ANALYSIS OF STRESS IN BEAMS 63 



Problem 68. In an inclined railway the angle of inclination with the horizontal 

 is 30. The stringers are 10 ft. 6 in. apart, inside measurement, and the rails are 

 placed 1 ft. inside the stringers. The ties are 8 in. deep and 6 in. wide, and the 

 maximum load transmitted by each rail to one tie is 10 tons. Calculate the maxi- 

 mum normal stress in the tie. 



Solution. The bending moment is the same for all points of the tie between the 

 rails, and is 20,000 ft. Ib. From Problem 42, S 2 = 64 in.* and S v = 48 in.. There- 

 fore, from equation (30), 



240,000 (-^\ 240,000 (*\ 

 - - + - -^ = 5744 



59. Eccentric loading. If the external forces acting on any cross 

 section reduce to a single force P, perpendicular to the plane of the 

 section, but not passing through its center of gravity, this force is 

 called an eccentric load. Let B denote the point of application of the 

 eccentric load /'. and let ./'//' denote the coordinates of B. Then the 

 eccentric force P acting at B can be replaced by an equal and parallel 

 force acting at the center of gravity C of the section, and a moment 



16 plane is perpendicular to the section. This moment can the i 

 be resolved into two components parallel to the principal axes, of 

 amounts /'//' and /':' respectively. Therefore, by the law of super- 



.'n. the intensity of the stress at any point (y y z) of the cross 



'ii is 



p /-.' p 



p-j+-r-*+-f*'> 



J. J. 



or, since / 



At the neutral axis tin- stress is zero, and consequently 1 + ~ + - 



must be zero ; or, since the semi-axes of the inertia ellipse are a = t y 



and I = /., this condition becomes 



(31) 5 + l3._i. 



This condition must be satisfied by every point on the neutral axis, 

 and is therefore the equation of the neutral axis. To each pair of values 

 md 2 ', that is, to each position of the point of application B of 

 the eccentric load, there corresponds one and only one position of the 

 neutral axis. 



