=126. Stress Conditions in the Arch-ring.=—Before proceeding to the construction of an actual line of resistance, a little consideration must be given to the stress conditions in the arch-ring. As the joints are considered capable of resisting no tension, the dimensions of the arch-ring must be finally so proportioned that pressure only will exist in each and every joint. If each centre of pressure, as _a_, _b_, etc., in Fig. 36, is found in the middle third of the joint, it is known from a very simple demonstration in mechanics that no tension will ever exist in that joint, although the pressure may be zero at one extremity and a maximum at the other. This is the condition usually imposed in designing an arch-ring to carry given dead or live loads. It is usually specified that “the line of resistance of the ring must lie in the middle third.” It must be borne in mind, however, that the stability of the ring is perfectly consistent with the location of the line of resistance outside of the limits of the middle third, provided it is not so far outside as to induce crushing of the ring-stones. Whenever that crushing begins the arch is in serious danger and complete failure is likely to result. =127. Applications to an Actual Arch.=—These principles will be applied to the arch-ring shown in Fig. 38, in which the clear span _TU_ is 90 feet. The radius _CO_ of the soffit (as the under surface of the arch is called) is 50 feet, the ring being circular and segmental. The uniform thickness of the ring shown at the various joints is assumed at 4 feet as a trial value. The loading above the ring to the level of the line _EʹO_ is assumed to be dry earth weighing, when well rammed in place, 100 pounds per cubic foot. The depth of this earth filling at the crown _n_ of the arch is taken at 4 feet. The ring-stones are assumed to be of granite or best quality of limestone, weighing 160 pounds per cubic foot. The thickness or width of arch-ring of one foot is assumed, as each foot in width is like every other foot, and the loads are taken for that width of ring. The rectangle _EJJʹEʹ_ is supposed to represent a moving load covering one half of the span and averaging 500 pounds per linear foot; in other words, averaging 500 pounds per square foot of upper surface projected in the line _EʹO_. The total length of the arch-ring, measured on the soffit, is about 113 feet, and it is divided into ten equal portions for the purpose of convenient computation. The radial joints so located are as shown at _de_, _fg_, _hk_. From the points where these joints cut the extrados (as the upper surface of the arch-ring is called) vertical broken lines are erected, as shown in Fig. 38. [Illustration: FIG. 38.]