theory manual volume 2

Theory Manual Volume 2 LUSAS Version 14 : Issue 2

LUSAS Forge House, 66 High Street, Kingston upon Thames,

Surrey, KT1 1HN, United Kingdom

Tel: +44 (0)20 8541 1999 Fax +44 (0)20 8549 9399 Email: [email protected] http://www.lusas.com

Distributors Worldwide

Table of Contents

i

Table of Contents 7 Element Formulations .................................................................................1

7.1 Bar Elements (BAR2, BAR3, BRS2, BRS3) ...........................................1 7.1.1 Formulation.......................................................................................1 7.1.2 Evaluation and Output of Stresses/Forces.......................................1 7.1.3 Nonlinear Formulation ......................................................................2

7.2 Beam Elements.......................................................................................7 7.2.1 2-D Straight Beam (BEAM) ............................................................10 7.2.2 2-D Straight Grillage (GRIL) ...........................................................12 7.2.3 2-D Ribbed Plate Beam (BRP2).....................................................14 7.2.4 3-D Straight Beam (BMS3).............................................................17 7.2.5 2-D Curved Thin Beam (BM3, BMX3) ............................................21 7.2.6 3-D Curved Thin Beam (BS3, BS4, BSX4) ....................................29 7.2.7 Semiloof Thin Beam (BSL3, BSL4, BXL4) .....................................37 7.2.8 3-D Straight Beam (BTS3)..............................................................45

7.3 Two-Dimensional Continuum Elements................................................55 7.3.1 Standard Isoparametric Elements ..................................................55 7.3.2 Enhanced Strain Elements (QPM4M, QPN4M, QAX4M)...............66 7.3.3 Incompatible Plane Membrane Element (PMI4).............................73 7.3.4 2D Explicit Dynamics Elements......................................................76 7.3.5 Two Phase Plane Strain Continuum Elements (TPN6P and QPN8P) ...................................................................................................87 7.3.6 Large-strain Mixed-type Elements (QPN4L, QAX4L).....................90

7.4 Three-Dimensional Continuum Elements .............................................95 7.4.1 Standard Isoparametric Elements (HX8, HX16, HX20, PN6, PN12, PN15, TH4, TH10)...................................................................................95 7.4.2 Enhanced Strain Element (HX8M) ...............................................102 7.4.3 3D Explicit Dynamics Elements (HX8E, PN6E, TH4E) ................105 7.4.4 Composite Solid Elements (HX8L,HX16L,PN6L PN12L).............113 7.4.5 Two Phase 3D Continuum Elements (TH10P, PN12P, PN15P, HX16P and HX20P)...............................................................................118

7.5 Space Membrane Elements................................................................122 7.5.1 Axisymmetric Membrane (BXM2, BXM3).....................................122 7.5.2 3-D Space Membrane (SMI4, TSM3)...........................................126

7.6 Plate Elements ....................................................................................128 7.6.1 Isoflex Thin Plate (QF4, QF8, TF3, TF6)......................................128 7.6.2 Isoflex Thick Plate (QSC4) ...........................................................135 7.6.3 Isoparametric Thick Mindlin Plate (QTF8, TTF6) .........................139 7.6.4 Ribbed Plate (RPI4, TRP3) ..........................................................144

7.7 Shell Elements ....................................................................................148 7.7.1 Axisymmetric Thin Shell (BXS3) ..................................................148 7.7.2 Flat Thin Shell (QSI4, TS3) ..........................................................155 7.7.3 Flat Thin Shell Box (SHI4) ............................................................159

Table of Contents

ii

7.7.4 Semiloof Thin Shell (QSL8, TSL6) ...............................................163 7.7.5 Thick Shells (TTS3, TTS6, QTS4, QTS8) ....................................176

7.8 Field Elements.....................................................................................189 7.8.1 Thermal Bar (BFD2, BFD3) ..........................................................189 7.8.2 Thermal Axisymmetric Bar (BFX2, BFX3) ....................................191 7.8.3 Thermal Link (LFD2, LFS2, LFX2) ...............................................192 7.8.4 Plane Field (QFD4, QFD8, TFD3, TFD6) .....................................194 7.8.5 Axisymmetric Field (QXF4, QXF8, TXF3, TXF6) .........................198 7.8.6 Solid Field (HF8, HF16, HF20, PF6, PF12, PF15, TF4, TF10) ....200 7.8.7 Solid Composite Field (HF8C, HF16C, PF6C, PF12C)................204

7.9 Joint Elements.....................................................................................208 7.9.1 Joints (JNT3, JPH3, JF3, JRP3, JNT4, JL43, JSH4, JL46, JSL4, JAX3, JXS3) ..........................................................................................208 7.9.2 Evaluation of Stresses/Forces......................................................209 7.9.3 Nonlinear Formulation ..................................................................209 7.9.4 Use of Joints With Higher Order Elements...................................212

7.10 Fourier Element Formulation (TAX3F, QAX4F, TAX6F, QAX8F).....215 7.10.1 Global and Local Coordinate Systems .......................................215 7.10.2 Standard Isoparametric Elements ..............................................215 7.10.3 Strain-Displacement Relationships.............................................217 7.10.4 Constitutive Relationships ..........................................................218 7.10.5 Element Loading.........................................................................219 7.10.6 Inertial Loading ...........................................................................221 7.10.7 Evaluation of Stresses................................................................225

7.11 Interface Elements (INT6, INT16) .....................................................225 7.11.1 Definition and interpolation .........................................................225 7.11.2 Internal force vector and stiffness matrix....................................226

Appendix A ..................................................................................................229 Quadrature Rules ......................................................................................229

Appendix B ..................................................................................................239 Restrictions On Element Topology............................................................239

Mid-Length and Mid-Side Nodes ...........................................................239 Warping of Flat Elements ......................................................................239

References...................................................................................................241

Notation

iii

Notation Standard matrix notation is used whenever possible throughout this manual and the expressions are defined as follows:

Basic Expressions Vector

Matrix or second order tensor

Fourth order tensor

: Matrix scalar product

| | Determinant of a matrix

|| || Norm of a vector

trb g Trace of a matrix

b gT Transpose of a vector of matrix

b g−1 Inverse of a matrix

db g Variation

δb g Virtual variation

•e j Rate

Δb g Increment

b g∑ Summation

diag , Diagonal matrix with terms given

< >, Orthogonality condition

∧e j Variables defined in local axes

Theory Manual 2

iv

Subscripts cr Critical value

g Ground displacements, velocities and accelerations

i Component i

max Maximum value

n Normal component in slideline analyses

o Initial components (initial strains)

t Thermal components (thermal strains)

x Tangential component in slideline analyses

x,y,z Components in the local x,y,z Cartesian system

X,Y,Z Components in the global X,Y,Z Cartesian system

Z Zienkiewicz constants

,x Differentiation with respect to x (or other variable)

Superscripts i Iteration I

l Local quantities (co-rotational for continuum elements)

t Values at time t

t t+ Δ Values at time t+Δt

Scalars aR Rayleigh damping coefficient (multiplies mass matrix M)

B Bulk modulus

bR Rayleigh damping coefficient (multiplies stiffness matrix K)

C0 neo-Hookean constant

C1 , C2 Mooney-Rivlin constants

c Cohesion in friction based material models

c Wave speed

Da Maximum distance between two adjacent contact nodes

Notation

v

E Young’s Modulus

fi Slideline interface force on contact node i

F Yield surface

g Initial gap for nonlinear joint models

G Shear Modulus

Gf Fracture energy in concrete model

h Transfer coefficient in field analysis

I1 First stress invariant

I1,I2,I3 Strain invariants

I I1 2, Modified strain invariants

J Volume ratio (det F)

J2 Second deviatoric stress invariant

J3 Third deviatoric stress invariant

k Interface stiffness coefficient

k Bulk modulus

K Thermal conductivity

Kc Spring stiffness when in contact for nonlinear joint models

K1 Spring stiffness after liftoff for nonlinear joint models

l Length of local contact segment

lo Initial chord length of beam element

ln Current chord length of beam element

M Moment

N Stress resultant

p Number of required eigenvalues

P Participation factor in spectral response analysis

P Axial force

Theory Manual 2

vi

q Field variable flux in field analysis

q Number of starting iteration vectors for subspace iteration

Q Rate of internal heat generation in field analysis

Qhg Hourglass constant

Q1,Q2 Constants for shock wave smoothing

rz Contact zone radius

Sd Spectral displacement in special response analysis

Sv Spectral velocity in spectral response analysis

Sa Spectral acceleration in spectral response analysis

t Thickness of local contact segment

T Period of oscillation for transient and dynamic analysis

T Temperature in structural applications

T Torque

u Axial stretch

V Element volume

w Crack width in concrete model

W Work

X Normal penetration distance

α Radial overlap constant

α Coefficient of thermal expansion

α Softening Parameter in concrete model

α Constant used in dynamic recurrence algorithms

β Constant used in dynamic recurrence algorithms

β Shear retention factor in concrete model

γ Constant used in dynamic recurrence algorithms

γ d Displacement norm used for convergence

Notation

vii

γ ψ Residual norm used for convergence

γ w Work norm used for convergence

γ ψ1 Root mean square of residuals convergence criterion

γ ψ2 Maximum absolute residual convergence criterion

εi Error estimate in subspace iteration

η Step length multiplier for line search

e Lode’s Angle

e Angle between old and new displacement vector in arch-length method

e Angle of orthotropy

e Angle defining crack directions in concrete model

e Local slope at a node

κ Strain hardening parameter

λ Load factor

λ Plastic strain rate multiplier

λi Eigenvalue (ith)

λi Principal stretches

μ Eigenvalue shift in subspace iteraction

μ Friction coefficient

μ αp p, Ogden constants

ν Poisson’s ratio

ξ Modal Damping ratio

ρ Model coefficient for CQC combination in spectral response analysis

ρ Mass density

σ Effective stress

Theory Manual 2

viii

φ Interface stiffness scale factor

φ Friction angle in friction based models

φ Structural damping in harmonic response analysis

Φ Field variable in field analysis

ψ Potential energy

ω Circular frequency in transient and dynamic analysis

Ω Circular frequency of load in harmonic response analysis

Ω1 Local spin at the centroid

Vectors a Nodal displacement vector

ei Unit vectors forming the co-rotated base axes

E Green-Lagrange strain vector

f Vector of master slideline surface forces

f Vector of nodal body forces

g g gξ η ζ, , Covariant base vectors

Mm Vector of master slideline surface mass

n Vector of unit segment normal

P Global internal force vector

P Local internal force vector

~q Euler parameters

R External force vector

ri Unit vectors defining the beam cross section at a gauss point

s Deviatoric Cauchy stress

S Second Piola-Kirchhoff stress vector

t Vector of surface tractions

Notation

ix

t qi i, Unit vectors defining the beam cross section at a node

x Vector of unit segment tangent

Y Generalised displacement vector in spectral response analysis

ε Logarithmic strain vector

θ Displacement gradient vector (geometric nonlinearity)

θ Unscaled pseudovector (co-rotational formulation - section 3.5.1)

ψ Pseudovector of rotation

λ Lagrangian multiplier vector

λi Incompatible modes for enhanced elements

σ Cauchy stress vector

d Jσ Jaumann variation of Cauchy stress

Φi i,ω Eigenvector

ψ Residual force vector

Ω1 Local spin at the centroid

Matrices/Tensors A Matrix of slopes

B Strain-displacement matrix

B0

Linear strain-displacement matrix

B1 Displacement dependent strain-displacement matrix

C Damping matrix in dynamic analysis

C Matrix of constrain constants

C Green deformation tensor

C Compliance matrix of material moduli

Theory Manual 2

x

D Rate of deformation tensor

D Material modulus matrix

F Deformation gradient matrix

G Matrix of shape functions

K Stiffness matrix

KT

Tangent stiffness matrix

Kσ

Stress stiffness matrix

M Mass matrix

N Shape function array

Ni Principal directions of the Lagragian triad

Q Vector of constraint constants

R Rotation tensor

S θa f Skew symmetric matrix of the vector θ

$S Matrix containing Second Piola-Kirchoff stresses

T Transformation matrix for co-rotational formulation

U Right stretch tensor

α Angular acceleration tensor

Λ Matrix of eigenvalues

ε Local engineering strain tensor

ρ Density matrix

$σ Matrix containing Cauchy stresses

$$σ Matrix containing Cauchy stresses

Notation

xi

σ Biot stress tensor

Φ Matrix of eigenvectors

τ Kirchhoff stress tensor

Ω Angular velocity tensor

Theory Manual 2

xii

Bar Elements (BAR2, BAR3, BRS2, BRS3)

1

7 Element Formulations This section of the Theory Manual covers the basic theoretical assumptions made for each element formulation. Appropriate references are included when full details of the element derivation are not provided.

7.1 Bar Elements (BAR2, BAR3, BRS2, BRS3) 7.1.1 Formulation The bar elements are 2-node and 3-node isoparametric elements that can only transmit longitudinal force (fig.7.1-1).

The nodal variables are:-

BAR2 and BAR3 U and V BRS2 and BRS3 U, V and W

The element strain-displacement relationship and thermal strain vector are defined in the local Cartesian system as

∈ =xux

∂∂

and ∈ =o t Tb g α

The elastic constitutive relationship is defined as

σx xE= ∈

A complete description of the element stiffness formulation is given in [B1].

The consistent and lumped mass matrices are evaluated using the procedures defined in (section 2.7).

7.1.2 Evaluation and Output of Stresses/Forces The element output can be obtained at both the element nodes and Gauss points and consists of

Fx - the axial force, (tension +ve)

∈x - the axial strain. (tension +ve)

Element Formulations

2

The forces and strains are output in the local element coordinate system defined by

BAR2 and BAR3 elements The element local x-axis lies along the element axis in the direction in which the element nodes are specified (fig.7.1-3). The local y and z axes form a right-hand set with the x-axis such that the y-axis lies in the global XY-plane and the z-axis is parallel to the global Z-axis (up out of page).

BRS2 and BRS3 elements The local x-axis lies along the element axis in the direction in which the element nodes are specified (for a curved element it is tangent to the curve at the point concerned).

For a curved element the local xy-plane is defined by the element nodes (fig.7.1-4). Local y is perpendicular to local x and +ve on the convex side of the element.

For a straight element parallel to the global x-axis, the local z-axis is defined by the unit vector z = j x x where j is a unit vector defining the Global Y-axis and x is a unit vector defining the local x-axis (fig.7.1-4).

For a straight element not parallel to the global x-axis, the local z-axis is defined by the unit vector z = i x x where i is a unit vector defining the global X-axis (fig.7.1-4).

The local y-axis forms a right-hand set with the local x and z-axes for all three cases.

7.1.3 Nonlinear Formulation The bar elements can be employed in

Materially nonlinear analysis utilising the elastoplastic constitutive laws [O1] (section 4.2). Geometrically nonlinear analysis. Geometrically and materially nonlinear analysis utilising the nonlinear

material laws specified in 1. Nonlinear dynamics utilising the nonlinear material laws specified in 1. Linear eigen-buckling analysis.

Note. The geometric nonlinearity is a Total Lagrangian formulation which accounts for large displacements but small strains. The nonlinear strain-displacement relationship is defined by

BAR2 and BAR3

∈ = + LNMOQP + LNM

OQPx

ux

ux

vx

∂∂

∂∂

∂∂

12

12

2 2

BRS2 and BRS3


3


OQP + LNM

OQPx

ux

ux

vx

wx

∂∂

∂∂

∂∂

∂∂

12

12

12

2 2 2

with reference to the local x-axis.

The forces and strains output with the geometrically nonlinear analysis will be the 2nd Piola-Kirchhoff forces and Green-Lagrange strains respectively, referred to the undeformed configuration. The loading is conservative.

Y

X

3

1

1

2

2

VU

U U

U

U

V

V

V

V

(a) 2-D Bar Elements

BAR2

BAR3

Y

X

Z

U

2

(b) 3-D Bar Elements

U

V

BRS2

BRS3

V

U

U

UV

V

V

U

W

W

W

W

W

2

1

1

3


4

Fig.7.1-1 Nodal Freedoms For BAR Elements

2-D Roof Truss Excavation Supports

Pressure

Struts represented withBAR2 elements

Continuum elements

Fig.7.1-2 Examples Illustrating Use Of BAR Elements

Y

X

31

1

2

2

xy

yx

y

y

x

x

Fig 7.1-3 Local Cartesian System For BAR2 And BAR3 Elements


5

Y

X

Z(a) Curved Element

x

y x

z

x

x-y plane

y

y

z

2

1

3

z

Y

X

Z

(b) Straight Element Parrallel With Global X-axis

xz

x

yy

21

z

Y

X

Z

(c) Arbitrarily Orientated Straight Element

x

z

xy

y

2

1

z

Fig. 7.1-4 Local Cartesian System For BRS2 And BRS3 Elements


6

Beam Elements

7

7.2 Beam Elements The family of explicit straight beams are derived by restraining various degrees-of-freedom of the full 3D beam. The stiffness and mass matrices of these reduced elements may be obtained by deleting the appropriate rows and columns of the full stiffness and mass matrices.

The nodal forces/moments and degrees of freedom (in local coordinates) for the 3D beam are

F P P P M M M P P P M M MTx y z x y z x y z x y z= 1 1 1 1 1 1 2 2 2 2 2 2, , , , , , , , , , ,

a u v w u v wTx y z x y z= 1 1 1 1 1 1 2 2 2 2 2 2, , , , , , , , , , ,θ θ θ θ θ θ

The corresponding stiffness and mass matrices are

Element stiffness matrix

KK KK K=LNMM

OQPP

11 21

12 22

where submatrices are defined:

K

EAL

SymmetricEI

LEI

LGJL

EI

L

EIL

EIL

EI

L

z

y

y

z

y

z

z y

z

z

y

y z

y

11

3

3

2

2

0 121

0 012

1

0 0 0

0 06

10

41

0 61

0 0 04

1

=

+

+

−

+

+

+

+

+

+

L

N

MMMMMMMMMMMMMMMM

O

Q

PPPPPPPPPPPPPPPP

Φ

Φ

Φ

Φ

Φ

Φ

Φ

Φ

d i

b g

b gb gb g

d id id i


8

K

EAL

SymmetricEI

LEI

LGJL

EI

L

EIL

EIL

EI

L

z

y

y

z

y

z

z y

z

z

y

y z

y

22

3

3

2

2

0 121

0 012

1

0 0 0

0 06

10

41

0 61

0 0 04

1

=

+

+

+

+

+

−

+

+

+

L

N

MMMMMMMMMMMMMMMM

O

Q

PPPPPPPPPPPPPPPP

Φ

Φ

Φ

Φ

Φ

Φ

Φ

Φ

d i

b g

b gb gb g

d id id i

K K

EAL

EIL

EIL

EI

L

EI

LGJL

EI

L

EIL

EIL

EI

L

T

z

y

z

y

y

z

y

z

y

z

z y

z

z

y

y z

y

12 21

3 2

3 2

2

2

0 0 0 0 0

0 121

0 0 0 61

0 012

10

6

10

0 0 0 0 0

0 06

10

21

0

0 61

0 0 02

1

= =

−

−

+

−

+−

+ +−

−

+

−

+

+

−

+

L

N

MMMMMMMMMMMMMMMM

O

Q

PPPPPPPPPPPPPPPP

Φ Φ

Φ Φ

Φ

Φ

Φ

Φ

Φ

Φ

d i d i

b g b g

b gb gb g

d id id i

and where

Φyz

s

EIGA L

y

=12

2 and Φzy

s

EI

GA Lz

=12

2

Asy and Asz

are the cross-sectional areas effective in shear about the respective bending axis.

Element mass matrix

M ALM MM M=LNMM

OQPPρ 11 21

12 22

where submatrices are defined

Beam Elements

9

M

SymmetricI

ALI

ALJA

L IAL

L IA

L IAL

L IA

z

y

x

y y

z z

11

2

2

2

2

130 13

356

5

0 0 1335

65

0 0 03

0 0 11210 10

0105

215

0 11210 10

0 0 0105

215

=

+

+

− − +

+ +

L

N

MMMMMMMMMMMMMM

O

Q

PPPPPPPPPPPPPP

M

SymmetricI

ALI

ALJA

L IAL

L IA

L IAL

L IA

z

y

x

y y

z z

22

2

2

2

2

130 13

356

5

0 0 1335

65

0 0 03

0 0 11210 10

0105

215

0 11210 10

0 0 0105

215

=

+

+

+ +

− − +

L

N

MMMMMMMMMMMMMM

O

Q

PPPPPPPPPPPPPP

M M

IAL

IAL

IAL

IAL

JA

IAL

L IA

IAL

L IA

T

z z

y y

x

y y

z z

21 12

2

2

2

2

16

0 0 0 0 0

0 970

65

0 0 0 13L420 10

0 0 970

65

0 13L420 10

0

0 0 06

0 0

0 0 13L420 10

0140 30

0

0 13L420 10

0 0 0140 30

=

− −

− − +

− − −

− + − −

L

N

MMMMMMMMMMMMMM

O

Q

PPPPPPPPPPPPPP

The lumped mass matrix contains terms only the following terms,

M AL AL AL112

2 22

3 32

, , ,a f a f a f= = =ρ ρ ρ M M


10

M J L I L I Lx y z4 42

52

6 62

, ,5 ,a f a f a f= = =ρ ρ ρ M M

7.2.1 2-D Straight Beam (BEAM) Formulation This element is a 2-D, 2-noded straight beam formulated by superimposing the bending, shear and axial behaviour derived directly from the differential equations for beam displacements, used in engineering beam theory.

The nodal degrees of freedom are (fig.7.2.1-1)

U, V and θz at each node

The displacement variations along the length of the beam are linear axial, linear rotation and cubic transverse displacements. The stress resultant variations are constant axial force, linear moments and linear shear forces.

The nodal forces due to the thermal strains are assumed to be constant within each element, and are evaluated explicitly using

FM

EA T

EI Tdy

x

z

e

zz

eRSTUVW = L

NMOQP

RS|

T|

UV|

W|

α

α

Δ

Δ

a f

where ( )ΔT e and ( )ΔT dz e are average element values.

See [P1] for further element details.

Evaluation of stresses/forces The element output obtained at the nodes consists of

Fx , Fy and Mz - +ve forces and moments are in the directions of the positive local Cartesian system.

The forces are output in the local Cartesian system which is defined as having its local x-axis along the element axis in the direction in which the element nodes are specified. The local y and z-axes form a right-hand set with the x-axis, such that the y-axis lies in the global XY plane, and the z-axis is parallel to the global Z-axis (up out of page) (fig.7.2.1-3).

The nodal forces F are evaluated directly using

F Ka=

in the local Cartesian system.

Beam Elements

11

The local Cartesian forces may also be output at eleven equally spaced points along the beam. These values are evaluated by combining the nodal values with the local element forces and moments calculated explicitly.

Nonlinear formulation The element does not possess any nonlinear capability, but may be utilised in a nonlinear environment. The element cannot be employed for linear buckling analyses.

Y

X

V

U

U

V

2

1

θz

θz

Fig.7.2.1-1 Nodal Freedoms For BEAM Element

Load

Load

Plane FrameCantilever Beam

Fig.7.2.1-2 Examples Illustrating Use Of Beam Elements


12

Y

X

y

x 2

1

Fig.7.2.1-3 Local Cartesian System For BEAM Element

7.2.2 2-D Straight Grillage (GRIL) Formulation This element is a 2-D, 2-noded straight beam formulated by superimposing the bending, shear, and torsional behaviour derived directly from the differential equations for beam displacements used in engineering beam theory.


W, and x yθ θ

The displacement variations along the length of the beam are linear axial, linear rotation and cubic transverse displacements. The stress resultant variations are constant axial, and linear moment and linear shear.


M EI Tdzyy

e

y = LNMOQPα

Δ

where ( )ΔT dz e is the average element value.



Fz , Mx and My +ve forces and moments are in the directions of the positive local Cartesian system.

Beam Elements

13

The forces are output in the local Cartesian system which is defined as having its local x-axis along the element axis in the direction in which the element nodes are specified. The local y and z-axes form a right-hand set with the x-axis, such that the y-axis lies in the global XY plane, and the z-axis is parallel to the global Z-axis (up out of page) (fig.7.2.2-3).


F Ka=


The local Cartesian forces may also be output at eleven equally spaced points along the bar. These values are evaluated by combining the nodal values with the local element forces and moments calculated explicitly.

Nonlinear formulation The element does not possess any nonlinear capability, but may be utilised in a nonlinear environment.

The element cannot be employed for linear buckling analyses.

Y

X

Z

θy

θy

w 1

θx

θx

w

Fig.7.2.2-1 Nodal Freedoms For GRIL Element


14

Y

X

Z

Problem Defintion Finite Element Mesh

Y

X

PointLoad

Fig.7.2.2-2 Example Illustrating Use Of GRIL Elements

Y

X

1

2

y

x

Fig.7.2.2-3 Local Cartesian System For GRIL Element

7.2.3 2-D Ribbed Plate Beam (BRP2) Formulation This element is a 2-D, 2-noded, straight eccentric beam formulated by superimposing the bending, shear, torsional and axial behaviour derived directly from the differential equations for beam displacements used in engineering beam theory.


U X Y, V, W, and θ θ at each node

The displacement variations along the length of the beam are linear axial, linear rotation and cubic transverse displacements. The stress resultant variations are constant axial, linear moment and linear shear.


Beam Elements

15

FM

EA T

EI Tdz

x

y

e

yy

eRSTUVW

= LNMOQP

RS|T|

UV|W|

α

α

ΔΔa f

where (ΔT) and (ΔT/dy) are average element values.



Fx , Fy , Fz , Mx , My +ve forces and moments are in the directions of the positive local cartesian system.

The forces are output in the local Cartesian system which is defined as having its local x-axis along the element axis in the direction in which the element nodes are specified. The local y and z-axes form a right-hand set with the x-axis, such that the y-axis lies in the global XY-plane, and the z-axis is parallel to the global Z-axis (up out of page) (fig.7.2.3-3).


F Ka=


The local Cartesian forces may also be output at eleven equally spaced points along the beam. These values are evaluated by combining the nodal values with the local element forces and moments calculated explicitly.


The element cannot be employed for linear buckling analysis.


16

Y

X

Z

θy

θy

W 1

θx

θx

W

V

U

U

V 2

Fig.7.2.3-1 Nodal Freedoms For BRP2 Element

Y

X

Z

RPI4 elements

Problem Definition Finite Element Mesh

Y

X

BRP2 elements

Fig.7.2.3-2 Ribbed Plate Illustrating Use Of BRP2 Element

Beam Elements

17

Y

X

1

xy

2

Fig.7.2.3-3 Local Cartesian System For BRP2 Element

7.2.4 3-D Straight Beam (BMS3) Formulation This element is a 3-D two noded straight beam formulated by superimposing the bending, shear, torsional and axial behaviour derived directly from the differential equations for beam displacements used in engineering beam theory.


U X Y, , , ,V W θ θ and θZ at each node

The displacement variations along the length of the beam are linear axial, linear rotation and cubic transverse displacements. The stress resultant variations are constant axial, constant torsion and linear moment and linear shear.


FMM

EA T

EI Tdz

EI Tdy

x

y

z

e

yy

e

zz

e

RS|T|

UV|W|

= LNMOQP

LNMOQP

R

S

||||

T

||||

U

V

||||

W

||||

α

α

α

ΔΔ

Δ

a f

where ( )ΔT e and ( )ΔT dz e are average element values.



18


F , F , Fx y z Forces in the local Cartesian system.

M , M , Mx y z Moments in the local Cartesian system.

The local x-axis lies along the element axis in the direction in which the element nodes are specified. The local xy-plane is defined by the third element node and the element x-axis. The local y and z-axes form a right-hand set with the local x-axis (fig.7.2.4-3).


F Ka=


The local Cartesian forces may also be output at eleven equally spaced points along the bar. These values are evaluated by combining the nodal values with the local element forces and moments calculated explicitly.


The element cannot be utilised for linear buckling analysis.

Y

X

Z

θY

θY

1

θX

θX

WU

V

W

θZ

θZ

2

Fig.7.2.4-1 Nodal Freedoms For BMS3 Element

Beam Elements

19

Y

X

Z

1

2

3

x

y

z

Fig.7.2.4-2 Local Cartesian System For BMS3 Element


20

(a) 3-D Frame Structure

(a) 3-D Frame Structure

Fig.7.2.4-3 Examples Illustrating The Use Of BMS3 Elements

Beam Elements

21

7.2.5 2-D Curved Thin Beam (BM3, BMX3) Formulation The BM3 and BMX3 elements are thin, curved, non-conforming beam elements formulated using the constraint technique.

The global displacements and rotations are initially quadratic and are interpolated independently using linear Lagrangian shape functions for the end nodes and a hierarchical quadratic function for the central node. Therefore, the initial degrees of freedom are (fig.7.2.5-1)

U, V, j at the end nodes

Δu, Δv, Δj at the mid-length node.

The Kirchhoff condition of zero shear strain is applied at the two integration points, by forcing

∂∂

∂∂

∂∂

θvx

uz

vx z+ = − = 0

and eliminating the local transverse translational and rotational degrees of freedom at the central node. The final degrees of freedom for the element are (fig.7.2.5-1)

U, V, zθ at the end nodes,

Δu at the mid-length node

where Δu is the local axial relative (departure from linearity) displacement.

The infinitesimal strain-displacement relationship is defined in the local Cartesian system as

∈ =xux

∂∂

ψ∂∂z

vx

= −2

2

The elastic rigidity (resultant modulus) and modulus matrices are defined as

Explicit $DEA EIEI EI

z

z zz=LNM

OQP

Numerically Integrated DEb Eyb

Eyb Ey bh=LNM

OQPz 2 dy

The thermal strain vector is defined as


22

ψαΔ

αα

0e j a ft

T

d Tdy

T ddT

=

+LNMOQP

R

S||

T||

U

V||

W||Δ

Δ

A complete description of the element formulation is given in [M1,S1]. The consistent and lumped mass matrices are evaluated using the procedures defined in (section 2.7).

