Seismic Response and Design of Liquid Storage Tanks Dr. Mohanad Al affach 2014-2015

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PART. 1 : Introduction and Basic Concepts about Liquid Storage Tanks. PART. 2 : Requirements for Seismic Design of Liquid Storage Tanks. PART 3 : Simplified Methods for Seismic Design of Liquid Storage Tanks and Numerical Applications. PART. 4 : Review of Code Requirements on Seismic Analysis of Liquid Storage Tanks.

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PART. 1: Introduction and Basic Concepts about Liquid Storage Tanks 1.1. Introduction Liquid storage tanks are vital facilities in lifelines and environmental engineering systems. They also play an important role in the rescue work after disasters like earthquakes. Any damage of such tanks after a disaster may cause consequential loss to the society. Therefore, elevated tanks should remain functional in the post-earthquake period to ensure water supply is available in earthquake-affected regions. On the other hand, several elevated tanks were damaged or collapsed during past earthquakes as presented in the next section. As a result, the seismic behavior of elevated tanks should be known and understood, and they should be designed to be earthquake-resistant.

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The problem related to liquid storage tanks involves many fundamental subjects which are of concern. One of them is the interaction between the fluid and structures under seismic loading. Due to the fluid-structure-soil/foundation interactions, the seismic behavior of elevated tanks has the characteristics of complex phenomena. In this course, the following important requirements have been incorporated: a) Analysis of ground supported tanks. b) Analysis of elevated tanks, the single degree of freedom idealization of tank is done away with; instead a two-degree of freedom idealization is used for analysis. c) The effect of convective hydrodynamic pressure is included in the analysis. d) The distribution of impulsive and convective hydrodynamic pressure is represented graphically for convenience in analysis; a simplified hydrodynamic pressure distribution is also suggested for stress analysis of the tank wall.

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e) Effect of vertical ground acceleration on hydrodynamic pressure is considered. 1.2. Types of Liquid Storage Tanks 1.2.1. Elevated Liquid Storage Tanks Elevated liquid storage tanks are used to deliver water either through large distribution systems or through stand-pipes located at or near the source or at other common watering points. Elevated tanks can serve either a large community or a small group of families such tanks do not have as large a capacity as ground storage due to the need for a tower structure to support the tank. This technical note discusses the design of elevated storage tanks and offers suggestions for choosing the appropriate tank design and construction materials. .

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The choice of the best type of elevated tank to use depends on the type of system being installed, the materials available for construction and availability of skilled labor . Elevated storage tanks are useful for providing water to standpipes. One of the cheapest and easiest materials to use for small capacity tank is steel . Pressed steel plates can be bolted together on top of a tower with little or no skilled labor . If steel tanks are impossible to obtain or are very costly, a storage tank can be constructed using reinforced concrete (RC ). Large capacity elevated tanks for more populated communities are generally made of steel or reinforced concrete (RC). Fig.1. shows a typical reinforced tank while tank Fig.2. is steel tank. Steel tanks raised on steel platforms are generally paid for by governments or national authorities.

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They are very expensive and usually must be imported. The choice of use this type of tank depends on whether they are manufactured in the country or can be acquired at an appropriate price. If a tank of standard design is manufactured, the forms used in building the first tank can be reused in constructing later tanks and will result in lower construction costs. Furthermore, local employment opportunities will be created and the local economy will be helped by increased incomes. The decision to use an elevated tank is a result of the need to provide sufficient head in the attached piping system. 1.2.2. Ground Supported Tanks (Cylindrical & Rectangular Tanks) The cylindrical practical storage limit is affected by hoop stress. The larger its diameter and height, the greater the rupture strength required of the round tank's structure to safely contain its contents .

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Rectangular tanks are modest in height, ranging up to several meters. Since wall pressure relates to the height of the liquid contained, the structural members of low-profile tanks can be fabricated of lighter-gauge, less-costly steel. Fewer bolts are required, and special, huge foundations are not needed. However, rectangular tanks obviously have larger footprints than do higher cylindrical tanks of comparable volumes. If space is at a premium, a cylindrical tank may be the only practical choice. Furthermore, depending on the storage application, a cylindrical tank might be more suitable for extended long-term permanent containment. Accordingly, the cost-efficiency of the rectangular tank could recommend it for shorter term high-volume service involving millions of gallons of liquids.

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1.2.3. Some Simple Technical Facts to Consider (Steel and Concrete Tanks) Recent steel tank does not leak or develop leaks. Concrete water storage tanks are somewhat problematic, since they crack and leak. It may a surprise to know that concrete tanks are designed to allow for an acceptable amount of leakage. Concrete storage tanks for storing water and wastewater do not stand up to the test of time. Concrete dome covers worth no value in today’s advanced storage technology. If a storage tank component does not worth value in its application, you can typically assume that it may be a problem. Concrete covers are a major weak bond in liquid storage applications.

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Concrete construction in the field is subject to numerous uncontrolled factors from outside influences. From mixture specifications to joint connections, concrete construction in the field is subject to a high level of varying quality control even with severe quality control procedures in place. Wire-wound pre-stressed concrete with a steel diaphragm embedded in the tank wall is the premier concrete storage tank. A steel diaphragm is utilized for a watertight barrier and to assure a leak proof tank wall. (In other words, a very thin steel tank is utilized inside the concrete to keep it from leaking). Bolted steel tanks are easily repaired if damaged in the field. Field repairs on bolted tanks make the tank as good as new .

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Concrete tanks are never easily repaired, but rather only patched and sealed. Crack repairs on concrete tanks are considered patches only and are never ending. Concrete tank construction costs over twice as much as steel tank construction. The life cycle cost savings of bolted steel construction over concrete is significant. The obvious advantages of steel tanks are light weight, anti-corrosion , heat-resistant , anti-seepage and strong impact resistance and shock resistance, easy installation , dispense with maintenance, convenient cleaning.

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1.2.4. Spherical Liquid Storage Tanks Spherical storage tanks (Fig.5) are widely used for various types of liquids, including hazardous contents; consequently these storage tanks must be satisfactorily designed for seismic actions. A sphere is the most efficient pressure container because it offers the maximum volume for the least surface area and the required thickness of a sphere is one-half the thickness of a cylinder of the same diameter. The stresses in a sphere are equal in each of the major axes, ignoring the effects of supports. In terms of weight, the magnitude are similar. When compared with a cylindrical container, for a given volume, a sphere would weigh approximately only half as much. However, spheres are more expensive to fabricate, so they aren't used extensively until larger sizes. In the larger sizes, the higher costs of fabrication are balanced out by larger volumes. Spheres are typically utilized as "storage “ vessels rather than "process" container.

