ANALYSIS OF STRESS IN BEAMS 47 



Problem 51. For a Carnegie I-beam, No. B 7, 15 in. deep and weighing 

 42 lb./ft., the principal radii of gyration are 5.05 in. for an axis perpendicular to 

 web at center, and 1.08 in. for an axis coincident with web at center. Construct 

 the inertia ellipse. 



Problem 52. For a Cambria channel, No. C 21, depth of web 7 in., width of 

 flanges 2.51 in., thickness of web .63 in., the radius of gyration about an axis per- 

 pendicular to the web at center is 2.39 in. ; the distance of the center of gravity 

 from outside of web is .58 in., and the radius of gyration about an axis through 

 the center of gravity parallel with center line of web is .56 in. Construct the 

 inertia ellipse. 



Problem 53. In Problems 47, 48, and 49 determine graphically the radii of 

 gyration about an axis through the center of gravity and inclined at 30 to the 

 major axis of the inertia ellipse. 



50. Vertical reactions and shear. Under the assumptions of the 

 common thenry >f tlrxure. the external forces acting on a beam all 

 lie in the same vertical plane. Therefore, since the beam is assumed 

 tn In- in equilibrium, the sum 

 nf the reactions of the sup- L 



ports must equal the total j* " 1 

 l;:d n the beam. 



instance, consider a 

 simple leaiu AB of length /, 

 which is supported at the Fio.33 



ends and bears a single con- 



;ited load P at a distance d from A (Fig. 33). Let R l and R z 

 the reactions at A and B respectively. Then, from the above, 



R\ + fit = p - 

 T< tind the values of R l and R v take moments about either end, say A. 



Tlu ' n 

 whence 



*~T" 



Also, since .R, + R t = P, 



l 



If any cross section of a beam is taken, the stresses acting on this 

 section must reduce to a single force and a moment, as explained in 



