FEM for continuum statics

FEM for continuum statics

Piotr Pluciński

e-mail: Piotr.Plucinski@pk.edu.pl

Jerzy Pamin

e-mail: Jerzy.Pamin@pk.edu.pl

Chair for Computational Engineering

Faculty of Civil Engineering, Cracow University of Technology URL: www.CCE.pk.edu.pl

Computational Methods, 2020 J.Paminc

Lecture contents

1 Equilibrium state

2 FEM discretization

3 Plane stress

4 Example

Equilibrium state

Y Z

X

S

n t

u(x, y, z) ρb(x, y, z)

P

P

⁰

V

Body force density vector [N/m

³

] ρb = ρ



 0 0

−g





Traction vector [N/m

²

] t =



 t x

t y

t z





Displacement vector, strain tensor, stress tensor (Voigt’s notation)

u =

 _u

_x

u

y

u

z



,  =

 _

_xx

_

_xy

_

_xz



xy



yy



yz



xz



yz



zz



→











x



y



z

γ

xy

γ

yz

γ

zx









, σ =

 _σ

_xx

_σ

_xy

_σ

_xz

σ

xy

σ

yy

σ

yz

σ

xz

σ

yz

σ

zz



→









σ

x

σ

y

σ

z

τ

xy

τ

yz

τ

zx









Computational Methods, 2020 J.Paminc

Equilibrium state

Y Z

X

S

n t

u(x, y, z) ρb(x, y, z)

P

P

⁰

V

Body force density vector [N/m

³

] ρb = ρ



 0 0

−g





Traction vector [N/m

²

] t =



 t x

t y

t z





Displacement vector, strain tensor, stress tensor (Voigt’s notation)

u =

 _u

_x

u

y

u

z



,  =

 _

_xx

_

_xy

_

_xz



xy



yy



yz



xz



yz



zz



→











x



y



z

γ

xy

γ

yz

γ

zx









, σ =

 _σ

_xx

_σ

_xy

_σ

_xz

σ

xy

σ

yy

σ

yz

σ

xz

σ

yz

σ

zz



→









σ

x

σ

y

σ

z

τ

xy

τ

yz

τ

zx









Equilibrium state

Y Z

X

S

n t

u(x, y, z) ρb(x, y, z)

P

P

⁰

V

Body force density vector [N/m

³

] ρb = ρ



 0 0

−g





Traction vector [N/m

²

] t =



 t x

t y

t z





Displacement vector, strain tensor, stress tensor (Voigt’s notation)

u =

 _u

_x

u

y

u

z



,  =

 _

_xx

_

_xy

_

_xz



xy



yy



yz



xz



yz



zz



→











x



y



z

γ

xy

γ

yz

γ

zx









, σ =

 _σ

_xx

_σ

_xy

_σ

_xz

σ

xy

σ

yy

σ

yz

σ

xz

σ

yz

σ

zz



→









σ

x

σ

y

σ

z

τ

xy

τ

yz

τ

zx









Computational Methods, 2020 J.Paminc

Equilibrium state

Y Z

X

S

n t

u(x, y, z) ρb(x, y, z)

P

P

⁰

V

Body force density vector [N/m

³

] ρb = ρ



 0 0

−g





Traction vector [N/m

²

] t =



 t x

t y

t z





Displacement vector, strain tensor, stress tensor (Voigt’s notation)

 _

x



  _σ

x

σ



Equilibrium state

Y Z

X

S

n t

u(x, y, z) ρb(x, y, z)

P

P

⁰

V

Equilibrium equations for a body

Z

S

tdS + Z

V

ρbdV = 0

Static boundary conditions t = σn where σ – stress tensor

Using Green–Gauss–Ostrogradsky theorem

Z

S

σndS = Z

V

L ^T σdV where L – differential operator matrix

Computational Methods, 2020 J.Paminc

Equilibrium state

Y Z

X

S

n t

u(x, y, z) ρb(x, y, z)

