300 STRENGTH OF MATERIALS 



Euler's Formula. Structural members in compression whose 

 ratio of slenderness exceeds 150 should preferably not be used. 

 Sometimes, however, long columns cannot be avoided, and 



I 

 when - exceeds the limits for which the preceding formulas may 



be applied, Euler's formula should be used. This formula is as 

 follows: 



in which E is the modulus of elasticity of the material and n 

 is a' constant depending on the end condition, having the 

 value of 1 for columns with both ends pivoted and 4 for 

 columns with both ends fixed. The preceding table gives the 

 values of nv 2 E, expressed in millions of pounds. 



Formula for Wooden Columns. The formula for determining 

 the strength of wooden columns having flat or square ends was 

 deduced from exhaustive tests of full-size specimens, made at 

 the Watertown Arsenal, Mass., and may be expressed as follows: 



in which 5 is the ultimate strength of column, per square inch 

 of section; U, the ultimate compressive strength of the material, 

 per square inch; /, the length of the column, in inches; d, the 

 dimension of the least side of the column, in inches. 



This formula may be applied to all wooden columns, the 

 length or height of which is not under 10 times nor over 45 



times the dimension of the least side. In other words, 



d 



should not be less than 10 nor more than 45. If the length 

 is less than 10 times the least side, the direct compressive 

 strength of the material per square inch, multiplied by the 

 sectional area of the column, in square inches, will give the 

 strength of the column. If the length is over 45 times the 

 least side, Rankine's formula should be used. 



