The work of finding the stresses in an elastic arch having fixed ends is so long that numerous approximations have been used.
One of these consists in dividing the span into equal parts instead of dividing the neutral line into parts such that ds _ I is constant (§ 1311-12). If the span is divided into equal parts, the arch ring is divided into parts whose lengths increase as the secant of the angle with the horizontal; and consequently the divisions increase in length from the crown toward the abutment, but not as required to make ds _ I constant. This method is most nearly correct for a very flat arch having a nearly uniform depth. The method of dividing the neutral line as explained in § 1312 is so simple as not to make the above approximation of any great advantage.
Another approximation consists in assuming the loads at points on the arch independent of the division of the neutral line, which introduces errors in scaling the intercepts ac (see § 1316 and § 1329) ; but this approximation is necessary when the weight of the roadway and of the live load is transferred to the main arch by spandrel arches or by columns.
Sometimes, when the spandrel filling- is earth, the horizontal components of the earth filling are included; but this violates one of the fundamental principles upon which the method of solution is based, viz.: that the bending moment is proportional to the vertical intercept in the equilibrium polygon, which principle is true only with vertical forces. Therefore including the horizontal components adds accuracy in one respect but introduces error in another; and on the whole is not wise, since usually the horizontal components are not included, and including them prevents a comparison of the re sults by this method with those by the ordinary method.
The preceding solution is quite long and complicated; but it is shorter and simpler than an algebraic solution. Further, the graphic solution is self checking at various intermediate steps; any errors in the graphic solution, being visible to the eye, are more easily detected than in an algebraic solution; and great errors are less likely in a graphic than in an algebraic solution.