- Email: [email protected]
A displacement-based pipe elbow element
t-KM-7949188 S3.00 + 0.00 0 i9~S~mon~~
computers & ~luctwes Vol. 29, No. 2, PP. 339-343, 1988 Printed in Gnat Britain.
A DISPLACEMENT-BASED
PIPE ELBOW ELEMENT
C. MILM-ELLO and A. E. HUBPE INGAR, Institute de Desarrollo y Diseiio, CONICET, Fundacibn ARCIEN, Avellaneda 3657, 3000 Santa Fe, Argentina (Received 6 August 1987) Abstract-A new element for stress analysis of piping is presented. The element can handle interaction effects and can be seen as an enhan~ment of the element from Bathe and Almeida (J. a&. Me&. 47, 93-100, 1980). The necessary inter-element continuity for the elbow skin is assured through Hermitian interpolation of the ovahzation pattern in the axial direction. The inclusion of warping for the elbow cross-section increases the element’s capability. The main characteristics of the Co elbow element, the easy handling of boundary conditions and the satisfaction of the rigid-body mode criterion, are retained,
for this. In this way the displa~ment
In a series of papers, Bathe and Almeida [I, 2] presented what they call a simple pipe elbow element. The element is a beam element enhanced to support ovalization. The element is actually a simple one and can handle boundary conditions in an easy way. The necessary inter-element continuity for the tube wall deformation is imposed by penalty matrices and only the most relevant terms of linear shell theory are retained to compute the ovalization strains. The model assumes that the elbow sections originally planar and normal to the neutral axis remain planar after defo~ation. As has been pointed out by other authors, this assumption is a serious drawback in a pipe element, In this paper we propose an improvement. With the present formulation no penalty matrices are needed and distortion of the cross-sections is allowed. Nevertheless the simplicity of the element and main characteristics of the former model are retained.
of a skin midsurface point can be described by the b.d.o.f. plus the field (u, v, w) related with the (q, 5. [) coordinate system (Fig. 1). The proposed expressions for u and v are:
k=l
+
u(r,#)=
[ m=2
d&k(rm(m4)
j,
(1)
z(c$ff&)
i
k-1 [ +
1
m=1
&hk(r))sin(2m4)
+ :
(d%%(r)
m=i
+_fthk(r))cos(2m#)
.
1
(2)
FIJXD OF DISPLACEMENTS AND STRAINS The element formulation starts from a third-order C’ beam element (four modes), with planar sections normal to the neutral axis remaining planar after deformation. The beam element has three displacements a,, uv, w, and three rotations 0,, fl,, 0, at each node. From these degrees of freedom, hereinafter called b.d.o.f., we can compute the strains at the elbow skin in the curvilinear system (g, <, 0. The procedure to obtain the system is clearly developed in [l]. Following Bathe and Almeida [l], we also introduce an ovalization pattern through the addition of a displacement field v and w (Fig. I), but the axial variation of the pattern obeys a Hermitian inte~lation of order 2 (cubic). We add to this pattern a displacement ti in the direction q to allow the warping of the cross-section. We select a thirdorder Lagrangian interpolation in the axial direction
w(r, 4) is obtained from the condition that the circumferential strains are assumed to vanish at the middle surface of the pipe wall: t3V
w=-drb’ ci and di are the degrees of freedom (d.0.f.) of the ovalization modes, and ai and fi are the d.o.f. of their derivatives in the r direction, respectively, at both ends. Here 0 < r < 8, R, where O,R is the length of the elbow axis. The values of NCand Nd depend on the pipe factor 1 = RGla’. We cannot yet establish which are the most adequate values for NP and N,, although we use N,, N4 = 4 in this work. We do not use m = 0 and m = 1 in the circumferential expansion for u because they are introduced by the b.d.o.f. through the beam theory. 339
C. MILITELLOand A. E.
340
HLJESPE
%t =
_clcosm
R
(g-u
sin6)+:$
(12)
L
Kv’=(R - a cos d)a ab au aea4 +~cos9-ZJ
a0
(-
’
)
2 sin 4 (13)
+ (R - a cos(#l)2 We must take (l/R)a( Pai-.
into
account
that
a( )/a0 =
CONTINUITY CONDITION OF PIPE SKIN DEFORMATION
Fig. 1. Pipe elbow model-coordinate parameters.
