Panel buckling=Bay buckling
• bay (skin between stiffeners) buckles when
___


___
 
( AR) 4 
2
2
 2 D11 skin k  2D12 skin  2D66 skin ( AR)  D22 skin
a 
k2 


2
N xskin
equation from before for buckling of ss
plate under compression
___
a
( AR) 
ds
ds
b
a
(5.4.2.3)
Panel Buckling = Bay Buckling
• panel as a whole buckles when
Nx 
 2 
D11 skin 
a 

2
( AR ) 
Nx 
a
b
EI stif 
ds
 m  2D12 skin  2D66 skin ( AR)  D22 skin

2
2
( AR) 4 


m2 

(5.4.2.4)
stiffener contribution
FTOT
b
(5.4.2.5)
it is assumed the stiffener has
an open cross-section and,
therefore, GJ≈0
ds
b
a
Panel Buckling = Bay Buckling
• combining (5.4.2.2) with (5.4.2.3) and dropping the subscript
“skin” from Dij:
FTOT

EA 
 A11 

___

2 
___
d
( AR) 4 
s 

2
2

D11 k  2D12   2D66 ( AR)  D22  2
b
A11
a2 
k 


(5.4.2.6)
• combining (5.4.2.4) with (5.4.2.5) and dropping the subscript
“skin” from Dij :
FTOT  b
 2 
D11  
a2 

EI stif 
ds
( AR) 4 

 m  2D12   2D66 ( AR)  D22 
2 
m



2
2
(5.4.2.7)
Panel Buckling = Bay Buckling
• combining (5.4.2.6) with (5.4.2.7) and rearranging a bit

EA 
___
 A11 
 

___
EI stif
d
( AR ) 4  
s  

2
2
D11k  2D12  2 D66 ( AR )  D22

D

 11

A11
k2  
ds


 2
( AR ) 4
2
 m  2D12  2 D66 ( AR )  D22
m2

(5.4.2.8)
• to proceed, we need to decide on the values of k and m
• recall that k and m are integers that minimize the
respective buckling loads in eqs (5.4.2.3) and (5.4.2.4)
Panel Buckling = Bay Buckling
• consider the continuous function
___
y
( AR) 4
y  D11 x  D22
x2
2
x
• this function has one and only one minimum
determined by
1/ 4
D 
dy
 0  xmin   22 
dx
 D11 
 ___ 
 AR 
 
Panel Buckling = Bay Buckling
• therefore, the discontinuous function
___
y,y*
y
y*
( AR) 4
y  D11k  D22
k2
*
2
• is minimized when
 D 1 / 4  ___ 
k  int  22   AR 

 D11  
or
 D 1 / 4  ___ 
k  int  22   AR   1

 D11  
whichever makes y* lowest, with one important
note:
if
 D 1 / 4  ___ 
int  22   AR   0

 D11  
then k (or m) = 1
x
Panel Buckling = Bay Buckling
• for the case of m,
xmin


D22


EI
 D11 
ds







1/ 4
a
 
b
• for typical applications, xmin<1
because D11>D22 and b>a
• so setting m=1 covers most cases of interest
Panel Buckling = Bay Buckling
• now approximate k with its corresponding xmin
1/ 4
 D22 

k  
 D11 
 ___ 
 AR 
 
• and substitute for m and k in eq. (5.4.2.8)
D11 
EI stif
ds

EA  
 A11 
 
___
d
D 22 ___ 2
s  

2
4
 2D12  2 D66 ( AR )  D 22 ( AR ) 
D11
AR  2D12  2 D66 ( AR ) 2  D 22

A11
D11


=λ, λ>1


( AR ) 4 
D 22 ___ 2 
AR 
D11

___
• solving for (EI)stiff and dropping the subscript “stiff”
___ 2
 D22 

D22
2D12  2D66 )   ___ 2
4
2




EI  D11 d s 
  AR   AR    1
 2 AR  D  AR   
D
D
11 
11
11


 

