Page 1
University of Liège
Aerospace & Mechanical Engineering
Aircraft Structures
Instabilities
Aircraft Structures - Instabilities
Ludovic Noels
Computational & Multiscale Mechanics of Materials – CM3
http://www.ltas-cm3.ulg.ac.be/
Chemin des Chevreuils 1, B4000 Liège
[email protected]
Page 2
Elasticity
• Balance of body B– Momenta balance
• Linear
• Angular
– Boundary conditions• Neumann
• Dirichlet
• Small deformations with linear elastic, homogeneous & isotropic material
– (Small) Strain tensor , or
– Hooke’s law , or
with
– Inverse law
with
b
T
n
2ml = K - 2m/3
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• General expression for unsymmetrical beams
– Stress
With
– Curvature
– In the principal axes Iyz = 0
• Euler-Bernoulli equation in the principal axis
– for x in [0 L]
– BCs
– Similar equations for uy
Pure bending: linear elasticity summary
x
z f(x) TzMxx
uz =0
duz /dx =0 M>0
L
y
zq
Mxx
a
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• General relationships
–
• Two problems considered
– Thick symmetrical section
• Shear stresses are small compared to bending stresses if h/L << 1
– Thin-walled (unsymmetrical) sections
• Shear stresses are not small compared to bending stresses
• Deflection mainly results from bending stresses
• 2 cases
– Open thin-walled sections
» Shear = shearing through the shear center + torque
– Closed thin-walled sections
» Twist due to shear has the same expression as torsion
Beam shearing: linear elasticity summary
x
z f(x) TzMxx
uz =0
duz /dx =0 M>0
L
h
L
L
h t
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• Shearing of symmetrical thick-section beams
– Stress
• With
• Accurate only if h > b
– Energetically consistent averaged shear strain
• with
• Shear center on symmetry axes
– Timoshenko equations
• &
• On [0 L]:
Beam shearing: linear elasticity summary
ht
z
y
z
t
b(z)
A*
t
h
x
z
Tz
dx
Tz+ ∂xTz dx
gmax
gg dx
z
x
g
qy
qy
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• Shearing of open thin-walled section beams
– Shear flow
•
• In the principal axes
– Shear center S
• On symmetry axes
• At walls intersection
• Determined by momentum balance
– Shear loads correspond to
• Shear loads passing through the shear center &
• Torque
Beam shearing: linear elasticity summary
x
z
y
Tz
Tz
Ty
Ty
y
z
S
Tz
TyC
q
s
y
t
t
h
b
z
C
t
S
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• Shearing of closed thin-walled section beams
– Shear flow
•
• Open part (for anticlockwise of q, s)
• Constant twist part
• The q(0) is related to the closed part of the section,
but there is a qo(s) in the open part which should be
considered for the shear torque
Beam shearing: linear elasticity summary
y
z
T
Tz
Ty
C
q s
pds
dAh
y
z
T
Tz
Ty
C
q s
p
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• Shearing of closed thin-walled section beams
– Warping
•
• With
– ux(0)=0 for symmetrical section if origin on
the symmetry axis
– Shear center S
• Compute q for shear passing thought S
• Use
Beam shearing: linear elasticity summary
y
z
T
Tz
Ty
C
q s
pds
dAh
y
z
S
Tz
C
q s
p ds
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• Torsion of symmetrical thick-section beams
– Circular section
•
•
– Rectangular section
•
•
• If h >> b
– &
–
–
Beam torsion: linear elasticity summary
t
z
y
C
Mx
r
h/b 1 1.5 2 4 ∞
a 0.208 0.231 0.246 0.282 1/3
b 0.141 0.196 0.229 0.281 1/3
z
y
C
tmaxMx
b
h
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• Torsion of open thin-walled section beams
– Approximated solution for twist rate
• Thin curved section
–
–
• Rectangles
–
–
– Warping of s-axis
•
Beam torsion: linear elasticity summary
y
z
l2
t2
l1
t1
l3
t3
t
t
z
y
C
Mx
t
ns
t
R
pR
us
q
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• Torsion of closed thin-walled section beams
– Shear flow due to torsion
– Rate of twist
•
• Torsion rigidity for constant m
– Warping due to torsion
•
• ARp from twist center
Beam torsion: linear elasticity summary
y
z
C
q s
pds
dAh
Mx
y
z
R
C p
pRY
us
q
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• Panel idealization
– Booms’ area depending on loading
• For linear direct stress distribution
Structure idealization summary
b
sxx1
sxx2
A1 A2
y
z
x
tD
b
y
z
xsxx
1sxx
2
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• Consequence on bending
– If Direct stress due to bending is carried by booms only
• The position of the neutral axis, and thus the second moments of area
– Refer to the direct stress carrying area only
– Depend on the loading case only
• Consequence on shearing
– Open part of the shear flux
• Shear flux for open sections
• Consequence on torsion
– If no axial constraint
• Torsion analysis does not involve axial stress
