1. INTRODUCTION
These notes supplement class lectures on "thin shell" pressure vessel stress analysis.
The use of the simplified thin shell methods are illustrated by application to a pressure vessel that has many of the geometric and operational features of a pressurized water reactor (PWR) reactor vessel. More detailed analyses (e.g., by using a finite element computer code applied to a more realistic geometry) would undoubtedly be used in a final design. However, the simplified techniques can be used to give approximate answers (and answers that are easily understood) for many actual stresses of interest.
The reactor vessel is only one of a large number of nuclear reactor plant components for which stress analyses must be performed. Hence, in one sense, the analyses here are being used to represent many other calculations. But the reactor vessel is also a component of very special significance. That is, the reactor vessel is, for most purposes, considered to be designed, constructed, and operated so that a catastrophic (rapid or brittle) failure is incredible. Calculations analogous to those of this note and corresponding detailed analyses are used to support statements of incredibility.
2. DESCRIPTION OF REPRESENTATIVE VESSEL
The representative reactor vessel shown in Fig C-1 is chosen as a specific example. The overall height of the vessel (including both closure heads) is 13.3 m and the inside diameter is 4.4 m. Subregions1 of interest, with some approximate dimensions, are:
-the lower head region (approximately a hemispherical shell of thickness 120 mm);
-the beltline region (a cylindrical shell of thickness 220 mm);
These subregions are mostly constructed of a ductile low-carbon steel (such as type SA533B). However, an austenitic stainless steel (such as SS304) covers all portions of the low-carbon steel that are adjacent to the coolant. The purpose of the stainless steel clad is corrosion protection. It has about a 3 mm minimum thickness and about a 5 mm average thickness.
1
Notes C: Pressure Vessel Stresses Page C-2
-the nozzle shell course region (a thick cylindrical shell of thickness 380 mm with large penetrations for two hot leg pipes (1.1 m inside diameter) and four cold leg pipes (0.8 m inside diameter);
- the closure flange (a heavy ring that is welded to the closure head and contains holes through which closure studs pass); and
-the closure head (approximately a hemispherical shell with penetrations for instrumentation and for control element assemblies).
Potential structural limits are listed as follows, as are
locations of major severity for Figure C-1: A Representative PWR Reactor Vessel each: (adapted from Ref 1) Dimensions in millimeters.
Structural LimitLocations of Major Severity
Pressure stress beltline cylinder, lower head hemisphere, and closure head hemisphere (away from joints with adjacent regions in beltline cylinder)
Thermal stress in the thick portion of the nozzle shell course region adjacent to a main coolant pipe penetration
Notes C: Pressure Vessel Stresses Page C-3
Discontinuity stress in vicinity of joints between beltline cylinder and lower head hemisphere, between beltline cylinder and nozzle shell course region cylinder, and between closure head and the closure flange
Radiation embrittlement in the beltline cylinder adjacent to the axial center of the reactor core
3. PRESSURE STRESS
The pressure stress limits may be discussed by considering a vessel that is constructed of a thin cylindrical shell of length L that is capped by a hemisphere at either end. The mean radius of the cylinder (and the caps) is denoted by R. The cylinder has a uniform thickness equal to tc; each cap has a uniform thickness equal to ts. The vessel is subjected to an internal pressure (p) and a zero external pressure. No other external forces act. The vessel walls are at a uniform temperature and are constructed of a single material.
3.1 Long Cylinder
In a region of the cylinder that is far from the ends, three normal stresses (sr, sq, and sz) may be calculated to characterize the thin shell stress state. These stress components are, respectively, stresses in the radial, hoop, and axial directions. The stresses sq, and sz are found from equations of static equilibrium. The stress sr is obtained by averaging the pressures on the inner and outer wall. Therefore:
s
=ÁÊ pR ˜ˆ ; (1)
q
Ë t¯
c
Á
s
= Ê pR ˜ˆ ; and (2)
z
Ë 2 t¯
c
Ê 1 ˆ
sr=-Ë 2p¯ . (3)
An elastic calculation of strain in the hoop direction (q direction) can be converted to wc, the radially outward displacement of the center surface of the cylinder.
wc = ÁÊ p R2 ˆ È2 -n + nË Ê tc ˆ ˘ (4)
Ë 2 E t ¯ ˜ ÎÍ R ¯ ˙˚ ;
c
where: the first term on the right hand side of Eq 4 gives the scontribution to wc; the second term,
q
the sz contribution; and the last term, the sr contribution.
