ARCHES AND ARCHED RIBS 



171 



the linear arch is the broken line joining these centers of pressure. 

 In a concrete arch the linear arch becomes a continuous curve. With 

 each change of loading the same shifting of the linear arch occurs 

 as hi the case of the model with curved joints, the only difference 

 being that with flat joints this action is not visible. To assure sta- 

 bility, however, the linear arch must be restricted to lie within the 

 middle third of the arch ring, as will be proved in Article 140. 



If we consider a single voussoir of a masonry arch, or a portion of 

 a concrete arch bounded by two plane sections, as shown in Fig. 128, 

 the resultant joint pressures It and R', and the weight P of the 

 block and the material directly above it, form a system of forces in 

 equilibrium. Consequently, if the amount, direction, and point of 

 application of one of these 

 ivsultunt joint pressures 

 are known, tin- amount, 

 direction, and point of ap- 

 plication of the other can 

 be found by construct- 

 ing a triangle of forces. 

 Therefore, if one result- 

 ant joint pressure is com- 

 pletely known in position, 

 amount, and diivction, the others can be successively found as above, 

 thus dftmiiining the linear arch as an equilibrium polygon for the 

 i:ivMi system of loads. 



Since an equilibrium polygon may be drawn to any given scale, 

 if no one joint pressure is completely known, which is usually the 

 case, there will be, in general, an in finite number of equilibrium 

 polygons corresponding to any given system of loads. The linear 

 arch may, however, be denned as that particular equilibrium polygon 

 which coincides with the pressure line, and the question then arises 

 how to determine the equilibrium polygon so that it shall coincide 

 with the pressure line. This problem will be discussed more fully in 

 Articles 142, 143, and 144. 



When the linear arch has been determined, the resultant pressure 

 on a joint having any inclination to the vertical can easily be 

 ol .taint" 1. Thus, in Fig. 129, let R be the resultant pressure on a 



