Calculation method of bearing capacity of long-span concrete-filled steel tubular arch

Abstract

The application relates to the technical field of building engineering, in particular to a large-span steel pipe concrete arch bearing capacity calculation method. The method comprises the following steps: S1, dividing a steel pipe concrete arch into a plurality of steel pipe concrete beam units and subunits; S2, obtaining a stiffness matrix of the steel pipe concrete beam units; S3, applying a reference load to the steel pipe concrete beam units, and calculating the node pure deformation displacement of the steel pipe concrete beam units in a structural coordinate system; S4, establishing a co-rotation coordinate system, and calculating the stress vectors of the subunits in the co-rotation coordinate system; S5, calculating the rod end resistance and unbalanced force of the steel pipe concrete beam units; S6, judging whether the unbalanced force meets the convergence requirement and whether the material damage standard is reached, and outputting the corresponding applied load as the limit bearing capacity when the conditions are met. The large-span steel pipe concrete arch bearing capacity calculation method greatly reduces the calculation workload through the condensation of degrees of freedom and the conversion of the co-rotation coordinate system, and provides an accurate and efficient algorithm.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of building engineering, in particular to a large-span steel pipe concrete arch bearing capacity calculation method. BACKGROUND

[0002] When the bearing capacity of a large-span steel pipe concrete arch is calculated, the influence of geometric and material nonlinearities must be considered. For geometric nonlinearity, most existing researches use the total Lagrangian method (T.L method) or the updated Lagrangian method (U.L method). The T.L method can only solve moderate nonlinear problems, and has large errors when solving large-angle problems. To ensure calculation accuracy, the U.L method needs to divide the structure into fine blocks, and its calculation efficiency is low due to its complex nonlinear strain relationship.

[0003] The co-rotational coordinate method (C.R method) developed in recent years can solve the problem of large-angle by establishing a co-rotational coordinate system attached to the element and translating and rotating with the element. The total displacement is used to remove the rigid body displacement of the element with the co-rotational coordinate system to obtain pure deformation. Compared with the T.L and U.L methods, the C.R method has the following advantages: 1. The calculation of geometric nonlinearity and material nonlinearity is independent in the C.R method, which can introduce a particularly complex material constitutive relationship without affecting the convergence; 2. The co-rotational column can solve the problem of spatial large-angle of beam, plate and shell elements; 3. In the C.R method, elements containing the same node degrees of freedom can use the same co-rotational column for nonlinear analysis, which is convenient for the promotion between various elements. Therefore, the C.R method is more suitable for program development and secondary development of high-performance elements than the U.L and T.L methods.

[0004] For material nonlinearity, the existing researches often use the following two methods. The first method borrows the nonlinear algorithm based on the fiber layering method commonly used in reinforced concrete structures, divides the steel pipe concrete section into many fiber layers, simulates each fiber layer with a plane beam element, and finally condenses the degrees of freedom. Although this method is convenient for introducing geometric nonlinearity and can reduce the calculation amount, the two-dimensional plane element cannot simulate the constraint effect of the outer steel pipe on the internal concrete deformation in multiple directions, i.e., the "sleeving effect" of the steel pipe concrete arch, and cannot describe the stress-strain relationship of the concrete and steel pipe materials in three-dimensional space.

[0005] The other method divides the section into a fine grid of fiber, and each grid is simulated by a solid element. Existing commercial software modeling often uses this method. Since the steel pipe concrete arch mostly adopts a multi-leg truss (lattice) section, a large number of elements and nodes need to be divided when performing finite element modeling, and the section grid generally needs to be divided into thousands to ensure accuracy, making the calculation amount of this method huge and the calculation efficiency low. SUMMARY

[0006] The present application aims to solve the problems in the prior art and provide a large-span steel pipe concrete arch bearing capacity calculation method.

[0007] The technical scheme of the present application is as follows: a large-span steel pipe concrete arch bearing capacity calculation method, comprising:

[0008] S1, divide the steel pipe concrete arch into a plurality of steel pipe concrete beam units, and divide the steel pipe concrete beam units into a plurality of sub-units according to material properties;

[0009] S2, obtain the stiffness matrix of the steel pipe concrete beam unit;

[0010] S3, apply a reference load to the steel pipe concrete beam unit, and calculate the node displacement vector of the steel pipe concrete beam unit in the structure coordinate system based on the stiffness matrix of the steel pipe concrete beam unit;

[0011] S4, establish a co-rotation coordinate system, and calculate the stress vector of each sub-unit in the co-rotation coordinate system based on the node displacement vector of the steel pipe concrete beam unit in the structure coordinate system;

[0012] S5, calculate the end resistance of the steel pipe concrete beam unit based on the stress vector of each sub-unit in the co-rotation coordinate system, and calculate the unbalanced force based on the end resistance and the applied load;

[0013] S6, judge whether the unbalanced force meets the convergence requirement, if not, take the unbalanced force as the external load and calculate according to the above steps S2, S3, S4 and S5; if the convergence requirement is met, further judge whether the material failure criterion is reached, if the material failure criterion is reached, output the total load of all load steps before the failure criterion is reached as the ultimate bearing capacity of the steel pipe concrete beam unit; if the material failure criterion is not reached, continue to add the load and calculate according to the above steps S2, S3, S4 and S5.

[0014] According to the large-span steel pipe concrete arch bearing capacity calculation method provided by the present application, in the step S1, the method for dividing the steel pipe concrete arch into a plurality of steel pipe concrete beam units and sub-units is as follows: first, divide the steel pipe concrete arch into a plurality of columnar steel pipe concrete beam units, and divide the steel pipe concrete beam units into a plurality of hexahedral sub-units along the radial direction and the circumferential direction of the steel pipe concrete beam units.

[0015] According to the large-span steel pipe concrete arch bearing capacity calculation method provided by the present application, in the step S4, the method for establishing the co-rotation coordinate system comprises: establishing a co-rotation coordinate system that translates and rotates with the steel pipe concrete unit, taking one side node i of the steel pipe concrete beam unit as the origin, taking the direction of the line connecting the two side nodes of the steel pipe concrete beam unit as the x-axis, taking the direction parallel to the side section of the steel pipe concrete and perpendicular to the x-axis as the y-axis, and taking the direction parallel to the side section of the steel pipe concrete and perpendicular to the x-axis and the y-axis as the z-axis.