Evaluation of stresses/forces The element output obtained at the nodes or Gauss points consists of

Fx - axial force (+ve tension) Mz - moment ex - axial strain ψz - flexural strain

The forces and strains are output in the local x-axis which lies along the element axis in the direction in which the element nodes are specified. The local y and z-axes form a right-hand set with the x-axis, such that the y-axis lies in the global XY plane, and the z-axis is parallel to the global Z-axis (up out of page) (fig.7.2.5-6).

Note. The moments are +ve for tension in the top fibre of the element (hogging). The the fibre lies on the +ve local y side of the element.

Force and stress output may be obtained at either the nodes or element Gauss points. Greatest accuracy is obtained at the Gauss points.

Three options for interpreting the forces and moments within an element are available

The axial force and moment are computed at the two Gauss points using numerical integration. The true nodal moments for a beam element between supports is then obtained by adding the fixed end moments to the end node values, and the sagging moment to the mid-node value (fig.7.2.5-7). This is the default technique and must be used for nonlinear analyses. The axial force and moment are computed at the two end nodes by using

F T KaendT=

where T is the global-local transformation matrix. The values at the centre point are then interpolated from these end values and the values at the Gauss points assuming a cubic variation (fig.7.2.5-7). This method can only be used for linear analyses and is invoked via OPTION 136.

Beam Elements

23

This method is similar to (b) except that the stress resultants at the centre node are also computed by considering equilibrium and is invoked via OPTION 137.

Nonlinear formulation The beam elements can be employed in


material laws specified in 1. Nonlinear dynamics utilizing the nonlinear material laws specified in 1. Linear eigen-buckling analysis.

Notes

BM3 and BMX3 may be used in conjunction with the stress resultant plasticity model (section 4.2). BMX3 may be used with the concrete model and continuum-based plasticity models (section 4.2). The geometric nonlinearity may be either A Total Lagrangian formulation which accounts for large displacements but

small strains. The nonlinear strain-displacement relationship is defined by


OQPx

ux

ux

vx

∂∂

∂∂

∂∂

12

12

2 2

ψ∂∂

∂∂

∂∂

∂∂

∂∂z

vx

ux

vx

vx

ux

= − − +2

2

2

2

2

2

with reference to the local element x-axis. The force and strain output with the geometrically nonlinear analysis will be

the 2nd Piola-Kirchhoff stress resultants and Green-Lagrange strains respectively, referred to the undeformed configuration. The loading is conservative. An Updated Lagrangian formulation takes account of large displacements and

large rotations but small strains, provided that the rotations are small within a load increment. The output approximates to the true Cauchy stress resultants and logarithmic strains. The loading approximates to being non-conservative. The initial assumptions used in deriving the BM3 and BMX3 elements limit

the rotations to one radian in a Total Lagrangian analysis and rotation increments of one radian in an Updated Lagrangian analysis (section 3.5). The BMX3 elements are valid for rotations (TL) or rotation increments (UL)

greater than one radian. As rotations become large, ∂ ∂u x/ may no longer be interpreted as axial strain. The axial force distribution from a simple problem is given in fig.7.2.5-8.


24

Y

X

1 U

V

θZ

V

U

Δθ Z

θZ

3

2ΔU

ΔV

Initial Variables

Y

X

1 U

V

θZ

V

U

ΔU

θZ

3

2

Final Variables

Fig.7.2.5-1 Nodal Freedoms For BM3 And BMX3 Elements

Quadrature pointscoincide withframe joints

Quadrature Points

Fig.7.2.5-2 Portal Frame Showing Locations Of Quadrature Points With A 3-Point Newton-Cotes Rule

Beam Elements

25

Y

Z

1

2 3

4

Fig.7.2.5-3 Local Cartesian Axes For Cross-Section

Y

Z

1

2 3

4

1

2 3

4

1

2 3

4

Element 1 orQuadrilateral 1



Fig.7.2.5-4 Cross-Section Of I-Beam Represented By Superimposing Three BMX3 Elements Or By Defining Three Quadrilaterals


26

Y

Z

3-Point Newton-Coates

Y

Z

5-Point Newton-Coates

Fig.7.2.5-5 Quadrature Rules For Cross-Section Integration

3

1

2

X

Y

yx

y

y

xx

Fig.7.2.5-6 Local Cartesian System For BM3 And BMX3 Elements

Beam Elements

27

UDL

Support

wl/24 wl/12

wl/12

(a) Adding Fixed End Moments

True moment distributionLUSAS + fixed end moments

Values evaluated atGauss points and

extapolated to nodes

(b) Cubic Fit Through Gauss and Nodal Values

Nodal values computeddirectly from F = K a

Gauss points values

(c) Quadratic Fit Through Nodal and Mid-length Values

Mid-point momentevaluated using

equilibrium

Nodal values computeddirectly from F = K a

Fig.7.2.5-7 Interpretation Of Results Obtained Using BM3 And BMX3 Elements


28

Load

(a) Problem Definition

BM3

(b) Axial Force Distribution

Axi

al F

orce

Fig.7.2.5-8 Axial Force Distributions Obtained For A Geometrically Nonlinear Analysis Of A Cantilever Beam

Beam Elements

29

7.2.6 3-D Curved Thin Beam (BS3, BS4, BSX4) Formulation The BS3, BS4, and BSX4 elements are 3-D thin, curved, non-conforming beam elements formulated using the constraint technique.

The global displacements and rotations are initially quadratic and are independently interpolated using linear Lagrangian shape functions for the end nodes and a hierarchical quadratic function for the central node. This provides C(0) continuity of the in-plane displacement. The initial freedoms are (fig.7.2.6-1)

U X Y Z, , , , ,V W θ θ θ at the end nodes

Δ Δ Δ Δθ Δθ ΔθU V W X Y Z, , , , , at the mid-side node

The Kirchhoff condition of zero shear strain is applied at the two integration points, by forcing

∂∂

∂∂

∂∂

θvx

uy

vx z+ = − = 0

∂∂

∂∂

∂∂

θwx

uz

wx y+ = + = 0

and eliminating the local transverse translational and bending rotational freedoms at the central node. The final degrees of freedom for the element are (fig.7.2.6-1)

U X Y Z, , , , ,V W θ θ θ at the end nodes

Δu and ΔθX at the mid-side node

where Δu and ΔθX are the local relative (departure from linearity) axial displacement and torsional rotation of the central node.

The infinitesimal strain-displacement relationship is

∈ =xux

∂∂

ψ∂∂y

ux

= −2

2

ψ∂∂z

vx

= −2

2

ψ∂∂ ∂xy

wx y

= −2


30

ψ∂∂ ∂xz

wx y

= −2

Note. ψ ψ ψxy xz z+ = the total torsional strain

The elastic rigidity (resultant modulus) and modulus matrices are defined as

Explicit

$D

EA EI EIEI EI EIEI EI EI

GIGI

GA

y z

y yy yz

z yz zz

yy

zz

=

L

N

MMMMMMMM

O

Q

PPPPPPPP

0 0 00 0 00 0 0

0 0 0 0 00 0 0 0 00 0 0 0 0

Alternatively, if Kt has a non-zero value in the element geometric properties data section, the resultant torsional moduli GIyy and GIzz are replaced with GKt / 2 where Kt is a torsional constant (typically, for circular cross-sections K Jt = , the polar second moment of area).

Numerically integrated

$D

E Ey EzEy Ey EyzEz Eyz Ez

GyGz

G

h b=

L

N

MMMMMMMM

O

Q

PPPPPPPP

z z0 0 00 0 00 0 0

0 0 0 0 00 0 0 0 00 0 0 0 0

2

2

2

2

dydz


ψ

αΔ

αα

αα

0

00

e j

a f

a ft

Td T

dzT d

dTd T

dyT d

dT=

+LNMOQP

+LNMOQP

R

S

||||

T

||||

U

V

||||

W

||||

ΔΔ

ΔΔ

A description of the element formulation is given in [M2].


Beam Elements

31


Fx axial force My z, M moments Ty, Tz torques ∈x axial strain Ψ Ψy z, flexural strain Ψ Ψxy, xz torsional strain The forces and strains are output in the local Cartesian system which is defined by

BS3 For a curved element the local xy-plane is defined by the three element nodes. Local y is perpendicular to local x and +ve on the convex side of the element. The local y and z-axis form a right-hand set with the local x-axis (Fig.7.2.6-7a). For a straight element parallel to the global X-axis, the local z-axis is given

by the unit vector z = j x x (j is a unit vector along the global Y-axis) (fig.7.2.6-7b) For a straight element not parallel to the global X-axis, the local z-axis is

given the unit vector z = i x x (i is a unit vector along the global X-axis) (fig.7.2.6-7c) The local y-axis forms a right-hand set with the local x and z axes. BS4, BSX4 The local xy-plane is defined by all four element nodes which

are assumed to be coplanar. The local y-axis is perpendicular to the local x-axis and +ve on the side of the element where the fourth node lies. The local y and z-axis form a right-hand set with the local x-axis (fig.7.2.6-6)

Note. The torques are +ve for anti-clockwise rotations at first node and clockwise rotations at third node.



Materially nonlinear analysis utilising the elasto-plastic constitutive laws [O1] (section 4.2). Geometrically nonlinear analysis. Geometrically and materially nonlinear analysis utilising the nonlinear

material laws specified in 1.


32

Nonlinear dynamics utilising the nonlinear material laws specified in 1. Linear eigen-buckling analysis.

Notes

BS3, BS4 and BSX4 may be used in conjunction with the stress resultant plasticity model (section 4.2). BSX4 may be used with the concrete model and continuum based plasticity models (section 4.2). All continuum based nonlinear material models do not consider nonlinear

torsional effects. The geometric nonlinearity utilises a Total Lagrangian formulation which

accounts for large displacements but small strains. The nonlinear strain-displacement relationship is defined by


OQPx

ux

ux

vx

∂∂

∂∂

∂∂

12

12

2 2

ψ∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂y

wx

ux

wx

wx

ux

wy

vx

= − − + +2

2

2

2

2

2

2

2

ψ∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂z

vx

ux

vx

vx

ux

wy

wx

= − − + −2

2

2

2

2

2

2

2

ψ∂∂ ∂

∂∂

∂∂ ∂

∂∂

∂∂xz

wx y

ux

wx y

wx

vx

= − − −2 2 2

2

ψ∂∂ ∂

∂∂

∂∂ ∂

∂∂

∂∂xy

wx y

ux

wx y

vx

vx

= − − −2 2 2

2

γ∂∂

∂∂yz

vx

wx

= −

with reference to the local element x-axis. The force and strain output for a geometrically nonlinear analysis will be 2nd

Piola-Kirchhoff stress resultants and Green-Lagrange strains respectively, referred to the undeformed configuration. The loading is conservative. The initial assumptions in deriving the BS3, BS4 and BSX4 elements limit

the rotations to one radian in a Total Lagrangian (TL) analysis, and rotation increments of one radian in an Updated Lagrangian (UL) analysis (Section 3.5).

Beam Elements

33

Y

X

Z

Δθ Y

θY

1

Δθ X

θX

ΔW

U

V

W

Δθ Z

θZ

2

θY

θXW θZ

3

ΔV

ΔU

V

U

Y

X

Z

θY

1

Δθ X

θX

Δu

U

V

W θZ

2

θY

θXW θZ

3

V

U

Initial Variables Final Variables

Fig.7.2.6-1 Nodal Freedoms For BS3, BS4 And BSX4 Elements

Quadrature pointscoincide withframe joints

Quadrature Points

Fig.7.2.6-2 Portal Frame Showing Locations Of Quadrature Points With A 3-Point Newton-Cotes Rule


34

Y

Z

1

2 3

4


Y

Z

1

2 3

4

1

2 3

4

1

2 3

4




Fig.7.2.6-4 Cross-Section Of An I-Beam Represented By Superimposing Three Bsx4 Elements Or By Defining Three Quadrilaterals

Beam Elements

35

3*3 Newton-Cotes 5*5 Newton-Cotes


Y

X

Z

1

2

3

x

y

z

4

x

yx

y

Fig.7.2.6-6 Local Cartesian System For Bs4 And BSX4 Elements


36

Y

X

Z(a) Curved Element

x

y x

z

x

x-y plane

y

y

z

2

1

3

Y

X

Z


xz

x

yy

31

z

Y

X

Z


x

z

xy

y

3

1

z

2

2

Fig.7.2.6-7 Local Cartesian System For The BS3 Element

Beam Elements

37

7.2.7 Semiloof Thin Beam (BSL3, BSL4, BXL4) Formulation The BSL3,BSL4 and BXL4 elements are 3-D thin, curved beam elements based on the Kirchhoff constraint technique. Their formulation and nodal configuration has been specifically designed to provide an element compatible with the Semiloof shell element QSL8. Initially, the displacements and rotations are interpolated using quadratic and cubic shape functions respectively, where the cubic variation is provided by the rotational degrees of freedoms of the 'loof' nodes, which are located at the quadrature points of the 2 point Gauss rule (fig.7.2.7-1).

Unlike the thick beam formulation presented by Irons [I1], the present formulation utilises Kirchhoff constraints of zero shear strain applied at the 2-point Gauss quadrature locations, by forcing

∂∂

∂∂

∂∂

θvx

uy

vx z+ = − = 0

∂∂

∂∂

∂∂

θwx

uz

wx y+ = + = 0

which provides four constraint equations and permits elimination of the two flexural degrees of freedoms at these positions. The final degrees of freedom for the element are (fig.7.2.7-1)

U z, , V, W, , X yθ θ θ at nodes 1 and 3 U, V, W at node 2

and

θX at nodes 4 and 5 Note. The rotations at the 'loof' nodes are local, but are not relative rotations (departures from linearity) as with the other LUSAS beam elements based on Kirchhoff constraints.

The infinitesimal strain-displacement relationship is

∈ =xux

∂∂

ψ∂∂y

wx

= −2

2

ψ∂∂z

vx

= −2

2


38

ψ∂∂ ∂xy

wx y

= −2

ψ∂∂ ∂xz

wx y

= −2

Note. ψ ψ ψxy xz z+ = the total torsional strain The elastic rigidity (resultant modulus) and modulus matrices are defined as

Explicit

$D

EA EI EIEI EI EIEI EI EI

G I Ae

G I AeGA

y z

y yy yz

z yz zz

yy z

zz y

=+

+

L

N

MMMMMMMM

O

Q

PPPPPPPP

0 0 00 0 00 0 0

0 0 0 0 0

0 0 0 0 00 0 0 0 0

2

2e j

e j

Alternatively if Kt has a non-zero value in the element geometric properties data

section, the resultant torsional moduli GIyy and GIzz are replaced with GKt / 2 where Kt is a torsional constant (typically, for circular cross-sections K Jt = , the polar second moment of area).

Numerically integrated

D

E E EE Ez EyzE Eyz Ey

GzGy

G

h b

z y

z

y=

L

N

MMMMMMMM

O

Q

PPPPPPPP

z z0 0 00 0 00 0 0

0 0 0 0 00 0 0 0 00 0 0 0 0

2

2

2

2

dydz

The thermal strain vector is defined by

ψ

αΔ

αα

αα

0

00

e j

a f

a ft

Td T

dzT d

dTd T

dyT d

dT=

+LNMOQP

+LNMOQP

L

N

MMMMMMMM

O

Q

PPPPPPPP

ΔΔ

ΔΔ

A more detailed description of the element formulation is given in [A1,I1,M1].

Beam Elements

39



Fx axial force My z, M moments Ty z, T torques ∈x axial strain ψ ψy, z flexural strain ψ ψxy, xz torsional strain The forces and strains are output in the local Cartesian system which is defined by

BSL3

For a curved element the local xy-plane is defined by the three element nodes. Local y is perpendicular to local x and +ve on the convex side of the element. The local y and z-axis form a right-hand set with the local x-axis (fig.7.2.7-6a).

For a straight element parallel to the global X-axis, the local z-axis is given by the unit vector z j x= ∗ (j is a unit vector along the global Y-axis) (fig.7.2.7-6b).

For a straight element not parallel to the global X-axis, the local z-axis is given the unit vector z i x= ∗ (i is a unit vector along the global X-axis) (fig.7.2.7-6c).

The local y-axis forms a right-hand set with the local x and z axes.

BSL4

The local xy-plane is defined by all four element nodes which are

BXL4

assumed to be coplanar. The local y-axis is perpendicular to the local x-axis and +ve on the side of the element where the fourth node lies. The local y and z-axis form a right-hand set with the local x-axis (fig.7.2.7-7)

Note. The torques are +ve for anti-clockwise rotations at the first node and clockwise rotations at the third node.



40




Notes

BSL3, BSL4 and BXL4 may be used in conjunction with the stress resultant plasticity model (section 4.2). BXL4 may be used with the concrete model and continuum based plasticity models (section 4.2). All continuum based nonlinear material models ignore nonlinear torsional

effects. The geometric nonlinearity utilises a Total Lagrangian formulation which

accounts for large displacements but small strains. The nonlinear strain-displacement relationship is defined by


OQPx

ux

ux

vx

∂∂

∂∂

∂∂

12

12

2 2

ψ∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂y

wx

ux

wx

wx

ux

wy

vx

= − − + +2

2

2

2

2

2

2

2

ψ∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂z

vx

ux

vx

vx

ux

wy

wx

= − − + −2

2

2

2

2

2

2

2

ψ∂∂ ∂

∂∂

∂∂ ∂

∂∂

∂∂xz

wx y

ux

wx y

wx

vx

= − − −2 2 2

2

ψ∂∂ ∂

∂∂

∂∂ ∂

∂∂

∂∂xy

wx y

ux

wx y

vx

vx

= − − −2 2 2

2

γ∂∂

∂∂yz

vx

wx

= −

with reference to the local element x-axis. The force and strain output with the geometrically nonlinear analysis will be

the 2nd Piola-Kirchhoff stress resultants and Green- Lagrange strains respectively, referred to the undeformed configuration. The loading is conservative.

Beam Elements

41

Y

X

Z

θY

1 θX

W

U

V

W θZ

2

θY

θXW θZ

3

V

U

V

U

Initial Variables Final Variables

θY

4 θXθZ

θY

5 θXθZ

Y

X

Z

θY

1 θX

W

U

V

W θZ

2

θY

θXW θZ

3

V

U

V

U

4θX

5 θX

Fig.7.2.7-1 Nodal Freedoms For BSL3, BSL4 And BXL4 Elements


QSL8 elements

BSL3 elements

Fig.7.2.7-2 Stiffened Shell Illustrating Use Of BSL3 Element


42

Y

Z

1

2 3

4


Y

Z

1

2 3

4

1

2 3

4

1

2 3

4




Fig.7.2.7-4 Cross-Section Of An I-Beam Represented By Superimposing Three BSL4 Elements Or By Defining Three Quadrilaterals

Beam Elements

43

3*3 Newton-Cotes 5*5 Newton-Cotes



44

Y

X

Z(a) Curved Element

x

y x

z

x

x-y plane

y

y

z

2

1

3

Y

X

Z


xz

x

yy

31

z

Y

X

Z


x

z

xy

y

3

1

z

2

2

Fig.7.2.7-6 Local Cartesian System For BSL3 Element

Beam Elements

45

Y

X

Z

1

2

3

x

y

z

4

x

yx

y

Fig.7.2.7-7 Local Cartesian System For BSL4 And BXL4 Elements

7.2.8 3-D Straight Beam (BTS3) Formulation This element is a 3-D, two noded, straight beam formulated using Timoshenko beam theory so that shear deformations are accounted for. In essence, this element is formulated in a very straight forward manner, using linear shape functions and standard degrees of freedom. The complexities in this formulation arise in the consistent derivation of the geometric tangent stiffness and in the treatment of the rotational degrees of freedom. The nodal degrees of freedom for BTS3 are identical to those of the BMS3 element, (fig.7.2.8-1). End releases may be applied to all the nodal freedoms, see section 7.2.8.4.

The nodal degrees of freedom are

U Y Z, V, W, , ,Xθ θ θ at each node.

All displacement and rotation variations along the length of the element are linear while all internal forces and moments are constant.

Consistent and lumped mass matrices are available which are evaluated using the procedures defined in section 2.7.1.

Evaluation of stresses/forces The element output consists of


46

Fx y z, F , F - Forces in the local Cartesian system

Mx y z, M , M - Moments in the local Cartesian system

Element strains and curvatures are also available but nodal values are not output. The local x-axis lies along the element axis in the direction in which the element nodes are specified. The local xy-plane is defined by the third element node and the element x-axis. The local y and z-axes form a right-hand set with the local x-axis. These axes are consistent with those of the BMS3 element (fig.7.2.8-2).

The formulation is such that engineering strain measures are used in both linear and geometrically nonlinear applications. These strains always relate to a local Cartesian system.

The internal forces are computed using:

P D= ∈

Where P are the local internal forces, ∈ are the local strains and curvatures and D is the modulus matrix given by (terms not shown are zero)

D

EA EA eGA

GAG J A e

EA e E I A eEI

xx xx z

sy

sz

xx xx z

xx z yy xx z

zz

= +

+

L

N

MMMMMMMM

O

Q

PPPPPPPP

2

2e j

e j

Nonlinear formulation This element can be employed in

Materially nonlinear analysis utilising the stress resultant plasticity model (section 4.2). Geometrically nonlinear analysis. Geometrically and materially nonlinear analysis utilising the nonlinear


Geometric nonlinearity is accounted for using a co-rotational formulation. In this approach local strains are computed in a local Cartesian frame which is 'fixed' to the element and follows the element as it rotates in 3-D space. Total local strains are computed using the current configuration and local frame. In other words, the strains computed at the end of one increment do not depend on the strains computed at the

Beam Elements

47

end of a previous increment. Solutions obtained using this element will not be load step size dependant. The local strains for the element are given by

∈ = ∈xT

, , , , , X X Xy z x y zγ γo t

where:

∈ =x u lo/

γ θ θy = − +2 5 2e i

γ θ θz = − +3 6 2e i

Xx = −θ θ4 1 10e i

Xy = −θ θ2 5 10e i

Xz = −θ θ3 6 10e i

Where, lo is the initial element length, u is the axial stretch measured in the co-rotated frame, and θi , i=1,6 are the local gradients at the nodes or 'curvature producing' rotations relative to the co-rotated frame, (fig. 7.2.8-3).

The current local gradients at the nodes are computed from

2 1 2 3 3 2θ = −t e t eT T

2 2 1 2 2 1θ = −t e t eT T

2 3 1 3 3 1θ = −t e t eT T

2 4 2 3 3 2θ = −q e q eT T

2 5 1 2 2 1θ = −q e q eT T

2 6 1 3 3 1θ = −q e q eT T

Where ei are unit vectors defining the co-rotated base frame and ti , qi, i=1,3 are the

cartesian sets at nodes 1 and 2 respectively defining the orientation of the beam cross-section. These expressions may be thought of as being a means of computing an 'average' value for a local gradient at a node. This is easily visualised in two dimensions where, for example, t e t eT T

1 2 2 1= − .This is not true for the three dimensional case and a stricter derivation of the expressions for local gradients would involve the polar decomposition theorem. The approach then taken is to decompose


48

the total rotation into a rigid body component and a local gradient. The expressions described above are the result of applying these principles. The variation of these expressions is used in the virtual work equation to relate variations in local strains to variations in global nodal displacements.

The local frame ei , i=1,3 is easily established for a two dimensional problem. In three dimensions defining the local frame is more difficult. The approach described by Crisfield [C7] has been used for this purpose. The following expressions are used for defining e2 and e3 .

e rr e

e rT

2 22 1

1 12= − +n s

e rr e

e rT

3 33 1

1 12= − +n s

The local frame is established at the centre of the element and the vectors ri , i=1,3 represent the 'average' of the nodal cartesian sets, ti and q

i. These expressions are

approximations to the exact expressions for defining the 'smallest' rotation between vectors r1 and e1. These expressions have been used with a view to obtaining less costly derivatives in a consistent derivation of the tangent stiffness. This lack of orthogonality has been shown to be 0.25 degrees for a local gradient of 15 degrees and 1.9 degrees for a gradient of 30 degrees [C7]. As these values actually represent the 'curvature producing' rotations in a single element the deformation would need to be very severe to reach these values.

The axial stretch may be taken as

u lo= −ln

where ln is taken as the current element length (or chord length). By defining the vectors

x x x21 2 1= −

d d d21 2 1= −

and then by rearranging Pythagoras's theorem this may be expressed as

ulo

x d dT

=+

+RSTUVW

2 1221 21 21ln

The vector x defines the nodes in the initial configuration while d21 is the 'net' translational displacement vector, (fig. 7.2.8-4). The variation of this expression reduces to

Beam Elements

49

δ δu e dT= 1 21

Differentiation of the above equations relating to axial stretch and local gradients allows virtual variations of local strains to be related to virtual variations in global nodal displacements via a strain displacement matrix B

δ δ∈= B a

where a are the global nodal displacements. Using this expression in the virtual work equation allows global internal forces at the nodes to be expressed in terms of local internal forces as

P B PT=

The out of balance force vector is then given by

ψ = −B P RT

where R is the applied nodal loading.

The variation of this equation gives the tangent stiffness matrix. Assuming conservative loading this gives

d B dP dB PT Tψ = +

d B DBda dB PT Tψ = +

The first term on the right hand side of this equation may be recognised as the material or standard linear stiffness matrix. The second term gives rise to the geometric stiffness.

Notes

This geometrically nonlinear formulation is consistently formulated and displays a quadratic rate of convergence in the limit. A consequence of this consistency is the ability of the element to cope with larger load increments. The total strains are computed from the current configuration and local frame

only. Therefore, results obtained using this element are not load step size dependant. This element incorporates rotational degrees of freedom. As explained in

section 3.5, large rotations in three dimensions are non-vectorial in nature and therefore may not be summed as vectors. To overcome this problem the rotation variables are never added to establish the current orientation of the element. A set of Cartesian axes are established at each node to define the orientation of the beam cross section. These axes, which are used in the computation of internal forces and the stiffness matrix, are updated correctly


50

using the iterative increments in nodal rotations, section 3.5. The procedure for this operation is outlined as Extract the Euler parameters from the initial nodal Cartesian set. Form an incremental Euler parameter from the iterative rotation increment. Update the Euler parameter by manipulating the previous and incremental

parameter using quarternion algebra. Form the updated Cartesian set from the updated Euler parameter. In view of the non-vectorial nature of these rotations it should be noted that

the nodal rotation output represents approximate values which should be treated with caution. However, the translational displacements and internal force output will be correct for problems involving arbitrary large nodal rotations. A nonsymmetric stiffness matrix will result if a follower load is specified.

This arises as additional terms are added to the stiffness matrix to account for the variation in the load direction between iterations. Full details of the derivation of these additional terms are given in [C6]. A more detailed derivation of this element formulation may be found in [C6]

and [C8]. End releases Many structures which are modelled with three-dimensional beam elements require joints at the nodes which follow the axes of the rotating system. Examples include deployable space structures, robots and rotating machinery.

Prismatic (sliding), revolute (hinges), spherical and cylindrical joints can be modelled by releasing the appropriate degrees of freedom at a node. These freedoms relate to the local beam axes and a master-slave procedure has been adopted to model the release [J2]. At present, this facility is restricted to static analyses.

Consider a node which is initially shared by a number of elements, one of which is not fully connected to the others. In the deformed configuration the node is no longer completely shared and from (fig.7.2.8-5) the following relationships can be established:

d dm= + ρ

Q Q Qm

= *

where dm and Qm

define the displacement vector and rotation matrix of the master

node, and d and Q define the displacement vector and rotation matrix of the

disconnected (at least partially) slave node. It should be noted that, following conventional beam theory assumptions, the origins of the vectors dm and d coincide, with the gap in (fig.7.2.8-5) drawn for illustrative purposes only.

Beam Elements

51

The columns of the rotation matrices Qm

and Q consist of orthonormal base vectors

qm1

, qm2

, qm3

and q1, q

2, q

3 (fig.7.2.8-5):

Q qm m

=1, q , q

m2 m3

Q q=1, q , q

2 3

The rotation matrix Q* is the matrix that defines the rotation of the master triad Qm

,

on to the slave triad Q.

When modelling different types of joints, the master variables, dm and Qm

, are

generally not entirely independent from the slave variables, d and Q. Depending on

the type of joint, some of the components of the displacement vectors, dm and d , and/or parameters of the rotation matrices Q

m and Q, can be the same. Different

types of joints are defined by releasing displacements and/or rotations around chosen axes. In a geometrically nonlinear analysis these axes rotate together with the structure. For translational joints, the 'difference vector', ρ (with local components), between the master and slave variables is, when transformed into coordinates defined by the master triad, equal to the vector of released displacements (fig.7.2.8-6):

s QmT= ρ

where the vector of released displacements, s , has zero components in non-released directions. In a similar manner, if the rotational pseudovector β* , is extracted from the 'rotation difference matrix', Q* and it is transformed to the master triad, the rotational

pseudovector of released rotations is obtained (fig.7.2.8-7):

ψ β= QmT *

Using these equations a relationship can be established between the variations of the master, slave and released freedoms. This relationship can then be used to derive a modified stiffness matrix and internal force vector which accounts for any released freedoms. Full details of this derivation can be found in [J2] while (fig.7.2.8-6) and (fig.7.2.8-7) illustrate a prismatic (sliding) and revolute (hinge) release.


52

Y

X

Z

θY

θY

1

θX

θX

WU

V

W

θZ

θZ

2

Fig.7.2.8-1 Nodal Freedoms For BTS3 Element

Y

X

Z

1

2

3

x

y

z

Fig.7.2.8-2 Local Cartesian System For BTS3 Element

Beam Elements

53

Y

X

Z

t2

e2

e1

e3

t3

t1

q2

q1

q3

θ3

θ1

θ2

θ6 θ4

θ5

Fig.7.2.8-3 Local Gradients BTS3 Element

lnFinal Configuration

Initial Configuration

lo

d2

d1

x1

x2Y

XZ

x21

Fig.7.2.8-4 Axial Stretch BTS3 Element


54

e3

e2

e1

qm1

qm2

qm3

q3

q2

q1

dm

d

ρ

Fig.7.2.8-5 General Displacements At A Node With Released Freedoms

e3

e2

e1

qm1

qm2

qm3

ρ

S

Fig.7.2.8-6 Prismatic (Sliding) Release

Two-Dimensional Continuum Elements

55

qm3= q1

q2

qm3

ϕ

qm2

q3

e3

e2

e1

Fig.7.2.8-7 Revolute (Hinged) Release

7.3 Two-Dimensional Continuum Elements 7.3.1 Standard Isoparametric Elements Isoparametric finite elements utilise the same shape functions to interpolate both the displacements and geometry, i.e.

displacement U N Uii

n

i==∑ ξ η,b g

1

geometry X ==∑N Xi ii

nξ η,b g

1

where Ni ξ η,b g is the element shape function for node i and n is the number of nodes. Fig.7.3.1-1 shows the nodal configurations available within LUSAS.

