Lecture #4
Buckling in aircraft structures
PLATES IN UNI-AXIAL COMPRESSION
Object
Buckling factor k
0.9  kE
 cr 
2
b  
2
PLATES IN SHEAR
Object
kE
 cr 
2
b  
Buckling factor k
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CYLINDRICAL RODS IN COMPRESSION
Object
Buckling factor k
0.3
kEh
 cr 
R
k
0.25
0.2
0.15
0.1
analytically k=0.6!
0.05
R/h
0
0
500
1000
1500 4
SHPERICAL SHELL UNDER EXTERNAL PRESSURE
Object
Buckling factor k
0.2
k
kEh
 cr 
R
0.15
0.1
0.05
R/h
0
0
500
1000
1500
analytically k=0.6!
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FACTORS WHICH AFFECT THE ACCURACY OF
BUCKLING ANALYSIS
• Initial imperfections (the effect is especially large for
cylindrical and spherical shells);
• Curved geometry (very actual for aircraft panels);
• Uncertain supporting conditions;
• Inelastic behavior of material (the effect is crucial for
lightweight and high-loaded aircraft structures).
Ways to increase the accuracy of calculations:
• Use empirical methods and formulas (much testing
required);
• Use numerical methods.
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COMPARISON BETWEEN CIRCULAR AND SQUARE
CROSS SECTIONS
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WAYS TO FIND THE CRITICAL STRESS WHICH IS
BEYOND THE YEILD LIMIT
• Empirical determination of critical stress (Southwell
method).
• Usage of plasticity correction factors calculated by
empirical formulas.
• Finite element analysis (FEA).
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SOUTHWELL METHOD FOR PLATES
The deformed shape of panel is
m x
n y
w  x , y    Amn  sin
 sin
a
b
m 1 n 1
C mn  N x
The view of coefficients is
Amn 
f  m, n  N x
Near the buckling mode corresponding
Cmn  N x
to m and n, the displacement is wmax 
N x cr  N x

Thus, the graph of

wmax
against
straight line with a slope of
wmax
N x cr .
will form a
Nx
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PLASTICITY CORRECTION – EMPIRICAL FORMULAS
Kan and Sverdlov proposed the following formula to
take into account the plastic behavior of material:
1 
 cr   u 
2
1  
where
u

 cr , el
,
;
 cr , el - elastic critical stress (calculated using linear
physical law);
 u - material ultimate stress.
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PLASTICITY CORRECTION – EMPIRICAL FORMULAS
According to Gerard, the nonlinear material behavior
could be taken into account using the plasticity
correction factor h:
0.9  h  k  E
 cr 
2
b  
1   E s  1 1 1 3 Et 
h
   
  
1 
E  2 2 4 4 Es 
Et , E s - tangent and secant moduli;
 el , pl - elastic and inelastic Poisson’s ratios.
2
el
2
pl
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BUCKLING MODES OF AIRCRAFT PANEL
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POST-BUCKLING BEHAVIOR
The post-buckling
behavior of aircraft
panels is usually
studied using the
concept of “attached
skin” having the
width of 2c:
2c
 str
 cr , skin
2c  t
 str
 cr , skin
t
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FLOWCHART OF NONLINEAR BUCKLING FEA
Linear
compression
Linear buckling
Transfer of initial
imperfections
Nonlinear analysis
Linear physical law is used
Probable modes of buckling are
found, dangerous one is chosen
depending on the area of interest
The model is slightly modified
according to buckling mode,
nonlinearities are specified
The critical force corresponds to
the loss of convergence
Results
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FLOWCHART OF NONLINEAR BUCKLING FEA
Dimensions of the panel:
a = 60 mm, h = 22 mm, 1 = 4 mm, 2 = 6 mm.
Length is 300 mm.
Supporting conditions: clamped at all sides.
Problem is to find a critical stress for this panel.
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FLOWCHART OF NONLINEAR BUCKLING FEA
Linear
compression
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FLOWCHART OF NONLINEAR BUCKLING FEA
Linear buckling
1st mode
884 MPa
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FLOWCHART OF NONLINEAR BUCKLING FEA
Linear buckling
2nd mode
996 MPa
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FLOWCHART OF NONLINEAR BUCKLING FEA
Linear buckling
3rd mode
1210 MPa
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FLOWCHART OF NONLINEAR BUCKLING FEA
Transfer of initial
imperfections
1) Rename *.rst file to
"buckling.rst“, move it from
buckling to structural analysis directory.
2) Insert a command object in the environment:
/prep7
upgeom,0.1,1,1,buckling,rst
/solu
3) Set nonlinearities:
large deflection -> on
nonlinear effects -> yes
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FLOWCHART OF NONLINEAR BUCKLING FEA
Nonlinear analysis
485 MPa –
last substep
converged
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FLOWCHART OF NONLINEAR BUCKLING FEA
Results
35
30
Shortening, mm
25
Nonlinear FEA
predicted 509 MPa
which is 1.73 times
smaller than elastic
solution.
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15
10
5
Stress, MPa
0
0
100
200
300
400
500
600
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EFFECT OF BUCKLING ON A STRUCTURAL LAYOUT
Structural layout
dimension
Determinative factor
Distance between ribs and
fuselage frames
Primary buckling of panels
Distance between stringers
Local buckling of skin
between stringers
Distance between spar web Buckling of spar web under
stiffeners
shear loads
Distance between rivets in
longitudinal joints
Local buckling of skin
between rivets
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EFFECT OF BUCKLING ON A STRUCTURAL LAYOUT
The stringer cross section is dramatically affected by
buckling, showing the compromise between buckling
resistance and technological simplicity:
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WHERE TO FIND MORE INFORMATION?
Megson. An Introduction to Aircraft Structural Analysis. 2010
Chapter 9
… Internet is boundless …
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TOPIC OF THE NEXT LECTURE
The concept of thin-walled beam.
Normal stresses
All materials of our course are available
at department website k102.khai.edu
1. Go to the page “Библиотека”
2. Press “Structural Mechanics (lecturer Vakulenko S.V.)”
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