from structural analysis to femevents.iitgn.ac.in/2013/fem-course/handouts/from... · method”,...
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From Structural Analysis to FEM
Dhiman Basu
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AcknowledgementAcknowledgement
Following text books were consulted whileFollowing text books were consulted while preparing this lecture notes:
• Zienkiewicz O C and Taylor R L (2000) “The Finite ElementZienkiewicz, O.C. and Taylor, R.L. (2000). The Finite Element Method”, Vol. 1: The Basis, Fifth edition, Butterworth‐Heinemann.
• Yang, T.Y. (1986). “Finite Element Structural Analysis”, Prentice‐Hall Inc.
• Jain A K (2009) “Advanced Structural Analysis” Nem ChandJain, A.K. (2009). Advanced Structural Analysis , Nem Chand& Bros.
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IntroductionIntroductionStructural Modeling• Line element• Line element• Refined Line Element• Detailed Finite Element
Line elementLine element
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LiRefined lineLine
elementline element
FEM: Discretization over entire volume in general
Analysis
•Conventional Structural AnalysisLine elementRefined line element
•FEMVolume discretization
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OrganizationOrganization
• Conventional Structural AnalysisConventional Structural Analysis
• Revisit to Conventional Analysis
i f C l i f S l• Brief Conceptual Review of FEM Structural Analysis
• Similitude between both Analyses
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Element EquilibriumElement Equilibrium
{ } { }e e eK⎡ ⎤
2 2
12 6 12 6L L L L
⎡ ⎤⎢ ⎥−⎢ ⎥⎢ ⎥⎧ ⎫ ⎧ ⎫
EA EA⎡ ⎤⎢ ⎥
{ } { }e e eq K a⎡ ⎤= ⎢ ⎥⎣ ⎦
1 1
1 1
2 22 2
6 64 2
12 6 12 6
Y vM EI L LY vL
L L L LM
θ
θ
⎢ ⎥⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪⎢ ⎥⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪−⎢ ⎥⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎢ ⎥⎪ ⎪ ⎪ ⎪=⎨ ⎬ ⎨ ⎬⎢ ⎥⎪ ⎪ ⎪ ⎪⎢ ⎥⎪ ⎪ ⎪ ⎪− − −⎪ ⎪ ⎪ ⎪⎢ ⎥⎪ ⎪ ⎪ ⎪⎢ ⎥⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪
3 2 3 21
1
0 0 0 0
12 6 12 60 0
6 4 6 20 0
EA EAL L
EI EI EI EIX L L L LY EI EI EI EI
⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥
−⎢⎧ ⎫⎪ ⎪⎪ ⎪ ⎢⎪ ⎪ ⎢⎪ ⎪⎪ ⎪ ⎢⎪ ⎪ −⎪ ⎪ ⎢
1
1
uv
⎥ ⎧ ⎫⎪ ⎪⎪ ⎪⎥⎪ ⎪⎥⎪ ⎪⎪ ⎪⎥⎪ ⎪⎪ ⎪⎥2 26 62 4
L L L LM
L L
θ⎢ ⎥⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭⎢ ⎥⎢ ⎥−⎢ ⎥⎣ ⎦
2 21
2
2
23 2 3 2
0 0
0 0 0 0
12 6 12 60 0
M L L L LX EA EA
L LYEI EI EI EIM
L L L L
−⎪ ⎪ ⎢⎪ ⎪⎪ ⎪ ⎢=⎨ ⎬ ⎢⎪ ⎪ ⎢⎪ ⎪ −⎪ ⎪ ⎢⎪ ⎪⎪ ⎪ ⎢⎪ ⎪ ⎢⎪ ⎪⎪ ⎪ ⎢⎪ ⎪ − − −⎩ ⎭ ⎢
1
2
2
2
uv
θ
θ
⎪ ⎪⎥⎪ ⎪⎪ ⎪⎥⎨ ⎬⎥⎪ ⎪⎥⎪ ⎪⎪ ⎪⎥⎪ ⎪⎪ ⎪⎥⎪ ⎪⎥⎪ ⎪⎪ ⎪⎥⎪ ⎪⎩ ⎭⎥( ),eK i j Force along j‐th dof when unit 3 2 3 2
2 2
6 2 6 40 0
L L L LEI EI EI EIL L L L
⎢⎢⎢⎢ −⎢⎣ ⎦
⎥⎥⎥⎥⎥
( )jdisplacement is applied i‐th dofwhile all others are restraint
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Local and Global Coordinate SSystems
Local coordinate
l b l
Non‐orthogonally aligned element axis
Global coordinate
Coordinate transformation by rotationby rotation
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Orthogonal TransformationOrthogonal Transformation
