Arch foot elastic deformation calculation method and system in overall arch rib lifting construction process

Abstract

The application discloses an arch rib overall lifting construction process arch foot elastic deformation calculation method and system, wherein the method steps comprise: based on the stress characteristics in the arch rib overall lifting process, the relationship between the arch rib stress and the arch foot displacement is obtained; based on the relationship between the arch rib stress and the arch foot displacement, an equivalent stress model is constructed; and based on the equivalent stress model, the elastic deformation of the arch foot is calculated. The application aims at the stress characteristics of the arch rib in the overall lifting construction process, constructs an equivalent stress model, and based on the equivalent stress model, establishes an arch foot elastic deformation theoretical calculation method considering the coupling influence of the self weight of the arch rib, the lifting height, the horizontal stiffness of the support, the material and cross section characteristics of the transverse cable, and the transverse pre-tension. The method can be used for the research and calculation of the deformation law in the arch rib overall lifting process, fills the blank of the theoretical calculation method in the related field, and has guiding significance for the related structure design and construction.

Description

Technical Field

[0001] This application relates to the field of overall lifting construction of arch ribs in bridge engineering, specifically to the calculation method and system for elastic deformation of arch foot during the overall lifting construction of arch ribs. Background Technology

[0002] In the construction process of arch ribs for arch bridges, the integral lifting construction method involves using lifting supports to raise the assembled arch ribs to the closing position via vertical cables, followed by welding. Compared to the cable-stayed segmental assembly method, the integral lifting method avoids the oblique hook-and-loop assembly of arch rib segments, better ensuring the arch alignment and offering faster construction speed. However, during the integral lifting construction of long-span arch bridges, due to the significant weight of the arch ribs and their less favorable stress state compared to the completed bridge stage, the arch feet may experience substantial horizontal displacement, severely affecting the closing accuracy of the arch ribs. Therefore, the key technology in integral lifting construction lies in controlling the horizontal displacement of the arch feet. Currently, a relatively effective method for controlling the horizontal displacement of the arch feet is to apply pre-tension to the arch feet using transverse cables to counteract the horizontal thrust generated by the arch ribs under their own weight. However, due to the impossibility of achieving absolute precision in actual construction, the arch rib inevitably experiences an initial horizontal displacement of the arch foot the instant it is lifted by the vertical cables. At this moment, the vertical cables will have a certain inclination angle, generating a horizontal component force acting on the arch foot. When the self-weight of the lifted arch rib is large, the influence of this horizontal component force becomes significant. Since this horizontal component force originates from the horizontal displacement of the arch foot but also tends to reduce the horizontal displacement of the arch foot, it has a coupled relationship with the arch foot displacement. However, as the lifting height increases, the length of the vertical cables will decrease, resulting in a larger inclination angle of the vertical cables and a corresponding increase in the horizontal component force, which in turn reduces the arch foot displacement. Therefore, at different lifting heights, the vertical cables and the arch rib are in different coupled equilibrium states. In fact, since the upper end of the vertical cables is connected to the support frame, and the arch foot is subjected to the pre-tension of the transverse cables, both the horizontal stiffness of the support frame and the transverse cables will participate in influencing the coupled equilibrium relationship between the vertical cables and the arch rib. Summary of the Invention

[0003] To study the coupling balance between the arch rib and the vertical cables, support structure, and transverse cables at different lifting heights during the overall lifting construction of the arch rib, this application proposes a method for calculating the elastic deformation of the arch foot during the entire lifting process of the arch rib, taking into account the lifting height, the horizontal stiffness of the support structure, the material and cross-sectional properties of the transverse cables, and the magnitude of the transverse preload.

[0004] To achieve the above objectives, this application provides a method for calculating the elastic deformation of the arch foot during the overall lifting construction of the arch rib, the steps of which include:

[0005] Based on the stress characteristics during the overall lifting process of the arch rib, the relationship between the stress on the arch rib and the displacement of the arch foot is obtained;

[0006] An equivalent force model is constructed based on the relationship between the force on the arch rib and the displacement of the arch foot.