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Spheres are economical for the storage of volatile liquids and gases under pressure, the design pressure being based on some marginal allowance above the vapour pressure of the contents. A sphere is a very strong structure. The even distribution of stresses on the sphere's surfaces, both internally and externally, generally means that there are no weak points. Moreover, they have a smaller surface area per unit volume than any other shape of container. This means, that the quantity of heat transferred from warmer surroundings to the liquid in the sphere, will be less than that for cylindrical or rectangular storage containers. Thus less pressurization due to external heat.

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Spherical tank inspection can be carried out while the tank remains in-service and offers the client the following economic benefits: Minimal pre-inspection tank preparation. Tanks that are not structurally deficient remain in service. Structurally deficient tanks are ranked in terms of damage and prioritized for maintenance. Leaks are detected early, minimizing environmental damage. While very detailed and specific seismic design rules for cylindrical tanks are provided by several codes, such rules are missing for spherical tanks.

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1.3. Failures Modes of The Liquid Storage Tanks during Past Earthquakes Liquid storage tanks are very important components of lifelines and industrial facilities. The damage of such tanks may not mean loss of economic value of tank and content. Without the assured water supply, some consequential damage such as uncontrolled fires may occur. In addition, as these structures are used widely for the storage of variety of liquids and liquid-material such, oil, liquefied gas, chemical fluids, and wastes from different forms, any collapse during earthquake can be a disaster to the environment. This will lead to the even worse condition in which the loss may be more than earthquake itself.

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The failure modes of liquid storage tanks in past earthquakes can be summarized into the following categories: For steel storage tanks , it may be: Elephant-foot buckling of bottom shell in which large axial compressive stresses and lead to beam-like bending of tank wall “elephant-foot” buckling of the wall. Fig.6. shows the elephant foot deformation of one steel tank after San Fernando, California Earthquake in 1971. Diamond-shaped buckling of tanks with very thin shells . Fig.7. shows such deformation in winery after Livermore, California Earthquake in 1980, (all these filled stainless steel tanks buckled).

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3. Damage to the roof caused by sloshing of the liquid or failure of frangible joints between wall and roof. Fig. 8. Shows such collapse of roof occurred regularly in San Fernando, California Earthquake, 1971. 4. Fracture of wall-base connection in tanks partially restrained or tanks unrestrained against up-lift. Base shear can overcome friction causing the tank to slide. Base uplifting in unanchored or partially anchored tanks can damage the piping connections that are unable of accommodating vertical displacements. Fig. 9. Shows a fracture failure at the bottom of large tank in Kobe, Japan Earthquake in 1995. 5. The rocking of tanks resulting from the sloshing of liquid inside may pull out or cause tension failure of the anchor bolts. Fig. 10. Shows an anchor bolts pulled out about 33 cm in Joseph Jensen Filtration Plant wash-water tank in San Fernando, California Earthquake in 1971. 6. Failure of tanks support system for elevated tanks . Fig.11. shows the Collapse of steel tank, Imperial Valley County, California during the 1979 , Imperial Valley Earthquake.

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( B) For concrete liquid storage tanks , they have suffered severe damages to the roofs, columns and wall systems due to extreme initial forces in past earthquake. In San Fernando, California Earthquake 1971, a finished underground water reservoir of Joseph Jensen Filtration Plant was subjected to a maximum horizontal inertial force estimated about 0.4g. The concrete support system is commonly used in elevated liquid storage tanks. The failure of such system ; for example the failure of beam and column plastic hinges in the circular frames , can also frequently be found after strong earthquakes. Fig.12. shows the damaged 700 m3 capacity elevated RC water tank during Chile Earthquake in 1960.

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Also, Fig.13. shows the damaged tank during the recent earthquake of Izmit , Turkey in 1999. Moreover, there are many other types of damages to both steel and concrete liquid storage tank such as foundation failure due to liquefaction of soil beneath the tank and failure of connection between the tank and piping or other accessory systems. Based on observation from previous earthquakes, it is concluded that liquid storage tanks can be subjected to large hydrodynamic pressure during earthquakes. As a result, high stress can cause buckling failure in the steel tanks. In concrete tanks, due to large inertial mass of concrete, the stresses could be large and result in cracking, leakage or even collapsing of the structure.

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PART. 2 : Requirements for Seismic Design of Liquid Storage Tanks. 2.1 – General Dynamic analysis of liquid containing tank is a complex problem involving fluid-structure interaction. Based on numerous analytical, numerical, and experimental studies, simple spring mass models of tank liquid system have been developed to evaluate hydrodynamic forces 2.2 – Spring Mass Model For Seismic Analysis When a tank containing liquid with a free surface is subjected to horizontal earthquake ground motion , tank wall and liquid are subjected to horizontal acceleration. The liquid in the lower region of tank behaves like a mass that is rigidly connected to tank wall.

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This mass is termed as impulsive liquid mass which accelerates along with the wall and induces impulsive hydrodynamic pressure on tank wall and similarly on base. Liquid mass in the upper region of tank undergoes sloshing motion. This mass is termed as convective liquid mass and it exerts convective hydrodynamic pressure on tank wall and base. Thus, total liquid mass gets divided into two parts, i.e., impulsive mass and convective mass. In spring mass model of tank-liquid system, these two liquid masses are to be suitably represented . A qualitative description of impulsive and convective hydrodynamic pressure distribution on tank wall and base is given in Fig.14. Sometimes, vertical columns and shaft are present inside the tank. These elements cause obstruction to sloshing motion of liquid. In the presence of such obstructions, impulsive and convective pressure distributions are likely to change. At present, no study is available to quantify effect of such obstructions on impulsive and convective pressures .

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These elements cause obstruction to sloshing motion of liquid. In the presence of such obstructions, impulsive and convective pressure distributions are likely to change. At present, no study is available to quantify effect of such obstructions on impulsive and convective pressures. However, it is reasonable to expect that due to presence of such obstructions , impulsive pressure will increase and connective pressure will decrease . Briefly, we may say that When a tank containing liquid vibrates, the liquid exerts impulsive and convective hydrodynamic pressure on the tank wall and the tank base in addition to the hydro-static pressure.

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However, it is reasonable to expect that due to presence of such obstructions , impulsive pressure will increase and connective pressure will decrease . Briefly, we may say that When a tank containing liquid vibrates, the liquid exerts impulsive and convective hydrodynamic pressure on the tank wall and the tank base in addition to the hydro-static pressure. In order to include the effect of hydrodynamic pressure in the analysis, tank can be idealized by an equivalent spring mass model, which includes the effect of tank wall – liquid interaction. The parameters of this model depend on geometry of the tank and its flexibility .