[0016] According to the large-span steel pipe concrete arch bearing capacity calculation method provided in the application, in step S4, the method for calculating the stress vector of each sub-unit in the co-rotational coordinate system comprises: calculating the displacement vector of the steel pipe concrete beam unit in the co-rotational coordinate system based on the x-axis direction pure deformation displacement and the rotation pure deformation displacement in the node displacement vector of the steel pipe concrete beam unit in the structural coordinate system, and calculating the pure deformation vector of each sub-unit in the co-rotational coordinate system through the displacement vector of the steel pipe concrete beam unit in the co-rotational coordinate system; the strain vector of each sub-unit in the co-rotational coordinate system can be obtained based on the pure deformation vector of each sub-unit in the co-rotational coordinate system; and the stress vector of each sub-unit in the co-rotational coordinate system can be obtained based on the strain vector of each sub-unit in the co-rotational coordinate system and the strain matrix of each sub-unit.

[0017] According to the large-span steel pipe concrete arch bearing capacity calculation method provided in the application, the method for calculating the displacement vector of the steel pipe concrete beam unit in the co-rotational coordinate system comprises:

[0018]

[0019] wherein d gl the displacement vector of the steel pipe concrete beam unit in the co-rotational coordinate system;

[0020] the pure deformation displacement of the steel pipe concrete beam unit in the x direction in the co-rotational coordinate system;

[0021] θ xil the rotation angle of the node i of the steel pipe concrete beam unit in the yz plane in the co-rotational coordinate system;

[0022] θ yil the rotation angle of the node i of the steel pipe concrete beam unit in the xz plane in the co-rotational coordinate system;

[0023] θ zil the rotation angle of the node i of the steel pipe concrete beam unit in the xy plane in the co-rotational coordinate system;

[0024] θ xjl the rotation angle of the node j of the steel pipe concrete beam unit in the yz plane in the co-rotational coordinate system;

[0025] θ yjl the rotation angle of the node j of the steel pipe concrete beam unit in the xz plane in the co-rotational coordinate system;

[0026] θ zjl the rotation angle of the node j of the steel pipe concrete beam unit in the xy plane in the co-rotational coordinate system.

[0027] According to the large-span steel pipe concrete arch bearing capacity calculation method provided in the application, the method for calculating the pure deformation vector of each sub-unit in the co-rotational coordinate system comprises: calculating the pure deformation vector of each sub-unit in the co-rotational coordinate system according to the following formula:

[0028] d qkl =T kd d gl

[0029] Wherein: d qkl is the pure deformation vector of each sub-unit in the co-rotational coordinate system;

[0030] d gl is the displacement vector of the steel pipe concrete beam unit in the co-rotational coordinate system;

[0031] T kd is the conversion matrix.

[0032] According to the large-span steel pipe concrete arch bearing capacity calculation method provided in the application, the method for calculating the strain vector of each sub-unit in the co-rotational coordinate system comprises: calculating the strain vector of each sub-unit in the co-rotational coordinate system according to the following formula:

[0033] ε qkl =B qk d qkl

[0034] Wherein: ε qkl is the strain vector of the kth sub-unit in the co-rotational coordinate system;

[0035] B qk is the strain matrix of the kth sub-unit in the co-rotational coordinate system;

[0036] d qkl is the pure deformation vector of the kth sub-unit in the co-rotational coordinate system.

[0037] According to the large-span steel pipe concrete arch bearing capacity calculation method provided in the application, the method for calculating the stress vector of each sub-unit in the co-rotational coordinate system comprises: calculating the stress vector of each sub-unit in the co-rotational coordinate system according to the following formula:

[0038] σ qk =D qk ε qkl

[0039] Wherein: σ qk is the stress vector of the kth sub-unit in the co-rotational coordinate system;

[0040] D qk is the stress matrix of the kth sub-unit in the co-rotational coordinate system;

[0041] ε qklThe stress matrix of the kth sub-unit in the corotational coordinate system.

[0042] According to the large-span steel pipe concrete arch bearing capacity calculation method provided in the application, in step S5, the method for calculating the bar end resistance of the steel pipe concrete beam unit comprises: calculating the tangent stiffness matrix of the sub-unit according to the following formula:

[0043] k qk =∫B qk T D qk B qk dV

[0044] Wherein: k qk The tangent stiffness matrix of the kth sub-unit in the corotational coordinate system;

[0045] B qk The transformation matrix of the node of the kth sub-unit in the corotational coordinate system;

[0046] D qk The stress matrix of the kth sub-unit in the corotational coordinate system;

[0047] The bar end resistance matrix of the sub-unit is calculated based on the tangent stiffness matrix of the sub-unit and the pure deformation vector of the sub-unit, and the bar end resistance of the steel pipe concrete beam unit is obtained by integrating the bar end resistance matrix of the sub-unit.

[0048] According to the large-span steel pipe concrete arch bearing capacity calculation method provided in the application, in step S5, the method for calculating the unbalanced force based on the bar end resistance and the applied load comprises: subtracting the applied load from the bar end resistance of the steel pipe concrete beam unit to obtain the unbalanced force.

[0049] According to the large-span steel pipe concrete arch bearing capacity calculation method provided in the application, in step S6, the method for judging whether the unbalanced force meets the convergence requirement comprises: the absolute value of the unbalanced force is less than a set value.

[0050] According to the large-span steel pipe concrete arch bearing capacity calculation method provided in the application, if the convergence requirement is not met, the reference load is added to the applied load, and the above steps S2, S3, S4 and S5 are performed for calculation; if the material failure criterion is not reached, the unbalanced force is added to the applied load, and the above steps S2, S3, S4 and S5 are performed for calculation.

