2.Theoretical Background
1.Introduction:
The Seismic analysis of Water Tanks is not automatically taken care of using the existing facilities of STAAD.Pro, unless the tank is empty or the water is an a solidified condition. When the tank is empty or the water is in a frozen condition - unable to apply the hydrodynamic loads during an earthquake acceleration – the dynamic facilities in STAAD.Pro can directly be applied.
When the tank contains water, and is accelerated during an earthquake, hydrodynamic conditions develop which STAAD cannot handle directly. However, we can have the program analyse the tank under dynamic conditions by adopting some special modeling techniques, which is intended to be discussed in this article.
2.Theoretical Background:
When a Water Tank is accelerated under seismic conditions a part of the liquid mass moves in unison with the tank. This part of the liquid mass is known as the impulsive mass. The dynamic modes set up by this mass are called the impulsive modes. Another part of the liquid mass undergoes vibration due to it’s inertia properties and cause sloshing. This part of the liquid mass is called the convective mass and the dynamic modes set up by this mass are called the convective modes.
G. W.Housner, in his paper “Dynamic Behaviour of Water Tanks”, has proposed a Spring-Mass Model for tanks resting on the ground and elevated tanks.The design codes around the world adopts the Housner model for analyzing liquid retaining structures under seismic conditions.
2.1.Tanks on Ground
In the aforementioned paper, G. W.Housner has proposed the following spring-mass model to represent tank-water system under seismic conditions.
Figure 1
The mass M0 represents the impulsive mass that moves rigidly with the tank. Hence, M0 is shown to be rigidly connected to than tank wall. The mass M1 represents the convective mass that is responsible for the sloshing action. This is represented as the mass being connected to the tank using flexible spring to enable a to and fro movement under dynamic condition. The heights h0 and h1 represents the centre of the resultant force being imparted by the impulsive and the convective masses respectively. The parameters M0, M1, K1, h0 and h1 can be determined from the equations that are given in the aforementioned paper.
As per the paper, “Simple Procedure for Seismic Analysis of Liquid Storage Tank” by Malhotra, Wenk and Wieland, which is for fully anchored, rigidly supported tanks, the impulsive and convective modes may be considered separately and then the results be combined.
2.2 Elevated Water Tanks
The following spring-mass model was proposed by G. W. Housner in his aforementioned paper for elevated water tanks.
Figure 2
Here M0’ is the equivalent mass of structure plus M0 (impulsive mass). As per the paper “Modified proposed provisions for aseismic design of liquid storage tanks : Part II –commentary and examples” by O. R. Jaiswal and Sudhir K. Jain, the structural mass contains of mass of container and one-third mass of staging. The same paper suggests that this be treated as two-degree of freedom system. However, this can also be treated as uncoupled single degree of freedom system if ratio of the period of the two uncoupled system exceeds the value of 2.5.
3. STAAD Modelling of Water Tanks for Seismic Analysis
3.1 Tanks on Ground
Though Malhotra, Wenk and Wieland has proposed that the impulsive and convective modes may be obtained separately and later combined using the Absolute Summation Method, we think that is due to the convenience of hand calculations. However, when we are looking to model that in STAAD, we can model the impulsive and convective masses in the same model and consider the first few modes to ensure that all significant modes has been duly captured.
Let us model a steel tank of radius 10 metres and a total height of 10metres. The tank is filled with water upto a height of 8 metres. Let the thickness of the tank walls 1 cm thick.
The total volume of water = π2Rh = π2 x 10 x 8 = 789.56 m3
Mass Density of Water = 1000 kg/m3
Thus, the total mass of water in the tank = M = 1000 x 789.56 = 789560 Kg
R/h = 10/8 = 1.25
h/R = 8/10 = 0.8
M0 = M * tanh (1.7 R/h) / (1.7 R/h) = 789560 * (0.972/2.125) = 361106. 75 Kgs
M1 = M * 0.6 * tanh (1.8 h/R)/(1.8 h/R) = 789560*(0.894/1.44) = 490019.43 Kgs
K1 = 5.4*(M12/M)*(gh/R2) = 5.4 * 304117.54 * 0.7848 = 1288825.81 N/m
h0 = (3/8)*h = 3 m (based on the assumption that dynamic fluid forces are exerted on the wall only and not on the floor).
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h1 = h * [ 1 – 0.185 (M/M1)(R/h)2 – 0.56(R/h) √(MR/3M1h)2-1 ] = 8 (1-0.466) = 4.272 m
For simplified modeling, let us consider the height h1 as 4 metres. The third term within the brackets in the expression is an imaginary value and hence, neglected.
The tank is assumed to be supported on a rigid foundation. So, the STAAD model would look like the following.
Figure 3
The masses M0 and M1 need to be specified as forces. Let us denote the corresponding forces by F0 and F1.
F0 = 361106. 75*9.81 = 3542 KN
This mass will be modeled at the centre of the tank at a height of 3 metres from the bottom of the tank. This mass will be connected to the tank using highly stiff beams to ensure the impulsive action of the liquid.