均布载荷下带硬芯的预应力环膜最大应力的确定方法

摘要

本发明公开了均布载荷下带硬芯的预应力环膜最大应力的确定方法：对中心区域带有硬芯的周边固定夹紧的预应力环形薄膜构造横向施加一个均布载荷q，其中环形薄膜的厚度为h、杨氏弹性模量为E、泊松比为ν、预应力为σ₀、外半径为a、内半径为b，环形薄膜中心区域所带硬芯的半径为b，基于这个轴对称变形问题的静力平衡分析，利用均布载荷q的测量值，则可确定出该环形薄膜变形后的最大应力σ_m。

主权项

均布载荷下带硬芯的预应力环膜最大应力的确定方法，其特征在于：对中心区域带有硬芯的周边固定夹紧的预应力环形薄膜构造横向施加一个均布载荷q，其中环形薄膜的厚度为h、杨氏弹性模量为E、泊松比为ν、预应力为σ₀、外半径为a、内半径为b，环形薄膜中心区域所带硬芯的半径也为b，测得所施加的均布载荷q的值，由以下公式确定该环形薄膜变形后的最大应力σ_m：${\sigma }_m=-\frac{1}{2}{\left(\frac{{\mathrm{Ea}}^2{q}^2}{h^2}\right)}^{1/3}\underset{i=0}{\overset{10}{\Sigma }}{d}_i{(\frac{b}{a}-\beta )}^i,$其中，β＝(a+b)/2a，$d_0=\frac{\beta }{c_1},$$d_1=\frac{1}{c_1^2}(-2{\mathrm{\beta c}}_2+c_1),$$d_2=\frac{1}{2{\beta }^2{c}_1^2}(6{\beta }^2{c}_2-3{\mathrm{\beta c}}_1+c_1^4),$$d_3=\frac{1}{6{\beta }^3{c}_1^2}(-24{\beta }^2{c}_2+4{\mathrm{\beta c}}_1^3{c}_2+12{\mathrm{\beta c}}_1-5c_1^4),$$d_4=\frac{1}{24{\beta }^5{c}_1^2}(24{\beta }^3{c}_1^2{c}_2^2+120{\beta }^3{c}_2-48{\beta }^2{c}_1^3{c}_2-60{\beta }^2{c}_1+33{\mathrm{\beta c}}_1^4-2c_1^7),$$\begin{array}{c}{d}_5=\frac{1}{60{\beta }^6{c}_1^2}(96{\beta }^4{c}_1{c}_2^3-288{\beta }^3{c}_1^2{c}_2^2-360{\beta }^3{c}_2+180{\beta }^2{c}_1+294{\beta }^2{c}_1^3{c}_2-22{\mathrm{\beta c}}_1^6{c}_2\\ -132{\mathrm{\beta c}}_1^4+17c_1^7)\end{array},$$\begin{array}{c}{d}_6=\frac{1}{360{\beta }^8{c}_1^2}(960{\beta }^6{c}_2^4-3936{\beta }^5{c}_1{c}_2^3+\mathrm{5940}{\beta }^4{c}_1^2{c}_2^2+2520{\beta }^4{c}_2-408{\beta }^3{c}_1^5{c}_2^2\\ -4038{\beta }^3{c}_1^3{c}_2-1260{\beta }^3{c}_1+678{\beta }^2{c}_1^6{c}_2+1281{\beta }^2{c}_1^4-282{\mathrm{\beta c}}_1^7+11c_1^{10})\end{array},$$\begin{array}{c}{d}_7=\frac{1}{2520{\beta }^9{c}_1^3}(11520{\beta }^7{c}_2^5-60480{\beta }^9{c}_1{c}_2^4+\mathrm{124128}{\beta }^5{c}_1^2{c}_2^3-7632{\beta }^4{c}_1^5{c}_2^3\\ -125100{\beta }^4{c}_1^3{c}_2^2-20160{\beta }^4{c}_1{c}_2+\mathrm{20160}{\beta }^3{c}_1^6{c}_2^2+63324{\beta }^3{c}_1^4{c}_2+10080{\beta }^3{c}_1^2\\ -17442{\beta }^2{c}_1^7{c}_2-15003{\beta }^2{c}_1^5+584{\mathrm{\beta c}}_1^{10}{c}_2+5010{\mathrm{\beta c}}_1^8-427c_1^{11})\end{array},$$\begin{array}{c}{d}_8=\frac{1}{20160{\beta }^{11}{c}_1^4}(161280{\beta }^9{c}_2^6-1036800{\beta }^8{c}_1{c}_2^5+2718720{\beta }^7{c}_1^2{c}_2^4-150912{\beta }^6{c}_1^5{c}_2^4\\ -3722112{\beta }^6{c}_1^3{c}_2^3+555840{\beta }^5{c}_1^6{c}_2^3+\mathrm{2815020}{\beta }^5{c}_1^4{c}_2^2+181440{\beta }^5{c}_1^2{c}_2\\ -752400{\beta }^4{c}_1^7{c}_2^2-1131876{\beta }^4{c}_1^5{c}_2-90720{\beta }^4{c}_1^3+21960{\beta }^3{c}_1^{10}{c}_2^2+444984{\beta }^3{c}_1^8{c}_2\\ +210087{\beta }^3{c}_1^6-34584{\beta }^2{c}_1^{11}{c}_2-97905{\beta }^2{c}_1^9+13440{\mathrm{\beta c}}_1^{12}-292c_1^{15})\end{array},$$\begin{array}{c}{d}_9=\frac{1}{181440{\beta }^{12}{c}_1^5}(2580480{\beta }^{10}{c}_2^7-19676160{\beta }^9{c}_1{c}_2^6+63106560{\beta }^8{c}_1^2{c}_2^5\\ -3200256{\beta }^7{c}_1^5{c}_2^5-110246400{\beta }^7{c}_1^3{c}_2^4+\mathrm{15258240}{\beta }^6{c}_1^6{c}_2^4+113274720{\beta }^6{c}_1^4{c}_2^3\\ -28549152{\beta }^5{c}_1^7{c}_2^3-68581080{\beta }^5{c}_1^5{c}_2^2-1814400{\beta }^5{c}_1^3{c}_2+741024{\beta }^4{c}_1^{10}{c}_2^3\\ +26215056{\beta }^4{c}_1^8{c}_2^2+22867921{\beta }^4{c}_1^6{c}_2+907200{\beta }^4{c}_1^4-1851552{\beta }^3{c}_1^{11}{c}_2^2\\ -11836062{\beta }^3{c}_1^9{c}_2-3469230{\beta }^3{c}_1^7+1517496{\beta }^2{c}_1^{12}{c}_2+2114559{\beta }^2{c}_1^{10}\\ -28384{\mathrm{\beta c}}_1^{15}{c}_2-408768{\mathrm{\beta c}}_1^{13}+20504c_1^{16})\end{array},$$\begin{array}{c}{d}_{10}=\frac{1}{1814400{\beta }^{14}{c}_1^6}(46448640{\beta }^{12}{c}_2^8-410296320{\beta }^{11}{c}_1{c}_2^7+1559900160{\beta }^{10}{c}_1^2{c}_2^6\\ -73036800{\beta }^9{c}_1^5{c}_2^6-3330132480{\beta }^9{c}_1^3{c}_2^5+429684480{\beta }^8{c}_1^6{c}_2^5+4362206400{\beta }^8{c}_1^4{c}_2^4\\ -1035268992{\beta }^7{c}_1^7{c}_2^4-3588896160{\beta }^7{c}_1^5{c}_2^3+24303168{\beta }^6{c}_1^{10}{c}_2^4+\mathrm{19958400}{\beta }^6{c}_1^4{c}_2^4\\ +1307134368{\beta }^6{c}_1^8{c}_2^3+1812822120{\beta }^6{c}_1^6{c}_2^2-84591072{\beta }^5{c}_1^{11}{c}_2^3-912287016{\beta }^5{c}_1^9{c}_2^2\\ -516903120{\beta }^5{c}_1^7{c}_2-9979200{\beta }^5{c}_1^5+108652752{\beta }^4{c}_1^{12}{c}_2^2+334076706{\beta }^4{c}_1^{10}{c}_2\\ +66432150{\beta }^4{c}_1^8-1792752{\beta }^3{c}_1^{15}{c}_2^2-61072440{\beta }^3{c}_1^{13}{c}_2-50333643{\beta }^3{c}_1^{11}\\ +2756784{\beta }^2{c}_1^{16}{c}_2+12690723{\beta }^2{c}_1^{14}-1047372{\mathrm{\beta c}}_1^{17}+14192c_1^{20})\end{array},$而中间参量c₁和c₂的值由方程$\underset{i=2}{\overset{10}{\Sigma }}i(i-1)c_i{(1-\beta )}^{i-2}-(2-v)\underset{i=2}{\overset{10}{\Sigma }}{\mathrm{ic}}_i{(1-\beta )}^{i-1}=\frac{2h^{2/3}(1-v)}{q^{2/3}{a}^{2/3}{E}^{1/3}}{\sigma }_0{\left[\underset{i=1}{\overset{10}{\Sigma }}{\mathrm{ic}}_i{(1-\beta )}^{i-1}\right]}^2$和${\left(\frac{b}{a}\right)}^2\underset{i=2}{\overset{10}{\Sigma }}i(i-1)c_i{(\frac{b}{a}-\beta )}^{i-2}-\frac{b}{a}(2-v)\underset{i=1}{\overset{10}{\Sigma }}{\mathrm{ic}}_i{(\frac{b}{a}-\beta )}^{i-1}=\frac{2h^{2/3}(1-v)}{q^{2/3}{a}^{2/3}{E}^{1/3}}{\sigma }_0{\left[\underset{i=1}{\overset{10}{\Sigma }}{\mathrm{ic}}_i{(\frac{b}{a}-\beta )}^{i-1}\right]}^2$确定，其中，$c_3=\frac{1}{6}\frac{1}{{\beta }^3{c}_1}(8{\beta }^3{c}_2^2-10{\beta }^2{c}_1{c}_2+3{\mathrm{\beta c}}_1^2-c_1^5),$$c_4=\frac{1}{24}\frac{1}{{\beta }^4{c}_1^2}(48{\beta }^4{c}_2^3-120{\beta }^3{c}_1{c}_2^2+90{\beta }^2{c}_1^2{c}_2-16{\mathrm{\beta c}}_1^5{c}_2-21{\mathrm{\beta c}}_1^3+8c_1^6),$$\begin{array}{c}{c}_5=\frac{1}{120}\frac{1}{{\beta }^6{c}_1^3}(384{\beta }^6{c}_2^4-1440{\beta }^5{c}_1{c}_2^3+1896{\beta }^4{c}_1^2{c}_2^2-232{\beta }^3{c}_1^5{c}_2^2-1032{\beta }^3{c}_1^3{c}_2\\ +312{\beta }^2{c}_1^6{c}_2+198{\beta }^2{c}_1^4-101{\mathrm{\beta c}}_1^7+8c_1^{10})\end{array},$$\begin{array}{c}{c}_6=\frac{1}{720}\frac{1}{{\beta }^7{c}_1^4}(3840{\beta }^7{c}_2^5-19200{\beta }^6{c}_1{c}_2^4+36720{\beta }^5{c}_1^2{c}_2^3-3552{\beta }^4{c}_1^5{c}_2^3\\ -33360{\beta }^4{c}_1^3{c}_2^2+8296{\beta }^3{c}_1^6{c}_2^2+14340{\beta }^3{c}_1^4{c}_2-6408{\beta }^2{c}_1^7{c}_2\\ -2340{\beta }^2{c}_1^5+344{\mathrm{\beta c}}_1^{10}{c}_2+1489{\mathrm{\beta c}}_1^8-204c_1^{11})\end{array},$$\begin{array}{c}{c}_7=\frac{1}{2510}\frac{1}{{\beta }^9{c}_1^5}(23040{\beta }^9{c}_2^6-144000{\beta }^8{c}_1{c}_2^5+362880{\beta }^7{c}_1^2{c}_2^4-29664{\beta }^6{c}_1^5{c}_2^4\\ -469800{\beta }^9{c}_1^3{c}_2^3+100944{\beta }^5{c}_1^6{c}_2^3+328140{\beta }^5{c}_1^4{c}_2^2-124644{\beta }^4{c}_1^7{c}_2^2\\ -116810{\beta }^4{c}_1^5{c}_2+5416{\beta }^3{c}_1^{10}{c}_2^2+66120{\beta }^3{c}_1^8{c}_2+16605{\beta }^3{c}_1^6\\ -7468{\beta }^2{c}_1^{11}{c}_2-12708{\beta }^2{c}_1^9+2524{\mathrm{\beta c}}_1^{12}-86c_1^{15})\end{array},$$\begin{array}{c}{c}_8=\frac{1}{20160}\frac{1}{{\beta }^{10}{c}_1^6}(322560{\beta }^{10}{c}_2^7-2419200{\beta }^9{c}_1{c}_2^6+7580160{\beta }^8{c}_1^2{c}_2^5-543744{\beta }^7{c}_1^5{c}_2^5\\ -12821760{\beta }^7{c}_1^3{c}_2^4+2455488{\beta }^6{c}_1^6{c}_2^4+12602520{\beta }^6{c}_1^4{c}_2^3-4317408{\beta }^5{c}_1^7{c}_2^3\\ -7175700{\beta }^5{c}_1^5{c}_2^2+156816{\beta }^4{c}_1^{10}{c}_2^3+3689808{\beta }^4{c}_1^8{c}_2^2+2186730{\beta }^4{c}_1^6{c}_2\\ -357560{\beta }^3{c}_1^{11}{c}_2^2-1531056{\beta }^3{c}_3^9{c}_2-274995{\beta }^3{c}_1^7+265844{\beta }^2{c}_1^{12}{c}_2\\ +246591{\beta }^2{c}_1^{10}-7136{\mathrm{\beta c}}_1^{15}{c}_2-64454{\mathrm{\beta c}}_1^{13}+4508c_1^{16})\end{array},$$\begin{array}{c}{c}_9=\frac{1}{181440}\frac{1}{{\beta }^{12}{c}_1^7}(5160960{\beta }^{12}{c}_2^8-45158400{\beta }^{11}{c}_1{c}_2^7+169344000{\beta }^{10}{c}_1^2{c}_2^6\\ -10916352{\beta }^9{c}_1^5{c}_2^6-354654720{\beta }^9{c}_1^3{c}_2^5+61770240{\beta }^8{c}_1^6{c}_2^5+452571840{\beta }^8{c}_1^4{c}_2^4\\ -142403904{\beta }^7{c}_1^7{c}_2^4-359432640{\beta }^7{c}_1^5{c}_2^3+4495680{\beta }^6{c}_1^{10}{c}_2^4+171002304{\beta }^6{c}_1^8{c}_2^3\\ +173093760{\beta }^6{c}_1^6{c}_2^2-14640192{\beta }^5{c}_1^{11}{c}_2^3-112679604{\beta }^5{c}_1^9{c}_2^2-46131120{\beta }^5{c}_1^7{c}_2\\ +17513808{\beta }^4{c}_1^{12}{c}_2^2+38590524{\beta }^4{c}_1^{10}{c}_2+5205060{\beta }^4{c}_1^8-392216{\beta }^3{c}_1^{15}{c}_2^2\\ -9120816{\beta }^3{c}_1^{13}{c}_2-5362677{\beta }^3{c}_1^{11}+547624{\beta }^2{c}_1^{16}{c}_2+1744380{\beta }^2{c}_1^{14}\\ -188706{\mathrm{\beta c}}_1^{17}+3568c_1^{20})\end{array},$$\begin{array}{c}{c}_{10}=\frac{1}{181440}\frac{1}{{\beta }^{13}{c}_1^8}(92897280{\beta }^{13}{c}_2^9-92897280{\beta }^{12}{c}_1{c}_2^8+4058449920{\beta }^{11}{c}_1^2{c}_2^7\\ -238970880{\beta }^{10}{c}_1^5{c}_2^7-10149027840{\beta }^{10}{c}_1^3{c}_2^6+1630158336{\beta }^9{c}_1^6{c}_2^6\\ +15980146560{\beta }^9{c}_1^4{c}_2^5-4676078592{\beta }^8{c}_1^7{c}_2^5-16397095680{\beta }^8{c}_1^5{c}_2^4\\ +131654016{\beta }^7{c}_1^{10}{c}_2^5+7304043456{\beta }^7{c}_1^8{c}_2^4+109426400{\beta }^7{c}_1^6{c}_2^3\\ -564117120{\beta }^6{c}_1^{11}{c}_2^4-6702770736{\beta }^6{c}_1^9{c}_2^3-4571471520{\beta }^6{c}_1^7{c}_2^2\\ +949114656{\beta }^5{c}_1^{12}{c}_2^3+3610117836{\beta }^5{c}_1^{10}{c}_2^2+1083264840{\beta }^5{c}_1^8{c}_2\\ -18404544{\beta }^4{c}_1^{15}{c}_2^3-783565920{\beta }^4{c}_1^{13}{c}_2^2-1055710800{\beta }^4{c}_1^{11}{c}_2\\ -110837160{\beta }^4{c}_1^9+41460120{\beta }^3{c}_1^{16}{c}_2^2+317333772{\beta }^3{c}_1^{14}{c}_2\\ +129215439{\beta }^3{c}_1^{12}-30678936{\beta }^2{c}_1^{17}{c}_2-50420286{\beta }^2{c}_1^{15}\\ +484984c_1^{20}{c}_2+7457610{\mathrm{\beta c}}_1^{18}-316628c_1^{21})\end{array},$中间参量c₁和c₂没有单位，其它所有参量皆采用国际单位制。