• So torsion is unaffected by the structural idealization
Structure idealization summary
Tz
y
z
x
dx
Ty
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• Virtual displacement
– In linear elasticity the general formula of virtual displacement reads
• s (1) is the stress distribution corresponding to a (unit) load P(1)
• DP is the energetically conjugated displacement to P in the direction of P(1) that
corresponds to the strain distribution e
– Example bending of semi cantilever beam
•
– In the principal axes
– Example shearing of semi-cantilever beam
•
Deflection of open and closed section beams summary
x
z Tz
uz =0
duz /dx =0 M>0
L
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• Torsion of a built-in end closed-section beam
– If warping is constrained (built-in end)
• Direct stresses are introduced
• Different shear stress distribution
– Example: square idealized section
• Warping
• Shear stress
Structural discontinuities summary
z
y
Mxx
L
b
h
dx
qb
qh
uxm
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• Shear lag of a built-in end closed-section beam
– Beam shearing
• Shear strain in cross-section
• Deformation of cross-section
• Elementary theory of bending
– For pure bending
– Not valid anymore because of the
cross section deformation
– Example
• 6-boom wing
• Deformation of top cover
Structural discontinuities summaryz
y
x
L
d
h
Tz/2d
A1 A2A1 qh
qd
Tz/2
d
y
x
d
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• Torsion of a built-end open-section beam
– If warping is constrained (built-in end)
• Direct stresses are introduced
• There is a bending contribution
to the torque
– Examples
• Equation for pure torque
with
• Equation for distributed torque
Structural discontinuities summary
Mx Mx
Mx
x
z
ymx
Mx+∂xMx dx
dx
Mx
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• 2 kinds of buckling
– Primary buckling
• No changes in cross-section
• Wavelength of buckle ~ length of element
• Solid & thick-walled column
– Secondary (local) buckling
• Changes in cross-section
• Wavelength of buckle ~ cross-sectional dimensions
• Thin-walled column & stiffened panels
– Pictures:– D.H. Dove wing (max loading test)– Automotive beam – Local buckling
Column instabilities
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• Assumptions
– Perfectly symmetrical column (no imperfection)
– Axial load perfectly aligned along centroidal axis
– Linear elasticity
• Theoretically
– Deformed structure should remain symmetrical
– Solution is then a shortening of the column
– Buckling load PCR is defined as P such that if a
small lateral displacement is enforced by a
lateral force, once this force is removed
• If P = PCR, the lateral deformation is constant
(neutral stability)
• If P > PCR, this lateral displacement increases &
the column is unstable
• If P < PCR, this lateral displacement disappears &
the column is stable
• Practically
– The initial lateral displacement is due to
imperfections (geometrical or material)
Euler buckling
P
P P
F
P
P < PCR
P
P > PCR
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• Euler critical axial load
– Bending theory
•
– Solution
• General form: with
• BCs at x = 0 & x = L imply C1 = 0 &
• Non trivial solution with k = 1, 2, 3, …
• In that case C2 is undetermined and can → ∞
– Euler critical load for pinned-pinned BCs
• with k = 1, 2, 3, …
Euler buckling
Puz
x
z
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• Euler critical axial load (2)
– Euler critical load for pinned-pinned BCs (2)
• with k = 1, 2, 3, …
• Buckling will occur for lowest PCR
– In the plane of lowest I
– For the lowest k k = 1
– In case modes 1, .. k-1 are prevented, critical load becomes the load k
Euler buckling
Puz
x
z
Px
z
Px
z
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• Euler critical axial load (3)
– For pinned-pinned BCs
•
• (compression) with gyration radius
– General case
• Euler critical loads , (compressive)
• With le the effective length
Euler buckling
Puz
x
z
P
le = L/2
le = L/2
P
le = 2L
P
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• Practical case: initial imperfection
– Let us assume an initial small curvature of the beam
•
– As this curvature is small the equation of bending
for straight beam can still be used, but with the
change of curvature being considered for the strain
•
• The general form of the initial deflection satisfying the BCs is
the deflection equation becomes
– Solution
• With
Initial imperfection
uz0
x
z
Puz
x
z
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• Practical case: initial imperfection (2)
– Solution for an initial small curvature of the beam
•
• With
• BCs at x = 0 & x = L imply C1 = C2 = 0, & as
