Laterally loaded Struts and Tie-rods. 717 
At the origin 
ztfEI id 
-M = - p 
wl /El K/P n «N 
and the bending moment of greatest magnitude is at the fixed 
ends where 
M 
«?EI wl /EI ,1 /V „_. 
i= T^yVp cot 2Vi- • • ( 17 > 
If the forms (16) and (17) are expanded they show a 
similar relation to those in cases I. and II. between the 
exact method of calculating the bending moment and method 
(6) if deflexions above or below the points of inflexion are 
used. These points as mentioned above vary somewhat 
in position according to the value of P and method (b) 
becomes less simple than the exact method. 
Case IY. 
Uniform straight strut with central load W and ends firmly 
fixed in the direction of the axis of x. 
The equation is 
SPy - P W (I \ M, , 1R . 
where M 1 is the bending moment at the fixed ends. The 
solution gives the bending moment with sign reversed, 
W /EI c l /IF /"P" /~P~ i 
M= yv p- i tan 4y ei cos v M*~ sin v ei • • ] ' ( 19 > 
which vanishes for all values of P when %=. -. 
4 
M 
-Tv^-iVi—. • • '-'»> 
which becomes infinite for P= — ™ — . 
And expanding (20), 
47T 2 EI 
where P e = — ™ — , Euler's limiting value of P for the ideal 
strut. 
This again shows the relation of the method (b) to the more 
exact method, the deflexions being reckoned above or below 
