Int. J. Pres. Ves. & Piping 13 (1983) 65-83 The Influence of Flanged End Constraints on Smooth Curved Tubes Under In-Plane Bending G. Thomson Ferranti, Edinburgh, Scotland, Great Britain & J. Spence University of Strathclyde, Department of Mechanics of Materials, James Weir Building, 75, Montrose Street, Glasgow G1 IX3, Scotland, Great Britain (Received: 19 April, 1982) ABSTRACT A theoretical solution is presentedjor the in-plane bending linear elastic dejbrmation behaviour of smooth circular cross-section, constant thickness pipe bends whose ends are terminated by rigid flanges. The technique employs the theorem of minimum total potential energy with suitable kinematieally admissible displacements in the form of trigonometric series. Results are given covering a fairly wide spectrum of practical bend geometries. These are compared with previous theoretical predictions and with various published experimental data. Some test results which were obtained during the present investigation are also given. NOMENCLATURE A, B, C, D, F, G with subscripts: Displacement coefficients. (A, B, C, D, F, G) (Non-dimensionalised). Et/(1 - v2). Et3/(l - v2)12. Young's Modulus. nr3t. with subscripts: Curvature strain. 65 Int. J. Pres. Ves. & Piping 0308-0161/83/$03"00 © Applied Science Publishers Ltd, 1983. Printed in Great Britain C D E I K 66 G. T h o m s o n , K M R R' J. S p e n c e Flexibility factor. In-plane bending moment. Radius of bend centreline (4) = 0). R + r sin qS. Rigid cross-section centreline displacements. Total potential energy (non-dimensionalised). u,,v, Subscripts Series subscripts. Mid-surface radius of pipe cross-section. Bend wall thickness. Rigid cross-section displacements. Distortion displacements. j, In, H F t H R , UR , W R H D, UD, W D Greek symbols )'o~ 7 Bend angle. with subscript: Shear strain. Rotation between ends of bend under bending moment, M. 70 MRo~/E1. 0 2 v a q5 with subscript" Strain. Angle along bend from centre, circumferential coordinate. Pipe factor, Rt/r 2. Poisson's ratio. with subscript: Stress. Meridional angle, measured around cross-section from midway between intrados and extrados. ~"J = 1, j even. = 0, j odd. )~= 0, ./even. = 1, j odd. JT, M T , N T Total number of terms in series. INTRODUCTI ON With present trends in the petrochemical and power industries towards higher temperatures and pressures, problems associated with the design Flanged end constraints on smooth curved tubes 67 and safety assessment of pipework systems have become correspondingly more complex. In the analysis of such systems the smooth curved pipe bend merits special attention. Until relatively recently most of the published analytical work has considered the bend in isolation, 1-3 usually under a pure bending moment, ignoring the influence of connections to other components in the piping system. Furthermore, there exists a considerable variance in the results of published experiments on smooth bends with end constraints. In what follows an attempt will be made to further the investigation of the effect of end constraints and to provide the pipework with some information on the behaviour of bends with rigid flange terminations. Over the last 15 years, several attempts at an analytical solution have been published, including those by Thailer and Cheng, 4 Axelrad, 5 Findlay and Spence 6 and Whatham and Thompson. 7 However, divergence between their results is apparent but comment will be reserved until later, in order that comparison with the present work can be given at the same time. THEORETICAL ANALYSIS General considerations Details of the pipe bend geometry, applied loading and associated notation are shown in Fig. 1. The bend is considered to be part of a thin toroidal shell using the shells equation given by Novozhilov. 