Method and system for calculating safety factor of main cable of suspension bridge based on secondary stress effect

Abstract

The present invention relates to a method and system for calculating the safety factor of the main cable of a suspension bridge based on the secondary stress effect. The method comprises the following steps: establishing a finite element model of the bridge; determining, based on the finite element model of the bridge, the most unfavorable stress of the main cable under the most unfavorable live load based on cable structure theory, as the cable unit stress response of the main cable; calculating and evaluating the magnitude of the secondary stress of the main cable based on the cable unit stress response of the main cable according to the secondary stress effect; evaluating the magnitude of the most unfavorable stress response of the main cable by integrating the cable unit stress response and the secondary stress effect, establishing a limit state judgment model using the ultimate strength of the main cable as the failure judgment condition; establishing a probability distribution model of the structure; setting an initial design safety factor for the main cable, calculating the failure probability and reliability index of the main cable; and iterating through an inverse reliability method to change the design number of main cable strands and solve for the reliability index. Compared with the existing technology, the present invention has the advantages of more accurate calculation and strong engineering practicality.

Description

Technical Field

[0001] The present invention relates to the field of architectural design, and in particular to a method and system for calculating the safety factor of a main cable of a suspension bridge based on a secondary stress effect. Background Art

[0002] The main cable is the most critical component of a super-long-span suspension bridge, bearing live loads, stiffening beams, hangers, and its own dead loads. It cannot be replaced throughout the bridge's service life. Therefore, ensuring the safety of the main cable is crucial for designers. As the span of a suspension bridge continues to increase, the diameter and bending stiffness of the main cable also increase accordingly, resulting in considerable secondary stress in the main cable.

[0003] In traditional suspension bridge design, the safety of the main cables is generally ensured by setting a relatively high total safety factor for the main cables. However, as the span of suspension bridges increases, continuing to use the safety factor values ​​commonly used in smaller spans will inevitably result in excessive safety of the main cables and waste of materials, which in turn will lead to increased costs for structures such as the main tower and foundation. Scholars have also begun to study methods to determine reasonable design safety factors, such as discussing the dead load / live load ratio based on reliability indicators. However, existing studies have only considered the stress behavior of the main cables based on cable units, ignoring the portion exceeding the average stress. There is room for improvement in the rationality of the methods for calculating and determining safety factors. Summary of the Invention

[0004] The purpose of the present invention is to overcome the deficiencies in the calculation assumptions of the above-mentioned prior art and to provide a method and system for calculating the safety factor of the main cable of a suspension bridge based on the secondary stress effect.

[0005] The purpose of the present invention can be achieved by the following technical solutions:

[0006] A method for calculating the safety factor of a main cable of a suspension bridge based on a secondary stress effect comprises the following steps:

[0007] S1: Based on the bridge background project, establish the finite element model of the bridge;

[0008] S2: Based on the established bridge finite element model, determine the most unfavorable stress of the main cable based on cable structure theory under the most unfavorable live load, and use it as the cable unit stress response of the main cable;

[0009] S3: Based on the stress response of the main cable unit, the magnitude of the secondary stress of the main cable is evaluated according to the secondary stress effect;

[0010] S4: Comprehensively analyze the stress response of cable units and the secondary stress effect, evaluate the most unfavorable stress response of the main cable, and establish a limit state judgment model based on the ultimate strength of the main cable as the failure judgment condition;

[0011] S5: Establish a probability distribution model of the structure based on material properties, manufacturing errors and load conditions;

[0012] S6: Set the initial main cable design safety factor and calculate the failure probability and reliability index of the main cable based on the limit state judgment model and the structural probability distribution model;

[0013] S7: Iterate through the inverse reliability method, change the number of main cable strands, and repeat S6 to solve the reliability index.

[0014] Furthermore, the finite element model of the bridge includes cable elements, beam elements, simulated main cables, bridge towers and stiffening beams.

[0015] Furthermore, the secondary stress effect includes the general secondary stress effect of the main cable, the secondary stress effect at the saddle, the secondary stress effect at the cable clamp, the special secondary stress effect at the saddle outlet, and the secondary stress effect at the anchor span.

