Abstract
The paper is devoted to the investigation of regularities inherent to optimal geometrically non-linear trusses. The single static loading case is considered, a single structural material is used (except specially indicated cases) and buckling effects are neglected. The so-called small strains and large rotations case is investigated.
Some regularities inherent to the kinematic and static variational principles for geometrically non-linear trusses are considered. Then the strain compatibility conditions resulting from the static variational principle are obtained and explored. It is shown that 1) the conditions are linear with respect to subcomponents of rod Green strains such as rotations and geometrically linear strains, 2) strains (in particular, rotations and geometrically linear strains) within rods which are not members of the so-called basic structure are fully determined by geometrically linear strains in rods of the basic structure.
Extensions of some theorems (Maxwell’s theorem, Michell’s theorem, theorems on the stiffness properties of equally-stressed structures, etc.) known for geometrically linear structures are proved.
Conditions assuring better or worse quality of equally-stressed geometrically non-linear truss as compared to geometrically linear ones are obtained.
It is shown that in numerical optimization of geometrically non-linear trusses in the case of negligible rotations of compressed rods some updated analytical optimization algorithms (derived earlier for geometrically linear case) are monotonic. A simple numerical example confirming the features is presented.
Similar content being viewed by others
References
Buhl, T.; Pedersen, C.B.W.; Sigmund, O. 2000: Stiffness design of geometrically non-linear structures using topology optimization. Struct Multidisc Optim19(2), 93–104
Dems, K.; Mroz, Z. 1978: Multiparameter structural shape optimization by the finite element method. Int J Numer Methods Eng13, 247–263
Gea, H.C.; Luo, J. 2001: Topology optimization of structures with geometrical nonlinearities. Comput Struct79, 1977–1985
Gierlinski, J.; Mroz, Z. 1981: Optimal design of elastic plates and beams taking large deflections and shear forces into account. Acta Mechanica39, 77–92
Khot, N.S.; Kamat, M.P. 1985: Minimum weight design of truss structures with geometric nonlinear behavior. AIAA J23(1), 139–144
Lurie, A.I. 1980: Non-linear theory of elasticity (in Russian). Moscow: Nauka Publ., p. 512
Maxwell, C. 1890: Scientific Papers. II. Camb. Univ. Press, pp. 175–177
Michell, A.G.M. 1904: The limit of economy of material in frame structures. Phil Mag Ser6, 8(47), 589–597
Mroz, Z.; Kamat, M.P.; Plaut, R.H. 1985: Sensitivity analysis and optimal design of nonlinear beams and plates. J Struct Mech13(3–4), 245–266
Novozhilov, V.V. 1948: Foundations of the non-linear theory of elasticity (in Russian). Leningrad, Moscow: Gostekhizdat Publ., p. 211
Orozco, C.E.; Ghattas, O.N. 1997: A reduced SAND method for optimal design of non-linear structures. Int J Numer Methods Eng40, 2759–2774
Pedersen, P.; Taylor, J.E. 1993: Optimal design based on power-law non-linear elasticity. In: Pedersen, P. (ed.), Optimal Design with Advanced Materials. Amsterdam: Elsevier Science, pp. 51–66
Saka, M.P.; Ulker, M. 1992: Optimum design of geometrically nonlinear space trusses. Comput Struct42(3), 289–299
Selyugin, S.V. 1995: On optimal physically nonlinear trusses. Struct Optim10(3–4), 159–166
Selyugin, S.V. 2000: On topology aspects of optimal bi-material physically non-linear structures. In: Rozvany, G.I.N.; Olhoff, N. (eds.), Topology optimization of structures and composite continua. Dordrecht, Boston, London: Kluwer Academic, pp. 327–336
Selyugin, S.V. 2001: Optimization algorithms for physically non-linear structures. Int J Numer Methods Eng50, 2211–2232
Selyugin, S.V., Chekhov, V.V. 2001: Multi-material design of physically non-linear structures. Struct Optim21, 209–217
Washizu, K. 1982: Variational methods in elasticity and plasticity. Oxford: Pergamon
Washizu, K. 1974: A note on the principle of stationary complementary energy in nonlinear elasticity. In: Mechanics Today, Vol. 5
Zubov, L.M. 1970: Complementary work stationarity principle in non-linear theory of elasticity (in Russian). PMM34(2)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Selyugin, S. On optimal geometrically non-linear trusses. Struct Multidisc Optim 29, 113–124 (2005). https://doi.org/10.1007/s00158-004-0462-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-004-0462-4