BUCKLING STRENGTH OF MONOCOQUE CYLINDERS
C8 4
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avengers XANGMEPÀ: @ NOUN ORS HEM vot ~ đe ZOE Ít Fone Bat zxŠvrz£ ĩ ra Ke 9 ” oe oe ~“ z Fig C8.5 r/t over 2, 000 Fig C8 2-C8.5 (Ref 1) Compressive Buckling Coefficients for Unpressurized Circular Cylinders L0, CRATE 8 SCHWARTZ “zð— To a7 38 Fig C8.6 (Ref 1) Compressive Buckling Coefficients as a Function of r/t 2500 “$608 a Fao — sống, lơ L Nx 3] Ser TRE [tit TT <e S112) iu Ke z.šv rat lo 10 “ ° + ros
Fig C8.7 (Ref 1) Compressive Buckling Stress Coefficients for Unpressurized Circular Cylinders
Trang 2ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES C8 5 Fig C8 8a UNPRESSURIZED, UNSTIFFENED, CIRCULAR CYLINDERS TN AXIAL COMPRESSION (Clamped Ends)
(Taken by Permission from General Dynamics - Fort Worth Structures Manual)
NOTE:
Most of the test data for these curves fall within cylinder
dimensions of L/r 2.0 and 300 = r/t = 1000 Caution
should be used when curves
Trang 3BUCKLING STRENGTH OF MONOCOQUE CYLINDERS Fig C8 8b UNPRESSURIZED, UNSTIFFENED, CIRCULAR CYLINDERS IN AXIAL COMPRESSION (Clamped Ends)
(Taken by Permaisaion from General Dynamics - Fort Worth
Structures Mamai)
NOTE:
‘Moat of the test data for these curves fall within cylinder
dimenatons of L/r~ 2,0 and 300 r/t ©1000 Caution
sbould be used wheo curves tor L/r = 2 are used
90% PROBABILITY CURVES Confidence Levei = 95%
Trang 4
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-
bility values from Pig C&.7 were used AS
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
19 THEORETICAL CURVE
Sat met Kẻ TỶ
fìg C8.10 Compressive Buckling Stress for Pressurized Cireular Cylinders (Ref 1) < PB rs ry Fig C8.11 Increase in Compressive to Internal Pressure
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
3 ay on
Trang 5LL 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
C8.7 Available Design Curves for Bending Based on Experimental Results,
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 BUCKLING STRENGTH OF MONOCOQUE CYLINDERS PRORAELITY CORE 35004000
Fig C8.12 Unpressurized Bending Buckling Stress Co- efficients as a Function of r/t, percent probability and confidence level of 95 percent These curves are from the structures
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
C8 8 Buckling Strength of Circular Cylinders in Bending with Internal Pressure,
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
Trang 6ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES Pig 8.13 UNPRESSURIZED, UNSTIFFENED, CIRCULAR CYLINDERS IN BENDING
(Taken by Permission (rom General Dynamics + Fort Worth Structures Manual) NOTE: Based on test data for cylinder sizes of „35 = L/r= $ and 300 ~ r/t^= 1800 99%, PROBA EILITY CURVES Confidence Level = 3%
Dashed curves are extrapolated into untested regions
1xioÊ rt axo3
Fig C813
trội
Trang 7c3 10, BUCKLING STRENGTH OF MONOCOQUE CYLINDERS Pig C8 13a UNPRESSURIZED, UNSTIFFENED, CIRCULAR CYLINDERS IN BENDING (From General Dynamics - Fort Worth Structures Manual) NOTE: Based on test data for cylinder stzes of «25 = L⁄? ~ 5 and 300 = z/t « 1500, 90% PROBABILITY CURVES Confidence Level = 95%
Trang 8ANALYSIS 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-
pressive stress
The buckling compressive stress for this type of loading is given by the following equation (Ref 12):- kp "n®E : ty - °1£1T- 1Ð TT TT TT TT” c8.7 Foor
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
Trang 11C3.14 ap ae NS Fsor oe? k mE gt ye Fa) TU) # 6 pee = ` | -——— ta Fig C8, 19 Nondimensional-Buckling-Stress Curves for Long Cylinders in Torsion TỊ= Œ/Đ[d - Ua%/ -U*)]*⁄^
