Inelastic Column Buckling of Internally Pressurized Tubes Experimentally determined buckling loads of internally pressurized, axially compressed tubes substantiate the predictions of the incremental theory of plasticity and cast further doubt on the use of the deformation theory by John B, Newman

ABSTRACT The incremental theory and the deformation theory of plasticity are used to analyze column buckling of internally pressurized tubes subjected to axial thrust, and the results are compared to the buckling loads determined in tests of annealed-aluminum tubes. By using stress-strain curves obtained from tensile tests of tubing samples, reliable predictions are obtained with the incremental theory. However, the results of the deformation theory are so conservative as to cast doubt on the usefulness of this theory for buckling analyses.

Symbols E -- modulus of elasticity S -~ plastic portion of strain in tensile test I = centroidal m o m e n t of inertia of cross section of tube I = length of flexible section of b u c k l i n g specimen Io = length of rigid section of buckling specimen P -- internal pressure r -= radial coordinate in r, 0, z coordinates, or radius of end of specimen R = radius of cup r~ = internal radius of t u b e ro = e x t e r n a l radius of t u b e T -- axial thrust u -- displacement of column f r o m straight line z -- axial coordinate in r, 0, z coordinates 8 - - i n c r e m e n t of following variable, or m e a s u r e d deflection e = strain. Components are er, co, ez ea = strain at n e u t r a l axis in S h a n l e y analysis of colu m n buckling ~b = bending strain in S h a n l e y analysis of column buckling eP = plastic portion of strain 0 = t a n g e n t i a l coordinate in r, 8, z coordinates -- stress. Components are ~r, ~0, ~z aa "- stress at n e u t r a l axis in S h a n l e y analysis of colu m n buckling

~b----bending stress in S h a n l e y analysis of column buckling cg = generalized stress

Introduction Plastic buckling of columns is a d e q u a t e l y d e s c r i b e d by the w e l l - k n o w n S h a n l e y " t a n g e n t modulus theory."l In this theory, t h er e is no need to distinguish b e t w e e n the " i n c r e m e n t a l t h e o r y of plasticity ''2 in w h i ch the stress depends on the history of plastic strains, and t h e t h e o r e t i c a l l y less precise " d e f o r m a tion t h e o r y of plasticity, ''8 w h e r e the stress depends only on the c u r r e n t strain and not on t h e previous history. For, t h e t w o theories reduce to the same form w h e n applied to t h e p r o b l e m of Shanley's t h e ory. H o w e v e r , for column buckling of a pressurized tube, t h e two theories of plasticity give substantially different results. This p ap er summarizes the results obtained w i t h the two theories, and presents the r e sults of e x p e r i m e n t s substantiating the i n c r e m e n t a l t h e o r y as compared to the d ef o r m at i o n theory.

Inelastic Column Buckling In Sh an l ey 's t an g en t modulus t h e o r y of inelastic buckling, strain i n c r e m e n t s in the e x t r e m e fibers of a column or unpressurized tube are assumed to be essentially t h e same as the strain increments at the centroidal axis. Thus the stress-strain relation t h r o u g h o u t the cross section can be a p p r o x i m a t e d by a linearized Taylor series expansion of the s t r e s s strain relation for t h e centroid. If the stress increm e n t at the centroid is 8r -- 8r and t h e resulting strain i n c r e m e n t is 8~z = 8ca, the stress i n c r e m e n t associated w i t h the strain i n c r e m e n t 5~ = ~}ea Jr 5eb is 8,rz ~ 8,ra Jr 5~b(O,Tz/O~z), w h e r e Or is calculated w i t h all v ar i ab l es except ~ and ez r e m a i n i n g fixed. Let 8~b = -- x8 ( 02u ~ be a b e n d i n g - s t r a i n i n c r e -

\ Oz2 /

John B. N e w m a n is Senior Engineer, Westinghouse Electric Corporation. Bettis Atomic Power Laboratory, West Mifflin, PA 15122, Paper was presented at 1972 SESA Fall Meeting held in Seattle, W A on O a o b e r 17-20.

m e n t in a measuring the z axis increment

column w i t h a s y m m e t r i c a l cross section, x f r o m the axis of s y m m e t r y and u f r o m to the axis of sy m m et r y . Then, the stress is

Experimental Mechanics I 265

k Oz2 /

(1)

O~"

Hence, w h e n the cross section has a m o m e n t of i n ertia I about the axis of symmetry, the increase in b e n d i n g m o m e n t associated with an increase i n u is

5ezP = ~ez

8r

-E

~= =

d=z

dg

= g'(~o)

= $'(~,) 6~

= J'(r r

co M

(2)

