Building Design and Construction Handbook - Merritt - Ventech!

5.14.2 Two-Hinged Arches
FIGURE 5.95 Two-hinged arch.
STRUCTURAL THEORY 5.117
When an arch has hinges at the supports only (Fig. 5.95), it is statically indeterminate,
and some knowledge of its deformations is required to determine the reactions.
One procedure is to assume that one of the supports is on rollers. This
makes the arch statically determinate. The reactions and the horizontal movement
of the support are computed for this condition (Fig. 5.95b). Then, the magnitude
of the horizontal force required to return the movable support to its original position
is calculated (Fig. 5.95c). The reactions for the two-hinged arch are finally found
by superimposing the first set of reactions on the second (Fig. 5.95d).
For example, if �x is the horizontal movement of the support due to the loads,
and if �x� is the horizontal movement of the support due to a unit horizontal force
applied to the support, then
�x � H�x� � 0 (5.167)
�x
H �� (5.168)
�x�
where H is the unknown horizontal reaction. (When a tie rod is used to take the
thrust, the right-hand side of Eq. (5.167) is not zero, but the elongation of the rod,
HL/AE.) The dummy unit-load method [Eq. (5.96)] can be used to compute �x and
�x�:
My N dx
EI AE
B
�x � �
B
ds � � (5.169)
A A