Geometric Nonlinear Analytical Method for Parabolic Two-Hinged Arch Surface under Concentrated Force at Mid-Span

Abstract

A geometric nonlinear analytical method for a parabolic two-hinge arch under a concentrated force at mid-span is disclosed. This method derives the nonlinear equilibrium differential equations in the horizontal and vertical directions of the parabolic arch structure and the conservative system under a concentrated force load at mid-span in Cartesian coordinates based on the principle of virtual work. According to the mechanical analysis and geometric boundary of a parabolic arch with a symmetrical section in Cartesian coordinates, an expression for the nonlinear vertical displacement of the parabolic two-hinge arch under a concentrated force at mid-span is obtained. Based on the principle that the curve integral of the arch structure's compressive strain along the arch axis is equal to the arch axis's compressive deformation, the geometric nonlinear equilibrium equation for the parabolic two-hinge arch under a concentrated force at mid-span is obtained, and thus the nonlinear deformation analysis of the parabolic two-hinge arch under a concentrated force at mid-span is derived. This method can quickly determine the geometric nonlinear equilibrium differential equations and approximate deformation analysis of a parabolic two-hinge arch under a concentrated force at mid-span.

Description

Technical Field

[0001] This invention relates to a geometric nonlinear analytical method for a parabolic two-hinged arch surface under concentrated force at mid-span, belonging to the field of civil engineering technology. Background Technology

[0002] The analytical methods for the geometric nonlinear deformation within the plane of a parabolic arch structure can be divided into two types: the analytical method for the nonlinear deformation of a circular arch under the action of a radial concentrated force at the arch top in polar coordinates, and the numerical analysis method for the nonlinear deformation of a parabolic two-hinged arch based on finite element theory.

[0003] (1) Analytical method for nonlinear deformation of a circular arch under radial concentrated force at the top of the arch in polar coordinate system.

[0004] This method investigates the in-plane nonlinear elastic stability of a circular arch under a concentrated load at the arch crown. Based on the principle of virtual work, it establishes the nonlinear equilibrium conditions and buckling equilibrium equations for the circular arch in polar coordinates. By utilizing the boundary conditions of the arch structure to solve the eigenvalues ​​of the differential equations, analytical solutions for the antisymmetric bifurcation buckling and positive symmetric leap buckling loads of the circular arch under a concentrated load at the arch crown are obtained. Furthermore, it reveals that shallow circular arches under concentrated loads exhibit multiple deformations and extreme points. While this method is simple and feasible, the analytical method for the nonlinear deformation of parabolic arches under a concentrated load at mid-span in rectangular coordinates remains unsolved.

[0005] (2) Numerical Solution Method for Nonlinear Deformation of Parabolic Two-Hinged Arch Based on Finite Element Theory. Within the framework of the finite element method, the parabolic arch is divided into a finite number of elements, which are interconnected through a finite number of nodes. This constructs the element shape and nonlinear stiffness matrix of the parabolic arch. After applying external loads to the nodes and considering boundary conditions, the nonlinear matrix equation based on the finite element method is obtained. The numerical solution for the nonlinear deformation of the parabolic two-hinged arch is then obtained using the arc-length method. While the finite element method is widely used in engineering, it cannot provide analytical results for the nonlinear deformation of a parabolic two-hinged arch, thus hindering the analysis of its nonlinear deformation characteristics.

[0006] The classic Euler-Bernoulli beam theory assumes that the cross-section remains perpendicular to the central axis before and after deformation, neglecting the influence of shear deformation. Currently, analytical solutions for the nonlinear deformation of parabolic arches under uniformly distributed loads along the arch length, based on Euler-Bernoulli beam theory, have been developed. However, the in-plane geometric nonlinear analytical method for parabolic arches with two hinges under concentrated forces at mid-span remains unsolved.

[0007] Existing analytical methods for in-plane geometric nonlinear deformation of arch structures face certain difficulties when applied to approximate analytical methods for in-plane geometric nonlinear deformation of parabolic structures in Cartesian coordinates. These difficulties are mainly manifested in the following aspects:

[0008] 1) The nonlinear deformation analysis method of a circular arch under radial concentrated force at the top of the arch in polar coordinate system is mainly used for circular arch structures in polar coordinate system, but cannot be applied to the analysis of geometric nonlinear deformation in the parabolic arch surface in Cartesian coordinate system.

