CN112858001A - 一种确定均布载荷下圆形预应力薄膜最大应力的方法 - Google Patents

一种确定均布载荷下圆形预应力薄膜最大应力的方法 Download PDF

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CN112858001A
CN112858001A CN202110065301.2A CN202110065301A CN112858001A CN 112858001 A CN112858001 A CN 112858001A CN 202110065301 A CN202110065301 A CN 202110065301A CN 112858001 A CN112858001 A CN 112858001A
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何晓婷
王晓光
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Abstract

本发明公开了一种确定均布载荷下圆形预应力薄膜最大应力的方法：对半径为a、厚度为h、杨氏弹性模量为E、泊松比为v、预应力为σ0的周边夹紧的圆形预应力薄膜施加一个横向均布载荷q，让该圆形预应力薄膜产生轴对称变形，那么基于该圆形预应力薄膜轴对称变形问题的解析求解结果，利用载荷q的测量值就可以确定该圆形预应力薄膜轴对称变形后的最大应力σm。

Description

一种确定均布载荷下圆形预应力薄膜最大应力的方法
技术领域
本发明涉及一种确定横向均布载荷作用下周边固定夹紧的圆形预应力薄膜的最大应力的方法。
背景技术
横向均布载荷作用下周边夹紧的圆形预应力薄膜轴对称变形问题的解析解，在各类仪器、仪表和传感器的研制、以及薄膜/基层系统力学性能测试研究等领域都有较为广泛的应用。从文献查新的结果来看，在解析求解横向均布载荷作用下周边夹紧的圆形预应力薄膜的轴对称变形问题的过程中，目前仅在建立面外平衡方程时放弃了通常所谓的薄膜小转角假设(即假设薄膜转角θ很小，因而可以让sinθ近似等于tanθ)，而在建立面内平衡方程和几何方程时仍然采用了薄膜小转角假设。例如，发明专利“横向均布载荷下预应力圆薄膜最大应力的确定方法”(专利号：ZL201610803498.4)中所采用的解析解，就是在建立面外平衡方程时放弃薄膜小转角假设、而在建立面内平衡方程和几何方程时采用薄膜小转角假设的条件下获得的。显而易见，在薄膜产生较大转角的情形下，面内平衡方程和几何方程的建立过程中所采用的薄膜小转角假设，仍然会给所获得的解析解带来一定的计算误差。并且随着施加在薄膜上的均布载荷的增大，薄膜的转角也会不断地增大，因而也会给解析解带来越来越大的计算误差。而在大多数工程和技术应用领域，通常都希望解析解能够有一个尽量高的计算精度，这就要求我们必须建立更精确的面内平衡方程和几何方程。
发明内容
本发明致力于横向均布载荷作用下周边夹紧的圆形预应力薄膜的轴对称变形问题的解析研究，基于所建立的更精确的面内平衡方程和几何方程，得到了该圆形预应力薄膜轴对称变形问题的更精确的解析解，并在此基础上给出了一种确定均布载荷下圆形预应力薄膜最大应力的方法。
一种确定均布载荷下圆形预应力薄膜最大应力的方法：对半径为a、厚度为h、杨氏弹性模量为E、泊松比为ν、预应力为σ0的周边夹紧的圆形预应力薄膜施加一个横向均布载荷q，让该圆形预应力薄膜产生轴对称变形，那么对该圆形预应力薄膜轴对称变形问题进行更精确的解析求解，就可以得到所施加的均布载荷q与圆形预应力薄膜轴对称变形后的最大应力σm之间的解析关系为
Figure BDA0002903928220000021
其中，
Figure BDA0002903928220000022
Figure BDA0002903928220000023
Figure BDA0002903928220000024
Figure BDA0002903928220000025
Figure BDA0002903928220000031
Figure BDA0002903928220000041
Figure BDA0002903928220000051
Figure BDA0002903928220000052
而其中的b0的值由方程
Figure BDA0002903928220000053
确定，其中，
d0＝b0、
Figure BDA0002903928220000054
Figure BDA0002903928220000055
Figure BDA0002903928220000056
Figure BDA0002903928220000061
Figure BDA0002903928220000071
Figure BDA0002903928220000081
Figure BDA0002903928220000091
这样，只要测得均布载荷q的值，就可以把该圆形预应力薄膜轴对称变形后的最大应力σm确定下来，其中，a、h的单位均为毫米(mm)，E、q、σm、σ0的单位均为牛顿每平方毫米(N/mm2)，而ν、b0、b2、b4、b6、b8、b10、b12、d0、d2、d4、d6、d8、d10、d12、Q均为无量纲的量。
附图说明
图1为横向均布载荷作用下周边夹紧的圆形预应力薄膜的轴对称变形的示意图，其中，1是轴对称变形后的圆形预应力薄膜，2是夹紧装置，3是轴对称变形前的圆形预应力薄膜，而a表示圆形预应力薄膜的半径和夹紧装置的内半径，q表示横向均布载荷，wm表示圆形预应力薄膜轴对称变形后的最大挠度。
具体实施方式
下面结合具体案例对本发明的技术方案作进一步的说明：
如图1所示，对半径a＝20mm、厚度h＝0.5mm、杨氏弹性模量E＝7.84N/mm2、泊松比ν＝0.47、预应力σ0＝0.001N/mm2的周边夹紧的圆形预应力薄膜施加一个横向均布载荷q，测得均布载荷q＝0.2N/mm2，采用本发明所给出的方法，由方程
Figure BDA0002903928220000092
Figure BDA0002903928220000101
Figure BDA0002903928220000102
Figure BDA0002903928220000103
Figure BDA0002903928220000104
Figure BDA0002903928220000111
Figure BDA0002903928220000121
Figure BDA0002903928220000131
d0＝b0、
Figure BDA0002903928220000132
Figure BDA0002903928220000133
Figure BDA0002903928220000134
Figure BDA0002903928220000141
Figure BDA0002903928220000151
Figure BDA0002903928220000161
Figure BDA0002903928220000171
Figure BDA0002903928220000172
得到b0＝0.412257以及b2＝0.195318、b4＝0.050791、b6＝-0.030318、b8＝-0.029665、b10＝0.006068、b12＝0.019009、d0＝0.412257、d2＝-0.045468、d4＝-0.045201、d6＝-0.020407、d8＝-0.001987、d10＝-0.000118、d12＝-0.005982，然后由
Figure BDA0002903928220000173
确定出该圆形预应力薄膜在均布载荷q＝0.2N/mm2作用下的最大应力为σm＝4.887924N/mm2。

