080] V x =V a (4)

[0081] Bending moment at a distance x from the top of the column:

[0082] M x =V a x+M a (5)

[0083] In formula (3), L C is the column height, and x is the distance between a point in the column and the top of the column. According to formula (2), the shear force V at the bottom of the column can be obtained from the known shear force at the top of the column. b , in the horizontal force balance of the node, it is the bottom shear force V of the column element connected above the node a ,like Figure 6 (a), the only unknown quantity is N2, and then according to formula (1) we can get

[0084] The axial force N2 of the beam element on the right side of the g-type node. For the d, e, and h-type nodes, in formula (1), the axial force N2 of the beam element on the right side of the node can be obtained by simply setting the internal forces corresponding to the missing elements around the node to 0.

[0085] According to the above process, the axial forces of all beam elements and the shear forces at the ends of column elements in the frame structure can be obtained. Next, the bending moments and shear forces at the ends of beam elements will be calculated by using the beam axial forces, the basic unknown quantities of curvature at the beam ends, and the internal lateral loads of the beam elements. For beam elements with basic unknown quantities of curvature at both ends, the beam element is taken as a detached body, and the internal force calculation diagram is as follows: Figure 5 .

[0086] Figure 5 Where F1 and F2 are concentrated loads, f is the full-span uniform load, V1 and V2 are the shear forces at the left and right ends of the beam element, respectively, and M1 and M2 are the bending moments at the left and right ends of the beam element, respectively. According to the regulations on the positive and negative signs of the internal forces and external loads of the structure, the following equilibrium equation can be obtained:

[0087] Transverse force balance of beam element:

[0088] F1+F2+fL b -V1+V2=0 (6)

[0089] Bending moment equilibrium about the left endpoint:

[0090]

[0091] In formula (6) and formula (7), L b is the beam element length, L F1 , L F2are the distances from the left endpoint of the beam, respectively, for the concentrated load. For beam elements with curvature as a fundamental unknown at both ends, the beam element is treated as a detached body. The beam end moments M1 and M2 are calculated using the N-φ-M relationship based on the fundamental unknown curvature and the calculated beam element axial force. V1 and V2 are then calculated from the simultaneous equations based on the known external loads. This allows calculation of the end moments and shear forces for all beam elements with fundamental unknown curvature at both ends.

[0092] According to the obtained bending moment and shear force at the beam end, the node is taken as a detached body, and the bending moment at the top of the column unit below the node can be obtained through the node bending moment balance. Figure 7 The column element is separated from the body and the lateral force and moment balance is used to calculate the column bottom end bending moment. The calculation of the column end bending moment is still started from the top of the frame, from left to right and from top to bottom. Figure 6 .

[0093] For general nodes in the frame, that is, nodes that are connected to units on all sides, such as Figure 6 (a), M1 is the bending moment at the left side of the node, M2 is the bending moment at the right side of the node, M a is the bending moment at the bottom of the column above the node, M b is the bending moment at the top of the column below the node. Since the node is considered as an ideal rigid node, according to the regulations on the positive and negative signs of the bending moment, the node bending moment equilibrium equation can be obtained:

[0094] M1+M a -M2-M b =0 (8)

[0095] Starting from the top node of the frame, the calculation is performed from left to right. The first node to be calculated is the type A node. The force condition of the node detached from the body is as follows Figure 6 (b), at this time M a = 0, M1 = 0, M2 is the bending moment at the beam end calculated based on the beam axial force and the curvature of the beam end. According to formula (8), the bending moment M at the top of the column below the node can be obtained. b ; Then calculate the type B node, the node is separated from the body and the force is as follows Figure 6 (c), at this time M a =0, M1 and M2 are obtained by the beam axial force and the curvature of the beam end. According to formula (8), the bending moment M at the top of the column below the node can be obtained. b ; For type C nodes, the force conditions of the nodes separated from the body are as follows Figure 6 (d), at this time M a = 0, M2 = 0, M1 is the bending moment at the beam end calculated based on the beam axial force and the curvature of the beam end. According to formula (8), the bending moment M at the top of the column below the node can be obtained. b Through the above process, the bending moment at the top of the column below the top node of the frame can be obtained, and then according to the obtained column end shear force, the bending moment at the bottom of the column can be obtained through formula (3).

[0096] For general floor nodes, the moment balance of the node separation body is as follows: Figure 6 (a), M1 and M2 are the bending moments at the beam ends calculated based on the beam axial force and the curvature of the beam ends, M a is the calculated bending moment at the bottom of the column above the node, and the only unknown quantity is M b According to formula (8), the top of the column below the g-type node M can be obtained b For type h and type i nodes, in formula (8), the bending moment M at the top of the column below the node can be obtained by simply setting the bending moment corresponding to the missing elements around the node to 0. b The bending moment calculation at the ends of the beam-column element is completed.

[0097] Finally, according to the obtained beam end shear force and the vertical external load at the node, the column axial force is solved by the vertical force balance of the node detached body. Still starting from the top of the frame, the calculation is carried out from left to right and from top to bottom. The simplified diagram of the detached body calculation is as follows Figure 7 .

[0098] For general nodes in the frame, that is, nodes that are connected to units on all sides, such as Figure 7 (a), V1 represents the shear force at the bottom of the beam element on the left side of the node, V2 represents the shear force at the top of the beam element on the right side of the node, N a Represents the axial force of the column element above the node, N b represents the axial force of the column element below the node, F y Represents the vertical external load at the node. According to the regulations on the positive and negative signs of the internal force and external load of the structure, the following equilibrium equation can be obtained:

[0099] F y +N b +V2-N a -V1=0 (9)

[0100] Starting from the top node of the frame, the calculation is performed from left to right. The first node to be calculated is the type A node. The force condition of the node detached from the body is as follows Figure 7 (b), at this time N a =0, V1=0, V2 is the obtained beam end shear force, F y is the vertical concentrated load at the node. According to formula (9), the column axial force N below the node can be obtained. b ; Then calculate the type B node, the node is separated from the body and the force is as follows Figure 7 (c), at this time N a =0, V1, V2 are the calculated beam end shear forces, F y is the vertical concentrated load at the node. According to formula (9), the column axial force N below the node can be obtained. b ; For type C nodes, the force conditions of the nodes separated from the body are as follows Figure 7 (d), N a=0, V2=0, V1 is the obtained beam end shear force, F y is the vertical concentrated load at the node. According to formula (9), the column axial force N below the node can be obtained. b Through the above process, the axial force of the column below the top node of the frame can be obtained.

