STRUCTURAL DESIGN NOTES TOPIC C PRESSURE VESSEL STRESS ANALYSIS J. E. Meyer reviion of Augut 1996 1. INTRODUCTION Thee note upplement cla lecture on "thin hell" preure veel tre analyi. The ue of the implified thin hell method are illutrated by application to a preure veel that ha many of the geometric and operational feature of a preurized water reactor (PWR) reactor veel. More detailed analye (e.g., by uing a finite element computer code applied to a more realitic geometry) would undoubtedly be ued in a final deign. However, the implified technique can be ued to give approximate anwer (and anwer that are eaily undertood) for many actual tree of interet. The reactor veel i only one of a large number of nuclear reactor plant component for which tre analye mut be performed. Hence, in one ene, the analye here are being ued to repreent many other calculation. But the reactor veel i alo a component of very pecial ignificance. That i, the reactor veel i, for mot purpoe, conidered to be deigned, contructed, and operated o that a catatrophic (rapid or brittle) failure i incredible. Calculation analogou to thoe of thi note and correponding detailed analye are ued to upport tatement of incredibility.. DESCRIPTION OF REPRESENTATIVE VESSEL The repreentative reactor veel hown in Fig C-1 i choen a a pecific example. The overall height of the veel (including both cloure head) i 13.3 m and the inide diameter i 4.4 m. Subregion 1 of interet, with ome approximate dimenion, are: - the lower head region (approximately a hemipherical hell of thickne 10 mm); - the beltline region (a cylindrical hell of thickne 0 mm); 1 Thee ubregion are motly contructed of a ductile low-carbon teel (uch a type SA533B). However, an autenitic tainle teel (uch a SS304) cover all portion of the low-carbon teel that are adjacent to the coolant. The purpoe of the tainle teel clad i corroion protection. It ha about a 3 mm minimum thickne and about a 5 mm average thickne.
Note C: Preure Veel Stree Page C- - the nozzle hell coure region (a thick cylindrical hell of thickne 380 mm with large penetration for two hot leg pipe (1.1 m inide diameter) and four cold leg pipe (0.8 m inide diameter); - the cloure flange (a heavy ring that i welded to the cloure head and contain hole through which cloure tud pa); and - the cloure head (approximately a hemipherical hell with penetration for intrumentation and for control element aemblie). Potential tructural limit are lited a follow, a are location of major everity for each: Figure C-1: A Repreentative PWR Reactor Veel (adapted from Ref 1) Dimenion in millimeter. Structural LimitLocation of Major Severity Preure tre beltline cylinder, lower head hemiphere, and cloure head hemiphere (away from joint with adjacent region in beltline cylinder) Thermal tre in the thick portion of the nozzle hell coure region adjacent to a main coolant pipe penetration
Note C: Preure Veel Stree Page C-3 Dicontinuity tre Radiation embrittlement in vicinity of joint between beltline cylinder and lower head hemiphere, between beltline cylinder and nozzle hell coure region cylinder, and between cloure head and the cloure flange in the beltline cylinder adjacent to the axial center of the reactor core 3. PRESSURE STRESS The preure tre limit may be dicued by conidering a veel that i contructed of a thin cylindrical hell of length L that i capped by a hemiphere at either end. The mean radiu of the cylinder (and the cap) i denoted by R. The cylinder ha a uniform thickne equal to t c ; each cap ha a uniform thickne equal to t. The veel i ubjected to an internal preure (p) and a zero external preure. No other external force act. The veel wall are at a uniform temperature and are contructed of a ingle material. 3.1 Long Cylinder In a region of the cylinder that i far from the end, three normal tree ( r, q, and z ) may be calculated to characterize the thin hell tre tate. Thee tre component are, repectively, tree in the radial, hoop, and axial direction. The tree q, and z are found from equation of tatic equilibrium. The tre r i obtained by averaging the preure on the inner and outer wall. Therefore: = ÁÊ p R q ˆ ; Ë t c (1) = Á Ê p R z ˆ ; and () Ë tc Ê 1 ˆ r = - Ë p. (3) An elatic calculation of train in the hoop direction (q direction) can be converted to w c, the radially outward diplacement of the center urface of the cylinder. w c = ÁÊ p R ˆ È Ê t - n + n c ˆ Ë E t Î Í Ë R ; c (4) where: the firt term on the right hand ide of Eq 4 give the contribution to w c ; the econd term, q the z contribution; and the lat term, the r contribution.