[0051] The advantages of the application are: 1. The steel pipe concrete beam arch is divided into a plurality of steel pipe concrete beam units in the application, each steel pipe concrete beam unit is divided into a plurality of sub-units, each sub-unit is analyzed, and finally the rod end resistance of the steel pipe concrete beam unit is calculated, the unbalanced force is calculated based on the rod end resistance and the applied load, the unbalanced force is analyzed and judged, and finally the ultimate bearing capacity of the steel pipe concrete beam unit is output, which greatly reduces the calculation amount of the bearing capacity of the steel pipe concrete beam arch, reduces the calculation difficulty of the bearing capacity of the steel pipe concrete beam arch, greatly improves the calculation efficiency, and provides great convenience for the design of the steel pipe concrete beam arch;

[0052] 2. The steel pipe concrete beam arch is divided into a plurality of steel pipe concrete beam units in the application, each steel pipe concrete beam unit is divided into a plurality of hexahedral sub-units according to the radial direction and the circumferential direction, and the eight nodes of each hexahedral sub-unit are analyzed, so that all sub-units can be unified, the construction of the calculation model is more convenient, and subsequent calculation and analysis are facilitated;

[0053] 3. The steel pipe concrete beam unit is analyzed in the co-rotational coordinate system in the application, the pure deformation displacement of the steel pipe concrete beam unit in the co-rotational coordinate system is simpler, only the pure deformation displacement of the x-axis and the three-direction rotation angle pure deformation displacement, the displacement change items are greatly reduced, subsequent calculation and analysis are facilitated, and the calculation amount is reduced;

[0054] 4. The method for obtaining the displacement vector of the steel pipe concrete beam unit in the co-rotational coordinate system through the node pure deformation displacement of the steel pipe concrete beam unit in the structural coordinate system is very simple, the steel pipe concrete beam unit in the co-rotational coordinate system only has displacement changes in the x direction and three rotation angle displacement changes, such operation greatly reduces the items of subsequent calculation and analysis, and reduces the calculation amount;

[0055] 5. The calculation methods of the pure deformation vector, the strain vector and the stress vector of the sub-unit are very simple, the stress vector of the sub-unit can be obtained step by step through formula conversion calculation, which is equivalent to obtaining the stress decomposition of the sub-unit under the condition of applying load to the steel pipe concrete beam unit, and the rod end resistance of the steel pipe concrete beam unit is calculated subsequently;

[0056] 6. The method for calculating the rod end resistance is simple, the tangent stiffness matrix of the sub-unit is calculated based on the stress vector of the sub-unit, then the rod end resistance matrix of the sub-unit is obtained based on the tangent stiffness matrix, and finally the rod end resistance of the steel pipe concrete beam unit can be obtained through integration, the calculation amount is small, and the calculation result is accurate.

[0057] The calculation method of the application firstly subdivides the steel pipe concrete arch along the section into a plurality of 8-node hexahedral subunits, establishes the conversion matrix between the nodes of the subunits and the main node displacement of the section center, so as to condense the degrees of freedom in the section; secondly, based on the co-rotational coordinate method (C.R method), the pure deformation displacement of each subunit node is extracted, and the strain and stress matrix is calculated by using the three-dimensional geometry and physical equation of the solid element; finally, based on the virtual displacement principle, the tangent stiffness matrix of the spatial steel pipe concrete beam unit in the structure coordinate system is calculated through the coordinate system conversion. By introducing the subunit method and the C.R method, the spatial steel pipe concrete beam model considering the geometric and material nonlinearities is established, and the calculation workload is greatly reduced, which provides an accurate and efficient algorithm for the calculation of the bearing capacity of the large-span steel pipe concrete arch.

[0058] The calculation method of the large-span steel pipe concrete arch bearing capacity of the application is simple, high in calculation efficiency, greatly reduces the calculation amount, can quickly obtain the ultimate bearing capacity of the large-span steel pipe concrete arch, provides great convenience for the design and construction of the large-span steel pipe concrete arch, and has great popularization value. BRIEF DESCRIPTION OF DRAWINGS

[0059] Figure 1 : side view of the steel pipe concrete beam unit of the application;

[0060] Figure 2 : front view of the steel pipe concrete beam unit of the application;

[0061] Figure 3 : division schematic diagram of the subunit in the structure coordinate system of the application;

[0062] Figure 4 : subunit schematic diagram of the application;

[0063] Figure 5 : division schematic diagram of the subunit in the co-rotational coordinate system of the application;

[0064] Figure 6 : calculation flow schematic diagram of the application. DETAILED DESCRIPTION

[0065] The embodiments of the application are described in detail below, wherein the same or similar reference numerals represent the same or similar elements or elements with the same or similar functions throughout. The embodiments described below by referring to the accompanying drawings are exemplary and are intended to explain the application, and cannot be understood as a limitation of the application.

[0066] In the description of the present application, it needs to be understood that the orientation or positional relationship indicated by the terms "longitudinal", "transverse", "length", "width", "thickness", "upper", "lower", "front", "rear", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", "outer" and the like is based on the orientation or positional relationship shown in the drawings, which is only for the convenience of describing the present application and simplifying the description, and does not indicate or imply that the device or element indicated must have a particular orientation, be constructed and operated in a particular orientation, and therefore cannot be understood as a limitation on the present application.

[0067] In addition, the terms "first", "second" are only for descriptive purposes, and cannot be understood as indicating or implying relative importance or implicitly indicating the number of the technical features indicated. Therefore, the features defined with "first", "second" can explicitly or implicitly include one or more of the features. In the description of the present application, the meaning of "multiple" is at least two, such as two, three, etc., unless otherwise explicitly specified.

[0068] The present application will be further described in detail below in conjunction with the drawings and specific embodiments.

[0069] The present application relates to a large-span steel pipe concrete arch bearing capacity calculation method, first, the steel pipe concrete arch is divided into a plurality of 8-node hexahedral sub-units along the cross section, the transformation matrix between the nodes of these sub-units and the displacement of the cross-sectional centroid master node is established, so as to condense the degrees of freedom in the cross section; secondly, based on the co-rotation coordinate method (C.R method), the pure deformation displacement of each sub-unit node is extracted, and the three-dimensional geometry and physical equation of the solid element is used to calculate the strain and stress matrix; finally, based on the virtual displacement principle, the tangent stiffness matrix of the spatial steel pipe concrete beam element in the structure coordinate system is calculated through the transformation of the coordinate system. By introducing the sub-unit method and the C.R method, a spatial steel pipe concrete beam model considering both geometric and material nonlinearities is established, and the calculation workload is greatly reduced, providing an accurate and efficient algorithm for the calculation of the bearing capacity of large-span steel pipe concrete arch.

