seismic design of circular liquid-containing n.a. legates · pdf fileseismic design of...

Seismic design of circular liquid-containing

structures

N.A. Legates

Abstract

The ability of large, liquid-containing structures to resist earthquakes without damage is asubject of considerable interest not only to the engineering profession but also to thecommunity at large. This is because these structures often constitute an essential part of acommunity's lifeline and must therefore be maintained viable during emergencies. Also, insome applications the stored contents may be hazardous in which case their accidental releasemust be prevented.

The behavior of structures subjected to earthquake-induced fluid pressures was firststudied in the early 30's by Westergaard and others. In 1949 and 1951 Jacobsen analyzed arigid cylindrical liquid-containing tank and a cylindrical pier surrounded by liquid, subjectedto horizontal accelerations. Subsequently, Housner simplified the method of analysis andintroduced the concept of the two components of dynamic pressures, impulsive andconvective.

In subsequent work, Haroun, Housner, Veletsos and others modified the tank modelto account for the flexibility of the tank walls. Essentially, this acknowledges that the tankwall (together with the impulsive component of the stored liquid) responds independently of,i.e. at a different frequency than, the accelerating ground.

Design standards and codes currently being drafted in the US have evolved from,and built upon, these principles.

This paper highlights the seismic analysis and design of concrete liquid-storagecontainers, and offers an overview of current US practice in this field. To assist thepracticing engineer, detailed step-by -step design procedures are presented based on severalUS design standards recently published or currently under preparation.

1 Introduction

Current methods for analyzing and designing liquid-containing structuressubjected to earthquakes are derived primarily from work originated byWestergaard , Jacobsen^^ and others, and further developed by Housner .Housner's work was summarized in a 1963 U.S. Atomic Energy CommissionReport titled, "Nuclear Reactors and Earthquakes" ^. That report, betterknown as TED (Technical Information Document) -7024, has become theclassic reference for the analysis of circular and rectangular liquid-containingstructures subjected to earthquakes.

TID-7024 treats ground-supported, flat-bottomed tanks of uniformrectangular or circular section. The tanks are assumed to behave as rigid

Transactions on the Built Environment vol 20, © 1996 WIT Press, www.witpress.com, ISSN 1743-3509

374 Earthquake Resistant Engineering Structures

bodies, rigidly attached to the ground. Consequently, during a horizontalground acceleration, the tank wall and floor respond as part of, and in unisonwith, the moving ground. The horizontal acceleration of the rigid wall and floorgenerates inertia forces directly proportional to the ground acceleration.

When the accelerating tank is full, the lower portion of the containedliquid, Wi, acts as if it were a solid mass rigidly attached to the tank wall (Fig.1). As this mass accelerates, it exerts a horizontal force against the wall directlyproportional to the maximum acceleration in the tank bottom. This force isidentified as an impulsive force, Pj. Under the same accelerations, the upperportion of the contained liquid responds as if it were a solid oscillating massflexibly connected to the tank wall (We). This portion, which oscillates(sloshes) at is own natural frequency, exerts on the wall an additional force thatis proportional to the square of that frequency, as well as to the groundacceleration. This portion is defined as the convective component PC. Theconvective component oscillations are characterized by the "sloshing" actionwhereby the liquid rises above the static level on one side of the tank, and dropsbelow that level on the other.

The rigid-body concept, however, does not adequately represent theactual behavior of most large liquid-storage tanks. Work by Haroun, Housner,Veletsos ™ and others demonstrated that "the hydrodynamic effectsinduced by earthquake ground motions in flexible tanks may be appreciablygreater than those in rigid tanks of the same dimensions." <">

Equations for the natural frequency vary depending on the mode oflateral deformation of the tank wall (e.g. cantilever shear beam; oscillating ring:or a series of cantilever strips in flexure), which in turn depends on thediameter-to-height ratio (D/Ht) of the tank \ In the interest of simplicity,however, and in view of the fact that this ratio commonly ranges between 1 and8, a single Equation is adopted for all tanks (Eq's (12) or (13)).

