Az increment of height between sections at which properties and forces are known.

=

4, x', B', y', C, d', and " abutment functions as defined in chapter III.

α

=

angular movement of foundation in vertical radial plane due to one unit of bending moment, MA; per unit length normal to plane of cantilever.

γ radial movement of foundation due to one unit of radial shear force, VA; per unit length normal to plane cantilever.

tangential movement of foundation due to one unit of tangential shear force, HA; per unit length normal to plane of cantilever.

angular movement of foundation in horizontal plane due to one unit of twisting moment, MA; per unit length normal to plane of cantilever.

angular movement of foundation in radial plane due to one unit of radial shear force, VA; per unit length normal to plane of cantilever; or radial movement of foundation due to one unit of bending moment, MA; per unit length normal to plane of cantilever.

68. Foundation Movements. In applying results obtained in chapter III to movements of cantilever elements, the following formulas are used:

Raxis

r

The factor in the equations corrects the movements derived for a cantilever of unit width at the center line to the movements of a radialside cantilever one unit wide at the axis. Foundation movements for a radial-side cantilever are assumed to be those for a parallel-side cantilever of the same cross sectional area.

69. Radial Deflections. Radial cantilever deflections required for a trial load analysis are those caused by radial loads and vertical water loads. Radial deflections are produced by radial bending moments and radial shear forces. They are determined by the ordinary theory for cantilever beams with contributions from bending, shear, and foundation yielding included.

The radial deflection at any horizontal section due to bending moment is the integral of the slope of the center line from the base to the elevation of the given section. The slope of the center line at any horizontal section is the angular foundation rotation, normal to the foundation surface, plus the integral of the curvature of the center line from the base to the elevation of the given section. The curvature of the center line at any point is M/EcI.

The radial deflection at any horizontal section due to radial shear forces is the foundation deformation plus the integral, from the base to that section, of the detrusion or change in horizontal position of the center line. The detrusion of the center line of a differential height is (VK/AG) Az.

Actual calculations for radial deflections are performed by summating the quantities noted above, which are determined at sufficient horizontal sections to render the summation reasonably accurate. Operations followed in evaluating radial deflections may be expressed by the formula,

Land RAC =

=Σ (M2a + v2 a2+ Σ M = Az | Az + (V2+*+ Ma2++ XKAZ) (59)

in which terms designated by an asterisk apply only to the maximum cantilever or one that does not set on the end of an arch, see chapter III, section 56. In the other case, where the cantilever rests on an arch abutment, these terms are replaced by the equivalent movements of the arch abutment. In the application of the formula, the convention of signs to be used is given in figure 19, which shows directions of positive movements,

FIGURE 19 CONVENTION OF SIGNS, RADIAL-SIDE CANTILEVER

forces, moments, and loads, and directions of forces and moments due to positive loads. Figure 19 also gives the convention for tangential and twist loads and movements.

70. Tangential Deflections. Tangential cantilever deflections are calculated for tangential loads. These loads, which represent tangential shear forces, produce deflections only by shear detrusions. Movements due to bending are omitted because tangential bending moments are negligible. Tangential deflections are determined by the method used for radial deflections due to radial shear forces. Expressed as formulas, they are

in which terms designated by an asterisk apply only to the maximum cantilever or one that does not set on the abutment of an arch. In the alternate case they are replaced by the equivalent movements of the arch abutment. In applying the formulas to the evaluation of tangential movements, the value of K is taken as unity because tangential shear is assumed to be distributed uniformly at each horizontal section.

71. Angular Movements.-Angular movements or rotations of a cantilever are calculated for twist loads. These loads represent twisting

moments in horizontal planes, producing rotations about the cantilever center line. The angular movement at any horizontal section is the angular foundation rotation in a horizontal plane, plus the integral, from the base to that section, of the angular movements of the differential heights about the center line. Since the summation method is used to evaluate quantities. to be integrated, and since the angular movement of a differential height is (M/2GI)Az, formulas for the left and right sides of the dam may be written as

in which the terms designated by an asterisk apply only to cantilevers that do not set on the abutment of an arch. In the other case they are replaced by the equivalent movements of the arch abutment.

Since cantilevers are units of a continuous structure, shears set up by twisting moments are assumed to act in tangential directions and to have a linear variation from the upstream face to the downstream face of the cantilever. With these assumptions, the equations correspond to formulas for twist in a continuous slab.

72. Secondary Movements. Many secondary movements are produced by radial, tangential, and twist loads applied to the cantilevers in trial load adjustments. Most of these have have been investigated, and those having small effects are usually neglected. Only two secondary effects have been found large enough to consider in ordinary cases. They are radial deflections due to twist loads on the arch, which are discussed in chapter VII, and angular movements in horizontal planes due to tangential loads, which are discussed in the following paragraph.

Angular movements in horizontal planes exist because a tangential force, having an eccentricity from the center of gravity of each horizontal section below, causes twisting moments in those horizontal sections. In considering this effect, it is convenient to find the effect for each unit tangential load, and to establish the total effect by the use of these separate effects. It is first necessary to find the twisting moments due to each unit tangential load. These are obtained by multiplying the thrust of a unit load by the horizontal distance between the center line of the cantilever at the load peak elevation and the center of gravity of the section at which the moment is desired. After these twisting moments have been deter

mined, angular movements due to unit tangential loads can be calculated by equations 62 and 63.

73.

Tabulations.-Deflections due to the unit radial, tangential, and twist loads are tabulated for use in trial load adjustments. The tabulations are made for each cantilever and are usually arranged with all deflections for one type of load on one sheet. For example, radial deflections for all unit radial loads are tabulated on one sheet.

ANALYSIS OF CRACKED CANTILEVER

74. General Statement. In a thin arch dam, radial loads carried by a cantilever may cause tensile stress on a horizontal section at either face. Although the cantilever is capable of withstanding a small amount of tension, it is deemed advisable to assume that such cantilevers are cracked to the point of zero stress. For this case, the resultant of the forces above a horizontal section is equal to the total stress on the uncracked area. Stresses are assumed to vary as a straight line from zero at the end of the crack to a maximum at the uncracked face.

If a horizontal crack occurs at the face of a cantilever below the water surface, the pressure of the water exerts an uplift or wedging force which may equal full water pressure near the face of the dam and may act with varying intensity over a large part of the area of the crack. This uplift pressure deflects the cantilever and causes cracking to progress farther toward the opposite face. However, the cantilever is restrained at some point of its movement by the resistance of the arches. The accurate inclusion of uplift effects in a trial load analysis is not usually feasible because the amount and variation of uplift pressure intensities is not definitely known and because the amount of cracking at each horizontal section would have to be determined by trial for each load adjustment. Although uplift effects are somewhat uncertain, equations and curves in the following sections include uplift forces and moments, so that they may be included when considered necessary.

75. Notation.-Additional symbols used in considering cracked cantilevers are given below. The symbols refer to the base of the cantilever or to the horizontal section under consideration.

eccentricity of a resultant force, measured from the center of gravity of the total horizontal section, positive upstream.