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Rev 2 to Calculation 52182-C-039, Seismic Margins HCLPF Calculations for Refueling Water Storage Tank
	ML18151A651
Person / Time
Site:	Surry, North Anna
Issue date:	04/05/1997
From:	; EQE ENGINEERING CONSULTANTS (FORMERLY EQE ENGINEERING
To:	;
Shared Package
ML18151A652	List: ML18151A651; ML18152A083;
References
	52182-C-039, 52182-C-039-R02, 52182-C-39, 52182-C-39-R2, NUDOCS 9810070209
	Download: ML18151A651 (89)
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4. General Description The Refueling Water Storage Tank (RWST) is a large flat bottomed water storage tank that is mounted in the yard. The foundation consists oflow strength reinforced concrete (1000

!171.Li"-~C-/Y.-8 -rl))#- psi) that has been poured to bedrock (DWG.J:r:11 Y.:P@ 12 B.J.:i). The tank is anchored to the foundation through the use of 44 - 2" diameter anchor bolts (A325), and carbon steel chairs that are welded to the tank. The tank is 57'-6" to the spring line, and has a domed roof that extends another 5'-1" above the springline for a total height of 62'-7" (DWG. l l 715-FV-44A-3). The tank was manufactured by Nooter Corp. It has a nominal capacity of 480, 000 gallons. The weight of the tank when empty is 102,000 lbs, and it is 4,272,000 lbs when full. These calculations determine the most likely failure mode, or limiting capacity of the RWST

to seismic input motion. Separate check are made for: I) anchor bolt yield, 2) concrete pullout, 3) top-plate bending, 4) weldrnent of the bolt chair to the tank waH, and 5) overturning moment capacity. A sketch of the tank is included.

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~*. : :.. ~ Refueling Water Storage Tank 1-QS-TK-1, 2-QS-TK-1 North Anna HCLPE Cafcufations 52182-C-039, Rev. 1 Refueling Water Storage Tank (1-QS-TK-1; 2-QS-TK-1) Response Computations RWSTRSP.MCD 12/21/9 Attachment A, 1/1 l This MA THCAD template computes the response parameters which are needed in performing a tank evaluation per EPRI NP-6041 methodology for vertical tanks. Inputs required are an assumed earthquake, and the necessary tank parameters. Two non-dimensional parameters are also needed for the calculation of the diamond-buckling capacity, and for the calculation of the C!Jmpressive buckling capacity (elephant's foot buckling). Base units are feet, seconds, and pounds. Derived Units: kip= I 000* lbf hz= l*sec* 1 ksi = lOOO*psi Define Tank Geometry: R := 19*ft Nominal Inner Tank Radius Rd := 38.*ft Dome Radius (estimated) H := 56.5*ft Height to Water Elevation t b := 0.25*in Bottom Plate thickness t d := 0.3125*in Dome thickness h d : =Rd" ( 1 - cos( asin( td))) Clearance between peak of dome to spring line nrings :=4 Number of different diameter rings composing the tank shell 0.375 0.3125 *in t := 0.25 Shell Thickness at each ring from bottom of the tank to the top. 0.1875 8.0 Hr:= 8.0 *ft Height of each ring measured from the bottom of the tank to-the top. 10.0 31.5 Define Anchorage Details; n :=44 cj, :=2.00*in Number of equally spaced anchor bolts Anchor bolt diameter *
. -~ Refueling Water Storage Ta~k 1-QS-TK-1, 2-QS-TK-1 52182-C-039, Rev. 1 Define Material Properties; E 5 :=27.7*106*psi
Young's Modulus for Shell Material Eb := 29* I06*psi aye :=40*103*psi lbf
'YI :=62.4*3 ft )bf y 5 : = 0.284*-:--j m 1C J := 3.25* I05*psi Young's Modululs for Bolt Material Effective yield stress for shell material Unit Weight for liquid Unit Weight for shell material Bulk Modulus of fluid, 3.25x10**5 psi for water RWSTRSP.MCD 12/21/9E Attachment A, 2/10
. '\\... ~ 11:_ Refueling Water Storage Tank 1..QS-TK-1, 2-QS-TK-1 Define Earthquake Amplification Parameters: 52182-C-039,Rev. 1 Note that this template was created for perfonning response calculations according to the methodology contained in Reference 1 - This assumes a NUREG CR/0098 type response spectrum. This is the place where the amplification factors are input pga := 0.30*g vmatio :=36*[ i:i] The varatio is 36 for rock, 48 for soil v :=varatio*pga 6*v2 d *-.-- NEP :=50 (non exceedance probability, 50%, or 84%) pga RWSTRSP.MCD 12/21/95 Attachment A, 3/10 amp 8(P) :=if(NEP=84,4.38-l.04*ln(P),3.21- 0.68*ln(P)) Definition for Spectral Amplification. ampv(P) :=if(NEP::84,3.38- 0.67*ln(~),2.31- 0.4J.ln(~)) Factors (NUREG CR/0098) amp d(~) := if(NEP::84,2.73 - 0.45*ln(~), 1.82- 0.27*ln(~)) amp 8(5) =2.116 amp y(5) = 1.65 amp d( 5) = 1.385 Example: amplification factors for 5% damping. Note that these values check with those given in NUREG/0098 for non-exceedance level selected. f 3 := 8*hz f 4 := 33*hz (Equation 1)
Refueling Water Storage Tank 1-QS-TK-1, 2-QS-TK-1 52182-C-039, Rev. 1 Compute average shell thickness, and total shell height i := 1.. nrings Hs :=LHri i tavg := Hs t avg= 0.242*in RWSTRSP.MCD 12/21/95 Attachment A. 4/1 O t avg+ min(t) t eff := 2 t eff = 0.215 *in The effective thickness should be more weighted toward the top of the tank, where the deflections are greatest (Ref. GIP) Define Dimensionless Parameters from Refs. 2 and 4: Obtain Cwi from Reference 2. Table 7.4: Parameters needed for table 7.4: H R =2.974 t eff R =9.41&10 4 Read-off value for Cwi: C WI := 0.079 Tank Weight and C.G. Components: Note that the distance to the component C.G. is measured from the bottom of the tank. (Shell) j := 1.. nrings wi := 2*1t*R*y 5*Hri"ti Ws:=I:wi i W 5 = 67.893 *kip Hr. ~ c*.> J C&j :: ~Hr( J!,J - 2 i (Bottom Plate) W b := (1t*R2)*t b"'Y s W b = 11.595 *kip X 5 =24.S28*ft
52182-C-039, Rev. 1 d"h d )*t d"'Y s kip (Liquid} 3*a R teff
1 _ 2*sin(a)*(I - _teff +-1 -)]
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W w := 7t*R2*H*YJ w w = 3.998* l o3*kip H Xw:=2 X w = 28.25*ft Maximum fluid pressure occurs at base of tank RWSTRSP.MCD 12/21/95 Attachment A. 5/10 iI;t;t<);*... *. fr).:... </:/*:*.-. I; :' ~* ' ',, * *.. ',
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~~ Refueling Water Storage_ Tank 1..QS-TK-1, 2..QS-TK-1 Compute Horizontal tmpulstye Mode Response: 52182-C-039, Rev. 1 RWSTRSP.MCD 12/21/95 Attachment A. 6110 (Reference 1, equation H-2} f1 =3.597*hz Compute Spectral Acceleration at this frequency, damping for the impulsive mode may be taken as about 5% (this is considered to be a median estimate of damping). S 8 (f1,S) =0.635*g. s ah:= s 8(fz,5) (See Equation #1) Compute Weight of fluid effective in the Impulsive Mode, and its corresponding C.G.:
- *{H 3 tanh( 1.732*~)
Rl Wi.-1 R~' R ,1.0- 0.436-H *Ww (Ref. 3, Eqn. C3500-1,-2,-3,-4) 1.732*H X .~H 3 R) i :=1.\\R~,0.375,0.5-o.1ss-8 *H (Ref. 3, Eqn. C3500-1,-2,-3,-4) W i = 3.412* I03*kip Xi = 24.678*ft _Compute Impulsive Mode Base Shear and Overturning Moment sah V1 :=g*(Wb+ W s+ Wi) (Ref. 1, Eqn. H-3) sah Mz :=g*(Wh-Xh+ W 5*X 5:r Wj*Xi) (Ref. 1, Eqn. H-4) VI =2.219"I03*kip MI =S.Sl*I04*kip*ft Estimate hydrodynamic fluid pressure on the tank at the bottom plate sah Wi"Xj*- pi := 1.36*R* Pi -= 4.499-psi (Ref. 1, eqn. H-8: Note this is conservative at fluid depths less than about 0.1 s*H) ~: ~: ;*_:*:* ~* :*
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~* Refueling Water Storage Tank 1-QS-TK-1, 2-QS-TK-1 52182-C-039,Rev.1 Compute Horizontal conyectjye (Sloshing) Mode Response: Convective Mode frequency {Ref. 1, eqn. H-10) f C = 0.281 *hz Compute Spectral Acceleration at this frequency, damping for the convective mode response is primarily fluid controlled and is estimated to be about 0.5%. S a(f c,0.5) =0.098*g Sac := S a(f c* 0.5) (See Equation #1) Compute Weight of Fluid acting in the convective mode and its C. G. location W c :: [ ( 0.46*:)*tanh( 1.835*~) }w w W c =618.497*kip X c =46.234*ft (Ref. 1. eqn. H-11) (Ref. 1, eqn. H-12) Compute Convective Mode Base Shear and Overturning Moment: Sac V c :=-*W c (Ref. 1, eqn. H-13) g Sac Mc:=-*Wc*Xc g {Ref. 1, eqn. H-14) V c = 60.549 *kip Mc =2.799-lo3.ftkip Compute Hydrodynamic Convective Pressure at fluid depth "y" y := H This maximizes the hydrodynamic convective pressure 0.267*W *S cosh(l.835* HR-y) P *- w ac C.-
----..----=-""'-
g*R*H cosh( 1.835*:) (Ref. 1, eqn. H-16) Pc a: 0.006*psi RWSTRSP.MCD 12/21/95 AttachmentA, 7/10
. Refueling Water Storage Tank 1-QS-TK-1, 2-QS-TK-1. 52182-C-039,Rev.1 Compute the fundamental mode fluid slosh height Sac h 5 :=0.837*R*- g h 5 = 1.557*ft Compute Vertical fluid Mode Response: (Ref. 1, eqn. H-17) Compute the vertical fluid mode fundamental frequency I l [YI ( 2*R 1 )]- 2 fv := 4*H* s*i tetrEs +KI (Ref. 3, eqn. C3500-13) f V = 4.269 *hz Compute the hydrodynamic vertical fluid response mode pressure, based on a tank on a rigid foundation, note this pressure is also at y=H, which maximizes p. p v := O.S*y i-H-S av(; v,5).cos( ;.(H~ y)] y =56.S*ft p V = 8.287 *psi RWSTRSP.MCD 12/21/95 Attachment A. 8/1 O ~ *.... :..
