تجزیه و تحلیل حساسیت از بارهای بحرانی همزمان با توجه به نقص جزئی

A new formulation is presented for sensitivity analysis of a coincident critical load factor. Only symmetric elastic structures subjected to symmetric loads are considered, and sensitivity coefficients are found for a symmetric design modification which corresponds to a minor imperfection. It is shown that the formulation for sensitivity analysis of multiple linear buckling load factor can be successfully combined with that of nonlinear buckling to develop a formulation for coincident nonlinear buckling load factor. The proposed formulation is verified by analytical examples of simple spring–bar systems.

Sensitivity of buckling load factor of an elastic structure with respect to an asymmetric imperfection has been extensively investigated and general forms of imperfection sensitivity analysis have been presented for distributed parameter structures (Koiter, 1945) and for finite dimensional structures (Thompson, 1969; Thompson and Hunt, 1973). It has also been pointed out that imperfection sensitivity can increase due to interaction of buckling modes if two or more critical points coincide or are closely spaced (Ho, 1974; Huseyin, 1975; Thompson and Hunt, 1973; Hutchinson and Amazigo, 1967). Most of all the studies on sensitivity analysis of buckling load factor, however, concern reduction factor of the critical load due to asymmetric imperfection that is classified as major imperfection. Evaluation of sensitivity with respect to a major imperfection is very important to estimate maximum load factor of a structure that has an unavoidable manufacturing or construction error. In the engineering practice, structures to be built often have symmetry properties, and variation of critical load factor with respect to a symmetric design modification is to be evaluated in the process of redesign or design modification. For a symmetric structure subjected to a set of symmetric loads, which is called symmetric system for brevity, symmetric design modification is conceived as minor imperfection. Roorda (1968) derived analytical formulation of sensitivity analysis of bifurcation load factor with respect to a minor imperfection, and showed that the sensitivity coefficient is bounded for such an imperfection. Ohsaki and Uetani (1996) presented three approaches for sensitivity analysis of bifurcation load factors of symmetric systems, one of which is an explicit form of the formulation by Roorda (1968). In the field of optimum design, on the other hand, extensive research has been made for computing sensitivity coefficients of linear buckling load factor with respect to design variables such as stiffnesses and nodal locations. Such sensitivity coefficients are called design sensitivity coefficients which are used for optimum design and simply for redesigning process. Note that imperfection sensitivity and design sensitivity are virtually identical, and same formulations should be developed. Ohsaki and Nakamura (1994) incorporated the method of imperfection sensitivity analysis for finding optimum designs for specified limit point load factor. It is well known that optimum designs for specified linear buckling load factor often have multiple or repeated buckling load factors that are nondifferentiable; only directional derivatives or subgradients (Mistakidis and Stavroulakis, 1998) can be defined (Masur, 1984; Haug et al., 1986). Several algorithms have been presented for design sensitivity analysis of multiple buckling load factors (Seyranian et al., 1994), and optimum designs have been obtained for plates and shells under constraints on linear buckling load factor. It has been well discussed in the field of stability analysis that optimization for nonlinear buckling results in coincident buckling that may dramatically increase imperfection sensitivity (Huseyin, 1975; Thompson and Lewis, 1972). Ohsaki (2000) demonstrated that optimization does not always increase imperfection sensitivity even if the optimal design has coincident critical points. In this paper, a new formulation is presented for sensitivity analysis of coincident buckling load factor of symmetric systems with respect to symmetric design modification which is classified as minor imperfection. The validity of the proposed formulation is demonstrated through the examples of two- and three-degree-of-freedom spring-bar systems.

A new formulation has been presented for sensitivity analysis of coincident buckling load factor of symmetric systems with respect to symmetric design modification. A weak condition is introduced for defining a symmetric system. In this definition, the prebuckling deformation should be orthogonal to all the buckling modes corresponding to the coincident critical point. A new concept called completely minor imperfection has also been introduced for symmetric systems with coincident bifurcation points. For a completely minor imperfection, the condition for the minor imperfection is to be satisfied by all the coincident buckling modes, and the sensitivity coefficients of a coincident critical load factor are bounded if a completely minor imperfection is considered. A symmetric design modification corresponds to a completely minor imperfection if the system is symmetric. The formulation for sensitivity analysis of a simple bifurcation load factor with respect to minor imperfection has been extended to the case of coincident buckling. It has been shown that sensitivity analysis of coincident bifurcation load factor can be carried out in a similar manner as multiple linear buckling load factor. The directional derivatives are computed by solving an eigenvalue problem for obtaining the proper choice of the coefficients for the eigenmodes. Detailed formulations have been presented for a two-degree-of-freedom spring-bar system to illustrate the validity of the proposed formulations. It has been shown that subgradients may be found by incorporating arbitrary set of the coefficients into the Rayleigh's quotient. Since this is a special case where all the buckling load factors coincide, applicability for a general case has been demonstrated by a three-degree-of-freedom spring-bar system. Application of the proposed method to finite dimensional models with moderately large degree of freedom is to be in a similar manner as shown in the examples in Ohsaki and Uetani (1996). If the undeformed state is taken as the reference configuration for defining the strains, the third order differential coefficients of the total potential energy with respect to the displacements can be easily obtained by using appropriate package of symbolic computation, if necessary.