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 was: - Ret? + Rp 2b rer crt tere C8.10 where, applied torsion shear stress =
Ret = ratio of siyowable torsion shear Stress
applied internal pressure
= E
Rpsratio of sxrsrmal aydrostatic puckling pressure Note that Rp has 4 negative sign The
value of the external hydrostatic buckling pres- sure can be determined by use of Fig 28.16 Fig C8.22 shows a plot from Equation C8.10 and
its comparison with test data
€8.14 Buckling of Circular Cylinders Under Transverse Shear and Internal Pressure
For this load system, the derived inter- action equation is Similar to Eq 04.10
Rg t+ Rp Fl - 7-25 - re Œ8.11
where,
a 2 —apolied transverse shear stress 3 er erase”
allowable transverse shear stress
Rp 18 same as explained in Article C9.13
Trang 133, 18 BUCKLING STRENGTH OF MONOCOQUE CYLINDERS Pig C8.21 UNPRESSURIZED, UNSTIFFENED, CIRCULAR CYLINDERS IN TORSION {From General Dynamics - Fort Worth Structures Macual) -8 tao NoTE:
Trang 14
ANALYSIS AND DESIGN OF FLIGHT VEHICLE
STRUCTURES C8, 17
Table C8 L
Summary of Interaction Equations tor Buckling of Pressurized and unpressurized Circular Cylinders Shear Combined Equation | ‘eae | “Saunton | Condition | a For Macgia of Satety | | Tongituainal | [ L \ Compression and| Re«Rb=t M 5 *ECRp at Pure Sending aw Longitudinal +Rst?=1 2 2 \ Compression and| 7 BS M.S- ©RCV/R_ THn Foster Kế (See Fig C8.23} Re+VRệ+ 4Rất | y T 1 ksi Longinudingt [ge d= | | | Torsion (See Nate 1} | Pure Bending Ì Rụt* + Rạp” sL | | and Torsion [Be Fig C3 20) | t
Pure Bending Ry? + Rel ° 3 and Transverse | (he Fig 8.23) T Longitudinal | l Compression, 3 | Pure Bending Reo/Re +RR *! l Transverse Shear Pint ~P ext crit.’ psi Longitudinal 3 " Compression, ~ Pure Bendi Re+Rp+Bậy #1 ˆ MS = TM Hà at —— 5 + +ải
Fig C8.22 Effect of Internal Pressure on Torsionai- Buckling Stress Par eon
| R + Re vý/(Re v Ra) ` ARat of Long Cylinders Test Results are from Ref (14) Shear and Torsion | | | Longtmdinal Compression, | Pure Bending, 3
pare eee [Ret Retry Rà + Rẻ~L
C8.15 Buckling of Circular Cylinders Under Combined
Load Systems appiled tensile stress
NOTE 1 Re tông niien buckling 2lowaBLe
it is very seldom that 4 circular cylinder
used in an aerospace structure such ag 4 L—
R, <0 9 _]
missile is subjected to 4 load system causing
only one internal stress system such as axial
stress, vending stress, and shearing stress Therefore it is necessary to De able to safely determine the buckling strength under the practical cases of combined stress systems his problem ts handled by the use of inter-
action equations
Table C8.1 sumarizes the interaction
equations that appear in the structures design manuals of several aerospace companies These
aquations no doubt nave been proven reliable
by checking against test data Fig C8.23
gives a plot of taree interaction curves
mese curves are useful in quickly observing
amather cylinder is weak and to determine
margins of safety
C8.16 Tlustrative problems for Finding the Buckling
Strength of Circular Monocoque Cylinders
problem `- axial Compressive Strength A circular cylinder has 4 radius rT = 50 imcnes, 4 length L = 75 inches, and 2 wall
thickness t of 05 inches The material is
aluminum alloy 2024-13, for which Ey
10,700,000 psi What compressive load will 1t
