Transcript Diapositiva 1 - Politecnico di Milano

Finite elements for plates and shells
Advanced Design for Mechanical
System LEC 2008/11/04
1
Plates and shells
A
Midplane
•The mid-surface of a plate is plane, a curved geometry make it a shell
•The thickness is constant.
•May be isotropic, anisotropic, composite and layered.
A
•The isotropic case is considered.
•Thin and thick cases are take into account.
Advanced Design for Mechanical
System LEC 2008/11/04
2
Comparison plate-beam
•A plate can be regarded as the two-dimensional analogue of a beam:
- both carry transverse loads by bending action;
- but they have significant differences;
- a beam typically has a single bending moment; a plate has two bending
moments (and two twisting moments)
- a deflection of a beam need not strain its axis; deflection of a plate will
strain its midsurface
Et3
Bending stiffness EI 
12
Flessione di una trave
Bending of a beam
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3
Thin plates (Kirchhoff)
t<1/10 the span of the
plate
w=w(x,y)
Transverse shear
deformation is neglected
Arbitrary point P
displacement
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 yz   zx  0
 xy  0
w
u  z
x
w
v  z
y
4
The strain displacement relations give
u
2w
x 
 z 2
x
x
v
2w
y 
 z 2
y
y
 xy
u v
2w


 2 z
y x
xy
the first of these equation is the only one used in the beam theory.
In a thin plate the stress normal to the midplane is considered negligible
z  0
Accordingly, the plane stress equation gives
  D

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1 
E 
 1
 D 
2 
1  
0 0
0 

0 
1   
2 
5
Stress distribution
 x 
E 1    2 w x 2 
2w
 xy  2 zG
   z
 2

2 
2
1   1  w y 
xy
 y 
•Like in a beam, stresses vary linearly with distance from the midsurface
•Transverse shear stresses yz and zx are also present, even though
transverse deformation is neglected
•Transverse shear stresses vary quadratically through the thickness
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Moments
•The stresses give rise to bending moment Mx and My and twisting
moment Mxy
•Moments are function of x and y and are computed for unit length in the
plane of the plate
M x  M x ( x, y )
dM x  z ( x dA)
dA  1  dz
M y  M y ( x, y )
dM y  z ( x dA)
dA  1  dz
M xy  M xy ( x, y )
dM xy  z ( xy dA)
t/2
Mx 

t/2
x
t / 2
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zdz M y 

t / 2
dA  1  dz
t/2
y
zdz M xy 

xy
zdz
t / 2
7
Maximum magnitude of stresses
 x (t / 2)   x
 y (t / 2)   y
 xy (t / 2)   xy
z
Mx
  x  2 x
 x  6 2
t
t
My
z
  y  2 y
 y  6 2
t
t
M xy
z
  xy  2 xy
  xy  6 2
t
t
•Formula for x can be regarded as the flexure formula x=Mxc/I applied
to a unit width of the plate with c=t/2
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2w
Mx  2  0
x
2w
0
y 2
2w
 0   y   x
y 2
EJ
Et3
Bendingstiffness D 

2
(1   ) 12(1   2 )
Bending of a thin plate
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Bending of a plate
• A plate has a wide cross section
- top and bottom edge of a cross section remain straight y-parallel axis
when Mx is applied
• When a plate is bent to a cylindrical surface, only Mx acts:
2w
Mx  2  0
x
2w
0
y 2
2w
 0   y   x
y 2
• The flexural stress x is accompanied by the stress y
• Stress y constrains the plate against the deformation y, thereby
stiffening the plate
• The amount of stiffening is proportional to 1/(1-2), so that a unit width
of the plate has
Et3
EI

Bending stiffness D 
2
(1   ) 12(1   2 )
Advanced Design for Mechanical
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10
Thick plates (Mindlin)
 yz  0  zx  0
 xy  0
•Accounts for transverse shear deformation
•The assumption that right angle in a cross section are preserved
must be abandoned
•The planes initially normal to the midsurface experience rotations
different from rotations of the midsurface itself (x, y)
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Strain-displacements relations
u  z y
v   z x
If
w
x 
y
x  z
 y
x
θ
εy  z x
y
  y  x 

