ELSEVIER
`
`PII:SO308-0161(97)00067-7
`
`J. Pres. Ves. & Piping
`ht.
`0 1998 Elsevier
`Science
`Limited.
`Printed
`
`(1997)
`73
`rights
`All
`in Northern
`0308-0161/97/$17.00
`
`183-190
`reserved
`Ireland
`
`factors of flat end to
`Stress concentration
`cylindrical shell connection with a fillet or stress
`relief groove subjected to internal pressure
`
`Institute for pressure vessel and plant
`
`Reinhard Preiss
`technology, Vienna University of Technology, Gusshausstrafie, 30 A-1040 Vienna, Austria
`
`(Received 20 July 1997; accepted 7 August 1997)
`
`For fatigue assessment of pressure vessels (parts), as proposed by the CEN Technical
`Committee 54, the knowledge of some stress quantities in the structure is necessary,
`e.g. equivalent stresses according to Tresca’s yield criterion and principal stresses at
`welds. In this article, these quantities are given in the form of stress concentration
`factors for the flat end to cylindrical shell connection with a fillet or stress relief
`groove subjected to internal pressure. In this context, a stress concentration factor is
`defined as the ratio of the ‘true’ stress (obtained by FE analysis) to the analytically
`determined stress corresponding to the linear elastic idealized shell plate model. For
`different values of the ratio of fillet/groove-radius to plate thickness, approximation
`functions for those stress concentration factors are obtained, covering the range of
`conventional flat end to cylindrical shell connection geometries. 0 1998 Elsevier
`Science Limited.
`
`1 INTRODUCTION
`
`To apply the ‘detailed fatigue assessment method for pres-
`sure vessels’ ’ proposed by the Working Group C of the CEN
`Technical Committee 54, the following information on the
`following stress quantities is beneficial:
`
`1. for unwelded regions, the equivalent stress according
`to the Tresca criterion and the arithnetic mean of
`the principal stresses corresponding to the Tresca cri-
`terion;
`2. for welded regions, the equivalent stress according to
`the Tresca criterion and the maximum principal stress
`which acts closest to the normal to the weld, if the
`considered fatigue crack initiation site is the weld toe
`or at the weld surface.To have these quantities
`available for flat end to cylindrical shell connections
`with a fillet or a stress relief groove subjected to
`internal pressure, stress concentration factors which
`are defined as the ratio of the wanted stress to the
`analytically calculated stress of the idealized shell
`plate model are given in this paper. Additionally,
`the stress concentration factors for the equivalent
`stress according to the von Mises criterion are
`presented.
`
`183
`
`The geometric parameters influencing the stress concen-
`tration factors are, as shown in Fig. 1:
`
`1. the mean diameter of the shell d;
`2. the shell thickness t,;
`3. the plate thickness t,;
`4. the radius r of the fillet or the stress relief groove,
`respectively.
`
`To enable a parameter study with the finite element
`method the following non-dimensional parameters were
`chosen:
`Ap = 4,
`‘P
`
`A, =
`
`‘,
`z
`
`.fl = 5
`
`P
`in the case of a fillet, and
`
`f2=
`
`1
`
`in the case of a stress relief groove.
`
`(34
`
`(3b)
`
`Page 1 of 8
`
`

