Seismic design of circular liquid-containing

Transcription

1 Seismic design of circular liquid-containing structures N.A. Legates Abstract The ability of large, liquid-containing structures to resist earthquakes without damage is a subject of considerable interest not only to the engineering profession but also to the community at large. This is because these structures often constitute an essential part of a community's lifeline and must therefore be maintained viable during emergencies. Also, in some applications the stored contents may be hazardous in which case their accidental release must be prevented. The behavior of structures subjected to earthquake-induced fluid pressures was first studied in the early 30's by Westergaard and others. In 1949 and 1951 Jacobsen analyzed a rigid cylindrical liquid-containing tank and a cylindrical pier surrounded by liquid, subjected to horizontal accelerations. Subsequently, Housner simplified the method of analysis and introduced the concept of the two components of dynamic pressures, impulsive and convective. In subsequent work, Haroun, Housner, Veletsos and others modified the tank model to account for the flexibility of the tank walls. Essentially, this acknowledges that the tank wall (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-storage containers, and offers an overview of current US practice in this field. To assist the practicing engineer, detailed step-by -step design procedures are presented based on several US design standards recently published or currently under preparation. 1 Introduction Current methods for analyzing and designing liquid-containing structures subjected to earthquakes are derived primarily from work originated by Westergaard^, Jacobsen^^ and others, and further developed by Housner^^. Housner's work was summarized in a 1963 U.S. Atomic Energy Commission Report titled, "Nuclear Reactors and Earthquakes" ^. That report, better known as TED (Technical Information Document) -7024, has become the classic reference for the analysis of circular and rectangular liquid-containing structures subjected to earthquakes. TID-7024 treats ground-supported, flat-bottomed tanks of uniform rectangular or circular section. The tanks are assumed to behave as rigid

2 374 Earthquake Resistant Engineering Structures bodies, rigidly attached to the ground. Consequently, during a horizontal ground acceleration, the tank wall and floor respond as part of, and in unison with, the moving ground. The horizontal acceleration of therigidwall and floor generates inertia forces directly proportional to the ground acceleration. When the accelerating tank is full, the lower portion of the contained liquid, 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 directly proportional to the maximum acceleration in the tank bottom. This force is identified as an impulsive force, Pj. Under the same accelerations, the upper portion of the contained liquid responds as if it were a solid oscillating mass flexibly connected to the tank wall (We). This portion, which oscillates (sloshes) at is own natural frequency, exerts on the wall an additional force that is proportional to the square of that frequency, as well as to the ground acceleration. This portion is defined as the convective component PC. The convective component oscillations are characterized by the "sloshing" action whereby the liquid rises above the static level on one side of the tank, and drops below that level on the other. The rigid-body concept, however, does not adequately represent the actual behavior of most large liquid-storage tanks. Work by Haroun, Housner, Veletsos^ ^ ^ and others demonstrated that "the hydrodynamic effects induced by earthquake ground motions inflexibletanks may be appreciably greater than those inrigidtanks of the same dimensions." <"> Equations for the natural frequency vary depending on the mode of lateral 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 the diameter-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 and 8, a single Equation is adopted for all tanks (Eq's (12) or (13)). Once the structural and fluid models are thus established, the ground motion is represented by a design response spectrum which is either derived from the actual earthquake record for the site, or is constructed by analogy to sites with known soil and seismic characteristics (Fig. 2). Alternatively, in lieu of a seismic design response spectrum, design codes usually define the seismicity of a region or locality in terms of either (a) the peak ground acceleration (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 specified value represents the maximum effective peak acceleration (EPA) corresponding to a site-dependent ground motion having a 90% probability of not being exceeded in a 50-year period. Special structures, such as those containing highly hazardous liquids, may have to be designed for ground accelerations with higher probabilities of non-exceedance. This paper utilizes the seismic coefficient Z as reflected in U.S. regional seismic maps ^\

3 Earthquake Resistant Engineering Structures 375 TABLE 1 SEISMIC ZONE FACTOR Z SEISMIC MAP 1 2A ZONE FACTOR Z PEAK GROUND ACCELERATION, A, g 0 isg 0 20g 0 30g 0 40g Having thus defined the structure and the imposed ground motions, one proceeds to compute the various inertia forces, and from these derive the total lateral base shear. 2 General Approach 2.1 For a typical single-degree-of-freedom (SDOF) system subjected to a lateral ground acceleration, the general equations for the base shear, V, and the overturning moment, M, are: C V = ZISx *W (1) R w C M = Z/Sx xaxmt (2) R w Where, C = lateral force coefficient (a function of the natural period of vibration of the structure) W = effective mass of the structure and its contents I = Importance Factor (which depends on how important it is to have the structure continue to function safely during and after an earthquake) (Table 2) S = Soil Profile Coefficient (Table 3) All other symbols are defined in Section 5, Notations

