The proper value to be adopted for t is not easy to determine. Often the upper surface of the arch is covered with earth, and con sequently does not vary much, if any, in temperature; and usually the lower surface of the arch is not exposed to the direct rays of the sun, and consequently the range of temperature of that surface is only that of the atmosphere. Owing to its high thermal conduc tivity steel will readily acquire the temperature of the air; but con crete is a very poor conductor of heat, and consequently the tempera ture of the interior of a concrete arch ring does not vary as much as the surface. Observations made under the author's direction * seem to show that concrete 4 to 6 inches below the surface does not follow the diurnal variations of atmospheric temperature. Observations for two years upon the width of cracks in the masonry near the top of the New Croton Dam (§ 964) seem to show that the coefficient of expansion was 0.000,003,1, or that the range of temperature of cut-stone and rubble masonry approximately 30 ft. thick, exposed to the atmosphere on both sides and the top, was only about half or two-thirds of that of the monthly mean of the atmosphere. "Expansion joints in the most exposed cases do not show over inch motion per 100 ft., which assuming a coefficient of expansion of 0.000,006 is equivalent to a maximum change of temperature of not more than 35°' F." $ "A self-recording thermometer placed in the ring of a reinforced concrete bridge having earth filling indicated that the total range of temperature did not exceed about 20° F. in some ten or twelve months."¶ In the design of the 280-ft. concrete arch now (1909) in process of construction in Cleveland, Ohio (see § 1346), the range of temperature was taken at ±30° F., the arch ring being 6 ft. thick at the crown and 11 ft. at the springing, and being exposed on both the intrados and the extrados. From a limited number of ob servations extending over nearly two years with thermophones em bedded in the masonry of the Boonton (N. J.) Dam, before the
water was admitted behind it, the following formula was deduced.** in ,which R is the total range of temperature on Fahrenheit degrees at any point within the mass, the numerator is the total atmospheric range, and D is the distance in feet to the nearest exposed face of the dam. The above formula is true only for values of D between 0.5 ft. and 20 ft.
In the example under consideration, t will be assumed to be 20° Fahr. above a mean temperature of 60°, and 30° below. The sufficiency of this allowance will depend, of course, upon the locality and the exposure of the arch ring.
Substituting the above values in equation 27, gives for a maximum • rise of temperature That is, a rise of temperature of 20° F. in an arch ring 1 foot long exerts an outward thrust of 2,550 pounds upon the abutments; and similarly a fall of 30° F. will exert an inward pull upon the abutments 1336. Temperature Stresses. The fiber stress due to temperature changes is, in the nomenclature of § 1328, in which is the component of Q parallel to the tangent of the neutral line at the point where the stress is desired. The total fiber stress due to the combined bending and the thrust caused by a change of temperature is The first term is + for a rise, and — for a fall of temperature. To aid in interpreting the character of the stresses given by the second term, consider only the left-hand half of the arch; and conceive that the right-hand half is removed and that its effect is replaced by the force Q along acting toward the left for a rise and toward the right for a fall. Then, if the point lies below the line as for example a„ for a rise the second term gives tension at the intrados and compression at the extrados, and for a fall gives compression at the intrados and tension at the extrados; and when the point lies above the line the above stresses are reversed. Table 96 shows the temperature in the arch ring of Fig. 233.