[0007] Based on the equivalent stress model, the elastic deformation of the arch foot is calculated.

[0008] Preferably, the method for constructing the equivalent force model includes: calculating the horizontal component of the vertical cable force, the preload of the transverse cable force, and the additional force generated by the elongation of the transverse cable due to the displacement of the arch foot during the overall lifting of the arch rib, then:

[0009] F = F0 + T + T l

[0010] Where F represents the external load; F0 represents the horizontal component of the vertical cable force; T is the preload of the transverse cable; T l This refers to the additional force generated after the transverse cable elongates.

[0011] Preferably, during the process of constructing the equivalent force model, the following mechanical equilibrium relationship exists:

[0012]

[0013] In the formula, This indicates the horizontal displacement of the upper end of the vertical cable; K1 represents the horizontal displacement of the lower end of the vertical cable; x represents the horizontal stiffness of the support. 1p Let F0 represent the arch foot displacement; L represent the length of the vertical cable; finally, let F0 be expressed as x. 1p Relationship:

[0014]

[0015]

[0016] Where G represents the self-weight of the arch rib.

[0017] Preferably, during the process of constructing the equivalent force model, T l The size depends on the elongation of the lateral cables, that is, on the horizontal displacement of the arch foot, and T l Represented as about x 1p Relationship:

[0018]

[0019] In the formula, E l Indicates the elastic modulus of the transverse cable; A l L0 represents the cross-sectional area of ​​the transverse cable; L0 represents the length of the transverse cable.

[0020] Preferably, during the process of constructing the equivalent stress model, the external load F on the arch structure is ultimately expressed as:

[0021]

[0022] Preferably, the expression for the displacement result calculated by the equivalent force model includes:

[0023]

[0024] In the formula, M p F represents the bending moment at a point (x, y) on the basic structure under the action of an external force; Np M1 represents the axial force at a point (x, y) on the basic structure under the action of an external force; M2 represents the bending moment at a point (x, y) on the basic structure under the action of a unit force; F represents the bending moment at a point (x, y) on the basic structure under the action of a unit force. N1 The axial force at a point (x, y) on the basic structure under a unit force; E represents the elastic modulus of the arch structure material; I represents the moment of inertia of the arch structure section; A represents the cross-sectional area of ​​the arch structure.

[0025] This application also provides a system for calculating the elastic deformation of the arch foot during the overall lifting construction of the arch rib, including: an analysis module, a construction module, and a calculation module;

[0026] The analysis module is used to obtain the relationship between the force on the arch rib and the displacement of the arch foot based on the force characteristics during the overall lifting process of the arch rib.

[0027] The construction module is used to construct an equivalent force model based on the relationship between the force on the arch rib and the displacement of the arch foot.

[0028] The calculation module calculates the elastic deformation of the arch foot based on the equivalent stress model; the calculation module includes: a vertical cable calculation unit, a horizontal cable calculation unit, and an external load calculation unit.

[0029] Preferably, the workflow of the vertical cable calculation unit includes: performing a force balance analysis on the vertical cable.

[0030]

[0031] In the formula, This indicates the horizontal displacement of the upper end of the vertical cable; K1 represents the horizontal displacement of the lower end of the vertical cable; x represents the horizontal stiffness of the support. 1p Let F0 represent the arch foot displacement; L represent the length of the vertical cable; finally, let F0 be expressed as x. 1p Relationship:

[0032]

[0033]

[0034] Where G represents the self-weight of the arch rib.

[0035] Preferably, the workflow of the transverse cable calculation unit includes: calculating T l Represented as about x 1p Relationship:

[0036]

[0037] In the formula, E l Indicates the elastic modulus of the transverse cable; A l L0 represents the cross-sectional area of ​​the transverse cable; L0 represents the length of the transverse cable.

