166 | APPLIED MECHANICS 
become a column fixed at one end and loaded at the others as shown at 
2 
(t), Fig. 235, where L=3L, Hence, P=7 1-7 | which shows 
Li 41 
that a column fixed at one end and loaded at the other has only one- 
fourth the strength of the same column with hinged or rounded ends. 
159. Empirical Formule for Struts.—The Euler formule for struts 
is rational, but it is only applicable to struts which are very long com- 
pared with their least transverse dimension, and when applied to struts 
which are common in practice they give values which are too high for 
their strength. Numerous empirical formule have been devised by 
different authorities to give the strength of ordinary struts, the constants 
or coefficients in these formule being derived from the results of experi- 
ments on struts. The empirical formula which has been most used is 
that known as the Rankine or Rankine-Gordon formula. 
160. Rankine-Gordon Formula for Struts.—The Rankine-Gordon 
formula is p= > Matas, 9» Where 
A 1+a(7) 
P=crushing or crippling load on strut in tons. 
p=crushing or crippling load on strut in tons per square inch of 
cross section. . 
f=direct crushing strength of the material of the strut in tons 
per square inch. 
A =area of cross section of strut in square inches. 
L=length of strut in inches. 
k =least radius of gyration of section of strut in inches. 
a == constant. 
Values of and @ commonly taken for different materials in different 
cases are given in the following table :— 
Values of a. 
Material. Ff 
Case I. Case II, Case III. 
Cast-iron . 3 : 36 pay 1.4: = 1 16 = 1 
6,400 6,400 1,600 9 x 6,400 3,600 
Wrought-iron . -| 16 : - = shin ee = : 
36,000 | 36,0007 9,000 9 x 36,000 20,250 
: 1 a i eng 16 1 
Mild-steel . ; 21 = = 
30,000 | 30,000 7,500 9x 30,000 16,875 
i , 1 4 ik 16 1 
Dry timber(strong kind 3:2 ines Ss Lae kt ee peanh eee ee Pe 
7 io 3,000 | 3,000~ 750 | 9x3,000 ~ 1,687 
Case I. Fixed ends. Case II. Hinged ends. Case III. One end 
fixed and the other hinged. 
It will be observed that the values of a for Cases II. and IIL. are 
obtained by multiplying the values for Case I. by 4 and ao 
9 respectively. 
