Static Analysis: Truss Element Equations

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Static Analysis:Truss Element Equations


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Objectives

The objective of this module is to show how the equations developed in Modules 1 and 2 convert to matrix equations for a typical element.

Notation familiar with upper division undergraduate students is used instead of the more compact indicial notation introduced at the graduate level.

The various differentials, variations, and time derivatives of Green’s strain used in the Lagrangian rate of virtual work are developed.

A one-dimensional truss element is used to demonstrate the process It contains most of the features of multi-dimensional elements, The integrations can be carried out manually.

Section II – Static Analysis

Module 3 – Truss Element Equations

Page 2


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Configurations

x,x*

y,y*

z,z*

Body in Reference Configuration

Body in Current Configuration

P

P*

Q

Q*

),,(

),,(

),,(

*

*

*

zyxhz

zyxgy

zyxfx

f, g & h are functions that relate

the coordinates of a point in the current configuration to the coordinates in reference configuration.

An arbitrary line element is defined by points P & Q in the reference configuration.

The same points are defined by P* and Q* in the current configuration.



Page 3


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Deformation Gradient

dzzhdy

yhdx

xhdz

dzzgdy

ygdx

xgdy

dzzfdy

yfdx

xfdx

*

*

*

Differential Changes Matrix Form

dzdydx

zh

yh

xh

zg

yg

xg

zf

yf

xf

dzdydx

***

The Deformation Gradient is defined by the array [F].

It is the Jacobian of the transformation between the current and reference configurations.

zh

yh

xh

zg

yg

xg

zf

yf

xf

F



Page 4

A differential change in the reference configuration and current configuration coordinates are related through the deformation gradient.


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Deformation Gradient Components

),,(*

),,(*),,(*

zyxuzz

zyxuyyzyxuxx

z

y

x

),,(

),,(),,(

zyxuzh

zyxuygzyxuxf

z

y

x

The displacements ux, uy, and uz in the x, y and z directions can be used to determine the mapping functions f, g and h.

.

1

1

1

zu

yu

xu

zu

yu

xu

zu

yu

xu

F

zzz

yyy

xxx

Using these functions, the components of the deformation gradient become



Page 5

Current Configuration

Reference Configuration

u

xu

yuzu

zyxP ,,

**** ,, zyxP

x

y

z


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Green’s Strain

2

22

21

o

o

dSdSdSE



Page 6

Green’s strain is defined by the equation

Green’s strain uses the length squared of a differential line element instead of the differential length.

For small displacements and rotations, Green’s strain and the small strain give similar results.

Small strain definition

o

o

dSdSdSe


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Green’s Strain and Deformation Gradient

2222 dzdydxdSo

Reference ConfigurationCurrent Configuration2222 *** dzdydxdS

dzdydx

dzdydxdSo100010001

2

***

***2

dzdydx

dzdydxdS

dzdydx

FFdzdydxdS T2

The Deformation Gradient is the fundamental building block needed to find the components of Green’s strain

Deformation Gradient



Page 7

dzdydx

IdzdydxdSo2


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Components of Green’s Strain

Green’s Strain

Components of Green’s Strain

zzzyzx

yzyyyx

xzxyxxT

EEEEEEEEE

IFFE21

o

o

o

ooo

T

o

dSdzdSdydSdx

EdSdz

dSdy

dSdx

dzdydx

IFFdS

dzdydxE 221



Page 8

The combination of equations on the previous slide gives the equation

where

This equation is used extensively in subsequent slides.


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Green’s Strain – Displacement Equations

zzzyzx

yzyyyx

xzxyxxT

EEEEEEEEE

IFFE21

zw

yw

xw

zv

yv

xv

zu

yu

xu

F

1

1

1

Carrying out the matrix operations yields

222

21

xw

xv

xu

xuExx

222

21

yw

yv

yu

yvEyy

222

21

zw

zv

zu

zwEzz

zv

zu

yv

yu

xv

xu

xv

yuExy 2

121

zw

zv

yw

yv

xw

xv

yw

zvEyz 2

121

zu

zw

yu

yw

xu

xw

zu

xwExz 2

121



Page 9

The Deformation Gradient can be used to find the equations for the component of Green’s strain commonly found in textbooks.


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Truss Element Geometry

Reference Configuration

Current Configuration

I

J

K

iu

ju

3

2

1

uuu

uuu

uiz

iy

ix

i

Node i

Node j

6

5

4

uuu

uuu

ujz

jy

jx

j

xy

z X

Y

Z

X,Y, Z Global Coordinate System

x,y,z Element Coordinate System

Global Coordinate System

Element Coordinate System

Node i

Node j

Element matrices are derived using the element coordinate system. They can be transformed to the global coordinate system at the end.