8 The general limitations of thin shell theory apply in particular r/t > 10. Results will be given later for r/t > 5 since many 'real' bends are at, or just below, the thin shell limitation. The solution will be obtained using the theorem of minimum total potential energy on suitable kinematically admissible displacements. Displacement series formulation The displacement field is decomposed into two sets. The first set refers to 'rigid cross-section displacements' which are associated with the movement of the tube cross-section but with no change in its configuration. The second set describes 'distortion displacements' which are associated with deformation of the cross-sections relative to the particular section centre. The kinematic conditions imposed by a flange sin Geometry of the smooth pipe bend. Meridional angle e Circumferential angle Fig. I. LI .=R.I,~, ]~I R d Fig. 2. J ~ I 1 ~÷ "'v~ WR y¢=lR ( U c - .6 ~@ :) Rigid section displacements. Displacements are exaggerated for clarity. 0 / .,~ // I ' . ~. ~ profile e~ G~ oc Flanged end constraints on smooth curved tubes 69 are automatically satisfied by the rigid cross-section displacements and therefore need only be applied to the distortion displacements which considerably simplifies the displacement formulation. Rigid cross-section displacements uR, %, wR can be derived from the bend centreline (4~ = 0) displacement components, U,. and Vc,9 Fig. 2, using: r . ( c3Vc~ u~ = u , . + ~ s m o ~ u,,- ~0/ v~ = V, cos ¢ wR = Vc sin ~b (1) The following series equations: U,.= ~ F ~ ( O - G ~ ) s i n ( J ~ - ) ) j=l V~.= - X DjsinZ\2aj j= 1,2,3...,JT (2) j=l were selected for Uc and V~ since they satisfy the required boundary conditions of anti-symmetry and symmetry, respectively at 0 = 0 and those of a free end at 0 = +_a/2. Unknown coefficients Dj and Fj will be determined from the minimisation of the total potential energy. Substitution of eqn. (2) into eqn. (1) gives the rigid cross-section shell displacements as: JT =~F 1/ ~ sin ( ~ 0 - ) ) ( 1 + R s i n ~b) j=l ,IT 1~ I/jzr\ . (jrcO'~r . j=l JT 2 j~O j=l JT w. = - ~ D~sin2(jrcO'~ \ 2 ~ j sin~b j=l (3) G, Thomson, J. Spence 70 Distortion displacements uo, vo and w o are associated with deformation of the normally plane, circular cross-sections, vo and wo being ovalisation of the sections in their own plane and u o being distortion of the plane. To satisfy the conditions of a rigid flange u o, vo, w o and Own~dOare required to be zero at 0 = _ ~/2. The following series can be derived 9 using the flange conditions, together with appropriate symmetry: MT NT /'/D = , R ~ ~Amn(~len~ cOSFI¢ + Ooo~-sin.¢ m=l smi~-- ) n=2 MT NT Vo= ~ Z B m n ( - ~ e n ! s i n n ( P + ~ o n ! C ° s n ¢ m=l ) n=2 MT NT WD= ~ZCmn(OenC°sn~p+~tonSinn~p ) m=ln=2 (re:o) Hj / + 2 /jTtO\ L_, j=l Total displacement series for u, v and w can then be found from the summation of the rigid cross-section displacements and the distortion displacements. Strains Strains can now be found by substituting the total displacements into the shell strain displacement .relationships given by Novozhilov. s ira, ] Lra~ a~w] Flanged end constraints on smooth curved tubes 1 [-Ou 71 ] ,% = ~ - Lifo + v cos 4) + w sin 4) ~ s i n 4~ - ~0i- + - - r K°= 1 [~,v v- R'~,u 1 ~'o, = k T L~o - u c°s 4~ + r K°~=r~'-[_~sln~q~ e~ j a~bT~ ~ R' \c~0-usinq~/_] (5) Total potential energy The total potential energy for a bend subjected to an in-plane bending moment can be found from: V=~flf_~z[(e,4~+%)2-2(1-v)(%eO-¼724,)]rR'dOd(o C 2n a/2 + 2- - ..Io ,3-~/2 [(K,~ + Ko) 2 - 2(1 - v)(KoK ~ - K~4,)lrR'dO ddp - M 7 (6) The rotation between the ends of the bend 7 can be obtained from: (7) as: JT ~'= ~ j~ sin ( 2 ) ) + D,/(J~--n)sin ( 2 ) ) (8) j=l Non-dimensionalisation of the total potential energy is achieved using: V= V /( :M o ) (1 - v2)n~J 7o- E1 I=nr3t (9) G. Thomson, J. Spence 72 After some re-arranging this gives' 1-2 (g4, + gO)2 -- 2(1 - v)(ioi ~ - ~Y0,) = ,J - W 2 + i 2 {(k~ + ko) 2 - 2(1 - v)(KoK~-k ~ ) } 1 + Rsin 40d0dq~ JT j~ - (1 - v2)g~ j=l )~ Rt r2 R~ r~ R g----- e - r)~0 K (10) Note that the integration limits have been changed to make use of the symmetry. The displacement coefficients are non-dimensionalised using: ,4,, = Am, r~~° /~,, = Bm, r7~° 1 = Dr RTo Solution Om =Cm, rTo 1 1 RTo = H~ RT0 technique Equation (10) was solved for the displacement coefficients using numerical integration and minimisation on a computer. Numerical integration was performed using Simpson's rule in two dimensions. In general 17 and 9 integration points were found to