RELIABILITY OP ELASTIC THEORY. The chief sources of error in applying the elastic theory to a plain concrete arch are: 1. The uncertainty as to the coefficient of elasticity. The coefficient varies with the quality and the age of the concrete, and also with the unit stress; but not according to any definite law. 2. The uncer tainty as to the temperature variations. The effect of temperature changes were entirely neglected by the older builders and by nearly all the modern builders; and as the older voussoir arches have stood for thousands of years without signs of distress, and as the newer concrete arches show no signs of approaching failure, it may be con cluded that the stresses due to temperature changes are proportion ally not very great, probably because the variation of the mean tem perature of the arch ring is not very large. Accurate experiments on this phase of the subject are very much needed. 3. The uncer tainty as to the coefficient of expansion of concrete (see § 1334). Accurate experiments are v 3ry much needed in this field. 4. The uncertainty as to the fixedness of the ends of the arch. The un certainty as to the fixity of the ends can be greatly reduced or be entirely eliminated by taking the springing line for purposes of analysis at, a plane where the ends of the arch are virtually fixed; and whenever there is no pronounced change of resisting section from abutments to arch ring, or whenever the abutments are so high or of such a form that the ends of the arch are not really fixed, then the analysis should include the whole structure down to the founda tion, where the unit pressure will likely be, or can be made, so low that the distortion due to the live load on the arch will be inappreciable.
5. When the spandrel filling is earth, the omission of the horizontal components of the pressure makes the computed stability less than the actual.
If the arch ring is built monolithic, the elastic theory applies reasonably well, and a small amount of tension may be permitted, say 50 lb. per sq. in. (see § 405-06); but if the arch ring is built in voussoirs, the bond between the two adjacent voussoirs is likely to be much less than the tensile strength of the concrete (see § 345), and consequently it is unwise to permit any tension in the arch ring.
The dimensions deduced by the elastic theory do not differ greatly from those for the thrust theory, particularly if the moving load is comparatively light; but the elastic theory permits a more accurate determination of the maximum live-load stresses.
An elaborate series of instructive experiments on arches of various spans up to 75 ft., by the Austrian Society of Engineers and Architects * showed that the deflections of voussoir arches well within working limits conformed to the law of elasticity, and therefore the elastic theory is applicable to a voussoir arch provided the curve of pressure always lies within the middle third of the depth of the arch ring, i.e., provided there is no tension; but owing to the uncertain ties of the properties of the composite arch ring, the degree of accuracy is not as great as in a plain concrete arch.