Temos Hen Kobe Am

Plates and Shells: Theory and Computation

- 4D9 -

Dr Fehmi Cirak (fc286@)

Office: Inglis building mezzanine level (INO 31)

F. CirakPage 2

Outline -1-

! This part of the module consists of seven lectures and will focus on finite

elements for beams, plates and shells. More specifically, we will consider

! Review of elasticity equations in strong and weak form

! Beam models and their finite element discretisation

! Euler-Bernoulli beam

! Timoshenko beam

! Plate models and their finite element discretisation

! Shells as an assembly of plate and membrane finite elements

! Introduction to geometrically exact shell finite elements

! Dynamics

F. CirakPage 3

Outline -2-

! There will be opportunities to gain hands-on experience with the

implementation of finite elements using MATLAB

! One hour lab session on implementation of beam finite elements (will be not marked)

! Coursework on implementation of plate finite elements and dynamics

F. CirakPage 4

Why Learn Plate and Shell FEs?

! Beam, plate and shell FE are available in almost all finite element software

packages

! The intelligent use of this software and correct interpretation of output requires basic

understanding of the underlying theories

! FEM is able to solve problems on geometrically complicated domains

! Analytic methods introduced in the first part of the module are only suitable for computing plates

and shells with regular geometries, like disks, cylinders, spheres etc.

! Many shell structures consist of free form surfaces and/or have a complex topology

! Computational methods are the only tool for designing such shell structures

! FEM is able to solve problems involving large deformations, non-linear

material models and/or dynamics

! FEM is very cost effective and fast compared to experimentation

F. CirakPage 5

Literature

! JN Reddy, An introduction to the finite element method, McGraw-Hill (2006)

! TJR Hughes, The finite element method, linear static and dynamic finite element

analysis, Prentice-Hall (1987)

! K-J Bathe, Finite element procedures, Prentice Hall (1996)

! J Fish, T Belytschko, A first course on finite elements, John Wiley & Sons (2007)

! 3D7 - Finite element methods - handouts

F. CirakPage 6

Examples of Shell Structures -1-

! Civil engineering

! Mechanical engineering and aeronautics

Masonry shell structure (Eladio Dieste) Concrete roof structure (Pier Luigi Nervi)

Fuselage (sheet metal and frame)Ship hull (sheet metal and frame)

F. CirakPage 7

Examples of Shell Structures -2-

! Consumer products

! Nature

Red blood cellsFicus elastica leafCrusteceans

F. CirakPage 8

Representative Finite Element Computations

Virtual crash test (BMW)

Sheet metal stamping (Abaqus)

Wrinkling of an inflated party balloon

buckling of carbon nanotubes

F. CirakPage 9

0.74 m

0.02

5 m

Shell-Fluid Coupled Airbag Inflation -1-

Shell mesh: 10176 elements

0.86 m

0.49

m

0.86 m

0.123 m

Fluid mesh: 48x48x62 cells

F. CirakPage 10

Shell-Fluid Coupled Airbag Inflation -2-

F. CirakPage 11

Detonation Driven Fracture -1-

! Modeling and simulation challenges

! Ductile mixed mode fracture

! Fluid-shell interaction

Fractured tubes (Al 6061-T6)

F. CirakPage 12

Detonation Driven Fracture -2-

F. CirakPage 13

Roadmap for the Derivation of FEM

! As introduced in 3D7, there are two distinct ingredients that are combined

to arrive at the discrete system of FE equations

! The weak form

! A mesh and the corresponding shape functions

! In the derivation of the weak form for beams, plates and shells the

following approach will be pursued

1) Assume how a beam, plate or shell deforms across its thickness

2) Introduce the assumed deformations into the weak form of three-dimensional elasticity

3) Integrate the resulting three-dimensional elasticity equations along the thickness direction

analytically

F. CirakPage 14

Elasticity Theory -1-

! Consider a body in its undeformed (reference) configuration

! The body deforms due to loading and the material points move by a displacement

! Kinematic equations; defined based on displacements of an infinitesimalvolume element)

! Axial strains

F. CirakPage 15


! Shear components

! Stresses

! Normal stress components

! Shear stress component

! Shear stresses are symmetric

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! Equilibrium equations (determined from equilibrium of an infinitesimal

volume element)

! Equilibrium in x-direction

! Equilibrium in y-direction

! Equilibrium in z-direction

! are the components of the external loading vector (e.g., gravity)

F. CirakPage 17


! Hookes law (linear elastic material equations)

! With the material constants Youngs modulus and Poissons ratio

F. CirakPage 18

Index Notation -1-

! The notation used on the previous slides is rather clumsy and leads to very

long expressions

! Matrices and vectors can also be expressed in index notation, e.g.

! Summation convention: a repeated index implies summation over 1,2,3, e.g.

! A comma denotes differentiation

F. CirakPage 19

Index Notation -2-

! Kronecker delta

! Examples:

F. CirakPage 20

Elasticity Theory in Index Notation -1-

! Kinematic equations

! Note that these are six equations

! Equilibrium equations

! Note that these are three equations

! Linear elastic material equations

! Inverse relationship

! Instead of the Youngs modulus and Poissons ratio the Lame constants can be used

F. CirakPage 21

Weak Form of Equilibrium Equations -1-

! The equilibrium, kinematic and material equations can be combined into

three coupled second order partial differential equations

! Next the equilibrium equations in weak form are considered in preparation

for finite elements

! In structural analysis the weak form is also known as the principle of virtual displacements

! To simplify the derivations we assume that the boundaries of the domain are fixed (built-in, zero

displacements)

! The weak form is constructed by multiplying the equilibrium equations with test functions vi which

are zero at fixed boundaries but otherwise arbitrary

F. CirakPage 22


! Further make use of integration by parts

! Aside: divergence theorem

! Consider a vector field and its divergence

! The divergence theorem states

! Using the divergence theorem equation (1) reduces to

! which leads to the principle of virtual displacements

F. CirakPage 23


! The integral on the left hand side is the internal virtual work performed by the internal stresses due to virtual

displacements

! The integral on the right hand side is the external virtual work performed by the external forces due to virtual

displacements

! Note that the material equations have not been used in the preceding derivation.