Once the structural and fluid models are thus established, the groundmotion is represented by a design response spectrum which is either derivedfrom the actual earthquake record for the site, or is constructed by analogy tosites with known soil and seismic characteristics (Fig. 2). Alternatively, in lieuof a seismic design response spectrum, design codes usually define theseismicity of a region or locality in terms of either (a) the peak groundacceleration (PGA), A,, for that region or locality; or (b) a seismic coefficient,Z, which has a direct relationship to the PGA. In either case, the specifiedvalue represents the maximum effective peak acceleration (EPA) correspondingto a site-dependent ground motion having a 90% probability of not beingexceeded in a 50-year period. Special structures, such as those containinghighly hazardous liquids, may have to be designed for ground accelerations withhigher probabilities of non-exceedance. This paper utilizes the seismiccoefficient Z as reflected in U.S. regional seismic maps ^\


Earthquake Resistant Engineering Structures 375

SEISMIC MAP1

2A

2

3

4

TABLE 1SEISMIC ZONE FACTOR

ZONE FACTOR Z0.075

0.15

0.20

0.30

0.40

Z

PEAKACCELE

0.

0

0

0

0

GROUNDRATION, A,075g

isg

•20g

•30g

•40g

Having thus defined the structure and the imposed ground motions, oneproceeds to compute the various inertia forces, and from these derive the totallateral base shear.

2 General Approach

2.1 For a typical single-degree-of-freedom (SDOF) system subjected to alateral ground acceleration, the general equations for the base shear, V, andthe overturning moment, M, are:

CV = ZISx *W (1)

RwC

M = Z/Sx xAxMT (2)Rw

Where, C = lateral force coefficient (a function of the natural periodof vibration of the structure)

W = effective mass of the structure and its contentsI = Importance Factor (which depends on how important

it is to have the structure continue to function safelyduring and after an earthquake) (Table 2)

S = Soil Profile Coefficient (Table 3)All other symbols are defined in Section 5, Notations



Equation (1) is the typical base shear equation found in most codes or relateddocuments <

TABLE 2IMPORTANCE FACTOR, I

TANK USE FACTOR I

Tanks that must remain usable, with slight structural damage, 1.25for emergency purposes after an earthquake; or tanks whichare part of lifeline systems

Tanks that must remain usable, without significant leakage, but 1.0may suffer repair able structural damage

TYPE

TABLE 3SOIL PROFILE COEFFICIENT, S*

SOIL PROFILE DESCRIPTION

A A soil profile with either:( (a) A rock-like material characterized by a shear- wave

velocity greater than 762 m/s or by other suitablemeans of classification, or

(b) Medium-dense to dense or medium-stiff to stiff soilconditions, where soil depth is less than 60 960 mm

B A soil profile with predominantly medium-dense todense or medium-stiff to stiff soil conditions, where thesoil depth exceeds 60 960 mm

C A soil profile containing more than 6 096 mm of soft tomedium-stiff clay but not more than 12 192 mm of softclay

D A soil profile containing more than 12192 mm of softclay characterized by a shear wave velocity less than152.4 m/s

* References (12), (15), (16), (17) (18)

COEFF. S

1.0

1.2

1.5

2.0



The site factor should be established from properly substantiated geotechnicaldata. In locations where the soil properties are not known in sufficient detail todetermine the soil profile type, soil profile C should be used. Soil profile Dneed not be assumed unless the building official determines that soil profile Dmay be present at the site, or in the event that soil profile D is established bygeotechnical data.

2.2 When applied to liquid-containing structures, the total mass W is replacedby the four discreet mass components: The effective weight of the tankshell (wall), Ww, and tank roof, Wr ; the impulsive component of thecontained liquid, W; ; and the convective component, We As a result,Equations (1) and (2) take the following forms:

(7^=Z#x_^_x^ (3)

CV = ZIS x •—£- x W (4)c R e

we

= ^ x - x (5)wec

K = ZIS x — £- x W (6)V '

M. = ZIS x -Si- x (W. h. + eW hj + W,h,) (?)wi

CM = ZIS x — £- x W h (8)c R c c

we

Fig. 1 contains a physical representation of some of these terms.

