CHAPTER III-FOUNDATION AND ABUTMENT DEFORMATIONS INTRODUCTION 48. General Statement.-At the time the trial load method was first developed, foundations and abutments of arch dams were considered rigid. Later developments included effects of elastic movements in the foundation and abutment rock. This chapter deals with the development of curves and formulas for foundation deformation. The inclusion of yielding abutments increases deflections of the dam as determined by a trial load analysis. Stresses are usually decreased at the abutments and foundations, but may be increased in other parts of the structure. The changes required in the analysis are relatively slight. Elastic deformation of the foundation causes tilting, twisting, and displacement of arch abutments and cantilever bases. Since a cantilever normally rests on an arch abutment, the two move together as a single unit. It is not possible to compute movements of each separately, at least insofar as foundation loading and deformation are concerned. Foundation rotations and displacements must be computed separately for each trial combination of loads on the two elements. That is, any change in the loading of one element will affect foundation movements and hence the movements of both. It is necessary, therefore, to use the total abutment load from both members in computing foundation deformations. Being rigidly fastened to the foundation surface, both arch and cantilever are rotated and displaced as rigid bodies. These rigid body movements are added directly to the elastic deformations of the elements themselves, giving total rotations and displacements of the arch or cantilever at any point. Deformations of an element as used in the analysis are thus seen to be deformations of the element for a fixed base plus rigid body movements due to the foundation. 49. Assumptions. Those familiar with foundation research will note several deficiencies in the procedure given on the following pages. It is recognized that the method of foundation analysis given here is hardly more than an approximate approach to the problem. A short discussion of the major assumptions involved serves to indicate the departure from actual conditions. In addition to the usual assumptions of elasticity, the principal assumptions of the analysis are: 1. Foundation deformations are independent of the shape of foundation surface. 2. Movements of a unit foundation area are due only to loads applied directly on that area. 3. Water loads on the reservoir walls do not cause differential movements at the dam site. Assumption 1 is utilized in the analysis in the following manner. Consider the surface of contact between dam and foundation developed upon a plane, as shown in figure 10-b. Take this plane as the surface of an infinite elastic solid. Deformations of this surface, after applying forces existing at the boundary of the dam, are then computed by means of the well known Boussinesq and Cerruti formulas. Computed movements of the developed surface are, of course, referred back to the abutment slopes of the actual structure before being used in the analysis. Data on the effect of canyon shape on foundation deformations are meager, if not entirely lacking. It should be remembered that the loaded. area is of the same order of magnitude as the canyon. Consequently the influence of the curvature of the canyon on deformations can hardly be considered negligible. An approximate analysis of the effects of loads applied to the sides of a U-shaped canyon on displacements at the bottom and along the sides indicates that assumption 1 departs rather widely from actual conditions. This structural picture of the foundation is further modified by assumption 2 which makes each unit foundation area independent of the rest of the foundation. The dam is now supported by what is essentially a series of independent springs, or, more accurately, by a series of independent infinite foundations, if such can be imagined. The surface of each such foundation is the tangent plane at the point of contact. The effect of this modification on assumption 1 is, in general, to increase movements of the foundation near the ends of the dam and to reduce movements near the center. This is readily seen from a consideration of a series of uniform loads. For assumption 2, the foundation is depressed equally under all loads of the series; but for assumption 1, using the Boussinesq and Cerruti formulas, the depression under central loads is greater than under end loads, due to the additive effect of adjacent loads upon the depression under any central load. The third assumption merely disregards effects of reservoir water loads on foundations and abutments. In view of the degree of approxi mation involved in the first two assumptions, it appears hardly worth while to introduce further complications at the present time. When the direct foundation analysis is in a satisfactory state this secondary effect should be considered. Although these are the assumptions of major importance, others appear as details of