from structural analysis to femevents.iitgn.ac.in/2013/fem-course/handouts/from... · method”,...

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