ACI 350.3-06 Seismic Design of Liquid-Containing Concrete Structures and Commentary

ACI 350.3-06

Seismic Design of Liquid-ContainingConcrete Structures and Commentary

(ACI 350.3-06)An ACI Standard

Reported by ACI Committee 350

Seismic Design of Liquid-Containing Concrete Structuresand Commentary

ISBN 0-87031-222-7

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American Concrete Institute38800 Country Club DriveFarmington Hills, MI 48331U.S.A.Phone: 248-848-3700Fax: 248-848-3701

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Satish K. SachdevChair

Jon B. ArdahlVice Chair

John W. BakerSecretary

Walter N. Bennett Carl A. Gentry* Dennis C. Kohl Jerry ParnesLucian I. Bogdan Gautam Ghosh* Nicholas A. Legatos Andrew R. PhilipSteven R. Close Charles S. Hanskat Ramon E. Lucero* Narayan M. Prachand

Patrick J. Creegan* Keith W. Jacobson Andrew R. Minogue Risto Protic*

Ashok K. Dhingra* Dov Kaminetzky Lawrence G. Mrazek* William C. Sherman*

Robert E. Doyle M. Reza Kianoush* Javeed A. Munshi* Lawrence M. Tabat*

Anthony L. Felder David G. Kittridge

Subcommittee Members

Iyad M. Alsamsam Daniel J. McCarthyClifford T. Early, Jr. Carl H. Moon

Paul Hedli G. Scott RiggsJerry A. Holland John F. Seidensticker

Lawrence E. Kaiser Paul J. St. JohnSalvatore Marques Paul Zoltanetzky, Jr.

Seismic Design of Liquid-Containing Concrete Structures

and Commentary (ACI 350.3-06)AN ACI STANDARD

REPORTED BY ACI COMMITTEE 350Environmental Engineering Concrete Structures

Consulting Members

William IrwinTerry Patzias

Lawrence Valentine

*Members of Seismic Provisions Subcommittee.Seismic Provisions Subcommittee Chair.Seismic Provisions Subcommittee Secretary.

REPORTED BY ACI COMMITTEE 350

This standard prescribes procedures for the seismicanalysis and design of liquid-containing concretestructures. These procedures address the loading side ofseismic design and are intended to complement ACI 350-06,Section 1.1.8 and Chapter 21.

Keywords: circular tanks; concrete tanks; convective component; earth-quake resistance; environmental concrete structures; impulsive component;liquid-containing structures; rectangular tanks; seismic resistance;sloshing; storage tanks.

INTRODUCTIONThe following paragraphs highlight the development of thisstandard and its evolution to the present format:

From the time it embarked on the task of developing anACI 318-dependent code, ACI Committee 350 decided toexpand on and supplement Chapter 21, Special Provisionsfor Seismic Design, to provide a set of thorough andcomprehensive procedures for the seismic analysis and design ofall types of liquid-containing environmental concrete structures.The committees decision was influenced by the recognitionthat liquid-containing structures are unique structureswhose seismic design is not adequately covered by theleading national codes and standards. A seismic designsubcommittee was appointed with the charge to implementthe committees decision.

The seismic subcommittees work was guided by twomain objectives:1. To produce a self-contained set of procedures thatwould enable a practicing engineer to perform a full seismicanalysis and design of a liquid-containing structure. This meantthat these procedures should cover both aspects of seismic

ACI Committee Reports, Guides, Standards, and Commentaries are intendedfor guidance in planning, designing, executing, and inspecting construction.This Commentary is intended for the use of individuals who are competent toevaluate the significance and limitations of its content and recommendationsand who will accept responsibility for the application of the material it contains.The American Concrete Institute disclaims any and all responsibility for thestated principles. The Institute shall not be liable for any loss or damage arisingtherefrom. Reference to this commentary shall not be made in contract docu-ments. If items found in this Commentary are desired by the Architect/Engineerto be a part of the contract documents, they shall be restated in mandatorylanguage for incorporation by the Architect/Engineer.design: the loading side (namely the determination of theseismic loads based on the mapped maximum consideredearthquake spectral response accelerations at short periods(Ss) and 1 second (S1) obtained from the Seismic GroundMotion maps [Fig. 22-1 through 22-14 of ASCE 7-05,Chapter 22] and the geometry of the structure); and theresistance side (the detailed design of the structure inaccordance with the provisions of ACI 350, so as to resistthose loads safely); and

2. To establish the scope of the new procedures consistentwith the overall scope of ACI 350. This required the inclusion ofall types of tanksrectangular, as well as circular; andreinforced concrete, as well as prestressed.(Note: While there are currently at least two national standardsthat provide detailed procedures for the seismic analysis anddesign of liquid-containing structures (ANSI/AWWA1995a,b), these are limited to circular, prestressed concretetanks only).

As the loading side of seismic design is outside the scope ofACI 318, Chapter 21, it was decided to maintain this practicein ACI 350 as well. Accordingly, the basic scope, format, andmandatory language of Chapter 21 of ACI 318 were retainedwith only enough revisions to adapt the chapter to environmentalengineering structures. Provisions similar to Section 1.1.8of ACI 318 are included in ACI 350. This approach offersat least two advantages:

1. It allows ACI 350 to maintain ACI 318s practice oflimiting its seismic design provisions to the resistance sideSEISMIC DESIGN OF LIQUID-CONTAINING CONCRETE STRUCTURES 350.3-1

Seismic Design of Liquid-ContainingConcrete Structures andCommentary (ACI 350.3-06)only; and2. It makes it easier to update these seismic provisions so as to

keep up with the frequent changes and improvements in thefield of seismic hazard analysis and evaluation.

ACI 350.3-06 supersedes 350.3-01 and became effective on July 3,2006.

Copyright 2006, American Concrete Institute.All rights reserved including rights of reproduction and use in any

form or by any means, including the making of copies by any photoprocess, or by any electronic or mechanical device, printed or writtenor oral, or recording for sound or visual reproduction or for use in anyknowledge or retrieval system or device, unless permission in writingis obtained from the copyright proprietors.

350.3-2 ACI STANDARD/COMMENTARY

The seismic force levels and R-factors included in this standardprovide results at strength levels, such as those included forseismic design in the 2003 International Building Code (IBC),particularly the applicable connection provisions of 2003 IBC,as referenced in ASCE 7-02. When comparing these provisionswith other documents defining seismic forces at allowablestress levels (for example, the 1994 Uniform Building Code[UBC] or ACI 350.3-01), the seismic forces in this standardshould be reduced by the applicable factors to derive comparableforces at allowable stress levels.