be necessary in the meridional (~b) and circumferential (0) directions, respectively, to give the optimum between high accuracy and computer solution time. Minimisation was performed using the method outlined in the Appendix. The Fortran program takes less than 3rain to run on an ICL 2980 computer for a fully converged solution. Flexibility factors for all bend geometries were obtained from: JT ?, " - ,i~z . jTz j=l which represents the ratio of the end rotation of a bend to the rotation Flanged end constraints on smooth curved tubes 73 between the ends of a similar length (1 = Rot) of straight pipe under the same loading conditions. Stress factors were found in the form: where (Mr/l) is the peak stress in a straight pipe subjected to the bending moment, M. The meridional (64;) and circumferential (6o) stress factors were obtained from: 6ea= (g4~+vg°)-t-2 (Kea+ vK°) ( 1 - v z) 6o= (go+ rico) +_-2(Ko+ vK~) (1 where + and respectively. - v 2) refer to the outer and inner surfaces of the shell, Convergence Convergence of the displacement series was examined over a range of the major characterising parameters and the total number of terms found to give satisfactory convergence for each of the coefficient subscripts j, m and n was JT = 5, MT = 5 and NT = 6, respectively. This requires the use of 105 displacement coefficients and is suitable for 2 >__0.05, R/r < 10 and < 180 °. Although reasonable convergence could be obtained by fewer coefficients when parameters were inside the above ranges, this number was used for all results presented herein. RESULTS Flexibility factors Flexibility factors for flanged bends under in-plane bending obtained from the present theory are given in Fig. 3(a), (b) and (c) for bend angles of 180 o, 90 o and 45 °, respectively. The Figures show that flanged bends with smaller subtended angles and shorter radius have the lowest flexibility. The general trend is that shorter length bends have lower flexibility. Note G. Thomson, J. Spence 74 A L ~ ~ND ANGLE = 180 j los' " %~ io .~ ' .; ' ' B 2o BEND ANGLE= 90' o~ 10 R/r " 2 05 .1 .2 '5 1 C i 2l , ' 1 :'05 Fig. 3. r /10 i i i /5 ~'1" i 3 '1 "2 PIPE FACTOR (X) BEND , I p * , , AN~LE--~.5" "5 1 Flexibility factors for flanged bends. that these results indicate a significant variation not only with the pipe factor, 2 as in the Karman 1 case without end constraints, but also with bend angle, ~, and radius ratio, R/r. Furthermore, the present results cannot be approximated by straight lines on a log-log plot as was the case with the Karman solution. Figure 4 shows a comparison of present theory with the earlier work of Thailer and Cheng 4 and Findlay and Spence. 6 Thailer and Cheng only published results for ~ = 180 ° and R/r = 3 but these are considerably higher than the present work. A close examination for the Thailer and Cheng theory reveals that although they assumed the shear strain in the bend to be zero they did not enforce this condition on their displacements. Experimentation with the present theory has shown that had they done so they would have achieved results closer to the present ones. The work of Findlay and Spence (one of Flanged end constraints on smooth curved tubes , , . . . . , i RX:3IO , , i , , P~'esL~" Theory Whatham & Thompson ...... --.-.... ..... [] r ~ 0C:180oJ ~ . . , 75 A~etrad Finley & Sotnce ThaiIer & Che~g Finite Element Results ~ Ftanged Bends E. ~ I '05 I I i I ] -1 ,,2 i i I "5 h i I I I 1 Pipe Factor (,~) Fig. 4. Comparison of theoretical flexibilities. the current authors) has recently been re-examined and has been shown to be inadequate in certain respects. Experimentation with the present theory has shown that the main problem with the results in the work of Findlay and Spence 6 was that they assumed that the circumferential distortion displacement was insignificant and could be neglected. Although this displacement is generally small, its contribution to the shear strain is, in fact, significant. Note that by neglecting u o, Findlay and Spence's results show no variation with radius ratio. Figure 4 compares the present theory with flexibility factors calculated from a formula given by Axelrad 5 and with those given by Whatham and Thompson. 