[0009] 2) The numerical solution method for the nonlinear deformation of a parabolic two-hinged arch based on the finite element theory cannot obtain the analytical solution for the nonlinear deformation of the parabolic two-hinged arch, and cannot further study the geometric nonlinear deformation law in the parabolic arch surface. Summary of the Invention

[0010] The purpose of this invention is to address the problems existing in the analytical methods for geometric nonlinear deformation within the surface of existing arch structures, and to propose an analytical method for geometric nonlinear deformation within the surface of a parabolic two-hinged arch under the action of a concentrated force at mid-span.

[0011] The technical solution of this invention is as follows: a geometric nonlinear analytical method for a parabolic two-hinged arch under a concentrated force at mid-span. This method is based on Euler-Bernoulli beam theory. Through nonlinear compressive strain-displacement and nonlinear bending strain-displacement expressions within the arch structure in Cartesian coordinates, and based on the principle of virtual work, the nonlinear equilibrium differential equations in the horizontal direction for the parabolic arch structure and the conservative system under a concentrated force at mid-span in Cartesian coordinates are derived. The vertical nonlinear equilibrium differential equations of the parabolic arch structure and the conservative system under concentrated force load are derived. Based on the mechanical analysis and geometric boundary of the parabolic two-hinged arch with symmetrical cross-section in Cartesian coordinate system, the expression for the nonlinear vertical displacement of the parabolic two-hinged arch under concentrated force at mid-span in Cartesian coordinate system is obtained. Based on the principle that the curve integral of the compressive strain of the arch structure along the arch axis is equal to the compressive deformation of the arch axis, the in-plane geometric nonlinear equilibrium equation of the parabolic two-hinged arch under concentrated force at mid-span is obtained, and then the nonlinear deformation analysis of the parabolic two-hinged arch under concentrated force at mid-span is obtained.

[0012] The expressions for the nonlinear compressive strain-displacement and nonlinear bending strain-displacement in the parabolic two-hinged arch surface under a concentrated force at mid-span in the Cartesian rectangular coordinate system are as follows:

[0013]

[0014] Where, ε m Let ε be the in-plane geometrical nonlinear compressive strain at any point on the parabolic arch; b Let be the in-plane geometrical nonlinear bending strain at any point on the parabolic arch; y is the vertical coordinate of the parabolic arch, y = [z]. 2 -(L2) 2 ] / (2p), z is the x-coordinate of the Cartesian coordinate system, p=L 2 / 8f, L and f are the span and rise of the parabolic arch, respectively; w and v are the horizontal and vertical displacements of the infinitesimal element of the parabolic arch after deformation, respectively; y * The distance from any point on the cross-section of the main arch ring to the neutral axis of the cross-section is given by: ()′=d() / dz; ()″=d 2 () / dz 2 .

[0015] The nonlinear equilibrium differential equations for the parabolic arch structure and the conservative system under concentrated force at mid-span in the Cartesian rectangular coordinate system are as follows:

[0016]

[0017] Where δ() is the variational function; Π is the total energy of the arch structure and the conservative system of concentrated force load in the Cartesian rectangular coordinate system; E is the elastic modulus of the arch structure material; V is the volume of the arch structure in the Cartesian rectangular coordinate system; and ε is the nonlinear total strain at any point on the arch structure in the Cartesian rectangular coordinate system. For the Dirac function, and Q represents the concentrated force load at the top of the arch.

[0018] The horizontal nonlinear equilibrium differential equations for the parabolic arch structure and the conservative system under concentrated force at mid-span in the Cartesian rectangular coordinate system are as follows:

[0019] [EAε m / (1+y′ 2 ) 1 / 2 ]′=0;

[0020] The vertical nonlinear equilibrium differential equations of the parabolic arch structure and the conservative system of concentrated force load at mid-span under the Cartesian rectangular coordinate system are as follows:

[0021]

[0022] Where A is the cross-sectional area of ​​the arch structure; I x The moment of inertia resisted by the cross section of the main arch ring, I x =∫ A y *2 dA.

[0023] The expression for the nonlinear vertical displacement of the parabolic arch under a concentrated force at mid-span in the Cartesian rectangular coordinate system is as follows:

[0024]

[0025] Where μ is the horizontal reaction force parameter of the parabola's arch axis; θ is the dimensionless axial force stability parameter of the parabola; and H(z) is the Heaviside function.

[0026] The geometric nonlinear equilibrium equations within the parabolic two-hinged arch surface subjected to the concentrated force at mid-span are as follows:

[0027]

[0028] in, For dimensionless loads, I x The moment of inertia resisted by the cross section of the main arch ring, I x =∫ A y *2 dA.

