An arch is a beam curved in the plane of the loads to a radius that is very large relative to the depth of section. Loads induce both bending and direct compressive stress. Reactions have horizontal components, though all loads are vertical. Deflections, in general, have horizontal as well as vertical components. At supports, the horizontal components of the reactions must be resisted. For the purpose, tie rods, abutments, or buttresses may be used. With a series of arches, however, the reactions of an interior arch may be used to counteract those of adjoining arches.
A three-hinged arch is constructed by inserting a hinge at each support and at an internal point, usually the crown, or high point (Fig. 4.1). This construction is statically determinate.
There are four unknowns—two horizontal and two vertical components of the reactions— but four equations based on the laws of equilibrium are available.
1. The sum of the horizontal forces acting on the arch must be zero. This relates the horizontal components of the reactions:
HL = HR = H
2. The sum of the moments about the left support must be zero. For the arch in Fig. 4.1, this determines the vertical component of the reaction at the right support:
VR = Pk
where P load at distance kL from left support
L = span
3. The sum of the moments about the right support must be zero. This gives the vertical component of the reaction at the left support:
VL = P(1 - k)
4. The bending moment at the crown hinge must be zero. (The sum of the moments
about the crown hinge also is zero but does not provide an independent equation for determination of the reactions.) For the right half of the arch in Fig. 4.1, Hh - VRb = 0, from which
The influence line for H for this portion of the arch thus is a straight line, varying from zero for a unit load over the support to a maximum of ab/Lh for a unit load at C.
Reactions of three-hinge arches also can be determined graphically by taking advantage of the fact that the bending moment at the crown hinge is zero. This requires that the line of action of reaction RR at the right support pass through C. This line intersects the line of action of load P at X (Fig. 4.1). Because P and the two reactions are in equilibrium, the line of action of reaction RL at the left support also must pass through X. As indicated in Fig. 4.1b, the magnitudes of the reactions can be found from a force triangle comprising P and the lines of action of the reactions.
For additional concentrated loads, the results may be superimposed to obtain the final horizontal and vertical reactions. Since the three hinged arch is determinate, the same fourequations of equilibrium can be applied and the corresponding reactions determined for any other loading condition. It should also be noted that what is important is not the shape of the arch, but the location of the internal hinge in relation to the support hinges.
After the reactions have been determined, the stresses at any section of the arch can be found by application of the equilibrium laws (Art. 4.4).