March 2014 

NASA/TM–2014-218176 

Buckling Analysis for Stiffened Anisotropic 

Circular Cylinders Based on Sanders’ 

Nonlinear Shell Theory 

Michael P. Nemeth 

Langley Research Center, Hampton, Virginia 
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March 2014 

NASA/TM–2014-218176 

Buckling Analysis for Stiffened Anisotropic 

Circular Cylinders Based on Sanders’ 

Nonlinear Shell Theory 

Michael P. Nemeth 

Langley Research Center, Hampton, Virginia 
Available from: 

NASA Center for AeroSpace Information 

7115 Standard Drive 

Hanover, MD 21076-1320 

443-757-5802 


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Nonlinear and linear-bifurcation buckling equations for elastic, geometrically perfect, rightcircular cylindrical shells subjected to combined loads is presented. The loads include 

compression, shear, and uniform external and hydrostatic pressure. The analysis includes 

constitutive equations that are applicable to stiffened or unstiffened cylinders made from isotropic 

or laminated-composite materials. Complete sets of equations are presented for the nonlinear 

boundary-value problem of shell buckling and the corresponding prebuckling and linearbifurcation buckling problems that are based on Sanders’ shell theory for "small" strains and 

"moderately small" rotations. 

Based on these equations, a three-parameter approximate Rayleigh-Ritz solution and a 

classical solution to the buckling problem are presented for cylinders with simply supported 

edges. Extensive comparisons of results obtained from these solutions with published results are 

also presented for a wide range of cylinder constructions. These comparisons include laminatedcomposite cylinders with a wide variety of shell-wall orthotropies and anisotropies. Numerous 

results are also given that show the discepancies between the results obtained by using Donnell’s 

equations and variants of Sanders’ equations. For some cases, nondimensional parameters are 

identified and "master" curves are presented that facilitate the concise representation of results. 

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x

y

R

h

dr

ds

Shell wall

Ring

Stringer

)LJXUH6WLIIHQHUGHWDLOV

x

y

ζ

er

Shell wall

h

2

Centroid

F5LQJFURVVVHFWLRQ

D6WLIIHQHUOD\RXW

es

Shell wall

h

2

Centroid

E6WULQJHUFURVVVHFWLRQ


)LJXUH&LUFXPIUHQWLDOVWUHVVUHVXOWDQWGLVWULEXWLRQDQGORFDOL]HGEHQGLQJ]RQHV

LQDFRPSUHVVLRQORDGHGF\OLQGHUZLWKHQGVUHVWUDLQHGDJDLQVWUDGLDOGLVSODFHPHQW

Zone of localized

bending Zone of localized

bending

Central zone of uniform

stress and deformation

n = 1 n = 2 n = 3 n = 4

n = 5 n = 6 n = 7 n = 8

n = 9 n = 10 n = 11 n = 12

)LJXUH&LUFXPIHUHQWLDOZDYHIRUPVXVHGLQHTXDWLRQF


L

2πR mcr = 2

Axial half-wave length

)LJXUH7\SLFDOEXFNOHSDWWHUQRIDFRPSUHVVLRQORDGHGF\OLQGHU

Buckle

D7KUHHGLPHQVLRQDOUHQGHULQJRIDEXFNOLQJPRGH

E&RQWRXUSORWRIWKHUDGLDOGLVSODFHPHQW

ncr = 4


E&RQWRXUSORWRIWKHUDGLDOGLVSODFHPHQW

D7KUHHGLPHQVLRQDOUHQGHULQJRIDVNHZHGEXFNOLQJPRGH

2

 

πR

)LJXUH7\SLFDOVNHZHGEXFNOHSDWWHUQRIDFRPSUHVVLRQORDGHGDQLVRWURSLFF\OLQGHURUDQRUWKRWURSLF

F\OLQGHUVXEMHFWHGWRVKHDUORDGV

L

mcr = 1

ncr = 4

Nodal line

x

y

 

φ


)LJXUH5HSUHVHQWDWLYHF\OLQGHUVFRUUHVSRQGLQJWRWKHVDPHYDOXHRIWKH%DWGRUI=SDUDPHWHU=DERXW

Constant h

Constant L

Constant R

Aspect ratio, L/R

0.8

0.6

0.4

0.2

0.00 10 20 30 40 50

Nx

crRh

π2

D

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUFRPSUHVVLRQORDGHG1[LVRWURSLFF\OLQGHUVZLWKVLPSO\

VXSSRUWHGHGJHVIRU5K DQG/5

 

≤

 

ν 

Nx

crRh

π2

D

= 0.67

Sanders’ equations 

R

h

L Sanders’ equations, 

 

φn neglected 2

Donnell’s equations, 

Euler column-buckling equation

D = Eh3

12 1 − ν2

(25.5, 0.41)

m = n = 1


Aspect ratio, L/R

0.8

0.6

0.4

0.2

0.002468

Nx

crRh

π2

D

Nx

crRh

π2

D

= 0.67

R

h

L

Sanders’ equations 

Sanders’ equations, φn neglected 2

Donnell’s equations, 

D = Eh3

12 1 − ν2

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUFRPSUHVVLRQORDGHG1[LVRWURSLFF\OLQGHUVZLWKVLPSO\

VXSSRUWHGHGJHVIRU5K DQG/5≤ν 

Aspect ratio, L/R

100

60

40

20

0

0 10 20 30 40 50

)LJXUH'LIIHUHQFHLQEXFNOLQJORDGVZLWKUHVSHFWWRUHVXOWVEDVHGRQ6DQGHUV¶HTXDWLRQVIRUFRPSUHVVLRQORDGHG

LVRWURSLFF\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K DQG/5

 

≤

 

ν 

Percent

difference

in buckling

loads

80

Donnell’s equations 

Sanders’ equations, 

 

φn neglected 2

R

h

L


Aspect ratio, L/R

0.8

0.6

0.4

0.2

0.00 10 20 30 40 50

Nx

crRh

π2

D

(36.3, 0.41)

Nx

crRh

π2

D

= 0.67

Sanders’ equations 

R

h

L Sanders’ equations, φn neglected 2

Donnell’s equations, 

Euler column-buckling equation

D = Eh3

12 1 − ν2

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUFRPSUHVVLRQORDGHG1[LVRWURSLFF\OLQGHUVZLWKVLPSO\

VXSSRUWHGHGJHVIRU5K DQG/5≤ν 

m = n = 1

Aspect ratio, L/R

0.8

0.6

0.4

0.2

0.002468

Nx

crRh

π2

D

R

h

L

D = Eh3

12 1 − ν2

Sanders’ equations, 

 

φn neglected 2

Sanders’ equations 

Nx

crRh

π2

D Donnell’s equations, = 0.67

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUFRPSUHVVLRQORDGHG1[LVRWURSLFF\OLQGHUVZLWKVLPSO\

VXSSRUWHGHGJHVIRU5K DQG/5

 

≤

 

ν 


Aspect ratio, L/R

100

60

40

20

0

0 10 20 30 40 50

80

Percent

difference

in buckling

loads

Donnell’s equations 

Sanders’ equations, φn neglected 2

R

h

L

)LJXUH'LIIHUHQFHLQEXFNOLQJORDGVZLWKUHVSHFWWRUHVXOWVEDVHGRQ6DQGHUV¶HTXDWLRQVIRUFRPSUHVVLRQORDGHG

LVRWURSLFF\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K DQG/5≤ν 

Aspect ratio, L/R

0.8

0.6

0.4

0.2

0.00 10 20 30 40 50

Nx

crRh

π2

D

Nx

crRh

π2

D

= 0.67

Sanders’ equations 

R

h

L

Sanders’ equations, 

 

φn neglected 2

Donnell’s equations, 

D = Eh3

12 1 − ν2

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUFRPSUHVVLRQORDGHG1[LVRWURSLFF\OLQGHUVZLWKVLPSO\

VXSSRUWHGHGJHVIRU5K DQG/5

 

≤

 

ν 


Aspect ratio, L/R

0.8

0.6

0.4

0.2

0.002468

Nx

crRh

π2

D

R

h

L

D = Eh3

12 1 − ν2

Sanders’ equations, φn neglected 2

Sanders’ equations 

Nx

crRh

π2

D Donnell’s equations, = 0.67

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUFRPSUHVVLRQORDGHG1[LVRWURSLFF\OLQGHUVZLWKVLPSO\

VXSSRUWHGHGJHVIRU5K DQG/5≤ν 

Aspect ratio, L/R

80

60

40

20

0

0 10 20 30 40 50

Percent

difference

in buckling

loads

Donnell’s equations 

Sanders’ equations, 

 

φn neglected 2

R

h

L

)LJXUH'LIIHUHQFHLQEXFNOLQJORDGVZLWKUHVSHFWWRUHVXOWVEDVHGRQ6DQGHUV¶HTXDWLRQVIRUFRPSUHVVLRQORDGHG

LVRWURSLFF\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K DQG/5

 

≤

 

ν 


Aspect ratio, L/R

0.8

0.6

0.4

0.2

0.00 10 20 30 40 50

Nx

crRh

π2

D

Nx

crRh

π2

D

= 0.67

Sanders’ equations 

R

h

L

Sanders’ equations, φn neglected 2

Donnell’s equations, 

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUFRPSUHVVLRQORDGHG1[LVRWURSLFF\OLQGHUVZLWKVLPSO\

VXSSRUWHGHGJHVIRU5K DQG/5≤ν 

D = Eh3

12 1 − ν2

Aspect ratio, L/R

0.8

0.6

0.4

0.2

0.002468

Nx

crRh

π2

D

Nx

crRh

π2

D

= 0.67

Sanders’ equations 

R

h

L

Sanders’ equations, 

 

φn neglected 2

Donnell’s equations, 

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUFRPSUHVVLRQORDGHG1[LVRWURSLFF\OLQGHUVZLWKVLPSO\

VXSSRUWHGHGJHVIRU5K DQG/5

 

