IN-PLANE BENDING OF SINGLE, UNREINFORCED MITRED PIPE BENDS R. KITCMNG Department of Mechanical Engineering, University of Manchester Institute of Science and Technology W. J. THOMPSON Department of Mechanical Engineering, University of Manchester Znstitute of Science and Technology Measurements of strain, transverse deflection, and change of radius have been taken on single unreinforced pipe bends subjected to in-plane bending moments. Specimens having three different mitre angles for each of two sizes of pipe have been used. Observations were made with different combinations of leg lengths for each specimen. Stresses and flexibilities are compared with those calculated by the formulae given in the A.S.A. Piping Code B.31.1 and the concept of equivalent smooth bend is examined. Alternative methods of assessment are considered and improved methods are suggested. INTRODUCTION PIPINGSYSTEMS often include mitred pipe bends of both the multi- and single-mitre types. A multi-mitre bend usually acts as a substitute for a smooth bend and consists of a number of closely spaced mitred joints (I)*, while a single-mitre bend is formed when two pipes, having their axes inclined, are welded together. The joint so formed may or may not be reinforced. The present investigation concerns single unreinforced mitred joints. At any point in a pipe line the internal forces may have normal or shear components, the bending moments may have components which bend the pipe in, or perpendicular to, the plane containing the pipe axes, and twisting moments may also exist. The forces and moments depend on the loading system upon the pipe run, and on its capability of absorbing thermal movements, which in turn depends on its flexibility. The type of loading considered here is that of in-plane bending only. When designing piping systems which incorporate mitred bends, it is important to know the flexibility of the pipe in that vicinity as well as the stresses which would arise because of different types of load. The experimental information available on the in-plane bending of single unreinforced mitres is small (2)-(4); analytical stress and flexibility analyses are few, and necessarily approximate. The behaviour of a single mitred bend under in-plane bending is similar to that of a smooth bend since flattening of the pipe section occurs (Fig. 1) and this gives rise to a greater flexibility than that which would be predicted by simple-bending theory. The flattening produces stresses higher than simple bending stresses, and a further increase of stress arises because of the sharp discontinuity at the joint. Second moment of area of pipe section about a diameter. Flexibility factor for smooth bend = Flexural rigidity (EZ)of straight length of [pipe calculated by simple bending theory 1 1 Flexural rigidity of same length of pipe, having same diameter and thickness, but forming a smooth bend [ Flexibility factor for equivalent smooth bend. Variable flexibility factor for any section along pipe. Value of Klat mitred joint. Value of KOfor bend angle 2/?. p + p + (14)1/211/2. The M S . of this paper was received at the Institution of Mechanical Engineers on 22nd M ay 1969 and accepted for publication on 25th August 1969. 33 * References are given in the Appendix. Applied bending moment. RE/rfor a 90" mitre bend. Smooth-bend radius. Equivalent smooth-bend radius. Mean pipe radius. Stress-intensification factor for smooth bend. Stress-intensification factor for equivalent smooth bend. Pipe thickness. Leg lengths. Distance of pipe section from mitred joint. Mitre angle, see Fig. 1. (k-A)/2. Bend angle, see Fig. 1. Vertical deflection at end of pipe. Rotation at end of pipe. Cylindrical co-ordinate defining position on pipe. [3(1-~')r~/t~]~/~. Pipe factor for smooth bend. Pipe factor for equivalent smooth bend. Poisson's ratio. Normal stress at a point. Nominal maximum simple bending stress (= M / n r 2 t ) . 