A form-finding method for spatial arch bridges based on the inverse hanging method

Abstract

The present invention proposes a form-finding method for a spatial arch bridge based on the inverse hanging method, the process of which includes: establishing an initial arch bridge, applying external loads and constraints, solving, extracting the internal forces of the hangers, applying the internal forces in reverse to the arch, deleting the hangers, solving, extracting the node displacements of the nodes on the arch, updating the shape of the arch, and reducing the node displacements to 0 after multiple iterations to complete the form-finding process, i.e., obtaining the ideal arch axis. The form-finding method mentioned in the present invention is based on the theory of the inverse hanging method, and mainly overcomes the fact that the force density method ignores the influence of geometric nonlinearity on the form-finding results. The present invention adopts a dual-unit numerical model, takes into account the influence of key parameters on the optimization results, avoids the influence of the number of units by applying linear pressure, considers geometric nonlinearity, and applies the algorithm to spatial arch bridges of various shapes, proving that the form-finding method mentioned in the present invention is efficient and highly applicable.

Description

Technical Field

[0001] The invention relates to the technical field of building structure engineering, and in particular to a form-finding method for a space arch bridge based on a reverse hanging method. Background Art

[0002] Space arch bridges are increasingly being used in urban environments due to their novel appearance. When an arch structure is used to support a horizontally curved bridge deck, a space arch bridge emerges. The arch is one of the most critical components of an arch bridge, supporting the bridge deck.

[0003] Numerous scholars have conducted research on arches. Lewis proposed a mathematical model for predicting the geometry of rigid, double-pin, moment-free arches with uniform cross-sections. Bessini proposed a design tool that uses a multi-objective optimization strategy to generate efficient structural configurations for fixed active-bending tie arches. Jorquera-Lucerga used the force density method to obtain three-dimensional cableway arches. This method makes the arches and hangers resemble a cable net. The force density method is very effective in cable net form-finding, but it cannot account for geometric nonlinearity, which can lead to errors in large arch bridges.

[0004] There are many methods for structural optimization. The inverted hanging method is widely used for form-finding of spatial structures. In the absence of computational technology, optimization has traditionally been performed through experimental methods. Gaudí made significant contributions to the application of experimental form-finding to arch structures. Kolodziejczyk immersed a wire model in water and then used the surface tension of water to find the form of branching structures. Buelow used a main line model to find the shape of branching structures. The inverted hanging method can reduce or completely eliminate moments, thereby improving structural efficiency and increasing spans.

[0005] The numerical inversion method is used to explore the ideal arch axis of the spatial arch structure. The ideal arch axis can make the arch only bear the pressure without being affected by the moment. There are many form-finding methods for plane arches, but there are few studies on the form-finding of spatial arches. Summary of the Invention

[0006] In order to solve the existing arch form-finding problem, an effective form-finding method is proposed to make it applicable to different types of spatial arch bridges. The form-finding method proposed in this invention is based on the inverted hanging method theory and simulates the arch bridge by establishing a double-unit numerical model.

[0007] Step 1: Determine the initial shape and geometric parameters according to the design requirements. The geometric parameters include: arch cross-sectional area and moment of inertia, arch span and length, arch foot position, bridge deck cross-sectional dimensions, and bridge deck shape.

[0008] Step 2: Establish an initial arch bridge model and form an arch bridge structure with a dual-unit numerical model based on the initially set shape and parameters.

[0009] Step 3: Apply external loads and constraints to the bridge deck and perform static analysis;

[0010] Step 4: Extract the internal force and direction of each boom. The internal force of the boom is represented by F, and the components of F in the three directions are represented by F. x 、F y 、F z express;

[0011]

[0012]

[0013]

[0014] The symbol || is the absolute value symbol, and the last term in the formula is the reaction force.

[0015] Step 5: Reverse the internal forces and apply them to the arch through the hangers. Delete the bridge deck and the hangers.

[0016] Step 6: Perform nonlinear static analysis to extract the nodal displacements of the nodes on the arch and determine the displacements in three directions, namely Δx, Δy, and Δz.

[0017] Step 7: Change the position of the node according to the displacement of the node on the extracted arch in three directions.