Hence, the principle of virtual work is independent of material (valid for elastic, plastic,

)

! The internal virtual work can also be written with virtual strains so that the principle of

virtual work reads

! Try to prove

Finite Element Formulation for Beams

- Handout 2 -


Completed Version

F CirakPage 25

Review of Euler-Bernoulli Beam

Physical beam model

Beam domain in three-dimensions

Midline, also called the neutral axis, has the coordinate Key assumptions: beam axis is in its unloaded configuration straight Loads are normal to the beam axis

midline

F CirakPage 26

Kinematics of Euler-Bernoulli Beam -1-

Assumed displacements during loading

Kinematic assumption: Material points on the normal to the midline remain on the normal duringthe deformation

Slope of midline:

The kinematic assumption determines the axial displacement of the material points acrossthickness

Note this is valid only for small deflections, else

reference configuration

deformed configuration

F CirakPage 27

Kinematics of Euler-Bernoulli Beam -2-

Introducing the displacements into the strain equations of three-dimensional elasticity leads to Axial strains

Axial strains vary linearly across thickness

All other strain components are zero

Shear strain in the

Through-the-thickness strain (no stretching of the midline normal during deformation)

No deformations in and planes so that the corresponding strains are zero

F CirakPage 28

Weak Form of Euler-Bernoulli Beam

The beam strains introduced into the internal virtual work expressionof three-dimensional elasticity

with the standard definition of bending moment:

External virtual work

Weak work of beam equation

Boundary terms only present if force/moment boundary conditions present

F CirakPage 29

Stress-Strain Law

The only non-zero stress component is given by Hookes law

This leads to the usual relationship between the moment and curvature

with the second moment of area

Weak form work as will be used for FE discretization

EI assumed to be constant

F CirakPage 30

Beam is represented as a (disjoint) collection of finite elements

On each element displacements and the test function are interpolated usingshape functions and the corresponding nodal values

Number of nodes per element

Shape function of node K

Nodal values of displacements

Nodal values of test functions

To obtain the FE equations the preceding interpolation equations areintroduced into the weak form Note that the integrals in the weak form depend on the second order derivatives of u3 and v

Finite Element Method

F CirakPage 31

A function f: is of class Ck=Ck() if its derivatives of order j, where0 j k, exist and are continuous functions For example, a C0 function is simply a continuous function For example, a C function is a function with all the derivatives continuous

The shape functions for the Euler-Bernoulli beam have to be C1-continuousso that their second order derivatives in the weak form can be integrated

Aside: Smoothness of Functions

C1-continuous functionC0-continuous function

diffe

rent

iatio

n

F CirakPage 32

To achieve C1-smoothness Hermite shape functions can be used Hermite shape functions for an element of length

Shape functions of node 1

with

Hermite Interpolation -1-

F CirakPage 33

Shape functions of Node 2

with

Hermite Interpolation -2-

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According to Hermite interpolation the degrees of freedom for each element are thedisplacements and slopes at the two nodes Interpolation of the displacements

Test functions are interpolated in the same way like displacements

Introducing the displacement and test functions interpolations into weak form gives the element stiffness matris

Element Stiffness Matrix

F CirakPage 35

Load vector computation analogous to the stiffness matrix derivation

The global stiffness matrix and the global load vector are obtained by assembling theindividual element contributions The assembly procedure is identical to usual finite elements

Global stiffness matrix

Global load vector

All nodal displacements and rotations

Element Load Vector

F CirakPage 36

Element stiffness matrix of an element with length le

Stiffness Matrix of Euler-Bernoulli Beam

F CirakPage 37


Kinematic assumption: a plane section originally normal to the centroid remains plane, but inaddition also shear deformations occur Rotation angle of the normal: Angle of shearing: Slope of midline:

The kinematic assumption determines the axial displacement of the material points acrossthickness

Note that this is only valid for small rotations, else

Kinematics of Timoshenko Beam -1-



F CirakPage 38

Introducing the displacements into the strain equations of three-dimensional elasticity leads to Axial strain

Axial strain varies linearly across thickness

Shear strain

Shear strain is constant across thickness

All the other strain components are zero

Kinematics of Timoshenko Beam -2-

F CirakPage 39

The beam strains introduced into the internal virtual work expressionof three-dimensional elasticity give

Hookess law

Introducing the expressions for strain and Hookes law into the weak form gives

virtual displacements and rotations:

shear correction factor necessary because across thickness shear stresses are parabolicaccording to elasticity theory but constant according to Timoshenko beam theory

shear correction factor for a rectangular cross section

shear modulus

External virtual work similar to Euler-Bernoulli beam

Weak Form of Timoshenko Beam

F CirakPage 40

Comparison of the displacements of a cantilever beam analyticallycomputed with the Euler-Bernoulli and Timoshenko beam theories

Bernoulli beam Governing equation:

Boundary conditions:

Timoshenko beam Governing equations:

Boundary conditions:

Euler-Bernoulli vs. Timoshenko -1-

F CirakPage 41

Maximum tip deflection computed by integrating the differential equations

Bernoulli beam

Timoshenko beam

Ratio

For slender beams (L/t > 20) both theories give the same result For stocky beams (Lt < 10) Timoshenko beam is physically more realistic because it includes the shear

deformations

Euler-Bernoulli vs. Timoshenko -2-

F CirakPage 42

The weak form essentially contains and the correspondingtest functions C0 interpolation appears to be sufficient, e.g. linear interpolation

Interpolation of displacements and rotation angle

Finite Element Discretization

F CirakPage 43

Shear angle

Curvature

Test functions are interpolated in the same way like displacements and rotations Introducing the interpolations into the weak form leads to the element stiffness matrices

Shear component of the stiffness matrix

Bending component of the stiffness matrix

Element Stiffness Matrix

F CirakPage 44

Gaussian Quadrature The locations of the quadrature points and weights are determined for maximum accuracy

nint=1

nint=2

nint=3

Note that polynomials with order (2nint-1) or less are exactly integrated

The element domain is usually different from [-1,+1) and an isoparametricmapping can be used

Review: Numerical Integration

F CirakPage 45

Necessary number of quadrature points for linear shape functions Bending stiffness: one integration point sufficient because is constant Shear stiffness: two integration points necessary because is linear

Element bending stiffness matrix of an element with length le and one integrationpoint

Element shear stiffness matrix of an element with length le and two integration points

Stiffness Matrix of the Timoshenko Beam -1-

F CirakPage 46

Limitations of the Timoshenko Beam FE

Recap: Degrees of freedom for the Timoshenko beam

Physics dictates that for t0 (so-called Euler-Bernoulli limit) the shear anglehas to go to zero ( ) If linear shape functions are used for u3 and

Adding a constant and a linear function will never give zero! Hence, since the shear strains cannot be arbitrarily small everywhere, an erroneous shear strain

energy will be included in the energy balance In practice, the computed finite element displacements will be much smaller than the exact solution

F CirakPage 47

Shear Locking: Example -1-

Displacements of a cantilever beam

Influence of the beam thickness on the normalized tip displacement

2 point

2

4

1

# elem.

0.0416

8

0.445

0.762

0.927

Thick beam

0.00021

2

4

8

0.0008

0.0003

0.0013

# elem. 2 point

Thin beam

from TJR Hughes, The finite element method.

TWO integration points

F CirakPage 48

The beam element with only linear shape functions appears not to be ideal for verythin beams

The problem is caused by non-matching u3 and interpolation For very thin beams it is not possible to reproduce

How can we fix this problem? Lets try with using only one integration point for integrating the element shear stiffness matrix Element shear stiffness matrix of an element with length le and one integration points

Stiffness Matrix of the Timoshenko Beam -2-

F CirakPage 49

Shear Locking: Example -2-

Displacements of a cantilever beam

Influence of the beam thickness on the normalized displacement

ONE integration point

2

4

1

# elem.

0.762

0.940

0.985

0.9968

1 point

Thick beam

0.750

0.938

0.984

0.996

1

2

4

8

# elem. 1 point

Thin beam

from TJR Hughes, The finite element method.

F CirakPage 50

If the displacements and rotations are interpolated with the same shapefunctions, there is tendency to lock (too stiff numerical behavior)

Reduced integration is the most basic engineering approach to resolvethis problem

Mathematically more rigorous approaches: Mixed variational principlesbased e.g. on the Hellinger-Reissner functional

Reduced Integration Beam Elements

CubicShape functionorder

Quadrature rule

Linear

One-point

Quadratic

Two-point Three-point

Finite Element Formulation for Plates- Handout 3 -


Completed Version

F CirakPage 52

Definitions

A plate is a three dimensional solid body with one of the plate dimensions much smaller than the other two zero curvature of the plate mid-surface in the reference configuration loading that causes bending deformation

A shell is a three dimensional solid body with one of the shell dimensions much smaller than the other two non-zero curvature of the shell mid-surface in the current configuration loading that causes bending and stretching deformation

mid-surfaceor mid-plane

F CirakPage 53

For a plate membrane and bending response are decoupled

For most practical problems membrane and bending response can be investigated independentlyand later superposed

Membrane response can be investigated using the two-dimensional finite elements introduced in3D7

Bending response can be investigated using the plate finite elements introduced in this handout

For plate problems involving large deflections membrane and bendingresponse are coupled For example, the stamping of a flat sheet metal into a complicated shape can only be simulated

using shell elements

Membrane versus Bending Response

loading in the plane of the mid-surface(membrane response active)

loading orthogonal to the mid-surface(bending response active)

F CirakPage 54

Overview of Plate Theories

In analogy to beams there are several different plate theories

The extension of the Euler-Bernoulli beam theory to plates is the Kirchhoff plate theory Suitable only for thin plates

The extension of Timoshenko beam theory to plates is the Reissner-Mindlin plate theory Suitable for thick and thin plates As discussed for beams the related finite elements have problems if applied to thin problems

In very thin plates deflections always large Geometrically nonlinear plate theory crucial (such as the one introduced for buckling of plates)

physicalcharacteristics

transverse sheardeformations

negligible transverseshear deformations

geometrically non-linear

Lengt / thickness ~5 to ~10 ~10 to ~100 > ~100

thick thin very thin

F CirakPage 55


Kinematic assumption: Material points which lie on the mid-surface normal remain on the mid-surface normal during the deformation

Kinematic equations In-plane displacements

In this equation and in following all Greek indices take only values 1 or 2 It is assumed that rotations are small

Out-of-plane displacements

Kinematics of Kirchhoff Plate -1-

undeformed and deformed geometries along one of the coordinate axis

deformed

reference

F CirakPage 56

Kinematics of Kirchhoff Plate -2-

Introducing the displacements into the strain equations of three-dimensionalelasticity leads to Axial strains and in-plane shear strain

