ANALYSIS OF STRESS IN BEAMS 



39 



of the same section with respect to any parallel axis, c the distance 

 between the two axes, and F the area of the cross section. Then 



(19) !. = !. + Fc\ 



(B) Every section has two axes through its center of gravity, called 

 principal axes, such that for one of these the moment of inertia is 



a maximum, and for the other is a ^*~ x^ ^ n 



minimum. Let the principal axes be 



taken for the axes of Y and Z re- 

 spectively. Then if I v and / 2 denote 

 the moments of inertia of the section 

 with respect to these axes, and I a 



r i< ; . 22 



denotes the moment of inertia with 



respect to an axis inclined at an angle a to the axis of Z, 



(20) I a = /, cos 2 a + / sin 2 a. 



(C) The moment of inertia of a compound section about any axis is 

 equal to the sum of the moments of inertia about this axis of the 

 various parts of which the compound section is composed. 



(D) The moment of inertia of any section with respect to an axis 

 through its center of gravity and perpendicular to its plane is called 



the polar moment of inertia. The polar 

 moment of inertia is defined by the 

 equation 



',= 



where r is the distance of the infini- 

 tesimal area dF from the center of 

 gravity of the section. 



Since r 2 = y 2 + y 



Fio. 23 



Cv*dF = Ci?dF + CfdF 



whence 



(21) !, = !,+ 7 - 



(E) Let /j and / denote the moments of inertia of any section with 

 respect to its principal axes. Then I p = 1^ + I v and, consequently, 