Since each of these component forces (V;, VJ, or moments, (Mi, Me) aredecoupled (i.e. they oscillate at different frequencies, or "out of phase"with each other), they are usually combined by the square-root-of-the-sum-of-the-squares (SRSS) rule resulting in the following expressions: ^(16) (17) (19)



UNDISTURBED OSCILLATINGWATER SURFACE A WATER SURFACE

/ /\

(a) FLUID MOTION IN TANK

).5V

(b) DYNAMIC MODEL

M, +«^-Ph, +Plfc(c) DYNAMIC EQUILIBRIUM OF HORIZONTAL FORCES

Figure 1: Dynamic model of liquid-containing tank rigidly supported on theground

PERIOD, T

Figure 2: Normalized design response spectrum showing acceleration as afunction of period and damping



= ZIS x. (9)

= A/A/,-

i2

00)

The significance of the lateral coefficients Q and Cj can best beillustrated by reference to a design response spectrum such as in Fig. 2.Each of these coefficients represents the dynamic amplification factor(DAF) of a single-degree-of-freedom system with a natural period ofvibration T, where T>0.

Cc represents the DAF of the convective component of the storedliquid, which typically responds in a low-frequency, high-period (T>2.5s)mode of oscillation to a horizontal acceleration.

d represents the DAF of the impulsive component of the tankwall (together with the impulsive component of the liquid), whichconstitutes a more rigid structural system responding in a high-frequency,low-period mode of vibration (T<2.5s).

Coefficients R™ and Rwi (Table 4) represent the ductility andenergy-dissipating ability of the structure and are used to essentiallyconvert the elastic response spectrum into an inelastic < )U3)(i4)

Section 3 below presents a summary of all the Equations involvedin the seismic analysis of a liquid-containing circular tank; while Section 4contains a step-by-step procedure for calculating all the necessaryparameters to solve Equations (3) and (4), and for computing the resultingforces and stresses.



TABLE 4lw VALUES

TYPE OF STRUCTURE *

(a) Anchored, flexible-base tanks (Bl)

(b) Fixed or hinged-base tanks (A)

(c) Unanchored, contained oruncontained (B2 or B3)

(d) Elevated tanks

ON ORABOVEGRADE

4.5

2.75

2.0

3.0

fiffi

BURIED^

4.5

4.0

2.75

1.0

1.0

1.0

1. Buried tank is defined as a tank whose maximum water surface is at orbelow ground level.

2. Rwi - 4.5 is the maximum R*i value to be used for any liquid-containingstructure

3. Unanchored, uncontained tanks shall not be built in zones 2B or higher.*For tank types see Fig. 3

3 Design Equations

w.=

tanh(0.866DH

D

W~

'.W

H0230—— x tanh(3.68—M

il)

02)

CO = (13)



- TANK WALL (Typ)

r- FLOOR (Typ)

CLOSURESTRIP

A1-FIXED A2-HINGED OR PINNED

TYPE A - NON-FLEXIBLE BASE CONNECTIONS

r

SEISMIC CABLES)/OR ANCHORS (Typ)

A/| /- FLEXIBLE BASE PAD(Typ)

FLEXIBLE CONTAIN-

81-ANCHOREDFLEXIBLE BASE

B2-UNANCHORED, CONTAINED B3-UNANCHDRED, UNCDNTAINEDFLEXIBLE BASE FLEXIBLE BASE

(FOR SECTION A-A SEE FIG. 4)TYPE B - FLEXIBLE-BASE CONNECTIONS(CIRCULAR PRESTRESSED TANKS ONLY)

Figure 3: Types of liquid-containing structures classified on the basis of theirwall-to-foundation connection details

SEISMIC BASE CABLi r— WALL

SECTION A-A(FROM FIG. 3)

^— FOUNDATION

Figure 4: Details of seismic base cables used in conjunction with prestressed-concrete, flexible-base circular tanks



from Table 5)

Q = J3.68g x tanh(3.68— -)

TV For tank types Al and A2 (Fig. 3),

(14)

(15)

(16)

(17)

(18)

For tank type Bj_, B2, or B3 (Fig. 3),

k =a

_~

*j*5™* " , r P P P

1.25 ,. 5 < 2.75)

(19)

(20)

(21)

c =c

6.0(22)

cosh(3.68 )-l1 — - D

D

hi: For tall tanks (D/Ht <1.333),

D

/i = 05 - 0.093 75(—)HL J

x//

(23)

(24a)



For shallow tanks (D/Ht >1.333),hi = 0.375 HL (24b)

hw = 0.5OHw (for uniform-thickness wall) (25)

g = O.oi51(—) -0.1908(—)-f 1.021 (Reference 11) (26)

(References 20 and 21) (27)

(29)

(30)

(33)

Piy and Pcy represent the vertical distribution of the lateral impulsive andconvective forces respectively; and Pwy represents the vertical distributionof the inertia force of the tank wall. The distribution of these forces isdefined and illustrated in Fig. 5.



