IN-PLANE BENDING OF SINGLE,
UNREINFORCED MITRED PIPE BENDS
R. KITCMNG
Department of Mechanical Engineering, University of Manchester Institute of
Science and Technology
W. J. THOMPSON
Department of Mechanical Engineering, University of Manchester Znstitute of Science and Technology
Measurements of strain, transverse deflection, and change of radius have been taken on single unreinforced
pipe bends subjected to in-plane bending moments. Specimens having three different mitre angles for each
of two sizes of pipe have been used. Observations were made with different combinations of leg lengths for
each specimen.
Stresses and flexibilities are compared with those calculated by the formulae given in the A.S.A. Piping
Code B.31.1 and the concept of equivalent smooth bend is examined. Alternative methods of assessment are
considered and improved methods are suggested.
INTRODUCTION
PIPINGSYSTEMS often include mitred pipe bends of both
the multi- and single-mitre types. A multi-mitre bend
usually acts as a substitute for a smooth bend and consists
of a number of closely spaced mitred joints (I)*, while a
single-mitre bend is formed when two pipes, having their
axes inclined, are welded together. The joint so formed
may or may not be reinforced. The present investigation
concerns single unreinforced mitred joints.
At any point in a pipe line the internal forces may have
normal or shear components, the bending moments may
have components which bend the pipe in, or perpendicular to, the plane containing the pipe axes, and twisting
moments may also exist. The forces and moments depend
on the loading system upon the pipe run, and on its
capability of absorbing thermal movements, which in turn
depends on its flexibility. The type of loading considered
here is that of in-plane bending only.
When designing piping systems which incorporate
mitred bends, it is important to know the flexibility of the
pipe in that vicinity as well as the stresses which would
arise because of different types of load. The experimental
information available on the in-plane bending of single
unreinforced mitres is small (2)-(4); analytical stress and
flexibility analyses are few, and necessarily approximate.
The behaviour of a single mitred bend under in-plane
bending is similar to that of a smooth bend since flattening
of the pipe section occurs (Fig. 1) and this gives rise to a
greater flexibility than that which would be predicted by
simple-bending theory. The flattening produces stresses
higher than simple bending stresses, and a further increase
of stress arises because of the sharp discontinuity at the
joint.
Second moment of area of pipe section about a
diameter.
Flexibility factor for smooth bend =
Flexural rigidity (EZ)of straight length of
[pipe calculated by simple bending theory 1
1
Flexural rigidity of same length of pipe,
having same diameter and thickness, but
forming a smooth bend
[
Flexibility factor for equivalent smooth bend.
Variable flexibility factor for any section along
pipe.
Value of Klat mitred joint.
Value of KOfor bend angle 2/?.
p + p + (14)1/211/2.
The M S . of this paper was received at the Institution of Mechanical
Engineers on 22nd M ay 1969 and accepted for publication on 25th
August 1969. 33
* References are given in the Appendix.
Applied bending moment.
RE/rfor a 90" mitre bend.
Smooth-bend radius.
Equivalent smooth-bend radius.
Mean pipe radius.
Stress-intensification factor for smooth bend.
Stress-intensification factor for equivalent
smooth bend.
Pipe thickness.
Leg lengths.
Distance of pipe section from mitred joint.
Mitre angle, see Fig. 1.
(k-A)/2.
Bend angle, see Fig. 1.
Vertical deflection at end of pipe.
Rotation at end of pipe.
Cylindrical co-ordinate defining position on
pipe.
[3(1-~')r~/t~]~/~.
Pipe factor for smooth bend.
Pipe factor for equivalent smooth bend.
Poisson's ratio.
Normal stress at a point.
Nominal maximum simple bending stress
(= M / n r 2 t ) .
14
JOURNAL OF STRAIN ANALYSIS
Notation
C,C1,C' Constants.
E
Modulus of elasticity.
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VOL 5 NO I
1970
IN-PLANE BENDING OF SINGLE, UNREINFORCED MITRED PIPE BENDS
DESIGN CODE
(see Figs l and 2), and where a = 45", RE is equal to r.
For a smooth bend a useful non-dimensional quantity
known as the pipe factor h is frequently used and is defined
as
Rt
A=.