The nodal degrees of freedom are U and V.

All the isoparametric elements described in this section must be defined using only X and Y coordinates. For 3-D plane membrane elements see section 7.5 on space membrane elements A complete description of their formulation is given in [H1,B2].


Plane stress (QPM4, QPM8, TPM3, TPM6, QPK8, TPK6) The plane stress elements are formulated by assuming that the variation of out of plane direct stress and shear stresses is zero, i.e.


56

σ σ σz = = =0 0 0, , xz yz

The plane stress elements are suitable for analysing structures which are thin in the out of plane direction, e.g. thin plates subject to in-plane loading (fig.7.3.1-2).

Note. The thickness of the material is defined at each node and may vary over the element.

The infinitesimal strain-displacement relationship is defined as

∈ =XUX

∂∂

∈ =YVY

∂∂

γ∂∂

∂∂XY

UY

VX

= +

The isotropic and orthotropic elastic modulus matrices are

Isotropic D =− −

L

N

MMMM

O

Q

PPPPE

1

1 01 0

0 01

2

2υ

υυ

υe j a f

Orthotropic D =−

−

L

N

MMM

O

Q

PPP

−1 0

1 00 0 1

1/ /

/ //

E EE E

G

x xy x

xy x y

xy

υυ

where υyx has been set to υxy y xE E/ to maintain symmetry.

Note. To obtain a valid material

υxy x yE E< //d i1 2

∈ = − +z x yEυ

σ σd i for isotropic materials

∈ = − −zxz

xx

yz

yyE E

υσ

υσ for orthotropic materials

The thermal strain is defined by


57

isotropic D E=

+ −

−−

−

L

N

MMMM

O

Q

PPPP( )( )

( )( )

( )1 1 2

1 01 0

0 0 1 22

υ υ

υ υυ υ

υ

Orthotropic ( ) ,∈ =0 t x yT

TΔ α α α , xy

Plane strain (QPN4, QPN8, TPN3, TPN6, QNK8, TNK6) The plane strain elements are formulated by assuming that the variation of out of plane direct strain and shear strains is zero, i.e.

∈ = ∈ = ∈ =Z 0 0 0, , YZ XZ

The plane strain elements are suitable for analysing structures which are thick in the out of plane direction, e.g. dams or thick cylinders (fig.7.3.1-3). The infinitesimal strain-displacement relationship is defined as

∈ =XUX

∂∂

∈ =YVY

∂∂

γ∂∂

∂∂XY

UY

VX

= +


Isotropic D E=

+ −

−−

−

L

N

MMMM

O

Q

PPPP( )( )

( )( )

( )1 1 2

1 01 0

0 0 1 22

υ υ

υ υυ υ

υ

Orthotropic D

E EE E

E EE E

E EE E

E EE E

G

z xz x

x z

xy z xz yz y

y z

xy z yz xz x

x z

z yz y

y z

xy

=

− − −

− − −

L

N

MMMMMMMM

O

Q

PPPPPPPP

−υ υ υ υ

υ υ υ υ

2

2

1

0

0

0 0 1

where for symmetry

E E E E E Ey xy z yz xz x x xy z xz yz yυ υ υ υ υ υ+ = +d i d i



58

υ υ υxy x y xz x z yz y zE E E E E E< < </ / // / /d i b g d i1 2 1 2 1 2

σ υ σ σz X Y= +b g for isotropic materials

σ υ υz xzz

Xyz

z

y

EE

EE

= + for orthotropic materials


Isotropic ∈ = +0 1 0d itTT( ) ,υ α αΔ ,

Orthotropic ∈ =

+L

N

MMMM

O

Q

PPPP0d it

z

xxz z x

yz z

T

T

EE

Δ

υ α α

υ α α α

,

EE

+ , z

yy xy

Axisymmetric (QAX4, QAX8, TAX3, TAX6, QXK8, TXK6) The axisymmetric elements are formulated by assuming that the variation of out of plane shear stresses is negligible, i.e.

σ σXZ = =0 0, YZ

and the out of plane direct strain is defined as

∈ =ZUR

where R is the distance from the axis of symmetry.

The axisymmetric elements are suitable for analysing solid structures which exhibit geometric symmetry about a given axis, e.g. thick cylinders or circular plates (fig.7.3.1-4).

The elements are defined in the XY-plane and symmetry can be specified about either the X or Y axes. The infinitesimal strain-displacement relationship is defined as:

∈ =XUX

∂∂

∈ =YVY

∂∂

γ∂∂

∂∂XY

UY

VX

= +

∈ =ZUR

symmetry about the Y axis


59

or ∈ =ZVR

symmetry about the X axis

The isotropic and orthotropic linear elastic modulus matrices are defined as

Isotropic D E=

− −

−−

−−

L

N

MMMM

O

Q

PPPP( )( )

( )( )

( ) /( )

1 1 2

1 01 0

0 0 1 2 2 00 1

υ υ

υ υ υυ υ υ

υυ υ υ

Orthotropic D

E E EE E E

GE E E

x yx y zx z

xy x y zy z

xy

xz x yz y z

=

− −− −

− −

L

N

MMMMM

O

Q

PPPPP

−1 01 0

0 0 1 00 1

1/ / // / /

// / /

υ υυ υ

υ υ

where υ υ υyx, and zx zy are defined by

υ υyx xy y xE E= / υ υzx xz z x= E / E υ υzy = yz z yE E/

to maintain symmetry.


υxy x yE E< /d i1/2 υxz x z

1/2< E / Eb g υyz y zyE E< //d i1 2


Isotropic ∈ =0 0d itTTΔ α α α, , ,

Orthotropic ∈ =0d it x yT

TΔ α α α α, , , xy z

Evaluation of stresses The element output obtained at the element nodes and Gauss points consists of

Stress Output σ σ σ σX, , , Y XY Z the direct and shear stresses

σ σmax, min the maximum and minimum principal stresses β the angle between the maximum principal stress and the

positive X-axis σS the maximum shear stress σV von Mises equivalent stress

Strain Output ∈ ∈ ∈X, , , Y XY zγ the direct and shear strains


60

∈ ∈max, min the maximum and minimum principal strains β the angle between the maximum principal strain and the

positive X-axis ∈S the maximum shear strain ∈V von Mises equivalent strain

Stress resultant output, which accounts for the thickness of the element, is available as an alternative to stress output for the plane stress elements, i.e

Stress Resultant Output NX, , ,N N NY XY z the direct and shear stress resultants/unit length

Nmax, Nmin the maximum and minimum principal stress resultants/unit length

β the angle between the maximum principal stress resultant and the positive X-axis

NS the maximum shear stress resultant/unit length NV von Mises equivalent stress resultant/unit length

The sign convention for stress, stress resultants and strain output is shown in fig.7.3.1-6.

The Gauss point stresses are usually more accurate than the nodal values. The nodal values of stress and strain are obtained using the extrapolation procedures detailed in section 6.1.

Nonlinear formulation The 2-D isoparametric elements can be employed in:-

(Materially nonlinear analysis, utilising the elasto-plastic constitutive laws [O1] (section 4.2) and the concrete model (section.4.3) Geometrically nonlinear analysis. Geometrically and materially nonlinear analysis utilising the nonlinear


Notes

The plane stress elements can be used with the nonlinear concrete model (section 4.3). The plane stress and plane strain elements may be used with the nonlinear

interface model (section 4.2). The geometric nonlinearity may utilize: A Total Lagrangian formulation which accounts for large displacements but



61

Plane stress


OQPX

UX

UX

VX

∂∂

∂∂

∂∂

12

12

2 2


OQPY

VX

UY

VY

∂∂

∂∂

∂∂

12

12

2 2

γ∂∂

∂∂

∂∂

∂∂

∂∂

∂∂XY

UY

VX

UX

UY

VX

VY

= + + +

Plane Strain


OQPX

UX

UX

VX

∂∂

∂∂

∂∂

12

12

2 2


OQPY

VX

UY

VY

∂∂

∂∂

∂∂

12

12

2 2

γ∂∂

∂∂

∂∂

∂∂

∂∂

∂∂XY

UY

VX

UX

UY

VX

VY

= + + +

Axisymmetric


OQPX

UX

UX

VX

∂∂

∂∂

∂∂

12

12

2 2


OQPY

VX

UY

VY

∂∂

∂∂

∂∂

12

12

2 2

γ∂∂

∂∂

∂∂

∂∂

∂∂

∂∂XY

UY

VX

UX

UY

VX

VY

= + + +

∈ = + LNMOQPZ

UR

UR

12

2 symmetry about the Y axis

or ∈ = + LNMOQPZ

VR

VR

12

2 symmetry about the X axis

The output is now in terms of the 2nd Piola-Kirchhoff stresses and Green-Lagrange strains referred to the undeformed configuration. The loading is conservative.

An Updated Lagrangian formulation, which takes account of large displacements and moderately large strains provided that the strain increments are small. The output is now in terms of the true Cauchy stresses and the strains approximate to logarithmic strains. The loading approximates to being non-conservative.


62

An Eulerian formulation, which takes account of large displacements and large strains. The output is in terms of true Cauchy stresses and the strains approximate to logarithmic strains. The loading is non-conservative.

3 node triangle 6 node triangle

4 node quadrilateral 8 node quadrilateral

12

3

1 23

4

5

6

12

34

12 3

4

567

8

Fig.7.3.1-1 Nodal Configuration For Standard 2-D Isoparametric


63


(a) Plate subject to Inplane Loading


(b) Cantilever subject to a Point Loading

Fig.7.3.1-2 Examples Illustrating The Use Of Plane Stress Elements


64

(a) Embankment Dam


(b) Thick Cylinder

Problem Definition

Finite Element Mesh

QPN8 elements

TPN6 elements

QPN4 elements

Fig.7.3.1-3 Examples Illustrating The Use Of Plane Strain


65

(a) Thick Cylinder


(b) Circular Plate

Finite Element Mesh

r

r

QAX4 elements

Problem Definition

QAX8 elements

Fig.7.3.1-4 Examples Illustrating The Use Of Axisymmetric Solid Elements


66

Fig.7.3.1-5 Deformed Mesh Illustrating Formation Of Spurious Mechanisms

Y

X

σY

σX

σY

σX

σXY

σXY

σX, σY +ve tensionσXY +ve into XY quadrant

Fig.7.3.1-6 Sign Convention For Stress/Strain Output

7.3.2 Enhanced Strain Elements (QPM4M, QPN4M, QAX4M)

The lower order enhanced strain elements exhibit improved accuracy in coarse meshes when compared with their parent elements QPM4,QPN4 and QAX4, particularly if bending predominates. In addition, these elements do not suffer from 'locking' in the nearly incompressible limit. The elements are based on a three-field mixed formulation [S8] in which stresses, strains and displacements are represented by three independent functions in three separate vector spaces. The formulation is based on the inclusion of an assumed 'enhanced' strain field which is related to internal degrees of freedom. These internal degrees of freedom are eliminated at the element level before assembly of the stiffness matrix for the structure. The formulation provides for the following three conditions to be satisfied


67

Independence of the enhanced and standard strain interpolation functions. L2 orthogonality of the stress and enhanced strains. Capability of the element to model a constant state of stress after enforcing

the orthogonality condition, i.e. requirement for passing the patch test. In addition to ensuring that the element passes the patch test, these conditions also allow the stress field to be eliminated from the formulation.

Formulation The formulation requires that the total strain is expressed as the sum of a 'compatible' strain and an 'enhanced' strain

∈= ∈ + ∈c e

The compatible strain is directly related to the displacements of the element nodes in the standard manner. The enhanced strain is related to internal degrees of freedom which are eliminated using static condensation at the element level. The enhanced strains are therefore discontinuous between elements. The weak form of the three field variational equations for equilibrium, compatibility and constitutive relationship may be expressed as

δ σ δ∈ − =z cT TR a

ΩΩ d 0

δσΩ

Ωz ∈ =T

e d 0

δ σ∂∂

∈ − +∈

LNM

OQP

=zΩ ΩT W d 0

where R is the applied loading, W is the strain energy density, a are nodal displacements and σ is the stress vector.

By enforcing the so called L2 orthogonality condition between stress and enhanced strain, terms involving σT

e∈ will disappear. This allows the stress field to be eliminated from the formulation.

The compatible and enhanced strains are computed from

∈ =c Ba

∈ =e eGα

where G operates on the assumed strain parameters αe to provide the enhanced strains.


68

Substitution of these expressions into the two remaining field variational equations yields

δ α δ δ α∈ + − + ∈ + =z zcT

eT

eT

eD Bda Gd R a d D Bda GdΩ Ω

Ω Ω{ } { } d 0

The following matrices are defined for use in discretising this equation

K B D Bk Ta f = z dΩ

Ω (n nel el* matrix)

H G D Gk Ta f = z dΩ

Ω (m mel el* matrix)

Γ ΩΩ

k TG D Ba f = z d (m nel el* matrix)

nel is the dimension of the element displacement field, mel is the number element enhanced strain modes. D is the modulus matrix at loadstep k.

The internal force vectors are given by

P Bk k T ka f a f a f= z σΩ

Ω d

h Gk k T ka f a f a f= z σΩ

Ω d

where h(k) is the internal force vector relating to the incompatible modes which is subsequently eliminated at the element level.

Using standard finite element techniques for assembling the system of equations gives

K

Ha R P

h

k k T

k k

k

k

k

k

a f a fa f a f

a fa f

a fa f

Γ

ΓΔΔ

LNMM

OQPPRS|T|

UV|W|=

−−

RS|T|UV|W|

+

+

1

1 0α

This nonlinear system of equations is solved using a Newton-Raphson iteration scheme. However, for the linear case, no iterations are necessary as h will be 0 and P will not be considered.

Static condensation of this system of equations eliminates the equations included to enforce the orthogonality condition. The element stiffness and internal forces used to assemble the equations for the structure then become

K ak k ka f a f a fΔ + =1 ψ

where

K K Hk k k T k ka f a f a f a f a f= −−

Γ Γ1


69

ψ k kR Pa f a f= −

P P H hk k k T k ka f a f a f a f a f= −−

Γ1

In nonlinear analyses, the enhanced strain parameters are updated as

α αk k k k k kH a h+ −= − −1 1a f a f a f a f a f a fΓ Δ

The actual implementation of this formulation requires the orthogonality condition to be related to the isoparametric space. Transformations are therefore required to assemble matrices and vectors that relate to covariant strains and contravariant stresses. Standard transformations are applied and full details of this procedure are given in [S8].

It is postulated that the covariant enhanced strain field is given by

∈ =e eEα

where E is the equivalent of G in the isoparametric space.

Enhanced strain interpolation - plane elements (QPM4M, QPN4M) The incompatible displacement field is given by

U N N= +1 1 2 2ξ λ η λb g a f

where

N N12

221

21 1

21ξ ξ η ηb g e j a f e j= − = −,

and λ i represent the incompatible modes

λ λ1 1 2 2= =u uT T, , , v v1 2l q l q

The covariant base vectors associated with the isoparametric space are

gx Ny N

x ay a

x hy h

g gT

T

T

T

T

Tξξ

ξη η=

RS|T|UV|W|

=RS|T|UV|W|

+RS|T|UV|W|

= +,,

1

1 10

gx Ny N

x ay a

x hy h

g gT

T

T

T

T

Tξη

ηξ ξ=

RS|T|

UV|W|

=RS|T|UV|W|

+RS|T|UV|W|

= +,,

2

220

where

N a a a h= + + +0 1 2ξ η ξη


70

and

a T0

14

1 1 1 1=

a T1

14

1 1 1 1= − −

a T2

14

1 1 1 1= − −

h T= − −14

1 1 1 1

x x T= 1 x x x2 3 4

y y T= 1 y y y2 3 4

The initial enhanced strain field in isoparametric space is then given by

∈ =

+

R

S|||

T|||

U

V|||

W|||

i

T

T

T T

u g

u g

u g u g

,

,

, ,

ξ ξ

η η

η ξ ξ η

e jd i

d i e j

=L

NMMM

O

QPPP

−

−

−

−

−

−

R

S

||||

T

||||

U

V

||||

W

||||

=ξ ξη

η ξηξ η ξ η

λ

λ

λ

λ

λ

λ

α0 0 0 0

0 0 0 00 0 2 2

1 10

2 20

1 20

2 10

1

2

T

T

T

T

T

T

iei

g

g

g

g

gg

E

The stress field for the element is derived from the linear uncoupled stress field [P2]

σσστ

ξ ηξ η

ξ ηβ=

RS|

T|

UV|

W|=L

NMMM

O

QPPP

x

y

xy

11

1

The introduction of four internal degrees of freedom allows four of the nine stress parameters (β) to vanish. The remaining terms satisfy the equilibrium equations. By basing the formulation on natural coordinates the element is less sensitive when distorted and possesses no zero-energy deformation modes. Full details of the elimination of the four stress parameters is described in [P2] for a hybrid element. The final contravariant stress field using five β parameters is defined as


71

∑∑∑∑

=

RS||

T||

UV||

W||

=L

NMMM

O

QPPP

ξ

η

ξη

ηξ β

1 0 0 00 1 0 00 0 1 0 0

To satisfy the L2 orthogonality condition < ∑ , ∈ > L2

− −z z ∑∈ ≡1

1

1

10

r d dT ξ η

This condition is violated if the six initial enhanced strain parameters (α) are used. However, the condition is satisfied if

α α5 6= −

Forcing this equality, and hence L2 orthogonality, gives the final enhanced strain interpolation matrix as

E = −−

L

NMMM

O

QPPP

ξ ξηη ξη

ξ η ξ η

0 0 00 0 00 0 2 2

This matrix is used in linear analyses but for nonlinear applications four enhanced strain parameters are used with the final column of E deleted [S8]. The final interpolation functions E also allow condition (III) to be satisfied. This is a requirement for passing the patch test [S8] and is implied in the sense that:

− −z z ≡1

1

1

10

dE dξ η

Enhanced strain interpolation - axisymmetric element (QAX4M) The procedure for establishing the enhanced strain interpolation matrix for the axisymmetric element is similar to that used for the plane elements. The initial matrix is given by

Ei =

L

N

MMMM

O

Q

PPPP

ξη

ξ ηξη

0 0 0 00 0 0 00 0 00 0 0 0

For the axisymmetric case, a factor r(ξ,η) will be included in the integrand for enforcing orthogonality

− −z z ∑∈ ≡1

1

1

10

r d dT ξ η


72

where

∑ ∑ ∑ ∑ ∑= ξξ ηη ξη θθ T, ∈ = ∈ ∈ ∈ ∈ξξ ηη ξη θθ 2

r r NT=

z z NT=

r r= 1, , , r r r2 3 4

z z= 1, , , z z z2 3 4

Inclusion of the factor r(∈,η) means that the orthogonality condition is violated using this interpolation matrix. Simo and Rifai [S8] have derived interpolation functions which account for the factor r and satisfy this condition

E E E r di i= −

− −

− −z z z z1

1

1

1

1 1

1

1

1

r d d d

ξ ηξ η

=

−−

− −−

L

N

MMMMM

O

Q

PPPPP

ξ ξη η

ξ ξ η ηξη ξη

0 0 0 00 0 0 00 0 00 0 0 0

where

ξ η ξη= =13

13

1

0

2

0 0

r ar a

r ar a

r hr a

T

T

T

T

T

T, , = 19

a0 , a1, a2 and h vectors have been defined for the plane elements.

Evaluation of stresses The evaluation of stresses is identical to that described in section 7.3.1.5.

Nonlinear formulation The comments made in section 7.3.1.6 regarding the nonlinear capability of the standard elements are also applicable to these elements. The nonlinear formulation for the enhanced strain elements involves enforcing orthogonality between assumed Green-Lagrange strains and 2nd Piola-Kirchhoff stresses. The geometrically nonlinear performance of these elements is much improved in comparison with the standard elements.


73

7.3.3 Incompatible Plane Membrane Element (PMI4) Formulation This element is a high performance, non-conforming, 4-noded, plane membrane element. It is formed by adding two non-conforming modes to the standard isoparametric formulation presented for QPM4, i.e.

U N Ui ii

n=

=∑ ξ η,b g

1

is replaced with

U N U P ai ii

n

i ii

= += =∑ ∑ξ η ξ η, ,b g b g

1 1

2

where

P121ξ η ξ,b g = − and P2

21ξ η η,b g = −

and ai are nodeless degrees of freedom which are condensed out before element i assembly. The nodal configuration and non-conforming shape functions are shown in fig.7.3.3-1.

The element passes the patch test (ensuring convergence as the mesh is refined) and the displacement field is approximately an order higher than the QPM4 element (i.e. quadratic displacement accuracy).

The infinitesimal strain-displacement relationship is the same as QPM4, i.e.

∈ =XUX

∂∂

∈ =YVY

∂∂

γ∂∂

∂∂XY

UY

VX

= +

The isotropic and orthotropic elastic modulus matrices are defined as

Isotropic D E=

− −

L

N

MMMM

O

Q

PPPP1

1 01 0

0 01

2

2υ

υυ

υe j a f


74

Orthotropic Dyx y

xy x y=−

−L

NMMM

O

QPPP

−1 01 0

0 0 1

1υυ

// /

/

EE E

Gxy

where υyx is set to υxy x yE E/ to maintain symmetry.

Note. For a valid material υxy x yE E< //d i1 2


Isotropic ∈ =0 0d itTTΔ α α, ,


TΔ α α α, , xy

Full details of the formulation are presented in [T2,W2].

Only a lumped mass matrix is evaluated using the procedure defined in (section 2.7).

Evaluation of stresses The element output obtained at the element nodes consists of

Stress Resultant Output Nx y xy, N , N the direct and shear stress resultants/unit length Nmax min, N the maximum and minimum principal stress

resultants/unit length β the angle between the maximum principal stress resultant

and the positive X-axis NS the maximum shear stress resultant/unit length NV Von Mises equivalent stress resultant/unit length.

Strain Output ∈ ∈X Y XY, , γ the direct and shear strains ∈ ∈max min, the maximum and minimum principal strains β the angle between the maximum principal strain and the

positive X-axis ∈S the maximum shear strain ∈V Von Mises equivalent strain

The sign convention for stress resultant and strain output is shown in fig.7.3.3-4. The stress resultants are evaluated directly at the nodes.


75

Nonlinear formulation The element has no nonlinear capability, but may be utilised in a nonlinear environment.

The element cannot be used for linear buckling analyses.

Y,V

X,U

(a) Nodal Configuration

1

2

3

4

(b) Non-conforming shape functions

P1 = 1-ξ2 P2 = 1-η2

Fig.7.3.3-1 Nodal Configuration And Non-Conforming Shape Functions For The PMI4 Element


(a) Plate subject to Inplane Loading


(b) Cantilever Plate subject to Point Loading

Fig.7.3.3-2 Examples Illustrating The Use Of PMI4 Elements


76

Y

X

1

2

3

4

yx

FIG.7.3.3-3 LOCAL CARTESIAN SYSTEM FOR THE PMI4 ELEMENT

Y

X

σY

σX

σY

σX

σXY

σXY



7.3.4 2D Explicit Dynamics Elements Explicit time integration schemes have used simple linear elements rather than those of a higher order by virtue of their computational efficiency. A number of further advantages may also be obtained in explicit dynamic analyses

The use of higher order shape functions creates difficulties at the contact interface in the form of uncontrolled overlap. It has been shown that higher order continuum elements require a time step

reduced from that of linear elements because of the greater mass associated with the interior nodes.


77

The mass lumping formulations for higher order elements are currently impractical for modelling shock wave propagation since the resulting numerical noise pollutes or destroys the solution. The combination of mass lumping with linear elements, when applied in

conjunction with the central difference operator, increases accuracy in solutions by virtue of their respective compensatory spectral errors.

The linear explicit dynamics elements have been implemented to take advantage of these benefits. They are for use only with the explicit central difference time integration scheme.

The explicit dynamics elements are based upon the isoparametric approach in which the same shape functions are used to interpolate both the displacements and the geometry, i.e.

displacement U N Ui ii

n=

=∑ ξ η,b g

1

geometry X N Xi ii

n=

=∑ ξ η,b g

1

where Ni ξ η,b g is the element shape function for node and n is the number of nodes. Fig.7.3.1-1 shows the nodal configurations available within LUSAS. The nodal degrees of freedom are U and V. All the explicit dynamics elements described in this section must be defined using only X and Y coordinates.

Plane stress (QPM4E, TPM3E) The plane stress elements are formulated by assuming that the variation of the out of plane direct stress and shear stresses is negligible, i.e.

σ σ σZ = = =0 0 0, , XZ YZ

The plane stress elements are suitable for analysing structures which are thin in the out of plane direction, e.g. thin plates subject to in-plane loading (fig.7.3.4-2).

Note that the thickness of the material is defined at each node and may vary over the element.

A rate relationship is used to define the strain-displacement characteristics as

t UXx

t

t&

&

∈ =∂∂

t VYy

t

t&

&

∈ =∂∂


78

t UY

VXXY

t

t

t

t&

& &γ

∂∂

∂∂

= +

t UX

VYz

t

t

t

t&

& &∈ = − +LNMM

OQPP

υ∂∂

∂∂


Isotropic D E=

− −

L

N

MMMM

O

Q

PPPP1

1 01 0

0 01

2

2υ

υυ

υe j a f

Orthotropic Dyx y

xy x y=−

−L

NMMM

O

QPPP

−1 01 0

0 0 1

1υυ

// /

/

EE E

Gxy

where υyx is set to υxy y xE E/ to maintain symmetry.

Note. To obtain a valid material υxy x yE E< //d i1 2

The initial thermal strain is defined by

Isotropic ∈ =0d itTTΔ α α, , 0


TΔ α α α, , xy

Plane strain (QPN4E, TPN3E) The plane strain elements are formulated by assuming that the variation of the out of plane direct strain and shear strains is negligible, i.e.

∈ = ∈ = ∈ =Z 0 0 0, , YZ XZ

The plane strain elements are suitable for analysing structures which are thick in the out of plane direction, e.g. dams or thick cylinders (fig.7.3.4-3).


tX

t

tUX

&&

∈ =∂∂

tY

t

tVY

&&

∈ =∂∂


79

tXY

t

t

t

tUY

VX

&& &

γ∂∂

∂∂

= +

tY&∈ = 0


Isotropic

D E=

− −

−−

−

L

N

MMMMM

O

Q

PPPPP1 1 2

1 01 0

0 01 2

2

υ υ

υ υυ υ

υa f a f( )

( )( )

Orthotropic D

E EE E

E EE E

E EE E

E EE E

G

z xz x

x z

xy z xz yz y

y z

xy z yz xz x

x z

z yz y

y z

xy

=

− − −

− − −

L

N

MMMMMMMM

O

Q

PPPPPPPP

−υ υ υ υ

υ υ υ υ

2

2

1

0

0

0 0 1

where for symmetry

E E E E E Ey xy z yz xz x x xy z xz yz yυ υ υ υ υ υ+ = +d i d i


Isotropic ∈ = +0 1 0d i a ft x yT

Tυ α αΔ , ,

Orthotropic ∈ = +0 1d i a ft x y xyT

Tυ α α αΔ , ,

Axisymmetric (QAX4E, TAX3E) The axisymmetric elements are formulated by assuming that the variation of out of plane shear stresses is negligible, i.e.

σ σXZ YZ = =0 0,

and the out of plane direct strain rate is defined as

&

&∈ =Z

UR

where R is the distance from the axis of symmetry.


80

The axisymmetric elements are suitable for analysing solid structures which exhibit geometric symmetry about a given axis, e.g. thick cylinders or circular plates (fig.7.3.4-4).

The elements are defined in the XY-plane and symmetry can be specified about either the X or Y axes.

Standard axisymmetric isoparametric elements are formulated with the Galerkin weighted residual method, in which the governing differential equation is utilised directly to form a weighted residual statement, where the weighting functions are generally the element shape functions. For large strain axisymmetric analyses, the use of elements based on the Galerkin method leads to computational difficulties near the axis of symmetry. These difficulties may be overcome by formulating the elements with the Petrov-Galerkin method [G2]. This method is also a weighted residual method, however, the weighting functions are taken to be the product of the element shape functions and the inverse of the radius, i.e. eliminating the radial weighting in the governing equations.

The use of this particular formulation produces a time dependent mass matrix and as such must be computed each time.


tX

t

tUX

&&

∈ =∂∂

tY

t

tVY

&&

∈ =∂∂

tXY

t

t

t

tUY

VX

&& &

∈ = +∂∂

∂∂

tZ

t UR

&&

∈ = (symmetry about the Y axis)

or tZ

t VR

&&

∈ = (symmetry about the X axis)

The isotropic and orthotropic linear elastic modulus matrices are defined as

Isotropic

D E=

− −

−−

−−

L

N

MMMM

O

Q

PPPP( )( )

( )( )

( ) /( )

1 1 2

1 01 0

0 0 1 2 2 00 1

υ υ

υ υ υυ υ υ

υυ υ υ

Orthotropic


81

D

E E EE E E

GE E E

x yx y zx z

xy x y zy z

xy

xz x yz y z

=

− −− −

− −

L

N

MMMMM

O

Q

PPPPP

−1 01 0

0 0 1 00 1

1/ / // / /

// / /

υ υυ υ

υ υ

in which symmetry is maintained by defining

υ υ υ υ υ υyx xy y x yz z yE E E E= =/ / = E / E zx xz z x zy


υ υ υxy x y yz y zyE E E E< </ //d i b g d i1/2

xz x z < E / E 1 2

The initial thermal strain vector is defined as

Isotropic ∈ =0 0d it x yT

TΔ α α α, , , z


TΔ α α α α, , , xy z

Integration rule for the elements A one point quadrature integration rule is utilised. This provides elements that are efficient, do not lock when incompressible behaviour is being modelled, e.g. plastic straining with von Mises plasticity, and integrate the stresses at the most accurate location.