Note C: Preure Veel Stree Page C-4 3. Sphere In any portion of the hemiphere which act to give diplacement and tree that are the ame a thoe in a full phere, the following analogou tre and diplacement equation apply (the ubcript q1 and q refer to two orthogonal direction within the hell center urface): q1 = q = ÁÊ p R ˆ Ë t ; (5) r = - Ê 1 ˆ Ë p ; and (6) Ê t ˆ w = Ê Á p R ˆ È 1- n + n Ë R. (7) Ë E t Î Í 4. THERMAL STRESSES Thermal tre calculation may be illutrated by conidering a cylinder that i ubjected to a known temperature ditribution. The ditribution i a function only of x (the ditance meaured radially outward from a poition (x = 0) at the hell center urface). Thu, the inner urface of the hell i located at x = - _ t c and the outer urface i at x = + _ t c. The temperature ditribution (T(x)) i converted to a thermal train ditribution (e T (x)) by uing an integrated a T (the coefficient of linear thermal expanion), a follow: T e T = a T dt ; (8) Ú TR where T R i a convenient reference temperature (e.g., 0 C); and where thi integral provide e T a a function of T. The thermal tree may be expreed in term of the patially average thermal train ( e T ) a follow: 1 + t 1 c e T = (e T ) dx ; (9) Ú tc 1 - t c where, by evaluating elatic tre-train relation, by requiring that axial train are uniform (at a poition far from the end of the cylinder); and by invoking zero internal preure and zero axial force: Thee portion of the hemiphere would have no hell moment and no hell hear force.
Note C: Preure Veel Stree Page C-5 e z = e q = e T ; and (10) = = Ê E ˆ Ë 1- n (e - e z q T T ) ; (11) where e T indicate the local value of thermal train at a pecified x-poition. 5. BENDING (CURVATURE) OF A CYLINDRICAL SHELL In 3 and 4, the radial diplacement (w) i uniform. Now conider a portion of the cylindrical hell in which the hell may have an axial lope (non-zero value of f = dw/dz) and an axial curvature (nonzero value of d w/dz ). The exitence of uch portion of the hell mut be caued by hell moment and hell hear. Thoe moment and hear would typically develop near the end. The development that follow are baed on lope and curvature which may vary in the axial direction but have no change in the hoop direction. 5.1 Strain Relation The hell curvature reult in a bending train at any cro-ection given by e bz : t d w c e bz = - d z ; (1) The correponding tenile train variation through the hell thickne i taken to be linear (analogou to a tatement of beam theory that plane ection that are originally normal to the beam axi remain plane and normal to the beam axi in the deflected condition ). e = e + ÁÊ x ˆ z z e bz. (13) Ë tc The tenile train in the hoop direction are uniform at each axial poition: e q = e q = (w/r). (14) 5. Stre Relation
Note C: Preure Veel Stree Page C-6 The tree at each poition may be related to hell force and hell moment (N = normal force per unit center urface length; M = moment per unit center urface length; ubcript indicate the normal direction for the face on which N or M act) a follow: Ê N z ˆ - Á Ê 1 M z ˆ z = Á x ; Ë 3 tc Ë t c Ê Á N ˆ Ê Á 1 M q q ˆ q = - 3 x ; and Ë t c Ë t c (15) (16) Ê 1 ˆ r = - Ë p. (17) 5.3 Shell Variable Relation The hell hear force variable (V z = the hear force normal to the axial direction (per unit center urface length)) may be related to other hell variable by uing force and moment balance, a follow. Each equation i preceded by the name of an analogou beam theory equation. hear 3 dv z = p - dz N q ; and (18) R moment dm z = V z. (19) dz Additional analog of beam theory equation are a follow: df lope = M z/d ; (0) dz dw diplacement = f ; (1) dz where 4 D = E t 3 c 1(1-n ). () 3 The name hear implie that Eq 18 could, in principle, be integrated to obtain a hear ditribution. The hoop normal force N q i however a linear function of w(z). w(z) mut be known prior to integration. Thi feature identifie a beam on elatic foundation.