HL

0.0.0.0.0.0.0.

1.1.1.1.2.2.2.2.2.2.22223

(*) Based oBy A. S

/R

3456789012345678901234567890

n tw/R = 0.. Veletsos,

TABLE 5

COEFFICIENT

D/HL

6.675.004.003.332.862.502.222.001.821.671.541.431.331.251.181.111.051.000.950.910.870.830.800.770.740.710.690.67

01, PL/PC -0.40 andunpublished commun

000000000.0.0.00000000000000000

Poissicatioi

c*

1222129313661431148815371578161216391658167116791681167916731664165216381621160415841564154415221500147814561434

an's Ratio v = 0.17i



«g—<—e»«3-

— — - EXACT PRESSURE DISTRIBUTIONLINEAR APPROXIMATION

= »$:

P = m

4Hf - 6/1 - 6H, - 12/i, x^ ' ^ ^ '^

4//, -6h^-(6Hr -126L -0^-^D/7^-lZA7^

(Adapted from Reference 19)

F/^ = —— (Constant for uniform-thickness wall)

^

Figure 5: Vertical distribution of impulsive and convective hydrodynamicforces, and wall inertia forces, along the wall height



4 Step-By-Step Design Procedure

4.1 - Total Base Shear

1. Calculate the mass of the tank shell (wall), W*, and roof, W? Also,calculate coefficient e, Equation (26).

2. Calculate the effective mass of the impulsive component of the storedliquid, Wj and the convective component, We, using Equations (11) and(12).

3. Calculate the combined natural frequency of vibration, C0j, of thecontainment structure and the impulsive component of the stored liquid,Equations (13) and (14).

4. Calculate the frequency of vibration C0c, of the convective component of thestored liquid, Equations (15) and (16).

5. Using the frequency values determined in 3 and 4, calculate thecorresponding natural periods of vibration, Tj and TC , Equations (17),(18), (19), and (20).

6. Based on the periods determined in 5, calculate the corresponding lateralforce coefficients Q and Q, Equations (21) and (22).

7. Determine the seismic coefficient Z from the seismic zone maps or Table 1.NOTE:Where a site-specific response spectrum is available, substitute the site-specific spectral accelerations Ai and & for coefficients Q and Q (step 6),S (step 8) and coefficient Z (step 7) combined. Ai, representing theEffective Peak Acceleration, should be used for short-period structures(T<0.31 sec) and for a damping ratio P = 5%;, while AC, representing theEffective Velocity-Related Peak Acceleration, should be used for long-period structures or structural components for a damping ratio (3 = 0.50%(Reference 12).

8. Select an Importance Factor 1 and Soil Profile Coefficient S, Tables 2 & 3.9. Select the Factor R* specified for the class of structure being investigated,

Table 4.10. Compute the total base shear, V, Equation (9)11. Compute the vertical distribution of the impulsive and convective force

components, Fig. 5.

4.2 Overturning Moments

12. Calculate the heights h*, hr, h; and he to the center of gravity of the tankwall, roof, impulsive component and convective component respectively,Equations (23), (24) and (25)



13. Calculate the overturning moment M due to the impulsive and convectiveforce components, Equation (10).

4.3 Vertical Acceleration

14. Calculate the natural period of vertical acceleration, TV, Equation (27).15. Calculate the vertical force coefficient Cv as a function of TV, Equation

(28).16. To calculate the additional pressure on the tank wall caused by the vertical

"pounding" of the tank, multiply the hydrostatic load by a spectralamplification factor, Kv,

4.4 Stresses

17. Unit shear stresses at the base:

(35)

18. Dynamic hoop (circumferential) stresses in tank wall: Calculate thecircumferential forces, Njy, Ncy and N*y due to the impulsive, convective,and wall inertia forces respectively; and circumferential forces, Kv*Nhy, dueto the vertical acceleration, Equations (29), (30), (31), (32) and (34).Combine by the SRSS rule, and obtain the resultant hoop stress Oy:

'N- +N \ + N^ •f,y+»wy) +"cy , . ..,,

w19. Calculate the vertical stresses due to the overturning moments.

