Method for solving stiffness and consistent mass matrix of variable cross-section space beam element

Abstract

The application discloses a method for solving stiffness and consistent mass matrix of a variable cross-section space beam unit, and comprises the following steps: establishing a generalized coordinate system of a cantilever beam constraint; establishing a stiffness matrix without rigid body displacement; establishing a stiffness matrix containing rigid body displacement; establishing a consistent mass matrix when shear action is ignored; establishing a consistent mass matrix when shear action is considered; and forming a digital twin model of the variable cross-section space beam unit. The application uses the generalized coordinate system of the cantilever beam constraint, adopts force interpolation functions to establish the cross-section stiffness matrix and flexibility matrix without rigid body displacement, and uses the virtual work principle to respectively solve the element stiffness matrix under the conditions of considering shear effect and ignoring shear effect, obtains the displacement deformation expression in the generalized coordinate system, and further establishes the corresponding shape function matrix, so that the element consistent mass can be obtained by substituting the shape function matrix into the general integral expression of the consistent mass matrix, and the stiffness and consistent mass matrix calculation technology of the variable cross-section space beam unit is established.

Description

Technical Field

[0001] This invention belongs to the field of bridge engineering technology in the transportation industry, specifically relating to a method for solving the stiffness and uniform mass matrix of a variable cross-section spatial beam element. Background Technology

[0002] China's high-speed rail network is developing rapidly. By the end of 2021, the operational length of high-speed rail had reached 40,000 kilometers, with bridges accounting for more than 50% of that. A key issue that arises is how to scientifically and rationally design and manage high-speed rail bridges to ensure the long-term operational safety of high-speed rail.

[0003] Shrinkage, creep, and pier settlement during bridge construction and operation can cause track irregularities, thus affecting train safety. From a dynamic perspective, the natural frequency characteristics of the structure have a significant impact on vehicle-bridge coupled vibration, and these characteristics are closely related to the structure's stiffness and mass matrices. Therefore, in vehicle-bridge coupled analysis, the accuracy of the element used in describing the stiffness and mass matrices directly affects the analysis results.

[0004] Vehicle-bridge coupling analysis is an important means of quantifying safe operation indicators. Current vehicle-bridge coupling software often uses constant cross-section elements to simplify the processing of variable cross-section bridge components. However, the high-speed railway bridge design software commonly used in the design process cannot accurately simulate the mechanical properties of variable cross-section bridge components.

[0005] Therefore, in response to the above problems, it is necessary to develop a method for solving the stiffness and uniform mass matrix of variable cross-section spatial beam elements. This method can systematically solve the problem of accurately calculating the stiffness and uniform mass matrix of railway bridge piers in a real environment using twin models, so as to serve the development of independent finite element software and the construction of independent digital twin models of high-speed railway bridges. Summary of the Invention

[0006] This invention is proposed to solve the problems existing in the prior art, and its purpose is to provide a method for solving the stiffness and uniform mass matrix of a variable cross-section spatial beam element.

[0007] The technical solution of this invention is: a method for solving the stiffness and uniform mass matrix of a variable cross-section spatial beam element, comprising the following steps: A. Establish a generalized coordinate system for cantilever beam constraints; B. Based on the generalized coordinate system, establish a stiffness matrix without rigid body displacement according to the internal stress inside the beam element; C. Based on the coordinate system transformation, establish a stiffness matrix that includes rigid body displacement; D. Based on the displacement variables, establish a consistent mass matrix when shearing is ignored; E. Based on the consistent mass matrix ignoring shear effects, add a shear effect term to establish a consistent mass matrix considering shear effects; F. Based on the stiffness matrix including rigid body displacement, the uniform mass matrix neglecting shear action, and the uniform mass matrix considering shear action, a digital twin model of the variable cross-section spatial beam element is formed.

[0008] Furthermore, the generalized coordinate system established in step A for cantilever beam constraint has generalized degrees of freedom that are only related to the deformation of the element itself, and the generalized coordinates adopt the constraint form of cantilever beam constraint.