All other strain components are zero Out-of-plane shear

Through-the-thickness strain (no stretching of the mid-surface normal during deformation)

F CirakPage 57

Weak Form of Kirchhoff Plate -1-

The plate strains introduced into the internal virtual work expression ofthree-dimensional elasticity

Note that the summation convention is used (summation over repeated indices)

Definition of bending moments

External virtual work Distributed surface load

For other type of external loadings see TJR Hughes book

Weak form of Kirchhoff Plate

Boundary terms only present if force/moment boundary conditions present

F CirakPage 58


Moment and curvature matrices

Both matrices are symmetric

Constitutive equation (Hookes law) Plane stress assumption for thin plates must be used

Hookes law for three-dimensional elasticity (with Lam constants)

Through-the-thickness strain can be determined using plane stress assumption

Introducing the determined through-the-thickness strain back into the Hookes law yields theHookes law for plane stress

F CirakPage 59


Integration over the plate thickness leads to

Note the change to Youngs modulus and Poissons ratio The two sets of material constants are related by

F CirakPage 60

Finite Element Discretization

The problem domain is partitioned into a collection of pre-selected finiteelements (either triangular or quadrilateral)

On each element displacements and test functions are interpolated usingshape functions and the corresponding nodal values

Shape functions

Nodal values

To obtain the FE equations the preceding interpolation equations areintroduced into the weak form Similar to Euler-Bernoulli Beam the internal virtual work depends on the second order derivatives

of the deflection and virtual deflection C1-continuous smooth shape functions are necessary in order to render the internal virtual work

computable

F CirakPage 61

Review: Isoparametric Shape Functions -1-

In finite element analysis of two and three dimensional problems theisoparametric concept is particularly useful

Shape functions are defined on the parent (or master) element Each element on the mesh has exactly the same shape functions

Shape functions are used for interpolating the element coordinates and deflections

parent element

physical element

Isoparametric mapping of a four-node quadrilateral

F CirakPage 62

Review: Isoparametric Shape Functions -2-

In the computation of field variable derivatives the Jacobian of the mapping has to beconsidered

The Jacobian is computed using the coordinate interpolation equation

F CirakPage 63

Shape Functions in Two Dimensions -1-

In 3D7 shape functions were derived in a more or less ad hoc way

Shape functions can be systematically developed with the help of the Pascalstriangle (which contains the terms of polynomials, also called monomials, of variousdegrees)

Triangular elements Three-node triangle linear interpolation

Six-node triangle quadratic interpolation

Quadrilateral elements Four-node quadrilateral bi-linear interpolation

Nine-node quadrilateral bi-quadratic interpolation

It is for the convergence of the finite element method important to use only complete polynomials up to a certaindesired polynomial order

Pascals triangle(with constants a, b, c, d, )

F CirakPage 64

Shape Functions in Two Dimensions -2-

The constants a, b, c, d, e, in the polynomial expansions can beexpressed in dependence of the nodal values For example in case of a a four-node quadrilateral element

with the shape functions

As mentioned the plate internal virtual work depends on the secondderivatives of deflections and test functions so that C1-continuous smoothshape functions are necessary It is not possible to use the shape functions shown above

F CirakPage 65

Early Smooth Shape Functions -1-

For the Euler-Bernoulli beam the Hermite interpolation was used which has the nodaldeflections and slopes as degrees-of-freedom

The equivalent 2D element is the Adini-Clough quadrilateral (1961) Degrees-of-freedom are the nodal deflections and slopes Interpolation with a polynomial with 12 (=3x4) constants

Surprisingly this element does not produce C1- continuous smooth interpolation(explanation on next page)

monomials

F CirakPage 66


Consider an edge between two Adini-Clough elements For simplicity the considered boundary is assumed to be along the axis in both elements

The deflections and slopes along the edge are

so that there are 8 unknown constants in these equations

If the interpolation is smooth, the deflection and the slopes in both elements along the edge haveto agree

It is not possible to uniquely define a smooth interpolation between the two elements becausethere are only 6 nodal values available for the edge (displacements and slopes of the two nodes).There are however 8 unknown constants which control the smoothness between the twoelements.

Elements that violate continuity conditions are known as nonconformingelements. The Adini-Clough element is a nonconforming element. Despitethis deficiency the element is known to give good results

F CirakPage 67


Bogner-Fox-Schmidt quadrilateral (1966) Degrees-of-freedom are the nodal deflections, first derivatives and second mixed derivatives

This element is conforming because there are now8 parameters on a edge between two elements in order togenerate a C1-continuous function

Problems Physical meaning of cross derivatives not clear At boundaries it is not clear how to prescribe the cross derivatives The stiffness matrix is very large (16x16)

Due to these problems such elements are not widely used in present daycommercial software

monomials

F CirakPage 68

New Developments in Smooth Interpolation

Recently, research on finite elements has been reinvigorated by the use ofsmooth surface representation techniques from computer graphics andgeometric design

Smooth surfaces are crucial for computer graphics, gaming and geometric design

Fifa 07, computer game

F CirakPage 69

Splines - Piecewise Polynomial Curves

Splines are piecewise polynomial curves for smooth interpolation For example, consider cubic spline shape functions

Each cubic spline is composed out of four cubic polynomials; neighboring curve segments are C2

continuously connected (i.e., continuous up to second order derivatives)

An interpolation constructed out of cubic spline shape functions is C2 continuous

cubicpolynomial

cubicpolynomial

cubicpolynomial

cubicpolynomial

F CirakPage 70

Tensor Product B-Spline Surfaces -1-

A b-spline surface can be constructed as the tensor-product of b-splinecurves

Tensor product b-spline surfaces are only possible over regular meshes A presently active area of research are the b-spline like surfaces over irregular meshes