1st equation
(5.4.2.9)
Stiffener buckling = PB x skin buckling
beff beff
• in post-buckling regime, skin is replaced by the
effective skin portion; strain compatibility then gives
A11
Fskin 
b
2beff
ds
b
b
2 A11 beff  EA
ds
ds
Fstiffeners 
FTOT  Fskin 
A11 2beff
2 A11beff  EA
FTOT
(5.4.2.10)
EA
FTOT
2 A11beff  EA
• and for a single stiffener, dividing by ns=b/ds
Fstif 
ds
EA
FTOT
b 2 A11beff  EA
(5.4.2.11)
Stiffener buckling = PB x skin buckling
• single stiffener buckles when
Fstif 
 2 EI
(5.4.2.12)
a2
ds
b
a
• combining (5.4.2.11) and (5.4.2.12):
ds
EA
 2 EI
 2 EI b 2 A11beff  EA
FTOT 
 FTOT 
2
b 2 A11beff  EA
EA
a
a2 ds
(5.4.2.13)
Stiffener buckling = PB x skin buckling
• final failure occurs when the post-buckling factor (PB) is
reached; the force in the skin at that load is given by
Fskin  Fskin buckling (PB)
(5.4.2.14)
• with the skin buckling load given by
___


___
 
( AR) 4 
2
2
 b 2 D11 k  2D12   2D66 ( AR)  D22  2
a 
k 


2
Fskin buckling
(5.4.2.15)
• combining (5.4.2.10), (5.4.2.14), and (5.4.2.15)
FTOT
___
4
___
2 A11beff  EA  2 
(
AR
)
2
2

b 2 D11 k  2D12   2D66 ( AR)  D22  2 ( PB)
2 A11beff
a 
k 


(5.4.2.16)
Stiffener buckling = PB x skin buckling
• Equations (5.4.2.13) and (5.4.2.16) are combined to give
___


___
EI
( PB) 
( AR) 4 
2
2

D11k  2D12  2 D66 ( AR)  D22
d s EA 2 A11beff 
k2 


• use the definition of λ to simplify things
A11 

EA
ds
A11
 A11  A11 
EA
EA
EA
 A11 (  1) 

 (  1)d s
ds
ds
A11
• substitute in the above equation to obtain
EI    1( PB)d s
ds
2beff
___

4 
___
(
AR
)
2
2
 D11k  2D12  2 D66 ( AR)  D22

2

k 


(5.4.2.17)
Stiffener buckling = PB x skin buckling
• from the expression for beff derived earlier,



ds
A12 
A11
1 
1 

 21  21 

beff
A
(
PB
)
A

3
A
 11

11 
22 

(note that instead of a in the original beff expression we
have ds because this is the width of the plate that buckles)
ds
b
a
Results: Min EI required for stiffeners
• eqns (5.4.2.9) and (5.4.2.17) are used to determine the
minimum required bending stiffness for stiffeners
• EI is evaluated using both equations and the highest
value is used
• Note that this does not guarantee that there is no
crippling or inter-rivet buckling failure of the stiffener;
separate checks must be made for these and they
(usually) supersede the EI requirement (i.e. meeting the
crippling and/or inter-rivet buckling requirements usually
leads to higher EI than eqs (5.4.2.9) and (5.4.2.17))
Example of min stiffener EI required
• [(±45)/(0/90)/(±45)] skin
D11
D12
D22
D66
659.7
466.9
659.7
494.0
Nmm
Nmm
Nmm
Nmm
ds
b=762mm
a=508mm
A11
A12
A22
A66
28912.44
12491.43
28912.44
13468.58
N/mm
N/mm
N/mm
N/mm
Stiffener geometry and spacing
unknown. Determine minimum
EI for the stiffeners
Results: Min EI required for stiffeners
PB=5.0
500
EI/(D11ds)
______ panel buckling = bay
buckling
- - - - - - stiffener buckles at
PB x bay buckling
λ=1.5
400
300
λ=1.1
ds
b
200
a
100
0
50
100
150
200
250
300
350
Stiffener spacing (mm)
• between λ=1.1 and λ=1.5, stiffener buckling condition
becomes more critical (design driver changes)
why?
• note that as ds increases, the min req’d stiffness decreases
Results: Min EI required for stiffeners
(using the more critical of the two conditions)
9000
8000
7000
6000
5000
4000
3000
2000
1000
0
EI/(D11 ds)
λ=10
ds
b
a
λ=5
λ=3
λ=1.5
λ=1.1
50
PB=5.0
100
150
200
Stiffener spacing (mm)
250
300
Min stiffener EI: some notes
• comparing the min EI for PB=5 and PB=1.5 when λ=1.1
we see that they are identical; the reason is that the
“active” constraint is that bay buckling=panel buckling
which is independent of post-buckling factor PB
• when the axial stiffness EA of the stiffeners is low (λ=1.11.2) the min required bending stiffness for the stiffeners is
independent of the ratio of the final failure load to the skin
buckling load (PB)
• for higher axial stiffnesses (λ>1.2) the higher the PB the
higher the bending stiffness EI required for the stiffener