Notes C: Pressure Vessel Stresses Page C-4
3.2 Sphere
In any portion of the hemispheres which act to give displacements and stresses that are the same as those in a full sphere,2 the following analogous stress and displacement equations apply (the subscripts q1 and q2 refer to two orthogonal directions within the shell center surface):
sq1 =sq2 = ÁÊ pR ˜ˆ ; (5)
Ë 2 t¯
s
sr=-Ê 1 ˆ ; and (6)
Ë 2p¯
˜ Ê ts ˆ ˘
ws = ÊÁ pR2 ˆ È1-n + nË R ¯ ˙ . (7)
Ë 2E t ¯ ÎÍ ˚
s
4. THERMAL STRESSES
Thermal stress calculations may be illustrated by considering a cylinder that is subjected to a known temperature distribution. The distribution is a function only of x (the distance measured radially outward from a position (x = 0) at the shell center surface). Thus, the inner surface of the shell is located at x = - _ tc and the outer surface is at x = + _ tc. The temperature distribution (T(x)) is converted to a “thermal strain” distribution (eT(x)) by using an integrated aT (the coefficient of linear thermal expansion), as follows:
T
eT =aT dT ; (8)
Ú
TR
where TR is a convenient reference temperature (e.g., 20°C); and where this integral provides eTas a function of T. The thermal stresses may be expressed in terms of the spatially average thermal strain ( e T ) as follows:
1
+ t
2 c
1
eT = (eT ) dx ; (9)
Ú
t1
c -t
c
2
where, by evaluating elastic stress-strain relations, by requiring that axial strains are uniform (at a position far from the ends of the cylinder); and by invoking zero internal pressure and zero axial force:
These portions of the hemispheres would have no shell moments and no shell shear forces. 2
Notes C: Pressure Vessel Stresses Page C-5
=
eq
=
ez
eT ; and (10)
s
=s=Ê Ë1-E nˆ ¯ (eT -eT ) ; (11)
z q
where eT indicates the local value of thermal strain at a specified x-position.
5. BENDING (CURVATURE) OF A CYLINDRICAL SHELL
In §3 and §4, the radial displacement (w) is uniform. Now consider a portion of the cylindrical shell in which the shell may have an axial slope (non-zero value of f = dw/dz) and an axial curvature (nonzero value of d2w/dz2). The existence of such portions of the shell must be caused by shell moments and shell shears. Those moments and shears would typically develop near the ends.
The developments that follow are based on slopes and curvatures which may vary in the axial direction but have no changes in the hoop direction.
5.1 Strain Relations
e
bz:
The
shell curvature results in a “bending strain” at any cross-section given by
td2w
c
ebz= -2dz2 ; (12)
The corresponding tensile strain variation through the shell thickness is taken to be linear (analogous to a statement of beam theory that “plane sections that are originally normal to the beam axis remain plane and normal to the beam axis in the deflected condition”).
e
= e + ÁÊ 2x ˆ˜ ebz . (13)
zz
Ë t¯
c
The tensile strains in the hoop direction are uniform at each axial position:
eq
=
eq
=
(w/R) . (14)
5.2 Stress Relations
Notes C: Pressure Vessel Stresses Page C-6
The stresses at each position may be related to shell forces and shell moments (N = normal force per unit center surface length; M = moment per unit center surface length; subcripts indicate the normal direction for the face on which N or M acts) as follows:
Ê Nz ˜ˆ -ÁÊ 12 Mz ˆ x ;
Á
sz=
Ë tc ¯ Ë tc 3 ¯ (15)
Ê N ˆ ˜ Ê Á 12 M qˆ x ; and
Á q
sq=-
Ë tc ¯ Ë tc3 ¯ (16)
Ê 1 ˆ
sr=-Ë 2p¯ . (17)
5.3 Shell Variable Relations
The shell shear force variable (Vz = the shear force normal to the axial direction (per unit center surface length)) may be related to other shell variables by using force and moment balances, as follows. Each equation is preceded by the name of an analogous beam theory equation.
shear3 dVz = p -Nq ; and (18)
dz R
moment dMz =Vz . (19)
dz
Additional analogs of beam theory equations are as follows:
d
f
slope = Mz/D ; (20)
dz
dw
displacement = f ; (21)
dz
c
where4 D = E t3
12(1-n2) . (22)
The name “shear” implies that Eq 18 could, in principle, be integrated to obtain a shear
distribution. The hoop normal force N q is however a linear function of w(z). w(z) must be
known prior to integration. This feature identifies a “beam on elastic foundation.”