systems and main
The complete strain-displacement written as follows:
relations can be
=B”+b/B”]
1
-$,
[
At the intersection of two elbow elements (Fig. 2) the ovalization is continuous because the same ovalization degrees of freedom pertain to both elements. The other condition to be satisfied in the skin of an elbow union is: aw
(4)
aw
% I.=;i;;
I “+,’
As can be seen in Fig. 1, where ub is a vector containing b.d.0.f. and II* is a vector containing the ovalization and warping parameters. Bb is evaluated following Bathe and Almeida’s work [l]. The components of matrix B” can be obtained through the following expressions. They are a specialization of the equations from Washizu’s book [4] for a thin toroidal shell (6/a <<1). (5) (6) (7)
R
a( ) q=
R -a
aw
R, R,-acos4%
R II+1 .=R,+,-acos4dr
1 K”=(~ -UCOS~)~
+(R
azw au -++cos~ a82
R R -acos4’
(8)
) (9)
(10)
(11)
aw “+,’
(14)
Taking into account the fact that the proposed displacement automatically satisfies aw/arl, = for the common case of equal radius aw/arl,+,, elbows, the continuity condition is fulfilled. In joining a straight pipe with an elbow, we find that the error in fulfilling the condition is: Error = 1 -
g+vsinB-wcos4
a( ) cos $I ar
I Fig. 2. Elbow joint.
(15)
A displacement-based pipe elbow element
341
We cannot diminish this error by means of penalty matrices. It could be diminished introducing short elements at the junction with fictitious increasing radii. This last trick was not used in the present work. EFFECT OF THE CROSSSECTIONAL DISTORTION The importance of the elbow cross-section warping was pointed out by Ohtsubo and Watanabe [5]. To show this effect we are going to analyze two problems: an elbow subjected to out-of-plane and another with end effects subjected to in-plane bending, cases 3 and 1 rezpeztively of Fig. 3. The experimental values were taken from [6] (case 1) and [7] (cases 2 and 3). We name the element formulation without u Pi, and the full formulation P2. Figures 4a and b and 5a and b show the improvement of the numerical results modeling with the P2 formulation with respect to PI. As can be expected, the distortion effect in the out-of-plane case is more significant than in the in-plane case. In the graphs
,$%;
b+-%,m
@CC nom
0
30
tb)
60
90
120
150
160
+ (de$l)
Fig. 4. Case 1, section C. (a) ~n~tudin~ stress factor. (b) Circumferential stress factor.
effects. The problem consists of a 60” elbow with
with a,,,, = Mu/Z, where M is the moment at the section considered and I is the inertia of the elbow cross-section. END-EFFECTS
Case 2 was proposed in order to show the element behavior in analyzing bends with severe interaction
one end joined to a straight pipe and the other to a clamped flange. In the experimental work [7] the authors did not establish exactly the strain-gauge position for section A (we can quote from their work ‘section A close to the flange’). To carry out the analysis, we model the elbow with 12 elements. This model is necessary in order to obtain an acceptable continuity of the stresses near the flange. In Fig. 6a and b we depict the Core 3
II
R=1066.8 8-11.2 2am711.2 x*0.097 Material data
I
I
Flanges at A an? 6 Clamped at A Moment M, at 6 (in-plane bending)
End effects
Elements
~80.28 E = 20.000
6 in 49
I
I
vso.3 E=21?00
vao.3 E*l9900
Flanges at A and 8 Clamped at A Force 6C00 kg at 6 (In-plane bendlng)
Flanges at A and 8 Clamped at A Force F = loo0 kg at 6
6 In elbow
6 In elbow
3 in straight tube
3 in stmlght tubes
Fig. 3. Data for proposed problems.
(out-Of-plane
bending)
C. MILITELLOand A. E. HUFSPE
342
0
l5-
0
IO
-
0.5 o-05 -4 -
Experlmentol values (IMAMASA-URAGAMI) PI (3 ovalization modes)
0
---6,--
P2 (3 ovalization
1
0
-I .o -I .5
modes)
I
I
I
I
I
30
60
90
120
150
(al
-
1
.i
I
25FO)
30
60
-2.0
I
I
I
90 120 150 + ldeg) 0 Experlmevtalvalues (IMAMASA-UkAGAMI) --PI (2 ovaliratlon modes, 12 elements In elbow) 2 o - Z;T p’; (2 Ovalization modes)
(p (deg)
15-
-2 -4 -6l
I 30
0
I 60
I 90
I 120
I 150
I 60
-150 ^ h 0 30 (b)
# (deg)
(b)
section C. (a) Longitudinal stress factor. (h) Circumferential stress factor.
