349 
1911-12.] The Elastic Strength of Flat Plates. 
St Yenant, and others. It is still engaging the attention of mathematicians, 
and several papers of great analytical interest have recently been produced. 
Chief among these is Dr Dougall’s treatise “ An Analytical Theory of the 
Equilibrium of an Isotropic Elastic Plate.” * The problem is there treated 
as a purely mathematical one, the object being to deduce the approximate 
differential equations for a thin plate from the exact equations of elasticity. 
It makes, however, no contribution to the solution of the equations for 
particular boundaries. Another recent paper, “ Sur le probleme d’analyse 
relatif a l’equilibre des plaques elastiques encastrees,” by J. Hadamard, 
and crowned by the Paris Academy, is of much theoretical interest. 
The results, however, of these analytical researches cannot be reduced 
to figures, except in a few special cases, as follows : — 
1. The circular plate for free and fixed boundaries and for uniform and 
concentrated loadings. 
2. The elliptical plate for free and fixed boundaries and for uniform 
loading. For concentrated loads the solution becomes very cumbersome, 
and, in fact, almost impracticable. 
3. For certain forms of no practical interest, such as the lima 9 on of 
Pascal and in particular the cardioid. 
The above refers to rigorous mathematical analysis only. But, as so 
often occurs in engineering theory, approximate solutions bearing a 
likelihood of success have been deduced for special cases. Thus, for square 
and rectangular plates, with fixed edges and uniform loading, Grashof gave, 
many years ago, an approximate solution, and this has been extensively 
used by boiler-makers and others in fixing working values. He assumed 
for the deflection an expression which (a) satisfies the conditions at the 
edge, ( b ) has the right sort of symmetry for a rectangle, (c) reduces to the 
known value for the deflection along a diameter in the limiting case in 
which the perpendicular diameter is very long. As the expression does 
not satisfy the fundamental differential equation, it cannot be correct, 
though it may not be very far wrong for points on the shorter diameter of 
a rectangular plate. Summarising, then : — 
1. Rigorous solutions have been obtained for circular and elliptical 
plates, and for practically no others. 
2. An approximate solution has been given for square and rectangular 
plates. 
Very little experimental work seems to have been done on the subject 
of flat plates, and the results recorded are variable. It therefore seemed to 
the author that there was room for further work on this question. He 
* See Proceedings of the Royal Society of Edinburgh, 1904. 
