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University of Liège
Aerospace & Mechanical Engineering
Aircraft Structures
Plates – Reissner-Mindlin Theory
Aircraft Structures - Plates – Reissner-Mindlin Theory
Ludovic Noels
Computational & Multiscale Mechanics of Materials – CM3
http://www.ltas-cm3.ulg.ac.be/
Chemin des Chevreuils 1, B4000 Liège
[email protected]
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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
2m l = K - 2m/3
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• Description
– In the reference frame Ei
• The plate is defined by
with
– Mapping of the plate
• Neutral plane
• Cross section with t0 = E3
• Initial plate S0
– S0 = A x [-h0/2 h0/2], for a plate of initial thickness h0
–
• Deformed plate S
–
Plate kinematics
a =1 or 2, I = 1, 2 or 3
F = j(x1, x2)+x3 t(x1, x2)
E1
E2
E3
x1
x2 A
x2=cst
t S
x1=cst
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• Assumptions
– Small deformations/displacements
•
•
•
• with
– Kirchhoff-Love assumption (no shearing)
• Normal is assumed to remain
– Planar
– Perpendicular to the neutral plane
– Reissner-Mindlin (shearing is allowed)
• Normal is assumed to remain planar
• But not perpendicular to neutral plane
Plate kinematics
F = j(x1, x2)+x3 t(x1, x2)
E1
E2
E3
x1
x2 A
x2=cst
t S
x1=cst
z
x
g
qy
qy
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• Idea
– Avoiding discretization on the thickness
• u and t constant on the thickness
– Equations are integrated on the thickness
• Linear momentum equation
–
– Small transformations assumptions ( , , )
• Using
Resultant equilibrium equations
F = j(x1, x2)+x3 t(x1, x2)
E1
E2
E3
x1
x2 A
x2=cst
t S
x1=cst
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• Linear momentum equation (2)
– Inertial term
• As the main idea in plates is to consider u and t constant on the thickness
– Volume loading term
•
• With the loading per unit area
Resultant equilibrium equations
F = j(x1, x2)+x3 t(x1, x2)
E1
E2
E3
x1
x2 A
x2=cst
t S
x1=cst
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• Linear momentum equation (3)
– Stress term
• Gauss theorem
– Where n0 is the normal to the plate
surface (3D volume) in the reference
configuration
– On top/bottom faces n0 = ± E3
– On lateral surface: n0 = na Ea
– Let us define the resultant stresses:
Resultant equilibrium equations
F = j(x1, x2)+x3 t(x1, x2)
E1
E2
E3
x1
x2 A
x2=cst
t S
x1=cst
∂NA
n
n0 = na Ea E1
E2
E3
A ∂DA
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• Linear momentum equation (4)
– Resultant stresses
• These are two vectors
–
–
• Which correspond to the integration of the
surface traction on the thickness
–
– Symmetric 2x2 matrix +
Out of plane component
Resultant equilibrium equations
F = j(x1, x2)+x3 t(x1, x2)
E1
E2
E3
x1
x2 A
x2=cst
t S
x1=cst
E1
E2
E3
A
ñ11
ñ22
ñ12
ñ21
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• Linear momentum equation (5)
– From
•
•
•
• Applying Gauss theorem on last term leads to
– Resultant linear momentum equation
Resultant equilibrium equations
F = j(x1, x2)+x3 t(x1, x2)
E1
E2
E3
x1
x2 A
x2=cst
t S
x1=cst
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• Linear momentum equation (6)
– Defining
• Density per unit area
• Volume loading per unit area
• Resultant loading
– The linear momentum equation
•
becomes after being integrated on the volume
• Is rewritten in the Cosserat plane A as
• With the resultant stresses
• With the resultant loading
– But we have no equation for bending (yet)
Resultant equilibrium equations
F = j(x1, x2)+x3 t(x1, x2)
E1
E2
E3
x1
x2 A
x2=cst
t S
x1=cst
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• Angular momentum equation
–
– Small transformations assumptions ( , , )
• Using
Resultant equilibrium equations
F = j(x1, x2)+x3 t(x1, x2)
E1
E2
E3
x1
x2 A
x2=cst
t S
x1=cst
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• Angular momentum equation (2)
– Small transformations assumptions (2)
• As second order terms can be neglected
• With
Resultant equilibrium equations
F = j(x1, x2)+x3 t(x1, x2)
E1
E2
E3
x1
x2 A
x2=cst
t S
x1=cst
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• Angular momentum equation (3)