The location of the integration point is given in Appendix A.

Element stabilisation The utilisation of one point Gauss quadrature has a limitation in that zero energy deformation or hourglass modes are generated (see fig.7.3.4-5). The effects of such modes are minimised by the viscous damping technique [H7].

The technique provides a damping force capable of preventing the formation of spurious modes but which has negligible influence on the true structural modes. This is possible since the spurious modes are orthogonal to the real deformations.

The rate of diagonal drifting is defined by the velocity at which the mid-points of the element are separating. This is utilised as the basis for hourglass detection, giving the hourglass velocities as

h xij ik

j

==∑ & i = 1,2jkΓ a f

1

4

The viscous hourglassing forces are


82

f hik hg ijj

= −=∑1 4

1

4/ Q A c 1/2

jkρ Γd i

in which A is the current element area, Qhg is a constant which is modified via the SYSTEM command and is usually set to a value between 0.05 and 0.15, and &xi

k is the nodal velocity of the kth node in the ith direction. ρ is the current element density, while c, the material sound speed is defined from

cE2 1

1 1 2=

−+ −

υρ υ υa f

a fa f

The hourglass base vectors for the four node quadrilateral are defined as:-

ΓiT= 1 -1 1 -1

these viscous forces are included directly into the element force vector.

Shock wave smoothing The shock discontinuities that occur in impact problems may promote numerical instabilities which must be smoothed out. This is achieved using an artificial bulk viscosity method. The salient characteristic of the method is the augmentation of element pressure with an artificial viscous term (q) prior to the evaluation of the element internal force. This is zero in expanding elements and non-zero in contracting elements. The algorithm has the effect of spreading the shock front over a small number of elements.

The exact form of artificial viscosity is somewhat arbitrary and the method used is based on the formulation originally proposed in [V1]

q Q= +ρ L D Q L D cc kk 1 c kk 2

where Q1 and Q2 are dimensionless constants which default to 1.5 and 0.06 respectively, and may be modified as necessary via the SYSTEM command. Dkk is the trace of the velocity strain tensor and Lc is the characteristic length of the element which is related to the smallest element diagonal as

L ALc

D=

2

where

L MAX y x y xD = + +1 2 1 2 1 2 1 2242

422

312

132/ / , / / e j

in which the distance between any two nodal points i,j is given as


83

x x xij i j= −

The quadratic term in strain rate is chosen to be small except in regions of very large gradients. The linear term, however, is included to control the small spurious oscillations following the shock waves in which the gradients are insufficient to make the quadratic term effective. Care should be taken with the linear term since there is a danger of distorting the solution.

In converging geometries, the centred strain rate term is negative and the q term is then non-zero. This occurs even though no shocks are generated and results in a non-physical generation of pressure. In view of the abundance of excellent results, however, it is generally agreed that the effect is negligible.

Force calculations

The direct stresses at time t+Δt are modified by the addition of the artificial viscosity pressure q as follows

σ σx x q= + and σ σy = +y q

The contribution to the force vector due to the element stresses is evaluated from the equilibrium equations of Timoshenko as

F x y rx x xy r= + + − =∂σ ∂ ∂τ ∂ σ σθ/ / /b g 0

F y x ry y xy xy= + + =∂σ ∂ ∂τ ∂ τ/ / / 0

Note that the terms σr r and τxy r from these two equations are not typically included in static analyses and occur as a result of the inertial effects. The hourglass forces are included to give the final force vector. The mass matrix is computed as each node i as

t A A v vMt t o t t o

xi= =1 4 1 4/ / /ρ ρ e j

t A A v vMt t o t t o

yi= =1 4 1 4/ / /ρ ρ e j

where t v is the current volume and o v is the initial volume of an element.


Stress Output σ σ σ σX, , , Y XY Z the direct and shear stresses

σ σmax, min the maximum and minimum principal stresses β the angle between the maximum principal stress and the

positive X-axis


84

σS the maximum shear stress σV von Mises equivalent stress

Strain Output

∈ ∈ ∈X, , , Y XY Zγ the direct and shear strains ∈ ∈max, min the maximum and minimum principal strains β the angle between the maximum principal strain and the

positive X-axis ∈S the maximum shear strain ∈V von Mises equivalent strain

Stress resultant output which accounts for the thickness of the element is available as an alternative to stress output for the plane stress elements, i.e.

Stress Resultant Output NX, , ,N N NY XY Z the direct and shear stress resultants/unit length

Nmax, Nmin the maximum and minimum principal stress max min resultants/unit length

β the angle between the maximum principal stress resultant and the positive X-axis

NS the maximum shear stress resultant/unit length NV von Mises equivalent stress resultant/unit length.

The sign convention for stress, stress resultants and strain output is shown in fig.7.3.4-5. The Gauss point stress is usually more accurate than the nodal values. The nodal values of stress and strain are obtained using the extrapolation procedures detailed in section 6.1.

Nonlinear formulation The 2-D explicit dynamics elements can be employed in

Materially nonlinear dynamic analysis utilising the elasto-plastic constitutive laws [O2] (section 4.2). Geometrically nonlinear dynamic analysis. Geometrically and materially nonlinear analysis utilising the nonlinear

material laws specified in 1. Notes

The plane stress elements may not be used with nonlinear material model 75. Plain strain and axisymmetry are, however, supported. All explicit dynamics elements may be used with nonlinear material models

61, 64, 72. Eulerian geometric nonlinearity is always invoked with the use of the explicit

elements in which the velocity strain measure is utilised. The Green-Naghdi stress rate formulation is used to refer the constitutive variables to an unrotated


85

configuration prior to the stress integration. The output is in terms of true Cauchy stresses and the strains approximate to logarithmic strains. The loading is non-conservative.

12

3

1

2

34

Fig.7.3.4-1 Nodal Configuration For 2d Explicit Dynamics Elements


Plate subject to Inplane Loading

Fig.7.3.4-2 Example Illustrating The Use Of Plane Stress Elements


86


Thick Cylinder

Fig.7.3.4-3 Example Illustrating The Use Of Plane Strain

Thick Cylinder

Finite Element Mesh

r

r

Problem Definition

Fig.7.3.4-4 Example Illustrating The Use Of Axisymmetric Solid Elements


87

Fig.7.3.4-5 Deformed Mesh Illustrating Formation Of Spurious Mechanisms

Y

X

σY

σX

σY

σX

σXY

σXY



7.3.5 Two Phase Plane Strain Continuum Elements (TPN6P and QPN8P)

Formulation These isoparametric finite elements utilise the same shape functions to interpolate the displacements and geometry, i.e.

displacements U N Uii

n

i==∑ ξ η,b g

1


nξ η,b g

1


88

where Ni ξ η,b g is the element shape function for node i and n is the number of nodes. However, for consideration of stability, the pressures are only interpolated using the corner nodes

pressures P N Pii

n

i==∑ ξ η,b g

1

corner

where ncorner is the number of corner nodes. Fig.7.3.5-1 shows the nodal configurations available within LUSAS.

The nodal degrees of freedom are U, V and P at the corner nodes and U and V at the midside nodes.


The plane strain assumptions and details of elastic modulus matrices applicable for these elements are described in section 7.2.1.2.

These elements are used to model the behaviour of a two phase medium such as soil. In this instance the two phases comprise the soil skeleton and the pore water fluid. Separate equations are derived for each phase, coupled by the interaction of the pore pressure and the soil deformation. The soil skeleton is analysed in terms of effective stress (total stress minus pore water pressure), taking into account the loading due to the pore pressure; whilst the pore fluid analysis takes account of the volumetric strain due to the soil skeleton deformation.

The finite element method is used to solve the coupled equations in terms of nodal displacements and pore pressures. Two plane strain elements QPN8P (quadrilateral) and TPN6P (triangular) based on a mixed displacement-pressure formulation are available in LUSAS to solve these problems.

Undrained/fully drained conditions In this type of analysis no consolidation is assumed to take place and the coupled governing equations for static undrained conditions can be expressed as:

K L

L SUP

F F

L U SPT T

ext intL

NMMOQPPRSTUVW =

−

− −

RS|T|

UV|W|

ΔΔ

where the matrices K L, and S are defined as:

K B D BT

v

= z dv'

L B mNT

v

= z- dv


89

SK

N Ne

T

v

= z - dv1

K is the tangent stiffness matrix

L is the coupling matrix

S is the compressibility matrix, where Ke is the equivalent bulk modulus of the soil (see section 7.2.5.4) and D' the ‘effective’ soil modulus matrix.

ext F and int F are external and internal forces Under static fully drained conditions the above coupled governing equations can be further simplified as

K

IUP

F F

ext0

0 0LNMOQPRSTUVW =

−RSTUVW

ΔΔ

int

where I is a unit matrix block.

Drainage/consolidation process In the drainage/consolidation process, fluid flow in/out from the soil needs to be considered. For linear transient consolidation the coupled governing equations can be expressed as:

K L

L THUP

K L

L THUP

FQT

t tT

t

βδ β δδ

LNMM

OQPPRSTUVW =

−

LNMM

OQPPRSTUVW +RSTUVW+ ( )1

ΔΔ

where:

ΔF is the incremental load ΔQ the incremental flow β the time stepping scheme parameter (set to1.0 for backward Euler scheme) H the permeability matrix

The permeability matrix H is defined in terms of the shape function derivatives and a permeability matrix of the soil, K

p, as:

H N K NTp

v

= ∇ ∇z dv

For nonlinear consolidation process, the coupled governing equations can be written as


90

K LL t H

UP

F Ft Q Q tH P L U UT

k

k

n k n k

n n k T n k nΔΔΔ Δ Δβ βΔ

LNM

OQPRSTUVW

=−

− + − − −RS|T|

UV|W|+

+

+ext int

+

1 1

1 1c h c h

where the superscript on the left/right hand side represents the increment/iteration number.

Material assumptions The bulk modulus of the soil particle Ks is very large compared to the bulk modulus of the pore fluid Kf . Therefore the overall compressibility of the soil mass is approximated to be that of the pore fluid.

1 1K K K Ke f s f

= +−

≡η η η( )

where:

Ke is the equivalent bulk modulus of the soil Kf the bulk modulus of the pore fluid Ks the bulk modulus of the solid soil particle η the porosity of the soil

In practical geotechnical applications it is usually difficult to determine Kf and Ks so a large value of the equivalent modulus Ke is usually assumed, 1012>Ke>109.

Nonlinear formulation The two phase plane strain continuum elements can be employed in:-

Materially nonlinear drained/undrained/consolidation analysis utilising the elasto-plastic constitutive laws [O1] (section 4.2). Geometrically nonlinear drained/undrained/consolidation analysis. Geometrically and materially nonlinear drained/undrained/consolidation

analysis utilising the nonlinear material laws specified in 1. Geometrically and materially nonlinear dynamic drained/undrained analysis

utilising the nonlinear material laws specified in 1.

7.3.6 Large-strain Mixed-type Elements (QPN4L, QAX4L) Nonlinear formulation These elements are based on a mixed displacement/pressure formulation, which overcomes the problems of near-incompressibility and effective incompressibility in standard plane-strain and axisymmetric elements. The formulation utilises a nonlinear (spatial) Eulerian formulation, based on the logarithmic strain tensor, associated with the polar decomposition of the deformation gradient F VR= , where V is the left


91

stretch tensor and R is the rotation of the axes of the stretches λ i . The Kirchhoff (nominal) stress tensor τ is related to the (true) Cauchy stress σ via τ σ= J , where J F= =det λ λ λ1 2 3 .The deformation gradient is given as:

F xX

=∂∂

where X and x denote the material and spatial position vector of a material particle. The elements are currently available with Hencky and Ogden matrial models described in section 4.10, so that the principal Kirchhoff stresses τ λi i

i=

∂ψ∂λ

[C16] are

obtained from the corresponding stored-energy function ψ as

τ λi iG kJ J= + −2 1ln a f for the Hencky material model, where G is the shear modulus, k is the bulk modulus and λ λi i J= / 3 are the “deviatoric” stretches, and as

τ μ λ λ λ λα α α α

i pp

N

ip p p p kJ J= − + + + −

=∑

11 2 3

13

1[ ( )] a f

for the Ogden material model, where N is the number of pairs of Ogden parameters μp and αp , while k and λ i have the same meaning as for the Hencky model. By introducing the independent pressure variable as

p k J= − − 1a f and by transforming τi from the principal directions the Kirchhoff stress tensor τ is obtained as

τ λ= −2Gn n pJTln

for the Hencky material model, where λ is the diagonal matrix of deviatoric stretches and n n n n= [ , , ]1 2 3 is the Eulerian triad (spatial orientation of the principal directions) and as

τ μ λ λα α

= − −=

∑n tr I n pJpp

NTp p

1

13

[ ( ) ]

for the Ogden material model.

Element equilibrium (mixed formulation) The element equilibrium equations are given as


92

g P R

f J pk

dVV

≡ − =

≡ − − +LNM

OQP =z

0

1 00

0a f

where the first equation is the conventional nodal equilibrium equation, where R is the vector of applied loading and P is the vector of nodal internal forces, and the second equation follows from p k J= − − 1a f. By expressing the stress tensor in the vector form, the vector of nodal internal forces can be written as

P B x dVV

T= z

0

0b gτ

where, in line with the adopted spatial approach, x is the spatial and not the material position vector. Note that the formulation is defined in terms of the Kirchhoff and not the Cauchy stresses, hence integration is still performed over the initial rather than the current volume.

Linearisation of the equilibrium - tangent stiffness matrix By expanding the element equilibrium into a Taylor’s series, the following linearised equilibrium is obtained

δδ

δδ

gf

ap

gf

K KK KT

RSTUVW

≡

L

N

MMM

O

Q

PPPRSTUVW

= −RSTUVW

11 12

12 22

where a is the vector of nodal displacements, and the entries in the tangent stiffness matrix are obtained by the consistent linearisation of the element equilibrium.

In order to derive the subvector K12 and (in particular) submatrix K11

it helps to regard the vector of nodal internal forces P as coming from the internal virtual power via

& &:a P dVT

V

= z ε τ

0

0

where &a dadt

= is the time rate of the nodal displacements, and & ( )ε = +12

L LT is the

strain-rate tensor with

L d ux

= =∂∂

& &


93

with L being the so-called velocity gradient and d ux

=∂∂

being only introduced for the

sake of convenience during the following derivation. Also &: ( &) &ε τ τ ε ε τ= =tr Tij ij , where

the repeated indices indicate summation over the dimension of the space. For configuration-independent loads, &a gTδ is equal to &a PTδ , hence

& ( ) ( & & ) &a K a K p dV

ppdVT

Vij ij ij

ij

klkl ij

ij

V11 12 0 0

0 0

δ δ δε τ ε δε ε δ+ = +∂τ

∂ε+

∂τ

∂z z

where, for both material models, ∂τ

∂= −ij

ijpJδ so, by introducing standard FE

matrix/vector notation whereby & ( )&ε = B x a , subvector K12 immediately follows as

K B x iJdVV

T12 0

0

= − z ( )

with i =RS|T|

UV|W|

110

for the plane strain element QPN4L and i =

RS||

T||

UV||

W||

1101

for the axisymmetric

element QAX4L. By noting the relationship between Kirchhoff stress τ and second Piola-Kirchhoff stress S via τ = FSFT and bearing in mind that δ δFF d− =1 we obtain ∇ = ∇ + + = + +ε τδε δ δ τ τδ δτ δ τ τδF S E F d d d dE

T TT

T( ) , or in indicial notation

∂τ

∂ε= + + = + +ij

klkl T ij ik kj ik jk ijkl

tTKkl ik kj ik jkd d D d dδε δτ δ τ τ δ δε δ τ τ δ,

where δτT

is called the Truesdell rate of Kirchhoff stress (which is often used in rate-dependent constitutive models; here it is introduced because it enables a straightforward formation of the material part of the stiffness matrix) and Dijkl

tTK is the tangent constitutive matrix relating the strain-rate tensor to the Truesdell rate of Kirchhoff stress. By using & ( & & )εij ij jid d= +

12

and noting the symmetry of the Kirchhoff

stress tensor τ τij ji= , the product &ε δεijij

klkl

∂τ

∂ε can be written as

& & & &ε δε ε δε δ τ δ τij

ij

klkl ij ijkl

tTKkl ij ik kj ji ik kjD d d d d

∂τ

∂ε= + +


94

By noting that the above-mentioned & &FF d− =1 yields ∂∂

=& &uX

dF , the variation of which

gives ∂∂

∂∂FHGIKJ = +

XuX

X dF d F& & &δ δ δ , and by noting that the variation of the material

position vector δX is equal to zero, we obtain δ δ δ& & &d d FF d d= − = −−1 , which finally gives

δε δ δ δ δ& ( & & ) ( & & )ij ij ji ik kj jk kid d d d d d= + = − +12

12

so that, after noting the symmetry of the Kirchhoff stress tensor, the product δε τ& ij ij reduces to

δε τ δ τ& &ij ij ik kj ijd d= −

The symmetry of the Kirchhoff stress tensor further implies

& & & &ε δε δε τ ε δε τ δij

ij

klkl ij ij ij ijkl

tTKkl ij kj ikD d d

∂τ

∂ε+ = +

so that eventually the submatrix K11

follows from

& (& & )a K a D d d dVT

Vij ijkl

tTKkl ij kj ik11 0

0

δ ε δε τ δ= +z

Following the standard FE notation, the submatrix K11

is then given as

K B x D B x G x G x dVV

TtTK

T11 0

0

= +z [ ( ) ( ) ( )$ ( )]τ

where the tangent constitutive matrix DtTK

, which relates the strain-rate to the Truesdell rate of Kirchhoff stress can be defined in different ways. An easy way to define it is by rotating the constitutive matrix D

tTKE, which relates the strain-rate with

the Truesdell rate of Kirchhoff stress, where both of these are given with components in the Eulerian frame, via

D n n n n DijkltTK

ia jb kc ld abcdtTKE=

where nij denotes components of the Eulerian triad n . The components of the constitutive matrix D

tTKE follow from the stretches and the principal Kirchhoff

stresses. By dropping the summation convention, the “normal” components are defined as

Three-Dimensional Continuum Elements

95

DiijjtTKE

ji

ji ij=

∂τ∂λ

−λ τ δ2

where δij is the Kronecker symbol and

λ μ μδji

jijpJ∂τ

∂λ= − − +

23

2

for the Hencky model and

λμ α

λ δ λ λ λ λ λα α α α α α

ji

j

p p

p

N

i ij i jp p p p p p pJ∂τ

∂λ= + + + − − −

=∑ 3

3 131

1 2 3[ ( ) ]

for the Ogden model. The “shear” components are defined as (i j≠ )

D D D DijijtTKE

ijjitTKE

jijitTKE

jiijtTKE j i i j

i j= = = =

−

−

λ τ λ τ

λ λ

2 2

2 2

unless λ λi j= , in which case the shear components are given as

D D D DijijtTKE

ijjitTKE

jijitTKE

jiijtTKE i i

i

i

ji= = = =

∂τ∂λ

−∂τ∂λ

FHG

IKJ

−λ

τ2

which returns the result μ τ− i for the Hencky model and μ α

λ ταp p

p

N

i ip

21=∑ − for the

Ogden model. Varying the second equilibrium equation gives

δ δ δ δδf K a K p J pk

dVT

V

≡ + = − +LNMOQPz12 22 0

0

where K12 has already been defined and

K dVk

V22

0

0

= − z

7.4 Three-Dimensional Continuum Elements 7.4.1 Standard Isoparametric Elements (HX8, HX16,

HX20, PN6, PN12, PN15, TH4, TH10) Three dimensional isoparametric finite elements utilise the same shape functions to interpolate both the displacements and geometry, i.e.


96


n=

=∑ ξ η,b g

1

geometry X N Xi ii

n=

=∑ ξ η,b g

1

where Ni ξ η,b g is the element shape function for node i and n is the number of i nodes. Fig.7.4.1-1 shows the nodal configurations available within LUSAS. The nodal degrees of freedom are

U, V and W at each node

The infinitesimal strain-displacement relationship is fully 3-D and is defined as

∈ =XUX

∂∂

∈ =YVY

∂∂

∈ =ZUZ

∂∂

γ∂∂

∂∂XY

UY

VX

= +

γ∂∂

∂∂YZ

VZ

WY

= +

γ∂∂

∂∂XZ

UZ

WX

= +


Isotropic

D E=

− −

−

−

−

−

−

−

L

N

MMMMMMMMMMMMMM

O

Q

PPPPPPPPPPPPPP

1 1 2

1 0 0 0

1 0 0 0

1 0 0 0

0 0 01 2

20 0

0 0 0 01 2

20

0 0 0 0 01 2

2

υ υ

υ υ υ

υ υ υ

υ υ υ

υ

υ

υ

a fa f

a fa f

a fa f

a fa f


97

Orthotropic

D

E E EE E EE E E

GG

G

x yx y zx z

xy x y zy z

xz x yz y z

xy

yz

xz

=

− −− −− −

L

N

MMMMMMMM

O

Q

PPPPPPPP

−1 0 0 01 0 0 0

1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

1// / // / /

//

/

υ υυ υυ υ

where υ υyx, , zx and υzy are defined by







Isotropic ∈ =0d itTTΔ α α α, , , 0, 0, 0


TΔ α α α α α α, , , , , z xy yz xz

A complete description of their formulation is given in [H2,B1].


Evaluation of stresses The element output can be obtained at both the element nodes and Gauss points and consists of

Stress Output σ σ σ σ σ σX Z XY YZ XZ, , , , , Y the direct and shear stresses

Strain Output ∈ ∈ ∈X Z XY YZ XZ, , , , , Y γ γ γ the direct and shear strains

Principal stresses and strains and the corresponding direction cosines may also be output.

The sign convention for stress and strain output is shown in fig.7.4.1-3.


98

The Gauss point stresses are usually more accurate than the nodal values. The nodal values of stress and strain are obtained using the extrapolation procedures detailed in section 6.1.

Nonlinear formulation The 3-D isoparametric elements can be employed in



Notes

The nonlinear interface model (section 4.2) may be used with elements HX8, HX16, HX20, PN6, PN12. The geometric nonlinearity may utilise

A Total Lagrangian formulation which accounts for large displacements but



OQP + LNM

OQPX

UX

UX

VX

WX

∂∂

∂∂

∂∂

∂∂

12

12

12

2 2 2


OQP + LNM

OQPY

VY

UY

VY

WY

∂∂

∂∂

∂∂

∂∂

12

12

12

2 2 2


OQP + LNM

OQPZ

WZ

UZ

VZ

WZ

∂∂

∂∂

∂∂

∂∂

12

12

12

2 2 2

γ∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂XY

UY

VX

UX

UY

VX

VY

WX

WY

= + + + +

γ∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂YZ

VZ

WY

UY

UZ

VY

VZ

WY

WZ

= + + + +

γ∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂XZ

UZ

WX

UX

UZ

VX

VZ

WX

WZ

= + + + +

The output is now in terms of the 2nd Piola-Kirchhoff stresses and Green-Lagrange strains referred to the undeformed configuration. The loading is conservative. An Updated Lagrangian formulation, which takes account of large

displacements and moderately large strains provided that the strain increments


99

are small. The output is now in terms of the true Cauchy stresses and the strains approximate to logarithmic strains. The loading approximates to being non-conservative. An Eulerian formulation, which takes account of large displacements and

large strains. The output is in terms of true Cauchy stresses and the strains approximate to logarithmic strains. The loading is non-conservative.


100

1

2

3

4

5

6

78

512 4

3

768

91011

12

131415

16

51 2 4

3

768

9

10

111213 14

15

1617

18

1920

1

2

3

4

5

6

12

3

4

56

789

10

1112

12

3

4

567

8

910

1112

131415

1

2

3

4

12

3

4

56

78

910

HX8HX16

HX20

PN6

PN12 PN15

TH4 TH10

Fig.7.4.1-1 Nodal Configuration For Solid Elements


101

Fig.7.4.1-2 Tractor Brake Component

Y

X

Z

σXZ

σYZ

σZ

σXY

σYZ

σY

σXZ

σXY

σX

Arrows indicate +vestress directions



102

7.4.2 Enhanced Strain Element (HX8M) The low order enhanced strain element HX8M exhibits improved accuracy in coarse meshes when compared with the parent element HX8, particularly if bending predominates. In addition, the element does not suffer from 'locking' in the nearly incompressible limit. The element is based on a three-field mixed formulation [S8] in which stresses, strains and displacements are represented by three independent functions in three separate vector spaces. The formulation is based on the inclusion of an assumed 'enhanced' strain field which is related to internal degrees of freedom. These internal degrees of freedom are eliminated at the element level before assembly of the stiffness matrix for the structure. The formulation provides for the following three conditions to be satisfied

Independence of the enhanced and standard strain interpolation functions. L2 orthogonality of the stress and enhanced strains. Capability of the element to model a constant state of stress after enforcing

the orthogonality condition, i.e. requirement for passing the patch test. In addition to ensuring that the element passes the patch test, these conditions also allow the stress field to be eliminated from the formulation.

Formulation The general approach taken to formulate this element is identical to that described for the 2-D continuum elements in section 7.3.2.

Enhanced strain interpolation The incompatible displacement field is given by

u N N N= + +1 1 2 2 3 3ξ λ ξ λ ζ λb g b g b g

where

N N N12

22

321

21 1

21 1

21ξ ξ η η ζ ζb g e j a f e j b g e j= − = − = −, ,

and λ i , represent the incompatible modes

λ λ λ1 1 2 2 3 3= = =u u uT T T, , , , , , v v v1 2 3l q l q l q

The covariant base vectors associated with the isoparametric space are

gx ay az a

x hy hz h

x hy hz h

x ky kz k

g g g g

T

T

T

T

T

T

T

T

T

T

T

Tξ

η ζ ηζ η ζ ηζ=

RS|

T|

UV|

W|+

RS|

T|

UV|

W|+

RS|

T|

UV|

W|+

RS|

T|

UV|

W|= + + +

1

1

1

1

1

1

3

3

3

10

11

31


103

gx ay az a

x hy hz h

x hy hz h

x ky kz k

g g g g

T

T

T

T

T

T

T

T

T

T

T

Tη

ξ ζ ξζ ξ ζ ξζ=

RS|

T|

UV|

W|+

RS|

T|

UV|

W|+

RS|

T|

UV|

W|+

RS|

T|

UV|

W|= + + +

2

2

2

1

1

1

2

2

2

20

11

21

gx ay az a

x hy hz h

x hy hz h

x ky kz k

g g g g

T

T

T

T

T

T

T

T

T

T

T

Tζ

η ξ ξη ξ ξ ηξ=

RS|

T|

UV|

W|+

RS|

T|

UV|

W|+

RS|

T|

UV|

W|+

RS|

T|

UV|

W|= + + +

3

3

3

2

2

2

3

3

3

30

21

31

where

a T1

18

1 1 1 1 1 1 1 1= − − − −

a T2

18

1 1 1 1 1 1 1 1= − − − −

a T3

18

1 1 1 1 1 1 1 1= − − − −

h T1

18

1 1 1 1 1 1 1 1= − − − −

h T2

18

1 1 1 1 1 1 1 1= − − − −

h T3

18

1 1 1 1 1 1 1 1= − − − −

k T= − − − −18

1 1 1 1 1 1 1 1

x T= x x x x x x x x1 2 3 4 5 6 7 8

y T= y y y y y y y y1 2 3 4 5 6 7 8

z T= z z z z z z z z1 2 3 4 5 6 7 8

The enhanced covariant strains are given by

∈ = ∈ ∈ ∈ ∈ ∈ ∈ 2 2 2 ξξ ηη ζζ ξη ηζ ζξT

The enhanced strain field in isoparametric space can initially be expressed using 21-α parameter interpolation functions as follows


104

∈ =+

+

+

R

S

||||||

T

||||||

U

V

||||||

W

||||||

=

i

T

T

T

T T

T T

T T

u g

u g

u g

u g u g

u g u g

u g u g

,

,

,

, ,

, ,

, ,

ξ ξ

η η

ζ ζ

η ξ ξ η

ζ η η ζ

ξ ζ ζ ξ

ξ ξη ξζ ξηζη ηζ ηξ ξηζ

ζ ζξ ζη ξηζξ η

e jd ie jd i e je j d ie j e j

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0

2 2 2 2

2 2 2 2

2 2 2 2

ξ η ξζ ηζ ξ ζ η ξη ζ η ζ ηξ ζξ η ξ ζ ξ

ζ ξ ζ ξ ξη ζη ξ η ζ η

α

α

L

N

MMMMMMMM

O

Q

PPPPPPPP

=

ei

ieiE

An element stress field with 12-β parameters is considered:

∑ =

L

N

MMMMMMMM

O

Q

PPPPPPPP

1 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 1

12

η ζξ ζ

ξ ηβ

where the contravariant stresses are defined as

∑ ∑ ∑ ∑ ∑ ∑ ∑= ξξ ηη ζζ ξη ηζ ζξ T

This stress field is similar to the assumed five β stress field used by Pian [P2] for a hybrid stress quadrilateral element. The field satisfies both equilibrium and symmetry conditions.

The final enhanced strain interpolation matrix is assembled by enforcing the L2 orthogonality condition < ∑ , ∈ > L2

− − −z z z ∑∈ ≡1

1

1

1

1

10

T d d dξ η ζ


105

The final interpolation matrix involving eighteen β parameters is

E18 2 2 2 2

2 2 2 2

2 2 2 2

0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0

=

−−

−−

−−

L

N

MMMMMMMM

O

Q

PPPPPPPP

ξ ξη ξζ ξηζη ξη ηζ ξηζ

ζ ηζ ξζ ξηζξ η ξ η ξζ ηζ ξ ζ η ζ

η ζ η ζ ηξ ζξ η ξ ζ ξζ ξ ζ ξ ξη ζη ξ η ζ η

A further enhanced strain interpolation matrix is also derived which is similar to an interpolation field defined in [S8] for planar elements. This matrix is based on nine a parameters and is also orthogonal to the twelve β stress field

E9

0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

=

L

N

MMMMMMMM

O

Q

PPPPPPPP

ξη

ζξ η

η ζζ ξ

Both the final interpolation functions E9

and E18

also allow condition III to be satisfied. This is a requirement for passing the patch test [S8] and is implied in the sense that

− − −z z z ≡1

1

1

1

1

10

dE d dξ η ζ

Evaluation of stresses The evaluation of stresses is identical to that described in section 7.4.1.2.

