For example, the first one of the equations marked 48 may be interpreted as follows: Ill represents the transverse moment of the arch rib at any point of the arch rib. M is a variable, being sometimes positive, sometimes negative, and sometimes zero; E is the modulus of elasticity, and we shall here assume that this is also constant; ds represents the distance between any two consecutive sections of the arch rib. Theoretically, ds is assumed to be infinitely small, which means that we consider an infinite number of sections of the arch rib. I represents the moment of inertia of the arch rib at any section. In some cases this may be considered a constant; and it is a constant, provided the arch rib is of a uniform cross-section throughout its length. If, as is frequently the case, the arch rib is of variable cross section, then the value of I is variable for each section. It is assumed that the moment at each section is multiplied by the distance ds between the consecutive sections, and divided by the product of the modulus of elasticity and the moment of inertia at that section. All these quantities are positive, except M, which is sometimes positive, sometimes negative, and occasionally zero. Whenever any term has a constant value for each one of these small products, it may be placed outside of the summation sign, since the summation of a constant quantity times a variable is, of course, equal to that same constant quantity multiplied by the summation of the variables. As a corollary of this, we may also say that if the summation equals zero, we may even take the constant term out altogether; since, if a constant times a summation of positive and negative terms equals zero, then the sum mation of those positive and negative terms must of itself equal zero. There will be an illustration in the following sections, of the dropping of constant terms, and therefore the simplification of the mathematics. If such a product were obtained for each one of a very large number of cross-sections of the rib, we should have a series of products, some of which would be positive, some negative, and probably two of which would be zero. The algebraic sum of these terms would equal zero. The letters 0 and B near the top and bottom of the summation sign represent that sections arc made all the way from 0 to B in Fig. 22s. If the sections had been taken between two other points (as, for example, between 0 and C), the letter C would take the place of the letter B in the equation.
The three equations of Equation 48 are given without demon stration. The student must accept the equations as being mathe matically true, since their demonstration involves work in integral calculus which cannot be here given; but it should also be realized that the equations are only precisely true when the number of terms is infinitely large, and the distance ds is therefore infinitely small. When the sections are taken at a finite distance apart, as it is practi cally necessary to do, then there may be theoretically a slight error; but when the number of sections of an arch rib is made from 12 to 20 in the length of the span, the inac curacy involved because the num ber of terms is not infinite is so very small that it is of no practical importance.
424. Classification of Arch Ribs. Arch ribs may be classified in three ways: first, those which have fixed ends and no hinges; second, those which have a hinge or joint at each end; and third those which are hinged at both ends and in the center. The first class is by far the most common, and is the simplest and cheapest to construct; but, as will be developed later, it necessitates a very considerable allowance for temperature stresses which, under very unfavorable conditions, are even greater than the maximum stresses due to loading. The temperature stresses of a two-hinged arch are less severe, while those for a three-hinged arch may be neglected; but the construction of hinges in arch ribs adds considerably to the cost.
425. Mathematical Principles. In the following demonstration, the arch rib is considered as a single line OC'B (Fig. 22S), which is as sumed to have the properties of an arch rib—namely, the moment of inertia, modulus of elasticity of the material, and the consequent resist ing moment. The curved line PQR represents the special equilibrium polygon corresponding to some one condition of loading. Although
this line is drawn as a curved line, it is assumed to be a curve which is made up of a large number of correspondingly short lines, each of which corresponds to a section of an equilibrium polygon similar to those described under "Voussoir Arches." This equilibrium polygon is yet to be determined.
In Church's "Mechanics of Engineering," Chapter XI, is given the mathematical proof of three, general equations which apply to this problem. No demonstration will here be made of these three equa tions, which are as follows; The practical meaning of the first of these equations may be described as follows (see Fig. 22S) : ds represents one of an even number of very short sections into which the length OCR of the arch rib has been divided. M represents the transverse moment acting on the arch rib at that section under the particular condition of loading which is being considered. E is the modulus of elasticity of the material, and I is the moment of inertia of the section. At some of the sections the moment is positive, and at some it is negative. The product of Mand ds, divided by the product of E and I, is therefore sometimes positive and sometimes negative. According to this equation, the suntinatio-n of these various products for each short section (ds) of the rib equals zero; or, in other words, the summation of the positive products will exactly equal numerically the summation of the negative products.
The other two parts of Equation 48 must be interpreted similarly, Mds i the only difference being that in each case the term is multiplied El by the corresponding value of y for one of the equations, and by x for the other. This group of three equations (48) has nothing to do with the form of the special equilibrium polygon PQR.
It may also be proved by analytical mechanics, that if the curve PQR represents the special equilibrium polygon corresponding to some system of loading, and z represents the vertical distance between the arch rib and the special equilibrium polygon at any section, then the moment M at that section a of the rib, equals in which II is a constant which may be determined from the force diagram. The curve PQR represents a typical special equilibrium polygon which crosses the arch rib at two points. These points of intersection indi cate points of contraflexure, where the transverse moment changes its direction of rotation, and where it is therefore zero. When the special equilibrium polygon is above the rib curve, we call the MO merit positive; and when it is below, we call it negative. When it is positive, it means that there is tension in the lower part of the rib, and compression in the upper part. The conditions are, of course, the re verse of this when the curve is below the rib. We may therefore sub stitute Hz for the value of M in the group of Equations 48; and since Hand E are both constant for all points, from the principle enunciated in Article 423, we may not only place them outside of the sign of sum mation, but may even drop them altogether, since the summation equals zero; and we may therefore transform Equations 48 to the following: Whenever we are investigating the mechanics of an arch rib which has a constant moment of inertia, we may simplify Equations 49 by dropping out altogether the I of the denominators of those equations; but since arch ribs are usually made with deeper sections near the abutments, the I will be greater near the abutments. Calling the I at the center I„ then I equals n4, in which n is a variable. If we substitute this value of I in the denominators of Equations 49, then, since is a constant quantity, it may be placed outside of the summation sign, and even dropped altogether, which practically means that we substitute n for I in Equations 49. We shall also substitute for z its value z" — z' (see Fig. 228), and shall rewrite Equations 49 as follows, by making the substitutions: It will later be shown how we can draw a line (marked in Fig. 228) which will satisfy the following equations: Since the arch rib (represented by the curve OCB) is assumed to be symmetrical about its center C, and since via is horizontal, any tion of vat which will satisfy the first of Equations 51 will also satisfy the second.