444
Le~Adr
ENGINEERING: W. HOVGAARD
PRO(, N.A. S.
BENDING OF CURVED TUBES'
By WILLIAM HOVGAARD MASSACHUSUTTS INST...
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444
Le~Adr
ENGINEERING: W. HOVGAARD
PRO(, N.A. S.
BENDING OF CURVED TUBES'
By WILLIAM HOVGAARD MASSACHUSUTTS INSTITUTS O0 TmCHNoLOGY
DUPARTMINT oF NAVAL ARCHITCtURZ, MASSACHUSS INSTITUTS Or TECHNOLOGY Read before the Academy, April 29, 1930
In an earlier paper2 an account has been given of formulas which the author developed for calculating the displacements and deformations of curved tubes subject to bending by compressive end forces, as indicated in figure 1. Formulas were given also for determining the longitudinal stresses in the pipe wall. It was explained that when a curved pipe is subject to a bending couple, tending for instance to increase the curvature, the stress forces will cause a deformation of the transverse section, which becomes an oval with its minor axis in the plane of bending, see figure 2.
P0
z*
C'
OC
I~~~~~~~~~
Ad
P
Hereby the material farthest from the neutral axis is largely relieved of longitudinal stresses and the maximum values of these stresses, instead of occurring at the top and bottom of the section, as in bending of a solid bar, occur at points much nearer to the neutral axis. The result is that equilibrium between the external bending moment and the internal stress couples will not be established until the bend has taken an angular deflection much greater than in the absence of deformation of the section. Hence, also, the linear displacement of the ends of the pipe are much greater, a point which is of great importance to the engineer in the socalled expansion bends used in steam pipes to provide for expansion due to a rise in temperature. Tests made at the Massachusetts Institute of Technology in 1926 on two pipe bends, one of 41/2 in. and one of 6 in. diameter, were described and an analysis was given of these as well as certain other tests. The result was a good confirmation of the theory. Since that time the author has developed the theory further and made a number of additional full scale tests as recorded in table 1. Detailed reports of these tests, with the exception of No. 10, are given in four papers published in the Journal of Mathematics and Physics,
VOL. 16, 1930
ORIGIN OP PIPE NO.
BEND
ENGINEERING: W. HOVGAARD TABLE 1 NOMINALR 2r WHEN R DIAM. IN.
TESTED
Boston Navy Yard 41/2 1926 Walworth Mfg. Co. 6 1926 Boston Navy Yard 91/2 1928 Boston Navy Yard 14 1928 Walworth Mfg. Co. 6 1928 National Tube Co. 8 1929 National Tube Co. 10 1929 12 National Tube Co. 1929 1929 Walworth, Stone & Webster 6 Tube Turns Co. 12 1929 Now: The symbols used in this table are defined
1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
445
I
39.2 24.2 36.5 35.6 24.2 15.2 19.5 27.8 18.0 33.5
2r
3.0 4.7 4.6 5.9 4.7 4.4 3.2 4.8 4.9 1.42
K
5.24 2.07 3.19' 2.44 2.07 1.53 1.61 2.32 1.59 7.69
SHAPE
..r 5,
A
in the Appendix.
M. I. T.3 We shall here give a brief outline of the tests, together with the principal results of their analysis, and in the Appendix we give the formulas as finally adopted, accompanied by a brief explanation. The primary object of the 1926 tests was to determine the displacement Ax of the ends of the pipe bend when subject to compressive end forces. The longitudinal strains in the pipe wall were measured by Berry strain gauges and it was attempted also to measure the transverse strains by this means, but on account of the strong curvature of the pipes and the relatively large span (2") of the gauges, the result was unsatisfactory. It was decided therefore to test pipes of larger diameter. In 1928 a 91/2in. and a 14in. pipe were tested, but as it was still found difficult to make corrections for the curvature of the section, a new method of determining the transverse strains was developed. The diameters all around the section were measured with a micrometer caliper in the unstrained and the strained condition and from the changes in diameter so obtained the changes in curvature and hence the transverse strains were calculated. This method gave more consistent and in general more accurate results than obtainable by direct strain measurements. Later in 1928 these tests were supplemented by one on the lyreshaped 6in. pipe, listed as No. 2, which had been but slightly overstrained. This test is listed as No. 5 in the table. The principal objects of the 1928 tests were first, to get further confirmation of the formulas, and second, to obtain a criterion for the strength of pipe bends. For this purpose the pipes were tested to destruction, that is, until they were unable to hold the load to which the hydraulic machine was set. The point at which breakdown occurred was fairly well defined, being characterized by a decided drop in pressure in the machine and by a progressive reduction in the length of the base line OL in figure 1. Another point carefully observed was that at which the shortening of OL ceased to be proportional to the increment of the pressure P exerted
446
ENGINEERING: W. HOVGAARD
PROc. N. A. S.
by the machine. This point is here referred to as the elastic limit of the pipe as a whole. So far the tests had been made on pipes designed for moderate pressures, but in view of the increase in steam pressures during recent years, it was considered desirable to test some highpressure pipe bends, especially for the purpose of obtaining a more generally applicable strength criterion. Hence, in 1929, when further tests were made on five bends, three of these, viz., No. 6, 7 and 9, were for working pressures of 600 lb. per sq. in.
F/G. 3 /2 /r TUBE TURN
CuRvEsO rfoa
pi

