366

- UNITED STATES
EXPERIMENTAL MODEL BASIN

NAVY YARD, WASHINGTON, D.C.

THE CRITICAL EXTERNAL PRESSURE OF CYLINDRICAL TUBES

UNDER UNIFORM RADIAL AND AXIAL LOAD

BY R.VON MISES

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TRANSLATED AND ANNOTATED BY 0.F. WINDENBURG DOC iJ favs :
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as*

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TAL MO
cRIMen DEL 845
ERECTED 1898 ‘N

CONSTRUCTION AND REPAIR
NAVY DEPARTMENT

AUGUST 1933 REPORT NO. 366

Translation &

36G

THE CRITICAL EXTERNAL PRESSURE OF CYLINDRICAL TUBES
UNDER UNIFORM RADIAL AND AXIAL LOAD
by R. von Mises.

Translated and Annotated by D. F. Windenburg.

U.S. Experimental Model Basin
Navy Yard, Washington, D.C.
August, 1933. Report No. 366.

7

aK

THE CRITICAL EXTERNAL PRESSURE OF CYLINDRICAL
TUBES UNDER UNIFORM RADIAL AND AXIAL LOAD.
by R. von Mises,
Berlin
Stodola's Festschrift, Zurich, 1929, pp. 418-430.

Translated and Annotated
by D. F. Windenburg*

The magnitude of buckling pressure and of the deformetion of a cylindrical
tube closed at both ends and subjected to external pressure is calculated on the
basis of elasticity equations for thin shells. The results of the theory are com—
pared with tests on two series of models. Complete agreement is shown with regard
to the lobe formation; the problem of determining the buckling pressure has not yet
been completely solved.

More than 14 years ago, in view of the conditions prevailing with regard
to the design of fire tubes of boilers, I calculated the buckling pressure of
cylindrical tubes which, with fixed ends, are subjected to external pressure. A-
somewhat different problem is presented when a circular cylinder, closed at the
ends by means of bulkheads, is deeply submerged in water and thereby exposed to
pressure from all sides, as is approximately the case of submarines. The changes
in theory caused by the introducticn of axial load are not very extensive, but the
altered numerical ratios result in the occurrence of phenomena that necessarily
have a certain intrinsic interest, especiaily concerning the large number of waves
or bulges that becomes visible on the circumference of the cylinder at the moment
of buckling; see Fig. 6, F

the lines just indicated. The fundamental equations, which, by the way, are to be

In the following, I present a supplement to my previous publication” along
found in every text book on the theory of elasticity, and the detailed calculations
carried out in the previous article, are not repeated; reference to that article
is indicated by the symbol Z.

(Translator's Note: The translation of the previous publication, of which this
is the supplement, is given in U. S. Experimental Model Basin Report No. 309. It
should be referred to in connection with this article,]

* Assistant Physicist, U. S. Experimental Model Basin, Washington, D. C.
tyeits id) Ver, deutsch. Ingen.) 1914, p. 750.

1. DERIVATION of the FUNDAMENTAL FORMULA.

Let us consider a thin-walled, circular, cylindrical, hollow body, closed
at the ends by flat heads, and subjected to the pressure p on its shell and to
the compressive force P' on its
heads; Fig. 1.

Let the axial pressure on a unit

OTT LLL hh hhh hh hhh hhhbobacokahebed

ZA

WZ

' cross—section of the circular ring hav—
ing a diameter 2a and thickness 2h have

WLLL hh hhh hhdicdea

a value p'. If the pressure on the
heads is due to the fact that the end
surfaces are subjected to the external
pressure p, it follows from:

FIG. 1 PY = Wa?* p Tr 2a 2h p!
that Di= p B/4R 5c a cose een)

In order to take into consideration the presence of an end pressure p' the condi-
tions of equilibrium of Z must be supplemented. For those equations are incom-
plete inasmuch as only terms of the first order in stress and deformation magni—
tudes are retained in them. Products of stresses and deformations of the second
order of magnitude are omitted, with the exception of the products of p by w and
re (Z, Eq. (9)) since p, unlike all other stresses, is not of the order of
magnitude of the deformations accompanying buckling but possesses a finite value

independent of them. We must now, in an analogous manner, include also products
of p' with deformation magnitudes in our equations.

