In early days as in the 1940s, a partial cylindrical concrete shell (often referred to as long shell) was frequently used as a barrel roof structure. A simple guide was proposed as to the buckling of a long shell in the ASCE Manuals of Engineering Practice (1952), by making use of the theoretical buckling load of a circular cylinder subjected to the action of uniform axial pressure.
It is to be noted that a strict membrane stress field is difficult to be maintained in a barrel-roof-type shell, because the number of stress unknowns is unfit for the number of available boundary conditions. Accordingly, for a long shell, use has been made of approximate membrane stress states in a “shallow” partial cylinder under external lateral pressure in not only a linear stability analysis but also a non-linear large deformation analysis.
In a previous paper, the present author has shown that a cantilever-supported shell segment cut out from a circular cylinder with particular shapes and boundaries is able to satisfy the condition of a membrane stress field, under the action of a variety of loads; by assuming strain-displacement relations stay linear until buckling takes place, the stability of the shell segment can be analyzed as an eigenvalue problem.
To be examined in this paper are two examples that are regarded as being typical and comprehensive. Each loading of self-weight and earthquake excitation is considered for varying thicknesses and shapes. Initiated with nonlinear equilibrium equations which are known as Love's Formulae and referred to as “general case” by Timoshenko, simplification is made by estimating non-linear terms with reference to the order of magnitude. Based on the assumption that the linearity of load-displacement relations remains valid until buckling takes place, above equations are led to equilibrium relations valid for an early post-buckling stage. These are made discrete through FEM formulation in Galerkin type weak form, being led to a system of equations in conformity with a linear eigenvalue problem. Further study is made to see how each parameter influences on partial matrices appearing in the overall FEM formulation.
(In appendices at the end of the paper, the results of an analysis on an example of eigenvalue problem are compared with the results from the classical theory of elastic buckling, leading to perfect coincidence.)
Linear buckling analyses are then conducted for each of the two load cases as eigenvalue problems by varying thickness and shapes as parameters. Results are shown for the load factors of buckling as contour lines in a plane of parameters; radial displacements of free edge lines are drown overlapped, representing modes of buckling for each combination of the parameters.
It is intended to show general buckling characteristics of the shell in the form of graphics over the parameters varying widely.
Finally, a large deformation step-by-step analysis is carried out, by converting the non-linear weak form led from the calculus of variation into a matrix form by making use of the predetermined values in the preceding step. An analysis is repeated for each load level, modifying the fore-going matrix step-by-step, until the displacement converges.
Consequently, it is shown diagrammatically that buckling occurs at the minimum load level that the displacement does not converge but diverges, and that a large displacement appears suddenly with the same mode as that of linear buckling. Material behavior is assumed to be linearly elastic, throughout the paper.
