CN106469256A - The determination method of the prestress annular film maximum stress of uniform load lower band hard core - Google Patents

Abstract

The invention discloses the determination method of the prestress annular film maximum stress of uniform load lower band hard core：The prestressing force annular film configuration that the periphery carrying hard core to central area fixes to clamp is laterally applied to uniform load q, wherein the thickness of annular membrane be h, Young's modulus of elasticity be E, Poisson's ratio be ν, prestressing force be σ₀, outer radius be a, inside radius be b, the radius of the carried hard core in annular membrane central area is b, the standing balance analysis based on this On Axisymmetric Deformation of A, using the measured value of uniform load q, then can determine that this annular membrane deform after maximum stress σ_m.

Description

The determination method of the prestress annular film maximum stress of uniform load lower band hard core

Technical field

The present invention relates to the prestressing force annular film configuration that the periphery with hard core for the central area fixes to clamp is laterally equal The determination method of the thin film maximum stress under the effect of cloth load.

Background technology

The annular membrane construction that the periphery with hard core for the central area fixes to clamp is thin under horizontal Uniform Loads The analytic solutions of film On Axisymmetric Deformation of A, the development to sensor and instrument and meter is significant.However, from document Look into new result to see, the prestressing force annular film configuration that the periphery with hard core for the central area fixes to clamp is in horizontal uniform load The analytic solutions of the thin film On Axisymmetric Deformation of A under effect are not also presented, not pre- only be applied to annular membrane at present The analytic solutions of stress conditions, for example, the patent of invention (" annular of center band rigid plate under uniform load declared before applicant The determination method of thin film maximum stress ", number of patent application：201610263954.0) employed in analytic solutions it is simply that suppose Do not have in annular membrane prestressed under the conditions of obtain, thus it is only applicable to do not have prestressed situation.

Content of the invention

This invention address that the analysis research of thin film problem, Analytical Solution central area carries the periphery fixation clamp of hard core Tight prestressing force annular thin film On Axisymmetric Deformation of A under horizontal Uniform Loads for the film configuration, obtains this problem Analytic solutions, and on this basis, give the prestressing force annular that under horizontal uniform load, the periphery with hard core fixes to clamp thin Film maximum stress σ_mDetermination method.

The determination method of the prestress annular film maximum stress of uniform load lower band hard core：Central area is carried with the week of hard core While the prestressing force annular film configuration fixing to clamp is laterally applied to uniform load q, the wherein thickness of annular membrane is h, poplar Family name's elastic modelling quantity is E, Poisson's ratio is ν, prestressing force is σ₀, outer radius be a, inside radius be b, annular membrane central area is carried firmly The radius of core is also b, after the standing balance analysis based on this On Axisymmetric Deformation of A is it is possible to obtain the deformation of this annular membrane Maximum stress σ_mParsing relation with uniform load q being applied

Wherein,

β=(a+b)/2a,

And centre parameter c₁And c₂Value by equation

With

Determine, wherein,

So, as long as accurately measuring the value of applied uniform load q it is possible to determine this prestressing force annular membrane Maximum stress σ after deformation_m.Wherein, middle parameter c₁And c₂There is no unit, other all parameters all adopt the International System of Units.

Brief description

Centered on Fig. 1, region carries prestressing force annular film configuration and its loading signal that the periphery of hard core fixes to clamp Figure, wherein, 1- annular membrane, 2- hard core, 3- clamping device, and a represents the inside radius of clamping device and the outer of annular membrane Radius, b represents the radius of hard core and the inside radius of annular membrane, and r represents radial coordinate, and w (r) represents the lateral coordinates at point r, Q represents horizontal uniform load, w_mRepresent the maximum defluxion of thin film.