,5=g .= g' (o'a) 8=a.

dvg

Thus 5 ~ / r H' is given b y

"~H'

~ M = _ I 0 ~ 5 [ 02u ~ 0~ t Oz2 / "

d]

--

--

Taking this as the definition of 8 v/z H', 8ez is 1

If the column has a stress-strain relation ~ : ~(~) = ,~/~ + ~(~,)

(3)

for increasing u n i a x i a l stress, where ] &) ~ 0 is zero before yielding, eq (2) becomes 6 M = - - I E ~Uzz/[1 + E f ' ( ~ a ) ]

(4)

+ (~-

0 . 5 ~ r - 0.5~0)$'(r

~

O'g

Consider a simply supported column of cross-sectional area A subjected to axial thrust T -- Ar The column "buckles" once equilibrium states u ( z ) ~ 0 can exist. For deflection u, the t h r u s t causes a mom e n t M = Tu. Thus, after deflection 5u = u away from the straight configuration, e q u i l i b r i u m of moments requires T u + Z ~ u = / [ 1 + g f ' ( ~ ) ] = 0.

(5)

+ (~-

~2IE/l ~ [1 + El' (*a)],

(6)

b u t no nonzero solution for T < Ts. Hence, b u c k l i n g occurs w h e n T : Ts. For column buckling of a pressurized tube, eqs (4), (5) and (6) no longer apply. This is so because plastic flow can Occur in a pressurized tube even though the thrust is small.

Incremental-theory Analysis For column buckling of an axially compressed, i n t e r n a l l y pressurized tube, assume the material of the tube displays the stress-strain relation (3) i n simple tension. Applying the yon Mises criterion of yielding for an isotropic material, yield occurs w h e n

8, H'

where ~ ~/~ H' = 8 ~ / ( r d H / d e av) can be determined from the uniaxial tensile test. Upon considering cz -" c z / E + 5(r = r + g(~g), use of the fact that ,a reduces to Cz for u n i a x i a l stress leads to o Here, and throughout this paper, the superscript prime indicates differentiation with respect to the independent variable.

266 I July

1973

(10)

A

A

tional stresses (~r, ~0, ~ ) assumed very small compared to those in the history ( cr, ~o, =z) u n d e r consideration and to be applied at much smaller rates. Since the incremental stress-strain relations (8) are linear functions of 5r 5~0, and 5~z, the additional strains resulting from the p e r t u r b a t i o n a l stresses are A

A

A

linear functions of ~r, ~0 and *z, and relation (8) applies for c o l u m n - b u c k l i n g analyses of pressurized tubes. This can be seen from the following. For an i n c r e m e n t of pressure 5P and an i n c r e m e n t of thrust 5T the u n p e r t u r b e d stress increments in a thin-walled, axially compressed, pressurized tube are 5 c,z -- 5P~ri~ -- ~T = ( r J - ri2)

(n) ~

riSP

8~0 =-. attains a value ~g ----r H e r e ' - - - - H(dea P) is a function of the generalized plastic strain. 4 According to the i n c r e m e n t a l theory, w h e n plastic flow occurs, the plastic-strain increments are characterized b y

V2~- V2~0)8~>0.

When eq (10) is not satisfied, the stress-strain r e l a tion is elastic. Consider a history of i n t e r n a l pressure and axial thrust satisfying eq (10). To investigate the stability of this loading process, subject the body to p e r t u r b a A

For a column of length 1 this has a nonzero solution for

~z v = (~z -- 0.Svr -- 0.5r

(9)

I n c r e m e n t a l stress-strain relations (8) apply only for

w h e n df/d~zl=z=~a is w r i t t e n as *J' (r

:

(8)

Here E is Young's modulus of elasticity, v is Poisson's ratio and 5~a is the i n c r e m e n t of Zo given b y

~*r(ar -- 0.5~0 -- 0.5~) + 8~0(~0 -- 0.5r -- 0.Szr) + 5Z~(r -- 0.5~ -- 0.5~0)

T : Ts : A (r

.

