In-plane nonlinear deformation calculation method of arch structure with axis defects under rectangular coordinate system

Abstract

A method for calculating the in-plane nonlinear deformation of an arch structure with axial defects in a Cartesian coordinate system is presented. This method is based on two states of the arch structure's curve element in the Cartesian coordinate system: an ideal axis and an actual axial defect. It derives the changes in the arc length and curvature of the curve element before and after deformation, obtaining expressions for the nonlinear compressive and bending strains in the in-plane of the arch structure with axial defects in the Cartesian coordinate system. The method also derives the nonlinear equilibrium differential equations in the in-plane of the arch structure with axial defects in the Cartesian coordinate system, obtaining solutions that satisfy the mechanical and geometric boundaries of the arch structure with axial defects. Finally, the method derives the nonlinear equilibrium governing equations in the in-plane of the arch structure with axial defects, ultimately obtaining the solution to the nonlinear mechanical problem in the in-plane of the arch structure with axial defects in the Cartesian coordinate system. This method can obtain the essence of the nonlinear mechanical problem of the arch structure with axial defects in the Cartesian coordinate system, filling the gap in the nonlinear strain-displacement expression for the in-plane of the arch structure with axial defects in the Cartesian coordinate system.

Description

TECHNICAL FIELD

[0001] The present application relates to a kind of axis defect arch structure in-plane nonlinear deformation calculation method under rectangular coordinate system, belong to arch structure in-plane nonlinear mechanics analysis technical field;It is mainly used for arch structure in-plane nonlinear mechanics analysis under Cartesian rectangular coordinate system. BACKGROUND

[0002] Arch structure in-plane nonlinear strain expression is the key of nonlinear deformation calculation, these strains mainly have three kinds of arch structure in-plane nonlinear strain under polar coordinate system without defect circular arch, arch structure in-plane nonlinear strain under polar coordinate system with defect circular arch, arch structure in-plane nonlinear strain under Cartesian rectangular coordinate system without defect.

[0003] (1) arch structure in-plane nonlinear strain under polar coordinate system without defect circular arch.The nonlinear strain includes compression strain and bending strain, compression strain Bending strain Wherein v, w are circular arch radial displacement and tangential displacement, R is circular arch radius, y is the radial distance of certain point on cross section to center axis.This expression does not consider arch axis defect, and cannot solve the problem of arch structure in-plane nonlinear mechanics.

[0004] (2) arch structure in-plane nonlinear strain under polar coordinate system with defect circular arch.In polar coordinate system, the strain of circular arch containing defect includes compression strain and bending strain, wherein compression strain is Bending strain is Wherein D is initial axis defect.This strain is the expression type under polar coordinate, and cannot solve the problem of arch structure in-plane nonlinear mechanics under Cartesian rectangular coordinate system with axis defect.

[0005] (3) arch structure in-plane nonlinear strain under Cartesian rectangular coordinate system without defect.The strain includes compression strain And bending strain Wherein w, v are respectively horizontal displacement and vertical displacement of arch structure under Cartesian rectangular coordinate system, y is arch axis equation under Cartesian rectangular coordinate system, y * The normal distance of arbitrary point on arch cross section to cross section center axis.This strain does not consider the effect of defect, and cannot solve the problem of arch structure in-plane nonlinear mechanics under Cartesian rectangular coordinate system with axis defect.

[0006] Based on the existing arch structure in-plane nonlinear strain expression, for Cartesian rectangular coordinate system under the axis defect arch structure in-plane nonlinear mechanics analysis all have certain difficulty, mainly in the following points: 1) Polar coordinate system under defect-free circular arch structure in-plane nonlinear strain, because of not considering the arch axis defect, cannot solve the Cartesian rectangular coordinate system under the axis defect arch structure in-plane nonlinear mechanics problem.2) Polar coordinate system under defect circular arch structure in-plane nonlinear strain, because the polar coordinate expression type is different from Cartesian rectangular coordinate system, cannot solve the Cartesian rectangular coordinate system under the axis defect arch structure in-plane nonlinear problem.3) Cartesian rectangular coordinate system under defect-free arch structure in-plane nonlinear strain, because of not considering the arch axis defect, cannot solve the Cartesian rectangular coordinate system under the axis defect arch structure in-plane nonlinear mechanics problem. SUMMARY

[0007] The purpose of the present application is to solve the problem of the calculation method of the Cartesian rectangular coordinate system under the axis defect arch structure in-plane nonlinear mechanics analysis, and to provide a Cartesian rectangular coordinate system under the axis defect arch structure in-plane nonlinear deformation calculation method.