`

`184
`
`R. Preiss
`
`Fig. 1. Geometric parameters influencing the stress concentration
`factors.
`
`2 IDEALIZED
`
`SHELL PLATE MODEL
`
`The free body diagram of this model is shown in Fig. 2. The
`longitudinal bending moment m, and the transverse shear
`force qr (both generalized stresses) for the linear elastic
`constitutive law can be determined by means of the influ-
`ence function for the infinitely
`long cylinder, i.e. from
`Young’, Table 29, Case 8 and 10, the influence function
`for the annular plate, Table 24, Case 5a, the deformation
`of the single bodies under pressure action, Table 28, Case lb
`and Table 24, Case 2a and the transition conditions at the
`junction foot, i.e. the radial displacements and the rotations
`of the shell and the plate.
`The following equations result for the bending moment
`m, and the transverse shear force qr:
`
`+ C242
`m, = 0.25pd2. C1B22
`BHBZZ -%
`
`+ C241
`qr = OSpd - C1B12
`B:2 - BI$%z
`
`,
`
`3
`
`(5)
`
`using the abbreviations3 given, for Poisson’s ratio v = 0.3,
`below
`
`c, = 0.1313$,
`
`C2 = 0.4250h, +0.1313X;,
`
`@a)
`
`(6b)
`
`Bll = 1.5013h;5 + 1.0500X;,
`
`B12 =0.8261X,2 - l.OSOO?$,
`
`(6~)
`
`(64
`
`Bz2 = 0.9089X;.5 + 1.0500X,.
`
`(6e)
`Therefore, the longitudinal stress at the inner side of the
`cylinder at the junction foot is:
`
`u,, =ph, 0.25 + 1.5X, W22
`BIIBZZ
`
`+ C242
`
`-
`
`$2
`
`(7)
`
`which is the maximum principal stress in the structure.
`The circumferential membrane force no and the circum-
`ferential bending moment me can be expressed as:
`
`(8)
`
`mg = urns,
`
`(9)
`where u, is the radial displacement at the junction foot and
`E is Young’s modulus.
`Eqn (8) can be given in terms of the non-dimensional
`parameters:
`
`W22
`ng =p$ 1 + 1.6523X,
`[
`
`BI&Z
`
`+ CzB12
`
`-%
`
`(10)
`
`_
`
`1 8178xo.5
`
`z
`
`- G&z
`
`- C241
`
`This leads to the hoop stress on the inner side of the
`
`$2 - BllBzz 1.
`
`discrete
`caiculated
`finite
`
`valbes
`by
`
`element
`
`met
`
`.hod
`
`or^
`
`the
`
`domobn
`approxlmotion
`functions
`
`I
`
`5
`
`10
`
`15
`
`20
`
`)
`
`h*
`
`Fig. 2. Free-body diagram.
`
`Fig. 3. Domain of the approximation
`
`functions.
`
`Page 2 of 8
`
`

`

`Stress concentration
`
`factors
`
`185
`
`ANSYS 5.3
`JUL 23 1997
`08:39:09
`PLOT NO.
`ELEMENTS
`TYPE NUM
`
`1
`
`=l
`zv
`DIST=220
`XF
`=110.526
`YF
`-200
`Z-BUFFER
`
`WIND=2
`zv
`=l
`*DIST=41.615
`=198.343
`*XF
`=49.548
`*YF
`Z-BUFFER
`
`Fig. 4. Example of an FE-model
`
`for the connection with a fillet.
`
`cylinder at the junction foot:
`
`use =p
`
`0.425h, + 1.2761A; ;;;Z+$‘2
`L
`
`12
`
`(11)
`
`- 2.5708 -C142-C2fh1
`B:2
`-
`
`B, 1B22
`
`1.
`
`Thus, the equivalent stresses at the inner side of the
`structure at the junction foot for plane stress conditions,
`i.e. neglecting the third principal stress - p (pressure),
`are given by:
`
`uVT =max( Ls,, - ~~~1, l(sssl, loeel),
`
`according to the Tresca criterion and
`
`%4=Jn>
`
`(12)
`
`(13)
`
`according to the von Mises criterion, respectively.
`
`with an increment of 2.5. The ratios of the plate thickness to
`the shell thickness were l:l, 2:1, 3:1,4:1,5:1 and6.1; there-
`fore, the parameter A, varied from 5 to 120. Hence, the
`number of finite element calculations per fixed parameter
`f2 is 42. The calculations were carried out forf,, 0.65, 1.00,
`1.35, 1.70, 2.05 and 2.40.
`The stress concentration factors K are calculated as the
`ratio of the maximum equivalent stress SM or ST at the fillet
`corresponding to the finite element analysis, to the analyti-
`cally calculated equivalent stress (TvT or (TvM from the
`idelized shell plate model:
`
`OVT
`
`KM=%
`
`(TVM
`
`(15)
`
`3 FLAT END TO CYLINDRICAL SHELL
`CONNECTION WITH A FILLET
`
`Within one set of finite element calculations, the parameter
`f, was kept constant; the parameter A, varied from 5 to 20
`
`where the index T corresponds to the Tresca criterion and
`the index M to the von Mises criterion.
`For every fixed value of the parameter f,, an analytic
`function K = K(h,, AP) was determined by means of regres-
`sion analysis. Fig. 3 shows the domain of this function as
`well as the discrete points, where the finite element calcula-
`tions were carried out. Within the domain of the function,
`
`Page 3 of 8
`
`