Cg

To ~ ~'t

and 8~r ~- 0. Here the tube has outer radius ro and i n n e r radius ri. W h e n this loading process is perturbed by the b e n d i n g stresses A

8r

and

(12) A

8r162

A

repeated application of inequality eq (10) indicates that eq (8) applies for the perturbed loading eq (I1) plus eq (12) so long as k is small, provided the stress increments eq (11) satisfy eq (10). Since initiation of b u c k l i n g occurs for a r b i t r a r i l y small k, the desired p e r t u r b a t i o n a l stress i n c r e m e n t - s t r a i n i n c r e m e n t re-

lation is Oez

1

(,= - - 0.5,r - - 0.5(70)2

O(Tz

E

E,g 2

-{- ('z--O'5(Tr--0"5"O)2 [ l~--}-s'(r

1

(To2

The r e q u i r e d b e n d i n g - m o m e n t - d i s p l a c e m e n t relation corresponding to eq (4) is

and Tc of the d e f o r m a t i o n t h e o r y shows Tc -- Tc* only when (Tr = "~ ---- 0 - - c o l u m n buckling in the absence of i n t e r n a l pressure. F o r nonzero values of -r and ,e, Tc* exceeds Tc. In fact, as will be seen b e low, the discrepancy can be large. The test results p r e s e n t e d h e r e substantiate t h e i n c r e m e n t a l theory, and show the d e f o r m a t i o n t h e o r y to be so conservative as to cast doubt on its practical value as well as its theoretical soundness.

-- EI 5Uzz ,0 2 8M --

((7,

3 i + Ef'(~g) +

4

,0) 2

3

(-z - - 0.5(7r-- 0.5-0) 2

j (=, - o . 5 , , -

0.5,~

a n d t h e critical t h r u s t f o r a p i n n e d - p i n n e d p r e s s u r ized t u b e of l e n g t h I is

Tc*

n2EI

= 12

3 1 + gI'(~o) + -~

(,~ -

(-~ - (7o)~ 0.5,, - 0.5,o)2

(r162

(13)

(792

Deformation-theory Analysis

Test Design

I n analyses using the d e f o r m a t i o n theory, plastic deformations in bending are governed by ae~P/0~ = OJ'(~g)/0~g, w h e r e ~a is defined as in eq (7) a n d 5((7~) can be i n f e r r e d f r o m eq (3). I n this case,

The tests w e r e designed to provide d a t a in the r e gion of greatest d i s a g r e e m e n t between the incremental t h e o r y and the d e f o r m a t i o n theory. P r e l i m i n a r y analyses indicated c l a m p e d buckling specimens w o u l d buckle at u n a c c e p t a b l y high thrusts and t h a t the r e suits w o u l d be subject to uncertainties arising from misalignment. Accordingly, tests w e r e p e r f o r m e d w i t h an assembly shown in Fig. 1 on specimens w i t h h e m i s p h e r i c a l ends. Pressurization was achieved t h r o u g h a hose attached to the end piece. A x i a l t h r u s t was applied b y the spherical cups indicated. P r e s s u r e - h o s e r e s t r a i n t and effective length of the

O,z Oez

:

E/[1 + Ey((Tg)],

and, in contrast to eq (13), the critical t h r u s t for an i n t e r n a l l y pressurized t u b e is Tc : ~2IE/F [1 + El'((Tg) ] .

(14)

Comparison b e t w e e n Tc* of the i n c r e m e n t a l t h e o r y

I-I/2"O.D. X 0.028" WALL 3005-HI4 ALUMINUM TUBtNG - ANNEALED BEFORE WE'LDING

0.75 DIA. SPHERICALSURFACE - 16 RMS FINISH

I

l

L~.

WELD - EACH END-'-~

\

ALOM,NOMB'./

~t/8" PIPE THD 0.50 DEEP -EACH END

/

14.55" NOMINAL

=" I

BUCKLING TEST SPECIMEN DIA

5/4 X I0 CLASS 2

t•2"

