196 



JOHN C. KOCH 



29. Take any beam loaded in any manner and imagine a ver- 

 tical plane cutting the beam at any section t-t as in figure 8, a, b. 

 At this section there are in action unknown stresses of various 

 directions and amounts. Suppose the beam separated into two 

 parts by this plane, and let the forces X, Y and Z, equivalent 

 to the internal stresses, be applied as shown in figure 8, c. Then 

 the equilibrium of each part will be unaffected, for each part 

 will be acted upon by a system of forces that are in equilibrium. 



30. From this we may deduce the fundamental principle that 

 ''The internal stresses in any cross section of a beam hold in 



n^.^-C 







Neutral Axis 



.1. 



Fic, dfcl 



w 



Cross SideVicivdBeam ^ 

 ^^^^'^" Fiq.&idl 



Fig. 8 c The three forces X, Y, and Z are indicated replacing the in- 

 ternal stresses (forces) in action in these beams to produce equilibrium in each 

 beam. The detailed analysis is presented in paragraphs 29-37. 



Fig. 8 d Cross section and side view of any rectangular beam. To the 

 right is a diagram showing the variation in the intensity of the horizontal inter- 

 nal stresses in such a beam. Explained in paragraph 38. 



equilibrium the external forces on each side of that section." 

 This applies to all beams of whatever cross section or nature of 

 loading. 



31. Considering either part of the beam, a system of forces in 

 equilibrium is seen, to which the three necessary and sufficient 

 conditions of statics for forces in one plane apply: 



Algebraic sum of all horizontal forces = 

 Algebraic sum of all vertical forces = 



Algebraic sum of moments of all forces = 



