CN111474038A - Method for determining maximum deflection of prestressed circular film under uniformly distributed load - Google Patents

Method for determining maximum deflection of prestressed circular film under uniformly distributed load Download PDF

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CN111474038A
CN111474038A CN202010322046.0A CN202010322046A CN111474038A CN 111474038 A CN111474038 A CN 111474038A CN 202010322046 A CN202010322046 A CN 202010322046A CN 111474038 A CN111474038 A CN 111474038A
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circular film
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prestressed circular
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许劲
龚河洛
李卷利
朱杰东
刘孟秋
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Abstract

The invention discloses a method for determining the maximum deflection of a prestressed circular film under uniformly distributed loads, which comprises the following steps: the pair radius is a, the thickness is h, the Young's modulus is E, the Poisson ratio is v, and the prestress is sigma0The prestressed circular film with the fixed and clamped periphery transversely applies an evenly distributed load q to generate axisymmetric deformation, and the maximum deflection w after the axisymmetric deformation of the prestressed circular film can be determined by utilizing the measured value of the load q based on the static balance analysis of the axisymmetric deformation problem of the prestressed circular filmm。

Description

Method for determining maximum deflection of prestressed circular film under uniformly distributed load
Technical Field
The invention relates to a method for determining the maximum deflection of a prestressed circular film with the periphery fixedly clamped under the action of transversely uniformly distributed loads.
Background
The analytical solution for the problem of axisymmetric deformation of a circular membrane structure fixedly clamped at the periphery under transversely uniform load is applied in many engineering technical fields, for example, for developing various instruments and sensors, and for studying adhesion energy measurement of a membrane/substrate system, etc. from the research results in the present, two analytical solutions are currently available, one is the analytical solution adopted by the invention patent "a method for determining maximum deflection of a prestressed circular membrane under uniform load" (patent No. Z L201410238568.7), and the other is the analytical solution adopted by the invention patent "a method for determining maximum deflection of a prestressed circular membrane under transversely uniform load" (patent No. Z L201610802706.9), both of which take into account the fact that a membrane has an initial stress (i.e. a so-called prestress), wherein the analytical solution adopted by the first patent is the analytical solution adopted under the assumption that a membrane is small in the corner (i.e. the assumption that a membrane corner theta is small, so as to satisfy the tan, and the analytical solution obtained under the assumption that the corner equation is not suitable for the initial stress (i.e. the corner equation) is obtained under the assumption that the analytical solution adopted by the initial stress is obtained under the assumption that the small in the corner equation, and the analytical solution adopted in the case that the corner equation is not adopted by the assumption that the case that the film is obtained under the assumption that the corner equation is found in the case that the initial stress is not solved under the corner equation, the case that the corner equation is found in the case that the corner equation is found that the film is small in the case that the initial stress is solved.
From the results of study, the problem of axisymmetrical deformation of a circular film which is fixedly clamped at the periphery under the action of transversely uniform load is not solved so far, and the analytical solution which gives up the assumption of small corner of the film and considers the influence of the film deflection on the establishment of an in-plane equilibrium equation and is suitable for the situation that the film has initial stress (namely prestress) is not provided.
Disclosure of Invention
The invention is dedicated to the analytical research of the axial symmetric deformation problem of the prestressed circular film with the periphery fixedly clamped under the action of the transversely uniformly distributed load, obtains the analytical solution of the axial symmetric deformation problem under the condition of giving up the assumption of a small corner of the film and simultaneously considering the influence of the deflection of the film on the establishment of an in-plane equilibrium equation, and provides the method for determining the maximum deflection of the prestressed circular film under the uniformly distributed load on the basis.
The method for determining the maximum deflection of the prestressed circular film under uniformly distributed load comprises the following steps: the pair radius is a, the thickness is h, the Young's modulus is E, the Poisson ratio is v, and the prestress is sigma0The prestressed circular film with the fixed and clamped periphery transversely applies an evenly distributed load q to generate axisymmetric deformation, and the maximum deflection w after the axisymmetric deformation of the prestressed circular film can be obtained based on the static balance analysis of the axisymmetric deformation problem of the prestressed circular filmmAnd c0The relationship between is wm＝ac0And c is0Is dependent on the applied load q and is given by the equation