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2.2.1- Ground Supported Tank The spring mass model for ground supported tank is based on work of Housner (1963). In the spring mass model of tank, hi is the height at which the resultant of impulsive hydrodynamic pressure on wall is located from the bottom of tank wall. On the other hand, hi * is the height at which the resultant of impulsive pressure on wall and base is located from the bottom of tank wall. Thus, if effect of base pressure is not considered, impulsive mass of liquid, mi will act at a height of hi and if effect of base pressure is considered, mi will act at hi * . Heights hi and hi * , are schematically described in Fig.14.a and 14.b . Heights hc and hc * are described in Fig.14.c and 14.d.

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Similarly, hc , is the height at which resultant of convective pressure on wall is located from the bottom of tank wall, while, hc * is the height at which resultant of convective pressure on wall and base is located. Heights hc and hc * are described in Fig.14.c and 14.d. Briefly we can say, Ground supported tanks can be idealized as spring-mass model shown in Fig 15. The impulsive mass of liquid, mi is rigidly attached to tank wall at height hi (or hi * ). Similarly , convective mass , mc is attached to the tank wall at height hc (or hc *) by a spring of stiffness Kc .

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Fig.15

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Circular and Rectangular Tank The parameters of spring mass model depend on tank geometry and were originally derived by Housner ( 1963a). The parameters shown in Fig.15 and 16 are slightly different from those given by Housner (1963a), and have been taken from ACI 350.3 (2001). Expressions for these parameters are given in Table .1. It may be mentioned that these parameters are for tanks with rigid walls. In the literature , spring mass models for tanks with flexible walls are also available (Haroun and Housner (1981) and Veletsos (1984)). Generally, concrete tanks are considered as tanks with rigid wall; while steel tanks are considered as tanks with flexible wall.

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Spring mass models for tanks with flexible walls are more cumbersome to use. Moreover, difference in the parameters ( mi , mc , hi ,) obtained from rigid and flexible tank models is not substantial ( Jaiswal et al. (2004 )). Hence in the present code, parameters corresponding to tanks with rigid wall are recommended for all types of tanks. Further, flexibility of soil or elastic pads between wall and base do not have appreciable influence on these parameters. It may also be noted that for certain values of h/D ratio, sum of impulsive mass ( mi ) and convective mass ( mc ) will not be equal to total mass ( m ) of liquid; however, the difference is usually small (2 to 3%). This difference is attributed to assumptions and approximations made in the derivation of these quantities.

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One should also note that for shallow tanks, values of hi * and hc * can be greater than h (Refer Figures 15a and 15b ) due to predominant contribution of hydrodynamic pressure on base . If vertical columns and shaft are present inside the tank, then impulsive and convective masses will change . Though, no study is available to quantify effect of such obstructions, it is reasonable to expect that with the presence of such obstructions, impulsive mass will increase and convective mass will decrease . In absence of more detailed analysis of such tanks, as an approximation, an equivalent cylindrical tank of same height and actual water mass may be considered to obtain impulsive and convective masses .

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In a few words, For circular tanks, parameters mi , mc , hi ,* hi , hc *, hc and Kc shall be obtained from Fig.15 . and for rectangular tanks these parameters shall be obtained from Fig.16. hi and hc account for hydrodynamic pressure on the tank wall only. hi* and hc * account for hydrodynamic pressure on tank wall and the tank base. Hence, the value of hi and hc shall be used to calculate moment due to hydro-dynamic pressure at the bottom of the tank wall. The value of hi* and hc * shall be used to calculate overturning moment at the base of tank.

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Fig.16. Parameters of the spring mass model for circular tank

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Fig.17. Parameters of the spring mass model for rectangular tank

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2.2.2- Elevated Tank Most elevated tanks are never completely filled with liquid. Hence a two-mass idealization of the tank is more appropriate as compared to a one mass idealization, which was used in IS 1893:1984. Two mass model for elevated tank was proposed by Housner (1963b) and is being commonly used in most of the international codes. Structural mass m s , includes mass of container and one-third mass of staging. Mass of container comprises of mass of roof slab, container wall, gallery, floor slab, and floor beams . Staging acts like a lateral spring and one-third mass of staging is considered based on classical result on effect of spring mass on natural frequency of single degree of freedom system ( Tse et al., 1983 ).

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The response of the two-degree of freedom system can be obtained by elementary structural dynamics. However, for most elevated tanks it is observed that the two periods are well separated . Hence , the system may be considered as two uncoupled single degree of freedom systems. This method will be satisfactory for design purpose, if the ratio of the period of the two uncoupled systems exceeds 2.5 ( Priestley et al. (1986)). If impulsive and convective time periods are not well separated, then coupled 2-DOF system will have to be solved using elementary structural dynamics. In this context it shall be noted that due to different damping of impulsive and convective components, this 2-DOF system may have non- proportional damping .

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Practically, For elevated tanks with circular container, parameters mi , mc , hi *, hi , hc *, hc and Kc shall be obtained from Fig.15. For elevated tanks with rectangular container, these parameters shall be obtained from Fig.16. In Fig.17, ms is the structural mass and shall comprise of mass of tank container and one-third mass of staging. For elevated tanks, the two degree of freedom system of Fig.17c. can be treated as two uncoupled single degree of freedom systems (Figure 17d), one representing the impulsive plus structural mass behaving as an inverted pendulum with lateral stiffness equal to that of the staging, Ks and the other representing the convective mass with a spring of stiffness, Kc .

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Parameters of spring mass models (i.e ., mi , mc, hi *, hi , hc *, hc and Kc ) are available for circular and rectangular tanks only. For tanks of other shapes, equivalent circular tank is to be considered. Joshi ( 2000) has shown that such an approach gives satisfactory results for intze tanks. Similarly, for tanks of truncated conical shape, Eurocode 8 (1998) has suggested equivalent circular tank approach .

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Fig. 18. Two mass idealization for elevated tank

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2.3- Time Period 2.3.1 – Impulsive Mode Ground Supported Circular Tank For a ground supported circular tank, wherein wall is rigidly connected with the base slab, time period of impulsive mode of vibration Ti , in seconds, is given by : where: Ci = Coefficient of time period for impulsive mode. Value of Ci can be obtained from Fig.19, h = Maximum depth of liquid, D = Inner diameter of circular tank, t = Thickness of tank wall , E = Modulus of elasticity of tank wall, and ρ = Mass density of liquid ..

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NOTE : In some circular tanks, wall may have flexible connection with the base slab . For tanks with flexible connections with base slab, time period evaluation may properly account for the flexibility of wall to base connection. Fig.19. Coefficient of impulsive ( C i ) and convective ( Cc ) mode time period for circular tank

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Ground Supported Rectangular Tank : For a ground supported rectangular tank, wherein wall is rigidly connected with the base slab, time period of impulsive mode of vibration, Ti in seconds, is given by : : Where d = deflection of the tank wall on the vertical center-line at a height of h ’ , when loaded by uniformly distributed pressure of intensity q , m w = Mass of one tank wall perpendicular to the direction of seismic force, and B = Inside width of tank . In case of tanks with variable wall thickness (particularly , steel tanks with step variation of thickness), thickness of tank wall at 1/3 rd height from the base should be used in the expression for impulsive time period .