14 JOURNAL OF STRAIN ANALYSIS Notation C,C1,C' Constants. E Modulus of elasticity. Downloaded from sdj.sagepub.com at RYERSON UNIV on June 19, 2015 VOL 5 NO I 1970 IN-PLANE BENDING OF SINGLE, UNREINFORCED MITRED PIPE BENDS DESIGN CODE (see Figs l and 2), and where a = 45", RE is equal to r. For a smooth bend a useful non-dimensional quantity known as the pipe factor h is frequently used and is defined as Rt A=. . r2 where R is the smooth-bend radius. For the equivalent smooth bend the pipe factor is given by (1+cot a)t A, = . . . * (3) 2r For pipe-flexibility calculations, the length of pipe forming a smooth bend is assumed to have a uniform flexural rigidity of EIIK, where EI is the flexural rigidity of a straight pipe of the same diameter and thickness; K is known as the flexibility factor of the bend and will be greater than unity because of the flattening effect (6) (7). For a smooth bend the code recommends the use of the formula X I Because of the similarity of mitre bends and smooth bends (for which analytical expressions are well established), the A.S.A. Piping Code (5) adopts the procedure of assuming that a mitre bend can be replaced by an equivalent smooth bend of radius RE for the purposes of calculating flexibilities and stresses. In the code RE is given by . Equivalent , smooth b e n d Fig. 2. Notation: single-mitre bend and equivalent smooth bend If the maximum stress to be used in the design calculations is u, which is based on an analysis by Clark and Reissner (8). For a smooth bend equivalent to a single mitre (Fig. 2) it is modified to the following empirical formula (5) 1.52 K -* . . * (5) - * The stress-intensification factor for a smooth bend is given in the code as 0.9 S = -h2'3 * ' * * (6) This is expressed with reference to the nominal simple bending stress [u, = (M/T#C)]which would be calculated for a straight pipe subjected to a bending moment of &f. u = sun For a smooth bend the formula for Sis exactly half of the elastic-stress factor given by Clark and Reissner (8). The reason for this is that S is a stress-intensification factor based on fatigue considerations (2). Elastic stresses present in commercial piping with welds and rigid attachments are approximately twice those which would occur in plain lengths of non-fabricated pipes. Design stresses in the code apply to commercial piping and allow for welds and rigid attachments, for which elastic stress-intensification factors of 2 (with reference to non-fabricated pipes) are already assumed to exist. Since design stresses are based on the assumption that the factor of 2 is already present, the stress-intensification factor for the design of smooth bends is only required to be half of the true theoretical elastic-stress factor, as given by Clark and Reissner, viz. p-i u D e f l e c t e d shape at s e c t i o n near junction For a single-mitred bend, and again using the concept of the equivalent smooth bend by modifying equation ( 6 ) for smooth bends, the code gives the following empirical formula for the stress-intensification factor This implies that the true elastic stress factor should be L M Fig. 1. Flattening of section under in-plane bending JOURNAL O F STRAIN ANALYSIS VOL 5 N O I 1970 Downloaded from sdj.sagepub.com at RYERSON UNIV on June 19, 2015 15 R. KITCHING AND W. J. THOMPSON AIM O F INVESTIGATION The object of the present investigation was to test the design formulae for flexibility and stress-intensification factors for a single-mitre bend, and to examine the concept of ‘equivalent smooth bend’ in the same context. Horizontal leg TEST RIG AND SPECIMENS The specimens were manufactured from aluminium-alloy tube to B.S. 1474 N.V.5 (9). In all, six specimens were manufactured. Details are given in Table 1, where the notation of Figs 1 and 2 is used. Each specimen was clamped at end A and loaded Table I Pipe specimen No. ;;; 1 1 1;; I ~~ 1 2 3 4 5 Mean Thickness, E, lbfx 1O6/in2 pipe in radius, in 28, degrees 90 1 3.250 p 3 5 1 2.188 I I 0.138 0.125 ~ . I I 10.5 10.3 V Denotes a knife-edge support LXI D e n o t e s a l o a d c e l l Fig. 3. Loading system ~ 1 ~ 0.36 0.33 through a screwjack and a suitable lever system with a pure in-plane bending moment at the end B, see Fig. 3. T h e load in each of the levers connected directly to the specimen was measured by a C-type dynamometer incorporating a simple dial gauge. Fig. 4. Photograph showing dejlection gauge