Art. 68. BENDING STRESS IN SOLID BEAMS. 



93 



4 ?.- 



k Fig. 



60. 



bending moment at a section (the sum of the moments of the ex- 

 ternal forces to the left of the section) is equal to the moment of 

 resistance at the section (the sum of the moments of the stresses 

 at the section). That the bending moment is equivalent to the 

 moment of a couple is shown in 

 Fig. 66, where R at c is the re-' 

 sultant of all the external forces 

 to the left of section mn ; this is 

 equivalent to a force E at the 

 section and a couple whose mo- 

 ment is Ra (34). To resist the 

 single force, the shearing stress, S = R (2 vert comps. = 0) ; to 

 resist the couple, it is only necessary that there be stresses at the 

 section forming an opposite couple of equal moment (30). Since 

 there may be many couples f ulfilling this condition, the distribu- 

 tion of the bending stresses is statically indeterminate. The dis- 

 tribution of the bending stresses over the cross section of a beam 

 is assumed to be in accordance with the theory of flexure. 



It is a matter of common experience that a load placed on a 

 beam will bend it. In some cases the bending may not be appar- 

 ent, but it nevertheless takes place on account of the elasticity 



of the material. Fig. 67 shows, 

 to an exaggerated degree, the 

 bending in the simple case of 

 Fig. 65. If before the beam is 

 loaded, two vertical lines were 

 marked on each face, BB and 

 B'B f , Fig. 67, it would be 

 found that when the beam is 

 bent, these lines are no longer 

 parallel to each other, but that 

 the distance between them (in 

 67 this case) is diminished in the 



upper part of the beam and increased in the lower c^ B 

 part. If the upper part of the beam is compressed and 

 the lower part elongated, there must be an intermedi- 

 ate surface in which the length of the fibers is un- ., . 

 changed ; this is called the neutral surface and is indi- -pig. 68. 

 cated by the line N^N^ The line in which the neutral surface 

 intersects any cross section as NN (Fig. 67) is called the neutral 

 axis. 