The user should note the following general design methodsused in this standard, which represent some of the key

differences relative to traditional methodologies, such asthose described in ASCE (1984):

1. Instead of assuming a rigid tank for which the accelerationis equal to the ground acceleration at all locations, thisstandard assumes amplification of response due to naturalfrequency of the tank;

2. This standard includes the response modification factor;3. Rather than combining impulsive and convective modes by

algebraic sum, this standard combines these modes bysquare-root-sum-of-the-squares;

4. This standard includes the effects of vertical acceleration;and

5. This standard includes an effective mass coefficient,applicable to the mass of the walls.

SEISMIC DESIGN OF LIQUID-CONTAINING CONCRETE STRUCTURES 350.3-3CONTENTS

CHAPTER 1GENERAL REQUIREMENTS ............................................................................ 51.1Scope1.2Notation

CHAPTER 2TYPES OF LIQUID-CONTAINING STRUCTURES ......................................... 112.1Ground-supported structures2.2Pedestal-mounted structures

CHAPTER 3GENERAL CRITERIA FOR ANALYSIS AND DESIGN................................... 133.1Dynamic characteristics3.2Design loads3.3Design requirements

CHAPTER 4EARTHQUAKE DESIGN LOADS .................................................................... 154.1Earthquake pressures above base4.2Application of site-specific response spectra

CHAPTER 5EARTHQUAKE LOAD DISTRIBUTION........................................................... 215.1General5.2Shear transfer5.3Dynamic force distribution above base

CHAPTER 6STRESSES....................................................................................................... 276.1Rectangular tanks6.2Circular tanks

CHAPTER 7FREEBOARD ................................................................................................... 297.1Wave oscillation

CHAPTER 8EARTHQUAKE-INDUCED EARTH PRESSURES .......................................... 318.1General8.2Limitations8.3Alternative methods

CHAPTER 9DYNAMIC MODEL ........................................................................................... 339.1General9.2Rectangular tanks (Type 1)9.3Circular tanks (Type 2)9.4Seismic response coefficients Ci , Cc, and Ct9.5Site-specific seismic response coefficients Ci , Cc, and Ct 9.6Effective mass coefficient 9.7Pedestal-mounted tanks

CHAPTER 10COMMENTARY REFERENCES .................................................................... 53


APPENDIX ADESIGN METHOD...........................................................................................55A.1General outline of design method

APPENDIX BALTERNATIVE METHOD OF ANALYSIS BASED ON 1997 Uniform Building Code...........................................................................57

B.1IntroductionB.2Notation (not included in Section 1.2 of this standard)B.3Loading side, general methodologyB.4Site-specific spectra (Section 1631.2(2))B.5Resistance sideB.6Freeboard

1.2Notation

As = cross-sectional area of base cable, strand,or conventional reinforcement, in.2 (mm2)

b = ratio of vertical to horizontal design accel-eration

B = inside dimension (length or width) of a rectan-gular tank, perpendicular to the direction ofthe ground motion being investigated, ft (m)

Cc, Ci , and Ct = period-dependent seismic response coeffi-

cients defined in 9.4 and 9.5.Cl, Cw = coefficients for determining the fundamental

frequency of the tank-liquid system (referto Eq. (9-24) and Fig. 9.3.4(b))

Cs = period-dependent seismic coefficientd, dmax= freeboard (sloshing height) measured from

the liquid surface at rest, ft (m)D = inside diameter of circular tank, ft (m)EBP = excluding base pressure (datum line just

above the base of the tank wall) Ec = modulus of elasticity of concrete, lb/in.2 (MPa)Es = modulus of elasticity of cable, wire, strand,SEISMIC DESIGN OF LIQUID-CON

STANDARD1.1Scope

This standard describes procedures for the design ofliquid-containing concrete structures subjected to seismicloads. These procedures shall be used in accordancewith Chapter 21 of ACI 350-06.

CHAPTER 1GENERAL REQUIRor conventional reinforcement, lb/in.2 (MPa)Fa = short-period site coefficient (at 0.2 second

period) from ASCE 7-05, Table 11.4-1Fv = long-period site coefficient (at 1.0 second

period) from ASCE 7-05, Table 11.4-2Gp = shear modulus of elastomeric bearing pad,

lb/in.2 (MPa)g = acceleration due to gravity [32.17 ft/s2

(9.807 m/s2)]TAINING CONCRETE STRUCTURES 350.3-5

COMMENTARYR1.1Scope

This standard is a companion standard to Chapter 21 of theAmerican Concrete Institute, Code Requirements for Environ-mental Engineering Concrete Structures and Commentary (ACI350-06) (ACI Committee 350 2006).

This standard provides directions to the designer of liquid-containing concrete structures for computing seismic forcesthat are to be applied to the particular structure. Thedesigner should also consider the effects of seismic forceson components outside the scope of this standard, such aspiping, equipment (for example, clarifier mechanisms), andconnecting walkways where vertical or horizontal movementsbetween adjoining structures or surrounding backfill couldadversely influence the ability of the structure to functionproperly (National Science Foundation 1981). Moreover,seismic forces applied at the interface of piping or walkwayswith the structure may also introduce appreciable flexural orshear stresses at these connections.

R1.2Notation

EMENTS

EBP refers to the hydrodynamic design in which it is necessaryto compute the overturning of the wall with respect to thetank floor, excluding base pressure (that is, excluding thepressure on the floor itself). EBP hydrodynamic design is

For Cs, refer to International Building Code (IBC)(International Code Council 2003), Section 1617.4.used to determine the need for hold-downs in nonfixed basetanks. EBP is also used in determining the design pressureacting on walls. (For explanation, refer to Housner [1963].)

ka = spring constant of the tank wall supportsystem, lb/ft2 per foot of wall width (N/m permeter of wall width)

Ka = active coefficient of lateral earth pressureKo = coefficient of lateral earth pressure at restL = inside dimension of a rectangular tank,

parallel to the direction of the ground motionbeing investigated, ft (m)

Lc = effective length of base cable or strandtaken as the sleeve length plus 35 times thestrand diameter, in. (mm)

Lp = length of individual elastomeric bearingpads, in. (mm)

m = total mass per unit width of a rectangularwall = mi + mw , lb-s2/ft per foot of wallwidth (kg per meter of wall width)

mi = impulsive mass of contained liquid per unitwidth of a rectangular tank wall, lb-s2/ft perfoot of wall width (kg per meter of wall width)

mw = mass per unit width of a rectangular tankwall, lb-s2/ft per foot of wall width (kg permeter of wall width)

Mb = bending moment on the entire tank cross350.3-6 ACI STANDARD

STANDARD

hc = height above the base of the wall to thecenter of gravity of the convective lateralforce for the case excluding base pressure(EBP), ft (m)

hc = height above the base of the wall to thecenter of gravity of the convective lateralforce for the case including base pressure(IBP), ft (m)

hi = height above the base of the wall to thecenter of gravity of the impulsive lateralforce for the case including base pressure(EBP), ft (m)

hi = height above the base of the wall to the centerof gravity of the impulsive lateral force for thecase including base pressure (IBP), ft (m)

hr = height from the base of the wall to thecenter of gravity of the tank roof, ft (m)

hw = height from the base of the wall to thecenter of gravity of the tank shell, ft (m)

HL = design depth of stored liquid, ft (m)Hw = wall height (inside dimension), ft (m)I = importance factor, from Table 4.1.1(a)IBP = including base pressure (datum line at the

base of the tank including the effects of thetank bottom and supporting structure)

k = flexural stiffness of a unit width of a rectilineartank wall, lb/ft per foot of wall width (N/mper meter of wall width)section just above the base of the tank wall,ft-lb (kN-m)

Mc = bending moment of the entire tank crosssection just above the base of the tank wall(EBP) due to the convective force Pc , ft-lb(kN-m)/COMMENTARY

COMMENTARYh = as defined in Section R9.2.4, ft (m)

IBP refers to the hydrodynamic design in which it is necessaryto investigate the overturning of the entire structure withrespect to the foundation. IBP hydrodynamic design is usedto determine the design pressure acting on the tank floor andthe underlying foundation. This pressure is transferreddirectly either to the subgrade or to other supportingstructural elements. IBP accounts for moment effects due todynamic fluid pressures on the bottom of the tank byincreasing the effective vertical moment arm to the appliedforces. (For explanation, refer to Housner [1963].)