– Inertial term
•
• As the main idea in plates is to consider u and t constant on the thickness
– With the density per unit area
– With the mass inertia
Resultant equilibrium equations
F = j(x1, x2)+x3 t(x1, x2)
E1
E2
E3
x1
x2 A
x2=cst
t S
x1=cst
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• Angular momentum equation (4)
– Loading term
•
– With the loading per unit area
Resultant equilibrium equations
F = j(x1, x2)+x3 t(x1, x2)
E1
E2
E3
x1
x2 A
x2=cst
t S
x1=cst
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• Angular momentum equation (5)
– Stress term
•
• Integration by parts
• Gauss theorem
Resultant equilibrium equations
F = j(x1, x2)+x3 t(x1, x2)
E1
E2
E3
x1
x2 A
x2=cst
t S
x1=cst
∂NA
n
n0 = na Ea E1
E2
E3
A ∂DA
djl
i
i
i
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as s symmetric
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• Angular momentum equation (6)
– Stress term (2)
•
– Where n0 is the normal to the plate
surface (3D volume) in the reference
configuration
– On top/bottom faces n0 = ± E3
– On lateral surface: n0 = na Ea
Resultant equilibrium equations
F = j(x1, x2)+x3 t(x1, x2)
E1
E2
E3
x1
x2 A
x2=cst
t S
x1=cst
∂NA
n
n0 = na Ea E1
E2
E3
A ∂DA
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• Angular momentum equation (7)
– Stress term (3)
•
• Resultant bending stresses & resultant stresses
– ,
Resultant equilibrium equations
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• Angular momentum equation (8)
– Resultant bending stress
•
• These are two vectors
–
–
• Which correspond to the integration of the
surface couple on the thickness
–
– Symmetric 2x2 matrix
Resultant equilibrium equations
E1
E2
E3
A
m11 ~ m21 ~
m22 ~
m12 ~
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• Angular momentum equation (9)
– Stress term (4)
•
• Applying Gauss theorem
Resultant equilibrium equations
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• Angular momentum equation (10)
– From
•
•
•
• It comes
Resultant equilibrium equations
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• Angular momentum equation (11)
– Resultant form
•
• But the resultant linear momentum equation reads
• So the angular momentum equation reads
Resultant equilibrium equations
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• Angular momentum equation (12)
– Resultant form (2)
•
• Defining the applied torque
• Term which is preventing from uncoupling the equations is
Resultant equilibrium equations
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• Angular momentum equation (13)
– Term
• Let us rewrite the Cauchy stress tensor in terms of its components
• As Cauchy stress tensor is symmetrical
• Using
• This new equation can be integrated on the thickness
• Defining the out-of-plane resultant stress
Resultant equilibrium equations
F = j(x1, x2)+x3 t(x1, x2)
E1
E2
E3
x1
x2 A
x2=cst
t S
x1=cst
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• Angular momentum equation (14)
– Resultant form (3)
•
•
• If l is an undefined pressure applied through the thickness, the resultant angular
momentum equation reads
• With
–
–
–
Resultant equilibrium equations
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• Resultant equations summary
– Linear momentum
•
• Resultant stresses
– Angular momentum
•
• Resultant bending stresses
– Interpretation
Resultant equilibrium equations
x
sXX
x
sXX
x
sXX
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• What is missing is the link between the deformations and the stresses
– Idea: as the discretization does not involve the thickness, the deformations
should be evaluated at neutral plane too
– Works only in linear elasticity
Material law
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• Deformations
– In small deformations, the tensor reads
• But defining
leads to
• A vector can always be written in terms of its components
Material law
F = j(x1, x2)+x3 t(x1, x2)
E1
E2
E3
x1
x2 A
x2=cst
t S
x1=cst
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• Deformations (2)
– In small deformations, the tensor reads (2)
• Relations on the normal
– By definition
– In small deformations
– Which implies &
• So relation
• Interpretation
– See next slide