Trang 15CB 18 BUCKLING STRENGTH OF earry using design curves based probability on 90 percent Solution: 50’,.05 = 1000, L/r = te te mg For = 12 (1 ~ xi —)” (See Eq C8.2) Ve To find the duckling coefficient Ke, we Fig 03.7 gee / aetpvi- vs" esa 73 50 x 05 Vì ~ .5°®= 2140 From Fig C8.7 using 2 = 2140 and r/t > 1000, we read kK, = 280, then 280 e 10,700,090 2 - x Fer “Te (1 - 3) The buckling axial compressive P = Qnrt For = 2n x 50 x 05 x 1205 (92)* = 110 pại -oad = 18950 lb I2 we use the design curves of Fig C8.8
based on 90 percent probability ani 95 percent
confidence, we read for r/t = 100¢ and L/r =
1.5, 6 Fogp/S = ,000121 Thus Poor =
10,700,000 x 000121 = 1295 pst If we multiply this value oy the 95 confidence value, we obtain 1230 psi which is practically the same ag obtained above using Fig C8.7
Pz = 1295 x en x 05 x 50 = 20350 lbs If we
95 percent
r⁄t = 1000
require $9 percent probability and confidence, we use Fig CS.8a For and L/r = 1.5, we read Fs, /E = 700,000 x O00082 = 877 an axial buckling load aris 4 99 percent the buckling loed
used in Problem 1 will roblem What bending
cylinder under 20
The same cylinder be used in this example moment will suckle this percent probacil 2 will de «3, We read ing duckling = 16x 10,700,000 x 05/50 = 1710 251 MONOCOQUE CYLINDERS
The bending moment developed at this
ouckling stress is,
=P Jot Fn r?
M Foon i/r = Foo ™ t
1710 nx 50° x 05 = 670,000 in lb
f we use Fig C8.13a based on 90 percent probability and 95 pereent confidence, we read
for r/t = 1000 and L/r = 1.5 that Pho /= is
.000160 Thus Poor = 10,700,000 x "000180 1710 psi Since it is difficult to read Fig C8.12, it is recommended that Fig C&.léa be
used in design
If we require 99 percent probability, we
use Fig €8.13 and obtain Poop/E = ,O00092, which z 10,700,000 x 00092 =
against 1710 for 90 percent probability
gives Poor 985 psi as
Problem 3 Torsional Strength
Same cylinder as in Problem 1 nhat torsional moment will this cylinder develop Solution Using design curve tn Fig ¢8.18
a 758
7 =— -1⁄_.* =———- = 3® = Ø1
š =frV 1~-g®* = say og V L¬ ‹3” = 2140
For this value of Z, we read the torsional
buckling coefficient k, from Fig C8.18 to be 170 # "=¬ (See Pa, 08.9) PSter F 12 (1= v 1 (See Eq, 08.9) 170nZ x 10,700,000 = + s T2 Pstcr 7“ 12 (1 - 25 CO” = 785 psi
Trang 16
ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES C8 18
bility, we read Fgt/E = 000060 which gives - =e 527+,481+V/(,587+.481)»4x.13% ee TP non, aa) edn Vue le Ls =e .055
Fst = 642 psi which, in turn, gives allowable
torque’ T = 504000 in Lbs
Problem 4, Combined Compression and Bending The cylinder of Problem 1 is subjected to an axial compressive load P of 10,700 lbs and a bending moment M of 282,000 in lbs what is the margin of safety under this combined load- ing for 90 percent probability and 95 percent
confidence
Solution, The interaction equation for this
type of combined loading from Table C8.1 is, Re + Rp =1 Re = P/Pg (Pg from Problem 1 solution is 20860 1b.) Ro = 10700/20350 3 527 Ry = M/M (From Problem 2 results:~ My = 670,000 in.ibs.) Rp = 282,000/670,000 = 421
Re + Rp = 527 + 421 = 948 Since the result
is less than 1.0, we have a small margin of safety =——— 2 = WS gop g 7 bt ygag 7 iF 2068 Problem 5 Combined Compression, Bending and Torsion
Same cylinder as in Problem 1 The load- ing {s the same as Problem 4 plus a torsional moment T of 89300 in, 1b Fing the M.S under this combined loading for 90% probability Solution
The interaction equation for this com- bined load system from Table C8.1 is, Re t RE + Rep FL - ete (a) From Problem 4, R, = 527, Rp Rot = T⁄Ta (From Problem 3 r Tạ 18 688,000 in ib.) Ret = 89500/688,000 = 1Z Subt in fq (A}:-
-527 + ,421 + 15'° = 965, (since this result
is less than 1.0, we have a small margin of safety From Table C8.1,
2
WS 5 E Tapr ites Ry) ee? 7?
