STRENGTH OF MATERIALS 293 



changes sign under the load d and the bending moment is 

 maximum at that point. It is equal to 18,170X13 10,000X7 



Formulas for the maximum bending moments and shears 

 for beams loaded and supported in different ways are given 

 in the accompanying table. 



For a beam supporting moving loads, the maximum bending 

 moment occurs: 



1. For a single load, when the load is at the middle of the 

 span. 



2. For two equal loads, under either load, when the two 

 loads are on opposite sides of the center and one of the loads 

 is at a distance from the center equal to one-fourth the dis- 

 tance between the loads. 



3. For two unequal loads, under the heavier load, when 

 that load and the center of gravity of the two loads are equi- 

 distant from the center of gravity of the beam. 



EXAMPLE. A beam 24 ft. long supports two moving loads 

 6 ft. apart. The left-hand load is 8,000 lb., and the right-hand 

 load is 4,000 lb. Find the maximum bending moment. 



SOLUTION. The center of gravity of the loads is 2 ft. from 

 the left-hand load. The maximum bending moment occurs under 

 the heavy load, and obtains when the latter is 1 ft. to the left 



of the center of the beam. The left reaction is, then, '- - 



= 5,500 lb., and the maximum bending moment is 5,500X11 

 = 60,500 ft.-lb. 



Designing of Beams. In every section of a carrying beam 

 there is induced an internal moment called the moment of 

 resistance, which is equal to the bending moment at that 

 section. As previously explained, the resisting moment is 



equal to /; and, if the maximum bending moment is denoted 



by M, M -/; whence, = -, which is the fundamental formula 



c f c 



for the designing of beams; / is the working stress in flexure, 

 which is the modulus of rupture divided by a suitable factor 