[0101] For general floor nodes, the moment balance of the node separation body is as follows: Figure 7 (a), V1 and V2 are the calculated beam end shear forces, N a is the calculated column axial force above the node, F y is the vertical concentrated load at the node, the only unknown quantity is N b According to formula (9), the column axial force N below the node can be obtained b For type h and type i nodes, in formula (9), the internal force corresponding to the missing elements around the node can be calculated by setting it to 0: b .

[0102] The calculation steps of the internal forces at the unit ends are as follows: Figure 8 In the figure, blue fonts represent known quantities, black fonts represent required internal forces, and lowercase letters represent the detached body equilibrium conditions or N-φ-M relationships used in the relevant steps.

[0103] At this point, the internal forces at the element ends of conventional nodes (beam element ends connected by nodes have a basic unknown curvature) are calculated, including the bending moment, axial force, and shear force at the ends of beam and column elements. Throughout the internal force calculation process, given the known N-φ-M relationship, the nodes or elements can be treated as separate entities and calculated using simple force balance conditions.

[0104] 3C) Calculation of special nodes and unit internal forces

[0105] For beam elements with unknown curvature at both ends, the bending moment at the ends of the beam elements can be directly calculated using the N-φ-M relationship through the unknown axial force and curvature at the ends of the beam elements. Figure 8 The calculation process sequentially calculates various types of internal forces at the element ends. However, for beam elements whose curvature at both ends is not fully considered as basic unknowns, the beam end bending moment and shear force cannot be calculated using the equilibrium conditions of a single beam element separated from the body. Instead, the combination of the beam element and column element must be separated. Based on the basic unknowns in the combination, external loads, and internal forces transmitted from adjacent elements, a system of equations is established using the force equilibrium conditions of the combination to calculate the unknown beam end bending moment and shear force. After calculating the beam end bending moment and shear force.

[0106] There are two cases where the curvature basic unknowns are not set at the beam ends:

[0107] (1) The beam element has an unknown curvature at one end and no unknown curvature at the other end.

[0108] Type d nodes. The curvatures on the left and right sides of this type of node are not basic unknowns, and the corresponding beam end bending moments cannot be directly calculated using the N-φ-M relationship. It is necessary to establish equilibrium equations with the assembly as a detached body to solve the unknown internal forces. The vertical direction and force conditions of the selected assembly are as follows: Figure 9 (a), take the nodes and beam elements in the assembly as separated bodies, and the internal force calculation diagram is as follows Figure 9 (b).

[0109] exist Figure 9 In (a), F y is the vertical load of the node, F1 and F2 are the resultant lateral loads of the beam elements on both sides, N c is the column axial force above the node, M c is the bending moment at the top of the column, N c and M c All of them are the calculated internal forces of the structure. M1 and M4 are the basic unknown quantities of the curvature at that location and are calculated by N-φ-M. V1 and V4 are the shear forces at the beam ends, which are unknown quantities that need to be solved by force balance. Figure 9 In (b), M5 can be c The unknown quantities are the bending moments M2 and M3 at the beam ends connected by node C, and the shear forces V1, V2, V3, and V4 at the beam ends. The others are all known quantities. Figure 9 The force on the detached body in (b) is listed as the following equilibrium equation:

[0110] From the vertical force balance and bending moment balance at node C, we can get:

[0111] F y -N c +V3-V2=0 (10)

[0112] M2+M5-M3=0 (11)

[0113] From the vertical force balance and bending moment balance of beam element AC, we can get:

[0114] F1-V1+V2=0 (12)

[0115] M1+M F1 -M2+V2L AC =0 (13)

[0116] From the vertical force balance and bending moment balance of beam element BC, we can get:

[0117] F2-V3+V4=0 (14)

[0118] M3+M F2 -M4+V4L BC =0 (15)

[0119] Among them, MF1 M is the bending moment generated by the resultant external load F1 on point A. F1 is the bending moment generated by the external load F2 on point C, L AC , L BC are the lengths of beam element AC and beam element BC, respectively. By combining equations (10)-(15), we can solve the unknown quantities M2, M3, V1, V2, V3, and V4.

[0120] (2) There is no unknown curvature at either end of the beam element, corresponding to

[0121] Type e and type f nodes are cantilevered in the frame. The beam elements connected by these nodes do not have basic unknowns of curvature at both ends, so the bending moment cannot be calculated using the N-φ-M relationship. Instead, it is necessary to establish equilibrium equations with the assembly as a detached body to solve the unknown internal forces. Taking type e nodes as an example, the vertical direction and force conditions of the selected assembly are as follows: Figure 10 (a), take the nodes and beam elements in the assembly as separated bodies, and the internal force calculation diagram is as follows Figure 10 (b) In Figure 10 In (a), F y is the vertical load of the node, F1 is the resultant lateral load of the beam element, N c is the column axial force above the node, M c is the bending moment at the top of the column, N c and M c The structural internal forces have been calculated. The beam end shear force V2 and bending moment M2 are unknown quantities that need to be solved through force balance. Figure 10 In (b), M3 can be obtained by M c The unknown quantities are the bending moments M1 and M2 at both ends of the beam unit and the shear forces V1 and V2, and the others are known quantities.

[0122] according to Figure 10 The force on the detached body in (b) is listed as the following equilibrium equation:

[0123] From the vertical force balance and bending moment balance at node A, we can get:

[0124] F y -N c +V1=0 (16)

[0125] M3-M1=0 (17)

[0126] From the vertical force balance and bending moment balance of the side beam unit AB, we can get:

[0127] F1-V1+V2=0 (18)

[0128] M1+M F1 -M2+V2L AB =0 (19)

[0129] Among them, M F1 M is the bending moment generated by the resultant external load F1 on point A. F2 is the bending moment generated by the external load F2 on point C, L AB is the length of beam element AB. By combining equations (16)-(19), we can solve the unknown quantities M1, M2, V1, and V2. Then, according to the obtained special nodes, element bending moments, and shear forces, we can calculate the unknown quantities M1, M2, V1, and V2. Figure 8 The required internal forces can be calculated by following the steps in the previous section. At this point, the axial forces, bending moments, and shear forces at both ends of all beam and column elements of the frame structure under the action of basic unknown quantities and external loads have been calculated.

