均布载荷下中心带刚性板的环形薄膜最大应力的确定方法

专利类型：发明专利

语言：中文

申请号：CN201610263954.0

申请日：20160426

发明人：孙俊贻练永盛杨志欣何晓婷蔡珍红郑周练杨鹏

申请人：重庆大学

申请人地址：400044 重庆市沙坪坝区沙正街174号

公开日：20160921

公开号：CN105956379A

代理人：张先芸

代理机构：重庆博凯知识产权代理有限公司 50212

摘　　要：本发明公开了均布载荷下中心带刚性板的环形薄膜最大应力的确定方法：采用内半径为a的夹紧装置，将厚度为h、杨氏弹性模量为E、泊松比为ν、中心带半径为b的刚性板的环形薄膜固定夹紧，形成一个外半径为a、内半径为b的周边固定夹紧的中心带刚性板的轴对称环形薄膜，对其横向施加一个均布载荷q，基于这个轴对称变形问题的静力平衡分析，并利用均布载荷q的测量值，则可确定出该环形薄膜的最大应力σ_m。

主权项：均布载荷下中心带刚性板的环形薄膜最大应力的确定方法，其特征在于：采用内半径为a的夹紧装置，将厚度为h、杨氏弹性模量为E、泊松比为ν、中心带半径为b的刚性板的环形薄膜固定夹紧，形成一个外半径为a、内半径为b的周边固定夹紧的中心带刚性板的轴对称环形薄膜，对其横向施加一个均布载荷q，并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的最大应力σ_m： $σ_{m} = - \frac{1}{2} {(\frac{{Ea}^{2} q^{2}}{h^{2}})}^{1 / 3} Σ_{n = 0}^{10} d_{n} {(\frac{b - a}{2 a})}^{n},$ 其中， $d_{0} = \frac{β}{c_{1}},$ $d_{1} = \frac{1}{c_{1}^{2}} (- 2 {βc}_{2} + c_{1}),$ $d_{2} = \frac{1}{2 β^{2} c_{1}^{2}} (6 β^{2} c_{2} - 3 {βc}_{1} + c_{1}^{4}),$ $d_{3} = \frac{1}{6 β^{3} c_{1}^{2}} (- 24 β^{2} c_{2} + 4 {βc}_{1}^{3} c_{2} + 12 {βc}_{1} - 5 c_{1}^{4}),$ $d_{4} = \frac{1}{24 β^{5} c_{1}^{2}} (24 β^{3} c_{1}^{2} c_{2}^{2} + 120 β^{3} c_{2} - 48 β^{2} c_{1}^{3} c_{2} - 60 β^{2} c_{1} + 33 {βc}_{1}^{4} - 2 c_{1}^{7}),$ $\begin{matrix} d_{5} = \frac{1}{60 β^{6} c_{1}^{2}} (96 β^{4} c_{1} c_{2}^{3} - 288 β^{3} c_{1}^{2} c_{2}^{2} - 360 β^{3} c_{2} + 180 β^{2} c_{1} + 294 β^{2} c_{1}^{3} c_{2} - 22 {βc}_{1}^{6} c_{2} \\ - 132 {βc}_{1}^{4} + 17 c_{1}^{7}) \end{matrix},$ $\begin{matrix} d_{6} = \frac{1}{360 β^{8} c_{1}^{2}} (960 β^{6} c_{2}^{4} - 3936 β^{5} c_{1} c_{2}^{3} + 5940 β^{4} c_{1}^{2} c_{2}^{2} + 2520 β^{4} c_{2} - 408 β^{3} c_{1}^{5} c_{2}^{2} \\ - 4038 β^{3} c_{1}^{3} c_{2} - 1260 β^{3} c_{1} + 678 β^{2} c_{1}^{6} c_{2} + 1281 β^{2} c_{1}^{4} - 282 {βc}_{1}^{7} + 11 c_{1}^{10}) \end{matrix},$ $\begin{matrix} d_{7} = \frac{1}{2520 β^{9} c_{1}^{3}} (11520 β^{7} c_{2}^{5} - 60480 β^{6} c_{1} c_{2}^{4} + 124128 β^{5} c_{1}^{2} c_{2}^{3} - 7632 β^{4} c_{1}^{5} c_{2}^{3} \\ - 125100 β^{4} c_{1}^{3} c_{2}^{2} - 20160 β^{4} c_{1} c_{2} + 20160 β^{3} c_{1}^{6} c_{2}^{2} + 63324 β^{3} c_{1}^{4} c_{2} + 10080 β^{3} c_{1}^{2} \\ - 17442 β^{2} c_{1}^{7} c_{2} - 15003 β^{2} c_{1}^{5} + 584 {βc}_{1}^{10} c_{2} + 5010 {βc}_{1}^{8} - 427 c_{1}^{11}) \end{matrix},$ $\begin{matrix} d_{8} = \frac{1}{20160 