Problem 6 Combined Compression, Bending,
Torsion and Transverse Shear
The same cylinder as in Problem 1 is sub- jected to the following loads:-
P 10700 lbs compression, M = 282,000 in.1bs., T = 715000 in.lb., V = 1790 lb., (transverse external shear), Will the cylinder carry this combined loading under 90 percent probability
Hu
Solution:
From Table C8.1 the interaction equation for this combined loading is,
Ro + Ret + V Rg? + Ry =1
R„ = 10700/20350 = 527 Rp = 282000/670000 = 421
Ret = 71500/688000 = 104
The equation for shear stress due to transverse shear load V is, Vv fs = T ⁄/ ydA I for cylinder section = mrt areas art /yQdA = rt x 6366r 225% _ (LP see - a a fs = ast (2 ret) = Fe fs = 1790/nx50x 05 = 228 pst
From Article C8.12 we use buckling co-
efficient for transverse shear to be 1.25 times
that for torsional buckling
From Problem 3 the torsional buckling stress calculated to be Fg = 876 psi Therefore the transferse shear buckling stress is 1.25 x 876 = 1095 pst, = Fs Rs = ïs/#s = 228/1095 = 208 Substituting in Equation (B), «527 + 1047 + ¥.208% + 421° = 975 This result ts less than 1.0 so a small margin of safety exists Problem 7 Combined Compression, Bending and Transverse Shear
Same cylinder and loads as in Problem 6,
but without tne torsional moment load
Trang 17BUCKLING STRENGTH OF Ro +V Rs? tr Rp FL 2 eee ee eek (0) Re = 527, Rp 2 421 and Rg = 208 (trom Problem 4) Subt in Equation (C), 527 + /,208% + 4217 = 964 (less than 1.0 ‘hus cylinder will not buckle.)
Problem 8 Combined Bending and Torsion Using the sane cylinder as in previous problems, the combined loading is:- M = 478,000 in.lbs., T = 358,000 in lb Will the cylinder buckle under this loading Solution: Rp = 470,000/870,000 = 701 Ret = 358,000/S88,000 = 520 The interaction equation from Table ¢8.1 is, Rp”'® + Rap =1
-701°°° + ,ö20* = 88 less than 1.0, therefore cylinder will not buckle
The margin of safety can be determined graphically by using Fig C3.21 as follows:-
Find point (a) as indicated on Pig, 08.21
using Rp = 701 and Reg = 520 Draw line from origin o through point (a) and extend it
to intersect interaction curve at point (b) Projecting downward and horizontally from point (b), we read Ry = 78 and Ret = 58
The factor of utilization U = 70/.78 = 52/.58 = 895, Therefore Margin of Safety =
(1U) ~ 1= (1⁄,895) ~ L3 ,12,
Problem 9 Combined Compression and Torston, Same cylinder as in previous problems The loads are P = 12900 lbs T = 358000 1n.1bs, What {s M.S Solution: The interaction equation trom Table 98.1 15, R„ + Rsry? =1 Ry *= 12900/20550 = s34 „520 +524 + 5207 = 3C4 (less than 1.0, there~ fore no buckling.) 2 eS syle M = 4S «634 + (.634* + 4 x 5207)472 7 - „08 MONOCOQUE CYLINDERS C8.17 Problems Involving Internal Pressure with External Loadings
The same cylinder and material as was used
in Problems 1 to 9 will be used in the following example problems, namely, r = 50, t= 05, Las
78, 210.7 x i0%
Problem 10, Axial Compression with Internal
Pressure
The cylinder is subjected to an axial com~ pressive load of 50000 lbs and an internal pressure of 5 lbs per sq in What is the margin of safety under thts combined load system with 90 percent probability
Solution, From Problem 1, the buckling com-
pressive stress Foop With no internal pressure was 1295 psi To obtain the increase in the
compressive buckling due to the internal pres—
Sure, we use Fig C8.11 The horizontal scale parameter is, Rimes 5 30,2_, tp * 10,790,006 (0g) = 4.87 From Fig C8.11, we raad on vertical scale, aFy 7 er Gp = 21, whence fy _ , ¬ aFeon * (2X 21x 10,700,000 x 05)/50 = 22507] pst