P

P

⁰

V

Equilibrium equations for a body

Z

S

tdS + Z

V

ρbdV = 0

Static boundary conditions t = σn where σ – stress tensor

Using Green–Gauss–Ostrogradsky theorem

Z

S

σndS = Z

V

L ^T σdV where L – differential operator matrix

Equilibrium state

Y Z

X

S

n t

u(x, y, z) ρb(x, y, z)

P

P

⁰

V

Equilibrium equations for a body

Z

S

tdS + Z

V

ρbdV = 0

Static boundary conditions t = σn where σ – stress tensor

Using Green–Gauss–Ostrogradsky theorem

Z

S

σndS = Z

V

L ^T σdV where L – differential operator matrix

Computational Methods, 2020 J.Paminc

Equilibrium equations

Navier’s equations

Z

V

L ^T σ + ρb
dV = 0 ⇐⇒L ^T σ + ρb = 0 ∀P ∈ V

σ ij,j + ρb i = 0

Weak formulation – weighting function w ≡ δu – kinematically admissible displacement variation

Z

V

(δu) ^T L ^T σ + ρb
dV = 0 ∀δu

Equilibrium equations

Navier’s equations

Z

V

L ^T σ + ρb
dV = 0 ⇐⇒L ^T σ + ρb = 0 ∀P ∈ V σ ij,j + ρb i = 0

Weak formulation – weighting function w ≡ δu – kinematically admissible displacement variation

Z

V

(δu) ^T L ^T σ + ρb
dV = 0 ∀δu

Computational Methods, 2020 J.Paminc

Equilibrium equations

Navier’s equations

Z

V

L ^T σ + ρb
dV = 0 ⇐⇒L ^T σ + ρb = 0 ∀P ∈ V σ ij,j + ρb i = 0

Weak formulation – weighting function w ≡ δu – kinematically admissible displacement variation

Z

V

(δu) ^T L ^T σ + ρb
dV = 0 ∀δu

Equilibrium equations

Navier’s equations

Z

V

L ^T σ + ρb
dV = 0 ⇐⇒L ^T σ + ρb = 0 ∀P ∈ V σ ij,j + ρb i = 0

Weak formulation – weighting function w ≡ δu – kinematically admissible displacement variation (complying with kinematic b.cs)

Z

V

(δu) ^T L ^T σ + ρb
dV = 0 ∀δu

− Z

V

(Lδu) ^T σdV + Z

S

(δu) ^T σndS + Z

V

(δu) ^T ρbdV = 0

Computational Methods, 2020 J.Paminc

Equilibrium equations

Navier’s equations

Z

V

L ^T σ + ρb
dV = 0 ⇐⇒L ^T σ + ρb = 0 ∀P ∈ V σ ij,j + ρb i = 0

Weak formulation – weighting function w ≡ δu – kinematically admissible displacement variation (complying with kinematic b.cs)

Z

V

(δu) ^T L ^T σ + ρb
dV = 0 ∀δu

− Z

V

(Lδu) ^T σdV + Z

S

(δu) ^T σn t

dS + Z

V

(δu) ^T ρbdV = 0

Equilibrium equations

Navier’s equations

Z

V

L ^T σ + ρb
dV = 0 ⇐⇒L ^T σ + ρb = 0 ∀P ∈ V σ ij,j + ρb i = 0

Weak formulation – weighting function w ≡ δu – kinematically admissible displacement variation (complying with kinematic b.cs)

Z

V

(δu) ^T L ^T σ + ρb
dV = 0 ∀δu

− Z

V

(Lδu) ^T σdV + Z

S

(δu) ^T tdS + Z

V

(δu) ^T ρbdV = 0

Computational Methods, 2020 J.Paminc

Equilibrium equations

Navier’s equations

Z

V

L ^T σ + ρb
dV = 0 ⇐⇒L ^T σ + ρb = 0 ∀P ∈ V σ ij,j + ρb i = 0

Weak formulation – weighting function w ≡ δu – kinematically admissible displacement variation – it is virtual work principle

