Static Analysis: Truss Element Equations
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Transcript of 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.
Section II – Static Analysis
Module 3 – Truss Element Equations
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
Section II – Static Analysis
Module 3 – Truss Element Equations
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
Section II – Static Analysis
Module 3 – Truss Element Equations
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
Section II – Static Analysis
Module 3 – Truss Element Equations
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
Section II – Static Analysis
Module 3 – Truss Element Equations
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
Section II – Static Analysis
Module 3 – Truss Element Equations
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
Section II – Static Analysis
Module 3 – Truss Element Equations
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.
Section II – Static Analysis
Module 3 – Truss Element Equations
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
Section II – Static Analysis
Module 3 – Truss Element Equations
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.
Section II – Static Analysis
Module 3 – Truss Element Equations
Page 12
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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.
Section II – Static Analysis
Module 3 – Truss Element Equations
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.
Section II – Static Analysis
Module 3 – Truss Element Equations
Page 14
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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.
Section II – Static Analysis
Module 3 – Truss Element Equations
Page 15
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1st Integral: Differential of Green’s Strain
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.
Section II – Static Analysis
Module 3 – Truss Element Equations
Page 16
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1st Integral: Differential of Green’s Strain
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
Section II – Static Analysis
Module 3 – Truss Element Equations
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
Section II – Static Analysis
Module 3 – Truss Element Equations
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
Section II – Static Analysis
Module 3 – Truss Element Equations
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
Section II – Static Analysis
Module 3 – Truss Element Equations
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
Section II – Static Analysis
Module 3 – Truss Element Equations
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
Section II – Static Analysis
Module 3 – Truss Element Equations
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
Section II – Static Analysis
Module 3 – Truss Element Equations
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
Section II – Static Analysis
Module 3 – Truss Element Equations
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
Section II – Static Analysis
Module 3 – Truss Element Equations
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
Section II – Static Analysis
Module 3 – Truss Element Equations
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.
Section II – Static Analysis
Module 3 – Truss Element Equations
Page 27
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Module SummarySection II – Static Analysis
Module 3 – Truss Element Equations
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.