positions 16 JOURNAL O F STRAIN ANALYSIS Downloaded from sdj.sagepub.com at RYERSON UNIV on June 19, 2015 VOL 5 NO I 1970 IN-PLANE BENDING OF SINGLE, UNREINFORCED MITRED PIPE BENDS Deflections and strains (see later for positions) were measured as the moment was applied in increments and then reduced to zero in steps. The maximum bending moment for all specimens was 5650 Ib in and the increments 942 Ib in. All strain gauges, dial gauges, and transducers were read during the first load cycle and the zero readings were checked. The maximum bending moment was such that all gauges behaved linearly. Deflection gauges were mounted as shown in Fig, 4 so that diameter changes and transverse movements of the pipe centre-line could be measured. In specimens 1, 2, and 3 the gauges were mounted on both legs of the specimen but in the other specimens they were confined to the horizontal leg, with a minimum of additional check gauges. The transducer probes (Mercer Parnum) were positioned close to the mitred joint where measurements were required at smaller pitch and where there was little room for dial gauges. Strain measurements were taken by means of electricalresistance foil gauges of in gauge length, in conjunction with a direct-strain measuring set. Past results (I) (3) have shown that maximum stresses in single- and multi-mitred bends occur at the junction but within the well defined region 90" < 0 < 135" (see Fig. l), and that they are in directions normal to the plane of the junction (a plane of symmetry). Strain gauges were therefore attached to the outer surfaces of the pipe in this region only. They were mounted in pairs and as close as possible to the weld. Some check gauges were used on the opposite side of the pipe but in equivalent positions. As a check on the nominal stress, pairs of gauges were mounted on the outside surface of the pipe at a section remote from the mitred joint and also from the ends of the specimen. The latter gauge readings confirmed values predicted by simple-bending theory. Readings of the strain gauges and of deflection gauges, disposed in a similar manner to that already described, were taken after the horizontal leg length (dimension X in Fig. 2) of the specimen had been shortened by 8 in. In-plane bending moments were applied as before. Further tests were carried out for the specimen with dimension X having a number of different values, the smallest being 6 in (see Table 2). Dimension Y remained the same throughout. The same procedure was adopted for all six specimens, the details of the different leg lengths X for each specimen being shown in Table 2. T o facilitate the continued removal, modification, and repositioning of a specimen in the rig, the ends of it were + fitted over carefully machined spigots, and were then secured in position by split clamps. RESULTS General observations The maximum applied moment (5650 lb in) for all deflections and stresses which are quoted was the same for all specimens, and all measurements were linear for the range zero to maximum load. A limited number of tests were carried out beyond this load and the tendency was for the flexibility of specimens 4,5, and 6 to become less at the higher moments, which were in a direction corresponding to the decrease of the angle 2p. The distributions of deflections and strains are quoted for a nominal bending moment of 5650 lb in, and, since this was in the linear range, they were computed in each case by plotting load-deflection or load-strain graphs and then using the best straight line to obtain an average. The elastic constants E and Y for the material of the pipes were determined by using tensile test pieces cut from the pipe walls and also by performing compression tests on some of the short tubular pieces cut from each of the specimens. Little scatter was registered and the results in tension agreed with those in compression. They are given, together with the relevant dimensions, in Table 1. Deflections and stresses Where possible dial gauges were arranged in pairs across a diameter and were used to obtain both transverse deflections of the pipe axis and the radial flattening of the pipe section. The latter