INTERNAL FORCES 63 



material, with equal tensile and compressive unit stresses. There- 

 fore the center of gravity of the beam section is the center of 

 gravity of the area of the section. For this reason a great many 

 students claim that the neutral plane is always in the center of 

 gravity, which is correct, but they insist that this section of 

 gravity is at mid-depth, which is not always correct. 



The total tension must equal the total compression. When a 

 beam is strong in compression and weak in tension, or vice versa, 

 the neutral plane will not be in the center of area of the cross 

 section if the beam is not proportioned for this condition, but it 

 will be in the center of gravity. We will now explain this state- 

 ment, which appears to be contradictory. Assume the stress 

 triangles in Fig. 54 to be equal, although the tensile fiber stress 

 may be lower than the compressive fiber stress. The fact that 

 the two forces must balance makes the height of one triangle 

 greater than the other, for " fiber stresses are directly propor- 

 tional to their distance from the neutral axis." We therefore have 

 two triangles of stress, one with a wide base (high fiber stress) 

 and the other with a narrow base (low fiber stress). The heights 

 obviously must vary in proportion in order that these " stress 

 areas" will be equal. 



The illustrations are of an imaginary beam section, a thin slice, 

 so the position of the neutral plane is represented by a line called 

 the " neutral axis " and the triangles represent the sides of wedges 

 across the beam. Much of the difficulty met with by students in 

 regard to the neutral axis arises from the fact that such illustra- 

 tions are used. The material is not actually stretched and short- 

 ened as indicated, the triangles being imaginary, each with a 

 base representing a force and not a length. 



The neutral axis is in the center of gravity of a section, sym- 

 metrical or unsymmetrical, provided such position is the center 

 of gravity of a couple in the section. If the stresses are different 

 but the section is symmetrical the neutral axis will be nearer the 

 side having the higher stress. If the section is unsymmetrical, 

 to balance the difference in the fiber stresses, the neutral axis will 

 be in the center of gravity of the section and also in the center 

 of area. An example of this is the cast iron beam. Before taking 

 it up a definition will be given of a couple. A couple consists of 

 two equal opposite forces acting in parallel lines about a point. 

 One illustration is a load on a beam and the reaction at the end 