[0038] Preferably, the workflow of the external load calculation unit includes: calculating the external load F on the arch structure:

[0039]

[0040] Compared with the prior art, the beneficial effects of this application are as follows:

[0041] This application addresses the stress characteristics of arch ribs during the overall lifting construction process by constructing an equivalent stress model. Based on this model, a theoretical calculation method for the elastic deformation of the arch foot is established, considering the effects of the arch rib's self-weight, lifting height, support horizontal stiffness, transverse cable material and cross-sectional properties, and the coupling effect of transverse preload. This method can be used to study and calculate the deformation law during the overall lifting of arch ribs, filling a gap in theoretical calculation methods in related fields and providing guidance for related structural design and construction. Attached Figure Description

[0042] To more clearly illustrate the technical solutions of this application, the drawings used in the embodiments are briefly introduced below. Obviously, the drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0043] Figure 1 This is a schematic diagram of the forces acting on the arch rib at different lifting heights during the overall lifting process of the arch rib in an embodiment of this application;

[0044] Figure 2 This is a schematic diagram of the stress on the arch rib and transverse cable structure in an embodiment of this application;

[0045] Figure 3 This is a schematic diagram of the equivalent force model of an embodiment of this application;

[0046] Figure 4 This is a simplified schematic diagram of the equivalent force model in the embodiments of this application;

[0047] Figure 5 This is a schematic diagram of the horizontal component of the vertical tension cable on the arch rib in an embodiment of this application;

[0048] Figure 6 This is a schematic diagram of the spring force in an embodiment of this application;

[0049] Figure 7 This is a schematic diagram illustrating the force relationship between external loads and arch foot displacement in an embodiment of this application.

[0050] Figure 8 This is a schematic diagram of the basic structure under stress in an embodiment of this application;

[0051] Figure 9 This is a schematic diagram of the system structure according to an embodiment of this application. Detailed Implementation

[0052] The technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, and not all embodiments. Based on the embodiments of this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0053] To make the above-mentioned objectives, features and advantages of this application more apparent and understandable, the application will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0054] Example 1

[0055] like Figure 1 As shown, this is a schematic diagram of the forces acting on the arch rib at different lifting heights during the overall lifting process. G is the self-weight of the arch rib (for convenience, the self-weight of the arch rib is represented in the form of a concentrated force), and T is the pre-tension force provided by the transverse cable to the arch foot. After the arch rib is lifted, the vertical cable becomes shorter, the inclination angle of the vertical cable changes from θ1 to θ2, and the tension of the vertical cable also changes from T1 to T2.

[0056] In this embodiment, Figure 1 If the arch ribs and transverse cables are considered as an independent system, then the structural force diagram of this system is as follows: Figure 2 As shown in the figure. Here, represents the horizontal component of the vertical cable force, and G is the self-weight of the arch rib. Then, based on the relationship between the force on the arch rib and the displacement of the arch foot, an equivalent force model is constructed.

[0057] Depend on Figure 3 The structural force equivalence shown can be used to... Figure 1 The arch rib and transverse cable force system in the middle is simplified as follows: Figure 4 The mechanical model shown is as follows. Figure 5 As shown, the horizontal displacement of one arch foot of the arch rib is x, and the length of the vertical cable is approximately L. Then, the horizontal component T of the vertical cable on the arch rib is...x for:

[0058]

[0059] Considering that the vertical cables only exert a horizontal component force on the arch rib when the arch foot undergoes horizontal displacement, a property similar to that of a spring, we consider using a spring to represent the horizontal force exerted by the vertical cables on the arch rib.

[0060] like Figure 6 As shown, Figure 4 T in the mechanical model x If we consider it as an equivalent spring force, then the spring stiffness K is defined as:

[0061]

[0062] Then we have:

[0063] T x =K·x.

[0064] Next, the elastic deformation of the arch foot is calculated based on the above model. Figure 6 Based on the equivalent mechanical model shown, the mechanical relationships during the overall lifting process of the arch rib are derived.

[0065] like Figure 7 As shown, the displacement of the arch foot of the arch structure under its own weight G and external load F is x. 1p Considering the horizontal component of the vertical cable force, the pre-tension of the transverse cable, and the additional force generated by the elongation of the transverse cable due to the displacement of the arch foot during the overall lifting of the arch rib, we have:

[0066] F = F0 + T + T l

[0067] Where F represents the external load; F0 represents the horizontal component of the vertical cable force; T is the preload of the transverse cable; T l This refers to the additional force generated after the transverse cable elongates.