Since the stress is below the proportional limit stress the plasticity correction is zero
or = 1.0 Thus aFonn ® 2250 psi
Then the total buckling stress with the internal pressure 1s 1285 + 2250 = 3545 psi
Now the internal pressure produces 4 longi- tudinal tensile stress in the cylinder wall, This tensile stress is,
P, = pr/2t = 5 x 50/2 x 05 = 2500 psi
This tensile stress must be cancelled before
walls can be subjected to 4 compressive stress due to an external compressive axial load The allowable failing load P, thus equals,
Pa = Z7t (Foo + Fenn * Pe)
= én x 50 x 05 (3545 + 2500) = 65000 15,
= (P,/P)-1 = (95000/50000) -2 = 30 Tats result shows the tremendous effect of
internal pressure in increasing the compressive strength of 4 thin walled unstiffened circular cylinder With no internal pressure, the failing compressive load as calculated in
Trang 18ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES
Problem 1 was cnly 20350 lbs as compared to $5000 los with 5 psi internal pressure
Problem ll Bending with Internal Pressure If the same cylinder as used in the pre- ceding example problems 1s subjected to 2
banding moment of 3,000,000 in lbs when
internal pressure is S psi, what is the M.S
use 90 percent probability
For bending without internal
ure, the bending buckling stress Foor was found in Problem 2 to be 1710 psi The in- crease in this bending buckling stress due to internal pressure is obtained from the curves in Fig C8.14 The horizontal scale parameter ts,
Bz, 5 50 j8 aS)" = 16,760,000 ‘cos? * 4-87 From fig C5.14, we read
(B = 51 for 90 sercent probability
= (.51x10,700,000 x 05)/50 =
5350 psi
AS calculated in Prodlem 11, the internal pressure of 5 psi oreduces a longitudinal tensiis stress in the cylinder wall equal to Fe = 2500 psi The allowable bending moment that cylinder can develop is,
=F F a
Ma 5(total)1⁄ ® Fac vai X KẾ
Po(total) = Fo, * APog * Fe F
1710 + S350 + 2506 = 9560 pst Mạ = 2560xnx50?x 06 = 2760000 in ibs
M.S 5 (Mg/M) -1 = (2750000/3000000} -1 = 25
in note the tremendous effect of
internal 2rassure on tha bending strength
with no internal pressure, the failing bending moment was 570000 as compared to 2740000 with 5 281 internal oressure C8 21 Solution:
Interaction equation is, Rg + Rp Fl
From Problem 10, the allowable load Py for compression plus 5 psi, internal pressure is $5000 lns
R, = P/Pg = 51000/35000 = 536
From Problem 11, the allowable bending moment Mg with an internal pressure is 2,760,000
in lbs
Ry = M/M, * 1475000/3760000 = 392
Ro + Rp = 536 + 592 = 928 (less than 1, thus will not buckle)
M.S = (1/,928) - 1 = 08
Problem 13, Torsion with Internal Pressure
If the same cylinder is subjected to an internal pressure of 5 psi, what pure torsional
moment Tz will it carry without buckling Solution:
With no internal pressure, the torsional
shearing buckling stress as calculated in
Problem 3 was 876 psi for SO percent proba- bility
The increase in shear buckling stress due to the internal pressure sill be determined by use of interaction curve in Fig C9.22 To use this curve, the external nydrostatic pressure
(Pext,) to cause buckling of the cylinder must
be determined This buckling stress Poop is
determined by equation Z8.7, which is, Ky ne ta 2 ° Tp ty (r- Foor * TEC = ⁄ẹ `) TL) The buckling coeffictant Ky 1s found by use of Fiz 03.15, a
Ze ea ~ U,*)*/* = 2140 (See Prodlem 1)
From Fig C3.15, we read Ky = 60
Trang 19C8 22 BUCKLING STRENGTH OF MONOCOQUE CYLINDERS ™ not €8.17 Buckling Strength of Thin-Walled (Monocoque) Conical Shells
In a multi-stage missile, the uoper stages normally have smaller diameters than the lower