≤ν 

D = Eh3

12 1 − ν2


Aspect ratio, L/R

80

60

40

20

0

0 10 20 30 40 50

Percent

difference

in buckling

loads

Donnell’s equations 

Sanders’ equations, φn neglected 2

R

h

L

)LJXUH'LIIHUHQFHLQEXFNOLQJORDGVZLWKUHVSHFWWRUHVXOWVEDVHGRQ6DQGHUV¶HTXDWLRQVIRUFRPSUHVVLRQORDGHG

LVRWURSLFF\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K DQG/5≤ν 

Normalized cylinder length, L/LEuler

100

60

40

20

0

0 0.2 0.4 0.6 0.8 1.0

80

Percent

difference

in buckling

loads

R

h

L

Donnell’s equations 

Sanders’ equations, φn neglected 2

)LJXUH'LIIHUHQFHLQEXFNOLQJORDGVZLWKUHVSHFWWRUHVXOWVEDVHGRQ6DQGHUV¶HTXDWLRQVIRUFRPSUHVVLRQORDGHG

LVRWURSLFF\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K DQGν 

LEuler

Euler column-buckling mode

m = n = 1


Results obtained from 

)LJXUH1RQGLPHQVLRQDOEXFNOLQJVWUHVVUDWLRVIRUFRPSUHVVLRQORDGHGLVRWURSLFF\OLQGHUVZLWKVLPSO\

VXSSRUWHGHGJHVIRU5K DQGν 

σx

cr

σ*x

cr

Buckling

stress

ratio,

σ∗x

cr

σEuler

cr = 2

π

2

3 1 − ν2

h

R

L

R

σ*x

cr

= 1

3 1 − ν2

Eh

R σEuler

cr

σ*x

Euler column buckling, cr

Sanders’ equations 

(1.27, 0.61)

σEuler

cr

= π

2

ER2

2L2

L

R

h

L

Shell column-buckling modes, m = n = 1

σ∗x

cr

σEuler

cr

Radius−to−thickness ratio, R

h

Aspect ratio, L

R = 10

L

R = 1

)LJXUH9DOXHVRIWKHDEVFLVVDVKRZQLQILJXUHDVDIXQFWLRQRI5KDQGIRUVHOHFWYDOXHVRI/5ν 

σ∗x

cr

σEuler

cr = 2

π

2

3 1 − ν2

h

R

L

R

R

h

L


σ∗x

cr

σEuler

cr

Radius−to−thickness ratio, R

h

Aspect ratio, L

R = 10

)LJXUH9DOXHVRIWKHDEVFLVVDVKRZQLQILJXUHDVDIXQFWLRQRI5KDQGIRUVHOHFWYDOXHVRI/5ν 

σ∗x

cr

σEuler

cr = 2

π

2

3 1 − ν2

h

R

L

R

50

40

30

20

15

Value for column

buckling, 1.27

R

h

L


Aspect ratio, L/R

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUH[WHUQDOSUHVVXUHORDGHGLVRWURSLFF\OLQGHUVZLWKVLPSO\

VXSSRUWHGHGJHVIRU5K ν 

pcrR3

D

Nondimensional

buckling pressure,

R

h

L

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 


Aspect ratio, L/R

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUH[WHUQDOSUHVVXUHORDGHGLVRWURSLFF\OLQGHUVZLWKVLPSO\

VXSSRUWHGHGJHVIRU5K ν 

pcrR3

D

Nondimensional

buckling pressure,

R

h

L

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 


Aspect ratio, L/R

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUH[WHUQDOSUHVVXUHORDGHGLVRWURSLFF\OLQGHUVZLWKVLPSO\

VXSSRUWHGHGJHVIRU5K ν 

pcrR3

D

Nondimensional

buckling pressure,

R

h

L

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 


Aspect ratio, L/R

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUH[WHUQDOSUHVVXUHORDGHGLVRWURSLFF\OLQGHUVZLWKVLPSO\

VXSSRUWHGHGJHVIRU5K ν 

pcrR3

D

Nondimensional

buckling pressure,

R

h

L

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 


Aspect ratio, L/R

50

30

20

10

0

0 10 20 30 40 50

)LJXUH'LIIHUHQFHVLQH[WHUQDOEXFNOLQJSUHVVXUHZLWKUHVSHFWWRUHVXOWVEDVHGRQ6DQGHUV¶OLYHSUHVVXUHHTXDWLRQV

IRULVRWURSLFF\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

Percent

difference

in buckling

pressures

40

R

h

L

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Aspect ratio, L/R

50

30

20

10

0

0 10 20 30 40 50

)LJXUH'LIIHUHQFHVLQH[WHUQDOEXFNOLQJSUHVVXUHZLWKUHVSHFWWRUHVXOWVEDVHGRQ6DQGHUV¶OLYHSUHVVXUHHTXDWLRQV

IRULVRWURSLFF\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

Percent

difference

in buckling

pressures

40

R

h

L

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 


Aspect ratio, L/R

50

30

20

10

0

0 10 20 30 40 50

)LJXUH'LIIHUHQFHVLQH[WHUQDOEXFNOLQJSUHVVXUHZLWKUHVSHFWWRUHVXOWVEDVHGRQ6DQGHUV¶OLYHSUHVVXUHHTXDWLRQV

IRULVRWURSLFF\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

Percent

difference

in buckling

pressures

40 R

h

L

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Aspect ratio, L/R

50

30

20

10

0

0 10 20 30 40 50

)LJXUH'LIIHUHQFHVLQH[WHUQDOEXFNOLQJSUHVVXUHZLWKUHVSHFWWRUHVXOWVEDVHGRQ6DQGHUV¶OLYHSUHVVXUHHTXDWLRQV

IRULVRWURSLFF\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

Percent

difference

in buckling

pressures

40 R

h

L Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 


Normalized cylinder length, L/L(1, 2)

40

20

10

0

0.0 1.0 2.0 3.0 4.0

30

Percent

difference

in buckling

pressures

R

h

L

)LJXUH'LIIHUHQFHLQWKHH[WHUQDOEXFNOLQJSUHVVXUHVZLWKUHVSHFWWRUHVXOWVEDVHGRQ6DQGHUV¶OLYHSUHVVXUH

HTXDWLRQVIRULVRWURSLFF\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K DQGν 

L(1,2)

m = 1, n = 2 buckling mode

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

)LJXUH1RQGLPHQVLRQDOEXFNOLQJSUHVVXUHUDWLRVIRUH[WHUQDOSUHVVXUHORDGHGLVRWURSLFF\OLQGHUVZLWKVLPSO\

VXSSRUWHGHGJHVIRU5K DQGν 

Buckling

pressure

ratio,

p ∞

cr

pcr

p ∞

cr = 1

4 1 − ν2

Eh3

R3

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 

0.0 2.0 4.0 6.0 8.0

1.2

1.0

0.8

0.6

0.4

0.2

0.0

L

R

h

R

R

h

L


Aspect ratio, L/R

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUK\GURVWDWLFSUHVVXUHORDGHGLVRWURSLFF\OLQGHUV

ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

pcrR3

D

Nondimensional

buckling pressure,

R

h

L

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 


Aspect ratio, L/R

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUK\GURVWDWLFSUHVVXUHORDGHGLVRWURSLFF\OLQGHUV

ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

pcrR3

D

Nondimensional

buckling pressure,

R

h

L

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 


Aspect ratio, L/R

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUK\GURVWDWLFSUHVVXUHORDGHGLVRWURSLFF\OLQGHUV

ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

pcrR3

D

Nondimensional

buckling pressure,

R

h

L

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 


Aspect ratio, L/R

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUK\GURVWDWLFSUHVVXUHORDGHGLVRWURSLFF\OLQGHUV

ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

pcrR3

D

Nondimensional

buckling pressure,

R

h

L

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 


Aspect ratio, L/R

60

40

20

10

0

0 10 20 40 50

)LJXUH'LIIHUHQFHVLQH[WHUQDOK\GURVWDWLFEXFNOLQJSUHVVXUHZLWKUHVSHFWWRUHVXOWVEDVHGRQ6DQGHUV¶

OLYHSUHVVXUHHTXDWLRQVIRULVRWURSLFF\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

Percent

difference

in buckling

pressures

50

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

30

R

h

L

30

Aspect ratio, L/R

60

40

20

10

0

0 10 20 40 50

)LJXUH'LIIHUHQFHVLQH[WHUQDOK\GURVWDWLFEXFNOLQJSUHVVXUHZLWKUHVSHFWWRUHVXOWVEDVHGRQ6DQGHUV¶

OLYHSUHVVXUHHTXDWLRQVIRULVRWURSLFF\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

Percent

difference

in buckling

pressures

50

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

30

R

h

L

30


Aspect ratio, L/R

60

40

20

10

0

0 10 20 40 50

)LJXUH'LIIHUHQFHVLQH[WHUQDOK\GURVWDWLFEXFNOLQJSUHVVXUHZLWKUHVSHFWWRUHVXOWVEDVHGRQ6DQGHUV¶

OLYHSUHVVXUHHTXDWLRQVIRULVRWURSLFF\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

Percent

difference

in buckling

pressures

50

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

30

R

h

L

30

Aspect ratio, L/R

60

40

20

10

0

0 10 20 40 50

)LJXUH'LIIHUHQFHVLQH[WHUQDOK\GURVWDWLFEXFNOLQJSUHVVXUHZLWKUHVSHFWWRUHVXOWVEDVHGRQ6DQGHUV¶

OLYHSUHVVXUHHTXDWLRQVIRULVRWURSLFF\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

Percent

difference

in buckling

pressures

50

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

30

R

h

L

30


Normalized cylinder length, L/L(1, 2)