Z

V

(δu) ^T L ^T σ + ρb
dV = 0 ∀δu

Z

V

(Lδu) ^T σdV = Z

S

(δu) ^T tdS + Z

V

(δu) ^T ρbdV

Equilibrium equations

Navier’s equations

Z

V

L ^T σ + ρb
dV = 0 ⇐⇒L ^T σ + ρb = 0 ∀P ∈ V σ ij,j + ρb i = 0

Weak formulation – weighting function w ≡ δu – kinematically admissible displacement variation – it is virtual work principle

Z

V

(δu) ^T L ^T σ + ρb
dV = 0 ∀δu

Z

V

(Lδu) ^T σdV = Z

S

(δu) ^T tdS + Z

V

(δu) ^T ρbdV

work of internal forces work of external forces

Computational Methods, 2020 J.Paminc

FEM discretization (n=NNE, N =NDOF, E=NE)

Displacement field approximation u ^eh =

n

X

i=1

N _i ^e (ξ, η, ζ)d ^e _i = N ^e d ^e

N ^e

[3×3n] =





N ₁ ^e 0 0 . . . N _n ^e 0 0 0 N ₁ ^e 0 . . . 0 N _n ^e 0 0 0 N ₁ ^e . . . 0 0 N _n ^e



 d ^e

[3n×1] =



 d ^e ₁ . . . d ^e _n





y z

2 8

9

11 14

17

η ζ

η ζ

i

j l

m n

o p d ^e

[3n×1]

= T I ^e

[3n×N ]

d

[N ×1]

Equilibrium equation of discretized structure

Equilibrium equation (ρb ^e = f ^e – body force vector)

E

X

e=1

Z

V

^e

(L ^e δu ^e ) ^T σ ^e dV ^e − Z

S

^e

(δu ^e ) ^T t ^e dS ^e − Z

V

^e

(δu ^e ) ^T f ^e dV ^e



= 0

Computational Methods, 2020 J.Paminc

Equilibrium equation of discretized structure

Equilibrium equation

E

X

e=1

Z

V

^e

(L ^e δu ^e ) ^T σ ^e dV ^e − Z

S

^e

(δu ^e ) ^T t ^e dS ^e − Z

V

^e

(δu ^e ) ^T f ^e dV ^e



= 0

E

X

e=1

Z

V^e

(L

^e

N

^e

δd

^e

)

^T

σ

^e

dV

^e

−

Z

S^e

(N

^e

δd

^e

)

^T

t

^e

dS

^e

−

Z

V^e

(N

^e

δd

^e

)

^T

f

^e

dV

^e



= 0

Equilibrium equation of discretized structure

Equilibrium equation

E

X

e=1

Z

V

^e

(L ^e δu ^e ) ^T σ ^e dV ^e − Z

S

^e

(δu ^e ) ^T t ^e dS ^e − Z

V

^e

(δu ^e ) ^T f ^e dV ^e



= 0

E

X

e=1

Z

V^e

(L

^e

N

^e

B

^e

δd

^e

)

^T

σ

^e

dV

^e

−

Z

S^e

(N

^e

δd

^e

)

^T

t

^e

dS

^e

−

Z

V^e

(N

^e

δd

^e

)

^T

f

^e

dV

^e



= 0

Computational Methods, 2020 J.Paminc

Equilibrium equation of discretized structure

Equilibrium equation

E

X

e=1

Z

V

^e

(L ^e δu ^e ) ^T σ ^e dV ^e − Z

S

^e

(δu ^e ) ^T t ^e dS ^e − Z

V

^e

(δu ^e ) ^T f ^e dV ^e



= 0

E

X

e=1

Z

V^e

(B

^e

δd

^e

)

^T

σ

^e

dV

^e

−

Z

S^e

(N

^e

δd

^e

)

^T

t

^e

dS

^e

−

Z

V^e

(N

^e

δd

^e

)

^T

f

^e

dV

^e



= 0

Equilibrium equation of discretized structure

Equilibrium equation

E

X

e=1

Z

V

^e

(L ^e δu ^e ) ^T σ ^e dV ^e − Z

S

^e

(δu ^e ) ^T t ^e dS ^e − Z

V

^e

(δu ^e ) ^T f ^e dV ^e



= 0

E

X

e=1

Z

V^e

(B

^e

δd

^e

)

