FEM for continuum statics

FEM for continuum statics

Piotr Pluciński e-mail: pplucin@L5.pk.edu.pl

Jerzy Pamin

e-mail: jpamin@L5.pk.edu.pl

2 FEM discretization

3 Plane stress

4 Example

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

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





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





This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

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



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

Z

S

σndS = Z

V

L ^T σdV where L – differential operator matrix

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

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 (in Voigt’s notation)

Using Green–Gauss–Ostrogradsky theorem Z

S

σndS = Z

V

L ^T σdV where L – differential operator matrix

Y Z

X

n u(x, y, z)

ρb(x, y, z)

P ⁰ V

S V

Static boundary conditions t = σn

where σ – stress tensor (in Voigt’s notation)

Using Green–Gauss–Ostrogradsky theorem Z

S

σndS = Z

V

L ^T σdV where L – differential operator matrix

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

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

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

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

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

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

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

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

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

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

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

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

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

Displacement field approximation u ^eh =

n

X

i=1

N _i ^e (ξ, η, ζ)q ^e _i = N ^e q ^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



 q ^e

[3n×1]

=



 q ^e ₁ . . . q ^e _n





y z

2 8

9

11 14

17

η ζ

η ζ

i

j l

m n

o p q ^e

[3n×1]

= T I ^e

[3n×N ] Q

[N ×1]

e=1

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

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 δq ^e ) ^T σ ^e dV ^e − Z

S ^e

(N ^e δq ^e ) ^T t ^e dS ^e − Z

V ^e

(N ^e δq ^e ) ^T f ^e dV ^e



= 0

E

X

e=1

Z

V ^e

(L ^e N ^e B ^e

δq ^e ) ^T σ ^e dV ^e − Z

S ^e

(N ^e δq ^e ) ^T t ^e dS ^e − Z

V ^e

(N ^e δq ^e ) ^T f ^e dV ^e



= 0

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

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 δq ^e ) ^T σ ^e dV ^e − Z

S ^e

(N ^e δq ^e ) ^T t ^e dS ^e − Z

V ^e

(N ^e δq ^e ) ^T f ^e dV ^e



= 0

E

X

e=1

Z

V ^e

(B ^e δq ^e ) ^T σ ^e dV ^e − Z

S ^e

(N ^e δq ^e ) ^T t ^e dS ^e − Z

V ^e

(N ^e δq ^e ) ^T f ^e dV ^e



= 0

E

X

e=1

(δq ^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

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

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 δq ^e ) ^T σ ^e dV ^e − Z

S ^e

(N ^e δq ^e ) ^T t ^e dS ^e − Z

V ^e

(N ^e δq ^e ) ^T f ^e dV ^e



= 0

E

X

e=1

( δq ^e I T ^e δQ

) ^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

Z

V ^e

(B ^e δq ^e ) ^T σ ^e dV ^e − Z

S ^e

(N ^e δq ^e ) ^T t ^e dS ^e − Z

V ^e

(N ^e δq ^e ) ^T f ^e dV ^e



= 0

E

X

e=1

( I T ^e δQ) ^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

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

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 δq ^e ) ^T σ ^e dV ^e − Z

S ^e

(N ^e δq ^e ) ^T t ^e dS ^e − Z

V ^e

(N ^e δq ^e ) ^T f ^e dV ^e



= 0

E

X

e=1

( I T ^e δQ) ^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

(δQ) ^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

E

X

e=1

Z

V ^e

(B ^e δq ^e ) ^T σ ^e dV ^e − Z

S ^e

(N ^e δq ^e ) ^T t ^e dS ^e − Z

V ^e

(N ^e δq ^e ) ^T f ^e dV ^e



= 0

E

X

e=1

( I T ^e δQ) ^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

(δQ) ^T

∀δQ

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

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

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 δq ^e ) ^T σ ^e dV ^e − Z

S ^e

(N ^e δq ^e ) ^T t ^e dS ^e − Z

V ^e

(N ^e δq ^e ) ^T f ^e dV ^e



= 0

E

X

e=1

( I T ^e δQ) ^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

Z

V ^e

(B ^e δq ^e ) ^T σ ^e dV ^e − Z

S ^e

(N ^e δq ^e ) ^T t ^e dS ^e − Z

V ^e

(N ^e δq ^e ) ^T f ^e dV ^e



= 0

E

X

e=1

( I T ^e δQ) ^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



This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

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 δq ^e ) ^T σ ^e dV ^e − Z

S ^e

(N ^e δq ^e ) ^T t ^e dS ^e − Z

V ^e

(N ^e δq ^e ) ^T f ^e dV ^e



= 0

E

X

e=1

( I T ^e δQ) ^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 _eT Z

eT e e

 E

X _eT Z

eT e e

Z

eT e 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



Equilibrium equation

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

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 q ^e = D ^e B ^e T I ^e Q

Equilibrium equation

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 q ^e = D ^e B ^e T I ^e Q

Equilibrium equation

E

X

e=1

I T ^eT

Z

V ^e

B ^eT D ^e B ^e T I ^e QdV ^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



This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

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 q ^e = D ^e B ^e T I ^e Q

Equilibrium equation

E

X _eT (

Z

eT e e e

)

e E

X _eT (

Z

eT e e Z

eT e 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 q ^e = D ^e B ^e T I ^e Q

