Currently water-retaining structures in South Africa are only designed for static loading and in order to determine the influence of seismic loading on these structures, the results from the static analyses were compared with the results obtained for the seismic analyses in terms of the hoop stress and bending moment in the tank wall. Two numerical methods were used for the seismic analysis of a structure, namely Eurocode
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8: Part 4 (2006) and Veletsos (1997). This section provides more information on the seismic analysis of a structure with the use of Eurocode 8: Part 4 (2006).
A series of steps is defined in Eurocode 8: Part 4 (2006) to determinate the hydrodynamic loads and resulting forces acting on a water-retaining structure due to seismic excitation. This section provides information on the individual steps with step 1 to 7 applicable to the ultimate limit state and step 8 to the serviceability limit state. The steps include:
Step 1: Tank dimensions and material properties
The tables provided in Eurocode 8: Part 4 (2006) were compiled for tanks with H/R ratio’s ranging between 0.3 and 3. Restrictions on the H/R ratio’s of tanks are therefore provided. In this study the height/radius ratio was varied between 0.3 and 1.5.
Step 2: Sloshing and wall frequency
The frequency of the sloshing motion is required to determine the pseudoacceleration experienced by this component during seismic excitation. The convective component is considered to be elastic and the elastic response spectrum in Eurocode 8: Part 1 (2004) is applicable which is dependent on the peak ground acceleration. For the serviceability limit state the reduction factor must also be considered, depending on the importance class of the structure as defined in Eurocode 8: Part 4 (2006).
The frequency of the tank-liquid system is different from that of the ground motion in the case of flexible tanks. The frequency of the tank-liquid system is required to determine the pseudoacceleration experienced by the structure either from the design response spectrum for the ultimate limit state or the elastic response spectrum for the serviceability limit state as provided in Eurocode 8: Part 1 (2004). Even though local plastic deformations may occur during the ultimate limit state, the overall response of the tank-liquid system may be considered to be elastic and therefore the response spectrum is used for the ultimate limit state in conjunction with a behaviour factor equal to 1.5. The structure remains elastic under serviceability loads due to restrictions on
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the allowable crack width and the elastic response spectrum is used with a behaviour factor equal to 1.0 and consideration of the reduction factor appropriate for the importance class assigned to the structure.
Step 3: Effects of impulsive component
The impulsive component is subjected to the peak ground acceleration and moves rigidly with the tank wall. The mass associated with this component is determined along with the height at which this mass is situated to produce the correct base shear force and overturning moment. Due to the acceleration of the impulsive mass, a horizontal shear force Qi, at the base of the tank wall is
produced along with an overturning moment Mi about the global horizontal axis, resulting from the eccentricity of the impulsive mass with respect to an axis perpendicular to the direction of excitation. Eurocode 8: Part 4 (2006) and Veletsos (1997) provides information on the calculation of two overturning moments. Mi refers to the rotation of the tank alone about an axis
perpendicular to the direction of excitation while Mi’ refers to the overturning
of the structure including the foundation, both resulting from the hydrodynamic pressure on the wall. A height hi’ is used to calculate this overturning moment
Mi’. Both of these concepts are illustrated in Figure 2.11 and Figure 2.12.
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Figure 2.12: Overturning moment associated with impulsive component
The ratio of the impulsive mass to total liquid mass increases as the H/R ratio increases, reaching nearly total liquid mass for tall tanks. This can be attributed to the restriction of the convective component to the liquid surface in tall tanks. The height hi, at which the impulsive mass is placed to produce an overturning
moment above base remains relatively constant with an increase in the H/R ratio and is a little less than midheight of the tank wall. Figure 2.13 illustrates the increase of the impulsive mass as well as impulsive height with H/R ratio with the solid line referring to hi and the dashed line to hi’. The impulsive mass
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Figure 2.13: Variance of mi and hi with H/R ratio (Eurocode 8: Part 4, 2006)
Step 4: Effects of convective component
The respective frequencies of the sloshing component are calculated with the use of the corresponding roots λ, defined as the first derivative of the Bessel function of the first kind and first order. For design purposes it is only necessary to consider the first mode of vibration and therefore λ1=1.841 (Eurocode 8: Part
4, 2006).
The convective component has a significant influence, in broad tanks, on the hydrodynamic loads down to the tank bottom, while it is mostly restricted to the liquid surface for tall tanks. The mass associated with the convective component decreases with increased H/R ratios which correspond to an increase in the impulsive mass. This is consistent with step 3. The variance of the convective mass as well as the height, at which the convective mass is placed, is presented in Figure 2.14. The solid line in Figure 2.14 is used for parameters associated with the first mode of vibration while the dashed line is associated with the second mode of vibration.
mi/m 1 0.8 0.6 0.4 0.2 0 0 1.0 2.0 3.0 γ=H/R 3.0 2.5 2.0 1.5 1.0 0.5 0 0 1.0 2.0 3.0 γ=H/R hi/H
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Figure 2.14: Variance of mc and hc with H/R ratio (Veletsos, 1997)
The convective mass is not rigidly attached to the tank wall, since this component is by no means influenced by the flexibility of the wall and the frequency differs from that of the tank-liquid system or ground motion. In a finite element model, the convective component may be attached to the wall with the use of springs with a prescribed stiffness which corresponds with the frequency of the sloshing motion. This is illustrated in Figure 2.15 and Figure 2.16.
Figure 2.15: Base shear force associated with convective component
0.8 0.6 0.4 0.2 0 0 1 2 3 mc/m γ=H/R γ=H/R hc/h 4 3 2 1 0 0 1 2 3 1A 2A 1B 2B
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Figure 2.16: Overturning moment associated with convective component
Step 5: Effects of flexibility of the wall
In the case of rigid structures, step 5 is ignored since only the impulsive and convective components exists and needs to be considered for design purposes.
For design purposes it is sufficient to only consider the fundamental or first mode of vibration in the case of flexible walls (Eurocode 8: Part 4, 2006). The tank is subjected to horizontal excitation from the earthquake but due to the flexibility of the wall, the tank wall does not experience the same acceleration as the ground but is rather subjected to its relevant pseudoacceleration, which may be highly amplified with respect to the peak ground acceleration. Due to the increased value of the pseudoacceleration with respect to the ground peak acceleration, the contribution of the “flexibility” may be the governing contribution to the hydrodynamic pressure exerted on a wall during horizontal seismic excitation.
Pressure resulting from the flexibility of the wall produces a base shear force Qf,
as well as an overturning moment Mf, above the base. The flexibility mass mf
and height of placement hf can be calculated and the layout is similar to that of
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Step 6: Effect of wall inertia
The inertia effects of the tank wall are small in comparison to the hydrodynamic forces and may be neglected in the case of steel tanks. However, for concrete tanks the loads resulting from wall inertia may be significant and needs to be considered for design purposes.
Pressure resulting from the wall inertia is added to that of the impulsive component. In a similar way the base shear force and overturning moment resulting from the inertia effect of the tank wall is added to that of the impulsive component.
Step 7: Combination of the respective components
Eurocode 8: Part 4 (2006) suggests when combining the action effects of the various components, affecting the hydrodynamic pressure, the absolute values should be added to provide an upper bound estimate. The “square root of sums” rule may be unconservative due to the wide separation in frequency between the different components.
Step 8: Height of the convective wave
It is important to provide adequate freeboard when considering the serviceability limit state, in order to avoid damage to the roof or spillage of the liquid in the absence of a roof. The freeboard height is determined by the height of sloshing wave and only first mode of vibration needs to be considered (Eurocode 8: Part 4, 2006).