^T

σ

^e

dV

^e

−

Z

S^e

(N

^e

δd

^e

)

^T

t

^e

dS

^e

−

Z

V^e

(N

^e

δd

^e

)

^T

f

^e

dV

^e



= 0

E

X

e=1

(δd

^e

)

^T

Z

V^e

B

^eT

σ

^e

dV

^e

− Z

S^e

N

^eT

t

^e

dS

^e

− Z

V^e

N

^eT

f

^e

dV

^e



= 0

Computational Methods, 2020 J.Paminc

Equilibrium equation of discretized structure

Equilibrium equation

E

X

e=1

Z

V

^e

(L ^e δu ^e ) ^T σ ^e dV ^e − Z

S

^e

(δu ^e ) ^T t ^e dS ^e − Z

V

^e

(δu ^e ) ^T f ^e dV ^e



= 0

E

X

e=1

Z

V^e

(B

^e

δd

^e

)

^T

σ

^e

dV

^e

−

Z

S^e

(N

^e

δd

^e

)

^T

t

^e

dS

^e

−

Z

V^e

(N

^e

δd

^e

)

^T

f

^e

dV

^e



= 0

E

X

e=1

( δd

^e

I T

^e

δd

)

^T

Z

V^e

B

^eT

σ

^e

dV

^e

− Z

S^e

N

^eT

t

^e

dS

^e

− Z

V^e

N

^eT

f

^e

dV

^e



= 0

Equilibrium equation of discretized structure

Equilibrium equation

E

X

e=1

Z

V

^e

(L ^e δu ^e ) ^T σ ^e dV ^e − Z

S

^e

(δu ^e ) ^T t ^e dS ^e − Z

V

^e

(δu ^e ) ^T f ^e dV ^e



= 0

E

X

e=1

Z

V^e

(B

^e

δd

^e

)

^T

σ

^e

dV

^e

−

Z

S^e

(N

^e

δd

^e

)

^T

t

^e

dS

^e

−

Z

V^e

(N

^e

δd

^e

)

^T

f

^e

dV

^e



= 0

E

X

e=1

( I T

^e

δd)

^T

Z

V^e

B

^eT

σ

^e

dV

^e

− Z

S^e

N

^eT

t

^e

dS

^e

− Z

V^e

N

^eT

f

^e

dV

^e



= 0

Computational Methods, 2020 J.Paminc

Equilibrium equation of discretized structure

Equilibrium equation

E

X

e=1

Z

V

^e

(L ^e δu ^e ) ^T σ ^e dV ^e − Z

S

^e

(δu ^e ) ^T t ^e dS ^e − Z

V

^e

(δu ^e ) ^T f ^e dV ^e



= 0

E

X

e=1

Z

V^e

(B

^e

δd

^e

)

^T

σ

^e

dV

^e

−

Z

S^e

(N

^e

δd

^e

)

^T

t

^e

dS

^e

−

Z

V^e

(N

^e

δd

^e

)

^T

f

^e

dV

^e



= 0

E

X

e=1

( I T

^e

δd)

^T

Z

V^e

B

^eT

σ

^e

dV

^e

− Z

S^e

N

^eT

t

^e

dS

^e

− Z

V^e

N

^eT

f

^e

dV

^e



= 0

(δd)

^T

E

X

e=1

I T

^eT

Z

V^e

B

^eT

σ

^e

dV

^e

− Z

S^e

N

^eT

t

^e

dS

^e

− Z

V^e

N

^eT

f

^e

dV

^e



= 0

Equilibrium equation of discretized structure

Equilibrium equation

E

X

e=1

Z

V

^e

(L ^e δu ^e ) ^T σ ^e dV ^e − Z

S

^e

(δu ^e ) ^T t ^e dS ^e − Z

V

^e

(δu ^e ) ^T f ^e dV ^e



= 0

E

X

e=1

Z

V^e

(B

^e

δd

^e

)

^T

σ

^e

dV

^e

−

Z

S^e

(N

^e

δd

^e

)

^T

t

^e

dS

^e

−

Z

V^e

(N

^e

δd

^e

)