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52182-C-039, Rev. 1 RWSTRSP.MCD 12/21/95 Attachment A, 9/10 combine Individual Mode Responses to get Total Seismic Demand: Base Shear: v tot:= j(vi2+ v c2). V tot =2.219"103*kip Overturning Moment Mtot := j(M 12+ M c2) M tot = S.S 1 7* I04*kip* ft Fluid Pressures: Psh :=jPi2+Pc2 P cmax := P st+ P sh+ 0.4*P v P cmin : = P st + P sh - 0.4* P v Pavg :=Pst- 0.4*Pv P cmax = 32.298 *psi P cmin = 2S.668 *psi P tmin = 16.669 *psi P avg =21.168*psi Total Horizontal Seismic Response Maximum and minimum compression zone pressures at the time of maximum base moment. (Ref. 1, eqn. H-22} Minimum tension zone fluid pressure at the time of maximum base moment (Ref. 1, eqn. H-23) Minimum average fluid presssure on the base plate at the time of maximum base shear (Ref. 1. eqn H-14) *.. Expected minimum total effect weight of the tank shell acting on the base at the time of the maximum moment and base shear: ( 2 pga) W te := (Wh + W s)* 1- 0.4**rg (Ref. 1, eqn. H-26) W tc = 76. 753 *kip W h = 1S.S3S *kip W s = 67.893 *kip
Refueling Water Storage Tank 1-QS-TK-1, 2-QS-TK-1
References:
52182-C-039,Rev.1 RWSTRSP.MCD 12/21/95 Attachment A. 10/1 O
1. A Methodology for Assessment of Nuclear Power Plant Seisml Margin (Revision 1),
EPRI NP-6041-SL, Final Report, Electric Power Research Institute, Palo Alto, CA, August, 1991.
2. AS. Veletsos, "Seismic Response and Design of Liquid Storage Tanks", Chapter 7, Guidelines for the Seismic Design of Oil and Gas Pipeline Systems, ASCE, 1984.

3. ASCE Standard and Commentary - Seismic Analysis of Safety-Related Nuclear Structures, ASCE 4-86, ASCE, September 1986.

4. Buckling of Thin-Walled Circular Cylinders, NASA SP-8007, National Aeronautics and Space Administration, August 1986.

5. Newmark, N.M., and Hall, W.J., Development of Criteria for Seismic Review of Selected Nuclear Power Plants, NUREG-CR 0098, U.S. Nuclear Regulatory Commission, 1978.
Refueling Water Storage Tank 1-QS-TK-1, 2-QS-TK-1
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- ~-, *:-** - - ~~~ 52182-C-039, Rev 1 TOPPLATE.MCD 12/21/95 Attachment B, 1/2 Maximum Pennissible load in Plate. This template is based on yield line theory as it applies to the top plate bending in bolt chairs. Bolt chairs are typically used in the anchorage of large, flat bottomed storage tanks to distribute shears in tanks to the anchor bolt. Large bolt projections provide for ductile response under seismic loading. The moment capacity of these tanks is limited by the compressive buckling capacity on the compression side of the tank, and by the allowable bolt hold down tensile capacity on the tension side of the tank. The allowable tensile capacity is, in tum, limited to the smallest of the: 1) force required to yield the anchor bolt, 2) force required to pull the bolt from the concrete embedment (concrete failure), 3) force required to bend the top plate to a maximum allowable deflection, and 4) force required to tear the bolt chair to tank wall weld. This template computes the force required to collapse the top plate based on top plate bending. define unit variables kip = 1000* lbf ksi = I 000* psi define plate dimensions: a :=2.0*in b :=4.0*in dimension of top plate from tank wall to C. Line of bolt hole dimension of top plate from tank wall to outer extent g := 6.0*in dimension of top plate adjoining tank wall f := 0.875*in dimension of top plate from edge of bolt hole to free edge t pl:= 0.75*in top plate thickness t vp :=.375*in vertical (gusset) plate thickness t tn1c := 0.375*in tank wall thickness ~ h := 1.I3*in radius of the bolt hole Reff<= 1.5625*in effective location of the applied bolt load - may be taken as equal to one-half of the width of the nut, for Heavy Hex 2" Nuts the distance across the flats is 3-1/8. define plate material properties Etn1c :=51.7*ksi Evp :=30*ksi E pl:= 30*ksi Yield Stress for tank wall material (SA 240, Grade 304L) For this material, the effective yield stress may be taken to be equal to 2/3 of the ultimate stress plus 1/3 of the minimum specified yield. Stainless steels have a very flat stress strain curve. Thus, the outer fiber must go to ultimate, while the inner fiber goes to yield. Minimum yield is 25 ksi, Minimum ultimate is 65 ksi. Yield Stress for vertical plate material (A283, Grade C). For low carbon steels, with a well defined yield limit, the effective yield is taken to be the minimum specified yield. Yield Stress for top plate material
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Refueling Water Storage Tank 1-QS-TK-1, 2-QS-TK-1 52182-C-039, Rev 1 calculate plastic moment capacities for yield lines shown io figure 1, Mwa11 :=.5*t 1n1{EtJ1k M pl := 0.2S*t p{ E pl M pl := if(M wan>M pl,M pl,M wall) M p2 :=if(M vp>M pl,M pJ,Mvp) Mp3 :=Mp] Mp4:=Mp3 M wall = 3.635 *kip*~ m _Mpl =4.219*kip*: M vp = 1.055 *kip* !n m M pl = 3.635 *kip* ! 0 m M p2 = 1.055 *kip* ! 0 m M 3 =4.219*kip-! 0 p m M p4 = 4.219*kip* ~n ID calculate load corresponding to collapse for yield lines shown io figure 1, i :=1..25 ~i := [ (i - I)*~~ }deg
- a - R h*sin(~i)
].. - . ( ) I SID aj Pu1 c= ( g 2 r[ M prlcot( a,) + M p2*b + M p3" r.-*lll( a,) + ~ + M p4* l;*cos( a,) *cot( a,) l 2-Reff' CI :=min{Pu) CI =47.I94*kip 49 Pui kip -49 5 JO Pi clei TOPPLATE.MCD 12/21/95 Attachment B, 1/2
Refueling Water Storage Tank 1-QS-TK-1, 2-WS-TK-1 North Anna HCLPF Calculations 52182-C-039, Rev. 1 Refueling Water Storage Tank (1-QS-TK-1, 2-QS-TK-1 ), Fluid Hold Down Forces Fluid Hold-Down Forces RWSTHD2.MCD 12/21/9 Attachment C, 1/6 For tanks with minimum anchorage, hold-down forces resulting from fluid pressure acting on the tank bottom will contribute significantly to the overturning capacity of the tank. EPRI TR-103959, Methodology for Developjng Seismic Fragilities, Section 7, contains procedures for calculating these fluid hold-down forc~s as a function of tank uplift This Mathcad template follows the procedure in EPRI TR-103959 for determining the fluid hold-down forces. Inputs are tank parameters, the output is a plot of the fluid hold-down tension as a function of both uplift displacement and uplift length. Derived Units: kip= I 000* lbf ksi = I OOO*psi Define Tank Geometry <input); R := 19*ft Tank Radius t 5 := 0.375*in Shell Thickness H := 56.S*ft Height to Fluid Surface t b := 0.25*in Baseplate Thickness A max:= 0.07-inMaximum permissible uplift height, this is a function of the failure mode 0 n := 2.S*rad Assumed angle to the neutral axis of the tank Define Tank Material Properties <input}: E 5 := 27.7* J06*psi Young's Modulus o := 0.30 Poisson's Ratio au := 65*ksi Ultimate Strength of Tank Material a y := 25*ksi Minimum Yield Strength of Material a ye := 25*ksi Effective Yield Strength of Material, the effective yield stress of minimum is recommended in EPRI 103959 Define Fluid Pressure Parameters <input): P cmin := 25.7*psi P cmax := 32.3*psi Ptmin := 16.7*psi P tmax := 23;29*psi Defined pressures to be used in analysis; Pc :=P cmin
Refueling Water Storage Tank 1-QS-TK-1, 2-WS-TK-1 52182-C-039, Rev. 1 Calculate Additional Parameters needed for analysis: K: 3 Es*ts 12* (1 - u2) I K =[ ~-~3* (t - u 2 ) r K = 133.768 *kip*in IC =31.695 K 5 =37.191 *kip (EPRI TR-103959 calls this term Mf/P) . : ~* RWSTHD2.MCD 12/21/9 Attachment C, 2/6
