Bruhn - Analysis And Design Of Flight Vehicles Structures Episode 3 part 7 pptx

One of the first experimental and theoretical investigations of the effect of internal pressure upon the puckling compressive stress of monocoque cylinders was by Lo, Crate and Schwartz

BUCKLING STRENGTH OF MONOCOQUE CYLINDERS

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Fig C8.7 (Ref 1) Compressive Buckling Stress Coefficients

for Unpressurized Circular Cylinders

(90 Percent Probability)

should be used when curves

tor L/r = 2 are used

ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES

cylinders in missile structures has become &

common type of missile structural design The

famous Atlas missile was one of the first to

use @ pressurized monocoque type of structure

The major reason for the large discrepancy

between the actual test strength and the

theoretical strength by the linear small

deflection theory that is generally accepted

4s that the discrepancy is due to geometrical

imperfections and the associated stress con-

centrations Now large internal pressures

should smooth out such imperfections and

approach a perfect cylinder and thus the re~

sulting buckling strength should approach that

given by the linear small deflection theory

However, much of the available test data for

pressurized cylinders gives values below that

given by the linear small deflection theory

(See Fig Œ8.1a for buckling action of

pressurized cylinder.)

One of the first experimental and

theoretical investigations of the effect of

internal pressure upon the puckling compressive

stress of monocoque cylinders was by Lo, Crate

and Schwartz (Ref 5) They analyzed the

problem of long pressurized cylinders using an

extensicn of the large deflection theory of

Yon Karman and Tsien (Ref 6) Plotting their

results in terms of the non-dimensional

parameters (p/E)(r/t)* and (Foy/B) (n/t), they

found that the buckling coefficient C increased

*rom the Tsien value (Ref 7) of 0.375 at zero

tnternal pressure to the maximum classical

value of 0.605 at (p/E)(r/t)*= 0.169 Fig

Œ8.10 taken from (Ref 1) shows the large

deflection theoretical curve Also shown are

the experimental values obtained in (Ref 5)

as well as those obtained oy the investigators

in (Ref 1) and dy other investigators in

(Res 3), From Fig C6.10 it ts apparent that

large discrepancies exist between the theo-

retical predicted values and the experimental

values Lo, Crate and Schwartz suggested that

better correlation with test results could be

obtained 1f the increment in the buckling

stress parameter (APer/E) (r/t) were plotted as

a function of the pressure parameter as shown

in Fig C8.11 The increase in the critical

stress OFor due to the internal pressure

directly represents the beneficial effact of

the internal pressure The total critical

stress is thus obtained by adding the critical

gtress for the unpressurized cylinder to the

increase in the critical stress due to the

internal pressure In order to plot the test

points in Fig C8.il, it was necessary to

determine the unpressurized critical buckling

stress for each test The $0 percent ?reba-

shown in Fig Cé.1l, the general trend of the

test data agrees fairly well with the theo~

retical curve Tie investigations in Ref 1

Ca.7

nave shown a best fit curve and 4 80 percent

probability curve obtained by a statistical approach and this 90 percent curve in Fig

8.11 is recommended as 3 design curve for taking into account the effect of internal

pressure

10

BUCHY (NAA) FUNG & SEGHLER

LO, CRATE & SCHWARTZ

m———aNÀA

Sat met Kẻ

TỶ fìg C8.10 Compressive Buckling Stress for Pressurized

Cireular Cylinders (Ref 1)

Buckling Stress Due

BUCKLING OF MONOCOQUE CIRCULAR CYLINDERS UNDER PURE SENDING

C8.6 Introduction

Flight venicles are subjected to forces in

flight and in ground operations that cause bend-

ing action on the structure, thus it is necessary

to know the bending strength of cylinders Two

rather extremes fave been used in past design

practice One design assumption takes the value

of the bending buckling stress as 1.3 times the buckling stress under axial compression The other assumption is to assume the bending buckling