[0016] Furthermore, the sources of the general secondary stress effect of the main cable include the main cable elastic modulus, length manufacturing error and main cable installation error. The calculation expression of the secondary stress effect caused by the main cable elastic modulus is:

[0017]

[0018] Where σ1 is the secondary stress effect caused by the elastic modulus of the main cable, E is the elastic modulus of the main cable, and ΔE is the deviation value of the elastic model. is the primary stress of the main cable;

[0019] The calculation expression of the secondary stress effect caused by length manufacturing error is:

[0020]

[0021] Where σ2 is the secondary stress effect caused by the length manufacturing error, l is the designed length of the steel wire in the main cable, and Δl is the deviation value of the main cable steel wire length;

[0022] The calculation expression of the secondary stress effect caused by the main cable installation error is:

[0023]

[0024] n ssr =f / L

[0025] Δn ssr =Δf / L

[0026] Where σ3 is the secondary stress effect caused by the main cable installation error, n ssr is the span ratio of the bridge, L is the span, Δn ssr is the span ratio error caused by the sag error Δf.

[0027] Furthermore, the calculation expression of the secondary stress effect at the saddle is:

[0028]

[0029] Where R is the design radius of the saddle, E is the elastic modulus of the main cable, and r is the radius of the steel wire in the main cable.

[0030] Furthermore, the secondary stress effect at the cable clamp includes the secondary stress due to porosity change and the secondary stress caused by the local rotation of the cable clamp. The calculation expression of the secondary stress due to porosity change is:

[0031]

[0032] Where R sj is the bending radius of the steel wire at the cable clamp outlet, E is the elastic modulus of the main cable, and r is the radius of the steel wire in the main cable;

[0033] The calculation expression of the secondary stress caused by the local rotation of the cable clamp is:

[0034]

[0035] Where Δφ is the angle at the end of the cable clamp, The primary stress of the main cable.

[0036] Furthermore, the special secondary stress effect at the saddle outlet includes the secondary stress caused by the angle change and shape change of the main cable at the saddle outlet. The calculation expression of the secondary stress caused by the angle change is:

[0037]

[0038] Where λ be is the effective length of the cable clamp, λ b is the length of the cable clamp, is the primary stress of the main cable, r is the wire diameter, E is the elastic modulus of the main cable, J cb is the main cable bending stiffness, C b is the distance between the cable clamp and the saddle, Δφ1 is the angle change of the main cable out of the saddle;

[0039] The calculation expression of the secondary stress caused by shape change is:

[0040]

[0041] Where, L AA′ , L BB′ are the lengths of the steel wire corresponding to the end points and the midpoints of the saddle shape, L is the length of the section with changed cross section, and r is the radius of the steel wire.

[0042] Furthermore, the secondary stress effect at the anchor span includes the secondary stress due to the tension error and the secondary stress inside the cable strand. The calculation expression for the secondary stress due to the tension error at the anchor span is:

[0043] σ9=ΔT / A

[0044] Where, σ9 is the secondary stress of tension error, ΔT is the tension error; A is the cross-sectional area of ​​the main cable;

[0045] The calculation method for the secondary stress inside the cable strands at the anchor span is as follows: the cable strand stress caused by the deformation of the main cable and the cable strand stress caused by the friction of the cable strand adjustment are combined to obtain the maximum stress in each cable strand. The difference between the maximum cable strand stress and the primary stress is the secondary stress inside the cable strand, and the corresponding expression is:

[0046]

[0047] Where, σ 10 A is the secondary stress of the anchor span cable under uneven load, s is the area of ​​a single cable strand, k is the corresponding σ 10 The largest strand number, S kP is the corresponding axial deformation cable force, S kQ is the corresponding vertical deformation cable force, F k is the corresponding stock adjustment friction force.

[0048] Furthermore, the expression of the limit state judgment model is:

[0049]

[0050] Where, σ y is the ultimate strength of the main cable material, i is the number of the secondary stress effect involved in the model calculation, when g≤0, it means the main cable fails, and when g>0, it means the main cable is safe.

[0051] The second aspect of the present invention is a system for calculating the safety factor of the main cable of a suspension bridge based on the secondary stress effect, comprising a memory, a processor, and a system program stored in the memory. When the processor executes the program, it implements any of the above methods for calculating the safety factor of the main cable of a suspension bridge based on the secondary stress effect.

