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- Propagation of ovalization a long straight pipes and elbows

MILLARD A., ROCHE R. 6. International conference on structural mechanics in reactor technology. 6. SMIRT. Paris, France,August 17 - 21, 1981. CEA - CONF 5950

A. MILLARD - 1 - M 10/1

Summary

In most of the calculations of piping systems using simplified methods, the flexibility of elbows is accounted for by means of the well-known flexibility factors. These factors are determined for elbows, considered as a part of an axi-symmetrical torus assuming an uniform behaviour along the bend. In reality, adjacent straight pipes or even flanges tend to restrict the deformation of the elbow cross-section, thus lowering the flexibility factor and stress intensification factors.

The aim of this paper is to present analytical solutions for the propagation of ovaliza-tion and the variation of the flexibility factor along pipe bends terminated by straight pipes or flanges, under in-plane bending, assuming an elastic material behaviour.

In the first part, the influence of the various strains is analysed in the simple case of a straight pipe, subjected to an elliptical cross-section shape deformation at one end. The results enlighten the very important part played by the distorsion in the propagation. They have been compared with finite elements solutions and with simple expérimente.

In the second part, the solution is developed for an elbow terminated by a straight pipe or a flange, following the Von Karman's approach : local displacements are expanded in Fourier series, the coefficients of which vary along the curvilinear abscissa, like the rotation of the cross-section as a whole ; the differential equations as well as the boundary conditions are found by minimization of the total potential energy of the assembly.

The solutions are compared to existing and experimental results.

A. MILLARD - 2 - M 10 /

1. Introduction The stress concentrations which occur in elbows are one of the most important problems

set to the piping systems designer. The search for stress intensification factors and flexi­bility factors was originated by Von KARMAN [ 1 ] in 1911, who introduced a certain number of simplifying hypothesis. Since then, many refinements were proposed but most of them still require the assumption of a uniform behaviour along the bend. However, the presence of adja­cent stiffening structural elements like flanges or straight parts will cause a variation of ovalization along the elbow, and tend to reduce the stress intensification factors as well as the flexibility factor. The most straightforward approach to cope with this problem for a given assembly, is to perform a detailed numerical analysis using for example shell finite elements [1 - 6], but it remains limited to several geometries and it is not much helpful in the understanding of the phenomenon. In the same sense, experimental studies have been per­formed [6 - 9] . Up to now, few analytical or semi-analytical solutions have been published. KALNINS [ 10 ] has proposed a general method based on a finite-difference solution of the complete set of the shell problem differential equations. The flexibility of bends with flanged constraints has been investigated by THAILER and CHENG for U-bends [ll] and byFINDLAY and SPENCE for arbitrary angles [12] using Fourier series expansions in both circumferential and longitudinal directions. A solution to the more general problem of pipe bends with flanged tangents has been proposed by WH/.THAM and THOMPSON [ 131 . The present paper does not claim to bring the complete solution to the problem of end-effects, but rather to derive ana­lytical solutions in most simple cases in order to appraise the merits of various simplifying assumptions, the extension to more complicated cases being considered as a straightforward application of the method. The frame of the study is the elastic linear theory of thin shells, assuming small displacements. The propagation of ovalization along a straight pipe is first considered. The conclusions are then applied to the cases of elbows with flanges or straight parts.

2. Propagation of ovalization along a straight pipe 2.1 Solution for the propagation of a mode 2 A thin straight pipe of length i , thickness e, mean radius r (e < r) is considered in

the following. The displacement field of any point M of its mean surface is characterized by three local components (see Fig. 1) : u parallel to the axis of the pipe, v tangential in the plane of the cross-section and w normal to the pipe ; they are functions of circumferential and longitudinal parameters x.

At one end, the pipe is subjected to prescribed displacements :

w(0,O) = a cos 20

v(0,O) - y- sin 20

It corresponds to a mode 2 ovalization of the section, which propagates along the straight pipe.