Refueling Water Storage Tank 1-QS-TK-1, 2-WS-T_K-1 52182-C-039, Rev. 1 RWSTHD2.MCD 12/21/9 Attachment C, 3/6 Pertoun tterative solution for uplift height, tank shell hoJd-down tension, end moment, and maximum
posjtiye moment in the base plate as a function of the uplift length and fluid pressure:
p L2 M f(L,P) M :f(L,P) [ 2] MmaxCL,P) :=2* 4-p + ( P*L ) ( 6(L,P)) T :f(L, P) := T eCL, P) + F b(L, P)* -L - Horizontal force in the tank wall at the intersection with the tank bottom Hold down tension as a function of uplift length, the term Fh(L) is the additional correction which accounts for membrane effects EPRI IR-103959 approximates the relationship between fluid-hold down forces and uplift displacement with a linear expression, This simplification is useful in the overtuminq moment capacity evaluation. Solve for the minimum and maximum length of plate effective in resisting uplift (corresponding to zero uplift, and maximum uplift at the extreme outer tensile fiber). The fluid pressure is a function of the distance from the neutral axis of the tank:
** Reference 1, eqn. 7-38 Assume an angle to the neutral axis in order to calculate the minimum hold down force, this angle may be changed later.
P(8 n) =24.80S*psi
.*~*~:. Refueling Water Storage Tank 1-QS-TK-1, 2-WS-TK-1 ~.. 52182-C-039,Rev.1 RWSTHD2.MCD 12/21/9 Attachment C, 4/6 Calculate fluid hold down force at various locations about the circumference of the tank. Note that at the neutral axis, the fluid hold down forces will be minimized, since there is zero tank uplift. At the point of maximum tension the fluid hold down force will be maximized: (cos(8)- cos(8 n)) A(8).-Amax* 1-cos(8n) N:=20 x := lO*in (Initial guess, note that the above derivation it is assumed that delta is less than 2/10 of L ) i := l.. N ( i - 1 ) J3i:=8n*N-l L is the length of plate effective in resisting uplift and is calculated here as a function of the angle beta measured about the tank circumference Tf is the fluid hold-down pressure and is calculated here as a function of the angle beta measured about the tank circumference. Print out the length of baseplate effective in resisting uplift and the corresponding fluid hold-down forces at both the neutral axis and at the point of maximum tension in the tank in order to check published solutions: L1 =10.761*in Tf1 =O.I22*~ip m ~ =6.993*in ki Tf,N = 0.13 -~ m Point 1 is at the point of maximum tension in the tank Point N is at the neutral axis.
Refueling Water Storage Tank 1-QS-TK-1, 2-WS-TK-1 52182-C-039, Rev. 1
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EPRI TR-103959 recommends fitting a best-flt linear relationship for the fluid hold-down force as a function of the angle beta: ai := cos(J3i). a : = intercept( a, TI) AT r:=slope(a, Tt) ki AT r=--0.006*~ m T fil :=AT fCOS(~) + a ki T fit =0.135*~ m Plot fluid hold-down force versus the angle beta for the tank: 0.135,---~---.---..--~---,----,----,-----.---.------. 0.13 T, (Kips/in} 0.125 0.12 L---'::-:--....L..--.t.-----L---L---...1....--'---.....I.--...J..-__J -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Cos (J3)
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... ** =*.'. -*: .. ~.:.~. 52182-C-039,Rev. 1 RWSTHD2.MCD 12/21/9 Attachment C, 6/6 The Upper limit on this equation is based on a fully plastic moment at both ends of the uplifted zone of the base plate. and should be limited to the following values: check :=max(M) check = 0.341 *kip* !n m M pb = 0.807 *kip*~ JD The maximum moment in the tank shell shou@ be limited to 90% of the fully plastic moment capacity. Mall :=0.90*M pb M all = 0. 727 *kip* !n m (OK), Mall is greater than check.
Refueling Water Storage Tank 1-QS-TK-1, 2-0S-TK-1 Oyertumjng Moment CapacitY 52182-C-039, Rev. 1 RWSTCAP2.MCD 12/27/9 Attachment D, 1/4 The overturning moment capacity of anchored tanks is computed in an iterative process. EPRI NP-6041, A Methodology for Assessment of Nuclear Power Plant Seismic Margin <Revision 1 ), appendix H, contains procedure for calculating the overturning moment capacity. This template has been updated to incorporate the recommendations given in EPRI 103959. Updated 12/2/94. This Mathcad template follows the procedure in EPRI NP-6041 for determining the overturning 111oment capacity of tanks. This template is intended to be used in conjunction with the terT)plate FLUIDHD. The FLUIDHD template calculates the fluid hold-down forces which lead to slightly increased capacities for marginal tanks. This template may be used with or without consideration of the FLUID hold-down forces. Derived Units: kip= 1000* lbf ksi = 1 OOO*psi Define Tank Parameters R := 19*ft WTE :=76.75*kip N:=44 o eo :=0.2S*in Es :=27.7-l<f*ksi t 5 := 0.37S*in P cmin := 25.7*psi P cmax := 32.3-psi P avg := 21.17-psi er ye:= 2S*ksi Tank Radius Expected Minimum Total Effective Tank Weight Number of equally spaced anchor bolts Permissible uplift elongation Young's Modulus for the Tank Material Tank wall thickness at bolt chair location Minimum pressure in compression zone at the time of maximum moment Maximum pressure in compression zone at the time of maximum moment Average pressure across the bottom plate at the time of maximum moment Effective yield stress for tank material Define Anchor Bolt Parameters h a := 36*in Depth to embedded anchor bearing surface h c := 11.S*in Height of anchor bolt chair above tank foundation Ab := 3.14*in2 Nominal Area for each anchor E b : = 29* 103
ksi Anchor modulus of elasticity (ksi)
Tbp :=O.O*kip Anchor bolt pre-load (kips) T BC:= 47-kip Maximum anchor bolt load (anchor capacity)
Refueling Water Storage Tank 1-QS-TK-1, 2-0S-TK-1 52182-C-039, Rev. 1 Obtain diamond bucldjna coefficient from Reference 4, Figure 6: Parameter needed for Figure 6: ~- - =0.343 P * (R)2 Es ts Read-off value for delta-gamma /J.y := 0.16 Input Unear Approximation to fluid hold::down forces <from template FLUIDHD) kip T fh := 0.135*-.- fluid hold-down force at estimated neutral axis ID ki /J. T e : = -0.006*....! Slope of linear approximation ID Compute the Buckling Capacity of the Tank Shen: axial stress limit at the onset of elephant's foot buckling: S l := (~)* 4~0 S l = 1.52 diamond buckling capacity based on NASA SP-8007: ~ := /6.;, y := 1- 0.73*(1-c-+) E s*t s a.cb :=(0.6*y+ t.y)*--,r-a cb = I.894*104*psi Compressive shell capacity: CB :=i~a cb<(0.9*a p),a cb* (0.9*a p)Jt 5 kip C B== 2.068 *-. ID RWSTCAP2.MCD 12/27/9 Attachment D, 2/4 (Ref. 1, eqn. H-27)
Refueling Water Storage Tank 1-QS-TK-1, 2-0S-TK-1 52182-C-039, Rev. 1 Compute dimensionless parameters for solution scheme Note: the angle Beta represents the angle to the neutral axis, it is assumed prior to the start of the algorithm. C
=
1 + cos(J3) 1(J3)
sin(f3)+(7t-f3)*cos(f3)
. C (J3) :- sin(f3)*cos(f3)+7t-f3. 2 1 + cos(J3) C
=
sin(J3)- f3*cos(f3) *(l +cos(f3)) 3(J3)
sin(f3) + (7t-f3)*cos(f3) 1-cos(f3)
C ( f3) := J3-sin( f3)*cos( f3) 4 1 - cos(J3) Compute Anchor Tension in Each Anchor Bolt as a Function of Location i :=1,2.. N cxi := [ (i-1 )*~!~}deg T
=T
+ ~eo-Ab-Eb*(cos(cxi)- cos(f3)) Bi bp ha+hc 1-cos(J3) TB. :=if(TB.:STBc,TB.,TBc) I I I TB.:= if(TB)::O.O*lbf, TB.,O.O*lbf\\ I
1.