.
r2
where R is the smooth-bend radius.
For the equivalent smooth bend the pipe factor is given
by
(1+cot a)t
A, =
. . . * (3)
2r
For pipe-flexibility calculations, the length of pipe
forming a smooth bend is assumed to have a uniform
flexural rigidity of EIIK, where EI is the flexural rigidity
of a straight pipe of the same diameter and thickness; K is
known as the flexibility factor of the bend and will be
greater than unity because of the flattening effect (6) (7).
For a smooth bend the code recommends the use of the
formula
X
I
Because of the similarity of mitre bends and smooth
bends (for which analytical expressions are well established), the A.S.A. Piping Code (5) adopts the procedure
of assuming that a mitre bend can be replaced by an
equivalent smooth bend of radius RE for the purposes of
calculating flexibilities and stresses. In the code RE is
given by
.
Equivalent ,
smooth b e n d
Fig. 2. Notation: single-mitre bend and equivalent
smooth bend
If the maximum stress to be used in the design calculations is u,
which is based on an analysis by Clark and Reissner (8).
For a smooth bend equivalent to a single mitre (Fig. 2)
it is modified to the following empirical formula (5)
1.52
K -* . . * (5)
-
*
The stress-intensification factor for a smooth bend is given
in the code as
0.9
S = -h2'3 * ' * *
(6)
This is expressed with reference to the nominal simple
bending stress [u, = (M/T#C)]which would be calculated
for a straight pipe subjected to a bending moment of &f.
u
= sun
For a smooth bend the formula for Sis exactly half of the
elastic-stress factor given by Clark and Reissner (8). The
reason for this is that S is a stress-intensification factor
based on fatigue considerations (2). Elastic stresses present
in commercial piping with welds and rigid attachments
are approximately twice those which would occur in plain
lengths of non-fabricated pipes. Design stresses in the
code apply to commercial piping and allow for welds and
rigid attachments, for which elastic stress-intensification
factors of 2 (with reference to non-fabricated pipes) are
already assumed to exist. Since design stresses are based
on the assumption that the factor of 2 is already present,
the stress-intensification factor for the design of smooth
bends is only required to be half of the true theoretical
elastic-stress factor, as given by Clark and Reissner, viz.
p-i
u
D e f l e c t e d shape
at s e c t i o n
near junction
For a single-mitred bend, and again using the concept
of the equivalent smooth bend by modifying equation ( 6 )
for smooth bends, the code gives the following empirical
formula for the stress-intensification factor
This implies that the true elastic stress factor should be
L
M
Fig. 1. Flattening of section under in-plane bending
JOURNAL O F STRAIN ANALYSIS
VOL
5 N O I 1970
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15
R. KITCHING AND W. J. THOMPSON
AIM O F INVESTIGATION
The object of the present investigation was to test the
design formulae for flexibility and stress-intensification
factors for a single-mitre bend, and to examine the concept
of ‘equivalent smooth bend’ in the same context.
Horizontal leg
TEST RIG AND SPECIMENS
The specimens were manufactured from aluminium-alloy
tube to B.S. 1474 N.V.5 (9). In all, six specimens were
manufactured. Details are given in Table 1, where the
notation of Figs 1 and 2 is used.
Each specimen was clamped at end A and loaded
Table I
Pipe
specimen
No.
;;;
1 1
1;;
I
~~
1
2
3
4
5
Mean Thickness, E, lbfx
1O6/in2
pipe
in
radius,
in
28,
degrees
90
1
3.250
p 3 5 1
2.188
I
I
0.138
0.125
~
.
I
I
10.5
10.3
V
Denotes a knife-edge support
LXI D e n o t e s a l o a d c e l l
Fig. 3. Loading system
~
1
~
0.36
0.33
through a screwjack and a suitable lever system with a
pure in-plane bending moment at the end B, see Fig. 3.
T h e load in each of the levers connected directly to the
specimen was measured by a C-type dynamometer incorporating a simple dial gauge.
