finite elements for plates and shellslibvolume3.xyz/civil/btech/semester8/finiteelement... ·...

Advanced Design for Mechanical

System LEC 2008/11/04 1

Finite elements for plates and shells


System LEC 2008/11/04 2

Plates and shells

A

A

•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.

•The isotropic case is considered.

•Thin and thick cases are take into account.

Midplane


System LEC 2008/11/04 3

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

Flessione di una trave

Bending of a beam

12 stiffness Bending

3Et

EI =


System LEC 2008/11/04 4

Thin plates (Kirchhoff)

w=w(x,y)

y

wzv

x

wzu

xyzxyz

∂∂

−=∂∂

−=

≠==

0 0 γγγ

t<1/10 the span of the

plate

Transverse shear

deformation is neglected

Arbitrary point P

displacement


System LEC 2008/11/04 5

The strain displacement relations give

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

Accordingly, the plane stress equation gives

0

2 2

xy2

2

y2

2

≈

∂∂∂

−=∂∂

+∂∂

=∂∂

−=∂∂

=∂∂

−=∂∂

=

z

xyx

wz

x

v

y

u

y

wz

y

v

x

wz

x

u

σ

γεε

σσσσ ==== Dεεεε D[[[[ ]]]] ====E

1−−−− ν 2

1 ν 0

ν 1 0

0 01−−−−ν(((( ))))

2


System LEC 2008/11/04 6

yx

wzG

yw

xwEz xy

y

x

∂∂∂

−=

∂∂

∂∂

−−=

2

22

22

22

1

1

1τ

ν

ν

νσ

σ

Stress distribution

•Like in a beam, stresses vary linearly with distance from the midsurface

•Transverse shear stresses τyz and τzx are also present, even though transverse deformation is neglected

•Transverse shear stresses vary quadratically through the thickness


System LEC 2008/11/04 7

zdzMzdzMzdzM

dzdAdAzdMyxMM

dzdAdAzdMyxMM

dzdAdAzdMyxMM

t

t

xyxy

t

t

yy

t

t

xx

xyxyxyxy

xyyy

xxxx

∫∫∫−−−

===

⋅===

⋅===

⋅===

2/

2/

2/

2/

2/

2/

1 )( ),(

1 )( ),(

1 )( ),(

τσσ

σ

σσ

•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

Moments


System LEC 2008/11/04 8

2

2

2

6 2 )2/(

6 2 )2/(

6 2 )2/(

t

M

t

z t

t

M

t

z t

t

M

t

z t

xy

xyxyxyxyxy

y

yyyyy

xxxxxx

=→=→=

=→=→=

=→=→=

τττττ

σσσσσ

σσσσσ

Maximum magnitude of stresses

•Formula for σx can be regarded as the flexure formula σx=Mxc/I applied

to a unit width of the plate with c=t/2


System LEC 2008/11/04 9

)1(12)1( stiffness Bending

0 0 0

2

3

2

2

2

2

2

2

2

νν

νσσ

−=

−=

−=→=∂∂

=∂∂

≠∂∂

→

EtEJD

y

w

y

w

x

wM xyx

Bending of a thin plate


System LEC 2008/11/04 10

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:

• The flexural stress σx is accompanied by the stress σy

• Stress σy constrains the plate against the deformation εy, thereby

stiffening the plate

• The amount of stiffening is proportional to 1/(1-ν2), so that a unit width

of the plate has

)1(12)1( stiffness Bending

0 0 0

2

3

2

2

2

2

2

2

2

νν

νσσ

−=

−=

−=→=∂∂

=∂∂

≠∂∂

→

EtEID

y

w

y

w

x

wM xyx


System LEC 2008/11/04 11

Thick plates (Mindlin)

0 0 0 ≠≠≠ xyzxyz γγγ•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 (θx, θy)


System LEC 2008/11/04 12

y

θzεzv

xzzu

xyx

y

xy

∂∂

−=−=

∂

∂==

θ

θεθ

yxz

xyz

xy

xy

x

w

y

w

xyz

θγ

θγ

θθγ

+∂∂

=

−∂∂

=

∂∂

−∂

∂=

If x

w

y

wx ∂

∂−=

∂∂

= y θθ

Transverse shear deformations vanish and the equations reduce to the

equations for thin plates.

Strain-displacements relations


System LEC 2008/11/04 13

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.