[0009] Furthermore, step B, based on a generalized coordinate system and the internal stress within the beam element, establishes a stiffness matrix without rigid body displacement. The specific process is as follows: First, establish the expression for the internal forces on any section inside the spatial beam element; Then, the sectional forces on any internal cross section are calculated; Next, the cross-sectional deformation at any internal cross-section was calculated; Then, based on the cross-sectional force and cross-sectional deformation, the relationship between the cross-sectional force and cross-sectional deformation is obtained; Then, the element compliance matrix in the generalized coordinate system is obtained by solving; Finally, the stiffness matrix without rigid body displacement is obtained by inverting the element flexibility matrix.

[0010] Furthermore, step C involves establishing a stiffness matrix that includes rigid body displacements based on coordinate system transformation. The specific process is as follows: First, establish the transformation expression from the generalized coordinate system to the global coordinate system; Then, the specific explicit expressions for each transformation matrix are established.

[0011] Furthermore, step D establishes a consistent mass matrix ignoring shear effects based on the displacement variables. The specific process is as follows: First, establish the displacement and deformation quantities at any position of the element in the generalized coordinate system; Then, establish the displacement relationships in the unit's local coordinate system; Then, the expressions related to spatial displacement and nodal displacement are obtained by sorting them out; Then, the general integral expression for the uniform mass matrix of the unit is obtained; Finally, the uniform mass matrix of the variable cross-section spatial beam element is obtained when shear action is neglected.

[0012] Furthermore, step E, based on the consistent mass matrix ignoring shear effects, adds a shear effect term to establish a consistent mass matrix considering shear effects. The specific process is as follows: First, the expression for displacement deformation in the displacement deformation amount is modified by adding a shear effect term; Then, establish the expression for the shear deformation coefficient; Then, the interpolation function considering shear action is derived. N 1~ N 20 expression; Finally, a uniform mass matrix considering shear effects is established.

[0013] Furthermore, step F generates a digital twin model of the variable cross-section spatial beam element based on the stiffness matrix including rigid body displacement, the uniform mass matrix neglecting shear action, and the uniform mass matrix considering shear action. The specific process is as follows: The stiffness matrix obtained in step C, the uniform mass matrix obtained in step D when ignoring shear action, and the uniform mass matrix obtained in step E when considering shear action are used as twin models for the variable cross-section spatial beam element of the bridge structure to characterize its mechanical features.

[0014] The beneficial effects of this invention are as follows: This invention addresses the problem of accurate simulation and calculation of the stiffness and uniform mass matrix of variable cross-section spatial beam elements in bridge structures. Utilizing a generalized coordinate system constrained by a cantilever beam, a force interpolation function is used to establish the cross-sectional stiffness and flexibility matrices without rigid body displacement. The principle of virtual work is used to solve for the element flexibility matrix considering and neglecting shear effects. Expressions for each displacement deformation in the generalized coordinate system are obtained using the unit force method in mechanics of materials, thus establishing the corresponding shape function matrices. Substituting the shape function matrices into the general integral expression of the uniform mass matrix yields the element uniform mass, thereby establishing a technique for calculating the stiffness and uniform mass matrix of variable cross-section spatial beam elements in bridge structures.

[0015] This invention can accurately simulate the stiffness and consistent mass matrix of variable cross-section spatial beam elements in bridge structures in the transportation field, and solves the underlying key technical problems of accurate mapping between the physical model and the data-driven model of variable cross-section spatial beam elements for modeling, design analysis, diagnosis and decision-making. Attached Figure Description

[0016] Figure 1 This is a schematic diagram of the steps of the present invention; Figure 2 This is a schematic diagram of the overall coordinate system of the present invention; Figure 3 This is a schematic diagram of a partial coordinate system of the present invention; Figure 4 This is a schematic diagram of the generalized coordinate system of the present invention; Figure 5 This is a schematic diagram of a box-shaped variable cross-section simply supported beam in this invention; Figure 6 This is a comparison diagram of the free vibration frequencies of the uniform mass matrix in this invention; Figure 7 This is a schematic diagram of the elevation measuring point layout for a variable cross-section bridge structure in this invention; Figure 8 This is a comparison diagram of the calculated elevation and the measured elevation of the variable cross-section bridge structure in this invention. Detailed Implementation