{ } [ ]{ }'
' '
'
cos sin 0sin cos 00 0 1
x xy y
θ θθ θ δ λ δ
θ θ
⎧ ⎫⎧ ⎫ ⎡ ⎤⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎢ ⎥⎪ ⎪⎪ ⎪⎪ ⎪ ⎪ ⎪⎢ ⎥= − ⇒ =⎨ ⎬ ⎨ ⎬⎢ ⎥⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎢ ⎥⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎣ ⎦⎪ ⎪
[ ] [ ] 1
0 0 1T
θ θ
λ λ −
⎪ ⎪⎩ ⎭ ⎣ ⎦⎪ ⎪⎩ ⎭
=
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Element Equilibrium in Gl b l C diGlobal Coordinate
Transformation of displacement and force vectors
{ }
1 11 1
1 11 1
cos sin 0 0 0 0sin cos 0 0 0 0
L
u Xu Xv Yv Y
φ φφ φ
⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥−⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ { }G
Transformation of displacement and force vectors
{ }1 11 1
2 22 2
2 22 2
0 0 1 0 0 00 0 0 cos sin 00 0 0 sin cos 0
eL ea TMMu Xu Xv Yv Y
θθφ φφ φ
⎢ ⎥⎢ ⎥ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ = ⎢⎣⎢ ⎥ ⎢ ⎥⎢ ⎥= ⇒⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥−⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥
{ }{ } { }
eG
eL e eG
a
q T q
⎥⎦⎡ ⎤= ⎢ ⎥⎣ ⎦
2 22 2 0 0 0 0 0 1 MM θθ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦
Transformation of Equilibrium Equation
{ } { } { } { } { } { }{ } { } and
TeL eL eL e eG eL e eG eG e eL e eG
TeG eG eG eG e eL e
q K a T q K T a q T K T a
q K a K T K T
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤= ⇒ = ⇒ =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦{ } { }q ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦
Size of the problem remains same
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Direct Stiffness MethodDirect Stiffness MethodStep‐1: Element Equilibrium in Local Coordinate
{ } { }6 1 6 16 6
ii ieL eL eLq K a× ××
⎡ ⎤= ⎢ ⎥⎣ ⎦
i f h fi d d f d l dinegative of the fixed end forces due to span loading
Step‐2: Element Equilibrium in Global Coordinate
{ } { }
{ } { } { } { }6 1 6 16 6
6 1 6 1 6 1 6 16 6 6 6 6 6 6 6 6 6 6 6
, ,
ii ieG eG eG
i T i T Ti i i ieG ei eL ei eG ei eL eG ei eL
q K a
K T K T q T q a T a
× ××
× × × ×× × × × × ×
⎡ ⎤= ⎢ ⎥⎣ ⎦⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤= = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦
Step‐3: Element Equilibrium in Expanded Global Coordinate
{ } { }ii iE G E G E G⎡ ⎤{ } { }3 1 3 13 3
ii iExp eG Exp eG Exp eG
N NN Nq K a
× ××⎡ ⎤= ⎢ ⎥⎣ ⎦ Assuming a plane frame of N nodes
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Step‐4: Assemble in Element Equilibrium in Expanded Global Coordinate
{ } { }( )3 1 3 13 3
M M ii iExp eG Exp eG Exp eG
N NN Nq K a
× ××⎡ ⎤= ⎢ ⎥⎣ ⎦∑ ∑{ } { }( )3 1 3 13 3
1 1N NN N
i i× ××
= =⎣ ⎦∑ ∑
{ } { } { }*
3 1 3 1 3 13 3
G G G
N N NN Nq q K a
× × ××⎡ ⎤+ = ⎢ ⎥⎣ ⎦
Accounting for directly applied nodal concentrated forces
Step‐5: Effect of Restraints
{ } [ ] { }1 1S S S Sq K a
× × ×=
Step‐6: Solution for Displacement
{ } [ ] { } { }1
1 1 3 1
GS S S S N
a K q a−
× × ×= ⇒{ } [ ] { } { }1 1 3 1S S S S N
q× × × ×
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Step‐7: Solution for Element Response
{ } { }