was calculated by assuming that the radial deformation at the position 8 = 0" was the same as that diametrically opposite at 8 = 180". It is appreciated that at positions near the mitre this procedure is not necessarily justified but it was adopted in the absence of an alternative. In positions close to the joint it was impossible (see Fig. 4) to position pairs of gauges across a diameter so the radial deflections in that vicinity have been based on single-gauge readings. Each experimental point shown in Fig. 5 represents the transverse deflection at end B for one test length (X)used in the series of tests on specimen 1. For clarity the deflections at other points along the leg have been omitted, H O R I Z O N T A L DISTANCE F R O M JOINT-inches Table 2 Y,in I I Leg length X, in Test No. 46.0 38.0 30.0 22.0 14.0 10.0 6.0 51.50 42.25 34.25 24.25 26.25 _.-. 16.25 18.25 8.25 10.25 4.25 6.25 48.75 40.25 32.25 ~ 1 1 1 I I I 49.5 41.5 33.5 25.5 17.5 46.0 38.0 30.0 22.0 13.5 9.5 51.25 42.25 34.25 26.25 18.25 14.25 10.25 JOURNAL OF STRAIN ANALYSIS VOL 5 NO 17- L7_j_o1 Fig. 5. Transverse dejections in specimen 1 I 1970 Downloaded from sdj.sagepub.com at RYERSON UNIV on June 19, 2015 17 R. KITCHING AND W. J. THOMPSON but it was observed that at any such point the deflection was the same for all possible leg lengths which included that point, except for the smallest leg length. Figs 6 and 7 show experimental points plotted in the same way for specimens 2 and 3. Radial deformations at sections are plotted in Fig. 8 for the horizontal leg of specimen 1 across diameter 0 = 0" to 180". I t should be noted that in Fig. 8 the position of a section has been defined by its distance from the extremity, G, of the horizontal leg (see Fig. 2) and not from the join of the pipe centre lines, J, which is indicated in Fig. 8. Figs 9 and 10 show radial deformations for HORIZONTAL DISTANCE FROM JOINT-inches 60 50 40 30 20 10 I I I 1 I I 00 A Calculated w i l h code specimens 2 and 3. Fig. 10 also gives radial deformations for the inclined leg (diameters 0 = 90" to 270" and fl= 0" to 180"). Edge stresses, derived from the measured strains, are plotted against angular positions 0 for specimen 1 in Fig. 11 and in Figs 12 and 13 for specimens 2 and 3. Stresses have been shown non-dimensionally as stress ratios where Stress ratio = (Stress)/(Maximum simplebending stress u,). The different distributions are for different horizontal leg lengths indicated in the key. (Some test lengths, see Table 2, have been omitted for clarity.) Similar, although less detailed, information is given for specimens 4, 5, and 6 in Tables 3 and 4. Distribution patterns of transverse deflection, radial deformation, and stress were found to be similar to those for specimens 1, 2, and 3 respectively. Table 3 Specimen I /.' , 1 ~ ~ ~ d d p l l p number r e p r e s e n t e d b C l i Experiment 1 1 2 3 60 I 6 ----r Modified code (equation (13)) I I 0.620 0.722 I I 0.873 . _ - 0.765 0.857 - 0.187 0.191 0.204 0.621 0.610 0.700 Table 4 HORIZONTAL DI STANCE FROM JOINT -i n c h es 50 40 30 T 1 Code 0.275 0.550 I 5 Fig. 6. Transverse deflections in specimen 2 0.210 0.220 0.245 I 4 I End deflection, in ~ Specimen -~ 2s -r Stress-intensification factor Calculated w i t h c o d e [equations (l1,(31,(511 (equation 1 2 3 I - 04 I 1 I ';;1 23.5 135" :ol 19.0 10.0 1 :;:': 10.2 I p_______ 1 1 1 --(from 9.4 (L) 7.6 (L) 7.1 (L) ~ 6.2 (L) 12.8 5.6 (L) 8.30 9.4 8.23 6.9 8.1 5.31 (10)) Fig. 7. Transverse deflections in specimen 3 25 HORIZONTAL DISTANCE FROM J O I N T ( M E A S U R E D FROM P O I N T GI -inches 20 15 10 5 0 Fig. 8. Radial deformations in specimen I 18 JOURNAL O F STRAIN ANALYSIS Downloaded from sdj.sagepub.com at RYERSON UNIV on June 19, 2015 VOL -j NO I 1970 IN-PLANE BENDING OF SINGLE, UNREINFORCED MITRED PIPE BENDS H O R I Z O N T A L D I S T A N C E F R O M J O I N T ( M E A S U R E O FROM P O I N T GI-inches 20 IS 10 5 25 0 c u E ._ I z 0 -5 F 4 I e 0 -10 tn: 2 -I5 P 2 rr x IO-~ F%. 9. Radial deformations in specimen 2 H O R I Z O N T A L D I S T A N C E FROM J O I N T ( M E A S U R E 0 FROM P O I N T G I - i n c h e s 20 25 10 15 5 L+-- r 4 . I 1 1-1 _. 