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Also, h’ is the height of combined center of gravity of half impulsive mass of liquid ( mi /2), and mass of one wall ( m’w ). For tanks without roof, deflection, d can be obtained by assuming wall to be free at top and fixed at three edges (Fig.20a). ACI 350.3 (2001) and NZS 3106 (1986) have suggested a simpler approach for obtaining deflection, d for tanks without roof. As per this approach, assuming that wall takes pressure q by cantilever action, one can find the deflection, d, by considering wall strip of unit width and height h’ , which is subjected to concentrated load, P = q h (Fig.20b and C-2c). Thus, for a tank with wall f uniform thickness, one can obtain d as follows: The above approach will give quite accurate results for tanks with long walls (say, length greater than twice the height). For tanks with roofs and/or tanks in which walls are not very long, the deflection of wall shall be obtained using appropriate method .

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Fig.20. Description of deflection d , of rectangular tank wall

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Elevated Tank Time period of impulsive mode, Ti in seconds, is given by : where ms = mass of container and one-third mass of staging, and Ks = lateral stiffness of staging . Lateral stiffness of the staging is the horizontal force required to be applied at the center of gravity of the tank to cause a corresponding unit horizontal displacement. NOTE: The flexibility of bracing beam shall be considered in calculating the lateral stiffness, Ks of elevated moment resisting frame type tank staging. Also, Time period of elevated tank can also be expressed as:

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where, Δ is deflection of center of gravity of tank when a lateral force of magnitude ( ms +mi)g is applied at the center of gravity of tank. Center of gravity of tank can be approximated as combined center of mass of empty container and impulsive mass of liquid. The impulsive mass mi acts at a height of hi from top of floor slab . For elevated tanks with moment resisting type frame staging, the lateral stiffness can be evaluated by computer analysis or by simple procedures (Sameer and Jain, 1992), or by established structural analysis method. In the analysis of staging, due consideration shall be given to modeling of such parts as spiral staircase , which may cause eccentricity in otherwise symmetrical staging configuration. For elevated tanks with shaft type staging, in addition to the effect of flexural deformation, the effect of shear deformation may be included while calculating the lateral stiffness of staging.

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2.3.2 – Convective Mode Time period of convective mode, in seconds, is given by : The values of mc and Kc can be obtained from Fig.16, for circular and rectangular tanks.Since the expressions for mc and Kc are known, the expression for Tc can be alternatively expressed as: Circular Tank: Time period of convective mode, Tc in seconds, is given by: Where : Cc = Coefficient of time period for convective mode. Value of Cc can be obtained from Fig.19, and D = Inner diameter of tank . Rectangular Tank: Time period of convective mode of vibration, Tc in seconds, is given by:

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Where Cc = Coefficient of time period for convective mode. Value of Cc can be obtained from Fig.21, and L = Inside length of tank parallel to the For rectangular tank, L is the inside length of tank parallel to the direction of loading, as described in direction of seismic force. For tanks resting on soft soil, effect of flexibility of soil may be considered while evaluating the time period . Generally , soil flexibility does not affect the convective mode time period. However, soil-flexibility may affect impulsive mode time period .

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Fig.21. Coefficient of convective mode time period ( C c ) for rectangular tank Fig.22. Description of length, L and breadth, B of rectangular tank

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2.4- Damping Damping in the convective mode for all types of liquids and for all types of tanks shall be taken as 0.5 % of the critical. Damping in the impulsive mode shall be taken as 2% of the critical for steel tanks and 5 % of the critical for concrete or masonry tanks (used in most of the international codes ) . 2.5- Design Horizontal Seismic Coefficient Design horizontal seismic coefficient, A h shall be obtained by the following expression, subject to previous sections herein . Where: Z = Zone factor given in Table 2 of IS 1893(Part 1): 2002, I = Importance factor given in Table 2 herein. R = Response reduction factor here, and Sa/g = Average response acceleration coefficient as here .

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Importance factor (I), is meant to ensure a better seismic performance of important and critical tanks. Its value depends on functional need, consequences of failure, and post earthquake utility of the tank. The requirements presented here, liquid containing tanks are put in three categories and importance factor to each category is assigned (Table 2). Highest value of I =1.75 is assigned to tanks used for storing hazardous materials. Since release of these materials can be harmful to human life, the highest value of I is assigned to these tanks. For tanks used in water distribution systems, value of I is kept as 1.5, which is same as value of I assigned to hospital, telephone exchange, and fire station buildings in IS 1893 ( Part 1 ):2002. Less important tanks are assigned I = 1.0.

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Table.2. Importance factor, I

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Response reduction factor (R), represents ratio of maximum seismic force on a structure during specified ground motion if it were to remain elastic to the design seismic force. Thus , actual seismic forces are reduced by a factor R to obtain design forces. This reduction depends on over strength, redundancy, and ductility of structure. Generally, liquid containing tanks posses low over strength, redundancy, and ductility as compared to buildings. In buildings, non-structural components substantially contribute to over-strength; in tanks, such non structural components are not present. Buildings with frame type structures have high redundancy; ground supported tanks and elevated tanks with shaft type staging have comparatively low redundancy . Moreover , due to presence of non structural elements like masonry walls, energy absorbing capacity of buildings is much higher than that of tanks. Based on these considerations, value of R for tanks needs to be lower than that for buildings.

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All the international codes specify much lower values of R for tanks than those for buildings. It is seen that for a building with special moment resisting frame value of R is 8.0 whereas, for an elevated tank on frame type staging (i.e., braced legs ), value of R is 3.0. Further , it may also be noted that value of R for tanks varies from 3.0 to 1.5 . Values of R given here are based on studies of Jaiswal et al. (2004a, 2004b). In this study, an exhaustive review of response reduction factors used in various international codes is presented. 2, the highest value of R is 2.5 and lowest value is 1.3 .

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Table 3. Response reduction factor, R

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On the other hand, If time period is less than 0.1 second, the value of Sa /g shall be taken as 2.5 for 5 % damping and be multiplied with appropriate factor, for other damping . Response acceleration coefficient (Sa /g) as :

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2.6 – Base Shear 2.6.1 - Ground Supported Tank Base shear in impulsive mode, at the bottom of tank wall is given by: V i = ( A h ) i ( m i + m w + m t ) g and base shear in convective mode is given by: V c =( A h ) c m c g Where: ( Ah)i = Design horizontal seismic coefficient for impulsive mode , ( Ah)c = Design horizontal seismic coefficient for convective mode , m i = Impulsive mass of water, m w = Mass of tank wall, m t = Mass of roof slab, and g = Acceleration due to gravity. 2.6.2 – Elevated Tank Base shear in impulsive mode, just above the base of staging (i.e. at the top of footing of staging) is given by : V i = ( A h ) i (m i + m s ) g

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and base shear in convective mode is given by: V c =( A h ) c m c g Where ms = Mass of container and one-third mass of staging. Total base shear V , can be obtained by combining the base shear in impulsive and convective mode through Square root of Sum of Squares (SRSS) rule and is given as follows : Live load on roof slab of tank is generally neglected for seismic load computations. However, in some ground supported tanks, roof slab may be used as storage space. In such cases, suitable percentage of live load should be added in the mass of roof slab, m t . For concrete/masonry tanks, mass of wall and base slab may be evaluated using wet density of concrete/masonry.

