Stress intensification factors in elbows subjected to in-plane bending moments at elevated temperature

Int. J. Pres. Ves. & Piping 26 (1986) 1-52

Stress Intensification Factors in Elbows Subjected to In-plane Bending Moments at Elevated Temperature F. G. Cesari Nuclear Engineering Laboratory, University of Bologna, Bologna, Italy

and S. Menghini Societa' per l'Analisi di Strutture e Sistemi, Via A. Capecelatro, 55-20148, Milan, Italy (Received: 10 October, 1985)

ABSTRACT This paper reports the results of a static stress analysis in piping elbows, either with or without stiffening effects due to the sectional ovalization restraint provided by tangent straight pipes. The physical and geometrical aspects together with the loading conditions used in this study are typical of those of pipings for main cooling systems of Liquid Metal Fast Breeder Reactors (LMFBR). The aim of the study is directed towards comparing stress intensification factors in elbows subjected to pure in-plane bending moments obtained by experimental, analytical (using the A S M E Section III Subsection NB 3685 detailed analysis procedures) and numerical tests. The numerical calculation was performed taking into account some nonlinearity effects due to the particular geometry of sections or load-induced phenomena, and neglecting the plastic response of the material (elastic behaviour).

NOMENCLATURE B Do

Constant, in accepted units, for use in N o r t o n type creep law O u t s i d e diameter 1

2

E F I k L r

R t

W

w Gt

7

0 2 v trbj ffs

F.G. Cesari, S. Menghini

Young's modulus Force Stress intensification factor in j-direction Moment of inertia Flexibility factor Straight pipe length In-plane bending moment Mean radius Bend radius Thickness; time Section modulus Thickness section modulus Meridional angle Bend angle Strain Strain rate End section mutual rotation (elbow rotation) Elbow factor = tR/r 2 Poisson's ratio Elbow normal stress in j-direction Reference normal stress (from the beam theory) Parallel angle 1. I N T R O D U C T I O N

Elbows are used in piping systems, whatever the type of installation, in order to fulfil layout needs and to correspond to plant process requirements. In the case of thin components, the curved regions can help the whole structure to be more flexible. Because such curved regions are connected to straight pipes or to some other fittings, the straight pipes transfer axial thermal expansion on the elbows. Primary importance is therefore attached to the knowledge of how these elements behave, particularly when operating at high temperatures and under low pressures. As a consequence of such conditions and of piping geometrical and restraint configurations, particularly high stress/strain states of the loop are concentrated on the elbows. Thus, it was necessary to provide a more comprehensive analysis of the stress distribution in the bend sections, with special attention to the

SIFs in elbows subjected to in-plane bending moments

3

implications resulting from the use of particular computing methods and/or the application of Codes and Standards. Clearly, the exhaustive discussion and clarification of all debatable aspects of piping elbow behaviour goes far beyond the scope of this paper. Particularly, creep-plasticity and creep-fatigue interaction analyses are neglected, as is cyclic change of the applied load. The purpose of the present work is rather to bring together in a systematic way certain peculiarities of the problem and some parameters which specially affect bend operating performance, such as stiffening influence at the ends connected to straight pipes, geometrical non-linearities (large displacement) and secondary creep (Norton's type law) effects. The study consists of five parts (Sections 2-6): (a) Section 2 consists of a properly qualified idealization and discretization of the problem, with a detailed comparison within and between experimental, analytical and numerical data, in order to describe the best model for the subsequent calculations on restrained and free elbows; (b) Section 3 considers time-independent and time-dependent (creep) investigations on simple and isolated elbow models loaded by a pure in-plane bending moment; (c) Section 4 examines the end restraint ovalization on an elbow joined to tangent pipes at both ends; (d) Section 5 discusses all the preceding sections; (e) Section 6 brings together the conclusions from this work. It must be pointed out that in reality the piping system complexity generally yields to multidirectional loadings. In our case, data on the flexibility response of an elbow with typical geometrical ratios of liquid metal fast breeder reactor fittings, obtained from a basic research point of view, are reported.

2.

ANALYSES FOR ISOLATED ELBOWS

2.1. Statement of the problem

The elbow model discussed in this section has been experimentally evaluated I in order to determine the typical stress intensification factor values for bends, when loaded under different conditions.

4

F.G. Cesari, S. Menghini

The stress intensification factor in the j-direction, i~, is defined as ij = ab----~J

(1)

O"s

It should be pointed out that the (a) (b) (c) (d)

ij

values are related to:

applied load type; physical and geometrical features of the elbow; j-direction of the sustained stress; reference value in a straight pipe under the same geometric and loading conditions.

It is clear that, once this simplified computing method has been properly verified and qualified, it can be used to determine the elbow stress state through the stress analysis of an equivalent straight pipe. The design procedures contained in the ASME Section III Subsection NB 3600 rules, 2 to which reference will be made throughout this paper, are based on the above assumption. In the present case, a s is given by °'s

Mf W

(2)

Therefore, eqn (1) provides a quantity normalized to a s, that can be used as a reference value, being independent of the actual value of the applied bending moment. The pattern of the hoop and axial stress intensification factors was derived by comparing experimental stresses 1 with those calculated from the finite element computer program application and from the detailed analysis method suggested by the ASME code. 2.2. Geometrical features The elbow under evaluation has the geometrical features given in Table 1. The values there reported, which are taken as terms of comparison, correspond to the BEND2 model of Ref. 1. 2.3. Experimental values of the

ij

factors

The experimental data, derived by Imamasa and Uragami 1 from the BEND2 model subjected to a non-uniform in-plane bending moment,

S I F s in elbows subjected to in-plane bending m o m e n t s

BEND2

5

TABLE 1 Elbow Model: Geometrical Features (from Ref. 1)

Bend angle (~) Bend radius (R) Thickness (t) Outside diameter (Do) Mean radius (r) Elbow factor (2)

90 ° 1 066.8 mm 21 '0 mm 711.2 mm 345.0 mm 0-188

are reported in Figs 1 and 2, which represent the ij polar patterns on the outside surface of the median section (ct = 45°). 2.4. Calculation of the ii factors according to ASME Table NB 3685.1-2 The ASME Table 3685.1-2 collates formulae for detailed analyses of some piping fittings and has been used in this case. It should be noted that the Value of the elbow factor for the actual model geometry is* 2=0.1973 The procedures of Subsection NB 3680 of ASME are acceptable only for 2 > 0.2. On the other hand, it is known that the 2 value, which is mainly governed by geometrical configuration, may change the i~ factor value, even though sometimes the variation is only slight. 3 However, the exact value of the elbow factor was used here. The results are shown in Table 2 and reported in Figs 3 and 4, comparing them with experimental data. 2.5. Calculation of the ij factors using the TRICO computer code 2.5.1. Geometrical discretization The mesh used for the numerical calculations was obtained by applying the automatic-generation code COCO. 4 Altogether, 680 three-node triangular elements were derived. * The ,~ value considered here is higher than that given in Table 1. It results from the ASME formula which is slightly different from that given in the Nomenclature, as follows: 2 = tR/(r2x/1 - v2)

-6

-2

-'

4-

8

9

-T5

Fig. 1.

-~o

- ~ h - - -

4~

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-is

internal

o

45

surr . . . . .

surf.

so

4.5

aa

c

60

'~

~('.)

BEND2 hoop stress intensification factor experimental data.

-Go

/"I

%

~,__. ~

external

--i

~

Fig. 2.

~

6

6

z

4

5

O

I

I

-

.

_L~

i

r

[

I

|

i

E

Oe) BEND2 axial stress intensification factor experimental data.

"~

-

I

rl

qlb

--45

\

/

/ !j

,\

.~i.i/ ,tl

45

~"'d

75

go

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~ ,

Fig. 3.

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1

2

3

4

5

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-15

0

/

15

,,J 30

60

B E N D 2 hoop stress intensification factors ih by A S M E Section III NB 3685.1-2 rules.