0 '+0 u XIO-3 Horizontal leg - I 0"- 180" 1 G o ' b Inclined leg 10-3 Fig. 10. Radial deformations in specimen 3 @-degrees 60 50 40 I 30 60 20 I 1 Longitudinal Hoop N -2 t- 4 I 7 -6 Fig. 11. Edge stresses in specimen 1 J O U R N A L OF S T R A I N A N A L Y S I S VOL 5 N O I I970 Downloaded from sdj.sagepub.com at RYERSON UNIV on June 19, 2015 19 R. KITCHING AND W. J. THOMPSON -2 10 - 1 0 I- ," Longitudinal ~--2 m m W ---4 : m -~--6 --a Fig. 12. Edge. stresses in specimen 2 T 4 J -, 2 /L 18 Fig. 13. Edge stresses in specimen 3 DISCUSSION O F RESULTS Flattening effect Fig. 8, which shows the change of pipe radius along the horizontal leg for specimen 1, illustrates that, for leg lengths greater than 6.5 in (one diameter), neither the pattern nor the magnitude of deformations varies significantly for different leg lengths. Significant differences were only experienced when the leg lengths became about one diameter. It is clear that for the longest leg lengths the flattening effect was greatest at the joint, and persisted for a considerable length (about 4 diameters) along the pipe. For leg lengths between 6-5in and 24 in, deformation patterns are similar to those for longer leg lengths except for a region of restraint (about 3 in long) close to the end fitting where the load was applied. For the shortest leg length used (6.5 in) there was a considerable reduction in the flattening at sections close to the joint in comparison with all other leg lengths. Similar effects were noted for specimens 2 and 3 which had different mitre angles. Changes of radius near the joint were about the same as those for specimen 1 where leg lengths were greater than 6.5 in, but they appeared to 20 be marginally greatest for the 90" bend and marginally smallest for the 135" bend. Transverse deflection of horizontal leg Fig. 14a is a diagram showing the pipe axes for any specimen, JA being the horizontal leg before it has been cut. JB in Fig. 146 shows the axes after the horizontal leg has been shortened. If all local effects (e.g. restriction of flattening) changing the flexural rigidity of the pipe are ignored, the same in-plane bending moment R applied at A in Fig. 14a will produce the same vertical deflection at B as it does when applied at B in Fig. 146. This means that, for that applied moment, the deflection curve for any unshortened specimen will be the same as the locus of the end deflections of the shortened specimens. Fig. 5 shows this to be substantially so, provided that the leg length is greater than about one diameter. However, it may be expected that the end restraint provided by the attachment through which the load is applied will affect the local flexural rigidity and this would most affect the end deflection when the leg is short, with the restraint close to the joint. Figs 5 , 6 , and 7 show that this is the case. JOURNAL OF STRAIN ANALYSIS Downloaded from sdj.sagepub.com at RYERSON UNIV on June 19, 2015 VOL 5 NO I I970 IN-PLANE BENDING OF SINGLE, UNREINFORCED MITRED PIPE BENDS axis from the joint. A suitable expression for the reciprocal of the flexural rigidity would be x’ / o Uncut With equivolent smooth bend c where EI is the flexural rigidity of a straight pipe under simple bending, K1is the flexibility factor at the point, and KOis a factor for a given combination of rjt and a. The index a2 is a suitable one (I) for cylinders subjected to flattening and is given by k-A Q2 = -- 2 where r2 A4 = 3(1-v2)b k = [8+{64+A4}1’2]1‘2 22’ Cut a Original leg lengths. b Horizontal leg shortened. c With equivalent smooth bend. Fig. 14. Diagram showing pipe axes If the equivalent smooth-bend concept is assumed the configuration in Fig. 14a would be replaced by that in Fig. 14c. The preceding remarks would still apply but the geometry is significantly incorrect in the region CD, so that if vertical deflections of the horizontal leg are calculated on the assumption of an equivalent smooth bend, unreasonable results would be likely if the end of the leg is close to the tangent point C . For a small equivalent-bend radius such as that predicted by equation ( l ) , the deflections would only be unrealistic for a small portion of the pipe layout