4.5 Sloshing

20. Sloshing of the convective portion of the stored liquid causes verticaldisplacement of the liquid surface which represents the amplitude ofoscillation, Fig. 1. The maximum displacement, or sloshing height, dmax, canbe calculated using the following approximate relationship:

CJ P?)



5 Notation

Aa = seismic coefficient representing the Effective Peak AccelerationA. = cross-sectional area of the base cable, strand, or conventional

reinforcingB = fraction of horizontal acceleration to be used as vertical accelerationCc = lateral convective force coefficient (Equation 22)Ci = lateral impulsive force coefficient (Equation 21)d = a factor for adjusting coefficient C* for liquids other than water

(Equation 14)Cv = a vertical seismic coefficient (Equation 28)Cw = a coefficient for determining the fundamental frequency of the tank

liquid system, Table 5.d = freeboard (sloshing height) (Equation 37)D = tank diameterEC = modulus of elasticity of concreteES = modulus of elassticity of steelg = acceleration due to gravityGp = shear modulus of elastomeric bearing padshe, hj, hr, hw = Vertical distance from the base of the wall to the centers of

gravity of We, W;, W,, and W*HL = depth of stored liquidHW = wall heightI = importance Factor, Table 2ka = spring constant of the tank wall system (Equation 20)Kv = coefficient applied to the hydrostatic force to account for the effect of

vertical acceleration (Equation 34)Lp = length of individual elastomeric bearing padsLS = effective length of base cable taken as the sleeve length plus 35 times

the cable diameterNcy = hoop (circumferential) force due to the convective component of the

acclerating liquid at height y from the base (Equation 30)Nhy = hydrostatic hoop force at the level being investigated (Eq. 32)Niy = hoop (circumferential) force due to the impulsive component of the

accelerating liquid at height y from the base (Equation 29)Nwy = hoop (circumferential) force due to the inertia of the accelerating wall,

at height y from the base (Equation 31)Pcy, Piy and P*y = lateral impulsive, convective and wall inertia forces acting on

the wall at a height y from the wall base, Fig. 5Q = a coefficient defined in Equation (16)R = radius of circular tank



Rw = response-modification factor representing the energy-dissipating abilityof the structure (Rwc for the convective component of the acceleratingliquid; R*i for the impulsive component), Table 4

S = site profile representing the soil characteristics as they pertain to thestructure, Table 3

Sp = spacing of elastomeric bearing padsSs = spacing between individual base cable loopstp = thickness of elastomeric bearing padstw = wall (shell) thicknessTC = natural period of the first (convective) mode of vibration (Eq. 17)Ti = fundamental period of oscillation of the tank wall (plus the impulsive

component of the tank contents( (Equations 18 & 19)TV = natural period of vertical acceleration, sec. (Equation 27)iiv = vertical accelerationV = total horizontal base shearVc = horizontal shear due to convective component of the stored liquidVj = horizontal shear due to impulsive component of the stored liquidVr = horizontal shear due to the inertial force of the accelerating tank roofVw = horizontal shear due to the inertial force of the accelerating tank wallWp = width of elastomeric bearing padsWe = equivalent mass of the convective component of the stored liquid,

(Equation 12)Wi = equivalent mass of the impulsive component of the stored liquid

(Equation 11)WL = total mass of the stored liquidWr = mass of the tank roofWw = mass of the tank wall (shell)y = liquid level at which tank wall is being investigated (measured from

tank base)Z = seismic Zone Factor, from Table 1a = angle of base cable or strand with the horizontalP = percent of critical dampingYL = unit mass of watere = ratio of effective mass of the tank shell (wall) to the actual mass,

(Equation 26)PC = mass density of concretePL = mass density of the contained liquidGy = combined hoop stresses in wall at height y from base, (Equation 36)T = radial shear stress through wall, (Equation 35)C0c = circular frequency of oscillation of the first (convective) modeCD; = circular frequency of oscillation of the tank wall (plus the impulsive

component of the tank contents)



6 References

1. Westergaard, H.M. Water pressures on dams during earthquakes,Transactions of the American Society of Civil Engineers, Vol. 98, 1933.

