Building Design and Construction Handbook - Merritt - Ventech!

5.118 SECTION FIVE
where M � moment at any section resulting from loads
N � normal thrust on cross section
A � cross-sectional area of arch
y � ordinate of section measured from A as origin, when B is on rollers
I � moment of inertia of section
E � modulus of elasticity
ds � differential length along axis of arch
dx � differential length along horizontal
y cos � dx
EI AE
B
�x� ���
2 B
ds � �
2
(5.170)
A A
where � � the angle the tangent to the axis at the section makes with the horizontal.
Unless the thrust is very large and would be responsible for large strains in the
direction of the arch axis, the second term on the right-hand side of Eq. (5.169)
can usually be ignored.
In most cases, integration is impracticable. The integrals generally must be evaluated
by approximate methods. The arch axis is divided into a convenient number
of sections and the functions under the integral sign evaluated for each section. The
sum is approximately equal to the integral. Thus, for the usual two-hinged arch,
B
�A (My �s/EI)
� �
H � B B
(5.171)
2 2
(y �s/EI) � (cos � �x/AE)
A A
(S. Timoshenko and D. H. Young, ‘‘Theory of Structures,’’ McGraw-Hill Book
Company, New York; S. F. Borg and J. J. Gennaro, ‘‘Modern Structural Analysis,’’
Van Nostrand Reinhold Company, Inc., New York.)
5.14.3 Stresses in Arch Ribs
When the reactions have been found for an arch (Arts. 5.14.1 to 5.14.2), the principal
forces acting on any cross section can be found by applying the equations of
equilibrium. For example, consider the portion of an arch in Fig. 5.96, where the
FIGURE 5.96 Interior stresses at X hold portion LX of an
arch rib in equilibrium.