[0130] 3D) beam-column element micro-segment internal force calculation

[0131] Taking a beam or column element as a detached unit, the bending moment, shear force, and axial force at any point within the element can be calculated based on the calculated element axial force, element end bending moment, shear force, and external loads. Based on the micro-segment division rules, the bending moment and axial force at the midpoint of each micro-segment can be calculated.

[0132] In beam elements, in addition to the internal forces at the ends of the elements, there are also lateral external loads. The schematic diagram of the calculation of the internal forces of the micro-segment of the beam element is as follows: Figure 11 .

[0133] like Figure 11 , assuming that the i-th beam element is divided into n micro-segments, the distance between the right end of the j-th micro-segment and the left end of the beam element is X j , use x ij N represents the distance between the midpoint of the jth micro-segment of the i-th beam element and the left end of the beam element. ij is x ij Axial force at the section, M ij is x ij Bending moment at the section, V ij is x ij Shear force at the cross section. i represents the uniformly distributed load on the i-th beam element, F ik and d ik Indicates the magnitude of the kth concentrated load on the i-th beam element and its distance from the left end of the beam element. N i 、M i and V i They represent the axial force, bending moment and shear force at the left end of the i-th beam element, which have been calculated in the previous section. ij 、M ij and V ij is the internal force at the midpoint of the micro-segment that needs to be calculated, and the others are known quantities.

[0134] According to the basic assumption of internal force calculation, the axial force of each unit in the frame structure is the same along the length direction of the unit, so the axial force at the midpoint of the micro segment is the end axial force of the unit.

[0135] N ij =N i (20)

[0136] The bending moment and shear force of the beam element micro-segment are related to the bending moment and shear force of the rod end and the lateral external load. ij When the bending moment and shear force at x ij The beam element is cut at , and the left part is taken as the detached body for calculation. The uniformly distributed load is equal along the entire beam, and is f i For concentrated loads, d ik with x ij For comparison, if d ik <x ij This means that the concentrated load acts on the detached body. Assuming that there are l concentrated loads acting on the detached body, according to the equilibrium condition of the detached body, the shear force and bending moment at the midpoint of the jth micro-segment of the i-th beam element are respectively

[0137]

[0138]

[0139] For column elements, the calculation process of micro-segment internal forces is the same as that of beam elements. Since there is no lateral load in the middle of the column element, the calculation is simpler than that of the beam element. ij N represents the distance between the midpoint of the jth micro-segment of the i-th column unit and the upper end of the column unit. ij y ij Axial force at the section, M ij y ij Bending moment at the section, V ij y ij Shear force at the section. N i 、M i and V i They represent the axial force, bending moment and shear force at the top of the i-th column element, which have been calculated in the previous section. ij 、M ij and V ij is the internal force at the midpoint of the micro-segment that needs to be calculated, and the others are known quantities.

[0140] The axial force at the midpoint of the jth micro-segment of the i-th column element is

[0141] N ij =N i (twenty three)

[0142] There is no lateral load in the column element. The micro segment bending moment and shear force are only related to the element end bending moment and shear force. ij When the bending moment and shear force at y ij The beam element is cut at , and the upper part is taken as the detached body for calculation. According to the equilibrium condition of the detached body, the shear force and bending moment at the midpoint of the jth micro-segment of the i-th column element are

[0143] V ij =V i (twenty four)

[0144] M ij =M i +V ij y ij (25)

[0145] At this point, the calculation of the internal forces of the unit micro-segments in the frame structure under the action of basic unknown quantities and external loads is completed.

[0146] 4) Calculate the component deflection curve and target point displacement using the deflection curve method. The curvature φ of any section is calculated using the NM-φ relationship. When calculating member deformation, consider the additional bending moment generated by the axial force during component deflection. The sum of this additional moment and the bending moment solved by the force balance equation is used to calculate the curvature φ.

[0147] Deformation and displacement calculation instructions

[0148] The curvature, rotation and flexural deformation of the components in the frame structure, as well as the axial tension or compression deformation of the beams and columns are solved according to the internal forces of the frame. Starting from the basic unknown quantity of the initial displacement, the rotation angle θ, horizontal displacement x and vertical displacement y at any position of the structural component are calculated along the horizontal and vertical paths respectively, thereby obtaining the target displacement (θ, x, y) at the boundary conditions.

[0149] (1) Rules for the sign of displacement and deformation

[0150] During the calculation process, there is no need to establish a coordinate system. It is assumed that the displacement of the structure in the unloaded state is 0. It is only necessary to calculate the relative deformation of each member under the load. These relative deformations are accumulated into the change of displacement. The positive and negative signs of the displacement are used to indicate its movement direction. In this section, the positive and negative signs of displacement and deformation are defined as follows:

[0151] The rotation angle θ of beam and column members is positive clockwise and negative counterclockwise; horizontal displacement x is positive to the right and negative to the left; vertical displacement y is positive downward and negative upward; axial deformation is positive for tension and negative for compression; deflection is positive downward and rightward and negative upward and leftward. Curvature itself has no positive or negative sign, but when calculating rotation and deflection using the deflection curve method, the positive or negative sign of the calculated result is related to the direction of the bending moment. To unify the calculation formulas, the positive or negative sign of the curvature is specified to be consistent with the bending moment. The above assumptions about deformation and displacement maintain consistency with the directions of bending moment, shear force, and axial force.

[0152] (2) Basic assumptions in the calculation process

[0153] ① Shear deformation is not considered.

[0154] Unlike the elastic-plastic displacement method, which requires the application of the previous step's beam-column internal force calculations for iterative shear deformation calculations, this method simultaneously calculates various internal forces, bending, and shear deformations within the same iteration. Given that beam-column shear deformation is dependent on multiple factors, including shear force magnitude, bending-shear ratio, and web reinforcement configuration, a unified, mature analytical formula for calculating shear deformation in reinforced concrete components is currently lacking. In slender beams and columns, shear deformation accounts for a relatively small proportion of the total deformation, so this paper temporarily ignores its influence. This assumption will result in slightly larger errors in the deflection curves of beams and columns with small span-to-height ratios or shear spans, but for general frame structures, this error is acceptable.

[0155] ②All nodes in the frame structure are considered as ideal rigid nodes

[0156] That is, the ends of the elements connected at the same node do not rotate relative to each other. Figure 12 Schematic diagram of the node state before and after displacement.

[0157] ③The axial deformation of beams and columns is calculated based on elasticity.