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For ground supported tanks, to obtain base shear at the bottom of base slab/plate, shear due to mass of base slab/plate shall be included. If the base shear at the bottom of tank wall is V then, base shear at the bottom of base slab, V', will be given by : V '= V + (A h ) i m b where , m b is mass of base slab/plate 2.7 – Base Moment 2.7.1 – Ground Supported Tank Bending moment in impulsive mode, at the bottom of wall is given by: Mi = ( Ah ) i ( mi hi+ mw hw + mt ht ) g and bending moment in convective mode is given by: Mc = ( Ah ) c mc hc g Where: hw = Height of center of gravity of wall mass, and ht = Height of center of gravity of roof mass . Overturning moment in impulsive mode to be used for checking the tank stability at the bottom of base slab/plate is given by :

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and overturning moment in convective mode is given by : Where : m b = mass of base slab/plate, and t b = thickness of base slab/plate 2.7.2- Elevated Tank Overturning moment in impulsive mode, at the base of the staging is given by : and overturning moment in convective mode is given by :

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where h s = Structural height of staging, measured from top of footing of staging to the bottom of tank wall, and h cg = Height of center of gravity of empty container, measured from top of footing. Total moment shall be obtained by combining the moment in impulsive and convective modes through Square of Sum of Squares (SRSS) and is given as follows : The design of elevated tanks, shall be worked out for tank empty and tank full conditions ..

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For tank empty condition, convective mode of vibration will not be generated. Thus, empty elevated tank has to be analyzed as a single degree of freedom system wherein, mass of empty container and one-third mass of staging must be considered . As such, ground supported tanks shall also be analyzed for tank empty condition. However , being very rigid, it is unlikely that tank empty condition will become critical for ground supported tanks Structural mass m s , which includes mass of empty container and one-third mass of staging is considered to be acting at the center of gravity of empty container. Base of staging may be considered at the top of footing . 2.8 – Direction of Seismic Force Ground supported rectangular tanks shall be analyzed for horizontal earthquake force acting non-concurrently along each of the horizontal axes of the tank for evaluating forces on tank walls .

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Base shear and stresses in a particular wall shall be based on the analysis for earthquake loading in the direction perpendicular to that wall. For elevated tanks, staging components should be designed for the critical direction of seismic force. Different components of staging may have different critical directions . 2.9 – Hydrodynamic Pressure When a tank containing liquid is subjected to lateral earthquake ground motion, the liquid exerts impulsive and convective hydrodynamic pressure on the tank wall in addition to the hydrostatic pressure . The impulsive component is the liquid mass in the lower region of the tank that moves in harmony with the tank structure at a relatively short period (under 1 second). The convective component in the upper region undergoes a sloshing motion with a period that can exceed 10 seconds. The effect of hydrodynamic pressure on the structures has been studied for long time by a number of researchers .

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The research work in this area has mostly focused on studying the cylindrical steel tanks which were more commonly used in civil engineering especially in the oil and water supply industry. Also, slight concentration was drawn to the seismic response of concrete tanks, particularly to rectangular tanks . 2.9.1 – Impulsive Hydrodynamic Pressure The impulsive hydrodynamic pressure exerted by the liquid on the tank wall and base is given by: For Circular Tank Lateral hydrodynamic impulsive pressure on the wall, p iw , is given by :

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where ρ = Mass density of liquid, φ = Circumferential angle, and y = Vertical distance of a point on tank wall from the bottom of tank wall. Coefficient of impulsive hydrodynamic pressure on wall, Q iw ( y ) can also be obtained from Fig.23a . Impulsive hydrodynamic pressure in vertical direction, on base slab (y = 0) on a strip of length l', is given by : Where x = Horizontal distance of a point on base of tank in the direction of seismic force, from the center of tank. For Rectangular Tank Lateral hydrodynamic impulsive pressure on wall piw , is given by:

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where, Qiw ( y ) is same as that for a circular :, with h / L being used in place of h / D . Impulsive hydrodynamic pressure in vertical direction, on the base slab ( y = 0 ), is given by : 2.9.2 – Convective Hydrodynamic Pressure The convective pressure exerted by the oscillating liquid on the tank wall and base shall be calculated as follows: Circular Tank Lateral convective pressure on the wall p cw is given by:

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Convective pressure in vertical direction, on the base slab ( y = 0) is given by:

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The value of Q cb (x) may also be read from designated Figures . Rectangular Tank The hydrodynamic pressure on the wall p cw , is given by: The pressure on the base slab ( y = 0 ) is given by :

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In circular tanks, hydrodynamic pressure due to horizontal excitation varies around the circumference of the tank . However , for convenience in stress analysis of the tank wall, the hydrodynamic pressure on the tank wall may be approximated by an outward pressure distribution of intensity equal to that of the maximum hydrodynamic pressure.

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Fig.24. Geometry of (a) Circular tank and (b) Rectangular tank

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Fig.25. Hydrodynamic pressure distribution for wall analysis

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2.10- Sloshing Wave Height The problem of water sloshing in closed containers has been the subject of many studies over the past few decades . This phenomenon can be described as a free surface movement of the contained fluid due to sudden loads. Olsen classified the free surface fluid motion in three different slosh modes consisting of: i ) lateral sloshing, ii) vertical sloshing, and iii) rotational sloshing (Swirling). Sloshing is a phenomenon that can be found in a wide variety of industrial applications such as Liquefied Natural Gas (LNG ) carriers and their new design, rockets and airplanes fuel reservoirs and road tankers . Sloshing can be the result of external forces due to acceleration/deceleration of the containment body. Of particular concern is the pressure distribution on the wall of the container reservoir and its local sequential peaks that can reach as in road tankers twice the rigid load value. problems .

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In road tankers, the free liquid surface may experience large expeditions for even very small motions of the container leading to stability problems. Liquid sloshing is associated with various engineering problems, such as the behavior of liquid in containers , the liquid oscillations in large storage tanks caused by earthquakes. Large amplitude sloshing flows considered in engineering applications are usually followed by impact of liquid at the side wall and top surface of fluid containers. However , some Japanese researchers continued the studies on the sloshing phenomenon and several interesting results have been reported. The traditional approaches that have been used to assess sloshing loads include linear and nonlinear potential flow theory, direct experimentation on scaled models and more recently the use of Computational Fluid Dynamics (CFD) investigated by Godderidge et al. The results showed that the sloshing natural frequency and the inertia of the system are affected by the fluid level.

