Masonry Magazine August 1975 Page. 15
For minor segmental arches
For minor segmental arches, the angle between the line of resistance and the normal to the joint is greatest at the skewback. This will also be true for jack arches if the joints are radial about a center at the intersection of the planes of the skewbacks. However, if the joints are not radial about this center, each joint should be investigated for resistance to sliding. This can be done most easily by constructing an equilibrium polygon, assuming that the crown thrust is applied at the top of the middle third and the reaction at the skewback is applied at the bottom of the middle third of the section.
For segmental arches with radial joints, the angle (y) between the skewback and the vertical is
y = tan(4rS / (S-4r))
or in terms of the radius of curvature
y = sin(S / 2R)
For jack arches in which the skewback equals 1½ in. per ft of span for each 4 in. of arch depth, the angle (y) that the skewback makes with the vertical is
y = tan(S / 8)
In equations 2, 3 and 4:
S = span,
r = rise,
R = radius of curvature.
(c) Crushing
# (1) Segmental Arch
Figure 3 is a graphic representation of thrust coefficients (H/W) for segmental arches subjected to uniform load over the entire span. Once the thrust coefficient is determined for a particular arch, the horizontal thrust (H) may be determined as the product of the thrust coefficient and the total load (W). To determine the proper thrust coefficient, one must first determine the characteristics of the arch, S/r and S/d:
where:
S = the clear span,
r = the rise of the soffit and
d = the depth of the arch.
In these ratios and in the ratios and equations that follow, all terms of length must be expressed in the same units; for example, in computing S/r and S/d, if S is in feet, r and d must be in feet also.
If the applied load is triangular or concentrated, the above method may still be used, but the horizontal thrust coefficient must be increased by ½ for triangular loading and doubled for concentrated loads.
Once the horizontal thrust has been determined, the maximum compressive stress in the masonry is determined by the following formula:
f = 2H / bd
In this equation:
f = maximum compressive stress in the arch in pounds per square inch,
H = horizontal thrust in pounds,
b = breadth of the arch in inches and
d = depth of the arch in inches.
This value is twice an axial compressive stress on the arch, due to a load H, because the horizontal thrust is located at the third point of the arch depth.
# (2) Jack Arch
The common rule for jack arches is to provide a skewback (K, measured horizontally) of ½ in. per ft of span for each 4 in. of arch depth. Jack arches are commonly constructed in depths of 8 and 12 in. with a camber of 1/8 in. per ft of span.
For jack arches, applying the same assumptions as previously outlined, the horizontal thrust at the spring line may be determined by the following formulae:
For uniform loading over full span,
H = 3WS / 8d
For triangular loading over full span,
H = WS / 2d
Maximum compressive stress (f) in the jack arch may be determined from the following formulae:
f = 2H / bd