AIRFRAME STATIC ANALYSIS
– BUCKLING ON STIFFENED
PANELS
Objectives for this session:
Understand critical cases for an aircraft structure
Able to perform static analysis in order to check margin
of safety of stiffened panels under critical load cases
Assumption
Majority of modern aircraft structure is highly
indeterminate structure due to its complex configuration
Accurate solution requires Finite Element Approach
For quick analysis, there are simpler approaches using
empirical data to support the theory
This empirical approach is very beneficial for analysis or
preliminary sizing at design stage; or as a quick check
toward the computer result from Finite Element calculation
or result from Experimental Test
Critical Loads
Critical Loads
Type of Stiffened Panels
Compression Panels Efficiency
Buckling
Failure Modes of Stiffened Panels
Initial Buckling Inter-Rivet Buckling Flexural Buckling
Failure Modes of Stiffened Panels
Flexural and Torsional Instability Skin WrinklingTorsional Instability
Column Buckling
Plate Buckling
2
62.3
b
tEfcr
22
22
)1(12 L
tkEfcr
2
b
tKEfcror
2
9.0
L
tEfcr
2
6.3
L
tEfcr
if,
All edges: Simply supported
Buckling Coefficients for Different Support
Conditions
Effective Width, be
st
ce
f
EKtb
Where:
fst = Stringer Compression buckling stress
Kc = Skin compression buckling coeff.
E = Young’s Modules (use Et in inelastic
range)
Effective Width: Compression Buckling
Constant, Kc
For large panel with thin skin,
(e.g wing panel near tip) as
shown in fig (a) the torsional
stiffness of a stringer is large in
Comparison to the force tending
to twist it.
This effect produces a fixed edge
condition for the panel and the
compression buckling constant,
Kc = 6.32
A narrow panel with heavy skin
(e.g. wing panels near wing root),
as shown in fig (b) produces buckling
forces so great that the stringer will
twist locally.
This panel will act as if it had
hinged edges and the buckling
constant, Kc = 3.62
Effective Width: Compression Buckling
Constant, Kc
Kc = 3.62 for b/t < 40
Kc = 6.32 for b/t >110
Between the above two
values, Kc is plotted in
the left figure:
Effective Width: an Example
"24.225000
105.108.405.0
6
1
st
ce
F
EKtb
"58.225000
105.1032.605.0
6
1
st
ce
F
EKtb
"41.22
24.258.2
2
)( 21
ee bb
Assume the allowable crippling
stress of the stringers,
Fst = 25000 psi
Determine the skin effective
width Stringer no.2.
For:
(b/t)
= 160 Kc = 6.32
(b/t)
= 60 Kc = 4.8
The effective width is
The total effective width of the no.2 stringer is:
ESDU METHODS
Engineering Science Data Unit (ESDU) for
Buckling Checks on Stiffened Panels
Local Buckling
Panels with un-flanged Integral Stiffeners
Ref: ESDU 7003
Panels with Flanged Stringers
Ref: ESDU 71014
Inter Rivet Buckling
Ref: ESDU 02.01.09
Crippling of Stringer
Ref: ESDU 78020
Local Buckling of Compression Panels with un-
Flanged Integral Stiffeners
Average elastic compressive
stress in panel at which local
buckling first occurs, fb
fb = h (fb)e
Where:
(fb)e = KE (t/b)2
Ref: ESDU 70003
Notation
Ref: ESDU 70003
Example
Ref: ESDU 70003
Ref: ESDU 70003
Local Buckling of Compression Panels with
Flanged Stringers
Average elastic compressive
stress in panel at which local
buckling first occurs, fb
fb = h (fb)e
Where:
(fb)e = KE (t/b)2
Ref: ESDU 71014
Notation
Ref: ESDU 71014
Example
Ref: ESDU 71014
Ref: ESDU 71014
Ref: ESDU 71014
Ref: ESDU 71014
Ref: ESDU 71014
Ref: ESDU 71014
Exercise
Find local buckling stress for build-up ‘Z’ stringer-
skin panel. Use the same data as in previous
example.
Inter Rivet Buckling
22
22
)1(12 s
tKEfir
Type of attachment Fixity
coefficient at
rivets, K
Universal/Flathead
rivets
4
Spotwelds 3½
Roundhead/Mushro
om or snaphead
rivets
3
Countersunk or
dimpled rivets
1 or 1½
Note that: The effective width is important in the interest of structural efficiency
and weight economy. However, If the skin buckles between rivets, it can not
carry the compression load and the calculated effective width will be erroneous
and the structure is much less efficient.
Inter Rivet Buckling
Normally the skin-stringer
construction will be designed
so that rivet spacing is derived
from the crippling stress of the stringer.
However when the inter rivet
buckling stress of the skin is reached
before the crippling stress of the stringer,
the skin exhibit the ability to maintain
the inter rivet buckling stress while
the stringer continues to take load.
Inter Rivet Buckling
Inter Rivet Buckling: - an Example
Question:
Obtain the rivet.spacing for countersunk head rivets from the following given data:
Stringer crippling stress, Fcc = 32 ksi
Skin thickness, t = 0.05”; material is 7075-T6 bare (non-clad material).
Answer:
Using Fig. 14.3.2 with Fir = Fcc = 32 ksi, go across horizontally to curve (8) for 7075-T6 material.
Go down vertically to read the rivet spacing ratio s/t = 33.5 (for universal head rivets, c = 4.0).
s = 33.5 x 0.05 = 1.68”
For countersunk head rivets, c = 1.0.
The rivet spacing of countersunk head rivet is:
s= 1.68(1.0/4.0) = 0.84”
Crippling of Stringer
fc = (c2 fb)1/2
Where:
fb = h fbe
fbe = KE (th/h)2
Ref: ESDU 78020
Notation
Ref: ESDU 78020
Example
Ref: ESDU 78020
Ref: ESDU 78020
Ref: ESDU 78020
Exercise
Calculate the crippling stress for z section. Use the
same material and the associated dimensions
Flexural Buckling
tNE
LfF
The Farrar’s efficiency
factor (F) accounts for
A pure flexural instability
(assume flexural-torsional
Coupling is small):
Where:
f – failure stress of skin stringer panel
N – end load per inch width of skin stringer panel
Et – tangent modulus
L – Length of the panel (rib or frame spacing
Flexural (Euler) Buckling