Page 10


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Notation

Quantities that have an over score are associated with a node point (i.e. node i or node j).

3

2

1

uuu

uuu

uiz

iy

ix

i

In the array, is equal to the x displacement measured at node i.1u



Page 11


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Truss Interpolation Functions

6231

5221

4211

uNuNu

uNuNuuNuNu

z

y

x

LxN 2

LxLN

1

6231

5221

4211

vNvNv

vNvNvvNvNv

z

y

x

The displacement or virtual velocity components at any point along the length of the element can be found using interpolation functions.

where

Note that at x = L, N1 = 0 and N2 = 1, at x = 0, N1 = 1 and N2 = 0.



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Tangent Stiffness Matrix Equations

.~1

~

IEEI WWuWWu

dVdfvdSdtvdVTEddVdTEWWdoooo V

oTo

S

oT

VVEI

1st Integral 2nd Integral 3rd Integral 4th Integral

The Newton-Raphson equations developed in Module 2 are

1~1

~

EIEI WWduWWu

The left hand side of this equation can be written as

where

The first two integrals are evaluated in subsequent slides. The third and fourth integrals are not as important for common problems and are not evaluated.



Page 13


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1st Integral: Stress Increment

dVdTEoV

The first integral that contributes to the tangent stiffness matrix is

The differential of the matrix containing the components of the 2nd Piola stress tensor can be related to the differential of Green’s strain via material constitutive equations.

dEMdT

For a truss element made from a linear elastic material this equation becomes

xxxx dEYdT

Where Y is Young’s Modulus.



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1st Integral: Differential of Green’s Strain

FdFFFdEd TT 21

xu

xdu

xu

xdu

xu

xdu

xdudE zzyyxxx

xx

yu

ydu

yu

ydu

yu

ydu

ydu

dE zzyyxxyyy

zu

zdu

zu

zdu

zu

zdu

zdudE zzyyxxz

zz

xu

ydu

xu

ydu

xu

ydu

yu

xdu

yu

xdu

yu

xdu

xdu

ydudE zzyyxxzzyyxxyx

xy 21

21

yu

zdu

yu

zdu

yu

zdu

zu

ydu

zu

ydu

zu

ydu

zdu

ydudE zzyyxxzzyyxxyz

yz 21

21

xu

zdu

xu

zdu

xu

zdu

zu

xdu

zu

xdu

zu

xdu

xdu

zdudE zzyyxxzzyyxxzx

xz 21

21

The first integral also requires equations for the differential of the virtual rate of Green’s strain.

All six components are given, but only dExx is needed for the truss element.



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xu

xdu

xu

xdu

xu

xdu

xdudE zzyyxxx

xx

6231

5221

4211

uNuNu

uNuNuuNuNu

z

y

x

6231

5221

4211

udNudNdu

udNudNduudNudNdu

z

y

x

414

21

1 11 udL

udL

udxNud

xN

xdux

525

22

1 11 udL

udL

udxNud

xN

xdu y

636

23

1 11 udL

udL

udxNud

xN

xduz

4142

11 11 u

Lu

Lu

xNu

xN

xux

5245

12 11 u

Lu

Lu

xNu

xN

xu y

6346

13 11 u

Lu

Lu

xNu

xN

xu y

The truss element interpolation functions can be used to evaluate dExx for the truss element.



Page 16


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uduBBdE NLLxx

001001

LLBL

236

225

214

263

252

241

Luu

Luu

Luu

Luu

Luu

LuuBNL

6

5

4

3

2

1

udududududud

ud

Combining the equations from the previous slide and writing them in matrix notation yields

where

The array BL contains the linear terms and array BNL contains the displacement dependent non-linear terms.

1x1 1x6 6x1



Page 17


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1st Integral: Virtual Rate of Green’s Strain

FFFFE TT 21

xu

xv

xu

xv

xu

xv

xvE zzyyxxx

xx

xu

yv

xu

yv

xu

yv

yu

xv

yu

xv

yu

xv

xv

yvE zzyyxxzzyyxxyx

xy

21

21

yu

yv

yu

yv

yu

yv

yv

E zzyyxxyyy

zu

zv

zu

zv

zu

zv

zvE zzyyxxz

zz

yu

zv

yu

zv

yu

zv

zu

yv

zu

yv

zu

yv

zv

yvE zzyyxxzzyyxxyz

yz

21

21

xu

zv

xu

zv

xu

zv

zu

xv

zu

xv

zu

xv

xv

zvE zzyyxxzzyyxxzx

xz

21

21



Page 18

The first integral also requires equations for the virtual rate of Green’s strain.