The new approaches developed will most likely be available in next generation finite element software

twodimensional

onedimensional

onedimensional

irregular mesh

spline like surfacegenerated onirregular mesh

Finite Element Formulation for Plates- Handout 4 -


Completed Version

F CirakPage 72

The extension of Timoshenko beam theory to plates is the Reissner-Mindlinplate theory

In Reissner-Mindlin plate theory the out-of-plane shear deformations arenon-zero (in contrast to Kirchhoff plate theory)

Almost all commercial codes (Abaqus, LS-Dyna, Ansys, ) use Reissner-Mindlin type plate finite elements


Kinematic assumption: a plane section originally normal to the mid-surface remains plane, but inaddition also shear deformations occur

Kinematics of Reissner-Mindlin Plate -1-

reference

deformed

undeformed and deformed geometries along one of the coordinate axis

F CirakPage 73

Kinematic equations In plane-displacements

In this equation and in following all Greek indices only take values 1 or 2

It is assumed that rotations are small

Rotation angle of normal:

Angle of shearing:

Slope of midsurface:

Out-of-plane displacements

The independent variables of the Reissner-Mindlin plate theory are therotation angle and mid-surface displacement

Introducing the displacements into the strain equation of three-dimensionalelasticity leads to the strains of the plate

Kinematics of Reissner-Mindlin Plate -2-

F CirakPage 74

Axial strains and in-plane shear:

Out-of-plane shear:

Note that always

Through-the-thickness strain:

The plate strains introduced into the internal virtual work of three-dimensional elasticity give the internal virtual work of the plate

with virtual displacements and rotations:

Weak Form of Reissner-Mindlin Plate -1-

F CirakPage 75

Definition of bending moments:

Definition of shear forces:

External virtual work Distributed surface load

Weak form of Reissner-Mindlin plate

As usual summation convention applies


F CirakPage 76

Constitutive equations For bending moments (same as Kirchhoff plate)

For shear forces

Note that the curvature is a function of rotation angleand the shear angle is a function of rotation angle and the mid-surface displacement


F CirakPage 77

The independent variables in the weak form are and thecorresponding test functions

Weak form contains only so that C0-interpolation is sufficient Usual (Lagrange) shape functions such as used in 3D7 can be used

Nodal values of variables:

Nodal values test functions:

Interpolation equations introduced into the kinematic equations yield

Finite Element Discretization -1-

four-node isoparametricelement

F CirakPage 78

Constitutive equations in matrix notation

Bending moments:

Shear forces:

Element stiffness matrix of a four-node quadrilateral element Bending stiffness (12x12 matrix)


F CirakPage 79

Shear stiffness (12x12 matrix)

Element stiffness matrix

The integrals are evaluated with numerical integration. If too few integration points are used,element stiffness matrix will be rank deficient. The necessary number of integration points for the bilinear element are 2x2 Gauss points

The global stiffness matrix and global load vector are obtained byassembling the individual element stiffness matrices and loadvectors The assembly procedure is identical to usual finite elements


F CirakPage 80

As discussed for the Bernoulli and Timoshenko beams with increasing plateslenderness physics dictates that shear deformations have to vanish(i.e., )

Reissner-Mindlin plate and Timoshenko beam finite elements have problems to approximatedeformation states with zero shear deformations (shear locking problem)

1D example: Cantilever beam with applied tip moment

Bending moment and curvature constant along the beam Shear force and hence shear angle zero along the beam Displacements quadratic along the beam

Discretized with one two-node Timoshenko beam element

Shear Locking Problem -1-

F CirakPage 81

Shear Locking Problem -2-

Deflection interpolation:

Rotation interpolation:

Shear angle:

For the shear angle to be zero along the beam, the displacements and rotations have to be zero. Hence, ashear strain in the beam can only be reached when there are no deformations!

Similarly, enforcing at two integration points leads to zero

displacements and rotations!

However, enforcing only at one integration point (midpoint of the

beam) leads to non-zero displacements

In the following several techniques will be introduced to circumvent theshear locking problem Use of higher-order elements Uniform and selective reduced integration Discrete Kirchhoff elements Assumed strain elements

F CirakPage 82

Constraint ratio is a semi-heuristic number for estimating an elementstendency to shear lock Continuous problem

Number of equilibrium equations: 3 (two for bending moments + one for shear force) Number of shear strain constraints in the thin limit: 2

Constraint ratio:

With four-node quadrilateral finite elements discretized problem Number of degrees of freedom per element on a very large mesh is ~3

Number of constraints per element for 2x2 integration per element is 8

Constraint ratio:

Number of constraints per element for one integration point per element is 2

Constraint ratio:

Constraint Ratio (Hughes et al.) -1-

F CirakPage 83

Constraint ratio for a 9 node element Number of degrees of freedom per element on a very large mesh is ~ 4x3 =12 Number of constraints per element for 3x3 integration is 18

Constraint ratio:

Constraint ratio for a 16 node element Number of degrees of freedom per element on a very large mesh is ~ 9x3=27 Number of constraints per element for 4x4 integration is 32

Constraint ratio:

As indicated by the constraint ratio higher-order elements are less likely toexhibit shear locking

Constraint Ratio (Hughes et al.) -2-

F CirakPage 84

Uniform And Selective Reduced Integration -1-

The easiest approach to avoid shear locking in thin plates is to usesome form of reduced integration

In uniform reduced integration the bending and shear terms are integrated with thesame rule, which is lower than the normal

In selective reduced integration the bending term is integrated with the normal rule andthe shear term with a lower-order rule

