R

This result, showing that the angle a is simply related to v, was proposed by Cornu [38] for the determination of Poisson's ratio (Fig. 1.55). Since v < 1/2, we always have arctan a >V2 i.e. a > 54.735°.

Because the four edges of the plate do not remain straight lines in the bending, for a more accurate application of the boundaries the optical interferences are generally obtained with a long plate where the moments Ma and -Ma are applied to the short edges. Cornu's method with a He-Ne laser source has become of classical use for accurate Poisson's ratio measurements.

Fig. 1.55 Interferogram of a bent rectangular plate for the determination of Poisson's ratio. This method was proposed by Cornu [38] (after Timoshenko [158])

Fig. 1.55 Interferogram of a bent rectangular plate for the determination of Poisson's ratio. This method was proposed by Cornu [38] (after Timoshenko [158])

• Plate bent by a load: Because of the discontinuity at the boundaries due to the corners of the plate, the flexure of rectangular plates submitted to a load q is a complex problem. The general solution of Poisson's equation without second member cannot be dissociated from that of the particular solution.

For a simply supported rectangular plate where the edges remain straight lines, a first and classical solution of the problem is due to Navier [114] who showed that if both the load q and the flexure w are represented by a double trigonometric series of the form

m=1 n=1 a b then, Poisson's equation can be satisfied.

An extended account of the analytical developments of rectangular plate flexure using the double series representation and other representations is given in Timo-shenko and Woinowsky-Krieger [155], who also consider cases with various edge conditions and treat continuous floor-slab plates. For active optics methods, the rectangular plate case is of very limited interest.

1.13.10 Axisymmetric Bending of Circular Plates of Constant Thickness

The flexure w of the mid-surface of a thin circular plate resulting from an axisym-metric load is derived from Poisson's equation. This equation is now only relative to the radial distance r from the center and becomes, from (1.164b),

0 0