Nonlinear formulation The comments made in section 7.4.1.3 regarding the nonlinear capability of the standard isoparametric element are also applicable to this element. The nonlinear formulation for the enhanced strain element involves enforcing orthogonality between assumed Green-Lagrange strains and 2nd Piola-Kirchhoff stresses. The geometrically nonlinear performance of this element is much improved in comparison with HX8.

7.4.3 3D Explicit Dynamics Elements (HX8E, PN6E, TH4E)

Explicit time integration schemes have used simple linear elements rather than those of a higher order by virtue of their computational efficiency. A number of further advantages may also be obtained in explicit dynamic analyses


106

The use of higher order shape functions creates difficulties at the contact interface in the form of uncontrolled overlap. It has been shown that higher order continuum elements require a time step

reduced from that of linear elements. The mass lumping formulations for higher order elements are currently

impractical for modelling shock wave propagation since the resulting numerical noise pollutes or destroys the solution. The combination of mass lumping with linear elements, when applied in

conjunction with the central difference operator, increases accuracy in solutions by virtue of their respective compensatory spectral errors.

The linear explicit dynamics elements have been implemented to take advantage of these benefits. They are for use only with the explicit central difference time integration scheme.

The explicit dynamics elements are based upon the isoparametric approach in which the same shape functions are used to interpolate both the displacements and geometry, i.e.


n=

=∑ ξ η,b g

1

geometry X N Xi ii

n=

=∑ ξ η,b g

1

where Ni ξ η,b g is the element shape function for node i and n is the number of nodes. Fig.7.4.3-1 shows the nodal configurations available within LUSAS. The nodal degrees of freedom are U, V and W at each node.

Evaluation of current strain increments The velocity strain rates e t+Dt/2are defined from the midpoint velocity ij gradients in the global axis system. A rate relationship is used to define the strain-displacement characteristics as

t UXx

t

t&

&

∈ =∂∂

t VYy

t

t&

&

∈ =∂∂

t WZz

t

t&

&

∈ =∂∂

t UY

VXXY

t

t

t

t&

& &γ

∂∂

∂∂

= +


107

t VZ

WYYZ

t

t

t

t&

& &γ

∂∂

∂∂

= +

t UZ

WXXZ

t

t

t

t&

& &γ

∂∂

∂∂

= +

Evaluation of modulus matrices The isotropic and orthotropic elastic modulus matrices are as follows

Isotropic

D E=

− −

−

−

−

−

−

−

L

N

MMMMMMMMMMMMMM

O

Q

PPPPPPPPPPPPPP

1 1 2

1 0 0 0

1 0 0 0

1 0 0 0

0 0 01 2

20 0

0 0 0 01 2

20

0 0 0 0 01 2

2

υ υ

υ υ υ

υ υ υ

υ υ υ

υ

υ

υ

a fa f

a fa f

a fa f

a fa f

Orthotropic

D

E E EE E EE E E

GG

G

x yx y zx z

xy x y zy z

xz x yz y z

xy

yz

xz

=

− −− −− −

L

N

MMMMMMMM

O

Q

PPPPPPPP

−1 0 0 01 0 0 0

1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

1// / // / /

//

/

υ υυ υυ υ

to maintain symmetry the following relations are utilised


Note that a valid material is obtained only if




Isotropic ∈ =0d itTTΔ α α α, , , 0, 0, 0


108


TΔ α α α α α α, , , , , z xy yz xz

The lumped mass matrix is computed as each node i as

tx

t tM Vi

= 1 8/ ρ

ty

t tM Vi

= 1 8/ ρ

where t v is the current volume of an element.

Integration rule for the elements A one point quadrature integration rule is utilised. This provides elements that are efficient and do not lock when incompressible behaviour is being modelled, e.g. plastic straining with von Mises plasticity. The stresses are integrated at the most accurate location.

The location of the integration point is given in Appendix I.

Element stabilisation The utilisation of one point Gauss quadrature has a limitation in that zero energy deformation or hourglass modes are generated (see Fig.7.3.3-5). The effects of such modes are minimised by the viscous damping technique [H7].

The technique provides a damping force capable of preventing the formation of spurious modes but which has negligible influence on the true structural modes. This is possible since the spurious modes are orthogonal to the real deformations.

The rate of diagonal drifting is defined by the velocity at which the mid-points of the element are separating. This is utilised as the basis for hourglass detection, giving the hourglass velocities as

h xij ik

j

==∑ & i = 1,3jkΓ a f

1

4

The viscous hourglassing forces are

f h hik hg ijj

hg ijj

= − +L

NMM

O

QPP

= =∑ ∑ Q v c / 4 Q e

2/3jk jkρ Γ Γd i d i

1

4

1

41 100

in which ve is the current element volume, Qhg is a constant which is modified via the SYSTEM command and is usually set to a value between 0.05 and 0.15, and &xi

k is the nodal velocity of the kth node in the ith direction. ρ is the current element density, while c, the material sound speed is defined from


109

cE2 1

1 1 2=

−+ −

υρ υ υa f

a fa f

The hourglass base vectors Γij for the 8 node solid elements are given as

Γij

T

=

− − − −− − − −

− − − −− − − −

L

N

MMMM

O

Q

PPPP

1 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 1

these viscous forces are included directly into the element force vector.

Shock wave smoothing The shock discontinuities that occur in impact problems may promote numerical instabilities which must be smoothed out. This is achieved using an artificial bulk viscosity method. The salient characteristic of the method is the augmentation of element pressure with an artificial viscous term (q) prior to the evaluation of the element internal force. This is zero in expanding elements and non-zero in contracting elements. The algorithm has the effect of spreading the shock front over a small number of elements.

The exact form of artificial viscosity is somewhat arbitrary and the method used is based on the formulation originally proposed in [V1]

q Q= +ρ L D Q L D cc kk 1 c kk 2

where Q1 and Q2 are dimensionless constants which default to 1.5 and 0.06 respectively, and may be modified as necessary via the SYSTEM command. Dkk is the trace of the velocity strain tensor and Lc is the characteristic length of the element which is related to the smallest element diagonal as

L VAc

f=

where V is the current element volume and Af the current largest face area of the element. The face area is evaluated by considering each face in turn and using

A Jf = 4 0 0,a f The surface Jacobian J may be evaluated from

J x x0 0, *a f =∂∂ξ

∂∂η

in which the differentials are evaluated explicitly.


110

The quadratic term in strain rate is chosen to be small except in regions of very large gradients. The linear term, however, is included to control the small spurious oscillations following the shock waves in which the gradients are insufficient to make the quadratic term effective. Care should be taken with the linear term since there is a danger of distorting the solution.

In converging geometries the centred strain rate term is negative and the q term is then non-zero. This occurs even though no shocks are generated and results in a non-physical generation of pressure. In view of the abundance of excellent results, however, it is generally agreed that the effect is negligible.

The direct stresses at time t+Δt are modified by the addition of the artificial viscosity pressure q as follows

σ σii ii q= +

Nonlinear formulation The 3-D explicit dynamics elements can be employed in

Materially nonlinear dynamic analysis (see note 1.) Geometrically nonlinear dynamic analysis. Geometrically and materially nonlinear analysis utilising the nonlinear

material laws specified in note I. Notes

The 3D explicit dynamics elements may be used with nonlinear material models 61 to 64, 72 and 75 (section 4.2). Eulerian geometric nonlinearity is always invoked with the use of the explicit

elements in which the velocity strain measure is utilised. The Jaumann stress rate formulation is used to eliminate rigid motion prior to stress integration. The output is in terms of true Cauchy stresses and the strains approximate to logarithmic strains. The loading is non-conservative.

Evaluation of stresses The element output can be obtained at both the element nodes and Gauss points and consists of

Stress Output σ σ σ σ σ σX Z XY YZ XZ, , , , , Y the direct and shear stresses

Strain Output ∈ ∈ ∈X Z XY YZ XZ, , , , , Y γ γ γ the direct and shear strains

Principal stresses and strains and the corresponding direction cosines may also be output.


111

The sign convention for stress and strain output is shown in fig.7.4.3-3.

The Gauss point stress is usually more accurate than the nodal values.

The nodal values of stress and strain are obtained using the extrapolation procedures detailed in section 6.1.

1

2

3

4

5

6

78

1

2

3

4

5

6

1

2

3

4

HX8E

PN6E

TH4E

Fig.7.4.3-1 Nodal Configuration For 3d Explicit Dynamics Elements


112

PN6E Elements

HX8E Elements

Fig.7.4.3-2 Compact Tension Fracture Specimen

Y

X

Z

σXZ

σYZ

σZ

σXY

σYZ

σY

σXZ

σXY

σX

Arrows indicate +vestress directions



113

7.4.4 Composite Solid Elements (HX8L,HX16L,PN6L PN12L)

If brick elements are used for an analysis of composite structures the number of degrees of freedom even for small laminate structures rapidly becomes very large leading to prohibitively excessive computer costs. To overcome this difficulty layered brick elements were developed where several laminae are included in a single element. For these elements the three degrees of freedom per node are used to interpolate a displacement field that varies linearly over the thickness and quadratically in-plane for the higher order elements. Each layer is specified by an orthotropic material stiffness matrix.

In order to speed up the computation, the elements are restricted to reasonably constant layer thicknesses [H13]. This limitation requires the calculation of only a 2x2 Jacobian matrix. For the integration of the element stiffness matrix, the material stiffnesses are summed layerwise through the thickness, while the strain-displacement matrices by default are integrated using a plane 2x2 (for HX8L and HX16L), or a single point (for PN6L), or a 3 corner point quadratic (for PN12L) gauss integration scheme outside the through thickness loop.

The shape functions for the top and bottom surfaces of the composite elements can be considered to be single membrane element shape functions, see figure 7.4.4-1. The shape functions N N Ni top i bot i( ) ( )= = , are defined in terms of natural coordinates ξ and η, for the HX16L element these are given by:

Ni i i i i= + + + −14

1 1 1ξ ξ η η ξ ξ η ηb gb gb g for corner nodes

Ni i i i i= − − + +12

1 12 2 2η ξ ξ η ξ ξ η ηe jb g for mid-side nodes

The PN12L, HX8L, and PN6L elements use the equivalent shape functions for 6, 4 and 3 noded membranes. To form the complete shape functions for the brick element Nbr , linear interpolation is used between the functions for the top and bottom surfaces:

N N NbrT

i botT

i topT= − + +

12

1 1ζ ζb g b ga f b g;

The in-plane and through-thickness shape functions can then be separated to give:

NbrT T T= +φ ζψ

where

φTiT

iTN N=

12

;


114

ψTiT

iTN N= −

12

;

The displacement field, U, can now be interpolated as:

Uuvw

T T T T

T T T T

T T T T=

+

+

+

L

N

MMMM

O

Q

PPPP

RS|T|

UV|W|

φ ζψ

φ ζψ

φ ζψ

0 00 00 0

U Ha=

with the displacement vectors in terms of the nodal degrees of freedom:

u u u unT= 1 2, ,..............l q

v v v vnT= 1 2, ,..............l q

w w w wnT= 1 2, ,...........l q

The three-dimensional strain vector ∈ is defined as

∈ = + + +RST

UVWT u

xvy

wz

uy

vx

vz

wy

uz

wx

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

, , , , ,

The strain displacement relationship is given by:

∈= Ba

where B is the strain displacement matrix.

B

x x

y y

c

y y x x

c y y

c x x

T TT T

TT T

T

T T T

T T T TT

T TT T

T TT T

=

+

+

+ +

+

+

L

N

MMMMMMMMMMMMMMMMM

O

Q

PPPPPPPPPPPPPPPPP

∂φ

∂ζ

∂ψ

∂∂φ

∂ζ

∂ψ

∂

ψ

∂φ

∂ζ

∂ψ

∂

∂φ

∂ζ

∂ψ

∂

ψ∂φ

∂ζ

∂ψ

∂

ψ∂φ

∂ζ

∂ψ

∂

0 0

0 0

0 0 2

0

0 2

2 0

B can be split into two matrices combining in-plane and through thickness terms:


115

B B B= +1 2

ζ

where

B

x

y

c

y x

c y

c x

TT T

TT

T

T T T

T TT

T TT

T TT

1

0 0

0 0

0 0 2

0

0 2

2 0

=

L

N

MMMMMMMMMMMMMMMMM

O

Q

PPPPPPPPPPPPPPPPP

∂φ

∂∂φ

∂

ψ

∂φ

∂

∂φ

∂

ψ∂φ

∂

ψ∂φ

∂

B

x

y

y x

y

x

TT T

TT

T

T T T

T TT

T TT

T TT

2

0 0

0 0

0 0 0

0

0 0

0 0

=

L

N

MMMMMMMMMMMMMMMMM

O

Q

PPPPPPPPPPPPPPPPP

∂ψ

∂∂ψ

∂

∂ψ

∂

∂ψ

∂∂ψ

∂∂ψ

∂

The restriction of constant layer thicknesses provides an uncoupling between the in-plane coordinates and the through-thickness coordinate. Consequently for the transformation of the cartesian derivatives into the natural derivatives only a 2 dimensional Jacobian matrix is required.

∂∂ξ∂

∂η

∂∂ξ

∂∂ξ

∂∂η

∂∂η

∂∂∂

∂

RS||

T||

UV||

W||

=

L

N

MMMM

O

Q

PPPP

RS||

T||

UV||

W||

x y

x yx

y

or inverted

∂∂

∂∂ξ

∂∂ηx

J J= +− −11

112

1

∂∂

∂∂ξ

∂∂ηy

J J= +− −21

122

1

and an integration constant for the thickness is computed from:

z cz c

= → =2

2ζ

∂∂

∂∂ζ

.

where c is the depth of the element see figure 7.4.4-1. The differential of the volume is given by

dV c J=2

d dζ ξ η


116

where |J| is the Jacobian determinant.

The element stiffness matrix in basic form may be defined as

K B DBTV

= z dV

where D is the modulus matrix for an orthotropic material.

D

E E EE E EE E E

GG

G

x yx y zx z

xy x y zy z

xz x yz y z

xy

yz

xz

=

− −− −− −

L

N

MMMMMMMM

O

Q

PPPPPPPP

−1 0 0 01 0 0 0

1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

1// / // / /

//

/

υ υυ υυ υ

where υ υ υyx zx, and zy are defined by



As the matrices B1 and B

2 are independent of ζ, only D varies from layer to layer.

Therefore the strain-displacement matrices can be left out of the integration through the thickness:

K

B D d B B D d B

B D d B B D d B

c J d d

Tnlay

n

nlayT

nlayn

nlay

Tnlay

n

nlayT

nlayn

nlay=

LNMM

OQPP +

LNMM

OQPP

+LNMM

OQPP +

LNMM

OQPP

F

H

GGGGGG

I

K

JJJJJJ

z∑ z∑

z∑ z∑zz = =

= =

11

1 11

2

21

1 22

12

2

ζ ζ ζ

ζ ζ ζ ζ

η ξζ ζ

ζ ζ

ηξ

with B1 and B

2as:


117

B

J J

J J

c

J J J J

cJ J

cJ J

T TT T

TT T

T

T T T

T T T TT

T TT T

T TT T

1

111

121

211

221

211

221

111

121

211

221

111

121

0 0

0 0

0 0 2

0

0 2

2 0

=

+

+

+ +

+

+

L

N

MMMMMMMMMMMMMMMMM

O

Q

PPPPPPPPPPPPPPPPP

− −

− −

− − − −

− −

− −

∂φ

∂ξ

∂φ

∂η∂φ

∂ξ

∂φ

∂η

ψ

∂φ

∂ξ

∂φ

∂η

∂φ

∂ξ

∂φ

∂η

ψ∂φ

∂ξ

∂φ

∂η

ψ∂φ

∂ξ

∂φ

∂η

B

J J

J J

J J J J

J J

J J

T TT T

TT T

T

T T T

T T T TT

T TT T

T TT T

2

111

121

211

221

211

221

111

121

211

221

111

121

0 0

0 0

0 0 0

0

0 0

0 0

=

+

+

+ +

+

+

L

N

MMMMMMMMMMMMMMMMM

O

Q

PPPPPPPPPPPPPPPPP

− −

− −

− − − −

− −

− −

∂ψ

∂ξ

∂ψ

∂η∂ψ

∂ξ

∂ψ

∂η

∂ψ

∂ξ

∂ψ

∂η

∂ψ

∂ξ

∂ψ

∂η∂ψ

∂ξ

∂ψ

∂η∂ψ

∂ξ

∂ψ

∂η

The through thickness dependency is condensed in the integration of the material modulus matrix which makes the assembly of the element stiffness matrix more efficient. The strain displacement matrices only have to be computed in-plane. This is possible by restricting the element to a reasonably uniform thickness for a single layer.

Nonlinear formulation The 3-D solid composite elements can be employed in

Materially nonlinear analysis utilising the elastoplastic constitutive laws [O1] (section 4.2). Geometrically nonlinear analysis utilising the corotational formulation

(section 3.5.2). Geometrically and materially nonlinear analysis utilising the nonlinear



118

Nonlinear dynamics utilising the nonlinear material laws specified in 1. Linear eigen-buckling analysis.

Fig.7.4.4-1 Topology Of 3-D Layered Isoparametric Bricks

7.4.5 Two Phase 3D Continuum Elements (TH10P, PN12P, PN15P, HX16P and HX20P)

Formulation These isoparametric finite elements utilise the same shape functions to interpolate the displacements and geometry, i.e.

displacements U N Uii

n

i==∑ ξ η,b g

1


nξ η,b g

1


119

where Ni ξ η,b g is the element shape function for node i and n is the number of nodes. However, for consideration of stability, the pressures are only interpolated using the corner nodes

pressures P N Pii

n

i==∑ ξ η,b g

1

corner

where ncorner is the number of corner nodes. Fig.7.4.5-1 shows the nodal configurations available within LUSAS.

The nodal degrees of freedom are U, V, W and P at the corner nodes and U, V and W at the midside nodes.


The details of elastic modulus matrices applicable for these elements are described in section 7.3.1.

These elements are used to model the behaviour of a two phase medium such as soil. In this instance the two phases comprise the soil skeleton and the pore water fluid. Separate equations are derived for each phase, coupled by the interaction of the pore pressure and the soil deformation. The soil skeleton is analysed in terms of effective stress (total stress minus pore water pressure), taking into account the loading due to the pore pressure; whilst the pore fluid analysis takes account of the volumetric strain due to the soil skeleton deformation.

The finite element method is used to solve the coupled equations in terms of nodal displacements and pore pressures. Five 3D elements TH10P, PN12P, PN15P, HX16P and HX20P based on a mixed displacement-pressure formulation are available in LUSAS to solve these problems.

Undrained/fully drained conditions In this type of analysis no consolidation is assumed to take place and the coupled governing equations for static undrained conditions can be expressed as:

K L

L SUP

F F

L U SPT T

extL

NMMOQPPRSTUVW =

−

− −

RS|T|

UV|W|

ΔΔ

int

where the matrices K L, and S are defined as:

K B D BT

v

= z dv'

L B mNT

v

= z- dv


120

SK

N Ne

T

v

= z - dv1

K is the tangent stiffness matrix

L is the coupling matrix

S is the compressibility matrix, where Ke is the equivalent bulk modulus of the soil (see section 7.3.5.4) and D' the ‘effective’ soil modulus matrix.

ext F and int F are external and internal equivalent nodal forces Under static fully drained conditions the coupled governing equations can be further simplified as

K

IUP

F F

ext0

0 0LNMOQPRSTUVW =

−RSTUVW

ΔΔ

int

where I is a unit matrix block.

Drainage/consolidation process In the drainage/consolidation process, fluid flow in/out from the soil needs to be considered. For linear transient consolidation the coupled governing equations can be expressed as:

K L

L THUP

K L

L THUP

FQT

t tT

t

βδ β δδ

LNMM

OQPPRSTUVW =

−

LNMM

OQPPRSTUVW +RSTUVW+ ( )1

ΔΔ

where:

ΔF is the incremental load ΔQ the incremental flow β the time stepping scheme parameter (set to 1.0 for backward Euler scheme) H the permeability matrix

The permeability matrix H is defined in terms of the shape function derivatives and a permeability matrix of the soil, K

p, as:

H N K NTp

v

= ∇ ∇z dv

For nonlinear consolidation, the coupled governing equations can be written as


121

K LL t H

UP

F Ft Q Q tH P L U UT

k

k

n k n k

n n k T n k nΔΔΔ Δ Δβ βΔ

LNM

OQPRSTUVW

=−

− + − − −RS|T|

UV|W|+

+

+ext int

+

1 1

1 1c h c h

where the superscript on the left/right hand side represents the increment/iteration number.

Material assumptions The bulk modulus of the soil particle Ks is very large compared to the bulk modulus of the pore fluid Kf . Therefore the overall compressibility of the soil mass is approximated to be that of the pore fluid.

1 1K K K Ke f s f

= +−

≡η η η( )

where:

Ke is the equivalent bulk modulus of the soil Kf the bulk modulus of the pore fluid Ks the bulk modulus of the solid soil particle η the porosity of the soil

In practical geotechnical applications it is usually difficult to determine Kf and Ks so a large value of the equivalent modulus Ke is usually assumed, 1012>Ke>109.

Nonlinear formulation The two phase 3-D continuum elements can be employed in

Materially nonlinear drained/undrained/consolidation analysis utilising the elastoplastic constitutive laws [O1] (section 4.2). Geometrically nonlinear drained/undrained/consolidation analysis. Geometrically and materially nonlinear drained/undrained/consolidation

analysis utilising the nonlinear material laws specified in 1. Geometrically and materially nonlinear dynamic drained/undrained analysis

utilising the nonlinear material laws specified in 1.


122

7.5 Space Membrane Elements 7.5.1 Axisymmetric Membrane (BXM2, BXM3) Formulation BXM2 and BXM3 elements are axisymmetric, isoparametric membrane elements. They are defined in the XY-plane and symmetry may be specified about either the X or Y axes. The nodal degrees of freedom are (fig.7.5.1-1)

U and V at each node

The infinitesimal strain-displacement relationship is defined in the local Cartesian system by

∈ =xux

∂∂

∈ =zUR

The elastic modulus matrix is defined by

D E=

−LNMOQP1

112υ

υυ


∈ =o tTTb g Δ α α,



Stress Output σx Meridional stress (+ve tension)

σz Circumferential stress (+ve tension)

Strain Output ∈x Meridional strain (+ve tension)

∈z Meridional stress (+ve tension)

The element local x-axis lies along the element axis in the direction in which the element nodes are specified. The local y and z axes form a right-hand set with the x-

Space Membrane Elements

123

axis such that the y-axis lies in the global XY-plane and the z-axis is parallel to the global Z-axis (up out of page) (fig.7.5.1-4).

The Gauss point stresses are usually more accurate than the nodal values.

The nodal values of stress and strain are obtained using the extrapolation procedures detailed in section 6.1.

Nonlinear formulation The axisymmetric membrane elements can be employed in

Materially nonlinear analysis utilising the elasto-plastic constitutive laws [O1] (section 4.2). Geometrically nonlinear analysis. Geometrically and materially nonlinear analysis utilising the nonlinear


Notes

The geometric nonlinearity utilises a Total Lagrangian formulation which accounts for large displacements but small strains. The nonlinear strain-displacement relationship is defined by


OQPx

ux

ux

ux

∂∂

∂∂

∂∂

12

12

2 2

∈ = + LNMOQPx

UR

UR

12

2


Y,V

X,U

1

2

1

2

3

BXM2

BXM3


124

Fig.7.5.1-1 Nodal Configuration For BXM2 And BXM3 Elements


(b) Circular Plate


(b) Circular Pipe

Fig.7.5.1-2 'Stand-Alone' Applications For BXM2 And BXM3 Elements


125


QAX8 elements

BXM3 elements

Fig.7.5.1-3 Fibre Reinforced Cylinder Illustrating Coupling Between QAX8 And BXM3 Elements

Y

X

1

2

xy

3

2

yx

y

y

x

xx

y

1

Fig.7.5.1-4 Local Cartesian System For Bxm2 And Bxm3 Elements


126

7.5.2 3-D Space Membrane (SMI4, TSM3) Formulation SMI4 and TSM3 elements are membrane elements that function in 3-D. They are formulated in 2-D, by forming a local Cartesian system in the plane of the element (using a least squares fit through the element nodes). Once the element matrices have been formed they are then transformed to the global Cartesian basis.

Their formulations are exactly equivalent to their 2-D conterparts given in table 7.5.2-1

Space Membrane Plane Membrane

SMI4 PMI4

TSM3 TPM3

TABLE 7.5.2-1 Space Membrane Elements And Equivalent Plane Elements

The nodal configurations are shown in fig.7.5.2-1. The nodal degrees of freedom are

U, V and W at each node

Only a lumped mass matrix is evaluated using the procedure defined in (section 2.7).

Evaluation of stresses The element output obtained at the element nodes consists of

Stress Resultant Output Nx , Ny , Nxy the direct and shear stress resultants/unit length Nmax,Nmin the maximum and minimum principal stress resultants/ unit

length b the angle between the maximum principal stress resultant and

the positive X-axis

Strain Output ∈ ∈x y xy, , γ the direct and shear strains ∈ ∈max min, the maximum and minimum principal strains β the angle between the maximum principal strain and the

positive X-axis The sign convention for stress resultant and strain output is shown in fig.7.5.2-4. The stress resultants are evaluated directly at the nodes.

Nonlinear formulation The element has no nonlinear capability, but may be utilized in a nonlinear environment. The element cannot be used for linear buckling analyses.


127

Y, V

X, U

Z, W

TSM31

3

2 1 2

34

SMI4

Fig.7.5.2-1 Nodal Configuration For SMI4 And TSM3 Elements

Thin membrane

Stiffening members

Problem definition

QSI4 elements

SMI4 Elements

Finite Element Mesh

Fig.7.5.2-2 Box Structure Illustrating The Use Of Space Membrane Elements


128

Y

X

Z

12

3

4

x

yz

Fig.7.5.2-3 Local Cartesian System For SMI4 And TSM3 Elements

Y

X

σY

σX

σY

σX

σXY

σXY



7.6 Plate Elements 7.6.1 Isoflex Thin Plate (QF4, QF8, TF3, TF6) Formulation The Isoflex family of thin plate elements are formed by applying Kirchhoff constraints within elements formulated using Mindlin plate assumptions. The displacements and rotations are considered independent and the unconstrained nodal configurations are (fig.7.6.1-1)

Plate Elements

129

w, , x yθ θ at the corner

Δ Δθ Δθw, , x y at the mid side nodes of the quadrilateral,

Δθ Δθx y , at the central node of the triangle.

where Δθ and θ re the relative (departure from linearity) and absolute rotations of the through-thickness normals after deformation. These rotations include the transverse shear deformations (fig.7.6.1-2). An element with thin plate performance is then produced by constraining the shear strains to zero at discrete points within the element. These constraints provide extra equations that permit certain nodal degrees of freedom to be discarded. The final nodal configurations are (fig.7.6.1-3)

w, , x yθ θ at the corner

Δθ at the mid side nodes

where Δθ is the relative rotation about a tangent to the element edge. This removes 8 and 11 degrees of freedom for the 6 and 3 noded triangles and 11 and 15 degrees of freedom for the 8 and 4 noded quadrilaterals respectively. This is achieved by using the following constraints, originally proposed by Irons for the Semiloof shell [I1]

γ∂∂

θt ywx

= − = 0

At the points shown in fig.7.6.1-4, Where γ t is the through-thickness shear strain tangential to the element edges. This provides 6 and 8 constraints respectively for the triangles and quadrilaterals which are suitable for eliminating the mid-side translation and normal rotation.

γ XZAdAz = 0 , γ YZA

dAz = 0

Where the integral is performed using 2*2 Gauss quadrature. This provides 2 constraints for both the triangles and quadrilaterals, which are suitable for removing the rotations at the central node.

γ nSdSz = 0

Where γ n is the transverse shear strain normal to the element sides and the integral is performed using 2-point quadrature along each side. This provides 1 constraint suitable for removing the central translation of the quadrilaterals.

These constraints are sufficient for the higher order elements and the extra constraints required for the lower order elements are provided by enforcing a linear variation of tangential rotation along the element sides.


130

The infinitesimal strain-displacement relationship is derived from the 3-D continuum relationship [Section 7.4] by neglecting ∈Z which is zero in the Mindlin plate assumptions, and γ XZ and γ YZ which have been constrained to zero, so that

∈ =XUX

∂∂

∈ =YVY

∂∂

γ∂∂

∂∂XY

UY

VX

= +

The continuum displacements for plates of varying thickness are related to the original degrees of freedom of the plate using

U zt

Nii

n

i Yi=

=∑ ξ η θ,b g

1

t

V zt

Nii

n

i Xi= −

=∑ ξ η θ,b g

1

t

W Nii

n

i==∑ ξ η,b g

1

W

where t and ti are the thicknesses of the plate at the integration and nodal points respectively, and N(ξ,η) are the element shape functions. Therefore the discretised, generalised, flexural strain-displacement relationship is

ψψ

ψ

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

θθ

X

Y

XY

i i

i i

i i i i

i

X

Yi

n ttX

NX

tt

NX

ttY

NY

tt

NY

ttY

NX t

tX

NY

tt

NY

tt

NY

W

i

i

L

NMMM

O

QPPP

= −

+ −

L

N

MMMMMM

O

Q

PPPPPP

L

N

MMM

O

Q

PPP=∑

1 0

1 0

1 1

1

1

1 11

where the terms involving ∂ ∂t X/ and ∂ ∂t Y/ are the small strain contributions due to thickness variations. For flat plates

ψψ

ψ

∂∂∂∂∂

∂ ∂

X

Y

XY

WXW

YW

X Y

L

NMMM

O

QPPP

=

−

−

−

L

N

MMMMMMM

O

Q

PPPPPPP

2

22

22

2

Plate Elements

131

The isotropic and orthotropic elastic resultant modulus or rigidity matrices are

Isotropic ψ α α0

0 0 0e jt

Td Tdz

=( ) , , , ,Δ

Orthotropic ψ α α α α α0e jt x y xy yz xz

Td Tdz

=( ) , , , ,Δ


Note. For a valid material υxy x yE E< ( / ) /1 2


Isotropic ψ α α0

0e j a ft

Td Tdz

=Δ

, ,

Orthotropic ψ α α α0e ja f

tx y

Td Tdz

=Δ

, , xy

Full details of the element formulation are given in [L1].