, ' i

FIG. 2

Fig. 2 shows an element of the shell in a longitudinal plane through the
axis of the tube, in the deformed condition. The generatrix, originally rectilin—
ear and parallel to the x-axis, has acquired a curvature, which, except for terms
of higher order, is measured by

2 2w
Ox"

This expression, in the case of Fig. 2, is negative. It is seen that

‘the pressure p' 2h which acts on the surface of the cross-section of the element,
produces a resultant in the w direction (radially inward, hence negative) equal to

This expression, with opposite sign, must therefore be added on the right
side of equation (9) in Z, if we would take into account a general end loading.
Substituting the value of Eq. (1) in (2) we have:

ea ote ee as ot (3)

2
We must, therefore, instead of — apr in the parenthesis on the right hand side

of Eq. (9) in Z, write
2 2w a? 22w

The same remarks apply to Eq. (9") and (III) in Z. The end loading has no further
influence on the expressions for the determination of the buckling pressure, as
long as we remain within the limits of the simplifying assumptions introduced in Z
or presupposed.

It is not difficult to follow the influence of the supplementary term,
equation (3), through to the final formula in Z.

From Eq. (11) of Z and the abbreviation a& given in Eq. (12) of Z, we have

a? aye —0c?w, a= Pt te ee te TLS (4)

(1 = effective length or frame spacing; see below, Section 2).
Therefore,

ye Si ae)
is to be substituted on the right hand side of Eq. (III') of Z in place of y(1 — n?);
h represents the number of lobes. This same substitution is valid for the last
member in the determinant (16) of Z. Since now y is the only quantity in the final
equation that contains p, and, therefore, is directly proportional to p, it follows

that: The buckling pressure of a tube, including the effect of end load, may be
obtained by multiplying the pressure determined from Z by the factor

This statement applies primarily to the complete result of the calculation
carried out in Z, Eq. (A) or (At), but it may be directly applied also to the
simplified expression (B), since the assumptions that led to B — — smallness of

x and y defined by Eq. (14) of Z — — continue to hold generally. If we write (B)
somewhat differently (by setting the expression in the parenthesis over a denomi-
nator and substituting partially for p its value from Eq. (18) of Z), we get from
the foregoing expression:

y- =P (ay x

apreinn Vie roa [at satya nt + pe. - (6)
- —

a4 Goo
m 2
where

peafi+asoe] [2+G-mp] = 141.650 + 0.45508
a= (1 - fit (r+ zee - (1 - 07) + PE ep?!

sage 1. 3¢ 1-392 — 1.417 3 + 0.50707

Here as in Z

2 2
x= aon, y=pa/tehtat, ie a rr)

and E and 60 are the elastic constants (1 = 0.3). The determination of y from
Fq. (6) is not particularly cumbersome once the two functions of (, Pu and fz,
have been computed, and placed in the form of tables or curves for P, and py .-
Fig. 3 contains the two curves for the entire range = 0 to 1 sufficiently ade—
quate for all practical cases.

[Translator's Note: The error noted in equation (B) of Z which was carried
through to the final formula (D) in Z has been corrected in the derivation of
formula (6). Hence, formula (6) is correct despite the error in the original
equation (B) in Z.]

FIG. 3. Auxiliary Functions yp, and p, for
Use in Computations by Formula (6).

