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UBC Theses and Dissertations

The strongest column : a matrix approach Kerr, Peter A.


The strongest column problem is defined, for this thesis, as the determination of the column shape which gives the maximum Euler buckling load for a given length, volume of material, type of cross-section and type of tapering. This thesis presents a new method - the matrix method - for solving some strongest column problems. In the matrix method a member is approximated by a number of uniform sub-members. A structure stiffness matrix, with the effect of axial force on deflections included, is generated from the sub-members. By setting the determinant of this matrix equal to zero the critical buckling load and the buckled shape of the member in the first mode are found. The section properties of the sub-members are then altered, according to the constant stress criterion, so that the extreme fibre bending stress, determined from the first mode, is the same in each sub-member. The process is repeated until the stresses are sufficiently close to being equal so that no further alterations are required. The optimum shape is taken from the last iteration. The constant stress criterion is based on the fact that, when certain conditions are satisfied, the extreme fibre bending stress at any section is constant along the length of the strongest column when it is buckled in the first mode. The matrix method gives results in very close agreement with those found by previous authors for cases where the constant stress criterion is valid. For the one example presented where the constant stress criterion was not valid, the column buckling under its own weight, the matrix method gave poor results.

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