I j RWSTCAP2.MCD 12/27/f, 3/4 kip AT c =-0.006*-.- m Slope is based on best linear fit to fluid hold down forces M sc(J3) := C mCJ3)*C 2(J3)*R2+ ~TBi*R*cos(~) + T fii"R2*2*sin(J3) + AT c*C 4(J3)*R2 J
Refueling Water Storage Tank 1-QS-TK-1, 2-QS-TK-1 52182-C-039, Rev. 1 RWSTCAP2.MCD 12/27/9 Attachment D, 4/4 Assume a beta value to begjn solution: Note that the angle beta represents the angle between the maximum tension side of the tank and the neutral axis. For a lightly anchored tank the angle beta should approach pi. For anchored tanks, the angle should be between zero and pi. The assumed beta and the calculated angle below should be identical for the final solution. f3=1.365 assumed beta Solve for beta. Note that this is an iterative process. The beta printed below should be equal to the assumed beta for the final solution. Therefore, the user must input a new assumed beta which is equal to the assumed beta for the final run. ~:=1,1.1..3 ft~) :=Cm(~)- CB cp := 1.365 angle := root(ft cp ), cp) angle= 1.365 Final Results: Moment Capacity: M 5c(angle) =2.803*104*kip*ft Shear Capac;:ity: The plot below shows the difference between the compressive stress in the tank wall and the calculated buckling stress as a funciton of an assumed angle. The point at which the curves intersect is the correct angle beta. tU;> s*106 0_kip ft -s*10-.._ __ 2 3 4 COF :=0.70 Recommended value of 0.70 in EPRI NP 6041-SL, for steel tanks on a sand cushion. V hclpf = 3.098* Ia3*kip where: LT B. = 892.227 *kip I I W TE= 76.75*kip P avg*(1t*R2) =3.457*la3*kip
~--~fj - I :::,::,o ii:) : :;:) I ""av] r"'u1*1 rv- ~-.,~.:..1.,11 , 1..,,, o,:::J:::1 11-,4.. 'S p~ I/ 3 ,.-. I M y l J;* * )1,1'/;~ STRUCTURAL TITLE 7 :'JI f'G: CONSULTING ~ )nr /t:fl'rn ~,/jll,(' p~ MECHANICS 8 fK DATE_~ - 8Y'---J..J._...i..... __ ,Aca!Lo, i > Job No / 0 J couMENreSI £fl~._ f;,,,J. CHKD. IY--= DATI I I C pr#! A,t;n.-'-qc:... 1.. {'rirl<./ fh~J~/.f"fowh} ~ *.-'. *Ji T ~fc_l ..e J I>
I
~ LJM:~
~@
~~ p?"" W12. i f4
1 ~
Jj_
---~... ______...
/Vvf /,.~./ ()r, /frfe f( o/i./'fr;fl~ C or d wA,*r ~ ~Pr /r1CJ11.~J /,,J./ ~ e - e - ~re1J~ 7~/l ( { o/i - ~.r/n-@) ~ .r;,, f) =- e - /(.r{0J ?c -= 1,-{!tf, b + ). -l~A fr In).,:;. I-,# I- (,'3 +-~,. ;,. ')} C, = r d~.:rr z.. UirJ...~ ~.r/ - -1. 2.. CT,;t )IA
q;?f M
I' C :#1 A1 *-= ~°e.;/ < M /YI-I-4- \\ z.. 4 I 4. - + cl'" -r--*---*------- r-----..,,, J-{ ~l~1-tJ tv,(~ t:/e1c,('fr iwo..1-;~.rf) {; /;"f; ()-/1--r,-,;,-4,1;/t;" o. ~ a;,, J/~ :;/,.//..r/,,,.( >f 1/fvtr :11vrr (n() l,.lr/( t:fp.(.;,,r..:./ 7:,.ld) tr, = ~ + -j- ( o; - o;,) ti; = r;//i,,,rf~ Jt~;(i At" c-J, PrK Jh f>t:r ('","r; I/ of ev~M lo.f'fnk
52182.05 ksi := 1000-psi b := 6-in e :=2-in a :=3.S*in ts := 0.375-in t C := 0.75*in t g := 0.375-in er ye :=30*ksi 2-er 55-t / M *=---
s.
4 Cl kip : = 1000* lbf C :=0.44*in d := 1.44-in Rb := l.13*in I 3 := 0.875-in J er 55 := 52-ksi Mp = 50.625*ft-1 *in*kip M s = 43.87S*ft-1 *in-kip Mg= 12.656*ft-1 *in*kip By: >Hwl:> Date: "3-'a/.Cf 7 Chk'd 17~ Date:~ Shell is A204 SS, chair is A283 carbon Hinge capacity of the top plate Hinge capacity of the shell -/ M I = if(M 5>M p,M p,M 5) M 1 =43.875*ft-1 *in*kip Hinge capacity of chair to shell yield line M 2 := if(M g>M p,M p,M g) Hinge capacity of chair to gussett yield line M 2 = 12.656*ft-1 *in*kip 8=30*degu
52182.05 [ e - R b*cos(8) l P(8) :=atan....,,(b,...,.) ___ 2 - R b*sin(8) C =Q.44*in
c 1(8) := if(c<d*tan(P(8)), c, d*tan(P(8) ))
e - R b*cos(8) 14(0) := sin(P(8)) By: Mw~ Date: ~-31-'ll Chk'd O\\',ti Date: 1i1r!.1 P c<S) :=- 1-*[M 1 *b + 2*tan(P(8))*[M 2*a+ M p*(l 3 + l 4(8)-sin(P(S)) ))) c'(8) Pc< 30-deg) = 72.15 *kip i:=0,1..10 P cC7*deg) =65.326°kip P c(i-deg)
k.
