Art. 70. MOMENT OF RESISTANCE OF A BEAM. 97 



the unit stress at unit distance from the neutral axis is or 



Vl V2 



and at a distance v. s = si or s* (<)\ 



Vl V2 * / 



If an infinitesimal area dA be considered, the stress over 

 it will be constant and equal to sdA, and its moment about the 

 neutral axis will be vsdA. (The center of moments may be 

 taken at any point since the moment of resistance is equivalent 

 to the moment of a couple). Summing up the moments of all 



the elementary stresses, M R = I vsdA. 



Opposite signs for the parts above and below the neutral axis 

 simply denote that the stresses are of opposite sign, as they 

 should be, to make the moments of the same sign 1 (Fig. 68). 



Substituting the value of s from equation (9), 



in which 1= [ MA (11) 



-V2 



I is called the moment of inertia of the cross section ; it is a 

 ''second moment," while J vdA is the first or statical moment. 



The moment of inertia accounts for the fact that the stress 

 varies as the distance from the neutral axis, and that the moment 

 of this stress about the neutral axis varies in the same way. 



Since v appears as a square, the moment of inertia of the 

 low^er as well as the upper part of the cross section is positive, 

 and they are added together. 



It is evident that the more the area is disposed away from 

 the neutral axis, the greater will be the moment of inertia, and 

 the greater the moment of resistance. For this reason steel 

 beams and girders have T-shaped cross sections ; the average unit 

 stress is much greater than in rectangular cross sections of the 

 same height and area. Fig. 70 shows four different ways in 

 which four angles and a plate may be riveted together to from a 



J It does not follow that the moment of the resultant of the tensile 

 stresses equals that of the compressive stresses, about the neutral axis; this 

 is true for sections symmetrical about the neutral axis, but it is not neces- 

 sary in order to satisfy the conditions of equilibrium ; it is only necessary 

 that the total tension shall equal the total compression, and that the mo- 

 ment of resistance shall equal the bending moment. 