In viow of the symmetry, displacement u is taken as :

u(0,:<) - b(x) COG 20 (2)

i-ollov.'i ii'i !,ovu-Kirchhoff assumptions, the «train and curvature variation tensors of the mean

A. MILLARD surface of the pipe can be written

E00 r t W + a o ' 30

: = 2» 'xx 3x v . 1 3u T0x r 30 *£ k - i ( K0O r2 I

32w 302

a 2 V

k - W

xx ~ 3x2

• . * / 3 w

(3)

3v\ 0x r I 30 3x

Assuming a plane state of stresses, strain energy per unit surface is

2 W a = JL?^ T e2 • e 2 + d i_v2 I 00 xx

12(1-V2) L

'00 xx

k 2 + k 2 + 2v krv. k 00 xx 00 xx

4c

J 12 K 0 x (4)

where E is Young's modulus, G shear's modulus and V Poisson's ratio. e2

Since e r , the term — k£ is negligible in front of y 2 '0x In other respects, inextensibility Is assumed all along the pipe, i.e.

£00 W ~ " 30 (5)

M 11,, i

The minimization of potential energy leads to the differential system

a - 6 f.SL + l=v [r 2 4e 2 a" - 6 Cl-V) b , + 9 a = 0

e 2r

Solutions of this system are ir the form

(6)

a(x) = a e Xx , b(x) = 3 e Xx (7)

where X is given by the characteristic equation

A6-A" f3(l-v) 2U+2V) 1 . 2 ri2v(l-v)+9l 18(1-LTT" + ~^~J + x L—F—J- — •v) = 0 (8)

and a and 3 are constants given by the boundary conditions. The pipe is supposed long enough to keep only the roots X with negative real part. Boundary conditions then reduce to :

a(o)

a"(o) 3v n(o) = 0 (9)

I:' (O)

A. MII.LARD - 4 -which enable the determination of constants A 1, A , A , so that :

A* X A_X A—X a(x) = A. e + A_ e cos y x + A, e sin y_ x (10)

M 10/1 " —I

( x x < o, x 2 < o). Figures 2 and 3 show this solution calculated for i. - 400, r = 1 0 , V = 0 . 3 and e = 0.5

or e = 0.05. Corresponding X values are :

e = 0.5 | Xj = - 2.06 I X 2 = - 3.05 10" 2 u 2 = 2.82 10~ 2

e = 0.05 ( *j = - 20.50 ( X 2 = - 1.05 in"2 u 2 = 8.24 10" 3

It may be noticed that the first solution function is decaying very quickly, X varying quite like — . e

2.2 Approximate Solutions 2.2.1 Neglect of Distorsions If Ypw a n ^ kflv a r e n e g l e c t e ^ i° t n e strain energy, the same method supplies the

solution : X.x

a(x) = a e cos p. x (11) o l

where X ^ - i V f d + v ) ^ - i f f Ô ^ )

This solution is independent of the thickness e and is much too stiff as shown on figures 2 and 3.

2.2.2 Neglect of Displacement u The displacement u introduced corresponds to the warping of the cross-section. If it

is neglected, the solution is of the form : X.x X-x

a(x) = A j e + A 2 e (12)

Numerical a p p l i c a t i o n g i v e s :

(X, = - 2.05 i \ 1 => - 20.50

(X„ = - 1.46 1 0 " 2 e = ° ' 0 5 )A„ = - 1.46 10" e = 0 ' 5 ' ' \ 2 = -1.46 10-2 e = ° - 0 5 K - ~ 3

The rapidly decaying solution is well calculated, but neither the shape nor the decay length of the predominant function arr yielded as shown on figures 2 and 3.