-60

/

----

Fig. 4.

-6

=4

-z

-t

o

~j

2

.5

4

5

6

111 :t

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IHAHASA-URAGAI4

9

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8

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40

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I

I

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,,

0

.

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L/

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I |MAHAS&

/I/

I

-J,---

-L

B E N D 2 axial stress intensification factors ia by A S M E Section III N B 3685.1-2 rules.

,

/

A s M E I NB ta.b I'IB31MI5 1-2

8

F.G. Cesari, S. Menghini TABLE

2

B E N D 2 Elbow Model: Stress Intensification Factors ij Calculated Using A S M E Section III Table NB 3685.1-2 Procedures

49 angle (°)

Stress intensification factors

- 90

-75 -60 -45 - 30 -15 0 15 30 45 60 75 90

Axial (i~)

Hoop (i.)

0-09 -0"21 - 1-16 -2'36 -2"67 -1-10 1-63 3-45 2'99 1"06 -0"47 -0"83 -0"75

-- I '01 - 1"71 -2.91 -2.74 -0"26 3'38 5.62 4.70 1"38 - 1"75 -2.71 - 1.89 - 1.25

In Fig. 5 the positions of the general reference axes are shown. The perfect symmetries of the problem make it possible to use only half of a modelling structure in the calculations. 2.5.2. Loads and constraints

The elbow under evaluation was loaded by two closing in-plane bending moments, applied at both ends by means of an equivalent line distribution

~.~.

lJ / Iljane°t symrnetry ~ ' ~!.9o 1#

/

_2_A_

Fig. 5.

Load, geometrical and constraint conditions for the simple elbow model subjected to an in-plane bending m o m e n t .

-40

-t~

-4J$

-'~

-45

o

l/! .

t~

/i" ~

~

48

~o

i

"~5

--1

~ !

~o

Fig. 6. BEND2 calculated longitudinal stress intensification factors by shell type TRICO finite element code; external surface. Calculated (* ihext•• , (2) i ~ i n t A - - - A ) from Spence (pers. comm., J. Spence, May 1981).

-i;o

'

-a

I

~

.,~I~

-4

-,

o

fX

° ~. \,

,

II

! Sl~ j \ ~

~I~_

Fig. 7. BEND2 calculated axial stress intensification factors by shell type TRICO finite element code. Calculated (* i, ext • • , © i~ int A - - - A ) from Spence (pers. comm., J. Spence, May 1981).

-!t

-6

-4

-'

o

"

\\

"

t

6

el

t

-10

~o

F. G. Cesari, S. Menghini

10

TABLE 3 BEND2 Elbow Model: Stress IntensificationFactors iI Calculated Using TRICO Finite Element Computer Code

ck angle (°) 81 63 45 27 9 -9 -27 -45 -63 -81

Internal SIF

External SIF

Axial

Hoop

Axial

Hoop

0"15 0-76 1.56 0'58 -0'36 -2'68 -2-91 -0"75 -2"16 -2"52

1.67 2.16 1.90 0"18 -4.92 --6.55 -3.13 - 1.65 -2.86 --3'33

-0'32 0"10 1'27 2"77 2-91 0"48 -2-70 -3"52 -1'86 -0'25

- 1.71 -2.15 - 1-90 0.41 3'92 5'14 1'98 -2.01 -3-10 -2.38

of nodal forces. In this calculation M r was assumed to be equal to 344 610 lb in. The conditions of symmetry, shown in Fig. 5, were imposed on the median section of the elbow (~ = 45°). Other constraint configurations were considered in order to determine the optimized b o u n d a r y conditions, which generally provide the major difficulties in the threedimensional structural analysis.

2.5.3. Numerical calculations of the ij factors The numerical stress analysis was performed using the T R I C O shell type finite element computer program, s The Mohr's circles were applied and the hoop and axial stresses were obtained for each mesh element. The stress intensification factors are shown in Table 3. The results are reported in Figs 6 and 7 versus the experimental patterns.

3. A N A L Y S E S F O R S I N G L E E L B O W M O D E L S 3.1. General discussion

L o a d and geometrical conditions different from those previously discussed will be considered. For this purpose, comparative calculations were made using the analytical procedures suggested in Ref. 2 and

SIFs in elbows subjected to in-plane bending moments

11

numerical analysis by TRICO computer program. In the latter case, perfectly elastic material behaviour in time-independent conditions of equilibrium was assumed. The time-dependent results (creep effects) can therefore be interpreted as being free from any creep--plasticity type interactions. This operating procedure was obtained by imposing a very high yield stress value in such a way that alternative options available exclusively for non-linear computer analysis could be used (large displacements and creep law programming, in our case). Moreover, the TRICO output was elaborated by means of a special post-processor which made it possible to transcend the computer program internal algorithm and to preserve, for the triangular element thickness section modulus, a value of t2

We = ~-

(3)

which is typical of the linear analysis. Thus, the hoop and axial stresses were calculated on an elastic basis, answering the Spence specifications (pers. comm., J. Spence, May 1981), by processing the membrane and bending (per unit length) stress on the thickness, as determined through simple equilibrium conditions, by means of the above-mentioned post-processor. The creep law considered here is of the Norton type and can be written as follows: = Ba"

(h - 1)

(4)

where B = 3 × 10 -26 and n = 4. It should be noted that this expression and the quoted values of the constants are associated with the use of USA measurements (e.g. psi) which have therefore been adopted in the following calculations. It should be pointed out that the stress intensification factor data are in a dimensionless form. 3.2. Geometrical and material features

The data for the recurring geometrical models are summarized in Table 4. The material data, following the specification of Spence (pers. comm., J. Spence, May 1981), are E = 2 3 x 106psi, v =0.29.

12

F. G. Cesari, S. Menghini

TABLE 4 Geometrical Simple Elbow Models Considered in the Analytical Computation Following the ASME Section III Detailed Analysis Procedures Model

R (in) D o (in) t (in)

SPENCE1

9'0 6"5 0' 134 0-125

MOD1

MOD2

9.0 6.5 0.268 0-260

9.0 5.0 0.268 0"450

SPENCE2

9.0 6"0 0.1 0'108

3.3. Calculations using the ASME Section III NB 3680 procedures As in Section 2, the stresses and stress intensification factors were calculated using the ASME rules. Different geometric models and consequently different 2 values were considered in order to apply the formulae; 2 was within the proper range of validity (2 > 0.2). Table 5 reports the bending load values applied to the different calculation cases. Figures 8-15 show the patterns of the stress intensification factors, whereas Figs 16 and 17 present the stress intensification factor curves versus 2, evaluated on the outside surfaces of the models.

3.4. SPENCE1 model: TRICO finite element calculations 3.4.1. Mesh features The mesh used is shown in Fig. 18. Figures 19 and 20 show an element and nodal numeration split next to the median section (~ = 45°). In this case, too, 385 nodes and 680 triangular shell type elements were used. In view of the results reported in Section 2, the mesh has proved to be sufficiently accurate. 3.4.2. Loads and constraints The constraint conditions in the median section and loads were imposed TABLE 5 In-plane Bending Moments Applied to the Geometrical Models in the Analytical Calculations Model

SPENCEI

MODI

MOD2

SPENCE2

Mf (Ib in)

80 000

160 000

123 000

80 000

-90

Fig. 8.

-,t50

-'tO0

5O

4¢0

L~

a~

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,_

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,

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,

~

/'\

I

45

"--,._

_

\

\

........

60

*

Nrf.

/5

--

90

formulae, M r = 80 000 lb in.

SPENCE1 calculated hoop stress by ASME Section Ili

-t5

,J

/

e~* A - - --~k ~Mm'rml

~')1

Fig. 9.

•

- (.0

"-45

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W--" - ~/ m e , . 4.- - - & . internal

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.

.

s.rr.

60

.

