MOVEMENTS OF UNIT ELEMENTS 53. Notation.-The following list gives additional notation needed in considering movements of unit horizontal and vertical elements of a dam with a loaded foundation surface as shown in figure 10. Symbols refer to the left side of the dam. Reference planes and directions are indicated in figures 11, 12, and 13. ,, and are at right angles to the directions of the foundation deformations not designated by the circles, see figure 13-b. = = = = vertical movement. angle between vertical plane and plane of foundation surface at its intersection with the unit element. perpendicular to foundation surface. parallel to, or in plane of foundation surface. Note: Positive Forces, Moments, and Movements have same direction with respect to right abutment except (*) which are opposite; for example, at right abutment, positive H acts toward the abutment. FIGURE 11-UNIT HORIZONTAL ELEMENT AT LEFT ABUTMENT (a) FORCES AND MOMENTS (b) MOVEMENTS Note: Positive Forces, Moments, and Movements have same direction with respect to right abutment except (*) which are opposite, for example, at right abutment, positive H acts toward the abutment. FIGURE 12-UNIT VERTICAL ELEMENT AT LEFT ABUTMENT Deformation equations of the preceding section contain elastic constants ER and μ, the ratio b/a, and the load intensity applied either normally or tangentially to the foundation plane. Of these, elastic constants are ordinarily determined by direct experimental methods. As formulated, conditions of the problem do not fix a definite value for the ratio b/a. An assumption of some kind is necessary. At present, the value of b/a, is determined in the following manner. Consider the surface of contact between dam and foundation developed and plotted, as in figure 10-b, with the axis as a straight line. This is replaced by a rectangle of the same area and approximately the same proportions, called the "equivalent developed area." The ratio of length to width of this rectangle is taken as the b/a ratio for the foundation in question. The value of b/a is, therefore, a constant for any one dam. In computing deformations, the width a' in b/a is made equal to T, the thickness of the dam at the element considered, making b' equal to (b/a)T, see a'b' in figure 10-b. The reasoning behind this is simple. Movements of the equivalent developed area are assumed equal to movements of the actual foundation. Then, by taking the ratio, length/width, of this equivalent area as the b/a value throughout, the average of the deformation of all elements equals the average deformation of the equivalent developed area. This follows directly from the fact that the only variables in the equations, exclusive of moment effects, are the b/a ratio and the load intensity. In this discussion, loads have been considered as acting normally or tangentially to the foundation surface. Forces and moments in the dam, however, have all been measured with reference to radial, tangential, and horizontal planes, see figure 13-a. It is necessary, then, to resolve these forces and moments into normal and tangential components acting on the foundation. For this purpose, contributions of horizontal and vertical elements are considered separately, see figures 11-a and 12-a. Forces and moments transmitted by the unit element to the foundation surface are resolved into forces and moments on the unit differential area, acting both normal to and in the abutment plane. These resolved forces and moments produce deformations and rotations of the unit differential area, both normal to and in the abutment plane, as shown in figure 13-b. Deformations and rotations are resolved into components in principal planes of (a) POSITIVE FORCES AND MOMENTS (b) FOUNDATION DEFORMATIONS (c) POSITIVE MOVEMENTS Note: Positive Forces, Moments, and Movements have same direction with respect to right abutment except (*) which are opposite, for example, at right abutment, positive H acts toward the abutment. FIGURE 13-ABUTMENT OF LEFT SIDE OF DAM movements, and the summation of the components in any plane or direction is the net movement of the element in that plane or direction. |