Page 202 - Handbook of Materials Failure Analysis

198 CHAPTER 8 Seismic risk of RC water storage elevated tanks

Table 8.5 Evaluation of Different Masses

Mass of the water M e 104,404.08 kg
Mass of the vessel M c 44,327.83 kg
Total mass M tot 148,731.91 kg
Mass of the supporting system M t 12,870.00 kg
Inert mass M i 79,704.25 kg
M 1 124,032.08 kg
M 0 22,677.45 kg

5.3.3 Evaluation of stiffnesses
The fundamental pulsation of the water vibration in the vessel is given by:
!
r ffiffiffiffiffi r ffiffiffiffiffi
g 27
2 27 H e
0
R 8 8 R
ω ¼ th (8.5)
Stiffness K 1 of mass M 0 is given by:
K 1 ¼ M 0 ω 2 0 (8.6)
For the calculation of the spring constant K 0 , we start from the fundamental period
value of a structure that can be estimated from empirical formulas, or calculated by
analytical or numerical method. Empirical relations proposed by the Algerian seis-
mic code are only applicable for buildings. In the case of an elevated tank, it can be
considered as an inverse pendulum in a realistic way. For this reason, we used the
Rayleigh method, which considers the elevated tank as a console. It allows the cal-
culation of the fundamental period of the first mode of structure vibration, assimi-
lated to a concentrated mass resting on a support of significant mass and a
constant cross-section. The tank is modeled mechanically into a single mass sup-
ported by a tower (see Figure 8.4). The period is given by:
s ffiffiffiffiffiffiffiffiffiffiffiffiffi
P l 3
0
(8.7)
g 3EI
T ¼ 2π

33
P ¼ M total + M tower g (8.8)
0
140
M total : concentrated mass (mass of the concrete vessel and the water it contains);
M tower : mass of the supporting system; I: moment of inertia of the supporting system
cross section; E: elasticity modulus of concrete; l: height of gravity center of the
oscillating mass to the embedding.
This relation implicitly assumes that it is not coupled planar oscillations with
other modes of oscillations. That is to say, there are the oscillations in which the var-
ious masses of structure move parallel to the same plane, without exciting perpen-
dicular oscillations to this plane. This condition is satisfied by the structures having a
vertical plane of symmetry, as it is in the case of circular tanks. In addition to the