The further simplification which was effected in Z by the assumption of
only small values of p cannot be used here since the ratio of the radius to the
frame spacing is sometimes too large. On the other hand, in most calculations of
submarine hulls we have to do with a larger number of lobes, n. It is possible,
therefore, if n equals at least 8 or 10, to neglect the last two terms in the
‘prackets of Eq. (6) in comparison with the first term (n? + a*)?. In the same way,
we may neglect 1 in the denominators in comparison with n*, by which the error made
by the previous assumption is partly compensated. We have, then, with sufficient—
ly close approximation for all cases of practical importance, the final formula:

eee a ae + GF sate | eS Ya (7)
This is even simpler than the equations (C) and (D) in Z. If we substitute here
as before 0 = 0.3, and further for x and y the values from Eq. (6'), we obtain
for the required buckling pressure p, if 2a designates the diameter, 2h the wall
thickness and 1 the effective frame spacing:

4
p = aes Lethe + 0.73 (n? + 0c?)? wr © (eve Te, leh Aahlvey © (8)
n —
2

where A =Ta/1

For n use tnaat whole number which makes the expression for p a minimum for
a given a, h, and 1. E designates the elastic modulus of the material as long
as the stresses in the shell are below the proportional limit. Concerning the
applicability of the equations above the proportional limit, similar conditions
apply as in the case of ordinary buckling of rods (see below).

Eg. (7) and (8) do not give accurate values for a tube of infinite length
Since here, as is well known, tne number of lobes, which we have assumed to be
large, decreases to two.

2- DISCUSSION of the FINAL FORMULA. TABLES.

Eq. (6), as well as the simplified Eq. (7), represents a straight line in
an x-—y coordinate system. Each’ wave number, n, represents a straight line for
a given @/1, as shown in Z. Hence, y as a function of x will be represented by
the ordinates of a straight line polygon. The vertexes of each one of these pol—
ygons increase in number towards the origin of the coordinate systen.

The significance of « requires more detailed explanation. According to Z,
Eq. (12),

COLT YG eto Ping Me ts, A tat LL 6 ie ea (9)
_ The "effective" length of the tube is to be substituted here for 1; that is, the

spacing of two successive nodal points of the lobe line appearing in a longitudi—-
nal section of the tube. Jf, as is the case in the fire tubes of a boiler, the —
tube ends can be regarded as fixed radially, then 1 can be taken equal to the total
length between the ends of the tube. If the end supports are yielding, a somewhat
higher value of 1 must be used. If the tube is divided into several belts by
sufficiently rigid frames, the effective length 1 is an average value of the in—
dividual frame spacings. It is possible to obtain more exact data for determining
the average value from the known calculated results for the buckling of rods with
several bays. However, since the frames in submarine hulls are usually equally
spaced, no difficulty is presented.

It is a question whether in many cases a higher number of waves between
two successive frames, that is to say dividing 1 into halves, thirds, etc., may
not result in a smaller value of the buckling pressure. This assumption seems not
improbable when we consider that the smallest possible number of lobes, n = 2, in
a circumferential belt between two frames, does not always result in the smallest
critical pressure. In reality Eq. (7) indicates, in contrast to (C) and (D) of Z,
that for fixed n and x the ordinate y does not increase unconditionally with &.
For sufficiently small values of x, y decreases; for example, if « is made three
times as large while n remains constant. However, it does not follow from this
that the foregoing assumption is true since it is not yet certain whether a lower
buckling pressure occurs by changing n while & is increased. The question to de—
cide is whether the polygons that are constructed for a single & intersect in the
region of very small x or whether the polygons always lie above each other and
meet only in the origin. The answer is apparent if we consider that with small x
on the one hand the number of lobes increases without limit, while on the other
hand the vertexes of the polygons increase without limit. From the first condi-
tion it follows that Eq. (7), by expanding in powers of 1/n? and neglecting the
higher powers, can be simplified to:

4
y= GaGa" (, 50552) one (143 S2) a (10)

From the second condition it follows that we may with sufficient accuracy, in the
neighborhood of the origin, consider the polygon as the envelope of the family of
straight lines determined by Eq. (10) for successive values of n. This envelope is
obtained through Eq. (10) in conjunction with the following equation obtained by
the differentiation of Eq. (10) with respect to n?*:

2) 7" 2
o9--tU-Pw wee oe ee Py eee ELE re pecs. Calito)

Elimination of n? from Eq. (10) and (11) gives the equation of the envelope:
4 : 4 4
y= ig ax? + § (2+ 2x=1.N4 exo +5 (x? + 2)x 2. (12)