[0070] Specifically, as shown in the drawings, Figure 6 The large-span steel pipe concrete arch bearing capacity calculation method of the present application is carried out according to the following steps:

[0071] S1, the steel pipe concrete arch is divided into a plurality of steel pipe concrete beam elements, and the steel pipe concrete beam elements are divided into a plurality of sub-units according to the material properties;

[0072] Because the steel pipe concrete arch has a complex "sleeving effect", it cannot be modeled and analyzed as a whole, the sub-unit method used in the present application can greatly reduce the calculation amount while ensuring the calculation accuracy, and improve the calculation efficiency;

[0073] S2, obtain the stiffness matrix of the concrete-filled steel tube beam unit;

[0074] In the initial calculation, the stiffness matrix of the concrete-filled steel tube beam unit adopts the linear stiffness matrix of the concrete-filled steel tube beam unit. Since the present application is an iterative calculation method, the stiffness matrix of the concrete-filled steel tube beam unit in the subsequent calculation adopts the tangent stiffness matrix of the concrete-filled steel tube beam unit in the previous iteration calculation;

[0075] S3, apply a reference load to the concrete-filled steel tube beam unit, and calculate the node displacement vector of the concrete-filled steel tube beam unit in the structure coordinate system based on the stiffness matrix of the concrete-filled steel tube beam unit and the applied load;

[0076] The present application adopts the Newton-Raphson method for iterative calculation. The calculated unbalanced force is subjected to convergence judgment at each iteration step, and then it is judged whether iteration needs to be continued based on the judgment result. The initial applied load is the reference load, which is set according to the requirements;

[0077] S4, establish a co-rotational coordinate system, and calculate the stress vector of each sub-unit in the co-rotational coordinate system based on the node displacement vector of the concrete-filled steel tube beam unit in the structure coordinate system;

[0078] The structure of each sub-unit can be unified, so that the analysis of each sub-unit can be carried out separately, and the analysis results can be applied to all sub-units. By calculating the stress vector of the sub-unit, the stress distribution in the sub-unit can be analyzed, which facilitates the subsequent stress analysis of the concrete-filled steel tube beam unit;

[0079] S5, calculate the end resistance of the concrete-filled steel tube beam unit based on the stress vector of each sub-unit in the co-rotational coordinate system, and calculate the unbalanced force based on the end resistance and the applied load;

[0080] The stress vectors of the sub-units are integrated to obtain the end resistance of the concrete-filled steel tube beam unit;

[0081] S6, judge whether the unbalanced force meets the convergence requirement. If the unbalanced force does not meet the convergence requirement, the unbalanced force is taken as the external load and the steps S2, S3, S4 and S5 are calculated. If the convergence requirement is met, it is further judged whether the material failure criterion is met. If the material failure criterion is met, the total load of all load steps before the failure criterion is output as the ultimate bearing capacity of the concrete-filled steel tube beam unit. If the material failure criterion is not met, the load is added and the steps S2, S3, S4 and S5 are calculated;

[0082] The unbalanced force is obtained based on the end resistance and the applied load, and reflects the relationship between the end resistance and the applied load. The end resistance represents the resistance generated by the steel pipe concrete beam unit under the current displacement deformation. If the unbalanced force is not converged, the unbalanced force is applied to the structure as an external load.

[0083] The application judges whether the structure is damaged at the end of each load step. If the stress of the structure reaches the damage criterion at this time, it proves that the structure reaches the limit state, and the total load applied to the structure at this time is the ultimate bearing capacity of the structure.

[0084] In some embodiments of the application, the step S1 described above is optimized, and specifically, as shown in Figures 1-4 The steel pipe concrete arch is divided into a plurality of columnar steel pipe concrete beam units, and the nodes of the steel pipe concrete beam units are the centers of the two sides in the axial direction. The steel pipe concrete beam units are divided into a plurality of hexahedral sub-units along the radial direction and the circumferential direction of the steel pipe concrete beam units, and the nodes of the sub-units are eight corner points of the hexahedral structure (as shown in Figure 4 ).

[0085] That is, the steel pipe concrete beam unit is a cylindrical structure with two nodes, and the sub-unit is a hexahedral structure with eight nodes. When actually splitting the steel pipe concrete beam unit, the steel pipe concrete beam unit needs to be split into a steel pipe sub-unit and a concrete sub-unit according to the material, because the two materials correspond to different stress-strain relationships. However, whether it is a steel pipe sub-unit or a concrete sub-unit, both are hexahedral eight-node structures, and their structural forms are consistent. When performing specific analysis, only the specific stress-strain relationship needs to be substituted.

[0086] In further embodiments of the present application, the step S2 is optimized, and the stiffness matrix of the concrete-filled steel tubular beam unit is obtained in the following manner: in the initial calculation (the first load step), the stiffness matrix of the concrete-filled steel tubular beam unit is a linear stiffness matrix of the concrete-filled steel tubular beam unit, and the linear stiffness matrix of the concrete-filled steel tubular beam unit is obtained by construction (according to the deformation compatibility relationship of the spatial concrete-filled steel tubular beam unit and the plane cross-section assumption, the displacement vector of any hexahedral sub-unit (taking the kth sub-unit as an example) and the displacement vector of the concrete-filled steel tubular beam unit can be obtained, and based on the above relationship, the linear stiffness matrix of the concrete-filled steel tubular beam unit can be obtained by combining the virtual displacement principle); in the subsequent load steps, the stiffness matrix of the concrete-filled steel tubular beam unit is obtained by calculation, which is actually the tangent stiffness matrix of the concrete-filled steel tubular beam unit obtained by calculating the end resistance, that is, in the iteration steps other than the first load step, the stiffness matrix of the concrete-filled steel tubular beam unit is the tangent stiffness matrix of the concrete-filled steel tubular beam unit calculated according to the total displacement after the last load step.

