FRAMED STRUCTURES UNDER STRESS. 653 
is applied or removed. To give definiteness to the discus- 
sion, let us assume 4B to be 10 feet, BC 14 feet, ABC 45°, 
and the stress to be 5 tons per square inch on 4B and ABC, 
and the direct modulus of elasticity of the material in either 
tension or compression 30,000,000 lbs. per square inch. 
The elongation of AB will then be ;,/-, of its whole length, 
or 5}, -foot or .4,-inch very nearly, and by a similar calcula- 
tion the shortening of BC will be ;,-inch very nearly. 
Now, suppose that from 4, as a centre, we draw a circle 
10 feet plus 45-inch radius, and from B one of 14 feet less 
z'g-inch radius, the intersection of these circles will give 
the new position of the point B. But practically this can- 
not be satisfactorily done, for on any ordinarily convenient 
scale, such as say | inch to 1 foot, the whole movement of 
B is so smal] that the slightest spring or yielding of the 
compasses used, or even the thickness of lines drawn by 
ordinary pencil or drawing-pen will be sufficient to entirely 
mask and overpower the dimension we seek to obtain. 
Even were we at great inconvenience to make a full-size 
drawing of the bracket, we could hardly, with the most 
scrupulous care, obtain more than an extremely rough 
approximation to the result sought. To carry out the 
operation satisfactorily, we need to do our plotting on a 
scale at least 10 times the full size of the structure; but this 
is out of the question, requiring in the present case a sheet 
of paper more than 100 feet square. 
At this point comes in the artifice upon which the method 
depends. The arcs of circles used are in every instance 
excessively short compared with their radii—-in the case at 
present postulated they are not much more than 1 inch long 
for radii of 00*and 140 feet respectively. Such short arcs 
may, with but infinitesimal and perfectly negligeable error, 
be replaced by straight lines perpendicular to the radii, and 
if this be done there is no need to draw but a very minute 
portion of the gigantic diagram previously referred to. 
In Fig. 2 OB , represents to a scale of ten times the full 
size the elongation of AB in Fig. 1, and OB, to the same 
scale the shortening of BC. OB.,is to the same scale the 
total movement of & when the load is imposed, and BB, 
the deflection of the structure, also magnified ten times. 
As a matter of fact, B., should be by an excessively minute 
amount lower and more to the left than shown, as 4,», 
- should have been an arc of 100 feet radius with its centre 
to the left, and B.,, Bone of 140 feet radius with its centre 
in the direction OB,. But the error is imperceptible, being 
by calculation less than ;,'59-inch, and therefore absolutely 
imperceptible on an ordinary drawing. 