Specific embodiment

With reference to Fig. 1, technical scheme is described in further detail：

The prestressing force annular rubber film construction that the periphery carrying hard core to central area fixes to clamp is laterally applied to one Uniform load q, wherein, the thickness h=0.6mm of ring-shaped rubber thin film, Young's modulus of lasticity E=7.84MPa, Poisson's ratio ν= 0.47th, prestressing force σ₀=0.1MPa, outer radius a=20mm, inside radius b=10mm, and record q=0.005MPa.Using this Bright given method, by equation

With

C then can be obtained₁=-0.710280067, c₂=-0.6068478064, wherein,

β=(a+b)/2a,

Finally, by formula

The maximum stress σ after the deformation of this annular membrane then can be obtained_m=0.361163862Mpa, wherein,

Claims (1)

1. the prestress annular film maximum stress of uniform load lower band hard core determination method it is characterised in that：To central area band Prestressing force that the periphery having hard core fixes to clamp annular film configuration is laterally applied to uniform load q, wherein annular membrane Thickness is h, Young's modulus of elasticity is E, Poisson's ratio is ν, prestressing force is σ₀, outer radius be a, inside radius be b, annular membrane center The radius of the carried hard core in region is also b, records the value of applied uniform load q, determines that this annular membrane becomes by below equation Maximum stress σ after shape_m：

${\mathrm{\σ}}_m=-\frac{1}{2}{\left(\frac{{\mathrm{Ea}}^2{q}^2}{h^2}\right)}^{1/3}\underset{i=0}{\overset{10}{\Σ}}{d}_i{(\frac{b}{a}-\mathrm{\β})}^i,$

Wherein,

β=(a+b)/2a,

$d_0=\frac{\mathrm{\β}}{c_1},$

$d_1=\frac{1}{c_1^2}(-2{\mathrm{\βc}}_2+c_1),$

$d_2=\frac{1}{2{\mathrm{\β}}^2{c}_1^2}(6{\mathrm{\β}}^2{c}_2-3{\mathrm{\βc}}_1+c_1^4),$

$d_3=\frac{1}{6{\mathrm{\β}}^3{c}_1^2}(-24{\mathrm{\β}}^2{c}_2+4{\mathrm{\βc}}_1^3{c}_2+12{\mathrm{\βc}}_1-5c_1^4),$

$d_4=\frac{1}{24{\mathrm{\β}}^5{c}_1^2}(24{\mathrm{\β}}^3{c}_1^2{c}_2^2+120{\mathrm{\β}}^3{c}_2-48{\mathrm{\β}}^2{c}_1^3{c}_2-60{\mathrm{\β}}^2{c}_1+33{\mathrm{\βc}}_1^4-2c_1^7),$

$\begin{array}{c}{d}_5=\frac{1}{60{\mathrm{\β}}^6{c}_1^2}(96{\mathrm{\β}}^4{c}_1{c}_2^3-288{\mathrm{\β}}^3{c}_1^2{c}_2^2-360{\mathrm{\β}}^3{c}_2+180{\mathrm{\β}}^2{c}_1+294{\mathrm{\β}}^2{c}_1^3{c}_2-22{\mathrm{\βc}}_1^6{c}_2\\ -132{\mathrm{\βc}}_1^4+17c_1^7)\end{array},$

$\begin{array}{c}{d}_6=\frac{1}{360{\mathrm{\β}}^8{c}_1^2}(960{\mathrm{\β}}^6{c}_2^4-3936{\mathrm{\β}}^5{c}_1{c}_2^3+\mathrm{5940}{\mathrm{\β}}^4{c}_1^2{c}_2^2+2520{\mathrm{\β}}^4{c}_2-408{\mathrm{\β}}^3{c}_1^5{c}_2^2\\ -4038{\mathrm{\β}}^3{c}_1^3{c}_2-1260{\mathrm{\β}}^3{c}_1+678{\mathrm{\β}}^2{c}_1^6{c}_2+1281{\mathrm{\β}}^2{c}_1^4-282{\mathrm{\βc}}_1^7+11c_1^{10})\end{array},$

$\begin{array}{c}{d}_7=\frac{1}{2520{\mathrm{\β}}^9{c}_1^3}(11520{\mathrm{\β}}^7{c}_2^5-60480{\mathrm{\β}}^9{c}_1{c}_2^4+\mathrm{124128}{\mathrm{\β}}^5{c}_1^2{c}_2^3-7632{\mathrm{\β}}^4{c}_1^5{c}_2^3\\ -125100{\mathrm{\β}}^4{c}_1^3{c}_2^2-20160{\mathrm{\β}}^4{c}_1{c}_2+\mathrm{20160}{\mathrm{\β}}^3{c}_1^6{c}_2^2+63324{\mathrm{\β}}^3{c}_1^4{c}_2+10080{\mathrm{\β}}^3{c}_1^2\\ -17442{\mathrm{\β}}^2{c}_1^7{c}_2-15003{\mathrm{\β}}^2{c}_1^5+584{\mathrm{\βc}}_1^{10}{c}_2+5010{\mathrm{\βc}}_1^8-427c_1^{11})\end{array},$