3
Notes C: Pressure Vessel Stresses Page C-7
The moment in the hoop direction may be considered to be induced by the moment in the axial direction since:
M
=nM. (23)
q z
Other equations that relate shell variables are:
N
q
=
Etc w R
+ nNz
-
ntc p 2
;
and
(24)
(24)
N
z
=
pR 2
.
(25)
4
d w 4dz 4b w = 4 4b pw ;
(26)
where w = wp
5.4
Differential Equation Combine many of the above equations to obtain a differential equation for w as a function of z:
+
4
is
a particular solution to the differential equation and is a result identical to Eq 4:
wp = ÁÊ p R2 ˆ È2 -n + nË Ê tc ˆ ˘ (27)
Ë 2 E t ¯ ˜ ÎÍ R ¯ ˙˚ ;
c
and where b is given by:
2
b
= ÁÊ 3(1-n )ˆ
1
4
Ë R2 t2 ˜ . (28)
¯
c
5.5 Solution The general solution of Eq 26 is:
The symbol D denotes “flexural rigidity” and plays a role that is similar to the product (EI) in beam theory.
4
Notes C: Pressure Vessel Stresses Page C-8
Ï wp +¸
Ô Ô w =Ì exp(-b z) [c1cos(bz)+ c2 sin(bz)]+ ˝ ; (29) Ô
Óexp(-b( L
-
z
)) [
c3 cos(b(L
-
z))+ c4 sin(b(L
-
z))]Ô
˛
where the first line on the right hand side is the particular solution of Eq 27; the second line has a decaying envelope that dimishes as z increases from zero; and the final line has a decaying envelope that dimishes as z decreases from z = L. The envelopes reach a small value (exp (-3) = 0.050) as [(distance from end) = (3/b)] so that disturbances caused by end moments or by end shears are not felt at large distances.
6.
DISCONTINUITY STRESSES
6.1
End of Long Cylinder
If occurrences near z = L decay in the manner indicated above, then displacements near z = 0 can be considered to depend only on the shear & moment at z = 0. That is, if the subscript “o” denotes
w
o = w pc + 1 2 3b D Ê Ë Á ˆ ¯ oV + 1 2 2b D Ê Ë Á ˆ ¯ oM
;
and (30)
occurrences
at z = 0, then the end radial displacement and the end slope are:
ˆ
fo= -ÊÁ 1 V -o ÊÁ 1 ˆ Mo. (31)
2
Ë 2 b D ¯ Ë bD ¯
6.2 Edge of Hemisphere
A similar, but more involved, equation development leads to the following equations for the hemisphere that is joined to the cylinder at z = 0:
2
wo= wps -ÁÊ 2R l˜ˆ Vo + Ë ÊÁ 2E l t ¯ ˆ˜ Mo ; and (32)
Ë E t ¯
ss
23
Á
fo= -Ê 2l ˆ˜ Vo + ÊÁ 4l ˆ˜ Mo ; where (33)
Ë Et ¯ Ë REt ¯
ss
Notes C: Pressure Vessel Stresses Page C-9
l
= b R ; and (34)
s
2
b
= ÁÊ 3(1-n )˜ˆ 14 . (35)
sR22
Ë t¯
s
6.3
Combination
We now have four linear equations (Eqs. 30-33) in four unknowns (wo, fo, Mo, and Vo) and therefore can solve for conditions at the joint (z = 0) that give continuity. Subsequently, corresponding values of c1 and c21 (based on Eq. 29; c3 = 0; and c4 = 0) can be found and used to determine conditions in the cylinder for other values of z.
7.
RELATED INFORMATION
Related information on pressure vessel stress analyses can be found as follows:
-thin shell pressure stresses (Ref. 2, pp. 27-32; Ref. 3, pp. 33-45);
-thick shell pressure stresses (Ref. 2, pp. 32-37; Ref. 3, pp. 56-64);
-thermal stresses (Ref. 2, pp. 37-44; Ref. 3, pp. 74-88);
-discontinuity stresses (Ref. 3, pp. 159-185); and
-combination information for a cylindrical shell (Ref. 4) and for a spherical shell (Ref. 5).
