Economy of Material in Frame-structures. D9L 
as any elementary line in the deformed space, 7. e. if e=e in 
all parts, the signs of inequality may be replaced by that of 
equality, and 
cad a nS lita 
so that > .ly [fj] is a minimum, and consequently from 
(3), Vn, the volume of material in the frame M, is also a 
minimum. 
A frame therefore attains the limit of economy of material 
possible in any frame-structure under the same applied 
forces, if the space occupied by it can be subjected to an 
appropriate small deformation, such that the strains in all 
the bars of the frame are increased by equal fractions of their 
lengths, not less than the fractional change of length of any 
element of the space. 
If the space subjected to the deformation extends to infinity 
in all directions, the volume of the frame is a minimum re- 
latively to all others, otherwise it will have been shown to be 
a minimum only relatively to those within the same assigned 
finite boundary. 
The condition e=e can evidently be satisfied when all the 
bars of a frame have stresses of the same sign. ‘The test 
deformation to be applied is then a uniform dilatation or con- 
traction (according as the frame is in tension or compression) 
of a space enclosing the frame and extending to any finite: 
boundary or to infinity at pleasure. 
The simplest minimum frames of this special class are :— 
I. Ties and struts subjected to a single pair of equal and 
opposed forces. 
II, Triangular and tetrahedral frames under forces applied 
at the angles of the figure, and acting along lines 
which intersect within the figure. 
III. Catenaries in general, the points of application of the 
applied forces, as well as the forces themselves, being 
given. 
In all these cases the minimum volume of the frame is 
ee Zalig ps OW Sel cos 8 
Pi wi ee le 
where F is one of the applied forces, 7 the distance of its 
point of application, R, from an arbitrary fixed point O, 
2T 2 