Material law
F = j(x1, x2)+x3 t(x1, x2)
E1
E2
E3
x1
x2 A
x2=cst
t S
x1=cst
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• Deformations (3)
– In small deformations, the tensor reads (2)
• Interpretation
Material law
F = j(x1, x2)+x3 t(x1, x2)
E1
E2
E3
x1
x2 A
x2=cst
t S
x1=cst
Motion of the neutral plane
in its plane (2D in-plane
problem): membrane
mode
Motion of the neutral plane
out-of-its-plane: bending
mode
Change of the neutral plane
direction resulting from
1) Bending
2) Out-of-plane shearing
z
x
g1 qy = Dtx
-uz,x
qy = Dtx
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• Deformations (4)
– In small deformations, the tensor reads (3)
• From
• As
and as only the last term depends on x3, the gradient reads
• So the deformations tensor reads
Material law
F = j(x1, x2)+x3 t(x1, x2)
E1
E2
E3
x1
x2 A
x2=cst
t S
x1=cst
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• Deformations (5)
– Deformation modes
•
• Interpretation: membrane mode
–
– Location in the deformation tensor
– Corresponds to the in-plane
deformations of the neutral plane
Material law
E1
E2
E3
A
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• Deformations (6)
– Deformation modes (2)
•
• Interpretation: bending mode
–
– Location in the deformation tensor
– Corresponds to the final
curvature of the neutral plane
Material law
E1
E2
E3
A 1/k11
t
t
1/Dt,1
Dt Dt
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• Deformations (7)
– Deformation modes (3)
•
• Interpretation: Through-the-thickness shearing
–
– Location in the deformation tensor
– Corresponds to the average angle between
the neutral plane normal and the direction
vector t (initially the same)
Material law
z
x
g1 qy = Dt1
-uz,1
qy = Dt1
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• Deformations (8)
– Deformation modes (4)
•
• Interpretation: Through-the-thickness elongation
– In this model there is no through-the-thickness elongation
– Actually the plate is in plane-s state, meaning there is such a deformation
» To be introduced: x3 t should be substituted by lh(x3) t in the shell
kinematics
» We have to introduce it to get the plane-s effect
» In small deformations this term would lead to second order effects on
other components
Material law
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• Deformations (9)
– Final expression
• We had
– With
• And through-the-thickness elongation lh(x3) t
Material law
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• Hooke’s law
– Small strain tensor
– There are 4 contributions
• Membrane mode
– as frame is orthonormal Hijkl = Hijkl (notation abuse)
with
– As the non-zero components are
Material law
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• Hooke’s law (2)
– There are 4 contributions (2)
• Bending mode
–
with
– As the non-zero components are
Material law
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• Hooke’s law (3)
– There are 4 contributions (3)
• Shearing mode
–
with
– As
we can deduce the non-zero components – As we did for beams, we have to account for the non-uniformity of g by a shear section reduction
Material law
x
z
Tz
dx
Tz+ ∂xTz dx
gmax
g
g dx
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• Hooke’s law (4)
– There are 4 contributions (4)
• Through-the thickness elongation
–
with
– We can deduce the non-zero components
Material law
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• Resultant stresses
– Plane-s state
• Contributions
• Elongation depends on x3
– Part is stretched and
part is compressed
– For pure bending the change
of sign is on the neutral axis
• Average trough the thickness elongation
– Depends on the membrane mode only
Material law
x
z
h L y
z
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• Resultant stresses (2)
– Plane s-stated (2)
• Values ab can now be deduced
–
with
• Can be rewritten as a through-the-thickness constant term and a linear term
–
– With
• Out-of-plane shearing remains the same
–
Material law
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• Resultant stresses (3)
– From stress fields
•
•
– Membrane resultant stresses
•
Material law
In plane membrane stress Out-of plane shear stress