ISEISMIC DESIGN OF LIQUID-CONTA

STANDARDMc = overturning moment at the base of the tank,

including the tank bottom and supportingstructure (IBP), due to the convective forcePc, ft-lb (kN-m)

Mi = bending moment of the entire tank crosssection just above the base of tank wall (EBP)due to the impulsive force Pi, ft-lb (kN-m)

Mi = overturning moment at the base of the tank,including the tank bottom and supportingstructure (IBP), due to the impulsive forcePi, ft-lb (kN-m)

Mo = overturning moment at the base of the tank,including the tank bottom and supportingstructure (IBP), ft-lb (kN-m)

Mr = bending moment of the entire tank crosssection just above the base of the tank wall(EBP) due to the roof inertia force Pr, ft-lb(kN-m)

Mw = bending moment of the entire tank crosssection just above the base of the tank wall(EBP) due to the wall inertia force Pw, ft-lb(kN-m)

Ncy = in circular tanks, hoop force at liquid level y,due to the convective component of theaccelerating liquid, lb per foot of wall height(kN/m)

Nhy = in circular tanks, hydrodynamic hoop forceat liquid level y, due to the effect of verticalacceleration, lb per foot of wall height (kN/m)

Niy = in circular tanks, hoop force at liquid level y,due to the impulsive component of theaccelerating liquid, lb per foot of wall height(kN/m)

Nwy = in circular tanks, hoop force at liquid level y,due to the inertia force of the acceleratingwall mass, lb per foot of wall height (kN/m)

Ny = in circular tanks, total effective hoop force atliquid level y, lb per foot of wall height (kN/m)

Pc = total lateral convective force associatedwith Wc, lb (kN)

Pcy = lateral convective force due to Wc, per unitheight of the tank wall, occurring at liquidlevel y, lb per foot of wall height (kN/m)

pvy = unit equivalent hydrodynamic pressure dueto the effect of vertical acceleration, at liquidlevel y, above the base of the tank (pvy = v qhy), lb/ft2 (kPa)Peg = lateral force on the buried portion of a tankwall due to the dynamic earth and ground-water pressures, lb (kN)NING CONCRETE STRUCTURES 350.3-7

COMMENTARY

pcy = unit lateral dynamic convective pressure distributedhorizontally at liquid level y, lb/ft2 (kPa)

piy = unit lateral dynamic impulsive pressure distributedhorizontally at liquid level y, lb/ft2 (kPa)

pwy = unit lateral inertia force due to wall dead weight,distributed horizontally at liquid level y, lb/ft2 (kPa)

350.3-8 ACI STANDARD/

logical Survey (USGS) database at website (http://STANDARD

r = inside radius of circular tank, ft (m)R = response modification factor, a numerical

coefficient representing the combined effect ofthe structures ductility, energy-dissipatingcapacity, and structural redundancy (Rc forthe convective component of the acceleratingliquid; Ri for the impulsive component) fromTable 4.1.1(b)

Ph = total hydrostatic force occurring on length Bof a rectangular tank or diameter D of acircular tank, lb (kN)

Phy = lateral hydrostatic force per unit height ofthe tank wall, occurring at liquid level y, lbper foot of wall height (kN/m)

Pi = total lateral impulsive force associated withWi , lb (kN)

Piy = lateral impulsive force due to Wi, per unitheight of the tank wall, occurring at liquidlevel y, lb per foot of wall height (kN/m)

Pr = lateral inertia force of the accelerating roofWr, lb (kN)

Pw = lateral inertia force of the accelerating wallWw, lb (kN)

Pw = in a rectangular tank, lateral inertia force ofone accelerating wall (Ww ), perpendicular tothe direction of the earthquake force, lb (kN)

Pwy = lateral inertia force due to Ww, per unitheight of the tank wall, occurring at level yabove the tank base, lb per foot of wallheight (kN/m)

Py = combined horizontal force (due to theimpulsive and convective components ofthe accelerating liquid; the walls inertia,and the hydrodynamic pressure due to thevertical acceleration) at a height y abovethe tank base, lb per foot of wall height (kN/m)

qhy = unit hydrostatic pressure at liquid level y abovethe tank base [qhy = L (HL y)], lb/ft2 (kPa)gravity g from a site-specific spectrum. (S0 isequivalent to a PGA having a 2% probability ofexceedance in 50 years, as given in the U.S. Geo-COMMENTARY

COMMENTARY

q, qmax= unit shear force in circular tanks, lb/ft (kN/m)Q = total membrane (tangential) shear force at the

base of a circular tank, lb (kN)Qhy = in circular tanks, hydrostatic hoop force at liquid

level y (Qhy = qhy R), lb per foot of wallheight (kN/m)

For a schematic representation of Ph, refer to Fig. R5.3.1(a)and (b)

S0 = effective peak ground acceleration (at T = 0)related to the maximum considered earthquake;expressed as a fraction of the acceleration due toeqhazmaps.usgs.gov)

ASEISMIC DESIGN OF LIQUID-CONT

STANDARDS1 = mapped maximum considered earthquake

5% damped spectral response acceleration;parameter at a period of 1 second, expressedas a fraction of the acceleration due to gravityg, from ASCE 7-05, Fig. 22-1 through 22-14

SaM = maximum considered earthquake spectralresponse acceleration, 5% damped, atperiod Ti or Tv , taken from a site-specificacceleration response spectrum

Sc = center-to-center spacing between individualbase cable loops, in. (mm)

ScM = maximum considered earthquake spectralresponse acceleration, 0.5% damped, atperiod Tc , taken from a site-specificacceleration response spectrum

SD1 = design spectral response acceleration, 5%damped, at a period of 1 second, asdefined in 9.4.1, expressed as a fractionof the acceleration due to gravity g

SDS = design spectral response acceleration, 5%damped, at short periods, as defined in9.4.1, expressed as a fraction of the accel-eration due to gravity g

Sp = center-to-center spacing of elastomericbearing pads, in. (mm)

Ss = mapped maximum considered earthquake5% damped spectral response accelerationparameter at short periods, expressed as afraction of the acceleration due to gravity g,from ASCE 7-05, Fig. 22-1 through 22-14

tp = thickness of elastomeric bearing pads,in. (mm)

tw = average wall thickness, in. (mm)Tc = natural period of the first (convective) mode

of sloshing, sTi = fundamental period of oscillation of the tank

(plus the impulsive component of thecontents), s

TS = SD1/SDSINING CONCRETE STRUCTURES 350.3-9

COMMENTARY

SD = spectral displacement, ft (m)

Sa = generalized design spectral response accelerationcorresponding to a given natural period T,expressed as a fraction of the acceleration due togravity g

TS: In Appendix B,

Ts

Cv

2.5Ca

--------------- 0.40C

v

Ca

-------= = where Ca and Cv are defined in Appendix B.