`

`186
`
`R. Preiss
`
`circumference
`
`of
`
`fillet
`
`Cmml
`
`4
`
`5.3
`1997
`
`ANSYS
`JUL
`23
`09:02:19
`PLOT NO.
`POST1
`STEP-1
`SUB =1
`TIME=1
`PATH PLOT
`NODl=536
`NOD2488
`
`-1
`zv
`DISTa.75
`XF
`=.5
`YF
`-.5
`ZF
`0.5
`Z-SUFFER
`
`5
`
`5.3
`1997
`
`ANSYS
`JUL
`23
`09:04:03
`PLOT NO.
`EQSTl
`STEP=1
`SUB
`-1
`TIME-1
`PATH PLOT
`NOW-536
`NOD2=189
`
`=1
`zv
`DIST-.75
`XF
`=.5
`YF
`-.S
`ZF
`-.5
`Z-BIJFEZR
`
`Fig. 5. Typical equivalent stress curves for the connection with a fillet.
`
`the area of usual geometries is covered; the use of the func-
`tion outside the domain is not recommended.
`For a practical design, one can determine the K-values for
`thef[- parameters, which are surrounding the actualfl, by
`means of the regression functions given below and then
`calculate the actual K-value by linear interpolation.
`The finite element calculations were performed with
`the program ANSYS@ 5.3, the elements used were the
`
`eight-node isoparametric ‘PLANE 82’-elements. All analy-
`ses were carried out in a rotational symmetric arrangement,
`with a linear elastic constitutive law (E = 200 GPa, v = 0.3)
`and with an internal pressure of 0.1 MPa. To achieve a
`sufficient mesh fineness of the FE-model, a self-written
`macro was used, Fig. 4 shows the FE-model for the para-
`meters X, = 10, X, = 20, fi = 1 .OO, as example.
`On the very point on the fillet surface where the
`
`Page 4 of 8
`
`

`

`Stress concentration
`
`factors
`
`187
`
`A
`
`0.264
`0.229
`0.059
`0.098
`0.060
`0.053
`0.017
`
`A
`
`0.140
`0.174
`0.087
`0.040
`0.000
`- 0.026
`- 0.032
`
`B
`
`0.232
`0.191
`0.231
`0.191
`0.186
`0.169
`0.169
`
`B
`
`0.293
`0.219
`0.226
`0.219
`0.213
`0.204
`0.191
`
`Table 1. Coefficients
`
`for KT in the case of a fillet
`
`c
`
`- 0.0040
`- 0.0027
`- 0.0033
`- 0.0020
`- 0.0017
`- 0.0009
`- 0.0008
`
`D
`
`6.312
`2.619
`1.676
`1.180
`0.703
`0.130
`0.590
`
`E
`
`- 0.233
`- 0.018
`- 0.026
`- 0.018
`0.042
`0.166
`0.065
`
`F
`
`G
`
`0.0372
`0.0121
`0.0521
`- 0.0008
`- 0.0086
`- 0.0191
`- 0.0194
`
`75.783
`- 29.586
`- 18.001
`- 6.458
`7.382
`21.511
`24.822
`
`Table 2. Coefficients
`
`for KM in the case of a fillet
`
`c
`
`- 0.0055
`- 0.0032
`- 0.0030
`- 0.0024
`- 0.0018
`- 0.0013
`- 0.0007
`
`D
`
`9.537
`3.878
`2.57
`1.706
`1,021
`0.735
`0.504
`
`E
`
`- 0.858
`- 0.217
`- 0.156
`- 0.062
`0.050
`0.093
`0.128
`
`F
`
`G
`
`0.055 1
`0.0161
`0.0074
`- 0.0027
`- 0.0130
`- 0.0191
`- 0.0241
`
`98.311
`- 31.609
`- 17.226
`- 0.833
`15.411
`26.254
`34.400
`
`H
`
`18.026
`7.132
`4.484
`1.528
`- 2.188
`- 6.210
`- 7.689
`
`H
`
`24.222
`8.631
`5.351
`1.254
`- 2.988
`- 6.114
`- 8.505
`
`I
`
`0.344
`0.557
`0.520
`0.587
`0.743
`0.942
`1.043
`
`I
`
`0.076
`0.502
`0.486
`0.607
`0.783
`0.941
`1.066
`
`fl
`0.30
`0.65
`1.00
`1.35
`1.70
`2.05
`2.40
`
`fl
`0.30
`0.65
`1 .oo
`1.35
`1.70
`2.05
`2.40
`
`55
`,621 a I
`
`io
`
`is
`
`io
`
`is
`
`jo
`
`is
`
`40
`
`45
`
`io
`
`$5
`
`80
`
`Fig. 6. Contour plot of KT withf,
`
`= 1.00 for the connection with a fillet.
`
`-
`
`.62
`
`,587
`
`Page 5 of 8
`
`