• Clearly near buckling, so P→PCR, and the dominant term of the solution is for n = 1
Initial imperfection
uz0
x
z
Puz
x
z
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• Practical case: initial imperfection (3)
– Solution for an initial small curvature of the beam
• near buckling
– If central deflection is measured vs axial load
• As uz0(L/2) ~ A1
•
– Southwell diagram
•
• Allows measuring buckling loads without
breaking the columns
• Remark
– Critical Euler loading depends on BCs
Initial imperfection
uz0
x
z
Puz
x
z
D
D/P
A1
1/PCR
1
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• Thin-walled column under critical flexural loads
– Can twist without bending or
– Can twist and bend simultaneously
Flexural-torsional buckling of open thin-walled columns
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• Kinematics
– Consider
• A thin-walled section
• Centroid C
• Cyz principal axes
• Shear center S
– Section motion (CSRD)
• Translation
– Shear center is moved
– By uyS & uz
S
– To S’’
• Rotation around shear (twist) center
– We assume shear center=twist center
– By q
• Centroid motion
– To C’ after section translation
– To C’’ after rotation
– Resulting displacements uyC & uz
C
– Same decomposition for other
points of the section
Flexural-torsional buckling of open thin-walled columns
y
z
S
C
q
C’
uyS
uzS
uyS
uzS
S’’
C’’
a
a
uyC
uzC
y
z
S
C
q
C’
uyS
uzS
uyS
uzS
S’’
C’’
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• Kinematics (2)
– Relations
• Centroid
• Other points P of the section
• Considering axial loading
– If q remains small, the induced momentums are
– As we are in the principal axes (Iyz=0), and
as motion resulting from bending is uS
Flexural-torsional buckling of open thin-walled columns
Puz
x
z
Puy
x
y
y
z
S
C
q
C’
uyS
uzS
uyS
uzS
S’’
C’’
a
a
uyC
uzC
P
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• Torsion
– Any point P of the section
– As torsion results from axial loading,
this corresponds to a torque with
warping constraint
• See previous lecture
– Analogy between
beam bending/pin-ended column
– As
– The momentum at point P can be substituted by lateral loading
– Contributions on ds
Flexural-torsional buckling of open thin-walled columns
y
z
S
C
q
uyS
uzS
S’’
C’’P P’’
ds
dfz
dfy
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• Torsion (2)
– Lateral loading analogy
• Contributions on ds
• As axial load leads to uniform
compressive stress on section
of area A
• Resulting distributed torque (per unit length) on ds
–
Flexural-torsional buckling of open thin-walled columns
y
z
S
C
q
Mxx
uzS
uyS
uzS
S’’
C’’P P’’
ds
dfz
dfy
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• Distributed torque
– As
– As C is the centroid
Flexural-torsional buckling of open thin-walled columns
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• Distributed torque (2)
– The analogous torque by unit length resulting from the bending reads
– Polar second moment of area around S:
– For a built-in end open-section beam
• Warping is constrained
– Bending contribution to the torque
• New equation
• For a constant section
Flexural-torsional buckling of open thin-walled columns
Mx
x
z
ymx
Mx+∂xMx dx
dx
Mx
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• Equations
– In the principal axes
•
•
•
Flexural-torsional buckling of open thin-walled columns
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• Example
– Column with
• Deflection and rotation around x
constrained at both ends
– uy(0) = uy(L) = 0 & uz(0) = uz(L) = 0
– q(0) = 0 & q(L) = 0
• Warping and rotation around y & z allowed at both ends
– Twist center = shear center
– My(0) = My(L) = 0 uy,xx(0) = uy,xx(L) = 0
– Mz(0) = Mz(L) = 0 uz,xx(0) = uz,xx(L) = 0
– As warping is allowed
are equal to zero
q,xx(0) = 0 & q,xx(L) = 0
Flexural-torsional buckling of open thin-walled columns
x
z
y
P
L
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• Resolution
– Assuming the following fields satisfying the BCs
•
•
•
– The system of equations
Flexural-torsional buckling of open thin-walled columns
x
z
y
P
L
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• Resolution (2)
– Non trivial solution leads to buckling load P
• Buckling load is the minimum root
Flexural-torsional buckling of open thin-walled columns
x
z
y
P
L
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• If shear center and centroid coincide
– System becomes
– This system is uncoupled and leads to 3 critical loads
– Buckling load is the minimum value
Flexural-torsional buckling of open thin-walled columns
x
z
y
P
L
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• Example
– Column
• Length: L = 1 m
• Young: E = 70 GPa
• Shear modulus: m = 30 GPa
– Buckling load?