~

(t

_

/ -

J

~ - - SPHERICALSURFACE 16 RMS FINISH 0,62,5 + 0.025 RADIUS OEPTH 0.30

MATERIAL: MACHINABLESTAINLESS STEEL

Fig. 1--Buckling specimen, loading cup and buckling-test assembly

Experimental Mechanics

I 267

specimens were d e t e r m i n e d on elastic specimens 24.5in. long. This is discussed later u n d e r "Effective Length of Buckling Specimens." Test specimens were fabricated from commercially available 3003-H14, 0.5-in.-diam, 0.028-in.-wall alum i n u m tubing. All 14.55-in. buckling specimens were recrystallization a n n e a l e d as prescribed in the "Alcoa Handbook"5--Heat to 775~ ("Time i n the furnace need not be longer t h a n is necessary to b r i n g all parts of load to the a n n e a l i n g temperature. Rate of cooling is u n i m p o r t a n t . " ) Stress-strain data on the m a t e r i a l were obtained from tensile specimens of the type shown i n Fig. 2. The specimen was designed to ensure u n i a x i a l loading in the gage region. One tensile specimen was cut from each length of t u b i n g used in m a k i n g buckling specimens, in the expectation that this would provide adequate characterization of the material properties. The tensile specimens were annealed with the buckling specimens. When substantial data scatter was observed, additional tests were performed on modified buckling specimens (results labelled 1A2, 2B2, and 4D2 in Fig. 5). Stress-strain curves were obtained at an extension rate of 0.05 i n . / m i n on an I n s t r o n TTC tensile-test machine. Strain m e a s u r e ments were made with a W i e d e m a n n Baldwin B3M 1-in. gage-length LVDT extensometer. The stressstrain curves were recorded on the autographic r e corder, Data on the b u c k l i n g loads of the specimens were obtained by applying Southwell's method. ~ I n this method, the growth of the first F o u r i e r component of the deflection is used to d e t e r m i n e the buckling load. Thus, lateral deflections of the rod were measured by four dial gages a r r a n g e d as shown in Fig. 1. The gages were fitted with 3 / 8 - i n . - d i a m flat tips. I n t e r n a l pressure was generated by a pressure-gage tester with a s c r e w - d r i v e n hydraulic r a m and a Heise 5000-psi precision pressure gage calibrated against p r i m a r y - p r e s s u r e standards. The hose was h u n g from v e r y flexible springs to m a x i m i z e its flexibility. Thrust was applied and measured b y a B a l d w i n 60,000-1b universal testing machine.

machine was raised to enclose the specimen in the greased load cups, and the cups were aligned to within 0.02 in., as judged from a machinists square resting on the r a m of the test machine. With the specimen u n d e r a thrust of 20 lb, the dial gages were positioned as in Fig. 1 and adjusted to contact the specimen at midspan. Once all adjustments had been completed, the specimen was subjected to axial loading u n d e r one of the following t h r u s t - p r e s s u r e programs: (1) zero pressure; (2) pressure and thrust increased in a constant proportion ("radial loading", the condition closest to the assumptions of the deformation t h e o r y ) ; (3) pressure and thrust increased in proportion up to some pressure, t h e n thrust i n creased at constant pressure; (4) pressurization before a n y axial thrust was applied; and, (5) thrust increased at constant pressure, t h e n pressure increased at constant thrust, followed by increasing thrust at constant pressure. U n d e r each of these programs the load a n d / o r thrust was increased b y i n c r e m e n t s while the test machine was rapped lightly with a mallet to minimize the effects of friction in the cups, then load and pressure were held constant and the lateral deflection of the specimen was recorded. The test ended when the deflection increased continuously at constant load and pressure.

Analysis Used in Interpretation of Data Consider the strut shown in Fig. 3. The central portion, - - l / 2 ~ x ~ I/2, has flexural rigidity EI, and the ends, of length lo, are infinitely stiff. The supports offer no resistance to rotation of the ends. I n the u n loaded state, the central portion has shape [ ~lo ~z "~ wo(x) = q ~--~ + cosT) .

I n response to axial thrust T, the beam assumes a shape u ( z ) satisfying

T u " ( z ) + EI[uZV(z) -- uoZV(z)] = 0.

(15)

S y m m e t r y of the structure a n d deflections implies u'(0) = 0 (16)

Test Procedure

u"(0) = 0

The hose was attached to one end of the specimen and, after the specimen was filled with water, the tap i n the other end was closed w i t h a plug. The ends were greased a n d the pressurizer was moved so that the specimen r e m a i n e d vertical at the height and location of the load cups. The r a m of the test

The conditions at z = 1/2 reflect the fact that the deflection, and moment, at the support are zero:

u ( l / 2 ) + lou'(l/2) = 0 (17)

EI[u"(l/2)

END FITTING I/2" DIAWELDABLE ALUMINUM BAR

Uo"(I/2) ] + T u ( I / 2 )

/

,___~ACH

Fig. 2--Tensile-test specimen used to obtain stress-strain data on material used in fabrication of ,buckling specimens - - 4 "

NOMINAL

]'-9/16" NOMINAL

268 t Ju~

1973

---

O.

I/2"O.D. X 0,028" WALL 3003-H 14 ALUMINUM TUBING CUT FROM SAME STOCK AS BUCKLING SPECIMENS AND ANNEALED WITH THEM

/

I/2 X 13 CLASS 2 THD~

--

END

Solving eqs (15), (16) and (17), the a p p a r e n t deflection u~(0) = u ( 0 ) -- uo(O) measured at z = 0 is Q r k212 i - k212/~ 2 L ~2

u~(O) :

~lo +

1

~

1

~lo

'

(xr ~

R--r

r2

(18)

J -- Q - - ~

Here, k2 = T / E I has been introduced. I n a p i n n e d - p i n n e d strut with initial shape Q cos ~ z / L , the a p p a r e n t deflection, 5, is QTLB/nBEI 1 -- TL2/n2EI "

As Southwell showed, this can be r e a r r a n g e d to ,2

: 5 ~2E---[" + ~2EI .