[0052] Compared with the prior art, the present invention has the following beneficial effects:

[0053] 1) The present invention primarily considers the secondary stress effect, assesses the most unfavorable stress state of the main cable, and determines the probability distribution model of various key parameters. It fully accounts for the stresses caused by bending moment and deformation, as well as the portion of the main cable exceeding the average stress due to uneven axial tension between steel wires. It meticulously calculates the secondary stress effect of the main cable and infers the appropriate main cable design safety factor based on reliability indicators.

[0054] 2) The present invention calculates the failure probability and reliability index of the main cable through the limit state judgment model and the probability distribution model of the structure, taking into account the relevant structural manufacturing errors and the randomness of materials and loads, avoiding the situation where various uncertain and complex factors lead to unreasonable design safety factors, and has stronger engineering practicality. BRIEF DESCRIPTION OF THE DRAWINGS

[0055] Figure 1 is a calculation flow chart of the present invention;

[0056] Figure 2 This is the finite element model diagram of the suspension bridge;

[0057] Figure 3 Schematic diagram for calculating secondary stress at the saddle;

[0058] Figure 4 Schematic diagram of calculation of secondary stress caused by void ratio change at the cable clamp;

[0059] Figure 5 This is a schematic diagram for calculating the secondary stress caused by the rotation angle at the cable clamp;

[0060] Figure 6 Schematic diagram of calculation of secondary stress at the saddle outlet;

[0061] Figure 7 This is a schematic diagram of reliability calculation;

[0062] Figure 8 Schematic diagram for calculating the reasonable safety factor. DETAILED DESCRIPTION

[0063] The present invention is described in detail below with reference to the accompanying drawings and specific embodiments. This embodiment is implemented based on the technical solution of the present invention, and provides a detailed implementation method and specific operation process, but the protection scope of the present invention is not limited to the following embodiments.

[0064] Example 1

[0065] like Figure 1 As shown, the present invention is a method and system for calculating the safety factor of the main cable of a suspension bridge based on the secondary stress effect.

[0066] Taking a super-long span suspension bridge as an example, the implementation steps are as follows:

[0067] S1: Based on the bridge background project, cable elements and beam elements are used to simulate the main cables, bridge towers, stiffening beams and other components to establish a finite element model of the bridge, such as Figure 2 As shown in the figure, it is suggested to use parametric modeling method to realize automatic adjustment of bridge model under given different design parameters.

[0068] S2: Based on the number of designed lanes of the bridge, the span length, and the site wind conditions, the live load is determined and input into the finite element model established in S1 to determine the primary stress of the main cable according to the cable structure theory.

[0069] S3: Based on the finite element model and the most unfavorable stress value, further evaluate the magnitude of the secondary stress of the main cable, including:

[0070] 1. General secondary stress effect of main cable: including secondary stress effect caused by main cable elastic modulus, length manufacturing error and main cable installation error;

[0071] The secondary stress caused by the difference in elastic modulus of the main cable can be calculated by the following formula:

[0072]

[0073] Where σ1 is the secondary stress effect caused by the elastic modulus of the main cable, E is the average elastic modulus of the main cable, ΔE is the deviation value of the elastic model, is the primary stress of the main cable;

[0074] The calculation expression of the secondary stress effect caused by length manufacturing error is:

[0075]

[0076] Where σ2 is the secondary stress effect caused by the length manufacturing error, l is the designed length of the steel wire in the main cable, and Δl is the deviation value of the main cable steel wire length;

[0077] The calculation expression of the secondary stress effect caused by the main cable installation error is:

[0078]

[0079] n ssr =f / L

[0080] Δn ssr =Δf / L

[0081] Where σ3 is the secondary stress effect caused by the main cable installation error, n ssr is the span ratio of the completed bridge, L is the span, E is the elastic modulus of the main cable, Δn ssr is the span ratio error caused by the sag error Δf.

[0082] 2. Secondary stress effect at the saddle: The main cable at the saddle bends around the saddle, generating secondary bending stress, such as Figure 3 As shown, the calculation expression is:

[0083]

[0084] Where σ4 is the secondary stress effect at the saddle, R is the design radius of the saddle, and r is the radius of the steel wire in the main cable.

