CN103983380B - A kind of method determining uniformly distributed load lower prestress circular membrane maximum stress value - Google Patents

Abstract

The invention discloses a kind of method determining uniformly distributed load lower prestress circular membrane maximum stress value: be a by radius, thickness is h, Young's modulus of elasticity is E, Poisson ratio is ν, prestress is σ ₀circular membrane, fix to clamp in its border, and to its apply a horizontal uniformly distributed load q, so based on the statical equilibrium analysis of this circular membrane On Axisymmetric Deformation of A, utilize load measurement value q, then accurately can determine the maximum stress value σ of film _m.

Description

A kind of method determining uniformly distributed load lower prestress circular membrane maximum stress value

Technical field

The present invention relates to the defining method of the prestress circular membrane maximum stress value of Uniform Loads following peripheral clamping.

Background technology

Membrane structure is all widely used at many engineering fields, because film is very thin, thus larger amount of deflection can be presented under load, its problem on deformation belongs to GEOMETRICALLY NONLINEAR usually, the accurate Analysis of these nonlinear deformation problems solves, usually very difficult, in most cases only can provide numerical result.But, for many design problems such as such as diaphragm gauge, instrument etc., must design based on the Exact of these film problems, and numerical result is difficult to meet this requirement usually.

The loading structure of the prestress circular membrane of Uniform Loads following peripheral clamping, can be used for developing various sensor, and also have important application in the Mechanics Performance Testing of film/basic unit's system.In fact, after membrane structure making is shaping, due to factors such as temperature variation, humidity change, manufacture crafts, under most situation, there is certain stretching or compression primary stress (i.e. prestress) in film after making can be caused shaping, prestressed existence changes the mechanical behavior of film.But solve with prestressed circular membrane problem is not resolved at present, if but do not consider prestressed impact, inherently loss of significance is brought to problems such as instrument, instrument design and Mechanics Performance Testings.

Summary of the invention

The present invention, for the actual demand of engineer applied problem, the accurate Analysis being devoted to the prestress circular membrane problem of Uniform Loads following peripheral clamping solves, and give the defining method of film maximum stress value, BROAD SUMMARY is as follows:

As shown in Figure 1, be a by radius, thickness is h, Young's modulus of elasticity is E, Poisson ratio is ν, prestress is σ ₀circular membrane, fix to clamp in its border, and to its apply a horizontal uniformly distributed load q, so based on the statical equilibrium analysis of this circular membrane On Axisymmetric Deformation of A, easily obtain the maximum stress value σ of film _mwith the pass of load value q be:

${\mathrm{\σ}}_m={\left(\frac{a^{2}E}{2^5{h}^{2}c}\right)}^{1/3}{q}^{2/3},$

Wherein the value of c is by equation

${2\mathrm{cf}}^{\′}\left(c\right)-f\left(c\right)(v-1)+(v-1)\frac{2^{5/3}{h}^{2/3}{c}^{1/3}}{a^{2/3}{q}^{2/3}{E}^{1/3}}{\mathrm{\σ}}_0=0$

Determine, and f (c) and f ' (c) is respectively:

$\begin{array}{c}f\left(c\right)=1-\frac{1}{2}c-\frac{1}{6}{c}^2-\frac{13}{144}{c}^3-\frac{17}{288}{c}^4-\frac{37}{864}{c}^5-\frac{1205}{36288}{c}^6-\frac{219241}{8128512}{c}^7-\\ \frac{6634069}{292626432}{c}^8-\frac{51523763}{2633637888}{c}^9-\frac{998796305}{57940033536}{c}^{10}-\frac{118156790413}{7648084426752}{c}^{11}\end{array}$

With

$\begin{array}{c}{f}^{\′}\left(c\right)=-\frac{1}{2}-\frac{1}{3}c-\frac{13}{48}{c}^2-\frac{17}{72}{c}^3-\frac{185}{864}{c}^4-\frac{1205}{6048}{c}^5-\frac{219241}{1161216}{c}^6-\\ \frac{6634069}{35678304}{c}^7-\frac{51523763}{292626432}{c}^8-\frac{4993981525}{28970016768}{c}^9-\frac{118156790413}{695280402432}{c}^{10}\end{array}.$

Like this, as long as accurately measure applied horizontal uniformly distributed load value q, the maximum stress value σ of film can just accurately be determined _m.

Accompanying drawing explanation

Fig. 1 is the loading organigram of the prestress circular membrane of Uniform Loads following peripheral clamping, in FIG, 1-circular membrane, 2-clamp device, symbol is wherein, a represents the radius of circular membrane, r represents radial coordinate, w (r) represents the lateral coordinates at some r place, and q represents horizontal uniformly distributed load, w _mrepresent the maximum defluxion of circular membrane.