Figure BDA0002461798430000021
Determining, wherein,
Figure BDA0002461798430000022
Figure BDA0002461798430000023
Figure BDA0002461798430000024
Figure BDA0002461798430000025
Figure BDA0002461798430000026
Figure BDA0002461798430000027
Figure BDA0002461798430000028
and b0Is given by the equation
Figure BDA0002461798430000029
Determining, wherein,
Figure BDA0002461798430000031
Figure BDA0002461798430000032
Figure BDA0002461798430000033
Figure BDA0002461798430000034
Figure BDA0002461798430000035
Figure BDA0002461798430000036
d0＝b0，
Figure BDA0002461798430000037
Figure BDA0002461798430000038
Figure BDA0002461798430000039
Figure BDA0002461798430000041
Figure BDA0002461798430000042
Figure BDA0002461798430000043
thus, the maximum deflection w after the axial symmetric deformation of the prestressed circular film can be obtained by measuring the value of the load qmIs determined, wherein a, h, wmThe units of (A) are all millimeters (mm), E, q, sigma0All units of (2) are Newton per square millimeter (N/mm)2) And v, b0、b2、b4、b6、b8、b10、b12、c0、c2、c4、c6、c8、c10、c12、d0、d2、d4、d6、d8、d10、d12And Q are dimensionless quantities.
Drawings
FIG. 1 is a schematic view showing the axisymmetrical deformation of a prestressed circular thin film whose periphery is fixedly clamped under the action of a transversely uniformly distributed load, wherein 1 is the prestressed circular thin film after the axisymmetrical deformation, 2 is a clamping device, 3 is the prestressed circular thin film before the deformation, a represents the radius of the prestressed circular thin film and the inner radius of the clamping device, q represents the transversely uniformly distributed load, w represents the transverse uniformly distributed load, andmshowing the maximum deflection after axisymmetric deformation of the prestressed circular film.
Detailed Description
The technical scheme of the invention is further explained by combining the specific cases as follows:
as shown in fig. 1, the radius a is 20mm, the thickness h is 0.06mm, and the young's modulus E is 7.84N/mm2Poisson ratio v is 0.47, prestress sigma0＝2N/mm2Transversely applying an evenly distributed load q to the prestressed circular film fixedly clamped at the periphery, and measuring the load q to be 0.1N/mm2Using the inventionMethod, by the equation
Figure BDA0002461798430000051
Figure BDA0002461798430000052
Figure BDA0002461798430000053
Figure BDA0002461798430000054
Figure BDA0002461798430000055
Figure BDA0002461798430000056
Figure BDA0002461798430000057
Figure BDA0002461798430000058
Figure BDA0002461798430000059
Figure BDA0002461798430000061
Figure BDA0002461798430000062
Figure BDA0002461798430000063
Figure BDA0002461798430000064
Figure BDA0002461798430000065
d0＝b0，
Figure BDA0002461798430000066
Figure BDA0002461798430000067
Figure BDA0002461798430000068
Figure BDA0002461798430000069
Figure BDA00024617984300000610
Figure BDA0002461798430000071
Figure BDA0002461798430000072
To obtain b01.371399 and c01.126905 and b2＝1.279175、b4＝0.673654、b6＝-0.543665、b8＝-0.948286、b10＝0.695263、b12＝2.178050、c2＝-0.775066、c4＝-0.104131、c6＝0.211316、c8＝0.147241、c10＝-0.263354、c12＝-0.342911、d0＝1.371399、d2＝0.542178、d4＝0.294528、d6＝-0.060732、d8＝-0.275129、d10＝-0.009216、d120.483802, and then by equation wm＝ac0Determining that the horizontal uniform load q of the prestressed round film is 0.1N/mm2Maximum deflection w under influencem＝22.54mm。
Furthermore, under the same conditions, if the method of the invention patent "method for determining maximum deflection of circular film under uniform load" (application No. 202010189493.3) is adopted to give up the assumption of small corner of film and simultaneously consider the effect of film deflection on the establishment of in-plane equilibrium equation, but not the initial stress (i.e. prestress) of film, the maximum deflection w can be obtainedmCompared with the invention, the maximum deflection w of the circular film after axial symmetric deformation is obviously influenced by the prestress, and if the method for determining the maximum deflection value of the prestressed circular film under uniformly distributed load (patent number: Z L201410238568.7) provided by the invention patent is adopted, the method for considering the initial stress (namely the prestress) of the film and simultaneously considering the assumption of small corners of the film but not considering the influence of the deflection of the film on the establishment of an in-plane equilibrium equation can obtain the maximum deflection wmCompared with the invention, the method for establishing the in-plane balance equation by giving up the assumption of the small corner of the film and considering the influence of the deflection of the film has obvious influence on the maximum deflection of the prestressed circular film after the axial symmetric deformation, and if the method for determining the maximum deflection of the prestressed circular film under the transversely uniform load (patent number: Z L201610802706.9) is adopted, the method for considering the initial stress (namely the prestress) of the film and giving up the assumption of the small corner of the film at the same time but not considering the influence of the deflection of the film on the establishment of the in-plane balance equation can obtain the maximum deflection wmWhen compared with the present invention, it can be seen that considering the effect of the film deflection on the establishment of the in-plane equilibrium equation, the maximum deflection after the axial symmetric deformation of the prestressed circular film still has an effect. The comparison under the same conditions shows that the invention has better technical effect.