stages, thus 2 conical shell provides 4 type of Structure to permit changing the missile
diameter between the various stages
AS for the case of the cylindrical
monocoque shell, the theoretical analysis for
Pers So ti inv ion ct ¢ in Pig, - + - “ %
2 one bon ng HH t8 the buckling strength of conical shells oy alos the vs, s 30 on 35 9erosat pro5ab111+y| either the small or large deflection theories
belou the curve, A 9 vee peak prose “| givas results considerably above those en
WOuld give a curve considerably below the :
curve shom vould seen ta bs in ordsr or os xaed02 eso ost A correction factor of around 8 oe ees es 7 by tests, thus design of conical shells 2 present is based crimarily on results obtained by tests The material presented in this
a“
chapter will thus be limited to tresenting a
: far gn buckling curves
calculated sy w design buckling curvas
C8.18 Allowable Compressive Buckling Stress for Thin-Walled Conical Shelis,
rig C3.24 shows design curves for the Compr2ssi7e buckling stress 9£ ä conical shell
as derived from a statistical study of tast 2240 psi data by Hausrath and Dittoa (Re? 1€) The
expression for the buckling compressive stress
Trang 20ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES Fer 4 aa) 30 100 500 1,000 5,000 10,000 ~l©i Tìg C8 24 (Ref 16) The "Á" and "B8" curves are recommended for
practical design Tne "A" level curve is
recs for use for those structures where # re would result in catastrophic
18: r injury to personnel The stress level 3i the small end of the cone should de checked to precludes the pessibility of an early failure
precipitated by inelastic stresses
C8.19 Additional Design Buckling Curves for Thin- Walled Conical Shells
Figs C8.25, 26, 27 gives curves for the allowable buckling stress for comical shells in compression,
torsion respectively These curves
produced by permission of the Boeing
y from their report 02-3617 Fig C8.28
gn curve for the allowable reassure for thin-salled so from same Boging Renort used for minuteman interstage and the conical srells had ad by method in 2 ma 3 1 shells, curves were irt design 5 1 ners des DEPINITION OF TERMS :- a allowable comprass {pst} 1owab1e 3112waD1e T ct 4w 0à Hộ HỆ BỊ oO ow wc o Oo O00 8.23 der Combi ned Loading
From Boeing Report (D2-3617) the of simultaneous compression, ending, +
overpressure can be considered using * action equation:- (Re + By + Ry)*** # Rgh"? = 1 where, Ro = f¿/F¿, Rp = fp/fb, Ry = mm Rq = WGer C8.20 Example Problem
A conical snell with wall thickness t = 05 and other dimensions shown in Pig (a) is
fabricated
alloy (EZ = 10,700,000) from aluminum
Determine the following values :- Je-20" | — (1) A11oweble compres~ T † sive load Py „ I-90" i | (2) Allowable bending ™ _—— monent May | 1.17 (3) A11owable torsional Fig a moment Tor External duckling pressure der Solution: 10/cos 10° = 10 10.16/.05 = 203 20/cos 10° = 20.42/10.16 = 2 L/o = To find Pe:
From Fig, C8.25 with p/t = 203 and L/o =
we read Fo x 10°/E equals 1.20 Whence 1.20 ¥10,700,000/1000 = 12,920 psi Then Pe 2nrt 12820 x £n x œ *Ú *J tà Oo 10 x 05 = 40,500 lbs To find Mar:-
Trang 23
C8 26 BUCKLING STRENGTH OF MONOCOQUE CYLINDERS
To find external buckling pressure dori Ore Sor Use is made of Fig C8.28 - 10+ 13.52 „ Pave ~ Z cos 1o™ * 1.92 in Pavg/t = 11.92/.05 = 238 The lower scale parameter of Fig C8.28 is, 2 ơ=.đ= Pave 20,327 a 2= Toe x 08 VÌ - -3* = 660
From Fig C8.28 for Z = 660, we read Ky = 27.5 The equation for buckling pressure
from Fig C8.28 Is, Ky Eton? der = Pave L?ias 1) 27.8 x 10,700,000 x 05° xn? _ “T.32 x 20.22 x12 (1-55) 7 5:37 pst PROBLEMS