[0158] Compared with the lateral flexural deformation of components in ordinary frame structures, the axial deformation is relatively small, and the approximate elastic calculation can basically reflect the deformation state of the entire structure.

[0159] ④ The displacement at the node is transmitted independently in the horizontal and vertical directions respectively.

[0160] Starting from the initial node, the beam displacement is calculated from left to right, and the column displacement is calculated from top to bottom. This will result in different displacements of the same node calculated along different paths. When the displacement is transferred at the node where the two paths intersect (i.e., the node where the displacement coordination condition is located), this paper assumes that the displacement of the end of the beam unit on the left side of the node is transferred to the beam on the right side of the node, and the displacement of the end of the upper column unit is transferred to the column unit below the node, that is, "beam to beam, column to column", and the displacement is transferred along the bidirectional straight line, such as Figure 13 .

[0161] Figure 13 Point O is the intersection of the two calculation paths, that is, the node where the displacement coordination condition is located. B10} and {Z C10} represent the initial deformation of beam element B1 and column element C1, respectively, {Z B11} and {Z C11} (red dots in the figure) represent the calculated displacements at node O along the beam element B1 from left to right and along the column element C1 from top to bottom. Since the basic unknown quantity is not the true value under the load, {Z B11} and {Z C11}, and the actual elements B1, B2, C1, and C2 are connected by the same node. In order to establish a unified displacement transfer rule, it is stipulated here that the node displacement calculated by the beam element B1 is transferred to the beam element B2, and the node displacement calculated by the column element C1 is transferred to the column element C2, that is, {Z B20}={Z B11},{Z C20}={Z C11},{Z B20} and {Z C20} are used as the initial displacements of beam element B2 and column element C2 respectively.

[0162] Deformation and displacement calculation methods

[0163] The deflection curve method is still used to calculate the deflection curve of the beam-column component. Since bending moment and axial force act on both the frame beam and column, the calculation of the curvature φ of any section needs to refer to the axial force-bending moment-curvature relationship (NM-φ relationship). When calculating the deformation of the rod, the additional bending moment generated by the axial force during the deflection deformation of the component is taken into account, and the sum of this additional bending moment and the bending moment solved by the force balance equation in the previous section is used to calculate the curvature φ. Since bending moment and axial force are considered in both beams and columns, taking the beam element as an example, the schematic diagram of the deflection curve calculated by the deflection curve method is as follows Figure 14 .

[0164] Assume that the deformation of the initial end is θ0, x0, y0, the rod is divided into n micro-segments, and the end coordinate of the i-th micro-segment is defined as x i The lateral deformation of the midpoint of the i-th micro-segment relative to the left end point of the beam is ω i / 2 , bending moment is M i / 2 , the curvature is The bending moment without considering the second-order effect is M ai , the turning angle at the end of the i-th micro-segment is θ i , the lateral displacement relative to the beginning of the unit is ω i Then the lateral deformation and total bending moment at the midpoint of the first micro-segment are

[0165]

[0166] M1 / 2 =M a1 +N0ω 1 / 2 (27)

[0167] The curvature of the midpoint of the first micro-segment can be calculated by the NM-φ relationship, and the midpoint curvature is taken as the average curvature of the micro-segment

[0168]

[0169] f(M,N) is the NM-φ relationship function, that is, under the condition of known cross-section axial force and bending moment, the cross-section curvature can be obtained. The turning angle at the right end of the first micro segment can be obtained as

[0170]

[0171] The deflection of the right end of the first micro segment relative to the left end of the beam element is

[0172]

[0173] For the i micro-segment (i>1), the deflection and total bending moment of the midpoint relative to the left end point of the beam are:

[0174]

[0175] M i-1 / 2 =M ai +N0ω i-1 / 2 (32)

[0176] The curvature of the midpoint of the i-th micro-segment can be calculated by the NM-φ relationship, and the midpoint curvature is taken as the average curvature of the micro-segment

[0177]

[0178] The right corner of the i-th micro segment is

[0179]

[0180] The deflection of the right end of the i-th micro-segment relative to the left end of the beam element is

[0181]

[0182] The ω obtained above i are the deflections of the unit micro-segment relative to the left end point of the unit, and the actual displacement is

[0183] ω is =ω i +y0 (36)

[0184] y0 is the vertical displacement at the beginning of the beam element.

[0185] Since the axial force of the beam is considered, the axial deformation of the beam unit should also be considered. For the reinforced concrete axially loaded beam unit, the axial deformation calculation method in the elastic stage is as follows

[0186]

[0187] In formula (37), F N is the axial force of the beam element, pressure is negative and tension is positive. Figure 15 In beam elements, F N =N0; L is the length of the beam element; E c is the elastic modulus of concrete; A d is the concrete cross-sectional area equivalent to the reinforced concrete cross-sectional area,

[0188]

[0189] Formula (38) E s is the elastic modulus of steel bar, A s is the cross-sectional area of ​​the steel bar, A c For the concrete area, A c is taken as the cross-sectional area of ​​the beam element.

[0190] Starting from the left end of the beam element, the rotation angle, horizontal displacement and vertical displacement of the right end of the beam element are calculated by numerical integration and are θ n , x0+ΔL, ω ns Using this as the initial rotation angle and displacement of the "downstream" unit, we continue to perform numerical integration to obtain the displacement of all positions in the frame.

[0191] When calculating the displacement of a frame structure, the calculation is performed along two paths starting from the node where all the basic unknown quantities of the initial deformation are located.

[0192] 5) Establish the incremental correlation equation between the increments of the basic unknowns and the target displacement increments, and calculate the increment matrix. In the incremental mode, the target displacement increments caused by small increments of different basic unknowns are independent of each other and can be linearly superimposed.

[0193] The initial basic unknown quantity is recorded as

[0194]

[0195] n is the number of basic unknowns. By assuming the initial values ​​of each basic unknown, the internal force and displacement of the frame structure can be calculated in sequence, thereby obtaining the target displacement of each node, which is recorded as

[0196]

[0197] Keep the initial values ​​of the remaining unknowns unchanged, and apply increment Δx to each basic unknown in turn. 0i. Get the basic unknown unit increment matrix

[0198]

[0199] The i-th column (i=1, 2, ..., n) of formula (41) is used as the basic unknown quantity to calculate the corresponding target displacement

[0200]

[0201] Then the target displacement increment caused by the increment of the i-th basic unknown quantity is

[0202]

[0203] Thus, the incremental matrix under the current basic unknown conditions is obtained

[0204]

[0205] Then the incremental correlation equation is

[0206]

[0207] If {ΔZ Y} and {ΔZ X} represent the arrays on both sides of the equation, and the incremental matrix is ​​simply written as The superscript 0 indicates that the incremental matrix is ​​calculated under the initial basic unknown conditions, so Equation (45) can be simplified as

[0208]

[0209] In formula (46), {ΔZ Y} is the difference between the target displacement calculation value and the boundary condition under the current basic unknown quantity condition, {ΔZ X} is the basic unknown quantity correction amount that needs to be calculated during the iteration process.