(19)

This equation is the basis for the w e l l - k n o w n Southwell plot c o m m o n l y used to d e t e r m i n e the critical thrust from thrust-deflection data. For, the slope 0 (o~ from eq ( 1 9 ) i s the inverse of the Euler 08 \ T / critical load. 6 At first glance, eq (18) does not appear suitable for reduction to a form like eq (19) from which the critical thrust can be inferred from thrust-deflection data. But thrust-deflection data obtained from eq (18) can be analyzed by the Southwell method. F o r values of l, lo, E and I appropriate to the elastic-buckling specimens used here, Southwell's method predicted the actual critical thrust w i t h i n 0.5 percent, e v e n though the computations were based on the slope of the 5, ( 8 / T ) plot at a v a l u e of T equal to about 2 percent of the critical thrust! This proved Southwell's method to be suitable for analyzing the elastic calibration tests. Moreover, the stability of the predictions suggested analysis of the inelastic tests b y the same method.

Effective Length of Buckling Specimens Since the buckling load of a s t r u t is a sensitive function of its length, i n t e r p r e t a t i o n of test results requires a n accurate estimate of the effective length of the specimens. Consider the kinematics of the rigid mechanism, sketched in Fig. 3. W h e n the small circle rolls without slipping, through a n angle a, the contact point moves on the large circle from the bottom to a point

'~!~ ~~H~T~'~:~OU~

-

For small a, the point on the center line of the specim e n through which the thrust acts lies at a height

cos k l / 2 - - k l o s i n k l / 2

5 = Uc (0) =

o

R--r above the center of curvature. Thus, the effective length of the specimen is s = 2 ( r 4 - Y ) = 2 R r / ( R -r) less t h a n the total length of the specimen. For the spherical end of the specimen and the hemispherical load cups used here, R = 0.625, r = 0.375 and s becomes s = 1.875. Consider a specimen of total length 24.5 in. with flexible mid-section of length 21.125, and lo = 0.75. Use of the dimensions ro = 0.5, t = 0.028, and the modulus of elasticity E = 10.3 X 106 psi leads to a critical thrust T = 235.2 lb. The actual critical t h r u s t and effective length are inferred from the elasticbuckling tests discussed in the following section. Thrust loading at the spherical-bearing surfaces changes the effective length of the tubes after the first loading but a p p a r e n t l y not on the first loading w h e n greased cups are used.

Elastic-buckling Tests Three specimens of 24.5-in. length were buckled nine times (one twice, one three times, and one four times). The first time a specimen was tested, the effective length was greater t h a n on retesting by about 5 percent, b u t there was no decrease after the second test. This was a t t r i b u t e d to plastic deformation of the spherical end of the specimen d u r i n g the first test. One virgin specimen was tested in greased cups while the other two were tested in d r y cups. All of the retests were r u n i n dry cups. Three of the retests were r u n w i t h o u t the pressure hose and three were r u n with it. T h e hose was f o u n d to have no m e a s u r a b l e effect o n the b u c k l i n g thrust, b u t greased cups were found to have a substantial effect, lowering the critical t h r u s t b y about 10 percent. Since the effective length of the specimen tested in the greased cup agreed most closely with the predictions, it was decided to test the i n e l a s t i c - b u c k l i n g specimens in greased cups. Also, in order to e n h a n c e the effect of the lubrication, it was decided to tap the testing m a chine d u r i n g loading so t h a t the specimen would shake down before the deflection m e a s u r e m e n t s were made.

IW /--FLEXIBLESECTION Fig. 3--Idealizations of loading cup and buckling specimen used in analysis of test results

OFACTI OFNE APPLI EDO~ FORCE

Experimental Mechanics I 269

~

70

~ .60

g"

I

/

F

.,o

.•,/•P%'-

THIRD TEST

,35

.s

~

50

~ 25

J

20

g

TESTS OF SPECIMEN 245443 x - SECOND TEST - HOSE O-THIRD TEST-NO HOSE + - FOURTH TEST- HOSE

~ ,o

~.10

P

a

| 0

CIMEN 24561T x SECOND CUP TEST -HOSE 0 THIRD CUP TEST -NO HOSE

/

I

0

20

I

J

40

I

J

[

p

l

J ~ 05 D ~ ' FOURTH TEST

]

O

60 80 I00 120 140 DEFLECTION- I0 -~ INCHES

160

]

20

180

]

I

I

I

I

I

40 60 80 I00 120 DEFLECTION-10 -3 INCHES

Fig. 4--Southwell plots for elastic-buckling tests conducted to calibrate test assembly 9 8O

!