LL a

C8.8

Stress 18 equal to the

duckling stress axial compressive

The first assumption is con

Sidered by some designers as somewhat uncon-

Servative while the second assumption is no

doubt somewhat conservative

It is relatively recent (Ref 10) that a

small deflection approach has been completely

solved for a cylinder in bending Tests of

cylinders in bending show that the theoretical

result 1s lower than the test results but

higher than the buckling stress in axial com

Pression No large deflection analysis which

involves a consideration of initial lmper~

fections of the cylinder has deen formulated

to date for the buckling strength in bending,

Since the stress in bending varies trom zero

at the neutral axis to a maximum at the most

remote élement, the lower probability of

imperfections occurring within the smaller

highest stressed region would lead one to

conclude that higher buckling stresses in

bending, as compared to the buckling stress in

axial compression, should be expected

The same investigators (Suer, Harris,

Skene and Benjamin) that carried out tests on

eylinder in axial compression (Ref 1), have

also carried on an extensive investigation of

the duckling strength of monocoque cylindrical

cylinders in pure Đending (see Rer 9)

AS originally developed by ?lugge (Re#.11)

for long cylinders, the buckling stress in

bending is expressed as:-

Poop = CpB(t/r)

The theoretical vilue of the bending

buckling cosffictent as found by Flugge was

about 30 percent higher than the corresponding

Classical buckling coefficient or 0.605 in

axi2l compression

Fig CS.12 (from Ref 9) gives a plot of

considerable test data and a plot of Ch versus (r/E)

A dest zit curve, a 90 percent proba-

Dility curve and a 39 percent probability curve,

are Shown The dashed curve is a plot of the

90 percent probability curve as previously

given in Fig ¢8.6 for Duckling in axial com-

pression, thus ziving a comparison between

bending and compressive Duckling streneths As

indicated by the plotted test points, the test

data above an r/t value of 1500 1s quite

limited, thus the accuracy of the curves is

Somewhat unknown,

Figs C8$.13 and Œ8.1Za give convenient

design curves for finding the bending buckling

Stress based on 99 percent probability and 90

manual of the General Dynamics Corp (Fort Worth.) heir manual states that most of the

test data upon which the curves are based fall

Within the range of cylinder dimensions 25 <

L/T <5, and 300 < r⁄t < 1500, and the curves are based on tests of steel, aluminum and brass cylinders only

The published information on the buckling Strength of circular eylinders in bending with internal pressure ts very limited and the status

of theoretical studies to date leave much un- known regarding this subject

Reference 9 gives the results of a series

of tests of circular cylinders in bending with internal pressure Fig C8.14 1s taken from that published report In Fig CS.14 the ax~

perimental data are plotted in terms or the increment ACpp to the buckling coefficient Ấp

The increase in the buckling stress coefficient

4Chp represents the beneficial effect of

internal pressure, The total value of the ouckling coefficient 1s obtained by adding the buckling coefficient for unpressurized cylinders

to the increase in the buckling coefficient due

to the internal pressure, in order to plot the

data, 1t was first necessary to determine the

umpressurized Duckling coefficient for gach Specimen The SO percent Probability design curve of Fig, C8.12 was used for this purpose

The direct benefit of lateral internal pressure

to the stability or cylinders in bending is indicated by those Specimens with no net axial stress (the balanced Specimens) represented by the circular Symbols at large values of the pressure parameter, the additional benefit of che axial pretention is Clearly demonstrated by

the large increase tn ACpp of the pretensioned

Pig C8 13a UNPRESSURIZED, UNSTIFFENED, CIRCULAR CYLINDERS

Dashed curves are extrapolated into untested regions

ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES C8.11

specimens, represented by the triangular

symbols, over that for balanced specimens The

limiting value of the increase in the buckling

stress coefficient for pretensioned pressurized

cylinders at very high values of the pressure

parameter is given by the line Say = pr/2t

The analysis of the pressurized cylinder

data was achieved by selecting a best fit curve

for those specimens in which the axial pre-

tension was balanced This curve (shown in

Fig C8,14) was selected by the investigators

as best indicating the general trend of the

experimental data At large values of the

pressure parameter, the curve is drawn to

approach an asymptote agreement between the

best fit curve and experimental data is

apparent in Fig C8.14 A statistical analysis

of the test data was performed for the speci-

mens with no axial pretension to establish the

90 percent probability design curve shown in

Fig C8.14 Because data were available only

from the tests made by the investigators, they

indicate the sample may not be representative

and a lower probability curve should perhaps

be used for design purposes The data was not

considered sufficient to permit a statistical

analysis of the pretensioned test data, and

therefore they suggest a lower bound curve be

used in the design of pretensitoned cylinder

Additional tests are needed too for unpres-

surized cylinders with r/t ratios greater than

1500 to verify the shape of the design curve

0 «Fa 248L PRETENSON STRESS + g/2?