Fig. 4. Photograph showing dejlection gauge positions
16
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VOL
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NO I
1970
IN-PLANE BENDING OF SINGLE, UNREINFORCED MITRED PIPE BENDS
Deflections and strains (see later for positions) were
measured as the moment was applied in increments and
then reduced to zero in steps. The maximum bending
moment for all specimens was 5650 Ib in and the increments 942 Ib in. All strain gauges, dial gauges, and transducers were read during the first load cycle and the zero
readings were checked. The maximum bending moment
was such that all gauges behaved linearly.
Deflection gauges were mounted as shown in Fig, 4 so
that diameter changes and transverse movements of the
pipe centre-line could be measured. In specimens 1, 2,
and 3 the gauges were mounted on both legs of the specimen but in the other specimens they were confined to
the horizontal leg, with a minimum of additional check
gauges. The transducer probes (Mercer Parnum) were
positioned close to the mitred joint where measurements
were required at smaller pitch and where there was little
room for dial gauges.
Strain measurements were taken by means of electricalresistance foil gauges of in gauge length, in conjunction
with a direct-strain measuring set. Past results (I) (3) have
shown that maximum stresses in single- and multi-mitred
bends occur at the junction but within the well defined
region 90" < 0 < 135" (see Fig. l), and that they are in
directions normal to the plane of the junction (a plane of
symmetry). Strain gauges were therefore attached to the
outer surfaces of the pipe in this region only. They were
mounted in pairs and as close as possible to the weld. Some
check gauges were used on the opposite side of the pipe but
in equivalent positions. As a check on the nominal stress,
pairs of gauges were mounted on the outside surface of the
pipe at a section remote from the mitred joint and also
from the ends of the specimen. The latter gauge readings
confirmed values predicted by simple-bending theory.
Readings of the strain gauges and of deflection gauges,
disposed in a similar manner to that already described,
were taken after the horizontal leg length (dimension X in
Fig. 2) of the specimen had been shortened by 8 in. In-plane
bending moments were applied as before. Further tests
were carried out for the specimen with dimension X having
a number of different values, the smallest being 6 in (see
Table 2). Dimension Y remained the same throughout.
The same procedure was adopted for all six specimens,
the details of the different leg lengths X for each specimen
being shown in Table 2.
T o facilitate the continued removal, modification, and
repositioning of a specimen in the rig, the ends of it were
+
fitted over carefully machined spigots, and were then
secured in position by split clamps.
RESULTS
General observations
The maximum applied moment (5650 lb in) for all
deflections and stresses which are quoted was the same for
all specimens, and all measurements were linear for the
range zero to maximum load. A limited number of tests
were carried out beyond this load and the tendency was
for the flexibility of specimens 4,5, and 6 to become less at
the higher moments, which were in a direction corresponding to the decrease of the angle 2p.
The distributions of deflections and strains are quoted
for a nominal bending moment of 5650 lb in, and, since
this was in the linear range, they were computed in each
case by plotting load-deflection or load-strain graphs and
then using the best straight line to obtain an average.
The elastic constants E and Y for the material of the
pipes were determined by using tensile test pieces cut
from the pipe walls and also by performing compression
tests on some of the short tubular pieces cut from each
of the specimens. Little scatter was registered and the
results in tension agreed with those in compression. They
are given, together with the relevant dimensions, in
Table 1.
Deflections and stresses
Where possible dial gauges were arranged in pairs across a
diameter and were used to obtain both transverse deflections of the pipe axis and the radial flattening of the pipe
section. The latter was calculated by assuming that the
radial deformation at the position 8 = 0" was the same as
that diametrically opposite at 8 = 180". It is appreciated
that at positions near the mitre this procedure is not
necessarily justified but it was adopted in the absence of
an alternative. In positions close to the joint it was
impossible (see Fig. 4) to position pairs of gauges across a
diameter so the radial deflections in that vicinity have been
based on single-gauge readings.
Each experimental point shown in Fig. 5 represents the
transverse deflection at end B for one test length (X)used
in the series of tests on specimen 1. For clarity the
deflections at other points along the leg have been omitted,
H O R I Z O N T A L DISTANCE F R O M JOINT-inches
Table 2
Y,in
I
I
Leg length X, in
Test No.
