﻿Collapse of Tubes by External Pressure. 73 



A simple expression fulfilling these requirements is 



£ = 2*,—^ (12) 



±i_ Et* 



As the thickness is reduced, it approaches the limiting value 

 2E £ 3 /^ 3 , which is slightly less than that o£ Bryan's formula 

 (8) ; and since the ratio d/t cannot he less than unity, its 

 value is in all cases appreciably less than that which is given 

 by (11). Hence, provided that a cube is accurate in form 

 and of uniform material, it is clear that the pressure given 

 by this equation would be insufficient either to collapse it or 

 to impair its elasticity, so that the only modification required 

 for purposes of design is the insertion of a factor of safety 

 which shall make proper allowance for practical imperfec- 

 tions. Since the equation as it stands gives an estimate of 

 the collapsing pressure which errs on the side of safety, 1 

 believe that 2 would be amply sufficient as a factor of safety; 

 but this point can be investigated by a comparison with the 

 experimental results hitherto obtained. 



The similarity of (12) with the Rankine-Gordon formula 

 for columns will be at once remarked. Dr. Lilly* was the 

 first to recommend the employment of an equation of this 

 form, but in place of y c and E he suggests the insertion 

 of constants which are to be determined empirically from 

 the results of a complete series of tube-collapsing tests. 

 In general, when the comparison can be made, I imagine 

 that the difference between his formula and my own will be 

 negligible ; but the latter seems to me preferable, first, in 

 that it expressly provides against overstrain of the tube, and 

 secondly, owing to its greater scope : the appropriate factor 

 of safety may be determined from experiments on tubes of 

 any material, and when the corresponding formula for any 

 other material is required, it will be necessary only to change 

 the values of y c and E, which are quantities determinable 

 from one simple test f. 



It may be remarked here that in the case of the lap-welded 

 Bessemer steel tubes tested by Prof. Stewart J, the average 

 value of the stress at the yield-point was stated to be 37,000 

 pounds per square inch. If we assume a value of 30,000,000 

 pounds per square inch for the modulus of elasticity (which 



* Trans. Inst. Civ. Eng, Ireland, vol. xxxvi. pp. 138-164 (1910). 



t For practical purposes it will be sufficiently accurate to substitute ;/: . 

 the yield-point in tension, for y c ; and an ordinary tensile test is therefore 

 all that is necessary. 



\ Trans. American Soc. Mech. Eug., vol. xxvii. pp. 780-822 (1905 6). 



