142 
MR. L. N. C4. riT/)X ON AN APPROXIMATE SOLUTION FOR BENDING A 
§ 37, Apiylicafion of Solutions of § 33 to the Case of Tension Proclueed hy Shearing 
Stress Applied to the Edges. 
In practice test-pieces for tension are nsnally strained by pressure , applied to 
projecting collars, tlie latter transmitting this pressure to the body of the material in 
tlie shape of shear. In no case can we a})ply tension directly to the ends of a bar. 
It is therefore important to knoAv hou' far tlie eftect of the method of application of 
the total pull disturbs the usual solution for a uniform tension. 
bet us then consider the effect of liaAung two concentrated shears L, one as before 
acting at the point (0, h), and another e(^[ual and parallel to the first acting at the 
point (0, — h). By superimposing on one anotlier two solutions of the type 
olMained in § 33, we get the solution recpiired. It will he found that this solution 
gives a tension L/26 over the left-hand extremity of the beam and a pressure L/26 
over the right-liand end. 
If Ave require to liave no tension over the right-hand end, and a uniform 
tension L/6 at the left-iiand end, to balance the shears, Ave haA^e to introduce the 
uniform tension solution P = L/2/>, Q = 0, S — 0, U = L.r/2/>E, ^ = rjhyl'2l)Fj ; 
AA^e eventualK' find 
U = - 
X' -K 2/x ;ix' + 2ii Cf 
+ 
+ 
In 
]Gyu.(X' -\- (m) dh IG/x (X' + fx) ah ‘2t 
Y = 
» L 
"h — — 
1 rn a 
X' 
—) cosli vih — — sinh mh 
'K ^ fi / 
siiili + 2riih 
+ 
L 
-(cosh my cos mx — 1) 
())})/ cosh rnh) . , 
smh my cos mx 
L 
I vijxa sinh 2mh -f 2mh 
eosli rah 
_ hr,// Lr;,v *__ 
S/x{X' + /j) ah ** 2//E *1 rrifia sinh 2mh -f 2mti 
my cosh my sin mx 
_ V 
ran 
L X^ -j- fx 
cosh rati H- mh sinli rnh 
A 
sinh mif sin mx 
T. 
sinh 2mti + '2mh 
L 4 cosh rah — 2mh sinh mh 
s'_■ 
2ah ‘ ^ ~i u 
2L cosh 7/1A 
T’ = (« - ■^) - 
Q = 
a sinh 2ath -f 2mh 
2mh sinli mti 
s = 
_ s' 
* L 
s 
d a sinh 2mh -f 2mh 
>; 2L cosh ///// 
1 a sinli 2mh -f 2mh 
I,// « 2L cosh mh 
4. V __ - 
2ah i a sinh 2)/>h -f- 2)ah 
+ 
sinh 2mh + 2mh 
my sinh my sin mx 
cosh my sin mx 
my sinh my sin mx 
my cosli my cos mx 
cosh my sin mx 
+ N ■- 
1 a 
2L (cosli mh — vih sinh mh) 
sinh 2mh -(- 2mh 
sinh my cos mx. 
