172 



STRENGTH OF MATERIALS 



FIG. 120 



vertical section through B, and R' the resultant pressure on the. 

 inclined section AE through B. Since .R is due to the load on tin* 

 right of the vertical CF, and R' to the load on the right of tin- broken 

 line DAE, the difference between them must be due to the U-id 



ABCD minus the load BFE. I .. t 

 P denote the difference between 

 these two loads, represented by 

 the shaded portion in I i-. U ( .. 

 Then, since R, R f , and P must 

 be in equilibrium. A'' is found at 

 once by drawing a force trial 

 as shown in the figu: 



140. Conditions for stability. 

 A masonry arch may fail in any 

 one of three ua\ B: (1) by sliding 

 of one voussoir upon another; (2) by overturning: (.') by rnishing 

 of the material. 



These three methods of failure will now he cMii>iderrd in order. 



1. The first method of failure is caused by the shearing stress at 

 any joint exceeding the joint friction, or the adhe>i<.n .f the ni>n ; ir. 

 This kind of failure can only occur when th- an-jle which the n->ult- 

 ant pressure on any joint makes with a normal to the plane of 

 the joint exceeds the angle of repose for the material in <iu> 

 (Article 159). Ordinarily the resultant pressure on any joint is \.TV 

 nearly perpendicular to its plane, and since the an-jl- t repoa 

 masonry is very large, failure by sliding is not likely to occur. 



As a criterion for safety against failure of this kind, it may be 

 assumed that when the resultant makes an angle <>f less than 30 

 with the normal to the joint safety against sliding is assured. 



2. In order for an arch to fail by overturning, one or more of tin- 

 joints must open at one edge, the adjacent blocks rotating about 

 their center of pressure. For this to occur, one edge of the joint 

 must be in tension. Although in a well-laid masonry arch the 

 joints have considerable tensile strength, it is customary to disregard 

 this entirely, and in this case the condition necessary to assure 

 stability against rotation is that every joint shall be subjected to 

 compressive stress only. Assuming, then, a linear distribution of 