 xy  z 

x 
 y
w
 yz 
 x
y
w
 xz 
y
x
w
y  
x
Transverse shear deformations vanish and the equations reduce to the
equations for thin plates.
Advanced Design for Mechanical
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Loads
• Loads in the z direction, either distribuited or concentrated,to the
lateral surface z=t/2, or to the edge of the plate
• Bending moment whose vector is tangent to the edge
• At the point where concentrated lateral load (z direction) is applied:
- Kirchhoff theory predicts infinite bending;
- Mindlin theory predicts infinite bending and infinite displacement
• The infinities disappear if the “concentrated” load is applied over a
small area instead.
Advanced Design for Mechanical
System LEC 2008/11/04
13
Membrane forces
• Internal force resultant in the plane of the plate (membrane forces)
– can develop as a consequence of the deflection
– may be present because of loads component tangent to the
midsurface
– can significantly influence the response of the plate to load
• Membrane forces have a stress stiffening effect:
- if tensile they effectively increase the flexural rigidity
- if compressive they decrease it
• Compressive membrane forces may become large enough to
produce buckling
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Beam with immovable hinge supports
•Hinge support remains exactly a distance L apart, regardless of how
much load is applied.
•Beam develop the usual flexural stresses and also membrane force N
that support part of the applied load by spring effect.
• If the deflected shape is a parabola with center deflection wc the
uniform distributed load supported by spring action is
qs 
Advanced Design for Mechanical
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64 Ebt  wc 

4 
3 L  t 
4
3
15
• For a simple supported beam
5 qb L4
wc 
 qb
384 EI
• The total load supported by beam and spring action occurring
simultaneously is
3

Ebt
 wc 
 wc 
q  qs  qb  4 21.3   6.4 
L 
 t 
 t 
3
•For wc/t=0.5 (HP: t<<L) spring and beam action each support
about half of the total load
•This argument is of little value for beams because immovible
supports are not found in practice.
•The value of the argument is its implication for problems of thin
plates
Advanced Design for Mechanical
System LEC 2008/11/04
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Observation
•The counterpart of the spring action in a beam is strain of the
midsurface in a plate.
•Deflection of a plate w=w(x, y) produce no strain of the midsurface
only if w describes a developable surface, i.e. cylinder or cone.
•In general load produces a deflected shape that is not developable.
•Accordingly, in general there are strains at the midsurface, and
membrane forces appear that carry out part of the load.
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Finite elements for plates
•A plate is a thin solid and might be modeled by 3D solid elements
•A solid element is wasteful of d.o.f., as it computes transverse normal
stress and transverse shear stresses, all of which are considered
negligible in a thin plate.
•Also thin 3D elements invite numerical troubles because stiffness
associated with z is very much larger than other stiffnesses.
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•The plate element has half as many d.o.f. as the comparable 3D
element and omit z from its formulation
•Thickness appear to be zero, but the correct value is used in its
formulation
•Circular plates can be modeled by shell of revolution elements, simply
by making shell elements flat rather than cylindrical or conic.
•Each of such element is thus a flat annular ring, joined to adjacent
annular elements at its inner and outer radii.
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Patch tests for plate elements
•A plate element must be able to display states of constant x, y and
xy if it is to pass patch test.
•These states must be displayed in each z=constant layer.
•This means that a valid plate element must pass patch tests for states
of constant Mx, My, and Mxy..
2w
 y
x 2
x
 cost
Kirchoff theory
 cost
Mindlin theory
Patch test for constant Mx
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Kirchhoff plate elements
Nodal d.o.f.: w  w( x, y )
12 d.o.f.
w
x
w
x 
y
y 
•The assumed w field is a polynomial in x and y
•The stiffness matrix is
k   B DBdV
T
where D is the matrix of the flexural rigidities
B is contrived to produce curvature when it operates on nodal
d.o.f. that describe a lateral displacement field w=w(x,y)
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•This element is incompatible: that is , if n is a direction normal to an
element edge, w n is not continuous between elements for some
loading conditions
•Accordingly, the element cannot guarantee a lower bound on computed
displacements, so results may converge “from above” rather “from
below”.
•A compatible rectangular element with corner nodes only require that
twist 2w/xy also be used as a nodal d.o.f. which is undesiderable.
•Experience in formulation of plate elements has shown that the Midlin
formulation is more productive and Midlin plate elements are in common
use also for thin plates.
Advanced Design for Mechanical
System LEC 2008/11/04
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Mindlin plate elements
•A Mindlin element is based on three fields
w  w( x, y)
x  x ( x, y)
 y   y ( x, y)
each interpolated from nodal values.
•If all interpolations use the same polynomial, then for an element of
n nodes:
w
 wi 
N
0
0


