INTERNAL FORCES 65 



In Fig. 54 let the small / stand for the maximum fiber stress, that 

 is the unit stress, which is generally expressed in pounds per square 

 inch. Then/c = unit compressive stress and/< = unit tensile stress. 

 When the stresses are equal the letter / is used without a subscript. 



All forces must act through the center of gravity of bodies or 

 areas in order to effect a movement of the whole without turning 

 it, as about an axis. The center of gravity of a triangle is one- 

 third the distance from the base. In Fig. 54 the triangle on one 

 side of the neutral axis has a height equal to one-half the total 



height of the beam, therefore is represented by - The width of 



A 



the triangle at the base is represented by the unit stress at that 

 place, /. The triangle is a force triangle and the area is equal to 

 the total force exerted on it. This is found by the ordinary rule 

 for areas of triangles half the base multiplied by the height or 



A- f x*- f 

 -2 X 2~ 4 



The length j is known as the moment arm. The moment arm 

 is the distance between the centers of gravity of the tension and 

 compression members. The lower triangle is the tension member, 

 if we assume the beam to be a frame, and the upper triangle is 

 the compression member, the neutral plane being the dividing 

 line. Since the forces act at a point and this point is the center 

 of gravity of the member, then the total force in compression, 



^7-, acts to balance the total force in tension, -r t with a moment 



2 h 4fc 2h 

 arm = 2 x 3 X g = = T 



The moment arm times the compressive (or tensile) force gives 

 the moment of resistance per unit of breadth, 



M _?*v#-22--fi- 

 "3*4* 12 " 6* 



This reasoning has been based on a breadth equal to 1, or unity. 

 To make practical use of the expression the breadth, designated 

 by b, must be introduced. This gives the expression for the 

 moment of resistance of a beam of homogeneous material and 

 rectangular cross section commonly seen in text books, 



M W. 



Mr- -- 



