COLUMNS AND STRUTS 167 
d Most text-books give a so-called proof of the Rankine-Gordon 
formula, but the assumptions made are not warranted in the case of 
- actual struts. It is best to consider the formula as an empirical one, but 
it may be pointed out that when it is applied to very short struts it 
reduces to P= Af, which is correct, and when applied to very long struts it 
7 reduces to P= Vi JT x25 
ws L? aL?’ \ MILD STEEL COLUMNS. 
_ which is the form of the 429 6 HINGED ENDS. 
expression given by Euler’s ¢ NY, eC! 
theory of long columns. ec 15 Yor 
Fig, 236 shows graphi- @ ‘@ 
‘ NA 
cally the difference between 2 10 <N 
the Rankine-Gordon and § no 
the Euler formule applied , 5 ——— 
to mild steel columns with § Fe 
_ hinged ends. = 25 50 75 100 
If the results of the RATIO Lk oS og 
- Rankine-Gordon formula Fia. 236. 
be plotted for columns of 
different materials, instructive curves are obtained. Fig. 237 shows the 
- eurves for columns of cast- 
_ iron,wrought-iron, and mild COLUMNS. FIXED ENDS. 
_ steel with fixed ends. It RANKINE FORMULAE. 
_ will be seen that for very 
short columns cast-iron is 
- much stronger than either 
wrought-iron or mild steel, 
and that for ratios of L to 
k less than 85 the cast-iron 
_ columns are the stronger, 
but for ratios of L to k 
greater than 85 the mild 
steel columns are the 
stronger. Also for ratios 
of L to & less than 115 
the cast-iron columns are i 
stronger than the wrought- 25.50 75 100 125 150 175 200 
iron ones, but when the ah. RATIO Lik 
ratio of L to & is greater Fia. 237, 
than 115 the wrought-iron ; 
_ columns are stronger than the cast-iron ones. For all values of the 
ratio of L to & the mild steel columns are stronger than those made of 
wrought-iron. 
& 
7 
(4) 
Nn 
s 
— 
WRO : A 
Led ae 
PORN Stee . 
' eee! 
mR 
| 
ZK 
a 
[ 
l 
‘LOAD. TONS PER SQUARE INCH. 
ou & & 8 
Notr.—A defect in the theory of Arts, 155-158 is that the direct crushing 
stress is neglected; but in very long columns the direct stress is small com- 
pared with the bending stress, while in short columns the direct stress is large 
compared with the bending stress. 