i
  n 
 

 x     0 Ni 0   xi   Nd
  i 1  0 0 N   
i   yi 

 y
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•In the stiffness matrix
k   BT D
 BdV
5 x5
D include the 3by 3 matrix for plane stress and also shear moduli
associated with the two transverse shear strain.
•Integration with respect ot z is done explicitly.
•Integration in the plane of the element is done numerically if the
element is isoparametric.
•Four nodes and eight nodes quadrilater elements are popular based
on the same Ni used for a plane elements.
•In any layer z=constant, the behaviour of a Mindlin plate can be
deduced from the behaviour of the corresponding element provided
that the integrand are integrated by the same quadrature rule.
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Bending deformation
•Elements strains x are independent of x, any order o quadrature will
report the same strain energy of pure bending
•However this element displays spurious shear strain.
•If a/t is large, transverse shear strain zx becomes large and the element
is too stiff in bending, unless zx is evaluated at x=0, where it vanishes.
•But one-point quadrature for all strains will introduce four instability
modes.
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• This observation suggests “selective integration” in which one-point
quadrature is applied to transverse shear terms and four-point
quadrature is applied to bending terms.
• Two instability modes remain that are controlled by “stabilization”
matrices.
• Eight nodes Mindlin elements have the analogous shortcomings and
may also be treated by selective integration.
• Calculated stresses are usually more accurate at the Gauss points.
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Support conditions
• Support conditions are classed as clamps, simple, or free, in direct
analogy to the possible support conditions of a beam.
• Nodal d.o.f. that must be prescribed for these support conditions are
the following
Clamped
w=n= s=0
-----Simply supported
w=0
Mn=0
Free
-------
Q=Mn=Mns=0
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Bending of square plate
Et3
D
12(1  2 )
Maximum deflection
Distributed
load q
qL4
f 
D

Concentrated
load P
PL2
f 
D

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=0.00406 simply supported edges
=0.00126 clamped edges
=0.0116 simply supported edges
=0.00560 clamped edges
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Twisting test cae
E=107
=0.3
t=0.05
W3=0.0029 L
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Shells and shell theory
•The geometry of a shell is defined by its thickness and its midsurface,
which is a curved surface in space.
•Load is carried by a combination of membrane action and bending
action.
•A thin shell can be very strong if membrane action dominates, in the
same way that a wire can carry great load in tension but only small load
in bending.
•However, no shell is completely free of bending stresses.
•They appear or near point load, line loads, reinforcements, junctures,
change of curvature and supports.
•Any concentration of load or geometrical discontinuity can be
expected to produce bending stresses, often much larger, of membrane
stresses, but quite localized.
Example of load and geometry that produce bending
•Axial force G must be transferred through the structure to the support.
•The simple support around the base AA applies axially direct line load,
which has a shell normal component that causes bending.
•Around BB the cylindrical and conical part exert shell-normal load
component on one other.
•Shell-normal load is transferred across FF because the cylindrical and
spherical shells try to expand different amounts under internal pressure
•Line load EE is obviously shell-normal, as is the restrain provided by
reinforcing ring DD
Internal forces and moments associated with bending at a discontinuity
such as CC are the following
Flexural stress and bending moment in a shell are related in the same
way as for a plate
6M 0
M0  x  0
 x  2     x
t
V0  x  0
  0
6M
    x
2
t
Membrane stresses, constant through the thickness, would be
superposed on the flexural stress
V0  M  x  0
x 
How much is the boundary in which bending may be important?
A simple approximation can be obtained from the theory of a shell of
revolution.
Analytical solution for radial displacement and bending moment as
function of axial distance x from the end is
w  e  x C1 sin x  C2 cosx 
Where, given R the radius and t the thickness of the shell,


