seismic design of liquid-containing concrete structures...

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

First PrintingNovember 2006

ISBN 0-087031-222-7

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Anthony L. Felder David G. Kittridge Jerry Parnes

Seismic Design of Liquid-Containing Concrete Structures

and Commentary (ACI 350.3-06)AN ACI STANDARD

REPORTED BY ACI COMMITTEE 350Environmental Engineering Concrete Structures

SEISMIC DESIGN OF LIQUID-CONTAINING CONCRETE STRUCTURES 350.3-1

Seismic Design of Liquid-ContainingConcrete Structures andCommentary (ACI 350.3-06)

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 ta nks; convective component; earth-quake resistance; environmental concrete structures; impulsive component;liquid-containing structures; rect angular 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 committee’s 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 implem entthe committee’s decision.

The seismic subcommittee’s work was guided by twomain objectives:

1. To produce a self-contained set of p rocedures thatwould enable a practicing engineer to perfo rm a full seismicanalysis and design of a liquid-containing structure. This meantthat these procedures should cover both aspects of seismic

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 the“resistance 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 tanks—rectangular, 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 Chapte r 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 318’s p ractice oflimiting its seismic design provisions to the resistance sideonly; and

2. It makes it easier to update these seismic provisions so as tokeep up with the frequent changes and improvements in thefield of seismic hazard analysis and evaluation.

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 commentar y 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.

ACI 350.3-06 supersedes 350.3-01 a nd 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, th e 1994 Uniform Building Code[UBC] or ACI 350.3-01), the se ismic forces in this standardshould be reduced by the applicable factors to derive comparableforces at allowable stress levels.

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

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

1. Instead of assuming a rigid tank for which the accelerat ionis equal to the ground acceleration at all locations, thisstandard assumes amplification of response due to natura lfrequency 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 ef fects of vertical accelerat ion;and

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


CONTENTS

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

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

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

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

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

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

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

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

CHAPTER 9—DYNAMIC MODEL ........................................................................................... 339.1—General9.2—Rectangular tanks (Type 1)9.3—Circular tanks (Type 2)9.4—Seismic response coefficients Ci , Cc, and Ct9.5—Site-specific seismic response coefficients Ci , Cc, and Ct 9.6—Effective mass coefficient ε9.7—Pedestal-mounted tanks

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


APPENDIX A—DESIGN METHOD...........................................................................................55A.1—General outline of design method

APPENDIX B—ALTERNATIVE METHOD OF ANALYSIS BASED ON 1997 Uniform Building Code...........................................................................57

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


STANDARD COMMENTARY1.1—Scope

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

R1.1—Scope

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 pro vides 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 ef fects of seismic forceson components outside the scope of this standard, such aspiping, equipment (for example, clarifier mechanisms), andconnecting walkways where v ertical or horizontal mo vementsbetween adjoining structures or surrounding backf ill couldadversely influence the ability of the structure to functionproperly (National Science Foundation 1981). Moreover,seismic forces applied at the interf ace of piping or w alkwayswith the structure may also in troduce appreciable fle xural orshear stresses at these connections.

R1.2—Notation

CHAPTER 1—GENERAL REQUIREMENTS

1.2—Notation

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

b = ratio of vertical to hor izontal design accel-eration

B = inside dimension (length or width) of a rectan-gular tank, perpendicular to the direction of theground 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 (ref erto 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 cab le, wire, strand,

or 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 elastomer ic bearing pad,

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

(9.807 m/s2)]

EBP refers to the hydrodynamic design in which it is necess aryto compute the o verturning of the wall with respect to thetank floor , e xcluding base pressure (that is, e xcluding thepressure on the floor itself). EBP h ydrodynamic design isused 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].)

For Cs, refer to “International Building Code (IBC)”(International Code Council 2003), Section 1617.4.


STANDARD COMMENTARY

hc = height abo ve the base of the w all to thecenter of g ravity of the con vective later alforce for the case e xcluding base pressure(EBP), ft (m)

hc′ = height abo ve the base of the w all to thecenter of g ravity of the con vective later alforce for the case including base pressure(IBP), ft (m)

hi = height abo ve the base of the w all to thecenter of g ravity of the impulsiv e later alforce for the case e xcluding base pressure(EBP), ft (m)

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

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

hw = height from the base of the w all 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 eff ects of thetank bottom and supporting structure)

k = flexural stiffness of a unit width of a rectiline artank w all, lb/ft per f oot of w all width (N/mper meter of wall width)

ka = spring constant of the tank w all suppor tsystem, 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 cab le or str andtaken as the sleeve length plus 35 times thestrand diameter, in. (mm)

Lp = length of individual elastomer ic bear ingpads, in. (mm)

m = total mass per unit width of a rectangularwall = mi + mw , lb-s 2/ft per f oot of w allwidth (kg per meter of wall width)

mi = impulsive mass of contained liquid per unitwidth of a rectangular tank w all, lb-s 2/ft perfoot of wall width (kg per meter of wall width)

mw = mass per unit width of a rectangular tankwall, lb-s 2/ft per f oot of w all width (kg permeter of wall width)

Mb = bending moment on the entire tank crosssection 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 con vective force Pc , ft-lb(kN-m)

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

IBP refers to the h ydrodynamic design in which it is necess aryto in vestigate the o verturning 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 b yincreasing the effective vertical moment arm to the appliedforces. (For explanation, refer to Housner [1963].)


STANDARD COMMENTARYMc′ = overturning moment at the base of the tank,

including the tank bottom and suppor tingstructure (IBP), due to the con vective 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 suppor tingstructure (IBP), due to the impulsiv e forcePi, ft-lb (kN-m)

Mo = overturning moment at the base of the tank,including the tank bottom and suppor tingstructure (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 iner tia 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 w all iner tia force Pw, ft-lb(kN-m)

Ncy = in circular tanks, hoop force at liquid level y,due to the con vective component of theaccelerating liquid, lb per f oot of w all height(kN/m)

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

Niy = in circular tanks, hoop force at liquid level y,due to the impulsiv e 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 iner tia force of the acceler atingwall mass, lb per foot of wall height (kN/m)

Ny = in circular tanks , total eff ective hoop f orce atliquid level y, lb per foot of wall height (kN/m)

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)

Pc = total later al con vective f orce associatedwith Wc, lb (kN)

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

Peg = lateral force on the buried portion of a tankwall due to the dynamic ear th and g round-water pressures, lb (kN)

pvy = unit equivalent hydrodynamic pressure dueto the eff ect of v ertical acceleration, at liqu idlevel y, above the base of the tank (pvy = üv× qhy), lb/ft2 (kPa)

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


STANDARD 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, hydros tatic hoop force at liquid

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

r = inside radius of circular tank, ft (m)R = response modification f actor, a n umerical

coefficient representing the combined effect ofthe str ucture’s ductility , energ y-dissipatingcapacity, and str uctural redundancy ( Rc fo rthe convective component of the acceler atingliquid; Ri for the impulsiv e 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 h ydrostatic f orce per unit height ofthe tank w all, occurr ing at liquid le vel y, lbper foot of wall height (kN/m)

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

Piy = lateral impulsiv e f orce due to Wi, per unitheight of the tank w all, occurr ing at liquidlevel y, lb per foot of wall height (kN/m)

Pr = lateral iner tia force of the acceler ating roofWr, lb (kN)

Pw = lateral iner tia force of the acceler ating wallWw, lb (kN)

Pw′ = in a rectangular tank, later al iner tia f orce ofone accelerating wall (Ww′ ), perpendicular tothe direction of the earthquake force, lb (kN)

Pwy = lateral iner tia f orce due to Ww, per unitheight of the tank w all, occurring at level yabove the tank base , lb per f oot of w allheight (kN/m)

Py = combined hor izontal f orce (due to theimpulsive and con vective components ofthe acceler ating liquid; the w all’s iner tia,and the h ydrodynamic pressure due to thevertical acceler ation) at a height y abo vethe tank base, lb per foot of wall height (kN/m)

qhy = unit hydrostatic pressure at liquid level y abovethe tank base [qhy = γL (HL – y)], lb/ft2 (kPa)

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 earthquak e;expressed as a fraction of t he acceleration due togravity g from a site-specif ic spectrum. ( S0 isequivalent to a PGA having a 2% probability ofexceedance in 50 years, as gi ven in the U.S. Geo-logical Survey (USGS) database at website (http://eqhazmaps.usgs.gov)


STANDARD COMMENTARYS1 = mapped maxim um considered ear thquake

5% damped spectr al response acceler ation;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

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

SaM = maximum considered ear thquake spectr alresponse acceler ation, 5% damped, atperiod Ti or Tv , tak en from a site-specificacceleration response spectrum

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

ScM = maximum considered ear thquake spectr alresponse acceler ation, 0.5% damped, atperiod Tc , tak en from a site-specificacceleration response spectrum

TS: In Appendix B,

where Ca and Cv are defined in Appendix B.