(1) A monocoque circular eylinder {s fabri- cated from aluminum alloy for which RE 18 10,500,000 Wall thickness is 064, Cylinder diameter is 80 inches Length is 60 inches Using a probability factor of 90 percent, determine the following values:-
(a) Buckling compressive load for
cylinder
(0) Buckling bedding moment for cylinder {c) Buckling torsional moment for
cylinder
(2) The cylinder in Problem (1) 18 subjected te the following design loads, W111 the cylinder fail under these various design conditions:- Condition 1 P 30000 lbs compression M = 50000 in lbs Condition 2 P = 25000 lbs compression M = 45000 in lbs T = 50000 in lbs
(3) If the cylinder in Problem 1 ts
pressurized to 6 psi, what ultimate compressive load will it carry What ultimate bending moment wili it carry What ultimate torsional moment will it
carry
(4) A conical shell nas a small end diameter
of 30 inches, a length of 30 Inches, a Semi-vertex angle of 8 degrees and a wall
thickness of 057, Material is aluminum
alloy with E = 10,500,000 Determine the ultimate compressive load, the ultimate bending moment, and the ultimate torsional moment the cone will sustain when each load 1s acting Separately
Determine the external pressure that will buckle the cone
REFERENCES
(1) Harris, Seurer, Skeene and Benjamin The Stability of Thin-Walled Unstiffened Circular Cylinders Under Axial
Compression Jour Aero Sciences Vol 24, August, 1957
(3) Donnell, L H Stability of Thin-Walled Tubes Under Torsion NACA Report 479
(3) Batdorf, 3 B A Simplified Method of Elastic Stability
oon for Thin Cylindrical Shella NACA Report 874 1947)
(4) Weingarten, Morgan and Slede Final Report of Design Criteria for Elastic Stability of Thin Shell Structures
Contract AFO4(647)619 Air Force Ballistic Missile
Division) Space Tech, Labs Dec., 1960 (8) Lo, Crate and Schwartz Buckling of Thin Walled
Cylinders Under Axial Compression and Internal Pres- sure NACA T.N 2021, Jan 1950
(8) VonKarman and Tsien The Buckling of Thin Cylindrical Shells Under Axial Compression Jour of Aero Sciences Vol 8, June 1941
(T} Tsein, A Theory for Buckling of Thin Shells Jour of Aero Sciences Vol 9, August 1942,
(8) Fung and Sechier Buckling of Thin Walled Circular Cylinders Under Axial Compression and Internal Pres- sure, Jour of Aero Sciences Vol 24, May 1957, {9) Harris, Seurer, Skeene and Benjamin The Bending Stability of Thin Walled Unetiffened Circular Cylinders Including Effects of Internal Pregsure Jour of Aero Sciences, Vol 25, May 1958,
(10) Stede and Weingarten On the Bending of Circular Cylindrical Shells Under Pure Bending Jour of Applied Mechanics, Vol 28, March 1961 (11) Flugge Die Slabititat der Kruszylinderschole,
Tngenieur Archiv Vol 3, 1932,
(12) Gerard & Becker Handbook of Structural Stability Part IIL Buckling of Curved Plates and Shells NACA T.N 3783, Aug 1957
(13) Hopkins and Brown The Effect of Internal Pressure on the Initial Buckling of Thin Walled Circular Cylinders Under Torsion R& M No 2423, British A.R.C 1946, (14) Crate, Batdort and Baab The Effect of Internal
Pressure on the Buckling Stress of Thin Walled Circular Cylinders Under Torsion NACA WRL-67, 1940 (15) Batdorf, Stein and Schildcrout Critical Stress of Thin
Walled Cylinders in Torsion NACA T.N 1344 (1947) (16) Hausrath & Dittoe Development of Design Strength
Levels for the Elastic Stability of Monocoque Cones Under Axial Compression NASA T.N D-1510 (1962), (17) Heas and Garber Stability of Ring Stiffened Conical
Trang 24
CHAPTER Co
BUCKLING STRENGTH OF CURVED SHEET PANELS ULTIMATE STRENGTH OF STIFFENED CURVED SHEET STRUCTURES PART 1
