STABILITY AGAINST SLIDING. The horizontal pressure of the water tends to slide the dam forward, and is resisted by the friction due to the weight of the wall.
For possible additional forces tending to produce sliding, see 4 956.
The weight of a unit section of the dam, W, is equal to the area of the vertical cross section multiplied by the weight of a cubic unit of the masonry, w. Then the weight of the dam shown in Fig. 96 is The vertical pressure, V, of the water on the inclined face, I B, Fig. 96, is equal to the horizontal projection of that area multiplied by the distance of the center of gravity of that surface below the top of the water and also by the weight of a cubic unit of water.
If the earth rests against the heel of the dam (the bottom of the inside face), it will increase the pressure on the foundation, since earth weighs more than water; but the difference is not very great, and is compensated for in part by the fact that under such condition the horizontal pressure of the water will not be so much as assumed above, and hence it will be assumed that the water extends to the foundation of the dam.
If the permanent water level on the down-stream side is above the foundation, the back pressure of this water should be deducted from the value of H as computed above.
If the water finds its way under and around the foundation of the wall, even in very thin sheets, it will decrease the pressure of the dam on the foundation, and consequently decrease the stability of the wall. The effective weight of the submerged portion of the dam will be decreased 62i lb. per cu. ft. However, it is not likely that water in hydrostatic condition will find its way under or into a darn in appreciable quantities, and hence the effect of buoyancy will not be included.
For a discussion of a closely related phase of this subject, see Condition for Equilibrium. In order that the wall may not slide, it is necessary that the product found by multiplying the co efficient of friction by the sum of the weight of the dam and the vertical pressure of the water shall be greater than the horizontal pressure of the water; or in mathematical language, in order that the dam may not slide it is necessary that By stating H and V in terms of the height of the water and of the weight of a cubic unit of water (see equation 1 and 4, page 460) and giving W in terms of the dimensions of the dam, it is easy by means of equation 6 to determine the factor of safety in any particular case. Values of the coefficient of friction are given in Table 74, page 464.
However, it is not wise to attempt to compute the factor of safety against sliding, since the value obtained is dependent upon the value of the coefficient of friction assumed. • Therefore, it is better either (1) to state the relative values of the resisting forces and of the forces tending to produce sliding, or (2) to state the tangent of the angle which the resultant pressure makes with the normal to the base.
To secure economy of material in the construction of the dam, it is customary to make the up-stream surface nearly or quite vertical; and hence the vertical pressure of the water on the up stream face is comparatively small, and is usually neglected—an approximation which is on the safe side.* The preceding discussion assumes that a masonry dam may fail by the sliding of one stone upon another along a horizontal joint; but neglects two important elements of stability. First, the stones are laid with mortar which gives a considerable resistance of cohesion in addition to the frictional resistance. Second, masonry dams, at least high ones, are built of random rubble masonry, the stones of which interlock in every direction; and hence the tendency to slide is resisted by the shearing strength of the individual stones (˘ 20) as well as by friction. If the dam is built of coursed masonry, which is very improbable, the courses could be inclined downward toward the up-stream side.