Ts

Cv2.5Ca--------------- 0.40

CvCa-------= =

SD1 = design spectral response acceler ation, 5%damped, at a per iod of 1 second, asdefined in 9.4.1, expressed as a fr actionof the acceleration due to gravity g

SDS = design spectral response acceler ation, 5%damped, at shor t per iods, as defined in9.4.1, expressed as a fr action of the accel-eration due to gravity g

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

Ss = mapped maxim um considered ear thquake5% 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 elastomer ic bear ing 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 impulsiv e component of thecontents), s

TS = SD1/SDS


STANDARD COMMENTARYTv = natural per iod of vibr ation of v ertical liquid

motion, süv = effective spectr al acceler ation from an

inelastic v ertical response spectr um, asdefined b y Eq. (4-15) , that is der ived b yscaling from an elastic hor izontal responsespectrum, e xpressed as a fr action of theacceleration due to gravity g

V = total horizontal base shear, lb (kN)wp = width of elastomeric bearing pad, in. (mm)Wc = equivalent w eight of the con vective comp o-

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

(walls and roof) [We = (εWw + Wr)], lb (kN)Wi = equivalent weight of the impulsive component

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

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

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

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

Ww′ = in a rectangular tank, the equiv alentweight of one w all per pendicular to thedirection of the earthquake force, lb (kN)

y = liquid level at which the w all is being in vesti-gated (measured from tank base), ft (m)

α = angle of base cab le or strand with hor izontal,degree

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

for standard-weight concrete]γL = density of contained liquid, lb/ft3 (kN/m3)γw = density of water, 62.43 lb/ft3 (9.807 kN/m3)ε = effective mass coefficient (r atio of equiv alent

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

ηc , ηi = coefficients as defined in Section R4.2

For θ, refer to Fig. R5.2.1 and R5.2.2.θ = polar coordinate angle, degreeλ = coefficient as defined in 9.2.4 and 9.3.4σy = membrane (hoop) stress in w all of circular

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

(convective) mode of sloshing, radian/sωi = circular frequency of the impulsive mode of

vibration, radian/s


STANDARD COMMENTARY2.1—Ground-supported structures

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

2.1.1—Ground-supported liquid-containing str ucturesare 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 1—Rectangular tanksType 1.1—Fixed baseType 1.2—Hinged base

Type 2—Circular tanksType 2.1—Fixed base

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

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

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

2.2—Pedestal-mounted structures

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

R2.1—Ground-supported structures

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

CHAPTER 2—TYPES OF LIQUID-CONTAINING STRUCTURES

R2.1.1—The classif ications 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. W ith 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 standa rd, however, does not co verthe determination of seismic forces on the piles themselves.

The tank roof may be a free-span dome or column-supp ortedflat slab, or the tank may be open-topped.

Fig. R2.1—Typical tank configurations (adapted from ASCE[1984]).


COMMENTARY

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


STANDARD COMMENTARY3.1—Dynamic characteristics

The dynamic char acteristics of liquid-containingstructures shall be der ived in accordance with eitherChapter 9 or a more r igorous dynamic analysis thataccounts for the interaction between the structure andthe contained liquid.

3.2—Design loads

The loads gener ated by the design ear thquake shallbe computed in accordance with Chapter 4.

3.3—Design requirements

3.3.1—The 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 eff ects of theapplicable design static loads.

3.3.2—With regards to the hor izontal acceleration, thedesign shall take into account the effects of the transferof the total base shear between the wall and the footingand betw een the w all and the roof , and the dynamicpressure acting on the wall above the base.

3.3.3—Effects of maxim um hor izontal and v erticalacceleration shall be combined b y the square-root-sum-of-the-squares method.

CHAPTER 3—GENERAL CRITERIA FOR ANALYSIS AND DESIGN

R3.1—Dynamic characteristics

For an outline of the general steps in volved in the intera ctionbetween the structure and the contained liquid, refer toAppendix A.


Notes


STANDARD COMMENTARY4.1—Earthquake pressures above base

The w alls of liquid-containing str uctures 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 impulsiv e force Pi from the conta inedliquid;(c) Hydrodynamic convective force Pc from the containedliquid;(d) Dynamic ear th pressure from satur ated andunsaturated soils against the b uried por tion of thewall; and(e) The effects of vertical acceleration.

R4.1—Earthquake pressures above base

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

A more detailed discussion of the impulsi ve and convectivecomponents Wi and Wc is contained in Section R9.1.

The imposed ground motion is represented b y an elasticresponse spectrum that is either deri ved from an actualearthquake record for the site, or is constructed b y analogyto sites with kno wn soil and seismic characteristics. Theprofile of the response spectrum is defined by Sa, which is afunction of the period of vibration and the mapped accelerationsSS and S1 as described in Section R9.4.

Factor I provides a means for the engineer to increase thefactor of safety for the cate gories of structures described inTable 4.1.1(a). Engineering judgment may require a factor Igreater than tabulated in Table 4.1.1(a) where it is necess ary toreduce further the potential le vel 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 structure’ s ductility ,energy-dissipating properties, and redundanc y (A CI 350-06,Section R21.2.1).

Explanations of the impulsive and convective pressures Piand Pc are contained in Section R9.1 and Housner (1963).

CHAPTER 4—EARTHQUAKE DESIGN LOADS


STANDARD COMMENTARY4.1.1—Dynamic lateral forces

The dynamic later al f orces abo ve the base shall bedetermined as

(4-1)

(4-1a)

(4-2)

(4-3)

(4-4)

where Ci and Cc are the seismic response coefficientsdetermined in accordance with Sections 9.4 and9.5; I is the impor tance factor defined in Table 4.1.1(a);Ww and Wr are the weights of the cylindr ical tank wall(shell) and tank roof , respectiv ely, and Ww′ is theweight of one wall in a rectangular tank, perpendicularto the direction of the ground motion being investigated; εis a f actor defined in Section 1.2 and deter mined inaccordance with Section 9.6 ; Wi and Wc are theimpulsive and con vective components of the storedliquid, respectiv ely, 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 deter mined inaccordance with Table 4.1.1(b).

Where applicab le, the later al f orces due to thedynamic earth and groundwater pressures against theburied por tion of the w alls shall be computed inaccordance with the provisions of Chapter 8.

4.1.2—Total base shear

The base shear due to seismic f orces applied at thebottom of the tank wall shall be determined by

(4-5)

Where applicab le, the later al f orces due to dynamicearth and g roundwater pressures against the b uried

Pw Ci I=εWw

Ri------------

Pw′ CiIεWw′

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

Pr CiIWrRi-------=

Pi CiI=WiRi-------

Pc CcIWcRc--------=

V Pi Pw Pr+ +( )2 P 2c P 2

eg+ +=

R4.1.1—Dynamic lateral forces

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

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


STANDARD COMMENTARYportion of the walls shall be included in the deter minationof the total base shear V.

4.1.3—Moments 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)

(4-10)

Overturning moment at the base of the tank, includingthe tank bottom and supporting structure (IBP)

(4-11)

(4-12)

(4-13)

Where applicab le, the eff ect of dynamic ear th andgroundwater pressures against the b uried por tion ofthe walls shall be included in the deter mination of themoments at the base of the tank.

4.1.4—Vertical acceleration

4.1.4.1 The tank shall be designed f or the effects ofvertical acceleration. In the absence of a site-specificresponse spectr um, the r atio b of the v ertical-to-horizontal acceleration shall not be less than 2/3

4.1.4.2 The h ydrostatic load qhy from the tankcontents, at level y above the base, shall be multipliedby the spectral acceleration üv to account for the effectof the v ertical acceleration. The resulting h ydrodynamicpressure pvy shall be computed as

pvy = üvqhy (4-14)

Mb Mi Mw Mr+ +( )2 Mc2+=

Mi′ Pi hi′=

Mc′ Pchc′=

Mo Mi′ Mw Mr+ +( )2 M′c2+=

R4.1.4—Vertical acceleration

The effective fluid pressure will be increased or decreaseddue to the ef fects of v ertical acceleration. Similar changesin ef fective 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 f actor of 0.2SDS toaccount for the effects of vertical ground acceleration in thedefinition of seismic effects.


STANDARD COMMENTARYwhere

üv (4-15)

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

4.2—Application of site-specificresponse spectra

4.2.1—General

Where site-specific procedures are used, the maximumconsidered ear thquake spectr al response acceler ationshall be taken as the lesser of the probabilistic maxim umearthquake spectral response a cceleration as definedin Section 4.2.2 and the deterministic maximum spec-tral response acceleration as defined in Section 4.2.3.

4.2.2—Probabilistic maximum considered earthquake

The probabilistic maxim um considered ear thquakespectral response acceler ation shall be tak en as thespectral response acceler ation represented b y a 5%damped acceleration response spectrum having a 2%probability of exceedance in a 50-year period.

CtIbRi----- 0.2SDS≥=

R4.2—Application of site-specificresponse spectra

R4.2.1—General

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

R4.2.2—Probabilistic maximum considered earthquake

For probabilistic ground motions, a 2% probability of e xceed-ance in a 50-year period is equi valent to a recurrenceinterval of approximately 2500 years.

