XVI
269
FLOATING ELASTIC PLATES
according to Reusch,' E is equal to 236 kg/mm².
obtain for different thicknesses h
h=
=
10
20
α=
= 0.38
200 mm.
50 100
0.64 1.27 2.14 3.60 m.,
whence we easily get the depression produced by 100 kg.
20 = 86.4
30.5 7.72 2.73 0.96 mm.
2. The strain produced in the plate depends on the
second differential coefficients of z with respect to x and y;
hence it becomes infinite at the origin. This shows that the
greatest strain cannot be found without a knowledge of the
distribution of the weight. We shall calculate the maximum
tension in the simple case when the weight P is uniformly
distributed over a circular area of radius R, where R is
supposed small compared with a. For this purpose we
calculate 2% at the origin. If we call the distance from the
origin of the element at which dP rests p, then the portion of
2% due to this element is, by equation (3),
a'dP
2πS
(log ap-log 2+C),
where the terms which vanish with p have been neglected. A
simple integration now gives
v²%0 = 2
a2z
Əx²
a2z a¹P
= 2
=
dy2 2πs
(log aRlog 2+ C)
=
a*P
2πS
(log aR-0.6519).
The maximum tension at the centre of the curved plate
is p = (Eh/2)²z/dx²; by forming the expression for p and
substituting for at its value we find
p =
3(1 - µ³)P
2πh2
(log aR-0.6519).
It would be a mistake to attempt to apply this formula
even when R is of the order of the thickness of the plate or
1 Reusch, Wied. Ann. 9, p. 329, 1880.