Both consistent and lumped mass matrices are available and are evaluated using the procedures defined in (section 2.7).

Evaluation of stresses/strains The element output obtained at the element nodes or Gauss points consists of

Stress Resultant Output MX Y XY, M , M the moments/unit width in the global Cartesian system.

Strain Output ψ ψ ψX Y XY, , the flexural strains in the global Cartesian system.

The sign convention for stress resultant and strain output is shown in fig.7.5.2-6.

The Gauss point values are usually more accurate than the nodal values.

The nodal stress resultants are evaluated using the extrapolation procedures detailed in section 6.1.

Note. Approximate shear forces evaluated by differentiating the moments may also be output.

Nonlinear formulation The element has no nonlinear capability, but may be utilised in a nonlinear environment. The element cannot be used for linear buckling analysis.


132

Y

X

Z

W

θx

θy1

W

θx

θy

W

θx

θy

W

θx

θy

W

θy

θx

W

θx

θy

W

θx

θy

W

θy

θx

W

θx

θy

W

θx

θy

Y W

θx

θy

W

θx

θy

W

θx

θy

W

θx

θy

W

θx

θy

Quadrilateral Element Triangular Element

2 3

4

567

89

2 3

4

5

6

1

Fig.7.6.1-1 Initial Nodal Configuration For Isoflex Plate Element

Z

XθY

θY

γXZ

∂ ∂W X/Displacement of any

point a distance zalong normal is

U = z θYwhere

θ∂∂

γY XZWX

= −

Fig.7.6.1-2 Rotation Of The Through-Thickness Normal For A Thick Plate

Plate Elements

133

Z

Y

X

W

θx

θy

W

θx

θy

θxW

θx

θy

W

θx

θy

QF4

34

12

W

θx

θy

W

θx

θy

W

θx

θy

W

θx

θy

TF3

W

θx

θy

W

θx

θy

W

θx

θy

Δθ

Δθ

Δθ

Δθ

QF8

W

θx

θy

W

θx

θy

W

θx

θy

Δθ

ΔθΔθ

1 2

6

5

8

47

3

1

2

3

4

5

6

TF6

Fig.7.6.1-3 Final Nodal Configuration For Isoflex Thin Plate Elements

2

1 3/ 1 3/

1 3/

1 3/

1 3/

1 3/

2 2

1 3/ 1 3/

2

1 3/

1 3/

1 3/

1 3/

2

Fig.7.6.1-4 Locations Where The Transverse Shear Strain Tangential To The Element Edge Is Constrained To Zero


134


(b) Finite Element Mesh

Y

X

QF4 elements

Fig.7.6.1-5 Thin Cantilever Plate Illustrating Use Of QF4 Element

Plate Elements

135

Z

X

MY

Y

MXY

MXY

MY

MX

MXYMXY

MX

Fig.7.6.1-6 Sign Convention For Stress Resultant Output

7.6.2 Isoflex Thick Plate (QSC4) Formulation The Isoflex thick plate element QSC4 is formed by imposing an assumed shear strain field on the isoflex thin plate element QF4 [section 7.6.1]. This is accomplished by first forming the constrained flexural strain-displacement relationship in exactly the same manner as for the QF4 element, and then imposing a bilinear shear strain field defined using nodeless degrees of freedom, i.e.

∈ = ∈ + ∈XZ XZ XZP P1 1 3 3 and ∈ = ∈ + ∈YZ YZ YZP P2 2 4 4

where ∈XZi and ∈YZi are the transverse shear strains along the element sides and Pi are linear interpolation functions defined in fig.7.6.2-1.

The extra higher order degrees of freedom are condensed out before assembly so that the final nodal configuration is (fig.7.6.2-2)

w X Y, , θ θ at the corner nodes

The elastic resultant modulus or rigidity matrix is defined as


136

$$

$DD

Db

s

=L

NMM

O

QPP

0

0

where, for isotropic materials

$D Et=

− −

L

N

MMMM

O

Q

PPPP

3

212 1

1 01 0

0 01

2υ

υυ

υe j a f

$.4

D Ets

=+LNMOQP2 1

1 00 1υa f

and for orthotropic materials

$/ /

/ //

D tE E

E EG

b

x xy x

yx y y

xy

=−

−

L

N

MMM

O

Q

PPP

−3

1

12

1 01 0

0 0 1

υυ

and $.

D t GGs

yz

xz=LNM

OQP1 2

00




Isotropic ψ α α0

0 0 0e j a ft

Td Tdz

=Δ

, , , ,

Orthotropic ψ α α α α α0e ja f

tx y

Td Tdz

=Δ

, , , , xy yz xz

Full details of the element formulation are given in [C4].

Both consistent and lumped mass matrices are available and are evaluated using the procedures defined in section 2.7.

Evaluation of stresses/strains The element output obtained at the element nodes and Gauss points consists of

Stress Resultant Output MX Y XY, M , M the moments/unit width in the global Cartesian system,

Plate Elements

137

SX Y, S the shear forces/unit width in the global Cartesian system.

Strain Output ψ ψ ψX Y XY, , the flexural strains in the global Cartesian system,

γ γYZ XZ, the shear strains in the global Cartesian system.


The Gauss point values are usually more accurate than the nodal values. The nodal stress resultants are evaluated using the extrapolation procedures detailed in section 6.1.

Nonlinear formulation The element has no nonlinear capability but may be utilised in a nonlinear environment.

The element cannot be used for linear buckling analysis.

1 2

3

1 2

34

1 2

34

P5 P6

P7

P6

1 2

3

P8

Fig.7.6.2-1 Interpolation Functions For Nodeless Freedoms Of The QSC4 Element


138

W

θx

θy

W

θx

θy

θxW

θx

θy

W

θx

θy3

4

12Z

Y

X

Fig.7.6.2-2 Nodal Configuration For The QSC4 Element

Fig.7.6.2-3 Perferated Thick Plate Example Illustrating Use Of QSC4 Element

Plate Elements

139

Y

X

MY

Z

MXY

MXY

MY

MX

MXYMXY

MX

SX

SY

SX

SY


7.6.3 Isoparametric Thick Mindlin Plate (QTF8, TTF6) Formulation The QTF8 and TTF6 elements are isoparametric plate elements formulated using Mindlin plate theory [M3], which assumes that

Normal stress in the transverse stress is negligible in comparison with the in plane stresses, 'normals' to the mid-surface remain straight but not necessarily normal to

the mid-surface after deformation (fig.7.6.3-1). Thus the elements account for the transverse shear effects associated with thicker plates and the elements are termed 'thick' plate elements. The theory also permits treatment of lateral displacement and rotations as independent variables, producing elements which only require C(0) continuity.


W X Y, , θ θ at each node

where θX and θY are the rotation of the normals to the mid-surface and include the effects of shear deformations. The infinitesimal, generalized, flexural strain-displacement relationship is derived from the 3-D continuum strain-displacement relationship by neglecting the out of plane strain, so that


140

ψ∂θ∂X

YX

=

ψ∂θ∂Y

XY

= −

ψ∂θ∂

∂θ∂XY

Y YY X

= −

γ∂∂

θYZ XWY

= −

γ∂∂

θXZ YWY

= +

The elastic resultant modulus or rigidity matrix is defined as

$$

$DD

Db

s

=L

NMM

O

QPP

0

0

where, for isotropic materials

$( ) ( ) /

D Etb

=−

−

L

NMMM

O

QPPP

−3

212 1

1 01 0

0 0 1 2

1

υ

υυ

υ

and

$.

D t GGs

yz

xz=LNM

OQP1 2

00

and for orthotropic materials

$/ /

/ //

D tE E

E EG

b

x xy x

xy x y

xy

=−

−

L

N

MMM

O

Q

PPP

−3

12

1 01 0

0 0 1

1υ

υ

and

$.

D t GGs

yz

xz=LNM

OQP1 2

00



Plate Elements

141


Isotropic ψ α α0

0 0 0e j a ft

Td Tdz

=Δ

, , , ,

Orthotropic ψ α α α α α0e ja f

tx y

Td Tdz

=Δ

, , , , xy yz xz

Both consistent and lumped mass matrices are available and are evaluated using the procedures defined in section 2.7.


Stress Resultant Output MX Y XY, M , M - the moments/unit width in the global Cartesian system,

SX Y, S - the shear forces/unit width in the global Cartesian system.

Strain Output ψ ψ ψX Y XY, , - the flexural strains in the global Cartesian system,

γ γYZ XZ, - the bending strains in the global Cartesian system.


The Gauss point values are usually more accurate than the nodal values.



The elements cannot be used for linear buckling analysis.


142

Z

XθY

θY

γXZ

∂ ∂W X/Displacement of any

point a distance zalong normal is

U = z θYwhere

θ∂∂

γY XZWX

= −

Fig.7.6.3-1 Rotation Of The Through-Thickness Normal For A Thick Plate

Y

X

Z

W

θx

θy1

W

θx

θy

W

θx

θy

W

θx

θy

W

θy

θx

W

θx

θy

W

θx

θy

W

θy

θx

W

θx

θy

Y W

θx

θy

W

θx

θy

W

θx

θy

W

θx

θy

W

θx

θy

QTF8 TTF6

2 3

4

567

8

2 3

4

5

6

1

Fig.7.6.3-2 Nodal Configuration For QTF8 And TTF6 Elements

Plate Elements

143

Fig.7.6.3-3 Perforated Thick Plate Illustrating Use Of The TTF6 Element

Y

X

MY

Z

MXY

MXY

MY

MX

MXYMXY

MX

SX

SY

SX

SY



144

7.6.4 Ribbed Plate (RPI4, TRP3) Formulation The 2-D flat ribbed plate elements are formulated by superimposing standard isoparametric plane stress elements (QPM4,TPM3,PMI4) on the isoflex thin plate elements (QF4,TF3). The membrane and bending stiffnesses are formed independently, and combined to give

KK

aa

RR

membrane

bending

membrane

bending

membrane

bending

00

LNMM

OQPPRS|T|

UV|W|=RS|T|

UV|W|

The component elements are listed in table 7.6.4-1

Element Membrane Bending

RPI4 PMI4 QF4

TRP3 TPM3 TF3

Table 7.6.4-1 Component Elements Used To Form Ribbed Plate Elements

The element is formulated in a local Cartesian basis and then transformed to the global Cartesian system. The final nodal variables are (fig.7.6.4-1)

U X Y, V, W, , θ θ at each node

The strain-displacement relationship, resultant modulus matrix and thermal strains are defined in section 7.3 (in-plane) and section 7.6 (bending).

For further details of the element formulation see Section 7.3, Section 7.6 and [Z1,L1].

A lumped mass matrix is evaluated using the procedures presented in section 2.7.

Evaluation of stresses/strains The element output obtained at the element nodes consists of

Stress Output σ σ σx y xy, , direct and shear stresses in the local Cartesian system,

σ σmax min, the maximum and minimum principal membrane stresses, β the angle between the maximum principal membrane stress and

the local x-axis.

Stress Resultant Output Nx y xy, , N N the membrane stress resultants/unit width in the local

Cartesian system,

Plate Elements

145

Mx y xy, , M M the moments/unit width in the local Cartesian system,

Strain Output ∈ ∈x, , y xyγ the membrane strains in the local Cartesian system,

∈ ∈max, min the maximum and minimum principal membrane strains, β the angle between the maximum principal membrane strain and

the local x-axis, ψ ψ ψx, , y xy the flexural strains in the local Cartesian system,

ψ ψmax, min the maximum and minimum principal bending strains, β the angle between the maximum principal bending strain and

the local x-axis The sign convention for stress resultant and strain output is shown in fig.7.6.4-3.

The local x-axis is defined as being a line joining the first and second element nodes. The xy-plane is defined by the third element node and the local x-axis. The local y and z-axes are defined by a right hand screw rule (fig.7.6.4-4).


The stress resultants are most easily interpreted if the local Cartesian axes are all parallel. Also, the presence of eccentricity requires that the forces and moments are examined at the mid-points of the element sides by averaging the nodal values.



RPI4

34

1 2

TRP3

W

θx

θyV

U

W

θx

θyV

U

W

θx

θyV

U

W

θx

θyV

U

12

3

W

θx

θyV

U

W

θyV

U

θx

W

θx

θyV

U

Z

Y

X

Fig.7.6.4-1 Nodal Configuration For Ribbed Plate Elements


146

RPI4 elements

Finite Element Mesh

Y

XBRP2 elements

Y

X

Z

Fig.7.6.4-2 Ribbed Plate Illustrating Use Of RPI4 Element

Y

X

σY

σX

σY

σX

σXY

σXY


Y

X

MY

Z

MXY

MXY

MY

MX

MXYMXY

MX

Stresses Stress Resultants

Fig.7.6.4-3 Sign Convention For Stress And Stress Resultant Output

Plate Elements

147

Z

X

Y

z

x

y

1

2

3

4

x

y

z

1

23

Fig.7.6.4-4 Local Cartesian System For Ribbed Plate Elements


148

7.7 Shell Elements 7.7.1 Axisymmetric Thin Shell (BXS3) Formulation The BXS3 element is a thin, curved, non-conforming axisymmetric shell element formulated using the constraint technique.

The global displacements and rotations are initially quadratic and are interpolated independently using linear Lagrangian shape functions for the end nodes and a hierarchical quadratic function for the central node. Therefore, the initial degrees of freedom are (fig.7.7.1-1)

U, V, θ at the end nodes

and Δu, Δv, Δθ at the mid-length node.

The Kirchhoff condition of zero shear strain is applied at the two integration points by forcing

∂∂

∂∂

∂∂

θvx

uz

vx z+ = − = 0

and eliminating the local transverse translational and rotational degrees of freedom at the central node. The final degrees of freedom for the element are (fig.7.7.1-1)

U z, V, θ at the end nodes,

and Δu at the mid-length node

where Δu is the local axial relative (departure from linearity) displacement.

The infinitesimal strain-displacement relationship is defined in the local cartesian system as

∈ =xux

∂∂

∈ = −zUR

VR

cos sinφ φ

ψ∂∂x

vx

= −2

2

ψ∂∂

φz Rvx

=1 cos

Shell Elements

149

where R and φ are the radius and angle between the local and global Cartesian systems (fig.7.7.1-2)

The elastic modulus and resultant modulus (or rigidity) matrices are defined as

Explicit $$

$DD

Dm

b

=L

NMM

O

QPP

0

0

where Isotropic $ $D Et D Et

m b=

−LNMOQP =

−

LNMOQP1

11 12 1

112

3

2υ

υυ υ

υυe j

Orthotropic $ $D t EE

D t EEm xz

x xz

xz z bxz

x xz

xz z=

−LNM

OQP =

−

LNM

OQP1 12 12

3

2υ

υυ υ

υυe j

Numerically Integrated

D Ey yy y

y y y yy y y y

t=

−

L

N

MMMM

O

Q

PPPPz 1

11

2 2 2

2 2υ

υ υυ υ

υ υυ υ

dy


Isotropic ψ

αα

α

α

0e ja f

a ft

TT

d Tdy

d Tdy

=

L

N

MMMMMMM

O

Q

PPPPPPP

ΔΔΔ

Δ

Orthotropic ψ

αα

α

α

0e ja f

a ft

x

z

x

z

TT

d Tdy

d Tdy

=

L

N

MMMMMMM

O

Q

PPPPPPP

ΔΔΔ

Δ

Further information on the element formulation is given in [S1,C1,Z1].

The consistent and lumped mass matrices are evaluated using the procedures defined in section 2.7.


150

Evaluation of stresses/strains Element output is available at both the nodes and Gauss points and consists of (fig.7.7.1-5)

Stress Resultants Nx z, N the meridional and circumferential forces/unit width in the

local Cartesian system,

Mx z, M the meridional and circumferential moments/unit width in the local Cartesian system.

Strains ∈ ∈x, z - the meridional and circumferential membrane strains,

ψ ψx, z - the meridional and circumferential bending strains.

The forces and strains are output in the local Cartesian system, defined as having its x-axis lying along the element axis in the direction in which the element nodes are specified. The local y and z-axes form a right-hand set with the x-axis, such that the y-axis lies in the global XY plane, and the z-axis is parallel to the global Z-axis (up out of page) (fig.7.7.1-4).

The top fibre lies on the +ve local y side of the element and +ve values define tension.

The forces have greatest accuracy at the Gauss points.

Note Layer stress output is also available when the nonlinear continuum plasticity model is utilised.

Nonlinear formulation The axisymmetric shell element may be employed in


material laws specified in 1. Nonlinear dynamics utilising the nonlinear material laws specified in 1. Linear and nonlinear buckling analysis.

Notes

The BXS3 element may be used in conjunction with the stress resultant plasticity model (section 4.2). Geometric nonlinearity utilises either

Shell Elements

151

A Total Lagrangian formulation which accounts for large displacements but small strains. The nonlinear strain-displacement relationship is defined by


OQPx

ux

ux

vx

∂∂

∂∂

∂∂

12

12

2 2

∈ = − + + −zuR

vR

uR

vR

uvR

cos sin cos sin sinφ φ φ φ φ2

22

2

22

22 2 22

ψ∂∂

∂∂

∂∂

∂∂

∂∂x

vx

ux

vx

vx

ux

= − − +2

2

2

2

2

2

ψ∂∂

φ∂∂

φ∂∂

φ φz Rvx

uR

vx

vR

vx

= − +1

22

2cos cos cos sin

where R is the radius and φ is the angle between the local and global Cartesian systems.

The forces and strains output with the geometrically nonlinear analysis will be the 2nd Piola-Kirchhoff forces and Green-Lagrange strains respectively, referred to the undeformed configuration. The loading is conservative. or An Updated Lagrangian formulation which takes account of large

displacements and large rotations but small strains, provided that the rotations are small within a load increment. The output now approximates to the true Cauchy stresses and logarithmic strains. The loading approximates to being non-conservative. The initial assumptions used in deriving the BXS3 element limit the

rotations to one radian in a Total Lagrangian analysis and rotation increments of one radian in an Updated Lagrangian analysis (section 3.5).


152

Y

X

1 U

V

θZ

V

U

Δθ Z

θZ

3

2ΔU

ΔV

Initial Variables

Y

X

1 U

V

θZ

V

U

ΔU

θZ

3

2

Final Variables

Fig.7.7.1-1 Nodal Configuration For The BXSs3 Element

v, y

u, x

R

Axis ofRevolution

φ

Fig.7.7.1-2 Definition Of R And Φ For The Axisymmetric Shell Element

Shell Elements

153

Plan

AA

Section A - A


(a) Spherical Shell


(b) Circular Shell

Fig.7.7.1-3 Examples Illustrating The Use Of BXS3 Element


154

Y

X

yx

y x

yx

Fig.7.7.1-4 Definition Of Local Cartesian System For BXS3 Element

X

Y

Mz

y

x

y

xz

z

NxNz

Mx


Shell Elements

155

7.7.2 Flat Thin Shell (QSI4, TS3) Formulation These flat shell elements are formulated in a local Cartesian system by superimposing standard isoparametric plane stress elements (QPM4,TPM3,PMI4) and the isoflex thin plate elements (QF4,TF3). The xy-plane of the local Cartesian system is evaluated using a least squares fit through the element nodes. The membrane and bending stiffnesses are then formed independently, and combined to give

KK

K

aa

RR

M

membrane

bending

art

membrane

bending

z

membrane

bending

z

0 00 00 0

L

N

MMMM

O

Q

PPPP

RS|

T|

UV|

W|=

RS|T|

UV|W|θ

where the component elements are listed in Table 7.7.2-1

Element Membrane Bending

QSI4 PMI4 QF4

TS3 TPM3 TF3

Table 7.7.2-1 Primary Elements Used To Form Flat Thin Shell Elements

Initially, the membrane stiffness is formed in terms of u and v, the in-plane displacements. An artificial in-plane rotational stiffness K

art is then added to prevent

singularities occurring when elements are co-planar. Kart

is defined as

Triangles K k E E tAart ip x y= +

− −− −− −

L

NMMM

O

QPPP

d i1 0 0 5 0 50 5 1 0 0 50 5 0 5 1 0

. . .

. . .

. . .

Quadrilaterals K k E E tAart ip x y= +

− − −− − −− − −− − −

L

N

MMMM

O

Q

PPPPd i

1 0 1 3 1 3 1 31 3 1 0 1 3 1 31 3 1 3 1 0 1 31 3 1 3 1 3 1 0

. / / // . / // / . // / / .

The in-plane stiffness parameter kip has a default value of 0.02 which may be changed by using the SYSTEM command (variable STFINP).

Once the local element matrices have been evaluated they are transformed to the global Cartesian system. The final nodal variables are (fig.7.7.2-1)

U x y z, V, , , θ θ θ at each node


156

The strain-displacement relationship is defined in section 7.3 (in-plane) and section 7.6 (bending).

Note. The incompatible terms in the strain-displacement matrix are not used to evaluate nodal loads due to initial Gauss point stresses, e.g. thermal loading, initial stresses.

For further details of the element formulation see section 7.3, section 7.6, [Z1,L1]

A lumped mass matrix is evaluated using the procedures presented in section 2.7.


Stress Output σ σ σx y xy, , direct and shear stresses in the local Cartesian system, σ σmax min, the maximum and minimum principal membrane

stresses, β the angle between the maximum principal membrane

stress and the local x-axis.

Stress Resultant Output Nx y xy, N , N the membrane stress resultants/unit width in the local

Cartesian system, Mx y xy, M , M the moments/unit width in the local Cartesian system.

Strain Output ∈ ∈ ∈x xy, , y the membrane strains in the local Cartesian system, ∈ ∈max, min the maximum and minimum principal membrane strains, β the angle between the maximum principal membrane

strain and the local x-axis, ψ ψ ψx xy, , y the flexural strains in the local Cartesian system, ψ ψmax, min the maximum and minimum principal bending strains, β the angle between the maximum principal bending strain

and the local x-axis The sign convention for stress resultant and strain output is shown in fig.7.7.2-3.

The xy-plane of the local Cartesian system is evaluated using a least squares fit through the element nodes. The local x-axis is defined as being a line joining the first and second element nodes, and the local y and z-axes are defined by a right hand screw rule (fig.7.7.2-4)

Shell Elements

157

The nodal stress resultants are evaluated by extrapolating the strain-displacement relationship at the Gauss point to the nodes. The nodal stress is computed at each node directly.

The stress resultants are most easily interpreted if the local Cartesian axes are all parallel. Average nodal stresses are in the global Cartesian system.

Nonlinear formulation The elements have no nonlinear capability but may be utilised in a nonlinear environment.


QS4/QSI4

34

12

TS3

W

θx

θyV

U

W

θx

θy

V

U

W

θxθy

V

U

W

θx

θyV

U

12

3

W

θx

θyV

U

θyV

U

W

θx

W

θx

θyV

U

Z

Y

X

θz

θzθz

θz

θz

θz θz

Fig.7.7.2-1 Nodal Configuration For Flat Thin Shell Elements

Problem Description Finite Element Model

Fig.7.7.2-2 Cylindrical Roof Example Illustrating Use Of Thin Flat Shell Elements


158

Y

X

σY

σX

σY

σX

σXY

σXY


Y

X

MY

Z

MXY

MXY

MY

MX

MXYMXY

MX

Stresses

Stress Resultants


Z

X

Y

z

x

y

1

2

3

4

x

y

z

1

23

Fig.7.7.2-4 Local Cartesian System For Thin Flat Shell Elements

Shell Elements

159

7.7.3 Flat Thin Shell Box (SHI4) Formulation The flat shell box element is formulated in a local Cartesian system by superimposing a non-standard isoparametric plane membrane element on the isoflex thin plate element. The xy-plane of the local Cartesian system is evaluated using a least squares fit through the element nodes. The membrane and bending stiffnesses are then formed independently and combined to give the total element stiffness in the local Cartesian system, i.e.

KK

aa

RR

membrane

bending

membrane

bending

membrane

bending

00

LNMM

OQPPRS|T|

UV|W|=RS|T|

UV|W|

The component bending stiffness and force vector used for this element is from the QF4 element [Section 7.6.1]. The elements use a non-standard plane membrane formulation which is more effective for modelling the in-plane bending in the web of box structures than the standard plane membrane formulation. The initial nodal configuration (fig.7.7.3-1) has 4 nodes with 3 in-plane degrees of freedom at each node

u, v and ∂ ∂ ξv x/

where ∂ ∂ ξv x/ is the rotation of a line η = constant at each node and approximates to θz .

In addition, incompatible displacement modes are utilised so that typically

U N Pii

n

i i i==∑ ∑ξ η ξ η, ,b g b g

1

U + ai=1

2

where

P12

221 1ξ η ξ ξ η ξ, ,b g b g= − = − and P

and ai are nodeless degrees of freedom which are condensed out before element assembly.

The extra incompatible modes are condensed out and the element matrices are then transformed to the global Cartesian system. This provides an element with the following nodal degrees of freedom (fig.7.7.3-2)

U x y z, V, W, , , θ θ θ - at the corner nodes

Δu - the relative (departure from linearity) local x-displacement for the mid-side nodes


160

The strain-displacement relationship is defined in section 7.3 (in-plane) and section 7.6 (bending).

Note. No artificial in-plane rotational stiffnesses are required for this element. For further details of the element formulation see section 7.6, [L1,T2]. A lumped mass matrix is evaluated using the procedures presented in section 2.7.


Stress Output σ σ σx y xy, , direct and shear stresses in the local Cartesian system, σ σmax min, the maximum and minimum principal membrane

stresses, β the angle between the maximum principal membrane


Stress Resultant Output Nx y xy, N , N the membrane stress resultants/unit width in the local

Cartesian system, Mx y xy, M , M the moments/unit width in the local Cartesian system.

Strain Output ∈ ∈ ∈x y xy, , the membrane strains in the local Cartesian system, ∈ ∈max min, the maximum and minimum principal membrane strains, β the angle between the maximum principal membrane

strain and the local x-axis, ψ ψ ψx y xy, , the flexural strains in the local Cartesian system, ψ ψmax min, the maximum and minimum principal bending strains,

β the angle between the maximum principal bending strain and the local x-axis.


The xy-plane of the local Cartesian system is evaluated using a least squares fit through the element nodes. The local x-axis is defined as being a line joining the first and second element nodes, the local y and z-axes are defined by a right hand screw rule (fig.7.7.3-5).

The nodal stress resultants are evaluated by extrapolating the strain displacement relationship at the Gauss points to the nodes, and then computing the nodal stress at each node directly.

The stress resultants are most easily interpreted if the local Cartesian axes are all parallel.

Shell Elements

161

Note. The averaged nodal stresses are output in the global Cartesian system.

Nonlinear formulation The elements have no nonlinear capability but may be utilised in a nonlinear environment.


SHI4

Z

Y

4

V

Uθz

1

V

Uθz 2

V

Uθz

V

Uθz

3

Fig.7.7.3-1 Initial In-Plane Nodal Configuration

SHI4

34

12

W

U

V

θy

θx

W

θxθy

V

U

W

V

U θy

θx

W

θx

θyV

U

Z

Y

X

θzθz

θz

θz

Fig.7.7.3-2 Final Nodal Configuration For Flat Thin Box Shell ElemENTS


162

Box Girder Box Girder Bridge

Fig.7.7.3-3 Structures Suitable For Analysis With Flat Box Shell Elements

Y

X

σY

σX

σY

σX

σXY

σXY


Y

X

MY

Z

MXY

MXY

MY

MX

MXYMXY

MX

Stresses

Stress Resultants


Shell Elements

163

Z

X

Y

z

x

y

1

2

3

4

Fig.7.7.3-5 Local Cartesian System For Thin Flat Box Shell Elements

7.7.4 Semiloof Thin Shell (QSL8, TSL6) Formulation The Semiloof shell element is a thin, doubly curved, isoparametric element formed by applying Kirchhoff constraints to a three dimensional degenerated thick shell element. The displacements and rotations are considered independent and the unconstrained nodal configurations are (fig.7.7.4-1)

U, V, W - at the corner and mid-side nodes,

θ θx y, - at the loof nodes,

and w x y, , θ θ - at the central node,

where θx and θy are the rotations of the through-thickness normals. These rotations include transverse shear deformations.

An element with thin shell performance is then produced by constraining the shear strains to zero at discrete points within the element, i.e. by ensuring that [I1]

γ∂∂

θt ywx

= − = 0

at the points shown in fig.7.7.4-2. Where γ t is the through-thickness shear strain tangential to the element edges. This provides 6 and 8 constraints respectively for the triangles and quadrilaterals which are suitable for eliminating the tangential rotations at the loof nodes.

γ γxzA yzdA dAz z= =0 0, A


164

where the integral is performed using 2*2 Gauss quadrature. This provides 2 constraints for both the triangles and quadrilaterals which are suitable for removing the rotations at the central node.

γ ndS =z 0S

where γ n is the transverse shear strain normal to the element sides and the integral is performed using 2-point quadrature along each side. This provides 1 constraint suitable for removing the central translation of the quadrilaterals. These constraints provide extra equations that permit certain nodal degrees of freedom to be discarded. The final nodal configurations are (fig.7.7.4-3)

U, V, W - at the corner and mid-side nodes,

and θ - at the loof nodes.