β^{11} c_{1}^{4}} (161280 β^{9} c_{2}^{6} - 1036800 β^{8} c_{1} c_{2}^{5} + 2718720 β^{7} c_{1}^{2} c_{2}^{4} - 150912 β^{6} c_{1}^{5} c_{2}^{4} \\ - 3722112 β^{6} c_{1}^{3} c_{2}^{3} + 555840 β^{5} c_{1}^{6} c_{2}^{3} + 2815020 β^{5} c_{1}^{4} c_{2}^{2} + 181440 β^{5} c_{1}^{2} c_{2} \\ - 752400 β^{4} c_{1}^{7} c_{2}^{2} - 1131876 β^{4} c_{1}^{5} c_{2} - 90720 β^{4} c_{1}^{3} + 21960 β^{3} c_{1}^{10} c_{2}^{2} + 444984 β^{3} c_{1}^{8} c_{2} \\ + 210087 β^{3} c_{1}^{6} - 34584 β^{2} c_{1}^{11} c_{2} - 97905 β^{2} c_{1}^{9} + 13440 {βc}_{1}^{12} - 292 c_{1}^{15}) \end{matrix},$ $\begin{matrix} d_{9} = \frac{1}{181440 β^{12} c_{1}^{5}} (2580480 β^{10} c_{2}^{7} - 19676160 β^{9} c_{1} c_{2}^{6} + 63106560 β^{8} c_{1}^{2} c_{2}^{5} \\ - 3200256 β^{7} c_{1}^{5} c_{2}^{5} - 110246400 β^{7} c_{1}^{3} c_{2}^{4} + 15258240 β^{6} c_{1}^{6} c_{2}^{4} + 113274720 β^{6} c_{1}^{4} c_{2}^{3} \\ - 28549152 β^{5} c_{1}^{7} c_{2}^{3} - 68581080 β^{5} c_{1}^{5} c_{2}^{2} - 1814400 β^{5} c_{1}^{3} c_{2} + 741024 β^{4} c_{1}^{10} c_{2}^{3} \\ + 26215056 β^{4} c_{1}^{8} c_{2}^{2} + 22867921 β^{4} c_{1}^{6} c_{2} + 907200 β^{4} c_{1}^{4} - 1851552 β^{3} c_{1}^{11} c_{2}^{2} \\ - 11836062 β^{3} c_{1}^{9} c_{2} - 3469230 β^{3} c_{1}^{4} + 1517496 β^{2} c_{1}^{12} c_{2} + 2114559 β^{2} c_{1}^{10} \\ - 28384 {βc}_{1}^{15} c_{2} - 408768 {βc}_{1}^{13} + 20504 c_{1}^{16}) \end{matrix},$ $\begin{matrix} d_{10} = \frac{1}{1814400 β^{14} c_{1}^{6}} (46448640 β^{12} c_{2}^{8} - 410296320 β^{11} c_{1} c_{2}^{7} + 1559900160 β^{10} c_{1}^{2} c_{2}^{6} \\ - 73036800 β^{9} c_{1}^{5} c_{2}^{6} - 3330132480 β^{9} c_{1}^{3} c_{2}^{5} + 429684480 β^{8} c_{1}^{6} c_{2}^{5} + 4362206400 β^{8} c_{1}^{4} c_{2}^{4} \\ - 1035268992 β^{7} c_{1}^{7} c_{2}^{4} - 3588896160 β^{7} c_{1}^{5} c_{2}^{3} + 24303168 β^{6} c_{1}^{10} c_{2}^{4} + 19958400 β^{6} c_{1}^{4} c_{2} \\ + 1307134368 β^{6} c_{1}^{8} c_{2}^{3} + 1812822120 β^{6} c_{1}^{6} c_{2}^{2} - 84591072 β^{5} c_{1}^{11} c_{2}^{3} - 912287016 β^{5} c_{1}^{9} c_{2}^{2} \\ - 516903120 β^{5} c_{1}^{7} c_{2} - 9979200 β^{5} c_{1}^{5} + 108652752 β^{4} c_{1}^{12} c_{2}^{2} + 334076706 β^{4} c_{1}^{10} c_{2} \\ + 66432150 β^{4} c_{1}^{8} - 1792752 β^{3} c_{1}^{15} c_{2}^{2} - 61072440 β^{3} c_{1}^{13} c_{2} - 