I 90
I 120
I 150
180
+ ldegl
Fig. 5. Case 3,
results obtained at the flange-elbow joint (0 = 0’) and the results obtained at 8 = 2” (approximately 30 mm away from the joint). The computed response
0
Experimental values-section close to the flange (IMAMASA-URAGAMI 1 P2 (2 ovallzatlon modes) P2 (2 ovaliration modes 12 elements in elbow)
A-A
Fig. 7. Case 2, section C. (a) Longitudinal stress factor. (b) Circumferential stress factor.
exhibits a high stress variation near the joint. The correspondence between the computed and experimental values for longitudinal stresses is good, but the correspondence for the circumferential stresses is less good. In Fig. 7a and b, we can see the results obtained for section C (0 = 30”). The same results can be computed using six or 12 elements in the elbow. This is due to the smooth stress variation there. Again we use models Pl and P2 to analyze the effect of considering the warping of the cross-section. Finally, we can see in Fig. 8 that the longitudinal stress measured along the bend at 0 = 180” compares very well with the computed stress. THE INFLUENCE OF THE ODD TERMS
0 2
30
60
(al I
90
120
150
I60
In expression (2), sin(2mq5) accounts for in-plane problems. Through formula (3) it is translated to w
+ (deg) 0
4
-21 0
(b)
I 30
I 60
I 90
I I20
I I50
I I60
4 (deg)
Fig. 6. Case 2. (a) Longitudinal stress factor (close to the flange). (b) Circumferential stress factor (close to the flange).
-l
0
0
Experimental
I IO
values
I 20
(IMAMASA-URAGAMI)
I 30
I 40
,
I so
8 (deg)
Fig. 8. Longitudinal stress distribution along the axis bend at I#J= 180”.
A displacement-based
as cos(2m4). If the series for w contains only even terms, radial deflections are symmetrical about 4 = 90”. For an in-plane problem this is true only when a/R is small. The inclusion of odd terms for the in-plane problem allows the possibility of non-symmetrical ovalization. Using the terms sin(n4) with II = 1,2,3,4, instead of sin(2mb), we analyze again cases 1 and 2. The results obtained, i.e. the P3 curve, are depicted in Figs 4a and b, 6a and b and 7a and b. For case 2 at the elbow-flange joint, both stresses change, from a maximum at intrados to a maximum at extrados. For cases 1 and 2, at section C, the main effects are in the circumferential stress. CONCLUSIONS
After adding the warping displacement u, the present formulation could be seen as a midway between Bathe and Almeida’s [ 1,2] and Ohtsubo and Watanabe’s [5] elements. Our element retains the simplicity of the Co beam element to impose boundary conditions, to satisfy the rigid body mode criterion and to be linked with other beam elements or displacement-based finite elements. As in Ohtsubo and Watanabe’s element [5], no
343
pipe elbow element
penalty matrices are needed due to the Hermitian interpolation for the ovalization pattern. The decision about what terms of the Fourier series must be used for the ovalization pattern remains with the analyst. With expressions (1) and (2) for u and u, taking N,, = N, = 4 and N, = Nd= 3, an element with 72 degrees of freedom results. REFERENCES
1. K. J. Bathe and C. A. Almeida, A simple and effective pipe elbow element, linear analysis. J. uppl. Mech. 47, 93-100 (1980). 2. K. J Bathe and C. A. Almeida, A simple and effective pipe elbow element, interaction effects. J. uppl. Mech. 49, 165-173 (1982). 3. K. J. Bathe, Finite Element Procedures in Engineering Analysis. Prentice-Hall, Englewood Cliffs, NJ (1982). 4. K. Washizu, Variational Methods in Elasticity &d Plasticitv. 1st edn. Peraamon Press. Oxford (1968). 5. H. Oht&o and 0. Witanabe, Stress analysk of pipe bends by ring elements. J. Press. Vess. Tech. 100, 112-122 (1978). 6. J. F. Whatham, In-plane bending of flanged elbows. Proc. Metal. Struct. Conf. The Institution of Engineers, Australia, Perth (1978). 7. J. Imamasa and K. Uragami, Experimental study of flexibility factors and stresses of welding elbows with end effects. Second Int. Conf. on Pressure Vessel Tech., San Antonio, TX (1973).
computers & ~luctwes Vol. 29, No. 2, PP. 339-343, 1988 Printed in Gnat Britain.