[0008] The technical scheme realized by the present application is as follows: a Cartesian rectangular coordinate system under the axis defect arch structure in-plane nonlinear deformation calculation method, the method is to analyze the change of the length and curvature of the curve microelement deformation of the arch structure under the Cartesian rectangular coordinate system considering the axis defect, and to deduce the nonlinear expression of the compression strain-displacement and bending strain-displacement at this time; based on the virtual work principle, the in-plane nonlinear equilibrium differential equation of the arch structure under the Cartesian rectangular coordinate system is deduced; according to the mechanical and geometric boundary conditions of the arch structure, the nonlinear vertical displacement expression of any point of the arch structure under the Cartesian rectangular coordinate system is deduced; based on the principle that the compression strain along the arch axis is equal to the length reduction of the arch axis, the in-plane nonlinear equilibrium control equation of the arch structure under the Cartesian rectangular coordinate system is deduced, and then the solution of the in-plane nonlinear mechanics problem of the arch structure under the Cartesian rectangular coordinate system is obtained.

[0009] The nonlinear compression strain-displacement expression is as follows:

[0010]

[0011] The nonlinear bending strain-displacement expression is as follows:

[0012]

[0013] In the formula, ε mw is the horizontal displacement of the arch structure curve microelement in the Cartesian rectangular coordinate system; v is the vertical displacement of the arch structure curve microelement in the Cartesian rectangular coordinate system; z is the horizontal coordinate of the Cartesian rectangular coordinate system, and is positive to the right; y is the vertical coordinate of the Cartesian rectangular coordinate system, and is positive vertically downward; D y is the vertical component of the axis defect in the Cartesian rectangular coordinate system; ε b is the nonlinear bending strain of the axis defect arch structure in the Cartesian rectangular coordinate system; y * is the distance of an arbitrary point on the main arch ring cross section from the neutral axis of the cross section.

[0014] The nonlinear equilibrium differential equation of the arch structure and the load conservative system in the Cartesian rectangular coordinate system is:

[0015]

[0016] In the formula, δ() is a variation function; ∏ is the total energy of the arch structure and the load conservative system in the Cartesian rectangular coordinate system; V is the volume of the arch structure in the Cartesian rectangular coordinate system; ε is the nonlinear total strain of an arbitrary point on the arch structure in the Cartesian rectangular coordinate system; σ is the nonlinear total stress of an arbitrary point on the arch structure in the Cartesian rectangular coordinate system; L is the span of the arch structure in the Cartesian rectangular coordinate system; and q(z) is the load at an arbitrary point on the arch structure in the Cartesian rectangular coordinate system.

[0017] The vertical nonlinear equilibrium differential equation of the arch structure and the self-weight load conservative system in the Cartesian rectangular coordinate system is:

[0018]

[0019] In the formula, E is the elastic modulus of the material of the arch structure in the Cartesian rectangular coordinate system; I x is the bending moment of inertia of the cross section of the arch; H is the nonlinear horizontal thrust at the arch foot in the Cartesian rectangular coordinate system; a is the catenary arch coefficient; g is the self-weight load intensity of the catenary arch; and ch() is a hyperbolic cosine function.

[0020] The nonlinear vertical displacement expression of an arbitrary point of the axis defect arch structure in the Cartesian rectangular coordinate system is:

[0021]

[0022] In the formula, D max is the axis deviation amplitude; θ is a dimensionless axis force stability parameter; n is the axis deviation modulus; is a dimensionless load; L is the span of the arch structure in the Cartesian rectangular coordinate system; and a is the catenary arch coefficient.