As can be seen from Eq. (12), since the ordinates increase with increase of @,
no intersection occurs, and we come to the conclusion: The minimum buckling
pressure always corresponds to a deformation without nodal points between the
points of support in the longitudinal plane, and, therefore, to the smallest possi—
ble value of @ ,
[translators Note: Equation (12) derived to justify the conclusion in the preced—
ing paragraph is a good approximation to the envelope of Eq. (7) in the neighbor—
hood of the origin, that is, for small values of x and “. The exact equation of
the envelope has a very important property: It can be used in place of the original
equation (7) to determine y with sufficient accuracy for any given values of x
and %@. Graphically, this fact is evident from the proximity of the envelope and
those parts of the family of straight lines used to determine y represented by Eq.
(7) with n as a parameter. Analytically it is evident from the fact that the
equation of the envelope, obtained by eliminating n between Eq. (7) and (11), is
in effect Eq. (7) with that value of n substituted which for any given x and &
makes y a minimun.
Eq. (12) is an approximate equation of the envelope of Eq. (7), valid only
in the neighborhood of the origin, because of the nature of the assumptions which
led to Eq. (10). The equation of the envelope of Eq. (7) should be derived directly
from Eq. (7) itself. When so derived, its usefulness is no longer limited to the
neighborhood of the origin. This equation can be very accurately derived as follows:
Differentiating Eq. (7) with respect to either n, (n? + @), or fe and
equating to zero we get
(n? + a@?P x — (n? + aw? ox — 3 (n? + x?) x4(1 — 72) + X91 — 62) = 0 (Ha)
The solution of (11a) for n gives that value of n which will make Eq. (7) a
minimum for any given & and x. A very approximate solution can be readily ob—
tained. Rewriting Eq. (11a)
(n? + a2) [ (n2 + a?) x — ax] Eee (| =. 0-7) [ 2? ance tS 27/3] =0

Cee ete sie Ween ue eetace*/3)

nx

4 4 4
ae ape dae = oc\) 3 — 0°) \/1 “ Soct/n? = 2G = ©) thys Pea iie)
Daven) We SS Sih Dp EA oat Maes 5. ERE Pa RCAR Cea. 1 ae aes OO eer seat ee ete WCE (11¢)

Substituting (11b) in (7) we obtain

Blo

Lek Nf = 0?) a x
a xt

~ 2 YK — 5?)

In the majority of practical cases, the value of the ratio #/n lies between
2/3 and 1/3. Placing w#/n = 1/2 in Eq. (11c) and substituting (11c) in (12a) we ob-—
tain the simple expression

4
_ 1.718 x Vx2

ee ee (a)

1 - 0.3740 Vx

2/3 we obtain the slightly altered equation

45
1.724 « Yx (b)

With @/n

ae eS So ooo tee Srrorore te
1 — 0.365 Vx
while a/n = 1/3 gives
gq
orl ib® eG Vee (
re ee rib Me acts OTS ce kee oy ole)
1 — 0.382“ Vx

Any number of similar equations may be obtained by using different values of
the ratio aw/n. As a limiting case o/n = 0 gives the equation

_ 174 « Vx

ele ee we ee nee eae . (a)
1 — 0.389 cx

which differs but slightly from (c).

If equation (d) be expanded in a series, the first two terms are identical
with the first two terms of equation (12). - Moreover, the expansion of any of the
equations (a) to (d) yields a series all terms of which are positive. Therefore,
the same conclusion can be drawn from these equations as from Eq. (12) concerning
the impossibility of nodal points between points of support in the longitudinal
plane. It is evident that Eq. (a) — (c) are much better approximations to the
envelope than Eq. (12). Each gives extremely accurate values in that particular
region of a&/n for which it was developed and good values for a considerable range
of an.

To determine just how closely these approximate equations check with Kq.
(6), about 70 values of y, covering the region x = 4/3.10°° to 12.10°° (t/a = 0.002
to 0.006) and & = 1/4 to 16 (1/d = 6.28 to 0.1), were computed by each of the six
equations (6), (7), (a), (b), (c), and (d). These values and their percentage
variation from the exact values computed by Eq. (6) are given in Tables II and rE!
page 19.

