The middle point of the arch axis in each division is now located, and the values of x and y are determined with reference to the middle of the crown section. These values and their squares are tabulated in Table XXI for'use in the computations.
168. Analysis.—If vertical lines be drawn through the points of division of the arch axis, the weight of the portion of masonry and spandrel filling included between each pair of lines may be considered as the dead load resting upon the included division. The live load is similarly divided for the portion of the arch over which it is considered as acting. In Fig. 91 the live load is taken as extending over the left half of the arch, and the loads are as indicated. The values of and are now computed and placed in Table XXI, and include in each instance the moments of all loads between the division con sidered and the crown section about the center of division. The quantities 711Lx, MR X, ?nLJ, and mRJ may now be computed and placed in the table, and the summations of the various columns obtained. These substituted in Formulas (15), (16) and (17) give, Having obtained the thrusts and moment at the crown, we may now proceed to find the thrusts and moments at any other section desired. The thrusts are obtained graphically by drawing the line of pressure. The load line is first constructed, as shown by the ver tical line a—u. 1", and 11, are laid off from the mid-point k of this line, thus locating the pole 0. The force diagram is then completed by drawing connections from 0 to the extremities of the various loads.
The equilibrium polygon is now drawn, beginning with the crown thrust (0—K), the point of application of which is at a point e= + 1247 +2671a— +0.05 foot above the center of the crown section. The thrust upon each section is now shown in amount by the length of the line from 0 to the division in the load line, and its line of action by the corresponding line in the equilibrium polygon (or line of pressure).
The bending moment at any section may be found by multiplying the thrust upon the section by its perpendicular distance from the center of section, or it may be computed by the use of Formula (10) or (11). Usually the formula is employed and the measurement of the eccentricity used as a check.
and the measured thrust =40300 pounds, giving an eccentricity of — 20S99 40300 = — .5 foot. This may be checked by measure ment. Then for thrust, this gives at extrados f = 93 lb., in?, and at intrados L= 127 lb. "in? At point SR, in the same manner, we have at extrados, f =oS and at intrados f =1o0 lb.,'in 2 Full the live load extends across the whole span of the arch, the loading is symmetrical and the values given in C for become equal to those for We then have The force diagram is now drawn for one-half of the arch, and the equilibrium polygon may be drawn as in the case of partial load ing. To avoid confusion it is not drawn in Fig. 91. The
in the crown section due to this loading are This gives at extrados, total lb. and at intrados. total L=131-25=103 lb. At section 5, If = 1662+29030X2.23— Q0605.= —3629 ft.-lb. The thrust is 30.350 pounds, and the resulting unit stresses at extra dos L= 120 —30=S4 lb. and at intrados f = 156 lb. At the support in the same manner, the thrust is 12,600 pounds, and the moment, _if=1662+29030X11. —349930 = —5617 ft. lb. from which at extrados, f =57 lb. and at intrados f =111 lb. Temperature Strcses.—If we assume that a rise in temperature of above the normal may take place, Formula (20) gives The equilibrium polygon is a horizontal line 1.71 feet below the center of the crown section, and the bending moment at any section of the arch ring is equal to 3770 times the vertical distance from the center of section to the line of thrust. At point S the moment clue to change of temperature is The normal thrust on section at point 5 is the component of H, normal to the section, given in diagram in Fig. 11S, =3345 pounds. At section a, thrust =2.510. These thrusts and moments give at 8, for extrados L= 14+2-1=35 lb. for intrados, h= 14-24 = —10 lb. at support; extrados L= + 12 7 lb. intrados, f = lb. For a fall in temperature the stresses are equal and opposite to those for rising temperature.
Arch Shortening.—The effect of direct thrust in shortening the span of the arch, taking average unit compression as 100 lb. in.2 average of stresses at crown, point S and support under one-half live load by Formula (23), This is applied on the same line as the temperature thrust and the stresses are therefore equal to 1710/3770=.4d of those for falling temperature.
Table XXII shows the computed stresses upon the sections at crown, at point 8 and at supports. Examination of this table shows that the unit compression is nowhere excessive. Tension of 34 occurs at the intrados in the crown section at low temperature. This is too small to cause cracking in the reinforced section. The tension of 150 at the extrados of the support section would possibly crack the concrete. The compression at the intrados under the same conditions would be 331 and the reinforced section would be capable of bearing the load if the steel be assumed to carry all the tension. It might be desirable, however, to introduce additional reinforcement at this point t-o lessen the unit tension in the steel and prevent cracking, and these negative stresses might also be eliminated by slightly modifying the form of the arch, increasing the radius at the crown and decreasing those at the ends, although the form as shown agrees fairly well with the lines of thrust.