{ } { } { }6 1 6 16 6
i ieL ei eG
ii i ieL eL eL eL
a T a
F K a q
× ××⎡ ⎤= ⎢ ⎥⎣ ⎦⎡ ⎤= ⎢ ⎥
Displacement in Local coordinate
Member end forces in Local coordinate{ } { } { }6 1 6 1 6 16 6
F K a q× × ××
⎡ ⎤= −⎢ ⎥⎣ ⎦ Member end forces in Local coordinate
Step‐8: Calculation of Reaction Forcesp f
0r rr rs rr rs s
s sr ss s
q K K aq K a
q K K a⎧ ⎫ ⎡ ⎤⎧ ⎫=⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎢ ⎥= ⇒ =⎨ ⎬ ⎨ ⎬⎢ ⎥⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎣ ⎦⎩ ⎭s sr ss s⎩ ⎭ ⎣ ⎦⎩ ⎭
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Numerical ExampleNumerical Example
EA=8000 kN/m2 and EI= 20000 kNm2
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Solution vector: {‐0.00356, 0.00275, ‐0.0058, 0.00178}T.
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Member end forces but in global coordinate
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Revisit to Stiffness MatrixRevisit to Stiffness Matrix4
4 0v∂=
∂Equilibrium of a beam element (constant EI) in the unloaded region4x∂ unloaded region
( ) 2 31 2 3 4v x x x xα α α α= + + + Assumed solution
1 1
2 2
and at 0
and at
vv v xxvv v x L
θ
θ
∂= = =
∂∂
= = =
Boundary conditions
2 2 and at v v x Lx
θ∂
1 0 0 0v α⎧ ⎫ ⎧ ⎫⎡ ⎤⎪ ⎪ ⎪ ⎪ 3 0 0 0L⎡ ⎤⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪1 1
1 22 3
2 32
1 0 0 00 1 0 010 1 2 3
v
v L L LL L
αθ α
αθ α
⎡ ⎤⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎢ ⎥⎪ ⎪ ⎪ ⎪⎢ ⎥⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎢ ⎥=⎨ ⎬ ⎨ ⎬⎢ ⎥⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎢ ⎥⎪ ⎪ ⎪ ⎪⎢ ⎥⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎣ ⎦
{ } [ ]{ }
31 1
32 1
3 2 23 2
0 0 00 0 013 2 32 2
vLL
H avL L L L L
L L
αα θ
ααα θ
⎡ ⎤⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎢ ⎥⎪ ⎪ ⎪ ⎪⎢ ⎥⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎢ ⎥= ⇒ =⎨ ⎬ ⎨ ⎬⎢ ⎥⎪ ⎪ ⎪ ⎪− − −⎪ ⎪ ⎪ ⎪⎢ ⎥⎪ ⎪ ⎪ ⎪⎢ ⎥⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎢ ⎥⎪ ⎪ ⎪ ⎪⎣ ⎦2 40 1 2 3L Lθ α⎢ ⎥⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎣ ⎦⎩ ⎭ ⎩ ⎭ 4 22 2L Lα θ−⎪ ⎪ ⎪ ⎪⎢ ⎥⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭⎣ ⎦
Solution for coefficients
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( ) ( ) ( ) ( ) ( )1 1 1 2 2 3 2 4v x v f x f x v f x f xθ θ= + + + Displacement profile
( )2 3
1
2
1 3 2x xf xL L⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜= − +⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠
⎡ ⎤⎛ ⎞ ⎛ ⎞( )
( )
2
2 3
1 2
3 2
x xf x xL L
x xf
⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥⎟ ⎟⎜ ⎜= − +⎟ ⎟⎜ ⎜⎢ ⎥⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜( )
( )
3
2
4
3 2f xL L
x xf x xL L
⎟ ⎟⎜ ⎜= −⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥⎟ ⎟⎜ ⎜= − +⎟ ⎟⎜ ⎜⎢ ⎥⎟ ⎟⎜ ⎜( )4f L L⎢ ⎥⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦
Specific case
θ θ
( ) ( )1 1 2 2
1
1.0, 0, 0, 0,v vv x f x
θ θ= = = =
⇒ =
Displacement profile associated with first column of stiffness matrix
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Application of Castigliano’s Theorempp f g
ii
UPa
∂=
∂
22
