IMPACT UPON BEAMS. 
105 
of beams, so as best to resist concussions from a given load 
passing, upon an uneven road, over it. The formulae would 
apply too to impacts upon some other elastic structures, q being 
the pressure from the weight of the structure, or its inertia, in 
the point of impact. 
Theorem. The locus of ultimate curvature of all the points 
in a slightly flexible beam, whose depth is equal throughout, is 
a parabola. 
Let A D B be the natural F 
form of the beam, supported 
at its ends A, B; A C B its [_± c 
form when bent at C to the 
extent of its elastic force; C F perpendicular to the curve at 
C ; and A F, B E, perpendiculars from A B upon C F. 
Then, since it is shown by writers on mechanics that the 
ultimate deflection of a beam is as the curvature multiplied by 
the square of the length, we have 
C F (= deflection of the part A C) as (AC) 2 x curvature, 
C E (= deflection of the part B C) as (B C) 2 x curvature. 
But a beam of uniform depth will bear the same curvature in 
every part. We have therefore C F to (A C) 2 , and C E to 
(B C) 2 , in a constant ratio. 
Putting then l — A C B the length of the beam, x = A C, and 
c— a constant quantity such that C F = c (A C) 2 , CE = c(B C) 2 , 
we have C F = c.r 2 , C E = c (l — x) 2 — c (Z 2 — 2 l x -f x 2 ). 
Whence EF = CF - CE = d(2i- l). 
But the right-angled triangles A F D, BED are similar, 
BE : DE :: AF : FD; 
and AF:FD::AF:FD, 
AF + BE : FB+DE :: AF : FD. 
Here FD + DE ~ EF = c / (2 ^ ; and as the deflection 
is small, AF + BE = ACB — / nearly. In this case A F = x, 
BE = l — x , and the last proportion above becomes 
l : c l (2 x ~ l) : : x : F D. Whence F D = c x (2 x —■■ l). 
If we subtract the value of F D, just found, from that of C F 
= c x 2 , we obtain C D = c x 2 — c x (2 x — /) = c x (/ — x) ; 
where C D may be taken for the deflection of C, it being nearly 
perpendicular to A D B. Whence it appears that the ultimate 
deflection at any point C is as the rectangle of the segments 
into which the beam is divided at that point. The locus of these 
points is therefore a parabola. 
In the preceding theorem we have supposed the ultimate cur- 