[0023] The in-plane nonlinear equilibrium control equation of the axis defect arch structure under the Cartesian rectangular coordinate system is:

[0024]

[0025] Wherein,

[0026]

[0027]

[0028] In the formula, λ is the modified slenderness ratio, and the expression of λ is:

[0029]

[0030] The beneficial effects of the present application are as follows: the present application obtains the in-plane nonlinear compression strain-displacement expression and the nonlinear bending strain-displacement expression of the arch structure under the Cartesian rectangular coordinate system by analyzing the length, curvature and cross-section angle changes of the arch structure curve microelement before and after deformation, obtains the in-plane nonlinear equilibrium control equation of the axis defect arch structure by the principle that the curve integral of the compression strain along the arch axis is equal to the arch axis shortening amount, and finally obtains the solution of the in-plane nonlinear mechanical problem of the axis defect arch structure under the Cartesian rectangular coordinate system.

[0031] The present application has clear mechanical concept and simple method, can quickly determine the solution of the nonlinear mechanical problem of the arch structure under the Cartesian rectangular coordinate system, and greatly reduces the workload of bridge designers; meanwhile, based on the in-plane nonlinear equilibrium differential equation and approximate solution of the axis deviation arch structure under the Cartesian rectangular coordinate system, bridge researchers can further explore the law of the in-plane nonlinear deformation of the arch structure under the Cartesian rectangular coordinate system under the consideration of the initial axis deviation, and fill the gap of the research on the nonlinear mechanics related problems of the axis arch structure under the Cartesian coordinate system commonly used in actual engineering. BRIEF DESCRIPTION OF DRAWINGS

[0032] Figure 1 It is a flowchart of the in-plane nonlinear mechanical analysis method of the axis defect arch structure under the Cartesian rectangular coordinate system;

[0033] Figure 2 It is a schematic diagram of the nonlinear deformation of the axis defect arch structure under the Cartesian rectangular coordinate system;

[0034] Figure 3 It is a schematic diagram of the arch structure curve microelement before and after deformation under the Cartesian rectangular coordinate system;

[0035] In the figure, 1 represents the ideal arch axis in a Cartesian coordinate system; 2 represents the self-weight load of the arch structure; 3 represents the boundary constraint of the two-hinged arch; 4 represents the actual arch axis with axial defects in a Cartesian coordinate system; 5 represents the nonlinear deformation of the arch structure; 6 represents the Cartesian coordinate system; 7 represents the arc length ds of the arch structure curve element before deformation under the ideal axis condition; 8 represents the vertical length dy of the arch structure curve element ds under the ideal axis condition; 9 represents the horizontal length dz of the arch structure curve element ds under the ideal axis condition; 10 represents the arc length ds0 of the arch structure curve element before deformation under the actual axial defect condition; and 11 represents the vertical length dy+dD of the arch structure curve element ds0 under the axial defect condition. y ;12 represents the horizontal length of the infinitesimal element ds0 of the arch structure curve under the axial defect, dz+dD. z ;13 represents the arc length ds of the micro-element of the arch structure curve after deformation due to the axial defect. * ;14 represents the micro-element ds of the arch structure curve under the axial defect. * The vertical length after deformation is dy+dD y +dv; 15 represents the micro-element ds of the arch structure curve under the axial defect. * The horizontal length after deformation is dz+dD z +dw. Detailed Implementation

[0036] The invention will be further described below with reference to the accompanying drawings.

[0037] like Figure 1 As shown in the figure, this embodiment presents a method for calculating the nonlinear deformation within the plane of an axially defective arch structure in a rectangular coordinate system. The steps are as follows:

[0038] (1) As Figure 2 As shown, the arch structure undergoes nonlinear deformation under load in a Cartesian coordinate system. Since the theoretical arch axis is difficult to achieve in actual construction, errors and defects in the arch axis are inevitable. Existing nonlinear strain techniques for arch structures cannot solve the in-plane nonlinear mechanical problems of arch structures in a Cartesian coordinate system. Therefore, the defective arch axis should be used as the basis for analyzing the in-plane nonlinear strain of the arch structure.