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• Resultant stresses (4)
– Membrane resultant stresses (2)
•
• In plane component: membrane stress
– As is cst with x3
– Defining the membrane Hooke tensor
»
» As
– 2D plane-s problem
Material law
E1
E2
E3
A
ñ11
ñ22
ñ12
ñ21
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• Resultant stresses (5)
– Membrane resultant stresses (3)
•
• Out-of-plane component: shear stress
– As is cst with x3
– Defining the shearing Hooke tensor
»
» As
– Corresponds to the average
shearing of a beam
Material law
z
x
g1 qy = Dt1
-uz,1
qy = Dt1
q3
q3
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• Resultant stresses (6)
– Out-of-plane resultant stresses
• We defined
– Owing to previous definitions
»
» With
– Corresponds to the shearing
symmetrical to the out-of-plane
shearing
Material law
y
z
A*
x
t
t
b(z)
z
x
g1 qy = Dtx
-uz,x
qy = Dtx
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• Resultant stresses (7)
– From stress fields (2)
•
•
– Bending resultant stresses
•
Material law
Bending stress
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• Resultant stresses (8)
– Bending resultant stresses (2)
•
– As
– Defining the bending Hooke tensor
»
» As
– 2D bending problem
Material law
E1
E2
E3
A
1/k11
t
t
Dt Dt
m11 ~ m21 ~
m22 ~
m12 ~
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• Deformations (small transformations)
– In plane membrane
•
– Curvature
•
– Out-of-plane sliding
•
Reissner-Mindlin equations summary
E1
E2
E3
A
E1
E2
E3
A 1/k11
t
t
1/Dt,1
Dt Dt
z
x
g1 qy = Dt1
-uz,1
qy = Dt1
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• Resultant stresses in linear elasticity
– Membrane stress
•
– Bending stress
•
– Out-of-plane shear stress
•
Reissner-Mindlin equations summary
E1
E2
E3
A
ñ11
ñ22
ñ12
ñ21
E1
E2
E3
A
1/k11
t
t
Dt Dt
m11 ~ m21 ~
m22 ~
m12 ~
z
x
g1 qy = Dt1
-uz,1
qy = Dt1
q3
q3
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• Resultant equations
– Membrane mode
•
•
– with
– with
• Clearly, the solution can be directly computed in plane Oxy (constant Hn)
–
– Boundary conditions
» Dirichlet
» Neumann
• Remaining equation along E3:
Reissner-Mindlin equations summary
∂NA
n
n0 = na Ea E1
E2
E3
A ∂DA
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• Resultant equations (2)
– Bending mode
•
• &
– with
– with
• Solution is obtained by projecting into the plane Oxy (constant Hq, Hm)
–
– 2 equations (a=1, 2) with 3 unknowns (Dt1, Dt2, u3)
– Use remaining equation
Reissner-Mindlin equations summary
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• Resultant equations (3)
– Bending mode (2)
• 3 equations with 3 unknowns
–
–
• To be completed by BCs
– Low order constrains
» Displacement or
» Shearing
– High order
» Rotation or
» Bending
Reissner-Mindlin equations summary
∂NA
T
n0 = na Ea E1
E2
E3
A ∂DA
p
∂MA
M
n0 = na Ea E1
E2
E3
A ∂TA
p
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• Membrane problem
– Similar to 2D elasticity
• Bending problem
– 3 degree of freedom by nodes
– Shear locking
• For reduced thickness (h/L→0) the structure is too stiff
• This results from the fact that for thin thickness u3,a → Dta physically
– Bernoulli assumption for beams
• Then we have extra constrains but no new degree of freedom
• The solution found is then zero deformation
– In order to avoid shear locking
• Different techniques
• High order elements
• Shear strains g evaluated at particular points (assumed strain method)
– These values can be formulated in terms of the displacements/rotations
degrees of freedom
• Internal degrees of freedom (enhanced assumed strain method)
• ….
Finite-element implementation
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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
– Papers
• Simo JC, Fox DD. On a stress resultant geometrically exact shell model. Part I:
formulation and optimal parametrization. Computer Methods in Applied Mechanics
and Engineering 1989; 72:267–304.
• Simo JC, Fox DD, Rifai MS. On a stress resultant geometrically exact shell model.
Part II: the linear theory, computational aspects. Computer Methods in Applied
Mechanics and Engineering 1989; 73:53–92.
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