Dejardin's formulas, which are frequently employed by French engineers, are as follows: For circular arches, For elliptical and basket-handled arches, By Dejardin's formulas the thickness at the crown decreases as the rise decreases, which seems contrary to reason.
Croizette-Desnoyers, a French authority, recommends the following formulas: Notice that in none of the above formulas does the char acter of the material of the arch ring enter as a factor. Notice also that none of them has a factor depending upon the amount of the load—either live or dead.
Thickness of the Arch at the Springing. If the loads are vertical, the horizontal component of the compression on the arch ring 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, which is frequently employed: 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. In algebraic language, this rule is in which 1 is the length of the joint, d the depth at the crown, and a the angle the joint makes with the vertical.
Trautwine gives a formula for the thickness of the abut ment, which determines also the thickness of the arch at the springing (see 4 1258) .
"The. augmentation of the thickness at the springing line is made, by the Stephenson, from r to 30 per cent; and by the Rennes about 100 per cent." Thickness of the Abutment. Trautwine's formula is 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 cir cular or elliptical, and whatever the proportions of rise and span— and to any height of abutment. It applies also to all the usual methods of filling above the arch, whether with solid masonry to the level of the top of the crown, or entirely with earth. It gives a thickness of abutment which is safe in itself without any backing of earth behind it, and also against the pressure of the earth when the bridge is unloaded. It gives abutments which alone are safe when
the bridge is loaded; but for small arches, the formula supposes that earth will be deposited behind the abutments to the height of the roadway. In small bridges and large culverts on first-class railroads, subject to the jarring of heavy trains at high speed, the compara tive cheapness with which an excess of strength can be thus given to important structures has led, in many cases, to the use of abutments from one fourth to one half thicker than those given by the preceding rule. If the abutment is of rough rubble, acid 6 inches to the thick ness by the above formula, to insure full thickness in every part."* To find the thickness of the abutment at the bottom, lay off, in Fig. 203, on = t as computed by the above equation; vertically above n lay off an m half the rise; and horizontally from a lay off ab = one forty-eighth of the span. Then the line bn prolonged gives the back of the abutment, provided the width at the bottom, sp, is not less than two thirds of the height, os. "In practice, os will rarely ex ceed this limit, and only in arches of considerable rise. In very high abutments, the abutment as above will be too slight to sustain the earth pressure safely. "* To find the thickness of the arch, compute the thickness ce by equation 10, page 643, draw a curve through e parallel to the intrados, and from b draw a tangent to the extrados; and then will bfe be the top of the masonry filling above the arch. Or, instead of drawing the extrados as above, find by trial a circle which will pass through b, P, and b', the latter being a point on the left abutment corresponding to b on the right.
Trautwine's rule, or a similar one, for proportioning the abutment and the backing is frequently employed.
Rankine says that in some of the best examples of bridges the thickness of the abutment ranges from one third to one fifth of the radius of curvature of the arch at its crown.
The following formula is said to represent German and Russian practice, in which h is the distance between the springing line and the top of the foundation.