CD. However, Fig. 5 indicates that equation ( l ) , in conjunction with equations ( 3 ) and (9, predicts deflections of the pipe which are too high, and this is confirmed for specimens 4 , 5 , and 6 (Table 3), though not for specimens 2 and 3 (see Figs 6 and 7 ) . When the code gives too high a deflection, pipes have been assumed more flexible than in practice they are and this means that pipe reactions (and hence stresses) at vessels will be underestimated. Previous workers (3) (10) have also reported over-estimation of the flexibility factor by equation (5) and it has been suggested (3) that the equivalent radius is better given by 2r RE = tan CI . . . The larger values of equivalent radius resulting from equation (10) would make the concept of more limited use because the region of the pipe CD would be fictitious for many practical geometries of pipe layout. It is theoretically possible to calculate an equivalent smooth-bend radius R, and its corresponding flexibility factor KEto fit any of the experimental results obtained, This was attempted but the equations to be solved are invariably ill-conditioned so that R, and KE are open to large errors. An alternative approach in the calculations is to dispense with the use of an equivalent smooth bend to replace the mitre bend, and to revert to the true configuration as in Fig. 14a. For leg lengths greater than about one diameter it is reasonable to assume that the flexibility of the pipe is dependent upon the flattening effect which decreases with the distance x’ measured along the pipe JOURNAL O F STRAIN ANALYSIS VOL 5 This expression gives the flexibility factor as unity at large values of x’, and 1 + K O at the joint. Values of KO determined from experimental data can be tabulated for different combinations of r / t and Q or it may be possible to connect them by simple formulae. Flexibility calculations for a pipe layout are quite straightforward and the transverse end deflection for the pipe specimens used in the present investigation is given by The end rotation is EI ff2 I n both expressions X and Y have been assumed greater than ar/a2. By using the experimental values for Av and the relevant dimensions of specimens 1-6, the various KOvalues have been calculated and are plotted in Fig. 15. Also shown is the KO value given by the only other available results which allow it to be calculated, those of Jones and Kitching (3). Here it could be calculated from readings of both Av and A+, and the two values agreed within 3 per cent. The result fits in well with those from the present experiments. In Fig. 15 the KOvalue for a straight pipe (28 = 18O0)is inserted as zero and the results suggest that if KO,is the value of KO for a bend angle of 28 and K o . 4 5 is that for a 90” bend, the following formula would be reasonable: I -\,,,, f = 17.5 I “90 0 120 \#\ 150 I80 28-degrees Fig. 15. Variation of KOwith CY and rjt NO I 1970 Downloaded from sdj.sagepub.com at RYERSON UNIV on June 19, 2015 21 R. KITCHING AND W. J. THOMPSON Stresses Before the horizontal leg length of specimen 1 was reduced, the stress patterns at the edge were similar to those measured by Jones and Kitching (3) for a singlemitre bend and by Lane and Rose (11) and Kitching (12) for multi-mitre bends. For the range of leg lengths used, the angular position of the maximum recorded stress ratio remained in the region 105" < 8 < 135". As the horizontal leg was shortened the maximum longitudinal stress ratio increased slightly until it reached 8.5 at a leg length of 10 in. It corresponded to an angular position of 8 = 115". Further shortening of the horizontal leg gave rise to a more rapid change of stress pattern and a further movement of the maximum longitudinal-stress position towards the crotch by 7". For the same region hoop-stress ratios formed similar patterns, which moved like those for longitudinal-stress ratios. The maximum hoop stress was always less than the longitudinal stress. Corum (13) has investigated the stresses arising in a pipe mitred to a rigid flat plate. When subjected to an inplane bending moment it was said to give maximum longitudinal stresses at '6 = 0".A mitred bend with