G J

Here the first of Eqs. (3.81) and the final boundary condition from Eqs. (3.77) are used to derive the third boundary condition.

The exact solution for Eqs. (3.83) and (3.84) has been obtained by Diederich and Budiansky (1948). Its behavior is quite complex, having multiple branches, and it is not easily used in a design context. However, a simple approximation of one branch is presented next and compared with plots of the exact solution.

Approximate Solution for Bending—Torsion Divergence

In view of the complexity of the exact solution, it is fortunate that there are various approximate methods for treating such equations, one of which is the application of the method of Ritz to the virtual work principle (see Section 2.4). In this special case, the kinetic energy is zero, and the resulting algebraic equations are a special case of the generalized equations of motion (see Section 2.1.6), termed generalized equations of equilibrium. Determination of such an approximate solution is left as an exercise for the reader; see Problems 11-15.

Here we consider instead an approximation of one branch of the analytical solution for the bending-torsion divergence problem. Fortunately, the most important branch from a physical point of view behaves quite simply. Indeed, if one defines r = ^ cos2(A), GJ

qcal

then, as shown by Diederich and Budiansky (1948), the divergence boundary can be approximately represented within a certain range in terms of a straight line

7T2 3?r2

+1 0

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