CONDITIONS FOR AN ARCH HAVING FIXED ENDS. If the physical conditions are such as to fix the ends of the arch, then the three following mathematical conditions will be satisfied, viz.: 1. The inclination of the tangents at the ends of the neutral axis will not change when the load is applied. 2. The relative elevations of the two abutments will remain unchanged. 3. The length of span of the neutral axis of the arch ring will not change.
The problem of testing an arch according to the elastic theory consists in finding a line of resistance or a linear arch (§ 1195) that will satisfy the above conditions and at the same time give safe values for the stresses in the arch ring.
Conditions Stated Mathemati cally. To make the above conditions available as instruments in the investi gation of an arch, they must be stated in mathematical terms. To state them in algebraic form proceed as follows: First Condition. In Fig. 227, let CDKH be an element of a curved beam ds long, whose end faces CD and HK are at right angles to the neutral axis FG. In the original position, the tangents to the neutral axis at the points F and G make an angle with each other of dB. Let d = the change of angle between the end faces, or between the tangents at the ends, due to the bending caused by the load; da = an element of the area of the cross section; E = the coefficient of elasticity of the material; I = the moment of inertia of the cross section about the neutral line; M = the total bending moment of the external forces on one side of any section HK about G; s = the length of the neutral line of the arch ring; z = the distance of any fiber from the neutral line; The change in length of a fiber at a distance z from the neutral axis will be equal to z d˘, and the resulting stress per unit of area, in which ds is the original length of the fiber.* The integral of equation c gives the total change of the angle between the tangents at the two ends of the arch AB, Fig. 228; but for an arch having fixed ends this change is zero, and hence Equation 1 is an algebraic statement of the first condition given in § 1300.
angles GAK and AQA', we have AQ: AA' AK : AG, or dy : AG. d ˘ :: i 1 + x : AG; and therefore dy = al + x) 4, and substi tuting the value of d˘ from equation c and passing to the integral as in § 1302, we have for the equation for the second condition: The integral of equation d between the limits A and B, Fig. 228, is the total change of elevation along the arch ring due to the effect of the load; but for an arch having fixed ends, this change is zero, and hence which is the algebraic statement of the second condition in § 1300.
1304. Third Condition. In Fig. 228, from similar triangles we have QA' : AA' :: GK : AG, or dx : AG . y : AG, and therefore dx = y dIt; and by substituting the value of do from equation c, § 1302, we have The integral of equation e between the limits A and B is the change of span due to the effect of the load; but for an arch having fixed ends, this change is zero, and hence which is the algebraic statement of the third condition in § 1300.
Simplification of the Equations of Condition. To adapt the preceding equations of condition, equations 1, 2, and 3, to graphical computations, it is necessary to make certain modifications, as follows: To pass from Infintesimals to Finites. To adapt the equations of condition, equations 1, 2, and 3, to graphical computa tions, it is necessary to use finite increments instead of differentials. Each of the equations of condition contains the term ds _ 1. The value of ds varies from point to point according to the curvature of the arch ring; and I, the moment of inertia of the cross section, usually increases from the crown toward the springing, since the arch ring usually is deeper at the springing than at the crown,—as it should be, since the thrust in the arch increases toward the springing. Therefore, if the neutral line of the arch ring is divided into a number of short sections, 4s, such that ds - 1 is constant, we may sub stitute in the equations of condition the finite and constant quantity ds -I 1 for the infinitessimal and variable quantity ds - I. A method of dividing the arch ring so as to make ds ?- I constant will be explained later (see § 1311).