When the available site-specific response spectrum is for adamping ratio β other than 5% of critical, the period-depen-dent spectral acceleration SaM g iven by s uch s ite-specificspectrum should be modified by the factor η i to account forthe influence of damping on the spectral amplif ication asfollows (Newmark and Hall 1982)

For 0 seconds < (Ti or Tv) < Ts

For Ts < (Ti or Tv) < 4.0 seconds

For β = 5%, ηi = 1.0

When the available site-specific response spectrum is for adamping ratio β other than 0.5% of critical, the period-dependent spectral acceleration ScM given by that spectrum maybe modif ied b y the ratio ηc to account for the influence ofdamping on the spectral amplification as follows

ηi2.706

4.38 1.04 βln–------------------------------------=

ηi2.302

3.38 0.67 βln–------------------------------------=

ηc3.043

2.73 0.45 βln–------------------------------------=


STANDARD COMMENTARY

4.2.3—Deterministic maximum considered earthquake

The deter ministic maxim um considered ear thquakespectral response acceler ation at each per iod shall betaken as 150% of the largest median 5% dampedspectral response acceleration computed at that per iodfor characteristic earthquakes on all kno wn active faultswithin the region. The deterministic value of the spectralresponse acceler ation shall not be tak en lo wer than0.6Fv /T e xcept that the lo wer limit of the spectr alresponse acceleration shall not e xceed 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 maxim um considered ear thquake spectr alresponse accelerations SaM and ScM shall be determinedin accordance with Section 9.5 using the site-specificacceleration response spectr um as defined in Section4.2.1.

For β = 0.5%, ηc = 1.0

For site-specif ic response spectra dra wn 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-specif ic spectrum in therange Tc > 1.6/Ts.

R4.2.3—Deterministic 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 ηc

SD

g------ 2π

Tc

------⎝ ⎠⎛ ⎞ 2

ηc

1.226SD

Tc2

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


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 whos e 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.


STANDARD COMMENTARY

CHAPTER 5—EARTHQUAKE LOAD DISTRIBUTION

5.2.1—Rectangular tanks

The wall-to-floor, wall-to-wall, and wall-to-roof joints ofrectangular tanks shall be designed for the earthquakeshear f orces on the basis of the f ollowing shear-transfer mechanism:

• Walls per pendicular to the direction of the g roundmotion being investigated shall be analyzed as slabssubjected to the hor izontal 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 f orces computed inSection 5.3.

5.2.2—Circular tanks

The w all-to-footing and w all-to-roof joints shall bedesigned for the earthquake shear forces.

R5.2.2—Circular tanks

In fixed- and hinged-base circular tanks (Types 2.1 and 2.2), theearthquake base shear is transmitted partially b y membrane(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 th e earthquak e shear force istransmitted by the radial base reaction to v ertical bending. Theremaining 80% is transmitted by tangential shear transfer Q. Totransmit this tangential shear Q, a distrib uted shear force q isrequired at the wall/footing interface, where

The distribution is illustrated in Fig. R5.2.2.

The maximum tangential shear occurs at a point on the tankwall oriented 90 de grees to the design earthquak e directionbeing evaluated and is given by

qQ

πr----- θsin=

R5.2— Shear transfer (NZS 1986)

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

R5.2.1—Rectangular tanks

Typically, the distribution of forces and wall reactions inrectangular tank w alls will be similar to that sho wn inFig. R5.2.1.

5.1—General

In the absence of a more rigorous analysis that takesinto account the complex vertical and horizontalvariations in hydrodynamic pressures, liquid-containi ngstructures shall be designed for the following dynamicshear and pressure distributions in addition to thestatic load distributions:

5.2—Shear transfer


STANDARD COMMENTARY

The radial shear is created b y the flexural response of the w allnear 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 w all oriented zeroand 180 de grees to the ground motion and should bedetermined using c ylindrical shell theory and the tankdimensions. The design of the w all-footing interf aceshould 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, fle xible-base, circular tanks (T ype 2.3(1)) it isassumed that the entire base sh ear is transmitted b y 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 b y friction only. If friction between thewall base and the footing, or between the w all base and thebearing pads, is insuf ficient to resist the earthquak e shear,some form of mechanical restraint such as do wels, galvanizedsteel cables, or preformed slots may be required.

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

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

The roof-to-w all joint is subj ect to earthquak e shear fromthe horizontal acceleration of the roof. Where do wels areprovided to transfer this shear , the distrib ution 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 o verhangs, the concrete lip can bedesigned to withstand the earthquak e 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 w all reactions in circular tanks

qmaxQ

πr----- 0.8V

πr-----------= =

qmaxQ

πr----- V

πr-----= =

qmax

0.8Pr

πr-------------=


STANDARD COMMENTARYwill be similar to that sho wn in Fig. R5.2.1, but reacting ononly half of the circumference. The maximum reactionforce will be given by

qmax

2.0Pr

πr-------------=

Fig. R5.2.1—Hydr odynamic pr essure distrib ution in tankwalls (adapted from Housner [1963] and NZS [1986]).

Fig. R5.2.2—Membrane shear tr ansfer at the base of cir culartanks (adapted from NZS [1986]).


STANDARD COMMENTARY5.3—Dynamic force distribution above base


Walls per pendicular to the g round motion being in vesti-gated and in the leading half of the tank shall beloaded perpendicular to their plane (dimension B ) bythe wall’s own inertia force Pw′ , one-half the impulsiveforce Pi , and one-half the convective force Pc .

Walls per pendicular to the g round motion beinginvestigated and in the trailing half of the tank shall beloaded perpendicular to their plane (dimension B) bythe wall’s own inertia force Pw′ , one-half the impulsiveforce Pi; one-half the con vective f orce Pc, and thedynamic ear th and g roundwater pressure against theburied portion of the wall.

Walls par allel to the dire ction of the g round motionbeing in vestigated shall be loaded in their plane(dimension L) b y: (a) the w all’s o wn in-plane iner tiaforce Pw′ and the in-plane forces corresponding to theedge reactions from the abutting wall(s).

Superimposed on these later al unbalanced f orcesshall be the later al hydrodynamic force resulting fromthe hydrodynamic pressure due to the effect of verticalacceleration pvy acting on each wall.

5.3.2—Combining d ynamic f orces f or rectangulartanks

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

(5-1)

Where applicable, the effect of the dynamic ear th andgroundwater pressures against the b uried por tion ofthe walls shall be included.

Py Piy Pwy+( )2 Pcy2 pvyB( )+ 2+=

R5.3—Dynamic force distribution above base

R5.3.1—Rectangular tanks

The v ertical distrib ution, per foot of w all height, of thedynamic forces acting perpendicular to the plane of the wallmay be assumed as sho wn belo w (adapted from NZS[1986], Section 2.2.9.5), and Fig. R5.3.1(b).

The horizontal distribution of the dynamic pressures acrossthe wall width B is

pvy = üvqhy

The dynamic force on the leadi ng half of the tank will beadditive to the h ydrostatic force on the w all, and the dynamicforce on the trailing half of the tank will reduce the e ffects ofthe hydrostatic force on the wall.

pwyPwy

B--------= pcy

Pcy

B-------=

piyPiy

B-------=

Fig. R5.3.1(a)—Vertical force distribution: rectangular tanks.


COMMENTARY

Fig. 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 COMMENTARY5.3.3—Circular tanks

The cylindrical walls of circular tanks shall be loaded bythe wall’s own inertia force distributed uniformly aroundthe entire circumf erence; one-half the impulsiv e forcePi applied symmetr ically about θ = 0 degrees and actingoutward on one half of the wall’s circumference, and one-half Pi symmetr ically about θ = 180 degrees and actinginward on the opposite half of the w all’s circumference;one-half the convective force Pc acting on one-half of thewall’s circumference symmetr ically about θ = 0 degreesand one-half Pc symmetrically about θ = 180 degreesand acting inw ard on the o pposite half of the w all’scircumference; and the dynamic ear th and groundwaterpressure against the trailing half of the buried portion ofthe wall.

Superimposed on these later al unbalanced f orcesshall be the axisymmetr ic lateral hydrodynamic forceresulting from the h ydrodynamic pressure pvy actingon the tank wall.

R5.3.3—Circular tanks

The v ertical distrib ution, per foot of w all height, of thedynamic forces acting on one half of the w all may beassumed as shown below and in Fig. R5.3.3 and Fig. R5.2.1.

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

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

pvy = üvqhy

pwyPwy

πr--------= pcy

16Pcy

9πr-------------- θcos=

piy2Piy

πr---------- θcos=


STANDARD COMMENTARY

6.1—Rectangular tanks

The v ertical and hor izontal bending stresses andshear stresses in the w all and at the w all base due tolateral ear thquake f orces 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.

6.2—Circular tanks

The v ertical bending stress es 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 an y 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 le vel being investigated(liquid level y).

