Building Design and Construction Handbook - Merritt - Ventech!

5.116 SECTION FIVE
FIGURE 5.93 Components of an arch.
The necessity of resisting the horizontal components of the reactions is an important
consideration in arch design. Sometimes these forces are taken by the tie
rods between the supports, sometimes by heavy abutments or buttresses.
Arches may be built with fixed ends, as can straight beams, or with hinges at
the supports. They may also be built with a hinge at the crown.
5.14.1 Three-Hinged Arches
An arch with a hinge at the crown as well as at both supports (Fig. 5.94) 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 must be zero. (2) The sum of the
moments about the left support must be zero. (3) The sum of the moments about
the right support must be zero. (4) The bending moment at the crown hinge must
be zero (not to be confused with the sum of the moments about the crown, which
also must be equal to zero but which would not lead to an independent equation
for the solution of the reactions).
Stresses and reactions in three-
FIGURE 5.94 Three-hinged arch.
hinged arches can be determined graphically
by taking advantage of the fact
that the bending moment at the crown
hinge is zero. For example, in Fig.
5.94a, a concentrated load P is applied
to segment AB of the arch. Then, since
the bending moment at B must be zero,
the line of action of the reaction at C
must pass through the crown hinge. It
intersects the line of action of P at X.
The line of action of the reaction at A
must also pass through X. Since P is
equal to the sum of the reactions, and
since the directions of the reactions have
thus been determined, the magnitude of
the reactions can be measured from a parallelogram of forces (Fig. 5.94b). When
the reactions have been found, the stresses can be computed from the laws of statics
(see Art. 5.14.3) or, in the case of a trussed arch, determined graphically.