STANDARD COMMENTARYTv = natural period of vibration of vertical liquidi = circular frequency of the impulsive mode ofvibration, radian/se w rWi = equivalent weight of the impulsive component

of the stored liquid, lb (kN)WL = total equivalent weight of the stored liquid,

lb (kN)Wr = equivalent weight of the tank roof, plus

superimposed load, plus applicable portionof snow load considered as dead load, lb (kN)

Ww = equivalent weight of the tank wall (shell),lb (kN)

Ww = in a rectangular tank, the equivalentweight of one wall perpendicular to thedirection of the earthquake force, lb (kN)

y = liquid level at which the wall is being investi-gated (measured from tank base), ft (m)

= angle of base cable or strand with horizontal,degree

= percent of critical dampingc = density of concrete, [150 lb/ft3 (23.56 kN/m3)

for standard-weight concrete]L = density of contained liquid, lb/ft3 (kN/m3)w = density of water, 62.43 lb/ft3 (9.807 kN/m3) = effective mass coefficient (ratio of equivalent

dynamic mass of the tank shell to itsactual total mass), Eq. (9-44) and (9-45)

= polar coordinate angle, degree = coefficient as defined in 9.2.4 and 9.3.4y = membrane (hoop) stress in wall of circular

tank at liquid level y, lb/in.2 (MPa)c = circular frequency of oscillation of the first

(convective) mode of sloshing, radian/swp = width of elastomeric bearing pad, in. (mm)Wc = equivalent weight of the convective compo-

nent of the stored liquid, lb (kN)We = effective dynamic weight of the tank structure

(walls and roof) [W = (W + W )], lb (kN)motion, sv = effective spectral acceleration from an

inelastic vertical response spectrum, asdefined by Eq. (4-15), that is derived byscaling from an elastic horizontal responsespectrum, expressed as a fraction of theacceleration due to gravity g

V = total horizontal base shear, lb (kN)c , i = coefficients as defined in Section R4.2

For , refer to Fig. R5.2.1 and R5.2.2.

TC

SEISMIC DESIGN OF LIQUID-CON

STANDARD2.1Ground-supported structures

Structures in this category include rectangular andcircular liquid-containing concrete structures, on-gradeand below grade.

2.1.1Ground-supported liquid-containing structuresare classified according to this section on the basis ofthe following characteristics:

General configuration (rectangular or circular); Wall-base joint type (fixed, hinged, or flexible

base); and Method of construction (reinforced or prestressed

concrete).

Type 1Rectangular tanksType 1.1Fixed baseType 1.2Hinged base

Type 2Circular tanksType 2.1Fixed base

2.1(1)Reinforced concrete2.1(2)Prestressed concrete

Type 2.2Hinged base2.2(1)Reinforced concrete2.2(2)Prestressed concrete

Type 2.3Flexible base (prestressed only)2.3(1)Anchored2.3(2)Unanchored, contained2.3(3)Unanchored, uncontained

2.2Pedestal-mounted structures

Structures in this category include liquid-containingstructures mounted on cantilever-type pedestals.

CHAPTER 2TYPES OF LIQUID-

AINING CONCRETE STRUCTURES 350.3-11

COMMENTARYR2.1Ground-supported structures

For basic configurations of ground-supported, liquid-containing structures, refer to Fig. R2.1.

ONTAINING STRUCTURES

R2.1.1The classifications of Section 2.1.1 are basedon the wall-to-footing connection details as illustrated inFig. R2.1.1.

The tank floor and floor support system should be designedfor the seismic forces transmitted therein. With any one ofthe tank types covered under this standard, the floor may bea membrane-type slab, a raft foundation, or a structural slabsupported on piles. This standard, however, does not coverthe determination of seismic forces on the piles themselves.

The tank roof may be a free-span dome or column-supportedflat slab, or the tank may be open-topped.Fig. R2.1Typical tank configurations (adapted from ASCE[1984]).


COMMENTARYFig. R2.1.1Types of ground-supported, liquid-containing structures classified on the basis of their wall-to-footing connection details(base waterstops not shown).

SEISMIC DESIGN OF LIQUID-CONTAINING CONCRETE STRUCTURES 350.3-13

STANDARD COMMENTARY3.1Dynamic characteristics

The dynamic characteristics of liquid-containingstructures shall be derived in accordance with eitherChapter 9 or a more rigorous dynamic analysis thataccounts for the interaction between the structure andthe contained liquid.

3.2Design loads

The loads generated by the design earthquake shall

CHAPTER 3GENERAL CRITERIA FOR ANALYSIS AND DESIGN

R3.1Dynamic characteristics

For an outline of the general steps involved in the interactionbetween the structure and the contained liquid, refer toAppendix A.be computed in accordance with Chapter 4.

3.3Design requirements

3.3.1The walls, floors, and roof of liquid-containingstructures shall be designed to withstand the effects ofboth the design horizontal acceleration and the designvertical acceleration combined with the effects of theapplicable design static loads.

3.3.2With regards to the horizontal acceleration, thedesign shall take into account the effects of the transferof the total base shear between the wall and the footingand between the wall and the roof, and the dynamicpressure acting on the wall above the base.

3.3.3Effects of maximum horizontal and verticalacceleration shall be combined by the square-root-sum-of-the-squares method.


Notes


STANDARD COMMENTARY

CHAPTER 4EARTHQUAKE DESIGN LOADS4.1Earthquake pressures above base

The walls of liquid-containing structures shall bedesigned for the following dynamic forces in addition tothe static pressures in accordance with Section 5.3.1:

(a) Inertia forces Pw and Pr ;(b) Hydrodynamic impulsive force Pi from the containedliquid;(c) Hydrodynamic convective force Pc from the containedliquid;(d) Dynamic earth pressure from saturated andunsaturated soils against the buried portion of thewall; and(e) The effects of vertical acceleration.Section R21.2.1).

Explanations of the impulsive and convective pressures Piand Pc are contained in Section R9.1 and Housner (1963).Table 4.1.1(a). Engineering judgment may require a factor Igreater than tabulated in Table 4.1.1(a) where it is necessary toreduce further the potential level of damage or account for thepossibility of an earthquake greater than the design earthquake.

The response modification factors Rc and Ri reduce the elasticresponse spectrum to account for the structures ductility,energy-dissipating properties, and redundancy (ACI 350-06,SS and S1 as described in Section R9.4.

Factor I provides a means for the engineer to increase thefactor of safety for the categories of structures described incomponents Wi and Wc is contained in Section R9.1.

The imposed ground motion is represented by an elasticresponse spectrum that is either derived from an actualearthquake record for the site, or is constructed by analogyto sites with known soil and seismic characteristics. Theprofile of the response spectrum is defined by Sa, which is afunction of the period of vibration and the mapped accelerationsR4.1Earthquake pressures above base

The general equation for the total base shear normallyencountered in the earthquake-design sections of governingbuilding codes V = CsW is modified in Eq. (4-1) through(4-4) by replacing the term W with the four effectiveweights: the effective weight of the tank wall Ww and roofWr; the impulsive component of the liquid weight Wi andthe convective component Wc; and the term Cs with Ci, Cc,or Cv as appropriate. Because the impulsive and convectivecomponents are not in phase with each other, practice is tocombine them using the square-root sum-of-the-squaresmethod (Eq. (4-5)) (New Zealand Standard [NZS] 1986;ASCE 1981; ANSI/AWWA 1995a).

A more detailed discussion of the impulsive and convective

/Ww and Wr are the weights of the cylindrical tank wall(shell) and tank roof, respectively, and Ww is theweight of one wall in a rectangular tank, perpendicularto the direction of the ground motion being investigated; is a factor defined in Section 1.2 and determined inaccordance with Section 9.6; Wi and Wc are theimpulsive and convective components of the storedliquid, respectively, as defined in Section 1.2 anddetermined in accordance with Sections 9.2.1 and9.3.1; and Ri and Rc are the response modificationfactors defined in Section 1.2 and determined inaccordance with Table 4.1.1(b).

Where applicable, the lateral forces due to thedynamic earth and groundwater pressures against theburied portion of the walls shall be computed inaccordance with the provisions of Chapter 8.