[

"~5

t

9o

~.~1-

e

J

SPENCEI calculated axial stress by ASME Section III formulae, Mf = 80 000 lb in.

-~

a

-~__

I

"[k'.i~ :

~

3 ~

~.

r~

14

F. G. Cesuri, S. icpenghini

Fig.

- :loo

-5o

0

Jco

,lr--~o

~oo

~

12.

. O't,,, I .

-60

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~----~,

45

,,I

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surf.

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r

~

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M O D 2 calculated h o o p stress using A S M E Section III formulae, Mf = 123 000 lb in.

, ~ '

\

/,

,

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Fig.

13.

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._.__,

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. ;nternat

i

4X-

e

,,

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I

~-

[

M O D 2 calculated axial stress using A S M E Section III formulae, M r = 123 000 lb in.

,

-

~----,~

,

I • - -

~o

t~

5"

t~

F. G. Cesari, S. Menghini

-~-5

-'50

-2

o

t

2

3

4

5

6

Fig. 16. Calculated hoop stress intensification factors vs 2 elbow factor using ASME Section III detailed analysis formulae.

_-~

-6

.-~o

-~5

i

I

-6o

--

-4-5

-5o

-~5

A= 0 . 4 5 0

~--Zx

~= 0 . 1 2 5

),= 0 . 1 0 8

~,= 0 , 2 6 0

I

o

t----'/t

o

0

t5

3o

4.5

/ ""~~

r,o

t5

aJO

Fig. 17. Calculated axial stress intensification factors vs 2 elbow factor using ASME Section III detailed analysis formulae.

.6

-5

-4-

-~

A--~

8

9

-4

-'~

~, = 0 . 4 5 0

• --~

i i

4O

-3

~o

), = 0 . 2 6 0

•~ e

J/

~, = 0 . 1 0 8

X = 0.125

O--o

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-5

-?.

-'1

0

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2-

3

÷

5

6

&

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...J

t~

r~

%

e~

F. G. Cesari, S. Menghini

18

z

me~

moctJon

/ Fig. 18.

Fig. 19.

SPENCEI simple elbow model.

Mean region mesh of the elbow: element topology and local reference.

SIFs in elbows subjected to in-plane bending moments

19

~ra

Fig. 20. Nodal numeration in the mean region of the simple elbow mesh. in conformity with the structural symmetry. Bending moments were induced in the form of nodal forces with equivalent polar distribution. 6 The remaining system labilities were avoided by fixing in space a node of the symmetry section (~b = 0 ° and ~t = 45°). 3.4.3. Elastic stress calculation The elbow was subjected to the preset load and the values obtained are shown in Table 6. The stress diagrams, compared with the ASME prescription values, are shown in Figs 21 and 22. 3.4.4. Calculation under a geometrical non-linearity regime Whenever the geometry, load or constraint conditions make the structure especially sensitive to geometric deformations (such as from variation of the inertia and stiffness parameters or misalignment of the applied loads to the restrained section), it is necessary to get over the limits imposed by classical mechanics by the 'small displacements' assumption. 7 This is very important in the cases of cyclic load and local plasticity, as discussed in paragraphs 2.5 and 3.6 of Ref. 8.

-t'5

-(,O

-4-5

-3o

-~ff

!

/

0

...,

e x ter n al

i tff

N\

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i

:

\

4~7

1

T i~x 0"~

!.

gO

J

I

I

I£

results

gO

Fig. 21. SPENCEI simple elbow model: calculated external stress using shell type T R I C O finite element computer code, M t = 80 000 lb in.

[50

iso

d~

(70

.

ClO

! -IS

I Go

-45

e~*

-30

"-~

~

: -45

"

'

internal

0

i

"(5

i

~C,

;

ASME

I

4S"

L

relult$

6o

I

[

"~I

ir''£

L

:

'3o

~'ig. 22. S P E N C E I simple elbow model: calculated internal ,tress using shell type T R I C O finite element computer code, M r = 80 000 lb in.

-gO

r

~[Wii]

'b

,J

SIFs in elbows subjected to in-plane bending moments

21

It is known that the behaviour o f thin-walled elbows is different when they are subjected to a closing or opening in-plane bending moment, respectively. 9, ~o F o r our purpose, a special iterative computational procedure is used in the T R I C O program which verifies the conditions o f equilibrium for each increase in load (F + AF) on the deformed structure, as obtained in the previous calculation step (F).I TABLE 6

SPENCEI Simple Elbow Model: Stresses (ksi) and Stress Intensification Factors for the Inside and Outside Surfaces Calculated by TRICO Finite Element Code dp angle

Inside surface

(°)

ah

81 63 45 27 9 -9 -27 -45 -63 -81

ih

ffa

28.61 1.49 -1.05 46.65 2.44 9.94 60-00 3.13 33.91 9.02 0.47 43-82 -117-11 -6-12 -2.15 -182.80 -9-55 -75.08 -88-69 -4.63 -83.06 37.25 1.95 -17.49 85.38 4.46 47.69 102.91 5.36 79.98

Outside surface i

-0-05 0.52 1.77 2.29 -0.11 -3.92 -4.34 -0-91 2.49 4-18

a.

-29.73 -46.55 -58.52 - 12.46 96.77 146.97 57.58 -49.57 -74.25 -69.92

ih

ffa

ia

-1-55 -2.43 -3-06 -0.65 5.06 7.68 3.01 -2.59 -3.88 -3.65

-4.35 -3-14 17-09 60.35 77.70 19.47 -63.18 -77.51 -40.97 -21.31

-0.23 -0.16 0.89 3.15 4.06 1.02 -3.30 -4.05 -2.14 -1-11

The results are reported in Figs 23 and 24 and in Table 7 for the median cross section. Figure 25 clarifies the pattern of the external stress for the elbow median longitudinal section (~b = 0°).

3.4.5. Computation under a creep regime The behaviour of the structure under a creep regime was analyzed in conformity with specifications contained in Spence (pers. comm., J. Spence, M a y 1981), and in Subsection 3.1 of the present paper. The law adopted has already been mentioned in that subsection. Altogether, the time accumulated in the creep field is 3000h. The elementary computational time range was set in conformity with the suggestions of the program given in Ref. 5 and was chosen equal to 3000 h. The analysis was performed considering geometrical non-linearity effects.

22

F. G. Cesari, S. Menghini

I

~

7

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(

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T

SIFs in elbows subjected to in-plane bending moments

23

TABLE 7 SPENCEI Simple Elbow Model: Stresses (ksi) and Stress Intensification Factors Calculated Using TRICO Finite Element Code with Geometrical Non-linearity Effects

c~ angle

Inside surface

(°)

Gh

81 63 45 27 9 -9 -27 -45 -63 -81

ih

Outside surface

O'a

27"39 1'43 -2.70 46'91 2'45 7"19 67'59 3"53 32"58 22"44 1-17 47"25 -121"79 - 6 - 3 6 -0.54 - 2 0 9 ' 5 9 - 10"95 - 8 1 - 4 8 -98"30 -5"14 - 8 5 . 8 0 51'01 2"66 -10"91 95-69 5'00 53.12 107"24 5"60 80-21

/

ia

o',

ih

Ga

-0'14 0"38 1"70 2-47 -0"03 -4"26 -4'48 -0'57 2'77 4'19

-27"66 -45"66 -64-91 -26'09 100"67 175'99 70"22 -62"27 -81"99 -70"02

- 1-44 -2'39 -3"39 - 1'36 5'26 9"19 3'67 -3'25 -4.28 -3"66

-4-68 -5"62 11'27 56'96 87"00 34-72 -59'97 -83-41 -45'29 -25"41

-

\-

/,

400

Lo:'

~ }external surf.

L

-0"24 -0"29 0"59 2'98 4-54 1-81 -3.13 -4.36 -2"37 -1'33

I

_

I

/

/

/

S* %%

i

i

J

\

/

I"" a¢

\--

[d'~l \ i

0

40

tO

60

~0

@o

Fig. 25. SPENCE1 simple elbow model: calculated external stress with geometrical non-linearity effects using TRICO finite element code, on the median longitudinal section (~b = 0°), M r --- 80 000 Ib in.

