The Reciprocal Theorem for the Finite Displacement Theory and Its Applications

FU Bao-lian

doi:10.3879/j.issn.1000-0887.2015.10.002

Volume 36 Issue 10

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2015 > 36(10): 1019-1034

FU Bao-lian. The Reciprocal Theorem for the Finite Displacement Theory and Its Applications[J]. Applied Mathematics and Mechanics, 2015, 36(10): 1019-1034. doi: 10.3879/j.issn.1000-0887.2015.10.002

Citation:

FU Bao-lian. The Reciprocal Theorem for the Finite Displacement Theory and Its Applications[J]. Applied Mathematics and Mechanics, 2015, 36(10): 1019-1034. doi: 10.3879/j.issn.1000-0887.2015.10.002

Citation:

FU Bao-lian. The Reciprocal Theorem for the Finite Displacement Theory and Its Applications[J]. Applied Mathematics and Mechanics, 2015, 36(10): 1019-1034. doi: 10.3879/j.issn.1000-0887.2015.10.002

PDF( 873 KB)

The Reciprocal Theorem for the Finite Displacement Theory and Its Applications

doi: 10.3879/j.issn.1000-0887.2015.10.002

FU Bao-lian

College of Architectural Engineering and Mechanics, Yanshan University,Qinhuangdao, Hebei 066004, P.R.China

Received Date: 2015-04-14
Rev Recd Date: 2015-06-05
Publish Date: 2015-10-15

Abstract

Abstract

The reciprocal theorem of 3D linear elasticity for the finite displacement theory was proposed. On the basis of the theorem, the reciprocal theorem of rectangular plates in large defection was derived. Meanwhile, the reciprocal theorem of plate strips in large deflection was directly obtained through simplification of the theorem of the rectangular plates. For applications, the bending of a plate strip in large deflection with 2 ends fixed under uniformly distributed load and the bending of a rectangular plate in large deflection with 4 edges fixed under uniformly distributed load were calculated. The calculation shows, on the basis of the reciprocal theorem of bending thin plates in large deflection, the bending rectangular plates in large deflection can easily be sovled with the aid of the basic solution corresponding to the smalldeflection case.
- linear elasticity,
- finite displacement theory,
- large deflection,
- reciprocal theorem

FullText(HTML)

References(24)