Ny Niy Nwy+( )2 Ncy2 Nhy

2+ +=

σyNy

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

σyNytw------=

R6—General

In calculating the v ertical bending moments in the w alls ofrectangular and circular tank s, the boundary conditions atthe wall-to-base and w all-to-roof joints should be properlyaccounted for . T ypical earthquak e force distrib utions inwalls of rectangular and circular tanks are presented inR5.3.1 and R5.3.3, respectively.

CHAPTER 6—STRESSES

R6.2—Circular tanks

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

Niy = piy r = for (at θ = 0)

Ncy = pcy r = for (at θ = 0)

Nwy = pwy r =

Nhy = üvQhy

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

π----------

16Pcy

9π--------------

Pwy

π--------


Notes


STANDARD COMMENTARY7.1—Wave oscillation

Provisions shall be made to accommodate the maxim umwave oscillation dmax gener ated b y ear thquakeacceleration.

The maxim um v ertical displacement dmax to beaccommodated shall be calculated from the f ollowingexpressions:

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=

R7.1—Wave 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 v ary. Wh ere o vertopping is tolerable, nofreeboard 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 pi pes, 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), ηc , and SD are as defined in Section R4.2.2.

dmaxL

2---⎝ ⎠

⎛ ⎞ CcI( ) L

2---⎝ ⎠

⎛ ⎞ I ηc( )0.667SD( )

g------------------------- 2π

Tc

------⎝ ⎠⎛ ⎞ 2

= =

dmaxD

2----⎝ ⎠

⎛ ⎞ CcI( ) D

2----⎝ ⎠

⎛ ⎞ I ηc( )0.667SD( )

g------------------------- 2π

Tc

------⎝ ⎠⎛ ⎞ 2

= =

CHAPTER 7—FREEBOARD


Notes


STANDARD COMMENTARY8.1—General

Dynamic ear th pressures sha ll be tak en into accountwhen computing the base shear of a par tially or fullyburied liquid-containing str ucture and when designingthe walls.

The effects of g roundwater tab le, if present, shall beincluded in the calculation of these pressures.

The coefficient of lateral earth pressure at rest Ko shallbe used in estimating the ear th pressures unless it isdemonstrated by calculation that the structure deflectssufficiently to lo wer the coefficient to some v aluebetween Ko and the activ e coefficient of later al ear thpressure Ka.

In a pseudostatic analysis, the resultant of the seismiccomponent of the ear th pressure shall be assumed toact at a point 0.6 of the ear th height abo ve 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.

8.2—Limitations

In a b uried tank, the dynamic bac kfill forces shall notbe relied upon to reduce t he dynamic eff ects of thestored liquid or vice versa.

8.3—Alternative methods

The provisions of this chapter shall be permitted to besuperseded by recommendations of the project geo-technical engineer that are appro ved b y the b uildingofficial having jurisdiction.

R8.1—General

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

CHAPTER 8—EARTHQUAKE-INDUCED EARTH PRESSURES


Notes


STANDARD COMMENTARY9.1—General

The dynamic char acteristics of g round-supported liquid-containing structures subjected to ear thquake accelerationshall be computed in accordan ce with Sections 9.2 , 9.3and 9.5.

The dynamic char acteristics of pedestal-mountedliquid-containing str uctures shall be computed inaccordance with Section 9.7.

R9.1—General

The lateral seismic pressures and forces determined inaccordance with this Standard are based on v ertical tankwalls and v ertical w all 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 b y the acceleration of the structureitself; and the hydrodynamic forces generated by the horizontalacceleration of the contained liquid.

According to Housner (1963), the pressures associated withthese forces “can be separated into impulsive and convectiveparts. The impulsive pressures are not impulses in the usualsense b ut are associated with inertia forces produced b yaccelerations of the w alls of the container and are directlyproportional to these accelerations. The con vective press uresare those produced b y the osci llations 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 w alls. This model hasbeen accepted by the profession since the early 1960s. Inthis model, Wi represents the resultant ef fect of the impuls iveseismic pressures on the tank w alls. Wc represents theresultant of the sloshing (convective) fluid pressures.

In the model, Wi is rigidly f astened to the tank w alls at aheight hi above the tank bottom, which corresponds to thelocation of the resultant impulsi ve force Pi. Wi moves withthe tank w alls as the y respond to the ground shaking (thefluid is assumed to be incompressible and the fluid displace-ments small). The impulsive pressures are generated b y theseismic accelerations of the tank w alls so that the force Pi isevenly divided into a pressure force on the w all accelerating intothe fluid, and a suction force on the w all accelerating away fromthe fluid. During an earthquak e, the force Pi changes directionseveral times per second, corresponding to the change in directionof the base acceleration; the overturning moment generated by Piis thus frequently ineffective in tending to overturn the tank.

Wc is the equi valent weight of the oscillating fluid thatproduces the con vective pre ssures on the tank w alls withresultant force Pc, which acts at a height of hc above the tankbottom. In the model, Wc is fastened to the tank walls by springs

CHAPTER 9—DYNAMIC MODEL


STANDARD COMMENTARY

9.2—Rectangular tanks (Type 1)

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

(9-1)

(9-2)

9.2.2—Height to center s of gra vity 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<

that produce a period of vibration corresponding to the period offluid sloshing. The sloshing pr essures on the tank w alls resultfrom the fluid motion associated with the w ave oscillation. Theperiod of oscillation of the sloshing depends on the ratio of fluiddepth to tank diameter , and is usually se veral seconds. Theoverturning moment e xerted by Pc (Fig. R9.1 [on p. 43])acts for a sufficient time to tend to uplift the tank wall if there isinsufficient restraining weight. The forces Pi and Pc actindependently and simultaneous ly on the tank. The force Pi(and its associated pressures) primarily act to stress the tankwall, whereas Pc acts primarily to uplift the tank w all.The vertical vibrations of the ground are also transmitted to thefluid, thus producing pressures that act on the tank w alls. Theyact to increase or decrease the hoop stresses.

The pressures and forces on a cylindrical tank are similar to, butnot the same as, those acting on a rectangular tank.

The rapid fluctuations of the force Pi mean that the bendingmoments and stresses in the wall of a rectangular tank also varyrapidly (the effect is not like a constant force acting on the wall).The duration of the fluctuations is 10 to 15 seconds for earth-quakes of magnitude 6.5 to 7.5.

The force Pc fluctuates sinusoidally with a period of vibrationthat depends on the dimensions of the tank, and can be se veralseconds or longer. The duration of sloshing can be 20 to40 seconds for earthquakes of magnitude 6.5 to 7.5. Note thatthe damping of the sloshing w ater is small: approximately0.5 to 1% of critical damping. The sloshing increases anddecreases the fluid pressure on the w all. Normally, this issmaller than the impulsive effect, but if there is not enoughdead load, the tank will tend to uplift.

R9.2—Rectangular tanks (Type 1)

All equations in Section 9.2 e xcept Eq. (9-9), (9-10), and (9-11)were originally developed by Housner (1963), and subsequentlyused by other authors (Housner 1956; NZS 1986; Haroun 1984;ASCE 1981; V eletsos and Shi vakumar 1997; ANSI/A WWA1995a,b; Haroun and Ellaithy 1985).

Equations (9-9), (9-10), and (9-11) were adapted from NZS(1986).


STANDARD COMMENTARY

R9.2.4—Dynamic properties

The following equations are pr ovided as e xamples for thespecial case of a wall of uniform thickness. (Note that massis equal to weight divided by the acceleration due to gravity.)

mw Hwtw

12------

γc

g----⎝ ⎠

⎛ ⎞=

(9-3)

For tanks with

(9-4)

For all tanks

(9-5)

9.2.3—Heights to center of gra vity IBP ( Fig. 9.2.3[on p. 46])

For tanks with

(9-6)

For tanks with

(9-7)

For all tanks

(9-8)

9.2.4—Dynamic properties

The str uctural stiffness k shall be computed on thebasis of correct boundary conditions.

(9-9)

m = mw + mi (9-10)

hiHL------- 0.5 0.09375 L

HL-------⎝ ⎠

⎛ ⎞–=

LHL------- 1.333≥

hiHL------- 0.375=

hcHL------- 1

3.16 HLL

-------⎝ ⎠⎛ ⎞ 1–cosh

3.16 HLL

-------⎝ ⎠⎛ ⎞ 3.16 HL

L-------⎝ ⎠

⎛ ⎞sinh--------------------------------------------------------------------–=

LHL------- 0.75<

hi′

HL------- 0.45=

LHL------- 0.75≥

hi′

HL-------

0.866 LHL-------⎝ ⎠

⎛ ⎞

2 0.866 LHL-------⎝ ⎠

⎛ ⎞tanh--------------------------------------------------- 1

8---–=

hc′

HL-------- 1

3.16 HLL

-------⎝ ⎠⎛ ⎞ 2.01–cosh

3.16 HLL

-------⎝ ⎠⎛ ⎞ 3.16 HL

L-------⎝ ⎠

⎛ ⎞sinh--------------------------------------------------------------------–=

ωikm-----=


STANDARD COMMENTARY

(9-11)

(9-12)

where

(9-13)

(9-14)

Ti2π

ωi------ 2π

mk-----= =

ωcλ

L-------=

λ 3.16g 3.16 HLL

-------⎝ ⎠⎛ ⎞tanh=

Tc2π

ωc------ 2π

λ------⎝ ⎠

⎛ ⎞ L= =

2π

λ------⎝ ⎠

⎛ ⎞ from Fig. 9.2.4

[ in the SI system]

where hw = 0.5Hw, and hi is obtained from Eq. (9-3) and (9-4),and Fig. 9.2.2 (on p. 44).