24

F. G. Cesari. S. Menghini

-90

-

-

-~r5

•

i

!

_6o

~

-

+"J !

+

-

-45

-30

- -

i

-,t5

~

I

-

0

J

-f5

r

___~

l a t time= 3 0 0 0 h r

I

i

_

50

%-_

[

_

45

l

Go

:~: ~

:[5

90

.'-J==

Fig. 28. SPENCEI simple elbow model: calculated external axial stress intensification factors with geometrical non-linearity and creep effects using TRICO finite element code, Mr = 80 000 lb in.

-t

-5

--1.

o

2

5

4-

6

8

to

-a~O

.~

r

I

I

r

I

!

-60

-45

-30

-45

0

I

4K

30

I

I

45

e--e

l

I

60

~

at hrne= 0 hr

~0

--

Fig. 29. SPENCE1 simple elbow model: calculated external hoop stress intensification factors with geometrical non-linearity and creep effects using TRICO finite element code, Mr = 80 000 lb in.

-2

°

+

6

+

4O

r~

t%

1%

1%

1%

!

I

t

/

-7,0

++f~

]

--

/'

]]i

4~-

/

. . . .

!

I

!

I +

,..--,- .,.

I

~

!

5o

l

I

i

~

I

i

/--.-r--..'+

+ +

I

f-

Ohr

-<-'~/

I

r

,

i

4-~

l

l

f

I

~

+

+

I

~

I

i

~0

~

i

~

+

+

~

90

4-~;° _

I~,,] ~S-

l

-T-

,

I

,

I

Fig. 30. SPENCE1 simple elbow model: calculated internal axial stress intensification factors with geometrical non-linearity and creep effects using TRICO finite element code, M r = 80 000 lb in.

-~-5

i

J

-60

~.i

.t

I

+' , - ~

~

:

i

-~

¢

: ~%,.._J)

-1~

~

+

,

-+

-Oo

~

+V, ~

f

I

i

'

I

i I

+

'

--

X

'.

! +

1

!

~,~

'i,

",, .,

",,,+

!

o

" " - . _\

<4

-i

+

to

[

~ i

i

I

,

i

!

i

,!

i

L

r

!i

i

~-

I

i

f-

NJ

4

7

)

I

/ l ~

!

i I

, aO

30

• Ohr

i

•

!

'%

45-

!

I

- -

,

t

\

¥5

90

r,+,3

-

Fig. 31. SPENCEI simple elbow model: calculated internal hoop stress intensification factors with geometrical non-linearity and creep effects using TRICO finite element code, Mf = 80 000 lb in.

-

I I

- ' i i

'

,

-4-

I

o~

,.-.

¢3

t'~

!

!

iI

!

iI

I/I

/

/

/

//

/,

.°

l-

~'E ~ e x t e r n a l

jf-

at 3 0 0 0 h r

~

I

'

Fig. 32. SPENCEI single elbow model: calculated external stresses with geometrical non-linearity and creep effects using TRICO finite element code, along the median longitudinal section (~b = 0°), Mf = 80000 lb in.

~o

./

~0

40

9.0

~aO

hr

/~

40

I r'raO

~o®~ ~ 7

at 0

_

~°

60

~rO

/ _ ~

L

llO

,

.L gO

Fig. 33. SPENCEI simple elbow model: calculated external hoop stress intensification factors with geometrical non-linearity and creep effects using TRICO finite element code, along the median longitudinal section (qb =0°), Mf = 800001bin.

0

,- - - . ,

tl>

t~ "--4

r,o

F. G. Cesari, S. Menghini

28

TABLE

8

S P E N C E I Simple Elbow Model: Stresses (ksi) and Stress Intensification Factors Calculated Using T R I C O Finite Element Code with Geometrical Non-linearity and Creep Effects (t = 3 000 h)

0 angle

Inside surface

(°)

o"h 81 63 45 27 9 -9 -27 -45 - 63 -81

ih

Outside surface i

oh

-0.21 0.40 2-38 3.46 --0.27 -5.83 -4-48 1.62 4.53 5.60

--50.08 -71-77 - 104'96 -81"91 66"38 175-55 44"70 -123'49 - 123.19 -98-41

O"a

2"58 -3"96 49'45 3'79 7.71 72"52 5"67 45.54 108-58 4.33 66.17 82"97 -5.17 -84"79 -4-43 -210'08 -10'97 -111.56 - 7 5 " 7 4 - 3 " 9 6 -85"70 5"83 31"05 111"51 7"23 86.68 138"38 7"42 107.20 142.13

t

r--q

T

]

ih

oa

-2'62 --6.76 -3'75 -12.29 -5-48 -8.54 -4"28 40.90 3.47 102"80 9.17 82"79 2"34 -70-08 -6.45 -119-56 -6.44 -81.06 -5.14 -54.46

T

ia -0-35 -0.64 -0.45 2.14 5.33 4.32 -3'66 -6.25 -4.23 -2-85

[

. . . .

l

i

i, I

0

#0

~0

50

i

4o

~0

60

~0

80

~o

Fig, 34. SPENCE1 simple elbow model: calculated external axial stress intensification factors with geometrical non-linearity and creep effects using T R I C O finite element code, along the median longitudinal section (0 = 0°), Mr = 800001bin.

SIFs in elbows subjected to in-plane bending moments

29

Table 8 reports the values o f stresses and stress intensification factors time t = 3000 h. The same values are also reported in Figs 26-31. Figure 32 shows the patterns of the external stresses along the mean meridian (4 = 0°). Figures 33 and 34 show the corresponding ij factors and compare them with those for t = 0. It is worth noting that, to determine such factors, a constant value

ij at

a s = 19.1433 ksi was used in eqn (1), which is equal to the maximum longitudinal stress calculated in an elastic manner for an equivalent straight pipe at time t---0.

4.

ELBOW C O N N E C T E D TO S T R A I G H T PIPES

4.1. General discussion In this Section the cases considered are the load cases examined in Section 3 applied to a geometric configuration as shown in Fig. 35. The lz IB

_\_.!_._._ ...........

i

\

MI =80000 Ib in

E UE=Wc= ~y . 0

LJ "F I

/

I I

/

I I J

t

I I

t

I

i 8.73"

I

! , la

.A~c

I

i l,,if = 80000

Fig. 35.

Ib in

SPENCEI elbow model with tangent pipes: load, geometrical and constraint conditions.

F. G. Cesari, S. Menghini

30

purpose was to assess the effects of the tangent pipes on the stresses induced in the elbow under different computational conditions. Also, the stress intensification factors were assessed in order to obtain as accurate information as possible referring to both the simple elbow model and the ASME procedure data. Loads and constraints were the same as in the previous section. The total number of finite elements and nodes, used for the simple elbow model, was also the same. Computations were made on a finer longitudinal mesh in order to see how it ultimately affected the results obtained using different mesh arrangements. 4.2. Calculation in linear elastic field The model was evaluated assuming perfectly elastic behaviour. The results are summarized in Table 9 and are shown in Figs. 36 and 37. The stress intensification factor values are plotted in Figs 38 and 39 and compared with the simple elbow model data. For comparative purpose, a calculation using a finer mesh (924 degrees of freedom) was performed and the results are given in Table 10. They show that a tighter mesh ultimately affects the stresses being calculated. However, such differences are greater when the values are TABLE 9 SPENCE1 Jointed Elbow Model: Stresses (ksi) and Stress Intensification Factors Calculated Using T R I C O Finite Element Code in the Elastic Field