50333643 β^{3} c_{1}^{11} \\ + 2756784 β^{2} c_{1}^{16} c_{2} + 12690723 β^{2} c_{1}^{14} - 1047372 {βc}_{1}^{17} + 14192 c_{1}^{20}) \end{matrix},$ 而β＝(a+b)/2a，c₁和c₂的值由方程 $ν = 2 - \frac{b}{a} [Σ_{i = 2}^{10} i (i - 1) c_{i} {(\frac{b}{a} - β)}^{i - 2}] / [Σ_{j = 1}^{10} {jc}_{j} {(\frac{b}{a} - β)}^{j - 1}]$ 和 $ν = 2 - \frac{b}{a} [Σ_{i = 2}^{10} i (i - 1) c_{i} {(1 - β)}^{i - 2}] / [Σ_{j = 1}^{10} {jc}_{j} {(1 - β)}^{j - 1}]$ 确定,其中， $c_{3} = \frac{1}{6} \frac{1}{β^{3} c_{1}} (8 β^{3} c_{2}^{2} - 10 β^{2} c_{1} c_{2} + 3 {βc}_{1}^{2} - c_{1}^{5}),$ $c_{4} = \frac{1}{24} \frac{1}{β^{4} c_{1}^{2}} (48 β^{4} c_{2}^{3} - 120 β^{3} c_{1} c_{2}^{2} + 90 β^{2} c_{1}^{2} c_{2} - 16 {βc}_{1}^{5} c_{2} - 21 {βc}_{1}^{3} + 8 c_{1}^{6}),$ $\begin{matrix} c_{5} = \frac{1}{120} \frac{1}{β^{6} c_{1}^{3}} (384 β^{6} c_{2}^{4} - 1440 β^{5} c_{1} c_{2}^{3} + 1896 β^{4} c_{1}^{2} c_{2}^{2} - 232 β^{3} c_{1}^{5} c_{2}^{2} - 1032 β^{3} c_{1}^{3} c_{2} \\ + 312 β^{2} c_{1}^{6} c_{2} + 198 β^{2} c_{1}^{4} - 101 {βc}_{1}^{7} + 8 c_{1}^{10}) \end{matrix},$ $\begin{matrix} c_{6} = \frac{1}{720} \frac{1}{β^{7} c_{1}^{4}} (3840 β^{7} c_{2}^{5} - 19200 β^{6} c_{1} c_{2}^{4} + 36720 β^{5} c_{1}^{2} c_{2}^{3} - 3552 β^{4} c_{1}^{5} c_{2}^{3} \\ - 33360 β^{4} c_{1}^{3} c_{2}^{2} + 8296 β^{3} c_{1}^{6} c_{2}^{2} + 14340 β^{3} c_{1}^{4} c_{2} - 6208 β^{2} c_{1}^{7} c_{2} \\ - 2340 β^{2} c_{1}^{5} + 344 {βc}_{1}^{10} c_{2} + 1489 {βc}_{1}^{8} - 204 c_{1}^{11}) \end{matrix},$ $\begin{matrix} c_{7} = \frac{1}{2520} \frac{1}{β^{9} c_{1}^{5}} (23040 β^{9} c_{2}^{6} - 144000 β^{8} c_{1} c_{2}^{5} + 362880 β^{7} c_{1}^{2} c_{2}^{4} - 29664 β^{6} c_{1}^{5} c_{2}^{4} \\ - 469800 β^{6} c_{1}^{3} c_{2}^{3} + 100944 β^{5} c_{1}^{6} c_{2}^{3} + 328140 β^{5} c_{1}^{4} c_{2}^{2} - 124644 β^{4} c_{1}^{7} c_{2}^{2} \\ - 116910 β^{4} c_{1}^{5} c_{2} + 5416 β^{3} c_{1}^{10} c_{2}^{2} + 66120 β^{3} c_{1}^{8} c_{2} + 16605 β^{3} c_{1}^{6} \\ - 7468 β^{2} c_{1}^{11} c_{2} - 12708 β^{2} c_{1}^{9} + 2524 {βc}_{1}^{12} - 86 c_{1}^{15}) \end{matrix},$ $\begin{matrix} c_{8} = \frac{1}{20160} \frac{1}{β^{10} c_{1}^{6}} (322560 β^{10} c_{2}^{7} - 2419200 β^{9} c_{1} c_{2}^{6} + 7580160 β^{8} c_{1}^{2} c_{2}^{5} - 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