• Deflection and rotation around x
constrained at both ends
– uy(0) = uy(L) = 0 & uz(0) = uz(L) = 0
– q(0) = 0 & q(L) = 0
• Warping and rotation around y & z allowed
at both ends
Flexural-torsional buckling of open thin-walled columns
y
t = 2 mm
h=
10
0 m
m
b = 100 mm
z
y’
z’
C
t = 2 mm
t = 2 mm
S(y,0 ) O’
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• Centroid position
–
– By symmetry on Oy
• Second moment of area
–
–
– By symmetry
Flexural-torsional buckling of open thin-walled columns
y
t = 2 mm
h=
10
0 m
m
b = 100 mm
z
y’
z’
C
t = 2 mm
t = 2 mm
S(yS,0 ) O’
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• Shear center
– On Cy by symmetry
– Consider shear force Tz
• As Iyz = 0
• Lower flange, considering frame O’y’z’
• Upper flange by symmetry
• As Tz passes through the shear center: no torsional flux
Flexural-torsional buckling of open thin-walled columns
y
t = 2 mm
h=
10
0 m
m
b = 100 mm
z
y’
z’
C
t = 2 mm
t = 2 mm
S(yS, 0) O’
s
Tz
q
q
q
MO’
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• Uncoupled critical loads
– Using following definitions
• These values would be the critical loads
for an uncoupled system (if C = S)
?
Some values are missing
Flexural-torsional buckling of open thin-walled columns
y
t = 2 mm
h=
10
0 m
m
b = 100 mm
z
y’
z’
C
t = 2 mm
t = 2 mm
S(yS,0 ) O’
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• Uncoupled critical loads (2)
–
•
•
•
•
– Requires ARp(s)
Flexural-torsional buckling of open thin-walled columns
y
t = 2 mm
h=
10
0 m
m
b = 100 mm
z
y’
z’
C
t = 2 mm
t = 2 mm
S(yS, 0) O’
s
Tz
q
q
q
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• Uncoupled critical loads (3)
– Evaluation of the ARp(s)
• Lower flange
• Web
• Upper flange
Flexural-torsional buckling of open thin-walled columns
y
t = 2 mm
h=
10
0 m
m
b = 100 mm
z
y’
z’
C
t = 2 mm
t = 2 mm
S(yS, 0) O’
+
<0
+
<0>0
y
t = 2 mm
h=
10
0 m
m
b = 100 mm
z
y’
z’
C
t = 2 mm
t = 2 mm
S(yS, 0) O’
+
<0>0
y
t = 2 mm
h=
10
0 m
m
b = 100 mm
z
y’
z’
C
t = 2 mm
t = 2 mm
S(yS, 0) O’
<0
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• Uncoupled critical loads (4)
–
• Lower flange:
–
–
–
–
Flexural-torsional buckling of open thin-walled columns
y
t = 2 mm
h=
10
0 m
m
b = 100 mm
z
y’
z’
C
t = 2 mm
t = 2 mm
S(yS, 0) O’
s
Tz
q
q
q
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• Uncoupled critical loads (5)
–
• Web:
–
–
–
–
Flexural-torsional buckling of open thin-walled columns
y
t = 2 mm
h=
10
0 m
m
b = 100 mm
z
y’
z’
C
t = 2 mm
t = 2 mm
S(yS, 0) O’
s
Tz
q
q
q
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• Uncoupled critical loads (6)
–
• Upper flange:
–
Flexural-torsional buckling of open thin-walled columns
y
t = 2 mm
h=
10
0 m
m
b = 100 mm
z
y’
z’
C
t = 2 mm
t = 2 mm
S(yS, 0) O’
s
Tz
q
q
q
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• Uncoupled critical loads (7)
–
• Upper flange (2):
–
–
Flexural-torsional buckling of open thin-walled columns
y
t = 2 mm
h=
10
0 m
m
b = 100 mm
z
y’
z’
C
t = 2 mm
t = 2 mm
S(yS, 0) O’
s
Tz
q
q
q
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• Uncoupled critical loads (8)
–
• All contributions
–
–
–
Flexural-torsional buckling of open thin-walled columns
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• Uncoupled critical loads (9)
–
•
•
•
•
Flexural-torsional buckling of open thin-walled columns
y
t = 2 mm
h=
10
0 m
m
b = 100 mm
z
y’
z’
C
t = 2 mm
t = 2 mm
S(yS, 0) O’
s
Tz
q
q
q
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• Critical load – As the uncoupled critical loads read
, &
& as zS = 0, the coupled system is rewritten
Flexural-torsional buckling of open thin-walled columns
y
t = 2 mm
h=
10
0 m
m
b = 100 mm
z
y’
z’
C
t = 2 mm
t = 2 mm
S(yS, 0) O’
s
Tz
q
q
q
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• Critical load (2)