Note C: Preure Veel Stree Page C-7 The moment in the hoop direction may be conidered to be induced by the moment in the axial direction ince: M q = n M z. (3) Other equation that relate hell variable are: N q = (4) E t c w R + n N z - n t c p ; and (4) N z = pr. (5) 5.4 Differential Equation Combine many of the above equation to obtain a differential equation for w a a function of z: 4 d w 4 dz + 4 4 b w = 4 4 b w p ; (6) where w = w p i a particular olution to the differential equation and i a reult identical to Eq 4: w p and where b i given by: = ÁÊ p R ˆ È Ê t - n + n c ˆ Ë E t Î Í Ë R ; c (7) b = ÁÊ 3(1-n )ˆ 1 4 Ë R t. (8) c 5.5 Solution The general olution of Eq 6 i: 4 The ymbol D denote flexural rigidity and play a role that i imilar to the product (EI) in beam theory.
Note C: Preure Veel Stree Page C-8 Ï w p + Ô Ô w = Ì exp(-b z) [c 1 co(b z) + c in(b z)] + ; (9) Ô Ó exp(-b( L -z))[c co(b (L - z)) + c 3 4 in(b (L - z))]ô where the firt line on the right hand ide i the particular olution of Eq 7; the econd line ha a decaying envelope that dimihe a z increae from zero; and the final line ha a decaying envelope that dimihe a z decreae from z = L. The envelope reach a mall value (exp (-3) = 0.050) a [(ditance from end) = (3/b)] o that diturbance caued by end moment or by end hear are not felt at large ditance. 6. DISCONTINUITY STRESSES 6.1 End of Long Cylinder If occurrence near z = L decay in the manner indicated above, then diplacement near z = 0 can be conidered to depend only on the hear & moment at z = 0. That i, if the ubcript o denote occurrence at z = 0, then the end radial diplacement and the end lope are: w o = w pc + Ê Á Ë 1 ˆ b D 3 V o + Ê Á 1 ˆ Ë b D M o ; and (30) ˆ f o = - Ê Á 1 Ë b D V o - Ê Á 1 ˆ Mo. (31) Ë bd 6. Edge of Hemiphere A imilar, but more involved, equation development lead to the following equation for the hemiphere that i joined to the cylinder at z = 0: w o = w p - ÁÊ R l ˆ Ê Vo + Á l ˆ M o ; and (3) Ë E t Ë E t 3 f o = - Á Ê l ˆ V o + Ê Á 4 l ˆ M o ; where (33) Ë Et Ë R E t
Note C: Preure Veel Stree Page C-9 l = b R ; and (34) b = ÁÊ 3(1-n ) ˆ 1 4 Ë R t. (35) 6.3 Combination We now have four linear equation (Eq. 30-33) in four unknown (w o, f o, M o, and V o ) and therefore can olve for condition at the joint (z = 0) that give continuity. Subequently, correponding value of c 1 and c 1 (baed on Eq. 9; c 3 = 0; and c 4 = 0) can be found and ued to determine condition in the cylinder for other value of z. 7. RELATED INFORMATION Related information on preure veel tre analye can be found a follow: - thin hell preure tree (Ref., pp. 7-3; Ref. 3, pp. 33-45); - thick hell preure tree (Ref., pp. 3-37; Ref. 3, pp. 56-64); - thermal tree (Ref., pp. 37-44; Ref. 3, pp. 74-88); - dicontinuity tree (Ref. 3, pp. 159-185); and - combination information for a cylindrical hell (Ref. 4) and for a pherical hell (Ref. 5). Be aware that in thi note and in the reference, a variety of approximation and notation are ued. For example, the thin hell approach adopted herein doe take ome account for the hell "queezing" caued by preure action in the radial direction. Other reference adopt an aumption that the effect i negligible. The definition of the radiu R provide a econd example. It i ued herein a the radiu of the hell center urface but i ued elewhere a the hell inide radiu. Other reference-to-reference difference exit; therefore each reference hould be ued with care. However, each reference provide a unique viewpoint and derivation approach. It can be ued a a ueful upplement to the information of cla lecture and thi note.