[0210] 6) The basic unknowns are corrected according to the difference between the target displacement and the boundary conditions, and the corrected basic unknowns are used as new initial values ​​for iterative calculations. The target displacement is recalculated until the target displacement calculation value meets the boundary conditions.

[0211] Since the boundary condition values ​​selected in the framework calculation process are all zero, the target displacement itself is the difference between the target displacement calculation value and the boundary condition. According to formula (45), we can get

[0212]

[0213] Since the target displacement increments caused by the small increments of each basic unknown are independent of each other, the increment matrix must be a non-singular matrix. Solving equation (47), we can get the correction value of the first basic unknown:

[0214]

[0215] Then after the first correction, the basic unknown quantity is

[0216]

[0217] With {x 11 x 12 …x 1n} T As the new basic unknown quantity, the target displacement is recalculated and the target displacement {y 11 y 12 …y 1n} T If the boundary conditions are still not met, it is necessary to calculate the basic unknown quantity {x 11 x 12 …x 1n} T Incremental matrix under conditions Then pass and {y 11 y 12 …y 1n} T Make a second correction to the basic unknown quantity and continue the calculation. If the target displacement {y k1 y k2 …y kn} T If the boundary conditions are met, the calculation ends. k1 x k2 …x kn} T That is the actual value of the basic unknown quantity under the action of load.

[0218] During the iterative calculation process, a new incremental matrix must be calculated each time the basic unknown quantity is adjusted. Although this approach will increase the computational workload, it will greatly reduce the number of iterations and enable the calculation results to converge quickly.

[0219] Example 2:

[0220] The main contents of this embodiment are the same as those of embodiment 1, wherein each unit is divided into 10 micro-segments, and all information of each unit micro-segment is compiled into a TXT file database.

[0221] Example 3:

[0222] The main contents of this embodiment are the same as those of embodiment 1 or 2, wherein, in step 3), the axial force of the beam unit is positive in tension and negative in pressure. The shear force of the beam unit is positive when the cut micro-segment rotates in the clockwise direction, and negative otherwise. The bending moment of the beam unit is positive when the lower edge of the beam is in tension and the upper edge is in compression, and negative otherwise. The axial force of the column unit is positive in tension and negative in pressure. The shear force of the column unit is positive when the cut micro-segment rotates in the clockwise direction, and negative otherwise. The bending moment of the column unit is positive when the left edge of the column is in tension and the right edge is in compression, and negative otherwise. The vertical load is positive downward and negative upward. The horizontal load is positive to the right and negative to the left.

[0223] Nodes with basic unknowns in curvature at the ends of connected beam elements are marked as regular nodes. Beam elements with curvature at both ends that are not all basic unknowns are marked as special nodes. During the calculation of internal forces at the ends of elements connected by regular nodes, under the condition that the N-φ-M relationship is known, the node or element is taken as a detached body and the force balance condition is used to calculate the internal forces. During the calculation of internal forces at the ends of elements connected by special nodes, the combination of beam elements and column elements is taken as a detached body. Based on the basic unknowns in the combination, external loads, and internal forces transmitted from adjacent elements, the unknown bending moments and shear forces at the beam ends are calculated using the force balance condition of the combination.

[0224] Example 4:

[0225] The main contents of this embodiment are the same as any one of embodiments 1 to 3, wherein, see Figure 15 This example uses a four-story, seven-span, and complex special-shaped frame to introduce the algorithm process. This frame also includes the situations of withdrawing beams, withdrawing columns, setbacks, and cantilevers.

[0226] CS=4; ZS=8;

[0227] SPAN=[2400,2800,3000,2800,3200,2800,3000];

[0228] SH=[4200,3600,3600,3600];

[0229]

[0230] Input the above five parameters and number the nodes, beam elements and column elements in the frame structure according to the established numbering rules to uniquely determine the frame form and draw the frame diagram. According to the numbering rules from bottom to top and from left to right, number the nodes, column elements and beam elements layer by layer. The numbering results are as follows Figure 2 .

[0231] Analysis shows that the structure has 42 redundant constraints. To establish a corresponding number of deformation coordination equations, it is necessary to first determine 42 quantities related to internal forces as basic unknowns. When calculating the deformation curve of a frame structure, to avoid the problem of the beam being unable to resolve the curvature due to excessive bending moments after a plastic hinge appears in the frame, this embodiment simultaneously selects the curvature of the beam element end at a specific location and the shear force at the top of the column element as basic unknowns for the mixed section reaction. The principle for determining the location and number of basic unknowns for the section reaction is that when calculating the frame deformation curve using the deflection curve method, the internal forces of all beams and columns can be calculated using a minimum number of initial variables. In this embodiment, according to the different support conditions of the columns at both ends, the beams can be divided into two categories. One category is that the lower parts of both ends of the beam are supported by columns, and the other category is that one end is supported by a column and the other end is a cantilever. When the curvature of the cross-section of the supported end of the beam is determined, the bending moment of the cross-section of the supported end can be obtained through the NM-φ relationship. Combined with the load on the beam, the bending moment distribution of the entire length of the beam can be obtained only through the static equilibrium condition. This is the same as the calculation method of the continuous beam. Therefore, for all frame beams, it is preferred that the cross-section curvature of the end supported by the column under the beam is the basic unknown quantity of the cross-section reaction; at the top end node of the top column, when the bending moment of the beam is determined, the bending moment of the cross-section of the upper end of the column can be obtained according to the node equilibrium condition. When the internal force of the column is calculated from top to bottom, the shear force of the cross-section of the upper end of the column must be known, so it is necessary to determine the shear force of the cross-section of the upper end of the column of each layer as the basic unknown quantity of the cross-section reaction. In addition, under the action of horizontal load, the shear force at the rightmost column end of each layer can be calculated based on the total shear force of the layer and the shear force of the upper end sections of the remaining columns. Therefore, the shear force at the rightmost column end of each layer is not set as the basic unknown quantity of the section response. The distribution of all basic unknown quantities is as follows: Figure 15 In the figure, the yellow short line represents the basic unknown quantity of the curvature at the end of the beam element, and the purple short line represents the basic unknown quantity of the shear force at the top of the column element.