C9.1 Introduction
Curved sheet panels represent a common
external part of flight vehicle structures Examples are the skin of the fuselage and the missile If the curved sheet has no longi- tudinal stiffeners, failure will occur when buckling occurs If the curved sheet has stiffening elements attached then the composite unit will not fail on the development of sheet buckling, that is, the composite unit has an ultimate strength much greater than the load which caused initial buckling of the curved Sheet panels between the stiffening units commonly referred to as sheet stiffeners In some flight vehicle designs it may be specified that under a certain percentage of the limit loads that no buckling of the curved sheet
panels shall occur Thus {t is necessary to be able to determine what stresses will cause curved sheet panels to buckle and also to determine what external loads will cause a stiffened sheet panel to fail
C9.2 State of the Theory
In Chapter A18, the small deflection linear theory was used to determine the com- pressive buckling stress of flat sheet panels The theoretical results compare favorably with
experimental test results However, when the
same theory is applied to unstiffened curved sheet panels or thin-walled cylinders, the theoretical results are considerably above the test results on such units A large deflection
theory gives closer comparison to test results, but this theory {nvolves a consideration of
initial imperfections in the sheet and thus an unknown quantity For the application of both small and large deflection theortes to the buckling of curved sheet and cylinders, the
Teader is referred to the references listed at
the end of Chapter CS, particularly those
veports which give the results of such in-
vestigators as Donnell, Batdorf and Gerard C9.3 Compressive Buciling Stress of Curved
Sheet Panels
The expression for the buckling stress
under axial compression 1s of the same form as
for flat sheets, the value of the buckling coefficient Kp naving a higher value
_ ke wee
Poor = 12 (1 - YQ")
2
củ) T~~~—=~ 09.1
BUCKLING STRENGTH OF CURVED SHEET PANELS Fig C9.1 gives curves for determining the buckling coefficient Ke The theoretical derived curve is shown along with the recom- mended design curves which were dictated by test results These curves apply for buckling
stresses in the elastic range
C9.4 Shear Buckling Stress of Curved Sheet Panels
The equation for the buckling shear stress is,
_ Kg n* ta
Fsor "42 (l= Van (vo?
Pigs C9.2 to C9.5 gives curves for finding
the buckling stress coefficient K, to use in
Equation C9.2 These curves apply when buckling
stresses are in the elastic range
C9.5 Buckling Strength of Curved Sheet Panels Under Combined Axial Compression and Shear
The studies by Schildcrout and Stein
{Ref 1) gave the following interaction equation
for combined longitudinal compression and shear on curved sheet panels:-
Rg* + RE FL ~ + ee ee re etre fs > RLF fe
where, Rg =
F scr Peer
If the longitudinal stress is tension instead of compression, then Ry, is considered as 4 negative compression using the compression allowable The equation for margin of safety
18:—
- 2
mẽ =: bt TTT c9.4 C9.6 Compressive Buckling Stress of Curved Panels
with Internal Pressure
As for the case of monocoque cylinders,
internal outward pressure increases the axial buckling compressive stress of the curved sheet
panel Rafel and Sandlin (Ref 3) and Rafel
(Ref 4) performed tests on curved panels under
axial compression and internal outward pressure The results correlate with the interaction equation:-