Claims (1)

1.一种确定均布载荷下圆形预应力薄膜最大应力的方法，其特征是：对半径为a、厚度为h、杨氏弹性模量为E、泊松比为ν、预应力为σ0的周边夹紧的圆形预应力薄膜施加一个横向均布载荷q，让该圆形预应力薄膜产生轴对称变形，那么基于该圆形预应力薄膜轴对称变形的静力平衡分析，利用载荷q的测量值，由方程
Figure FDA0002903928210000011
Figure FDA0002903928210000012
Figure FDA0002903928210000013
Figure FDA0002903928210000014
Figure FDA0002903928210000021
Figure FDA0002903928210000031
Figure FDA0002903928210000041
Figure FDA0002903928210000051
d0＝b0、
Figure FDA0002903928210000052
Figure FDA0002903928210000053
Figure FDA0002903928210000054
Figure FDA0002903928210000061
Figure FDA0002903928210000071
Figure FDA0002903928210000081
Figure FDA0002903928210000091
Figure FDA0002903928210000092
确定b0以及b2、b4、b6、b8、b10、b12、d0、d2、d4、d6、d8、d10、d12的值，最后由方程
Figure FDA0002903928210000093
确定该圆形预应力薄膜轴对称变形后的最大应力σm，其中，a、h的单位均为毫米(mm)，E、q、σm、σ0的单位均为牛顿每平方毫米(N/mm2)，而ν、b0、b2、b4、b6、b8、b10、b12、d0、d2、d4、d6、d8、d10、d12、Q均为无量纲的量。
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CN115266333A (zh) * 2022-07-08 2022-11-01 重庆大学 均布载荷下最大挠度受限的圆薄膜的最大应力确定方法

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