66 lp -t-64._ __ ---,10
Refueling Water Storage Tank 1-QS-TK-1, 2-WS-TK-1 North Anna HCLPF Calculations 52182-C-039, Rev. 2 Refueling Water Storage Tank (1-QS-TK-1, 2-QS-TK-1), Fluid Hold Down Forces Fluid Hold-Down Forces Rwsthd3.mcd, 1/ 6 For tanks with minimum anchorage, hold-down forces resulting from fluid pressure acting on the tank bottom will contribute significantly to the overturning capacity of the tank. EPRI TR-103959, Methodology for Developing Seismic Fragilities, Section 7, contains procedures for calculating these fluid hold-down forces as a function of tank uplift. This Mathcad template follows the procedure in EPRI TR-103959 for determining the fluid hold-down forces. Inputs are tank parameters, the output is a plot of the fluid hold-down tension as a function of both uplift displacement and uplift length. Derived Units: kip= I 000* lbf ksi= IOOO*psi Define Tank Geometry {input): R := I9*ft Tank Radius H := 56.5*ft Height to Fluid Surface t b := 0.25*in Baseplate Thickness ts : = 0.375-in Shell Thickness I!> max:= 0.25*in Maximum permissible uplift height, this is a function of the failure mode 0 n : = 2.5*rad Assumed angle to the neutral axis of the tank Define Tank Material Properties {input): Es := 27.7* 106*psi u :=0.30 eru :=65*ksi er y := 25*ksi er ye :=25*ksi
Young's Modulus Poisson's Ratio Ultimate Strength of Tank Material Minimum Yield Strength of Material Effective Yield Strength of Material, the effective yield stress of minimum is recommended in EPRI 103959 Define Fluid Pressure Parameters (input):
P cmin := 25.7*psi Ptmin := 16.7*psi P cmax : = 32.3*psi P tmax := 23.29*psi Defined pressures to be used in analysis: Pc :=P cmin
Refueling Water Storage Tank 1-QS-TK-1, 2-WS-TK-1 52182-C-039, Rev. 2 Calculate Additional Parameters needed for analysis: K = 133.768 *kip*in. I 0 [,~-J3.(1-u 2)r K =31.695 K 5 =37.191 *kip R*t s ( R ) f = j (
2). l - H*K 12* 1 - u (EPRI TR-103959 calls this term Mf/P)
Rwsthd3.mcd, 2/ 6
Refueling Wajer Storage Tank 1-QS-TK-1, 2-WS-TK-1 52182-C-039, Rev. 2 Rwsthd3.mccl, 3/ 6 Perform iterative solution for uplift height. tank shell hold-clown tension, end moment, and maximum positive moment in the base plate as a function of the uplift length and fluid pressure: ( K *L ) F(L) := I+ 2*E :.J b L4 I Ks*L f*L2 P [ ( 5 ) l o(L,P) := 24-F(L). 72*E 5*1b +-r 0 E 5*1b p K 5*L ( 3 ) M :f(L,P) := F(L). I2*E s*I b + f (o(L,P)) T :f(L,P) = T eCL,P) + F b(L,P)* - 1 - Horizontal force in the tank wall at the intersection with the tank bottom Hold down tension as a function of uplift length, the term Fh(L) is the additional correction which accounts for membrane effects EPRI TR-103959 approximates the relationship between fluid-hold down forces and uplift displacement with a linear expression. This simplification is useful in the overturning moment capacity evaluation. Solve for the minimum and maximum length of plate effective in resisting uplift (corresponding to zero uplift, and maximum uplift at the extreme outer tensile fiber). The fluid pressure is a function of the distance from the neutral axis of the tank: ( I - cos(8)) P( 8) = (P c - P_ t). 2 + p t Reference 1, eqn. 7-38 Assume an angle to the neutral axis in order to calculate the minimum hold down force, this angle may be changed later. 8 n =2.5*rad P(en) =24.805*psi
Refueling Water Storage Tank 1-QS-TK-1, 2-WS-TK-1 52182-C-039, Rev. 2 Rwsthd3.mcd, 4/ 6 Calculate fluid hold down force at various locations about the circumference of the tank. Note that at the neutral axis, the fluid hold down forces will be minimized, since there is zero tank uplift. At the point of maximum tension the fluid hold down force will be maximized: (cos(8)-cos(0 0 )) A(0).-Amax* ( ) 1-cosen N:=20 x := lO*in (Initial guess, note that the above derivation it is assumed that delta is less than 2/10 of L ) i *= l.. N Pi :=P(~i) Li =root[ (a(~i)- c5(x,pi)),x J L is the length of plate effective in resisting uplift and is calculated here as a function of the angle beta measured about the tank circumference Tf is the fluid hold-down pressure and is calculated here as a function of the angle beta measured about the tank circumference. Print out the length of baseplate effective in resisting uplift and the corresponding fluid hold-down forces at both the neutral axis and at the point of maximum tension in the tank in order to check published solutions: L1 = 14.I64*in Tf1 =0.159*~ip JD ~ =6.993*in kip Tf.N = 0.13 *-.- JD Point 1 is at the point of maximum tension in the tank Point N is at the neutral axis.
Refueling Water Storage Tank 1-QS-TK-1, 2-WS-TK-1 52182-C-039, Rev. 2 Rwsthd3.mcd, 5/ 6 EPRI TR-103959 recommends fitting a best-fit linear relationship for the fluid hold-down force as a function of the angle beta: a := intercept(a, Tf) ,ff (=slope(a,Tf) kip Hr=0.01 *-. m T fu :=,ff rcos(~N) + a kip T fu =0.146*m Plot fluid hold-down force versus the angle beta for the tank: 0.17 ~---------~-----~----~--~-- T1 (Kips/in) Cos (13)
Refueling Water Storage Tank 1-QS-TK-1, 2-WS-TK-1 52182-C-039, Rev. 2 Rwsthd3.mcd, 6/ 6 The Upper limit on this equation is based on a fully plastic moment at both ends of the uplifted zone of the base plate, and should be limited to the following values: check :=max(M) check = 0.541 *kip*!" m M b = 0.807 *kip*!" p m The maximum moment in the tank shell should be limited to 90% of the fully plastic moment capacity. Mall:= 0.90*M pb
k. in Mall= 0.727* Ip*-.-
m (OK), Mall is greater than check.
Refueling Water Storage Tank 1-QS-TK-1, 2-QS-TK-1 Overturning Moment Capacity 52182-C-039, Rev. 2 Rwstcap3.mcd, 1/4 The overturning moment capacity of anchored tanks is computed in an iterative process. EPRI NP-6041, A Methodology for Assessment of Nuclear Power Plant Seismic Margin (Revision 1), appendix H, contains procedure for calculating the overturning moment capacity. This template has been updated to incorporate the recommendations given in EPRI 103959. Updated 12/2/94. This Mathcad template follows the procedure in EPRI NP-6041 for determining the overturning moment capacity of tanks. This template is intended to be used in conjunction with the template FLUIDHD. The FLUIDHD template calculates the fluid hold-down forces which lead to slightly increased capacities for marginal tanks. This template may be used with or without consideration of the FLUID hold-down forces. Derived Units: kip= 1000* lbf ksi = 1 OOO*psi Define Tank Parameters R := 19*ft W TE:= 76.75*kip N =44 lieo :=0.07*ina E 5 = 27.7* 103-ksi t 5 = 0.375*in P cmin.=25.7*psi P cmax :=32.3*psi P avg =21.17-psi cr ye : = 25
ksi Tank Radius Expected Minimum Total Effective Tank Weight Number of equally spaced anchor bolts Permissible uplift elongation Young's Modulus for the Tank Material Tank wall thickness at bolt chair location Minimum pressure in compression zone at the time of maximum moment Maximum pressure in compression zone at the time of maximum moment Average pressure across the bottom plate at the time of maximum moment Effective yield stress for tank material Define Anchor Bolt Parameters ha:= 36-in h c := 11.S*in Ab := 3.l4*in2 Depth to embedded anchor bearing surface Height of anchor bolt chair above tank foundation Nominal Area for each anchor Eb : = 29* l 03, ksi Anchor modulus of elasticity (ksi)
T bp : = 0.0* kip Anchor bolt pre-load (kips) T BC:= SO*kip Maximum anchor bolt load (anchor capacity)
Refueling Water Storage Tank 1-QS-TK-1, 2-QS-TK-1 52182-C-039, Rev. 2 Obtain diamond buckling coefficient from Reference 4, Figure 6: Parameter needed for Figure 6: Pcmin.(R)2 =0.343 Es ts Read-off value for delta-gamma /:J.y :=0.16 Input Linear Approximation to fluid hold-down forces {from template FLUIDHD) kip T fu : = 0.146*-. - fluid hold-down force at estimated neutral axis ID kip /:J.Te :=0.01*-.- m Slope of linear approximation Compute the Buckling Capacity of the Tank Shell: axial stress limit at the onset of elephant's foot buckling: S 1 = ( ~)
4~0 S 1 = 1.52

- 0.6*E s [ (P cmax*R) 2
] ( 1