`

`R. Preiss
`
`ANSYS 5.3
`JUL 23 1997
`09:48:47
`PLOT NO.
`ELEMENTS
`TYPE NUM
`
`1
`
`=l
`ZV
`DIST=220
`XF
`=110.526
`YF
`=200
`Z-BUFFER
`
`WIND=2
`zv
`=l
`'DIST=19.255
`=190.889
`*XF
`*YF
`=34.645
`Z-BUFFER
`
`Fig. 7. Example of an FE-model for the connection with a stress relief groove.
`
`maximum principal stress occurred, the other principal
`stress value (plane-stress state) was always positive. There-
`fore, the equivalent stress according to the Tresca criterion
`is equal to the maximum principal stress in that point. The
`direction of the maximum principal stress was found to be
`approximately tangential to the surface of the fillet in all
`calculated cases. Fig. 5 shows the typical forms of the
`curves:
`
`1. Tresca’s equivalent stress versus the circumference of
`the fillet,
`2. von Mises’ equivalent stress versus the circumference
`of the fillet,
`
`for the parameters h, = 20, h, = 10,fl = 1.00.
`The approximation functions for the stress concentration
`factor according to the Tresca or von Mises equivalent stress
`can, in general, be written as:
`
`given for KT in Table 1 and for KM in Table 2. For a
`quick use of the approximation functions, they can be
`drawn as contour plots as Fig. 6 shows for KT andfr = 1 .OO.
`The ratio of the ‘exact’ K-values (obtained by FE-
`analysis) and the approximated ones (obtained the regres-
`sion functions) is between 0.95 and 1.05 at all evaluated
`points except for:
`
`fi = 0.30, A, = hp = 5.0
`
`fi = 2.05, X, = hp = 5.0
`
`f, = 2.40, X, = hp = 5.0,
`
`where it is between 0.90 and 1.10.
`Due to the plane-stress state and the positive value of the
`smaller principal stress in the interesting points, the arith-
`metic mean of the principal stresses corresponding to the
`Tresca criterion is equal to 0.5 times the equivalent stress.
`
`4 FLAT END TO CYLINDRICAL SHELL
`CONNECTION WITH A STRESS RELIEF GROOVE
`
`The coefficients A to I for the different values of fr are
`
`Principally, the assumption is made that the centre of the
`
`Page 6 of 8
`
`

`

`Stress concentration
`
`factors
`
`189
`
`2
`
`LNSYS 5.3
`JUL 23 1997
`09:54106
`Pw? MO.
`PO921
`STLP-1
`SOS -1
`rwE=1
`PAT” PLOT
`NDDl-26
`NDD2'734
`
`3
`
`PHSYS 5.3
`JUL 23 1997
`09:55:25
`PLO? HO.
`POST1
`STEP-1
`SM
`-1
`TIMt-1
`PATH PLOT
`NODI-26
`NODZ-134
`
`-1
`S"
`DI(IT=.7,
`XP
`1.5
`lP
`1.5
`EP
`1.5
`I-tumm
`
`4
`
`ANSYS 5.3
`Jut
`23 1997
`10:00:50
`'PLOTHO.
`POST1
`STCP-1
`SUB
`-1
`TIPIt-
`PmT
`Pm"
`NODl-26
`NDDZ-4733
`
`-1
`tv
`l DIET=.'),
`*XI
`=.5
`VP
`1.5
`021
`1.5
`L-WIPER
`
`Fig. 8. Typical stress curves for the connection with a stress relief.
`
`Page 7 of 8
`
`