[0068] If the horizontal displacement of the upper end of the vertical cable is x0, the horizontal displacement of the lower end of the vertical cable is x1, and the length of the vertical cable is L, then the vertical cable has the following force equilibrium relationship:

[0069]

[0070] Considering that the upper end of the vertical cable is connected to the support, and the horizontal stiffness of the support is K1, then x0 can be expressed as:

[0071]

[0072] Given that the horizontal displacement x1 at the lower end of the vertical cable is the displacement of the arch foot on one side, x1 can be expressed as:

[0073]

[0074] Finally, F0 is represented as... 1p Relationship:

[0075]

[0076]

[0077] Due to T l The magnitude depends on the elongation of the transverse cables, which in turn depends on the horizontal displacement of the arch foot, and therefore can also be expressed as x. 1p Relationship:

[0078]

[0079] In the formula, E l Indicates the elastic modulus of the transverse cable; A l L0 represents the cross-sectional area of ​​the transverse cable; L0 represents the length of the transverse cable.

[0080] Therefore, the external load F on the arch structure can be expressed as:

[0081]

[0082] Finally, according to Figure 8 (a) The basic structure, using the basic knowledge of structural mechanics to solve... Figure 7 Displacement x of the arch foot of the central arch structure 1p :

[0083]

[0084] In the formula, M p F represents the bending moment at a point (x, y) on the basic structure under the action of an external force; Np M1 represents the axial force at a point (x, y) on the basic structure under the action of an external force; M2 represents the bending moment at a point (x, y) on the basic structure under the action of a unit force; F represents the bending moment at a point (x, y) on the basic structure under the action of a unit force. N1 The axial force at a point (x, y) on the basic structure under a unit force; E represents the elastic modulus of the arch structure material; I represents the moment of inertia of the arch structure section; A represents the cross-sectional area of ​​the arch structure.

[0085] according to Figure 8 As shown in (b), under the action of external force, the bending moment at point (x, y) on the arch rib is:

[0086]

[0087] according to Figure 8 As shown in (b), under the action of external forces, the axial force at point (x, y) on the arch rib is:

[0088]

[0089] according to Figure 8 As shown in (c), under the action of a unit force, the bending moment at point (x, y) on the arch rib is:

[0090] M1 = -y

[0091] according to Figure 8 As shown in (c), under the action of a unit force, the axial force at point (x, y) on the arch rib is:

[0092]

[0093] Then we have:

[0094]

[0095] make:

[0096]

[0097]

[0098] Then we have:

[0099]

[0100]

[0101] The values ​​of F0, x0, and x1 can be obtained accordingly, and the elastic deformation calculation of the arch foot can be completed.

[0102] Example 2

[0103] like Figure 9 The diagram shown illustrates the system structure of this embodiment, including an analysis module, a construction module, and a calculation module. The analysis module is used to obtain the relationship between the force on the arch rib and the displacement of the arch foot based on the force characteristics during the overall lifting process of the arch rib. The construction module is used to construct an equivalent force model based on the relationship between the force on the arch rib and the displacement of the arch foot. The calculation module calculates the elastic deformation of the arch foot based on the equivalent force model. The calculation module includes a vertical cable calculation unit, a horizontal cable calculation unit, and an external load calculation unit.

[0104] The following will, in conjunction with this embodiment, explain in detail how this application solves technical problems in real life.

[0105] First, the force analysis of the arch rib is performed using the analysis module, such as... Figure 1As shown, this is a schematic diagram of the forces acting on the arch rib at different lifting heights during the overall lifting process. G is the self-weight of the arch rib (for convenience, the self-weight of the arch rib is represented in the form of a concentrated force), and T is the pre-tension force provided by the transverse cable to the arch foot. After the arch rib is lifted, the vertical cable becomes shorter, the inclination angle of the vertical cable changes from θ1 to θ2, and the tension of the vertical cable also changes from T1 to T2.

