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## Lecture 1

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**Lecture 1**January 17, 2006**In this lecture**• Types of tanks • IS codes on tanks • Modeling of liquid E-Course on Seismic Design of Tanks/ January 2006**Types of tanks**• Two categories • Ground supported tanks • Also called at-grade tanks; Ground Service Reservoirs (GSR) • Elevated tanks • Also called overhead tanks; Elevated Service Reservoirs (ESR) E-Course on Seismic Design of Tanks/ January 2006**Types of tanks**• Ground supported tanks • Shape: Circular or Rectangular • Material : RC, Prestressed Concrete, Steel • These are ground supported vertical tanks • Horizontal tanks are not considered in this course E-Course on Seismic Design of Tanks/ January 2006**Types of tanks**• Elevated tanks Two parts: • Container • Staging (Supporting tower) E-Course on Seismic Design of Tanks/ January 2006**Types of tanks**• Elevated tanks Container: • Material: RC, Steel, Polymer • Shape : Circular, Rectangular, Intze, Funnel, etc. Staging: • RC or Steel frame • RC shaft • Brick or masonry shafts • Railways often use elevated tanks with steel frame staging • Now-a-days, tanks on brick or stone masonry shafts are not constructed E-Course on Seismic Design of Tanks/ January 2006**Use of tanks**• Water distribution systems use ground supported and elevated tanks of RC & steel • Petrochemical industries use ground supported steel tanks E-Course on Seismic Design of Tanks/ January 2006**Indian Codes on Tanks**• IS 3370:1965/1967 (Parts I to IV) • For concrete (reinforced and prestressed) tanks • Gives design forces for container due to hydrostatic loads • Based on working stress design • BIS is considering its revision E-Course on Seismic Design of Tanks/ January 2006**Indian Codes on Tanks**• IS 11682:1985 • For RC staging of overhead tanks • Gives guidelines for layout & analysis of staging • More about this code later • IS 803:1976 • For circular steel oil storage tanks E-Course on Seismic Design of Tanks/ January 2006**Indian Codes on Tanks**• IS 1893:1984 • Gives seismic design provisions • Covers elevated tanks only • Is under revision • More about other limitations, later • IS 1893 (Part 1):2002 is for buildings only • Can not be used for tanks E-Course on Seismic Design of Tanks/ January 2006**Hydrodynamic Pressure**• Under static condition, liquid applies pressure on container. • This is hydrostatic pressure • During base excitation, liquid exerts additional pressure on wall and base. • This is hydrodynamic pressure • This is in additional to the hydrostatic pressure E-Course on Seismic Design of Tanks/ January 2006**Hydrodynamic pressure**h • h • Hydrostatic pressure • Varies linearly with depth of liquid • Acts normal to the surface of the container • At depth h from liquid top, hydrostatic pressure = h Hydrostatic pressure E-Course on Seismic Design of Tanks/ January 2006**Hydrodynamic pressure**• Hydrodynamic pressure • Has curvilinear variation along wall height • Its direction is opposite to base motion Hydrodynamic pressure Base motion E-Course on Seismic Design of Tanks/ January 2006**Hydrodynamic pressure**• Summation of pressure along entire wall surface gives total force caused by liquid pressure • Net hydrostatic force on container wall is zero • Net hydrodynamic force is not zero E-Course on Seismic Design of Tanks/ January 2006**Hydrodynamic pressure**Circular tanks (Plan View) Hydrostatic pressure Hydrodynamic pressure Base motion Net resultant force ≠ zero Net resultant force = zero Note:- Hydrostatic pressure is axisymmetric; hydrodynamic is asymmetric E-Course on Seismic Design of Tanks/ January 2006**Hydrodynamic pressure**Rectangular tanks (Plan View) Hydrostatic pressure Hydrodynamic pressure Base motion Net resultant force = zero Net resultant force ≠ zero E-Course on Seismic Design of Tanks/ January 2006**Hydrodynamic pressure**• Static design: Hydrostatic pressure is considered • Hydrostatic pressure induces hoop forces and bending moments in wall • IS 3370 gives design forces for circular and rectangular tanks • Net hydrostatic force is zero on container wall • Hence, causes no overturning moment on foundation or staging • Thus, hydrostatic pressure affects container design only and not the staging or the foundation E-Course on Seismic Design of Tanks/ January 2006**Hydrodynamic pressure**• Seismic design: Hydrodynamic pressure is considered • Net hydrodynamic force on the container is not zero • Affects design of container, staging and foundation E-Course on Seismic Design of Tanks/ January 2006**Hydrodynamic pressure**• Procedure for hydrodynamic pressure & force: • Very simple and elegant • Based on classical work of Housner (1963a) • Housner, G. W., 1963a, “Dynamic analysis of fluids in containers subjected to acceleration”, Nuclear Reactors and Earthquakes, Report No. TID 7024, U. S. Atomic Energy Commission, Washington D.C. • We need not go in all the details • Only basics and procedural aspects are explained in next few slides E-Course on Seismic Design of Tanks/ January 2006**Modeling of liquid**• Liquid inbottom portion of the container moves with wall • This is called impulsive liquid • Liquid in top portion undergoes sloshing and moves relative to wall • This is called convective liquid or sloshing liquid Convective liquid (moves relative to tank wall) Impulsive liquid (moves with tank wall) E-Course on Seismic Design of Tanks/ January 2006**Modeling of liquid**• Impulsive liquid • Moves with wall; rigidly attached • Has same acceleration as wall • Convective liquid • Also called sloshing liquid • Moves relative to wall • Has different acceleration than wall • Impulsive & convective liquid exert pressure on wall • Nature of pressure is different • See next slide E-Course on Seismic Design of Tanks/ January 2006**Modeling of liquid**Impulsive Convective Base motion Base motion Hydrodynamic pressure E-Course on Seismic Design of Tanks/ January 2006**Modeling of liquid**• At this point, we will not go into details of hydrodynamic pressure distribution • Rather, we will first find hydrodynamic forces • Impulsive force is summation of impulsive pressure on entire wall surface • Similarly, convective force is summation of convective pressure on entire wall surface E-Course on Seismic Design of Tanks/ January 2006**Modeling of liquid**• Total liquid mass, m, gets divided into two parts: • Impulsive liquid mass, mi • Convective liquid mass, mc • Impulsive force = mi x acceleration • Convective force = mc x acceleration • mi & mc experience different accelerations • Value of accelerations will be discussed later • First we will find mi and mc E-Course on Seismic Design of Tanks/ January 2006**Modeling of liquid**• Housner suggested graphs for mi and mc • mi and mc depend on aspect ratio of tanks • Such graphs are available for circular & rectangular tanks • See Fig. 2a and 3a of Guidelines • Also see next slide • For taller tanks (h/D or h/L higher), mi as fraction of mis more • For short tanks, mc as fraction of m is more E-Course on Seismic Design of Tanks/ January 2006**Modeling of liquid**mi/m mi/m mc /m mc /m For circular tanks For rectangular tanks • See next slide for definition of h, D, and L E-Course on Seismic Design of Tanks/ January 2006**Modeling of liquid**D h L L Base motion Plan of Circular tank Elevation Base motion Plan of Rectangular tank E-Course on Seismic Design of Tanks/ January 2006**Modeling of liquid**Example 1: A circular tank with internal diameter of 8 m, stores 3 m height of water. Find impulsive and convective water mass. Solution: Total volume of liquid = /4 x 82 x 3 = 150.8 m3 • Total liquid mass, m = 150.8 x 1.0 = 150.8 t Note:- mass density of water is 1000 kg/m3; weight density of water is 9.81 x 1000 = 9810 N/m3. D = 8 m, h = 3 m h/D = 3/8 = 0.375. E-Course on Seismic Design of Tanks/ January 2006**mi/m**mc /m E-Course on Seismic Design of Tanks/ January 2006**Modeling of liquid**From graph, for h/D = 0.375 mi/m = 0.42 and mc/m = 0.56 mi = 0.42 x 150.8 = 63.3 t and mc = 0.56 x 150.8 = 84.5 t E-Course on Seismic Design of Tanks/ January 2006**Modeling of liquid**Kc/2 Kc/2 mc Rigid mi • Impulsive liquid is rigidly attached to wall • Convective liquid moves relative to wall • As if, attached to wall with springs Convective liquid (moves relative to wall) Impulsive liquid (moves with wall) E-Course on Seismic Design of Tanks/ January 2006**Modeling of liquid**• Stiffness associated with convective mass, Kc • Kc depends on aspect ratio of tank • Can be obtained from graph • Refer Fig. 2a, 3a of guidelines • See next slide E-Course on Seismic Design of Tanks/ January 2006**Modeling of liquid**E-Course on Seismic Design of Tanks/ January 2006**Modeling of liquid**Example 2: A circular tank with internal diameter of 8 m, stores 3 m height of water. Find Kc. Solution: Total liquid mass, m = 