4.1.2Total base shearSTANDARD4.1.1Dynamic lateral forcesThe base shear due to seismic forces applied at thebottom of the tank wall shall be determined by

(4-5)V P P P+ +( )2 P 2c P 2+ +=(4-3)

(4-4)

where Ci and Cc are the seismic response coefficientsdetermined in accordance with Sections 9.4 and9.5; I is the importance factor defined in Table 4.1.1(a);

Pi CiI= Ri-------

Pc CcIWcRc--------=Ri

Wi(4-2)Pr CiIr

-------=(4-1)

(4-1a)

Pw Ci I= Ri------------

Pw CiIWw

Ri--------------=

WThe dynamic lateral forces above the base shall bedetermined as

Ww350.3-16 ACI STANDARDWhere applicable, the lateral forces due to dynamicearth and groundwater pressures against the buried

i w r egCOMMENTARY

COMMENTARYR4.1.1Dynamic lateral forces

A model representation of Wi and Wc is shown in Fig. R9.1.

Energy Method: An energy method of dynamic analysismay be used instead of the base-shear approach of Section 4.1for sizing earthquake cables and base pad for flexible basejoints (Housner 1963; John A. Blume & Associates 1958;Medearis and Young 1964; Uang and Bertero 1988; Bertero1995; Scarlat 1997).


R4.1.4Vertical acceleration

The effective fluid pressure will be increased or decreaseddue to the effects of vertical acceleration. Similar changesin effective weight of the concrete structure may alsobe considered.

IBC (2003), Section 1617.1.1, and Building Seismic SafetyCouncil (2000), Section 5.2.7, use a factor of 0.2SDS toaccount for the effects of vertical ground acceleration in thedefinition of seismic effects.4.1.4.2 The hydrostatic load qhy from the tankcontents, at level y above the base, shall be multipliedby the spectral acceleration v to account for the effectof the vertical acceleration. The resulting hydrodynamicpressure pvy shall be computed as

pvy = vqhy (4-14)(4-13)Mo Mi Mw Mr+ +( )2 Mc2+=(4-10)Mb Mi Mw Mr+ +( )2 Mc2+=Where applicable, the effect of dynamic earth andgroundwater pressures against the buried portion ofthe walls shall be included in the determination of themoments at the base of the tank.

4.1.4Vertical acceleration

4.1.4.1 The tank shall be designed for the effects ofvertical acceleration. In the absence of a site-specificresponse spectrum, the ratio b of the vertical-to-horizontal acceleration shall not be less than 2/3Overturning moment at the base of the tank, includingthe tank bottom and supporting structure (IBP)

(4-11)

(4-12)

Mi Pi hi=

Mc Pchc=STANDARDportion of the walls shall be included in the determinationof the total base shear V.

4.1.3Moments at base, general equation

The moments due to seismic forces at the base of thetank shall be determined by Eq. (4-10) and (4-13).

Bending moment on the entire tank cross section justabove the base of the tank wall (EBP)

Mw = Pwhw (4-6)

Mr = Prhr (4-7)

Mi = Pihi (4-8)

Mc = Pchc (4-9)COMMENTARY


STANDARDwhere

v (4-15)

where Ct is the seismic response coefficient determinedin accordance with Sections 9.4 and 9.5.

4.2Application of site-specificresponse spectra

4.2.1General

Where site-specific procedures are used, the maximumconsidered earthquake spectral response accelerationshall be taken as the lesser of the probabilistic maximumearthquake spectral response acceleration as definedin Section 4.2.2 and the deterministic maximum spec-tral response acceleration as defined in Section 4.2.3.

4.2.2Probabilistic maximum considered earthquake

CtIbRi----- 0.2SDS=

For = 5%, i = 1.0

When the available site-specific response spectrum is for adamping ratio other than 0.5% of critical, the period-dependent spectral acceleration ScM given by that spectrum maybe modified by the ratio c to account for the influence ofdamping on the spectral amplification as followss i v

i2.302

3.38 0.67 ln------------------------------------=The probabilistic maximum considered earthquakespectral response acceleration shall be taken as thespectral response acceleration represented by a 5%damped acceleration response spectrum having a 2%probability of exceedance in a 50-year period.COMMENTARY

COMMENTARY

R4.2Application of site-specificresponse spectra

R4.2.1General

In locations with Ss 1.5 or S1 0.60 and sites with weak soilconditions, site-specific response spectra are normally used.

R4.2.2Probabilistic maximum considered earthquake

For probabilistic ground motions, a 2% probability of exceed-ance in a 50-year period is equivalent to a recurrenceinterval of approximately 2500 years.

When the available site-specific response spectrum is for adamping ratio other than 5% of critical, the period-depen-dent spectral acceleration SaM given by such site-specificspectrum should be modified by the factor i to account forthe influence of damping on the spectral amplification asfollows (Newmark and Hall 1982)

For 0 seconds < (Ti or Tv) < Ts

For T < (T or T ) < 4.0 seconds

i2.706

4.38 1.04 ln------------------------------------=c3.043

2.73 0.45 ln------------------------------------=


STANDARD COMMENTARY

4.2.3Deterministic maximum considered earthquake

The deterministic maximum considered earthquake

For = 0.5%, c = 1.0

For site-specific response spectra drawn on a tripartitelogarithmic plot, the design spectral acceleration ScM canalso be derived by using the relationship

where SD is the spectral displacement corresponding to Tcobtained directly from the site-specific spectrum in therange Tc > 1.6/Ts.

R4.2.3Deterministic maximum considered earthquake

For deterministic ground motions, the magnitude of acharacteristic earthquake on a given fault should be thebest estimate of the maximum magnitude capable for thatfault, and should not be less than the largest magnitude thathas occurred historically on the fault.

ScM cSD

g------

2

Tc------

2 c1.226SD

Tc2

--------------------= =0.6Fv /T except that the lower limit of the spectralresponse acceleration shall not exceed 1.5Fa. The sitecoefficients Fa and Fv shall be obtained from ASCE 7-05,Tables 11.4-1 and 11.4-2, respectively.

4.2.4 The maximum considered earthquake spectralresponse accelerations SaM and ScM shall be determinedin accordance with Section 9.5 using the site-specificacceleration response spectrum as defined in Section4.2.1.spectral response acceleration at each period shall betaken as 150% of the largest median 5% dampedspectral response acceleration computed at that periodfor characteristic earthquakes on all known active faultswithin the region. The deterministic value of the spectralresponse acceleration shall not be taken lower than


STANDARDTable 4.1.1(a)Importance factor I

Tank use Factor IIII Tanks containing hazardous materials* 1.5

IITanks that are intended to remain usable for emergency purposes after an earthquake, or

tanks that are part of lifeline systems1.25

I Tanks not listed in Categories II or III 1.0*In some cases, for tanks containing hazardous materials, engineering judgmentmay require a factor I > 1.5.

Table 4.1.1(b)Response modification factor R

Type of structure

Ri

Rc

On or above grade Buried*

Anchored, flexible-base tanks 3.25 3.25 1.0Fixed or hinged-base tanks 2.0 3.0 1.0

Unanchored, contained, or uncontained tanks 1.5 2.0 1.0

Pedestal-mounted tanks 2.0 1.0*Buried tank is defined as a tank whose maximum water surface at rest is ator below ground level. For partially buried tanks, the Ri value may be linearlyinterpolated between that shown for tanks on grade and for buried tanks.Ri = 3.25 is the maximum Ri value permitted to be used for any liquid-con-taining concrete structure.Unanchored, uncontained tanks shall not be built in locations where SS 0.75.