[0085] 3. Secondary stress effect at the cable clamp: At the cable clamp, the void ratio of the main cable will change, such as Figure 4 As shown in Figure 2, the calculation expression of the secondary stress caused by the change of porosity is:

[0086]

[0087] Where σ5 is the secondary stress of porosity change, R sj is the bending radius of the steel wire at the exit of the cable clamp, and r is the radius of the steel wire in the main cable.

[0088] When the main cable of the clamp undergoes local rotation, secondary stress will also be generated due to the rigidity constraint of the clamp, such as Figure 5 As shown, the calculation expression is:

[0089]

[0090] Where σ6 is the clamp stiffness stress, Δφ is the clamp end rotation angle, which can be calculated by the second-step finite element method, and E is the elastic modulus of the main cable.

[0091] 4. Special secondary stress effect at the saddle outlet. At the saddle outlet, due to the change in the main cable line shape under the action of variable load, the angle of the main cable at the cable support changes, thus generating secondary stress in the main cable at the saddle outlet. Figure 6 As shown, the calculation expression is:

[0092]

[0093] Where, σ7 is the secondary stress of the main cable at the saddle outlet, λ be is the effective length of the cable clamp, λ b is the length of the cable clamp, is the primary stress of the main cable, r is the wire diameter, E is the elastic modulus of the main cable, J cb is the main cable bending stiffness, C b is the distance between the cable clamp and the saddle, Δφ1 is the angle change of the main cable out of the saddle, which can be determined by the finite element results of the second step.

[0094] The main cable at the saddle position is in the shape of a regular hexagon, while the free section of the main cable outside the saddle is circular. Due to the change in shape, secondary stress will also be generated, which can be calculated using the following formula:

[0095]

[0096] Where, σ8 is the secondary stress of deformation, L AA′ , LBB′ The lengths of the steel wires corresponding to the endpoints and midpoints of the hexagon can be calculated according to the following formula:

[0097]

[0098] Where L is the length of the section with cross-section change, and r is the radius of the wire.

[0099] 5. When the anchor span cable is tensioned, the tension error will produce a secondary stress that exceeds the average stress. The calculation expression is:

[0100] σ9=ΔT / A

[0101] Where σ9 is the secondary stress of the anchor span cable strand, ΔT is the tension error, which is selected according to relevant specifications; A is the cross-sectional area of ​​the main cable, which can be obtained from the design data.

[0102] In addition, there are two cases of uneven stress inside the cable strands at the anchor span: ① uneven deformation of the cable strands caused by the deformation of the main cable; ② uneven stress on the cable strands caused by the friction force of the cable strands.

[0103] For case ①, the displacement Δ of the IP point along the tangent line of the loose cable saddle is decomposed into the component Δ along the central axis of the cable strand. P The component perpendicular to the central axis Δ Q , according to the force-deformation coordination equation, the cable forces generated by the axial deformation and vertical deformation of the i-th cable are calculated as follows:

[0104]

[0105] Where S iP is the axial deformation cable force, S iQ is the vertical deformation cable force, P is the main cable tension component along the cable center axis, and Q is the main cable tension component perpendicular to the cable center axis. P and Q can be obtained by designing the main cable force and the anchor span saddle; α i is the angle between the axis of the i-th cable strand and the cable centerline, α 0i is the angle between the projection of the axis of the i-th cable strand on the anchor surface and the vertical axis of the anchor surface; i With α 0i It can be calculated from the position of the cable anchor point.

[0106] For case ②, for the i-th cable strand, considering the friction force of the adjustment of the i+1th to n-th cables, the total friction force it bears is:

[0107]

[0108] Where, F iis the total friction force, T0 is the cable force when the cable is empty, which can be calculated by the finite element model; n is the number of cable strands, which is obtained from the design data; β is the angle between the force and the axis of the loose cable saddle, which is obtained from the design data; μ is the friction coefficient, which is selected based on engineering experience.

[0109] Since the maximum strand forces generated by items ① and ② do not act on the same strand, the two strand forces are superimposed to find the strand with the largest strand force (e.g., the kth strand). The maximum strand stress minus the primary stress is the maximum anchor span strand stress secondary stress.