202
LEI vU dxx
⎛ ⎞∂ ⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜∂⎝ ⎠∫ Assuming only flexural deformation
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
2 2'' '' '' '' ''
1 1 1 1 2 2 3 2 4 12 21 10 0
'' '' '' '' '' '' '' ''
L L
L L L
U v vY EI dx EI v f x f x v f x f x f x dxv x v x
θ θ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂⎟ ⎟⎜ ⎜ ⎡ ⎤⎟ ⎟= = = + + +⎜ ⎜⎟ ⎟ ⎢ ⎥⎣ ⎦⎜ ⎜⎟ ⎟⎜ ⎜∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡
∫ ∫
∫ ∫ ∫L
⎤∫( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )'' '' '' '' '' '' '' ''1 1 1 1 1 2 2 1 3 2 1 4
0 0 0
v EI f x f x dx EI f x f x dx v EI f x f x dx EI f x f xθ θ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡= + + +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦∫ ∫ ∫0
11 1 12 1 13 2 14 2
dx
K v K K v Kθ θ
⎤⎢ ⎥⎣ ⎦
= + + +
∫
First equation of equilibrium in local coordinate
( ) ( )'' ''
0
L
ij i jK EI f x f x dx⎡ ⎤= ⎢ ⎥⎣ ⎦∫ ij‐th element of stiffness matrix
For example,'' '' 22 3 2 3
11 2 3 30 0
6 12 121 3 2 1 3 2L Lx x x x x EIK EI dx EI dx
L L L L L L L
⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎟ ⎟ ⎟ ⎟⎜ ⎜ ⎜ ⎜ ⎢ ⎥= − + − + = − + =⎟ ⎟ ⎟ ⎟⎜ ⎜ ⎜ ⎜⎢ ⎥ ⎢ ⎥⎟ ⎟ ⎟ ⎟⎜ ⎜ ⎜ ⎜ ⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦∫ ∫
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Application of Rayleigh Ritz Method
( ) ( )
( ) ( )
2 3 ''1 2 3 4 3 4
2 2 2 2 33 4 3 3 4 4
2 6
2 6 2 6 62
L
v x x x x v x x
EIU x dx EI L L L
α α α α α α
α α α α α α
= + + + ⇒ = +
= + = + +∫ Strain energy( ) ( )02 ∫
{ } { } { }
1
2
0 0 0 00 0 0 01 1 T k
αα⎧ ⎫⎡ ⎤⎪ ⎪⎪ ⎪⎢ ⎥⎪ ⎪⎢ ⎥⎪ ⎪⎪ ⎪⎪ ⎪ ⎡ ⎤⎢ ⎥ Q d i f{ } { } { }2
1 2 3 4 23
2 34
2
1 10 0 4 62 20 0 6 12
TU kEIL EILEIL EIL
Uk
α α α α α ααα
⎪ ⎪⎪ ⎪ ⎡ ⎤⎢ ⎥= =⎨ ⎬ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎪ ⎪⎪ ⎪⎢ ⎥⎪ ⎪⎢ ⎥⎪ ⎪⎪ ⎪⎪ ⎪⎣ ⎦⎩ ⎭∂
=
Quadratic form
iji j
kα α
=∂ ∂
{ } [ ] [ ]( ){ }12
T TU a H k H a⎡ ⎤= ⎢ ⎥⎣ ⎦ { } { } [ ]{ }
1
11 1 T
YM
W Kθ θ
⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬
External work done
{ } [ ] [ ]( ){ }2 ⎢ ⎥⎣ ⎦ { } { } [ ]{ }1
1 1 2 22
2
2 2W v v a K a
YM
θ θ ⎪ ⎪= =⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎩ ⎭
T ⎡ ⎤[ ] [ ] [ ]TK H k H⎡ ⎤= ⎢ ⎥⎣ ⎦
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[ ]
3 3
3 3
23 32 2 2 2
0 0 0 00 0 0 0 0 00 0 0 00 0 0 0 0 01 10 0 4 63 2 3 3 2 3
TL L
L LK
EIL EILL LL L L L L L L L
⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥= ⎢ ⎥ ⎢ ⎥⎢ ⎥[ ] 23 32 2 2 2
2 3
2 2
0 0 4 63 2 3 3 2 30 0 6 122 2 2 2
12 6 12 6
EIL EILL LL L L L L L L LEIL EILL L L L
L L L L
⎢ ⎥ ⎢ ⎥⎢ ⎥− − − − − −⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥− −⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦
⎡⎢ −⎢⎢⎢
⎤⎥⎥⎥⎥
2 2
6 64 2
12 6 12 6
6 6
EI L LL
L L L L
⎢⎢
−⎢=
− − −
⎥⎥⎥
⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
Same as before
6 62 4L L
−⎣⎢ ⎥⎢ ⎥⎦
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FEM: A Preliminary RevisitFEM: A Preliminary Revisit
Nodal
Displacement function
{ }T
i xi yia u u=
Nodal displacement
( ) ( ){ }, ,T
xi yiu u x y u x y=
Displacement at any point
e⎧ ⎫
ˆ ....