For w alls of nonuniform thickness, special analysis isrequired to determine mw, mi, and h.

For fixed-base, free-top cantilever walls, such as in open-toptanks, flexural stiffness for a unit width of w all k may beapproximated using the following equation

[ in the SI system]

Flexural stiffness formulas may be developed for other wallsupport conditions. Such spring constants will generally fallwithin the low period range (less than about 0.3 seconds) fortanks of normal proportions.

As an alternati ve to computing the natural period of vibration,particularly for end conditions other than cantile ver, it isreasonable to assume the wall rigid. In such a case, Eq. (9-32)may be conserv atively used to calculate the impulsi veforces regardless of the actual boundary conditions of thestructure or structural components being analyzed.

R9.3—Circular tanks (Type 2)

All equations in Section 9.3, e xcept Eq. (9-23) through(9-28), were originally de veloped by Housner (1963), andsubsequently used by other authors (Housner 1956; NZS 1986;Haroun 1984; ASCE 1981; V eletsos and Shi vakumar 1997;ANSI/AWWA 1995a,b; Haroun and Ellaithy 1985).

Equations (9-23) through (9-28) were adapted from NZS(1986).

mw Hw

tw

103

--------γc

g----⎝ ⎠

⎛ ⎞=

mi

Wi

WL

--------⎝ ⎠⎛ ⎞ L

2---⎝ ⎠

⎛ ⎞HL

γL

g-----⎝ ⎠

⎛ ⎞=

hhwmw himi+( )mw mi+( )

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

kEc

48------

tw

h----⎝ ⎠

⎛ ⎞3

=

kEc

4 103×-----------------

tw

h----⎝ ⎠

⎛ ⎞3

=

9.3—Circular tanks (Type 2)

9.3.1—Equivalent weights of accelerating liquid(Fig. 9.3.1 [on p. 48])

(9-15)

(9-16)

WiWL--------

0.866 DHL-------⎝ ⎠

⎛ ⎞tanh

0.866 DHL-------⎝ ⎠

⎛ ⎞-----------------------------------------------=

WcWL-------- 0.230 D

HL-------⎝ ⎠

⎛ ⎞ 3.68 HLD-------⎝ ⎠

⎛ ⎞tanh=


STANDARD COMMENTARY

R9.3.4—Dynamic properties

Equations (9-23) and (9-24) are adapted from ASCE (1981)and Veletsos and Shivakumar (1997).

Equations (9-26) and (9-27) are adapted from ANSI/AWWA(1995a,b).

9.3.2—Heights to center s of gra vity (EBP [ Fig. 9.3.2;on p. 49])

For tanks with

(9-17)

For tanks with

(9-18)

For all tanks

(9-19)

9.3.3—Heights to center of gra vity (IBP [Fig. 9.3.3;on p. 50])

For tanks with

(9-20)

For tanks with

(9-21)

(9-22)

9.3.4—Dynamic properties

Ti: For tank Types 2.1 and 2.2:

(9-23)

DHL------- 1.333<

hiHL------- 0.5 0.09375 D

HL-------⎝ ⎠

⎛ ⎞–=

DHL------- 1.333≥

hiHL------- 0.375=

hcHL------- 1

3.68 HLD-------⎝ ⎠

⎛ ⎞ 1–cosh

3.68 HLD-------⎝ ⎠

⎛ ⎞ 3.68 HLD-------⎝ ⎠

⎛ ⎞sinh--------------------------------------------------------------------–=

DHL------- 0.75<

hi′

HL------- 0.45=

DHL------- 0.75≥

hi′

HL-------

0.866 DHL-------⎝ ⎠

⎛ ⎞

2 0.866 DHL-------⎝ ⎠

⎛ ⎞tanh--------------------------------------------------- 1

8---–=

hc′

HL-------- 1

3.68 HLD-------⎝ ⎠

⎛ ⎞ 2.01–cosh

3.68 HLD-------⎝ ⎠

⎛ ⎞ 3.68 HLD-------⎝ ⎠

⎛ ⎞sinh--------------------------------------------------------------------–=

ωi Cl12HL------- Ec

gγc-----=


STANDARD COMMENTARY

[ in the SI system]

(9-24)

[Cw from Fig. 9.3.4(a); on p. 51]

[ in the SI system]

(9-25)

For tank Type 2.3

(9-26)

but shall not exceed 1.25 seconds.

(9-27)

[in the SI system]

Tc:

(9-28)

where

(9-29)

(9-30)

[ from Fig. 9.3.4(b); on p. 52]

Tv: For circular tanks

(9-31)

ωi Cl1

HL------- 103Ec

gγc-----=

Cl Cw10 tw12r---------=

Cl Cwtw

10r--------- =

Ti2π

ωi------=

Ti8π Ww Wr Wi+ +( )

gDka-------------------------------------------------=

ka 144 AsEs α2cos

LcSc-------------------------------

⎝ ⎠⎜ ⎟⎛ ⎞ 2GpwpLp

tpSp-------------------------⎝ ⎠

⎛ ⎞+=

ka 103 AsEs α2cos

LcSc-------------------------------⎝ ⎠⎜ ⎟⎛ ⎞ 2GpwpLp

tpSp-------------------------⎝ ⎠⎜ ⎟⎛ ⎞

+=

ωcλ

D--------=

λ 3.68g 3.68 HLD-------⎝ ⎠

⎛ ⎞tanh=

Tc2π

ωc------ 2π

λ------⎝ ⎠

⎛ ⎞ D= =

2π

λ------⎝ ⎠

⎛ ⎞

Tv 2π YLDH2L

24gtwEc------------------------=

Equations (9-13) , (9-14), (9-29), and (9-30) are adapt edfrom Housner (1963).


STANDARD COMMENTARY

R9.4—Seismic response coefficients Ci, Cc,and Ct

In practice, Designations Ci, Cc, and Ct define the profile ofthe design response spectrum at periods Ti, Tc, and Tv,respectively. A plot of the seismic response coefficient Ci isshown in the design response spectrum in Fig. R9.4.1,which is adapted from IBC (2003).

Fig. R9.4.1—Design response spectrum.

[ in the SI system]

9.4—Seismic response coefficients Ci, Cc , and Ct

Tv 2πYLDH 2

L

2gtwEc----------------------=

R9.4.1 The mapped spectral response accelerations Ss

and S1 for any location can also be obtained from the late stdatabase of the U.S. Geological Surv ey (USGS), at http://eqhazmaps.usgs.gov, using the specif ic zip code or latitudeand longitude that identify the location.

In regions other than those sho wn in the maps in publicat ionsby IBC (2003), Building Se ismic Safety Council (1997,2000), and ASCE (2005), SS and S1 may be replaced bythe maximum considered earthquak e spectral responseaccelerations from 5% damped response spectra repres entingearthquakes with a 2% probability of e xceedance in a50-year period, equi valent to a recurrence interv al ofapproximately 2500 years.

9.4.1 Ci shall be determined as follows:

For T i ≤ TS

Ci = SDS (9-32)

For Ti > TS

(9-33)

where

TS = (9-34)

SDS = the design spectr al response acceler ation atshort periods

SDS = SsFa (9-35)

SD1 = the design spectral response acceleration at a 1 second period

CiSD1Ti

---------- SDS≤=

SD1SDS---------

23---


STANDARD COMMENTARY

R9.4.2 Factor 1.5 represents the approximate ratio of thespectral amplifications based on 0.5% damping to thosebased on 5% damping. 0.4SDS in Eq. (9-38) is anapproximation of the effective peak ground acceleration S0(at T = 0) reduced by a factor of 2/3.

R9.5—Site-specific seismic responsecoefficients Ci, Cc, and Ct

For damping ratios other than 5% of critical, refer to Section R4.2.

If the site-specif ic response spectrum does not e xtend into,or is not well defined in the Tc range, coefficient Cc may becalculated using the equation

Cc

6

2

3---S

0

T2

c

---------4S

0

T2

c

---------==

SD1 = S1Fv (9-36)

SS and S1 are the mapped spectr al response accel-erations at short periods (Ss) and 1 second (S1), respec-tively, and shall be obtained from the seismic g roundmotion maps in Fig. 22-1 through 22-14 of ASCE 7-0 5,Chapter 22; and Fa and Fv are the site coefficients andshall be obtained from T able 11.4-1 and 11.4-2,respectively, of ASCE 7-05, in conjunction withTable 20.3-1, “Site Classification,” of ASCE 7-05.

