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# Masonry Construction 99

## arch, crown, thickness, joint, rupture and arches

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MASONRY CONSTRUCTION 99 Table 13 gives the depth of keystone for semicircular arches, the second column being for hammer-dressed beds, the third for beds roughly dressed with the chisel, and the fourth for brick masonry.

## Thickness of Arch at the Springing.

Generally the thick ness of the arch at the springing is found by an application of theory.

If the loads are vertical, the horizontal component of the compression on the arch is constant; and hence, to have the mean pressure on the joints uniform, the vertical projection of the joints should be constant. This principle leads to the following formula: The length measured radially of each joint between the joint of rupture and the crown should be such that its vertical projection is equal to the depth of the keystone.

The length of the joint of rupture, i.e., the thickness of the arch at the practical springing line, can be computed by the formula z = d sec a in which z is the length of the joint, d the depth of the crown, a the angle the joint makes with the vertical.

The following are the values for circular and segmental arches : Thickness of the Abutments. The thickness of the abut ment is determined by the following formula: t = 0.2 p + 0.1 R 2.0 in which t is the thickness of the abutment at the springing, p the radius, and R the rise—all in feet.

The above formula applies equally to the smallest culvert or the largest bridge—whether circular or elliptical, and whatever the pro- . portions of rise and span—and to any height of abutment.

Table 14 gives the minimum thickness of abutments for arches of 120 degrees where the depth of crown does not exceed 3 feet. Calculated from the formula in which D = depth or thickness of crown in feet; H = height of abutMent to springing in 'feet; R = radius of arch at crown in feet; • T = thickness of abutment in feet.

Arches fail by the crown falling inward, and thrusting outward the lower portions, presenting five points of rupture, one at the key stone, one on each side of it which limit the portions that fall inward, and one on each side near the springing lines which limit the parts thrust outward. In pointed arches, or those in which the rise is

greater than half the span, the tending to yielding is, in some cases, different; and thrust upward and outward the parts near the crown.

The angle which a line drawn from the center of the arch to the joint of rupture makes with a vertical line is called the angle of rupture. This term is also used when the arch is stable, or where there is no joint of rupture, in which case it refers to that point about which there is the greatest tendency to rotate. It may also be defined as including that portion of the arch near the crown which will cause the greatest thrust or horizontal pressure at the crown. This thrust tends to crush the voussoirs at the crown, and also to overturn the abutments about some outer joint. In very thick arches rupture may take place from slipping of the joints.

In order to avoid any tendency of the joints to open, the arch should be so designed that the actual resistance line shall everywhere be within the middle third of the depth of the arch ring.

In general the design of an arch is reached by a series of approx imations. Thus, a form of arch and spandrel must be assumed in advance in order to find their common center of gravity for the pur pose of determining the horizontal thrust at the crown, and the reaction at the skewback.

## Backing.

The backing is masonry of inferior quality or con crete, laid outside and above the arch stones proper, to give additional security. Ordinarily, the backing has a zero thickness at or near the crown, and gradually increases to the upringing line.

## Spandrel Filling.

Since the surface of the roadway must not deviate from a horizontal line, a considerable quantity of material is required above the backing to bring the roadway level. Ordinarily this space is filled with earth, gravel, broken stone, cinders, etc. Sometimes to save filling small arches are built over the haunches of the main arch.

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