[0017] The present invention will now be described in detail with reference to the accompanying drawings and embodiments: like Figures 1 to 8 As shown, a method for solving the stiffness and uniform mass matrix of a variable cross-section spatial beam element includes the following steps: A. Establish a generalized coordinate system for cantilever beam constraints; B. Based on the generalized coordinate system, establish a stiffness matrix without rigid body displacement according to the internal stress inside the beam element; C. Based on the coordinate system transformation, establish a stiffness matrix that includes rigid body displacement; D. Based on the displacement variables, establish a consistent mass matrix when shearing is ignored; E. Based on the consistent mass matrix ignoring shear effects, add a shear effect term to establish a consistent mass matrix considering shear effects; F. Based on the stiffness matrix including rigid body displacement, the uniform mass matrix neglecting shear action, and the uniform mass matrix considering shear action, a digital twin model of the variable cross-section spatial beam element is formed.

[0018] The generalized coordinate system established in step A for cantilever beam constraint has generalized degrees of freedom that are only related to the deformation of the element itself. The generalized coordinate system adopts the constraint form of cantilever beam constraint.

[0019] Step B, based on the generalized coordinate system and the internal stress within the beam element, establishes a stiffness matrix without rigid body displacement. The specific process is as follows: First, establish the expression for the internal forces on any section inside the spatial beam element; Then, the sectional forces on any internal cross section are calculated; Next, the cross-sectional deformation at any internal cross-section was calculated; Then, based on the cross-sectional force and cross-sectional deformation, the relationship between the cross-sectional force and cross-sectional deformation is obtained; Then, the element compliance matrix in the generalized coordinate system is obtained by solving; Finally, the stiffness matrix without rigid body displacement is obtained by inverting the element flexibility matrix.

[0020] Step C involves establishing a stiffness matrix that includes rigid body displacements based on coordinate system transformation. The specific process is as follows: First, establish the transformation expression from the generalized coordinate system to the global coordinate system; Then, the specific explicit expressions for each transformation matrix are established.

[0021] Step D involves establishing a consistent mass matrix based on the displacement variables, neglecting shear effects. The specific process is as follows: First, establish the displacement and deformation quantities at any position of the element in the generalized coordinate system; Then, establish the displacement relationships in the unit's local coordinate system; Then, the expressions related to spatial displacement and nodal displacement are obtained by sorting them out; Then, the general integral expression for the uniform mass matrix of the unit is obtained; Finally, the uniform mass matrix of the variable cross-section spatial beam element is obtained when shear action is neglected.

[0022] Step E, based on the consistent mass matrix ignoring shear effects, adds a shear effect term to establish the consistent mass matrix considering shear effects. The specific process is as follows: First, the expression for displacement deformation in the displacement deformation amount is modified by adding a shear effect term; Then, establish the expression for the shear deformation coefficient; Then, the interpolation function considering shear action is derived. N 1~ N 20 expression; Finally, a uniform mass matrix considering shear effects is established.

[0023] Step F generates a digital twin model of the variable cross-section spatial beam element based on the stiffness matrix including rigid body displacement, the uniform mass matrix neglecting shear action, and the uniform mass matrix considering shear action. The specific process is as follows: The stiffness matrix obtained in step C, the uniform mass matrix obtained in step D when ignoring shear action, and the uniform mass matrix obtained in step E when considering shear action are used as twin models for the variable cross-section spatial beam element of the bridge structure to characterize its mechanical features.

[0024] Specifically, the generalized coordinate system for the cantilever beam constraint established in step A is based on the global coordinate system and the local coordinate system. The specific process is as follows: like Figure 2 , Figure 3 As shown, let the nodal forces of the element in the global coordinate system and the local coordinate system be respectively (1) In equation (1) above, the element nodal forces have 12 elements, corresponding to 12 degrees of freedom. These 12 degrees of freedom can be divided into two categories: Firstly, r(release class) r The (release) class is related to the transformation of the unit itself; Secondly, s (suppressed or supported) class, s The (suppressed or supported) class relates to the rigid body motion of the unit.