SEISMIC DESIGN OF LIQUID-CONTA

STANDARD

CHAPTER 5EARTHQUAKE LOA

(tangential) shear and the rest by radial shear that causes verticalbending. For a tank with a height-to-diameter ratio of 1:4 (D/HL= 4.0), approximately 20% of the earthquake shear force istransmitted by the radial base reaction to vertical bending. Theremaining 80% is transmitted by tangential shear transfer Q. Totransmit this tangential shear Q, a distributed shear force q isrequired at the wall/footing interface, where

5.1General

In the absence of a more rigorous analysis that takesinto account the complex vertical and horizontalvariations in hydrodynamic pressures, liquid-containingstructures shall be designed for the following dynamicshear forces on the basis of the following shear-transfer mechanism:

Walls perpendicular to the direction of the groundmotion being investigated shall be analyzed as slabssubjected to the horizontal pressures computed inSection 5.3. The shears along the bottom and sidejoints and the top joint in case of a roof-covered tankshall correspond to the slab reactions; and

Walls parallel to the direction of the ground motionbeing investigated shall be analyzed as shearwallssubjected to the in-plane forces computed inSection 5.3.

5.2.2Circular tanks

The wall-to-footing and wall-to-roof joints shall bedesigned for the earthquake shear forces.5.2.1Rectangular tanks

The wall-to-floor, wall-to-wall, and wall-to-roof joints ofrectangular tanks shall be designed for the earthquake

shear and pressure distributions in addition to thestatic load distributions:

5.2Shear transferINING CONCRETE STRUCTURES 350.3-21

COMMENTARY

D DISTRIBUTION

R5.2.2Circular tanks

In fixed- and hinged-base circular tanks (Types 2.1 and 2.2), theearthquake base shear is transmitted partially by membrane

R5.2 Shear transfer (NZS 1986)

The horizontal earthquake force V generates shear forcesbetween the wall and footing and the wall and roof.

R5.2.1Rectangular tanks

Typically, the distribution of forces and wall reactions inrectangular tank walls will be similar to that shown inFig. R5.2.1.The distribution is illustrated in Fig. R5.2.2.

The maximum tangential shear occurs at a point on the tankwall oriented 90 degrees to the design earthquake directionbeing evaluated and is given by

qQ

r----- sin=

350.3-22 ACI STANDARD

STANDARD/COMMENTARY

COMMENTARY

The radial shear is created by the flexural response of the wallnear the base, and is therefore proportional to the hydrodynamicforces shown in Fig. R5.2.1. The radial shear attains itsmaximum value at points on the tank wall oriented zeroand 180 degrees to the ground motion and should bedetermined using cylindrical shell theory and the tankdimensions. The design of the wall-footing interfaceshould take the radial shear into account.

In general, the wall-footing interface should have reinforcementdesigned to transmit these shears through the joint.Alternatively, the wall may be located in a preformed slot inthe ring beam footing.

In anchored, flexible-base, circular tanks (Type 2.3(1)) it isassumed that the entire base shear is transmitted by membrane(tangential) shear with only insignificant vertical bending.

Q = 1.0V

In tank Types 2.3(2) and 2.3(3) it is assumed that the baseshear is transmitted by friction only. If friction between thewall base and the footing, or between the wall base and thebearing pads, is insufficient to resist the earthquake shear,some form of mechanical restraint such as dowels, galvanizedsteel cables, or preformed slots may be required.

Failure to provide a means for shear transfer around thecircumference may result in sliding of the wall.

When using preformed slots, vertical bending momentsinduced in the wall by shear should be considered.

The roof-to-wall joint is subject to earthquake shear fromthe horizontal acceleration of the roof. Where dowels areprovided to transfer this shear, the distribution will be thesame as shown in Fig. R5.2.2 with maximum shear given by

where Pr is the force from the horizontal acceleration ofthe roof.

For tanks with roof overhangs, the concrete lip can be

qmaxQ

r-----

0.8V

r-----------= =

qmaxQ

r-----

V

r-----= =

qmax0.8Pr

r-------------=designed to withstand the earthquake force. Because theroof is free to slide on top of the wall, the shear transfer willtake place over that portion of the circumference where thelip overhang comes into contact with the wall. Typically, thedistribution of forces and wall reactions in circular tanks


STANDARD COMMENTARYFig. R5.2.1Hydrodynamic pressure distribution in tankwalls (adapted from Housner [1963] and NZS [1986]).will be similar to that shown in Fig. R5.2.1, but reacting ononly half of the circumference. The maximum reactionforce will be given by

qmax2.0Pr

r-------------=Fig. R5.2.2Membrane shear transfer at the base of circulartanks (adapted from NZS [1986]).


STANDARD5.3Dynamic force distribution above base

5.3.1Rectangular tanks

Walls perpendicular to the ground motion being investi-gated and in the leading half of the tank shall beFig. R5.3.1(a)Vertical force distribution: rectangular tanks.w

edge reactions from the abutting wall(s).

Superimposed on these lateral unbalanced forcesshall be the lateral hydrodynamic force resulting fromthe hydrodynamic pressure due to the effect of verticalacceleration pvy acting on each wall.

5.3.2Combining dynamic forces for rectangulartanks

The hydrodynamic force at any given height y from thebase shall be determined by

(5-1)

Where applicable, the effect of the dynamic earth andgroundwater pressures against the buried portion ofthe walls shall be included.

Py Piy Pwy+( )2 Pcy2 pvyB( )+ 2+=loaded perpendicular to their plane (dimension B ) bythe walls own inertia force Pw , one-half the impulsiveforce Pi , and one-half the convective force Pc .

Walls perpendicular to the ground motion beinginvestigated and in the trailing half of the tank shall beloaded perpendicular to their plane (dimension B) bythe walls own inertia force Pw , one-half the impulsiveforce Pi; one-half the convective force Pc, and thedynamic earth and groundwater pressure against theburied portion of the wall.

Walls parallel to the direction of the ground motionbeing investigated shall be loaded in their plane(dimension L) by: (a) the walls own in-plane inertiaforce P and the in-plane forces corresponding to theThe horizontal distribution of the dynamic pressures acrossthe wall width B is

pvy = vqhy

pwyPwy

B--------= pcy

Pcy

B-------=

piyPiy

B-------=COMMENTARY

COMMENTARYR5.3Dynamic force distribution above base

R5.3.1Rectangular tanks

The vertical distribution, per foot of wall height, of thedynamic forces acting perpendicular to the plane of the wallmay be assumed as shown below (adapted from NZS[1986], Section 2.2.9.5), and Fig. R5.3.1(b).The dynamic force on the leading half of the tank will beadditive to the hydrostatic force on the wall, and the dynamicforce on the trailing half of the tank will reduce the effects ofthe hydrostatic force on the wall.


COMMENTARYFig. R5.3.1(b)Distribution of hydrostatic and hydrodynamic pressures and inertia forces on the wall of a rectangular liquid-containing structure (adapted from Haroun [1984]). (For circular tanks, the vertical distribution of the impulsive and convectiveforces is identical to that shown above for rectangular tanks, while the horizontal distribution varies along the tank circumferenceas shown in Fig. R5.2.1.)


STANDARD COMMENTARY5.3.3Circular tanks R5.3.3Circular tanksThe cylindrical walls of circular tanks shall be loaded bythe walls own inertia force distributed uniformly aroundthe entire circumference; one-half the impulsive forcePi applied symmetrically about = 0 degrees and actingoutward on one half of the walls circumference, and one-half Pi symmetrically about = 180 degrees and actinginward on the opposite half of the walls circumference;one-half the convective force Pc acting on one-half of thewalls circumference symmetrically about = 0 degreesand one-half Pc symmetrically about = 180 degreesand acting inward on the opposite half of the wallscircumference; and the dynamic earth and groundwaterpressure against the trailing half of the buried portion ofthe wall.