9.4.2 Cc shall be determined as follows:

For Tc ≤ 1.6/Ts seconds

(9-37)

For Tc > 1.6/Ts seconds

(9-38)

9.4.3 Ct shall be determined as follows:

For circular tanks

For Tv ≤ TS

Ct = SDS (9-39)

For Tv >TS

Ct = (9-40)

For rectangular tanks, Ct = 0.4SDS.

9.5—Site-specific seismic responsecoefficients Ci, Cc, and Ct

When site-specific procedures are used, the maximumconsidered ear thquake spectr al response acceler ationsSaM and ScM shall be obtained from the site-specificacceleration spectrum as follows:

For per iods less than or equal to TS, SaM shall betaken as the spectral acceleration obtained from the site-specific spectra at a per iod of 0.2 seconds, except that itshall not be tak en less than 90% of the peak spectr alacceleration at an y per iod larger than 0.2 seco nds. For

23---

Cc1.5SD1

Tc-------------------- 1.5SDS≤=

Cc 60.4SDS

Tc2-------------------

2.4SDS

Tc2-------------------= =

SD1Tv

--------

R9.4.3 The period of vibration of v ertical liquid motion Tvfor a circular tank (upright c ylinder) is deri ved from theaxisymmetric pulsating (“breathing”) of the cylindrical walldue to the h ydrodynamic pressures resulting from the v ertical,piston-like “pounding” of the st ored liquid b y the v erticallyaccelerating ground.

This mode of vibration is relevant only to circular tanks, anddoes not apply to rectangular tanks. While the deri vation ofTv for circular tanks has been th e subject of se veral technicalpapers, the committee is not a ware of any work devoted tothe deri vation of this parameter for rectangular tanks.Therefore, for rectangular tanks, Ct is taken independent ofthe period of vibration.


STANDARD COMMENTARYwhere S0 is the effective site-specific peak ground accelerat ion(at T = 0) expressed as a fraction of the acceleration due togravity g.

The use of site-specific response spectra represents one specificcase of an “accepted alterna tive method of analysis” permitt edin Section 21.2.1.7 of ACI 350-06. Therefore, the 80%lower limit imposed in 9.5 should be considered the same asthe limit imposed in Section 21.2.1.7 of ACI 350-06.

periods g reater than TS, SaM shall be tak en as thespectral response acceler ation corresponding to Ti orTv , as applicab le. When a 5% damped, site-specificvertical response spectr um is a vailable, SaM shall bedetermined from that spectr um when used to deter mineCt ; and

ScM shall be taken as 150% of the spectr al responseacceleration corresponding to Tc, except that when a0.5% damped, site-specific horizontal response spectrumis a vailable, ScM shall be equal to the spectr alresponse acceleration from that spectr um correspondingto period Tc.

The seismic response coefficients Ci, Cc, and Ct shallbe deter mined from Eq. (9-41), (9-42), and (9-43),respectively.

For all periods, Ti

(9-41)

For all periods, Tc

(9-42)

For all periods, Tv

Ct = (9-43)

The values of Ci , Cc, and Ct used for design shall notbe less than 80% of the corresponding v alues asdetermined in accordance with Section 9.4.

9.6—Effective mass coefficient ε


(9-44)

9.6.2—Circular tanks

(9-45)

Ci23---SaM=

Cc23---ScM=

23---SaM

ε 0.0151 LHL-------⎝ ⎠

⎛ ⎞ 20.1908 L

HL-------⎝ ⎠

⎛ ⎞– 1.021+ 1.0≤=

ε 0.0151 DHL-------⎝ ⎠

⎛ ⎞ 20.1908 D

HL-------⎝ ⎠

⎛ ⎞– 1.021+ 1.0≤=

R9.6—Effective mass coefficient ε

The coefficient ε represents the ratio of the equi valent (orgeneralized) dynamic mass of the tank shell to its actualtotal mass. Equation (9-44) and (9-45) are adapted fromASCE (1981).

For additional information related to the ef fective masscoefficient ε, consult Veletsos and Shivakumar (1997).


STANDARD COMMENTARY9.7—Pedestal-mounted tanks

The equivalent weights, Wi and Wc , and heights to thecenters of gravity, hi, hc, hi′, and hc′ of a mounted tank,shall be computed using the corresponding Eq. (9-2)and (9-3) for rectangular and circular tanks, respectively.

The dynamic proper ties, including per iods of vibrationand later al coefficients , shall be per mitted to bedetermined on the basis of gener ally acceptab lemethods of dynamic analysis.

R9.7—Pedestal-mounted tanks

Housner (1963), Haroun and Ellaithy (1985), and A CICommittee 371 (1998) provide additional guidelines on thedynamic analysis of pedestal-mounted tanks.


STANDARD COMMENTARY

Fig. R9.1—Dynamic model of liquid-containing tank rigidly supported on the gr ound (adapted fr om Housner [1963] andASCE [1984]).


STANDARD COMMENTARY

Fig. 9.2.1—Factors Wi/WL and Wc/WL versus ratio L/HL for rectangular tanks.

(9-1)

(9-2)

WiWL--------

0.866 LHL-------⎝ ⎠

⎛ ⎞tanh

0.866 LHL-------⎝ ⎠

⎛ ⎞-----------------------------------------------=

WcWL-------- 0.264 L

HL--------⎝ ⎠

⎛ ⎞ 3.16HLL

--------⎝ ⎠⎛ ⎞tanh=

IMPULSIVE AND CONVECTIVE MASS FACTORS vs. L/HL RATIO


STANDARD COMMENTARY

Fig. 9.2.2—Factors hi/HL and hc/HL versus ratio L/HL for rectangular tanks (EBP).

hi /HL:

For tanks with (9-3)


hc /HL:

For all tanks

(9-5)

LHL------- 1.333<

hiHL------- 0.5 0.09375 L

HL-------⎝ ⎠

⎛ ⎞–=

LHL------- 1.333≥

hiHL------- 0.375=

hcHL------- 1

3.16 HLL

-------⎝ ⎠⎛ ⎞ 1–cosh

3.16 HLL

-------⎝ ⎠⎛ ⎞ 3.16 HL

L-------⎝ ⎠

⎛ ⎞sinh×

--------------------------------------------------------------------------–=

IMPULSIVE AND CONVECTIVE HEIGHT FACTORS vs. L /HL RATIO


STANDARD COMMENTARY

Fig. 9.2.3—Factors hi′/HL and hc′/HL versus ratio L/HL for rectangular tanks (IBP).

hi′ /HL:



hc′ /HL:

For all tanks

(9-8)

LHL------- 0.75<

hi′

HL------- 0.45=

LHL------- 0.75≥

hi′

HL-------

0.866 LHL-------⎝ ⎠

⎛ ⎞

2 0.866 LHL-------⎝ ⎠

⎛ ⎞tanh×

-------------------------------------------------------- 18---–=

hc′

HL-------- 1

3.16 HLL

-------⎝ ⎠⎛ ⎞ 2.01–cosh

3.16 HLL

-------⎝ ⎠⎛ ⎞ 3.16 HL

L-------⎝ ⎠

⎛ ⎞sinh--------------------------------------------------------------------–=

IMPULSIVE AND CONVECTIVE HEIGHT FACTORS vs. L /HL RATIO


STANDARD COMMENTARY

Fig. 9.2.4—Factor 2π/λ for rectangular tanks.

(9-12)

(9-13)

(9-14)

ωcλ

L-------=

λ 3.16g 3.16 HLL

-------⎝ ⎠⎛ ⎞tanh=

Tc2π

ωc------ 2π

λ------⎝ ⎠

⎛ ⎞ L= =

FACTOR (2π/λ)

Administrador

Text Box

"For results in the SI system, multiply the factors on the vertical axis by 1.811."


STANDARD COMMENTARY

Fig. 9.3.1—Factors Wi/ WL and Wc/WL versus ratio D/HL for circular tanks.