[0110]

[0111] Where, σ 10 A is the secondary stress of the anchor span cable under uneven load, s is the area of ​​a single cable strand, obtained from the design data.

[0112] S4. Comprehensively consider the stress response of cable units and the secondary stress effect, evaluate the most unfavorable stress response of the main cable, and establish a limit state judgment model based on the ultimate strength of the main cable as the failure judgment condition, as shown below:

[0113]

[0114] Where, σ y is the ultimate strength of the main cable material, i is the number of the secondary stress effect involved in the model calculation, when g≤0, it means the main cable fails, and when g>0, it means the main cable is safe.

[0115] S5: Considering key parameters such as various material properties, manufacturing errors, and load loading, use vector X to characterize these parameters. Based on literature research or actual measurement statistics, determine the probability distribution model f of X. X , specifically including the probability distribution types and characteristics of various parameters.

[0116] S6: Based on the initial main cable design safety factor, which is 2.5 in this embodiment, the most unfavorable stress value under cable structure theory Calculate the initial main cable area and complete the design of the number of strands. Use the first-order second moment method in reliability theory, such as Figure 7 As shown in Figure 3, the most likely failure point in the standard normal space is calculated to evaluate the failure probability and reliability index of the main cable under the initial design.

[0117] S7: Iterate by inverse reliability method, e.g. Figure 8 As shown, the number of main cable strands is changed and step 6 is repeated to solve the reliability index. In this embodiment, 4.7 is used as the reliability target to search for a reasonable main cable design safety factor.

[0118] This embodiment also includes a system for calculating the safety factor of the main cable of a suspension bridge based on the quadratic stress effect, including a memory, a processor, and a program stored in the memory. When the processor executes the program, it implements the method for calculating the safety factor of the main cable of a suspension bridge based on the quadratic stress effect provided by the present invention. The program code for implementing the method of the present invention can be written in any combination of one or more programming languages. These program codes can be provided to a processor or controller of a general-purpose computer, a special-purpose computer, or other programmable data processing device, so that when the program code is executed by the processor or controller, the functions / operations specified in the flowchart and / or block diagram are implemented. The program code can be executed entirely on the machine, partially on the machine, partially on the machine as a stand-alone software package and partially on a remote machine, or entirely on a remote machine or server.

[0119] The above describes in detail the preferred embodiments of the present invention. It should be understood that those skilled in the art can make numerous modifications and variations based on the concepts of the present invention without inventive effort. Therefore, any technical solutions that can be derived by those skilled in the art through logical analysis, reasoning, or limited experimentation based on the concepts of the present invention and the prior art should be within the scope of protection defined by the claims.

Claims

1. A method for calculating the safety factor of the main cable of a suspension bridge based on the secondary stress effect, characterized in that: The following steps are involved: S1: Based on the bridge background project, establish the finite element model of the bridge; S2: Based on the established bridge finite element model, determine the most unfavorable stress of the main cable based on cable structure theory under the most unfavorable live load, and use it as the cable unit stress response of the main cable; S3: Based on the stress response of the main cable unit, the magnitude of the secondary stress of the main cable is evaluated according to the secondary stress effect; S4: Comprehensively analyze the stress response of cable units and the secondary stress effect, evaluate the most unfavorable stress response of the main cable, and establish a limit state judgment model based on the ultimate strength of the main cable as the failure judgment condition; S5: Establish a probability distribution model of the structure based on material properties, manufacturing errors and load conditions; S6: Set the initial main cable design safety factor and calculate the failure probability and reliability index of the main cable based on the limit state judgment model and the structural probability distribution model; S7: Iterate through the inverse reliability method, change the number of main cable strands, and repeat S6 to solve the reliability index.

2. The method for calculating the safety factor of the main cable of a suspension bridge based on the secondary stress effect according to claim 1 is characterized in that: The finite element model of the bridge includes cable elements, beam elements, simulated main cables, bridge towers and stiffening beams.

3. The method for calculating the safety factor of the main cable of a suspension bridge based on the secondary stress effect according to claim 1 is characterized in that: The secondary stress effect includes the general secondary stress effect of the main cable, the secondary stress effect at the saddle, the secondary stress effect at the cable clamp, the special secondary stress effect at the saddle outlet, and the secondary stress effect at the anchor span; The special secondary stress effect at the saddle outlet includes the secondary stress caused by the angle change and the secondary stress caused by the shape change of the main cable at the saddle outlet.