one very short leg may be in a similar condition and it would seem that, if the leg had been further reduced in length, the stress patterns would have changed even more than they did in the present series of tests when short leg lengths were used. The foregoing comments also apply to the stress patterns for specimens 2 and 3. As the bend angle changed from 90" in specimen 1 to 135" in specimen 3 the patterns became smoother and changes stress with changes in 0 were less rapid; maximum values became less and the corresponding angular positions moved towards the crotch. Fig. 16 gives a comparison of the patterns of edgestress ratios for the three specimens when the horizontal legs had lengths corresponding to their maximum overall stresses. In Table 4 maximum stress ratios for all six specimens have been compared with those given by equation (9). Results from two other tests (3) (10) have also been included. Equation (9) is shown to over-estimate stresses in all cases but one. The experimental result for speci- OF 60 50 40 30 I I I I men 4 appears to be unduly low in comparison with the rest. It was checked a number of times, however, so it has been assumed that it was affected by the weld which was rather heavier than in the other specimens. For multi-mitred bends, in order to allow for discontinuity at the mitre, it is suggested that equation (9) should be modified to S=- 1.8C AE2'3 where C > 1. A similar consideration will be applied to single-mitre bends. For a 90" bend assume RE = nr then and When the experimental result for specimen 1 is used, the most suitable values for the constants are C = 1.316, C1 = 2.37 and n=3 From the results for specimens 2 and 3 with different bend angles a suitable general formula for stress-intensification factor for single mitres would be This gives 3r as a more realistic equivalent radius than r (the code value) for a 90" single-mitre bend. Fig. 17 and Table 4 give experimental stress-concentration factors compared with those calculated from equation (12), which, apart from specimen 4, is shown to be reasonable. If the same concept is used for other mitre angles an expression similar to equation (lo), suggested in (3), could be adopted, viz. 3r RE = - . . . (13) tan Q - 20 ///lo I I -0 60 50 40 I I I Test 30 /20 I 28 I Symbol -10 Fig. 16. Stress patterns for maximum overall stress 22 JOURNAL OF STRAIN ANALYSIS Downloaded from sdj.sagepub.com at RYERSON UNIV on June 19, 2015 VOL 5 N O I 1970 IN-PLANE BENDING OF SINGLE, UNREINFORCED MITRED PIPE BENDS REt A, = - where ;n 0 lp,.-l___, 90 150 120 28- 180 degrees Fig. 17. Variation of stress-intensification factors with a and rlt The same expression could be used to give a suitable pipe factor and hence a flexibility factor, as in the code. However, flexibility factors of multi-mitre bends are very similar to those of similar smooth bends so it is much more appropriate to use equation ( 4 ) with a modified constant. It would be logical to treat a single mitre in the same way and therefore use the expression C’ C’r tan CL K E ---=hE 3t . . (14) Deflection results from specimens l and 4 give C’ = 0.625 and 0.487, while those from (3) give C’ = 0.568. A suitable value would thus appear to be the average, which is 0.56. End deflections calculated by equations (13) and (14) for specimens 1 to 6 are given in Table 3 and are shown to be more satisfactory than those calculated by the code, equations (1) and (9,since reactions at the anchor points would not then be underestimated. CONCLUSIONS ( 1 ) The principle of using an equivalent smooth bend to replace a single-mitre bend for calculation purposes is justified, p.rovided that the length of straight pipe X between the joint and the next discontinuity is at least three diameters. The experimental measurements of transverse deflection (Figs 5-7) were consistent for all values of X except the smallest. Experimental stress- and radial-deformation patterns (Figs 8-10) only changed significantly from those for the greatest value of X when the leg length was less than 1.5 diameters, so the minimum length of three diameters specified above is considered to be very safe. (2) The equivalent smooth