Masonry Magazine August 1975 Page. 18
It is often assumed that the entire weight of masonry, above the soffit, presses vertically upon the arch. This certainly is not accurate, since even with dry masonry a part of the wall will be self-supporting. However, this assumption is certainly on the safe side. The passive resistance of the adjacent masonry materially affects the stability of an arch.
The designer must rely on empirical formulae, based on the performance of existing structures, to determine the loads on an arch. The dead load of masonry wall supported by an integral arch depends upon the arch rise and span and the wall height above the arch. It may be considered to be either uniform (rectangular) or variable (complementary parabolic) in distribution, or a combination thereof.
"Frames and Arches" gives solutions for arches with rise-to-span ratios (f/L) ranging from 0.0 to 0.6. The following recommended assumptions for loading of such arches are believed to be safe: For low rise arches, f/L 0.2 or less, a uniform load may be assumed. This load will be the weight of wall above the crown of the arch up to a maximum height of L/4.
For higher rise arches a dead load consisting of uniform plus complementary parabolic loading may be assumed. The maximum ordinate of the parabolic loading will be equal to a weight of wall whose height is the rise of the arch. The minimum ordinate of the parabolic loading will be zero. The uniform loading will be the weight of the wall above the crown of the arch up to a maximum height of L²/100.
Uniform floor and roof loads are applied as a uniform load on the arch. Small concentrated loads may be treated as uniform loads of twice the magnitude. Large concentrated loads may be treated as point loads on the arch.
Several major arches are shown in Fig. 5.
MAJOR ARCH DESIGN
General. "Frames and Arches" provides straightforward equations by which redundant moments and forces in arched members may be determined. The reader is referred to a discussion of this book which appears in Technical Notes, No. 31.
Without repeating the forementioned discussion here, let it suffice to say that, for relatively high-rise (f/L 0.2) constant-section arches, Method A of Section 22 yields the proper solutions. The recommendations for use of this section are:
1. Establish principal dimensions of the arch.
2. On this basis and depending upon the established shape and f/L ratio of the arch, obtain the corresponding k value of the arch (see Table 3).
3. Obtain the elastic parameters (α, β, y and 8), load constants and general constants.
4. Perform the algebraic operations with the given equations.
Equations
The equations are based upon a horizontal and vertical grid coordinate system with origin at the intersection of the arch axis and left skewback. Distances x and y are coordinates of the arch axis. The general equation for the parabolic arch axis is:
y=4f(1-)
Each set of equations depends upon the loading conditions. Among the solutions included with those in Section 22, Method A are the following:
For vertical complementary parabolic loading:
M₁ = M₂ = (JS-2T)
H₁ = H₂ = (K-2JT)
V₁ = V₂ = W/2
When x ≤ L/2:
M₁ = M₁ + [1-()]-Hy
N. = sin + H₂cos
Q. = cos + H₂sin
For vertical uniform load over the entire arch:
M and Q are zero at any section of the arch.
H₁ = H₂ = (K-2JT)
V₁ = V₂ = W/2
TABLE 3
Values of k
Arch rise-to
span-ratio f/L
Arch k value
0.2
0.3
0.4
0.5
0.6
1.28 1.56 1.90 2.40 2.80
Note: Adapted from Table 12. "Frames and Arches."
TABLE 4
Values of
Arch ratio
Values of where x =
F/L
O and L
0.1L and 0.9L
0.21 and 0.8L
0.31 and 0.7L
0.41 and 0.6L
0.5L
0.20
38°40′
32°37'
25 38
17°45
9°05'
0
0.30
50°12′
4350
35°45'
25°38′
13 30
0
0.40
58°00′
52°00′
43°50′
32°37'
17°45
0
0.50
63°26
58°00′
50°12′
38°40′
21°48'
0.60
67°23′
62°29'
55°13′
43°50′
25°38′
Note: From Table 10, "Frames and Arches."