4 376 Earthquake Resistant Engineering Structures Equation (1) is the typical base shear equation found in most codes or related documents < TABLE 2 IMPORTANCE FACTOR, I TANK USE FACTOR I Tanks that must remain usable, with slight structural damage, 1.25 for emergency purposes after an earthquake; or tanks which are part of lifeline systems Tanks that must remain usable, without significant leakage, but 1.0 may suffer repair able structural damage TABLE 3 SOIL PROFILE COEFFICIENT, S* TYPE SOIL PROFILE DESCRIPTION COEFF. S 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 suitable means of classification, or (b) Medium-dense to dense or medium-stiff to stiff soil conditions, where soil depth is less than mm 1.0 B A soil profile with predominantly medium-dense to dense or medium-stiff to stiff soil conditions, where the soil depth exceeds mm 1.2 C D A soil profile containing more than mm of soft to medium-stiff clay but not more than mm of soft clay A soil profile containing more than mm of soft clay characterized by a shear wave velocity less than m/s * References (12), (15), (16), (17) (18)

5 Earthquake Resistant Engineering Structures 377 The site factor should be established from properly substantiated geotechnical data. In locations where the soil properties are not known in sufficient detail to determine the soil profile type, soil profile C should be used. Soil profile D need not be assumed unless the building official determines that soil profile D may be present at the site, or in the event that soil profile D is established by geotechnical data. 2.2 When applied to liquid-containing structures, the total mass W is replaced by the four discreet mass components: The effective weight of the tank shell (wall), Ww, and tank roof, Wr ; the impulsive component of the contained liquid, W; ; and the convective component, We As a result, Equations (1) and (2) take the following forms: (7 ^=Z#x_^_x^ (3) C V = ZIS x - x W (4) c R e we =^x-x (5) we c K = ZIS x - x W (6) V ' M. = ZIS x -Si- x (W. h. + ew^ hj + W,h,) (?) wi C M = 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) are decoupled (i.e. they oscillate at different frequencies, or "out of phase" with each other), they are usually combined by the square-root-of-thesum-of-the-squares (SRSS) rule resulting in the following expressions: ^ (16) (17) (19)

6 378 Earthquake Resistant Engineering Structures UNDISTURBED OSCILLATING WATER SURFACE A WATER SURFACE / /\ (a) FLUID MOTION IN TANK (b) DYNAMIC MODEL ).5V M, +«^-Ph, +Plfc (c) DYNAMIC EQUILIBRIUM OF HORIZONTAL FORCES Figure 1: Dynamic model of liquid-containing tank rigidly supported on the ground PERIOD, T Figure 2: Normalized design response spectrum showing acceleration as a function of period and damping

7 Earthquake Resistant Engineering Structures 379 = ZIS x. (9) = A/A/,- i2 00) The significance of the lateral coefficients Q and Cj can best be illustrated 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 of vibration T, where T>0. Cc represents the DAF of the convective component of the stored liquid, 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 tank wall (together with the impulsive component of the liquid), which constitutes 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 and energy-dissipating ability of the structure and are used to essentially convert the elastic response spectrum into an inelastic <^)U3)(i4) Section 3 below presents a summary of all the Equations involved in the seismic analysis of a liquid-containing circular tank; while Section 4 contains a step-by-step procedure for calculating all the necessary parameters to solve Equations (3) and (4), and for computing the resulting forces and stresses.

8 380 Earthquake Resistant Engineering Structures TABLE 4 lw VALUES TYPE OF STRUCTURE * (a) Anchored, flexible-base tanks (Bl) ON OR ABOVE GRADE 4.5 fiffi BURIED^ 4.5^ 1.0 (b) Fixed or hinged-base tanks (A) (c) Unanchored, contained or uncontained (B2 or B3) (d) Elevated tanks Buried tank is defined as a tank whose maximum water surface is at or below ground level. 2. Rwi is the maximum R*i value to be used for any liquid-containing structure 3. Unanchored, uncontained tanks shall not be built in zones 2B or higher. *For tank types see Fig. 3 3 Design Equations w.= tanh(0.866 D H D W~ '.W il) H 0230 x tanh(3.68 M 02) CO = (13)

9 Earthquake Resistant Engineering Structures TANK WALL (Typ) r- FLOOR (Typ) CLOSURE STRIP A1-FIXED A2-HINGED OR PINNED TYPE A - NON-FLEXIBLE BASE CONNECTIONS SEISMIC CABLES )/OR ANCHORS (Typ) A/ /- FLEXIBLE BASE PAD r (Typ) FLEXIBLE CONTAIN- 81-ANCHORED FLEXIBLE BASE B2-UNANCHORED, CONTAINED FLEXIBLE BASE B3-UNANCHDRED, UNCDNTAINED 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 their wall-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 prestressedconcrete, flexible-base circular tanks

