Determination method of maximum radial bending moment of circular plate with variable thickness under non-uniform load

Abstract

The invention discloses a method for determining the maximum radial bending moment of a variable thickness circular plate under the action of a non-uniformly distributed load: a clamping device with an inner radius of a is used to fix and clamp the periphery of a variable thickness circular plate with a Young's elastic modulus of E, a Poisson's ratio of v, a thickness at a center point of h0, a thickness variation coefficient of α, and a thickness h that changes from the center along a radial coordinate r to the surroundings in a natural logarithmic function #imgabs0#, thereby forming a variable thickness circular plate structure with a radius of a and a fixed clamped periphery, and a load of q0 at a center point, a load variation coefficient of β, and a load that changes from the center along a radial coordinate r to the surroundings in a quadratic function q0(1+βr 2 / a 2 ) changes in the non-uniformly distributed load q, causing the variable thickness circular plate to produce axisymmetric deformation. Based on the static equilibrium analysis of the axisymmetric deformation problem of the variable thickness circular plate, the maximum radial bending moment M after the axisymmetric deformation of the variable thickness circular plate can be determined by using the measured value of the load q0 at the center point. rm .

Description

Technical Field

[0001] The invention relates to a method for determining the maximum radial bending moment of a circular plate with variable thickness under the action of a non-uniformly distributed load. Background Art

[0002] In the field of civil engineering, the foundation plates of special structures such as TV towers, oil storage tanks and chimneys are usually designed as variable thickness circular plates. Compared with equal thickness plates, variable thickness plates have the advantages of uniform force and material saving. However, according to the results of literature search, there are only analytical research results on the axisymmetric deformation of variable thickness circular plates with fixed clamps around the periphery under uniform loads, but no analytical research results on the axisymmetric deformation of variable thickness circular plates with fixed clamps around the periphery under non-uniform loads.

[0003] In actual engineering, the load borne by the structure is usually a non-uniform load, and a completely uniform load is difficult to achieve. In addition, for a variable thickness circular plate, its thickness varies radially, which means that its deadweight must also vary radially. Even if its external load is a uniform load, it will be converted into a non-uniform load after combining with its deadweight. Therefore, it is more in line with actual engineering conditions to study the axially symmetrical deformation problem of a variable thickness circular plate that is fixed and clamped at the periphery under a non-uniform load. However, due to the non-uniformity of the load and the variability of the plate thickness, the axially symmetrical deformation problem of a variable thickness circular plate that is fixed and clamped at the periphery under a non-uniform load has a strong nonlinearity, so in the process of analytical solution, it is inevitable to face nonlinear equations that are difficult to solve analytically, that is, considering the effect of non-uniform load will increase the difficulty of solving the analytical solution. In order to solve the above problems, the present invention provides the following technical solutions. Summary of the invention

[0004] The present invention is devoted to the analytical study of the axisymmetric deformation problem of a variable thickness circular plate with a peripheral fixed clamp under the action of a transverse non-uniform load. Based on the static equilibrium analysis, an analytical solution to the axisymmetric deformation problem is obtained, and on this basis, a method for determining the maximum radial bending moment of a variable thickness circular plate with a peripheral fixed clamp under the action of a transverse non-uniform load is given.

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

[0006] The method for determining the maximum radial bending moment of a circular plate with variable thickness under non-uniform load comprises the following steps:

[0007] For a Young's modulus of elasticity E, Poisson's ratio v, and a thickness at the center h 0 , the thickness variation coefficient is α and the thickness h is a natural logarithmic function from the center along the radial coordinate r to the surrounding The variable thickness circular plate is fixed and clamped at the periphery to form a variable thickness circular plate structure with a radius of a and fixed and clamped at the periphery.

[0008] The load q is applied laterally to the center point of the above variable thickness circular plate. 0 , the load variation coefficient is β and the load is a quadratic function q from the center along the radial coordinate r to the surrounding 0 (1+βr 2 / a 2 ) Varying non-uniform load q;

[0009] According to the equation Determine the maximum radial bending moment M of a circular plate with variable thickness after axisymmetric deformation rm ;

[0010] in:

[0011]

[0012]

[0013]

[0014]

[0015] Parameters a, h 0 The unit of parameter M is mm. rm The unit is N·mm, parameter q 0 The unit of , E is N / mm 2 , parameters α, β, v, A 1 , A 2 , A 3 , I are both dimensionless quantities.

[0016] Beneficial effects of the present invention:

[0017] In the technology of the present invention, by accurately measuring the lateral load acting on the center point of the variable thickness circular plate, the maximum radial bending moment M of the variable thickness circular plate after axial symmetrical deformation can be determined. rm It has good guiding significance for the axisymmetric deformation problem of a circular plate with variable thickness and fixed and clamped around the periphery under non-uniform load, and is more in line with actual engineering conditions. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] The present invention will be further described below in conjunction with the accompanying drawings.

[0019] Figure 1 Schematic diagram of the axisymmetric deformation of a variable thickness circular plate with peripheral clamping under a transverse non-uniformly distributed load.