`

`190
`
`fz
`
`0.10
`0.25
`0.40
`
`f2
`
`0.10
`0.25
`0.40
`
`f2
`
`0.10
`0.25
`0.40
`
`R. Preiss
`
`Table 3. Coefficients for KT in the case of a stress relief groove
`
`B
`
`0.036
`0.039
`0.043
`
`C
`
`- 0.0009
`- 0.0008
`- 0.0008
`
`D
`
`4.677
`3.097
`3.266
`
`E
`
`0.276
`0.120
`- 0.062
`
`F
`
`0.012
`- 0.031
`- 0.073
`
`G
`
`H
`
`- 18.346
`61.376
`165.155
`
`2.561
`- 21.340
`- 50.398
`
`Table 4. Coefficients for KM in the case of a stress relief groove
`
`B
`
`0.039
`0.012
`0.05
`
`C
`
`D
`
`- 0.0011
`0.0003
`- 0.0011
`
`6.404
`- 1.991
`4.144
`
`E
`
`- 0.087
`1.498
`- 0.260
`
`F
`
`0.024
`- 0.091
`- 0.066
`
`G
`
`H
`
`- 32.839
`109.820
`163.07
`
`7.359
`- 32.334
`- 48.26
`
`I
`
`1.495
`2.950
`5.073
`
`I
`
`1.269
`3.360
`4.937
`
`Table 5. Coefficients for Kw
`
`in the case of a stress relief groove
`
`B
`
`0.023
`0.030
`0.042
`
`C
`
`- 0.0006
`- 0.0008
`- 0.0011
`
`D
`
`0.189
`0.175
`1.842
`
`E
`
`0.4906
`0.291
`- 0.059
`
`F
`
`0.019
`0.015
`0.015
`
`G
`
`- 33.467
`- 22.702
`- 21.195
`
`H
`
`10.733
`7.212
`5.703
`
`I
`
`- 0.356
`- 0.290
`- 0.166
`
`A
`
`0.586
`0.446
`0.413
`
`A
`
`0.576
`0.559
`0.395
`
`A
`
`0.567
`0.395
`0.274
`
`stress relief groove (circle) is a point of the upper boundary
`plane of the plate. Calculations were carried out for semi-
`circular grooves and for straight end grooves. This showed
`that the influence of the groove’s geometric contour at the
`inner end of the groove, i.e. for the radial coordinate less
`than (d/2 - tz/2 - r), to the stress concentration factors is
`negligible.
`Within a set of FE calculations, the parameterfi was kept
`constant; the parameter X, varied from 5 to 20 with an
`increment of 2.5. The ratios of the plate thickness to the
`shell thickness were l:l, 2:1, 3:1, 4:1, 5:l and 6:l. There-
`fore, the parameter X, varies from 5 to 120. FE calculations
`were carried out forf2 = 0.25 and 0.40. Again, a self-written
`macro was used to provide sufficient mesh-fineness for all
`geometries, Fig. 7 shows the FE-model for the parameters
`f2=0.25, X, = 10, X, = 20, as an example.
`The stress concentration factors KT and KM are calculated
`according to eqns (14) and (15), whereas Sr and SM are the
`maximum values of the equivalent stresses on the groove
`surface (which are also the maximum equivalent stresses of
`the whole structure). The stress concentration factor KW is
`defined as the ratio of the maximum principal stress SW
`which acts closest to the generatrix of the shell at the
`upper end of the groove to the longitudinal (maximum)
`principal stress (T,, at the junction foot of the idealized
`shell plate model:
`
`(17)
`
`Fig. 8 shows the typical form of the curves:
`
`1. Tresca’s equivalent stress versus the circumference of
`the groove,
`2. von Mises’ equivalent stress versus the circumference
`of the groove,
`
`3. meridian stress at the groove’s upper end versus
`thickness of the cylinder
`
`for the parameters X, = 10, X, = 20,fz = 0.25.
`The approximation functions for the stress concentration
`factors KT, KM and Kw can, in general, be written as:
`
`The coefficients A-Z for the different values of fi are given
`for KT in Table 3, for KM in Table 4 and for Kw in Table 5.
`The ratio of the ‘exact’ K values as determined by the FE-
`analyses and the approximated ones is between 0.95 and
`1.05 for f2 = 0.10 and fi = 0.25 and between 0.90 and
`1.10 forfi = 0.40. Again, for a quick use the approximation
`function can be drawn as contour plots, as performed for all
`functions in Preiss4.
`
`REFERENCES
`
`1. CEN TC 54-Working Group C, Detailed Assessment
`Method of Fatigue Life for Pressure Vessels. Document no.
`1017, 1997.
`2. Young, W.C., Roark’s Formulas
`for Stress and Strain, 6th
`Edn, MacGraw-Hill, New York, 1989.
`3. Zeman, J. L. Die Verbindung Mantel-ebener Boden unter
`Druckeinwirkung. Techn. Ubenvachung, 1994,35( 12), 450-
`453.
`4. Preiss, R., Spannungserhohungsfaktoma fiir die Verbindung
`
`Zylindermantel-ebener Boden mit Ubergangsradius oder
`Entlastungnut. Institutsbericht No. 8, Technische Universit
`at Wien, Institut fur Apparate und Anlagenbau, 1997.
`
`Page 8 of 8
`
`