MECHANICS 105 



ing action at this place, the magnitude of which is equal to 

 the reaction RI. This shearing action is resisted by the 

 strength of the fibers of the beam at the section under con- 

 sideration. 



When the point of application of n is reached, the effect 

 of the upward force R\ is partly balanced by the downward 

 force of 10 Ib. due to the load n. Any section between the 

 points of application of n and m is therefore subject to a 

 shearing stress equal to the difference between the reaction RI 

 and the load , or 40 10=30 Ib. In the same way, it fol- 

 lows that the shearing stress for any section between m and o 

 is 40 (10+ 15) = 15 Ib. For any section, as cd, between 

 the points of application of o and p, the shearing stress is 

 40- (10 + 15 + 15) =0. In other words, it is a section in 

 which there is no shear. 



Positive and Negative Shear. For convenience, it is cus- 

 tomary to call the reactions, or forces, acting in an upward 

 direction, positive, and the loads, or downward forces, nega- 

 tive. Since the difference between the sums of the positive 

 and negative numbers representing a given set of values is 

 called their algebraic sum, it follows that the shear for any 

 section of a beam is equal to the algebraic sum of either 

 reaction and the loads between this reaction and the trans- 

 verse section under consideration. In speaking of the shear 

 at a certain section of a beam as being positive or nega- 

 tive, it is simply meant that the resultant of the forces 

 acting on the portion to the left of the section under con- 

 sideration is either positive or negative. 



If a transverse section of a simple beam is taken near 

 the left reaction and the forces acting on the part of the 

 beam at the left are considered, it will be seen that their 

 resultant acts upwards. The shear at this section is there- 

 fore called positive shear. If, however, a section near the 

 right reaction is taken, the resultant of the forces at the left 

 of this section is found to act downwards, and in conse- 

 quence the shear is called negative. It is also evident that 

 there is a section between the two where the resultant of 

 the forces changes from positive to negative. At such a 

 section the shear is said to change sign. 