. 70 .60

-

. 50

. 40

~ . 30

SPECIMEN 2 4 . 5 2 9 8

z , 20

5

~ ,10 o 0

Figure 4 presents the Southwell plot for the greased cup test and for five of the retests. The lines drawn on the plots are the straight-line fit of the data from which the critical thrust w a s inferred. The results of all the tests are summarized in Table 1. These results demonstrate the repeatability of retests, after account has been taken of peening of the ends during the first loading. The effective length in the first test of specimen 24.5298 was 22-5/8 in. This is equivalent to a flexible length, l, of 21-1/8 in., w i t h rigid end pieces of length lo ---- 3/4 in. This value of lo was the one used in the analysis of the inelastic-buckling specimens, as discussed in the next section. The absence of any m e a surable stiffening caused by the pressure hose made it possible to ignore any such stiffening in analyzing the inelastic buckling.

I 20

410

610 810 i I I I00 I20 140 DEFLECTION -I0 -~ INCHES

I 160

I

ISO

I07

9

8 3C

~T

36

~

282

106

i 0

2000

TABLE 1--ELASTIC-BUCKLING TESTS specimen Length 24.5298 24,5617

24.5443

270 I Ju/y 1973

Test 1. 2. 1. 2. 3. 1, 2. 3. 4.

Grease but no hose No hose or grease No hose or grease Hose but no grease No hose or grease No hose or grease Hose but no grease No hose or grease Hose but no grease

4000

6000

8000

I0000

12000

14000

16000

STRESS

critical Thrust 238.7 285.3 263.8 290.6 298.0 265.1 280.0 290.7 283.5

Fig. 5--Stress-tangent-modulus curves obtained from

tensile tests

Prediction of Critical Thrusts for Inelastic-buckling Tests Inelastic c o l u m n b u c k l i n g of a p r e s s u r i z e d t u b e can

be predicted by using the tangent modulus in place of E in calculating k 2 = T / E I for use in eq (18). Buckling occurs at the thrust for which ua(O) be-

140

400 "

m 300

400

~

TENSILE TEST 3C TEST 3C

~ 300 -

IAI 2BI

o_ i 200 ~IA2

~ 200

4DI

4DI

5

Fig. 6---Predicted buckling thrust-to-internal-pressure relations using deformation theory

~ I00

~ I00

0

0

I I 500 IOO0 1500 INTERNAL PRESSUREAT BUCKLINGPOUNDS PER SQUARE INCH

0

500 qO00 1500 LNTERNAL PRESSURE AT BUCKLINGPOUNDS PER SQUAREINCH

INCREMENTALTHEORY

DEFORMATIONTHEORY

SPECIMEN: 4 D I 4 . 5 3 9

SPECIMEN: l A I 4 . 5 5 8

TEST: 8 P S I / L B THRUST

TEST: 1250 PSI Tc = 187 LB

Tc = 146 LB

~

.80

-- 3::

/•

b o 4o bJ --

/

/

I.O

/

.50

.40

/x .30

/x x

/

.10

/X

X

/

20

/

~,?, o_.20

/

/X

(• 0

I

I

p

I

40

80

120

160

0 0

I 20

i 4O

.10

/ /x

~ .20

/

-- I

b~ LU --

.08

/

t% JO

/

wE,

x

-- .05

# 0

i 20

x/

x

.06

SPECIMEN: 2BI4.551 TEST: P=O Tc = 527 LB ~ 40

i 60

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.04

DEFLECTION-10 -3 INCHES

/

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,x SPECIMEN: 5 C I 4 . 5 4 2 3 9~ TEST: P = [ 2 5 0 PSI EC" Tc = 3 3 2 LB

.02 i 80

I 8O

DEFLECTION-IO -5 ~NCHES

DEFLECTION-IO -3 INCHES

.25

I 60

0 0

I

I

I

I

lO

20

50

40

DEFLECTION-IO -3 INCHES

Fig. 7--Southwell plots for typical buckling tests showing least-squares-fitted straight lines used to determine buckling loads