e
ie e
k k i j jk
au u N a N N a Na
⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎡ ⎤≈ = = =⎨ ⎬⎢ ⎥⎣ ⎦⎪ ⎪⎪ ⎪⎪ ⎪∑
An example of a plane‐stress problem
.
.⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎩ ⎭
Shape functions
( ) 1 i jN δ
⎧ =⎪⎪( ),0i j j ij
jN x y
i jδ ⎪= =⎨⎪ ≠⎪⎩
ˆ eu u Na≈ = In general
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Strain‐Displacement Relation
{ } { } [ ]{ }ˆ S uε ε≈ = For plane stress problem
0xu⎧ ⎫ ⎡ ⎤⎪ ⎪∂ ∂⎪ ⎪ ⎢ ⎥⎪ ⎪
{ }
0
0
x
xxxy
yyy
xy
x xuuuy y
εε ε
ε
∂⎢ ⎥⎪ ⎪⎪ ⎪ ⎢ ⎥∂⎪ ⎪ ∂⎧ ⎫ ⎢ ⎥⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎢ ⎥⎧ ⎫∂⎪ ⎪ ⎪ ⎪⎪ ⎪ ∂⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎢ ⎥= = =⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎢ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪∂ ∂⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩ ⎭⎢ ⎥⎪ ⎪ ⎪ ⎪ ⎢ ⎥⎪ ⎪ ⎪ ⎪⎩ ⎭ ∂ ∂ ∂⎪ ⎪∂
{ } { } [ ]{ } [ ][ ]{ } [ ]{ }[ ] [ ][ ]
ˆ e eS u S N a B aε ε≈ = = =
yyx uu
y xy x
⎢ ⎥⎪ ⎪ ⎪ ⎪⎩ ⎭ ∂ ∂ ∂⎪ ⎪∂ ⎢ ⎥⎪ ⎪+⎪ ⎪ ⎢ ⎥⎪ ⎪ ∂ ∂∂ ∂ ⎣ ⎦⎪ ⎪⎩ ⎭
[ ] [ ][ ]B S N=
Constitutive RelationFor plane stress problem
{ } [ ]{ } { }0 0Dσ ε ε σ= − +
For plane stress problem
{ } [ ]1 0xx E
σ ν⎧ ⎫ ⎡ ⎤⎪ ⎪⎪ ⎪ ⎢ ⎥⎪ ⎪⎪ ⎪ ⎢ ⎥{ } [ ]( )
2 and 1 01
0 0 1 2yy
xy
EDσ σ νν
τ ν
⎪ ⎪ ⎢ ⎥= =⎨ ⎬ ⎢ ⎥⎪ ⎪ −⎪ ⎪ ⎢ ⎥−⎪ ⎪ ⎢ ⎥⎪ ⎪ ⎣ ⎦⎩ ⎭
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External Loading
• Distributed body force• Distributed surface loading• Concentrated load directly acting on the nodes
Element Equilibrium (Using Virtual Work Principle)
{ }eaδ Virtual displacement at nodal points of an element
{ } [ ]{ } { } [ ]{ } and e eu N a B aδ δ δε δ= = At any point within the element
Equating External and Internal works (without the concentrated nodal loads)Equating External and Internal works (without the concentrated nodal loads)
{ } { } { } { } { } { } 0e e
T T Te e
V A
u b dV u t dAδε σ δ δ⎡ ⎤ ⎡ ⎤− − =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦∫ ∫
{ } { }[ ] [ ][ ]
e e e
Tee
q K a
K B D B dV
⎡ ⎤= ⎢ ⎥⎣ ⎦⎡ ⎤ =⎢ ⎥⎣ ⎦ ∫ Element
{ } [ ] [ ]{ } [ ] { } [ ] { } [ ] { }0 0
e
e e e e
V
T T T Tee e e e
V V V A
q B D dV B dV N b dV N t dAε σ ⎡ ⎤= − + + ⎢ ⎥⎣ ⎦∫ ∫ ∫ ∫Equilibrium in Local coordinate
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Overall Analysis
Nodal Displacement VectorConceptually, remaining steps followed in direct stiffness method will lead to the solution for nodal displacement vector of the whole structure
Stress at Any Point
