TABLE 1-MOVEMENTS OF HORIZONTAL ELEMENT DUE TO UNIT FORCES AND MOMENTS AT ABUTMENT OF ELEMENT.

54.

Unit Horizontal Element.-Table 1 gives components of the forces and moments of a horizontal element resolved to the unit abutment surface, the deformations and rotations of the unit abutment surface, and the resulting movements of the unit horizontal element.

Net movements of the unit horizontal element, written in descending order of magnitude, are given by the following equations:

Ax*=-HBcos3 + sin2 cos ]-My sin cos y]-w [(B-)sin cos2] & [℗sin ¥

(28)

Ay = v[r'cos ] + M2["cos2] + M, ["siny cos y]

(29)

Az = -w [~cos3 y + B'sin2 y cos ] + My [cos2 ] -H[(B-)sinycos2]

(30)

*Indicates movement has opposite sign for right side of dam, see figure 13.

When the unit horizontal element is one of a contiguous series in a dam, certain equations of compatibility or continuity that must be satisfied definitely exclude the use of some of the terms or movements. Others may be eliminated because of their negligibility in the practical analysis of stresses and deflections in a dam.

Since a vertical adjustment is not made, 0, and M, are neglected. This excludes equation 26.

0x and Az have a negligible effect on the behavior of a unit horizontal element and are neglected in the present methods of analysis. This excludes equations 25 and 30.

x

M in equations 27 and 29 and W in equation 28 have very little influence on the movement of a unit horizontal element and are neglected in the present methods of analysis.

With these eliminations, the equations are now,

unit differential area on one side of the dam be average values for the equivalent developed area of that side of the dam. If the dam site is approximately symmetrical about the maximum section, dimensions of the equivalent developed area for either or both sides of the dam are b/2

b

a

a and For this reason, ratios and are substituted for the ratio

2'

a

b/2

b/a in some cases in obtaining values from the curves in figures 5 to 9, inclusive. The ratios used are as follows:

By substituting symbols for bracketed quantities, and using the identity a"y", equations 31, 32, and 33 become

The application of these movements to the analysis of an arch is given in chapter V.

TABLE 2-MOVEMENTS OF VERTICAL ELEMENT DUE TO UNIT FORCES AND MOMENTS AT ABUTMENT OF ELEMENT.

55. Unit Vertical Element. Table 2 gives components of forces and moments of a vertical element resolved to the unit abutment surface,

deformations and rotations of the unit abutment surface, and resulting movements of the unit vertical element.

Net movements of a unit vertical element, written in descending order of magnitude, are given by the following equations.

ex = Mx[a' sin3 + 8'sin cos2]+v[a"sin2 ] + M2 [(a'-8') sin2 cos ]

(37)

(38)

→2* = M2 [S′sin3y + a′siny cos2]+v [a"sin ¥ cos¥] + Mx (a-8') sin2 y cos ¥]

(39)

Ax*= -Hsin3y+B'sin y cos2] - My[~sin2]-w[(B-)sin2 cos]

(40)

▲y = v[r'sin y] + Mx ["sin2] + M2 ["siny cosy]

(41)

(42)

▲ z = -w [B'sin3 + sin y cos2 + My [sin cos ] - H [(B'-sin2 cos]

By fulfilling the same conditions of compatibility or continuity as those for a unit horizontal element and eliminating effects which are negligible, these equations are materially reduced.

As before, M, and 0, are neglected, thus excluding equation 38. Az had a negligible effect on the behavior of a unit vertical element and is neglected in present methods of analysis. This excludes equation 42. W in equation 40, M in equations 37 and 41, and M, and V in equation 39 have very little influence on a unit vertical element and are omitted in the present methods of analysis.

With these eliminations, the equations now are

Foundation deformations of a unit differential area are obtained as indicated in the discussion of a unit horizontal element.

By substituting symbols for bracketed quantities and using the identity "y", equations 43, 44, 45, and 46 become

56. Application to Cantilevers. In an arch dam each vertical cantilever element, except the one at the maximum section, rests on the same strip of foundation surface as the end of an arch. In this case radial shear at the base of a cantilever corresponds with radial shear at the abutment of the corresponding arch; tangential thrust of the cantilever corresponds with normal thrust of the arch; and twisting moment of the cantilever corresponds with bending moment of the arch. For this reason radial shears, tangential thrusts, and twisting moments of the cantilevers are applied to the arches as concentrated radial, tangential, and twist loads, respectively. Then radial, tangential, and angular movements at foundations of cantilevers are equal to corresponding movements of arch abutments.

Concentrated loads take care of all foundation cantilever movements except rotation in a vertical plane, 0. Therefore equation 47 is the only equation used in computing movements of unit loads for cantilevers resting on arch abutments.