[0232] Under load, the deformation of a frame structure includes the beam and column rotation angle θ, horizontal displacement x, and vertical displacement y. When calculating the deflection curve using the deflection curve method, the rotation angle θ, horizontal displacement x, and vertical displacement y at all Class A nodes in the structure (nodes 27, 16, and 8) are first selected as the initial deformation unknowns at the starting point of the numerical integration. The deformation of each micro-element of the member is then calculated along a fixed regular path, starting from the initial node, from left to right along the beam and from top to bottom along the column. This approach allows calculation of all members in the frame.

[0233] exist Figure 15 In this framework, a total of 42 basic unknowns for cross-sectional reactions are set, plus three basic unknowns for initial deformation at each of the three numerical integration starting points, for a total of 51 basic unknowns. This requires 51 boundary conditions to establish the deformation coordination equations.

[0234] In continuous beams, support displacements are usually known, so support displacements are selected as boundary conditions. For frame structures, since the assumed basic unknowns are not the true values ​​under load, the displacements at the same node calculated along the two integral paths from left to right along the beam and from top to bottom along the column are not necessarily the same. There is a difference in displacement and rotation between the two, which is inconsistent with the actual node displacement and deformation difference being 0, such as Figure 16 The displacement of any node in the structural beam column is expressed as {θ,x,y} T , abbreviated as {Z}. Figure 5 Chinese B0} and {Z C0} represent the initial displacements of beam and column elements respectively, {Z B1} and {Z C1} are the element end displacements calculated along specified directions using the deflection curve method with the fundamental unknowns. The deformations calculated from the two paths to the node intersection must satisfy the coordination condition. Therefore, a zero displacement difference is selected as the boundary condition for all nodes except the top floor and the leftmost beam-column nodes on each floor. Additionally, the displacement at the base support node of the frame structure's bottom column is known and also serves as a displacement boundary condition.

[0235] {Z C1}-{Z B1} is the target displacement. Under the basic unknown quantity condition, when the target displacement does not meet the boundary conditions, that is, {Z C1}-{Z B1}≠0, the basic unknown quantities are corrected through incremental related equations and iterative calculations are performed.

[0236] like Figure 18 , the nodes marked with green circles are the nodes where the displacement coordination conditions are located. Since the calculation direction is from left to right and from top to bottom, all nodes connected to column elements above and beam elements on the left and all support nodes are target displacement points. Figure 18 It can be seen that there are 17 nodes with displacement coordination conditions. Each point has three displacement coordination conditions: rotation angle θ, horizontal displacement x, and vertical displacement y. There are a total of 51 displacement coordination conditions, which is exactly equal to the number of basic unknowns.

[0237] For Figure 2 The framework structure of the calculation path is as follows Figure 18 . Figure 18 The green arrow in the middle indicates the path and direction for calculating the displacement of the frame structure. Starting from the node where the basic unknown deformation quantity is located, the calculation proceeds from left to right and from top to bottom, simultaneously in two directions. The calculation path is like a stream, covering the entire frame.

[0238] In the calculation of component displacement transfer, for the element end nodes with basic unknown quantities of initial deformation, that is, type (a) nodes, Figure 18 For the corresponding nodes 27, 16, and 8, the beam and column elements B15 and C18, B7 and C8, and B1 and C1 connected by these nodes use the basic unknown quantities of the initial deformation at the corresponding nodes as the initial displacements of the numerical integration to calculate the internal micro-segment displacements of the elements, and the end displacements of the elements can be obtained. For element end nodes without basic unknowns for initial deformation, the calculated element end displacement to the left or above the node is used as the initial value for numerical integration. For nodes without displacement coordination conditions, there is only one displacement calculation path "upstream," and the element end displacement calculated along this path is used as the initial displacement for the "downstream" element. For example, at node 22, the lower end displacement of C18 calculated based on the basic unknowns for initial deformation is used as the initial displacement for B11. At node 28, the right end displacement of B15 is used as the initial displacement for B16 and C19. For nodes with displacement coordination conditions, the principle of "beams passing from beam to beam, columns passing from column to column" is followed. For example, at node 24, the right end displacement of B12 is used as the initial displacement for B13, and the lower end displacement of C19 is used as the initial displacement for C15. Similarly, according to this rule, displacements at all locations in a planar frame structure can be calculated based on the basic unknowns for initial deformation. Figure 18 The intersection of the two directional arrows and the support node are the nodes where the boundary conditions are located, that is, the target displacement points.

[0239] Example 5:

[0240] This embodiment adopts any one of Examples 1 to 3 to compare and analyze the test results of the mechanical properties of ordinary strength reinforced concrete plane frames with the finite element simulation results. An overall analysis of the calculation results shows that the calculation results of this algorithm program are in good agreement with the finite element simulation results, and the error between the two is small, which shows that this method has high accuracy.

[0241] Example 6:

[0242] This embodiment provides a computer-readable storage medium storing a computer program for implementing any one of the elastic-plastic analysis methods for RC frame structures based on the deflection curve method and the cross-sectional response increment correlation equations described in Embodiments 1 to 3. The design method of the computer program includes the following steps:

[0243] A. Determination of framework form and micro-segment division

[0244] According to the five basic parameters input, the frame form is determined, and a series of information about the nodes, beams, and other components in the frame structure is obtained. The results are presented and stored in the form of a matrix, which can be used in subsequent calculations and the frame diagram can be drawn. The program flow chart is as follows: Figure 19As shown. According to the cross-section information database, the beam-column unit in the frame structure is divided into micro-segments, and the size and position of the micro-segments are obtained. The cross-section information in the database is assigned to the corresponding micro-segments. The results are presented and stored in the form of a matrix and called in subsequent calculations. The program flow chart is shown as follows Figure 20 shown.

[0245] B. Determination of basic unknowns and target displacement

[0246] According to the selection rules of basic unknowns and target displacements, determine the basic unknowns and target displacement related information, including the type, number, position, initial value of the basic unknowns and the type and position of the target displacement. The results are presented and stored in the form of a matrix and called in subsequent calculations. The program flow chart is as follows Figure 21 shown.