10 382 Earthquake Resistant Engineering Structures from Table 5) (14) (15) Q = J3.68g x tanh(3.68 -) (16) (17) TV For tank types Al and A2 (Fig. 3), (18) For tank type Bj_, B2, or B3 (Fig. 3), (19) k = a *j*5 * ", r P P P (20) _ ~ 1.25,. 5 < 2.75) (21) c = c 6.0 (22) cosh(3.68^)-l 1 - D (23) D D hi: For tall tanks (D/Ht <1.333), /i = ( ) H L J x// (24a)

11 Earthquake Resistant Engineering Structures 383 For shallow tanks (D/Ht >1.333), hi = HL (24b) hw = 0.5OHw (for uniform-thickness wall) (25) g = O.oi51( )^ ( )-f (Reference 11) (26) (References 20 and 21) (27) (29) (30) (33) Piy and Pcy represent the vertical distribution of the lateral impulsive and convective forces respectively; and Pwy represents the vertical distribution of the inertia force of the tank wall. The distribution of these forces is defined and illustrated in Fig. 5.

12 384 Earthquake Resistant Engineering Structures TABLE 5 COEFFICIENT HL/R D/HL c* (*) Based on tw/r = 0. 01, PL/PC and Poiss an's Ratio v = 0.17 By A. S. Veletsos, unpublished communicatioii

13 Earthquake Resistant Engineering Structures 385 «g < e»«3- - EXACT PRESSURE DISTRIBUTION LINEAR APPROXIMATION =»$: 4Hf - 6/1-6H, - 12/i, x ^ ' ^ ^ '^ P = m 4//, L -6h^-(6Hr -0^-^D/7^-lZA7^ -126 (Adapted from Reference 19) F /^ = (Constant for uniform-thickness wall) ^ Figure 5: Vertical distribution of impulsive and convective hydrodynamic forces, and wall inertia forces, along the wall height

14 386 Earthquake Resistant Engineering Structures 4 Step-By-Step Design Procedure 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 stored liquid, Wj and the convective component, We, using Equations (11) and (12). 3. Calculate the combined natural frequency of vibration, C0j, of the containment 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 the stored liquid, Equations (15) and (16). 5. Using the frequency values determined in 3 and 4, calculate the corresponding natural periods of vibration, Tj and TC, Equations (17), (18), (19), and (20). 6. Based on the periods determined in 5, calculate the corresponding lateral force 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 sitespecific spectral accelerations Ai and & for coefficients Q and Q (step 6), S (step 8) and coefficient Z (step 7) combined. Ai, representing the Effective Peak Acceleration, should be used for short-period structures (T<0.31 sec) and for a damping ratio P = 5%;, while AC, representing the Effective Velocity-Related Peak Acceleration, should be used for longperiod 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 & Select the Factor R* specified for the class of structure being investigated, Table Compute the total base shear, V, Equation (9) 11. Compute the vertical distribution of the impulsive and convective force components, Fig Overturning Moments 12. Calculate the heights h*, hr, h; and he to the center of gravity of the tank wall, roof, impulsive component and convective component respectively, Equations (23), (24) and (25)

15 Earthquake Resistant Engineering Structures Calculate the overturning moment M due to the impulsive and convective force 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 spectral amplification factor, Kv, 4.4 Stresses 17. Unit shear stresses at the base: (35) 18. Dynamic hoop (circumferential) stresses in tank wall: Calculate the circumferential forces, Njy, Ncy and N*y due to the impulsive, convective, and wall inertia forces respectively; and circumferential forces, Kv*Nhy, due to the vertical acceleration, Equations (29), (30), (31), (32) and (34). Combine by the SRSS rule, and obtain the resultant hoop stress Oy: 'N- f,y+»wy) +N \ +"cy + N^,...,, w 19. Calculate the vertical stresses due to the overturning moments. 4.5 Sloshing 20. Sloshing of the convective portion of the stored liquid causes vertical displacement of the liquid surface which represents the amplitude of oscillation, Fig. 1. The maximum displacement, or sloshing height, dmax, can be calculated using the following approximate relationship: CJ P?)