[0039] (2) Figure 3 As shown, in the Cartesian coordinate system, the arc length of the arch structure curve element before and after deformation, considering the initial axis deviation, can be expressed as:

[0040]

[0041] In the formula, w represents the horizontal displacement of the arch structure curve element in the Cartesian rectangular coordinate system; v represents the vertical displacement of the arch structure curve element in the Cartesian rectangular coordinate system; z represents the horizontal coordinate in the Cartesian rectangular coordinate system, with positive to the right; y represents the vertical coordinate in the Cartesian rectangular coordinate system, with positive downward; Dy D represents the vertical component of the axial defect in a Cartesian coordinate system. z s is the horizontal component of the axial defect in the Cartesian rectangular coordinate system; s0 is the length of the infinitesimal element of the axial defect arch structure curve before deformation in the Cartesian rectangular coordinate system; s* is the length of the infinitesimal element of the axial defect arch structure curve after deformation in the Cartesian rectangular coordinate system.

[0042] According to the definition of compressive strain, the compressive strain ε of an arch is... m It can be represented as:

[0043]

[0044] By approximating the compressive strain using equivalent infinitesimals and substituting the arc element, we can obtain:

[0045]

[0046] Based on the deformation analysis of the arch structure, its horizontal deformation is much smaller than its vertical deformation, and the initial axis deviation is mainly vertical with approximately zero in the horizontal direction. Therefore, the nonlinear deformation term (dw / dz) in the horizontal direction is not significant. 2 / 2、dD z / dz and (dD) z / dz) 2 If negligible, then the nonlinear compressive strain ε of the arch is... m It can be simplified to:

[0047]

[0048] (3) Figure 3 As shown, in a Cartesian coordinate system, the curvature of an arch structure curve element before and after deformation, considering the initial axis deviation, can be expressed as:

[0049]

[0050] Where ρ1 is the curvature of the curve element with axial defects before deformation in the Cartesian coordinate system; ρ * Let be the curvature of a curve element with an axial defect before deformation in a Cartesian coordinate system.

[0051] Based on this, the radius of curvature at any point before deformation of the arch structure can be deduced as:

[0052]

[0053] The radius of curvature at any point after the arch structure deforms is:

[0054]

[0055] Based on Euler-Bernoulli beam theory, the bending strain of the arch axis in the Cartesian coordinate system can be expressed as:

[0056]

[0057] By substituting the bending strain expression, based on the deformation analysis of the arch structure, the horizontal deformation is much smaller than the vertical deformation, and the initial axis deviation is mainly vertical, and the horizontal direction is approximately zero, so the horizontal nonlinear deformation terms dw / dz, (dw / dz) 2 , dD z / dz and (dD z / dz) 2 can be ignored, and the nonlinear bending strain of the axis defect arch structure in the Cartesian coordinate system is obtained as:

[0058]

[0059] (4) Based on the nonlinear compression strain and bending strain expressions of the arch structure in the Cartesian coordinate system in steps (2), (3), the virtual work principle is used for the arch structure and the load conservative system to obtain the balance differential equations of the system in the horizontal and vertical directions:

[0060] The horizontal balance differential equation is:

[0061] The vertical balance differential equation is:

[0062]

[0063] In the formula, E is the elastic modulus of the arch structure in the Cartesian coordinate system; ε m is the nonlinear compression strain of the arch; I x is the bending moment of inertia of the cross section of the arch; H is the nonlinear horizontal thrust at the arch foot in the Cartesian coordinate system; v is the vertical displacement of the arch structure curve element in the Cartesian coordinate system; z is the horizontal coordinate of the Cartesian coordinate system, and the right is positive; y is the vertical coordinate of the Cartesian coordinate system, and the vertical downward is positive; a is the arch coefficient of the cable; g is the self-weight load intensity of the cable arch; D y is the vertical component of the axis defect in the Cartesian coordinate system; A is the cross-sectional area of the main arch.