Uniform reduced integrated elements have usually rank deficiency(i.e. there are internal mechanisms; deformations which do not needenergy) The global stiffness matrix is not invertible Not useful for practical applications

F CirakPage 85

Uniform And Selective Reduced Integration -2-

Shear refers to the integration of the element shear stiffness matrix Bending refers to the integration of the element bending stiffness matrix

BicubicShape functions

Selective reducedintegration

Bilinear

1x1

Biquadratic

2x2 3x3Uniform reduced

integration

1x1 shear2x2 bending



F CirakPage 86

Discrete Kirchhoff Elements

The principal approach is best illustrated with a Timoshenko beam

The displacements and rotations are approximated with quadratic shapefunctions

The inner variables are eliminated by enforcing zero shear stress at the twogauss points

Back inserting into the interpolation equations leads to a beamelement with 4 nodal parameters

F CirakPage 87

Assumed Strain Elements

It is assumed that the out-of-plane shear strains at edge centres are ofhigher quality (similar to the midpoint of a beam)

First, the shear angle at the edge centresis computed using the displacement and rotation nodal values

Subsequently, the shear angles from the edge centres are interpolated back

Note that the shape functions are special edge shape functions

These reinterpolated shear angles are introduced into the weak form and are for element stiffnessmatrix computation used

Finite Element Formulation for Shells- Handout 5 -


Completed Version

F CirakPage 89

Overview of Shell Finite Elements

There are three different approaches for deriving shell finiteelements Flat shell elements

The geometry of a shell is approximated with flat finite elements Flat shell elements are obtained by combining plate elements with plate stress elements

Degenerated shell elements Elements are derived by degenerating a three dimensional solid finite element into a shell

surface element

F CirakPage 90

Flat Shell Finite Elements

Example: Discretization of a cylindrical shell with flat shell finite elements

Note that due to symmetry only one eight of the shell is discretized

The quality of the surface approximation improves if more and more flat elements are used

Flat shell finite elements are derived by superposition of plate finite elements with plane stressfinite elements As plate finite elements usually Reissner-Mindlin plate elements are used As plane stress elements the finite elements derived in 3D7 are used

Overall approach equivalent to deriving frame finite elements by superposition of beam and truss finiteelements

Coarse mesh Fine meshCylindrical shell

F CirakPage 91

Four-Noded Flat Shell Element -1-

First the degrees of freedom of a plate and plane-stress finite element in alocal element-aligned coordinate system are considered

The local base vectors are in the plane of the element and is orthogonal to theelement

The plate element has three degrees of freedom per node (one out-of-plane displacement and tworotations)

The plane stress element has two degrees of freedom per node node (two in plane displacements) The resulting flat shell element has five degrees of freedom per node

Plate element Plane stress element Flat shell element

F CirakPage 92


Stiffness matrix of the plate in the local coordinate system:

Stiffness matrix of the plane stress element in the local coordinate system:

Stiffness matrix of the flat shell element in the local coordinate system

Stiffness matrix of the flat shell element can be augmented to include the rotations (seefigure on previous page)

Stiffness components corresponding to are zero because neither the plate nor the planestress element has corresponding stiffness components

F CirakPage 93


Transformation of the element stiffness matrix from the local to the globalcoordinate system Discrete element equilibrium equation in the local coordinate system

Nodal displacements and rotations of element Element force vector

Transformation of vectors from the local to the global coordinate system

Rotation matrix (or also known as the direction cosine matrix) Note that for all rotation matrices

Transformation of element stiffness matrix from the local to global coordinate system

Discrete element equilibrium equation in the global coordinate system

F CirakPage 94


The global stiffness matrix for the shell structure is constructred bytransforming each element matrix into the global coordinate system prior toassembly

The global force vector of the shell structure is constructed by transformingeach element force vector into the global coordinate system prior toassembly

Remember that there was no stiffness associated with the local rotationdegrees of freedom . Therefore, the global stiffness matrix will be rankdeficient if all elements are coplanar. It is possible to add some small stiffness for element stiffness components corresponding to

in order to make global stiffness matrix invertible

Add small stiffness in order to makestiffness matrix invertible

F CirakPage 95

Degenerated Shell Elements -1-

First a three-dimensional solid element and the corresponding parentelement are considered (isoparametric mapping)

In the following it is assumed that the solid element has on its top and bottom surfaces nine nodesso that the total number of nodes is eighteen The derivations can easily be generalised to arbitrary number of nodes

Coordinates of the nodes on the top surface are

Coordinates of the nodes on the bottom surface are

parent element solid element

F CirakPage 96


There are nine isoparametric shape functions for interpolating the top and bottom surfaces

with the natural coordinates Note that these shape functions are identical to the ones for two dimensional elasticity

The geometry of the solid element can be interpolate with

Definitions Shell mid-surface node

Shell director (or fibre) at node

The shell director is a unit vector and is approximately orthogonal to the mid-surface

F CirakPage 97


Using the previous definitions the solid element geometry can beinterpolated with

with the solid element thickness

The displacements of the solid element are assumed to be

The first component is the mid-surface displacement and the second component is the directordisplacement

Note that the deformed mid-surface nodal coordinates can be computed with and the deformed nodaldirector with

The director displacement has to be constructed so that the director can rotate but notstretch This was one of the of the Reissner-Mindlin theory assumptions

F CirakPage 98


The director displacements are expressed in terms of rotations at the nodes To accomplish this a local orthonormal coordinate system is constructed at each node

The definition of the orthonormal coordinate system in not unique.In a finite element implementation it is necessary to store at eachnode the established coordinate base vectors .