[0020] Among them, 1 is the variable thickness circular plate before axisymmetric deformation; 2 is the clamping device of the outer edge of the variable thickness circular plate before axisymmetric deformation; 3 is the support for fixing the clamping device of the outer edge of the variable thickness circular plate; a represents the outer radius of the variable thickness circular plate before axisymmetric deformation and the inner radius of the clamping device of its outer edge; o is the coordinate origin (located at the centroid of the geometric mid-surface of the variable thickness circular plate before axisymmetric deformation); r is the radial coordinate (used to represent the distance from a point on the variable thickness circular plate before or after deformation to the symmetry axis of the variable thickness circular plate before or after deformation); w is the transverse coordinate (used to represent the deflection of the variable thickness circular plate after axisymmetric deformation); q 0 represents the lateral load acting on the center point of the variable thickness circular plate; h 0 Indicates the thickness at the center point of a circular plate with variable thickness. DETAILED DESCRIPTION

[0021] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0022] A method for determining the maximum radial bending moment of a circular plate with variable thickness under non-uniform load is as follows: Figure 1 As shown:

[0023] A clamping device with an inner radius a is used to clamp a 0 , the thickness variation coefficient is α and the thickness h is a natural logarithmic function from the center along the radial coordinate r to the surrounding The variable thickness circular plate is fixedly clamped at its periphery, thereby forming a variable thickness circular plate structure with a radius of a and fixedly clamped at its periphery;

[0024] The load q is applied laterally to the center point of the variable thickness circular plate. 0 , the load variation coefficient is β and the load is a quadratic function q from the center along the radial coordinate r to the surrounding 0 (1+βr 2 / a 2 ) The non-uniformly distributed load q changes, causing the variable thickness circular plate to produce axisymmetric deformation;

[0025] Based on the static equilibrium analysis of the axisymmetric deformation problem of the variable thickness circular plate, the load q at the center point can be obtained: 0 The maximum radial bending moment M after the axisymmetric deformation of the variable thickness circular plate rm The analytical relationship between them:

[0026]

[0027] in:

[0028]

[0029]

[0030]

[0031]

[0032] In this way, as long as the load q at the center point is accurately measured 0 The maximum radial bending moment M of the variable thickness circular plate after axisymmetric deformation can be calculated by rm Determine, among which, the parameters a, h 0 The unit of the parameter M is millimeter (mm). rm The unit is Newton·millimeter (N·mm), parameter q 0 The unit of E is Newton per square millimeter (N / mm 2 ), and the parameters α, β, v, A 1 , A 2 , A 3 , I are both dimensionless quantities.

[0033] In a specific embodiment,

[0034] A clamping device with an inner radius a = 500 mm is used to clamp a Young's modulus of elasticity E = 200000 N / mm 2 , Poisson's ratio v = 0.3, thickness at the center h 0 = 50mm, thickness variation coefficient α = 0.1, and thickness h is a natural logarithmic function from the center along the radial coordinate r to the surroundings The variable thickness circular plate is fixedly clamped at the periphery, thereby forming a variable thickness circular plate structure with a radius a=500mm and fixedly clamped at the periphery;

[0035] The load q is applied laterally to the center point of the variable thickness circular plate. 0 , load variation coefficient β = 0.2 and the load is a quadratic function q from the center along the radial coordinate r to the surroundings 0 (1+βr 2 / a 2 ) The non-uniformly distributed load q changes, causing the variable thickness circular plate to produce axisymmetric deformation, and the load q at the center point is measured. 0 =40N / mm 2 , using the method provided by the present invention, from the equation

[0036]

[0037]

[0038]

[0039]

[0040] Get an A 1 =0.432, A 2 =-0.161, A 3 =0.101, and then from the equation

[0041]

[0042] Determine the load q at the center of the variable thickness circular plate 0 =40N / mm 2 , load variation coefficient β = 0.2 and the load is a quadratic function q from the center along the radial coordinate r to the surroundings 0 (1+βr 2 / a 2 ) Maximum radial bending moment M under the action of non-uniform load q rm =768740.6N·mm.

[0043] The above contents are merely examples and explanations of the present invention. Those skilled in the art may make various modifications or additions to the specific embodiments described or replace them in a similar manner. As long as they do not deviate from the invention or exceed the scope defined by the claims, they shall all fall within the protection scope of the present invention.

Claims

1. A method for determining the maximum radial bending moment of a circular plate with variable thickness under non-uniform load, characterized in that: The steps include: For a Young's modulus of elasticity E, Poisson's ratio v, thickness at the center point h0, thickness variation coefficient α, and thickness h from the center along the radial coordinate r to the surrounding natural logarithmic function The variable thickness circular plate is fixed and clamped at the periphery to form a variable thickness circular plate structure with a radius of a and fixed and clamped at the periphery. The load at the center of the above variable thickness circular plate is q0, the load variation coefficient is β, and the load is a quadratic function q0(1+βr 2 / a 2 ) Varying non-uniform load q; According to the equation Determine the maximum radial bending moment M of a circular plate with variable thickness after axisymmetric deformation rm ; in: The units of parameters a and h0 are both mm. rm The unit of is N·mm, and the units of parameters q0 and E are both N / mm 2 , parameters α, β, v, A1, A2, A3, and I are all dimensionless quantities.

Images