46.0
38.0
30.0
22.0
14.0
10.0
6.0
51.50
42.25
34.25
24.25 26.25
_.-.
16.25 18.25
8.25 10.25
4.25
6.25
48.75
40.25
32.25
~
1
1
1
I
I
I
49.5
41.5
33.5
25.5
17.5
46.0
38.0
30.0
22.0
13.5
9.5
51.25
42.25
34.25
26.25
18.25
14.25
10.25
JOURNAL OF STRAIN ANALYSIS
VOL
5 NO
17-
L7_j_o1
Fig. 5. Transverse dejections in specimen 1
I 1970
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17
R. KITCHING AND W. J. THOMPSON
but it was observed that at any such point the deflection
was the same for all possible leg lengths which included
that point, except for the smallest leg length. Figs 6 and 7
show experimental points plotted in the same way for
specimens 2 and 3.
Radial deformations at sections are plotted in Fig. 8 for
the horizontal leg of specimen 1 across diameter 0 = 0"
to 180". I t should be noted that in Fig. 8 the position of
a section has been defined by its distance from the
extremity, G, of the horizontal leg (see Fig. 2) and not
from the join of the pipe centre lines, J, which is indicated
in Fig. 8. Figs 9 and 10 show radial deformations for
HORIZONTAL DISTANCE FROM JOINT-inches
60
50
40
30
20
10
I
I
I
1
I
I
00
A
Calculated w i l h code
specimens 2 and 3. Fig. 10 also gives radial deformations
for the inclined leg (diameters 0 = 90" to 270" and fl= 0"
to 180").
Edge stresses, derived from the measured strains, are
plotted against angular positions 0 for specimen 1 in
Fig. 11 and in Figs 12 and 13 for specimens 2 and 3.
Stresses have been shown non-dimensionally as stress
ratios where Stress ratio = (Stress)/(Maximum simplebending stress u,). The different distributions are for
different horizontal leg lengths indicated in the key.
(Some test lengths, see Table 2, have been omitted for
clarity.)
Similar, although less detailed, information is given for
specimens 4, 5, and 6 in Tables 3 and 4. Distribution
patterns of transverse deflection, radial deformation, and
stress were found to be similar to those for specimens
1, 2, and 3 respectively.
Table 3
Specimen
I
/.'
,
1
~ ~ ~ d d p l l p
number r e p r e s e n t e d b
C
l
i
Experiment
1
1
2
3
60
I
6
----r
Modified code
(equation (13))
I
I
0.620
0.722
I
I
0.873
. _
-
0.765
0.857
-
0.187
0.191
0.204
0.621
0.610
0.700
Table 4
HORIZONTAL DI STANCE FROM JOINT -i n c h es
50
40
30
T
1
Code
0.275
0.550
I
5
Fig. 6. Transverse deflections in specimen 2
0.210
0.220
0.245
I
4
I
End deflection, in
~
Specimen
-~
2s
-r
Stress-intensification factor
Calculated w i t h c o d e
[equations (l1,(31,(511
(equation
1
2
3
I
-
04
I 1 I
';;1
23.5
135"
:ol
19.0
10.0
1
:;:':
10.2
I
p_______
1 1 1
--(from
9.4 (L)
7.6 (L)
7.1 (L)
~
6.2 (L)
12.8
5.6 (L)
8.30
9.4
8.23
6.9
8.1
5.31
(10))
Fig. 7. Transverse deflections in specimen 3
25
HORIZONTAL DISTANCE FROM J O I N T ( M E A S U R E D FROM P O I N T GI -inches
20
15
10
5
0
Fig. 8. Radial deformations in specimen I
18
JOURNAL O F STRAIN ANALYSIS
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VOL -j NO I
1970
IN-PLANE BENDING OF SINGLE, UNREINFORCED MITRED PIPE BENDS
H O R I Z O N T A L D I S T A N C E F R O M J O I N T ( M E A S U R E O FROM P O I N T GI-inches
20
IS
10
5
25
0
c
u
E
._
I
z
0
-5 F
4
I
e
0
-10
tn:
2
-I5
P
2
rr
x IO-~
F%. 9. Radial deformations in specimen 2
H O R I Z O N T A L D I S T A N C E FROM J O I N T ( M E A S U R E 0 FROM P O I N T G I - i n c h e s
20
25
10
15
5
L+--
r
4
.
