Governing Equation for Elastic Buckling |
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Consider a buckled simply-supported column of length L under an external axial compression force F, as shown in the left schematic below. The transverse displacement of the buckled column is represented by w. ![]() ![]() The right schematic shows the forces and moments acting on a cross-section in the buckled column. Moment equilibrium on the lower free body yields a solution for the internal bending moment M, ![]() Recall the relationship between the moment M and the transverse displacement w for an Euler-Bernoulli beam, ![]() Eliminating M from the above two equations results in the governing equation for the buckled slender column, ![]() |
Buckling Solutions | ||||||||||||
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The governing equation is a second order homogeneous ordinary differential equation with constant coefficients and can be solved by the method of characteristic equations. The solution is found to be,
![]() where ![]() The coefficient B is always zero, and for most values of m*L the coefficient A is required to be zero. However, for special cases of m*L, A can be nonzero and the column can be buckled. The restriction on m*L is also a restriction on the values for the loading F; these special values are mathematically called eigenvalues. All other values of F lead to trivial solutions (i.e. zero deformation). ![]() The lowest load that causes buckling is called critical load (n = 1). ![]() The above equation is usually called Euler's formula. Although Leonard Euler did publish the governing equation in 1744, J. L. Lagrange is considered the first to show that a non-trivial solution exists only when n is an integer. Thomas Young then suggested the critical load (n = 1) and pointed out the solution was valid when the column is slender in his 1807 book. The "slender" column idea was not quantitatively developed until A. Considère performed a series of 32 tests in 1889. The shape function for the buckled shape w(x) is mathematically called an eigenfunction, and is given by,
![]() Recall that this eigenfunction is strictly valid only for simply-supported columns.
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