60

20

10

0

0.0 1.0 2.0 3.0 5.0

30

Percent

difference

in buckling

pressures

R

h

L

)LJXUH'LIIHUHQFHLQEXFNOLQJORDGVZLWKUHVSHFWWRUHVXOWVEDVHGRQ6DQGHUV¶HTXDWLRQVIRUK\GURVWDWLFSUHVVXUH

ORDGHGLVRWURSLFF\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K DQGν 

L(1,2)

m = 1, n = 2 buckling mode

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

4.0

40

50

)LJXUH1RQGLPHQVLRQDOEXFNOLQJSUHVVXUHUDWLRVIRUK\GURVWDWLFSUHVVXUHORDGHGLVRWURSLFF\OLQGHUV

ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K DQGν 

Buckling

pressure

ratio,

p ∞

cr

pcr

p ∞

cr

= 1

4 1 − ν2

Eh3

R3

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 

0.0 2.0 4.0 6.0 8.0

2.5

2.0

1.5

1.0

0.5

0.0

L

R

h

R

R

h

L


Aspect ratio, L/R

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRULVRWURSLFF\OLQGHUVVXEHFWHGWRD[LDOFRPSUHVVLRQDQGLQWHUQDO

SUHVVXUHDQGZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

q = 0

q = 0.01

q = qint

E 3 1 − ν2 R

h

2

qint Nx

h

2R

L

Nx

crRh

π2

D

0.0

0.2

0.4

0.6

0.8

1.0

1.2 q = 1.0, Nx

crRh

π2

D

= 1.01

q = 0.1, Nx

crRh

π2

D

= 0.70

0 10 20 30 40 50

q = 0 and q = 0.01, Nx

crRh

π2

D

= 0.67

Sanders’ equations - live pressure 

Donnell’s equations - dead pressure 

Euler column-buckling equation

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRULVRWURSLFF\OLQGHUVVXEHFWHGWRD[LDOFRPSUHVVLRQDQGLQWHUQDO

SUHVVXUHDQGZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

q = 0

q = qint

E 3 1 − ν2 R

h

2

q = 0.005

Nx

crRh

π2

D

0.0

0.2

0.4

0.6

0.8

1.0

1.2

q = 0.1, Nx

crRh

π2

D

= 0.7

q = 1.0, Nx

crRh

π2

D

= 1.0

Aspect ratio, L/R

0 10 20 30 40 50

q = 0 and q = 0.005, Nx

crRh

π2

D

= 0.67

Euler column-buckling equation

Sanders’ equations - live pressure 

Donnell’s equations - dead pressure 


)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRULVRWURSLFF\OLQGHUVVXEHFWHGWRD[LDOFRPSUHVVLRQDQGLQWHUQDO

SUHVVXUHDQGZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

q = 0

q = qint

E 3 1 − ν2 R

h

2

qint Nx

h

2R

L

Nx

crRh

π2

D

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Aspect ratio, L/R

0 10 20 30 40 50

q = 1.0, Nx

crRh

π2

D

= 1.0

q = 0.1, Nx

crRh

π2

D

= 0.7

q = 0 and q = 0.005, Nx

crRh

π2

D

= 0.67

Sanders’ equations - live pressure 

Donnell’s equations - dead pressure 

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SUHVVXUHDQGZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

q = 0

q = qint

E 3 1 − ν2 R

h

2

Aspect ratio, L/R

0 10 20 30 40 50 0.0

0.2

0.4

0.6

0.8

1.0

1.2

Nx

crRh

π2

D

q = 1.0, Nx

crRh

π2

D

= 1.0

q = 0.1, Nx

crRh

π2

D

= 0.7

q = 0 and q = 0.005, Nx

crRh

π2

D

= 0.67

Sanders’ equations - live pressure 

Donnell’s equations - dead pressure 


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SUHVVXUHDQGZLWKVLPSO\VXSSRUWHGHGJHVIRU5K DQGν 

σx

cr

σ*x

cr

Buckling

stress

ratio,

σEuler

cr

σ*x

Column buckling, cr

q = qint

E 3 1 − ν2 R

h

2

R

h q = 0

0.25 0.5

R

h q = 1

σEuler

cr

= π

2

ER2

2L2 − qint

2

R

h

σ*x

cr

= 1

3 1 − ν2

Eh

R 1 + q

2

σ∗x

cr

σEuler

cr = L

R

h

R

2+q

π

2

3 1 − ν2− L

R

2

h

R q

qint Nx

h

2R

L

R

h

L


E6WULQJHUFURVVVHFWLRQ

D5LQJFURVVVHFWLRQ

0.90 in.

0.90 in.

0.10 in. 2.00 in.

0.10 in.

0.10 in.

0.10 in. 2.00 in.

0.10 in.

0.10 in. Dr

 = 4.00 in.

2.00 in.

α = 60 deg

hc = 0.16 in.

Cylinder radius, R = 200 in.

wc = 4.00 in.

Inside of cylinder

Outside of cylinder

wc

F&RUUXJDWHGZDOOFURVVVHFWLRQ

wc sinα 1

2

wc sinα 1

2

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E6WULQJHUGHWDLOV

0.33 in.

es = 0.165 in.

h

ws = 0.097 in.

R = 9.55 in.

Stringer spacing = 1.00 in.

D%XFNOHGF\OLQGHU


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Ds = 0.54 in.

es

h

hs

R = 15.92 in.

Stringer spacing = 1.24 in. wf2 = 0.22 in.

hs = 0.040 in.

hs

wf1 = 0.375 in.

Centroid

E6WULQJHUGHWDLOV

D%XFNOHGF\OLQGHU


Donnell’s equations, 

Sanders’ equations 

Sanders’ equations, φn neglected 2

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ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

A = Eh

1 − ν2

D = h2

A

12

R

h

Shell wall

Stringer

x

y

ζ

es

Shell wall

h

2

Centroid

Stringer properties:

Area, As

Centroidal moment of inertia, Is

Torsion constant, Js

Eccentricity, es

Young’s modulus, Es

c

Stringer spacing, ds

EsAs

dsA = 1 EsIs

c

dsD = 1

GsJs

2ds 1 − ν D

= 0 es

h = 0

Nx

crRh

π2

D

Nx

crRh

π2

D

= 0.77

Aspect ratio, L/R

0.8

1.0

0.6

0.4

0.2

0.00 10 20 30 40 50

Euler column-buckling equation

(30.7, 0.59)

m = n = 1


Aspect ratio, L/R

50

40

30

10

0

0 10 20 30 40 50

Percent

difference

in buckling

loads

Donnell’s equations 

Sanders’ equations, φn neglected 2

R

h

L

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VWULQJHUVWLIIHQHGLVRWURSLFF\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

20

EsAs

dsA = 1 EsIs

c

dsD = 1

GsJs

2ds 1 − ν D

= 0 es

h = 0


Aspect ratio, L/R

Donnell’s equations, 

Sanders’ equations 

Sanders’ equations, φn neglected 2

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ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

Euler column-buckling equation

R

h

Shell wall

Stringer

x

y

ζ

es

Shell wall

h

2

Centroid

Stringer properties:

Area, As

Centroidal moment of inertia, Is

Torsion constant, Js

Eccentricity, es

Young’s modulus, Es

c

Stringer spacing, ds

A = Eh

1 − ν2

D = h2

A

12

EsIs

c

dsD = 1

GsJs

2ds 1 − ν D

= 0 es

h = 0

EsAs

dsA = 2

0.8

1.0

0.6

0.4

0.2

0.00 10 20 30 40 50

Nx

crRh

π2

D

Nx

crRh

π2

D

= 0.78

(34.1, 0.72)

m = n = 1


Aspect ratio, L/R

50

40

30

10

0

0 10 20 30 40 50

Percent

difference

in buckling

loads

Donnell’s equations 

Sanders’ equations, φn neglected 2

R

h

L

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VWULQJHUVWLIIHQHGLVRWURSLFF\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

20

EsIs

c

dsD = 1

GsJs

2ds 1 − ν D

= 0 es

h = 0

EsAs

dsA = 2


Aspect ratio, L/R

Donnell’s equations, 

Sanders’ equations 

Sanders’ equations, φn neglected 2

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ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

Euler column-buckling equation

R

h

Shell wall

Stringer

x

y

ζ

es

Shell wall

h

2

Centroid

Stringer properties:

Area, As

Centroidal moment of inertia, Is

Torsion constant, Js

Eccentricity, es

Young’s modulus, Es

c

Stringer spacing, ds

A = Eh

1 − ν2

D = h2

A

12

EsIs

c

dsD = 1

GsJs

2ds 1 − ν D

= 0 es

h = 0

0.8

1.0

0.6

0.4

0.2

0.00 10 20 30 40 50

Nx

crRh

π2

D

Nx

crRh

π2

D

= 0.78

(42.6, 0.78)

m = n = 1

EsAs

dsA = 4


Aspect ratio, L/R

50

40

30

10

0

0 10 20 30 40 50

Percent

difference

in buckling

loads

Donnell’s equations 

Sanders’ equations, φn neglected 2

R

h

L

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VWULQJHUVWLIIHQHGLVRWURSLFF\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

20

EsIs

c

dsD = 1

GsJs

2ds 1 − ν D

= 0 es

h = 0

EsAs

dsA = 4


Donnell’s equations, 

Sanders’ equations 

Sanders’ equations, φn neglected 2

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ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

Euler column-buckling equation

R

h

Shell wall

Stringer

x

y

ζ

es

Shell wall

h

2

Centroid

Stringer properties:

Area, As

Centroidal moment of inertia, Is

Torsion constant, Js

Eccentricity, es

Young’s modulus, Es

c

Stringer spacing, ds

A = Eh

1 − ν2

D = h2

A

12

EsAs

dsA = 1

GsJs

2ds 1 − ν D

= 0 es

h = 0

EsIs

c

dsD = 10

Aspect ratio, L/R

0.8

1.0

0.6

0.4

0.2

0.00 10 20 30 40 50

Nx

crRh

π2

D

Nx

crRh

π2

D

= 0.91

(30.7, 0.59)

m = n = 1


Aspect ratio, L/R

60

50

40

10

0

0 10 20 30 40 50

Percent

difference

in buckling

loads

Donnell’s equations 

Sanders’ equations, φn neglected 2

R

h

L

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VWULQJHUVWLIIHQHGLVRWURSLFF\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

20

30

EsAs

dsA = 1

GsJs

2ds 1 − ν D

= 0 es

h = 0

EsIs

c

dsD = 10


Donnell’s equations, 

Sanders’ equations 

Sanders’ equations, φn neglected 2

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ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