The relationship between the director displacements and the two rotation angles in the localcoordinate system is

Rotations are defined as positive withthe right-hand rule

It is assumed that the rotation angles are small

so that the director length does not change

F CirakPage 99


Displacement of the shell element in dependence of the mid-surfacedisplacements and director rotations

The element has nine nodes There are five unknowns per node (three mid-surface displacements and two director rotations) This assumption about the possible displacements is equivalent to the Reissner-Mindlin

assumption

Introducing the displacements into the strain equation of three-dimensionalelasticity leads to the strains of the shell element

In computing the displacement derivatives the chain rule needs to be used

F CirakPage 100


The Jacobian is computed from the geometry interpolation

The shell strains introduced into the internal virtual work of three-dimensionalelasticity give the internal virtual work of the shell

For shear locking similar techniques such as developed for the Reissner-Mindlin plateneed to be considered

Dynamics of Beams, Plates and Shells- Handout 6 -


Completed Version

F. CirakPage 102

Equilibrium equations for elastodynamics (strong form)

Density Acceleration vector Stress matrix Distributed body force vector

Displacement initial condition (at time=0) Velocity initial condition (at time=0)

The weak form of the equilibrium equations for elastodynamics is known asthe dAlemberts principle It can be obtained by the standard procedure: Multiply the strong form with a test function,

integrate by parts and apply the divergence theorem

Virtual displacement vector Build-in boundaries are assumed so that no boundary integrals are present in the above equation

Strong and Weak Form of Elastodynamics

F. CirakPage 103

The weak form over one typical finite element element

Approximation of displacements, accelerations and virtual displacements

are the nodal displacements, accelerations and virtual displacements, respectively Shape functions are the same as in statics Nodal variables are now a function of time!

Element mass matrix is computed by introducing the approximations into kinetic virtual work

Element stiffness matrix and the load vector are the same as for the static case

FE Discretization of Elastodynamics -1-

F. CirakPage 104

Global semi-discrete equation of motion after assembly of element matrices

Global mass matrix Global stiffness matrix Global external force vector Initial conditions

Equation called semi-discrete because it is discretized in space but still continuous in time

Global semi-discrete equation of motion with viscous damping

Damping matrix Damping proportional to velocity

Rayleigh damping (widely used in structural engineering)

and are two scalar structure properties which are determined from experiments

FE Discretization of Elastodynamics -2-

F. CirakPage 105

Assumed deformations during dynamic loading

Kinematic assumption: a plane section originally normal to the centroid remains plane,but in addition also shear deformations occur Key kinematic relations for statics:

Corresponding relations for dynamics:

Dynamics of Timoshenko Beams



F. CirakPage 106

Kinetic Virtual Work for Timoshenko Beam

Introducing the beam accelerations and test functions into thekinetic virtual work of elastodynamics gives

Virtual axial displacements Virtual deflections and rotations

Kinetic virtual work due to rotation

Rotationary inertia (very small for thin beams)

Kinetic virtual work due to deflection

F. CirakPage 107

Interpolation with shape functions (e.g. linear shape functions)

Consistent mass matrix is computed by introducing the interpolations into the kineticvirtual work

For practical computations lumped mass matrix sufficient (the sum of the elements of each row of the consistentmass matrix is used as the diagonal element)

The components of the mass matrix are simply the total element mass and rotational inertia divided by two

Finite Element Discretization - Mass Matrix

F. CirakPage 108

Semi-discrete equation of motion

Time integration Assume displacements, velocities and accelerations are known

for ttn

Central difference formula for the velocity

Central difference formula for the acceleration

Discrete equilibrium at t=tn

Substituting the computed acceleration into the central difference formula for acceleration yields thedisplacements at t=tn+1

Most Basic Time Integration Scheme -1-

F. CirakPage 109

These equations are repeatedly used in order to march in time and to obtain solutionsat times t=tn+2, tn+3,

Provided that the mass matrix is diagonal displacements and velocities are computed withoutinverting any matrices. Such a scheme is called explicit. Matrix inversion is usually the most time consuming part of finite element analysis In most real world applications, explicit time integration schemes are used

Explicit time integration is very easy to implement. The disadvantage isconditional stability. If the time step exceeds a critical value the solution willgrow unboundedly Critical time step size

Characteristic element size Wave speed

Longitudinal wave speed in solids (material property)

Most Basic Time Integration Scheme -2-

exact solution

F. CirakPage 110

It is instructive to consider the time integration of the semi-discreteheat equation before attempting the time integration for elasticity

Temperature vector and its time derivative Heat capacity matrix Heat conductivity matrix Heat supply vector Initial conditions

- family of time integration schemes

with

Combining both gives

Introducing into the semidiscrete heat equation gives an equation for determining

Advanced Time Integration Schemes -1-

F. CirakPage 111

Common names for the resulting methods forward differences; forward Euler trapezoidal rule; midpoint rule; Crank-Nicholson backward differences; backward Euler

Explicit vs. implicit methods and their stability The method is explicit and conditionally stable for

Time step size restricted

The method is implicit and stable for Time step size not restricted. However, for large time steps less accurate.

