CHAPTER VIII. 



SPECIAL CASES OF BEAMS AND GIRDERS LOADED AND 

 SUPPORTED IN DIFFERENT WAYS. 



90. Explanation of Table. Figs. 94 to 113 illustrate all 

 of the cases of simple loading that an engineer will encounter 

 in ordinary practice. Accompanying these are equations giving 

 general and maximum values of the shears, the moments, and 

 the deflections, and diagrams showing how these vary. 



The beam is shown as a single heavy line in its bent condi- 

 tion, thus forming a diagram of deflections. The deflections are 

 determined from the equation of the elastic line, the effect of the 

 shear being neglected (87). The origin of coordinates is usually 

 taken at a support, and in such a manner as to make the equa- 

 tion of the elastic line as simple as possible. 



The moments and shears are determined by the application 

 of the equations of equilibrium by the method of sections, tho 

 section being taken where the stresses are wanted. The loads all 

 being vertical, there are no horizontal components except foi 

 stresses (70). From "the sum of the .vertical components 

 equals zero," 



Shear at a section = shearing stress at the section. 



Shear = 2| external forces on one side of section. 

 From "the sum of the moments equals zero," 



Bending moment at a section = moment of resistance. 



Bending moment = 2 moments of the external forces, on 

 one side of the section, about a point in the section. 



These formulas apply, of course, to any kind of a simple 

 beam or girder in which E and I are constant, and when these 

 quantities are not involved, to other beams, some applying to 

 trusses at certain sections. 



It is well in applying the formulas to use inches and pounds 

 in calculating deflections, because these correspond w r ith the 

 values of E usually given. 



144 