References

[1]	Maxwell J C. On calculation of the equilibrium and stiffness of frames[J].Philosophical Magazine Series 4, 1864,27(182): 294-299.
[2]	Betti E. Teoria della elasticita’[J].Nuovo Cimento,1872,7/8(1): 69-97.
[3]	Rayleigh L. More general form of reciprocal theorem[J].Scientific Papers,1873,1: 179-184.
[4]	Lamb H. On reciprocal theorem in dynamics[J].Proceedings of the London Mathematical Society,1887,1(1): 144-151.
[5]	Aндреев Н Н. Теорема взаимностн в теорий колебаний и акустике[J].Физ Словарь,1936,1: 458-459.(Andleev N N. Reciprocal theorem in theory of vibration and sound[J].Phy Dictionary,1936,1: 458-459.(in Russian))
[6]	胡海昌. 论弹性动力学中的倒易定理及它的一些应用[J]. 力学学报, 1957,1(1): 63-71.(HU Hai-chang. On reciprocal theorem in elasto-dynamics and its some applications[J].Mechanica Sinica,1957,1(1): 63-71.(in Chinese))
[7]	Payton R G. An application of the dynamic Betti-Rayleigh reciprocal theorem to moving point load in elastic media[J].Quarterly of Applied Mathematics,1964,21(4): 299-313.
[8]	Iesan D. On reciprocal theorems and variational theorems in linear elasto-dynamicis[J].Bulletin de l’Academie des Sciences, Serie des Sciences Techniques,1974,22: 273-281.
[9]	Айнола Л Я. К теореме взаимности для динамических задач теории упргости[J].Прикладная Математика и Mеханика,1967,31(1): 176-182.(Ainora L Y. On reciprocal theorem for dynamic problems of elasticity[J].Journal of Applied Mathematics and Mechanics,1967,31(1): 176-182.(in Russian))
[10]	付宝连. 关于功的互等定理与叠加原理的等价性[J]. 应用数学和力学, 1985,6(9): 813-816.(FU Bao-lian. On equivalents of the reciprocal theorem to superposition principles[J].Applied Mathematics and Mechanics,1985,6(9): 813-816.(in Chinese))
[11]	付宝连. 广义倒易定理及其应用[J]. 应用数学和力学, 2002,23(2): 188-194.(FU Bao-lian. Generalized reciprocal theorem and it is applications[J].Applied Mathematics and Mechanics,2002,23(2): 188-194.(in Chinese))
[12]	付宝连. 应用功的互等定理求解具有复杂边界条件的矩形板的挠曲面方程[J]. 应用数学和力学, 1982,3(3): 315-325.(FU Bao-lian. Applications of reciprocal theorem to solving the equations of deflection surface of rectangular plates with various edge conditions[J].Applied Mathematics and Mechanics,1982,3(3): 315-325.(in Chinese))
[13]	付宝连, 李农. 弹性矩形薄板受迫振动的功的互等定理法（Ⅰ）——四边固定的矩形板和三边固定的矩形板[J]. 应用数学和力学, 1989,10(8): 693-714.(FU Bao-lian, LI Nong. The method of the reciprocal theorem of forced vibration for the elastic thin rectangular plates(Ⅰ）——rectangular plates with four clamped edges and with three clamped edges[J].Applied Mathematics and Mechanics,1989,10(8): 693-714.(in Chinese))
[14]	付宝连, 李农. 弹性矩形薄板受迫振动的功的互等定理法（Ⅱ）——两邻边固定的矩形板[J]. 应用数学和力学, 1990,11(11): 977-988.（FU Bao-lian, LI Nong. The method of the reciprocal theorem of forced vibration for the elastic thin rectangular plates（Ⅱ）—rectangular plates with two adjacent clamped edges[J].Applied Mathematics and Mechanics,1990,11(11): 977-988.(in Chinese))
[15]	付宝连, 李农. 弹性矩形薄板受迫振动的功的互等定理法（Ⅲ）——悬臂矩形板[J]. 应用数学和力学, 1991,12(7): 613-620.(FU Bao-lian, LI Nong. The method of the reciprocal theorem of forced vibration for the elastic thin rectangular plates（Ⅲ）—cantilever rectangular plates[J].Applied Mathematics and Mechanics,1991,12(7): 613-620.(in Chinese))
[16]	付宝连. 应用功的互等定理法求解立方体的位移解[J]. 应用数学和力学, 1989,10(4): 297-308.(FU Bao-lian. Application of the method of the reciprocal theorem to finding displacement solutions of cubes[J].Applied Mathematics and Mechanics,1989,10(4): 297-308.(in Chinese))
[17]	付宝连. 关于求解弹性力学平面问题的功的互等定理法[J]. 应用数学和力学, 1989,10(5): 437-446.(FU Bao-lian. On the method of reciprocal theorem to finding solutions of the plane problems of elasticity[J].Applied Mathematics and Mechanics,1989,10(5): 437-446.(in Chinese))
[18]	付宝连.弯曲薄板的修正的功的互等定理及其应用[J]. 应用力学和数学, 2014,35(11):1197-1209.(FU Bao-lian. The corrected reciprocal theorem of bending of thin plates and its applications[J].Applied Mathematics and Mechanics,2014,35(11): 1197-1209.(in Chinese))
[19]	付宝连. 三维线弹性力学修正的功的互等定理及其应用[J]. 应用力学和数学, 2015,36(5): 523-538.(FU Bao-lian. The corrected reciprocal theorem of three dimensional linear elasticity and its application[J].Applied Mathematics and Mechanics,2015,36(5): 523-538. (in Chinese))
[20]	Novoshilov V V.Foundations of the Nonlinear Theory of Elasticity [M]. Roches-Ter, N Y: Graylook Press, 1955.
[21]	Reissner E. On a variational theorem for finite elastic deformations[J].Journal Mathematics and Physics,1953,32(2/3): 129-135.
[22]	HE Ji-huan. Family of generalized variational principles for nonlinear elasticity with finite displacement[J].Journal of China Univernsity of Mining & Technology,1999-2.
[23]	Teregllov I G. On a variational theorem of the nonlinear theory of elasticity[J].PMM,26(1): 167-171.
[24]	Timoshenko S P, Woinowsky-Krieger.Theory of Plates and Shells [M]. 2nd ed. New York: McGraw-Hill Inc, 1959.