– Resolution
Flexural-torsional buckling of open thin-walled columns
>
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• Thin plates
– Are subject to primary buckling
• Wavelength of buckle ~
length of element
– So they are stiffened
Buckling of thin plates
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• Primary buckling of thin plates
– Plates without support
• Similar to column buckling
– Same shape
– Use D instead of EIzz
– Supported plates
• Other displacement buckling shapes
• Depend on BCs
Buckling of thin plates
p
b a
E1
E2
E3
A
f
f
b a
E1
E2
E3
A
f
f
0
0.5
1
0
0.5
10
2
4
x 10-3
x/ay/a
u3 D
/(p
0 a
4)
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• Kirchhoff-Love membrane mode
– On A:
• With
– Completed by appropriate BCs
• Dirichlet
• Neumann
Buckling of thin plates
∂NA
n
n0 = na EaE1
E2
E3
A∂DA
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• Kirchhoff-Love bending mode
– On A:
• With
– Completed by appropriate BCs
• Low order
– On ∂NA:
– On ∂DA:
• High order
– On ∂TA:
with
– On ∂MA:
Buckling of thin plates
∂NA
T
n0 = na EaE1
E2
E3
A∂DA
p
∂MA
M
n0 = na EaE1
E2
E3
A∂TA
p
D
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• Membrane-bending coupling
– The first order theory is uncoupled
– For second order theory
• On A:
• Tension increases the bending
stiffness of the plate
• Internal energy
– In case of small initial curvature (k >>)
• On A:
• Tension induces bending effect
Buckling of thin plates
ñ11E1
E2
E3
A
ñ22
ñ12
ñ21
E1
E2
E3
A
ñ22
ñ12
ñ21
j03
u3
ñ11
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• Primary buckling theory of thin plates
– Second order theory
• On A:
– Simply supported plate
with arbitrary pressure
• Pressure is written in a Fourier series
• Displacements with these BCs can also be written
with
• There is a buckling load ñ11 leading to
infinite displacements for every couple (m, n)
– Lowest one?
Buckling of thin plates
ñ11E1
E2
E3
A
ñ22
ñ12
ñ21
p
b a
E1
E2
E3
A
ñ11
ñ11
2013-2014 Aircraft Structures - Instabilities 57
Page 58
• Primary buckling theory of thin plates (2)
– Simply supported plate
• Displacements in terms of
• Buckling load ñ11
– Minimal (in absolute value) for n=1
– Or again
with the buckling coefficient k
– Depends on ration a/b
Buckling of thin plates
p
b a
E1
E2
E3
A
ñ11
ñ11
2013-2014 Aircraft Structures - Instabilities 58
Page 59
• Primary buckling theory of thin plates (3)
– Simply supported plate (2)
• Buckling coefficient k
• Mode of buckling depends on a/b
• k is minimal (=4) for a/b = 1, 2, 3, …
• Mode transition for
• For a/b > 3: k ~ 4
– This analysis depends on the BCs, but same behaviors for
• Other BCs
• Other loadings (bending, shearing) instead of compression
• Only the value of k is changing (tables)
Buckling of thin plates
2013-2014 Aircraft Structures - Instabilities 59
0 2 4 60
2
4
6
8
10
20.5
60.5
120.5
a/b
k
m=1
m=2
m=3
m=4
𝑎/𝑏
𝑘
Page 60
• Primary buckling theory of thin plates (4)
– Shape of the modes for
• Simply supported plate
• In compression
• n=1
Buckling of thin plates
0
0.5
1
0
0.5
1-1
0
1
x/ay/a
u3m
=1
0
0.5
1
0
0.5
1-1
0
1
x/ay/a
u3m
=2
2013-2014 Aircraft Structures - Instabilities 60
Page 61
• Primary buckling theory of thin plates (5)
– Critical stress
• We found
• Or again
• This can be generalized to other loading
cases with k depending on the problem
– Picture for simply supported plate in
compression
– As
•
• k ~ cst for a/b >3
We use stiffeners to reduce b
to increase sCR of the skin
• As long as sCR < sp0
Buckling of thin plates
b
2013-2014 Aircraft Structures - Instabilities 61
0 2 4 60
2
4
6
8
10
20.5
60.5
120.5
a/b
k
m=1
m=2
m=3
m=4
𝑎/𝑏
𝑘
Page 62
• Primary buckling theory of thin plates (6)
– What happens for other BCs?