[0029] Where, λ s To correct the slenderness ratio, the expression is:

[0030] In the formula, L is the span of the parabolic arch; λ is the relative slenderness ratio of the parabolic arch, λ=2f / i x ; a is the arch shape coefficient of the suspension arch.

[0031] Thus, the geometric nonlinear control equations within the parabolic two-hinged arch surface under the action of a concentrated force at mid-span were obtained, and the analytical method for the geometric nonlinear deformation within the parabolic two-hinged arch surface under the action of a concentrated force at mid-span was obtained.

[0032] The beneficial effects of this invention lie in that it derives the analytical method for the geometrical nonlinear deformation of a parabolic arch surface under a concentrated force at mid-span by using the nonlinear strain-displacement expression within the parabolic arch surface in a Cartesian coordinate system. This invention features clear mechanical concepts and a simple method, enabling rapid and accurate determination of the analytical method for the nonlinear deformation of an arch structure in a Cartesian coordinate system. This allows engineers to avoid using finite element software, which is difficult to model and computationally complex, significantly reducing the workload of bridge designers. Furthermore, based on the geometrical nonlinear equilibrium differential equation and approximate analytical deformation of a parabolic arch surface under a concentrated force at mid-span, bridge researchers can further explore the laws governing the geometrical nonlinear deformation of a parabolic arch surface under a concentrated force at mid-span. Attached Figure Description

[0033] Figure 1 This is a schematic diagram of the analytical method for the geometric nonlinear mechanics of a parabolic arch with two hinges under the action of a concentrated force at mid-span in a Cartesian coordinate system.

[0034] Figure 2 This is a schematic diagram of the geometric nonlinear deformation within the plane of a parabolic arch with two hinges under the action of a concentrated force at mid-span in a Cartesian coordinate system.

[0035] Figure 3 This is a simplified calculation diagram of a parabolic two-hinged arch structure in a Cartesian rectangular coordinate system.

[0036] In the figure, 1 represents the boundary conditions of the arch foot of the two-hinged arch; 2 represents the ideal arch axis in the Cartesian coordinate system; 3 represents the concentrated force load at the mid-span of the arch structure; 4 represents the Cartesian coordinate system; 5 represents the span L between the arch feet; 6 represents the parabolic arch rise f; 7 represents the nonlinear deformation of the arch structure; 8 represents the boundary conditions of the parabolic two-hinged arch with half the structure; 9 represents the arch axis of the parabolic two-hinged arch with half the structure; and 10 represents the concentrated force load of the parabolic two-hinged arch with half the structure. Detailed Implementation

[0037] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0038] like Figure 1 As shown in the figure, this embodiment is a geometric nonlinear analytical method for a parabolic two-hinged arch surface under a concentrated force at mid-span. The steps are as follows:

[0039] (1) As Figure 2 As shown, a parabolic two-hinged arch undergoes nonlinear deformation under a concentrated force at mid-span in a Cartesian coordinate system. The geometric nonlinear deformation within the surface of the parabolic arch under the concentrated force at mid-span is calculated based on the principle of virtual work and the Euler-Bernoulli beam theory.

[0040] (2) Figure 3 As shown, in a Cartesian coordinate system, a sudden shear force change occurs at the crown of a parabolic two-hinged arch subjected to a concentrated force at mid-span. Analysis of the right half of the arch reveals:

[0041]

[0042] Analysis of the right semi-arch:

[0043]

[0044] Q represents the concentrated force load at the top of the arch.

[0045] (3) Based on the nonlinear compressive strain and bending strain expressions of the arch structure in the Cartesian coordinate system obtained in step (2), the principle of virtual work is applied to the arch structure and the conservative load system to obtain the nonlinear equilibrium differential equation of the system:

[0046]

[0047] In the formula, δ() is the variational function; E is the elastic modulus of the arch structure material; V is the volume of the arch structure in the Cartesian rectangular coordinate system; ε is the nonlinear total strain at any point on the arch structure in the Cartesian rectangular coordinate system; v is the vertical displacement of the infinitesimal element of the parabolic arch after deformation. For the Dirac function, and Q represents the concentrated force load at the top of the arch.