2.2.3 Neglect of Curvature Variation k The curvature variation is responsible for the fourth order of the differential

i.ysteni, vvhich reduces to two coupled second order differential equations if it is neglected. Solution i,. then :

X.x X x a(x) A. e co:; \\. x + A e sin \i.\ (13)

A. MILLARD - 5 - M 1 0 /

w i t h X,

ii - y ^ [ /3-2(^vr f ] = x i (14)

-2 -3 Numerical application : e = 0.5 =» ]x. = 2.94 10 ; e = 0.05 =* J^ = 9.31 10

It can be seen on figures 2 and 3 that this solution is a fairly good approximation to the exact solution.

As a conclusion to this first study, it can be stated that neither the warping of the cross-section nor the distorsion l . can be neglected, while suppressing k brings appre­ciable simplifications. 3. Propagation of Ovalization along an Elbow

3.1 Simplifying Assumptions The difficulty of the problem arises from the number of unknown functions. Therefore

following assumptions have been adopted : 3v

- inextensibility w = - -^

- r <^ R where R is the bend radius

- \ = ^ - 0.5 r

- the elbow is loaded by a constant in-plane bending moment - there is no pressure in the pipe.

Local displacements are expanded in Fourier series, which are truncated as follows, in view of the third hypothesis :

u(0,x) = U(x) + rij;(x) cos 0 + b(x) cos 20

v(0,x) = -W(x) sinO - ^ ~ rin 20 (15)

w(0,x) = W(x) cos0+ a(x) cos 20 The functions U(x), W(x), lfi(x) stand for the global, i.e., curved-beam like behaviour of the pipe, and a(x), b(x) stand for the ovalization and the warping of the cross-section respecti­vely. It may be noticed that all assumptions except r < R can be removed by adding new functions in the series, which present no theoretical difficulties but a tedious amount of numerical work.

3.2 Strains-displacements Relations - Strain Energy The strains and curvatures-variations tensors of the mean surface are given by :

eGG = 7 ( w + 30» e = __ + _ (w c o s 0 _ v s i n 0 ) xx dx R

1 3u , 3v u , 0 Yf.w = 7 3Ô + 37 + Ô s i n Q 'Ox r 30 dx R 1 3 2w 3v

d vi 2 du n cot;0 , A , „. k ••-- — r r - —• TT- cos 0 - — y — (v/ cos 0 - v H.I n () ) xx 3x x R ox n

L j 9 2 w 'éy. c o s P ^\ fjx """ r I DO dx " 9x R 3 0 /

r A. MILLAKD - 6 - M 10/ where x is the curvilinear abscissa along the axis of the pipe. The strain energy has the expression (4). The external work is : W = M. *(£) where I is the length of the elbow.

ext 1

3.3 Solutions Neglecting k 3.3.1

xx Basic Solutions

Having introduced the displacement fields, the application of the virtual work principle supplies a coupled system of five differential equations of second order :

D 2 q" + Dj q' + D Q q = 0 (17)

where q = [u W A a b] , and D_, D , D_ are matrices of constants which involve R, r, e. Solutions are of the form :

5 (18)

X. x -Ax q(x) = E A. e + B. e

i=l X

where X . are complex roots of the characteristic equation associated with (17) ; two of them correspond to the rigid body modes of the pipe :

X l = X 2 = i X i The otners are computed. The various constants A , B. are determined through the boundary con­ditions which are also supplied by the principle.

3.3.2 Case of an Elbow with Flanges By symmetry, half of the elbow is considered. Then, functions W(x) and a(x) must be

even in x and U(x), ljj(x), b(x) must be odd. At the flange, local displacements must be zero :

a(Jt) = 0 bU) = 0

and the global loads are known (Moment M and zero shear and normal loads). Figures 4 and 5 show the functions a(x) and l|)(x) for e = .1, r = 10, R = 50, X = 0.5 and

for a bend angle 2a - 90°. It may be noticed that the derivative of the ovalization function a(x) is not zero at

the flange, as a consequence of neglecting k in the strain energy. A global flexibility factor k can be defined for the elbow by :

k =.M*L (19)

where ty Ai) is the end rotation of a straight pipe of same characteristics under the same moment. Values for various bend angles are found higher than those of reference [12] (see Figure 6).