Superimposed on these lateral unbalanced forcesshall be the axisymmetric lateral hydrodynamic forceresulting from the hydrodynamic pressure pvy actingon the tank wall.The vertical distribution, per foot of wall height, of thedynamic forces acting on one half of the wall may beassumed as shown below and in Fig. R5.3.3 and Fig. R5.2.1.

Fig. R5.3.3Vertical force distribution: circular tanks.

The horizontal distribution of the dynamic pressure acrossthe tank diameter D may be assumed as follows

pvy = vqhy

pwyPwy

r--------= pcy

16Pcy

9r-------------- cos=

piy2Piy

r---------- cos=


CHAPTER 6STRESSESSTANDARD

6.1Rectangular tanks

The vertical and horizontal bending stresses andshear stresses in the wall and at the wall base due tolateral earthquake forces shall be computed on thebasis of slab action (Sections 5.2 and 5.3) using pressuredistribution consistent with the provisions of Section 5.3.1.

6.2Circular tanks

The vertical bending stresses and shear stresses inthe wall and at the wall base due to lateral earthquakeforces shall be computed on the basis of shell actionusing an acceptable pressure distribution.

Hydrodynamic membrane (hoop) forces in the cylindricalwall corresponding to any liquid level y above the tankbase shall be determined by

(6-1)

and hoop stress

(6-2)

[ in the SI system]

where tw = wall thickness at the level being investigated(liquid level y).

Ny Niy Nwy+( )2 Ncy2 Nhy2+ +=

yNy

12tw------------=

yNytw------=COMMENTARYR6General

In calculating the vertical bending moments in the walls ofrectangular and circular tanks, the boundary conditions atthe wall-to-base and wall-to-roof joints should be properlyaccounted for. Typical earthquake force distributions inwalls of rectangular and circular tanks are presented inR5.3.1 and R5.3.3, respectively.

R6.2Circular tanks

For free-base circular tanks (Type 2.3), the terms in Eq. (6-1)are defined as

Niy = piy r = for (at = 0)

Ncy = pcy r = for (at = 0)

Nwy = pwy r =

Nhy = vQhy

where

Qhy = qhy r

For fixed- or hinged-base circular tanks (Types 2.1 and 2.2),the terms in Eq. (6-1) should be modified to account for theeffects of base restraint. Similarly, the terms in Eq. (6-1)should be modified to account for the restraint of rigid wall-to-roof joints.

2Piy

----------

16Pcy9

--------------

Pwy

--------


Notes


STANDARD COMMENTARY7.1Wave oscillation

Provisions shall be made to accommodate the maximumwave oscillation dmax generated by earthquakeacceleration.

R7.1Wave oscillation

The horizontal earthquake acceleration causes the containedfluid to slosh with vertical displacement of the fluid surface.

The amount of freeboard required in design to accommodatethis sloshing will vary. Where overtopping is tolerable, no

CHAPTER 7FREEBOARDThe maximum vertical displacement dmax to beaccommodated shall be calculated from the followingexpressions:

Rectangular tanks

(7-1)

Circular tanks

(7-2)

where Cc is the seismic response coefficient ascomputed in Section 9.4.

dmaxL2---Cc I=

dmaxD2----Cc I=freeboard provision is necessary. Where loss of liquid shouldbe prevented (for example, tanks for the storage of toxic liquids)or where overtopping may result in scouring of the foundationmaterials or cause damage to pipes, roof, or both, then one ormore of the following measures should be undertaken:

Provide a freeboard allowance; Design the roof structure to resist the resulting uplift

pressures; and/or Provide an overflow spillway.

Where site-specific response spectra are used, the maximumvertical displacement dmax may be calculated from thefollowing expressions:

Rectangular tanks

Circular tanks

where Cc is defined in Section 9.5; and ScM (as in the equationfor Cc), c , and SD are as defined in Section R4.2.2.

dmaxL

2---

CcI( )L

2---

I c( )0.667SD( )

g-------------------------

2

Tc------

2= =

dmaxD

2----

CcI( )D

2----

I c( )0.667SD( )

g-------------------------

2

Tc------

2= =


Notes


STANDARD COMMENTARY8.1General

Dynamic earth pressures shall be taken into accountwhen computing the base shear of a partially or fullyburied liquid-containing structure and when designing

R8.1General

The lateral forces due to the dynamic earth and groundwaterpressures are combined algebraically with the impulsiveforces on the tank as in Eq. (4-5).

CHAPTER 8EARTHQUAKE-INDUCED EARTH PRESSURESthe walls.

The effects of groundwater table, if present, shall beincluded in the calculation of these pressures.

The coefficient of lateral earth pressure at rest Ko shallbe used in estimating the earth pressures unless it isdemonstrated by calculation that the structure deflectssufficiently to lower the coefficient to some valuebetween Ko and the active coefficient of lateral earthpressure Ka.

In a pseudostatic analysis, the resultant of the seismiccomponent of the earth pressure shall be assumed toact at a point 0.6 of the earth height above the base,and when part or all of the structure is below the watertable, the resultant of the incremental increase ingroundwater pressure shall be assumed to act at apoint 1/3 of the water depth above the base.

8.2Limitations

In a buried tank, the dynamic backfill forces shall notbe relied upon to reduce the dynamic effects of thestored liquid or vice versa.

8.3Alternative methods

The provisions of this chapter shall be permitted to besuperseded by recommendations of the project geo-technical engineer that are approved by the buildingofficial having jurisdiction.


Notes

SEISMIC DESIGN OF LIQUID-CONTA

STANDARD9.1General

The dynamic characteristics of ground-supported liquid-containing structures subjected to earthquake accelerationshall be computed in accordance with Sections 9.2, 9.3and 9.5.

The dynamic characteristics of pedestal-mountedliquid-containing structures shall be computed inaccordance with Section 9.7.

of the base acceleration; the overturning moment generated by Piis thus frequently ineffective in tending to overturn the tank.

Wc is the equivalent weight of the oscillating fluid thatproduces the convective pressures on the tank walls withresultant force Pc, which acts at a height of hc above the tank

CHAPTER 9DYNAMIC MODELthese forces can be separated into impulsive and convectiveparts. The impulsive pressures are not impulses in the usualsense but are associated with inertia forces produced byaccelerations of the walls of the container and are directlyproportional to these accelerations. The convective pressuresare those produced by the oscillations of the fluid and aretherefore the consequences of the impulsive pressures.

Figure R9.1 (on p. 43) shows an equivalent dynamic model forcalculating the resultant seismic forces acting on a ground-based fluid container with rigid walls. This model hasbeen accepted by the profession since the early 1960s. Inthis model, Wi represents the resultant effect of the impulsiveseismic pressures on the tank walls. Wc represents theresultant of the sloshing (convective) fluid pressures.

In the model, Wi is rigidly fastened to the tank walls at aheight hi above the tank bottom, which corresponds to thelocation of the resultant impulsive force Pi. Wi moves withthe tank walls as they respond to the ground shaking (thefluid is assumed to be incompressible and the fluid displace-ments small). The impulsive pressures are generated by theseismic accelerations of the tank walls so that the force Pi isevenly divided into a pressure force on the wall accelerating intothe fluid, and a suction force on the wall accelerating away fromthe fluid. During an earthquake, the force Pi changes directionseveral times per second, corresponding to the change in directionINING CONCRETE STRUCTURES 350.3-33

COMMENTARYR9.1General

The lateral seismic pressures and forces determined inaccordance with this Standard are based on vertical tankwalls and vertical wall elements, and the pressures andforces may need to be modified for sloping surfaces.