2. Jacobsen, L.S. Impulsive hydrodynamics of fluid inside a cylindrical tankand of fluid surrounding a cylindrical pier, Bulletin of the SeismologicalSociety of America, Vol. 39, No. 3, pp. 189-204, 1949.

3. Jacobsen, L.S. & Ayre, R.S. Hydrodynamic experiments with rigidcylindrical tanks subjected to transient motions, Bulletin of theSeismological Society of America, Vol. 41, 313-348, 1951.

4. Housner, G.W. Dynamic pressures on accelerated fluid containers,Bulletin of the Seismological Society of America, Vol. 47, No. 1, 15-35,1957.

5. Housner, G.W. The dynamic behavior of water tanks, Bulletin of theSeismological Society of America, Vol. 53, No. 20, pp. 381-387, 1963.

6. Housner. Dynamic Pressure on Fluid Containers, Technical InformationDocument (TID) 7024, Chapter 6 and Appendix F, U.S. AtomicEnergy Commission, 1963.

7. Veletsos, AS & Yang, J.Y. Dynamics of fixed-base liquid storagetanks, Proceedings, U.S.-Japan Seminar on earthquake with emphasison Lifeline Systems, Tokyo, pp. 317-341, November 1976.

8. Veletsos, AS & Yang, J.Y. Earthquake response of liquid-storage tanks,Advances in Civil Engineering through Engineering Mechanics,American Society of Civil Engineers, pp. 1-14, May 1977.

9. Haroun, MA & Housner, G.W. Earthquake response of deformableliquid-storage tanks, Proceedings, Pressure Vessel and PipingTechnology Conference, San Francisco, CA, August 1980.

10. Haroun, M.A. & Housner, G.W. Seismic design of liquid-storage tanks,Journal of the Technical Councils of the American Society of CivilEngineers, Vol. 107, No. TCI, pp. 191-207, 1981.



11. American Society of Civil Engineers, Guidelines for the Seismic Design ofOil and Gas Pipeline Systems, Prepared by the Committee on Gas andLiquid Fuel Guidelines of the Technical Council on Lifeline Engineering,Section 7, 1981.

12. International Conference of Building Officials, Uniform Building Code,Whittier, CA, 1994.

13. Applied Technology Council, Tentative Provisions for the Developmentof Seismic Regulations for Buildings. ATC 3-06, June 1978.

14. Federal Emergency Management Agency, NEHRP RecommendedProvisions for Seismic Regulations for New Buildings, Parts 1 & 2,FEMA 222A and 223A, Prepared by the Building Seismic SafetyCouncil, Washington, DC, May 1995.

15. American Water Works Association, Standard for Wire-Wound CircularPrestressed-Concrete Water Tanks, ANSI/AWWA Dl 10-1996.

16. American Water Works Association, Standard for Circular PrestressedConcrete Tanks with Circumferential Tendons, ANSI/AWWA D115-1996.

17. American Concrete Institute, Design and Construction of EnvironmentalConcrete Structures, Chapters 21 & 52, Seismic Provisions (Draft), 1996

18. BOCA (Building Officials and Code Administration International),National Building Code, 1996.

19. Standards Association of New Zealand, Code of Practice for ConcreteStructures for the Storage of Liquids, NZS 3106:1986.

20. Marchaj, T.J. Importance of vertical acceleration in the design of liquid-containing tanks, Proceedings of the 2nd U.S. National Conference onEarthquake Engineering, Stanford, University, Stanford, CA, pp. 146-155, August 1979.

21. Luft, R.W. Vertical acceleration in prestressed concrete tanks, Journal ofStructural Engineering of the American Society of Civil Engineers, Vol.110, No. 4, pp. 706-714, 1984


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