) (s 1 + 3:~:si) cr.---* 1-

1-

~~~-
p ( R) cr
t 1 12

S 11.5 S I + 1
)'C S T ts diamond buckling capacity based on NASA SP-8007: ,:= /6*~ y := 1- 0.73*(1-e-+) E s*t s cr cb := (0.6*y+ Ay)*---r-cr cb = 1.894* 104 *psi Compressive shell capacity: CB := i1 cr cb <(0.9*cr p),a cb* (o.9-cr p)}t s kip CB = 2.068 *-. m Rwstcap3.mcd, 2/4 (Ref. 1, eqn. H-27)
Refueling Water Storage Tank 1-0S-TK-1, 2-QS-TK-1 52182-C-039, Rev. 2 Compute dimensionless parameters for solution scheme Note: the angle Beta represents the angle to the neutral axis, it is assumed prior to the start of the algorithm. I + cos(p) C 1(P): ------ sin(P) + (1t-P)*cos(P) C 3(P) := sin(P)- P*cos(p)
(I+ cos(p))
sin(P) + (1t-P)*cos(P) I - cos(P) C 2(P): sin(P)*cos(P)+1t-p I+ cos(p) c 4 (P) : P-sin(P)*cos(P) I - cos(P) Compute Anchor Tension in Each Anchor Bolt as a Function of Location i := 1,2.. N ai := [ (i - I)* ~!~}deg 5e0*Ab,Eb (cos(ai)- cos(P)) TB.-Tb + i P h a + h c I - cos( p) TB. =if(TB.STBc,TB.,TBc) I I I TB. : = if(T B. 2".0.0* lbf, TB., 0.0* lbf\\ I I I j Rwstcap3.mcd, 3/4 k' ,ff e = 0.01 *~ m Slope is based on best linear fit to fluid hold down forces
Refueling Water Storage Tank 1-QS-TK-1, 2-QS-TK-1 52182-C-039, Rev. 2 Rwstcap3.mcd, 4/4 Assume a beta value to begin solution: Note that the angle beta represents the angle between the maximum tension side of the tank and the neutral axis. For a lightly anchored tank the angle beta should approach pi. For anchored tanks, the angle should be between zero and pi. The assumed beta and the calculated angle below should be identical for the final solution. ~::1.016 assumed beta 6 eo=0.2S*in Solve for beta. Note that this is an iterative process. The beta printed below should be equal to the assumed beta for the final solution. Therefore, the user must input a new assumed beta which is equal to the assumed beta for the flnal run. l; =1,1.1..3 f(l;) :=C' m(s)- CB ':=2.4 angle :=root(f(O,,) angle= 1.018 Final Results: Moment Capacity: M sc(angle) =3.357*104 *kip*ft Shear Capacity: The plot below shows the difference between the compressive stress in the tank wall and the calculated buckling stress as a funciton of an assumed angle. The point at which the curves intersect is the correct angle beta. f(r,) 6 kip s*10 o.lf COF =0.70 Recommended value of 0.70 in EPRI NP 6041-SL, for steel tanks on a sand cushion. V hclpf=3.222*103 *kip where: LT B. = 1.068* 103 *kip I I W TE = 76. 75 *kip Pavg*(1t*R2) =3.457*103 *kip
Refueling Water Storage Tank 1-QS-TK-1, 2-QS-TK-1 North Anna HCLPF Calculations 52182-C-039, Rev. 2 Refueling Water Storage Tank (1-QS-TK-1; 2-QS-TK-1) Response Computations This MATHCAD template computes the response parameters which are needed in performing Rwstrsp2.mcd, 1/10 -a tank evaluation per EPRI NP-6041 methodology for vertical tanks. Inputs required are an assumed earthquake, and the necessary tank parameters. Two non-dimensional parameters are also needed for the calculation of the diamond-buckling capacity, and for the calculation of the compressive buckling capacity (elephant's foot buckling). Base units are feet, seconds, and pounds. Derived Units: kip= I 000* lbf ksi = I OOO*psi Define Tank Geometry: R := 19*ft Nominal Inner Tank Radius Rd = 38.*ft Dome Radius (estimated) H := 56.5-ft Height to Water Elevation t b := 0.25-in Bottom Plate thickness t d := 0.3125*in Dome thickness h d :=Rd" ( I - cos( asin( td))) Clearance between peak of dome to spring line nrings :=4 Number of different diameter rings composing the tank shell 0.375 0.3125
in t =
0.25 Shell Thickness at each ring from bottom of the tank to the top. 0.1875 8.0 8.0 Hr - . ft Height of each ring measured from the bottom of the tank to the top. 10.0 31.5 Define Anchorage Details: n :=44 ~ := 2.00*in Number of equally spaced anchor bolts Anchor bolt diameter
Refueling Water Storage Tank 1-QS-TK-1, 2-QS-TK-1 52182-C-039, Rev. 2 Define Material Properties: E 5 := 27.7-106-psi Young's Modulus for Shell Material Eb :=29*106-psi aye :=40*103-psi lbf y I : = (?2.4*3 ft lbf y 5 : = 0.284*:---j" m KI := 3.25-105-psi Young's Modululs for Bolt Material Effective yield stress for shell material Unit Weight for liquid Unit Weight for shell material Bulk Modulus of fluid, 3.25x10**5 psi for water Rwstrsp2.mcd, 2/10
Refueling Water Storage Tank 1-QS-TK-1, 2-QS-TK-1 52182-C-039, Rev. 2 Define Earthquake Amplification Parameters: Note that this template was created for performing response calculations according to the methodology contained in Reference 1 - This assumes a NUREG CR/0098 type response spectrum. This is the place where the amplification factors are input: pga :=0.17*g v := varatio*pga 6*v2 d *- pga sec [ in l varat10 := 36* (g)° The varatio is 36 for rock, 48 for soil NEP := 50 (non exceedance probability, 50%, or 84%) amp a<P) = if(NEP::84,4.38 - l.04*ln(P),3.21- 0.68*1n(P)) Definition for Spectral Amplification ampv(P) := if(NEP::84,3.38- 0.67*1n(P),2.3I - 0.41-ln(P)) Factors (NUREG CR/0098) ampd(P) :=if(NEP::84,2.73- 0.45*ln(P), 1.82- 0.27*1n(P)) Rwstrsp2.mcd Attachment a, 3/1 O amp a(5) = 2.116 ampv(5) = 1.65 amp d( 5) = 1.385 Example: amplification factors for 5% damping. Note that these values check with those given in NUREG/0098 for non-exceedance level selected. ( f )(-1) m(P) =log( 1 )*log f 4 amp a<P) 3 ( V )-1 f2(P) :=ampa(P)* ampv(P)*-*2*7t pga f 3 := 8-hz f 4 := 33*hz [ 2 v2 i

__ s 1 ( ~. P) : = ( ~Sf 1 (P)) * (2*7t*~) *amp d(P)*6* pga J + [ ( ~>f 1 (P)) * (~sf 2(P)) * (2*7t*~) *amp v<P)*v J s 2< (, Pl * [ ( (>f 2<Pl) * ( (Sf 3) *amp a<P)*pga] + [ ( (>f 3 )* (S<f 4 )*amp a

f 4) *pg, (Equation 1) Refueling Water Storage Tank 1-QS-TK-1, 2-QS-TK-1 52182-C-039, Rev. 2 Compute average shell thickness, and total shell height i := 1.. nrings t avg:= Hs t avg = 0.242 *in Rwstrsp2.mcd, 4/10
- t avg+ min(t) t eff -

2 t eff=0.215*in The effective thickness should be more weighted toward the top of the tank, where the deflections are greatest (Ref. GIP) Define Dimensionless Parameters from Refs. 2 and 4: Obtain Cwi from Reference 2, Table 7.4: Parameters needed for table 7.4: H R =2.974 t eff -4 R =9.416*10 Read-off value for Cwi: C WI:= 0.079 Tank Weight and C.G. Components: Note that the distance to the component C.G. is measured from the bottom of the tank. (Shell) j = 1.. nrings wi *=2*11*R*y 5-Hri"ti Ws =LWi i W 5 = 67.893 *kip (Bottom Plate) W b := (11*R2)*t b"'Y 5 W b = 11.595 *kip X s = 24.528°ft Refueling Water Storage Tank 1-QS-TK-1, 2-QS-TK-1 (Dome) W h := (2*'1!*R d"h d)*t d"'Y s W h = 15.535 *kip a :=asin(td) X h = 60.861 *ft Fluid Hydrostatic Pressure: 52182-C-039, Rev. 2 (Liquid) W w :='1!*R2*H*y I W w = 3.998* 103 *kip H Xw:=2 Xw =28.25*ft Maximum fluid pressur.e occurs at base of tank P st = 24.483 *psi Rwstrsp2.mcd, 5/10 Refueling Water Storage Tank 1-QS-TK-1, 2-QS-TK-1 52182-C-039, Rev. 2 Compute Horizontal Impulsive Mode Response: Impulsive Mode Frequency: cu :=C wr Cu JE5g f1 := 2*7t*H* (rs) f1 =3.597*hz I (Reference 1, equation H-2) Compute Spectral Acceleration at this frequency, damping for the impulsive mode may be taken as about 5% (this is considered to be a median estimate of damping). S a(f1,5) = 0.36*g Sah :=S 3 (f1,S) (See Equation #1) Compute Weight of fluid effective in the Impulsive Mode, and its corresponding C.G.: 1 f R)

-. I H 3 tanh\\ 1.732*H R

W i.-1~ R ::;2, R , 1.0- 0.436*H *W w (Ref. 3, Eqn. C3500-1,-2,-3,-4) l 1.132.H W i = 3.412* 103 *kip Xi = 24.678*ft (Ref. 3, Eqn. C3500-1,-2,-3,-4) Compute Impulsive Mode Base Shear and Overturning Moment: sah V1 :=g*(Wh + W s+ Wi) (Ref. 1, Eqn. H-3) sah MI :=g*(Wh-xh + w s"Xs+ w rXi) (Ref. 1, Eqn. H-4) VI = 1.257* l 03 *kip MI= 3.122* l 04 *kip*ft Estimate hydrodynamic fluid pressure on the tank at the bottom plate sah W**X**-

1. I g