computations are developed. One might be mentioned here, since it determines the magnitude of the computed movements; that is, movements of any unit abutment area are average movements of a rectangular area on an infinite foundation surface. This rectangle has a width equal to the width of the dam at the point in question, a fixed ratio of length to width, and supports the load intensity, exerted by the dam, acting on the unit abutment area. The ratio of length to width of the loaded rectangle being fixed, it is seen later that the only variables in displacement equations are elastic constants and the load intensity, see equations 14 and 15. This assumption will be treated again in the second part of the chapter. The foregoing discussion can be restated simply. The dam is assumed supported by a series of independent springs. The elastic constants of the springs are determined by reference to deformations of an infinite foundation with a plane surface. 3 As stated previously, the assumptions listed are made for the purpose of utilizing known formulas for deformations of an infinite foundation with a plane surface. Basing his work on these formulas, Dr. Fredrik Vogt obtained equations for average deformations of a loaded rectangular area of the foundation surface. These equations express deformations due to a bending moment in a plane normal to the surface, a force normal to the surface, and a tractive or shear force in the plane of the surface. To supplement these equations, a formula has been derived for average deformation of the foundation due to a twisting moment in the plane of the surface. The next section is devoted to the evaluation of the deformation of a plane rectangular foundation surface for various types of loadings, For ease of computation, each fundamental foundation equation is transformed into an expression that includes a function of two variables, Poisson's ratio and the ratio of the dimensions of the loaded surface. This function is multiplied by two or more constants to give foundation deformation. In order that functions may be rapidly determined, curves have been developed covering the range of values usually encountered in concrete dams. The application of these deformation equations to the analysis. is covered in the remainder of the chapter. Movements of the abutments 3Vogt, Dr. Fredrik, “Ueber die Berechnung der Fundamentdeformation," Det Norske Videnskaps Akademi, Oslo, 1925. of unit horizontal elements and of the foundations of unit vertical elements are considered, and equations are derived for movements of the elements. The forms of the equations are then changed, so they can be included directly in computations for deflections of arches and cantilevers. DEFORMATION OF FOUNDATION SURFACE 50. Notation. The notation used in deformation equations for a plane rectangular foundation surface is given below. Numbers given in parentheses refer to equations in the paper cited above. μ ER Ρ a b = = = average rotation in a plane normal to foundation surface due to bending moment load, (16). average deformation normal to foundation surface due to normal load, (6). average deformation in plane of foundation surface due to tractive or shear load, (23). average rotation in plane of foundation surface due to twisting moment load. average rotation in a plane normal to foundation surface due to tractive load, (26). - average deformation in plane of foundation due to bending moment load, (19-a). = = = = Poisson's ratio for foundation material. modulus of elasticity of foundation material in direct stress. normal or tractive load per unit area. short dimension of loaded area ab. long dimension of loaded area ab. bending moment, in direction of a, per unit length of b = pa. shear force, in direction of a, per unit length of b pa. - pa2/6. 51. Deformation Equations.-The original development was based on a uniform distribution of load in the direction of the long dimension, b, of a loaded rectangular area. In the direction of the short dimension, a, the distribution of loading is linear for moment loading and uniform for shear and normal loadings. Satisfying this condition, the average deformation of the area for any type of load is a function of the shape of the area, the load per unit area, and the elastic properties of the foundation material. The following equations give average deformations in terms of these variables. The shape of the area is expressed by the dimensions a and b; the load per unit area, by p; and the elastic properties, by μ, Poisson's ratio, and ER, the modulus of elasticity in direct stress. ER π 3 b 9 a2 = (1-μ2) £.3 ± √a2 + b2-a_8 (√a2+b2)3- (a3 + 3⁄41⁄2 b3) 8 (√a2+b2)3-(a3+b3) 4 b la + +-- -·logel (1) a2 b 3 a la +√a2+ b2 \ (Va2+b2)3 - (a3 + b3)] +b.loge (2) These equations may be transformed to a function of two variables, μ and the ratio b/a multiplied by two or more constants. The constants are 1/ER, P, the dimension a, and such numerical constants as are present. The transformed equations, with the ratio b/a replaced by x for convenience, are |