Using the assumptions of thin shell theory, the strain-displacement relationship is written as

∈ =xux

∂∂

∈ =yvy

∂∂

γ∂∂

∂∂xy

uy

vx

= +

ψ∂∂x

wx

≈ −2

2

ψ∂∂y

wy

≈ −2

2

ψ∂∂ ∂xy

wx y

≈ −22

The isotropic and orthotropic modulus and resultant modulus (rigidity) matrices are defined as

Explicit

$$

$DD

Dmembrane

bending

=L

NMM

O

QPP

0

0

where, for Isotropic materials

Shell Elements

165

$D Emembrane

=− −

L

N

MMMM

O

Q

PPPP1

1 01 0

0 01

2

2υ

υυ

υa f

$D Etbending

=− −

L

N

MMMM

O

Q

PPPP

3

212 1

1 01 0

0 01

2υ

υυ

υe j a f

and for Orthotropic materials

$/ /

/ //

DE E

E EG

membrane

x xy x

xy x y

xy

=−

−

L

N

MMM

O

Q

PPP

−1 0

1 00 0 1

1υ

υ

$/ /

/ //

D tE E

E EG

bending

x xy x

xy x y

xy

=−

−

L

N

MMM

O

Q

PPP

−3

1

12

1 01 0

0 0 1

υυ


Notes

To obtain a valid material υxy x yE E< //d i1 2

A three dimensional orthotropic modulus matrix may be specified by using the appropriate data input. This 6 by 6 modulus matrix is the same as that given in section 7.4.1 and is reduced to the plane stress modulus matrix in the following way:

remove the γ YZ and γ XZ shear strain rows and columns, invert the matrix so that the stress-strain relationship is obtained, remove the s Z row and column since this stress is assumed to be zero, re-invert the matrix to obtain the stress-strain relationship (a 3 by 3 matrix).


166

Numerically Integrated

D E

z zz z

z

z z z zz z z z

z z

t=

−

− −

− −

L

N

MMMMMMMMM

O

Q

PPPPPPPPP

z 1

1 0 01 0 0

0 01

20 0

12

0 00 0

0 01

20 0

12

2 2 2

2 22

υ

υ υυ υ

υ υ

υ υυ υ

υ υ

a f a f

a f a f

dz


Isotropic ψ

αΔαΔ

αα

αα

0

0

0

e j a f

a ft

TT

d Tdz

T ddT

d Tdz

T ddT

= +LNMOQP

+LNMOQP

L

N

MMMMMMMMM

O

Q

PPPPPPPPP

ΔΔ

ΔΔ

Orthotropic ψ

ααα

αα

αα

αα

0e ja f

a f

a f

t

x

y

xy

xx

yy

xyxy

TTT

d Tdz

T ddT

d Tdz

TddT

d Tdz

TddT

=+LNM

OQP

+LNM

OQP

+LNM

OQP

L

N

MMMMMMMMMMMM

O

Q

PPPPPPPPPPPP

ΔΔΔ

ΔΔ

ΔΔ

ΔΔ

Full details of the element formulation are given in [I1].



Stress Output σ σ σx, , y xy direct and shear stresses in the local Cartesian system, σ σmax, min the maximum and minimum principal membrane stresses, β the angle between the maximum principal membrane


Shell Elements

167

Stress Resultant Output Nx, ,N Ny xy the membrane stress resultants/unit width in the local

Cartesian system, Mx, ,M My xy the moments/unit width in the local Cartesian system,

Strain Output ∈ ∈ ∈x y xy, , the membrane strains in the local Cartesian system, ∈ ∈max min, the maximum and minimum principal membrane strains, β the angle between the maximum principal membrane

strain and the local x-axis, ψ ψ ψx y xy , , the flexural strains in the local Cartesian system, ψ ψmax min, the maximum and minimum principal bending strains, β the angle between the maximum principal bending strain

and the local x-axis. The local Cartesian system varies over the element for curved elements. For the quadrilateral element, the local y-axis, at any point within the element, coincides with the curvilinear line ξ= constant (fig.7.7.4-8). The local x-axis is perpendicular to the local y-axis in the +ve ξ direction and is tangential to the shell mid-surface. For the triangular element, the local Cartesian system is formed by orientating the local y-axis parallel to a line joining the mid-point of the first side with the 5th node. The x-axis is then formed perpendicular to the y-axis and tangential to the shell mid-surface, with the +ve direction defined by the +ve ξ direction. The local z-axis forms a right-handed set with the x and y-axes. The +ve z-axis defines the top surface.



Notes

The Gauss point stresses are converted to the global Cartesian system before extrapolation. The average nodal stresses are in the global Cartesian system.

Nonlinear formulation The Semiloof shell element may be employed in

Materially nonlinear analysis utilising the elastoplastic constitutive laws [O1] (section 4.2) and the nonlinear concrete model (section 4.3). Geometrically nonlinear analysis. Geometrically and materially nonlinear analysis utilising the nonlinear



168

Nonlinear dynamics utilising the nonlinear material laws specified in 1. Linear and nonlinear buckling analysis.

Notes

Geometric nonlinearity may be represented with either A Total Lagrangian formulation which accounts for large displacements

but small rotations and strains. The nonlinear strain-displacement relationship is defined by


OQP + LNM

OQPx

ux

ux

vx

wx

∂∂

∂∂

∂∂

∂∂

12

12

12

2 2 2

∈ = +LNMOQP

+LNMOQP

+LNMOQPy

vy

uy

vy

wy

∂∂

∂∂

∂∂

∂∂

12

12

12

2 2 2

γ∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂xy

uy

vx

ux

uy

vx

vy

wx

wy

= + + + +

ψ∂∂x

wx

= −2

2

ψ∂∂x

wy

= −2

2

ψ∂

∂ ∂xyw

x y= −2

2

2


or An Updated Lagrangian formulation, which takes account of large

displacements and moderately large strains provided that the strain increments are small. The output is now in terms of the True Cauchy stresses and the strains approximate to logarithmic strains. The loading approximates to being non-conservative. The initial assumptions used in deriving the shell elements limit the

rotations to one radian in a Total Lagrangian analysis, and rotation increments of one radian in an Updated Lagrangian analysis (section 3.5).

Shell Elements

169

1

(a) QSL8

23

4

56

7

8

V

UW

V

UW

UW

V

UW

V

V

W

U

V

UW

V

W

UV

UW

θx

θy

θx

θy

θx

θy

θx

θy

θy

θx

W

Y

X

Z

V

UW

V

UW

V

U

W

V

UW

V

UW

V

UW

23

4

5

6

θy

θx

θx

θy

θx

θy

θy

θx

θy

θx

θx

θy

θx

θy

θx

θy

θy

θx

θy

θx1

Y

X

Z

(b) TSL6

Fig.7.7.4-1 Initial Nodal Configurations For QSL8 And TSL6 Elements


170

2

1 3/ 1 3/

1 3/

1 3/

1 3/

1 3/

2

2

1 3/ 1 3/

2

1 3/

1 3/

1 3/

1 3/

2

(a) QSL8

(b) TSL6

Fig.7.7.4-2 Locations Where Transverse Shear Strains Tangential To The Element Edge Are Constrained To Zero

Shell Elements

171

1

(a) QSL8

23

4

56

7

8

V

UW

V

UW

UW

V

UW

V

V

W

U

V

UW

V

W

UV

UW

θ2 θ1

θ1

θ1

θy

θx

W

Y

X

Z

V

UW

V

UW

V

U

W

V

UW

V

UW

V

UW

23

4

5

6

θ2

θ1

θ1

θ2

θ2

θ2

θ1

θ2

θ2θ1

1

Y

X

Z

(b) TSL6

Fig.7.7.4-3 Final Nodal Configuration For QSL8 And TSL6 Elements


172

Fig.7.7.4-4 Tubular Joint Example Illustrating Use Of QSL8 And TSL6 Elements

Fig.7.7.4-5 Pressure Vessel Example Illustrating Coupling Of HX20 And QSL8 Elements

Shell Elements

173


QSL8 elements

BSL3 elements

Fig.7.7.4-6 Stiffened Shell Illustrating Coupling Between QSL8 And BSL3 Elements

Fig.7.7.4-7 Bending Mechanism For QSL8 Element


174

1

2

3

4

5

6

7

8

xz

y

ξ

η

(a) QSL8 Element

xz

y

(b) TSL6 Element

1

2

3

45

6

Fig.7.7.4-8 Local Cartesian System

Shell Elements

175

Y

X

σY

σX

σY

σX

σXY

σXY


Y

X

MY

Z

MXY

MXY

MY

MX

MXYMXY

MX

Stress Resultants

Y

X

ΝY

ΝX

ΝY

ΝX

ΝXY

ΝXY

ΝX, ΝY +ve tensionΝXY +ve into XY quadrant

Stresses



176

7.7.5 Thick Shells (TTS3, TTS6, QTS4, QTS8) Formulation The formulation for this family of thick shell elements is based on the degeneration of a three dimensional continuum. In this approach, the displacements at any point in the shell are defined by the translation of the reference surface together with the rotation of a director. The director is subsequently referred to as the normal, however, the director need not be initially normal to the reference surface. The normal is considered to remain straight during deformation for computation of displacements through the element thickness. The triangular elements (TTS3, TTS6) are formulated using a standard isoparametric approach. The quadrilateral elements (QTS4,QTS8) adopt an assumed strain field for interpolation of the transverse shear strains. The inclusion of an assumed strain field prevents the element from 'shear locking' when used as a thin shell. The displacements and rotations are considered independent and the nodal degrees of freedom are (fig.7.7.5-1)

U, V, W, θ θα β, - at all nodes.

θα and θβ are the rotations of the through-thickness normals. These rotations include transverse shear deformations and relate to a set of 'local' axes set up at each node. To avoid singularities, the direction of these axes is dictated by the direction of the nodal normal. One of the global axes is chosen to define the θα rotation, the axis chosen corresponds with the smallest component of the nodal vector. The cross product of this axis and the nodal vector defines the second axis of rotation for θβ (fig.7.7.5-1). This definition of the rotations is used when a smooth surface configuration is to be modelled (fig.7.7.5-2). In the event of a discontinuity, connection with a beam element, or a branched shell junction, these rotations are transformed to relate to global axes, θx, θy, θz (fig.7.7.5-1).

The location of the transverse shear sampling points for defining the assumed strain fields are shown in fig.7.7.5-3. For the four noded quadrilateral (QTS4) the factors for interpolating from the sampling points to the gauss points are

R112

1= − ηa f

R212

1= + ηa f

while for the eight noded element (QTS8) the factors are

Ra

R1 514

1 1 14

= −LNMOQP − −

ξηa f

Shell Elements

177

Ra

R2 514

1 1 14

= −LNMOQP + −

ξηa f

Ra

R3 514

1 1 14

= +LNMOQP − −

ξηa f

Ra

R4 514

1 1 14

= +LNMOQP + −

ξηa f

Ra5

221

41 1= − LNMOQP

LNMM

OQPP −

ξηe j

where

a = 1 3/

and

S Ri i( , ) ( , )η ξ ξ η=

The covariant transverse shear strains at the gauss points are then given by

∈ = ∈=∑ξζ ξζξ ηRii

ni,b g

1

∈ = ∈=∑ηζ ηζξ ηSii

ni,b g

1

where ∈ξζ and ∈ηζ are the covariant transverse shear strains at the gauss points and ∈ξζ

i , ∈ηζi are the transverse shears at the sampling points.

Using this representation of shear strains allows

Correct representation of the six rigid body modes. Approximation of the Kirchhoff-Love thin shell hypothesis. No spurious zero energy modes using full numerical integration.

It is necessary to express the transverse shear strains in terms of covariant components so that interpolation can be carried out using the isoparametric map. The stress and strain terms are ultimately transformed to relate to a local orthogonal set of axes at each gauss point. The local axes are set up using

$ /e G G1 = ξ ξ

$ $ / $e e G e G3 1 1= x x η ηe j


178

$ $ $e e e2 3 1= x

where Gξ and Gη are the covariant base vectors at a gauss point.

Strains in the curvilinear system ∈lmb g may then be transformed to strains in the orthogonal local system $∈ijd i by using the contravariant base vectors

$ $ $∈ =∈ ⋅ ⋅ij lm i

mjG e G ed i d ie j1

The elements are formulated using the plane stress hypothesis so that σzz in the thickness direction is set to zero. The continuum strains are evaluated at integration points through the thickness, and for the geometrically linear case these strains are given by

$∈ =xx

ux

∂∂

$∈ =yy

vy

∂∂

$γ

∂∂

∂∂xy

uy

vx

= +

$γ

∂∂

∂∂yz

vz

wy

= +

$γ

∂∂

∂∂yz

uz

wx

= +

Material properties are specified in the local orthogonal axes. For a thick shell the modulus matrix is condensed so that the plane stress hypothesis is observed.

The isotropic modulus matrix is given by [Z1]

D E

Symm.

=−

−

−

−

L

N

MMMMMMMM

O

Q

PPPPPPPP

1

1 0 0 01 0 0 0

12

0 012

012

2υ

υ

υ

υ

υ.4

.4

If orthotropic properties are specified the modulus matrix becomes

Shell Elements

179

D

E E

E

GG

Symm. G

x

xy yx

x yx

xy yx

y

xy yx

xyyz

xz

=

− −

−

L

N

MMMMMMMMMMM

O

Q

PPPPPPPPPPP

1 10 0 0

10 0 0

0 0

1 20

1 2

υ υ

υ

υ υ

υ υ

d i d i

d i

.

.

Factors of 5/6 have been included in the transverse shear terms to take account of a parabolic distribution through the thickness.

As the material properties are specified in local element directions and the element formulation is based on covariant components of strain, the modulus matrix must be transformed.

The required transformation of the modulus matrix is

C G e G e G e G e Dijkl ia

jb

kc

ld abcd

= ⋅ ⋅ ⋅ ⋅d id id id i

where Gm m = ξ η ζ, , are the contravariant base vectors.

Full details of the element formulations may be found in [D4],[H9] and [S7].



Stress Output σ σ σx, , y xy direct and shear stresses in the local Cartesian σ σ σyz, , xz e system, together with von Mises equivalent stress

Three dimensional principal stresses and the corresponding direction cosines may also be output

Stress Resultant Output Nx, ,N Ny xy the membrane stress resultants/unit width in the local

Cartesian system, Mx, ,M My xy the moments/unit width in the local Cartesian system, Sx,Sy the shear stress resultants/unit width in the local Cartesian

system


180

Strain Output ∈ ∈ ∈x y xy, , , the direct and shear strains in the local Cartesian ∈ ∈ ∈yz xz e, , system, together with von Mises equivalent strain

The local cartesian systems are set up at the element reference surface. For curved elements, the local Cartesian system will vary over the reference surface. The local x-axis, at any point within the element, coincides with the curvilinear line η = constant in the direction of increasing ξ (fig.7.7.5-4). The direction of the local z-axis is defined by the vector product of the local x-axis and the curvilinear line ξ = constant (in the direction of increasing η). The local y-axis is defined by the vector product of the local z and local x-axes. The +ve z-axis defines the element top surface. The position of the origin of the curvilinear system for each element together with the directions of increasing values are shown in (fig.7.7.5-5).

The sign convention for stress and strain output is shown in fig.7.7.5-6 and fig.7.7.5-7.

The nodal stresses and strains are evaluated using the extrapolation procedures detailed in section 6.1.

Notes

The Gauss point stresses are converted to the global Cartesian system before extrapolation. The average nodal stresses are in the global Cartesian system.

Nonlinear formulation The thick shell elements may be employed in

Materially nonlinear analysis utilising the elastoplastic constitutive laws [O1] (section 4.2) and the nonlinear concrete model (section 4.3). Geometrically nonlinear analysis using a Total Lagrangian formulation. Geometrically and materially nonlinear analysis utilising the nonlinear

material laws specified in 1. Nonlinear dynamics utilising the nonlinear material laws specified in 1. Linear and nonlinear buckling analysis. Creep analysis

Note. The Total Lagrangian formulation used for these elements is valid for both large displacements and large rotations. However, the formulation is only valid for small strains. The nonlinear strain-displacement relationship is defined by


OQP + LNM

OQPx

ux

ux

vx

wx

∂∂

∂∂

∂∂

∂∂

12

12

12

2 2 2

Shell Elements

181

∈ = +LNMOQP

+LNMOQP

+LNMOQPy

vy

uy

vy

wy

∂∂

∂∂

∂∂

∂∂

12

12

12

2 2 2

γ∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂xy

uy

vx

ux

uy

vx

vy

wx

wy

= + + + +

γ∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂yz

vz

wy

uz

uy

vz

vy

wz

wy

= + + + +

γ∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂xz

uz

wx

uz

ux

vz

vx

wz

wx

= + + + +



182

V

θα

θβ

θα

θβV

Z,w

Y,v

X,u

(a) 5 degrees of freedom Definition of nodal rotations when global X defines θα.

θx

θz

(b) 6 degrees of freedom

Z,w

Y,v

X,u

θy

Fig.7.7.5-1 Nodal Variables For Thick Shell Elements

Shell Elements

183

Element 1 Element 21 3

2

V1

V2

V3

Default angle < 20o

Averaged nodal vector

(a) SMOOTH SURFACE (5 degrees of freedom)

Default angle > 20o

Separate nodal vector

Element 11

2V1

V21

Element 2 3

V3

V22

(b) DISCONTINUOUS SURFACE (5 degrees of freedom)

Fig.7.7.5-2 Smooth And Discontinuous Surface


184

1 2

34

1 2

η

ξ

Shear γηζ

1

2

Shear γξζ

η

ξ

1 2

34

(a) QTS4

1

η

ξ

4

2

3

1

5

a

a a

a

1 2 3

4

567

8

Shear γξζShear γηζ

21

43

321

8

567

a a

a a

η

ξ5

(b) QTS8

a = 3-1/2

Fig.7.7.5-3 Shear Sampling Points

Shell Elements

185

1

2

34

5

6

z

y

x

η = constant

ξ = constant

(a) TTS6 (TTS3 axes coincide when element is flat)

1

2

3

4

5

6

7

8

zy

x

η = constant

ξ = constant

(a) QTS8 (QTS4 axes coincide when element is flat)

Fig.7.7.5-4 Local Cartesian Systems


186

3

5

6

1

2

(a) TTS3

1

2

4

ξ

ηζ

3ξ

ηζ

(b) TTS6

1

2

3

4 ζ

ξη

(c) QTS4

1

2

3

4

(d) QTS8

5

6

7

8

ζ

ξη

Fig.7.7.5-5 Curvilinear Coordinates

Shell Elements

187

y

x

σy

σxσx

σxy

σxy

Direct stress (+ve) TensionShear stress (+ve) Shear into XY, YZ and ZX quadrantsNote: Positive values shown in figure

z

x

σxz

σxz

z

y

σyz

σyz

σy

Fig.7.7.5-6 Sign Convention For Continuum Stress Output


188

Y

X

Sy

Sy

Sx

SxMxy

Y

X

Mxy

Mxy

Mxy

My

My

Mx

Mx

Nxy

Y

X

Nxy

Nxy

Nxy

Ny

Ny

Nx

Nx

Membrane stress (+ve) Direct tension(+ve) In-plane shear into XY quadrant

Flexural stress (+ve) Hogging moment (producing +ve stresses onthe element top surface)

Shear stress (+ve) In-plane shear into YZ and XZ quadrants

Note: Positive values shown in figure


Field Elements

189

7.8 Field Elements 7.8.1 Thermal Bar (BFD2, BFD3) The thermal bar element (fig.7.8-1) has either 2 or 3 nodes and may transfer heat along its length. The bar is assumed to be perfectly insulated along its length and may transfer heat across its end areas via conduction, convection, radiation or applied heat flux.

The nodal degree of freedom is the field variable Φ. The cross sectional area of the material is defined at each node and may vary over the element.

The gradient-field variable relationship is defined as

gxx =

∂Φ∂

where x represents the local x-direction of the element. The field gradient is related to the flow by

q k gx x=

Element Output The element output consists of the gradients of the field variables and flow in the local element axis system at either the Gauss or nodal points, i.e.

gx field gradient

qx flow

The local x-axis lies along the element axis in the direction in which the element nodes are specified.

The Gauss point values are generally more accurate.


190

Y,V

X,U

3

1

2 BFD31

2BFD2

Fig.7.8-1 Nodal Configuration For Bar Field Elements

Heat transfer between members

Pressure

Continuum elements

Struts represented withBFD2 elements

Fig.7.8-2 Example Illustrating The Use Of Bar Field Elements

Field Elements

191

7.8.2 Thermal Axisymmetric Bar (BFX2, BFX3) The axisymmetric thermal bar element (fig.7.8-3) has either 2 or 3 nodes and may transfer heat along its length. The bar is assumed to be perfectly insulated along its length and may transfer heat across its end areas via conduction, convection, radiation or applied heat flux.

The nodal degree of freedom is the field variable Φ. The gradient-field variable relationship is defined as

gxx =

∂Φ∂

The field variable is related to the flow by

q k gx x=

Element output The element output consists of the gradients of the field variables and flows in the local element axis system at either the Gauss or nodal points, i.e.

gx field gradient

qx flow



Y,V

X,U

3

1

2 BFX31

2BFX2

Fig.7.8-3 Nodal Configuration For Axisymmetric Bar Field Elements


192

Problem Definition

Finite Element Mesh

Fig.7.8-4 Spinning Cylinder Example Illustrating The Use Of Axisymmetric Bar Field Elements

7.8.3 Thermal Link (LFD2, LFS2, LFX2) The thermal link element (fig.7.8-5) has 2 nodes and may transfer heat between two nodal points by either conduction, convection or radiation. The nodal degree of freedom is the field variable Φ. The cross sectional area of the material is defined at each node and may vary over the element.

The heat flow is positive in the direction of the local x-axis and is defined as

Conduction q K= −( )Φ Φ1 2

Convection q hc= −( )Φ Φ1 2

Radiation q hr= −( )Φ Φ14

24

where K is the gap conductance. hc and hr are the convective and radiative heat transfer coefficients.

Field Elements

193

Note that the material properties may be dependent on both the gap distance and the temperature evaluated at the centre of the element.

The element stiffness matrix may be derived from the nodal flows which are defined as

Q K A Qt1 1 2 2= − = −Φ Φb g

where Kt is the combined heat transfer coefficient for the element. The stiffness matrix is then rewritten as

K K Ko

= +1

where Ko

is the linear contribution to the stiffness matrix defined as follows

K K Ao t=

−−LNM

OQP

1 11 1

and K1 is the nonlinear contribution to the stiffness matrix defined as

K

K A K A

K A K A

t t

t t11

1 22

1 2

11 2

21 2

=− −

− − − −

L

N

MMMM

O

Q

PPPP

∂∂Φ

∂∂Φ

∂∂Φ

∂∂Φ

Φ Φ Φ Φ

Φ Φ Φ Φ

b g b g

b g b g

The nodal flows are evaluated directly using

Q K= Φ

where K is evaluated using the current properties in temperature dependent analyses.

Element output The element output consists of the gradients of the field variables and flows in the local element axis system at either the Gauss or nodal points, i.e.

gx field gradient

qx flow




194

Y,V

X,U

1

2LFD2LFX2LFS2

Fig.7.8-5 Nodal Configuration For Link Field Elements


QAX8 elements

LFX2 elements

Fig.7.8-6 Thermal Analysis Of An Interface Fit Illustrating The Use Of Link Field Elements

7.8.4 Plane Field (QFD4, QFD8, TFD3, TFD6) The plane field element (fig.7.8-7) is defined in the global XY-plane. The gradient-field variable relationship is defined as

Field Elements

195

gXX =

∂Φ∂

gYY =

∂Φ∂

The isotropic and orthotropic thermal conductivity modulus matrices are defined as follows

Isotropic kk

k=LNMOQP

00

Orthotropic kk

kx

y=LNM

OQP

00

Element output The element output consists of the gradients of the field variables or flows in the global axis system at either the Gauss or nodal points, i.e.

gX Y, g field gradient

qX, qY flow



196

Y

X

3

1

2

TFD3

12

34

QFD4

1 2 3

4

5

6

TFD6

12 3

4

567

8

QFD8

Fig.7.8-7 Nodal Configuration For Plane Field Elements

Field Elements

197



Fig.7.8-8 Cofferdam Example Illustrating The Use Of Plane Field Elements


198

7.8.5 Axisymmetric Field (QXF4, QXF8, TXF3, TXF6) The axisymmetric field elements (fig.7.8-9) are formulated using the axisymmetric quasi-harmonic equation (section 2.10) and are defined in the global XY-plane.


gXX =

∂Φ∂

gYY =

∂Φ∂


Isotropic kk

k=LNMOQP

00

Orthotropic kk

kx

y=LNM

OQP

00


gX Y, g field gradient

qX, qY flow


Field Elements

199

Y

X

3

1

2

TXF3

12

34

QXF4

1 2 3

4

5

6

TXF6

12 3

4

567

8

QXF8

Fig.7.8-9 Nodal Configuration For Axisymmetric Field Elements


200

Flow

Well


Point sink torepresent well


Fig.7.8-10 Groundwater Flow Example Illustrating The Use Of Axisymmetric Field Elements

7.8.6 Solid Field (HF8, HF16, HF20, PF6, PF12, PF15, TF4, TF10) The solid field elements (fig.7.8-11) are formulated using the 3-D quasi-harmonic equation (section 2.10). The nodal degree of freedom is the field variable Φ.


gXX =

∂Φ∂

gYY =

∂Φ∂

gZZ =

∂Φ∂

Field Elements

201


Isotropic kk

kk

=L

NMMM

O

QPPP

0 00 00 0

Orthotropic kk

kk

x

y

z

=L

NMMM

O

QPPP

0 00 00 0


gX Y Z, , g g field gradient

qX, , q qY Z flow



202

1

2

3

4

5

6

78

512 4

3

768

91011

12

131415

16

51 2 4

3

768

9

10

111213 14

15

1617

18

1920

1

2

3

4

5

6

12

3

4

56

789

10

1112

12

3

4

567

8

910

1112

131415

1

2

3

4

12

3

4

56

78

910

HF8HF16

HF20

PF6

PF12 PF15

TF4 TF10

Fig.7.8-11 Nodal Configuration For Solid Field Elements

Field Elements

203

Silicon Chips

Beryllia heatsinks

Stainless steelside walls

Copper Base

Fig.7.8-12 Thermal Analysis Of A Hybrid Power Assembly Illustrating Use Of Solid Field Elements


204

7.8.7 Solid Composite Field (HF8C, HF16C, PF6C, PF12C) If brick elements are used for a structural analysis of composites the number of degrees of freedom even for small laminate structures rapidly becomes very large leading to prohibitively excessive computer costs. To overcome this difficulty layered brick elements were developed where several laminae are included in a single element. Composite field elements are designed to compliment the structural elements in a thermo-mechanical analysis. For these elements the field variable at a node (temperature) is used to interpolate a temperature field that varies linearly over the thickness of the whole element and quadratically in-plane for the higher order elements.

As with the PN6L, PN12L, HX8L and HX16L structural elements, in order to speed up the computation the elements are restricted to constant layer thicknesses [H13]. This limitation requires the calculation of only a 2x2 Jacobian matrix. For the integration of the element thermal stiffness matrix, the thermal conductivity matrices are summed layerwise through the thickness, while the shape function derivative matrices are integrated using a plane 2x2 (for HF8C and HF16C), or a single point (for PF6C), or a 3 corner point quadratic (for PF12C) gauss integration scheme outside the through thickness loop. This indicates that the (in plane) shape function derivative matrix is independent of the non-dimensional coordinate through the thickness.

The shape functions for the top and bottom surfaces of the composite elements can be considered to be single membrane element shape functions, see figure 7.8-13. The shape functions N N Ni top i bot i( ) ( )= = , are defined in terms of natural coordinates ξ and η, for the HF16C element these are given by:

Ni i i i i= + + + −14

1 1 1ξ ξ η η ξ ξ η ηb gb gb g for corner nodes

Ni i i i i= − − + +12

1 12 2 2η ξ ξ η ξ ξ η ηe jb g for mid-side nodes

The PF12C, HF8C, and PF6C elements use the equivalent shape functions for 6, 4 and 3 noded membranes. To form the complete shape functions for the brick element Nbr , linear interpolation is used between the functions for the top and bottom surfaces:

N N NbrT

i botT

i topT= − + +

12

1 1ζ ζb g b ga f b g;

The in-plane and through-thickness shape functions can then be separated to give:

NbrT T T= +φ ζψ

where

Field Elements

205

φTiT

iTN N=

12

;

ψTiT

iTN N= −

12

;

The scalar temperature field, Φ, can now be interpolated as:

Φ Φ= +φ ζψT T l q

Φ Φ= H

with the field variable in terms of the nodal variables:

Φ Φ Φ Φ= 1 2, ,.............. nTl q

The field gradient vector g is defined as

gx y z

T =RST

UVW∂∂

∂∂

∂∂

Φ Φ Φ, ,

The field gradient - variable relationship is given by:

g B= Φ

where B is the field gradient - variable matrix.

B

x x

y y

c

T T

T T

T

=

+

+

L

N

MMMMMMMM

O

Q

PPPPPPPP

∂φ∂

ζ∂ψ∂

∂φ∂

ζ∂ψ∂

ψ2

and ‘c’ is the overall depth of the element shown in figure 7.8-13.


206

B can be split into two matrices combining in-plane and through thickness terms:

B B B= +1 2

ζ

where

B

x

y

c

T

T

T

1

2

=

L

N

MMMMMMMM

O

Q

PPPPPPPP

∂φ∂∂φ∂

ψ

B

x

y

T

T

T

2

0

=

L

N

MMMMMMM

O

Q

PPPPPPP

∂ψ∂

∂ψ∂

The restriction of constant layer thicknesses provides an uncoupling between the in-plane coordinates and the through-thickness coordinate. Consequently for the transformation of the cartesian derivatives into the natural derivatives only a 2 dimensional Jacobian matrix is required.

∂∂ξ∂

∂η

∂∂ξ

∂∂ξ

∂∂η

∂∂η

∂∂∂

∂

RS||

T||

UV||

W||

=

L

N

MMMM

O

Q

PPPP

RS||

T||

UV||

W||

x y

x yx

y

or inverted

∂∂

∂∂ξ

∂∂ηx

J J= +− −11

112

1

∂∂

∂∂ξ

∂∂ηy

J J= +− −21

122

1

and an integration constant for the thickness is computed from:

z cz c

= → =2

2ζ

∂∂

∂∂ζ

.

where c is the depth of the element see figure 7.8-13. The differential of the volume is given by

dV c J=2

d dζ ξ η

where |J| is the 2x2 Jacobian determinant.