The following commentary is adapted from Housner (1963):

The design procedures described in Chapter 4 recognizethat the seismic analysis of liquid-containing structuressubjected to a horizontal acceleration should include theinertia forces generated by the acceleration of the structureitself; and the hydrodynamic forces generated by the horizontalacceleration of the contained liquid.

According to Housner (1963), the pressures associated withbottom. In the model, Wc is fastened to the tank walls by springs

/COMMENTARY

COMMENTARY

9.2Rectangular tanks (Type 1)

9.2.1Equivalent weights of accelerating liquid(Fig. 9.2.1 [on p. 44])

(9-1)

(9-2)

9.2.2Height to centers of gravity EBP (Fig. 9.2.2[on p. 45])

For tanks with

WiWL--------

0.866 LHL-------

tanh

0.866 LHL-------

-----------------------------------------------=

WcWL-------- 0.264 L

HL-------

3.16 HLL

------- tanh=

LHL------- 1.333 Ts

[1997 UBC Eq. (30-4) and (34-2)]

For Tc > 1.6/Ts seconds

(B-10)

For Tc 1.6/Ts seconds

Cc

6C

a

Tc

2------=

Pw

CiI

Ri

--------Ww

=

Pr

CiI

Ri

--------Wr

=

Pi

CiI

Ri

--------Wi

=

Pc

CcI

Rc

--------Wc

=

Pi

2.5CaI

Ri

----------------Wi

=

Pi

CvI

RiT

i

----------Wi

0.56CaIW

i=

Pc

6CaI

RcT

c

2------------W

c

1.5 2.5Ca

( )I

Rc

-----------------------------Wc

=(B-11)Pc

1.5CvI

RcT

c

----------------Wc

=


COMMENTARYIn addition, for Seismic Zone 4

[1997 UBC Eq. (34-3)]

10. Total base shear VThe total base shear due to Pw , Pr, Pi , and Pc may be computed by combining these lateral loadsusing the square root of the sum of the squares method as in Section 4.1.2 of this Standard

11. Vertical load distributionThe vertical distribution of the lateral seismic forces may be assumed as shown in Section 5 ofthis Standard;

12. Vertical component of ground motionCompute the natural period of vibration of the vertical liquid motion Tv in accordance withSection 9.3.4 of this Standard.

Compute seismic response coefficient Ct as follows:

For all seismic zones:

For circular tanks

For Tv Ts

Ct = Ca (B-12)

For Tv > Ts

(B-13)

For rectangular tanks, for all periods Tv

Ct = Ca (B-14)

In addition, for Seismic Zone 4:

For rectangular and circular tanks

Ct 1.6ZNv (B-15)

Compute the spectral acceleration v as follows

v = (B-16)

13. Overturning momentsCompute overturning moments for the lateral loads described above using the procedures ofSection 4.1.3 of this Standard. Combine the computed moments using the square root of the sum of the squares method as inthe same section.

Pi

1.6ZNvI

Ri

---------------------Wi

V Pw

Pr

Pi

+ +( )2 Pc

2+=

Ct

Cv

Tv

------=

CtIb

Ri

-----B.4Site-specific spectra (Section 1631.2(2))

When a site-specific design response spectrum is available, the coefficients Ci and Ct are replaced by the actual spectral valuescorresponding to Ti and Tv, respectively, from the 5% damped site-specific spectrum, and Cc is replaced by the actual spectral


COMMENTARYvalue corresponding to Tc from the 0.5% damped site-specific spectrum.

If the site-specific response spectrum does not extend into or is not well defined in the Tc range, coefficient Cc may be calculated usingEq. (B-4), with Cv representing the effective site-specific peak ground acceleration expressed as a fraction of the accelerationdue to gravity g.

B.5Resistance side

1. The resistance side of the seismic design, including load combinations and strength reduction factors, may be computed inaccordance with the applicable provisions of 1997 UBC (ICBO 1997) or ACI 350-06, Chapter 9; and

2. Where the approved standard defines acceptance criteria in terms of allowable stresses (as opposed to strengths), the designseismic forces obtained from this appendix shall be reduced by a factor of 1.4 for use with allowable stresses, and allowablestress increases used in the approved standard are permitted. When such a standard is used, the following load combinationsare permitted to be used for design instead of the ASCE 7-05 load factor combinations (Haroun and Ellaithy 1985).

DL + LL + E/1.4

B.6Freeboard

(B-17)dmax

CcIL or D

2-----------------

=

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Seismic Design of Liquid-Containing Concrete Structuresand Commentary

CONTENTSCHAPTER 1 GENERAL REQUIREMENTS1.1Scope1.2Notation

CHAPTER 2 TYPES OF LIQUID-CONTAINING STRUCTURES2.1Ground- supported structures2.1.1

2.2Pedestal- mounted structures

CHAPTER 3 GENERAL CRITERIA FOR ANALYSIS AND DESIGN3.1Dynamic characteristics3.2Design loads3.3Design requirements3.3.13.3.23.3.3

CHAPTER 4 EARTHQUAKE DESIGN LOADS4.1Earthquake pressures above base4.1.1 Dynamic lateral forces4.1.2Total base shear4.1.3 Moments at base, general equation4.1.4Vertical acceleration4.1.4.14.1.4.2

4.2Application of site-specific response spectra4.2.1General4.2.2 Probabilistic maximum considered earthquake4.2.3 Deterministic maximum considered earthquake4.2.4

CHAPTER 5 EARTHQUAKE LOAD DISTRIBUTION5.1General5.2Shear transfer5.2.1 Rectangular tanks5.2.2Circular tanks

5.3Dynamic force distribution above base5.3.1 Rectangular tanks5.3.2 Combining dynamic forces for rectangular5.3.3Circular tanks

CHAPTER 6 STRESSES6.1Rectangular tanks6.2Circular tanks

CHAPTER 7 FREEBOARD7.1Wave oscillation

CHAPTER 8 EARTHQUAKE-INDUCED EARTH PRESSURES8.1General8.2Limitations8.3Alternative methods

CHAPTER 9 DYNAMIC MODEL9.1General9.2Rectangular tanks (Type 1)9.2.1 Equivalent weights of accelerating liquid (Fig. 9.2.1 [on p. 44])9.2.2Height to centers of gravity EBP (Fig. 9.2.2[on p. 45])9.2.3Heights to center of gravity IBP (Fig. 9.2.3[on p. 46])9.2.4 Dynamic properties

9.3Circular tanks (Type 2)9.3.1 Equivalent weights of accelerating liquid (Fig. 9.3.1 [on p. 48])9.3.2Heights to centers of gravity (EBP [Fig. 9.3.2;on p. 49])9.3.3Heights to center of gravity (IBP [Fig. 9.3.3;on p. 50])9.3.4 Dynamic properties

9.4 Seismic response coefficients9.4.19.4.29.4.3

9.5 Site- specific seismic response9.6Effective mass coefficient9.6.1 Rectangular tanks9.6.2Circular tanks

9.7Pedestal- mounted tanks

CHAPTER 10 COMMENTARY REFERENCESAPPENDIX A DESIGN METHODA.1General outline of design method

APPENDIX B ALTERNATIVE METHOD OF ANALYSIS BASED ONB.1IntroductionB1.1Scope

B1.2Design ground motionsB.2Notation (not included in Section 1.2 of this standard)B.3Loading side, general methodologyB.4Site-specific spectra (Section 1631.2(2))B.5Resistance sideB.6Freeboard

Documents

ACI 350.3-06 Seismic Design of Liquid-Containing Concrete Structures and Commentary