responds to J'(~) in eq (3), is used in eq (13) or eq (14) to compute Et or Et* in t e r m s of P, T and the g e o m e t r y of the specimen. The critical t h r u s t is comp u t e d b y finding successively m o r e precise u p p e r and l o w e r limits to the value of t h r u s t for which the l e f t - h a n d side of eq (20) changes from positive to negative. The computations p e r f o r m e d in this p r o g r a m have been checked b y comparison w i t h i n d e p e n d e n t calculations and b y detail checking of int e r m e d i a t e results. F i g u r e 5 displays the s t r e s s - t a n g e n t - m o d u l u s curves inferred from tensile tests of eight specimens (two each f r o m tubes "A", "B", a n d "D", and one each from tubes "C" and " E " - - t h e tubes from w h i c h the buckling specimens w e r e cut). As indicated in this figure, the s t r e s s - t a n g e n t - m o d u l u s curves d i s p l a y e d g r e a t variation. The fact t h a t two specimens m a d e from the same t u b e are noticeably different suggests t h a t the heat t r e a t m e n t was not a d e q u a t e l y controlled and t h a t a s i m i l a r - - o r , perhaps, g r e a t e r - - v a r i a t i o n of p r o p e r t i e s c a n be expected among the buckling specimens. F i g u r e 6 displays the i n t e r r e l a t i o n b e t w e e n i n t e r n a l p r e s s u r e and critical t h r u s t for specimens w i t h the s t r e s s - t a n g e n t - m o d u l u s curves of Fig. 5. Both d e f o r m a t i o n - and i n c r e m e n t a l - t h e o r y analyses used I = 11.125, lo ~ 0.75, and the values of inner and outer radius obtained b y m e a s u r i n g the t u b i n g from which the test specimens w e r e fabricated. The g r e a t disp a r i t y in s t r e n g t h b e t w e e n the i n c r e m e n t a l theory and the d e f o r m a t i o n t h e o r y results is a p p a r e n t from these plots.

Inelastic-buckling Tests comes infinite. That is, t h e t h r u s t for which cos k l / 2 - - k l o sin k l / 2 = 0.

(20)

Computations w e r e p e r f o r m e d b y m e a n s of a comp u t e r p r o g r a m t h a t accepts s t r e s s - s t r a i n d a t a as i n p u t and predicts the critical t h r u s t as a function of int e r n a l pressure. S t r e s s - s t r a i n data, obtained from a tensile test, a r e r e d u c e d to t r u e - s t r e s s - t r u e - s t r a i n form. Each set of t h r e e d a t a points is a n a l y z e d to give the best l e a s t - s q u a r e s i n c r e m e n t a l s t r e s s - s t r a i n relation, and these d a t a points a r e fitted b y use of a sixth degree polynomial. The resulting s t r e s s - i n c r e m e n t - p l a s t i c - s t r a i n - i n c r e m e n t function, w h i c h c o t -

T h i r t y - n i n e specimens were tested in the inelastic range, 11 w i t h no i n t e r n a l pressure, and 28 w i t h int e r n a l pressure at b u c k l i n g r a n g i n g b e t w e e n 800 and 1700 p s i - - a region in which the two theories provide distinctly different predictions. The critical thrust for each buckling test was i n f e r r e d b y Southwell's method. F o u r typical Southwell plots are shown in Fig. 7, w i t h the straight line p l o t t e d to show the slope from which the critical t h r u s t was calculated. F i g u r e 8 presents the critical t h r u s t vs. p r e s s u r e - a t - b u c k l i n g d a t a superimposed on the predictions of the d e f o r m a t i o n - t h e o r y analysis and the i n c r e m e n t a l - t h e o r y analysis. To minimize

Experimental Mechanics i 271

z•3( o to

Fig. 8--Comparisons between predicted and observed buckling thrust-to-internal-pressu re relations

INTERNAL PRESSURE AT BUCKLING-POUNOS PER SQUARE INCH

COMPARISON BETWEEN TESTS AND DEFORMATION THEORY

clutter, the areas b e t w e e n the highest p r e d i c t e d critical stress (3D) and the lowest (4D1, 4D2) have been indicated b y hatching. Close agreement b e t w e e n the test results and the i n c r e m e n t a l - t h e o r y analysis is evident. There was no u n i f o r m i t y in the b e h a v i o r of specimens cut from the same stock. F o r this reason the specimens are not identified. Discussion

