Wyboczenie Prętów Smukłych Przy Krótkotrwałym Obciążeniu - Kordecki, Z. (Engineering Transactions, 1964)
The object of the present considerations is the effect of rapidly growing short-duration longitudinal-compression force on a bar having an initial curvature. The influence of the inertia forces. due to the accelerations normal to the axis of the beam is taken into account. The relation between. the force and the time is assumed to be linear, which reflects to a considerable degree the conditions of catastrophic loads head frame structures.From the fundamental equation of the problem (1.1) a second order ordinary differential Eq. (2.7) is obtained by separating variables from which a function of time is obtained. This equation is accurate (within the range of small deflections) for a bar on hinged supports at both. ends and having a small curvature in the form of a sinusoidal half-wave. For other types of support it constitutes an approximate relation.In the sub-critical range, the solution of (2.7) is expressed by ordinary Bessel functions. For loads exceeding the Eulerian force it is expressed in modified Bessel functions with the same labels. However, for the value of the function of time at the instant where the load attains the value (3.13), may be obtained. These equations have of the Eulerian force simpler-approximate Eqs. (3.13), been derived by replacing Bessel functions with their asymptotic expansions. In the range important for catastrophic loads on head frame structures, the accuracy of these equations is very high. It is found that for elements of head frame structures, the influence of inertia forces is essential, if the maximum longitudinal force is greater than 70 percent of the Eulerian force. It is found, in addition, that if the ends are clamped the influence of the inertia forces is reduced, In conclusion stress is laid on the importance of the initial curvature for the problem under consideration.