System LEC 2008/11/04 14

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


System LEC 2008/11/04 15

•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

3

4

4

3

64

=t

w

L

Ebtq c

s

Beam with immovable hinge supports


System LEC 2008/11/04 16

• For a simple supported beam b

bc q

EI

Lqw →=

4

384

5

• The total load supported by beam and spring action occurring

simultaneously is

+

=+=

t

w

t

w

L

Ebtqqq cc

bs 4.63.21

3

4

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


System LEC 2008/11/04 17

•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.

Observation


System LEC 2008/11/04 18

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 εz is very much larger than other stiffnesses.


System LEC 2008/11/04 19

•The plate element has half as many d.o.f. as the comparable 3D

element and omit εz 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.


System LEC 2008/11/04 20

•A plate element must be able to display states of constant σx, σy and

τxy 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..

theoryKirchoffcost 2

2

=∂

∂x

w

eoryMindlin thcost =∂∂

xyθ

Patch tests for plate elements

Patch test for constant Mx


System LEC 2008/11/04 21

Kirchhoff plate elements

y

w

x

w

yxww

x

y

∂∂

=

∂∂

=

=

θ

θ

),(Nodal d.o.f.:

•The assumed w field is a polynomial in x and y

•The stiffness matrix is

dVDBBkT∫=

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)

12 d.o.f.


System LEC 2008/11/04 22

•This element is incompatible: that is , if n is a direction normal to an

element edge, 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.

nw ∂∂


System LEC 2008/11/04 23

),( ),(),( yxyxyxww yyxx θθθθ ===

Mindlin plate elements

•A Mindlin element is based on three fields

each interpolated from nodal values.

•If all interpolations use the same polynomial, then for an element of

n nodes:

Nd=

=

∑=

yi

xi

in

i

i

i

i

y

x

w

N

N

Nw

θθ

θθ

100

00

00


System LEC 2008/11/04 24

•In the stiffness matrix

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.

{

dVx

BDBkT

55∫=


System LEC 2008/11/04 25

Bending deformation

•Elements strains εx 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 γzx becomes large and the element

is too stiff in bending, unless γzx is evaluated at x=0, where it vanishes.

•But one-point quadrature for all strains will introduce four instability

modes.


System LEC 2008/11/04 26

• 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.


System LEC 2008/11/04 27

Support conditions

Clamped w=ϑn= ϑs=0 ------

Simply supported w=0 Mn=0

Free ------- Q=Mn=Mns=0

• 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


System LEC 2008/11/04 28

Bending of square plate

)1(12 2

3

ν−=

EtD

Maximum deflection

D

PLf

D

qLf

2

4

α

α

=

=Distributed

load q

Concentrated

load P

α=0.00406 simply supported edges

α=0.00126 clamped edges {

{ α=0.0116 simply supported edges

α=0.00560 clamped edges


System LEC 2008/11/04 29

Twisting test cae

E=107

ν=0.3

t=0.05

W3=0.0029 L

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

Membrane stresses, constant through the thickness, would be

superposed on the flexural stress

xx

xx

t

MxMV

xV

t

MxM

νσσσ

σ

νσσσ

θ

θ

θ

==≠→→

≠=→

===→

K

K

20

0

2

00

6 0

0 0

6 0


System LEC 2008/11/04 33

( )

( ) 4/1

22

2

21

13

cossin

−=

+= −

tR

xCxCew x

νλ

λλλ

For λx= 30.25 , e-λx=0.07 (ν=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.

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

Where, given R the radius and t the thickness of the shell,

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 θzi 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 εz is

very much larger than the other stiffness.

bcxbbcxbba

baybbcybba

wwt

vvt

vv

wwt

uut

uu

=+=−=

=+=−=

2

2

2

2

θθ

θθ

•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


System LEC 2008/11/04 40

• 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.

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 R o b L t E ν ∆A

Roof F=Peso

25 50 0.25 432 106 0.3 0.3024

Cylinder

F=1

300 600 3.00 3 106 0.3 0.1825 10-4

Hemisphere

F=2

10 -- 0.04 68.25 106 0.3 0.0924

Strip

F1=10-6

F2=10-6

1.1 12 0.0032 29 106 0.22 5256 10-6

1294 10-6

Shell of revolution

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

Aaa

A

Aaa

A

Aaa

A

Aaa

A

Aaa


System LEC 2008/11/04 44

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.

Examples


System LEC 2008/11/04 46

•From the beam behaviour

MPaI

PLR

mmEI

PLv

z 0713.0

373103

63

==

−== −

σ

•Shear center (x=-2R y=0)

•Torque about the shear center T= -2RP

•Rotation of the loaded end ϑ=-11.2 10-6 rad

•Total deflection v=-37310-6+2R ϑ=-933 10-6 mm

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