[0247] C. Internal Force Calculation

[0248] The internal force calculation includes two parts: the calculation of the internal force of the unit end and the unit micro segment. First, according to Figure 11 The calculation process shown here calculates the element end internal forces by detaching the node or element from the body. Based on these element end internal forces, micro-segment internal forces are then calculated for the beam and column elements. The resulting element end internal forces and micro-segment internal forces are stored in matrix form and used during displacement calculations. Figure 22 This is the flow chart of the internal force calculation program.

[0249] D. Displacement calculation

[0250] The goal of the displacement calculation is to determine the difference in calculated displacements along the two calculation paths and the calculated displacements at the supports. The rotation angle θ, horizontal displacement x, and vertical displacement y at the end of the member at the target displacement point must be calculated. The basic information required for the calculation includes the frame form, cross-sectional parameters, basic unknowns of the initial deformation, and structural internal forces. Since the effects of shear on deformation are not considered, the only internal forces required are micro-segment bending moments and axial forces. The displacement calculation is performed on a node-by-node basis, calculating the end displacements of the elements connected to each node from left to right and top to bottom. For nodes with basic unknowns of initial deformation, the initial deformation is the node displacement. For nodes without basic unknowns of initial deformation, the initial deformation is calculated using the calculated node displacements of the adjacent nodes above or to the left of the node, using the column elements above and the beam elements to the left as the initial deformation. The solution is numerically integrated using the deflection curve method. The structural internal forces and the NM-φ relationship are used in the solution process. In addition to calculating the member end displacements, the displacements of each micro-segment are also calculated. For nodes without boundary conditions, the calculated rod end displacement is a single value. For nodes with boundary conditions, the difference between the displacements of the element ends above and to the left of the node obtained along the two paths is the target displacement. The calculated target displacement is stored in a matrix for easy reference during the iterative calculation process. Figure 23 This is the flow chart of the displacement calculation program.

[0251] E. Iterative solution (main program)

[0252] Iterative calculation is the final step of the algorithm, and the calculation result is the final result of the MIR algorithm described in this article. Iterative calculation is used as the main program of the algorithm. The frame form, cross-section information, basic unknown quantity information, internal force calculation, and displacement calculation procedures described above are all subroutines, and the calculation results of the subroutines are called during the iterative calculation. The N-φ-M relationship and NM-φ relationship described in Chapter 4 are called as functions during the internal force and displacement calculation processes. In the main program, the internal forces and displacements of the structure are calculated sequentially based on the frame form information, cross-section parameter information, external load information, and basic unknown quantities. The calculation ends when the target displacement meets the boundary conditions. Otherwise, the incremental matrix is ​​calculated and the basic unknown quantities are corrected for iterative calculation. Figure 24 Main program flow chart.

[0253] Example 7:

[0254] This embodiment provides an RC frame structure analysis device based on the deflection curve method and incremental cross-sectional response equations, comprising a processor, an input unit, an output unit, and a storage unit. The input unit can receive basic frame parameters from an external source. The output unit can present external performance information in the form of a matrix. The storage unit is used to store instruction sequences executed by the processor for implementing any one of the RC frame structure elastic-plastic analysis methods based on the deflection curve method and incremental cross-sectional response equations described in Embodiments 1-3, as well as data during operation.

Claims

1. An elastoplastic analysis method for RC frame structures based on the deflection curve method and the section response increment correlation equation, characterized in that: The following steps are involved: 1) According to the form of the plane frame structure to be analyzed, select the basic unknown quantity and target displacement, and divide the rod micro-unit; 2) Under the conditions of basic unknown quantities and known external loads, the internal forces of the structure are determined according to the regular path based on the node force balance equation; then the axial forces, bending moments and shear forces of all the beam and column unit endpoints and all the intermediate micro-segments in the structure are solved; among them, the linear superposition analysis method of internal forces is adopted at the unit and structural level; in the process of calculating the internal forces of beam and column sections, the nonlinear analysis method is adopted to calculate the second-order effects under the influence of axial force-deflection; 3) Calculate the deformation deflection curve of the member and the displacement of the target point using the deflection curve method; The calculation of the curvature φ of any section refers to the NM-φ relationship; When calculating the deformation of the rod, consider the additional bending moment generated by the axial force when the member is flexed and deformed, and use the sum of the additional bending moment and the bending moment solved by the force balance equation to calculate the curvature φ; 4) Establish the incremental correlation equation between the basic unknown quantity increment and the target displacement increment, and obtain the incremental matrix; in the incremental mode, the target displacement increments caused by the small increments of different basic unknown quantities are independent of each other and can be linearly superimposed; 5) The basic unknown quantity is corrected according to the difference between the target displacement and the boundary conditions, and the corrected basic unknown quantity is used as the new initial value for iterative calculation, and the target displacement is recalculated until the calculated value of the target displacement meets the boundary conditions; the iterative calculation adopts the Newton Raphson method.

2. The elastoplastic analysis method of RC frame structure based on the deflection curve method and the section response increment correlation equation according to claim 1 is characterized in that: Step 1) specifically includes the following sub-steps: 1.1) Construct a plane frame; the plane frame structure is uniquely determined by the following parameters: the number of frame layers CS, the total number of frame column columns ZS and each layer information KJ; each layer information KJ includes the total number of column columns of the current layer, the position of the initial column in the total column column, the position of the missing column in the total column column, the position of the span where the missing beam is located in the overall layout, the span SPAN of each frame span and the height SH of each frame layer; wherein each layer information KJ is summarized into a matrix, the span SPAN is a matrix, and the height SH is a matrix; according to the numbering rule from bottom to top and from left to right, the nodes, column units and beam units are numbered layer by layer; with node 1 as the zero point, the right is the positive x direction, and the upward is the positive y direction, a plane rectangular coordinate system is established, and the coordinates of each node are calculated according to the input frame form parameters, and the following data are obtained: the number of nodes in the frame structure, the number of beam units, the number of column units, the node number and its corresponding coordinates, the column unit number and the nodes at both ends, the beam unit number and the nodes at both ends, the unit number connected around the node, and the number of nodes and their numbers in each layer and each column; 1.2) Each unit is divided into micro-segments; all section information in each micro-segment is the same; the information of each unit micro-segment includes section information and material information; the section information includes unit type, unit number, section size and concrete cover thickness; the material information includes concrete strength, steel bar strength, beam bottom reinforcement, top reinforcement and distribution information of different reinforcement along the length of the beam unit; the calculable loads on the frame structure include node horizontal loads, node vertical loads, beam unit full span uniformly distributed transverse loads and beam unit span uniformly concentrated loads; for node loads, record each node number, node vertical load and node horizontal load; for loads on beams, record the beam unit number, uniformly distributed dead load, uniformly distributed live load, number of concentrated loads, the size of each concentrated load and its distance from the left end point of the beam unit; 1.3) Determine basic unknowns and target displacements; wherein, the basic unknowns include initial deformation unknowns and cross-sectional response unknowns; the initial deformation unknowns are the initial deformations used in calculating displacements; the initial deformation unknowns are located at the starting point of numerical integration; the cross-sectional response unknowns are cross-sectional responses at a certain location inside the structure used in calculating displacements; the cross-sectional responses are generalized internal forces and generalized displacements related to internal forces; determine the selection rules for basic unknowns, using the end curvature of the beam unit and the top shear force of the column unit at a specific position as cross-sectional response unknowns; the total number of initial deformation unknowns and cross-sectional response unknowns is equal to the number of target displacements.