[0018] x i(j+1) =x ij +Δx j (4)

[0019] y i(j+1 )=y ij +Δy j (5)

[0020] z i(j+1) =z ij +Δz j (6)

[0021] where x ij 、y ij 、z ij is the node coordinate of the i-th node in the j-th iteration, Δx j , Δy j , Δz j is the node displacement at the jth iteration.

[0022] Step 8: Determine whether the iterative process is finished. When the iteration exceeds the number of algorithm executions, the node displacement is reduced to 0, and the number of iterations is less than or equal to the number of algorithm executions, that is, j≤J. If so, j=j+1, and return to step 2 to re-find the shape. Otherwise, end the shape-finding process and complete the determination of the final shape.

[0023] Beneficial technical effects:

[0024] 1. This invention uses dual elements to build a numerical model of the arch. Two elements are located at one location: a rod element that can only withstand axial forces, and a beam element that has no axial stiffness. The cross-sectional area of the rod element is much larger than that of the beam element, giving the beam element very little bending stiffness.

[0025] 2. The bridge deck is simulated using beam elements, and the external load is applied to the bridge deck as a linear pressure to avoid the influence of the number of elements and node movement on the load distribution.

[0026] 3. Compared with the force density form-finding method, which cannot consider the disadvantage of geometric nonlinearity, the form-finding method proposed in the present invention considers the influence of geometric nonlinearity on structural form-finding. The only thing that needs to be determined is the position of the arch foot and the shape of the bridge deck.

[0027] 4. This algorithm is an efficient and accurate method for designing spatial arch bridges, which can automatically determine the ideal arch axis according to various boundary conditions.

[0028] 5. The influence of key parameters on form-finding results was systematically studied. The larger the cross-sectional area of the arch, the smaller the axial deformation of the arch; the smaller the moment of inertia of the arch, the faster the optimization speed. However, when the stiffness matrix of the arch is irreversible, the moment of inertia cannot be too small. BRIEF DESCRIPTION OF THE DRAWINGS

[0029] Figure 1 This is a flow chart of a form-finding method for a spatial arch bridge based on the inverse hanging method in a specific implementation of the present invention;

[0030] Figure 2 Schematic diagram of components of an arch bridge in a specific embodiment of the present invention;

[0031] Figure 3 Schematic diagram of the internal force of the suspender in the specific implementation of the present invention;

[0032] Figure 4 This is a schematic diagram of a double unit used in the specific implementation of the present invention;

[0033] Figure 5 Schematic diagram showing the effect of cross-sectional area on optimization results in a specific implementation of the present invention;

[0034] Among them, (a) is the initial shape of the numerical model; (b) is the optimized shape of the numerical model; (c) is the displacement evolution diagram of node 46;

[0035] Figure 6 Schematic diagram of the effect of moment of inertia on optimization results in a specific implementation of the present invention;

[0036] Among them, (a) is the optimized shape of the numerical model; (b) is the displacement evolution diagram of node 46;

[0037] Figure 7 Schematic diagram of the initial shape of a straight deck arch bridge in a specific embodiment of the present invention;

[0038] Among them, (a) is a top view; (b) is a front view; (c) is a side view;

[0039] Figure 8 Schematic diagram of the shape of the front arch in the specific implementation of the present invention;

[0040] (a) is the front view; (b) is the isotropic view;

[0041] Figure 9 Schematic diagram of internal force comparison before and after shape finding in the specific implementation of the present invention

[0042] Among them, (a) is the schematic diagram of internal forces before form-finding; (b) is the schematic diagram of internal forces after form-finding;

[0043] Figure 10 Schematic diagram of the initial shape of the curved deck arch bridge in the specific implementation of the present invention;

[0044] Among them, (a) is a top view; (b) is a front view; (c) is a side view;

[0045] Figure 11 、 Figure 12 They are schematic diagrams of curved deck arch bridges with Ø=10m and Ø=30m after form-finding in the specific implementation of the present invention;

[0046] Among them, (a) is a top view; (b) is a front view; (c) is a side view;

[0047] Figure 13 This is a diagram showing the evolution of node displacement in a specific implementation of the present invention;

[0048] Among them, (a) is Ф = 10m; (b) is Ф = 30m;

[0049] Figure 14 Schematic diagram of internal force comparison before and after form-finding of a Ø=30m curved deck arch bridge in a specific embodiment of the present invention;

[0050] Among them, (a) is the schematic diagram of internal forces before form-finding; (b) is the schematic diagram of internal forces after form-finding;

