The third of Equations 48 must be satisfied, which practically means that Equation 54 must be satisfied. This means that we must find a trial equilibrium polygon, and increase or decrease its pole distance so that the summation of the products based on z" shall equal the summation of similar products based on z'. But in this case the special equilibrium polygon passes through the abutment points, and there is no moment at the abutment. Therefore, after having found the pole distance of the special equilibrium polygon, we may draw the special equilibrium polygon by commencing at one abutment point; and, as a check on the work, we should find that it passes through the other abutment point. The maximum moment due to temperature will be at the center of the arch rib, and will be based on an equation similar to Equation 60, which may be used by calling d = o.

Fig. 23-1. Berkley Bridge, Berks County, Pennsylvania. - Reinforced-Concrete Arch Rib.

Equation 61 does not apply, since the ends of the arch rib are free to turn at each abutment.

44G. Arch of Three Hinges. A three-hinged arch is a still more simple case, since none of the three fundamental equations (Equation 48) which are used for fixed arches needs to be satisfied. It is only necessary to find the special equilibrium polygon which will pass through the two abutment hinges and the center hinge. There are no temperature stresses and no stresses due to the shortening of the rib. It may thus be said that a three-hinged arch is much more simple to calculate, and its stresses are more definite. The construction of the hinges will of course add somewhat to the cost, and probably add more than any saving which might be made by a reduction in the cross-section of the arch. Probably the greatest advantage of three-hinged arches lies in their immunity from damage which may result from a settlement of the foundations. It has been assumed, in considering the theory of fixed arches, that the foundations are absolutely immovable. A settlement of either abutment of a fixed arch with reference to the other abutment, will inevitably result in stresses in the arch rib which might easily be greater than any stresses to which the arch rib would be subjected either on account of the loading or through change in temperature. The failure of many arches is unquestionably due to this cause. An arch rib with either two or three hinges is absolutely immune from any such danger; and there is therefore a strong argument for the use of hinged arches when the arch must be placed on foundations which are so uncertain that a settlement of either foundation is quite possible. Of course an equal settlement of both foundations would do no damage, but the equality of such a settlement could never be counted on.

Fig. 235. Reinforced-Concrete Oblique Arch. Graver's Lane Bridge, Philadelphia, Pa.