Note C: Preure Veel Stree Page C-10 REFERENCES 1. "Reactor Veel Information," MIT Nuclear Engineering Department Note L.7, from Pilgrim tation Unit, Preliminary Safety Analyi Report (PSAR).. L. Wolf, M.S. Kazimi, and N.E. Todrea, "Introduction to Structural Mechanic," MIT Nuclear Engineering Note L.4, reviion of Fall 1995. 3. J. F. Harvey, "Theory and Deign of Preure Veel," Van Notrand Reinhold Co., New York, 1985. 4. "Article A-000, Analyi of Cylindrical Shell," in Section III, Rule for Contruction of Nuclear Veel, ASME Boiler and Preure Veel Code," pp. 547-549, ASME, New York, 1968 Edition. 5. "Article A-3000, Analyi of Spherical Shell," in Section III, Rule for Contruction of Nuclear Veel, ASME Boiler and Preure Veel Code," pp. 553-556, ASME, New York, 1968 Edition.
APPENDIX C1 Stree Near Axial Center of Long Cylinder Comparion of Alternate Approximation J.E. Meyer reviion of July 1996 PRESSURE STRESSES (at r = a) (STRESS at r = a) / (PRESSURE) Equation for Stre at r = a ( / p) for (b / a ) = 1.1 radial hoop axial radial hoop axial Exact, Thick Shell - p 1b 1c -1.00 10.5 4.76 Thin Shell for Curvature - (p / ) b c -0.50 10.50 5.5 Ring Finite Element 3a 3b 3c = 1c -0.48 10.00 4.76 Ê b = Á + a ˆ Á Ê a ˆ (1b) (1c) q = p = p q z Ë b - a Ë b - a p R t (b) Ê a = p R (c) = - Ê a ˆ (3a) = ˆ z r q (3b) t Ë b + a p Ë b - a p Symbol are: p = inide preure; zero = outide preure; = tre; (r, q, z) = (radial, hoop, axial) coordinate direction; r = b = radiu of hell outide urface; r = a = radiu of hell inide urface; r = R = 0.5 (b + a) = radiu of hell central urface; t = b - a = hell thickne. THERMAL STRESSES (at r = a) radial direction hoop direction axial direction r = 0 q = E 1- n (e T -e Ta ) z = E 1 -n ( e T - e Ta ) Symbol are: E = Young Modulu; n = Poion Ratio; e T = volume averaged thermal train; e Ta = thermal train at r = a.
APPENDIX C (01-Mar-04) Category Primary Second- Peak ary General Local Bending Membr Membr Symbol P m P L P b Q F Normal Force X X Moment X Include X X X Dicontinuitie Include Stre X Concentration Caued by X X X X X Mechanical Load Caued by Thermal X X Stree Tenile Strain Bending Strain Fatigue Failure Mode Limit Control Limit Control Load Shake- Load (Shakedown) down) Limit from Concept S y S y 1.5 S y S y S-N Curve Code Limit S m 1.5 S m --------1.5 S m -------- ----------------3 S m ---------------- U F < 1 Inelatic Analyi e =1% (Elevated e = % Temperature) ------------ e = 5% ------------- U FT < D