86 
MR. L. N. G. FILON ON AN APPROXIMATE SOLUTION FOR BENDING A 
the Euler-Bernoulli deflection if I 
poses of numerical calculation 
we suppose uni-constant 
or, /< ’069 6, if for pur- 
isotropy and therefore 
E = 5/x/2. 
So that if I be less than about a^th of the height of the beam, the correction to 
be ajDplied to the Euler-Bernoulli deflection becomes negative. The critical point 
where, as we shorten the span, the correction passes from additive to subtractive 
corresponds to I slightly, but only very slightly, greater than ‘069 6, as in the 
neighbourhood of this value Xq is quite sufficiently small to make (sin iiXq/wXq)"^ = 1, 
a fair approximation for all the most important part of the range of integration 
of the integral in (63). 
We see therefore that when we have a beam loaded in this way, with a section 
of symmetry constrained to remain plane, the deflection at the centre, for all spans 
greater than ^^-th of the height, is larger than the one indicated by the Euler- 
Bernoulli theory. In the limit when the span is made very large, this additive 
correction is found to be of the same form as that given by de Saixt-T exaxt for 
a cantilever under special conditions of end fixing, but the coefficient is different, 
the correction being just under fths of de Saint-Venaxt’s value. For spans smaller 
than height the correction is negative. 
§ 14. The, Douhly-supported Beam under Central Load. Expressions for the Strains 
and Stresses lohen we remove the Sup)ports to the Two Extremities. 
Going back to the general expressions for U, V, P, Q, S given in § 10, if we have 
a beam as in § 13, but we make the two supports coincide with x = i a, Ave have 
— R — ~ 
“o — Po — 
W 
w 
-(- 1)“-. 
with the folloAving values for the displacements and stresses :— 
>/ ® W siiih 'Inirhla 
u = 
jjb 1 a sinh ■inirhla -|- 4:7'i7rhja 
V W co.sh(2/i- -|- V)iTljja 
/j, a Q siuh (4?i + 2)7rb(a — (4/i -1- 2)7rhla 
1 
sin 2mrx/a, sinh 'Imri/fa 
sin [ 2)1 fl- 1) TTxja cosh (2u -f 1) ny/i 
a 
AV 
cosh {2n-\-l)’7Thyi 
1 l)7rt , ,, 7 
-^^— smli ( 2)1 + 1) iTO, 
A 
!(l 
a 
rx 0 (2?i.-l-l)7r sinh(4?i -|- 2)7rbla — {+n -f 2)7Tbla 
1 
sin(2?i-l-l)7r.r/a sinh (2?i-l-1) Try/a 
W - 
__ V 
, ^ , sinli 2n7rb/a — — Cldf cosh 2)i'7rbja 
a W + /X ‘ fx a ' 
a 1 2mr 
3W«. 
sinh +)i'irbja 4- -InTrbja 
1 vlYx 
sin 'ImTxja cosh ^mryja 
