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From Structural Analysis to FEM
Dhiman Basu
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.
IntroductionIntroductionStructural Modeling• Line element• Line element• Refined Line Element• Detailed Finite Element
Line elementLine element
LiRefined lineLine
elementline element
FEM: Discretization over entire volume in general
Analysis
•Conventional Structural AnalysisLine elementRefined line element
•FEMVolume discretization
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
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
Local and Global Coordinate SSystems
Local coordinate
l b l
Non‐orthogonally aligned element axis
Global coordinate
Coordinate transformation by rotationby rotation
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 −
⎪ ⎪⎩ ⎭ ⎣ ⎦⎪ ⎪⎩ ⎭
=
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
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
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× × × ×
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⎩ ⎭ ⎣ ⎦⎩ ⎭
Numerical ExampleNumerical Example
EA=8000 kN/m2 and EI= 20000 kNm2
Solution vector: {‐0.00356, 0.00275, ‐0.0058, 0.00178}T.
Member end forces but in global coordinate
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
( ) ( ) ( ) ( ) ( )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
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
⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎟ ⎟ ⎟ ⎟⎜ ⎜ ⎜ ⎜ ⎢ ⎥= − + − + = − + =⎟ ⎟ ⎟ ⎟⎜ ⎜ ⎜ ⎜⎢ ⎥ ⎢ ⎥⎟ ⎟ ⎟ ⎟⎜ ⎜ ⎜ ⎜ ⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦∫ ∫
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⎡ ⎤= ⎢ ⎥⎣ ⎦
[ ]
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
−⎣⎢ ⎥⎢ ⎥⎦
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
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
⎪ ⎪ ⎢ ⎥= =⎨ ⎬ ⎢ ⎥⎪ ⎪ −⎪ ⎪ ⎢ ⎥−⎪ ⎪ ⎢ ⎥⎪ ⎪ ⎣ ⎦⎩ ⎭
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
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= − +
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
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
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
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
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
Thank You