^T

f

^e

dV

^e



= 0

E

X

e=1

( I T

^e

δd)

^T

Z

V^e

B

^eT

σ

^e

dV

^e

− Z

S^e

N

^eT

t

^e

dS

^e

− Z

V^e

N

^eT

f

^e

dV

^e



= 0

(δd)

^T

∀δd

E

X

e=1

I T

^eT

Z

V^e

B

^eT

σ

^e

dV

^e

− Z

S^e

N

^eT

t

^e

dS

^e

− Z

V^e

N

^eT

f

^e

dV

^e



= 0

Computational Methods, 2020 J.Paminc

Equilibrium equation of discretized structure

Equilibrium equation

E

X

e=1

Z

V

^e

(L ^e δu ^e ) ^T σ ^e dV ^e − Z

S

^e

(δu ^e ) ^T t ^e dS ^e − Z

V

^e

(δu ^e ) ^T f ^e dV ^e



= 0

E

X

e=1

Z

V^e

(B

^e

δd

^e

)

^T

σ

^e

dV

^e

−

Z

S^e

(N

^e

δd

^e

)

^T

t

^e

dS

^e

−

Z

V^e

(N

^e

δd

^e

)

^T

f

^e

dV

^e



= 0

E

X

e=1

( I T

^e

δd)

^T

Z

V^e

B

^eT

σ

^e

dV

^e

− Z

S^e

N

^eT

t

^e

dS

^e

− Z

V^e

N

^eT

f

^e

dV

^e



= 0

E

X

e=1

I T

^eT

Z

V^e

B

^eT

σ

^e

dV

^e

− Z

S^e

N

^eT

t

^e

dS

^e

− Z

V^e

N

^eT

f

^e

dV

^e



= 0

Equilibrium equation of discretized structure

Equilibrium equation

E

X

e=1

Z

V

^e

(L ^e δu ^e ) ^T σ ^e dV ^e − Z

S

^e

(δu ^e ) ^T t ^e dS ^e − Z

V

^e

(δu ^e ) ^T f ^e dV ^e



= 0

E

X

e=1

Z

V^e

(B

^e

δd

^e

)

^T

σ

^e

dV

^e

−

Z

S^e

(N

^e

δd

^e

)

^T

t

^e

dS

^e

−

Z

V^e

(N

^e

δd

^e

)

^T

f

^e

dV

^e



= 0

E

X

e=1

( I T

^e

δd)

^T

Z

V^e

B

^eT

σ

^e

dV

^e

− Z

S^e

N

^eT

t

^e

dS

^e

− Z

V^e

N

^eT

f

^e

dV

^e



= 0

E

X

e=1

I T

^eT

Z

V^e

B

^eT

σ

^e

dV

^e

− Z

S^e

N

^eT

t

^e

dS

^e

− Z

V^e

N

^eT

f

^e

dV

^e



= 0

E

X

e=1

I T

^eT

Z

V^e

B

^eT

σ

^e

dV

^e



=

E

X

e=1

I T

^eT

Z

S^e

N

^eT

t

^e

dS

^e

+

Z

V^e

N

^eT

f

^e

dV

^e



Computational Methods, 2020 J.Paminc

Equilibrium equation of discretized structure

Equilibrium equation

E

X

e=1

Z

V

^e

(L ^e δu ^e ) ^T σ ^e dV ^e − Z

S

^e

(δu ^e ) ^T t ^e dS ^e − Z

V

^e

(δu ^e ) ^T f ^e dV ^e



= 0

E

X

e=1

Z

V^e

(B

^e

δd

^e

)

^T

σ

^e

dV

^e

−

Z

S^e

(N

^e

δd

^e

)

^T

t

^e

dS

^e

−

Z

V^e

(N

^e

δd

^e

)

^T

f

^e

dV

^e



= 0

E

X

e=1

( I T

^e

δd)

