In a fixed arch, translation and rotation are prevented at the supports (Fig. 4.3). Such an arch is statically indeterminate. With each reaction comprising a horizontal and vertical component and a moment (Art. 4.1), there are a total of six reaction components to be determined. Equilibrium laws provide only three equations. Three more equations must be obtained from a knowledge of the elastic behavior of the arch.
One procedure is to consider the arch cut at the crown. Each half of the arch then becomes a cantilever. Loads along each cantilever cause the free ends to deflect and rotate.
To permit the cantilevers to be joined at the free ends to restore the original fixed arch, forces must be applied at the free ends to equalize deflections and rotations. These conditions provide three equations.
Solution of the equations, however, can be simplified considerably if the center of coordinates is shifted to the elastic center of the arch and the coordinate axes are properly oriented. If the unknown forces and moments V, H, and M are determined at the elastic center (Fig. 4.3), each equation will contain only one unknown. When the unknowns at the elastic center have been determined, the shears, thrusts, and moments at any points on the arch can be found from the laws of equilibrium.
Determination of the location of the elastic center of an arch is equivalent to finding the center of gravity of an area. Instead of an increment of area dA, however, an increment of length ds multiplied by a width 1/EI must be used, where E is the modulus of elasticity and I the moment of inertia of the arch cross section.
In most cases, integration is impracticable. An approximate method is usually used, such as the one described in Art. 4.2.
Assume the origin of coordinates to be temporarily at A, the left support of the arch. Let x’ be the horizontal distance from A to a point on the arch and y’ the vertical distance from A to the point. Then the coordinates of the elastic center are
If the arch is symmetrical about the crown, the elastic center lies on a normal to the tangent at the crown. In this case, there is a savings in calculation by taking the origin of the temporary coordinate system at the crown and measuring coordinates parallel to the tangent and the normal. Furthermore, Y, the distance of the elastic center from the crown, can be determined from Eq. (4.9) with y’ measured from the crown and the summations limited to the half arch between crown and either support. For a symmetrical arch also, the final coordinates should be chosen parallel to the tangent and normal to the crown.
For an unsymmetrical arch, the final coordinate system generally will not be parallel to the initial coordinate system. If the origin of the initial system is translated to the elastic center, to provide new temporary coordinates x1 = x’ - X and y1 = y’ - Y, the final coordinate axes should be chosen so that the x axis makes an angle , measured clockwise, with the x1 axis such that
The unknown forces H and V at the elastic center should be taken parallel, respectively, to the final x and y axes.
The free end of each cantilever is assumed connected to the elastic center with a rigid arm. Forces H, V, and M act against this arm, to equalize the deflections produced at the elastic center by loads on each half of the arch. For a coordinate system with origin at the elastic center and axes oriented to satisfy Eq. (4.10), application of virtual work to determine deflections and rotations yields