$\begin{array}{c}{d}_8=\frac{1}{20160{\mathrm{\β}}^{11}{c}_1^4}(161280{\mathrm{\β}}^9{c}_2^6-1036800{\mathrm{\β}}^8{c}_1{c}_2^5+2718720{\mathrm{\β}}^7{c}_1^2{c}_2^4-150912{\mathrm{\β}}^6{c}_1^5{c}_2^4\\ -3722112{\mathrm{\β}}^6{c}_1^3{c}_2^3+555840{\mathrm{\β}}^5{c}_1^6{c}_2^3+\mathrm{2815020}{\mathrm{\β}}^5{c}_1^4{c}_2^2+181440{\mathrm{\β}}^5{c}_1^2{c}_2\\ -752400{\mathrm{\β}}^4{c}_1^7{c}_2^2-1131876{\mathrm{\β}}^4{c}_1^5{c}_2-90720{\mathrm{\β}}^4{c}_1^3+21960{\mathrm{\β}}^3{c}_1^{10}{c}_2^2+444984{\mathrm{\β}}^3{c}_1^8{c}_2\\ +210087{\mathrm{\β}}^3{c}_1^6-34584{\mathrm{\β}}^2{c}_1^{11}{c}_2-97905{\mathrm{\β}}^2{c}_1^9+13440{\mathrm{\βc}}_1^{12}-292c_1^{15})\end{array},$

$\begin{array}{c}{d}_9=\frac{1}{181440{\mathrm{\β}}^{12}{c}_1^5}(2580480{\mathrm{\β}}^{10}{c}_2^7-19676160{\mathrm{\β}}^9{c}_1{c}_2^6+63106560{\mathrm{\β}}^8{c}_1^2{c}_2^5\\ -3200256{\mathrm{\β}}^7{c}_1^5{c}_2^5-110246400{\mathrm{\β}}^7{c}_1^3{c}_2^4+\mathrm{15258240}{\mathrm{\β}}^6{c}_1^6{c}_2^4+113274720{\mathrm{\β}}^6{c}_1^4{c}_2^3\\ -28549152{\mathrm{\β}}^5{c}_1^7{c}_2^3-68581080{\mathrm{\β}}^5{c}_1^5{c}_2^2-1814400{\mathrm{\β}}^5{c}_1^3{c}_2+741024{\mathrm{\β}}^4{c}_1^{10}{c}_2^3\\ +26215056{\mathrm{\β}}^4{c}_1^8{c}_2^2+22867921{\mathrm{\β}}^4{c}_1^6{c}_2+907200{\mathrm{\β}}^4{c}_1^4-1851552{\mathrm{\β}}^3{c}_1^{11}{c}_2^2\\ -11836062{\mathrm{\β}}^3{c}_1^9{c}_2-3469230{\mathrm{\β}}^3{c}_1^7+1517496{\mathrm{\β}}^2{c}_1^{12}{c}_2+2114559{\mathrm{\β}}^2{c}_1^{10}\\ -28384{\mathrm{\βc}}_1^{15}{c}_2-408768{\mathrm{\βc}}_1^{13}+20504c_1^{16})\end{array},$

