Friday, April 10, 2009

Irodov Problem 1.301

a) Consider the section of the cantilever OB such that point O is located at a distance x from the rigid end point to which the rod is attached.

If the moment generated at point O is N(x), we know that the shape of cantilever obeys the equation,

The derivation of this equation is given at this link.

For this part, the bending moment is independent of x and equal to No in the clockwise direction. In order for the cantilever part OB to be steady (no turning under the influence of the couple No), the moment generated at the end O i.e. N(x), must exactly counter-balance No. In order to generate an elastic moment in the counter-clockwise direction the upper surface OB must be stretched more than the lower surface of the beam O'B' as shown in the figure. This in turn means that the shape of the beam must be a concave function i.e. . Thus we have,

The negative sign
indicates depression.

b) For the second part since a force F acts at the end, the moment of this force about point O is given by in the clockwise direction. Hence, as in part a we have,

The negative sign indicates depression.

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