If the boundaries at the upstream and downstream faces in figure 15-b are assumed to be straight lines, the area of the section is

Likewise, the equation for the distance from the upstream face to the center of gravity is

The moment of inertia of the section about the center of gravity is

The right sides of equations 51, 52, and 53 depend on the value of RD/RU only. Curves have been drawn for these values and are given in figure 16. Using these curves, values of A, lg, and I may be evaluated by multiplying quantities from the curves by the denominators of the fractions at the left sides of the equations.

.70

60

50

40

.30

20

.10

62. Concrete and Vertical Water Loads.-A graphical method has been devised for determining weights and moments of horizontal slices of a radial-side cantilever due to concrete and vertical water load. The method consists of constructing a parallel-side cantilever equivalent to the given radial-side cantilever, from which the desired information is readily obtained by the use of a mechanical integrator. The construction of the equivalent cantilever, described in figure 17, depends on the fact

.RESERVOIR WATER
SURFACE EL. 1232

EL.1232 TOP OF DAM

EL.1100

HEL.1000

EL 900

CONSTRUCTION OF EQUIVALENT CANTILEVER

Draw axis of dam vertically and plot elevations desired for analysis.
Through lowest elevation, draw horizontal base ray perpendicular to axis.
From axis,measure horizontally the average Raxis for the base and
the elevation next above and plot the point.

Through this point and the point on the axis next above the base,
draw a ray.

On this ray, plot horizontally from the axis the average Raxis for
the elevation of that ray and the elevation next above.
Drow next ray and continue to top of dam.

To the ray through an elevation, measure horizontally the upstream
and downstream projections from the axis for that elevation, and
the center of gravity for that elevation from the upstream face.
Draw upstream and downstream faces.

On vertical lines from the rays at the faces, measure the vertical
water load ordinates as indicated and plot the points.
Draw water surfaces.

The parallel-side cantilever is one unit thick normal to paper.
Note: The rays represent the elevations to be analyzed.

UPSTREAM FACE

AXIS OF DAM

VERTICAL WATER LOAD

EL 800

DOWNSTREAM FACE

EL. 700

(19)700

EL 600

(US Proj)700

EL 520

CENTER OF
GRAVITY

(DS. Proj)700

(Raxis)700+ (Raxis)800

VERTICAL WATER LOAD.

BASE OF CANTILEVER-

FIGURE 17-PARALLEL-SIDE CANTILEVER EQUIVALENT TO

RADIAL-SIDE CANTILEVER

that, between any two consecutive horizontal sections, the volume of the cantilever, or vertical water load, and the radial position of the center of gravity of that volume are practically unchanged by the transformation.

63. Unit Triangular Loads.-Unit loads, triangular in shape, are used in analyzing uncracked cantilever elements. Basic unit loads, used to build up total radial, tangential, and twist loads, are shown in figure 18, together with illustrative calculations.

Each unit load is considered separately in the computations. Let AV represent the total portion of a triangular load above any given elevation. The value of AV is total shear at the given elevation, caused by the portion of the triangular load above that elevation, in the case of radial

and tangential loads; and total twisting couple at the elevation, caused by the portion of the triangular load above that elevation, in the case of twist loads. Let AM represent the moment of the portion of the triangular load above any given elevation, about the given elevation. Values of AM are used only in the case of radial loads. Then, in figure 18, AV is the area of the load diagram above an elevation, and AM, the product of AV and the distance of its centroid above an elevation.

For convenience, triangular loads may be designated 1, 2, 3, and so forth, at the elevation of P, peak elevation, starting at the top. P is generally 1,000 pounds per square foot for radial and tangential loads, and 1,000 foot-pounds per square foot for twist loads.

64. Unit Radial Loads.-Unit triangular radial loads are horizontal radial forces applied at the upstream or downstream faces of the cantilever. Values of AV and AM for these radial loads are obtained by multiplying values of AV and AM of the basic unit load by Ru/Raris, upstream face, or RD/Raxis, downstream face, for the elevation at which the unit load peaks. These factors adjust the basic unit loads, one unit in width, to the width of the face where they are applied as unit radial loads. Multiplications of the corrected values of AV and AM by the trial load ordinates at the respective peak elevations, determined by trial load. adjustments, give the shears and moments on the cantilever due to the loads applied.

Shears and moments at each elevation, due to unit loads above each elevation, are tabulated in columns for the different unit loads, as shown in figure 18. Values of the total shear and moment at each elevation, due to all loads above each elevation, are then obtained by adding quantities at each elevation and recording in a summation column at the right.

65. Unit Tangential Loads.-Unit tangential loads are tangential shear forces applied at the center line of the cantilever. Values of AH for a unit tangential load are obtained by multiplying values of AV of the basic unit load by r/Raris for the elevation at which the basic unit load peaks. Values of tangential thrust, H, are obtained by multiplying values of AH by the trial load ordinates at the load peak elevations. The total tangential thrust at each elevation is then obtained by summation, and recorded in a column at the right, as before.

66. Unit Twist Loads.-Unit triangular twist loads are twisting moments in horizontal planes, applied at the center line of the cantilever. Values of AM for a unit twist load are obtained by multiplying values of AV for the basic unit load by r/Raris for the elevation at which the basic unit load peaks. The twisting moment, M, is obtained by multiplying values of AM by the trial load ordinates at the load peak elevations; and the total twisting moment at each elevation obtained by summation, as before.

MOVEMENTS OF UNCRACKED CANTILEVER

67. Notation.-Additional symbols used in considering cantilever movements are:

Ar radial deflection of center line of cantilever.

As

0

=

tangential deflection of center line of cantilever.

angular movement, in horizontal plane, of horizontal section of cantilever about the center line.

Subscripts: A

Σ

Ec

G

K

and R

foundation; L = left part of dam, looking upstream; right part of dam, looking upstream.

summation.

modulus of elasticity of concrete in direct stress.

modulus of elasticity of concrete in shear stress

=

Ec 2(1+μ)°

ratio of detrusion due to actual shear distribution to detrusion due to equivalent shear distributed uniformly. K is usually assumed to be 1.25 for Ar and 1.00 for As.