The element thermal stiffness matrix in basic form may be defined as

Field Elements

207

K B k BT

V= z dV

where k is the principal thermal conductivity matrix for an orthotropic material.

kk

kk

x

y

z

=L

NMMM

O

QPPP

0 00 00 0

As the matrices B1 and B2 are independent of ζ, only k varies from layer to layer. Therefore the field gradient - variable matrices can be left out of the integration through the thickness:

As the matrices B1 and B2 are independent of ζ, only k varies from layer to layer. Therefore the strain-displacement matrices can be left out of the integration through the thickness:

KB k d B B k d B

B k d B B k d B

c J d d

Tnlay

n

nlayT

nlayn

nlay

Tnlay

n

nlayT

nlayn

nlay=

LNM

OQP

+LNM

OQP

+LNM

OQP +

LNM

OQP

F

H

GGGG

I

K

JJJJ

z∑ z∑

z∑ z∑zz = =

= =

11

1 11

2

21

1 22

12

2

ζ ζ ζ

ζ ζ ζ ζη ξ

ζ ζ

ζ ζ

ηξ

with B1 and B2 as:

B

J J

J J

c

T T

T T

T

1

111

121

211

221

2

=

+

+

L

N

MMMMMMMM

O

Q

PPPPPPPP

− −

− −

∂φ∂ξ

∂φ∂η

∂φ∂ξ

∂φ∂η

ψ

B

J J

J J

T T

T T

T

2

111

121

211

221

0

=

+

+

L

N

MMMMMMMM

O

Q

PPPPPPPP

− −

− −

∂ψ∂ξ

∂ψ∂η

∂ψ∂ξ

∂ψ∂η


208

The through thickness dependency is condensed in the integration of the thermal conductivity matrix which makes the assembly of the element thermal stiffness matrix more efficient. The field gradient - variable matrices only have to be computed in-plane. This is possible by restricting the element to a reasonably uniform thickness for a single layer.

Fig.7.8-13 Nodal Configuration For Solid Composite Field Elements

7.9 Joint Elements 7.9.1 Joints (JNT3, JPH3, JF3, JRP3, JNT4, JL43, JSH4,

JL46, JSL4, JAX3, JXS3) Joint elements are composed of translational and rotational springs that may be used to connect two nodes on adjacent finite elements (fig.7.9-1). The number of springs utilised in each element type is chosen to be compatible with a corresponding finite element type, e.g. two translational springs for plane elements or one translation and two rotations for plate bending elements (fig.7.9-2).

Joint Elements

209

The local element stiffness matrix is formulated directly from user input stiffness coefficients and is then transformed to the global Cartesian system.

The element mass matrix is lumped and formed directly from user input masses.

7.9.2 Evaluation of Stresses/Forces Element forces f are evaluated directly in the local Cartesian system from

f ka= '

where k and a are stiffness matrix and displacement vector in the local Cartesian system.

Element strain output is evaluated in the local element system as

∈∈∈

RS|T|

UV|W|

=−−−

RS|T|

UV|W|

x

y

z

u uv vw w

2 1

2 1

2 1

translational strain

and ψψψ

θ θθ θθ θ

x

y

z

x x

y y

z z

RS|T|

UV|W|

=−−−

RS|

T|

UV|

W|2 1

2 1

2 1

rotational strain

The element local axes are defined by

3 noded element The local x-axis is defined by a line joining the first and third element nodes. The local y-axis forms a right handed set with the x-axis such that the local z-axis is upwards (out of page) (fig.7.9-5).

4 noded element The local x-axis is defined by a line joining the first and third element nodes. The local xy-plane is defined by the fourth element node and the element x-axis. The local y and z-axes form a right handed set with the local x-axis (fig.7.9-6).

For convenience, element output may be obtained at either nodal or Gauss points.

7.9.3 Nonlinear Formulation The joint elements may be employed in

Materially nonlinear analysis using the nonlinear joint models (section 4.4). Note. The joint element possess no geometrically nonlinear terms, however they may be used as nonlinear support conditions in geometrically nonlinear analysis.


210

Y

X

Springs1

2

Fig.7.9-1 Springs Connecting Two Nodes In 2-D (No Rotational Stiffness)

xy

z

1

2

3

Rotational springs

Translational spring

Fig.7.9-2 Joint Element For Plate Bending Elements (One Translation And Two Rotations)

Joint Elements

211

Z

X

θy is given zero stiffnessto allow rotation

Y

zy

x

Fig.7.9-3 Modelling A Hinged Connection Between Shell Elements

Fig.7.9-4 Excavation Example Illustrating Use Of Joint Elements For Nonlinear Support Conditions


212

Y

X

xy

3

1, 2

Fig.7.9-5 Local Cartesian System For 3-Noded Joint Elements

Z

X

yx

z

3

4

1, 2

Y

Fig.7.9-6 Local Cartesian System For 4-Noded Joint Elements

7.9.4 Use of Joints With Higher Order Elements To illustrate the behaviour of joints with higher order elements, consider the problem of the beam on the elastic foundation shown in figure 7.9-7. A constant stress per unit length q along the top face is transmitted through the beam to the elastic foundation beneath. For equilibrium, the elastic foundation must apply an equal and opposite stress per unit length.

The beam is modelled by a single quadratic element of length two and the elastic foundation by three joint elements, as depicted in figure 7.9-8. For convenience, the coordinate system is centred at the mid-node of the element. The equivalent nodal loads are calculated using the principle of virtual work as

Joint Elements

213

q u dx q u x x dx u x dx x x dx

q u u

δ δ δ δ

δ δ δ

− − − −z z z z= − + −LNM + O

QP

+ +LNM

OQP

1 1 1 22

1 1

1 2

1 1 1

13

43

13

1 1 13

1

3

12

+ u 12

= u

a f e j a f

where the virtual work of the load is calculated from the perturbation of shape functions particular to each node. From equilibrium, the nodal loads calculated must be identical to the nodal loads developed by the discrete joint elements connected to the nodes on the lower surface. Note, the nodal loads are different since the virtual work arising from the perturbation of the mid-node is larger than that of the side nodes; there is a corresponding difference in the internal strain energy associated with each virtual perturbation.

If the elastic foundation has a stiffness per unit length of k, then the stress per unit length is related to k by

q = ku

For a constant deflection u along the lower face, q may be substituted in each of the nodal equilibrium equations resulting with discrete spring stiffnesses K of (see fig.7.9-9)

At node 1 and 3 K k=13

At node 2 K k=43

If the beam is modelled by two or more elements, then the spring stiffness at connecting nodes must be summed (fig.7.9-10).

For non-central midside nodes, the computer program may be used to calculate the ratio of joint stiffnesses. If the appropriate boundary nodes are restrained, and a unit face load applied, the resulting nodal reactions will correspond to the integrated shape functions; these are also the ratio of spring stiffnesses to be used. With a little ingenuity a variety of spring boundary conditions can be evaluated using appropriate loading and the program to calculate equivalent reactions.

However, it is recommended that for nonlinear contact problems, linear elements should be used if possible as higher order elements poorly represent the discontinuities in the boundary conditions. This may result in either poor convergence or divergence of the solution.


214

q

k

Fig.7.9-7 Beam On An Elastic Foundation

q

y

x1

2

34

5

6

78

l l

Fig.7.9-8 Finite Element Discretisation

q/3

y

x

4q/3q/3

k/3 4k/3 k/3

Fig.7.9-9 Discrete Joint Stiffnesses

Joint Elements

215

2k/3

q/3

k/3

q/3

k/3

4q/3

4k/3

q/3

k/3

q/3

k/3

4q/3

4k/3

Fig.7.9-10 Summation Of Joint Stiffnesses

7.10 Fourier Element Formulation (TAX3F, QAX4F, TAX6F, QAX8F)

These are axisymmetric solid elements that may be subjected to non-axisymmetric loading (fig.7.10-1).

7.10.1 Global and Local Coordinate Systems The structural mesh is specified in the global XY-plane and, together with the global Z-coordinate, the global coordinate axes form a right-hand coordinate system. In general the structure may be axisymmetric about either the X or Y axis, unless CBF loading is applied, in which case the structure is restricted to be axisymmetric about the X-axis.

The element axes are defined in the cylindrical coordinate system xyz, with associated displacements u,v,w. The tangential displacement w is positive in the direction of increasing j, where θ is the positive rotation defined by the right-hand coordinate system about the axis of symmetry. u and v are positive in the direction of increasing x and y respectively and may be either axial or radial displacements depending on the definition of the axis of symmetry.

7.10.2 Standard Isoparametric Elements The geometry of the body is defined using the shape functions


216

X N Xii

m

i==∑ ξ η,b g

1

and Y N Yii

m

i==∑ ξ η,b g

1

where Ni ξ η,b g are standard linear or quadratic isoparametric element shape functions for node i and m is the number of nodes. The nodal degrees of freedom of the element are

u,v,w at each node in the cylindrical coordinate system

Y,V

X,U

3

1

2

TAX3F

12

34

QAX4F

1 2 3

4

5

6

TAX6F

12 3

4

567

8

QAX8F

Figure 7.10-1 Nodal Configurations For Fourier Elements

Fourier Elements

217

where u,v,w are given by the Fourier expansions,

u u n u nns

na

n

m

n

m= +

==∑∑ cos sinθ θ

10

v v n v nns

na

n

m

n

m= +

==∑∑ cos sinθ θ

10

w w n w nns

na

n

m

n

m= −

==∑∑ sin cosθ θ

00

n=0,1,..,m represents the range of harmonics considered and superscripts 's' and 'a' denote the symmetric and asymmetric components. For each harmonic, the discretised displacement is defined as

u N a= '

where a is the nodal displacement vector and for any node i

a a a ui is

ia T

is

is

is

ia

ia

ia T

= = , v w u v w, , , , ,

and N' is the shape function matrix defined as

NN n N n

N n N nN n N n

i i

i i

i i

'cos sin

cos sinsin cos

=−

L

NMMM

O

QPPP

θ θθ θ

θ θ

0 0 0 00 0 0 00 0 0 0

where Ni are the standard isoparametric shape functions.

7.10.3 Strain-Displacement Relationships The infinitesimal strain-displacement relationships for structures symmetric about the global X-axis are given by

∈ =xux

∂∂

∈ =yvy

∂∂

∈ =∈ = +θ∂∂θz Yw v

Y1

γ∂∂

∂∂XY

uY

vX

= +


218

γ γ∂∂

∂∂θθY YZ

wY

WY Y

v= = − +

1

γ γ∂∂θ

∂∂θX XZ Y

u wX

= = +1

7.10.4 Constitutive Relationships The modulus matrix D, which for isotropic and orthotropic elasticity is defined as

Isotropic

D E=

+ −

−−

−−

−

−

L

N

MMMMMMMMMMM

O

Q

PPPPPPPPPPP

1 1 2

1 0 0 01 0 0 0

1 0 0 0

0 0 01 2

20 0

0 0 0 01 2

20

0 0 0 0 01 2

2

υ υ

υ υ υυ υ υυ υ υ

υ

υ

υ

a fa f

a fa f

a fa f

a fa f

Orthotropic

D

E E EE E EE E E

GG

G

x yx y zx z

xy x y zy z

xz x yz y z

xy

yz

xz

=

− −− −− −

L

N

MMMMMMMM

O

Q

PPPPPPPP

−1 0 0 01 0 0 0

1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

1/ / // / // / /

//

/

υ υυ υυ υ

where for symmetry

υ υ υ υ υ υxy

x

yx

y

xz

x

zx

z

yz

y

zy

zE E E E E E= = =, ,

In addition to the solid material definition of the element, a plane stress material model may also be utilised. The constitutive relationship is,

D

E EE E

G

x yx y

xy x y

xy=

−−L

N

MMMMMMMM

O

Q

PPPPPPPP

1 0 0 0 01 0 0 0 0

0 0 0 0 0 00 0 0 1 0 00 0 0 0 0 00 0 0 0 0 0

/ // /

/

υυ

Fourier Elements

219

The use of this material model results in a reduction of the active stresses from

σ σ σ σ σx y z xy yz, , , , and σxz

to

σ σx y, and σxy

The plane stress material option is intended to allow the modelling of fan blades for which the use of the full modulus matrix is inappropriate. Note that elements using this material model should be adequately restrained in the tangential w direction.

A complete description of the element formulation is given in [C1].

7.10.5 Element Loading Concentrated loads, constant body forces and body force potentials are applied in the global XYZ directions, in contrast, surface tractions, initial stresses, initial strains and thermal loading are applied in the local xyz directions. Note that concentrated loads (and nodal reactions) are applied as forces per unit length of the structural surface.

The resolution of the global loads into the local tangential and radial directions is performed using the following matrices. Two possible axes of symmetry must be considered.

(i) Axisymmetric about x-axis

From fig.7.10-2a, the global loads are related to the local loads via

PPP

PPP

x

y

z

XG

YG

ZG

1

1

1

1 0 000

L

NMMM

O

QPPP

=−

L

NMMM

O

QPPP

L

NMMM

O

QPPP

cos sinsin cos

θ θθ θ

Applying the appropriate virtual perturbations gives

For symmetric contributions

P nP nP n

nn nn n

PPP

x

y

z

XG

YG

ZG

1

1

1

0 000

coscossin

coscos cos sin cossin sin cos sin

θθθ

θθ θ θ θθ θ θ θ

L

NMMM

O

QPPP

=−

L

NMMM

O

QPPP

L

NMMM

O

QPPP

For asymmetric contributions

P nP nP n

nn nn n

PPP

x

y

z

XG

YG

ZG

1

1

1

0 000

sinsincos

sincos sin sin sinsin cos cos cos

θθθ

θθ θ θ θθ θ θ θ−

L

NMMM

O

QPPP

=−

L

NMMM

O

QPPP

L

NMMM

O

QPPP


220

(ii) Axisymmetric about y-axis

From fig.7.10-2b, the global loads are related to the local loads via

PPP

PPP

x

y

z

XG

YG

ZG

1

1

1

00 1 0

0

L

NMMM

O

QPPP

=−

− −

L

NMMM

O

QPPP

L

NMMM

O

QPPP

cos sin

sin cos

θ θ

θ θ

Applying the appropriate virtual perturbations gives

For symmetric contributions

P nP nP n

n nn

n n

PPP

x

y

z

XG

YG

ZG

1

1

1

00 0

0

coscossin

cos cos sin coscos

sin sin cos sin

θθθ

θ θ θ θθ

θ θ θ θ

L

NMMM

O

QPPP

=−

− −

L

NMMM

O

QPPP

L

NMMM

O

QPPP

For asymmetric contributions

P nP nP n

n nn

n n

PPP

x

y

z

XG

YG

ZG

1

1

1

00 0

0

sinsincos

cos sin sin sinsin

sin cos cos cos

θθθ

θ θ θ θθ

θ θ θ θ−

L

NMMM

O

QPPP

=−L

NMMM

O

QPPP

L

NMMM

O

QPPP

For dynamic and harmonic response analyses where the automatic evaluation of Fourier loads is not available, the global loads for the 'nth' harmonic must be converted to local loads using the above expressions.

Fourier Elements

221

Z

Yx

Pz

PZ Py

PY

θ

Z

XY

Pz

PZ

Px

PX

θ

(a) Axi-symmetric about the global X-axis

(b) Axi-symmetric about the global Y-axis

Fig 7.10-2 Local And Global Loads

7.10.6 Inertial Loading The inertial loading due to angular accelerations and rotations require explicit evaluation and are directly applied to the element.


222

Assuming an arbitrary origin for the XYZ coordinate system (as shown in fig.7.10-3), about which the angular accelerations and velocities are to be applied, results in the following definition for the displacement vector r

r X x i Y r j Z r k= + + + + +d i d i d icos sinθ θ

where X Y Z, , define the shift in the global coordinate system.

Definition of the acceleration vector The instantaneous acceleration of a particle in a rotating coordinate system

r( , Y, Z)X with respect to a fixed system r' ( ' , Y' , Z' )X is [S4]

&&' && &r r xr x xr xr= + + +2Ω Ω Ωb g α

where a dot signifies the derivative with respect to time and the vectors

Ω Ω Ω Ω= + + = + +X Y z X Y zi j k i j k α α α α

are the angular velocities and accelerations about the r'(X', Y', Z') axes and

&& && && &&r Xi Y j Zk= + +

are the linear accelerations in the global X', Y' and Z' directions. Substituting for r gives the accelerations with respect to the fixed system, X', Y' and Z'. To apply the resulting inertial forces in the cylindrical coordinate system, the accelerations are resolved (fig.7.10-4)

&& &&'&& &&' cos &&' sin&& &&' sin &&' cos

x xy y zz y z

L

L

=

= +

= − +

θ θ

θ θ

Fourier Elements

223

z

yx

zl

Zyl

Y

θ

Figure 7.10-3 Definition Of Rotation Axes Origin

X

Z

Y

o

z

x y

XY

Z

θxθ

r

ΩZ,αZ

ΩX,αX ΩY,αY

Figure 7.10-4 Resolution Of Global To Local Accelerations


224

which gives the following acceleration terms in cylindrical coordinates

&&' cos sin cos sinr c c c c c= + + + +1 2 3 4 52 2θ θ θ θ

where

c

x X x Y Z

r

r

Y Z X Y Z X Z Y

Y Z X

X

1

2 2

2 2 212

2=

− + + + − + +

− + + +FH

IK

+

L

N

MMMMM

O

Q

PPPPP

&&

&

&&

Ω Ω Ω Ω Ω Ω

Ω Ω Ω

e jd i b g b ge j

e j

α α

θ

θ α

c

r

Y X x Y Z

Z X x Y Z

Y X Z

X Y Z X Z Y Z X

X Z Y Y Z X X Y

22 2

2 2

2

=

+ −

+ + + − + + −

+ − + + + − +

L

N

MMMMM

O

Q

PPPPP

Ω Ω

Ω Ω Ω Ω Ω Ω

Ω Ω Ω Ω Ω Ω

&

&&

&&

θ α

α α

α α

e je jb gd i e j b gb gd i b g e j

c

r

Z X x Y Z

Y X x Y Z

Z X Y

X Z Y Y Z X X Y

X Y Z X Z Y Z X

32 2

2 2

2

=

+ +

+ − + + + − +

− − + + + + − −

L

N

MMMMM

O

Q

PPPPP

Ω Ω

Ω Ω Ω Ω Ω Ω

Ω Ω Ω Ω Ω Ω

&

&&

&&

θ α

α α

α α

e je jb gd i b g e jb gd i e j b g

c r Y Z

Y Z

42 21

2

0

2= −L

NMMM

O

QPPP

Ω ΩΩ Ω

c r Y Z

Z Y

52 2

12

02=

−

L

NMMM

O

QPPP

Ω ΩΩ Ω

Using d'Alembert's principle, the inertia force may be included as part of the load vector

R N f x dvbT

v= −z ' &&e j e jρ

= +R Rb f b a( ) ( )

where Rb f( ) is the element nodal and body forces and Rb a( ) is the inertial force vector which includes the effects of angular velocities and accelerations

R N x Jb aT

( ) ' &&= − zzz −

+

−

+ b g ρ ξ ηπ

Y d d0

2

1

1

1

1

Fourier Elements

225

Centripetal load stiffening Centripetal load stiffening has been applied to the n = 0 term, but there is no nonlinear stress stiffening contribution. The centripetal load stiffening matrix, contrary, to its name, actually decreases the stiffness of the structure. Centripetal forces are proportional to the angular rotation squared and the lever arm of the mass from the centre of rotation. As the body spins, the lever arm is lengthened by positive displacements, which increases the applied load. This may, conversely, be thought of as reducing the stiffness. The centripetal load stiffness is applied by default, but it may be omitted by setting OPTION 102.

7.10.7 Evaluation of Stresses The stresses are evaluated at the element Gauss points and are extrapolated to the nodal points. The output consists of

Stress Output σ σ σ σ σ σx y z xy yz xz, , , , , direct and shear stresses

Strain Output ∈ ∈ ∈x y z, , , , , xy yz xzγ γ γ direct and shear strains

where σx , σy are the coefficients of the Fourier series expansion of the stresses and strains. The principal stresses and strains are evaluated at θ = 0 only.

7.11 Interface Elements (INT6, INT16) 7.11.1 Definition and interpolation Interface elements INT6 (6-noded line element for 2D analyses) and INT16 (16-node surface element for 3D analyses) are used to model composite delamination in an incremental non-linear analysis. They may be inserted at planes of potential delamination to model inter-laminar failure, and crack initiation and propagation. These elements have no geometric properties and are assumed to have no thickness (see Fig. 7.11-1).

1

23

4

5

6 zero

1 2 3

7 6 5

9 108 411

14 13

12

15

16zero


226

Fig.7.11-1 Interface Elements INT 6 and INT16

The displacement field u for these elements contains the bottom displacement ub and the top displacement ut (number of components in ub and ut is two for INT6 and three for INT16) so that u u uT

bT

tT= . The bottom and the top displacement are

interpolated as

u Hpb b= and u Hpt t

=

where pb

and ptare the vectors of the bottom and the top nodal displacements

(number of components in these vectors is six for INT6 and twenty-four for INT16) and H is the matrix of shape functions of the type

INT6: H hh

T T

T T=LNMM

OQPP

00

or INT16: Hh

hh

T T T

T T T

T T T=

L

N

MMM

O

Q

PPP

0 00 00 0

where ith component in the vector h (i=3 for INT6 and i=8 for INT16) is the value of the ith shape function at a particular point. The actual constitution of the interface element is defined in terms of the relative displacements between the bottom and the top surfaces

ε = − =RS|T|UV|W|

=u u Bpp Bpt b

b

t

where p is the vector of the nodal displacements (which has twelve components for INT6 and forty-eight components for INT16) and the matrix Bfollows as

Bh h

h h

T T

T T

T T

T T=−

−

LNMM

OQPP0 0

0 0 for INT6 and

Bh h

h hh h

T T T

T T T

T T T

T T T

T T T

T T T=

−−

−

L

N

MMM

O

Q

PPP

00 00 0 0

0 0 00 0

0 for INT16.

7.11.2 Internal force vector and stiffness matrix The element equilibrium equation in a vector form is given as

g P R≡ − = 0

where R is the vector of applied loading and P is the vector of nodal internal forces. The vector of nodal internal forces can be written in a standard form as

Interface Elements

227

P B dAA

T= z σ

where the integration domain A is a line for INT6 and an area for INT16. For a given arbitrary constitutive relationship (note that ε is a relative displacement between the element’s surfaces rather than a strain measure)

σ σ ε= b g where σ is a stress vector, the stiffness matrix in a geometrically linear analysis follows as

K B D BdAA

T

t= z

where Dt is a tangent constitutive matrix, which follows from Dt ij

i

j, =

∂σ∂ε

for a given

material model. In order to eliminate spurious oscillations of the stress field along the element [H13], the internal force vector and the stiffness matrix are integrated using a Newton-Cotes integration rule rather than a reduced or full Gauss integration rule. The 3-point Newton-Cotes scheme is utilised for INT6 and the 3*3-point Newton-Cotes scheme is utilised for INT16. The interface elements INT6 and INT16 can currently be used only with the delamination damage model (non-linear material model 25).

Appendix A

229

Appendix A Quadrature Rules The locations and weights of the quadrature points used in integrating the element matrices are listed in table A-1 to table A-7 and are shown in fig.A-1 to fig.A-7.

ORDER LOCATION ξi WEIGHT WI

1 0.0000000000 2.0000000000

2 ±0.5773502692 1.0000000000

3 ±0.7745966692

0.00000000000

0.5555555555

0.8888888888

4 ±0.8611363116

±0.3399810436

0.3478548454

0.6521451549

TABLE A-1 SAMPLING POINTS AND WEIGHTS FOR BARS, BEAMS,QUADRILATERAL 2-D SOLIDS, PLATES, SHELLS AND 3-D

HEXAHEDRA AND PENTAHEDRA

RULE LOCATION WEIGHT

ει ηι

5 point ±0.592348877 ±0.592348877 0.95000000

0.000000000 0.000000000 0.20000000 TABLE A-2 SAMPLING POINTS AND WEIGHTS FOR 5-POINT RULE FOR 2-

D QUADRILATERALS AND SHELLS

Appendix A

230


A1 A2 A3

1-point 0.3333333333 0.3333333333 0.3333333333 1.0000000000

3-point 0.5000000000 0.0000000000 0.0000000000 0.3333333333

4-point 0.3333333333

0.6000000000

0.3333333333

0.2000000000

0.3333333333

0.2000000000

-0.5625000000

0.5208333333

7-point 0.3333333333

0.0597158717

0.7974269853

0.3333333333

0.4701420641

0.1012865073

0.3333333333

0.4701420641

0.1012865073

0.2250000000

0.1323941527

0.1259391805

TABLE A-3 SAMPLING POINTS AND WEIGHTS FOR TRIANGULAR 2-D SOLIDS, PLATES, SHELLS AND 3-D PENTAHEDRA


A1 A2 A3

3-Point 1.0000000000 0.0000000000 0.0000000000 0.3333333333

TABLE A-4 SAMPLING POINTS AND WEIGHTS FOR TRIANGULAR SEMILOOF SHELL


V1 V2 V3 V4

1-Point 0.25000000 0.25000000 0.25000000 0.25000000 1.00000000

2-Point 0.58541020 0.13819660 0.13819660 0.13819660 0.25000000

3-Point 0.50000000 0.50000000 0.00000000 0.00000000 0.16666666

TABLE A-5 SAMPLING POINTS AND WEIGHTS FOR 3-D TETRAHEDRA

Appendix A

231


ξi ηi ζi

13-Point 0.00000000 0.00000000 0.00000000 1.684210565

±0.88030430 ± -0.49584802 ± -0.49584802 0.54498736

±0.79562143 ±0.79562143 ±0.025293237 0.507644216

14-Point ±0.795822426 0.000000000 0.000000000 0.355555556

-0.758786911 -0.758786911 -0.758786911 0.335180055

0.758786911 -0.758786911 -0.758786911 0.335180055

0.758786911 0.758786911 -0.758786911 0.335180055

0.758786911 0.758786911 0.758786911 0.335180055 TABLE A-6 - SAMPLING POINTS AND WEIGHTS FOR SPECIAL RULES FOR

3-D SOLIDS


1-Point 0.000000000 2.000000000

2-Point ±1.000000000 1.000000000

3-Point ±1.000000000 0.166666667

0.000000000 1.333333333

4-Point ±1.000000000 0.250000000

±0.333333333 0.750000000

5-Point ±1.000000000 0.155555556

±0.500000000 0.711111111

0.000000000 0.266666667

TABLE A-7 - SAMPLING POINTS AND LOCATIONS FOR NEWTON-COTES RULES

Appendix A

232

(a) 1-Point Rule

(b) 2-Point Rule

(c) 3-Point Rule

(d) 4-Point Rule

1

1

2

2

1 2 3 4

3

11 2

1 2

1

21

2

FIG.A-1 GAUSS QUADRATURE RULES FOR BAR, BEAM AND AXISYMMETRIC SHELL ELEMENTS

Appendix A

233

1 3

5

4

67

8

2

1 2 3

456

789

1 2

34

1 2

34

1 3

5

4

67

8

2

1 2

34

5

1 3

5

4

67

8

2

1 2 3 4

5 6 7 8

9 10 11 12

13 14 15 16

(a) 2*2 Rule (b) 3*3 Rule

(a) 4*4 Rule (b) 5-Point Rule

FIG.A-2 GAUSS QUADRATURE RULES FOR QUADRILATERAL 2-D CONTINUUM, PLATE AND SHELL ELEMENTS

Appendix A

234

1

3

2

1

2

3

1 2

1

3

1

3

4

6

7

2

1

2

3

5

1

3

4

2

1

23

(a) 1-Point Rule (b) 3-Point Rule

(a) 4-Point Rule (b) 7-Point Rule

FIG.A-3 GAUSS QUADRATURE RULES FOR TRIANGULAR 2-D CONTINUUM, PLATE AND SHELL ELEMENTS

1 2

1

3

2

3

FIG.A-4 SPECIAL 3-POINT RULE FOR TRIANGULAR SEMILOOF SHELL ELEMENT

Appendix A

235

1

2

1

3

4

(a) 1-Point Rule

1

2

1

3

4

(b) 4-Point Rule

4 2

3

1

2

1

3

4

(c) 6-Point Rule

4

23

5

6

FIG.A-5 GAUSS QUADRATURE RULES FOR SOLID TETRAHEDRA ELEMENTS

Appendix A

236

1

3

5

6

78

2

1 2

34

56

78

1 2

3

5

4

(a) 3*2 Rule (b) 2*2*2 Rule

(c) 3*3*2 Rule (d) 3*3*3 Rule

12

3

456

9

10 11 12

13 14 15

16

1

3

5

6

78

2

1 2 3

4 5

6

7 8

1718

910 11 12

2725

1

3

5

6

78

2

1 23

4 5

6

26

1819

FIG.A-6 QUADRATURE RULES FOR SOLID PENTAHEDRA AND HEXAHEDRA ELEMENTS

Appendix A

237

1

(a) 1-Point Rule

(b) 2-Point Rule

(c) 3-Point Rule

(d) 4-Point Rule

1 2

1 2 3

1 2 3 4

(e) 5-Point Rule

FIG.A-7 NEWTON-COTES RULESAppendix B

Appendix A

238

Appendix B

239

Appendix B Restrictions On Element Topology Mid-Length and Mid-Side Nodes The mid-length and mid-side nodes of elements should be equidistant from the two end nodes, and the element curvature must satisfy the following requirements

(i) |a - b|/(a + b) < 0.05

(ii) (a + b)/c < 1.02

where a, b and c are defined in fig.B-1.

a

b

c

Fig.B-1 DEFINITION OF PARAMETERS FOR CURVATURE LIMITS

Warping of Flat Elements The four nodes defining a flat quadrilateral element in 3-D should be coplanar. However, a small amount of warping is permitted provided that

z < 0.01 a

where z is the distance of the out of plane node from the plane

and a is the length of the side between the first and second nodes.

Appendix B

240

References

241

References Contact [email protected] for details of all references stated in this manual.

mailto:[email protected]

References

242

theory manual volume 2

Documents