Large Deflection of Thin Plates Under Certain Mixed Boundary Conditions—Cylindrical Bending

Abstract

Considered in this paper is the large deflection of a thin beam. One end of the beam is fixed (clamped) to a rigid wall, while the other end is placed on a flat surface of arbitrary orientation. Under the assumption that the axial deformation of the neutral axis is negligible, a closed form analytical solution to the deflection curve is obtained in terms of elliptical functions. The analytical solution is shown to have certain scalability properties with respect to the beam length and cross-section. By using appropriate bending rigidity, the same solution can be used for a thin plate under large cylindrical bending. In addition, a finite element analysis using nonlinear shell elements is also conducted showing the axial strain of the neutral axis to be less than one percent of the overall deformation. Therefore, it is valid to assume that the axial strain is negligible.

Large deflection of thin plates in cylindrical bending – Non-unique solutions

“…Note that both components are unknown, and must be solved as part of the boundary value problem. We have demonstrated in a previous paper (Xue et al, 2002) that the axial deformation in problems of this nature is negligible. Hence, we assume e = 0 in our problem.…”

“…Note that a semi-analytical solution for the special case of F A = 0 was presented by Xue et al (2002). For dh dS 6 ¼ 0, the left hand side of (3.4) may be rewritten using the following relationship:…”

“…The derivation of the basic governing equations is described in detail in a previous paper (Xue et al, 2002), and summarized here for convenience. Fig.…”

“…The governing equations for large displacement, small strain deformation of simple beams were derived in a previous paper (Xue et al, 2002), following the approach of Frisch-Fay (1962) and Atanackovic (1997). The basic assumptions used are:…”

“…In our problem, similar governing equations are used except that one end of the plate is assumed fixed (clamped) to a rigid wall, while the other end, known as the free end, is placed on a flat surface of arbitrary orientation. Under the assumption that the axial deformation of the ''middle surface" is negligible, a closed form analytical solution to the deflection curve, for the specific case of a load that is normal to the beam at the free end, was derived by Xue et al (2002). Their solution forms a basis for the current work.…”