3.3.3 Case of an Elbow with Straight Tangents Symmetry considerations are still observed in the elbow. At the junction v/ith the

straight pipe, continuity of solution functions q(x) and associated loads is prescribed. In the straight pipe, the differential system uncouples into three differential subsystems of second order in (W,^) , (a,b) , and U. If the straight pipe is assumed sufficiently long in order to retain only decaying exponentials, the solutions are of the form :

2 3 W(x) = A + A x + A x + A x

ij'(x) A^ +?A2 x +3A x 2 (20)

A.MILLARD - 7 - M 10/1 À.x A.x

!

a(x) = A c e cos y. x + A, e sin y, x A.x A,x 1

b{x) = A_ e cos u. x + A_ e sin y. x (20) U(x) = A^C+ A 1 Q

The A. constants are determined by the continuity requirements at the junction with the elbow and by the global loads at the other end of the straight pipe.

Functions a(x) and ty(x), for the same characteristics as in 3.3.2, are shown on Figures 7 and 8 and for straight parts of length 20xr. It may be noticed that in the mid-section of the elbow, ovalization is superior to Von KARMAN*s solution. The global flexibility factor, calculated from the rotation of the elbow end-section is compared with results of reference (13] on Figure 9.

3.4 Solutions Accounting for k a xx

Accounting for k in the strain energy leads to a differential system of fourth order. There are seven basic solution functions to this system ; two of them correspond to the rigid body modes (see 3.3.1) and two others correspond to very local bending effects, the decay length varying like the thickness.

The system has been solved in the case of an elbow with flanges. At the flange, the boundary conditions associated with local displacements are :

a(Jt) = 0 a'U) = 0 b U ) = 0

The otherscorrespond to known global loads plus the additional condition :

IW" " 4" " | u' _ T {£) = ° (21)

which is associated with the virtual displacement ÔW {I) . The results are compared to the previous ones on Figures 4 and 5. The elbow is fou.id slightly stiffer, as a result of the boundary conditions on the derivatives. The difference on the global flexibility factors is inferior to 4 %.

4. Conclusion The propagation of ovalization has been first investigated in the case of a straight

pipe in order to estimate the importance of various simplifying assumptions. It has been shown that distorsions and warping terms must necessarily be accounted for in the solution. These considerations have been applied to the assembly of an elbow with flanges or straight parts, in the moyt simple cazz. Tho method can be 'xtendod to pipes with others characteristics.

References

[l] VON KARMAN, Th., "Ûber die Formânderung dûnnwandiger Rohre, insbesondere federnder Ausgleichrohre", Zeitschift Ver deut. Ing., Vol.55, p. 1889-1895, 1911.

[2] SOBEL, L.H., "In-plane bending of elbows", Computers and Structures, Vol. 7, p. 701-715 1977.

[3] KANO T., et al., "Stress distributions of an elbow with straight pipes", 4 SMIRT, .Sein Francisco, papar F 1/5, 1977.

A. MILLARD - 8 - H 10/1 NATARAJAN, R., BLOMPIELD, J.A., "Stress analysis of curved pipes with end restraints". Computers and Structures, vol. 5, p. 187-196, 1975.

RODABAUGH, E.C., ISKANDER, S.K., MOORE, S.E., "End effects on elbows subjected to moment loadings", ORNL/Sub.2913/7, 1978.

WRIGHT, W.B., RODABAUGH, E.C., THAILER, H.J., "Influence of end-effects on stresses and flexibility of a piping elbow with in-plane moment". Press Vess. and Pip., Analysis and C omP-, p. 95-106, 1974.

IMAhASA, J., URAGAMI, K., "Experimental study of flexibility factors and stresses of welding elbows with end effects". 2nd Int. Conf. Press Vess. Tech., p. 417-426, 1973.

