In equation 19, if d = 0, the load is symmetrical, and the pressure is uniform, as it should be. Notice that d is plus when r is on the same side of the center as A, i.e., as the point for which the pressure P is desired, and minus when on the other side of the center. For example, if the dam is a right-angled triangle with the right angle at B and the reservoir is empty, gB = it, d = —it, and the or P is twice the mean, which also is as it should be.
The last relation is known as the principle of the middle third; that is, as long as the center of pressure lies within the middle third of the joint, the maximum pressure is not more than twice the mean, and there is no tension in any part of the joint.
The first term of the right-hand side of equation 19 gives the uniform pressure on the base due to the weight of the dam and the vertical component of the water pressure, and the second term gives the effect upon the maximum pressure on the base of any system of forces that causes the centre of pressure to depart from the middle of the base, i.e., that causes the resultant pressure (R, Fig. 100, page 470) to intersect the base at a distance d from the center.
In other words, the term 6 Wd is the increase of pressure on the base due to the eccentricity of the center of pressure, r,—whether that eccentricity is due to an unsymmetrical vertical cross section or to the overturning effect of external forces, or to both.
Therefore, equation 19 is a perfectly general expression for the pressure between any two plane rectangular surfaces pressed together by any system of forces.
Equation 19 may be written thus: The -I- sign gives the maximum pressure at A, Fig. 100, page 470, for any given deviation of the center of pressure, r, from the middle of the base, m; and the — sign the corresponding minimum at B, d being taken without regard to algebraic sign. Equation 22 gives the maximum and minimum pressures at the two extremes of the base whether the deviation d is caused by the form of the wall or by forces tending to produce overturning, or by both.
pressure departs more than 1 from the center of the base, there will be a minus pressure, i.e., tension, at the opposite side of base; in other words, if d in equa tion 22, page 473, is more than i 1, the maximum pressure will be more than twice the mean and the minimum pressure will be minus, i.e., tension. If AK' represents the com pression at A when d is more than 1, and BL' represents the corresponding tension, i.e, the minus pressure at B, then the ordinates of the triangle ANK' represent the pressures at the several points along AN; and similarly the ordinates of the triangle BNL' represent the tensions at the several points along BN.
If a good quality of cement mortar is used, it is not unreasonable to count upon a little resistance from tension. As a general rule, it is more economical to increase the quantity of stone than the quality of the mortar; but in dams it is necessary to use a good mortar to prevent (1) leakage, (2) disintegration on the water side, and (3) crushing. Therefore, equation 22 will give the maximum pressure or maximum tension on the base AB up to the point at which the masonry fails either by tension or compression. It is customary to limit the maximum pressure in dams to twice the mean, which is equivalent to specifying that no part of the masonry shall be in tension or that the center of pressure shall not deviate more than it 1 from the center of any real or imaginary joint.
If the masonry be considered as incapable of resisting by tension, or if it is considered unwise to depend upon tension to help resist overturning, then when d in equation 22 exceeds } 1, the total pressure will be borne on AN', Fig. 101. If AK" represents the maximum pressure, P, then the area of the triangle AN'K" will represent the total normal pressure IV + V. The center of gravity of the triangle AN'K" must be under r', the center of pressure; and hence AN' = 3 Ar'. Then the area of AN'K" = AK" x AN' =iPX3Ar'=IP(H1—d) =W+V; or To illustrate the difference between equations 22 and 23, assume that the distance from the resultant to the center of the base is one quarter of the length of the base, i.e., assume that d = } 1. Then, by equation 22, the maximum pressure at A is Notice that equation 25 gives a larger value of P than equa tion 24.
Notice that equation 25 is not applicable when d is less than in that case, equation 24 must be used.
The discussion in the preceding section is, practically, not; applicable to masonry clams, since it is not wise to subject the masonry on the water side to tension; but the results are directly applicable in determining the maximum pressure on the joints of a voussoir arch (Chap. XXII).