$\begin{array}{c}{d}_{10}=\frac{1}{1814400{\mathrm{\β}}^{14}{c}_1^6}(46448640{\mathrm{\β}}^{12}{c}_2^8-410296320{\mathrm{\β}}^{11}{c}_1{c}_2^7+1559900160{\mathrm{\β}}^{10}{c}_1^2{c}_2^6\\ -73036800{\mathrm{\β}}^9{c}_1^5{c}_2^6-3330132480{\mathrm{\β}}^9{c}_1^3{c}_2^5+429684480{\mathrm{\β}}^8{c}_1^6{c}_2^5+4362206400{\mathrm{\β}}^8{c}_1^4{c}_2^4\\ -1035268992{\mathrm{\β}}^7{c}_1^7{c}_2^4-3588896160{\mathrm{\β}}^7{c}_1^5{c}_2^3+24303168{\mathrm{\β}}^6{c}_1^{10}{c}_2^4+\mathrm{19958400}{\mathrm{\β}}^6{c}_1^4{c}_2^4\\ +1307134368{\mathrm{\β}}^6{c}_1^8{c}_2^3+1812822120{\mathrm{\β}}^6{c}_1^6{c}_2^2-84591072{\mathrm{\β}}^5{c}_1^{11}{c}_2^3-912287016{\mathrm{\β}}^5{c}_1^9{c}_2^2\\ -516903120{\mathrm{\β}}^5{c}_1^7{c}_2-9979200{\mathrm{\β}}^5{c}_1^5+108652752{\mathrm{\β}}^4{c}_1^{12}{c}_2^2+334076706{\mathrm{\β}}^4{c}_1^{10}{c}_2\\ +66432150{\mathrm{\β}}^4{c}_1^8-1792752{\mathrm{\β}}^3{c}_1^{15}{c}_2^2-61072440{\mathrm{\β}}^3{c}_1^{13}{c}_2-50333643{\mathrm{\β}}^3{c}_1^{11}\\ +2756784{\mathrm{\β}}^2{c}_1^{16}{c}_2+12690723{\mathrm{\β}}^2{c}_1^{14}-1047372{\mathrm{\βc}}_1^{17}+14192c_1^{20})\end{array},$

And centre parameter c₁And c₂Value by equation

$\underset{i=2}{\overset{10}{\mathrm{\Σ}}}i(i-1)c_i{(1-\mathrm{\β})}^{i-2}-(2-v)\underset{i=2}{\overset{10}{\mathrm{\Σ}}}{\mathrm{ic}}_i{(1-\mathrm{\β})}^{i-1}=\frac{2h^{2/3}(1-v)}{q^{2/3}{a}^{2/3}{E}^{1/3}}{\mathrm{\σ}}_0{\[\underset{i=1}{\overset{10}{\mathrm{\Σ}}}{\mathrm{ic}}_i{(1-\mathrm{\β})}^{i-1}\]}^2$

With

${\left(\frac{b}{a}\right)}^2\underset{i=2}{\overset{10}{\mathrm{\Σ}}}i(i-1)c_i{(\frac{b}{a}-\mathrm{\β})}^{i-2}-\frac{b}{a}(2-v)\underset{i=1}{\overset{10}{\mathrm{\Σ}}}{\mathrm{ic}}_i{(\frac{b}{a}-\mathrm{\β})}^{i-1}=\frac{2h^{2/3}(1-v)}{q^{2/3}{a}^{2/3}{E}^{1/3}}{\mathrm{\σ}}_0{\[\underset{i=1}{\overset{10}{\mathrm{\Σ}}}{\mathrm{ic}}_i{(\frac{b}{a}-\mathrm{\β})}^{i-1}\]}^2$

Determine, wherein,

$c_3=\frac{1}{6}\frac{1}{{\mathrm{\β}}^3{c}_1}(8{\mathrm{\β}}^3{c}_2^2-10{\mathrm{\β}}^2{c}_1{c}_2+3{\mathrm{\βc}}_1^2-c_1^5),$

$c_4=\frac{1}{24}\frac{1}{{\mathrm{\β}}^4{c}_1^2}(48{\mathrm{\β}}^4{c}_2^3-120{\mathrm{\β}}^3{c}_1{c}_2^2+90{\mathrm{\β}}^2{c}_1^2{c}_2-16{\mathrm{\βc}}_1^5{c}_2-21{\mathrm{\βc}}_1^3+8c_1^6),$

$\begin{array}{c}{c}_5=\frac{1}{120}\frac{1}{{\mathrm{\β}}^6{c}_1^3}(384{\mathrm{\β}}^6{c}_2^4-1440{\mathrm{\β}}^5{c}_1{c}_2^3+1896{\mathrm{\β}}^4{c}_1^2{c}_2^2-232{\mathrm{\β}}^3{c}_1^5{c}_2^2-1032{\mathrm{\β}}^3{c}_1^3{c}_2\\ +312{\mathrm{\β}}^2{c}_1^6{c}_2+198{\mathrm{\β}}^2{c}_1^4-101{\mathrm{\βc}}_1^7+8c_1^{10})\end{array},$

