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This paper studies the critical issue of thin-walled beams with laterally fixed ends. The method for defining the formulae of twist moment for the beams subjected to combined loads was elucidated. Based on this, the governing differential equations of the beam were developed. The procedure for determining the critical state of the beam by the energy method was presented. With this procedure, the critical state of the beam concerned under three types of loadings such as axial force F , bending moment M and distributed load q (or concentrated load P) was examined deliberately. The outcomes were presented in explicit closed-form, which can be illustrated in 2D and 3D graphs. Also, the analytical solution obtained was in agreement with the numerical one obtained by the commercial software NX Nastran. Furthermore, the analytical solutions were applied straightforwardly to explore the stability and design optimization of the tooth-blade for the new frame-type saw machine under an eccentric load. The result can also be promisingly used to study problems of thin-walled beams with laterally fixed ends subjected to other types of loads. [ABSTRACT FROM AUTHOR]

The assessment of the design buckling resistance of single members is usually based either directly on the elastic buckling resistance of the member or indirectly on its non-dimensional slenderness computed from the elastic buckling resistance. Specifically, Eurocode 3 buckling curves define the buckling reduction factors as a function of non-dimensional slenderness and, according to EC3 "General Method", these curves may also be used for non-uniform members. In this context, a new procedure will be presented for the computation of the elastic critical moment of tapered members. As is well known, the elastic critical moment strongly depends on both the bending moment diagram and end support restrictions. For uniform members, elastic critical moments may be computed using a relatively simple formula in which the bending moment distribution is taken into account by an equivalent uniform moment factor, and the end support restrictions are introduced through the buckling length. Unfortunately, this formula has not been extended to tapered members and, as a consequence, the elastic critical moment for tapered beams must be obtained using numerical methods such as the finite element methods. Based on a comprehensive parametric study for the elastic critical moment of tapered beams with different moment diagrams, this paper offers a new procedure, called the Equivalent Moment Approach, for the substitution of a tapered beam with any moment diagram by an equivalent uniform beam. One advantage of the present procedure is that closed form expressions valid for uniform beam can be generalized and used for tapered beams. [ABSTRACT FROM AUTHOR]