bend so adopted should be used in conjunction with the following expressions replacing the A.S.A. code expressions (equations (l), (5), and (8)), which tend to over-estimate flexibility and grossly over-estimate stresses in the pipe: 3r R -- tan a 0.56 KE =hE JOURNAL OF S T R A I N A N A L Y S I S NO 5 VOL I r2 (i.e. half the elastic stress-intensification factor). (3) There is no reason why the equivalent-smooth-bend concept should be used for single-mitre-bend calculations. It has limitations for short lengths of straight pipe, although for lengths less than a diameter no reliable information is at the moment available. The following procedures are recommended where the pipe length between the joint and the next discontinuity is greater than a diameter. (a) For flexibility calculations the flexibility factor Kl . is,assumed to vary with the distance x’ from the joint as below: Kl= 1+ K Oe-(42x’/r) where k-A a2 = 2 r2 A4 = 3 ( 1 - ? ) ~ , k = [8+{64+A4}”2]”2 where K o . 4 5 is the value of KO for a single-mitre bend for a limited where 28 = 90”. Appropriate values of number of r / t ratios have been determined, but a comprehensive set of experiments is required to cover a large range of rlt ratios. (b) For stress-intensification factor an appropriate formula would be - 1 sin2a+l [ 1 This is based on the sinusoidal form (Fig. 17) of the S, = 1.14 - (:)2‘3 variation of stress-intensification factor with the mitre angle a, where 2a = 7r-2p. (4) The above expressions were found to be suitable for pipes with leg lengths of more than one diameter measured from the mitre. It is suggested, however, that, in the first instance, the expressions should only be assumed valid for leg lengths greater than two diameters. In the light of further experimental work this limit could then be modified. ACKNOWLEDGEMENTS The authors thank Professor W. Johnson, Head of the Mechanical Engineering Department, University of Manchester Institute of Science and Technology, in whose labofatories the work was carried out, and the members of the laboratory and workshop staff, particularly Mr W. E. Atkinson, Mr G. Robinson, and Mr J. Davies, for the assistance they have given in the project. They are also indebted to the Directors of the British Oxygen Company-Airco Limited, who provided the specimens. APPENDIX REFERENCES (I) KITCHING, R. ‘Mitre bends subjected to in-plane bending moments’, Znt.J. mech. Sci. 1965 7, 551. 1970 Downloaded from sdj.sagepub.com at RYERSON UNIV on June 19, 2015 23 R. KITCHING AND W. J. THOMPSON MAW, A. R. C. ‘Fatigue tests of piping components’, Trans. Am. SOC.mech. Engrs 1952 74, 287. (3) JONES,N. and KITCHING, R. ‘An experimental investigation of a right-angled single unreinforced mitred-bend subjected to various bending moments’, J. Strain Analysis 1966 1,248. (4) OWEN,B. S. and EMMERSON, W.C. ‘Elastic stresses in single mitred bends’,J. mech. Engng Sci. 1963 5, 303. ( 5 ) AMERICAN STANDARDS ASSOCIATION B31.1. American Standard Code f o r pressure piping (now contained in U.S.A.S. B.31.1.0 Power Piping, 1967). (6) DEN HARTOG, J. P. Advanced strength of materials 1952 234 (McGraw-Hill Book Co. Inc., New York and London). (7) GROSS,N. ‘Experiments on short-radius pipe-bends’, Proc. Instn mech. Engrs 1952-53 lB, 465. (2) 24 (8) CLARK, R. A. and REISSNER, E. ‘Bending of curved tube’, Adv. appl. Mech. 1951 2, 93. (9) BRITISHSTANDARDS INSTITUTION B.S. 1474: 1963 Wroughr aluminium and aluminium alloys for general engineering purposes. Extruded round tube and hollow sections (London). (10)OWEN,B. S., HOLLAND, M. ,and EMMERSON, W. C . ‘Stresses in and flexibility of mitred bends and lobster-back bends’, Proc. Instn mech. Engrs 1963-64 178 (Pt 3J),70. (11) LANE, P. H. R. and ROSE,R. T. ‘Experiments on fabricated pipe bends’, Brit. Weld.J. 1961 8, 323. (12) KITCHING,R. ‘In-plane bending of a 180’ mitred pipe bend’, Int. J . mech. Sci. 1965 7, 721. (13) CORUM, J. M. ‘A theoretical and experimental investigation of the stresses in a circular cylindrical shell with an oblique edge’, Nucl. Engng Design 1966 3, 256. JOURNAL OF STRAIN ANALYSIS Downloaded from sdj.sagepub.com at RYERSON UNIV on June 19, 2015 VOL 5 NO I 1 9 7 0