A DISPLACEMENT-BASED
PIPE ELBOW ELEMENT
C. MILM-ELLO and A. E. HUBPE INGAR, Institute de Desarrollo y Diseiio, CONICET, Fundacibn ARCIEN, Avellaneda 3657, 3000 Santa Fe, Argentina (Received 6 August 1987) Abstract-A new element for stress analysis of piping is presented. The element can handle interaction effects and can be seen as an enhan~ment of the element from Bathe and Almeida (J. a&. Me&. 47, 93-100, 1980). The necessary inter-element continuity for the elbow skin is assured through Hermitian interpolation of the ovahzation pattern in the axial direction. The inclusion of warping for the elbow cross-section increases the element’s capability. The main characteristics of the Co elbow element, the easy handling of boundary conditions and the satisfaction of the rigid-body mode criterion, are retained,
for this. In this way the displa~ment
In a series of papers, Bathe and Almeida [I, 2] presented what they call a simple pipe elbow element. The element is a beam element enhanced to support ovalization. The element is actually a simple one and can handle boundary conditions in an easy way. The necessary inter-element continuity for the tube wall deformation is imposed by penalty matrices and only the most relevant terms of linear shell theory are retained to compute the ovalization strains. The model assumes that the elbow sections originally planar and normal to the neutral axis remain planar after defo~ation. As has been pointed out by other authors, this assumption is a serious drawback in a pipe element, In this paper we propose an improvement. With the present formulation no penalty matrices are needed and distortion of the cross-sections is allowed. Nevertheless the simplicity of the element and main characteristics of the former model are retained.
of a skin midsurface point can be described by the b.d.o.f. plus the field (u, v, w) related with the (q, 5. [) coordinate system (Fig. 1). The proposed expressions for u and v are:
k=l
+
u(r,#)=
[ m=2
d&k(rm(m4)
j,
(1)
z(c$ff&)
i
k-1 [ +
1
m=1
&hk(r))sin(2m4)
+ :
(d%%(r)
m=i
+_fthk(r))cos(2m#)
.
1
(2)
FIJXD OF DISPLACEMENTS AND STRAINS The element formulation starts from a third-order C’ beam element (four modes), with planar sections normal to the neutral axis remaining planar after deformation. The beam element has three displacements a,, uv, w, and three rotations 0,, fl,, 0, at each node. From these degrees of freedom, hereinafter called b.d.o.f., we can compute the strains at the elbow skin in the curvilinear system (g, <, 0. The procedure to obtain the system is clearly developed in [l]. Following Bathe and Almeida [l], we also introduce an ovalization pattern through the addition of a displacement field v and w (Fig. I), but the axial variation of the pattern obeys a Hermitian inte~lation of order 2 (cubic). We add to this pattern a displacement ti in the direction q to allow the warping of the cross-section. We select a thirdorder Lagrangian interpolation in the axial direction
w(r, 4) is obtained from the condition that the circumferential strains are assumed to vanish at the middle surface of the pipe wall: t3V
w=-drb’ ci and di are the degrees of freedom (d.0.f.) of the ovalization modes, and ai and fi are the d.o.f. of their derivatives in the r direction, respectively, at both ends. Here 0 < r < 8, R, where O,R is the length of the elbow axis. The values of NCand Nd depend on the pipe factor 1 = RGla’. We cannot yet establish which are the most adequate values for NP and N,, although we use N,, N4 = 4 in this work. We do not use m = 0 and m = 1 in the circumferential expansion for u because they are introduced by the b.d.o.f. through the beam theory. 339
C. MILITELLOand A. E.
340
HLJESPE
%t =
_clcosm
R
(g-u
sin6)+:$
(12)
L
Kv’=(R - a cos d)a ab au aea4 +~cos9-ZJ
a0
(-
’
)
2 sin 4 (13)
+ (R - a cos(#l)2 We must take (l/R)a( Pai-.
into
account
that
a( )/a0 =
CONTINUITY CONDITION OF PIPE SKIN DEFORMATION
Fig. 1. Pipe elbow model-coordinate parameters.