7 The results of Axelrad are approximate, being for only one term in a series, and are more valid for higher ). values. However, his results show a similar trend to the current ones, particularly for higher 2 and R / r values. The absolute values are different, probably mainly due to a number of assumptions employed by Axelrad, most notably that of zero shear strain. Whatham and Thompson's results compare favourably with the present theory over a wide range of parameters. Generally, their flexibility factors are slightly higher than those of the present theory. Some of this difference is probably due to the present results being a lower bound. If the (1 - v2) term in the strain energy of the present method is G. Thomson, J. Spence 76 20 'o ' ~ND'AN&E-180 £ " 10 5 o o ~2 u ,,< 1 '~' 20 | , I I , , ! ! I I •1 -2 , I I , , i : i , , , , ; I .5 p , , , BEND ANGLE = 90 ° 10 tt t, + -- ,e. 1 PIPE 2 R/r 3 4, ~; o ]~ • + '" - - - Fig. 5. -~ t I tl I •2 FACTOR t I I "5 Present Expt. Vigness Oet Buono Findlay Whatham R/r=3 t Theory Par'due & Vissat & Rh'=2 Comparison of theory and experiment. neglected (as it has been by some investigators to ensure correct asymptotic behaviour at high 2), the results are much closer to those of Whatham. However, this term should be included for consistency. Comparison of flexibility factors with experiment Comparisons of the flexibility factors from the present theory with various experimental data are given in Fig. 5. It can be seen that there exists a considerable spread in the results. The most comprehensive set of published tests are those of Pardue and Vigness 1°'11 who tested nine bends with different pipe factors. As some of their early experiments indicated that different loadings would give Flanged end constraints on smooth curved tubes 77 different flexibilities they presented 'averaged' factors, together with a range of the extremes, making direct comparison slightly difficult. Furthermore, the relatively small thickness of the flanges they adopted for the tests (cut from ½in thick plate and soldered on to bends which already had tangent pipes) must cast some doubt on their comparability with the present theory which assumes rigid flanges. Vissot and Del-Buono have published results for eight flanged 180 ° bends. 12 However, they adopted a rather unusual definition for flexibility factor which included the flexibility of connected straight pipes. Some of their results are slightly odd, since the same bends without flanges gave lower flexibilities. Generally, their results are slightly higher than the present theory. Findlay and Spence provided experimental results from three flanged bends, two of which had 'adjustable' flanges. The factor for the normal 90 ° welded flange bend compares favourably with the present theory, as does the 180 ° results with the adjustable flanges. A higher result was obtained for the 90 o bend using the adjustable flanges, possibly because the circumferential distortion displacement, which is higher for lower angle bends, was not inhibited experimentally by the adjustable arrangement. Whatham 13 presented results for two 90 ° bends which show good comparison with the present theory. In the current programme two bends with flanges were tested and measured flexibilities are shown in Fig. 5. The 90 ° bend exhibited a higher flexibility than the theory predicted. This may have been due to the difficulty experienced during the test of measuring the comparatively small bend rotation. Good comparison with the theory was obtained from the more flexible 180 o bend. Stress factors In a pipe bend with end constraints, the stress factors vary in the meridional direction, circumferential directions and through the thickness, making it difficult to present a comprehensive set of results. The problem is further aggravated by the maximum stress not being at the same bend position for all bend geometries. Only typical results for selected geometries can be given here. For a more complete set of results refer to either reference 9 or reference 14. The latter is largely based on the present theory. QC 1¢ ~ 105 2 - - 2C b~ i . . . . , i i i 2 ".i , i , I .1 _ R/r J, ~L~0 5 .2' 10\ /5 R/r i ~ "2 /5 R/r , . . . . . , i • ~ • • i ' .; ' " 90° , i • i o BEND ANGLE = ~,5- ' BEND ANGLE= , i BEND ANGLE = 180° i 4 -2 "5 PIPE FACTOR (X) Fig. 6. M c r i d i o n a l stress factors. - , i 10 - - - z - - _ ~ 2(] , i 1 I ] •05 = , , , • -1 • R/r •S BEND ANGLE = 180 ° -z' ' F i g . 