2000 lb. Ob. Ca/c. /ong. ossY . A/V. Jsa Qb,, Cola. tmns ad9rOs 42 Obs tronSV 3sureSJ Eifr/ent strets Pe Iq AJd,
P

/AipM for
Re In tests No. 8, 9 and 10, a new strain meter, the Huggenberger tensometer, was tried. This instrument, of which nine were available, seemed to give fairly satisfactory results. The pipes from No. 1 to No. 9, inclusive, were bent to a radius of curvature of from three to six times the diameter of the pipe, but in No. 10, which was a socalled "TubeTurn" of 12 in. diameter, the radius of curvature was only one and onehalf times the diameter, the pipe being shaped
VOL. 16, 1930
447
ENGINEERING: W. HOVGAARD
by a special process. It was not expected that the formulas used for pipes of more easy curvature would be applicable to tubeturns, and attempts were made to develop special formulas, but it was not found possible to obtain a solution which was sufficiently simple to be of practical value. Contrary to expectations, however, the formulas used in previous analysis gave very good results in this case. The analysis of the tests led to the following general conclusions: 1. The observed displacements, Ax, of the ends of the bends relative to each other and the deflection of the point B at the top of the transverse section (Fig. 2), AYB, showed a close correspondence with the calculated values so long as the bend as a whole was within the elastic limit. 2. A good correspondence was fG. 4 found between the calculated
and observed stresses so long as the bends were well within the elastic limit, but when this point was passed, the observed maximum stresses generally exceeded the calculated values. This discrepancy is believed to be due to local flow of the material. Figure 3 shows curves for longitudinal and transverse stresses calculated by the formulas (5), (6) (8) and (11) (see the Appendix) for the 12in. TubeTurn for a load of
/2" 7aybe 7urn
Jeczhon
Top of Bend
o*
i2Istribuhon of Jlfrez'n ALges J
/
29
27. Ab lie.
NO i
59 Diav rru
Hi
20,000 pounds, which was close to 2 Lon the elastic limit of the bend. The 13 stresses at any given point are plotted along a line through the . point parallel with the neutral axis. 3. The displacements, Ax, which directly measure the elastic yielding capacity of the bends, seem to be dominated by the magnitude of the longitudinal stresses, while the transverse stresses, although generally much greater, seem to have little influence on the deflection of the pipe as a whole. This may perhaps be explained by the fact that the transverse stresses are of opposite sign on the inner and outer surface, changing sign at the middle surface. It is proposed, therefore, to use the longitudinal stress pi, calculated from formula (5) in the Appendix, as the strength criterion. It was found that so long as this stress did not exceed 20,000 lb. per sq. in., there was no appreciable permanent set in the bend. Yet it is likely that when this value is reached, local overstrains actually exist and repeated applications of the same load may in time produce a per
4484ENGINEERING: W. HOVGAARD
PRoc. N. A. S.
manent set of the whole bend. It is recommended, therefore, to design pipe bends for a calculated stress of 16,000 lb. per sq. in. A special study was made of the occurrence of plastic flow in the material. This phenomenon was directly evidenced by the appearance of strain lines (Luders' lines) on the surface of the pipes and in some cases by a movement of the pointers of the tensometers after the load had become stationary. It appears that plastic flow takes place along welldefined longitudinal zones and that the strain lines begin to appear a little after the bend as a
FIGURE 5
whole has reached its elastic limit. Figure 4 shows diagrammatically the distribution of the strain lines around the contour of the section of the 12in. Tube Turn. The strain lines were generally well defined, in some pipes showing on the bare metallic surface as rather fine lines. In other pipes, which were covered with mill scale, they were more coarse, being revealed by the cracking of the scale, as shown in figure 5. The lines were mostly diagonal, but in some cases they ran longitudinally, in others transversely. On figure 3 is shown also a curve marked p, giving the "equivalent"
VOL. 16, 1930
ENGINEERING: W. HOVGAARD
449
stress calculated by formula (13) in the Appendix. When P6 exceeds the yield point of the material, plasticity should occur according to theory. This point was probably reached in the lower part of the pipe section where p6 reached a value of 30,000 lb. per sq. in. Appendix. Formulas Used in the Analysis of the Bending of Curved Tubes Some of the following formulas, viz. (1) to (5) and (12), were given in the 1926 paper of these PROCUSDINGS. Let R = Radius of curvature of axis of pipe 2h r K
= Wall thickness = Radius of section of middle surface of pipe wal = Nondimensional coefficient 48 h2R2 + 10 to + 48 h2R2 + O4
K = E I
(1)
= Modulus of elasticity
of transverse section of pipe wall about its neutral axis M = Bending moment d4 = Angle of small arc of pipe axis Add = Change in d4& due to flexural strain Then the longitudinal strain at any point S (Fig. 2) in the pipe wall distant y from the neutral axis of the transverse section is approximately: = Moment of inertia of area
+ AAY) (y
C.,
(2)
where Ay is the deflection at S. It was found that the rate of change of angular deflection is:
Ad4,
KMR El
d4,w and
KMR = EI
As in the 1926 paper, the strain longitudinal stress pi and hence
=Eel =
el
120 r5sinO 480 h2R2 + 101,re
(4)
is regarded as a simple strain due entirely to the
KMr(.
I