[0025] For spatial beam elements r = s =6.

[0026] By appropriately selecting a coordinate system, the coordinate system in equation (1) can be optimized. s The term corresponding to the degree of freedom is equal to zero. At this point, only the deformation of the element itself is relevant. r The existence of degrees of freedom can be termed generalized degrees of freedom. The element nodal forces in this coordinate system, i.e., the generalized coordinate system, are denoted as: (2) As long as these 6 independent degrees of freedom can be represented, any coordinate system can be established.

[0027] Generally, for the convenience of deriving the formula, either a simply supported beam or a cantilever beam constraint can be used. This invention adopts the cantilever beam constraint.

[0028] Generalized coordinate system, such as Figure 4 As shown, the axial, transverse, longitudinal, and rotational degrees of freedom of element node 1 in three directions are fixed.

[0029] Specifically, step B, based on the generalized coordinate system and the internal stress within the beam element, establishes a stiffness matrix without rigid body displacement. The specific process is as follows: In the generalized coordinate system, the nodal forces corresponding to the deformation of the beam are expressed as: (3).

[0030] Then assume that any element within the unit... x The internal forces at the cross section are: (4).

[0031] Based on the boundary conditions of the cantilever beam, the equilibrium relationship can be obtained as follows: x Force F at the cross section x The formula is as follows: (5) In the formula, e( x ) is the force interpolation function.

[0032] The internal forces on any cross section can be obtained from the nodal forces at both ends of the beam element and the interpolation function, and their expressions are as follows: (6).

[0033] In a generalized coordinate system, within a unit x The deformation at the cross section is: (7) In the formula, ε( x ) represents the axial deformation of the centroid of the cross section. γ y ( x ), γ z ( x ) are respectively y and z Cross-sectional shear deformation in the direction of the direction, φ x ( x ), φ y ( x ), φ z ( x ) are respectively x , y and z The rotation angle of the central axis of the direction.

[0034] Combining the above cross-sectional forces and deformations, the relationship between cross-sectional forces and deformations can be obtained, specifically expressed as follows: (8) In the formula, k s ( x ) is the beam section stiffness matrix, and its matrix is: (9) In the formula, E For elastic modulus, G Shear modulus A For cross-sectional area, A sy 、A sz Let the shear area be assuming E , G It is a constant.

[0035] Cross-sectional compliance matrix f s ( x ) is the inverse matrix of the section stiffness matrix, expressed as: (10).

[0036] Using the principle of virtual work, the element flexibility matrix f in the generalized coordinate system can be obtained from the flexibility matrix of the cross section. r ( x The general formula for its calculation is: (11).

[0037] After derivation, the element flexibility matrix f r ( x The specific expression for ) is: (12).

[0038] After obtaining the flexibility matrix, the element stiffness matrix k can be obtained by inverting equation (12). r ( x ).

[0039] (13) In the formula, A sy 、A sz This represents the area of ​​the shear surface.

[0040] Specifically, step C involves establishing a stiffness matrix that includes rigid body displacements based on coordinate system transformation. The specific process is as follows: There are 12 displacements and 12 forces in the local and global coordinate systems, located at the nodes on both sides of the beam element.

[0041] Therefore, the transformation from the generalized coordinate system to the global coordinate system requires the multiplication of two transformation matrices, as shown in the following expression: (14) In the formula, K g T is the stiffness matrix in the global coordinate system. rg This is the transformation matrix from generalized coordinates to the global coordinate system.

[0042] Transformation matrix T from generalized coordinates to global coordinate system rg The expression is: (15) In the formula, T re Let T be the transformation matrix from generalized coordinates to local coordinates. eg This is the transformation matrix from local coordinates to the global coordinate system.

[0043] Transformation matrix T from generalized coordinates to local coordinate system re And the transformation matrix T from local coordinates to global coordinates. eg The expressions for each are: (16) (17) In the formula, matrix elements t for (18) In the formula, The transformation angle is between the two transformation matrices.