[0051] Figure 15 Schematic diagram of the final form of an arch bridge with different curved (sinusoidal) decks in a specific implementation of the present invention;

[0052] Among them, (a) is Case I; (b) is Case II; (c) is Case III; (d) is Case IV; DETAILED DESCRIPTION

[0053] The invention is further described below with reference to the accompanying drawings and specific implementation examples: A method for finding the shape of a spatial arch bridge based on the inverted hanging method, such as Figure 1 As shown, the process includes the following:

[0054] Step 1: Determine the initial shape and geometric parameters according to the design requirements. The geometric parameters include: arch cross-sectional area and moment of inertia, arch span and length, arch foot position, bridge deck cross-sectional dimensions, and bridge deck shape.

[0055] Step 2: Establish the initial arch bridge model, and form an arch bridge structure with a double-unit numerical model according to the initially set shape and parameters; Figure 2 The arch bridge shown consists of three parts: the deck, the hangers, and the arch. The hangers connect the arch and the deck, while the arch supports the deck. The deck and hangers are simulated using the beam188 and link180 elements in ANSYS, respectively.

[0056] Step 3: Apply external loads and constraints to the bridge deck and perform static analysis;

[0057] Step 4: Extract the internal force and direction of each suspender; the method for determining the magnitude and direction of the force at the arch is as follows Figure 3 As shown, node i is the connection node between the hanger and the arch, and node j is the connection node between the hanger and the bridge deck. The node coordinates can be obtained after the iterative algorithm is executed once. Then, the direction of the hanger can be determined; the internal force of the hanger is represented by F; the components of F in the three directions are represented by F x 、F y 、F z express;

[0058]

[0059]

[0060]

[0061] The symbol || is the absolute value symbol, and the last term in the formula is the reaction force;

[0062] Step 5: Reverse the internal force and apply it to the arch through the suspender. Delete the bridge deck and suspender.

[0063] Step 6: Perform nonlinear static analysis to extract the nodal displacements of the nodes on the arch and determine the displacements in three directions, namely Δx, Δy, and Δz;

[0064] Step 7: Change the position of the node according to the displacement of the extracted arch node in three directions;

[0065] x i(j+1)=x ij +Δx j (4)

[0066] y i(j+1) =y ij +Δy j (5)

[0067] z i(j+1) =z ij +Δz j (6)

[0068] where x ij 、y ij 、z ij is the node coordinate of the i-th node in the j-th iteration, Δx j , Δy j , Δz j is the node displacement at the jth iteration;

[0069] Step 8: Determine whether the iterative process is finished. When the iteration exceeds the number of algorithm executions, the node displacement is reduced to 0, and the number of iterations is less than or equal to the number of algorithm executions, that is, j≤J. If so, j=j+1, and return to step 2 to re-find the shape. Otherwise, end the shape-finding process and complete the determination of the final shape.

[0070] In the algorithm proposed above, parameters are the key factors that determine success or failure. The total length of the arch should be changed during the form-finding process. However, the shape of the arch can be changed freely. In other words, the axial deformation caused by the external load should be small enough to be negligible, as shown in Equation (7). To achieve this goal, the value of EA should be set large enough. On the other hand, the bending stiffness should be small enough to release the deformation caused by the moment, as shown in Equation (8). This determines that the value of EI should be small enough.

[0071]

[0072]

[0073] Where F, A, E, I, L arch are the axial force, area, elastic modulus, moment of inertia and axial length of the arch respectively.

[0074] In order to increase the axial stiffness and reduce the bending stiffness, a double element is used to establish the numerical model of the arch. There are two elements at one location, such as Figure 4 As shown in Figure 1. One element is a rod element that can only support axial forces, and the other element is a beam element with no axial stiffness. The axial stiffness of the rod element can be increased by increasing the cross-sectional area. The bending stiffness of the beam element can be reduced by reducing the moment of inertia.

[0075] For ordinary beam elements, the relationship between external force and node displacement can be expressed by equation (9). In the form-finding process, the external load is considered constant. When AE / L increases, u x and u x The value of ' will decrease. After form-finding analysis, the arch is only affected by axial force, so the influence of shear force is not considered.