The two m a i n sources of u n c e r t a i n t y about the r e sults a p p e a r to be the v a r i a b i l i t y of m a t e r i a l p r o p e r ties d i s p l a y e d b y the specimens, a n d the effects of plastic d e f o r m a t i o n of the ends of the buckling specimens. T h e v a r i a b i l i t y of m a t e r i a l p r o p e r t i e s m a y have arisen f r o m variations in the composition of the t u b ing or from some, as y e t unexplained, i n a d e q u a c y in the annealing process chosen to reduce the specimens to the "dead soft" recrystallized state. The supposition t h a t the a n n e a l i n g process was i n a d e q u a t e l y specified is s u p p o r t e d b y the loose specifications given the fabricator. No special care was t a k e n in specifying or, p r e s u m a b l y , in p e r f o r m i n g the annealing process. The significance of the v a r i a b i l i t y m u s t r e m a i n unassessed, b u t the close correlation b e t w e e n the ranges of p r e d i c t e d and m e a s u r e d b u c k l i n g loads suggests t h a t the tensile specimens displayed the same p r o p e r t i e s as t h e buckling specimens. Thus, while t h e scatter in the d a t a is s o m e w h a t l a r g e r t h a n is to be desired, the test results a p p e a r valid. Plastic d e f o r m a t i o n of t h e ends of the specimens can be seen upon e x a m i n a t i o n of the specimens tested. This w o u l d h a v e been diminished if the contact surfaces h a d been m a d e of a s o m e w h a t h a r d e r material. F o r t u n a t e l y , the elastic tests and the e x p e r i m e n t a l p r o c e d u r e a p p e a r to have minimized the u n c e r t a i n t y arising from this source. As shown in Fig. 8, the buckling loads for u n p r e s surized tubes fall in the r a n g e of loads p r e d i c t e d b y the analyses. This supports t h e conclusions t h a t the e l a s t i c - b u c k l i n g tests p r o v i d e d a d e q u a t e calibration of the e x p e r i m e n t a l a p p a r a t u s and t h e test p r o c e d u r e a n d t h a t t h e m a t e r i a l - p r o p e r t y r e p r e s e n t a t i o n and the computer analysis are valid. C l e a r l y the d e f o r m a t i o n t h e o r y results are exceedingly conservative c o m p a r e d to the buckling data p r e s e n t e d here. The g r e a t e r t h a n t h r e e - t o - o n e u n d e r -

272 ] July 1973

INTERNAL PRESSURE AT BUCKLING-POUNDS PER SQUARE INCH

COMPARISON BETWEENTESTS AND INCREMENTAL THEORY

estimate d i s p l a y e d at i n t e r n a l pressures above 1500 psi is p a r t i c u l a r l y striking. The discrepancy b e t w e e n t h e o r y and e x p e r i m e n t is all the m o r e r e m a r k a b l e for the fact t h a t a n u m b e r of the buckling tests involved " p r o p o r t i o n a l loading" in which t h r u s t and pressure were increased in proportion. Even in these cases, for which the d e f o r m a t i o n t h e o r y w o u l d be expected to be most accurate, the predictions w e r e grossly conservative. This discrepancy b e t w e e n predictions and test results illustrates the fact t h a t the limitations of the deformation t h e o r y are p r a c t i c a l as well as theoretical. By contrast, t h e r e is excellent a g r e e m e n t b e t w e e n the d a t a and the predictions m a d e using the increm e n t a l theory. The r a n g e of b u c k l i n g loads observed agrees closely w i t h the p r e d i c t e d range. No tube b u c k l e d below the lowest p r e d i c t e d critical load, and only one b u c k l e d above t h e highest p r e d i c t e d load. The d a t a display notable levelling off of the critical load w i t h increasing i n t e r n a l pressure as predicted. In conclusion, these data a p p e a r to confirm the v a l i d i t y of the i n c r e m e n t a l t h e o r y analysis of column buckling of i n t e r n a l l y pressurized tubes. In fact, the a g r e e m e n t between theoretical predictions and e x p e r i m e n t a l d a t a is so striking as to suggest more e x tensive application of the i n c r e m e n t a l t h e o r y in the analysis of buckling in shells and plates. Acknowledgments

The w o r k r e p o r t e d in this p a p e r was p e r f o r m e d as p a r t of the L W B R D e v e l o p m e n t P r o g r a m at the Bettis Atomic P o w e r Laboratory, o p e r a t e d b y Westinghouse Electric Corporation u n d e r contract to t h e United States Atomic E n e r g y Commission. The author wishes to t h a n k D. S. Griffin for his guidance and encouragement. ReJerences i . Shanley, F. R., "'Inelastic Column Theory," ]. o~" the Aeron. Sei., 14, 261-268 (1947). 2. Naghdi, P. M., "'Stress-Strain Relations in Plasticity and Thermoplasticity,'" Plasticity, Proc. 2nd Syrup. on Naval Structural Mech., Ed. by E. H. Lee and P. S. Symonds, Pergamon Press, New York, 121-169 (1960). 3. tIyushin, A. A., "'The Elasto-PIastlc Stability of Plates,'" N.A.C.A. TM Number 1188 (Dee. 1947). 4. Hill, R., The Mathematical Theory of Plasticity, Clarendon Press, Oxford, England (1956). 5. Alcoa Aluminum Handbook, Aluminum Company of America, Pittsburgh, PA (1967). 6. Timoshenko, S. P. and Gere, ]. M., Theory of Elastic Stability, McGraw-Hill Book Company, New York (1961).