• We cannot say anymore
• But buckling corresponds to a stationary point of the internal energy
(neutral equilibrium)
• So we can plug any Fourier series or displacement approximations in the form
and find the stationary point
Buckling of thin plates
2013-2014 Aircraft Structures - Instabilities 62
Page 63
• Primary buckling theory of thin plates (7)
– Energy method
• Let us analyze the simply supported plate
• Internal energy
• First term
– As the cross-terms vanish
Buckling of thin plates
2013-2014 Aircraft Structures - Instabilities 63
Page 64
• Primary buckling theory of thin plates (8)
– Energy method (2)
• Internal energy
– As
– And as cross-terms vanish
Buckling of thin plates
2013-2014 Aircraft Structures - Instabilities 64
Page 65
• Primary buckling theory of thin plates (9)
– Energy method (3)
• As
Buckling of thin plates
2013-2014 Aircraft Structures - Instabilities 65
Page 66
• Primary buckling theory of thin plates (10)
– Energy method (4)
• As
• At buckling we have at least for one couple (m, n)
• Most critical value for n=1
• In general
Buckling of thin plates
2013-2014 Aircraft Structures - Instabilities 66
Page 67
• Experimental determination of critical load
– Avoid buckling Southwell diagram
– Plate with small initial curvature
•
– Particular case of p = 0, tension ñ11,
simply supported edges
• For
with
• When ñ11 →
– Term bm1 is the dominant one in the solution
– Displacement takes the shape of buckling mode m (n=1)
Buckling of thin plates
E1
E2
E3
A
j03
u3
ñ11
2013-2014 Aircraft Structures - Instabilities 67
Page 68
• Experimental determination of critical load (2)
– Particular case of p=0, tension ñ11 , simply supported edges (2)
• When ñ11 →
– Term bm1 is the dominant one in the solution
– As
with
• Rearranging:
– m depends on ratio a/b
Buckling of thin plates
u3
-u3/ñ11
bm1
-1/ñ11CR
1
2013-2014 Aircraft Structures - Instabilities 68
Page 69
• Primary to secondary buckling of columns
– Slenderness ratio le/r with
• le: effective length of the column
– Depends on BCs and mode
• r: radius of gyration
– High slenderness (le/r >80)
• Primary buckling
– Low slenderness (le/r <20)
• Secondary (local) buckling
• Usually in flanges
– In between slenderness
• Combination
Secondary buckling of columns
le = L/2
2013-2014 Aircraft Structures - Instabilities 69
Page 70
• Example of secondary buckling
– Composite beam
– Design such that
• Load of primary buckling >
limit load >
web local buckling load
– Final year project
• Alice Salmon
• Realized by
• How to determine secondary buckling?