[0044] Specifically, step D establishes a consistent mass matrix ignoring shear effects based on the displacement variables. The specific process is as follows: The boundary conditions of the element in the generalized coordinate system are the same as those of a cantilever beam subjected to generalized nodal forces. The displacement and deformation at point x in the element in the generalized coordinate system can be obtained using the unit force method in mechanics of materials, as follows: (19) In the formula, N ( x 0) M z ( x 0), M y ( x 0), T ( x 0) and other variables are represented in the unit x Internal forces at section 0.

[0045] By combining the calculation with formula (5), we can obtain: (20) In the formula, H(·) is the Heaviside function, and its expression is: (twenty one) .

[0046] To avoid local discontinuities, it is stipulated that when x=x When the value is 0, the expression is: (twenty two) .

[0047] In a generalized coordinate system, the element nodal force equals the element stiffness matrix multiplied by the element nodal displacement, and its expression is: (twenty three) Combining formulas (5) and (19), the displacement relationship in the unit's local coordinate system can be obtained, expressed as: (twenty four) In the formula, e i ( x 0) ( i= 1 、 4 、 5、 6) are the first and second digits of the matrix in formula (6). i OK (25) Ignoring shear effects, equation (12) can be simplified to the matrix expression of equation (26), specifically: (26) Substituting the inverse matrices of formulas (6), (25), and (26) into formula (24) and rearranging, we obtain the expressions related to spatial displacement and nodal displacement: (27) After organizing it, we can conclude that: (28 a ) (28 b ) (28 c ) (28d) (28e) (28f) In the formula, The inverse matrix of expression (26) is in the i-th row and j-th column.

[0048] The corresponding expressions for φ1~φ10 are: (29 a ) (29 b ) Based on the above, the general integral expression for the uniform mass matrix of a unit cell is: (30) In the formula, ρ The mass density of the material, m s This is the cross-sectional mass matrix.

[0049] Cross-sectional mass matrix m s The expression is: (31) In the formula, r Radius of gyration r=sqrt ( J / A ).

[0050] The shape function matrix N is: (32) Thus, the uniform mass matrix of the variable cross-section spatial beam element without considering shear effects can be obtained by formula (30).

[0051] Specifically, step E, based on the consistent mass matrix ignoring shear effects, adds a shear effect term to establish a consistent mass matrix considering shear effects. The specific process is as follows: When the beam height is greater than 1 / 5 of the length, the effect of shear deformation must be considered.

[0052] At this time, the unit in the generalized coordinate system x The displacement deformations at each location can be modified based on formula (19), that is, it is necessary to modify the formula (19). θ y ( x ), θ z ( x Two expressions are modified, while the expressions for the other items remain unchanged. θ y ( x ), θ z ( x The two expressions are: (33) In the formula, γ y , γ z This represents the amount of shear deformation.

[0053] Similarly, the displacement relationship in the local coordinate system of the element can also be modified based on formula (24). The modified expression is as follows, while other sub-items remain unchanged.

[0054] (34) The expression related to spatial displacement and nodal displacement is the same as that in formula (27), assuming φ 1~ φ 10 The expression remains unchanged; now we need to consider the relevant effects of adding shear effect.

[0055] (35) In the formula, f y s , f z s They are respectively in xoy and xozWhen deformation occurs in a plane, the shear stress non-uniformity coefficient of the cross-section, also known as the shear deformation coefficient, is determined by the non-uniform distribution of shear stress along the cross-section. Q / GA The correction factor, which characterizes the ratio of the beam's cross-sectional area to the effective area resisting shear deformation, can be expressed as: (36) In formula (35), (37) (38).

