Abstract
The equations of the three-dimensional problem of elasticity for thin bodies (bars, beams, plates, shells) in dimensionless coordinates are singularly perturbed by a small geometrical parameter. The general solution of such a system of equations is a combination of the solutions of an internal problem and a boundary-layer problem.
The asymptotic orders of the stress tensor components and of the displacement vector in the second and mixed boundary value problems for thin bodies are established; the inapplicability of classical theory hypothesis for the solution of these problems is proved.
In the case of a plane first boundary value problem for a rectangular strip a connection of the asymptotic solution with the Saint-Venant principle is established and its correctness is proved.
Free and forced vibrations of beams, strips and possibly anisotropic and layered plates are considered by an asymptotic method. The connection of free-vibration frequency values with the propagation velocities of seismic shear and longitudinal waves is established. In a three-dimensional setting forced vibrations of two-layered, three-layered and multi-layered plates under the action of seismic and other dynamic loadings are considered and the resonance conditions are established.
At theoretical justification for the expediency of using seismoisolators in an aseismic construction is given.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Kilchevsky N.A. Grounds of analytical mechanics of shells. Kiev. Publish. House AS USSR, 1963. 354p.
Vekua I.N. Some general methods of constructing various variants of shell theory. M.: Nauka, 1982. 288p.
Timoshenko S.P., Voinovski-Kriger S. Plates and shells. M.: Physmathgiz, 1963. 636p.
Reissner E. On the theory of bending of elastic plates. J. Math. Phys. 1944. V. 23. pp. 184–191.
Ambartsumyan S.A. Theory of anisotropic shells. M.: Physmathgiz. 1961. 344p.
Ambartsumyan S.A. Theory of anisotropic plates. M.: Nauka. 1967. 266p.
Pelekh B.L. Theory of shells with finite shear stiffness. Kiev. Naukova Dumka. 1973. 248p.
Galimov K.Z. To non-linear theory of thin shells of Timoshenko type. Proc AS USSR. MTT. 1976. V. 4. pp. 155–166.
Vazov V. Asymptotic decompositions of solutions of ordinary differential equations. M.: Mir. 1968. 464p.
Nayfeh A.H. Perturbation methods. John Wiley and Sons, 1973. 455p.
Vasiljeva A.B., Boutuzov V.F. Asymptotic decompositions of solutions of singularly perturbed equations. M.: Nauka. 1973. 272p.
Lomov S.A. Introduction into the general theory of singular perturbation. M.: Nauka. 1981. 398p.
Iljin A.M. Concordance of asymptotic decompositions of solutions of boundary value problems. M.: Nauka. 1989. 336p.
Friedrichs K.O., Dressler R.F. A boundary-layer theory for elastic plates. Comm. Pure Appl. Math. 1961. V. 14. N 1.
Goldenveiser A.L. Construction of approximation theory of plate bending with the asymptotic integration method of equations of elasticity theory. J. Appl. Math. Mech. 1962. V. 26 Edition 4. pp. 668–686.
Green A.E. On the linear theory of thin elastic shells. Proc. Roy. Soc Ser. A. 1962. V. 266. N 1325.
Lekhnitsky S.G. Elasticity theory of anisotropic body. M.: Nauka. 1972. 416p.
Aghalovyan L.A. Asymptotic theory of anisotropic plates and shells. M.: Nauka. Fizmatlit, Moscow. 1997. 414p.
Aghalovyan L.A., Khachatryan A.M. Asymptotic analysis of stress-strain state of anisotropic layered beams. Proc. AS ARM SSR. Mech. 1986. V. 39. N 2. pp. 3–14.
Aghalovyan L.A. On bending equations of anisotropic plates. Proceeding of VII All-Union conference on theory of shells and plates. M.: Nauka 1970. pp. 17–21.
Aghalovyan L.A. On reduction of space problem of elasticity theory to two-dimensional for orthotropic shells and errors of some applied theories. Rep. AS ARMSSR. 1979. V. 69. N 3. pp. 151–156.
Aghalovyan L.A. On the structure of solution of one class of plane problems of elasticity theory of anisotropic body. Mechanics: Interuniversity Transactions: Yerevan University Publishing House. 1982. Edition 2. pp. 7–12.
Aghalovyan L.A., Gevorgyan R.S. On the asymptotic solution of mixed three-dimentional problems for two-layered anisotropic plates. Appl. Math Mech. 1986. V. 50. N 2. pp. 271–278.
Aghalovyan L.A., Gevorgyan R.S. On asymptotic solution of nonclassical boundary value problems for two-layered anisotropic thermoelastic shells. Proc. AS ARMSSR. Mech. 1989. V. 42. N 3. pp. 28–36.
Aghalovyan L.A., Gevorgyan R.S. Nonclassical boundary-value problems of anisotropic layered beams, plates and shells. Yerevan, Publish. House “Gitutjun” NAS of Armenia. 2005. 468p.
Aghalovyan L.A. To the asymptotic method of solution of dynamic mixed problems of anisotropic strips and plates. Publish. House of IHE of Russia. North-Caucasus region. Nat. Sci. 2000. N 3(111). pp. 8–11.
Aghalovyan L.A., Aghalovyan M.L. To the determination of frequencies and forms of orthotropic strip free vibrations. reports NAS RA. 2003. V. 103. N 4. pp. 296–301.
Aghalovyan M.L. On solution of the boundary layer in the problem on free vibrations of the strip. In collect. of conf.: Contemporary questions of optimal control of vibrations and stability of systems. Yerevan: Publish. House of Yerevan University. 1997. pp. 132–135.
Aghalovyan L.A., Gulgazaryan L.G. Asymptotic solutions of non-classical boundary-value problems of the natural vibrations of orthotropic shells. J. App. Math. Mech. 2006. 70. pp. 102–115.
Aghalovyan L.A. On one class of the problems on forced vibrations of anisotropic plates. Problems of mechanics of thin deformable bodies. Yerevan. Armenia. 2002. pp. 9–19.
Hovhannisyan R. Sh. The asymptotic form of forced vibrations of three-layered orthotropic plate in case of full contact conditions between layers. Elasticity, plastisity and creep selected topics. Yerevan, Armenia. 2006. pp. 242–248.
Aghalovyan L.A. Asymptotic of solution of classical and nonclassical boundary value problems of statics and dynamics of thin bodies. Int. Appl. Mech. 2002. V. 38. N 7. pp. 3–24.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer Science+Business Media B.V.
About this paper
Cite this paper
Aghalovyan, L.A. (2008). An Asymptotic Method for Solving Three-Dimensional Boundary Value Problems of Statics and Dynamics of Thin Bodies. In: Jaiani, G., Podio-Guidugli, P. (eds) IUTAM Symposium on Relations of Shell Plate Beam and 3D Models. IUTAM Bookseries, vol 9. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-8774-5_1
Download citation
DOI: https://doi.org/10.1007/978-1-4020-8774-5_1
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-8773-8
Online ISBN: 978-1-4020-8774-5
eBook Packages: EngineeringEngineering (R0)