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Potential flow theory has some limitations and cannot model fluid fragmentation or merging . Maximum sloshing wave height is given by : 2.11 – Effect of Vertical Ground Acceleration Due to vertical ground acceleration, effective weight of liquid increases, this induces additional pressure on tank wall, whose distribution is similar to that of hydrostatic pressure.

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Vertical ground acceleration induces hydrodynamic pressure on wall in addition to that due to horizontal ground acceleration . In circular tanks, this pressure is uniformly distributed in the circumferential direction. Hydrodynamic pressure on tank wall due to vertical ground acceleration may be taken as:

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Distribution of hydrodynamic pressure due to vertical ground acceleration is similar to that of hydrostatic pressure . This expression is based on rigid wall assumption. Effect of wall flexibility on hydrodynamic pressure distribution is described in Eurocode 8 (1998). Design vertical acceleration spectrum is taken as two-third of design horizontal acceleration spectrum , as per clause 6.4.5 of IS 1893 (Part1). To avoid complexities associated with the evaluation of time period of vertical mode, time period of vertical mode is assumed as 0.3 seconds for all types of tanks. However, for ground supported circular tanks, expression for time period of vertical mode of vibration (i.e ., breathing mode) can be obtained using expressions given in ACI 350.3 (2001) and Eurocode 8 (1998). While considering the vertical acceleration, effect of increase in weight density of tank and its content may also be considered.

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In absence of more refined analysis, time period of vertical mode of vibration for all types of tank may be taken as 0.3 sec . The maximum value of hydrodynamic pressure should be obtained by combining pressure due to horizontal and vertical excitation through square root of sum of squares (SRSS) rule, which can be given as :

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Fig. 26 – Impulsive pressure coefficient (a) on wall, Q iw (b) on base, Q ib

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Fig.27- Convective pressure coefficient for circular tank ( a) on wall, Q cw (b) on base, Q cb

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2.12 – Anchorage Requirement Circular ground supported tanks shall be anchored to their foundation ( Fig.28) When : (h/D )> 1/(A h ) i In case of rectangular tank, the same expression may be used with L instead of D . Fig.28 – Initiation of rocking of tank

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Example #1 Elevated Tank Supported on 4 Column RC Staging

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Problem Statement : Analyze the following tank for seismic loads: A RC circular water container of 50 m3 capacity has internal diameter of 4.65 m and height of 3.3 m ( including freeboard of 0.3 m). It is supported on RC staging consisting of 4 columns of 450 mm dia with horizontal bracings of 300 x 450 mm at four levels. The lowest supply level is 12 m above ground level . Staging adjusts to ductile detailing as per IS13920. Staging columns have isolated rectangular footings at a depth of 2m from ground level. Tank is located on soft soil in seismic zone II. Grade of staging concrete and steel are M20 and Fe415, respectively. Density of concrete is 25 kN /m3 .

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Fig. 1.1. Details of tank geometry

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Solution : Tank must be analysed for tank full and empty conditions . 1.1 . Initial Data Details of sizes of various components and geometry are shown in Table 1.1 and Figure 1.1. Table 1.1 Sizes of various components

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1 .2 . Weight Calculations Table 1.2 Weight of various components

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Note : i) Weights of floor finish and cover should be accounted, wherever applicable . ii ) Live load on roof slab and gallery is not considered for seismic load computations . iii ) Water load is considered as dead load . iv ) For seismic analysis, freeboard is not included in depth of water . From Table 1.2: Weight of staging = 186.1 + 185.2 = 371.3 kN. Weight of empty container = 60.1 + 251.4 + 100.2+ 38.1 + 52.3 = 502.1 kN. Hence, weight of container + one third weight of staging = 502.1 + 371.3 / 3 = 626 kN.

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1.3. Center of Gravity of Empty Container Components of empty container are: roof slab , wall , floor slab, gallery and floor beam. With reference to Figure 1.2, height of CG of empty container from top of floor slab will be = [(60.1 x 3.36) + (251.4 x 1.65 )– (100.2 x 0.1) – (52.3 x 0.055) – (38.1 x 0.4)] / 502.1= 1.18 m . Hence , height of CG of empty container from top of footing will be 14 + 1.18 = 15.18 m . Fig.1.2- CG of empty container

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1.4. Parameters of Spring Mass Model Weight of water = 499.8 kN = 4,99,800 N. Hence, mass of water, m = 4,99,800 / 9.81= 50,948 kg. Depth of water, h = 3.0 m. Inner diameter of the tank, D = 4.65m. Hence , for h / D = 3.0 / 4.65 = 0.65 , mi / m = 0.65; mi = 0.65 x 50,948 = 33,116 kg . mc / m = 0.35; mc = 0.35 x 50,948 = 17,832 kg hi / h = 0.375; hi = 0.375 x 3.0 = 1.13 m hi* / h = 0.64; hi* = 0.64 x 3.0 = 1.92 m h c / h = 0.65; h c = 0.65 x 3.0 = 1.95 hc * / h = 0.73; hc * = 0.73 x 3.0 = 2.19 m .

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Note that the sum of impulsive and convective masses is 50,948 kg which compares well with the total mass. However in some cases, there may be difference of 2 to 3%. Mass of empty container + one third mass of staging: ms = (502.1 + 371.3 / 3) x (1,000 / 9.81) = 63,799 kg . 1.5. Lateral Stiffness of Staging Lateral stiffness of staging is defined as the force required to be applied at the CG of tank to get a corresponding unit deflection. As per previously mentioned, CG of tank is the combined CG of empty container and impulsive mass. However , in this example, CG of tank is taken as CG of empty container .

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From the deflection of CG of tank due to an random lateral force one can get the stiffness of staging. Finite element software is used to model the staging (Refer Figure 1.3). Modulus of elasticity for M20 concrete is obtained as 5,000 ( f ck )^ 0.5 = 5,000 x (20 )^ 0.5 = 22,360 MPa or 22.36 x 106 kN /m2 . Since container portion is quite rigid, a rigid link is assumed from top of staging to the CG of tank. In FE model of staging, length of rigid link is = 1.18 + 0.3 = 1.48 m . Further , to account for the rigidity imparted due to floor slab, floor beams are modeled as T-beams. Here, stiffness of staging is to be obtained in X-direction (Refer Figure 1.1), hence, one single frame of staging can be analysed in this case .