[0056] Assuming the shear deformation on the element satisfies the Lagrange interpolation relation, then from equations (37) and (38), we can obtain: (39)

[0057] At this point, formulas (33) and (34) can be expressed as: (40)

[0058] After obtaining the expression for formula (40), the remaining derivation process is similar to the derivation process ignoring shear action, and finally we get... N 1~ N 20 The expression is described as follows: N 1~ N 12 The expression is the same as the one that ignores the shear effect. N 1~ N 12 The expression (28) a )~ (28 d The formula number can be recorded as (41). a )~ (41 d ). N 13 ~ N 20 The expression for this is different from the expression for ignoring shear effects; its specific formula is: (41 e ) (41 f ) In the formula, The inverse matrix of formula (12) represents the first... i Line 1 j List.

[0059] The cross-sectional mass matrix at this timem s Its expression is (42) In the formula, r=sqrt ( J / A ); r y =sqrt ( I y / A ); r z =sqrt ( I z / A ).

[0060] The general integral expression for the uniform mass matrix is ​​the same as that in formula (30), and the form function matrix is ​​the same as that in formula (32).

[0061] At this point, the uniform mass matrix considering shear action has been obtained.

[0062] Example 1 To verify the accuracy of the stiffness matrix and uniform mass matrix of the spatial beam element obtained by this invention, the relevant formulas obtained by this invention were incorporated into the autonomous finite element platform "Wolong" for modal frequency comparison analysis.

[0063] The test results were compared and analyzed with the commercial software Midas Civil. The calculation model is as follows: Figure 5 As shown, the cross-section is that of a box girder, which varies linearly along its length, is simply supported at both ends, and is 15m long. It is uniformly divided into 15 elements, and the elastic modulus of the concrete is taken as E = 3.15E7 KN / m. 2 Mass density ρ = 2549 kg / m³ 3 The variable cross-section group model is used, and the changes in the y-axis and z-axis are represented by a linear equation. The cross-sectional properties of the two ends of the simply supported beam are shown in Table 1.

[0064] Using the displacement shape function given in this invention, consistent mass matrices considering and not considering shear were obtained, and their free vibrations were analyzed. The comparison data of the first eight free vibration frequencies with the analysis results from Midas Civil are shown in Tables 2 and 3, respectively.

[0065] In summary, neglecting shear deformation, the maximum error between the self-developed program "Wolong" and Midas Civil is 0.053%; considering the shear effect, the maximum error between "Wolong" and Midas Civil is 0.154%, demonstrating high calculation accuracy. A comparison of the calculation results in Tables 2 and 3 also shows that the shear effect can influence the modal frequency values.

[0066] Plot the above calculation results on the same graph, such as... Figure 6 As shown in the figure, the calculation results are very close.

[0067] Example 2 The formulas for the spatial beam element stiffness matrix and uniform mass matrix obtained by this invention were incorporated into the autonomous finite element platform "Wolong". An elevation comparison test was conducted on the (75+3×135+75)m rigid continuous beam of the Xiangjiang Grand Bridge on the Changsha-Kunming Passenger Dedicated Line. This bridge is located on a circular curve and a transition curve with a radius R=7000m. The main pier foundation consists of underwater bored piles with a diameter of 2.5m, in an area with highly developed karst topography.

[0068] Figure 7 This is a schematic diagram showing the layout of measuring points near the main pier of the bridge. Figure 8 The graph shows a comparison between the calculated and measured elevations of the corresponding measuring points. Among all measuring points, the maximum elevation error is 1.476 cm, which meets the engineering requirements.

[0069] Therefore, it can be seen that the method for solving the stiffness and uniform mass matrix of the variable cross-section spatial beam element described in this invention meets the requirements of digital twin simulation in actual engineering.

[0070] This invention addresses the problem of accurate simulation and calculation of the stiffness and uniform mass matrix of variable cross-section spatial beam elements in bridge structures. Utilizing a generalized coordinate system constrained by a cantilever beam, a force interpolation function is used to establish the cross-sectional stiffness and flexibility matrices without rigid body displacement. The principle of virtual work is used to solve for the element flexibility matrix considering and neglecting shear effects. Expressions for each displacement deformation in the generalized coordinate system are obtained using the unit force method in mechanics of materials, thus establishing the corresponding shape function matrices. Substituting the shape function matrices into the general integral expression of the uniform mass matrix yields the element uniform mass, thereby establishing a technique for calculating the stiffness and uniform mass matrix of variable cross-section spatial beam elements in bridge structures.

