Hydraulic Structures: Fourth Edition

GRAVITY DAM ANALYSIS 147

and for the minor principal stress,

 3 

z

2

y

^ max, (3.36b)

where

max

z

2

y



2

^2 

1/2

. (3.36c)

The upstream and downstream faces are each planes of zero shear, and
therefore planes of principal stress. The boundary values for  1 and 3 are
then determined as follows: for the upstream face,

1uzu(1tan^2 u) pwtan^2 u, (3.37a)

3upw; (3.37b)

for the downstream face, assuming no tailwater,

1dzd(1tan^2 d), (3.37c)

3d0. (3.37d)

3.2.5 Permissible stresses and cracking

The compressive stresses generated in a gravity dam by primary loads are
very low, seldom exceeding 2.0–3.0 MN m^2 except in the largest struc-
tures. A factor of safety, Fc, with respect to the specified minimum com-
pressive strength for the concrete, c, is nevertheless prescribed; Fc3.0 is
a common but seldom critical criterion. Some authorities (USBR, 1976)
relate the values prescribed for Fcto the load combination applied, but
qualify them in terms of absolute maxima, as shown in Table 3.8. The table
also summarizes corresponding factors of safety, Fr, specified with regard
to the compressive strength of the rock foundation, r.

Table 3.8 Permissible compressive stresses (after USBR, 1976)

Load combination Minimum factor of safety on compressive strength

Fc(concrete) Fr(rock)

Normal 3.0 (max 10 MN m^2 )a 4.0
Unusual 2.0 (max 15 MN m^2 )a 2.7
Extreme 1.0 1.3

amaxis the maximum allowable compressive stress.