systems and main
The complete strain-displacement written as follows:
relations can be
=B”+b/B”]
1
-$,
[
At the intersection of two elbow elements (Fig. 2) the ovalization is continuous because the same ovalization degrees of freedom pertain to both elements. The other condition to be satisfied in the skin of an elbow union is: aw
(4)
aw
% I.=;i;;
I “+,’
As can be seen in Fig. 1, where ub is a vector containing b.d.0.f. and II* is a vector containing the ovalization and warping parameters. Bb is evaluated following Bathe and Almeida’s work [l]. The components of matrix B” can be obtained through the following expressions. They are a specialization of the equations from Washizu’s book [4] for a thin toroidal shell (6/a <<1). (5) (6) (7)
R
a( ) q=
R -a
aw
R, R,-acos4%
R II+1 .=R,+,-acos4dr
1 K”=(~ -UCOS~)~
+(R
azw au -++cos~ a82
R R -acos4’
(8)
) (9)
(10)
(11)
aw “+,’
(14)
Taking into account the fact that the proposed displacement automatically satisfies aw/arl, = for the common case of equal radius aw/arl,+,, elbows, the continuity condition is fulfilled. In joining a straight pipe with an elbow, we find that the error in fulfilling the condition is: Error = 1 -
g+vsinB-wcos4
a( ) cos $I ar
I Fig. 2. Elbow joint.
(15)
A displacement-based pipe elbow element
341
We cannot diminish this error by means of penalty matrices. It could be diminished introducing short elements at the junction with fictitious increasing radii. This last trick was not used in the present work. EFFECT OF THE CROSSSECTIONAL DISTORTION The importance of the elbow cross-section warping was pointed out by Ohtsubo and Watanabe [5]. To show this effect we are going to analyze two problems: an elbow subjected to out-of-plane and another with end effects subjected to in-plane bending, cases 3 and 1 rezpeztively of Fig. 3. The experimental values were taken from [6] (case 1) and [7] (cases 2 and 3). We name the element formulation without u Pi, and the full formulation P2. Figures 4a and b and 5a and b show the improvement of the numerical results modeling with the P2 formulation with respect to PI. As can be expected, the distortion effect in the out-of-plane case is more significant than in the in-plane case. In the graphs
,$%;
b+-%,m
@CC nom
0
30
tb)
60
90
120
150
160
+ (de$l)
Fig. 4. Case 1, section C. (a) ~n~tudin~ stress factor. (b) Circumferential stress factor.
effects. The problem consists of a 60” elbow with
with a,,,, = Mu/Z, where M is the moment at the section considered and I is the inertia of the elbow cross-section. END-EFFECTS
Case 2 was proposed in order to show the element behavior in analyzing bends with severe interaction
one end joined to a straight pipe and the other to a clamped flange. In the experimental work [7] the authors did not establish exactly the strain-gauge position for section A (we can quote from their work ‘section A close to the flange’). To carry out the analysis, we model the elbow with 12 elements. This model is necessary in order to obtain an acceptable continuity of the stresses near the flange. In Fig. 6a and b we depict the Core 3
II
R=1066.8 8-11.2 2am711.2 x*0.097 Material data
I
I
Flanges at A an? 6 Clamped at A Moment M, at 6 (in-plane bending)
End effects
Elements
~80.28 E = 20.000
6 in 49
I
I
vso.3 E=21?00
vao.3 E*l9900
Flanges at A and 8 Clamped at A Force 6C00 kg at 6 (In-plane bendlng)
Flanges at A and 8 Clamped at A Force F = loo0 kg at 6
6 In elbow
6 In elbow
3 in straight tube
3 in stmlght tubes
Fig. 3. Data for proposed problems.
(out-Of-plane
bending)
C. MILITELLOand A. E. HUFSPE
342
0
l5-
0
IO
-
0.5 o-05 -4 -
Experlmentol values (IMAMASA-URAGAMI) PI (3 ovalization modes)
0
---6,--
P2 (3 ovalization
1
0
-I .o -I .5
modes)
I
I
I
I
I
30
60
90
120
150
(al
-
1
.i
I
25FO)
30
60
-2.0
I
I
I
90 120 150 + ldeg) 0 Experlmevtalvalues (IMAMASA-UkAGAMI) --PI (2 ovaliratlon modes, 12 elements In elbow) 2 o - Z;T p’; (2 Ovalization modes)
(p (deg)
15-
-2 -4 -6l
I 30
0
I 60
I 90
I 120
I 150
I 60
-150 ^ h 0 30 (b)
# (deg)
(b)
section C. (a) Longitudinal stress factor. (h) Circumferential stress factor.
