collapse, as occurred with the 61 m high Malpasset cupola dam (France) in
1959.
The concepts of overturning and sliding stability applicable to gravity
or buttress analysis have little relevance to the arch or cupola. An arch
represents a stable structural form and, given that the integrity of the sup-
porting abutments is assured, failure can occur only as a result of over-
stress. Arch dam design is therefore centred largely upon stress analysis
and the definition of an arch geometry which avoids local tensile stress
concentrations and/or excessive compressive stress. In achieving this
objective it is frequently necessary to adopt varying curvatures and thick-
nesses between arch crown and abutment and also from crest level to base.
The arch and cupola dam offer great economies in volume of con-
crete. In the case of a slender cupola the saving in volume may exceed
80% of that necessary for an equivalent gravity profile. Associated savings
may also be realized in foundation excavation and preparation. As in the
case of buttress dams (Section 3.3), the sophisticated form of the cupola
leads to very much increased finished unit costs. In financial terms, there-
fore, the potential overall economies may be significantly diminished. In
the case of a geologically complex and difficult site they may be com-
pletely negated by stabilization costs associated with the paramount
requirement of ensuring abutment integrity under all conditions.
The structural interaction between a loaded arch or cupola shell and
its supporting abutments is extremely complex. This section is therefore
restricted to treating the preliminary elastic analysis of single-curvature
arch shells employing classical ring theory. The advanced mathematical
modelling techniques required for rigorous arch analysis are identified, but
their treatment lies outside the scope of this text and reference should be
made to USBR (1977) and Boggs, Jansen and Tarbox (1988). The applica-
tion of physical modelling methods to arch analysis is referred to in
Section 16.2.
3.4.2 Arch geometry and profile
The horizontal component of arch thrust must be transferred into the
abutment at a safe angle, i.e. one which will not promote abutment yield-
ing or instability. At any elevation the arch thrust may be considered to
enter the abutment as shown in Fig. 3.13. Horizontal thrust is then
assumed to distribute into the rock with an included angle of 60° as indi-
cated. In distributing through the abutment the thrust must not be aligned
too closely with the valley sound rock contours or with any major disconti-
nuity which may contribute to abutment instability. In general terms this
suggests an abutment entry angle, (Fig. 3.13), of between 45° and 70°. It
is apparent that the horizontal arch radius and therefore the arch stresses
158 CONCRETE DAM ENGINEERING