(9-15)

(9-16)

WiWL--------

0.866 DHL-------⎝ ⎠

⎛ ⎞tanh

0.866 DHL-------⎝ ⎠

⎛ ⎞-----------------------------------------------=

WcWL-------- 0.230 D

HL-------⎝ ⎠

⎛ ⎞ 3.68 HLD-------⎝ ⎠

⎛ ⎞tanh=

IMPULSIVE AND CONVECTIVE MASS FACTORS vs. D/HL RATIO


STANDARD COMMENTARY

Fig. 9.3.2—Factors hi / HL and hc/HL versus ratio D/HL for circular tanks (EBP).

hi /HL:



hc /HL:

For all tanks

(9-19)

DHL------- 1.333<

hiHL------- 0.5 0.09375 D

HL-------⎝ ⎠

⎛ ⎞–=

DHL------- 1.333≥

hiHL------- 0.375=

hcHL------- 1

3.68 HLD-------⎝ ⎠

⎛ ⎞ 1–cosh

3.68 HLD-------⎝ ⎠

⎛ ⎞ 3.68 HLD-------⎝ ⎠

⎛ ⎞sinh--------------------------------------------------------------------–=

IMPULSIVE AND CONVECTIVE HEIGHT FACTORS vs. D /HL RATIO


STANDARD COMMENTARY

Fig. 9.3.3—Factors hi′ /HL and hc′ /HL versus ratio D/HL for circular tanks (IBP).

hi′ /HL:



hc′ /HL:

For all tanks

(9-22)

DHL------- 0.75<

hi′

HL------- 0.45=

DHL------- 0.75≥

hi′

HL-------

0.866 DHL-------⎝ ⎠

⎛ ⎞

2 0.866 DHL-------⎝ ⎠

⎛ ⎞tanh--------------------------------------------------- 1

8---–=

hc′

HL-------- 1

3.68 HLD-------⎝ ⎠

⎛ ⎞ 2.01–cosh

3.68 HLD-------⎝ ⎠

⎛ ⎞ 3.68 HLD-------⎝ ⎠

⎛ ⎞sinh--------------------------------------------------------------------–=

IMPULSIVE AND CONVECTIVE HEIGHT FACTORS vs. D /HL RATIO


STANDARD COMMENTARY

COEFFICIENT CW

Fig. 9.3.4(a)—Coefficient Cw for circular tanks.

For D /HL > 0.667

Cw 9.375 10 2–× 0.2039 HL

D-------⎝ ⎠

⎛ ⎞ 0.1034 HLD-------⎝ ⎠

⎛ ⎞2

– 0.1253 HLD-------⎝ ⎠

⎛ ⎞3

– 0.1267 HLD-------⎝ ⎠

⎛ ⎞4

3.186 10 2– HLD-------⎝ ⎠

⎛ ⎞5

×–+ +=


STANDARD COMMENTARY

Fig. 9.3.4(b)—Factor 2π/λ for circular tanks.

(9-28)

(9-29)

(9-30)

ωcλ

D--------=

λ 3.68g 3.68 HLD-------⎝ ⎠

⎛ ⎞⎝ ⎠⎛ ⎞tanh=

Tc2π

ωc------ 2π

λ------⎝ ⎠

⎛ ⎞ D= =

FACTOR (2π/λ)

Administrador

Text Box

"For results in the SI system, multiply the factors on the vertical axis by 1.811."


COMMENTARY

ACI Committee 350, 2006, “Code Requirements for Environ-mental Engineering Concrete Structures and C ommentary(350-06),” American Concrete Institute, Farmington Hills,Mich., 484 pp.

ACI Committee 371, 1998, “Guide for the Analysis,Design, and Construction of Concrete-Pedestal W aterTowers (A CI 371R-98) (Reappro ved 2003), ” AmericanConcrete Institute, Farmington Hills, Mich., 36 pp.

American Society of Civil Engineers (ASCE), 1981, “Guide-lines for the Seismic Design of Oil and Gas Pipeline Systems,”Committee on Gas and Liquid Fuel Lifelines of the T echnicalCouncil on Lifeline Earthquake Engineering, Section 7.

American Society of Ci vil Engineering (ASCE), 1984,“Fluid/Structure Interaction During Seismic Excitation, ”Report by Committee on Seismic Analysis.

American Society of Ci vil Engineers (ASCE), 2005,“Minimum Design Loads for Buildings and Other Struct ures,”ASCE 7-05, Reston, Va.

ANSI/AWWA, 1995a, “ AWWA Standard for W ire- andStrand-Wound, Circular , Prestressed Concrete W aterTanks,” ANSI/AWWA D110-95.

ANSI/AWWA, 1995b, “ AWWA Standard for CircularPrestressed Concrete W ater T anks with CircumferentialTendons,” ANSI/AWWA D115-95.

Bertero, V. V., 1995, “Ener gy Based Design Approach, ”Performance-Based Seismic Engineering of Buildings ,SEAOC, Apr., pp. D-1 to D-12.

Building Seismic Safety C ouncil, 1997, “NEHRP Recom-mended Provisions for Seismic Regulations for New Buildingsand Other Structures—Part 1: Provisions (FEMA 302) and Part2: Commentary (FEMA 303),” Washington, D.C.

Building Seismic Safety C ouncil, 2000, “NEHRP Recom-mended Provisions for Seismic Regulations for New Buildingsand Other Structures—P art 1: Pro visions (FEMA 368)and Part 2: Commentary (FEMA 369),” Washington, D.C.

Haroun, M. A., 1984, “Stress Analysis of RectangularWalls Under Seismically Induced Hydrodynamic Loads, ”Bulletin of the Seismolo gical Society of America , V. 74,No. 3, June, pp. 1031-1041.

Haroun, M. A., and Ellaith y, H. M., 1985, “SeismicallyInduced Fluid Forces on Elevated Tanks,” Journal of Technical

Topics in Civil Engineering , ASCE, V. III, No. 1, Dec.,pp. 1-15.

Housner, G. W ., 1956, “Limit Design of Structures toResist Earthquak es,” Proceedings, W orld Conference onEarthquake Engineering, University of California, Berkeley,pp. 5-1 to 5-13.

Housner, G. W ., 1963, “Dynamic Pressure on FluidContainers,” Technical Information (TID) Document 7024,Chapter 6 and Appendix F, U.S. Atomic Energy Commission.

International Code Council, 2003, “International BuildingCode,” 656 pp.

International Conference of Building Of ficials (ICBO),1997, “Uniform Building Code,” V. 2, Whittier, Calif.

John A. Blume & Associates, 1958, “Report of T estingProgram on Earthquak e Cable Detail for the PreloadCompany, Inc.,” July.

Medearis, K., and Young, D. H., 1964, “Ener gy Absorptionof Structures Under Cyclic Loading, ” Journal of theStructural Division, ASCE, V. 90, ST1, Feb., pp. 61-89.

National Science F oundation, 1981, “Earthquak e DesignCriteria for Water Supply & Wastewater Systems,” NationalScience Foundation Report NSF/CE52-81079, Sept.

Newmark, N. M., and Hall, W. J., 1982, “Earthquak eSpectra and Design, ” Earthquake Engineering Resear chInstitute Monograph.

New Zealand Standard (NZS), 1986, “Code of Practicefor Concrete Structures for the Storage of Liquids, ” NZS3106, 77 pp.

Scarlat, A. S., 1997, “Design of Soft Stories—A Simplifi edEnergy Approach,” Earthquake Spectra, V. 1 3, N o. 2 , M ay,pp. 305-315.

Uang, C. M., and Bertero, V. V., 1988, “Use of Energy asa Design Criterion in Earthquak e-Resistant Design,” EERCReport No. UCB/EERC-88, Nov.

Veletsos, S. A., and Shivakumar, P., 1997, “DynamicResponse of Tanks Containing Liquids or Solids,” ComputerAnalysis and Design of Earthquake-Resistant Structures ,Earthquake Engineering Series V. 3, D. E. Beskos and S.A. Anagnostopoulos, eds., Computational MechanicsPublications, Billerica, Mass., 936 pp.

CHAPTER 10—COMMENTARY REFERENCES


Notes


COMMENTARY

A.1—General outline of design method

In the absence of a more rigorous method of analysis, the general procedures out lined below may be used to apply the pro visions ofChapters 1 through 9.

Basic seismic design parameters:

1. Establish the design depth of the stored liquid HL , the wall height Hw, and the tank length or diameter, L or D, respectively;

2. From the applicable seismic ground motion map of ASCE 7-05, Chapter 22, obtain the mapped maximum considered earth-quake spectral response accelerations at short periods and at 1 second (SS and S1, respectively). After selecting the site classi-fication from ASCE 7-05, T able 20.3-1, obtain coef ficients Fa and Fv using ASCE 7-05, T ables 11.4-1 and 11.4-2, andcalculate SDS and SD1 using Eq. (9-35) and (9-36);

3. Select an importance factor I from Table 4.1.1(a);

4. Select the factors Ri and Rc from Table 4.1.1(b) for the type of structure being investigated;

Tank dynamic properties:

5. Calculate the equivalent weight of the tank wall (shell) Ww, roof Wr, and the stored liquid WL. Also, compute the effectivemass coefficient ε;

6. Calculate the effective weight of the impulsive component of the stored liquid Wi, and the convective component Wc usingFig. 9.2.1 for rectangular tanks or Fig. 9.3.1 for circular tanks;

7. Calculate the heights hw, hr, hi, and hc (EBP) and h′i and h′c (IBP) to the center of gravity of the tank wall, roof, impulsivecomponent, and convective component, respectively (Fig. 9.2.2, 9.2.3, 9.3.2, and 9.3.3, or Sections 9.2 and 9.3);