Of the four approximate equations tried, equation (a) shows the best agree—.
ment with the exact equation (6). The agreement is excellent for all values of
1/d less than 2, The mean of the absolute deviations throughout the range of 1/d =
0.1 to 2 is 0.5 per cent. The maximum discrepancy is less than 3 per cent for
1/d = 2 and is considerably below 2 per cent for all values of 1/d equal to or less
than 1.6 and is less than 1 per cent for all values of 1/d less than 1. In fact,
Eq. (a) gives better agreement with Eq. (6) than does Eq. (7) from which it was
derived!

Equation (a) can be put in a more convenient form by substituting values of
y, X, and & from Eq. (6") and (9), whence

2.424 EB (nat
% 1- @?
Pee sal
Sq - 0-447 Vh/a

2.60 B (t/aye
: ~ 0.45 Vt/a

or (G@= 0.3) p

Again it might be pointed out that the usefulness of Eq. (a') is not con—
fined to the neighborhood of the origin. It is a simple equation, independent of
n, which determines p with a high degree of accuracy for a large range of values

of 1/d and t/d. Equation (a') may replace equation (6) in all practical computations

where 1/q is not greater than 2, When 1/d is less than 1 the discrepancy will be
less than 1 per cent.

In order to facilitate the practical application of the results derived
above, one may use to advantage a graph, Fig. 4, that contains the polygons (they
appear as curves) for the values W=2, 4, 6,. . . . . . to 20, within the
range up tox =6x 10°° corresponding to the ratio a/l from approximately 2/3 to
6 (1/d = 0.8 to 0.08) and to the ratio h/a up to about 0.0045. The straight lines
are computed throughout by the complete equation (6). In the neighborhood of the
origin, the polygons are made up of curves determined by Eq. (12). The corre—
sponding angles of the polygons are connected by dotted lines, forming quadrangular
areas for the various wave numbers. The separate vertical straight lines correspond
to the constant values of 1000 h/a written beneath; the values of the ratios
4fa/1 are written on the sides of the polygons. On the axis of ordinates, a second
scale is represented beside the y—scale which, under the assumption of an elastic
modulus E = 2,125,000 kg/cm? (30.22 x 10° 1b. per sq. in.), gives the value of

b= p/h, = ghee oy = 234 «Oy 2 es ERS (13)

up tok = 1900 kg/cm? (27,000 1b. per sq. in.) The values in the range of or—
dinates above this point correspond to the observations of Tetmajer on non—elastic

10

buckling of rods.. According to these, as is well known, the critical pressure for
a rod of slenderness ratio r/l is determined by means of two constants c and K by
the formula

eT -K (TPL 8B pr) &. VO eC se, Mp, asl tee gee ae eae (14)
where for mild steel K = 3100 kg/cm? (44,000 lb. per sq. in.) and c = 0. 00368.

902 o s as /Q /
oF kant USL.9/ ; ; By

20f frei J Se ra, yp n=the wo nels |

ae BRYSE AEE

FIG. 4. GRAPH FOR DETZRMINATION OF BUCKLING PRESSURE.

One might so interpret Eq. (14), as if in place of the actual elastic
modulus, E, the expression

J 2
(= KK)?

11

were to be substituted in the otherwise unaltered-
Euler buckling formula. We have employed this
expression for E in Eg. (13) for the conversion
of y into k for k= 1900, y 2 0.000809, and
thereby obtained the k-scale for the unelastic
region. The relation between y and k is repre—
sented in Fig. 5.

The application of the graphs is as follows.
For the given value of the ratio, wall thickness
to diameter, i.e. h/a, we seek the corresponding

ordinate, as well as the corresponding polygon,
for the given ratio of the radius to effective
FIG. 5. Relationship length a/l. The ordinate of the intersection of
between K and y. these two lines gives the value of y or of k from
which the buckling pressure p is obtained.

2h
p= 7p &y, OT Dae i ee ip a o 2 © © © e (16)

If no lines exist for the exact ratio given, y can be determined by interpolation.
In any event, the graoh indicates the number of lobes, n, which is determinative
for the given case. For a known n, it is then not difficult to compute the exact
value of y by equations (6), (7), or (8).