1/ 4
 3 1  
 2 2 
 Rt 
2
For x= 30.25 , e-x=0.07 (=0)
x=Rt
The end displacement and the bending moment declines to 7%.
This is the estimate value of the boundary layer. In a FE analysis al
least two element must be used to span the boundary layer.
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Shell tangent edge loads
Shell tangent edge loads produce actions that are not confined to
a boundary layer .
Axial loads act on end A of the unsupported cylindrical shell, but
the largest displacement appear at the end B in apparent
contradiction of Saint-Venant principle.
Saint-Venant principle is applicable to massive isotropic boides.
Thin-walled structure and high anisotropic structures may behave
quite differently.
Finite elements for shells
•The most direct mode to obtain a shell element is to combine a
membrane element and a bending element.
•A simple triangular element can be obtained by combining the plane
stress triangle with the plane bending triangle.
•The resulting element is flat and has five d.o.f. or six d.o.f. per node,
depending on whether or not the shell-normal rotation zi at node i is
present in the plane stress element.
•A quadrilater element can be produced in similar fashion, by combining
quadrilater plane and plate elements.
•A four node quadrilater element is in general a warped element
because its nodes are not all coplanar.
•A modest amount of warping can seriously degrade the performance of
an element.
Flat elements
•Advantages of a flat element include:
- simplicity of formulation,
- simplicity in the description of the geometry,
- element’s ability to represent rigid-body motion without strain.
•Disadvantage include:
- the representation of a smoothly curved shell surface by flat or slightly
warped facets.
There is a discretization error associated with the lack of coupling
between membrane and bending action within individual elements.
Common advice is that flat element should span no more than roughly
10° of the arc of the actual shell.
•Curved elements based on shell theory avoid some shortcomings of flat
elements but introduce other difficulties:
- more data are needed to describe the geometry of a curved element,
- formulation, based on shell theory, is complicated,
- membrane and bending action are coupled within the element
Isoparametric shell elements
•Isoparametric shell elements occupy a middle ground between flat
elements and curved elements based on shell theory.
•One begins with a 3d solid element
•The element can model a shell if thickness t is small in comparison
with other dimensions.
•However, such an element has the defect noted for plate bending:
invite numerical troubles because the stiffness associated with z is
very much larger than the other stiffness.
•Accordingly , the element is transformed, the number of nodes is
reduced from 20 to 8, by expressing translational d.o.f. of the 20-node
element in terms of translational and rotational d.o.f. of the 8-node
element.
•Node A and C appear in the solid element, but not in the shell
elements. Displacement at A and C are
t
ua  ub   yb
2
t
va  vb   xb
2
t
uc  ub   yb
2
t
vc  vb   xb
2
wa  wb
wc  wb
• With relations like these for all thickness direction lines of nodes,
shape function of the 20 node element are transformed so as to
operate on the three translations and two rotations at each node of
the 8-node element.
• A Mindlin shell element is obtained.
• Thickness direction normal stress is taken as zero, and stress-strain
relations reflect this assumption.
• The element matrix is integrated numerically.
• A reduced or selective integration scheme may be used to avoid
transverse shear locking and membrane locking.
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Test cases
a) Shell roof with q= 90 for unit area: membrane action dominates
b) Pinched cylinder, F=1: mambrane and bending action are active
c) Hemisphere, F=2: bending action dominates
d) Twisted strip: F1=10-6: sensitivity of the elements to warping can be
tested
Problem
Roof
F=Peso
Rob
25
L
50
t
0.25
E
432 106

0.3
A
0.3024
Cylinder
F=1
300
600
3.00
3 106
0.3
0.1825 10-4
Hemisphere 10
F=2
--
0.04
68.25 106
0.3
0.0924
Strip
F1=10-6
F2=10-6
12
0.0032
29 106
0.22
5256 10-6
1294 10-6
1.1
Shell of revolution
A
A
Aaa
A
Aaa
Aaa
A
Aaa
A
Aaa
In cross section, an element for a shell of revolution resembles a beam
element.
Like an element for a solid of revolution, an element for a shell of
revolution has nodal circle, rather than nodal points.
Typically there are two nodal circles for element.
A shell of revolution element may be flat (conical) or curved.
The simplest formulation resembles the 2D beam element , in that is
used:
- a cubic lateral displacement field,
- a linear meridianal displacement field,
- each nodal circle has two translational (radial and axial) and one
rotational as d.o.f.
Conical shell elements have advantages and disadvantages like those
of other flat element
Spherical shell loaded by internal
pressure and modeled by a
coarse mesh of conical shell
elements display spurious
bending moments.
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Examples
•From the beam behaviour
PL3
v
 37310 6 m m
3EI
PLR
z 
 0.0713MPa
I
•Shear center (x=-2R y=0)
•Torque about the shear center
•Rotation of the loaded end
T= -2RP
=-11.2 10-6 rad
•Total deflection v=-37310-6+2R =-933 10-6 mm
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