Embodiment

Below in conjunction with accompanying drawing, technical scheme of the present invention is described in further detail:

As shown in Figure 1, be a by radius, thickness is h, Young's modulus of elasticity is E, Poisson ratio is ν, prestress is σ ₀circular membrane, fix to clamp in its border, and to its apply a horizontal uniformly distributed load q, so based on the statical equilibrium analysis of this circular membrane On Axisymmetric Deformation of A, utilize load measurement value q, the maximum stress value σ of film _mcan be determined by following formula:

${\mathrm{\σ}}_m={\left(\frac{a^{2}E}{2^5{h}^{2}c}\right)}^{1/3}{q}^{2/3},$

Wherein the value of c is by equation

${2\mathrm{cf}}^{\′}\left(c\right)-f\left(c\right)(v-1)+(v-1)\frac{2^{5/3}{h}^{2/3}{c}^{1/3}}{a^{2/3}{q}^{2/3}{E}^{1/3}}{\mathrm{\σ}}_0=0$

Determine, and f (c) and f ' (c) is respectively:

$\begin{array}{c}f\left(c\right)=1-\frac{1}{2}c-\frac{1}{6}{c}^2-\frac{13}{144}{c}^3-\frac{17}{288}{c}^4-\frac{37}{864}{c}^5-\frac{1205}{36288}{c}^6-\frac{219241}{8128512}{c}^7-\\ \frac{6634069}{292626432}{c}^8-\frac{51523763}{2633637888}{c}^9-\frac{998796305}{57940033536}{c}^{10}-\frac{118156790413}{7648084426752}{c}^{11}\end{array}$

With

$\begin{array}{c}{f}^{\′}\left(c\right)=-\frac{1}{2}-\frac{1}{3}c-\frac{13}{48}{c}^2-\frac{17}{72}{c}^3-\frac{185}{864}{c}^4-\frac{1205}{6048}{c}^5-\frac{219241}{1161216}{c}^6-\\ \frac{6634069}{35678304}{c}^7-\frac{51523763}{292626432}{c}^8-\frac{4993981525}{28970016768}{c}^9-\frac{118156790413}{695280402432}{c}^{10}\end{array}.$

All parameters all adopt International System of Units.

Claims (1)

1. determine a method for uniformly distributed load lower prestress circular membrane maximum stress value, it is characterized in that: be a by radius, thickness is h, Young's modulus of elasticity is E, Poisson ratio is ν, prestress is σ ₀circular membrane, fix to clamp in its border, and to its apply a horizontal uniformly distributed load q, so based on the statical equilibrium analysis of this circular membrane On Axisymmetric Deformation of A, utilize load measurement value q, the maximum stress value σ of film _mcan be determined by following formula:

${\mathrm{\σ}}_m={\left(\frac{a^{2}E}{2^5{h}^{2}c}\right)}^{1/3}{q}^{2/3},$

Wherein the value of c is by equation

$2c{f}^{\′}\left(c\right)-f\left(c\right)(v-1)+(v-1)\frac{2^{5/3}{h}^{2/3}{c}^{1/3}}{a^{2/3}{q}^{2/3}{E}^{1/3}}{\mathrm{\σ}}_0=0$

Determine, and f (c) and f ' (c) is respectively:

$\begin{array}{c}f\left(c\right)=1-\frac{1}{2}c-\frac{1}{6}{c}^2-\frac{13}{144}{c}^3-\frac{17}{288}{c}^4-\frac{37}{864}{c}^5-\frac{1205}{36288}{c}^6-\frac{219241}{8128512}{c}^7-\\ \frac{6634069}{292626432}{c}^8-\frac{51523763}{2633637888}{c}^9-\frac{998796305}{57940033536}{c}^{10}-\frac{118156790413}{7648084426752}{c}^{11}\end{array}$

With

$\begin{array}{c}{f}^{\′}\left(c\right)=-\frac{1}{2}-\frac{1}{3}c-\frac{13}{48}{c}^2-\frac{17}{72}{c}^3-\frac{185}{864}{c}^4-\frac{1205}{6048}{c}^5-\frac{219241}{1161216}{c}^6-\\ \frac{6634069}{35678304}{c}^7-\frac{51523763}{292626432}{c}^8-\frac{4993981525}{28970016768}{c}^9-\frac{118156790413}{695280402432}{c}^{10}\end{array};$

All parameters all adopt International System of Units.