The following numerical table was obtained by readings from the graph. It
is valid both below and above the proportional limit, providing the cylindrical
shell is manufactured from medium steel plate.

Table I.

Critical pressure p in kg/cm* for various wall—thickness ratios,
h/a, and length ratios, a/1.

12

If we proceed purely analytically, without the use of tables, we must use
the equation

2h | Vi1- ¢? Fc}_ h- 0.011
K}i - =r 200? fA = Se fie ss eee < , eee
vy | el a

PS eS

‘in the unelastic region, which follows from the substitution of Eq. (15) in
Eq. (13) and the elimination of k. The value of y must here be determined by the
equations in Section 1. The numerical values in Eq. (17) are for medium steel.

3. COMPARISON with EXPERIMENTAL RESULTS.

Two series of experiments were carried out in 1918 with large medium steel
tubes welded along the longitudinal seam, and subjected to external water pressure.
The first series included tubes with the wall thickness—ratio

h/a = 1/400 = 0.0025; x = h2/3a? = 2.083 x 10°°
and the three length ratios

a/1 = 400/120, 400/180, 400/240 whence,

o& =Wa/l = 10.47, 6.98, 5.24

We obtain from the graph, Fig. 4, at the ordinate h/a = 0.0025, the values of
the number of lobes, n, in the three cases:
(a) n= 16 (b) n= 14 (c) n= 13
[Translator's Note: For case (c), n lies so close to the border line between 12
and 13 that practically it might be either. Actually, n = 12 is determinative. |
The experimental number of lobes from the tests on four different sizes of
models of diameters 800, 1200, 1600, 2400 mm (31.5 in., 47.2 in., 63.0 in.,
94.5 in.) are

Model Size Vase (a) (b) (c)
I (D = 800 mn) n= 17 n= 15 n=14
II (D = 1200 mm) 17 15 13 -— 14
IIT (D = 1600 mm) —_ 14-15 =
IV (D = 2400 mm) 16 14 - 15 -

We may conclude, therefore, that the agreement is almost exact. In addi-
tion it is worthy of note that the agreement appears to improve for the larger
models. The differences between the number of lobes observed in the various sizes ‘
of models, as well as the departures from the calculated number, hardly exceed the
limits of unavoidable experimental error. Fig. 6 shows the buckled model in
experiment IVa.

FIG. 6. A Portion of the Experimental Tube IVa with 16 lobes
Around the Circumference.

The situation is quite different in determining the buckling pressure it—
self. Here the experimental values obtained for the individual sizes of models
differ very materially from each other. Since the collapsing pressures increase
directly as the size of the model, it might be expected that the calculated values
should correspond to a limiting case of an infinitely large test body. According
to Eq. (6), using the approximation of Fig. 3, the computations show:

[Translator's Note: Numerous minor numerical errors, probably due to the use
of a slide rule, have been corrected by the translator without specific reference
in each case. Changes from the original text are indicated * a

Case (2) & = 10.47* n = 16 P= ee = 0.30
= 1.53 (es B28)

ee 309: poe Soo ~~ (365.67* — 2 x 153 x 256 + 1.23) x 10°

y = 0.000264* + 0.000894* = 0.00116

Since this value lies above the limit 0.00081, p must be determined from y by
Eq. (17).

Therefore, 0.011%
ji p = 0.0025 x 6200 {1 — 0 0341 = 10.5* kg/cm? = 10.2* atm.

14

[Translator's Note: Substituting the above values @= 10.47, x = 2.083 x 10¢

in (a) gives
~6
ye 1.718 x 10.47 x (2.083 x 10 2 = 0.00116
1 — 0.374 x 10.47 x (2.083 x 1076F*

Case (b) = 6,98* n= 14 f= 0.199 pus 1.35 P= 1.20

_ 0.91 x 0.0396* , 2.083 Re ae
Yi 219.4 + 219.4 (244. 7? 2 x 1.35 x 196 + 1.20) x 10

y = 0.000164 + 0.000564 = 0.00073
This value lies below the limit and therefore according to Eq. (13)

6
p = 0.005 alee le 0.00073 = 8.5 kg/cm? = 8.2* atm.