[0087] In preferred embodiments of the present application, the step S3 is optimized, and after the reference load is applied to the concrete-filled steel tubular beam unit, the nodal pure deformation displacement of the concrete-filled steel tubular beam unit in the structural coordinate system is calculated based on the stiffness matrix of the concrete-filled steel tubular beam unit obtained in the step S2, and the specific calculation formula is as follows:

[0088] P=K T ·d g

[0089] Wherein: P- the load applied to the concrete-filled steel tubular beam unit, the reference load is taken as the initial load in each load step, and then the iteration calculation is performed, the unbalanced force generated in each iteration step is applied to the structure as the load in the next iteration step, until convergence, and then the next load step is started;

[0090] K T - the stiffness matrix of the concrete-filled steel tubular beam unit, which is based on the linear stiffness matrix of the concrete-filled steel tubular beam unit in the first load, and is based on the tangent stiffness matrix of the concrete-filled steel tubular beam unit in the calculation of the previous iteration step in each subsequent iteration step, and the tangent stiffness matrix of the concrete-filled steel tubular beam unit needs to be updated in each iteration step;

[0091] d g - the displacement vector of the concrete-filled steel tubular beam unit.

[0092] The displacement vector d of the concrete-filled steel tubular beam unit g may be expressed as d g =[u i v i wi θ xi θ yi θ zi u j v j w j θ xj θ yj θ zj ] T ;

[0093] Where: u i - The displacement of node i in the x-direction of the structural coordinate system;

[0094] v i - The displacement of node i in the y-direction of the structural coordinate system;

[0095] w i - The displacement of node i in the z-direction of the structural coordinate system;

[0096] θ xi - The rotation angle of node i in the yz plane of the structural coordinate system;

[0097] θ yi - The rotation angle of node i in the xz plane of the structural coordinate system;

[0098] θ zi - The rotation angle of node i in the xy plane of the structural coordinate system;

[0099] u j - Node, displacement in the x-direction of the structural coordinate system;

[0100] v j - Node, displacement in the y-direction of the structural coordinate system;

[0101] w j - Node, displacement in the z-direction of the structural coordinate system;

[0102] θ xj - The rotation angle of node j in the yz plane of the structural coordinate system;

[0103] θ yj - The rotation angle of node j in the xz plane of the structural coordinate system;

[0104] θ zj - The rotation angle of node j in the xy plane of the structural coordinate system.

[0105] In some embodiments of this application, step S4 above has been optimized. In order to facilitate the analysis of concrete-filled steel tube beam elements and sub-elements, a co-rotating coordinate system is established, and the concrete-filled steel tube beam elements and sub-elements under the co-rotating coordinate system are analyzed. By condensing the degrees of freedom, the amount of calculation can be greatly reduced.

[0106] Establish a co-rotating coordinate system that moves with the translation and rotation of the concrete-filled steel tube element, with the nodes on one side of the concrete-filled steel tube beam element (e.g., ...) as the reference point. Figure 5 Point i shown is taken as the origin, and the line connecting the nodes on both sides of the steel-concrete composite beam element (as shown) is drawn from the origin. Figure 5 The line connecting points i and j is taken as the x-axis, the ray parallel to the side section of the steel-concrete composite tube and perpendicular to the x-axis is taken as the y-axis, and the ray parallel to the side section of the steel-concrete composite tube and perpendicular to both the x-axis and y-axis is taken as the z-axis.

[0107] In a co-rotating coordinate system, the steel-concrete composite beam element only exhibits pure deformation displacement along the x-axis. The coordinates in the co-rotational coordinate system and the structural coordinate system can be directly subtracted; only translational displacement exists in the y-axis and z-axis directions, with no deformation; the pure deformation displacement of the rotation angle can be calculated, and the specific method is as follows:

[0108] First, the spatial rotation vector ω of the cell is stored using the quaternion method:

[0109]

[0110] Where, ||ω||=(ω1) 2 +ω2 2 +ω3 2 ) 1 / 2 ω1, ω2, and ω3 are the components of the spatial rotation vector in the three coordinate axis planes under the co-rotation coordinate system; in the first iteration, the above quaternion matrix can be set as [1 0 0 0]. T ;

[0111] Let the rotation vector obtained in the i-th iteration be Δω, then the total number of quaternions at the end of the (i+1)-th iteration can be updated by the following formula:

[0112]

[0113] The rotation matrix corresponding to the rotation vector can be obtained from quaternion components:

[0114]

[0115] Using a second-order approximation method, the antisymmetric second-order tensor Ω(Δω) can be expressed in terms of the rotation matrix increment ΔR as follows:

[0116]

[0117] where tr(ΔR) is the trace of ΔR, and Ω(Δω) can be expressed as:

[0118]

[0119] The θ xil , θ yil and θ zil are the angular pure deformation of the i node of the steel tube concrete beam element, and the θ xjl , θ yjl and θ zjl of the j node of the steel tube concrete beam element can be obtained according to the above method, and thus the displacement vector of the steel tube concrete beam element in the co-rotational coordinate system can be obtained.

[0120]

[0121] wherein: - the pure deformation displacement of the steel tube concrete beam element in the x direction in the co-rotational coordinate system;

[0122] θ xil - the rotation angle of the i node of the steel tube concrete beam element in the yz plane in the co-rotational coordinate system;

[0123] θ yil - the rotation angle of the i node of the steel tube concrete beam element in the xz plane in the co-rotational coordinate system;

[0124] θ zil - the rotation angle of the i node of the steel tube concrete beam element in the xy plane in the co-rotational coordinate system;

[0125] θ xjl - the rotation angle of the j node of the steel tube concrete beam element in the yz plane in the co-rotational coordinate system;

[0126] θ yjl - the rotation angle of the j node of the steel tube concrete beam element in the xz plane in the co-rotational coordinate system;

[0127] θ zjl - the rotation angle of the j node of the steel tube concrete beam element in the xy plane in the co-rotational coordinate system.

[0128] After obtaining the displacement vector of the steel tube concrete beam element in the co-rotational coordinate system, the pure deformation vector of each sub-element in the co-rotational coordinate system is calculated through the displacement vector of the steel tube concrete beam element in the co-rotational coordinate system. Since the structure in the co-rotational coordinate system still satisfies the plane section assumption, the pure deformation vector of each sub-element in the co-rotational coordinate system can be calculated according to the following formula:

[0129] d qkl= T kd d gl

[0130] wherein: d qkl - the pure deformation vector of each sub-unit in the co-rotational coordinate system;

[0131] d gl - the displacement vector of the concrete-filled steel tube beam unit in the co-rotational coordinate system;

[0132] T kd - the transformation matrix.

[0133] T kd is the transformation matrix, which can be calculated by the following formula:

[0134]

[0135] wherein, t ki is the sub-matrix of the transformation matrix T kd , and x ki and y ki are the coordinates of the i-th node of the k-th sub-unit on the cross section of the concrete-filled steel tube beam.