I
1
1-1
_.
0
'+0
u
XIO-3
Horizontal leg
-
I
0"- 180"
1
G o '
b Inclined leg
10-3
Fig. 10. Radial deformations in specimen 3
@-degrees
60
50
40
I
30
60
20
I
1
Longitudinal
Hoop
N
-2
t- 4
I
7 -6
Fig. 11. Edge stresses in specimen 1
J O U R N A L OF S T R A I N A N A L Y S I S
VOL 5 N O I
I970
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19
R. KITCHING AND W. J. THOMPSON
-2
10
- 1
0
I-
,"
Longitudinal
~--2
m
m
W
---4
:
m
-~--6
--a
Fig. 12. Edge. stresses in specimen 2
T 4
J
-,
2
/L
18
Fig. 13. Edge stresses in specimen 3
DISCUSSION O F RESULTS
Flattening effect
Fig. 8, which shows the change of pipe radius along the
horizontal leg for specimen 1, illustrates that, for leg
lengths greater than 6.5 in (one diameter), neither the
pattern nor the magnitude of deformations varies significantly for different leg lengths. Significant differences
were only experienced when the leg lengths became about
one diameter.
It is clear that for the longest leg lengths the flattening
effect was greatest at the joint, and persisted for a considerable length (about 4 diameters) along the pipe. For leg
lengths between 6-5in and 24 in, deformation patterns are
similar to those for longer leg lengths except for a region
of restraint (about 3 in long) close to the end fitting where
the load was applied. For the shortest leg length used
(6.5 in) there was a considerable reduction in the flattening
at sections close to the joint in comparison with all other
leg lengths.
Similar effects were noted for specimens 2 and 3 which
had different mitre angles. Changes of radius near the
joint were about the same as those for specimen 1 where
leg lengths were greater than 6.5 in, but they appeared to
20
be marginally greatest for the 90" bend and marginally
smallest for the 135" bend.
Transverse deflection of horizontal leg
Fig. 14a is a diagram showing the pipe axes for any
specimen, JA being the horizontal leg before it has been
cut. JB in Fig. 146 shows the axes after the horizontal leg
has been shortened. If all local effects (e.g. restriction of
flattening) changing the flexural rigidity of the pipe are
ignored, the same in-plane bending moment R applied
at A in Fig. 14a will produce the same vertical deflection
at B as it does when applied at B in Fig. 146. This means
that, for that applied moment, the deflection curve for any
unshortened specimen will be the same as the locus of the
end deflections of the shortened specimens. Fig. 5 shows
this to be substantially so, provided that the leg length is
greater than about one diameter. However, it may be
expected that the end restraint provided by the attachment
through which the load is applied will affect the local
flexural rigidity and this would most affect the end
deflection when the leg is short, with the restraint close
to the joint. Figs 5 , 6 , and 7 show that this is the case.
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IN-PLANE BENDING OF SINGLE, UNREINFORCED MITRED PIPE BENDS
axis from the joint. A suitable expression for the reciprocal
of the flexural rigidity would be
x’
/
o
Uncut
With equivolent
smooth bend
c
where EI is the flexural rigidity of a straight pipe under
simple bending, K1is the flexibility factor at the point,
and KOis a factor for a given combination of rjt and a.
The index a2 is a suitable one (I) for cylinders subjected
to flattening and is given by
k-A
Q2
= --
2
where
r2
A4 = 3(1-v2)b
k = [8+{64+A4}1’2]1‘2
22’
Cut
a Original leg lengths.
b Horizontal leg shortened.
c With equivalent smooth bend.