[0071] This invention can accurately simulate the stiffness and consistent mass matrix of variable cross-section spatial beam elements in bridge structures in the transportation field, and solves the underlying key technical problems of accurate mapping between the physical model and the data-driven model of variable cross-section spatial beam elements for modeling, design analysis, diagnosis and decision-making.

Claims

1. A method for solving the stiffness and uniform mass matrix of a variable cross-section spatial beam element, characterized in that: Includes the following steps: (A) Establish a generalized coordinate system for cantilever beam constraints; In the generalized coordinate system, the generalized degrees of freedom are only related to the deformation of the element itself, and the generalized coordinate system adopts the constraint form of cantilever beam constraint; The axial, lateral, longitudinal, and rotational degrees of freedom of the element node are fixed; (B) Based on the generalized coordinate system, a stiffness matrix without rigid body displacement is established according to the internal stress inside the beam element; (C) Based on the coordinate system transformation, establish a stiffness matrix that includes rigid body displacement; (D) Based on the displacement variables, establish a consistent mass matrix when shearing is ignored; (E) Based on the consistent mass matrix when shearing is ignored, a shearing effect term is added to establish a consistent mass matrix that takes shearing into account. (F) Based on the stiffness matrix including rigid body displacement, the uniform mass matrix when shear action is ignored, and the uniform mass matrix when shear action is considered, a digital twin model of the variable cross-section spatial beam element is formed. The generalized coordinate system for cantilever beam constraints established in step (A) is based on the global coordinate system and the local coordinate system. The specific process is as follows: Let the nodal forces of the element in the global coordinate system and the local coordinate system be respectively (1)； In equation (1) above, there are 12 nodal forces in the element, which correspond to 12 degrees of freedom; these 12 degrees of freedom can be divided into two categories: Firstly, r kind, r The class is related to the deformation of the unit itself; Secondly, s kind, s The class is related to the rigid body motion of the unit; By selecting a coordinate system, the coordinates in equation (1) can be adjusted. s The term corresponding to the degree of freedom is equal to zero, at which point only the deformation of the element itself is relevant. r The existence of degrees of freedom can be referred to as generalized degrees of freedom; the element nodal forces in this coordinate system, i.e., the generalized coordinate system, are denoted as: (2)； Step B, based on the generalized coordinate system and the internal stress within the beam element, establishes a stiffness matrix without rigid body displacement. The specific process is as follows: In the generalized coordinate system, the nodal forces corresponding to the deformation of the beam are expressed as: (3)； Then assume that any element within the unit... x The internal forces at the cross section are: (4)； Based on the boundary conditions of the cantilever beam, the equilibrium relationship can be obtained as follows: x Force F at the cross section x The formula is as follows: (5)； In the formula, e( x ) is the force interpolation function; The internal forces on any cross section can be obtained from the nodal forces at both ends of the beam element and the interpolation function, and their expressions are as follows: (6)； In a generalized coordinate system, within a unit x The deformation at the cross section is: (7)； In the formula, ε( x ) represents the axial deformation of the centroid of the cross section. γ y ( x ), γ z ( x ) are respectively y and z Cross-sectional shear deformation in the direction of the direction, φ x ( x ), φ y ( x ), φ z ( x ) are respectively x , y and z The rotation angle of the central axis of the direction; Combining the above information on cross-sectional force and deformation, the relationship between cross-sectional force and deformation can be obtained, specifically expressed as follows: (8)； In the formula, k s ( x ) is the beam section stiffness matrix, and its matrix is: (9)； In the formula, E For elastic modulus, G Shear modulus A For cross-sectional area, A sy 、A sz Let the shear area be assuming E , G It is a constant; Cross-sectional compliance matrix f s ( x ) is the inverse matrix of the section stiffness matrix, expressed as: (10)； Using the principle of virtual work, the element flexibility matrix f in the generalized coordinate system can be obtained from the flexibility matrix of the cross section. r ( x The