6r
sixi3
24h2R2 + 54)
(5
The transverse strain produced by deformation of the section is likewise regarded as a simple strain and is determined from the calculated changes in curvature of the section. The transverse stress is found from: KMr l8r2 R cos 2 I
24h1R2 +5r4
(6)
450
ENGINEERING: W. HOVGAARD
PROC. N. A. S.
where z is the radial distance of the point under consideration from the middle surface of the pipe wall. At the outer and inner surface of the pipe z is replaced by  h, respectively. The stresses and strains as found from (5) and (6) are referred to as "calculated" to distinguish them from those found on the basis of measurements, which are referred to as "observed" stresses and strains and are denoted by the suffix (o). We distinguish simple strains, produced by a single stress, from those which result from the combined effect of two or more stresses, by placing a bar over the letter e as in (5) and (6). The observed longitudinal strains, being obtained by direct measurement, are clearly due to the combined effect of longitudinal and transverse stresses and are denoted by eol. The observed transverse strains are calculated from the measured diameters. First the radii of curvature are found on the assumption that the section is elliptic and the strain is then given by the formula:
0
h (__ 7r/ r
(7)
= 2h(
where r and r' are the radii of curvature before and after deformation, respectively. The strain so obtained is regarded as simple, that is, unaffected by the longitudinal stresses,4 so that the observed transverse stress is found simply from:
P02 = Eeo. The observed longitudinal stress if from the theory of elasticity: Pol =
E(o +m)
2
(8)
(9)
where m is Poisson's constant, and the combined strains are: e02

f01
= 6l
e2

E02
Eliminate fog from these equations, find observed longitudinal stress:
po,
=



m {
(10)
j
eo2 and substitute in
E oel + ).
(9). We find then the (1 1)
The flattening of the transverse section was characterized by the deflection AyB at the top of the section and was calculated from (4) by putting sin 0 = 1. The shortening of the base line OL, figure 1, due to the compressive end forces P, is found from: AX=
EI
jwy2ds
(12)
where the integration extends along the axis of the pipe from 0 through F to L. This formula enables the engineer to calculate the Pforces at the anchorages of the pipe, and hence the stresses in the bend, when a pipe line expands by a known amount Ax due to a rise in temperature. The condition of plasticity in twodimensional stress is expressed by:
VoL. 16, 1930
ENGINEERING: W. HOVGAARD
p12  p1P2 + p22 = p,2
451
(13)
where p. is a quantity, referred to in figure 3 as the "equivalent" stress. When this reaches the yield point of the material, plasticity should occur according to the theory. This formula is often referred to as the Hubervon MisesHencky equation, from the names of its authors, and is supported by the results of recent researches.6 1 A more detailed account will appear in the Proceedings Third Intern. Congress for Applied Mechanics, Stockholm, 1930. 2 W. Hovgaard, these PROCMUDINGS, 12, 365 (1926). 3 W. Hovgaard, M. I. T. J. M. & P.: First Paper, Vol. 6, No. 2, 1926. Second Paper, Vol. 7, No. 3, 1928. Third Paper, Vol. 7, No. 4, 1928. Fourth Paper, Vol. 8, No. 4, 1929. 4 In the analysis given in the third paper and the main body of the fourth paper (M. I. T., J. of M. and P.) all the strains, calculated as well as observed, were regarded as "combined," but as regards el, e2 and eo2 this may be open to question, and since a much simpler result is obtained by regarding these three strains as simple, this view was finally adopted. 5 Rog and Eichinger, Second Int. Congr. Appl. Mech., Zurich, 1926.