246 PRACTICAL STRUCTURAL DESIGN 



sary to determine the point through which the kinetic energy 

 acts, or determine what is essentially a center of gravity. The 

 moment of inertia is the sum of all the small units of a section 

 multiplied by the square of the distance of each unit from the 

 axis. Divide this by the area and extract the square root and 

 the radius of gyration is found to be the root mean square of the 

 distances of all the separate units of the mass from the axis. It 

 is actually a radius from the axis to the center of an imaginary 

 ring in which is assumed to be concentrated the mass of the sec- 

 tion. If this is not plain the author offers his apologies, for he 

 cannot make it any plainer without wandering off into a mathe- 

 matical demonstration which would defeat the objects aimed 

 at in writing this book. It is of little consequence, however, as 

 it is enough to accept the judgment of eminent men who have 

 worked the matter out satisfactorily. 



Straight-line Formula 



The Euler formula applies only to the thread in the vertical 

 axis of a very long and very slender column and the Rankine 

 formula is also to a large extent a theoretically correct formula, 

 the value of which is seriously affected by faults in workmanship 

 and design, as well as by defects in materials. It is a laborious 

 formula to use and engineers like simple formulas. In making 

 tests of full-size columns it was found by plotting the results 

 that the Rankine formula gives rather low stresses for columns 

 having a slenderness ratio under 80, and somewhat high stresses 

 for a slenderness ratio over 150. A straight line drawn through 

 the points, on the sheet on which the results of the tests were 

 plotted, in such a way that it passed through the center of mass 

 of the points, resulted in the formula: 



/ = 16,000 - 7Pi 



which is known as the "Straight-line" formula for steel columns. 

 It is used in Chicago, where the formulas for wrought iron and 

 cast iron are as follows: 



wrought iron, / = 12,000 - 60- 



T 



cast iron, / = 10,000 - 60- 