Claims (1)

1. The method for determining the maximum deflection of the prestressed circular film under uniformly distributed load is characterized by comprising the following steps of: the pair radius is a, the thickness is h, the Young's modulus is E, the Poisson ratio is v, and the prestress is sigma0Transversely applying a uniformly distributed load q to the prestressed circular film fixedly clamped at the periphery to generate axisymmetric deformation, and then carrying out static balance analysis on the prestressed circular film axisymmetric deformation problem by using the measured value of the load q and using the following equation
Figure FDA0002461798420000011
Figure FDA0002461798420000012
Figure FDA0002461798420000013
Figure FDA0002461798420000014
Figure FDA0002461798420000015
Figure FDA0002461798420000016
Figure FDA0002461798420000017
Figure FDA0002461798420000021
Figure FDA0002461798420000022
Figure FDA0002461798420000023
Figure FDA0002461798420000024
Figure FDA0002461798420000025
Figure FDA0002461798420000026
Figure FDA0002461798420000027
d0＝b0，
Figure FDA0002461798420000028
Figure FDA0002461798420000029
Figure FDA0002461798420000031
Figure FDA0002461798420000032
Figure FDA0002461798420000033
Figure FDA0002461798420000034
Figure FDA0002461798420000035
Determination of b0And c0And b2、b4、b6、b8、b10、b12、c2、c4、c6、c8、c10、c12、d0、d2、d4、d6、d8、d10、d12And finally, from the equation
wm＝ac0
Determining the maximum deflection w after the axial symmetric deformation of the prestressed circular filmmWherein, a, h, wmThe units of (A) are all millimeters (mm), E, q, sigma0All units of (2) are Newton per square millimeter (N/mm)2) And v, b0、b2、b4、b6、b8、b10、b12、c0、c2、c4、c6、c8、c10、c12、d0、d2、d4、d6、d8、d10、d12And Q are dimensionless quantities.
CN202010322046.0A 2020-04-22 2020-04-22 Method for determining maximum deflection of prestressed circular film under uniformly distributed load Pending CN111474038A (en)

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Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN112858000A (en) * 2021-01-18 2021-05-28 重庆大学 Method for determining maximum deflection of circular prestressed film under uniformly distributed load
CN112880950A (en) * 2021-01-18 2021-06-01 重庆大学 Method for determining deflection of circular prestressed film with limited maximum deflection under air pressure
CN113075048A (en) * 2021-03-30 2021-07-06 重庆大学 Method for determining maximum deflection of circular film under gas pressure

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CN110231215A (en) * 2019-06-05 2019-09-13 重庆大学 The determination method of the maximum defluxion for the prestressing force circular membrane that amount of deflection is limited by elasticity
CN110286031A (en) * 2019-06-05 2019-09-27 重庆大学 The determination method of the maximum stress for the prestressing force circular membrane that amount of deflection is limited by elasticity

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CN110231214A (en) * 2019-06-05 2019-09-13 重庆大学 The determination method of the elasticity energy for the prestressing force circular membrane that amount of deflection is limited by elasticity
CN110231215A (en) * 2019-06-05 2019-09-13 重庆大学 The determination method of the maximum defluxion for the prestressing force circular membrane that amount of deflection is limited by elasticity
CN110286031A (en) * 2019-06-05 2019-09-27 重庆大学 The determination method of the maximum stress for the prestressing force circular membrane that amount of deflection is limited by elasticity

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Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN112858000A (en) * 2021-01-18 2021-05-28 重庆大学 Method for determining maximum deflection of circular prestressed film under uniformly distributed load
CN112880950A (en) * 2021-01-18 2021-06-01 重庆大学 Method for determining deflection of circular prestressed film with limited maximum deflection under air pressure
CN113075048A (en) * 2021-03-30 2021-07-06 重庆大学 Method for determining maximum deflection of circular film under gas pressure

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Application publication date: 20200731