– Easy cases: particular sections
Secondary buckling of columns
2013-2014 Aircraft Structures - Instabilities 70
Page 71
• Secondary buckling of a L-section
– Represent the beam as plates
– Take critical plate and evaluate
k from plate analysis
• k = 0.43, mode m=1
– Deduce buckling load
•
• Check if lower than sp0
– This method is an approximation
• Experimental determination
Secondary buckling of columns
Loaded edges simply
supported
One unloaded edge free
one simply supported
(? Assumption)
0.43
b
a=3b
2013-2014 Aircraft Structures - Instabilities 71
Page 72
• Primary buckling of thin plates
– We found
•
• With k ~ constant for a/b >3
– In order to increase the buckling stress
• Increase h0/b ratio, or
• Use stiffeners to reduce effective b of skin
Buckling of stiffened panels
b
w
tst
tsk
bsk
bst
2013-2014 Aircraft Structures - Instabilities 72
Page 73
• Buckling modes of stiffened panels
– Consider the section
– Different buckling possibilities
• High slenderness
– Euler column (primary) buckling with cross-section depicted
• Low slenderness and stiffeners with high degree of strength compared to skin
– Structure can be assumed to be flat plates
» Of width bsk
» Simply supported by the (rigid) stringers
– Structure too heavy
• More efficient structure if buckling occurs in stiffeners and skin at the same time
– Closely spaced stiffeners of comparable thickness to the skin
– Warning: both buckling modes could interact and reduces critical load
– Section should be considered as a whole unit
– Prediction of critical load relies on assumptions and semi-empirical methods
– Skin can also buckled between the rivets
Buckling of stiffened panels
w
tst
tsk
bsk
bst
2013-2014 Aircraft Structures - Instabilities 73
Page 74
• A simple method to determine buckling
– First check Euler primary buckling:
– Buckling of a skin panel
• Simply supported on 4 edges
• Assumed to remain elastic
– Buckling of a stiffener
• Simply supported on 3 edges &
1 edge free
• Assumed to remain elastic
– Take lowest one (in absolute value)
Buckling of stiffened panels
w
tst
tsk
bsk
bst
0.43
2013-2014 Aircraft Structures - Instabilities 74
Page 75
• Shearing instability
– Shearing
• Produces compression in the skin
• Leads to wrinkles
– The structure keeps some stiffness
– Picture: Wing of a Boeing stratocruiser
Buckling of stiffener/web constructions
2013-2014 Aircraft Structures - Instabilities 75
Page 76
References
• Lecture notes
– Aircraft Structures for engineering students, T. H. G. Megson, Butterworth-
Heinemann, An imprint of Elsevier Science, 2003, ISBN 0 340 70588 4
• Other references
– Books
• Mécanique des matériaux, C. Massonet & S. Cescotto, De boek Université, 1994,
ISBN 2-8041-2021-X
2013-2014 Aircraft Structures - Instabilities 76
Page 77
• Example
– Uniform transverse load fz
– Pinned-pinned BCs
– Maximum deflection?
– Maximum momentum?
Annex 1: Transversely loaded columns
P
x
z
fz
2013-2014 Aircraft Structures - Instabilities 77
Page 78
• Equation
– Euler-Bernouilli
• This assumes deformed configuration ~ initial configuration
• But near buckling, due to the deflection, P is exerting a moment
• So we cannot apply superposition principle as the axial loading also produces a
deflection
– Going back to bending equation
•
Annex 1: Transversely loaded columns
P
x
z
fz
P
Mxx
fzL/2
x
z
fz
2013-2014 Aircraft Structures - Instabilities 78
Page 79
• Solution
– Going back to bending equation
•
– General solution
• with
• BC at x = 0:
• BC at x = L:
– Deflection
• Deflection and momentum are maximum at x = L/2
Annex 1: Transversely loaded columns
P
x
z
fz
2013-2014 Aircraft Structures - Instabilities 79
Page 80
• Maximum deflection
– Deflection is maximum at x = L/2
Annex 1: Transversely loaded columns
P
x
z
fz
2013-2014 Aircraft Structures - Instabilities 80
Page 81
• Maximum deflection (2)
– Deflection is maximum at x = L/2 (2)
• From Euler-Bernoulli theory
• As for plates, compression induces bending
due to the deflection (second order theory)
Annex 1: Transversely loaded columns
P
x
z
fz
2013-2014 Aircraft Structures - Instabilities 81
10-2
10-1
100
100
101
102
P/PCR
uz/u
z(P=
0)
𝑃/𝑃𝐶𝑅
𝑢𝑧/𝑢𝑧(𝑃 = 0)
Page 82
• Maximum moment