PARDUE, T.E., VIGNESS, I., "Properties of thin-walled curved tubes of short-bend radius". Trans. ASME, Vol. 73, p. 77-87, 1951.

BROUARD, D., TREMBLAIS, A., VRILLON, B., "In-plane and out-of-plane bending tests on carbon steel pipe bends", Proc. 5th SMIRT, Berlin, paper F 3/2, 1979.

KALNINS, A., "Stress analysis of curved tubes", Proc. 2 Int. Conf. Press Vess. Tech., paper 1-19, 1973.

THAILER, H.I., CHENG, D.H., "In plane bending of a U-shaped circular tube with end-constraints", Trans. ASME, J. of Eng. for Ind., p. 792-796, 1970.

FINDLAY, G.E., SPENCE, J., "Flexibility of smooth circular curved tubes with flanged end-constraints", Int. J. Press Vess. Pip., Vol. 7, p. 13-29, 1979.

HHATHAM, J.F., THOMPSON, J.J., "The bending and pressurizing of pipe bends with flanged tangents", Nucl. Eng. Des., Vol. 54, p. 17-28, 1979.

FIG. 1

1 - °1 ' M l M ' M I i I ' I ' M M I I I I I i I i I I I I I I I I I I I i

Q2' • ' • ' • ' . » • i . i . i . l . l . l . I I . I . 0 40 80 120 160 200 240 280 320 360 x 400

Exact solution v=0.3 • o •• Solution neglecting distorsions e=0.5

-x—x- Solution neglecting warping u I «400 Solution neglecting kxx r=10

00^=1 xx>j 6uno9i6au uoi^nios 0t= J 71 6uidJDM 6ui;oai69U uoj}rnos — * - * -

gOD=a suoisjo;sip 6utp9i6eu uoi^nios > • t*0=A uoi nios ;oox3

007 * 096 02e 083 072 002 09L Oil 08 F T I I I I I I I I | I | I | I | I | I | I | I | I | I | I | I | I

• I , I i I i I i I i I i I i I . I i I i I • I i 1 i I i I i » i 1 i I

ta

. i . i t i i i . i . i t t i i i » i i i » i i i » i I i » i I i l i » i I i

A 8 12 16 20 24 28 32 36 x 40

Elbow with flanges Bend angle =90°

e=1 r = 10

R=50 A =0.5

' h ' 1 ' 1 ' I ' I ' I ' 1 I > I » I > I « I i I ' I / ' I i I » I i I > H

/ Solution /

without kxx.

Solution without propagation

/ ^ s olution with kxx

A

i . » . i . i • i i i i i • i i » • i i i i i i i • i i I i I i i i i i t i « » » 0 8 12 16 20 24 28 32 36 x 40

refl12]

PRESENT SOLUTION

A=0.5 ref [12]

45° 90° 135° 180° BEND ANGLE

Fig.6

Variation of global flexibility factor with bend ongle.

2 a(x)

0

T 1 1 1 1 , 1 j 1 1 1 j 1 1 1 1 1 1 1 1 i 1 r

Straight pipe

Solution without propagation

J _ I L ! • ' » ' • ' I I I L 1 1 1 1 1 1 L 0 40 80 120 160 200 Propagation of ovalization along an elbow with straight parts Bend angle 2a=90° r = 10 R=50 Straight part lenght I = 2 0 x r e = 1 v=0.3

240

31 CO

30

Z6

2.2

T 1 1 1 1 1 1 < 1

[Solution without propagation

Straight pipe

0 40 80 Bend Angle 2a = 90° Straight part length I = 20 r

120 160 200 r = 10 R = 50 e= 1 v = 0.3

240

1

0.01 0.02 Q04 0.08 Û2 Û40.5 0.81 2 Bend characteristic A)

Equivalent flexibility of bends with infinite tangents

A s Present solution

Fig.9