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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 concrete arch
rib at any point of the concrete 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; the point
represents the distance between any two consecutive sections of the concrete
arch rib. Theoretically, it is assumed to be infinitely small, which means that
we consider an infinite number of sections of the concrete arch rib. The
concrete shown represents the moment of inertia of the concrete arch rib at any
section. In some cases this may be considered a constant; and it is a constant,
provided the concrete arch rib is of a uniform cross-section throughout its
length.

If, as is frequently the case, the concrete 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 0 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. The three
equations of Equation 48 are given without demonstration. The concrete
contractor 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 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 concrete arch rib is made from 12 to 20
in the length of the concrete span, the inaccuracy involved because the number
of terms is not infinite is so very small that it is of no practical importance.
Concrete arch ribs may be classified in
three ways: first, those which have fixed ends and no hinge; second and 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.

**Are You in ****Dudley Massachusetts****? Do You
Need Concrete Cutting?**

**We Are Your Local Concrete
Cutting Company**

**Call ****508-283-3135**

**We Service all
surrounding Cities & Towns.**