– Maximum moment at x = L/2
•
• With
Annex 1: Transversely loaded columns
P
x
z
fz
2013-2014 Aircraft Structures - Instabilities 82
Page 83
• Maximum moment (3)
– Maximum moment at x = L/2 (2)
• Remark: for large deflections the bending equation which assumes linearity is no
longer correct as curvature becomes
Annex 1: Transversely loaded columns
P
x
z
fz
2013-2014 Aircraft Structures - Instabilities 83
10-2
10-1
100
100
101
102
P/PCR
Mxx
/Mxx
(P=
0)
𝑃/𝑃𝐶𝑅
𝑀𝑥𝑥/𝑀𝑥𝑥(𝑃 = 0)
Page 84
• Spar of wings
– Usually not a simple beam
– Assumptions before buckling:
• Flanges resist direct stress only
• Uniform shear stress in each web
– The shearing produces compression
in the web leading to a-inclined wrinkles
– Assumptions during buckling
• Due to the buckles the web can only
carry a tensile stress st in the wrinkle
direction
• This leads to a new distribution of
stress in the web
– sxx & szz
– Shearing t
Annex 2: Buckling of stiffener/web constructions
T
b
dA
B
CD
ta
Thickness t
a
C
st
A
B
C
D
F
t
a
szz
st
st
A
B
D
Ft
a
sxx
st
Ex
Ez
2013-2014 Aircraft Structures - Instabilities 84
Page 85
• New stress distribution
– Use rotation tensor to compute in terms of st
Annex 2: Buckling of stiffener/web constructions
C
st
A
B
C
D
F
t
a
szz
st
st
A
B
D
Ft
a
sxx
st
Ex
Ez
2013-2014 Aircraft Structures - Instabilities 85
Page 86
• New stress distribution (2)
– From
– Shearing by vertical equilibrium
– Loading in flanges
• Moment balance around bottom flange
• Horizontal equilibrium
Annex 2: Buckling of stiffener/web constructions
T
b
dA
B
CD
ta
Thickness t
a
PT
tsxx
Ex
Ez
PBL-x
2013-2014 Aircraft Structures - Instabilities 86
Page 87
• New stress distribution (3)
– From
– Loading in stiffeners
• Assumption: each stiffener carries the loading
of half of the adjacent panels
• Stiffeners can be subject to Euler buckling if this load is too high
– Tests show that for these particular BCs, the equivalent length reads
Annex 2: Buckling of stiffener/web constructions
b
Ex
Ez
szz szz
P
2013-2014 Aircraft Structures - Instabilities 87
Page 88
• New stress distribution (4)
– From
– Bending in flanges
• In addition to the flanges loading PB & PT
• Stress szz produces bending
– Stiffeners constraint rotation
– Maximum moment at stiffeners
• Using table for double cantilever beams
Annex 2: Buckling of stiffener/web constructions
b
Ex
Ez
szz szz
P
b
Ex
Ez
P
szz szz szzszz
Mmax
2013-2014 Aircraft Structures - Instabilities 88
Page 89
• Wrinkles angle
– The angle is the one which minimizes
the deformation energy of
• Webs
• Flanges
• Stiffeners
– If flanges and stiffeners are rigid
• We should get back to a = 45°
– Because of the deformation of
flanges and stiffeners a < 45°
• Empirical formula for uniform material
– As non constant, a non constant
• Another empirical formula
Annex 2: Buckling of stiffener/web constructions
T
b
dA
B
CD
ta
Thickness t
a
Load in flange / Flange section
Load in stiffener / Stiffener section
2013-2014 Aircraft Structures - Instabilities 89
Page 90
• Example
– Web/stiffener construction
• 2 similar flanges
• 5 similar stiffeners
• 4 similar webs
• Same material
– E = 70 GPa
– Stress state?
Annex 2: Buckling of stiffener/web constructions
T = 5 kN
b = 0.3 m
d=
40
0 m
m
ta
t = 2 mm
a
AS = 300 mm2
AF = 350 mm2
(Iyy/x) = 750 mm3
Ex
Ez
Ixx = 2000 mm4
2013-2014 Aircraft Structures - Instabilities 90
Page 91
• Wrinkles orientation
–
Annex 2: Buckling of stiffener/web constructions
T = 5 kN
b = 0.3 m
d=
40
0 m
m
ta
t = 2 mm
a
AS = 300 mm2
AF = 350 mm2
(Iyy/x) = 750 mm3
Ex
Ez
Ixx = 2000 mm4
2013-2014 Aircraft Structures - Instabilities 91
Page 92
• Stress in top flange (> than in bottom one)
– 1st contribution: uniform compression
• Maximum at x = 0
– 2nd contribution: bending
• Maximum at stiffener at x=0
– Maximum compressive stress
Annex 2: Buckling of stiffener/web constructions
T
b
dA
B
CD
ta
Thickness t
a
PT
tsxx
Ex
Ez
PB
2013-2014 Aircraft Structures - Instabilities 92
Page 93
• Stiffeners
– Loading
– Buckling?
• As b < 3/2 d:
• Assuming centroid of stiffener lies in web’s plane
– We can use Euler critical load
• No buckling
Annex 2: Buckling of stiffener/web constructions
b
Ex
Ez
szz szz
P
2013-2014 Aircraft Structures - Instabilities 93