The ring theories consider water load only, self-weight stresses being
determined separately and superimposed if significant to the analysis.
Uplift load is not regarded as significant except for a thick arch, and there-
fore may normally be neglected. For analytical purposes the dam is con-
sidered to be subdivided into discrete horizontal arch elements, each of
unit height, and the important element of vertical cantilever action is thus
neglected. The individual rings are then analysed on the basis of the thick
ring or thin ring theories as considered most appropriate, and the horizon-
tal tangential arch stresses determined.(a) Thick ring stress analysis
The discrete horizontal arch elements are each assumed to form part of a
complete ring subjected to uniform external radial pressure, pw, from the
water load. The compressive horizontal ring stress, h, for radius Ris then
given byh (MN m^2 ) (3.44)whereRuandRdare respectively the upstream and downstream face radii
of the arch element considered.
Ring stress hhas a maximum at the downstream face. Ring thick-
nessTr, equal to Ru Rd, is assumed uniform at any elevation. For RRd,
equation (3.44) may consequently be rewritten in terms of h max, withpw (^) wz 1 , thus
h max (3.45)
(b) Thin ring stress analysis
If mean radius Rmis very large in comparison with Trit may be assumed
that RmRuRd and, consequently, that stress h through the ring
element is uniform. Equation (3.45) then simplifies to the classical thin
ring expression:
h (^) wz 1 Ru/Tr. (3.46)
In the upper reaches of a dam equations (3.44) and (3.46) agree closely,
the difference diminishing to under 2% when Ru/Tr 25.
Both variants of ring theory are inexact and of limited validity for
two principal reasons. First, the simplifying assumption of discrete,
independent horizontal rings which are free of any mutual interaction is
clearly untenable. Secondly, the assumption of uniform radial deformation
(^2) wz 1 R^2 u
Tr(RuRd)
pw(R^2 uR^2 uR^2 d/R^2 )
R^2 u R^2 d