BEAM OF KECTANGULAR CROSS-SECTION UNDER ANY SYSTEM OF LOAD. 
119 
^ 2W ^ mh cosh mb . . , 
^ ^ cos mi —■ -—- sin mx sinh mi/ 
a 1 sinh 2mh + 2vib 
2 W ” my sinh mb 
-i COS ml —- - sin mx cosh my 
(0 1 sinh zmb + 2mb 
2W * . mb sinh mb 
— N sin ml — — - --—r cos 'mx cosh mi/ 
a 1 sinh 2mb — 2mh ^ 
2W ® . mil cosh mb 
H-S sin ml ■ , „ , --— 
a 1 sinh 2mb — 2mh 
cos mx sinh 
my 
( 90 ), 
where A in the above is an arbitrary constant representing a rigid body rotation. If 
the conditions of fixing are that the two extremities of the horizontal axis are to 
remain at the same vertical height after strain, A is zero. 
If, on the other hand, we fix the beam in such a way that the sliears Wlja over 
the ends are each allowed to produce, at the extremities of the axis, the deflection 
which they would produce if the bar were clamped at its middle and the deflection 
were calculated on the Euler-Bernoulli theory, then we find Aa = Tbit; 
2 E &3 ■ 
appears to be the more natural method of fixing. We shall, therefore, in what 
follows, suppose A to have this value. 
§ 28 . Integral Expressio^is ivhen a is made Infinite. 
When we increase the length of the bar indefinitely, it is easy to show that, if we 
take the last given value of A, the displacements remain finite at a finite distance 
and the stresses remain finite throughout—excepting, of course, at the points where 
the concentrated loads act. 
We then obtain, as in § 15 , 
U= - 
TTr 1 1 I / I Slllll U Vj cosll Ih I , 
VV I 1 X ^ n. vl uy . v.r , 
, “ \ • , o—TTl-- / COS — cosh sm - du 
IT Jo r \ sinh 2e + hr / h b h 
id . , iry . ux 
W?/ r” siuli e 
— 7 - , - 7— COS — smh sm —da 
yuirb Jo smh 2iu -f '2u h b b 
f ( 1 u 
1 77 -cosh It — - sinh ii 1 , 
1 \X-f/i, /x 
vr Jo 
W 
sinh 2u — 2u 
j . (11- . wii ux / 1 
sm —smh Wcos— — 
X'hyjIvHr 
cosh u . vl uy ux 3 l \ , 
- -sm-coshycosy-—- 
sinh 2u. — 2n 
(97) 
'Ay > da 
