BEAM OF RECTANGULAK CROSS-SECTION UNDER ANY SYSTEM OF LOAD. 125 
or putting E = 5g/2 to simplify the aritlimetic 
2-80 — 4 - 9 G ' • 
Tr 
The slope of the strained form of tlie axis at the origin is therefore a maximum 
when 2-80 — 4-96 X v, = 0 , or lib = • 434 . 
For such a value of l/b the approximation will not be quite valid. fStill, it will be 
sufficient, even then, to give a rough idea of the values of the coefficients. 
Assuming the formula given for V to hold for this value of l/b, we see that this 
greatest slope is — (• 810 ). 
Now if the part of the beam between x- = ffi / Avere subjected to a uniform shear 
'W/2b giving the same total shear across the section, then, if the sections x = ffi / 
were kept vertical, we should have V = — X 1 - 25 , if E = 5u/2. This 
2 b ji hi) 
gives a slope nearly 3/2 of the preceding one. 
W.rl 
EU 
§ 31. Difitortiou of the Cross-section x = 0, and Shear in that C^'oss-section. 
II we Avork out in the same Avay the value of U for x = 0 we find 
•TT 0 
l' x + n j (2r + 1). n 
1 - 4 . 1 S 
Cv)\ 
If I be very small and yjb sufficiently small for 5 th and higlier poAvers to be 
neglected, this gives, assuming E = 5 p ,/2 to simplify the arithmetic. 
U ^ 
ttYII 
(4 11 — 2-5 G^) + (I Ey — lA 
^.e., 
r = 
wy 
7rK/r 
- 5-292 + '^(-492) 
We see, therefore, that the '}/ term is practically negligible, or, for a very large 
range of y, the mid-section remains sensibly plane. 
For the shear in this cross-section, Ave have 
^ S (jC\^ i . ; /), 
'rrl 0 -‘'V & / (2r) irh ^ / I 
tv+'l ;|^ 
(2r -f ]) : 
or 
® ~ “ b A (\f) - Slj - 
A A (m 
ttIj 52 V\ 2 } 
!S is therefore a numerical minimum at the centre if 7“ — > 0 