P**=---- 1. l.36*R*H2 P i = 2.55 *psi (Ref. 1, eqn. H-8: Note this is conservative at fluid depths less than about 0.1 S*H) Rwstrsp2.mcd, 6/10 Refueling Water Storage Tank 1-QS-TK-1, 2-QS-TK-1 52182-C-039, Rev. 2 Compute Horizontal Convective (Sloshing) Mode Response: Convective Mode frequency (Ref. 1, eqn. H-10) f c = 0.281 *hz Compute Spectral Acceleration at this frequency, damping for the convective mode response is primarily fluid controlled and is estimated to be about 0.5%. Sa(fc,0.5) =0.055*g Sac:= S a(f c,0.5) (See Equation #1) Compute Weight of Fluid acting in the convective mode and its C.G. location W c := [ ( 0.46*~)*tanh( 1.835*;) ]*W w J _ cosh( 1.835*;)- 1.0 Xe ll.O 'H' ( H) *H 1.835* \\R)*sinh l.835*R W c =618.497*kip X c = 46.234*ft (Ref. 1. eqn. H-11) (Ref. 1, eqn. H-12) Compute Convective Mode Base Shear and Overturning Moment: Sac Vc:=-*Wc g Sac Mc:=-*Wc*Xc g V c = 34.311 *kip Mc= 1.586*103*ft*kip (Ref. 1, eqn. H-13) (Ref. 1, eqn. H-14) Compute Hydrodynamic Convective Pressure at fluid depth "y" y : = H This maximizes the hydrodynamic convective pressure 0.267* w w*S ac cosh( 1.835*~) pc:= g*R*H ( H) (Ref. 1, eqn. H-16) cosh 1.835*R pc= 3.27* 10-J *psi Rwstrsp2.mcd, 7/10 Refueling Water Storage Tank 1-QS-TK-1, 2-QS-TK-1 52182-C-039, Rev. 2 Compute the fundamental mode fluid slosh height Sac h s := 0.837*R*- g h s = 0.882*ft Compute Vertical Fluid Mode Response: (Ref. 1, eqn. H-17) Compute the vertical fluid mode fundamental frequency I fv = 4IH-[ ~-(,:~,+,\\if (Ref. 3, eqn. C3500-13) fv =4.269*hz Compute the hydrodynamic vertical fluid response mode pressure, based on a tank on a rigid foundation, note this pressure is also at y=H, which maximizes p. P v := 0.8*y rH S av(: v,5) *cos[ i* (H~ y)] p V = 4.696 *psi y = 56.S*ft Rwstrsp2.mcd, 8/10 Refueling Water Storage Tank 1-QS-TK-1, 2-QS-TK-1 52182-C-039, Rev. 2 Combine Individual Mode Responses to get Total Seismic Demand: Base Shear: V tot:= ~(vi2 + V c2) V tot = 1.258* 103 *kip Overturning Moment Mtot :=~(M 12+ M /) M tot= 3.126* 104 *kip*ft Fluid Pressures: p sh = jP / + p / P cmax :=P st+ P sh+ 0.4-P v P cmin : = P st+ P sh - 0.4* P v Total Horizontal Seismic Response Maximum and minimum compression zone pressures at the time of maximum base moment. (Ref. 1, eqn. H-22) Minimum tension zone fluid pressure at the time of maximum base moment (Ref. 1, eqn. H-23) Minimum average fluid presssure on the base plate Rwstrsp2.mcd, 9/10

Pavg :=Pst- 0.4*Pv P cmax = 28.911 *psi P cmin =25.154*psi at the time of maximum base shear (Ref. 1. eqn H-14)

P tmin = 20.055 *psi P avg = 22.605 *psi Expected minimum total effect weight of the tank shell acting on the base at the time of the maximum moment and base shear: ( 2 pga) W te := (W h + W s)* 1- 0.4*3*g (Ref. 1, eqn. H-26) W te = 79.645 *kip wh =15.535*kip W s = 67.893 *kip Refueling Water Storage Tank 1-QS-TK-1, 2-QS-TK-1

References:
52182-C-039, Rev. 2
1. A Methodology for Assessment of Nuclear Power Plant SeismfMargin (Revision 1 ),
EPRI NP-6041-SL, Final Report, Electric Power Research Institute, Palo Alto, CA, August, 1991.
2. A.S. Veletsos, '"Seismic Response and Design of Liquid Storage Tanks", Chapter 7, Guidelines for the Seismic Design of Oil and Gas Pipeline Systems, ASCE, 1984.

3. ASCE Standard and Commentary - Seismic Analysis of Safety-Related Nuclear Structures, ASCE 4-86, ASCE, September 1986.
Rwstrsp2.mcd, 10/10
4. Buckling of Thin-Walled Circular Cylinders, NASA SP-8007, National Aeronautics and Space
Administration, August 1986.
5. Newmark, N.M., and Hall, W.J., Development of Criteria for Seismic Review of Selected Nuclear Power Plants, NUREG-CR 0098, U.S. Nuclear Regulatory Commission, 1978.
Refueling Water Storage Tank 1-QS-TK-1, 2-WS-TK-1 North Anna HCLPF Calculations 52182-C-039, Rev. 2 Refueling Water Storage Tank (1-QS-TK-1, 2-QS-TK-1}, Fluid Hold Down Forces Fluid Hold-Down Forces Rwsthd3.mcd, 1/ 6 For tanks with minimum anchorage, hold-down forces resulting from fluid pressure acting on the tank bottom will contribute significantly to the overturning capacity of the tank. EPRI TR-103959, Methodology for Developing Seismic Fragilities, Section 7, contains procedures for calculating these fluid hold-down forces as a function of tank uplift. This Mathcad template follows the procedure in EPRI TR-103959 for determining the fluid hold-down forces. Inputs are tank parameters, the output is a plot of the fluid hold-down tension as a function of both uplift displacement and uplift length. Derived Units: kip= IOOO*lbf ksi= IOOO*psi Define Tank Geometry (input): R := 19*ft Tank Radius H := 56.S*ft Height to Fluid Surface t b := 0.25*in Baseplate Thickness ts : = 0.375-in Shell Thickness A max :=0.25*in Maximum permissible uplift height, this is a function of the failure mode en := 2.5*rad Assumed angle to the neutral axis of the tank Define Tank Material Properties (input): Es :=27.7*I06*psi u := 0.30 au :=65*ksi cry :=25*ksi cr ye :=25*ksi Young's Modulus Poisson's Ratio Ultimate Strength of Tank Material Minimum Yield Strength of Material Effective Yield Strength of Material, the effective yield stress of minimum is recommended in EPRI 103959 Define Fluid Pressure Parameters (input): p cmin :=25.l*psi P tmin : = 20.l *psi P cmax = 28.9*psi -Ptmax :=25*psi Defined pressures to be used in analysis: Pc :=P cmin
Refueling Water Storage Tank 1-QS-TK-1, 2-WS-TK-1 52182-C-039, Rev. 2 Calculate Additional Parameters needed for analysis: 3 E s*t s K*=---~ . 12-(1 - /) K = 133.768 *kip*in I , "[ ~-,h-(1-u') r K =31.695 K 5 =37.191 *kip R*ts ( R) f = ~ (
2). l - H*K 12* 1 - u (EPRI TR-103959 calls this term Mf/P)
Rwsthd3.mcd, 2/ 6
Refueling Water Storage Tank 1-QS-TK-1, 2-WS-TK-1 52182-C-039, Rev. 2 Rwsthd3.mcd, 3/ 6 Perform iterative solution for uplift height. tank shell hold-down tension, end moment, and maximum positive moment in the base plate as a function of the uplift length and fluid pressure: ( K *L ) F(L) := 1 + 2*E :.1 b P[L2 Mf(L,P) (Mf(L,P)\\ 21 M max(L,P) :=-r 4-p + P*L ) " Fh(L,P) := p
R +

P
[ Mf(L,P)K t 5*K l ~12*[ I - (i/)] (o(L,P)) T f(L,P) :=T e(L,P) + F h(L,P)*,-L-Horizontal force in the tank wall at the intersection with the tank bottom Hold down tension as a function of uplift length, the term Fh(L) is the additional correction which accounts for membrane effects EPRI TR-103959 approximates the relationship between fluid-hold down forces and uplift displacement -with a linear expression. This simplification is useful in the overturning moment capacity evaluation. Solve for the minimum and maximum length of plate effective in resisting uplift (corresponding to zero uplift, and maximum uplift at the extreme outer tensile fiber}. The fluid pressure is a function of the distance from the neutral axis of the tank: ( I - cos(8)) P(8) :=(Pc""'*P*t)* . 2 .. +Pt ... Reference.1,.eqn. 7-38 Assume an angle to the neutral axis in order to calculate the minimum hold down force, this angle may be changed later. 8 n =2.S*rad P(e n) = 24.603 *psi
Refueling Water Storage Tank 1-QS-TK-1, 2-WS-TK-1 52182-C-039, Rev. 2 Rwsthd3.mcd, 4/ 6 Calculate fluid hold down force at various locations about the circumference of the tank. Note that at the neutral axis, the fluid hold down forces will be minimized, since there is zero tank uplift. At the point of maximum tension the fluid hold down force will be maximized: (cos(8)-cos(an)) A(8).-Amax* ( ) l-cos8n N:'=20 x := IO*in (Initial guess, note that the above derivation it is assumed that delta is less than 2/1 O of L ) i := l.. N ( i - I ) ~i :=8n* N-I Pi := P(~i) Li :=root[ (A(~i)- 6(x,pi)),x J L is the length of plate effective in resisting uplift and is calculated here as a function of the angle beta measured about the tank circumference Tf is the fluid hold-down pressure and is calculated here as a function of the angle beta measured about the tank circumference. Print out the length of baseplate effective in resisting uplift and the corresponding fluid hold-down forces at both the neutral axis and at the point of maximum tension in the tank in order to check published solutions: L1 = 13.575 *in Tf1 = o.ls4/iP m ~ =6.993*in kip Tt:N = 0.129*-.- m Point 1 is at the point of maximum tension in the tank Point N is at the neutral axis.