All six components are given, but only is needed for the truss element.

xxE


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1st Integral: Virtual Rate of Green’s Strain

vduBBE NLLxx

001001

LLBL

236

225

214

263

252

241

Luu

Luu

Luu

Luu

Luu

LuuBNL

6

5

4

3

2

1

vdvdvdvdvdvd

vd

In a manner similar to that used for the differential of Green’s strain, these equations can be written as

where

1x1 1x6 6x1



Page 19


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1st Integral: Final Form

udBLAYBvdVdTE TTo

Vo

61x 16x 61x 16x11x

udKvdVdTE To

Vo

where

BLAYBK T

Collecting terms from previous slides and carrying out the integration yields the equation for the 1st integral

or

NLL uBBB

Contribution to Tangent Stiffness Matrix

Relates element strain increments or rates to the node point displacement

increments or rates.

6x6 1x6



Page 20


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1st Integral: Material Stiffness Matrix

000000000000

0000000000000000

0000

LYA

LYA

LYA

LYA

KL

BLAYBK T

This 6x6 matrix is a function of Young’s modulus and is thus dependent on the material. It is also a function of the length and cross sectional area.

The matrix [B] contains two contributions. [BL] is linear and leads to the linear stiffness matrix, [KL] . [BNL] is a function of the displacements and changes as the truss element deforms. This gives rise to a non-linear stiffness contribution. This contribution to the tangent stiffness matrix is denoted as [K(u)].

Linear Stiffness Matrix



Page 21


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2nd Integral: Differential of Virtual Rate of Green’s Strain

.dVTEdoV

zdu

ydu

xdu

zdu

ydu

xdu

zdu

ydu

xdu

Fd

zzz

yyy

xxx

zv

yv

xv

zv

yv

xv

zv

yv

xv

F

zzz

yyy

xxx

zv

zv

zv

yv

yv

yv

xv

xv

xv

F

zyx

zyx

zyx

T

zdu

zdu

zdu

ydu

ydu

ydu

xdu

xdu

xdu

Fd

zyx

zyx

zyx

T

The second integral is



Page 22

FFdFdFEd TT 21

xdu

xv

xdu

xv

xdu

xvEd

zzyy

xxxx

Carrying out the matrix multiplications yields

This is the only component needed for a truss element.


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2nd Integral: Manipulations

xduT

xv

xdu

Txv

xduT

xvEdT z

xxzy

xxyx

xxx

xxxx

xduxduxdu

TT

T

xvxvxv

EdT

z

y

x

xx

xx

xx

T

z

y

x

xxxx

000000



Page 23

The integrand for a truss element is

The integrand can be written in matrix form as


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2nd Integral: Interpolation Functions

Using the interpolation functions the partial derivatives can be written in terms of node point values

LL

LL

LLG

100100

010010

001001

6

5

4

3

2

1

vvvvvv

G

xvxvxv

z

y

x

udGSGvEdT TTxxxx

udGSGvLAdVEdT TTo

Vxxxx

o

6

5

4

3

2

1

udududududud

G

xduxduxdu

z

y

x

where

The integrand becomes

and the integral becomes



Page 24


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2nd Integral: Initial Stress Stiffness Matrix

udLATvdVEdT xxT

oV

xxxx

o

100100010010001001100100

010010001001

Truss Element Initial Stress Stiffness Matrix

100100010010001001100100

010010001001

3 LATK xx

oV

T dVGSGKo

3

General Form for other Element Types



Page 25


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Restoring Force

The restoring force comes from the internal rate of virtual work term

oV

I dVETWo

The virtual rate of Green’s strain for the truss element is given on previous slides as

vBBE NLLxx

The truss element internal rate of virtual work becomes

vdVBBTW oNLLV

xxI

o



Page 26


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Restoring Force

TNLLxxTT BBTLAvRvW int

Since Txx and the components of BL and BNL are constant over the volume of the element, the previous equation reduces to

TNLLT

xxNLLxxI BBvTLAvBBTLAW

or

where

TNLLxx BBTLAR intThis is a 6 x 1 array that has units of force.



Page 27


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Module SummarySection II – Static Analysis


Page 28

This module has shown how to go from the incremental form of the rate of virtual work to the matrix equations for an element.

The deformation gradient and its variations and derivatives are key ingredients to this process.

A truss element was used because the element level integrations can be carried out by hand.

Matrix notation is used in-lieu of the more common indicial notation.

Documents

Static Analysis: Truss Element Equations