4. The method for calculating the safety factor of the main cable of a suspension bridge based on the secondary stress effect according to claim 3 is characterized in that: The sources of the general secondary stress effect of the main cable include the main cable elastic modulus, length manufacturing error and main cable installation error. The calculation expression of the secondary stress effect caused by the main cable elastic modulus is: Where, is the secondary stress effect caused by the elastic modulus of the main cable, is the elastic modulus of the main cable, is the deviation value of the elastic model, is the primary stress of the main cable; The calculation expression of the secondary stress effect caused by the length manufacturing error is: Where, is the secondary stress effect caused by length manufacturing error, Design the steel wire length in the main cable. The deviation value of the main cable wire length; The calculation expression of the secondary stress effect caused by the main cable installation error is: Where, The secondary stress effect caused by the main cable installation error, is the span-rise ratio of the completed bridge, is the span, is the sag error The resulting span ratio error.

5. The method for calculating the safety factor of the main cable of a suspension bridge based on the secondary stress effect according to claim 3 is characterized in that: The calculation expression of the secondary stress effect at the saddle is: In the formula is the design radius of the saddle, is the elastic modulus of the main cable, The radius of the steel wire in the main cable.

6. The method for calculating the safety factor of the main cable of a suspension bridge based on the secondary stress effect according to claim 3 is characterized in that: The secondary stress effect at the cable clamp includes the secondary stress caused by porosity change and the secondary stress caused by the local rotation of the cable clamp. The calculation expression of the secondary stress caused by porosity change is: Where, is the bending radius of the steel wire at the exit of the cable clamp, is the elastic modulus of the main cable, is the radius of the steel wire in the main cable; The calculation expression of the secondary stress caused by the local rotation of the cable clip is: Where, is the angle at the end of the cable clamp, The primary stress of the main cable.

7. The method for calculating the safety factor of the main cable of a suspension bridge based on the secondary stress effect according to claim 3 is characterized in that: The calculation expression of the secondary stress caused by the angle change is: Where, is the effective length of the cable clamp, is the length of the cable clamp, is the primary stress of the main cable, is the wire diameter, is the elastic modulus of the main cable, is the bending stiffness of the main cable, is the distance between the cable clamp and the saddle, The angle change value of the main cable out of the saddle section; The calculation expression of the secondary stress caused by the shape change is: Where, are the lengths of the steel wire corresponding to the end points and midpoints of the saddle shape, is the length of the section with changing cross section, is the wire radius.

8. The method for calculating the safety factor of the main cable of a suspension bridge based on the secondary stress effect according to claim 3 is characterized in that: The secondary stress effect at the anchor span includes the tension error secondary stress and the secondary stress inside the cable strand. The calculation expression of the tension error secondary stress is: Where, is the tensile error secondary stress, is the tension error; is the cross-sectional area of ​​the main cable; The calculation method of the secondary stress inside the cable strand is as follows: the cable strand stress caused by the deformation of the main cable and the cable strand stress caused by the friction of the cable strand adjustment are combined to obtain the maximum stress in each cable strand. The difference between the maximum cable strand stress and the primary stress is the secondary stress inside the cable strand, and the corresponding expression is: Where, is the secondary stress caused by uneven stress on the anchor span cable strands, is the area of ​​a single cable strand, To correspond The largest strand number, is the corresponding axial deformation cable force, is the corresponding vertical deformation cable force, is the corresponding stock adjustment friction force.

9. The method for calculating the safety factor of the main cable of a suspension bridge based on the secondary stress effect according to claim 3, characterized in that: The expression of the limit state judgment model is: Where, is the ultimate strength of the main cable material, i is the number of the secondary stress effect involved in the model calculation. When the main cable fails, When the main cable is safe.

10. A system for calculating the safety factor of a main cable of a suspension bridge based on a secondary stress effect, comprising a memory, a processor, and a system program stored in the memory, characterized in that: When the processor executes the program, the method for calculating the safety factor of the main cable of a suspension bridge based on the secondary stress effect as described in any one of claims 1 to 9 is implemented.

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