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From the analysis, deflection of CG of tank due to a random 10 kN force is obtained as 0.00330 m . Thus, lateral stiffness of one frame of staging = 10 /0.00330 = 3,030 kN /m. Since staging consists of two such frames, total lateral stiffness of staging , K s = 2 x 3,030 = 6,060 kN /m. Above analysis can also be performed manually by using standard structural analysis methods .

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Fig. 1.3 - FE model of staging Here, analysis of staging is being performed for earthquake loading in X-direction . 1.6. Time Period Time period of impulsive mode ,

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Time period of convective mode , For h / D = 0.65, Cc = 3.28 . 1.7 . Design Horizontal Seismic Coefficient Design horizontal seismic coefficient for impulsive mode, Z = 0.1 (IS 1893(Part 1 ):; Zone II ), I = 1.5 ( from previous Table).

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Table.2- Values of response reduction factor used in IBC 2000 Since staging has special moment resisting frames (SMRF), R is taken as 2.5 (Table 2)Herein, Ti = 0.80 sec , site has soft soil, Damping = 5%, Hence , ( Sa /g ) i = 2.09, ( A h ) i = (0.1/2)x(1.5/2.5)x 2.09=0.06 Design horizontal seismic coefficient for convective mode , Where, Z = 0.1 (IS 1893(Part 1): I = 1.5, R = 2.5

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For convective mode, value of R is taken same as that for impulsive mode as per previously mentioned part. Here, T c = 2.26 sec , Site has soft soil , Damping = 0.5%, Hence , ( S a /g ) c = 1.75 x 0.74 = 1.3 ((Multiplying factor of 1.75 is used to obtain S a /g values for 0.5% damping from that for 5 % damping)). ( A h ) c = ( 0.1/2)x(1.5/2.5)x 1.3= 0.04 1.8. Base Shear Base shear at the bottom of staging, in impulsive mode , Vi = ( Ah ) i ( mi + ms ) g = 0.06 x (33,116 + 63,799) x 9.81= 59.9 kN. Similarly, base shear in convective mode,

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V c = ( A h ) c m c g = 0.04 x 17,832 x 9.81= 7.0 kN. Total base shear at the bottom of staging, V = (( V i )^2 +( V c )^2)^0.5= = 60 kN . Total lateral base shear is about 5 % of total seismic weight (1,126 kN ). 1.9 . Base Moment Overturning moment at the base of staging, in impulsive mode , M i * = ( A h ) i [ m i ( h i * + h s ) + m s h cg ] g = 0.06 x [33,116 x (1.92 + 14) +( 63,799 x 15.18)] x 9.81 = 924 kN -m. Similarly, overturning moment in convective mode , Mc * = (Ah)c mc ( hc * + hs ) g = 0.04 x 17,832 x (2.19 +14) x 9.81= 113 kN -m .

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Total overturning moment at the base of staging , M* = (( Mi *)^ 2 + ( Mc*)^2)^0.5 = 931 kN -m . 1.10. Hydrodynamic Pressure 1.10.1. Impulsive Hydrodynamic Pressure Impulsive hydrodynamic pressure on wall p iw (y) = Q iw (y) (A h ) i ρ g h cos ф Q iw (y) = 0.866 [1 -( y / h)² ] tanh (0.866 D / h) Maximum pressure will occur at ф = 0. At base of wall, y = 0; Q iw (y= 0) = 0.866[1-(0/3.0)2]x tanh (0.866 x 4.65 /3.0)=0.76

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Impulsive pressure at the base of wall, p iw (y = 0) = 0.76 x 0.06 x 1,000 x 9.81 x 3.0 x 1 = 1.41 kN /m2. Impulsive hydrodynamic pressure on the base slab ( y = 0) Pi b = 0.866(A h )i ρ h sinh (0.866 x/L) /cosh (0.866 l‘/h) = 0.866 x 0.06 x 1,000 x 9.81 x 3.0 x sinh (0.866 x 4.65 /( 2 x 3.0)) / cosh ( 0.866 x 4.65 / 2 x 3.0 ) = 0.95 kN /m2 1.10.2. Convective Hydrodynamic Pressure Convective hydrodynamic pressure on wall , p cw = Q cw ( y ) ( A h ) c ρ g D [1- 1/3 cos 2 ф ] cos ф Q cw ( y ) = 0.5625 x cosh (3.674 y/D )/ cosh (3.674x h /D )

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Maximum pressure will occur at ф = 0. At base of wall, y = 0; Q cw ( y = 0) = 0.5625 x cosh (0) / cosh (3.674 x 3.0 / 4.65 ) =0.10 Convective pressure at the base of wall, p cw ( y = 0 )= 0.10 x 0.04 x 1,000 x 9.81 x 4.65 x 0.67 x 1 = 0.12 kN /m2 At y = h ; Q cw ( y = h ) = 0.5625 Convective pressure at y = h, p cw ( y = h ) = 0.5625 x 0.04 x 1,000 x 9.81 x 4.65 x 0.67 x 1 = 0.69 kN /m 2 . Convective hydrodynamic pressure on the base slab ( y = 0) p cb = Q cb ( x ) ( A h ) c ρ g D

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Q cb ( x ) = 1.125[ x / D – 4/3 ( x / D ) 3 ] sech (3.674 h / D ) = 1.125[ D /2 D – 4/3 ( D /2 D )3] sech (3.674 x 3/ 4.65 )= 0.07 Convective pressure on top of base slab ( y = 0) p cb = 0.07 x 0.04 x 1,000 x 9.81 x 4.65 = 0.13 kN /m2 1.11. Pressure Due to Wall Inertia Pressure on wall due to its inertia, p ww = ( A h ) i t ρ m g = 0.06 x 0.2 x 25= 0.32 kN /m2 . This pressure is uniformly distributed along the wall height . 1.12. Pressure Due to Vertical Excitation Hydrodynamic pressure on tank wall due to vertical ground acceleration , p v = ( A v ) [ ρ g h ( 1 - y / h )]

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Z = 0.1 (IS 1893(Part 1): Table 2; Zone II) I = 1.5, R = 2.5 Time period of vertical mode of vibration is recommended as 0.3 sec for 5% damping, Sa /g = 2.5 . Hence , A v =0.05 At the base of wall, i.e., y = 0, p v = 0.05 x [ 1 x 9.81 x 3 x ( 1 – 0 / 3 )] = 1.47 kN /m2. I n this case, hydrodynamic pressure due to vertical ground acceleration is more than impulsive hydrodynamic pressure due to lateral excitation . 1.13. Maximum Hydrodynamic Pressure Maximum hydrodynamic pressure, p = ( ( piw + pww ) 2 + ( pcw ) 2 + pv 2 )^0.5 At the base of wall, p = (1 . 41 + 0 . 32) 2 + 0 . 12 2 + 1 . 47 2 = 2.27 kN /m2 .