[0136] Based on the obtained pure deformation vector d qkl of each sub-unit in the co-rotational coordinate system, the strain vector of each sub-unit in the co-rotational coordinate system can be calculated, and the specific calculation formula is as follows:

[0137] ε qkl = Bd qk = [B qkl B qk1 … B qk2 ]d qk8 qkl

[0138] wherein: ε qkl - the strain vector of the k-th sub-unit in the co-rotational coordinate system;

[0139] B qk - the strain matrix of the k-th sub-unit in the co-rotational coordinate system;

[0140] d qkl - the pure deformation vector of the k-th sub-unit in the co-rotational coordinate system;

[0141] The calculation formula of the transformation matrix B qki of the node of the k-th sub-unit in the co-rotational coordinate system is as follows:

[0142]

[0143] wherein, N i，x , N i，y and N i，z ​N i The derivatives of x, y and z can be found in the book "Finite Element Method" (Wang Xu-cheng, Tsinghua University Press).

[0144] The stress vector of each sub-element in the co-rotational coordinate system can be obtained based on the strain vector of each sub-element in the co-rotational coordinate system and the strain matrix of each sub-element, and the specific calculation formula is as follows:

[0145] σ qk = D qk ε qkl

[0146] Wherein: σ qk - the stress vector of the kth sub-element in the co-rotational coordinate system;

[0147] D qk - the stress matrix of the kth sub-element in the co-rotational coordinate system;

[0148] ε qkl - the strain vector of the kth sub-element in the co-rotational coordinate system.

[0149] The calculation method of the stress matrix D qk of the kth sub-element in the co-rotational coordinate system is as follows:

[0150]

[0151]

[0152] D qk is the stress matrix of the kth sub-element in the co-rotational coordinate system, μ is the Poisson's ratio of the material, and E is the tangent elastic modulus of the material, which can be obtained by looking up the slope of the stress-strain curve of the corresponding material.

[0153] In the preferred embodiment of the present application, the step S5 is optimized, and after obtaining the stress vector of each sub-element in the co-rotational coordinate system in step S4, the bar end resistance of the concrete filled steel tube beam element can be calculated based on the stress vector of each sub-element in the co-rotational coordinate system. Specifically, first, the tangent stiffness matrix of each sub-element in the co-rotational coordinate system is calculated based on the stress matrix of each sub-element in the co-rotational coordinate system, and the calculation formula is as follows:

[0154] k qk = ∫B qk T D qk B qk dV

[0155] Wherein: k qk - the tangent stiffness matrix of the kth sub-element in the co-rotational coordinate system;

[0156] B qk - transformation matrix of the node of the kth sub-unit in the co-rotational coordinate system;

[0157] D qk - stress matrix of the kth sub-unit in the co-rotational coordinate system;

[0158] Then the bar end resistance of the sub-unit can be obtained based on the tangent stiffness matrix k qk and the pure deformation vector d qkl of each sub-unit in the co-rotational coordinate system, and the calculation formula is as follows.

[0159] f qk = k qk d qkl

[0160] Wherein: f qk - bar end resistance of the kth sub-unit in the co-rotational coordinate system;

[0161] k qk - tangent stiffness matrix of the kth sub-unit in the co-rotational coordinate system;

[0162] d 一 - pure deformation vector of the kth sub-unit in the co-rotational coordinate system.

[0163] The bar end resistance of each sub-unit is obtained, and the bar end resistance matrix of the sub-unit is integrated to obtain the bar end resistance of the concrete-filled steel tube beam unit, and the specific calculation formula is as follows.

[0164]

[0165] Wherein: f g - bar end resistance of the concrete-filled steel tube beam unit;

[0166] f gi - bar end resistance of the concrete-filled steel tube beam unit node i side;

[0167] f gj - bar end resistance of the concrete-filled steel tube beam unit node j side;

[0168] N qkn - axial force in the resistance matrix;

[0169] y kn - y coordinate of the nth node of the kth sub-unit on the concrete-filled steel tube beam section;

[0170] x kn - x coordinate of the nth node of the kth sub-unit on the concrete-filled steel tube beam section;

[0171] The bar end resistance f gAfter that, the load P applied and the bar end resistance f of the CFT beam unit are g The unbalanced force ΔZ is obtained by subtraction.

[0172] In a further embodiment of the present application, the above step S6 is optimized, specifically, after obtaining the unbalanced force ΔZ, it is first determined whether the unbalanced force ΔZ meets the convergence requirement. The unbalanced force ΔZ obtained in the embodiment is actually a vector, and it is determined whether the absolute value of the maximum item of the unbalanced force ΔZ vector is less than a set value (the set value in the embodiment is 10 -5 , other set values can also be selected as long as the requirement is met), if the absolute value of the maximum item of the unbalanced force ΔZ vector is less than the set value, it is determined that the current unbalanced force ΔZ meets the convergence requirement; if the absolute value of the maximum item of the unbalanced force ΔZ vector is not less than the set value, it is determined that the current unbalanced force ΔZ does not meet the convergence requirement.

[0173] If the unbalanced force ΔZ meets the convergence requirement, it is further determined whether the material of the CFT beam unit reaches the failure criterion. The failure criterion of the material in the embodiment refers to exceeding the ultimate deformation strain of the material, i.e., the yield strain of the steel pipe and the crushing strain of the concrete forming the CFT beam unit. By setting the material failure criterion, when the material failure criterion is exceeded, it can be considered that the load currently applied to the CFT beam unit will cause damage to the CFT beam unit, i.e., the failure criterion is reached. If the unbalanced force ΔZ meets the convergence requirement and the failure criterion is reached at this time, it is proved that the load applied in the load step will cause damage to the CFT beam unit, and the load iteration has reached the limit, so the load applied in the previous load step is output as the ultimate bearing capacity of the CFT beam unit; if the unbalanced force ΔZ meets the convergence requirement but the failure criterion is not reached, the reference load is increased on the basis of the load applied in the current load step for the next iteration calculation;

[0174] If the unbalanced force ΔZ does not meet the convergence requirement, the unbalanced force ΔZ is taken as the external load on the basis of the load applied in the current load step for the next iteration calculation;

[0175] Until the ultimate bearing capacity of the CFT beam unit, i.e., the bearing capacity of the CFT arch, is finally output.