Fig. 14. Diagram showing pipe axes
If the equivalent smooth-bend concept is assumed the
configuration in Fig. 14a would be replaced by that in
Fig. 14c. The preceding remarks would still apply but the
geometry is significantly incorrect in the region CD, so that
if vertical deflections of the horizontal leg are calculated on
the assumption of an equivalent smooth bend, unreasonable results would be likely if the end of the leg is close
to the tangent point C . For a small equivalent-bend radius
such as that predicted by equation ( l ) , the deflections
would only be unrealistic for a small portion of the pipe
layout CD. However, Fig. 5 indicates that equation ( l ) ,
in conjunction with equations ( 3 ) and (9, predicts
deflections of the pipe which are too high, and this is
confirmed for specimens 4 , 5 , and 6 (Table 3), though not
for specimens 2 and 3 (see Figs 6 and 7 ) .
When the code gives too high a deflection, pipes have
been assumed more flexible than in practice they are and
this means that pipe reactions (and hence stresses) at
vessels will be underestimated. Previous workers (3) (10)
have also reported over-estimation of the flexibility factor
by equation (5) and it has been suggested (3) that the
equivalent radius is better given by
2r
RE = tan CI
. . .
The larger values of equivalent radius resulting from
equation (10) would make the concept of more limited
use because the region of the pipe CD would be fictitious
for many practical geometries of pipe layout.
It is theoretically possible to calculate an equivalent
smooth-bend radius R, and its corresponding flexibility
factor KEto fit any of the experimental results obtained,
This was attempted but the equations to be solved are
invariably ill-conditioned so that R, and KE are open to
large errors.
An alternative approach in the calculations is to dispense with the use of an equivalent smooth bend to replace
the mitre bend, and to revert to the true configuration as
in Fig. 14a. For leg lengths greater than about one
diameter it is reasonable to assume that the flexibility of
the pipe is dependent upon the flattening effect which
decreases with the distance x’ measured along the pipe
JOURNAL O F STRAIN ANALYSIS
VOL
5
This expression gives the flexibility factor as unity at
large values of x’, and 1 + K O at the joint. Values of KO
determined from experimental data can be tabulated for
different combinations of r / t and Q or it may be possible to
connect them by simple formulae.
Flexibility calculations for a pipe layout are quite
straightforward and the transverse end deflection for the
pipe specimens used in the present investigation is given
by
The end rotation is
EI
ff2
I n both expressions X and Y have been assumed greater
than ar/a2.
By using the experimental values for Av and the relevant
dimensions of specimens 1-6, the various KOvalues have
been calculated and are plotted in Fig. 15. Also shown is
the KO value given by the only other available results
which allow it to be calculated, those of Jones and Kitching
(3). Here it could be calculated from readings of both
Av and A+, and the two values agreed within 3 per cent.
The result fits in well with those from the present
experiments.
In Fig. 15 the KOvalue for a straight pipe (28 = 18O0)is
inserted as zero and the results suggest that if KO,is the
value of KO for a bend angle of 28 and K o . 4 5 is that for
a 90” bend, the following formula would be reasonable:
I
-\,,,,
f = 17.5
I
“90
0
120
\#\
150
I80
28-degrees
Fig. 15. Variation of KOwith CY and rjt
NO I 1970
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21
R. KITCHING AND W. J. THOMPSON
Stresses
Before the horizontal leg length of specimen 1 was
reduced, the stress patterns at the edge were similar to
those measured by Jones and Kitching (3) for a singlemitre bend and by Lane and Rose (11) and Kitching (12)
for multi-mitre bends. For the range of leg lengths used,
the angular position of the maximum recorded stress ratio
remained in the region 105" < 8 < 135".
As the horizontal leg was shortened the maximum
longitudinal stress ratio increased slightly until it reached
8.5 at a leg length of 10 in. It corresponded to an angular
position of 8 = 115". Further shortening of the horizontal
leg gave rise to a more rapid change of stress pattern and a
further movement of the maximum longitudinal-stress
position towards the crotch by 7".
For the same region hoop-stress ratios formed similar
patterns, which moved like those for longitudinal-stress
ratios. The maximum hoop stress was always less than the
longitudinal stress.
Corum (13) has investigated the stresses arising in a
pipe mitred to a rigid flat plate. When subjected to an inplane bending moment it was said to give maximum
longitudinal stresses at '6 = 0".A mitred bend with one
very short leg may be in a similar condition and it would
seem that, if the leg had been further reduced in length,
the stress patterns would have changed even more than
they did in the present series of tests when short leg lengths
were used.