[0064] (5) Based on the boundary conditions of the two-hinged arch in the Cartesian coordinate system:

[0065] w(±L / 2)=0, v(±L / 2)=0 and

[0066] The vertical displacement expression of any point of the arch structure is obtained as:

[0067]

[0068] wherein: μ is the axial force parameter, is the dimensionless load, θ is the dimensionless axial force stability parameter; n is the axial deviation modulus; L is the span of the arch structure in the Cartesian rectangular coordinate system; a is the arch-shaped coefficient of the suspension cable.

[0069] (6) Based on the principle that the curve integral of the compressive strain along the arch axis is equal to the shortening amount of the arch axis, the vertical displacement result of step (5) is substituted to obtain the in-plane nonlinear equilibrium control equation of the axial defect arch structure in the Cartesian rectangular coordinate system:

[0070]

[0071] wherein,

[0072]

[0073]

[0074]

[0075] In the formula, λ is the modified slenderness ratio, and the expression of λ is:

[0076]

[0077] Thus, the analysis of the in-plane nonlinear mechanics problem of the axial defect arch structure in the Cartesian rectangular coordinate system is completely obtained.

Claims

1. A method for calculating in-plane nonlinear deformation of an arch structure with axis defects in a rectangular coordinate system, characterized in that, The method derives nonlinear compression strain-displacement and nonlinear bending strain-displacement expressions by analyzing the length and curvature changes of a curve microelement of the arch structure with axis defects before and after deformation in a Cartesian rectangular coordinate system; Based on the virtual work principle, vertical direction nonlinear equilibrium differential equations of a suspension cable arch structure and a self-weight load conservative system in the Cartesian rectangular coordinate system are derived; according to the mechanical and geometric boundaries of the arch structure, a nonlinear vertical displacement expression of any point of the arch structure with axis defects in the Cartesian rectangular coordinate system is derived; based on the principle that the compression strain along the arch axis is equal to the length reduction of the arch axis, nonlinear in-plane equilibrium control equations of the arch structure with axis defects in the Cartesian rectangular coordinate system are derived, and the in-plane nonlinear mechanical problem of the arch structure with axis defects in the Cartesian rectangular coordinate system is solved; The nonlinear compression strain-displacement expression is as follows: The nonlinear bending strain-displacement expression is as follows: wherein ε m is the in-plane nonlinear compressive strain of the axisymmetric imperfect arch structure in the Cartesian coordinate system; w is the horizontal displacement of the infinitesimal element of the arch structure in the Cartesian coordinate system; v is the vertical displacement of the infinitesimal element of the arch structure in the Cartesian coordinate system; z is the horizontal coordinate of the Cartesian coordinate system, which is positive to the right; y is the vertical coordinate of the Cartesian coordinate system, which is positive vertically downward; D y is the vertical component of the axisymmetric imperfection in the Cartesian coordinate system; ε b is the in-plane nonlinear bending strain of the axisymmetric imperfect arch structure in the Cartesian coordinate system; y * is the distance of an arbitrary point on the cross section of the main arch ring from the neutral axis of the cross section.

2. The method according to claim 1, wherein, The vertical direction nonlinear equilibrium differential equations of the suspension cable arch structure and the self-weight load conservative system in the Cartesian rectangular coordinate system are as follows: wherein E is the modulus of elasticity of the arch material in the Cartesian rectangular coordinate system; I x is the cross-sectional bending moment of inertia of the arch; H is the nonlinear horizontal thrust at the arch foot in the Cartesian rectangular coordinate system; a is the arching coefficient of the catenary; g is the self-weight load intensity of the catenary; and ch() is the hyperbolic cosine function.

3. The method according to claim 1, wherein, The nonlinear vertical displacement expression of any point of the arch structure with axis defects in the Cartesian rectangular coordinate system is as follows: In the formula, D max is the axial deviation amplitude; θ is the dimensionless axial force stability parameter; n is the axial deviation modulus; is the dimensionless load; L is the span of the arch structure in the Cartesian rectangular coordinate system; and a is the catenary line arch coefficient.

4. The method according to claim 3, wherein, The in-plane nonlinear equilibrium control equations of the arch structure with axis defects in the Cartesian rectangular coordinate system are as follows: wherein Wherein, sh() is a hyperbolic sine function. where λ is the modified slenderness ratio, and the expression for λ is:

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