0 angle (3 81 63 45 27 9 -9 -27 -45 -63 -81

Inside surlace

Outside surJaee

O"h

ih

O~a

i~,

O'h

ih

~a

i

11.08 24.61 40-03 8.44 -86-71 -123-00 -27.70 55"33 46.49 23.43

0"58 1'29 2.09 0"44 -4'53 -6"43 -1-45 2"89 2'43 1'22

-1"68 5"63 25-16 38"32 10'11 -44"59 -46.68 -1"27 21-46 20"71

-0.09 0'29 1.31 2.00 0"53 -2.33 -2.44 -0.07 1.12 1.08

-11.22 -24"02 -39.35 - 12.53 67"58 93-20 10.22 -58.20 -45-41 -21-78

-0"59 - 1.25 -2.06 -0'65 3"53 4.87 0.53 -3.04 -2.37 -I.14

-2"80 - 1.47 15.65 52'65 60"77 5'74 -58.06 -60.88 -27.82 -12.08

-0.15 -0'08 0-82 2.75 3.17 0"30 -3'03 -3.18 -1.45 -0.63

-~JO

-T~

-~

-45

-~ao

-~5

,I

L

I

;

0

"I~-

"3o

4~"

,~0

,°/-

:~5"

---:

# 90

---

./____ A _ "

Fig. 36. SPENCE1 elbow model with tangent pipes: calculated external stress in elastic field using TR|CO finite element code, Mf = 80 000 Ib in.

-4SO

i

-~0o

i

. . . . . ~.

-'"'~""~ / ~,~

(TL,o,

-50

o

4~

4~

2~

-'/~-

-6o

-~-s

-~Io

~J

"" -~-

~e# S

. °l]im.r..~

o

j

'

"~5

~o

i

e'3

Go

J

go

ir. :16

....

Fig. 37. SPENCEI elbow model with tangent'pipes: calculated internal stress in elastic field using TRICO finite element code, Mf = 80 000 lb in.

-oJo

i=s':. . . . .

•

~-

~

=.

~

N'

-43

-75

-60

-45

-30

-6 0

15

50

45 60

16

Fig. 38. SPENCEl elbow model with tangent pipes: calculated e.uternal stress intensification factors in elastic field using TRICO finite element code.

80

-30

-75

4.2

+5

-30

-m

0

IS

50

45

60

;5

Fig. 39. SPENCEI elbow model with tangent pipes: calculated inrernul stress intensification factors in elastic field using TRICO finite element code.

9D

33

SIFs in elbows subjected to in-plane bending moments

TABLE 10 SPENCEI Jointed Elbow Finer Mesh: Stresses (ksi) and Stress Intensification Factors Calculated Using TRICO Finite Element Code and Percentage Variation from Data of Table 9 q~ angle (°)

81 63 45 27 9 - 9 -27 -45 -63 -81

External surface ¢7h

ih

Var. (%)

t~

ia

Var. (%)

- 12.59 -26.54 -44.64 3.35 73.30 89.41 14.22 -59.34 -48.84 -18.94

-0.66 - 1.38 -2.33 0.17 3-83 4.67 0.74 -3.10 -2.55 -0.99

+ 10 + 10 + 12 - 3 +8 -4 +28 +2 +7 -14

-5.13 - 1.54 15.20 49.88 67.21 24.06 -59-16 -68.58 -28-40 -10-85

-0.27 -0.08 0.79 2.61 3.51 1-26 -3.09 -3-58 -!.48 -0.57

+45 +5 -3 - 5 +9 + 76 +2 + 11 +2 -11

very small while t h e y are r e d u c e d to 4 - 1 1 % at the peaks. T h e t e n d e n c y is f o r axial stress to yield g r e a t e r values a n d f o r h o o p stress to yield lesser ones.

4.3. Geometrical non-linearity effects A l s o in the S P E N C E 1 e l b o w m o d e l c o n n e c t e d to t a n g e n t pipes at the ends, a n u m e r i c a l analysis was m a d e o n the a s s u m p t i o n t h a t the s t r u c t u r e exhibits g e o m e t r i c a l n o n - l i n e a r b e h a v i o u r . T h e results are given in T a b l e 11 f o r the e l b o w m e d i a n cross section. T h e stress p a t t e r n s are s h o w n in Figs 40 a n d 41. T h e stress intensification f a c t o r s are c o m p a r e d with the s i m p l e e l b o w m o d e l values in Figs 42 a n d 43.

4.4. Calculation under a creep regime In c o n f o r m i t y with the p r o c e d u r e o f Section 3, the s t r u c t u r e w a s s u b j e c t e d to s i m u l t a n e o u s c r e e p a n d g e o m e t r i c a l n o n - l i n e a r i t y effects. T h e results are s u m m a r i z e d in T a b l e 12 f o r the e l b o w m e d i a n cross section. T h e s a m e values are r e p o r t e d in Figs 44 a n d 45 at t i m e t = 3000 h. F i g u r e s 46--49 s h o w the ij f a c t o r s at t i m e t = 0 a n d t = 3000 h.

F. G. Cesari, S. Menghini

34

TABLE 11 SPENCE1 Jointed Elbow Model: Stresses (ksi) and Stress Intensification Factors Calculated Using TRICO Finite Element Code with Geometrical Non-linearity Effects

cp angle (°)

Inside surface

Outside surface

81 63 45 27 9 -9 -27 -45 -63 -81

11-68 0.61 -2.27 -0-12 26.43 1.38 4-85 0.25 46.99 2.45 26.20 1.37 17.08 0.89 42.73 2.23 -101.02 - 5 . 2 8 8.79 0-46 -149,84 - 7 . 8 3 -53.48 - 2 . 7 9 -26.35 - 1 . 3 8 -48-73 - 2 . 5 5 69-90 3.65 4.81 0.25 51.22 2.69 25.51 1.33 25.13 1.31 22.70 1.19

-11-84 -25.75 -46.23 -20-88 82-52 119.73 8.51 -72.62 -49-97 -23.19

-0.62 -1.34 -2.41 -1.09 4.31 6.25 0.44 -3.79 -2.61 -1.21

-2.75 -2.35 13-47 54.63 76.09 15.05 -64.96 -68.25 -29.26 -13.19

-0-14 -0.12 0.70 2.85 3.97 0-79 -3.39 -3.57 -1.53 -0.69

TABLE 12 SPENCEI Jointed Elbow Model: Stresses (ksi) and Stress Intensification Factors Calculated Using TRICO Finite Element Code with Geometrical Non-linearity and Creep Effects (t = 3 000 h)

dp angle (°)

81 63 45 27 9 -9 -27 -45 -63 -81

Inside surface

16"38 0"86 -2"42 35"70 1"87 6'97 63"56 3"32 34'72 31-95 1.67 48'56 -95"71 -5"00 -8'00 -139"59 - 7 . 2 9 -66-98 -2"50 - 0 ' 1 3 -41.42 95'85 5.01 25-90 62"10 3'24 32.46 25"23 1"32 23.49

Outside surface

-0"13 0"36 1"81 2'54 -0-42 -3.50 -2'16 1.35 1.70 1.23

-16'86 -35'28 -62"51 -9.46 75'20 108,71 -14.16 -96-73 -63.12 --29"01

-0'88 - 1"84 -3'27 -0.49 3.93 5'68 -0"74 -5"05 -3"30 -1.52

-3"27 --2'64 11"80 51.60 72-32 34.85 -79.55 -78'82 -40.27 -19.88

-0.17 -0"14 0-62 2-70 3.78 1.82 -4.16 -4"12 -2"10 -1"04

-'~5

-CO

-45

-30

-~S

0

.IS

30

45

(,o

"~s

~o

SPENCEI elbow model with tangent pipes: calculated 'xternal stress with geometrical non-linearity effects using TRICO finite element code, Mr = 80 000 lb in.

-~X)

rig. 40.

BO

60

30

-~

-6o

-4~"

%

-3o

4

.~"

/

o

0"~,

/

/

~"

/

5o

j

45"

60

¥$

-

-

SPENCEI elbow model with tangent pipes: calculated internalstress with geometrical non-linearity effects using TRICO finite element code, M r = 80 000 lb in.