Be aware that in this note and in the references, a variety of approximations and notations are used.
For example, the thin shell approach adopted herein does take some account for the shell "squeezing" caused by pressure action in the radial direction. Other references adopt an assumption that the effect is negligible. The definition of the radius R provides a second example. It is used herein as the radius of the shell center surface but is used elsewhere as the shell inside radius.
Other reference-to-reference differences exist; therefore each reference should be used with care. However, each reference provides a unique viewpoint and derivation approach. It can be used as a useful supplement to the information of class lectures and this note.
Notes C: Pressure Vessel Stresses Page C-10
REFERENCES
1.
"Reactor Vessel Information," MIT Nuclear Engineering Department Notes L.7, from Pilgrim station Unit 2, Preliminary Safety Analysis Report (PSAR).
2.
L. Wolf, M.S. Kazimi, and N.E. Todreas, "Introduction to Structural Mechanics," MIT Nuclear Engineering Notes L.4, revision of Fall 1995.
3.
J. F. Harvey, "Theory and Design of Pressure Vessels," Van Nostrand Reinhold Co., New York, 1985.
4.
"Article A-2000, Analysis of Cylindrical Shells," in Section III, Rules for Construction of Nuclear Vessels, ASME Boiler and Pressure Vessel Code," pp. 547-549, ASME, New York, 1968 Edition.
5.
"Article A-3000, Analysis of Spherical Shells," in Section III, Rules for Construction of Nuclear Vessels, ASME Boiler and Pressure Vessel Code," pp. 553-556, ASME, New York, 1968 Edition.
APPENDIX C1
Stresses Near Axial Center of Long Cylinder Comparison of Alternate Approximations J.E. Meyer revision of July 1996
PRESSURE STRESSES (at r = a)
(STRESS at r = a) / (PRESSURE)
Equations for Stress at r = a
(
s / p) for (b / a ) = 1.1 radial
hoop
axial
radial
hoop
axial
Exact, Thick Shell
-p
1b
1c
-1.00
10.52
4.76 Thin Shell for Curvatures
-(p / 2)
2b
2c
-0.50
10.50
5.25 Ring Finite Element
3a
3b
3c
= 1c
-0.48
10.00
4.76
22
Ê b2 + a ˆ ÊÁ a
Á
(1c)
=
sq
pR
t
(2b)
ˆ
(1b)
¯ p
¯ p
s
q
=
s
z
=
Ë
2
b2
-a
Ë
2
b 2
-a
Ê a ˆ
s
= pR (2c) s=-Ê a ˆ (3a) s= (3b)
zr q
2 t Ëb +a ¯p Ë b -a ¯ p
Symbols are: p = inside pressure; zero = outside pressure; s = stress; (r, q, z) = (radial, hoop, axial) coordinate directions; r = b = radius of shell outside surface; r = a = radius of shell inside surface; r = R = 0.5 (b + a) = radius of shell central surface; t = b - a = shell thickness.
THERMAL STRESSES (at r = a)
radial direction hoop direction axial direction
s
r
= 0
=
sq
1
E
n
-
(
eT
-
eTa
)
s
z
=
E
(eT -eTa)
1 -n
Symbols
are: E = Young’s Modulus; n = Poisson’s Ratio; e T = volume averaged thermal strain; eTa = thermal strain at r = a.
--------
----------------
------------ -------------
APPENDIX C2 (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
Includes
X
X
X
Discontinuities
Includes
Stress
X
Concentrations
Caused
by
X
X
X
X
X
Mechanical Loads
Caused
by Thermal
X
X
Stresses
Tensile
Strain
Bending
Strain
Fatigue
Failure
Mode
Limit
Control
Limit
Control
Load
Shake-
Load
(Shakedown)
down)
Limit
from Concepts
S
y
2 S
y
1.5
Sy
2 S
y
S-N
Curves
Code
Limits
S
m
1.5
Sm
--------1.5
Sm
----------------3
Sm
U
F < 1
Inelastic
Analysis
e
=1%
(Elevated
e
= 2%
Temperature)
e = 5%
U
FT < D
.jpg)