The foregoing comments also apply to the stress patterns
for specimens 2 and 3. As the bend angle changed from
90" in specimen 1 to 135" in specimen 3 the patterns
became smoother and changes stress with changes in
0 were less rapid; maximum values became less and the
corresponding angular positions moved towards the
crotch. Fig. 16 gives a comparison of the patterns of edgestress ratios for the three specimens when the horizontal
legs had lengths corresponding to their maximum overall
stresses.
In Table 4 maximum stress ratios for all six specimens
have been compared with those given by equation (9).
Results from two other tests (3) (10) have also been
included. Equation (9) is shown to over-estimate stresses
in all cases but one. The experimental result for speci-
OF
60
50
40
30
I
I
I
I
men 4 appears to be unduly low in comparison with
the rest. It was checked a number of times, however, so it
has been assumed that it was affected by the weld which
was rather heavier than in the other specimens.
For multi-mitred bends, in order to allow for discontinuity at the mitre, it is suggested that equation (9)
should be modified to
S=- 1.8C
AE2'3
where C > 1.
A similar consideration will be applied to single-mitre
bends. For a 90" bend assume
RE = nr
then
and
When the experimental result for specimen 1 is used, the
most suitable values for the constants are
C = 1.316, C1 = 2.37
and
n=3
From the results for specimens 2 and 3 with different
bend angles a suitable general formula for stress-intensification factor for single mitres would be
This gives 3r as a more realistic equivalent radius than r
(the code value) for a 90" single-mitre bend.
Fig. 17 and Table 4 give experimental stress-concentration factors compared with those calculated from equation
(12), which, apart from specimen 4, is shown to be
reasonable.
If the same concept is used for other mitre angles an
expression similar to equation (lo), suggested in (3), could
be adopted, viz.
3r
RE = - . . .
(13)
tan Q
-
20 ///lo
I
I
-0
60
50
40
I
I
I
Test
30
/20
I
28
I
Symbol
-10
Fig. 16. Stress patterns for maximum overall stress
22
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VOL
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IN-PLANE BENDING OF SINGLE, UNREINFORCED MITRED PIPE BENDS
REt
A, = -
where
;n 0
lp,.-l___,
90
150
120
28-
180
degrees
Fig. 17. Variation of stress-intensification factors with
a and
rlt
The same expression could be used to give a suitable
pipe factor and hence a flexibility factor, as in the code.
However, flexibility factors of multi-mitre bends are very
similar to those of similar smooth bends so it is much
more appropriate to use equation ( 4 ) with a modified
constant. It would be logical to treat a single mitre in the
same way and therefore use the expression
C’ C’r tan CL
K E ---=hE
3t
. .
(14)
Deflection results from specimens l and 4 give
C’ = 0.625 and 0.487, while those from (3) give C’ =
0.568. A suitable value would thus appear to be the
average, which is 0.56.
End deflections calculated by equations (13) and (14)
for specimens 1 to 6 are given in Table 3 and are shown to
be more satisfactory than those calculated by the code,
equations (1) and (9,since reactions at the anchor points
would not then be underestimated.
CONCLUSIONS
( 1 ) The principle of using an equivalent smooth bend to
replace a single-mitre bend for calculation purposes is
justified, p.rovided that the length of straight pipe X between the joint and the next discontinuity is at least three
diameters.
The experimental measurements of transverse deflection
(Figs 5-7) were consistent for all values of X except the
smallest. Experimental stress- and radial-deformation
patterns (Figs 8-10) only changed significantly from those
for the greatest value of X when the leg length was less
than 1.5 diameters, so the minimum length of three
diameters specified above is considered to be very safe.
(2) The equivalent smooth bend so adopted should be
used in conjunction with the following expressions
replacing the A.S.A. code expressions (equations (l), (5),
and (8)), which tend to over-estimate flexibility and
grossly over-estimate stresses in the pipe:
3r
R -- tan a
0.56
KE =hE
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r2
(i.e. half the elastic stress-intensification factor).
(3) There is no reason why the equivalent-smooth-bend
concept should be used for single-mitre-bend calculations.
