The thrust at any section of the arch may be obtained from the thrust diagram as in the voussoir arch. The bending moment at any section is the moment of the thrust upon the section about the center of gravity of the section. The bending moment at any section may also be obtained by the use of Formula (10) or (11).
In analyzing an arch bridge subject to moving loads, it is necessary to assume different conditions of loading and find the thrust and mo ments resulting from each. For a small arch, it is usually sufficient to make the analysis for arch fully loaded and for moving load over one-half the arch. The maximum stresses will be mere accurately determined by dividing the moving load into thirds, and determining the stresses with span fully loaded, one-third loaded, two-thirds loaded, center third loaded, and with two end thirds loaded. If complete analysis be made for the arch under dead load alone, for live load over one end third, and live load over the middle third, the results of these three analyses may be combined to give the five conditions of loading above mentioned.
164. Effect of Changes of Temperature.—A rise in temperature tends to lengthen and a fall in temperature to shorten the span. If the ends of the arch ring are rigidly held in position, the tendency to change in length is resisted by moments and horizontal thrusts at the supports, which produce moments and thrusts throughout the arch ring.
If the arch ring were not restrained, a rise in temperature of t degrees would cause an increase in length = CtL; L being the length of span and C the coefficient of expansion of the material. The
moments throughout the arch ring are therefore those which cor respond to an actual change in length of span =CIL, or from For mula (3) As there are no exterior loads, mL, and V, are each equal to zero, and Formula (10) becomes Substituting this in (1S) and (19) and solving, we have The line of thrust consists of a single force H,, and is applied on a horizontal line at a distance, e= yin, below the middle of the crown section. The bending moment at any section due to II, is The direct thrust upon any 'section of the arch ring is the com ponent of II, normal to the section.
For temperatures below the normal, II, will be negative and may be found from Formula (20) by giving t the negative sign.
165. Effect of Direct Thrust.—Axial thrusts on the arch ring produce compressive stresses on the various sections and also tend to shorten the arch ring. As the span length does not change, this tendency to become shorter causes stresses in the arch ring in the same manner as does lowering temperature. If L lb./in 2 be the average unit compression due to axial thrust, the arch ring if unre strained would be shortened an amount from which, As the unit is not uniform through the arch ring, a value obtained by finding the stresses at several points and averaging them may be used.
The stresses due to shortening of the arch ring are comparatively small and are often neglected in the analysis of ordinary arches; in some instances, however, they may be considerable.