7. Circumferential ' .~ '" Z ~ stress factors at 0 = 0 ° PIPE FACTOR (X) 2~ B E N D ~ A N G L E ~ - '"~- %201 . . . . . . . . . . . . . / BEND ANGLE= 90° 1 2 5 1(] 20 t~ @ C~ Flanged end constraints on smooth curved tubes 20 , 79 . . . . BEND ANGLE = 1800 10 5 1-os _ 20 , , , .1 -i , i i . . -s. . . . , • • ' • , , BEN0 ANGLE = 90° rv- 10 o i-- ~s R,r ? t~ ~ .... I " 21- .i .~ 10 ~ R~ ~._~.--- - ~ PIPE Fig. 8. ' ' .; '" _ -BEN0-ANfiLE --.~__~ t FA£TOR (X) Circumferential stress factors at 0 = + c(/2. Figure 6 shows the maximum meridional stress factors for bend angles of 180 °, 90 ° and 45 °. The m a x i m u m meridional stress always occurs at the bend centre (0 = 0 °) but can be close to either the pipe centre (4, = 0 °) or the intrados (~b = - 9 0 ° ) , depending on the values of R/r, ~ and 2. Generally, bends of short length, i.e. low R/r and ~, are more likely to have a maximum at the intrados. Figures 7 and 8 give the peak circumferential stress factors at the bend centre (0 = 0 °) and at the flange 0 = + ~/2 for bend angles of 180 °, 90 ° and 45 °. The maximum circumferential stress can occur at almost any meridional position at either of these sections. Finite element results To complement the theory, some results for flanged bends have been attained using the finite element method. Figure 9 shows one of the G. Thomson, J. Spence 80 Fig. 9. Finite e l e m e n t m o d e l (using s y m m e t r i e s at 0 = 0 ° a n d ~b = + 90 °). models used to obtain the results. The models were run using the eight noded parabolic, six degree of freedom, isoparametric shell element (NSTIF = 7, I O R D E R = 2) with the SUPERB finite element code on the VAX 11/750 computer at Ferranti. As the mesh in Fig. 9 shows, the flanges were modelled by turning the shell elements on their sides. Experimentation with the models demonstrated the sensitivity of the bend flexibility factor and stresses to what might normally be thought of as reasonably thick flanges. For R = 9 in, r = 3 in, t = 0-28 in and ~ = 90 °, a flange thickness of 1 in gave a maximum meridional stress factor of 3.7 whereas a thickness of 4 in gave a factor of 2.4. This behaviour may help explain the wide variation in the results obtained by the early experiments. Figure 4 includes the flexibility factors given by the finite element models. The results are higher than both Whatham's and the present theory but show a similar trend. The higher results from the finite element model are probably mainly due to the 'non-rigid' flanges. CONCLUSIONS Both theory and experiment demonstrate that the incorporation of flanges on the ends of a bend causes a significant reduction in its inherent flexibility. The flexibility factor reduces with bend angle and radius ratio Flanged end constraints on smooth curved tubes 81 and increases as pipe factor reduces. The results exhibit considerable sensitivity to the radius ratio, unlike unflanged bends. Agreement of results from the present theory, Whatham's theory, the finite element models and the various experiments are affected somewhat by the flange boundary conditions. The flexibility factors from the theory presented herein provide a useful lower bound with the assumption of completely rigid flanges. The effect of flange flexibility on the bend stresses is more complex. Generally, the results presented provide a reasonable indication of the likely peak stresses, particularly at the bend centre, 0 = 0 °. APPENDIX: MINIMISATION The total potential energy (TPE) expression for linear elasticity is a quadratic function of the displacement coefficients. Therefore, when the T P E is differentiated with respect to each of the unknown coefficients, a set of linear equations is obtained. Since the displacement coefficients are not functions of the bend coordinates, integration of the T P E can be