-~o

Fig. 41.

,--

e---e

0o

/I

I~

-50

-+15

- ~ - - ~

- ,:I.5

analysls

.,

-(~o

I

-3o

~

L ~ /,'a

,,

~

e--.e

0

~5

.~'non-linearilies

~o

4-5

~;0

+

-oao

1'5

7

oJO

Fig. 42. SPENCEI elbow model with tangent pipes: calculated external stress intensification factors with geometrical nonlinearity effects using TRICO finite element code.

-4-

-2

o

1

2

.3

4-

5

8

.'10

-~JO

-75

[

r t

-~;0

I

i

I

I

i

i!

-4-5

_~.-4

\\

I "

i

%'

-10

x(-

I

~z~ /.~

-~5

i

L

!I

;

• ---

0

~

x!1

r

45"

i

• i

~

r

~

~0

]

i!

t

j

I

. ~ . ,-/'.. .~1~

• j r~n.l/n.

4-5

N

(;0

I

!

i

~5

'I

;

qO

46;--

Fig. 43. S P E N C E I elbow model with tangent pipes: calculated internal stress intensification factors with geometrical non-linearity effects using T R I C O finite element code.

-6

+

z

"~

2

+

inil)rsis

*90

-4¢3

-~o

,'

r

-,e~-

I

t

o

5o

!

I

,~5

,

+

i

P

'\

i

45

"+.

60

-f5

~JO

Cd,~l.

,~./

I

!+?I

j

-60

-~3

i -3o

-t~

O

J

J I

;it 3000

.(~r

11t

hr

') '/ ~

P¢~

3o

~'~

-,o

46

60

.i\

~

-

+

using TRICO finite element code, Mf = 80 000 lb in.

~O

i c,.~]

@.

SPENCE1 elbow model with tangent pipes: calculated

-'I'S

.,--~

internal

I

I

using TRICO finite element code, Mf = 80 lb in.

rig. 45.

-qo

•

i~t=

~ternal stress with geometrical non-linearity and creep effects

SPENCE! elbow model with tangent pipes: calculated

-60

,

j

..t....J I'--" °'¢l.t zooo,~ G

~

"~xternal stress with geometrical non-linearity and creep effects

-:r5

"~

:',~..,

Fig. 44.

rso

[0o

6o

oo

~o

5

38

F. G. Cesari, S. Menghini

-~o

-.T4S

-3o

-t~

0

"f5

3o

.4-'S

6.0

~Q,~,.~

~t5

¢Jo

J

~--~

~~

L --

[

!

'

/ at~ 3000 ' ~hf ~' 1 ~-o o -

-

-

e

-

"'"

•

SPENCEI elbow model with tangent pipes: calculated

/

linearity and creep effects using TRICO finite element code.

Fig. 49.

o

2

4-

£

linearity and creep effects using TRICO finite element code.

SPENCE1 elbow model with tangent pipes: calculated

-¢.o

• I

3000 hr

t

internal hoop stress intensification factors with geometrical non-

-~/5

k.---o,t

,

internal axial stress intensification factors with geometrical non-

Fig. 48.

_~'

'-6

-2

-~.

o

4

2

4-

8

9

n~

~"

~

425

~oo

5O

2~

0

-25 o

+

~'[+.] ++

+o

so

++

4o

0 [; . ] 4~

Fig. 50. SPENCE1 elbow model with tangent pipes: calculated external stress with geometrical non-linearity and creep effects (t = 0 h) using TRICO finite element code, along the median longitudinal section (q$ = 0°).

Jig

I ~,

e

i

I

I

t

I

I

I

I

t

I

I

t

I ! I

-~i'" ....,,,+o+I /

j !

5o

T

r

....

/ j_ l/"

-

~j] l

I

,

I

~

~

e

r

!; ° ! ....

~ J\

j

? ?--/'1-,,1' :l

T

"~

", !

- Z5

Fig. 51. SPENCEI elbow model with tangent pipes: calculated external stress with geometrical non-linearity and creep effects (t = 3000h) using TRICO finite element code, along the median longitudinal section (46 = 0:).

SIFs in elbows subjected to in-plane bending moments 4D

i

,

i

i

i

J

)

,

¢

i

i

i

i

r

t

i

[

,

I

i

t

i

i

i

i

r

)

,

r

I

¢

"t.s

41 )

I

I

.., .... , . ,

i

t

,

i'

~

--

f _ _ - ° - 000.

2.$

13

-25

I

~ q

i

i

i

i

i

i

i

i

i

I

i

i

•1 o

e[,',]

/ 4d

1

)

i

i

i

i

i

i I

, I

i I

i

i

i

~

i ~

o" ~s" ~o" ,)s" ~o" :ts* ~o. ,~ [a¢a3

i

i

i

~ [).]

i

t

i

i

40

45

Fig. 52. SPENCEI elbow model with tangent pipes: calculated external hoop stress intensification factors with geometrical non-linearity and creep effects using T R I C O finite element code, along the median longitudinal section (4~ = 0°).

. . . .

,

~

,

i

,

r

,

r

.... p .... i r ,

....

I ....

I ....

:15 f " 0 hr L.---3000 hr

external at

_ _....J

-25

i

i i i

i

r

J

~

l

40

e [ ,'),~

,

i i

I ~tl

i

J

t

i

~

i

i

4~[ O"

i

i

t i i

i 4-0

4~"

5O"

4"5"

£o"

"~5"

9o"

t 45

e C;,]

Fig. 53. SPENCE1 elbow model with tangen t pipes: calculated external axial stress intensification factors with geometrical non-linearity and creep effects using T R I C O finite element code, along the median longitudinal section (~b = 0°).

42

F. G. Cesari, S. Menghini

Finally, Figs 50 and 51 show the pattern of the external stresses on the median longitudinal section (~b = 0 °) for the H - G length in Fig. 35, and Figs 52 and 53 report the it factors for the same length.

5.

DISCUSSION

5.1. Comparison of analytical, numerical and experimental results In the first part of the analysis, the purpose was to obtain all the background knowledge that might prove useful in assessing such results as would be given in the other parts of the work. However, in spite of the fact that the study was not meant to compare in the first place the experimental results with those of the T R I C O program, some basis for such a comparison nevertheless exists. The structure used experimentally consisted of an elbow jointed to two straight pipes and loaded so as to produce a triangle-shaped diagram

S i

Fig. 54. BEND2 model: experimental load condition in Ref. 1 and bending moment diagram.

of the bending characteristic (see Fig. 54). It may be seen from the diagram that the experimental structure is equivalent to an elbow which is subjected to a bending m o m e n t

Mr=EL and is stiffer at its end edges. Thus, a situation of this kind could be compared neither to the situation shown in Fig. 5 nor to the one shown in Fig. 35. In the former case (Fig. 5) the added stiffnesses are not considered in the numerical calculations, whereas in the latter case (Fig. 35) the static regime is based on a different concept, even though the stresses on the median cross section of the structure could actually be

SIFs in elbows subjected to in-plane bending moments

43

comparable once all terms appearing in eqn (I) have been accurately determined. The main purpose of this work was, however, to compare the results obtained by numerical and theoretical analysis. Compared with the latter, the former show a misalignment of the positive hoop stress peak towards the negative values of the tp angle as well as non-axisymmetric values and patterns of the internal and external i factors; when theoretical analysis was used as opposed to numerical calculation, the results showed an antisymmetric profile with respect to the ~b angle. Comparing Figs 3 and 4 with Figs 6 and 7, it appears that the results obtained from numerical analysis are more satisfactory than those given by theoretical analysis when the hoop direction is considered. In particular, there is a peak position advance with reference to the tp = 0 ° origin which is not otherwise found if one applies the ASME formulae. For the axial direction, comparisons between the experimental and numerical results are still satisfactory, even though they show a discrepancy at the intrados (~b = -90°), and the finite element calculations tend to underestimate the maximum absolute value. However, it is clear that such data are ultimately affected by the actual static regime imposed. They provide further evidence that the maximum ii factor is located on the inside surface and has a hoop direction. Similar conclusions may be drawn from the numerical analysis reported in Ref. 12. As a general remark, it should be pointed out that the maximum deviation in the peak values of Figs 6 and 7 does not exceed 15 per cent and is in compliance with the indications of Ref. 13 for the stress distribution change between jointed and isolated elbows.