6
[Translator's Note: Substituting E = 2.125 x 10 kg. per sq. cm.

t/a = 0.0025 1/a = a = 0.225 directly in Eg. (a')

6 a
p = 2260 x 2.125 x 10° x (0.0025) _ 8.53 kg/om? = 8.26 ota
0.225 — 0.45 x Y 0.0025

Case (c). = 5.24, n = 13, f° = 0.140, fee = 12248 Pg = 12

y = GAT 0-016 4. 2,083 (196.58 — 2 x 1.24" x 169 + 1.2) x 10 °

181.7

0.000098 + 0.000438 = 0.000536.

2.125 x 10°

0.91 x 0.000536 = 6.26* kg. per sq.cm. = 6.06* atm.

p = 0.005

[translator's Note: The use of n = 12 instead of n = 13 in the above problem
gives a slightly smaller value of y.

W= 5.24, ne=al2, Pp = 0.160, P, = 1.28 My= 1.2

y = Scho peoe e+ $085 (171.5? — 2 x 1.28 x 144 + 1.2) x 107°

0.000149 + 0.000386 = 0.000535
However, in practice, collapse may occur in either 12 or 13 lobes.

tT]

The use of Eq. (a') gives

6 we }

= 2.60 x 2.125 x 10° x (0.0025) _ ¢ a9 kg. per sq.cm. = 6.02 atm.
5 - 0-45 x7/0.0025

2X AOO = mee

A comparison of the experimental and theoretical values shows

15

Theoretical Values: (a) p = 10.2* (b) p = 8.2% (c) p = 6.1*
Experimental Values: (a) (b) (c)
Model Size I (San Sho) 3.35
on 7.0 5.1 4.15
III — 6.0 —
IV 9.5 Wee —

The entirely disproportionate value of IIIa
is omitted here. Fig. 7 gives a general idea
of the distribution of these values. Here we
see that the calculated values may very well
be regarded as obtainable asymptotic limits.
There remains only the question as to what
causes the smaller models to collapse so far
below the theoretical collapsing pressure.

It seems most probable that the welding of
the thin plating disturbs the symmetry of the
circular form, or the homogeneity of the ma—
terial itself. It is certain that in the
case of buckling with higher number of lobes,

800 1200 1600

FIC. 7. Summary of Experimental

nabs even very slight irregularities may have a

decisive effect.
The second group of experiments concerns tubes with the wall thickness—ratio
h/a = 0.00406, = x = h?/3a® = 5.49*x 10°
and with frame spacing of 400, 500, and 600 mm (15.7 in., 19.7 in., 23.6 in.) and
a tube radius of 800 mm (31.5 in.). In both the first two cases the frames
failed under the test. In the third case, likewise, it must be assumed as certain,
because of the magnitude of the observed buckling pressure, that the limit of
strength was reached through failure of the frame.
(Translator's Note: It is understood that the testing arrangements were such

as to make it impossible to observe the inside of the model during the tests,]

As we know from a theory which will not be further discussed here, if the
frames are too weak, the buckling pressure must correspond to that of a tube with
an effective length of double the frame spacing or even higher. We can, therefore,
test the reliability of our theory by computing the buckling pressure for tubes of
the length 1 = 800, 1000 and 1200 mm and compare them with the test results. With
help of the graph we find:

Gime = 3.14 (2) Ce — "250i (3) A= 2.09
n=8 n=8 n=7

‘Further calculation shows:

GQ), @= 0.134 p= 1.24 pa= 1-2

16

_ 0.91 x 0.018 , 5.4 -
es (S7R ED MES (73.92 —2 x 1.24 x 64 + 1.2) x 10

0.000241 + 0.000428* = 0.000669*

; 6
p = 0.00812 x 2128 a x 0.000669 = 12.7* kg. per sq.cm. = 12.3* atm.