150.8 t (from Example 1) = 150.8 x 1000 = 150800 kg g = acceleration due to gravity = 9.81 m/sec2 D = 8 m, h = 3m h/D = 3/8 = 0.375. From graph, for h/D = 0.375; Kc h/mg = 0.65 E-Course on Seismic Design of Tanks/ January 2006**Modeling of liquid**• Kc= 0.65 mg/h • Kc = 0.65 x150800 x 9.81/3.0 = 320,525.4N/m • Note: - Unit of m is kg, hence unit of Kc is N/m. If we take m in ton, then unit of Kc will be kN/m. E-Course on Seismic Design of Tanks/ January 2006**Modeling of liquid**• Now, we know liquid masses mi and mc • Next, we need to know where these are attached with the wall • Like floor mass in building acts at centre of gravity (or mass center) of floor • Location of mi and mc is needed to obtain overturning effects • Impulsive mass acts at centroid of impulsive pressure diagram • Similarly, convective mass E-Course on Seismic Design of Tanks/ January 2006**Modeling of liquid**• Impulsive mass acts at centroid of impulsive pressure diagram • Location of centroid: • Obtained by dividing the moment due to pressure distribution by the magnitude of impulsive force • Similarly, location of convective mass is obtained • See next slide E-Course on Seismic Design of Tanks/ January 2006**Modeling of liquid**Resultant of impulsive pressure on wall Resultant of convective pressure on wall hc hi • hi, hc can be obtained from graphs • They also depend on aspect ratio, h/D or h/L • Refer Fig. 2b, 3b of guidelines • See next slide E-Course on Seismic Design of Tanks/ January 2006**Modeling of liquid**hc/h hc/h hi/h hi/h For circular tanks For rectangular tanks E-Course on Seismic Design of Tanks/ January 2006**Modeling of liquid**Example 3: A circular tank with internal diameter of 8 m, stores 3 m height of water. Find hi and hc. Solution: D = 8 m, h = 3m h/D = 3/8 = 0.375. E-Course on Seismic Design of Tanks/ January 2006**hc/h**hi/h E-Course on Seismic Design of Tanks/ January 2006**Modeling of liquid**• From graph,for h/D = 0.375; • hi/h = 0.375 • hi = 0.375 x 3 = 1.125 m • and hc/h = 0.55 • hc = 0.55 x 3 = 1.65 m • Note :- Since convective pressure is more in top portion, hc > hi. E-Course on Seismic Design of Tanks/ January 2006**Modeling of liquid**• Hydrodynamic pressure also acts on base • Under static condition, base is subjected to uniformly distributed pressure • Due to base motion, liquid exerts nonuniform pressure on base • This is in addition to the hydrostatic pressure on the base • See next slide E-Course on Seismic Design of Tanks/ January 2006**Modeling of liquid**Base motion Hydrostatic pressure on base Hydrodynamic pressure on base E-Course on Seismic Design of Tanks/ January 2006**Modeling of liquid**• Impulsive as well as convective liquid cause nonuniform pressure on base • Nonuniform pressure on base causes overturning effect • This will be in addition to overturning effect of hydrodynamic pressure on wall • See next slide E-Course on Seismic Design of Tanks/ January 2006**Modeling of liquid**hi Overturning effect due to wall pressure Overturning effect due to base pressure Note:- Both the overturning effects are in the same direction E-Course on Seismic Design of Tanks/ January 2006**Modeling of liquid**• Total overturning effect of wall and base pressure is obtained by applying resultant of wall pressure at height, hi* and hc*. • In place of hi and hc discussed earlier • For overturning effect due to wall pressure alone, resultant was applied at hi • For hi and hi*, see next slide E-Course on Seismic Design of Tanks/ January 2006**Modeling of liquid**h*i hi Location of Resultant of wall pressure when effect of base pressure is also included Location of resultant of wall pressure when effect of base pressure is not included E-Course on Seismic Design of Tanks/ January 2006**Modeling of liquid**hc • Similarly, hc and hc* are defined h*c Location of resultant of wall pressure when effect of base pressure is not included Location of Resultant of wall pressure when effect of base pressure is also included E-Course on Seismic Design of Tanks/ January 2006**Modeling of liquid**• hi and hi* are such that • Moment due to impulsive pressure on walls only = Impulsive force x hi • Moment due to impulsive pressure on walls and base = Impulsive force x hi* • hc and hc* are such that • Moment due to convective pressure on walls only = Convective force x hc • Moment due to convective pressure on walls and base = Convective force x hc* E-Course on Seismic Design of Tanks/ January 2006