5.2. Simple elbow model subjected to in-plane bending moment Using the ASME Section III NB 3680 procedures, it appears from Figs 8-15 and from Figs 16 and 17 that the stresses in an elbow are reduced as the elbow factor increases. This is also shown in Ref. 14. On the other hand, the ASME procedures provide an analysis method for calculating the flexibility of the isolated elbow, neglecting the added end stiffness resulting from jointed straight pipes.13 Looking at Figs 3 and 4, it is clear that the analytical values are higher than or equal to the experimental ones, which are themselves obtained on jointed elbows. Conversely, if one compares the results shown in Figs 21 and 22, one finds that numerical analysis shows the

44

F. G. Cesari, S. Menghini

internal hoop stresses to be about 37 per cent higher than those obtained by the ASME method, whilst the axial stresses are about 10 per cent lower. Numerical peak data are increased if one considers the geometrical non-linear behaviour, so that the previous deviations are now + 57 per cent and - 7 per cent respectively. Geometric non-linearities tend to enhance hoop stresses and to keep the axial stresses unchanged. The different behaviour of the two computational approaches is confirmed by analysis of the reciprocal rotation of the end sections (elbow rotation), which measures the elbow flexibility. This is expressed by the quantity 0 k= (5) IIOI1"1

which is also described by the flexibility factor. Here 0,om = ~-~ rfoZ Mf de

(6)

in radians represents 13 the rotation of a straight beam having the same length and cross section as the elbow under evaluation. The same formulae are contained in the ASME Section III Subsection NB 3687.2 rule. Following the procedures of this ASME rule, 0,om and k being known according to the relation 1.65 k= T (7) it is possible to calculate the 0 value; 0nom being equal to 0.2074 °, 0 becomes equal to 2.8754 ° . In numerical analysis, the 0 value is obtained using the procedure given in the Appendix. Thus, the data reported in Table 13 and shown in Fig. 55 are derived. To assess accurately the numerical results, it should be borne in mind that the value of the product k2 from eqn (7) is equal to 1.65. In the case of elastic computation, the previous product (equal to about 2.46) appears to be higher than that given by ASME formulae, whereas in conditions of geometric non-linearity this value increases to 2.66. Thus, one may conclude that under similar computational conditions the elbow has shown a more marked flexibility compared with the flexibility analysis performed following the ASME prescriptions.

SIFs in elbows subjected to in-plane bending moments

45

T A B L E 13

Elbow Rotations as Calculated for the SPENCEI Model Subjected to an In-plane Bending Moment of 80 000 lb in Case

1 2 3 4 5

Elbow rotation

Elbow model

2'875 4° 4-283° 4.642° 2.47° 2.815 3°

Remarks

ASME NB 3687.2 procedure TRICO elastic calculation Geometric non-linearities TRICO elastic calculation Geometric non-linearities

-Simple Simple Jointed Jointed

Moreover, when the elbow under evaluation was subjected to creep and geometric non-linearity effects, it is clear from Figs 28-31 that, if a constant external load is applied, the stress state in the median section is increased. However, the m a x i m u m absolute value is unchanged. A time-dependent increase occurs in axial stresses (Figs 25, 32 and 34) while a redistribution of the hoop stresses results in decreased values along the median cross section (~ = 45 °) and increased values in the regions next to the end edges. This pattern is compatible with the load conditions applied, but should be further investigated by additional theoretical, numerical and/or experimental evidence, if any can be obtained. However, as can easily be seen in Fig. 55, the elbow rotation

,12 4,1 40

~- = 4 . 6 6 3 7 + 0 . 0 0 2

t

6

4

elas t ;e

-

analysis

I

•

ASME

NB 3 6 8 7 . 2

__

t 4

- -

-[h,]

0

0

.4

$

4

~

~E03

Fig. 55. SPENCEI simple elbow model: end mutual rotation for the previous computer cases.

46

F. G. Cesari, S. Menghini 40

'8

I-

,__

elbow

ends

.~

°'"' ° t pipe

} geon'wtrical [ V P~n.rmelrRi¢$ • and cretp AASNIE

0

Fig. 56.

4

2

Sect. X

5

4

ends

_

NB 36/7.2

5

,I,go's

SPENCEI elbow model with tangent pipes: end mutual rotation for the previous computer cases.

increases according to a linear law, thus further increasing the flexibility factor (k2 = 6.03 at time t = 3000 h). The pattern of the latter is shown in Fig. 56. 5.3. Jointed elbow subjected to a bending moment Several authors have pointed out that the presence of constraints or added stiffness at the bend ends or next to them leads to reduced flexibility and, in consequence, to a stress reduction in the elbow. 18, ~2, ~3,15 Looking at Figs 38 and 39, it is clear that in the case of elastic computation the jointed elbow shows such decrease as mentioned above, which amounts to about 36 per cent on the maximum hoop stress intensification factor. Similar conclusions may be drawn if one calculates the elbow angle. Its value, reported in Fig. 56, is equal to 2.47 and is lower than the values obtained both with the same method in the simple elbow model ( - 4 2 per cent) and with the A S M E computing method ( - 14 per cent). In our case, k (by eqn (6)) being equal to 11"91, the flexibility product is k2 = 1.417

This result is in compliance with both what is indicated in Ref. 13 and what is stated in Ref. 16. Comparison between the stresses calculated here and those calculated by A S M E procedures is also worth noting briefly. Comparing Figs 8

SIFs in elbows subjected to in-plane bending moments

47

and 9 and Figs 36 and 37, it appears that on the whole the latter show less marked curves, as was to be expected from the previous assessment of the flexibility factors. The relative differences do not exceed the following limits: external internal external internal

hoop stress: hoop stress: axial stress: axial stress:

-30.6 -8"2 -44.6 -49.0

per per per per

cent cent cent cent

Such data are changed when the geometric non-linearity effects are considered. In this case, the elbow rotation is slightly higher than in the previous case and is equal to 0 = 2.8153 ( + 14 per cent) This value (see Fig. 56) is very close to, though lower than ( - 2 - 1 per cent) that which can be obtained with the theoretical analysis; the same is true for the product k2, which is equal to 1,615. As a consequence of the increased tendency to flexibility, Figs 42 and 43 show increased stresses (compared with the elastic field results) which are close to those in the analytical regime or even higher than the value of the maximum internal hoop stress from analysis ( + 12 per cent). The meridional distribution of the stress on the outside surface is shown in Fig. 50 and is compared with the result given by the same computational procedure for the simple elbow model (see Fig. 57). Looking at the latter, it appears that the increased length of the pipe being bent ultimately induces a decrease in the axial stress ( - 2 8 per cent) which is equal to the decrease in the rotation of the farthest straight pipe sections ( 0 = 3.3258 °) when compared with the value calculated for elbows alone (see Fig. 56). As is also shown in Ref. 13, the added stiffness, induced on the elbow edge by connected straight pipes, has the most effect on the ovalization of the section and therefore on the hoop stress, which in the case of a jointed elbow is reduced by about 27 per cent (22-5 per cent for ~ = 45°), almost independently of the 0~ angle. In this case, the introduction of creep shows a tendency to decrease hoop stresses and to increase maximum axial stresses, as is clear from Figs 46-49. For the former, the variation amounts to - 6 . 6 per cent in their maximum values while the peaks tend to become more marked at

F. G. Cesari, S. Menghini

48 2~o

'l

~external

•

l' I 'l'o

4(:0

50

; •

t

i

i

,

,

,

,

,

,

,

,

,

,,

,

L , ~ ,

iJ

J

,

35

4o O"

45"

"~"

~o"

~"

?s"

4O

46

e Iilv, l

9o"

,6 f,~,,a]

Fig. 57. SPENCEI elbow model: comparison between calculated external stresses for simple and jointed elbows, with geometrical non-linearity and creep effects at time t = 0, along median longitudinal section (4~ = 0°), Mf = 80 000 lb in.