[Translator's Note: This value of p is too high since n = 9 is determinative
instead of n= 8. Usingn=9, (=0.109, },= 1.19, P,= 1.1

ET OS9iLe Oo Oil 49 -6
v= aoe + 31-95 (90.92 — 2 x 1.19 x 81 + 1.1) x 10

= 0.000127 + 0.000522 = 0.000649

6
p = 0.00812 x S14 7 x 0.000649 = 12.31 kg. per sq.cm. = 11.9 atm.

The use of equation (a') gives

p = 2260 x 2.125 x 10° x (0.00406) _ 12.31 ke. porieq ential ete
0.5 — 0.45 V 0.00406
It will be noted that Eq. (a') gives the correct collapsing pressure independent
of the number of lobes, while the use of Eq. (6) with an incorrect value of n
gives a collapsing pressure which is too high.]
(2) = 0.0896" B= 1.16 Me = 1.1

3 —6
y = 221 5 0.00802" + 2A (70.38 — 2 x 1.16 x 64 + 1.1) x 10

0.000110* + 0.000398* = 0.000508*

p = 0.00812 x -— i x 0.000508* = 9.63* kg. per sq.cm. = 9.3* atm.

[transletor's Note: The use of Eq. (a") gives:

6 2
p= 2.60 Xtsiee ‘as = 2 = 9.73 kg. per sq.cm. = 9.4 ata

(3) p= 0.0820 M,= 1.13 M,= 1.1

in = + Ze (53.47 — 2 x 1.13 x 49 + 1.1) x 10°° ‘

0.000121 + 0.000300* = 0.000421*

6
= 0.00812 ee a x 0.000421* = 7.98% kg. per sq.cm. = 7.7* atm.

uo)
1

I

17

[Translator's Note: The use of Eq. (a') gives:

2.60 x 2.125 x 10°x (0.00406)

>
0.750 - 0.45 x Y 0.00406

= 8.04 kg. per sq.cm. = 7.8 ata]

The experimental pressures 12.2, 9.2, and 8.5 atmospheres are to be compared

with the three calculated values thus obtained, 11.9, 9.3, and 7.7 atmospheres.
[Translator's Note: These values are given as 12.8, 9.5, and 8.1 in the text.]
We see that there is good agreement.

The number of lobes, n = 14 — 15, observed in the third test as compared to
the theoretical number, n = 7, cannot be explained. Probably it is due to an error
made by the observer. In both the other tests it was impossible to determine the
number of lobes after failure.

Conclusions (added by translator).

Equation (6) gives a solution for the collapsing pressure of a circular
cylindrical vessel closed at the ends by flat heads and subjected to uniform hy—
drostatic pressure on both shell and heads. As will be observed, the determination
of the correct collapsing pressure depends not only upon the length, diameter and
thickness, but also upon the number of lobes, n, into which a circumferential belt
of the vessel, between stiffening rings, divides itself at collapse. Equation (6)
gives a different value of p for each assumed value of n and collapse occurs in
that number of lobes for which p is a minimum. The use of Eq. (6), therefore,
involves the determination of the value of n which gives a minimum value of p,
either by actual substitution in that equation or by the use of curves previously
prepared, such as given in Fig. 4. However, the determination of n by Fig. 4 is
not always accurate, as was shown in the solution of example 1, page 15, where the
author by using n = 8 instead of the correct value n = 9 obtained a collapsing
pressure which was considerably too high. Fig. 8 on page 18 is drawn to a differ-
ent scale and gives a very accurate determination of n.

Equation (7) is simpler than Eq. (6) and the collapsing pressures calculated
by it differ from those calculated by Eq. (6) by less than 1 per cent for all values
of 1/d below 0.5, as shown by Tables II and III, page 19. However, Eq. (7) like—
wise requires the determination of that value of n which gives the minimum collaps—
ing pressure and hence is somewhat cumbersome and indirect.

Equation (a) presents a striking contrast to equations (6) and (7). It is
very simple and gives the collapsing pressure directly without the use of n. More—
over, Eq. (a) checks Eq. (6) even more closely than does Eq. (7), and can replace
Eq. (6) in all practical calculations.

18

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