[0076] For beam elements without axial stiffness, the relationship between external force and node displacement can be expressed by equation (10), reducing I i The value can increase the bending deformation. The rod element can be regarded as a beam element without bending stiffness. The relationship between the external force and the node displacement can be expressed by Equation (11). Equation (9) is the sum of Equations (10) and (11). The beam element and the rod element are located at the same position, which means that the two elements share the same node. Therefore, the node displacements of Equations (10) and (11) are the same and can be directly added.

[0077]

[0078] Where: A, E, L, G, and J are cross-sectional area, elastic modulus, unit length, shear modulus, and torsional moment of inertia, respectively; f y (f z ) is related to I y (I z ) and I z (I y ) related parameters.

[0079]

[0080] I i is the moment of inertia about the i-axis, A s y(z) is the shear area perpendicular to the y-axis or z-axis.

[0081]

[0082]

[0083] Through the specific example 1, we analyze the influence of the two key parameters, arch cross-sectional area and moment of inertia, on the form-finding results. For the straight bridge deck, the cross-sectional area of the bridge deck is assumed to be H1×0.05×1×0.05m. The cross-sectional area of the suspender is assumed to be 0.004m 2 The analyzed arch bridge has 83 hangers. The magnitude of the linear pressure applied to the bridge deck is set to 100 kN / m. The proposed algorithm is repeated 2000 times to obtain the optimized arch shape. The value of A is set to 1×10 -5 , 0.01 and 10m 2 The value of I is set to 1×10 -5 , 2×10-5m4 Comparing the evolution of 46-node arch and vertical displacement, Figure 5 As shown in the results, it can be seen that when the A value is greater than 0.01m 2 When A=1×10 -5 m 2 When the axial deformation of the arch cannot be ignored, the total length of the arch increases. Therefore, the value of A should be set large enough to eliminate the axial deformation of the arch.

[0084] The influence of moment of inertia on optimization results is as follows Figure 6 As shown. When I changes from 2×10 -5 m 4 Reduced to 1×10 -5 m 4 When I increases from 1×10 -5 m 4 Reduced to 5×10 -6 m 4 0.2 m was added when the arch was centered. It can be seen that the value of I has little effect on the optimized arch shape. As I decreases, the node displacements quickly decrease to zero. In other words, a smaller I accelerates optimization convergence. However, given the irreversible stiffness matrix of the arch, I cannot be too small. An appropriate value for I can be determined through trial and error.

[0085] Through the specific example 2, the form-finding analysis of the straight deck arch bridge is carried out. The bridge deck shape is assumed to be a straight line, the arch shape is a circular arc, and the vector height h is 40m. Figure 7 In order to increase the axial stiffness of the arch, the cross-sectional area of the arch rod unit is set to 10.0m 2 The moment of inertia of the beam element is set to 1×10 -5 m 4 The cross-sectional area of the beam element is set to 1×10 -5 m 2 , which is sufficiently small compared to the rod unit. The deck cross section is assumed to be H1×0.05×1×0.05m. The cross-sectional area of the boom is assumed to be 0.004m 2 The analyzed arch bridge has 83 suspenders. The linear pressure applied to the bridge deck is set to 10 kN / m. All degrees of freedom at the nodes at the arch foot, including translational and rotational degrees of freedom, are fixed. Similarly, all degrees of freedom at both ends of the bridge deck are also fixed. The iterative form-finding algorithm is repeated 2000 times. The arch after form-finding is as follows Figure 8 As shown in the figure, the shape of the arch has changed significantly. The straight bridge deck will pass through the flat arch. The internal force comparison of the arch is as follows: Figure 9As shown in the figure. Through form-finding analysis, it can be seen from the results that the distribution of axial force has changed, and the magnitude of axial force has decreased after form-finding analysis. The maximum axial forces before and after form-finding are 1.67×10 4 kN and 7.56×10 3 kN. The arch bending moment is significantly reduced. Most of the arch bending moment is very small and can be ignored. The bending moment at the arch foot is relatively large. This is because the arch foot is fixed during the form-finding analysis. Generally speaking, the ideal arch axis can be found using the method proposed in this invention.