[0106] In this embodiment, Figure 1 If the arch ribs and transverse cables are considered as an independent system, then the structural force diagram of this system is as follows: Figure 2 As shown. Here, G represents the horizontal component of the vertical cable force, and G is the self-weight of the arch rib. Then, using the building module, an equivalent force model is constructed based on the relationship between the force on the arch rib and the displacement of the arch foot. The workflow is as follows:

[0107] Depend on Figure 3 The structural force equivalence shown can be used to... Figure 1 The arch rib and transverse cable force system in the middle is simplified as follows: Figure 4 The mechanical model shown is as follows. Figure 5 As shown, the horizontal displacement of one arch foot of the arch rib is x, and the length of the vertical cable is approximately L. Then, the horizontal component T of the vertical cable on the arch rib is... x for:

[0108]

[0109] Considering that the vertical cables only exert a horizontal component force on the arch rib when the arch foot undergoes horizontal displacement, a property similar to that of a spring, we consider using a spring to represent the horizontal force exerted by the vertical cables on the arch rib.

[0110] like Figure 6 As shown, Figure 4 T in the mechanical model x If we consider it as an equivalent spring force, then the spring stiffness K is defined as:

[0111]

[0112] Then we have:

[0113] T x =K·x

[0114] Then, the calculation module calculates the elastic deformation of the arch foot based on the above model. Figure 6 Based on the equivalent mechanical model shown, the mechanical relationships during the overall lifting process of the arch rib are derived. The workflow is as follows:

[0115] like Figure 7 As shown, the displacement of the arch foot of the arch structure under its own weight G and external load F is x. 1pConsidering the horizontal component of the vertical cable force, the pre-tension of the transverse cable, and the additional force generated by the elongation of the transverse cable due to the displacement of the arch foot during the overall lifting of the arch rib, we have:

[0116] F = F0 + T + T l

[0117] Where F represents the external load; F0 represents the horizontal component of the vertical cable force; T is the preload of the transverse cable; T l This refers to the additional force generated after the transverse cable elongates.

[0118] If the horizontal displacement of the upper end of the vertical cable is x0, the horizontal displacement of the lower end of the vertical cable is x1, and the length of the vertical cable is L, calculate the following force equilibrium relationship using the vertical cable calculation unit:

[0119]

[0120] Considering that the upper end of the vertical cable is connected to the support, and the horizontal stiffness of the support is K1, then x0 can be expressed as:

[0121]

[0122] Given that the horizontal displacement x1 at the lower end of the vertical cable is the displacement of the arch foot on one side, x1 can be expressed as:

[0123]

[0124] Finally, F0 is represented as... 1p Relationship:

[0125]

[0126]

[0127] Due to T l The size depends on the elongation of the transverse cables, which in turn depends on the horizontal displacement of the arch foot. Therefore, T is calculated using the transverse cable calculation unit. l Regarding x 1p Relationship:

[0128]

[0129] In the formula, E l Indicates the elastic modulus of the transverse cable; A l L0 represents the cross-sectional area of ​​the transverse cable; L0 represents the length of the transverse cable.

[0130] Finally, the external load F on the arch structure is calculated using the external load calculation unit:

[0131]

[0132] Finally, according to Figure 8 (a) The basic structure, using the basic knowledge of structural mechanics to solve... Figure 7 Displacement x of the arch foot of the central arch structure 1p :

[0133]

[0134] In the formula, M p F represents the bending moment at a point (x, y) on the basic structure under the action of an external force; Np M1 represents the axial force at a point (x, y) on the basic structure under the action of an external force; M2 represents the bending moment at a point (x, y) on the basic structure under the action of a unit force; F represents the bending moment at a point (x, y) on the basic structure under the action of a unit force. N1 The axial force at a point (x, y) on the basic structure under a unit force; E represents the elastic modulus of the arch structure material; I represents the moment of inertia of the arch structure section; A represents the cross-sectional area of ​​the arch structure.