It has limitations for short lengths of straight pipe,
although for lengths less than a diameter no reliable
information is at the moment available. The following
procedures are recommended where the pipe length
between the joint and the next discontinuity is greater
than a diameter.
(a) For flexibility calculations the flexibility factor Kl .
is,assumed to vary with the distance x’ from the joint as
below:
Kl= 1+ K Oe-(42x’/r)
where
k-A
a2 = 2
r2
A4 = 3 ( 1 - ? ) ~ , k = [8+{64+A4}”2]”2
where K o . 4 5 is the value of KO for a single-mitre bend
for a limited
where 28 = 90”. Appropriate values of
number of r / t ratios have been determined, but a comprehensive set of experiments is required to cover a large
range of rlt ratios.
(b) For stress-intensification factor an appropriate
formula would be
- 1 sin2a+l
[
1
This is based on the sinusoidal form (Fig. 17) of the
S, = 1.14 -
(:)2‘3
variation of stress-intensification factor with the mitre
angle a, where 2a = 7r-2p.
(4) The above expressions were found to be suitable for
pipes with leg lengths of more than one diameter measured
from the mitre. It is suggested, however, that, in the first
instance, the expressions should only be assumed valid
for leg lengths greater than two diameters. In the light of
further experimental work this limit could then be
modified.
ACKNOWLEDGEMENTS
The authors thank Professor W. Johnson, Head of the
Mechanical Engineering Department, University of
Manchester Institute of Science and Technology, in
whose labofatories the work was carried out, and the
members of the laboratory and workshop staff, particularly
Mr W. E. Atkinson, Mr G. Robinson, and Mr J. Davies,
for the assistance they have given in the project.
They are also indebted to the Directors of the British
Oxygen Company-Airco Limited, who provided the
specimens.
APPENDIX
REFERENCES
(I)
KITCHING,
R. ‘Mitre bends subjected to in-plane bending
moments’, Znt.J. mech. Sci. 1965 7, 551.
1970
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23
R. KITCHING AND W. J. THOMPSON
MAW, A. R. C. ‘Fatigue tests of piping components’,
Trans. Am. SOC.mech. Engrs 1952 74, 287.
(3) JONES,N. and KITCHING,
R. ‘An experimental investigation
of a right-angled single unreinforced mitred-bend subjected to various bending moments’, J. Strain Analysis
1966 1,248.
(4) OWEN,B. S. and EMMERSON,
W.C. ‘Elastic stresses in single
mitred bends’,J. mech. Engng Sci. 1963 5, 303.
( 5 ) AMERICAN
STANDARDS
ASSOCIATION
B31.1. American Standard Code f o r pressure piping (now contained in U.S.A.S.
B.31.1.0 Power Piping, 1967).
(6) DEN HARTOG,
J. P. Advanced strength of materials 1952 234
(McGraw-Hill Book Co. Inc., New York and London).
(7) GROSS,N. ‘Experiments on short-radius pipe-bends’, Proc.
Instn mech. Engrs 1952-53 lB, 465.
(2)
24
(8) CLARK,
R. A. and REISSNER,
E.
‘Bending of curved tube’,
Adv. appl. Mech. 1951 2, 93.
(9) BRITISHSTANDARDS
INSTITUTION
B.S. 1474: 1963 Wroughr
aluminium and aluminium alloys for general engineering purposes. Extruded round tube and hollow sections (London).
(10)OWEN,B. S., HOLLAND,
M. ,and EMMERSON,
W. C . ‘Stresses
in and flexibility of mitred bends and lobster-back bends’,
Proc. Instn mech. Engrs 1963-64 178 (Pt 3J),70.
(11) LANE,
P. H. R. and ROSE,R. T. ‘Experiments on fabricated
pipe bends’, Brit. Weld.J. 1961 8, 323.
(12) KITCHING,R. ‘In-plane bending of a 180’ mitred pipe
bend’, Int. J . mech. Sci. 1965 7, 721.
(13) CORUM,
J. M. ‘A theoretical and experimental investigation
of the stresses in a circular cylindrical shell with an oblique
edge’, Nucl. Engng Design 1966 3, 256.
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