general formula for its calculation is: (11)； After derivation, the element flexibility matrix f r ( x The specific expression for ) is: (12)； After obtaining the flexibility matrix, the element stiffness matrix k can be obtained by inverting equation (12). r ( x ); (13)； In the formula, A sy 、A sz The area of ​​the shear surface; Step C involves establishing a stiffness matrix that includes rigid body displacements based on coordinate system transformation. The specific process is as follows: There are 12 displacements and 12 forces in the local and global coordinate systems, located at the nodes on both sides of the beam element. Therefore, the transformation from the generalized coordinate system to the global coordinate system requires the multiplication of two transformation matrices, as shown in the following expression: (14)； In the formula, K g T is the stiffness matrix in the global coordinate system. rg This is the transformation matrix from generalized coordinates to the global coordinate system; Transformation matrix T from generalized coordinates to global coordinate system rg The expression is: (15)； In the formula, T re Let T be the transformation matrix from generalized coordinates to local coordinates. eg This is the transformation matrix from local coordinates to the global coordinate system; Transformation matrix T from generalized coordinates to local coordinate system re And the transformation matrix T from local coordinates to global coordinates. eg The expressions for each are: (16)； (17)； In the formula, matrix elements t for (18)； In the formula, The transformation angle between the two transformation matrices; Step D involves establishing a consistent mass matrix based on the displacement variables, neglecting shear effects. The specific process is as follows: The boundary conditions of an element in a generalized coordinate system are the same as those of a cantilever beam subjected to generalized nodal forces. The boundary conditions of the element in the generalized coordinate system can be determined using the unit force method in mechanics of materials. x The displacement and deformation amounts at each location are as follows: (19)； In the formula, N ( x 0) M z ( x 0), M y ( x 0), T ( x 0) and other variables are represented in the unit x Internal forces at section 0; By combining the calculation with formula (5), we can obtain: (20)； In the formula, H(·) is the Heaviside function, and its expression is: (21)； To avoid local discontinuities, it is stipulated that when x=x When the value is 0, the expression is: (22)； In a generalized coordinate system, the element nodal force equals the element stiffness matrix multiplied by the element nodal displacement, and its expression is: (23)； Combining formulas (5) and (19), the displacement relationship in the unit's local coordinate system can be obtained, expressed as: (24)； In the formula, e i ( x 0) ( i= 1 、 4 、 5 、 6) are the first and second digits of the matrix in formula (6). i OK (25)； Ignoring shear effects, equation (12) can be simplified to the matrix expression of equation (26), specifically: (26)； Substituting the inverse matrices of formulas (6), (25), and (26) into formula (24) and rearranging, we obtain the expressions related to spatial displacement and nodal displacement: (27)； After organizing it, we can conclude that: (28 a )； (28 b )； (28 c )； (28 d )； (28 e )； (28 f )； In the formula, The inverse matrix of expression (26) is the first i Line 1 j List; And the corresponding φ 1~ φ 10 The expression is: (29 a )； (29 b )； Based on the above, the general integral expression for the uniform mass matrix of a unit cell is: (30)； In the formula, ρ The mass density of the material, m s The cross-sectional mass matrix; Cross-sectional mass matrix m s The expression is: (31)； In the formula, r Radius of gyration r=sqrt ( J / A ); The shape function matrix N is: (32)； Thus, the uniform mass matrix of the variable cross-section spatial beam element without considering shear effects can be obtained by formula (30).

2. The method for solving the stiffness and uniform mass matrix of a variable cross-section spatial beam element according to claim 1, characterized in that: Step (F) involves generating a digital twin model of the variable cross-section spatial beam element based on the stiffness matrix including rigid body displacement, the uniform mass matrix neglecting shear action, and the uniform mass matrix considering shear action. The specific process is as follows: The stiffness matrix obtained in step (C), the uniform mass matrix obtained in step (D) ignoring shear action, and the uniform mass matrix obtained in step (E) considering shear action are used as twin models for characterizing the mechanical features of the variable cross-section spatial beam element of the bridge structure.

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