[0048] (4) Based on steps (2) and (3), and using the Euler-Bernoulli beam and parabolic arch theory, the equilibrium differential equations of the system in the horizontal and vertical directions are obtained:

[0049] The equilibrium differential equation in the horizontal direction:

[0050] Vertical equilibrium differential equations:

[0051]

[0052] (5) Boundary conditions for the two hinged arches in Cartesian coordinates:

[0053] w(±L / 2)=0, v(±L / 2)=0,

[0054] w and v represent the horizontal and vertical displacements of the infinitesimal element of the parabolic arch after deformation, respectively; L is the span of the parabolic arch; and z is the abscissa of the Cartesian coordinate system.

[0055] The expression for the vertical displacement at any point in the arch structure can be obtained as follows:

[0056]

[0057] Where: μ is the axial force parameter, θ is a dimensionless load and θ is a dimensionless axial force stability parameter.

[0058] (6) Based on the principle that the curve integral of the compressive strain of the arch structure along the arch axis is equal to the compressive deformation of the arch axis, the vertical displacement result of step (5) is substituted to obtain the in-plane geometric nonlinear equilibrium control equation of the arch structure under the action of the concentrated force at mid-span in the Cartesian coordinate system:

[0059]

[0060] in:

[0061] Thus, the analytical solution for the nonlinear mechanics problem of a parabolic two-hinged arch surface under a concentrated force at mid-span in Cartesian coordinates has been fully obtained.

Claims

1. A geometrically nonlinear analytical method for a parabolic two-hinged arch under a concentrated force at mid-span, characterized in that, The method, based on the principle of virtual work, derives the horizontal nonlinear equilibrium differential equations and the vertical nonlinear equilibrium differential equations of a parabolic arch structure under a concentrated force at mid-span in Cartesian coordinates, using the nonlinear compressive strain-displacement and nonlinear bending strain-displacement expressions within the arch structure plane. It also derives the horizontal nonlinear equilibrium differential equations of the conservative system of the parabolic arch structure under a concentrated force at mid-span in Cartesian coordinates. Based on the mechanical analysis and geometric boundary of a parabolic two-hinged arch with a symmetrical section in Cartesian coordinates, it obtains the expression for the nonlinear vertical displacement of the parabolic two-hinged arch under a concentrated force at mid-span in Cartesian coordinates. Based on the principle that the curve integral of the compressive strain along the arch axis is equal to the compressive deformation of the arch axis, it obtains the geometric nonlinear equilibrium equations within the plane of the parabolic two-hinged arch under a concentrated force at mid-span, and then calculates the nonlinear deformation analysis of the parabolic two-hinged arch under a concentrated force at mid-span. The expression for the nonlinear vertical displacement of the parabolic two-hinged arch under a concentrated force at mid-span in the Cartesian coordinate system is as follows: ； In the formula: μ is the axial force parameter; θ is the dimensionless axial force stability parameter; H(z) is the Heaviside function; For concentrated force loads at the arch crown; E is the elastic modulus of the arch structure material; The moment of inertia of the arch structure section; A represents the cross-sectional area of ​​the arch structure; y * z is the distance from any point on the cross-section of the main arch ring to the neutral axis of the cross-section; z is the abscissa of the Cartesian coordinate system. L and ƒ are the span and sag of the parabolic arch, respectively; the geometric nonlinear equilibrium equations of the parabolic two-hinged arch under the action of the concentrated force at mid-span are: ； In the formula: ; For dimensionless loads, in, To correct the slenderness ratio; The corrected slenderness ratio for: ； In the formula: λ is the relative slenderness ratio of the parabolic arch. ; ɑ is the arch coefficient of the suspension arch.

2. The geometric nonlinear analytical method for a parabolic two-hinge arch under a concentrated force at mid-span as described in claim 1, characterized in that, The horizontal nonlinear equilibrium differential equations for the parabolic arch structure and the conservative system under concentrated force at mid-span in the Cartesian rectangular coordinate system are as follows: ； In the formula: y represents the in-plane geometric nonlinear compressive strain at any point on the parabolic arch; y is the vertical coordinate of the parabolic arch. Let y be the first derivative.

3. The geometric nonlinear analytical method for a parabolic two-hinge arch under a concentrated force at mid-span as described in claim 1, characterized in that, The vertical nonlinear equilibrium differential equations of the parabolic arch structure and the conservative system of concentrated force load at mid-span under the Cartesian rectangular coordinate system are as follows: ； In the formula: v is the vertical displacement of the infinitesimal element of the parabolic arch after deformation; This is the Dirac function.

4. The geometric nonlinear analytical method for a parabolic two-hinge arch under a concentrated force at mid-span as described in claim 1, characterized in that, The Heaviside function H(z) is: 。

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