I 90
I 120
I 150
180
+ ldegl
Fig. 5. Case 3,
results obtained at the flange-elbow joint (0 = 0’) and the results obtained at 8 = 2” (approximately 30 mm away from the joint). The computed response
0
Experimental values-section close to the flange (IMAMASA-URAGAMI 1 P2 (2 ovallzatlon modes) P2 (2 ovaliration modes 12 elements in elbow)
A-A
Fig. 7. Case 2, section C. (a) Longitudinal stress factor. (b) Circumferential stress factor.
exhibits a high stress variation near the joint. The correspondence between the computed and experimental values for longitudinal stresses is good, but the correspondence for the circumferential stresses is less good. In Fig. 7a and b, we can see the results obtained for section C (0 = 30”). The same results can be computed using six or 12 elements in the elbow. This is due to the smooth stress variation there. Again we use models Pl and P2 to analyze the effect of considering the warping of the cross-section. Finally, we can see in Fig. 8 that the longitudinal stress measured along the bend at 0 = 180” compares very well with the computed stress. THE INFLUENCE OF THE ODD TERMS
0 2
30
60
(al I
90
120
150
I60
In expression (2), sin(2mq5) accounts for in-plane problems. Through formula (3) it is translated to w
+ (deg) 0
4
-21 0
(b)
I 30
I 60
I 90
I I20
I I50
I I60
4 (deg)
Fig. 6. Case 2. (a) Longitudinal stress factor (close to the flange). (b) Circumferential stress factor (close to the flange).
-l
0
0
Experimental
I IO
values
I 20
(IMAMASA-URAGAMI)
I 30
I 40
,
I so
8 (deg)
Fig. 8. Longitudinal stress distribution along the axis bend at I#J= 180”.
A displacement-based
as cos(2m4). If the series for w contains only even terms, radial deflections are symmetrical about 4 = 90”. For an in-plane problem this is true only when a/R is small. The inclusion of odd terms for the in-plane problem allows the possibility of non-symmetrical ovalization. Using the terms sin(n4) with II = 1,2,3,4, instead of sin(2mb), we analyze again cases 1 and 2. The results obtained, i.e. the P3 curve, are depicted in Figs 4a and b, 6a and b and 7a and b. For case 2 at the elbow-flange joint, both stresses change, from a maximum at intrados to a maximum at extrados. For cases 1 and 2, at section C, the main effects are in the circumferential stress. CONCLUSIONS
After adding the warping displacement u, the present formulation could be seen as a midway between Bathe and Almeida’s [ 1,2] and Ohtsubo and Watanabe’s [5] elements. Our element retains the simplicity of the Co beam element to impose boundary conditions, to satisfy the rigid body mode criterion and to be linked with other beam elements or displacement-based finite elements. As in Ohtsubo and Watanabe’s element [5], no
343
pipe elbow element
penalty matrices are needed due to the Hermitian interpolation for the ovalization pattern. The decision about what terms of the Fourier series must be used for the ovalization pattern remains with the analyst. With expressions (1) and (2) for u and u, taking N,, = N, = 4 and N, = Nd= 3, an element with 72 degrees of freedom results. REFERENCES
1. K. J. Bathe and C. A. Almeida, A simple and effective pipe elbow element, linear analysis. J. uppl. Mech. 47, 93-100 (1980). 2. K. J Bathe and C. A. Almeida, A simple and effective pipe elbow element, interaction effects. J. uppl. Mech. 49, 165-173 (1982). 3. K. J. Bathe, Finite Element Procedures in Engineering Analysis. Prentice-Hall, Englewood Cliffs, NJ (1982). 4. K. Washizu, Variational Methods in Elasticity &d Plasticitv. 1st edn. Peraamon Press. Oxford (1968). 5. H. Oht&o and 0. Witanabe, Stress analysk of pipe bends by ring elements. J. Press. Vess. Tech. 100, 112-122 (1978). 6. J. F. Whatham, In-plane bending of flanged elbows. Proc. Metal. Struct. Conf. The Institution of Engineers, Australia, Perth (1978). 7. J. Imamasa and K. Uragami, Experimental study of flexibility factors and stresses of welding elbows with end effects. Second Int. Conf. on Pressure Vessel Tech., San Antonio, TX (1973).