$\begin{array}{c}{c}_6=\frac{1}{720}\frac{1}{{\mathrm{\β}}^7{c}_1^4}(3840{\mathrm{\β}}^7{c}_2^5-19200{\mathrm{\β}}^6{c}_1{c}_2^4+36720{\mathrm{\β}}^5{c}_1^2{c}_2^3-3552{\mathrm{\β}}^4{c}_1^5{c}_2^3\\ -33360{\mathrm{\β}}^4{c}_1^3{c}_2^2+8296{\mathrm{\β}}^3{c}_1^6{c}_2^2+14340{\mathrm{\β}}^3{c}_1^4{c}_2-6408{\mathrm{\β}}^2{c}_1^7{c}_2\\ -2340{\mathrm{\β}}^2{c}_1^5+344{\mathrm{\βc}}_1^{10}{c}_2+1489{\mathrm{\βc}}_1^8-204c_1^{11})\end{array},$

$\begin{array}{c}{c}_7=\frac{1}{2510}\frac{1}{{\mathrm{\β}}^9{c}_1^5}(23040{\mathrm{\β}}^9{c}_2^6-144000{\mathrm{\β}}^8{c}_1{c}_2^5+362880{\mathrm{\β}}^7{c}_1^2{c}_2^4-29664{\mathrm{\β}}^6{c}_1^5{c}_2^4\\ -469800{\mathrm{\β}}^9{c}_1^3{c}_2^3+100944{\mathrm{\β}}^5{c}_1^6{c}_2^3+328140{\mathrm{\β}}^5{c}_1^4{c}_2^2-124644{\mathrm{\β}}^4{c}_1^7{c}_2^2\\ -116810{\mathrm{\β}}^4{c}_1^5{c}_2+5416{\mathrm{\β}}^3{c}_1^{10}{c}_2^2+66120{\mathrm{\β}}^3{c}_1^8{c}_2+16605{\mathrm{\β}}^3{c}_1^6\\ -7468{\mathrm{\β}}^2{c}_1^{11}{c}_2-12708{\mathrm{\β}}^2{c}_1^9+2524{\mathrm{\βc}}_1^{12}-86c_1^{15})\end{array},$

$\begin{array}{c}{c}_8=\frac{1}{20160}\frac{1}{{\mathrm{\β}}^{10}{c}_1^6}(322560{\mathrm{\β}}^{10}{c}_2^7-2419200{\mathrm{\β}}^9{c}_1{c}_2^6+7580160{\mathrm{\β}}^8{c}_1^2{c}_2^5-543744{\mathrm{\β}}^7{c}_1^5{c}_2^5\\ -12821760{\mathrm{\β}}^7{c}_1^3{c}_2^4+2455488{\mathrm{\β}}^6{c}_1^6{c}_2^4+12602520{\mathrm{\β}}^6{c}_1^4{c}_2^3-4317408{\mathrm{\β}}^5{c}_1^7{c}_2^3\\ -7175700{\mathrm{\β}}^5{c}_1^5{c}_2^2+156816{\mathrm{\β}}^4{c}_1^{10}{c}_2^3+3689808{\mathrm{\β}}^4{c}_1^8{c}_2^2+2186730{\mathrm{\β}}^4{c}_1^6{c}_2\\ -357560{\mathrm{\β}}^3{c}_1^{11}{c}_2^2-1531056{\mathrm{\β}}^3{c}_3^9{c}_2-274995{\mathrm{\β}}^3{c}_1^7+265844{\mathrm{\β}}^2{c}_1^{12}{c}_2\\ +246591{\mathrm{\β}}^2{c}_1^{10}-7136{\mathrm{\βc}}_1^{15}{c}_2-64454{\mathrm{\βc}}_1^{13}+4508c_1^{16})\end{array},$