{ } [ ][ ]{ } [ ]{ } { }eD B a Dσ ε σ= +{ } [ ][ ]{ } [ ]{ } { }0 0D B a Dσ ε σ= − +
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FEM: Without Assembling l ilib iElement Equilibrium
• Virtual work principle could have been applied directly on the whole structure
• Governing equation of equilibrium could be derived bypassing explicitly element equilibriumexplicitly element equilibrium
• Conceptually, similar to formation of stiffness matrix of the entire structure
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FEM: From the Minimization f i lof Potential Energy
Replace virtual quantities by ‘variation’ of real quantitiesReplace virtual quantities by variation of real quantities
{ } { } { } { } { } { }*T T T
V A
W a q u b dV u t dAδ δ⎛ ⎞⎟⎜ ⎟⎜− = + + ⎟⎜ ⎟⎜ ⎟⎝ ⎠
∫ ∫ Due to external load
V A⎝ ⎠
{ } { }T
V
U dVδ δ ε σ= ∫ Due to strain energy
( ) ( ) 0W U U Wδ δ δ δ Π− = ⇒ + = = Stationarity of total potential energy
0T
Π Π Π⎧ ⎫⎪ ⎪∂ ∂ ∂⎪ ⎪ Formulation of equilibrium equations1 2
. . 0a a aΠ Π Π∂ ∂ ∂⎪ ⎪= =⎨ ⎬⎪ ⎪∂ ∂ ∂⎪ ⎪⎩ ⎭
Formulation of equilibrium equations
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Example: FEM formulation of Stiffness f B El tof a Beam Element
St St i R l ti i li d f M t C t R l tiStress‐Strain Relation in generalized form Moment‐Curvature Relation
σ ε− M κ−2
2
2
d vdx
d
ε κ≡ =−
2
2
d vM EIdx
D EI
σ ≡ =−
≡
{ } { }T
Tei i i
dva v vdx
θ⎧ ⎫⎪ ⎪⎪ ⎪= =⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭
Nodal displacement vector at a typical node‐ith
( ) ( ) ( ) ( )1 2 3 4, , ,i jN f x f x N f x f x⎡ ⎤ ⎡ ⎤= =⎣ ⎦ ⎣ ⎦ Shape functions derived at two end nodes
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Formulation of Stiffness
( ) ( ) ( ) ( )
[ ] ( ) ( ) ( ) ( )
'' '' '' ''1 2 3 4
'' '' '' ''
, , ,i jB f x f x B f x f x
B B B f f f f
⎡ ⎤ ⎡ ⎤= − − = − −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎡ ⎤⎡ ⎤[ ] ( ) ( ) ( ) ( )1 2 3 4i jB B B f x f x f x f x⎡ ⎤⎡ ⎤⇒ = = − − − −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
[ ] [ ][ ] [ ] ( )[ ] ( ) ( )'' ''
e
T Tee i j
V L L
K B D B dV B EI B dx EI f x f x dx⎡ ⎤ = = =⎢ ⎥⎣ ⎦ ∫ ∫ ∫
Same as derived when revisiting direct stiffness method
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RemarksRemarks
• FEM when applied to beam element led toFEM when applied to beam element led to exactly same results
• This is not true in general• This is not true in general
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Thank You
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