横向均布载荷下预应力圆薄膜弹性能的确定方法与流程

技术特征：

1.横向均布载荷下预应力圆薄膜弹性能的确定方法，其特征在于：对周边固定夹紧的预应力圆形薄膜横向施加一个均布载荷q，其中圆形薄膜的厚度为h、半径为a、杨氏弹性模量为E、泊松比为ν、预应力为σ₀，测得所施加的均布载荷q的值，由以下公式确定该圆形薄膜变形后的弹性能U：

$U=\frac{{\mathrm{\πa}}^{10/3}{c}^{1/3}{q}^{4/3}}{2^{1/3}{E}^{1/3}{h}^{1/3}}\[g\left(c\right)-h\left(c\right)\],$

其中，π是圆周率，

$\begin{array}{c}g\left(c\right)=1+\left(\frac{1}{4}+\frac{1}{2}\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}\right)c+\left(\frac{5}{36}+\frac{11}{18}\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}+\frac{1}{2}\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}\right)c^2\\ +\left(\frac{55}{576}+\frac{49}{72}\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}+\frac{61}{48}\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}+\frac{5}{8}\frac{2a^2{q}^2}{h^2{E}^{2}c}\right)c^3+(\frac{7}{96}+\frac{2647}{3600}\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}\\ +\frac{4051}{1800}\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}+\frac{253}{100}\frac{2a^2{q}^2}{h^2{E}^{2}c}+\frac{7}{8}\frac{2^{4/3}{a}^{8/3}{q}^{8/3}}{h^{8/3}{E}^{8/3}{c}^{4/3}})c^4+(\frac{205}{3456}+\frac{101639}{129600}\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}\\ +\frac{443119}{129600}\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}+\frac{8521}{1350}\frac{2a^2{q}^2}{h^2{E}^{2}c}+\frac{7141}{1440}\frac{2^{4/3}{a}^{8/3}{q}^{8/3}}{h^{8/3}{E}^{8/3}{c}^{4/3}}+\frac{21}{16}\frac{2^{5/3}{a}^{10/3}{q}^{10/3}}{h^{10/3}{E}^{10/3}{c}^{5/3}})c^5\\ +(\frac{17051}{338688}+\frac{2639243}{3175200}\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}+\frac{866077}{181440}\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}+\frac{13308283}{1058400}\frac{2a^2{q}^2}{h^2{E}^{2}c}\\ +\frac{1904549}{117600}\frac{2^{4/3}{a}^{8/3}{q}^{8/3}}{h^{8/3}{E}^{8/3}{c}^{4/3}}+\frac{113623}{11760}\frac{2^{5/3}{a}^{10/3}{q}^{10/3}}{h^{10/3}{E}^{10/3}{c}^{5/3}}+\frac{33}{16}\frac{4a^4{q}^4}{h^4{E}^4{c}^2})c^6+(\frac{2864485}{65028096}\\ +\frac{5947093}{6773760}\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}+\frac{91705651}{14515200}\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}+\frac{1115512751}{50803200}\frac{2a^2{q}^2}{h^2{E}^{2}c}\\ +\frac{1366411591}{33868800}\frac{2^{4/3}{a}^{8/3}{q}^{8/3}}{h^{8/3}{E}^{8/3}{c}^{4/3}}+\frac{740195}{18816}\frac{2^{5/3}{a}^{10/3}{q}^{10/3}}{h^{10/3}{E}^{10/3}{c}^{5/3}}+\frac{33661}{1792}\frac{4a^4{q}^4}{h^4{E}^4{c}^2}\\ +\frac{429}{128}\frac{2^{7/3}{a}^{14/3}{q}^{14/3}}{h^{14/3}{E}^{14/3}{c}^{7/3}})c^7\end{array},$

$\begin{array}{c}h\left(c\right)=\frac{1}{2}+\frac{1}{3}\left(\frac{1}{4}+\frac{1}{2}\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}\right)c+\frac{1}{4}\left(\frac{5}{36}+\frac{11}{18}\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}+\frac{1}{2}\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}\right)c^2\\ +\frac{1}{5}\left(\frac{55}{576}+\frac{49}{72}\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}+\frac{61}{48}\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}+\frac{5}{8}\frac{2a^2{q}^2}{h^2{E}^{2}c}\right)c^3+\frac{1}{6}(\frac{7}{96}+\frac{2647}{3600}\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}\\ +\frac{4051}{1800}\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}+\frac{253}{100}\frac{2a^2{q}^2}{h^2{E}^{2}c}+\frac{7}{8}\frac{2^{4/3}{a}^{8/3}{q}^{8/3}}{h^{8/3}{E}^{8/3}{c}^{4/3}})c^4+\frac{1}{7}(\frac{205}{3456}+\frac{101639}{129600}\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}\\ +\frac{443119}{129600}\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}+\frac{8521}{1350}\frac{2a^2{q}^2}{h^2{E}^{2}c}+\frac{7141}{1440}\frac{2^{4/3}{a}^{8/3}{q}^{8/3}}{h^{8/3}{E}^{8/3}{c}^{4/3}}+\frac{21}{16}\frac{2^{5/3}{a}^{10/3}{q}^{10/3}}{h^{10/3}{E}^{10/3}{c}^{5/3}})c^5\\ +\frac{1}{8}(\frac{17051}{338688}+\frac{2639243}{3175200}\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}+\frac{866077}{181440}\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}+\frac{13308283}{1058400}\frac{2a^2{q}^2}{h^2{E}^{2}c}\\ +\frac{1904549}{117600}\frac{2^{4/3}{a}^{8/3}{q}^{8/3}}{h^{8/3}{E}^{8/3}{c}^{4/3}}+\frac{113623}{11760}\frac{2^{5/3}{a}^{10/3}{q}^{10/3}}{h^{10/3}{E}^{10/3}{c}^{5/3}}+\frac{33}{16}\frac{4a^4{q}^4}{h^4{E}^4{c}^2})c^6+\frac{1}{9}(\frac{2864485}{65028096}\\ +\frac{5947093}{6773760}\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}+\frac{91705651}{14515200}\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}+\frac{1115512751}{50803200}\frac{2a^2{q}^2}{h^2{E}^{2}c}\\ +\frac{1366411591}{33868800}\frac{2^{4/3}{a}^{8/3}{q}^{8/3}}{h^{8/3}{E}^{8/3}{c}^{4/3}}+\frac{740195}{18816}\frac{2^{5/3}{a}^{10/3}{q}^{10/3}}{h^{10/3}{E}^{10/3}{c}^{5/3}}+\frac{33661}{1792}\frac{4a^4{q}^4}{h^4{E}^4{c}^2}\\ +\frac{429}{128}\frac{2^{7/3}{a}^{14/3}{q}^{14/3}}{h^{14/3}{E}^{14/3}{c}^{7/3}})c^7\end{array},$