8. Calculate the combined natural frequenc y of vibration ωi of the containment structure and the impulsi ve component of thestored liquid (Eq. (9-9) for rectangular tanks or Eq. (9-23) for circular tank Types 2.1 and 2.2). The impulsive mode will generally fallinto the rigid range of the response spectra (that is, the constant spectral acceleration region of the design response spectrum inFig. R9.4.1) for common sizes of concrete tanks. Thus, if the maximum value of Ci is used (SDS), calculation of the naturalfrequency and natural period is not required;

9. Calculate the frequenc y of the vibration ωc of the con vective component of the stored liquid ( Eq. (9-12) for rectangulartanks or Eq. (9-28) for circular);

10. Using the frequency values determined in Steps 8 and 9, calculate the corresponding natural periods of vibration Ti and Tc.(Eq. (9-11) and (9-14) for rectangular tanks, or Eq. (9-25), (9-26), and (9-30) for circular tanks);

11. Based on the natural periods determined in Step 10 and the design spectral response acceleration values derived in Step 2, calcu-late the corresponding seismic response coefficients Ci and Cc (Eq. (9-32), (9-33), (9-37), and (9-38)). Note: Where a site-specific response spectrum is constructe d in accordance with Section 4.2.1, Ci and Cc are determined in accordance withSections 9.5 and R9.5;

Freeboard:

12. Where required, calculate the maximum v ertical displacement of liquid surf ace (wave height) in accordance with Chapter 7 .Adjust the wall height if required to meet freeboard requirements;

APPENDIX A—DESIGN METHOD


COMMENTARYBase shear and overturning moments:

13. Compute the dynamic lateral forces (Eq. (4-1) to (4-4)) and total base shear V (Eq. (4-5));

14. Calculate the bending and overturning moments (Eq. (4-10) and (4-13));

Vertical acceleration:

15. Compute the vertical amplification factor Ct in accordance with Section 9.4.3. For circular tanks, first calculate the naturalperiod of vibration of vertical liquid motion Tv (Eq. (9-31));

16. Calculate the hydrodynamic pressure pvy (Eq. (4-14));

Pressure distribution:

17. Compute the vertical distribution of the force components in accordance with Chapter 5;

Stresses:

18. In rectangular tanks, calculate the st resses in the w all due to the impulsi ve and con vective pressures, depending on thestructural system considered (Section 6.1) and the stresses associated with the increase in effective fluid density due to thevertical acceleration. In circular tanks, calculate the hoop stresses due to the impulsive and convective pressures and due to thevertical acceleration (Section 6.2); and

19. Calculate the o verall bending stresses due to the o verturning moments (from Step 14). Do wnward pressures on theneoprene bearing pads of free base circular tanks caused b y overturning moments should be considered. If uplift develops onthe heel side, then anchor cables must be provided.


COMMENTARY

B.1—Introduction

B1.1—Scope

The purpose of this appendix is to permit the user to adapt the pro visions of ACI 350.3 to the seismic provisions of the 1997edition of the Uniform Building Code (UBC) (ICBO 1997). The differences between the 1997 UBC and the 2003 IBC seismicprovisions as used in this Standard are primarily due to differences in the definition of design ground motions and theconstruction of the corresponding design response spectra as explained below.

NOTE: All section, table, figure and equation references are to 1997 UBC except as otherwise indicated.

B1.2—Design ground motions

This appendix presents an outline of the methodolog y to be followed when computing the loading side of seismic analysis inaccordance with 1997 UBC. In this case, the design ground motions are those with a 10% probability of exceedance in50 years.

B.2—Notation (not included in Section 1.2 of this standard)

Ca = seismic coefficient, as set forth in Table 16-Q Cv = seismic coefficient, as set forth in Table 16-RDL = dead loadE = earthquake load as defined in Section 1630.1LL = live loadSA, SB, SC, SD, SE, SF = soil profile types as set forth in Table 16-J Na = near-source factor used in the determination of Ca in Seismic Zone 4 related to the proximity of the structure to

known faults with magnitudes and slip rates as set forth in Tables 16-S and 16-UNv = near-source factor used in the determination of Cv in Seismic Zone 4 related to the proximity of the structure to

known faults with magnitudes and slip rates as set forth in Tables 16-T and 16-UTs = 0.40Cv /Ca

Z = seismic zone factors as given in Table 16-I

B.3—Loading side, general methodology

1. Select the seismic zone (1 through 4) where the site is located, using the seismic zone map (Fig. 16-2);

2. Using the seismic zone determined in Step 1, find the zone factor Z from Table 16-I;

3. Consulting paragraph 1636.2 and Table 16-J, select the soil profile type designation SA through SF that best represents thesoil at the site;

4. Using the zone factor Z and the soil profile designation from Steps 2 and 3, find the seismic coefficient Ca from Table 16-Qand seismic coefficient Cv from Table 16-R;

5. If the structure is in Seismic Zone 4, select a Seismic So urce Type A, B, or C from Table 16-U, and near-source factors Naand Nv from Tables 16-S and 16-T, respectively;

APPENDIX B—ALTERNATIVE METHOD OF ANALYSIS BASED ON 1997 Uniform Building Code


COMMENTARY

6. Compute ;

7. Using the values of Ca, Cv, and Ts, construct a design response spectrum as in Fig. 16-3 and B.1;

Fig. B.1—Modified UBC 1997 design response spectrum (ICBO 1997).

8. Seismic response coefficients Ci and Cc—

8.1 Ci (impulsive component): Compute period of vibration Ti in accordance with Eq. (9-11) of this Standard for rectangulartanks, and Eq. (9-25) or (9-26) for circular.

(Note 1: Section 1634.1.4 specif ies the method for determining th e fundamental period b y referencing 1630.2.2. F or liquid-retaining structures, however, the methods in Chapter 9 of this standard should be used.)

Compute seismic response coefficient Ci corresponding to T, using the above design response spectrum as follows

(Note 2: Section 1629.8, “Selection of Lateral-force Procedures,” allows three options for computing lateral forces, dependingon the type of structure being investigated: simplified static, static, or dynamic. For liquid-containing structures, this standarduses the static procedures in accordance with 1629.8.3, modified as indicated in this standard.)

For all seismic zones:

For Ti ≤ Ts

Ci = 2.5Ca (B-1)

For Ti > Ts

Ci = Cv /Ti (B-2)

In addition, for Seismic Zone 4

Ci ≥ 1.6ZNv (B-3)

8.2 Cc (convective component): Compute period of vibration Tc using Eq. (9-14) of this Standard for rectangular tanks,and (9-30) for circular tanks.


Cc = ≤ 3.75Ca (B-4)

Ts

Cv

2.5Ca-------------- 0.40

Cv

Ca

------= =

1.5Cv

Tc

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


COMMENTARYFor Tc > 1.6/Ts seconds

(B-5)

9. Base shears V—

• Compute Ww, Wr, and WL; • Compute Wi and Wc using the equations in Sections 9.2 and 9.3 of this Standard;• Select coefficient Ri and Rc from Table 4.1.1(b) of this Standard.• Select an importance factor I from Table 4.1.1(a) of this Standard.• Compute the component parts of the total lateral force, Pw, Pr , Pi, and Pc in accordance with Section 4.1.1 of this Standard

as follows.

For all seismic zones

(B-6)

(B-7)

(B-8)

(B-9)

Equation (B-8) and (B-9) take the following form depending on the period Ti and Tc.

For Ti ≤ Ts

[1997 UBC Eq. (30-5)]

For Ti > Ts

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

For Tc > 1.6/Ts seconds

(B-10)


(B-11)

Cc

6C

a

Tc

2------=

Pw

CiI

Ri

--------εWw

=

Pr

CiI

Ri

--------εWr

=

Pi

CiI

Ri

--------εWi

=

Pc

CcI

Rc

--------Wc

=

Pi

2.5CaI

Ri

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

=

Pi

CvI

RiTi

----------Wi

0.56CaIW

i≥=

Pc

6CaI

RcTc

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

c

1.5 2.5Ca

( )IR

c

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

≤=

Pc

1.5CvI

RcTc

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

=


COMMENTARYIn addition, for Seismic Zone 4

[1997 UBC Eq. (34-3)]

10. Total base shear V—The total base shear due to Pw , Pr, Pi , and Pc may be computed b y 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 distribution—The vertical distribution of the lateral seismic forces may be assumed as shown in Section 5 ofthis Standard;

12. Vertical component of ground motion—Compute 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 üv as follows

üv = (B-16)

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

B.4—Site-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

Pi

1.6ZNvI

Ri

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

≥

V Pw

Pr

Pi

+ +( )2 Pc

2+=

Ct

Cv

Tv

------=

CtIb

Ri

-----


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.5—Resistance side

1. The resistance side of the seismic desi gn, including load combinations and streng th 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 b y a factor of 1.4 for use with allowable stresses, and allowablestress increases used in the appro ved 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.6—Freeboard

(B-17)dmax

CcIL or D

2-----------------⎝ ⎠

⎛ ⎞=

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Seismic Design of Liquid-Containing Concrete Structuresand Commentary

seismic design of liquid-containing concrete structures...

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