Euler column-buckling equation

R

h

Shell wall

Stringer

x

y

ζ

es

Shell wall

h

2

Centroid

Stringer properties:

Area, As

Centroidal moment of inertia, Is

Torsion constant, Js

Eccentricity, es

Young’s modulus, Es

c

Stringer spacing, ds

A = Eh

1 − ν2

D = h2

A

12

EsAs

dsA = 1 EsIs

c

dsD = 1

es

h = 0

GsJs

2ds 1 − ν D

= 1

Aspect ratio, L/R

0.8

1.0

0.6

0.4

0.2

0.00 10 20 30 40 50

Nx

crRh

π2

D

Nx

crRh

π2

D

= 0.87

(30.6, 0.59)

m = n = 1


Aspect ratio, L/R

60

50

40

10

0

0 10 20 30 40 50

Percent

difference

in buckling

loads

Donnell’s equations 

Sanders’ equations, φn neglected 2

R

h

L

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VWULQJHUVWLIIHQHGLVRWURSLFF\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

20

30

EsAs

dsA = 1 EsIs

c

dsD = 1

es

h = 0

GsJs

2ds 1 − ν D

= 1


Donnell’s equations, 

Sanders’ equations 

Sanders’ equations, φn neglected 2

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ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

Euler column-buckling equation

R

h

Shell wall

Stringer

x

y

ζ

es

Shell wall

h

2

Centroid

Stringer properties:

Area, As

Centroidal moment of inertia, Is

Torsion constant, Js

Eccentricity, es

Young’s modulus, Es

c

Stringer spacing, ds

A = Eh

1 − ν2

D = h2

A

12

EsAs

dsA = 1 EsIs

c

dsD = 1

es

h = 0

GsJs

2ds 1 − ν D

= 10

Aspect ratio, L/R

0.8

1.0

0.6

0.4

0.2

0.00 10 20 30 40 50

Nx

crRh

π2

D

Nx

crRh

π2

D

= 0.97

(29.6, 0.63)

m = n = 1


Aspect ratio, L/R

60

50

40

10

0

0 10 20 30 40 50

Percent

difference

in buckling

loads

Donnell’s equations 

Sanders’ equations, φn neglected 2

R

h

L

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VWULQJHUVWLIIHQHGLVRWURSLFF\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

20

30

EsAs

dsA = 1 EsIs

c

dsD = 1

es

h = 0

GsJs

2ds 1 − ν D

= 10


Donnell’s equations 

Sanders’ equations 

Sanders’ equations, φn neglected 2

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ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

Euler column-buckling equation

R

h

Shell wall

Stringer

x

y

ζ

es

Shell wall

h

2

Centroid

Stringer properties:

Area, As

Centroidal moment of inertia, Is

Torsion constant, Js

Eccentricity, es

Young’s modulus, Es

c

Stringer spacing, ds

A = Eh

1 − ν2

D = h2

A

12 EsAs

dsA = 1 EsIs

c

dsD = 1

GsJs

2ds 1 − ν D

= 0 es

h =+1

Aspect ratio, L/R

1.6

2.0

1.2

0.8

0.4

0.00 10 20 30 40 50

Nx

crRh

π2

D

(30.5, 0.61)

m = n = 1


Aspect ratio, L/R

80

60

20

0

0 10 20 30 40 50

Percent

difference

in buckling

loads

Donnell’s equations 

Sanders’ equations, φn neglected 2

R

h

L

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VWULQJHUVWLIIHQHGLVRWURSLFF\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

40

EsAs

dsA = 1 EsIs

c

dsD = 1

GsJs

2ds 1 − ν D

= 0 es

h =+1


Donnell’s equations, 

Sanders’ equations 

Sanders’ equations, φn neglected 2

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ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

Euler column-buckling equation

R

h

Shell wall

Stringer

x

y

ζ

es

Shell wall

h

2

Centroid

Stringer properties:

Area, As

Centroidal moment of inertia, Is

Torsion constant, Js

Eccentricity, es

Young’s modulus, Es

c

Stringer spacing, ds

A = Eh

1 − ν2

D = h2

A

12

EsAs

dsA = 1 EsIs

c

dsD = 1

GsJs

2ds 1 − ν D

= 0 es

h = − 1

Aspect ratio, L/R

1.6

2.0

1.2

0.8

0.4

0.00 10 20 30 40 50

Nx

crRh

π2

D

Nx

crRh

π2

D

= 0.69

(30.9, 0.57)

m = n = 1


Aspect ratio, L/R

60

50

40

10

0

0 10 20 30 40 50

Percent

difference

in buckling

loads

Donnell’s equations 

Sanders’ equations, φn neglected 2

R

h

L

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VWULQJHUVWLIIHQHGLVRWURSLFF\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

20

30

EsAs

dsA = 1 EsIs

c

dsD = 1

GsJs

2ds 1 − ν D

= 0 es

h = − 1


Donnell’s equations 

Sanders’ equations 

Sanders’ equations, φn neglected 2

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ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

Euler column-buckling equation

R

h

Shell wall

Stringer

x

y

ζ

es

Shell wall

h

2

Centroid

Stringer properties:

Area, As

Centroidal moment of inertia, Is

Torsion constant, Js

Eccentricity, es

Young’s modulus, Es

c

Stringer spacing, ds

A = Eh

1 − ν2

D = h2

A

12

EsAs

dsA = 1 EsIs

c

dsD = 1

GsJs

2ds 1 − ν D

= 0 es

h = + 10

Aspect ratio, L/R

2.0

2.5

1.5

1.0

0.5

0.00 10 20 30 40 50

Nx

crRh

π2

D

(27.9, 0.93)

m = n = 1


Aspect ratio, L/R

100

80

60

20

0

0 10 20 30 40 50

Percent

difference

in buckling

loads

Donnell’s equations 

Sanders’ equations, φn neglected 2

R

h

L

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VWULQJHUVWLIIHQHGLVRWURSLFF\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

40

EsAs

dsA = 1 EsIs

c

dsD = 1

GsJs

2ds 1 − ν D

= 0 es

h = + 10


Donnell’s equations 

Sanders’ equations Sanders’ equations, φn neglected 2

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ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

Euler column-buckling equation

R

h

Shell wall

Stringer

x

y

ζ

es

Shell wall

h

2

Centroid

Stringer properties:

Area, As

Centroidal moment of inertia, Is

Torsion constant, Js

Eccentricity, es

Young’s modulus, Es

c

Stringer spacing, ds

A = Eh

1 − ν2

D = h2

A

12

EsAs

dsA = 1 EsIs

c

dsD = 1

GsJs

2ds 1 − ν D

= 0 es

h = − 10

Aspect ratio, L/R

2.0

2.5

1.5

1.0

0.5

0.00 10 20 30 40 50

Nx

crRh

π2

D

(33.2, 0.41)

m = n = 1


Aspect ratio, L/R

100

80

60

20

0

0 10 20 30 40 50

Percent

difference

in buckling

loads

Donnell’s equations 

Sanders’ equations, φn neglected 2

R

h

L

)LJXUH'LIIHUHQFHLQEXFNOLQJORDGVZLWKUHVSHFWWRUHVXOWVEDVHGRQ6DQGHUV¶HTXDWLRQVIRUFRPSUHVVLRQORDGHG

VWULQJHUVWLIIHQHGLVRWURSLFF\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

40

EsAs

dsA = 1 EsIs

c

dsD = 1

GsJs

2ds 1 − ν D

= 0 es

h = − 10


Aspect ratio, L/R

Donnell’s equations, 

Sanders’ equations 

Sanders’ equations, φn neglected 2

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUFRPSUHVVLRQORDGHGVWULQJHUVWLIIHQHGLVRWURSLFF\OLQGHUV

ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

A = Eh

1 − ν2

D = h2

A

12

R

h

Shell wall

Stringer

x

y

ζ

es

Shell wall

h

2

Centroid

Stringer properties:

Area, As

Centroidal moment of inertia, Is

Torsion constant, Js

Eccentricity, es

Young’s modulus, Es

c

Stringer spacing, ds

EsAs

dsA = 1 EsIs

c

dsD = 1

GsJs

2ds 1 − ν D

= 0 es

h = 0

Nx

crRh

π2

D

Nx

crRh

π2

D

= 0.77


Aspect ratio, L/R

Donnell’s equations, 

Sanders’ equations 

Sanders’ equations, φn neglected 2

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ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

R

h

Shell wall

Stringer

x

y

ζ

es

Shell wall

h

2

Centroid

Stringer properties:

Area, As

Centroidal moment of inertia, Is

Torsion constant, Js

Eccentricity, es

Young’s modulus, Es

c

Stringer spacing, ds

A = Eh

1 − ν2

D = h2

A

12

EsIs

c

dsD = 1

GsJs

2ds 1 − ν D

= 0 es

h = 0

EsAs

dsA = 2

0.8

1.0

0.6

0.4

0.2

0.00 10 20 30 40 50

Nx

crRh

π2

D

Nx

crRh

π2

D

= 0.78


Donnell’s equations, 

Sanders’ equations 

Sanders’ equations, φn neglected 2

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ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

R

h

Shell wall

Stringer

x

y

ζ

es

Shell wall

h

2

Centroid

Stringer properties:

Area, As

Centroidal moment of inertia, Is

Torsion constant, Js

Eccentricity, es

Young’s modulus, Es

c

Stringer spacing, ds

A = Eh

1 − ν2

D = h2

A

12

EsAs

dsA = 1

GsJs

2ds 1 − ν D

= 0 es

h = 0

EsIs

c

dsD = 10

Aspect ratio, L/R

0.8

1.0

0.6

0.4

0.2

0.00 10 20 30 40 50

Nx

crRh

π2

D

Nx

crRh

π2

D

= 0.90


Donnell’s equations, 

Sanders’ equations 

Sanders’ equations, φn neglected 2

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ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

R

h

Shell wall

Stringer

x

y

ζ

es

Shell wall

h

2

Centroid

Stringer properties:

Area, As

Centroidal moment of inertia, Is

Torsion constant, Js

Eccentricity, es

Young’s modulus, Es

c

Stringer spacing, ds

A = Eh

1 − ν2

D = h2

A

12

Aspect ratio, L/R

0.8

1.2

0.6

0.4

0.2

0.00 10 20 30 40 50

Nx

crRh

π2

D

1.0

Nx

crRh

π2

D

= 0.87

EsAs

dsA = 1 EsIs

c

dsD = 1

es

h = 0

GsJs

2ds 1 − ν D

= 1


Donnell’s equations, 

Sanders’ equations 

Sanders’ equations, φn neglected 2

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUFRPSUHVVLRQORDGHGVWULQJHUVWLIIHQHGLVRWURSLFF\OLQGHUV

ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

R

h

Shell wall

Stringer

x

y

ζ

es

Shell wall

h

2

Centroid

Stringer properties:

Area, As

Centroidal moment of inertia, Is

Torsion constant, Js

Eccentricity, es

Young’s modulus, Es

c

Stringer spacing, ds

A = Eh

1 − ν2

D = h2

A

12

EsAs

dsA = 1 EsIs

c

dsD = 1

es

h = 0

GsJs

2ds 1 − ν D

= 10

Aspect ratio, L/R

0.8

1.2

0.6

0.4

0.2

0.00 10 20 30 40 50

Nx

crRh

π2

D

Nx

crRh

π2

D

= 0.97

1.0


Donnell’s equations, 

Sanders’ equations 

Sanders’ equations, φn neglected 2

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUFRPSUHVVLRQORDGHGVWULQJHUVWLIIHQHGLVRWURSLFF\OLQGHUV

ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

R

h

Shell wall

Stringer

x

y

ζ

es

Shell wall

h

2

Centroid

Stringer properties:

Area, As

Centroidal moment of inertia, Is

Torsion constant, Js

Eccentricity, es

Young’s modulus, Es

c

Stringer spacing, ds

A = Eh

1 − ν2

D = h2

A

12

Aspect ratio, L/R

0.8

1.2

0.6

0.4

0.2

0.00 10 20 30 40 50

Nx

crRh

π2

D

1.0

EsAs

dsA = 1 EsIs

c

dsD = 1

GsJs

2ds 1 − ν D

= 0 es

h =+1


Donnell’s equations, 

Sanders’ equations 

Sanders’ equations, φn neglected 2

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUFRPSUHVVLRQORDGHGVWULQJHUVWLIIHQHGLVRWURSLFF\OLQGHUV

ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

R

h

Shell wall

Stringer

x

y

ζ

es

Shell wall

h

2

Centroid

Stringer properties:

Area, As

Centroidal moment of inertia, Is

Torsion constant, Js

Eccentricity, es

Young’s modulus, Es

c

Stringer spacing, ds

A = Eh

1 − ν2

D = h2

A

12

Aspect ratio, L/R

0.8

1.2

0.6

0.4

0.2

0.00 10 20 30 40 50

Nx

crRh

π2

D

1.0

EsAs

dsA = 1 EsIs

c

dsD = 1

GsJs

2ds 1 − ν D

= 0 es

h = − 1

Nx

crRh

π2

D

= 0.69


Donnell’s equations 

Sanders’ equations 

Sanders’ equations, φn neglected 2

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUFRPSUHVVLRQORDGHGVWULQJHUVWLIIHQHGLVRWURSLFF\OLQGHUV

ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

R

h

Shell wall

Stringer

x

y

ζ

es

Shell wall

h

2

Centroid

Stringer properties:

Area, As

Centroidal moment of inertia, Is

Torsion constant, Js

Eccentricity, es

Young’s modulus, Es

c

Stringer spacing, ds

A = Eh

1 − ν2

D = h2

A

12

EsAs

dsA = 1 EsIs

c

dsD = 1

GsJs

2ds 1 − ν D

= 0 es

h = + 10

Aspect ratio, L/R

1.5

2.0

1.0

0.5

0.00 10 20 30 40 50

Nx

crRh

π2

D


Donnell’s equations, 

Sanders’ equations 

Sanders’ equations, φn neglected 2

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUFRPSUHVVLRQORDGHGVWULQJHUVWLIIHQHGLVRWURSLFF\OLQGHUV

ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

R

h

Shell wall

Stringer

x

y

ζ

es

Shell wall

h

2

Centroid

Stringer properties:

Area, As

Centroidal moment of inertia, Is

Torsion constant, Js

Eccentricity, es

Young’s modulus, Es

c

Stringer spacing, ds

A = Eh

1 − ν2

D = h2

A

12

Aspect ratio, L/R

1.5

2.0

1.0

0.5

0.00 10 20 30 40 50

Nx

crRh

π2

D

EsAs

dsA = 1 EsIs

c

dsD = 1

GsJs

2ds 1 − ν D

= 0 es

h = − 10

Nx

crRh

π2

D

= 0.67


Donnell’s equations, 

Sanders’ equations 

Sanders’ equations, φn neglected 2

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUFRPSUHVVLRQORDGHGULQJVWLIIHQHGLVRWURSLFF\OLQGHUV

ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

A = Eh

1 − ν2

D = h2

A

12

Ring properties:

Area, Ar

Centroidal moment of inertia, Ir

Torsion constant, Jr

Eccentricity, er

Young’s modulus, Er

c

Nx

crRh

π2

D

Nx

crRh

π2

D

= 0.77

Aspect ratio, L/R

0.8

1.0

0.6

0.4

0.2

0.00 10 20 30 40 50

Euler column-buckling equation

(21.7, 0.59)

m = n = 1

ErAr

drA = 1 ErIr

c

drD = 1

er

h = 0

GrJr

2dr 1 − ν D

= 0

R

h

dr

Shell wall

Ring

x

y

ζ

er

Shell wall

h

2

Centroid


Donnell’s equations, 

Sanders’ equations 

Sanders’ equations, φn neglected 2

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUFRPSUHVVLRQORDGHGULQJVWLIIHQHGLVRWURSLFF\OLQGHUV

ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

A = Eh

1 − ν2

D = h2

A

12

Ring properties:

Area, Ar

Centroidal moment of inertia, Ir

Torsion constant, Jr

Eccentricity, er

Young’s modulus, Er

c

Nx

crRh

π2

D

Aspect ratio, L/R

0.8

1.0

0.6

0.4

0.2

0.00 10 20 30 40 50

Euler column-buckling equation

(21.8, 0.59)

m = n = 1

ErIr

c

drD = 1

er

h = 0 GrJr

2dr 1 − ν D

= 0

R

h

dr

Shell wall

Ring

x

y

ζ

er

Shell wall

h

2

Centroid

ErAr

drA = 2

Nx

crRh

π2

D

= 0.78


Donnell’s equations, 

Sanders’ equations 

Sanders’ equations, φn neglected 2

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUFRPSUHVVLRQORDGHGULQJVWLIIHQHGLVRWURSLFF\OLQGHUV

ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

A = Eh

1 − ν2

D = h2

A

12

Ring properties:

Area, Ar

Centroidal moment of inertia, Ir

Torsion constant, Jr

Eccentricity, er

Young’s modulus, Er

c

Nx

crRh

π2

D

Aspect ratio, L/R

0.8

1.0

0.6

0.4

0.2

0.00 10 20 30 40 50

Euler column-buckling equation

(17.5, 0.90)

m = n = 1

R

h

dr

Shell wall

Ring

x

y

ζ

er

Shell wall

h

2

Centroid

ErAr

drA = 1

er

h = 0

GrJr

2dr 1 − ν D

= 0

ErIr

c

drD = 10

Nx

crRh

π2

D

= 0.91


Donnell’s equations, 

Sanders’ equations 

Sanders’ equations, φn neglected 2

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUFRPSUHVVLRQORDGHGULQJVWLIIHQHGLVRWURSLFF\OLQGHUV

ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

A = Eh

1 − ν2

D = h2

A

12

Ring properties:

Area, Ar

Centroidal moment of inertia, Ir

Torsion constant, Jr

Eccentricity, er

Young’s modulus, Er

c

Nx

crRh

π2

D

Aspect ratio, L/R

0.8

1.0

0.6

0.4

0.2

0.00 10 20 30 40 50

Euler column-buckling equation

(21.7, 0.59)

m = n = 1

R

h

dr

Shell wall

Ring

x

y

ζ

er

Shell wall

h

2

Centroid

ErAr

drA = 1 ErIr

c

drD = 1

er

h = 0

GrJr

2dr 1 − ν D

= 1

Nx

crRh

π2

D

= 0.87


Donnell’s equations, 

Sanders’ equations 

Sanders’ equations, φn neglected 2

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUFRPSUHVVLRQORDGHGULQJVWLIIHQHGLVRWURSLFF\OLQGHUV

ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

A = Eh

1 − ν2

D = h2

A

12

Ring properties:

Area, Ar

Centroidal moment of inertia, Ir

Torsion constant, Jr

Eccentricity, er

Young’s modulus, Er

c

Nx

crRh

π2

D

Aspect ratio, L/R

0.8

1.0

0.6

0.4

0.2

0.00 10 20 30 40 50

Euler column-buckling equation

(20.9, 0.64)

m = n = 1

R

h

dr

Shell wall

Ring

x

y

ζ

er

Shell wall

h

2

Centroid

ErAr

drA = 1 ErIr

c

drD = 1

er

h = 0

Nx

crRh

π2

D

= 0.97

GrJr

2dr 1 − ν D

= 10


Donnell’s equations, 

Sanders’ equations 

Sanders’ equations, φn neglected 2

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUFRPSUHVVLRQORDGHGULQJVWLIIHQHGLVRWURSLFF\OLQGHUV

ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

A = Eh

1 − ν2

D = h2

A

12

Ring properties:

Area, Ar

Centroidal moment of inertia, Ir

Torsion constant, Jr

Eccentricity, er

Young’s modulus, Er

c

Nx

crRh

π2

D

Aspect ratio, L/R

1.0

1.2

0.6

0.4

0.2

0.00 10 20 30 40 50

Euler column-buckling equation

(16.8, 0.97)

m = n = 1

R

h

dr

Shell wall

Ring

x

y

ζ

er

Shell wall

h

2

Centroid

ErAr

drA = 1 ErIr

c

drD = 1

Nx

crRh

π2

D

= 0.97

er

h = 1

GrJr

2dr 1 − ν D

= 0

0.8


Donnell’s equations, 

Sanders’ equations 

Sanders’ equations, φn neglected 2

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUFRPSUHVVLRQORDGHGULQJVWLIIHQHGLVRWURSLFF\OLQGHUV

ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

A = Eh

1 − ν2

D = h2

A

12

Ring properties:

Area, Ar

Centroidal moment of inertia, Ir

Torsion constant, Jr

Eccentricity, er

Young’s modulus, Er

c

Nx

crRh

π2

D

Aspect ratio, L/R

1.0

1.2

0.6

0.4

0.2

0.00 10 20 30 40 50

Euler column-buckling equation

(16.9, 0.96)

m = n = 1

R

h

dr

Shell wall

Ring

x

y

ζ

er

Shell wall

h

2

Centroid

ErAr

drA = 1 ErIr

c

drD = 1

Nx

crRh

π2

D

= 0.97

GrJr

2dr 1 − ν D

= 0

0.8

er

h = 10


Donnell’s equations, 

Sanders’ equations 

Sanders’ equations, φn neglected 2

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUFRPSUHVVLRQORDGHGULQJVWLIIHQHGLVRWURSLFF\OLQGHUV

ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

A = Eh

1 − ν2

D = h2

A

12

Ring properties:

Area, Ar

Centroidal moment of inertia, Ir

Torsion constant, Jr

Eccentricity, er

Young’s modulus, Er

c

Nx

crRh

π2

D

Aspect ratio, L/R

0.8

1.0

0.6

0.4

0.2

0.00 10 20 30 40 50

Euler column-buckling equation

(20.2, 0.68)

m = n = 1

R

h

dr

Shell wall

Ring

x

y

ζ

er

Shell wall

h

2

Centroid

ErAr

drA = 1 ErIr

c

drD = 1

GrJr

2dr 1 − ν D

= 0

Nx

crRh

π2

D

= 0.69

er

h = − 1


Donnell’s equations, 

Sanders’ equations 

Sanders’ equations, φn neglected 2

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUFRPSUHVVLRQORDGHGULQJVWLIIHQHGLVRWURSLFF\OLQGHUV

ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

A = Eh

1 − ν2

D = h2

A

12

Ring properties:

Area, Ar

Centroidal moment of inertia, Ir

Torsion constant, Jr

Eccentricity, er

Young’s modulus, Er

c

Nx

crRh

π2

D

Aspect ratio, L/R

0.8

1.0

0.6

0.4

0.2

0.00 10 20 30 40 50

Euler column-buckling equation

(20.2, 0.67)

m = n = 1

R

h

dr

Shell wall

Ring

x

y

ζ

er

Shell wall

h

2

Centroid

ErAr

drA = 1 ErIr

c

drD = 1

GrJr

2dr 1 − ν D

= 0 er

h = − 10

Nx

crRh

π2

D

= 0.68


Donnell’s equations, 

Sanders’ equations 

Sanders’ equations, φn neglected 2

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUFRPSUHVVLRQORDGHGULQJVWLIIHQHGLVRWURSLFF\OLQGHUV

ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

A = Eh

1 − ν2

D = h2

A

12

Ring properties:

Area, Ar

Centroidal moment of inertia, Ir

Torsion constant, Jr

Eccentricity, er

Young’s modulus, Er

c

Nx

crRh

π2

D

Nx

crRh

π2

D

= 0.77

Aspect ratio, L/R

0.8

1.0

0.6

0.4

0.2

0.00 10 20 30 40 50

ErAr

drA = 1 ErIr

c

drD = 1

er

h = 0

GrJr

2dr 1 − ν D

= 0

R

h

dr

Shell wall

Ring

x

y

ζ

er

Shell wall

h

2

Centroid


Donnell’s equations, 

Sanders’ equations 

Sanders’ equations, φn neglected 2

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUFRPSUHVVLRQORDGHGULQJVWLIIHQHGLVRWURSLFF\OLQGHUV

ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

A = Eh

1 − ν2

D = h2

A

12

Ring properties:

Area, Ar

Centroidal moment of inertia, Ir

Torsion constant, Jr

Eccentricity, er

Young’s modulus, Er

c

Nx

crRh

π2

D

Aspect ratio, L/R

0.8

1.0

0.6

0.4

0.2

0.00 10 20 30 40 50

ErIr

c

drD = 1

er

h = 0

GrJr

2dr 1 − ν D

= 0

R

h

dr

Shell wall

Ring

x

y

ζ

er

Shell wall

h

2

Centroid

ErAr

drA = 2

Nx

crRh

π2

D

= 0.78


Donnell’s equations, 

Sanders’ equations 

Sanders’ equations, φn neglected 2

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUFRPSUHVVLRQORDGHGULQJVWLIIHQHGLVRWURSLFF\OLQGHUV

ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

A = Eh

1 − ν2

D = h2

A

12

Ring properties:

Area, Ar

Centroidal moment of inertia, Ir

Torsion constant, Jr

Eccentricity, er

Young’s modulus, Er

c

Nx

crRh

π2

D

Aspect ratio, L/R

0.8

1.0

0.6

0.4

0.2

0.00 10 20 30 40 50

R

h

dr

Shell wall

Ring

x

y

ζ

er

Shell wall

h

2

Centroid

ErAr

drA = 1 ErIr

c

drD = 1

er

h = 0

GrJr

2dr 1 − ν D

= 1

Nx

crRh

π2

D

= 0.87


Donnell’s equations, 

Sanders’ equations 

Sanders’ equations, φn neglected 2

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUFRPSUHVVLRQORDGHGULQJVWLIIHQHGLVRWURSLFF\OLQGHUV

ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

A = Eh

1 − ν2

D = h2

A

12

Ring properties:

Area, Ar

Centroidal moment of inertia, Ir

Torsion constant, Jr

Eccentricity, er

Young’s modulus, Er

c

Nx

crRh

π2

D

Aspect ratio, L/R

0.8

1.0

0.6

0.4

0.2

0.00 10 20 30 40 50

R

h

dr

Shell wall

Ring

x

y

ζ

er

Shell wall

h

2

Centroid

ErAr

drA = 1 ErIr

c

drD = 1

er

h = 0

GrJr

2dr 1 − ν D

= 10

Nx

crRh

π2

D

= 0.97


Aspect ratio, L/R

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUH[WHUQDOSUHVVXUHORDGHGVWULQJHUVWLIIHQHGLVRWURSLF

F\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

pcrR3

D

Nondimensional

buckling

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 

0 10 20 30 40 50

5

2

1

10

20

50

100

EsAs

dsA = 1 EsIs

c

dsD = 1

GsJs

2ds 1 − ν D

= 0 es

h = 0

A = Eh

1 − ν2

D = h2

A

12

pressure,

R

h

L

Stringer spacing, ds

Stringer properties:

Area, As

Centroidal moment of inertia, Is

Torsion constant, Js

Eccentricity, es

Young’s modulus, Es

c

es

Shell wall

h

2

Centroid

m = 1, n = 2

buckling modes


Aspect ratio, L/R

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUH[WHUQDOSUHVVXUHORDGHGVWULQJHUVWLIIHQHGLVRWURSLF

F\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

pcrR3

D

Nondimensional

buckling

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 

0 10 20 30 40 50

5

2

1

10

20

50

100

EsAs

dsA = 1 EsIs

c

dsD = 1

GsJs

2ds 1 − ν D

= 0 es

h = 0

A = Eh

1 − ν2

D = h2

A

12

pressure,

R

h

L

Stringer spacing, ds

Stringer properties:

Area, As

Centroidal moment of inertia, Is

Torsion constant, Js

Eccentricity, es

Young’s modulus, Es

c

es

Shell wall

h

2

Centroid


Aspect ratio, L/R

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUH[WHUQDOSUHVVXUHORDGHGULQJVWLIIHQHGLVRWURSLF

F\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

pcrR3

D

Nondimensional

buckling

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 

0 10 20 30 40 50

5

2

1

10

20

50

100

A = Eh

1 − ν2

D = h2

A

12

pressure,

m = 1, n = 2

buckling modes

ErAr

drA = 1 ErIr

c

drD = 1

er

h = 0

GrJr

2dr 1 − ν D

= 0

L

Ring spacing, dr

R

h

Ring properties:

Area, Ar

Centroidal moment of inertia, Ir

Torsion constant, Jr

Eccentricity, er

Young’s modulus, Er

c

er

Shell wall

h

2

Centroid


Aspect ratio, L/R

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUH[WHUQDOSUHVVXUHORDGHGULQJVWLIIHQHGLVRWURSLF

F\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

pcrR3

D

Nondimensional

buckling

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 

0 10 20 30 40 50

5

2

1

10

20

50

100

A = Eh

1 − ν2

D = h2

A

12

pressure,

m = 1, n = 2

buckling modes

L

Ring spacing, dr

R

h

Ring properties:

Area, Ar

Centroidal moment of inertia, Ir

Torsion constant, Jr

Eccentricity, er

Young’s modulus, Er

c

er

Shell wall

h

2

Centroid

ErAr

drA = 1 ErIr

c

drD = 1

er

h = 0

GrJr

2dr 1 − ν D

= 10


Aspect ratio, L/R

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUH[WHUQDOSUHVVXUHORDGHGULQJVWLIIHQHGLVRWURSLF

F\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

pcrR3

D

Nondimensional

buckling

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 

0 10 20 30 40 50

5

2

1

10

100

500

1000

A = Eh

1 − ν2

D = h2

A

12

pressure,

m = 1, n = 2

buckling modes

L

Ring spacing, dr

R

h

Ring properties:

Area, Ar

Centroidal moment of inertia, Ir

Torsion constant, Jr

Eccentricity, er

Young’s modulus, Er

c

er

Shell wall

h

2

Centroid

ErAr

drA = 1

er

h = 0 GrJr

2dr 1 − ν D

= 0

ErIr

c

drD = 10


Aspect ratio, L/R

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUH[WHUQDOSUHVVXUHORDGHGULQJVWLIIHQHGLVRWURSLF

F\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

pcrR3

D

Nondimensional

buckling

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 

0 10 20 30 40 50

5

2

1

10

100

500

1000

A = Eh

1 − ν2

D = h2

A

12

pressure,

L

Ring spacing, dr

R

h

Ring properties:

Area, Ar

Centroidal moment of inertia, Ir

Torsion constant, Jr

Eccentricity, er

Young’s modulus, Er

c

er

Shell wall

h

2

Centroid

ErAr

drA = 1 ErIr

c

drD = 1

er

h = 1

GrJr

2dr 1 − ν D

= 0


Aspect ratio, L/R

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUH[WHUQDOSUHVVXUHORDGHGULQJVWLIIHQHGLVRWURSLF

F\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

pcrR3

D

Nondimensional

buckling

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 

0 10 20 30 40 50

5

2

1

10

100

500

1000

A = Eh

1 − ν2

D = h2

A

12

pressure,

m = 1, n = 2

buckling modes

L

Ring spacing, dr

R

h

Ring properties:

Area, Ar

Centroidal moment of inertia, Ir

Torsion constant, Jr

Eccentricity, er

Young’s modulus, Er

c

er

Shell wall

h

2

Centroid

ErAr

drA = 1 ErIr

c

drD = 1

GrJr

2dr 1 − ν D

= 0 er

h = − 1


Aspect ratio, L/R

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUH[WHUQDOSUHVVXUHORDGHGULQJVWLIIHQHGLVRWURSLF

F\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

pcrR3

D

Nondimensional

buckling

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 

0 10 20 30 40 50

1

10

100

10000

A = Eh

1 − ν2

D = h2

A

12

pressure,

L

Ring spacing, dr

R

h

Ring properties:

Area, Ar

Centroidal moment of inertia, Ir

Torsion constant, Jr

Eccentricity, er

Young’s modulus, Er

c

er

Shell wall

h

2

Centroid

ErAr

drA = 1 ErIr

c

drD = 1

GrJr

2dr 1 − ν D

= 0 er

h = 10

1000


Aspect ratio, L/R

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUH[WHUQDOSUHVVXUHORDGHGULQJVWLIIHQHGLVRWURSLF

F\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

pcrR3

D

Nondimensional

buckling

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 

0 10 20 30 40 50

1

10

100

10000

A = Eh

1 − ν2

D = h2

A

12

pressure,

m = 1, n = 2

buckling modes

L

Ring spacing, dr

R

h

Ring properties:

Area, Ar

Centroidal moment of inertia, Ir

Torsion constant, Jr

Eccentricity, er

Young’s modulus, Er

c

er

Shell wall

h

2

Centroid

1000

ErAr

drA = 1 ErIr

c

drD = 1

GrJr

2dr 1 − ν D

= 0 er

h = − 10


Aspect ratio, L/R

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUH[WHUQDOSUHVVXUHORDGHGULQJVWLIIHQHGLVRWURSLF

F\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

pcrR3

D

Nondimensional

buckling

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 

0 10 20 30 40 50

5

2

1

10

20

50

100

A = Eh

1 − ν2

D = h2

A

12

pressure,

ErAr

drA = 1 ErIr

c

drD = 1

er

h = 0

GrJr

2dr 1 − ν D

= 0

L

Ring spacing, dr

R

h

Ring properties:

Area, Ar

Centroidal moment of inertia, Ir

Torsion constant, Jr

Eccentricity, er

Young’s modulus, Er

c

er

Shell wall

h

2

Centroid

m = 1, n = 2

buckling modes


Aspect ratio, L/R

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUH[WHUQDOSUHVVXUHORDGHGULQJVWLIIHQHGLVRWURSLF

F\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

pcrR3

D

Nondimensional

buckling

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 

0 10 20 30 40 50

5

2

1

10

100

500

1000

A = Eh

1 − ν2

D = h2

A

12

pressure,

L

Ring spacing, dr

R

h

Ring properties:

Area, Ar

Centroidal moment of inertia, Ir

Torsion constant, Jr

Eccentricity, er

Young’s modulus, Er

c

er

Shell wall

h

2

Centroid

ErAr

drA = 1

er

h = 0 GrJr

2dr 1 − ν D

= 0

ErIr

c

drD = 10

m = 1, n = 2

buckling modes


Aspect ratio, L/R

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUH[WHUQDOSUHVVXUHORDGHGULQJVWLIIHQHGLVRWURSLF

F\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

pcrR3

D

Nondimensional

buckling

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 

0 10 20 30 40 50

5

2

1

10

100

500

1000

A = Eh

1 − ν2

D = h2

A

12

pressure,

m = 1, n = 2

buckling modes

L

Ring spacing, dr

R

h

Ring properties:

Area, Ar

Centroidal moment of inertia, Ir

Torsion constant, Jr

Eccentricity, er

Young’s modulus, Er

c

er

Shell wall

h

2

Centroid

ErAr

drA = 1 ErIr

c

drD = 1

er

h = 1

GrJr

2dr 1 − ν D

= 0


Aspect ratio, L/R

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUH[WHUQDOSUHVVXUHORDGHGULQJVWLIIHQHGLVRWURSLF

F\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

pcrR3

D

Nondimensional

buckling

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 

0 10 20 30 40 50

5

2

1

10

100

500

1000

A = Eh

1 − ν2

D = h2

A

12

pressure,

m = 1, n = 2

buckling modes

L

Ring spacing, dr

R

h

Ring properties:

Area, Ar

Centroidal moment of inertia, Ir

Torsion constant, Jr

Eccentricity, er

Young’s modulus, Er

c

er

Shell wall

h

2

Centroid

ErAr

drA = 1 ErIr

c

drD = 1

GrJr

2dr 1 − ν D

= 0 er

h = − 1


Aspect ratio, L/R

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUH[WHUQDOSUHVVXUHORDGHGULQJVWLIIHQHGLVRWURSLF

F\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

pcrR3

D

Nondimensional

buckling

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 

0 10 20 30 40 50

1

10

100

10000

A = Eh

1 − ν2

D = h2

A

12

pressure,

L

Ring spacing, dr

R

h

Ring properties:

Area, Ar

Centroidal moment of inertia, Ir

Torsion constant, Jr

Eccentricity, er

Young’s modulus, Er

c

er

Shell wall

h

2

Centroid

ErAr

drA = 1 ErIr

c

drD = 1

GrJr

2dr 1 − ν D

= 0 er

h = 10

1000


Aspect ratio, L/R

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUH[WHUQDOSUHVVXUHORDGHGULQJVWLIIHQHGLVRWURSLF

F\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

pcrR3

D

Nondimensional

buckling

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 

0 10 20 30 40 50

1

10

100

10000

A = Eh

1 − ν2

D = h2

A

12

pressure,

m = 1, n = 2

buckling modes

L

Ring spacing, dr

R

h

Ring properties:

Area, Ar

Centroidal moment of inertia, Ir

Torsion constant, Jr

Eccentricity, er

Young’s modulus, Er

c

er

Shell wall

h

2

Centroid

1000

ErAr

drA = 1 ErIr

c

drD = 1

GrJr

2dr 1 − ν D

= 0 er

h = − 10


Aspect ratio, L/R

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUH[WHUQDOK\GURVWDWLFSUHVVXUHORDGHGVWULQJHUVWLIIHQHGLVRWURSLF

F\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

pcrR3

D

Nondimensional

buckling

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 

0 10 20 30 40 50

5

2

1

10

20

50

100

EsAs

dsA = 1 EsIs

c

dsD = 1

GsJs

2ds 1 − ν D

= 0 es

h = 0

A = Eh

1 − ν2

D = h2

A

12

pressure,

R

h

L

Stringer spacing, ds

Stringer properties:

Area, As

Centroidal moment of inertia, Is

Torsion constant, Js

Eccentricity, es

Young’s modulus, Es

c

es

Shell wall

h

2

Centroid


Aspect ratio, L/R

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUH[WHUQDOK\GURVWDWLFSUHVVXUHORDGHGULQJVWLIIHQHGLVRWURSLF

F\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

pcrR3

D

Nondimensional

buckling

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 

0 10 20 30 40 50

5

2

1

10

20

50

100

A = Eh

1 − ν2

D = h2

A

12

pressure,

ErAr

drA = 1 ErIr

c

drD = 1

er

h = 0 GrJr

2dr 1 − ν D

= 0

L

Ring spacing, dr

R

h

Ring properties:

Area, Ar

Centroidal moment of inertia, Ir

Torsion constant, Jr

Eccentricity, er

Young’s modulus, Er

c

er

Shell wall

h

2

Centroid


Aspect ratio, L/R

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUH[WHUQDOK\GURVWDWLFSUHVVXUHORDGHGULQJVWLIIHQHGLVRWURSLF

F\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

pcrR3

D

Nondimensional

buckling

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 

0 10 20 30 40 50

1

10

100

10000

A = Eh

1 − ν2

D = h2

A

12

pressure,

L

Ring spacing, dr

R

h

Ring properties:

Area, Ar

Centroidal moment of inertia, Ir

Torsion constant, Jr

Eccentricity, er

Young’s modulus, Er

c

er

Shell wall

h

2

Centroid

ErAr

drA = 1 ErIr

c

drD = 1

GrJr

2dr 1 − ν D

= 0 er

h = 10

1000


Aspect ratio, L/R

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUH[WHUQDOK\GURVWDWLFSUHVVXUHORDGHGULQJVWLIIHQHGLVRWURSLF

F\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

pcrR3

D

Nondimensional

buckling

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 

0 10 20 30 40 50

1

10

100

10000

A = Eh

1 − ν2

D = h2

A

12

pressure,

L

Ring spacing, dr

R

h

Ring properties:

Area, Ar

Centroidal moment of inertia, Ir

Torsion constant, Jr

Eccentricity, er

Young’s modulus, Er

c

er

Shell wall

h

2

Centroid

1000

ErAr

drA = 1 ErIr

c

drD = 1

GrJr

2dr 1 − ν D

= 0 er

h = − 10


Aspect ratio, L/R

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUH[WHUQDOK\GURVWDWLFSUHVVXUHORDGHGULQJVWLIIHQHGLVRWURSLF

F\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

pcrR3

D

Nondimensional

buckling

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 

0 10 20 30 40 50

1

10

100

10000

A = Eh

1 − ν2

D = h2

A

12

pressure,

L

Ring spacing, dr

R

h

Ring properties:

Area, Ar

Centroidal moment of inertia, Ir

Torsion constant, Jr

Eccentricity, er

Young’s modulus, Er

c

er

Shell wall

h

2

Centroid

ErAr

drA = 1 ErIr

c

drD = 1

GrJr

2dr 1 − ν D

= 0 er

h = 10

1000


Aspect ratio, L/R

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUH[WHUQDOK\GURVWDWLFSUHVVXUHORDGHGULQJVWLIIHQHGLVRWURSLF

F\OLQGHUVZLWKVLPSO\VXSSRUWHGHGJHVIRU5K ν 

pcrR3

D

Nondimensional

buckling

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 

0 10 20 30 40 50

1

10

100

10000

A = Eh

1 − ν2

D = h2

A

12

pressure,

L

Ring spacing, dr

R

h

Ring properties:

Area, Ar

Centroidal moment of inertia, Ir

Torsion constant, Jr

Eccentricity, er

Young’s modulus, Er

c

er

Shell wall

h

2

Centroid

1000

ErAr

drA = 1 ErIr

c

drD = 1

GrJr

2dr 1 − ν D

= 0 er

h = − 10


Aspect ratio, L/R

0.6

0.5

0.3

0.2

0.00 10 20 30 40 50

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUFRPSUHVVLRQORDGHG>

@VF\OLQGHUV

ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K DQG/5≤

Sanders’ equations 

Sanders’ equations, φn neglected 2

Donnell’s equations, 

Euler column-buckling equation

(30.8, 0.26)

m = n = 1

0.1

0.4

Nx

crRh

π2 D11D22

Nx

crRh

π2 D11D22

L = 0.39

R

h

 Fiber angle, θ

x

Typical 

fiber

lamina

Aspect ratio, L/R

0.6

0.5

0.3

0.2

0.00 10 20 30 40 50

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUFRPSUHVVLRQORDGHG>@V

F\OLQGHUV

ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K DQG/5≤

Sanders’ equations 

Sanders’ equations, φn neglected 2

Euler column-buckling equation

(16.5, 0.20)

m = n = 1

0.1

0.4

Nx

crRh

π2 D11D22

Nx

crRh

π2 D11D22

= 0.29

L

R

h

 Fiber angle, θ

x

Typical 

fiber

lamina

Donnell’s equations, 


Aspect ratio, L/R

0.6

0.5

0.3

0.2

0.00 10 20 30 40 50

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUFRPSUHVVLRQORDGHG>@VF\OLQGHUV

ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K DQG/5≤

Sanders’ equations 

Sanders’ equations, φn neglected 2

Euler column-buckling equation

(12.1, 0.26)

m = n = 1 0.1

0.4

Nx

crRh

π2 D11D22

Nx

crRh

π2 D11D22

= 0.39

L

R

h

 Fiber angle, θ

x

Typical 

fiber

lamina Donnell’s equations, 

Aspect ratio, L/R

0.6

0.5

0.3

0.2

0.00 10 20 30 40 50

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUFRPSUHVVLRQORDGHG>@VF\OLQGHUV

ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K DQG/5≤

Sanders’ equations 

Sanders’ equations, φn neglected 2

Donnell’s equations, 

0.1

0.4

Nx

crRh

π2 D11D22

Nx

crRh

π2 D11D22

= 0.39

L

R

h

 Fiber angle, θ

x

Typical 

fiber

lamina


Aspect ratio, L/R

0.6

0.5

0.3

0.2

0.00 10 20 30 40 50

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUFRPSUHVVLRQORDGHG>

@VF\OLQGHUV

ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K DQG/5≤

Sanders’ equations 

Sanders’ equations, φn neglected 2

Euler column-buckling equation

m = n = 1

0.1

0.4

Nx

crRh

π2 D11D22

L

R

h

 Fiber angle, θ

x

Typical 

fiber

lamina

Donnell’s equations, Nx

crRh

π2 D11D22

= 0.29

Aspect ratio, L/R

0.6

0.5

0.3

0.2

0.00 10 20 30 40 50

)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUFRPSUHVVLRQORDGHG>@V

F\OLQGHUV

ZLWKVLPSO\VXSSRUWHGHGJHVIRU5K DQG/5≤

Sanders’ equations 

Sanders’ equations, φn neglected 2

Donnell’s equations, 

Euler column-buckling equation

(39.1, 0.24)

m = n = 1

0.1

0.4

Nx

crRh

π2 D11D22

Nx

crRh

π2 D11D22

= 0.39

L

R

h

 Fiber angle, θ

x

Typical 

fiber

lamina


)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUH[WHUQDOSUHVVXUHORDGHG>@VF\OLQGHUVZLWK

VLPSO\VXSSRUWHGHGJHVIRU5K 

Nondimensional

buckling pressure,

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 

pcrR3

D11D22

Aspect ratio, L/R

10 20 30 40 50

5

2

1

10

20

50

100

L

R

h Fiber angle, θ

x

Typical 

fiber

lamina

0


)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUH[WHUQDOSUHVVXUHORDGHG>@VF\OLQGHUVZLWK

VLPSO\VXSSRUWHGHGJHVIRU5K 

Nondimensional

buckling pressure,

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 

pcrR3

D11D22

Aspect ratio, L/R

10 20 30 40 50

5

2

1

10

20

50

100

L

R

h Fiber angle, θ

x

Typical 

fiber

lamina

0


)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUH[WHUQDOSUHVVXUHORDGHG>@VF\OLQGHUVZLWK

VLPSO\VXSSRUWHGHGJHVIRU5K 

Nondimensional

buckling pressure,

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 

pcrR3

D11D22

Aspect ratio, L/R

10 20 30 40 50

5

2

1

10

20

50

100

L

R

h Fiber angle, θ

x

Typical 

fiber

lamina

0


)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUH[WHUQDOSUHVVXUHORDGHG>@VF\OLQGHUVZLWK

VLPSO\VXSSRUWHGHGJHVIRU5K 

Nondimensional

buckling pressure,

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 

pcrR3

D11D22

Aspect ratio, L/R

10 20 30 40 50

5

2

1

10

20

50

100

L

R

h Fiber angle, θ

x

Typical 

fiber

lamina

0


)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUH[WHUQDOSUHVVXUHORDGHG>@VF\OLQGHUVZLWK

VLPSO\VXSSRUWHGHGJHVIRU5K 

Nondimensional

buckling pressure,

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 

pcrR3

D11D22

Aspect ratio, L/R

10 20 30 40 50

5

2

1

10

20

50

100

L

R

h Fiber angle, θ

x

Typical 

fiber

lamina

0


)LJXUH1RQGLPHQVLRQDOEXFNOLQJORDGVIRUH[WHUQDOSUHVVXUHORDGHG>@VF\OLQGHUVZLWK

VLPSO\VXSSRUWHGHGJHVIRU5K 

Nondimensional

buckling pressure,

Sanders’ equations - dead pressure 

Donnell’s equations - dead pressure 

Sanders’ equations - live pressure 

pcrR3

D11D22

Aspect ratio, L/R

10 20 30 40 50

5

2

1

10

20

50

100

L

R

h Fiber angle, θ

x

Typical 

fiber

lamina

0


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REPORT DOCUMENTATION PAGE Form Approved

OMB No. 0704-0188

2. REPORT TYPE 

Technical Memorandum

 4. TITLE AND SUBTITLE

Buckling Analysis for Stiffened Anisotropic Circular Cylinders Based on 

Sanders’ Nonlinear Shell Theory 

5a. CONTRACT NUMBER

 6. AUTHOR(S)

Nemeth, Michael P.

 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Langley Research Center

Hampton, VA 23681-2199

 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

 8. PERFORMING ORGANIZATION

 REPORT NUMBER

L-20320

10. SPONSOR/MONITOR'S ACRONYM(S)

NASA

13. SUPPLEMENTARY NOTES

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Unclassified - Unlimited

Subject Category 39

Availability: NASA CASI (443) 757-5802

19a. NAME OF RESPONSIBLE PERSON

STI Help Desk (email: help@sti.nasa.gov)

14. ABSTRACT

Nonlinear and bifurcation buckling equations for elastic, stiffened, geometrically perfect, right-circular cylindrical, anisotropic 

shells subjected to combined loads are presented that are based on Sanders’ shell theory. Based on these equations, a 

three-parameter approximate Rayleigh-Ritz solution and a classical solution to the buckling problem are presented for 

cylinders with simply supported edges. Extensive comparisons of results obtained from these solutions with published results 

are also presented for a wide range of cylinder constructions. These comparisons include laminated-composite cylinders with a 

wide variety of shell-wall orthotropies and anisotropies. Numerous results are also given that show the discrepancies between 

the results obtained by using Donnell’s equations and variants of Sanders’ equations. For some cases, nondimensional 

parameters are identified and "master" curves are presented that facilitate the concise representation of results. 

15. SUBJECT TERMS

Anisotropic shells; Buckling; Shell theory; Stiffened cylinders

18. NUMBER

 OF 

 PAGES

413

19b. TELEPHONE NUMBER (Include area code)

(443) 757-5802

a. REPORT

U

c. THIS PAGE

U

b. ABSTRACT

U

17. LIMITATION OF 

 ABSTRACT

UU

Prescribed by ANSI Std. Z39.18

Standard Form 298 (Rev. 8-98)

3. DATES COVERED (From - To)

5b. GRANT NUMBER

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5e. TASK NUMBER

5f. WORK UNIT NUMBER

 869021.04.07.01.13 

11. SPONSOR/MONITOR'S REPORT

 NUMBER(S)

NASA/TM-2014-218176

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1. REPORT DATE (DD-MM-YYYY)

01- 03 - 2014