而中间参量c的值由方程

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

确定，其中，

$\begin{array}{c}f\left(c\right)=1-\frac{1}{2}c-\frac{1}{6}\left(1+2\×\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}\right)c^2-\frac{1}{144}(13+56\×\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}\\ +48\×\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}})c^3-\frac{1}{1440}(85+584\×\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}+1104\×\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}\\ +1152\frac{a^2{q}^2}{h^2{E}^{2}c})c^4-\frac{1}{21600}(925+8904\×\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}+27104\×\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}\\ +62592\frac{a^2{q}^2}{h^2{E}^{2}c}+23040\×\frac{2^{1/3}{a}^{8/3}{q}^{8/3}}{h^{8/3}{E}^{8/3}{c}^{4/3}})c^5-\frac{1}{907200}(30125+377688\×\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}\\ +1619968\×\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}+6038784\frac{a^2{q}^2}{h^2{E}^{2}c}+4907520\×\frac{2^{1/3}{a}^{8/3}{q}^{8/3}}{h^{8/3}{E}^{8/3}{c}^{4/3}}\\ +1382400\×\frac{2^{2/3}{a}^{10/3}{q}^{10/3}}{h^{10/3}{E}^{10/3}{c}^{5/3}})c^6-\frac{1}{203212800}(5481025+85400640\×\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}\\ +478992416\×\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}+2520947712\frac{a^2{q}^2}{h^2{E}^{2}c}+3305599488\×\frac{2^{1/3}{a}^{8/3}{q}^{8/3}}{h^{8/3}{E}^{8/3}{c}^{4/3}}\\ +2047057920\×\frac{2^{2/3}{a}^{10/3}{q}^{10/3}}{h^{10/3}{E}^{10/3}{c}^{5/3}}+928972800\frac{a^4{q}^4}{h^4{E}^4{c}^2})c^7\end{array},$

$\begin{array}{c}{f}^{\′}\left(c\right)=-\frac{1}{2}-\frac{1}{3}\left(1+2\×\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}\right)c-\frac{1}{48}(13+56\×\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}\\ +48\×\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}})c^2-\frac{1}{360}(85+584\×\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}+1104\×\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}\\ +1152\frac{a^2{q}^2}{h^2{E}^{2}c})c^3-\frac{1}{4320}(925+8904\×\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}+27104\×\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}\\ +62592\frac{a^2{q}^2}{h^2{E}^{2}c}+23040\×\frac{2^{1/3}{a}^{8/3}{q}^{8/3}}{h^{8/3}{E}^{8/3}{c}^{4/3}})c^4-\frac{1}{151200}(30125+377688\×\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}\\ +1619968\×\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}+6038784\frac{a^2{q}^2}{h^2{E}^{2}c}+4907520\×\frac{2^{1/3}{a}^{8/3}{q}^{8/3}}{h^{8/3}{E}^{8/3}{c}^{4/3}}\\ +1382400\×\frac{2^{2/3}{a}^{10/3}{q}^{10/3}}{h^{10/3}{E}^{10/3}{c}^{5/3}})c^5-\frac{1}{29030400}(5481025+85400640\×\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}\\ +478992416\×\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}+2520947712\frac{a^2{q}^2}{h^2{E}^{2}c}+3305599488\×\frac{2^{1/3}{a}^{8/3}{q}^{8/3}}{h^{8/3}{E}^{8/3}{c}^{4/3}}\\ +2047057920\×\frac{2^{2/3}{a}^{10/3}{q}^{10/3}}{h^{10/3}{E}^{10/3}{c}^{5/3}}+928972800\frac{a^4{q}^4}{h^4{E}^4{c}^2})c^6-\frac{1}{914457600}(165851725\\ +3108156432\×\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}+21730770208\×\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}\\ +149806363136\frac{a^2{q}^2}{h^2{E}^{2}c}+277407668736\×\frac{2^{1/3}{a}^{8/3}{q}^{8/3}}{h^{8/3}{E}^{8/3}{c}^{4/3}}+276724703232\×\frac{2^{2/3}{a}^{10/3}{q}^{10/3}}{h^{10/3}{E}^{10/3}{c}^{5/3}}\\ +274790154240\frac{a^4{q}^4}{h^4{E}^4{c}^2}+52022476800\×\frac{2^{1/3}{a}^{14/3}{q}^{14/3}}{h^{14/3}{E}^{14/3}{c}^{7/3}})c^7\end{array},$

中间参量c没有单位，其它所有参量皆采用国际单位制。