performed without values for the coefficients. Using these principles, together with the following procedure, provides a fast and efficient method of evaluating the coefficient. The method determines the matrices for the linear equations from the T P E function without any hand manipulation or integration. The simplest way to explain the technique is with an example. Consider the following typical T P E quadratic expression: V = ~ ( a l x2 q-a2x22 + a 3 x l x 2 + a 4 x l + a s x 2 d- a6)d~ where al, a2, a3, a4, a 5 and a 6 are functions of ~ and xx and x 2 are unknown coefficients. If all coefficients are given the value of 0 (x 1 = 0, x 2 = 0) then evaluation of V gives: a 6 d~ (A(1)) If each coefficient is given the value of 1 with the rest O, then evaluating V gives: ~(a 1 + a 4 + a 6 ) d ~ x 1= l,x 2=0 j" (a 2 + a 5 + a6) de x 1 = 0, x 2 = 1 (A(2)) 82 G. Thomson, J. Spence Similarly, making each coefficient the value negative 1 with the rest 0 gives: j'(a l - a 4 +a6)d~ X1 = - l,x2 = 0 (a 2 - a 5+a6)d~ x 1=0, x2=-1 (A(3)) Adding eqn. (A(2)) to the corresponding eqn. (A(3)), and subtracting twice, eqn. (A(1)) gives, respectively: 2yald¢ 2j"a 2 d~ (A(4)) which form the diagonal terms of the linear equation matrix. Subtracting eqn. (A(2)) from eqn. (A(3)) and dividing by two gives: -ja4d~ - f a s d~ (A(5)) which are the terms of the vector on the right-hand side of the matrix equation. Giving one coefficient the value of 1 and then evaluating V with one of the remainder, given the value of 1, the rest 0, gives: S(a1+a2+a3+a4+as+a6)d~ xl=l, x2=l (A(6)) The off-diagonal terms of the matrix are then found by subtracting eqn. (A(1)) and half of the relevant equations (A(4)) from (A(6)) and adding the relevant equations (A(5)) giving: S a 3 d~ (A(7)) Thus, the complete matrix equation can be formulated as: 2~a I d~ j'a3 d~ E j~a3d~lFx , q = [ - y a 4d~] 2j" a2 d{ ][_x2.] - S a s d ~ which can easily be solved to give the unknown coefficients using a standard matrix solution technique such as the Gauss algorithm. Further details of the technique, including a program, can be found in reference 9. REFERENCES 1. Karman, Von T. Uber die Formanderung Dunnwandiger Rohre insbesonders federner Ausgleichrohre. Zeits V.D.I., 55 (1911), pp. 1889-95. Flanged end constraints on smooth curved tubes 83 2. Rodabaugh, E. C. and George, H. H. Effect of internal pressure on flexibility and stress intensification factors of curved pipes or welding elbows. Trans. ASME, 79 (1957), pp. 939-48. 3. Spence, J. On the bounding of pipe bend flexibility factors. Nuc. Eng. Design, 12(1) (1970), pp. 39-47. 4. Thailer, H. J. and Cheng, D. H. In-plane bending of a U-shaped circular tube with end constraints. J. Eng. Industry, 92(4) (1970), pp. 792-6. 5. Axelrad, E. Flexible shell theory and buckling of shells and tubes. Ing. Archiv. (1978), pp. 95-104. 6. Findlay, G. E. and Spence, J. Flexibility of smooth circular curved tubes with flanged end constraints. Int. J. Pres. Ves. and Piping (1979). 7. Whatham, J. F. and Thompson, J. J., The bending and pressurising of pipe bends with flanged tangents. Nuc. Eng. and Design (1979), pp. 17-28. 8. Novozhilov, V. V., Thin shell theory, (Trans Ed. Lowe, P. G.). P. Noordhoff Ltd., Netherlands 1964. 9. Thomson, G. The influence oj end constraints on pipe bends. Doctoral Dissertation, University of Strathclyde, 1980. 10. Pardue, T. E. and Vigness, I. Characteristics of pipe bends under applied moments. Nat'al Research Lab. Rep. No. 111292, Dec., 1953. 11. Pardue, T. E. and Vigness, I. Properties of thin walled curved tubes of short bend radius. Trans. ASME, 73 (1951), pp. 77-84. 12. Vissat, P. L. and Del Buono, A. J. In-plane bending properties of welding elbows. Trans. ASME, 77 (1955), p. 161. 13. Whatham, J. F. In-plane bending of flanged pipe elbows. MetalStruct. Conf. Inst. oJ Eng., Perth, Australia, 1978. 14. ESDU, Flexibilities oj and stresses in thin unpressurised pipe bends with flanged ends under in-plane bending: Influence of bend angle. Item No. 81041, Nov. 1981. 15. Thomson, G. and Spence, J. The influence of end constraints on smooth pipe bends. 6th Int. Conj~ Struct. Mech. Reactor Tech., Paris, 1981.