^T

Z

V^e

B

^eT

σ

^e

dV

^e

− Z

S^e

N

^eT

t

^e

dS

^e

− Z

V^e

N

^eT

f

^e

dV

^e



= 0

E

X

e=1

I T

^eT

Z

V^e

B

^eT

σ

^e

dV

^e

− Z

S^e

N

^eT

t

^e

dS

^e

− Z

V^e

N

^eT

f

^e

dV

^e



= 0

E

X Z 

^E

X Z Z 

Equilibrium equation of discretized structure

Equilibrium equation

E

X

e=1

I T

^eT

Z

V^e

B

^eT

σ

^e

dV

^e



=

E

X

e=1

I T

^eT

Z

S^e

N

^eT

t

^e

dS

^e

+

Z

V^e

N

^eT

f

^e

dV

^e



Consideration of kinamatic and constitutive equations linear elasticity: σ = Dε

linear kinematic relation: ε = Lu

σ ^e = D ^e L ^e u ^e = D ^e L ^e N ^e d ^e = D ^e B ^e T I ^e d

Equilibrium equation

Computational Methods, 2020 J.Paminc

Equilibrium equation of discretized structure

Equilibrium equation

E

X

e=1

I T

^eT

Z

V^e

B

^eT

σ

^e

dV

^e



=

E

X

e=1

I T

^eT

Z

S^e

N

^eT

t

^e

dS

^e

+

Z

V^e

N

^eT

f

^e

dV

^e



Consideration of kinamatic and constitutive equations linear elasticity: σ = Dε

linear kinematic relation: ε = Lu

σ ^e = D ^e L ^e u ^e = D ^e L ^e N ^e d ^e = D ^e B ^e T I ^e d

Equilibrium equation

Equilibrium equation of discretized structure

Equilibrium equation

E

X

e=1

I T

^eT

Z

V^e

B

^eT

σ

^e

dV

^e



=

E

X

e=1

I T

^eT

Z

S^e

N

^eT

t

^e

dS

^e

+

Z

V^e

N

^eT

f

^e

dV

^e



Consideration of kinamatic and constitutive equations linear elasticity: σ = Dε

linear kinematic relation: ε = Lu

σ ^e = D ^e L ^e u ^e = D ^e L ^e N ^e d ^e = D ^e B ^e T I ^e d

Equilibrium equation

E

X

e=1

I T

^eT

Z

V^e

B

^eT

D

^e

B

^e

T I

^e

ddV

^e



=

E

X

e=1

I T

^eT

Z

S^e

N

^eT

t

^e

dS

^e

+

Z

V^e

N

^eT

f

^e

dV

^e



Computational Methods, 2020 J.Paminc

Equilibrium equation of discretized structure

Equilibrium equation

E

X

e=1

I T

^eT

Z

V^e

B

^eT

σ

^e

dV

^e



=

E

X

e=1

I T

^eT

Z

S^e

N

^eT

t

^e

dS

^e

+

Z

V^e

N

^eT

f

^e

dV

^e



Consideration of kinamatic and constitutive equations linear elasticity: σ = Dε

linear kinematic relation: ε = Lu

σ ^e = D ^e L ^e u ^e = D ^e L ^e N ^e d ^e = D ^e B ^e T I ^e d

Equilibrium equation

E

X T I

^eT

( Z

B

^eT

D

^e

B

^e

dV

^e

)

I T

^e

d =

E

X T I

^eT

( Z

N

^eT

t

^e

dS

^e

+

Z

N

^eT

f

^e

dV

^e

)