[0176] At the end of each iteration calculation, in order to facilitate the next iteration calculation, the tangent stiffness matrix of the CFT beam unit needs to be calculated based on the bar end resistance of the CFT beam unit calculated above, and the specific calculation method is as follows:

[0177] The tangent stiffness matrix K of the CFT beam unit is calculated according to the following formula T :

[0178]

[0179] wherein,

[0180]

[0181]

[0182] Pure deformation displacement of a steel tube concrete beam element in the co-rotational coordinate system and the total displacement d in the structural coordinate system g have the following relationship:

[0183]

[0184] The internal force vector f of a steel tube concrete beam element in the two coordinate systems gn and have the following relationship:

[0185]

[0186] wherein, E is a 12-order matrix with the co-rotational coordinate system base vectors as the diagonal sub-matrix; is the translational displacement in the co-rotational coordinate system and the rotational displacement and the translational displacement in the co-rotational coordinate system and the rotational vector have the following relationship:

[0187]

[0188] The above formula is transformed in combination with the above formula (assuming that a certain node a is taken as an example):

[0189]

[0190] is a projection matrix, and the expression is:

[0191]

[0192] is a 12-order unit matrix, is a conversion matrix composed of the antisymmetric tensors of each node of the element, reflects the influence of the change of the position of each node in the co-rotational coordinate system on the position in the co-rotational coordinate system.

[0193] and have the following expressions, respectively:

[0194]

[0195] wherein, l nis the distance between the two ends of the current CFT unit.

[0196]

[0197] η = b1 / b2, η 11 = b 11 / b2, η 12 = b 12 / b2, η 21 = b 21 / b2, η 22 = b 22 / b2

[0198]

[0199] where b, b1, b2 are the auxiliary vectors of the spatial beam element, and the specific expressions can be found in the literature (Crisfield MA. Non-Linear Finite Element Analysis of Solids and Structures, Volume 2: Advanced Topics [M]. Chichester: Wiley, 1997).

[0200] In actual calculation: such as Figure 6The steel pipe concrete arch is divided into a plurality of columnar steel pipe concrete beam units, and the steel pipe concrete beam units are divided into a plurality of hexahedral sub-units along the radial direction and the circumferential direction of the steel pipe concrete beam units; the linear stiffness matrix of the steel pipe concrete beam unit is obtained as the stiffness matrix of the steel pipe concrete beam unit at the first load loading; the tangent stiffness matrix of the steel pipe concrete beam unit obtained at the previous load step is taken as the stiffness matrix of the steel pipe concrete beam unit at the subsequent load step; the reference load is taken as the initial load at each load step, and then iterative calculation is performed, the unbalanced force generated at each iteration step is applied to the structure as the load of the next iteration step, until convergence is achieved, and then the next load step is started; the node pure deformation displacement of the steel pipe concrete beam unit in the structural coordinate system is calculated based on the stiffness matrix of the steel pipe concrete beam unit and the applied load; the co-rotation coordinate system is established, the displacement vector of the steel pipe concrete beam unit in the co-rotation coordinate system is obtained based on the node pure deformation displacement of the steel pipe concrete beam unit in the structural coordinate system, the pure deformation vector of each sub-unit in the co-rotation coordinate system is calculated through the displacement vector of the steel pipe concrete beam unit in the co-rotation coordinate system, the strain vector of each sub-unit in the co-rotation coordinate system is obtained based on the obtained pure deformation vector of each sub-unit in the co-rotation coordinate system, and the stress vector of each sub-unit in the co-rotation coordinate system is obtained based on the strain vector of each sub-unit in the co-rotation coordinate system and the strain matrix of each sub-unit; the end resistance of the steel pipe concrete beam unit is calculated based on the stress vector of each sub-unit in the co-rotation coordinate system, the end resistance of the sub-unit is obtained based on the tangent stiffness matrix of the sub-unit in the co-rotation coordinate system and the pure deformation vector of the sub-unit in the co-rotation coordinate system, and the end resistance of the steel pipe concrete beam unit is obtained by integrating the end resistance matrix of the sub-unit; the unbalanced force is obtained by subtracting the end resistance of the steel pipe concrete beam unit from the load applied at the current load step; whether the unbalanced force meets the convergence requirement is judged, that is, whether the absolute value of the maximum item of the unbalanced force vector is less than a set value, if the unbalanced force meets the convergence requirement, whether the material of the steel pipe concrete beam unit reaches the failure criterion is judged, that is, whether the effect of the current applied load on the steel pipe concrete beam unit exceeds the limit deformation strain of the material, if the failure criterion is reached, the total load of all load steps before the failure criterion is reached is output as the ultimate bearing capacity of the steel pipe concrete beam unit, if the failure criterion is not reached, the reference load is added to the current load step to perform the next iteration calculation; if the unbalanced force does not meet the convergence requirement, the unbalanced force is added to the current load step to perform the next iteration calculation.

[0201] The above shows and describes the basic principles, main features and advantages of the present application. Those skilled in the art should understand that the present application is not limited to the above-mentioned embodiments, and the above-mentioned embodiments and descriptions in the specification are only to illustrate the principles of the present application. Various changes and improvements can be made to the present application without departing from the spirit and scope of the present application, and these changes and improvements all fall within the scope of the claimed present application. The scope of protection of the present application is defined by the appended claims and their equivalents.