Implementation in a predictor-corrector form Simplifies the computer implementation. Does not change the basic method. Compute a predictor with known solution

Introducing into the semidiscrete heat equation gives an equation for determining

Advanced Time Integration Schemes -2-

F. CirakPage 112

Newmark family of time integration schemes are widely used instructural dynamics Assume that the displacements, velocities, and accelerations are known for ttn Displacements and velocities are approximated with

with two scalar parameters The two scalar parameters determine the accuracy of the scheme Unconditionally stable and undamped for

Implementation in a-form (according to Hughes) Simplifies the computer implementation. Does not change the basic method. Compute predictor velocities and displacements

Newmarks Scheme -1-

F. CirakPage 113

Displacement and velocity approximation using predictor values

Introducing the displacement approximation into the semidiscrete equation gives an equation forcomputing the new accelerations at time t=tn+1

The new displacements and velocities are computed from the displacement and velocity approximationequations

Newmarks Scheme -2-

Fehmi CirakPage 90

Elastodynamics - Motivation

Fehmi CirakPage 91

! The discrete elastodynamics equations can be derived from eitherHamiltonian, Lagrangian or principle of virtual work for (dAlembertsprinciple)

! with initial conditions

! Discretization with finite elements

! Element mass matrix

! The stiffness matrix and the load vector are the same as for the static case

Elastodynamics -1-

(build-in boundaries)

element mass matrix

Fehmi CirakPage 92

! Semi-discrete equation of motion

! Mass matrix

! Stiffness matrix

! External force vector

! Initial conditions

! Semi-discrete because it is discretized in space but continuous in time

! Viscous damping

! Rayleigh damping

Elastodynamics -2-

Fehmi CirakPage 93

! Kinetic virtual work

! Rotationary inertia (very small for thin beams)

Timoshenko Beam - Virtual Kinetic Work

reference

configuration

deformed

configuration

Fehmi CirakPage 94

! Discretization with linear shape functions

! Lumped mass matrix (lumping by row-sum technique)

! In practice the rotational contribution can mostly be neglected

! For the equivalent Reissner-Mindlin plate, the components of the mass matrix aresimply the total element mass divided by four

Timoshenko Beam - Mass Matrix

Fehmi CirakPage 95

! Semi-discrete equation of motion

! Discretization in time (or integration in time)! Assume displacements, velocities, and accelerations

for t!tn are known

! Central difference formula for the velocity

! Central difference formula for the acceleration

! Discrete equilibrium at t=tn

! Displacements at t=tn+1 follow from these equations as

Explicit Time Integration -1-

Fehmi CirakPage 96

! Provided that the mass matrix is diagonal the update of displacements andvelocities can be accomplished without solving any equations

! Explicit time integration is very easy to implement. The disadvantage isconditional stability. If the time step exceeds a critical value the solution willgrow unboundedly! Critical time step size

! Longitudinal wave speed in solids

Explicit Time Integration -2-

exact solution

Fehmi CirakPage 97

! Semi-discrete heat equation

! is the temperature vector and its time derivative

! is the heat capacity matrix

! is the heat conductivity matrix

! is the heat supply vector

! Initial conditions

! Family of time integrators

Semi-discrete Heat Equation -1-

Fehmi CirakPage 98

! Common names for the resulting methods

! forward differences; forward Euler

! trapezoidal rule; midpoint rule; Crank-Nicholson

! backward differences; backward Euler

! Explicit vs. implicit methods

! For the method is explicit

! For the method is implicit

! Implementation: Predictor-corrector form

Semi-discrete Heat Equation -2-

known

substituting in

Fehmi CirakPage 99

! For elastodynamics most widely used family of time integration schemes

! Assume that the displacements, velocities, and accelerations for t!tn are

known

! Unconditionally stable and undamped for

! Implementation: a-form (according to Hughes)

! Compute predictors

The Newmark Method -1-

Fehmi CirakPage 100

! To compute the accelerations at n+1 following equation needs to be solved

The Newmark Method -2-

CourseworkPlates and Shells: Analysis and Computation (4D9)

Dr Fehmi Cirak

Deadline: The deadline for the report and software is 11 March 2010, 5pm

Estimated time to complete: 10 hours

Introductory lab session: 3 March 2010, 11-12

Form of submission: A typed report of at least three pages and a working MATLABimplementation. The preferred form of submission is via email to fc286@. Make sure thatyou submit the entire CEAKIT directory on your computer and not just the functionsyou implemented.

Problem description

The objective of this coursework is to implement a finite element code for static and dy-namic analysis of plate structures. There is a MATLAB finite element library CEAKIT(Computational Engineering Analysis Kit) to be used for this coursework, which can beobtained from:http://www-g.eng.cam.ac.uk/csml/teaching/4d9/CEAKIT.tar.gz

The dowloaded file CEAKIT.tar.gz can be unpacked with tar -xzf CEAKIT.tar.gz,which will create the CEAKIT directory containing several *.m files. Running the driverfunction Ceakit in MATLAB will plot a deflected plate.

Tasks to be completed

1. Study the Ceakit.m driver and describe with few sentences the purpose of eachfunction called.

2. Implement a function which computes the load vector for uniform pressure loading.

3. Figure 1 shows the geometry of a plate to be analysed. The boundaries of the plateare simply supported and the plate thickness is t = 0.2. The Youngs modulusof the material is E = 35000 and the Poissons ratio is = 0.3. The plate isloaded by uniform pressure loading of p = 0.003. Study the convergence of themaximal displacements for fully and selectively reduced integrated finite elements.The meshes to be used should have 4 4, 8 8, 16 16 and 32 32 elements.

4. Implement a function for computing the element mass matrix of a plate finite ele-ment and extend Ceakit.m for assembling the global mass matrix.

5. Implement the implicit Newmark time integration scheme.

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Coursework 4D9 F Cirak

15.0

10.0

10.0

Figure 1:

6. The dynamics of the plate in Fig. 1 due to sudden uniform loading is to be studied.The mass density of the material is = 2000. Apply a sudden uniform loading ofp = 0.003 and plot the evolution of the maximal displacements over time.

Note, it is not sufficient just to submit a working MATLAB implementation.It is important that you submit a report, which addresses item by item eachpoint of the previous list. Do not forget to include the requested plots andthe implemented equations.

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