Studies of Nonlinear Theories for Thin Shells

Wu Shen-rong

Volume 4 Issue 2

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Abstract

References

Article Navigation > Applied Mathematics and Mechanics > 1983 > 4(2): 221-231

Wu Shen-rong. Studies of Nonlinear Theories for Thin Shells[J]. Applied Mathematics and Mechanics, 1983, 4(2): 221-231.

Citation:

Wu Shen-rong. Studies of Nonlinear Theories for Thin Shells[J]. Applied Mathematics and Mechanics, 1983, 4(2): 221-231.

Citation:

Wu Shen-rong. Studies of Nonlinear Theories for Thin Shells[J]. Applied Mathematics and Mechanics, 1983, 4(2): 221-231.

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Studies of Nonlinear Theories for Thin Shells

Wu Shen-rong

Fudan University, Shanghai

Received Date: 1982-06-10
Publish Date: 1983-04-15

Abstract

Abstract

In order to formulate the equations for the study here, the Fourier expansions upon the system of orthonormal polynomials areused.It may be considerably convenient to obtain the expressions of displacements as well as stresses directly from the solutions.Based on the principle of virtual work the equilibrium equations of various orders are formulated. In particular, the system of third-order is given in detail, thus providing the reference for accuracy analysis of lower-order equations. A theorem about the differentiation of Legendre series term by term is proved as the basis of mathematical analysis. Therefore the functions used are specified and the analysis rendered is no longer a formal one.The analysis will show that the Kirchhoff-Love's theory is merely of the first-order and the theory which includes the transverse deformation but keeps the normal straight is essentially of the first-order, too.

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References(24)

References

[1]	Clebsch A.,Théorie de l'Elasticitédes Corps Solides,avec des Notes Etendues de Saint-Venant(1883) 687-706.
[2]	Donnell,L.H.,A New Theory for the Buckling of Thin Cylinders under Axial Compression and Bending,Trans.ASME,56,11(1934) 795-806.
[3]	Von Kármán,T.and H.S.Tsien,The Buckling of Thin Cylindrical Shells under Axial Compression,J.Aeron;Sci.,8,8(1941),303-312.
[4]	Tsien,H.S.,The Theory for the Buckling of Thin Shells,J.Aeron.Sci.,Vol.9(1942) 373-384.
[5]	Chien,W.Z.,The Intrinsic Theory of Thin Shells and Plates,Quart.Appl..Math.,1.4(1944) 297-327;2,1(1944) 43-59.
[6]	Naghdi,P.M.and R.P.Nordgren,On the Nonlinear Theory of Elastic Shells under the Kirchhoff Hypothesis,Quart.Appl.Math.,21,1(1963,4),49-59.
[7]	Seide,P.,Some Inplication of Strain Energy Expressions for the Love-Kirchhoff Nonlinear Theory of Plates and Shells,Int.J.Nonlinear Mech.,6,3(1971,6) 361-376.
[8]	Basar,Y.,Eine Schalentheorie endlicher Verformungen und Ihre Anwendung zur Herleitung der Stabilitdtetheorie,Zeit-schrift für Ang.Math.& Mech.,52,5(1972,5) 197-211.
[9]	Sanders,Jr.,J.L.,Nonlinear Theory for Thin Shells,Quart.Appl.Math.,21,1(1963,4) 21-36.
[10]	Budiansky,B.,Notes on Nonlinear Shell Theory,J.Appl.Mech.,35,2(1968,6) 393-401.
[11]	Reissner,M.E.,On Transverse Bending of Plates,Including the Effect of Transverse Shear Deformation,Int.J.,Solids Struc.,11,5(1975,5) 569-573.
[12]	Schmidt,R.,A Refined Nonlinear Theory of Plates with Transverse Shear Deformation,Industrial Math.,27,1(1977) 23-38.
[13]	Berstein,E.L.,On large deflection theories of plates,J.Appl.Mech.,32,3(1965,9) 695-597(brief notes).
[14]	Ebicioglu,I.K.,Nonlinear Theory of Shells,Int.J.Non-LInear Mech.,6,4(1971,8) 469-478.
[15]	Biricikoglu,V.and A.Kalnins,Large Elastic Deformations of Shells with the Inclusion of Transverse Normal Strain,Int.J.Solids Structures,7,5(1971,5) 431-444.
[16]	Pietraszkiewicz.W.,Finite Rotations and Lagrangian Description in the Nonlinear Theory of Shells,(1979),Warszawa,Polish Scientific publishers.
[17]	Wempner G.,and C.M.Hwang,A Derived Theory of Elastic-Plastic Shells,Int.J.Solids Structures,13,11(1977,11) 1123-1132.
[18]	Abé,Hiroyuki,A Systematic Formulation of Nonlinear Thin Elastic-Plastic Shell Theories,Int.J.Enngg.Sci.,13,11(1975,11) 1003-1014(1975,11).
[19]	Kr?tzig,W.B.,Allgemeine Schalentheorie beliebiger Werkstoffe und Verformungen,Ingenieur-Archiev,40,5(1971,9) 311-326.
[20]	Yokoo Y.,and H.Matsunaga,A General Nonlinear Theory of Elastic Shells,Int.J.Solids Structures,10,2(1974,2) 261-274.
[21]	夏道行等,《实变函数论与泛函分析》,下册,人民教育出版社,(1979).
[22]	Sansone,G.,Orthogonal Polynomials,interscience Publishers,London,(1959).
[23]	И.П.那汤松,《函数构造论》中册,何旭初,唐述钊译,科学出版社,(1958).
[24]	W.弗留盖,《张量分析与连续介质力学》,白铮译,中国建筑工业出版社,(1980).

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