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This maximum hydrodynamic pressure is about 8 % of hydrostatic pressure at base ( ρ g h = 1,000x 9.81 x 3.0 = 29.43 kN /m2 ). In practice, container of tank is designed by working stress method. When earthquake forces are considered, permissible stresses are increased by 33%. Hence, hydrodynamic pressure in this case does not affect container design . 1.14.Sloshing Wave Height Maximum sloshing wave height , d max = ( A h ) c R D / 2 = 0.04 x 2.5 x 4.65 / 2= 0.23 m. Height of sloshing wave is less than free board of 0.3 m.

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1.15. Analysis for Tank Empty Condition For empty condition, tank will be considered as single degree of freedom system. Mass of empty container + one third mass of staging , ms = 63,799 kg. Stiffness of staging, Ks = 6,060 kN /m. 1.15.1 . Time Period Time period of impulsive mode , T= = 0.65 sec. Empty tank will not have convective mode of vibration. 1.15.2. Design Horizontal Seismic Coefficient Design horizontal seismic coefficient corresponding to impulsive time period T i ,

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Where, Z = 0.1, I = 1.5, R = 2.5, Here , Ti = 0.65 sec , Site has soft soil , Damping = 5 %, Hence , ( Sa /g ) i = 2.5, then : ( A h ) I =0.08. 1.15.3 . Base Shear Total base shear, V = Vi = ( A h ) i m s g = 0.08 x 63,799 x 9.81= 50 kN. 1.15.4 . Base Moment Total base moment , M * = ( A h ) i m s h cg g = 0.08 x 63,799 x 15.18 x 9.81 = 760 kN -m

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Since total base shear (60 kN ) and base moment ( 931 kN -m) in tank full condition are more than that total base shear (50 kN ) and base moment ( 760 kN -m) in tank empty condition, design will be governed by tank full condition.

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Ground Supported Circular Concrete Tank Problem Statement: A ground supported cylindrical RC water tank without roof has capacity of 1,000 m3. Inside diameter of tank is 14 m and height is 7.0 m (including a free board of 0.5 m). Tank wall has uniform thickness of 250 mm and base slab is 400 mm thick. Grade of concrete is M30. Tank is located on soft soil in seismic zone IV. Density of concrete is 25 kN/m3. Analyze the tank for seismic loads.

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Solution: Weight Calculations: Weight of tank wall : = π x (14 + 0.25) x 0.25 x 25 x 7.0 = 1,959 kN Mass of tank wall, mw: = 1,959 x 1,000 / 9.81 = 1,99,694 kg Mass of base slab, mb: = π x (7.25)2 x 0.4 x 25 x 1,000 / 9.81 = 1,68,328 kg. Volume of water = 1,000 m3 : Mass of water, m = 10,00,000 kg Weight of water = 9,810 kN Parameters of Spring Mass Model h = 6.5 m; D = 14 m For h / D = 6.5/14 = 0.46 mi / m = 0.511

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mi = 0.511 x 10,00,000 = 5,11,000 kg mc / m = 0.464; mc = 0.464 x 10,00,000 = 4,64,000 kg h i / h = 0.375; hi = 0.375 x 6.5 = 2.44 m hc / h = 0.593; hc = 0.593 x 6.5 = 3.86 m hi* / h = 0.853; hi * = 0.853 x 6.5 = 5.55 m hc* / h = 0.82; hc* = 0.82 x 6.5 = 5.33 m Note that about 51% of liquid is excited in impulsive mode while 46% participates in convective mode. Sum of impulsive and convective mass is about 2.5 % less than mass of liquid .

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Time Period : Time period of impulsive mode, Where, h = Depth of liquid = 6.5 m, ρ = Mass density of water = 1,000 kg/m3, t = Thickness of wall = 0.25 m, D = Inner diameter of tank = 14 m , E = Young’s modulus = 27,390 N/mm2 = 27,390 x 106 N/m2. For h / D = 0.46, Ci = 4.38 = 0.04 sec .

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Time period of convective mode, For h / D = 0.46, Cc = 3.38 Design Horizontal Seismic Coefficient Design horizontal seismic coefficient for impulsive mode, Where, Z = 0.24 (IS 1893(Part 1): Table 2; Zone IV)

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I = 1.5 ( Table 1) This tank has fixed base hence R is taken as 2.0. ( Table 2) Here, Ti = 0.04 sec, Site has soft soil, Damping = 5%, Since Ti < 0.1 sec ( Sa /g ) i = 2.5 Design horizontal seismic coefficient for convective mode,

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Where, Z = 0.24 (IS 1893(Part 1): Table 2; Zone IV) I = 1.5 ( Table 1) For convective mode, value of R is taken same as that for impulsive mode Here, Tc = 4.04 sec, Site has soft soil, Damping = 0.5%, ( Sa /g ) c = 1.75 x 0.413 = 0.72 Multiplying factor of 1.75 is used to obtain Sa /g values for 0.5% damping from that for 5% damping.

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Base Shear : Base shear at the bottom of wall in impulsive mode, Vi = ( Ah ) i ( mi + mw + mt ) g = 0.225 x (5,11,000 + 1,99,694 + 0) x 9.81 = 1,569 kN Similarly, base shear in convective mode, Vc = ( Ah ) c mc g = 0.065 x 4,64,000 x 9.81 = 296 kN Total base shear at the bottom of wall, = 1,597 kN. Total lateral base shear is about 14 % of seismic weight (11,769 kN) of tank.

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Moment at Bottom of Wall : Bending moment at the bottom of wall in impulsive mode, Mi = ( Ah ) i [ mi h i+ mw hw + mt h t ] g = 0.225 x [(5,11,000 x 2.44) + (1,99,694 x 3.5) + 0] x 9.81 = 4,295 kN-m Similarly, bending moment in convective mode, Mc = ( Ah ) c mc hc g = 0.065 x 4,64,000 x 3.86 x 9.81 = 1,142 kN-m Total bending moment at bottom of wall, = 4,444 kN-m.

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Overturning Moment Overturning moment at the bottom of base slab in impulsive mode, Mi* = ( Ah ) i [ mi ( hi* + tb ) + mw ( hw + tb ) + mt ( ht + tb ) + mb tb / 2] g =0.225x[(5,11,000x(5.55 + 0.4) + (1,99,694 x(3.5+ 0.4)) + 0 + (1,68,328 x 0.4 / 2)] x 9.81 = 8,504 kN-m. Similarly , overturning moment in convective mode , = 0.065 x 4,64,000 x (5.33 + 0.4) x 9.81 = 1,695 kN-m. Total overturning moment at the bottom of base slab ,

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Sloshing Wave Height : Maximum sloshing wave height, dmax = ( Ah ) c R D / 2 = 0.065 x 2.0 x 14 / 2 = 0.91 m Sloshing wave height exceeds the freeboard of 0.5 m. Anchorage Requirement :

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No anchorage is required.

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