[0086] The form-finding analysis of the curved deck arch bridge is carried out through the specific example 3. The initial shape of the arch bridge is as follows: Figure 10 As shown in Figure 1. The bridge deck is assumed to be sinusoidal in shape. The size of the sinusoidal curve is represented by Φ and is set to 10m and 30m. The distance between the arch feet is represented by L and is set to 100m. The arch feet and the bridge deck end are located in a vertical plane. The initial vector height of the arch is represented by h and is set to 40m. The arch bridge after shape finding is shown in Figure 1. Figure 11 and Figure 12 As shown in the results, if the bridge deck is curved, a spatial arch will be generated. The shape of the curved bridge deck directly affects the shape of the arch. The total length of the arch axis remains unchanged. The displacement evolution process of node 18 is shown in Figure 13 As shown in the results, it can be seen that when the number of iterations reaches about 500, the node displacement decreases to 0. This shows that the shape of the arch is determined after the iterative shape-finding algorithm is executed 500 times, verifying the effectiveness of the proposed method.

[0087] The internal force distribution of the space arch is as follows Figure 14 As shown in the figure, the bending moment of the arch is significantly reduced, especially at the haunch and crown. Compared to the axial force, the moment is very small. The ideal arch axis has been determined. Form-finding analysis has also reduced the axial force to a certain extent.

[0088] This algorithm is applicable to both planar and spatial arch bridges. All that needs to be done is to set the location of the arch foot and the shape of the bridge deck. The accuracy of the spatial arch can then be determined.

[0089] In order to demonstrate the applicability of the proposed algorithm, several arch bridges with curved decks are analyzed. The differences are in the deck shape and boundary conditions. Figure 15 As shown in the figure, the four cases are first assuming that the axis of the curved deck is half of a sine curve: Case I, Case II, Case III, and Case IV. The Φ value is assumed to be the same, set at 30 meters. The results show that the influence of boundary conditions and deck shape can be accurately considered in form-finding analysis. The arch shape, which combines the deck shape with the boundary conditions, is unique. This demonstrates the efficiency and strong adaptability of the proposed method.

Claims

1. A form-finding method for a spatial arch bridge based on the inverse hanging method, characterized in that: The following processes are included: Step 1: Determine the initial shape and geometric parameters according to the design requirements. The geometric parameters include: arch cross-sectional area and moment of inertia, arch span and length, arch foot position, bridge deck cross-sectional dimensions, and bridge deck shape. Step 2: Establish an initial arch bridge model and form an arch bridge structure with a dual-unit numerical model based on the initially set shape and parameters; Step 3: Apply external loads and constraints to the bridge deck and perform static analysis; Step 4: Extract the internal force and direction of each boom. The internal force of the boom is represented by F, and the components of F in the three directions are represented by F. x 、F y 、F z express; The symbol || is the absolute value symbol, and the last term in the formula is the reaction force; x i ,y i , z i , x j ,y j , z j are the coordinates of nodes i and j in the x, y and z directions; x ij (y ij , z ij ) is the x(y, z) coordinate of node i after the jth iteration; Step 5: Reverse the internal force and apply it to the arch through the suspender. Delete the bridge deck and suspender. Step 6: Perform nonlinear static analysis to extract the nodal displacements of the nodes on the arch and determine the displacements in three directions, namely Δx, Δy, and Δz; Step 7: Change the position of the node according to the displacement of the extracted arch node in three directions; x i(j+1) =x ij +Δx j (4) y i(j+1) =y ij +Δy j (5) with i(j+1) =with ij +Δz j (6) where x ij 、y ij 、z ij is the node coordinate of the i-th node in the j-th iteration, Δx j , Δy j , Δz j is the node displacement of the jth iteration; x i(j+1) (y i(j+1) , z i(j+1) ) is the node coordinate of node i after the j+1th iteration; Step 8: Determine whether the iterative process is finished. When the iteration exceeds the number of algorithm executions, the node displacement is reduced to 0, and the number of iterations is less than or equal to the number of algorithm executions, that is, j≤J. If so, j=j+1, and return to step 2 to re-find the shape. Otherwise, end the shape-finding process and complete the determination of the final shape.

2. The form-finding method for a spatial arch bridge based on the inverted hanging method according to claim 1, characterized in that: Considering the influence of load distribution, the bridge deck components were added to the numerical model; to avoid the influence of the number of elements, a linear load was applied on the bridge deck.

3. The form-finding method for a spatial arch bridge based on the inverted hanging method according to claim 1 is characterized in that: The effect of geometric nonlinearity on structural form-finding is studied by determining the position of the arch foot and the shape of the bridge deck.

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