Equilibrium equation of discretized structure

Equilibrium equation

E

X

e=1

I T

^eT

Z

V^e

B

^eT

σ

^e

dV

^e



=

E

X

e=1

I T

^eT

Z

S^e

N

^eT

t

^e

dS

^e

+

Z

V^e

N

^eT

f

^e

dV

^e



Consideration of kinamatic and constitutive equations linear elasticity: σ = Dε

linear kinematic relation: ε = Lu

σ ^e = D ^e L ^e u ^e = D ^e L ^e N ^e d ^e = D ^e B ^e T I ^e d

Equilibrium equation

E

X

e=1

I

T ^eT K ¯ ^e T I ^e d =

E

X

e=1

I T ^eT p ¯ ^e _b +

E

X

e=1

I T ^eT p ¯ ^e

Computational Methods, 2020 J.Paminc

Equilibrium equation of discretized structure

Equilibrium equation

E

X

e=1

I T

^eT

Z

V^e

B

^eT

σ

^e

dV

^e



=

E

X

e=1

I T

^eT

Z

S^e

N

^eT

t

^e

dS

^e

+

Z

V^e

N

^eT

f

^e

dV

^e



Consideration of kinamatic and constitutive equations linear elasticity: σ = Dε

linear kinematic relation: ε = Lu

σ ^e = D ^e L ^e u ^e = D ^e L ^e N ^e d ^e = D ^e B ^e T I ^e d

Equilibrium equation

E

X _eT ¯ ^{e e}

E

X _{eT e}

E

X _{eT e}

Equilibrium equation of discretized structure

Equilibrium equation

E

X

e=1

I T

^eT

Z

V^e

B

^eT

σ

^e

dV

^e



=

E

X

e=1

I T

^eT

Z

S^e

N

^eT

t

^e

dS

^e

+

Z

V^e

N

^eT

f

^e

dV

^e



Consideration of kinamatic and constitutive equations linear elasticity: σ = Dε

linear kinematic relation: ε = Lu

σ ^e = D ^e L ^e u ^e = D ^e L ^e N ^e d ^e = D ^e B ^e T I ^e d

Equilibrium equation

Kd = p b + p

Computational Methods, 2020 J.Paminc

Plane stress (σ _z = 0)

Displacement vector u = {u(x, y), v(x, y)}

Strain vector

ε = {ε x , ε y , γ xy }

Stress vector

σ = {σ x , σ y , τ xy }

Traction vector

Body force intensity vector f = {f x , f y }

Constitutive matrix

D = E 1 − ν ²





1 ν 0

ν 1 0

0 0 ^1−ν ₂





Differential operator matrix

L =







∂

∂x 0

∂







Plane stress (σ _z = 0)

Stiffness matrix k ^e =

Z

A

^e

B ^eT D ^e B ^e h ^e dA ^e

A ^e , h ^e – FE area and thickness, resp.

Element loading vector

p ^e = Z

A

^e

N ^eT f ^e h ^e dA ^e

Boundary loading vector

p ^e _b = Z

Γ

^e

N ^eT t ^e h ^e dΓ ^e

x ^e y ^e

Γ ^e A ^e

Computational Methods, 2020 J.Paminc

FEs for panels

Three-noded element

u ^e (x, y) = N ^e (x, y) d ^e

N ^e =  N _i ^e 0 N _j ^e 0 N _k ^e 0 0 N _i ^e 0 N _j ^e 0 N _k ^e

 , d ^e =















 d ₁ d ₂ d ₃ d ₄ d 5

d 6

















x ^e y ^e

i

k

j d 1 e

d 2

d ₃ d 4

d 5

d 6

FEs for panels

Three-noded element

u ^e (x, y) = N ^e (x, y) d ^e

N ^e =  N _i ^e 0 N _j ^e 0 N _k ^e 0 0 N _i ^e 0 N _j ^e 0 N _k ^e

 , d ^e =















 d ₁ d ₂ d ₃ d ₄ d 5

d 6

















x ^e y ^e

i

k

j d 1 e

d 2

d ₃ d 4

d 5

d 6

y

^e

N

i

(x

^e

, y

^e

)

x

^e

1 i

k j

y

^e

N

j

(x

^e

, y

^e

)

x

^e

1

i k

j y

^e

N

k

(x

^e

, y

^e

)

x

^e

1 i

k

j

Computational Methods, 2020 J.Paminc

FEs for panels

Four-noded element

u ^e (x, y) = N ^e (x, y) d ^e N

^e

=

 N

i^e

0 N

j^e

0 N

_k^e

0 N

_l^e

0 0 N

_i^e

0 N

_j^e

0 N

_k^e

0 N

_l^e



d ^e = {d ₁ , d ₂ , d ₃ , d ₄ , d ₅ , d ₆ , d ₇ , d ₈ }

x ^e y ^e

i

k

j l

e d ₁ d 2

d ₃ d 4

d ₅ d 6

d ₇

d 8