Refueling Water Storage Tank 1-QS-TK-1, 2-WS-TK-1 a : = intercept( a, Tf) AT r:=slope{a,Tf) kip ATr-0.025*-. m ki T fn =0.144*--.! m 52182-C-039, Rev. 2 Plot fluid hold-down force versus the angle beta for the tank: 0.2 ----.---....-,---,,--'"T"""---,---.---..-----.---'"T"""-- 0.18 T1 (Kips/in) 0.16 0.14 Cos (13) Rwsthd3.mcd, 5/ 6
Refueling Water Storage Tank 1-QS-TK-1, 2-WS-TK-1 52182-C-039, Rev. 2 Rwsthd3.mcd, 6/ 6
-:;-~.~---:-=---~*~-:--::--~"::;:",:;,j:_~.....==. ~------~"0..:.:~=--
._ -- - ~-;,-* ~--.,.... -..,..*--- _- ~:-~~~~:.~ ~~.=..:'='...:::~~-:.--.. - -...... ~ *--*~ - - ----~-;-;-=... _____ - -~... ---~-~*~~'c~~.. ~.:~:_:::,~~-:-:::?..,:.~~-::.~""~ The Upperlfrriifonffiisequatron is based on a fully plastic moment at both ends of the uplifted zone of the base plate, and should be limited to the following values: check:= max(M) check = 0.603 *kip* ! 0 ID M pb = 0.807*kip* ! 0 ID The maximum moment in the tank shell should be limited to 90% of the fully plastic moment capacity. Mall := 0.90*M pb Mall= 0.727*kip* '. 0 ID (OK), Mall is greater than check.
Refueling Water Storage Tank 1-QS-TK-1, 2-QS-TK-1 Overturning Moment Capacity 52182-C-039,Rev.2 Rwstcap4.mcd 0, 1/4 The -overtu~ng -~;~ent~p:;:;;. a=c:::::ity;;:t:. "'"""~tc::a.:, aii;i:** n~~=~;:;:::~r:::::;e.;;,a"ta~ks 7;-~~p't~~d-ir{~n iterative pr~~is~.,-.. EPRi'"-,,-.. ;_*c, --<_:: : ~:~-=-*........ NP-6041, A Methodology for Assessment of Nuclear Power Plant Seismic Margin {Revision 1 ), appendix H, contains procedure for calculating the overturning moment capacity. This template has been updated to incorporate the recommendations given in EPRI 103959. Updated 12/2/94. This Mathcad template follows the procedure in EPRI_ NP-6041 for determining the overturning moment capacity of tanks. This template is intended to be used in conjunction with the template FLUIDHD. The FLUIDHD template calculates the fluid hold-down forces which lead to slightly increased capacities for marginal tanks. This template may be used with or without consideration of the FLUID hold-down forces. Derived Units: kip= 1000* lbf ksi= lOOO*psi Define Tank Parameters R := 19*ft Wm :=76.75*kip N:=44 Ii eo := 0.07*ino E 5 =27.7*I03*ksi t 5 := 0.375*in p cmin :=25.l*psi P cmax := 28.9*psi P avg:= 22.6*psi cr ye:= 25*ksi Tank Radius Expected Minimum Total Effective Tank Weight Number of equally spaced anchor bolts Permissible uplift elongation Young's Modulus for the Tank Material Tank wall thickness at bolt chair location Minimum pressure in compression zone at the time of maximum moment Maximum pressure in compression zone at the time of maximum moment Average pressure across the bottom plate at the time of maximum moment Effective yield stress for tank material Define Anchor Bolt Parameters h 8 := 36*in Depth to embedded anchor bearing surface h c := 11.S*in Height of anchor bolt chair above tank foundation Ab:= 3.14*in2 Nominal Area for each anchor E b : = 29* I 03 ;ksi Anchor niodulus of elasticity (ksi) T bp : = 0.0* kip Anchor bolt pre-load (kips) T BC:= 60*kip Maximum anchor bolt load (anchor capacity)
Refueling Water Storage Tank 1-QS-TK-1, 2-QS-TK-1 p cmin. (~)2 = 0.335 Es ts Read-off value for delta-gamma /J.y := 0.16 52182-C-039, Rev. 2 Input Linear Approximation to fluid hold-down forces (from template FLUIDHD) kip T fu := 0.144*-.- fluid hold-down force at estimated neutral axis m kip /J.T e := 0.025*-.- Slope of linear approximation m Compute the Buckling Capacity of the Tank Shell: axial stress limit at the onset of elephant's foot buckling: S 1 : = ( ~)
4~0 S 1 = 1.52 er p = 8.095* l 03 *psi diamond buckling capacity based on NASA SP-8007:
~ = /6.;, y = 1- 0.73*(1-e-+) E s*t s ercb := (0.6*y+!J.y)*~ er cb = 1.894* l 04 *psi Compressive shell capacity: C B : = i1 er cb < ( 0.9*er p), er cb* ( 0.9*er p) }ts k" CB =2.732*~ m ./ / Rwstcap4.mcd 0, 2/4 (Ref. 1, eqn. H-27)
Refueling Water Storage Tank 1-QS-TK-1, 2-QS-TK-1 52182-C-039, Rev. 2 Compute dimensionless parameters for solution scheme Note: the angle Beta represents the angle to the neutral axis, it is assumed prior to the start of the algorithm. c I (P) := I + cos(P) sin(P) + ('It-P)*cos(p) C 3(P): sin(P)- P*cos(P) *(l + cos(p)) sin(P) + (1t-P)*cos(P) 1 - cos(p) C 2 (P) : sin(P)*cos(p) + 7t-P I+ cos(P) c 4(P) := P-sin(P)*cos(p) I - cos(P) Compute Anchor Tension in Each Anchor Bolt as a Function of Location i := 1,2.. N ai := [ (i-1 )*~!~}deg . ~e0*Ab-Eb (cos(ai)-cos(p)) TB.-Tbp-r . -~--- i ha+hc 1-cos(p) TB. : = if(T B. :C::0.0* lbf, TB., 0.0* lbf\\ I I I / Rwstcap4.mcd 0, 3/4 kip ~Te =0.025*-.- m Slope is based on best linear fit to fluid hold down forces
Refueling Water Storage Tank 1-QS-TK-1, 2-QS-TK-1 52182-C-039, Rev. 2 Rwstcap4.mcd 0, 4/4 Assume a beta value to begin solution: Note that the angle beta represents the angle between the maximum tension side of the tank and the neutral axis. For a lightly anchored tank the angle beta should approach pi. For anchored tanks, the angle should be between zero and pi. The assumed beta and the calculated angle below should be identical for the final solution. P= 1.441 assumed beta Solve for beta. Note that this is an iterative process. The beta printed below should be equal to the assumed beta for the-final solution. Therefore, the user must input a new assumed beta which is equal to the assumed beta for the final run. c_; =1,1.1..3 f( s) : = C' mC s) - C B + :=2.4 angle = root( f( +}, +) angle = 1.442 Final Results: Moment Capacity: M sc(angle) =3.622*104 *kip*ft Shear Capacity: The plot below shows the difference between the compressive stress in the tank wall and the calculated buckling stress as a funciton of an assumed angle. The point at which the curves intersect is the correct angle beta. 1.5*10'.-----.-----,----, r(r.) 6
k.
S*JO 0-~ ft -s*106.,.._--....-----.----'4 COF :=0.70 Recommended value of 0. 70 in EPRI NP 6041-SL, for steel tanks on a sand cushion. V hclpf = 3.445* l 03 *kip where: LTB. = 1.154*103 *kip I I W TE = 76. 75 *kip P avg* (1t*R2) = 3.691* 103 *kip
HCLPF Capacity Calculation for ECST Tanks Calculation No. 52182-C-038, Rev. 2 North Anna Power Station}}

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