40

'

,~

4-

.

.

.

I

I

1

'

i

]

+ ¢(.

-,

.

egi~,J

ce

!

~5 •

~o-

!

As.

:. }

!

~.

Is-

~

~-.

#

~. e L-i,3

Fig. 58. SPENCEI elbow model: comparison between results from different jointed elbow model meshes, considering geometrical non-linearity and creep effects and using TRICO finite element code, along median longitudinal section (4)=0"), M r = 80 000 lb in.

SIFs in elbows subjected to in-plane bending moments

49

the angle ~b = +45°; for the latter, their maximum values increase by about 20-25 per cent. A predictable consequence of creep is the increase in the elbow rotation, according to the law illustrated in Fig. 56. The increased flexibility of the elbow, demonstrated by the rising values of the O/M r ratio, does not bring about an increase in stress when a constant external load is applied, as it would under a time-independent regime. Thus, as is suggested by Ref. 15, the k factor is no longer enough to account for the elbow behaviour when creep is present. The pattern of the external stresses on the median longitudinal section (~b = 0 °) shows a redistribution of the stresses in an axial direction, as is clear from Figs 52 and 53, as well as in a circumferential direction, as discussed previously. In particular, a decrease in the hoop stress occurs for ~ = 45 ° as opposed to an increase in the axial stress. These findings are confirmed also when a finer mesh is used in the area of the elbow (see Fig. 58).

6.

CONCLUSIONS

Arising from this study, the following conclusions may be drawn: 1.

2.

3.

4.

The load scheme adopted here does not correspond to the actual operating conditions of piping in which the loads applied produce variable diagrams of the moments. This variability is due to either geometric effects or time-dependent phenomena; however, the adopted scheme does provide useful indications as to the behaviour of elbows. The computing methods and the schematizations adopted provide results that are sufficiently reliable enough and in agreement with the experimental findings obtained in comparable conditions. The ASME prescriptions, through the detailed analysis contained in the NB 3680 rule, indicate that stresses reach maximum values for t~ = 0 ° instead of q~ ~ - - 5 °, as is found experimentally. The same rules provide the same maximum values for both the inner and the outer surfaces, while differences are found experimentally and numerically between the two values; these differences may even be significant, with the values of the inner surface being higher.

50

F. G. Cesari, S. Menghini

5.

.

.

.

In the case considered here, even though the analysis was conducted on elbows not connected to other pipes and therefore tended to overestimate the stresses occurring in jointed elbows, the ASME prescriptions agree with the results of the computation made on the latter type of elbows if geometric non-linearities are considered. Further evidence is provided that the presence of straight pipe lengths connected to an elbow reduces the stresses in the elbow itself by about 30 per cent, resulting in a reduction of flexibility (k) of about 40 per cent. Under a creep regime, the elbow rotation varies in a linear way, provided that Norton's law can be applied to describe the evolution of the phenomenon in time. The state of stress seems to be disturbed also in straight pipes as a result of their being connected to the elbow; some authors maintain that such disturbances become zero after having spanned a length of pipe equal to about three times its diameter. ACKNOWLEDGEMENTS

The computing campaign was carried out by using the T R I C O shell type finite element program of the CASTEM (CEA) series. This research, performed on account of Commissione Normativa per Reattori Veloci (CNRV), arises from discussions in the Activity Group No. 2 of the Working Group on Codes and Standards created by the Fast Reactor Coordinating Committee of the European Economic Community (pers. comm., J. Spence, May 1981).

REFERENCES 1. Imamasa, J. and Uragami, K., Experimental study of flexibility factors and stresses of welding elbows with end effects, Proc. 2nd Int. Conf. on Pressure Vessel Technology, Part 1, I 30, American Society of Mechanical Engineers, San Antonio, TX, 1973. 2. ASME Boiler and Pressure Vessel Code, Section III, Nuclear power plant components--Div. 1, American Society of Mechanical Engineers, New York, 1980. 3. Roche, R., Hoffmann, A. and Mijllard, A., Inelastic analysis of piping systems. A beam type method for creep and plasticity, 5th S M I R T , Paper L 4/7, Berlin, 1979.

SIFs in elbows subjected to in-plane bending moments

51

4. Hoffmann, A. and Charras, T., Programme COCO--Maillage automatique de structures planes ou tridimensionnelles, Coco User Manual, Note CEA N. 1937, CEN, Saclay, France, 1976. 5. Hoffmann, A., Programme TRICO--Analyse des structures tridimensionelles compos6es de coques et de poutres, Trico User Manual, Note CEA N. 1930, CEN, Saclay, France, 1977. 6. Knapp, H. P. and Prij, J., Inelastic finite element analysis ofapipe--elbow assembly. Benchmark problem 2, IWGFR/27, International Working Group on Fast Reactors, International Atomic Energy Agency, Vienna, June 1979. 7. Jeanpierre, F., Syst6me CEASEMT--Ensemble de programmes de calcul de structures fi l'usage industriel, Coco User Manual, Note CEA N. 1938, Saclay, France, 1976. 8. Bung, H., Clement, G., Hoffmann, A. and Jakubowicz, H., Piping benchmark problems--Computer analysis with the CEASEMT finite element system, IWGFR/27 Paras 2.5 and 3.6, International Working Group on

Fast Reactors, International Atomic Energy Agency, Vienna, June 1979. 9. Hoffmann, A. and Roche, R., M6thodes simplifi6es d'analyse des tuyauteries--M6thode globale---Cas particulier des coudes, Journkes de confbrence du DEMT, paper 2/78, 6/1978. 10. Solal, P., Flexion en ouverture monotone et pincements enfermeture monotone sur un coude en Acier A 106 grade B, Report CEA/DEMT/77-29, Saclay, France, 1977. 11. Hoffmann, A., Jeanpierre, F., Charras, A. and Combescure, A., Syst~me CEASEMT--Aper¢us thboriques sur les :programmes .~lement-dynamiquenon linearitkes gkomk triques-flambag e-plasticitb, Report DEMT/SMTS/79/

22, CEA, Saclay, France, 1979. 12. Kano, T., Iwata, K., Asakura, J. and Takeda, H., Stress distribution of an elbow with straight pipes, 4th SMIRT, Paper F 1/5, San Francisco, 1977. 13. Thomas, K., Stiffening effects on thin-walled piping elbows of adjacent piping and nozzle constraints, J. Pres. Ves. Techn., 104(8) (1980), pp. 1807. 14. Mijilard, A., Hoffmann, A. and Roche, R., Programme TEDEL. Analyse blastique et plastique de coudes, Note CEA N. 2116, CEN, Saclay, France, 2/1980. 15. Boyle, J. and Spence, J., The flexibility of curved pipes in creep, J. Pres. Ves. Techn. (August 1977), pp. 444-53. 16. Natarajan, R. and Blomfield, J. A., Stress analysis of curved pipes with end restraints, Computers & Structures, 5 (1975), pp. 18%96.

APPENDIX

Numerical analysis procedure for 0 Analysis of the obtained numerical results showed that the nodal displacements of the model ends have an almost linear pattern versus

52

F. G. Cesari, S. Menghini

the pipe axis distance. Sectional buckles were considered to be normal for a finite element calculation. The equivalent end rotation was thus given by linear regression of the nodal value couples, and the 0 elbow rotation is the sum of the two single rotations of the model ends, with reference to the starting position.