This section is from the book "Cyclopedia Of Architecture, Carpentry, And Building", by James C. et al. Also available from Amazon: Cyclopedia Of Architecture, Carpentry And Building.
It has been found impracticable to develop the theory of elastic arches without employing some of the fundamental principles of integral calculus; but an effort will be made to explain each one of the equations which are used, in such a way that the application of calculus to this particular case may be understood. To facilitate this demonstration, a few of the fundamental principles of integral calculus will be briefly demonstrated. All of the calculus equations which are used are similar to Equation 48. In this, a character ∫, somewhat similar to the letter S (which may be considered to stand for the word summation), is placed in front of some mathematical quantities. The equation generally reads that this summation equals zero. The general meaning of the equation is that there is a group of quantities all of which are in general similar, but which have a variation in magnitude. In general, some of these quantities are positive, and some are negative, and the equation reads that the summation or the algebraic addition of all these positive and negative quantities just equals zero; or, in other words, that the sum of all the positive quantities is just equal to the sum of all the negative quantities.
For example, the first one of the equations marked 48 may be interpreted as follows: M 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 summation 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 O and B near the top and bottom of the summation sign represent that sections are made all the way from 0 to B in Fig. 228. 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 demonstration. The student must accept the equations as being mathematically 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 practically 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 inaccuracy involved because the number of terms is not infinite is so very small that it is of no practical importance.