3. The elastoplastic analysis method of RC frame structure based on the deflection curve method and the section response increment correlation equation according to claim 2 is characterized by: In step 1.2), each unit is divided into several micro-segments according to the calculation accuracy requirements; all information of each unit micro-segment is summarized into a TXT file database.

4. The elastoplastic analysis method of RC frame structure based on the deflection curve method and the section response increment correlation equation according to claim 1, characterized in that: In step 2), the axial force of the beam element is positive with tension and negative with pressure; The shear force of the beam element is positive when the cut micro-segment rotates clockwise, and negative when the cut micro-segment rotates clockwise; The bending moment of the beam element is positive when the lower edge of the beam is in tension and the upper edge is in compression, and vice versa. The axial force of the column element is positive when the tension is positive and negative when the pressure is negative. The shear force of the column element is positive when the cut micro-segment rotates clockwise, and vice versa. The bending moment of the column unit is positive when the left edge of the column is under tension and the right edge is under compression, and vice versa. The vertical load is positive when it is downward and negative when it is upward. The horizontal load is positive when it is to the right and negative when it is to the left.

5. The elastoplastic analysis method of RC frame structure based on the deflection curve method and the section response increment correlation equation according to claim 1, characterized in that: In step 2), the nodes with basic unknown curvature at the ends of the beam elements connected by the nodes are marked as regular nodes; the beam elements with no basic unknown curvature at both ends are marked as special nodes; In the process of calculating the internal force at the end of the unit connected by conventional nodes, under the condition that the relationship between the axial force N-curvature φ-bending moment M of the bar section is known, the node or unit is taken as a detached body and the result is obtained through the force balance condition; In the process of calculating the internal forces at the ends of units connected at special nodes, the combination of beam units and column units is taken out as a detached body. According to the basic unknown quantities in the combination, external loads, and internal forces transmitted from adjacent units, and through the force balance conditions of the combination, a set of equations is established to calculate the unknown beam end bending moment and shear force.

6. The elastoplastic analysis method of RC frame structure based on the deflection curve method and the section response increment correlation equation according to claim 1 is characterized by: Before step 2), there is also a step of considering all elastic-plastic stress states of the reinforced concrete rectangular section from the beginning of stress to the destruction process, and establishing an analytical expression for the relationship between axial force N-bending moment M-curvature φ.

7. The elastoplastic analysis method of RC frame structure based on the deflection curve method and the section response increment correlation equation according to claim 6 is characterized by: According to the basic calculation models of five different stress states, the analytical relationship of reinforced concrete rectangular section in the elastic-plastic range is established; among them, the basic calculation models of five different stress states are marked as model 1, model 2, model 3, model 4 and model 5; in model 1, the whole section is compressed, and the compressive strain at the edge of the concrete is greater than the peak strain; in model 2, the whole section is compressed, and the compressive strain at the edge of the concrete is less than the peak strain; in model 3, the section is partially tensile, and the compressive strain at the edge of the concrete is less than the peak strain; in model 4, the section is partially tensile, and the compressive strain at the edge of the concrete is greater than the peak strain; in model 5, the whole section is tensile; in N- In the φ-M relationship calculation, the limit curvature and limit axial force of the section under the characteristic stress state are first solved, and then the range of the curvature φ and the axial force N are determined in turn, and then the stress state of the section is determined, and the corresponding basic model is substituted to solve the bending moment M; in the NM-φ relationship calculation, the NM full failure curve enclosure range of the section is divided into regions, and the regional node coordinates and limit curve equations are solved according to the characteristic stress state. Each region corresponds to a certain stress state and calculation model. The section stress state is judged by the region where the coordinate point (M, N) is located, and the curvature φ is substituted into the corresponding basic model to solve.

8. The elastoplastic analysis method of RC frame structure based on the deflection curve method and the section response increment correlation equation according to claim 1 is characterized by: In step 4), the elastic-plastic deflection curves of the frame beams and columns are calculated according to a regular path from left to right and from top to bottom, and the displacement of each node position in each calculation path is extracted to obtain the difference with the target displacement; Under the force state, the increment of each basic unknown quantity is changed traversally to obtain the contribution to the target displacement difference, and the increment matrix and increment-related equations under the current basic unknown quantity conditions are formed; In step 5), the basic unknowns are corrected by the difference between the target displacement calculation value and the boundary conditions and the incremental matrix, and the corrected basic unknowns are used as new initial values ​​for iterative calculation, and the target displacement is recalculated until the target displacement calculation value meets the boundary conditions, thereby obtaining the true solution of the structural internal force and deformation.

9. A computer-readable storage medium, characterized in that: A computer program is stored for implementing the RC frame structure elastic-plastic analysis method based on the deflection curve method and the section response increment correlation equation as described in any one of claims 1 to 8.

10. An RC frame structure analysis device based on the deflection curve method and the section response increment correlation equation, characterized in that: It comprises a processor, an input unit, an output unit and a storage unit; the input unit can receive basic parameters of the frame from the outside; the output unit can present information reflecting external performance in the form of a matrix; the storage unit is used to store the instruction sequence and data during operation executed by the processor for implementing the RC frame structure elastic-plastic analysis method based on the deflection curve method and the section response increment correlation equation as described in any one of claims 1 to 8.

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