SEW DUE TO SæDRTIsING or ARCH RING. In deter mining the preceding stresses, the only effect of the tangential com ponent considered was that of a force T uniformly distributed over the cross section, which force produced a shortening of the arch ring, and being uniformly distributed over the cross section did not affect the bending due to R . ae or H H. ac; but the shortening of the arch ring produces a bending, which effect is now to be considered.
If the unit compression due to the thrust T were constant for all cross sections, the effect would be the same as a fall in temperature. If T - A represents the average unit compressive stress, the short ening of the span is A 1; and the shortening for a suppositious change of temperature l' is £'I e. Equating these two values, and solving we get: The proper value of T _ A to be used in the above equation is somewhat uncertain, since the unit thrust varies from point to point, and is quite different in the two halves of the arch ring. For the example in hand, the value used will be the average of the unit thrust at a„ a„ and (see the fifth column of Table 95, page 690), which is 71.8 lb. per sq. in. Substituting in equation 31, this value and also the values of E and e (see § 1334), we get: Therefore the shortening of the arch ring under the action of T, the tangential component due to the dead and live load, is equal to that due to a fall of temperature of 8.9° Fahr.; or the maximum stresses due to this shortening are 2911- per cent (= 8.9 _ 30) of those due to a fall of 30°. The values of the maximum stresses due to the above shortening are given in the fourth and the fifth columns of Table 97.
The stress due to the shortening caused by the tangential component of the dead and live loads is usually neglected; but it is unwise, particularly for flat arches, and especially as the above method of computing such stresses is so simple and brief.
There is a similar stress due to T,, the tangential com ponent of the abutment reaction for temperature stresses, which can be computed as in § 1336. For the example in hand, this
shortening is equivalent to a fall of temperature of 0.8° Fahr., which is almost exactly 11 per cent of the result in § 1336. Therefore the stresses due to this shortening are only about 11 per cent of the results in the tenth column of Table 97, page 697; and hence, in this case at least, they may be omitted.
Table 97, page 697, shows the max imum combined stresses due to dead and live loads and to temperature changes. The results are collected from Table 95 (page 690) and Table 96 (page 695). The stresses to be employed in checking the design of the arch ring are deduced as those in Table 97 using the maximum stresses for different positions of the live loads in stead of those for a single position as in Table 95. Table 97 does not show the shearing stress, partly because it would unduly extend the table. The shearing stress due to the dead and live loads is shown in Table 95, page 690; and that due to temperature changes in Table 96, page 695. The latter is really too small, in this case at least, to be considered; and hence the only shearing stress to be considered in checking the design is that in Table 95.