[0135] according to Figure 8 As shown in (b), under the action of external force, the bending moment at point (x, y) on the arch rib is:

[0136]

[0137] according to Figure 8 As shown in (b), under the action of external forces, the axial force at point (x, y) on the arch rib is:

[0138]

[0139] according to Figure 8 As shown in (c), under the action of a unit force, the bending moment at point (x, y) on the arch rib is:

[0140] M1 = -y

[0141] according to Figure 8 As shown in (c), under the action of a unit force, the axial force at point (x, y) on the arch rib is:

[0142]

[0143] Then we have:

[0144]

[0145] make:

[0146]

[0147]

[0148] Then we have:

[0149]

[0150]

[0151] The values ​​of F0, x0, and x1 can be obtained accordingly, and the elastic deformation calculation of the arch foot can be completed.

[0152] The embodiments described above are merely preferred embodiments of this application and are not intended to limit the scope of this application. Any modifications and improvements made to the technical solutions of this application by those skilled in the art without departing from the spirit of this application shall fall within the protection scope defined by the claims of this application.

Claims

1. A method for calculating elastic deformation of arch foot in the process of integral lifting construction of arch rib, characterized in that the steps of Comprise: Based on the stress characteristics of the overall lifting process of the arch rib, the relationship between the stress of the arch rib and the displacement of the arch foot is obtained; Based on the relationship between the stress of the arch rib and the displacement of the arch foot, an equivalent stress model is constructed; The method for constructing the equivalent stress model comprises: calculating the horizontal component of the vertical cable, the pre-tension of the transverse cable and the additional force generated after the transverse cable is elongated due to the displacement of the arch foot during the overall lifting process of the arch rib, then: F = F0+ T + T l Wherein, F represents external load; F0 represents horizontal component of vertical cable; T is pretension of transverse cable; T l is additional force generated by elongation of transverse cable; During the process of constructing the equivalent stress model, there is the following mechanical equilibrium relationship: wherein represents the horizontal displacement of the upper end point of the vertical cable; represents the horizontal displacement of the lower end point of the vertical cable; K1 represents the horizontal stiffness of the support; x 1p represents the displacement of the arch foot; L represents the length of the vertical cable; finally, F0 is expressed as a function of x 1p . Wherein, G represents the self weight of the arch rib; In the course of building the equivalent force model, T l depends on the lateral cable elongation, i.e. on the arch foot horizontal displacement, T l is expressed as a function of x 1p : wherein E l represents the transverse cable elastic modulus; A l represents the transverse cable cross-sectional area; L0the transverse cable length; Based on the equivalent stress model, the elastic deformation of the arch foot is calculated.

2. The method of claim 1, wherein, During the process of constructing the equivalent stress model, the external load F borne by the arch structure is finally represented as: 。 3. The method of claim 2, wherein the elastic deformation of the arch foot is calculated during the integral lifting construction of the arch rib. The displacement result expression calculated by the equivalent stress model comprises: where M p represents the bending moment at a point (x, y) in the basic structure under the action of external force; F Np represents the axial force at a point (x, y) in the basic structure under the action of external force; M1 represents the bending moment at a point (x, y) in the basic structure under the action of unit force; F N1 represents the axial force at a point (x, y) in the basic structure under the action of unit force; E represents the elastic modulus of the arch structure material; I represents the moment of inertia of the cross section of the arch structure; and A represents the cross-sectional area of the arch structure.

4. A system for calculating the elastic deformation of the arch foot during the integral lifting of the arch ribs, said system being used to implement the method according to any one of claims 1 to 3, characterized in that, Comprise: Analysis module, construction module and calculation module; The analysis module is used to obtain the relationship between the stress of the arch rib and the displacement of the arch foot based on the stress characteristics of the overall lifting process of the arch rib; The construction module is used to construct an equivalent stress model based on the relationship between the stress of the arch rib and the displacement of the arch foot; The calculation module calculates the elastic deformation of the arch foot based on the equivalent stress model; The calculation module comprises: a vertical cable calculation unit, a transverse cable calculation unit and an external load calculation unit.

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