Let Fig. 77 represent an inclined arch dam. A slice of the arch ring normal to the axis carries a water pressure which varies from the crown to the springing line, and also carries a portion of its own weight to the buttress. If a slice of the arch ring be divided into voussoirs as shown, the water pressures upon each voussoir — varies with the depth (0145) below the surface of the water. The weights of the voussoirs (G1—G5) may be considered as divided into components, (N=G cos 0) normal to the section and (TV=G sin 0) parallel to the section. The normal components are carried as longi tudinal thrusts to the foundation, while the parallel components arc carried by the arch ring to the buttress. Having determined these loads, an approximate line of thrust may be drawn by the method used for voussoir arches (see Section 162), from which stresses may be determined.
In designing such an arch, the required thickness at various depths may be approximately determined by finding the thickness for a horizontal arch at the same depth, then using this thickness in the analysis, modifying it as required. Practically an assumed thickness is given the ring at the top and tapered to the required thickness at some point below.
When the arch axis is vertical, the arch carries only the water pressure, which is uniformly distributed over the face. The weight of the arch, in this case, is normal to the arch section and is carried vertically to the foundation. The thickness required for the arch ring may be found from Formula (8), (Art. 138).
The stresses upon the buttresses of a multiple-arch clam may be found by the methods used for gravity dams. In Fig. 78 E—F is a section through the crown of an inclined arch; A BCD being the side projection of the buttress. The form of the buttress must be such that the resultant thrust upon any horizontal section A—B will act approximately at the middle of the section. The loads acting are: (1) The horizontal water pressure (11 = due to the depth of water above the plane A—B, upon a length of dam (L) equal to the distance between the middle points of adjacent arches.
(2) The vertical water pressure (V= II • tan • 0). The center of pressure for the vertical water pressure is at the center of gravity of a horizontal section of the water face of the arch at two-thirds the depth below the surface of the water.
(3) The weight of the two half arches upon each side of the buttress. The center of gravity for the weight of the arch is at the
center of gravity of the center line of a horizontal section of the arch ring which passes through the center of gravity of the vertical section (E—F) of the crown of the arch. If the centers of gravity of the center lines of the arch ring be determined for horizontal sections at the top and bottom of the arch, all intermediate centers will lie upon the line joining these points.
(4) The weight of the buttress itself, acting through its center of gravity.
The resultant (R) of these loads should cut the base A—B near its middle point, in order to secure uniform distribution of pressure over the section.
Buttresses, for clams of this type, are usually made very thin in comparison with their widths, and are therefore stiffened laterally by the use of horizontal struts from buttress to buttress, or by the use of cross walls. The design of these struts is purely a matter of judgment on the part of the designer.
In the design of multiple-arch dams, the general lay out is a matter which must depend upon local topography. Each clam is a problem by itself, and must be made to fit its location. It has been found that in some instances, where the conditions are favorable to the con struction of masonry clams of moderate height, multiple-arch dams may be built at much less cost than gravity structures. Forty to 60 or 70 feet between centers of buttresses are commonly found economical distances. Arches with axes making angles of or 40° with the vertical are apt to show some saving of material as compared with vertical axes, but this is not always the case. The unit cost of construction is usually somewhat greater for inclined arches.
In constructing gravity clams, a cheaper grade of masonry may be employed, and the form work costs less than for multiple-arch clams. Careful studies of local conditions, and tentative trial designs are necessary in each case for best results.
Temperature Stresses, clue to temperatures lover than those at which the arches are constructed, are to be expected in all structures. These produce shortening of the arch and give tensile stresses which may result in cracks when the clam is empty. Horizontal reinforce ment near the downstream face at the crown and near the upstream face at the springing line is desirable to resist this tendency to crack.