As the ends of the arch at ii and B are built into the sides of the valley and not free to move toward the center 0 when subjected to the water pressure, the lines of thrust of the arch will not be exactly axial as assumed in Formula (S), and bending stresses will develop in the arch, giving a maximum compression somewhat greater than the average value. This effect will usually be small as compared with the stress due to arch action, although French authorities recommend that the line of thrust be assumed at the outer edge of the middle third at the crown, thus making the maximum compression double the average. The use of vertical expansion joints through the dam, dividing it into voussoirs, has the effect of largely eliminating the bending stresses. In practice the bending stress is commonly neglected, very conservative values for being used.
When the length of the arch is small as compared with its thick ness, it becomes a curved wedge which acts as a beam between the abutments supporting its ends, and should be considered as a curved beam—a condition frequently occurring near the bottom of a curved dam, where the valley is narrow and the thickness of the clam con siderable. The thickness obtained by considering such a section as an arch is always sufficient.
A masonry structure cannot be considered to act as an arch when the thickness of the arch ring is more than from one-quarter to one third of the radius of its outer surface. The exact limitations within which such action may take place are not definitely known and are seldom of importance in a dam.
Rwissiltt of Vertical a dani is rigidly fastened to the foundation, it is evident that complete arch action cannot take place, and that in the lower part of the dam, the arch can carry very little of the load. A vertical section of the dam may be considered as a cantilever fixed at the bottom as in a gravity dam, and the resist ance of the cantilever to deflection will limit the extent to which arch action may occur.
Attempts have been made by estimating the relative deflections of the horizontal arch and the vertical cantilever at various heights upon the mid-section of the clam, to determine Ivhat portion of the load is resisted by each. Such studies have been made by Mr. Silas H. \Woodward 1 for the Lake Cleeseman dam, which is a curved dam of section (see Section 141) and by \Ir. Edgar T. for the Pathfinder dam, which was designed as an arch, and has a section considerably lighter than could have been employed in a gravity dam. The section of the darn has a width of 10 feet at the
top, a batter of .25 on the downstreiun and .15 on the upstream face.
These analyses, with accompanying discussions, are interesting as throwing light upon the probable action of such dams when sub jected to water pressure, but afford no means of determining the actual stresses occurring. The vertical cantilever has the effect of reducing the stresses in the arches, but it is not proposed to consider the combined actions in designing dams, or to attempt to use the actual stresses, as limited by the cantilever resistance in proportioning the arches. In practice, the arches are given sections which would enable them to carry the whole water pressure, and the vertical resistance is considered as a source of additional security.
Horizontal the dam is fixed at the bottom to the foun dation and the various horizontal slices are not free to act independ ently of each other, the thickness at any point should be sufficient to carry the total water pressure above as horizontal shear. If S he the safe unit shear per square foot, the thickness should not be less than h2 Such shearing stresses can exist only near the bottom of the dam, where it is rigidly attached to the foundation, and can never reach the assumed value if the water pressures toward the top of the clam are carried by arch action.
Weight of Masonry.—Each horizontal slice of an arch dam must carry the weight of the portion of the dani above as a vertical com pression. This compression is computed as in the gravity section when the dam is empty, and must not exceed a safe unit stress on any part of the section. The weight of masonry above also produces a distortion of the horizontal section. The value of Poisson's ratio for concrete may be taken as approximately one-fifth of the unit horizontal compression produced through the mass of masonry, if prevented from expanding laterally is approximately one-fifth of the unit vertical compression which causes it. The effect of this horizontal compression is to cause an expansion of the horizontal section, increasing the length of the arch ring, and deflecting the crown of the arch upstream. When water pressure is brought against the dam, a portion of the pressure, sufficient to produce com pression in the arch equal to the unit horizontal pressure due to the vertical load, will be used to bring the arch back to its initial position, and no deflection clue to arch action will occur until this pressure has been passed.