BEAM OF RECTANGULAR CROSS-SECTION UNDER ANY SYSTEM OF LOAD. 
65 
PART I. 
Establishment and General Solution of the Equations op the 
Problem Discussed. 
^ 1 . GcubtOjI SJcctcJi oj' the Pvohlem pToposed. 
'Ihe consideration pf the stresses and strains which occur in a rectangular 
parallelepiped of elastic material subjected to given surface forces over its six faces 
leads to one of the most general, as it is one of the oldest, problems in the Theory of 
Elasticity. Lame, in his ‘ Lecons sur I’Elasticite des Corps solides,’ published in 1852, 
describes it as ‘Ge plus difficile peut-etre de la theorie mathematique de lelasticitd.” 
In spite of repeated attempts, however, the problem remains still unsolved. 
In its complete form it may be stated as follows :_ 
Let the origin be taken at the centre of the parallelepiped and the axes Oa;, Oy, 0^ 
parallel to its edges. Let the lengths of these edges be 2a, 26, 2c. Let u, v, w 
denote the displacements of any point {x, y, z) parallel to the three axes,’ and, 
following the notation of Todhunter and Pearson’s ‘ History of Elasticity,’ let 7t 
denote the stress, parallel to c, across an elementary area perpendicular to t, then we 
have the six stresses 
dii 
/ dv 
+ 
dw\ 
dx 
yz 
— 
\dz 
¥/ 
dv 
/div 
+ 
dv\ 
dy 
zx 
= C 
[d^ 
dw 
(du 
+ 
dv\ 
dz 
xy 
= 1^ 1 
\dy 
Ixj 
where S 
xx = XS + 2/1 
yy = \8 -b 2/1 
zz = XS 2/1 
die dv div 
dy Th ’ ^ elastic constants of Lame. 
( 1 )> 
dx ' d>/ ' dz 
Also u, V, IV must satisfy, inside the material, the following differential equations. 
/ V , \ ciS _ 
(X + jx) — 4- fJiV'^u = 0 
(X + /i) -f /aVG; =: 0 
+ C) = 0 
(2), 
where 
d^ 
d^ 
+ 
d^ 
df~ 
“*“■ dz^ ’ foi’ce acting on the matter inside 
the block. It is required to find the values of n, v, lo at each point, subject to the 
condition that the stress across the outer faces x ±a, y = ± b, z — ±c shall be 
arbitrarily given at each point—regard being had, of course, to the conditions of rigid 
equilibrium of the block. ^ 
VOL. CCI.-A. 
K 