‘

Fig C8.14 Increase in Bending Buckling Stress Coefficients

Due to Internal Pressure

BUCKLING OF MONOCOQUE CIRCULAR CYLINDERS UNDER EXTERNAL PRESSURE

c8.9 External Hydrostatic Pressure

Under this type of loading, the cylinder

shell is placed in circumferential compressive

stress equal to twice the longitudinal com-

Values of the buckling coefficient kp are

given in Fig C8.15 Equation C8.7 is for buckling stresses below the proportional limit

stress of the material

C8, 10 External Radial Pressure

Under an inward acting radial pressure only

the circumferential compressive stress produced

is fg = pr/t where p is the pressure

The buckling stress under this type of loading from (Ref 12) is,

_ ky nt? E t2

Foor = TET

Values of the buckling coefficient ky are

given in Fig C8.16 Equation 8.8 is for buckling stresses within the proportional limit

stress of the material

C8.11 Buckling of Monocoque Circular Cylinders Under Pure Torsion

Fig C8.17 (from Ref 12) shows the results

of tests of thin walled circular cylinders under

pure torsion The theoretical curve in Fig

C8.17 is due to the work of Batdorf, Stein and Schildcrout (Ref 15) Their theoretical in- vestigation utilized a modified form of the single equilibrium form of Donneil (Ref 2) and

by use of Galerkins Method obtained the curve shown in Fig C8.17 This theoretical curve falls above the test results and thus for safety

a lowered curve should be used for design

purposes Fig C8.18 shows a design curve which

appears in the structural design manuals of 4 number of aerospace companies

The torsional buckling stress is given by the equation:-

Fig C8.19 gives the value of the torsion

buckling coefficient ky and applies for buckling

below the proportional limit stress

To correct for plasticity effect when buckling stress is above the proportional limit

stress, the non-dimensional chart of Fig C8.19 can be used Figs C3.20 and C8.21 give other convenient design curves involving duckling stresses for 90 and 99 percent probability

tưà tn yo

4 Fig C8.17 Comparison of Test Data and Theory for Simply Supported Circular Cylinders

DO Fig C8.18 Buckling of Simply Supported Circular Cylinders in Torsion

or Transverse Shear

C8, 13

C8.12 Buckling Under Transverse Shear

Shear stresses are also produced under

bending due to transverse loads These shear

stresses are maximum at the neutral axis and

zero at the most remote portion of the cylinder

wall, whereas the torsional shear stress is

uniform over the entire cylinder wall Limited

tests indicate a higher buckling shear stress

under a transverse shear loading as compared

to the torsional buckling stress A general

procedure in industry is to increase the shear

buckling stress under torsion by using 1.25

times kp Thus in Fig C&.18 find ky for

buckling under torsion and then multiply it by

1.25 in using Equation C8.9 to find buckling

stress under transverse shear

C8.13 Buckling of Circular Cylinders Under Pure Torsion

With Internal Pressure

Internal pressure places the cylinder

walls in tension, thus the torsional buckling

stress is increased as torsional buckling is

due to the compressive stresses that are

produced under shear forces

Hopkins and Brown (Ref 13}, using

Donnell’s equation, calculated the effect of

internal pressure on the buckling stress of

øirocular cylinders in torsion and the results

were in fair agreement with test results

Crate, Batdorf and Baab (Ref 14),

utilized an empirical interaction aquation to

git test data The derived interaction equation

Ret = ratio of siyowable torsion shear Stress

applied internal pressure

a 2 —apolied transverse shear stress 3 er erase”

allowable transverse shear stress

Rp 18 same as explained in Article C9.13

G00 c6 TH g

ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES C8 15

Fig C3 20 UNPRESSURIZED, UNSTIFPENED, CIRCULAR CYLINDERS

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Fig C8 20