$\begin{array}{c}{c}_9=\frac{1}{181440}\frac{1}{{\mathrm{\β}}^{12}{c}_1^7}(5160960{\mathrm{\β}}^{12}{c}_2^8-45158400{\mathrm{\β}}^{11}{c}_1{c}_2^7+169344000{\mathrm{\β}}^{10}{c}_1^2{c}_2^6\\ -10916352{\mathrm{\β}}^9{c}_1^5{c}_2^6-354654720{\mathrm{\β}}^9{c}_1^3{c}_2^5+61770240{\mathrm{\β}}^8{c}_1^6{c}_2^5+452571840{\mathrm{\β}}^8{c}_1^4{c}_2^4\\ -142403904{\mathrm{\β}}^7{c}_1^7{c}_2^4-359432640{\mathrm{\β}}^7{c}_1^5{c}_2^3+4495680{\mathrm{\β}}^6{c}_1^{10}{c}_2^4+171002304{\mathrm{\β}}^6{c}_1^8{c}_2^3\\ +173093760{\mathrm{\β}}^6{c}_1^6{c}_2^2-14640192{\mathrm{\β}}^5{c}_1^{11}{c}_2^3-112679604{\mathrm{\β}}^5{c}_1^9{c}_2^2-46131120{\mathrm{\β}}^5{c}_1^7{c}_2\\ +17513808{\mathrm{\β}}^4{c}_1^{12}{c}_2^2+38590524{\mathrm{\β}}^4{c}_1^{10}{c}_2+5205060{\mathrm{\β}}^4{c}_1^8-392216{\mathrm{\β}}^3{c}_1^{15}{c}_2^2\\ -9120816{\mathrm{\β}}^3{c}_1^{13}{c}_2-5362677{\mathrm{\β}}^3{c}_1^{11}+547624{\mathrm{\β}}^2{c}_1^{16}{c}_2+1744380{\mathrm{\β}}^2{c}_1^{14}\\ -188706{\mathrm{\βc}}_1^{17}+3568c_1^{20})\end{array},$

$\begin{array}{c}{c}_{10}=\frac{1}{181440}\frac{1}{{\mathrm{\β}}^{13}{c}_1^8}(92897280{\mathrm{\β}}^{13}{c}_2^9-92897280{\mathrm{\β}}^{12}{c}_1{c}_2^8+4058449920{\mathrm{\β}}^{11}{c}_1^2{c}_2^7\\ -238970880{\mathrm{\β}}^{10}{c}_1^5{c}_2^7-10149027840{\mathrm{\β}}^{10}{c}_1^3{c}_2^6+1630158336{\mathrm{\β}}^9{c}_1^6{c}_2^6\\ +15980146560{\mathrm{\β}}^9{c}_1^4{c}_2^5-4676078592{\mathrm{\β}}^8{c}_1^7{c}_2^5-16397095680{\mathrm{\β}}^8{c}_1^5{c}_2^4\\ +131654016{\mathrm{\β}}^7{c}_1^{10}{c}_2^5+7304043456{\mathrm{\β}}^7{c}_1^8{c}_2^4+109426400{\mathrm{\β}}^7{c}_1^6{c}_2^3\\ -564117120{\mathrm{\β}}^6{c}_1^{11}{c}_2^4-6702770736{\mathrm{\β}}^6{c}_1^9{c}_2^3-4571471520{\mathrm{\β}}^6{c}_1^7{c}_2^2\\ +949114656{\mathrm{\β}}^5{c}_1^{12}{c}_2^3+3610117836{\mathrm{\β}}^5{c}_1^{10}{c}_2^2+1083264840{\mathrm{\β}}^5{c}_1^8{c}_2\\ -18404544{\mathrm{\β}}^4{c}_1^{15}{c}_2^3-783565920{\mathrm{\β}}^4{c}_1^{13}{c}_2^2-1055710800{\mathrm{\β}}^4{c}_1^{11}{c}_2\\ -110837160{\mathrm{\β}}^4{c}_1^9+41460120{\mathrm{\β}}^3{c}_1^{16}{c}_2^2+317333772{\mathrm{\β}}^3{c}_1^{14}{c}_2\\ +129215439{\mathrm{\β}}^3{c}_1^{12}-30678936{\mathrm{\β}}^2{c}_1^{17}{c}_2-50420286{\mathrm{\β}}^2{c}_1^{15}\\ +484984c_1^{20}{c}_2+7457610{\mathrm{\βc}}_1^{18}-316628c_1^{21})\end{array},$

Middle parameter c₁And c₂There is no unit, other all parameters all adopt the International System of Units.