16 388 Earthquake Resistant Engineering Structures 5 Notation Aa = seismic coefficient representing the Effective Peak Acceleration A. = cross-sectional area of the base cable, strand, or conventional reinforcing B = fraction of horizontal acceleration to be used as vertical acceleration Cc = 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 diameter EC = modulus of elasticity of concrete ES = modulus of elassticity of steel g = acceleration due to gravity Gp = shear modulus of elastomeric bearing pads he, 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 liquid HW = wall height I = importance Factor, Table 2 ka = 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 pads LS = effective length of base cable taken as the sleeve length plus 35 times the cable diameter Ncy = 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 Nwy = accelerating liquid at height y from the base (Equation 29) 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. 5 Q = a coefficient defined in Equation (16) R = radius of circular tank

17 Earthquake Resistant Engineering Structures 389 Rw = response-modification factor representing the energy-dissipating ability of the structure (Rwc for the convective component of the accelerating liquid; R*i for the impulsive component), Table 4 S = site profile representing the soil characteristics as they pertain to the structure, Table 3 Sp = spacing of elastomeric bearing pads Ss = spacing between individual base cable loops tp = thickness of elastomeric bearing pads tw = wall (shell) thickness TC = natural period of thefirst (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 acceleration V = total horizontal base shear Vc = horizontal shear due to convective component of the stored liquid Vj = horizontal shear due to impulsive component of the stored liquid Vr = horizontal shear due to the inertial force of the accelerating tank roof Vw = horizontal shear due to the inertial force of the accelerating tank wall Wp = width of elastomeric bearing pads We = 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 liquid Wr = mass of the tank roof Ww = 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 1 a = angle of base cable or strand with the horizontal P = percent of critical damping YL = unit mass of water e = ratio of effective mass of the tank shell (wall) to the actual mass, (Equation 26) PC = mass density of concrete PL = mass density of the contained liquid Gy = 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 thefirst (convective) mode CD; = circular frequency of oscillation of the tank wall (plus the impulsive component of the tank contents)

18 390 Earthquake Resistant Engineering Structures 6 References 1. Westergaard, H.M. Water pressures on dams during earthquakes, Transactions of the American Society of Civil Engineers, Vol. 98, Jacobsen, L.S. Impulsive hydrodynamics of fluid inside a cylindrical tank and offluidsurrounding a cylindrical pier, Bulletin of the Seismological Society of America, Vol. 39, No. 3, pp , Jacobsen, L.S. & Ayre, R.S. Hydrodynamic experiments with rigid cylindrical tanks subjected to transient motions, Bulletin of the Seismological Society of America, Vol. 41, , Housner, G.W. Dynamic pressures on accelerated fluid containers, Bulletin of the Seismological Society of America, Vol. 47, No. 1, 15-35, Housner, G.W. The dynamic behavior of water tanks, Bulletin of the Seismological Society of America, Vol. 53, No. 20, pp , Housner. Dynamic Pressure on Fluid Containers, Technical Information Document (TID) 7024, Chapter 6 and Appendix F, U.S. Atomic Energy Commission, Veletsos, AS & Yang, J.Y. Dynamics offixed-baseliquid storage tanks, Proceedings, U.S.-Japan Seminar on earthquake with emphasis on Lifeline Systems, Tokyo, pp , November 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 Haroun, MA & Housner, G.W. Earthquake response of deformable liquid-storage tanks, Proceedings, Pressure Vessel and Piping Technology Conference, San Francisco, CA, August Haroun, M.A. & Housner, G.W. Seismic design of liquid-storage tanks, Journal of the Technical Councils of the American Society of Civil Engineers, Vol. 107, No. TCI, pp , 1981.

19 Earthquake Resistant Engineering Structures American Society of Civil Engineers, Guidelines for the Seismic Design of Oil and Gas Pipeline Systems, Prepared by the Committee on Gas and Liquid Fuel Guidelines of the Technical Council on Lifeline Engineering, Section 7, International Conference of Building Officials, Uniform Building Code, Whittier, CA, Applied Technology Council, Tentative Provisions for the Development of Seismic Regulations for Buildings. ATC 3-06, June Federal Emergency Management Agency, NEHRP Recommended Provisions for Seismic Regulations for New Buildings, Parts 1 & 2, FEMA 222A and 223A, Prepared by the Building Seismic Safety Council, Washington, DC, May American Water Works Association, Standard for Wire-Wound Circular Prestressed-Concrete Water Tanks, ANSI/AWWA Dl American Water Works Association, Standard for Circular Prestressed Concrete Tanks with Circumferential Tendons, ANSI/AWWA D American Concrete Institute, Design and Construction of Environmental Concrete Structures, Chapters 21 & 52, Seismic Provisions (Draft), BOCA (Building Officials and Code Administration International), National Building Code, Standards Association of New Zealand, Code of Practice for Concrete Structures for the Storage of Liquids, NZS 3106: Marchaj, T.J. Importance of vertical acceleration in the design of liquidcontaining tanks, Proceedings of the 2nd U.S. National Conference on Earthquake Engineering, Stanford, University, Stanford, CA, pp , August Luft, R.W. Vertical acceleration in prestressed concrete tanks, Journal of Structural Engineering of the American Society of Civil Engineers, Vol. 110, No. 4, pp , 1984