Claims

1. A method for calculating the load bearing capacity of a long-span concrete-filled steel tube arch, characterized in that: The method comprises the following steps: S1, dividing the steel pipe concrete arch into a plurality of steel pipe concrete beam units, and dividing the steel pipe concrete beam units into a plurality of sub-units according to material properties; S2, obtaining a stiffness matrix of the steel pipe concrete beam unit; S3, applying a reference load to the steel pipe concrete beam unit, and calculating a node displacement vector of the steel pipe concrete beam unit in a structure coordinate system based on the stiffness matrix of the steel pipe concrete beam unit; S4, establishing a co-rotation coordinate system, and calculating stress vectors of each sub-unit in the co-rotation coordinate system based on the node displacement vector of the steel pipe concrete beam unit in the structure coordinate system; S5, calculating a bar end resistance of the steel pipe concrete beam unit based on the stress vectors of each sub-unit in the co-rotation coordinate system, and calculating an unbalanced force based on the bar end resistance and the applied load; S6, judging whether the unbalanced force meets the convergence requirement, if the unbalanced force does not meet the convergence requirement, calculating the unbalanced force as an external load according to the steps S2, S3, S4 and S5, if the unbalanced force meets the convergence requirement, further judging whether a material failure criterion is reached, if the material failure criterion is reached, outputting total loads of all load steps before the material failure criterion is reached as a limit bearing capacity of the steel pipe concrete beam unit, if the material failure criterion is not reached, continuing to add loads and calculating according to the steps S2, S3, S4 and S5; In the step S1, the method for dividing the steel pipe concrete arch into the plurality of steel pipe concrete beam units and the sub-units comprises the following steps: firstly, dividing the steel pipe concrete arch into a plurality of columnar steel pipe concrete beam units, and dividing the steel pipe concrete beam units into a plurality of hexahedral sub-units along a radial direction and a circumferential direction of the steel pipe concrete beam units; In the step S4, the method for establishing the co-rotation coordinate system comprises the following steps: establishing a co-rotation coordinate system that moves and rotates with the steel pipe concrete unit, taking a node i on one side of the steel pipe concrete beam unit as an origin, taking a direction of a line connecting two nodes on opposite sides of the steel pipe concrete beam unit as an x-axis, taking a direction parallel to a side section of the steel pipe concrete and vertical to the x-axis as a y-axis, and taking a direction parallel to the side section of the steel pipe concrete and vertical to the x-axis and the y-axis as a z-axis; In the step S4, the method for calculating the stress vectors of each sub-unit in the co-rotation coordinate system comprises the following steps: calculating a displacement vector of the steel pipe concrete beam unit in the co-rotation coordinate system based on a pure deformation displacement in an x-axis direction and a rotation angle pure deformation displacement in the node displacement vector of the steel pipe concrete beam unit in the structure coordinate system, calculating pure deformation vectors of each sub-unit in the co-rotation coordinate system through the displacement vector of the steel pipe concrete beam unit in the co-rotation coordinate system, calculating strain vectors of each sub-unit in the co-rotation coordinate system based on the pure deformation vectors of each sub-unit in the co-rotation coordinate system, and calculating the stress vectors of each sub-unit in the co-rotation coordinate system based on the strain vectors of each sub-unit in the co-rotation coordinate system and strain matrices of each sub-unit.

2. The method for calculating the load carrying capacity of a long-span concrete-filled steel tubular arch according to claim 1, wherein: The method for calculating the displacement vector of the steel pipe concrete beam unit in the co-rotation coordinate system comprises the following steps: wherein: d gl —displacement vector of the steel tube concrete beam element in the co-rotational coordinate system; — the pure deformation displacement of the steel pipe concrete beam unit in the x direction under the common coordinate system; θ xil —the rotation angle of the node i of the CFST beam element in the yz plane under the common coordinate system; θ yil —the rotation angle of the node i of the CFST beam element in the xz plane under the common coordinate system; θ zil —the rotation angle of the node i of the CFST beam element in the xy plane under the common coordinate system; θ xjl —the rotation angle of the node j of the steel tube concrete beam element in the yz plane under the common coordinate system; θ yjl —the rotation angle of the node j of the steel tube concrete beam element in the xz plane under the common coordinate system; θ zjl —the rotation angle of the node j of the steel tube concrete beam element in the xy plane under the common coordinate system.

3. The method for calculating the load carrying capacity of a long-span concrete-filled steel tubular arch according to claim 1, wherein: The method for calculating the pure deformation vectors of each sub-unit in the co-rotation coordinate system comprises the following steps: calculating the pure deformation vectors of each sub-unit in the co-rotation coordinate system according to the following formula: d qkl = T kd d gl where: d qkl — the pure deformation vector of each sub-unit in the co-rotational coordinate system; d gl - displacement vector of the steel tube concrete beam element in the co-rotational coordinate system; T kd — conversion matrix.

4. The method for calculating the load carrying capacity of a long-span concrete-filled steel tubular arch according to claim 1, wherein: The method for calculating the strain vector of each sub-unit in the co-rotational coordinate system comprises: calculating the strain vector of each sub-unit in the co-rotational coordinate system according to the following formula: e qkl = B qk d qkl where: ε qkl — strain vector of the kth subunit in the corotational coordinate system; B qk — the strain matrix of the kth subunit in the corotational coordinate system; d qkl — the pure deformation vector of the kth sub-cell in the co-rotational coordinate system.

5. The method for calculating the load carrying capacity of a long span concrete filled steel tubular arch according to claim 1, wherein: The method for calculating the stress vector of each sub-unit in the co-rotational coordinate system comprises: calculating the stress vector of each sub-unit in the co-rotational coordinate system according to the following formula: σ qk = D qk ε qkl where: σ qk — stress vector of the kth subunit in the co-rotational coordinate system; D qk — the stress matrix of the kth subunit in the corotational coordinate system; ε gkl — strain vector of the kth subunit in the co-rotational coordinate system.

6. The method for calculating the load carrying capacity of a long span concrete filled steel tubular arch according to claim 1, wherein: In the step S5, the method for calculating the end resistance of the concrete-filled steel tube beam unit comprises: calculating the tangent stiffness matrix of the sub-unit according to the following formula: k qk = ∫B qk T D qk B qk dV where: k qk - the tangent stiffness matrix of the kth subunit in the co-rotational coordinate system; B qk - transformation matrix of the nodes of the kth subunit in the common coordinate system; D qk — the stress matrix of the kth subunit in the corotational coordinate system; The end resistance matrix of the sub-unit is calculated based on the tangent stiffness matrix of the sub-unit and the pure deformation vector of the sub-unit, and the end resistance of the concrete-filled steel tube beam unit is obtained by integrating the end resistance matrix of the sub-unit.

7. The method for calculating the load carrying capacity of a long span concrete filled steel tubular arch according to claim 1, wherein: If the convergence requirement is not met, the reference load is increased based on the applied load, and the calculation is performed according to the above steps S2, S3, S4 and S5; if the material failure criterion is not reached, the unbalanced force is increased based on the applied load, and the calculation is performed according to the above steps S2, S3, S4 and S5.

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