Equilibrium equation

E

X

e=1

I

T ^eT k ^e T I ^e Q =

E

X

e=1

I T ^eT p ^e _b +

E

X

e=1

I T ^eT p ^e

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

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 q ^e = D ^e B ^e T I ^e Q Equilibrium equation

E

X _{eT e e}

E

X _{eT e}

E

X _{eT 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 q ^e = D ^e B ^e T I ^e Q

Equilibrium equation

KQ = P b + P

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

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







∂

∂x 0

∂







A

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

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

FEs for panels

Three-noded element

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

 , q ^e =















 q ₁ q ₂ q ₃ q ₄ q ₅ q 6

















x ^e y ^e

i

k

j q 1 e

q 2

q 3

q ₄ q 5

q 6

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

 , q ^e =















 q ₁ q ₂ q ₃ q ₄ q ₅ q 6

















x ^e i

k

j q 1 e

q 2

q 3

q ₄

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

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

FEs for panels

Four-noded element

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



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

x ^e y ^e

i

k

j l

e q 1

q ₂

q 3

q ₄ q 5

q ₆ q 7

q ₈

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



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

x ^e

i j

e q 1

q ₂

q 3

q ₄

y ^e N i (x, y)

x ^e 1

i

j

k

l y ^e

N l (x, y)

x ^e i 1 j

k l y ^e

N j (x, y)

x ^e 1 j i

k

l y ^e

N k (x, y)

x ^e 1

i j

k l

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

Example

Statics of a panel

3 kN/m

7.5 kN/m

4 m 2 m

2 m

E = 18 GPa ν = 0.25 h = 0.2 m

X Y

4 i

5 j

2 k 1

l

Discretization

i j

3 k

1 2

Q 1

Q ₂

Q 3

Q ₄ Q 5

Q 6

Q 7

Q 8

Q 9

Q 10

elem. no. node numbers

1 4 5 2 1

2 5 3 2

4 m 2 m

2 m

ν = 0.25 h = 0.2 m

X Y

4 i

5 j

2 k 1

l

Discretization

i j

3 k

1 2

Q 1

Q 2

Q 3

Q 4

Q 5

Q 6

Q 7

Q 8

Q 9

Q 10

elem. no. node numbers

1 4 5 2 1

2 5 3 2

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

Example

Statics of a panel

3 kN/m

7.5 kN/m Y Discretization

1 Q2 Q3 2 Q4 3

Q5

Constitutive matrix

D = 18 · 10 ⁶ 1 − 0.25 ²





1 0.25 0

0.25 1 0

0 0 ^1−0.25 ₂



 [kPa]

D =





19.2 4.8 0 4.8 19.2 0 0 0 7.2



 · 10 ⁶ [kPa]

3 kN/m

7.5 kN/m

4 m 2 m

2 m

E = 18 GPa ν = 0.25

h = 0.2 m X

Y Discretization

4

i j

k 1

l 1

i j

3 k

2 2

5 Q1

Q2 Q3 Q4

Q5

Q6 Q7

Q8

Q9 Q10

D = 1 − 0.25 ²  0.25 1 0 0 0 ^1−0.25 ₂

 [kPa]

D =





19.2 4.8 0 4.8 19.2 0 0 0 7.2



 · 10 ⁶ [kPa]

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

Example

Statics of a panel

3 kN/m

7.5 kN/m y (1) Discretization

1 Q1 8 = Q2 Q1 6 = Q4 2 3

Shape functions – Element 1

N _i ¹ (x ⁽¹⁾ , y ⁽¹⁾ ) = x ⁽¹⁾ y ⁽¹⁾ − 2x ⁽¹⁾ − 4y ⁽¹⁾ + 8

8 , N _k ¹ (x ⁽¹⁾ , y ⁽¹⁾ ) = x ⁽¹⁾ y ⁽¹⁾ 8 N _j ¹ (x ⁽¹⁾ , y ⁽¹⁾ ) = − x ⁽¹⁾ y ⁽¹⁾ − 2x ⁽¹⁾

8 , N _l ¹ (x ⁽¹⁾ , y ⁽¹⁾ ) = − x ⁽¹⁾ y ⁽¹⁾ − 4y ⁽¹⁾ 8 N ¹ =

 N _i ¹ 0 N _j ¹ 0 N _k ¹ 0 N _l ¹ 0 0 N _i ¹ 0 N _j ¹ 0 N _k ¹ 0 N _l ¹



Example

Statics of a panel

3 kN/m

7.5 kN/m

4 m 2 m

2 m

E = 18 GPa ν = 0.25

h = 0.2 m x (1)

y (1) Discretization

4

i j

k 1

l 1

i j

3 k

2 2

5 Q1 7 = Q1

Q1 8 = Q2

Q1 5 = Q3 Q1 6 = Q4

Q1 1 = Q7

Q1 2 = Q8 Q1 3 = Q9

Q1 4 = Q10

Matrix K – Element 1

B ¹ (x ⁽¹⁾ , y ⁽¹⁾ ) =











y ⁽¹⁾

8 − ¹ ₄ 0 ¹ ₄ − ^y ₈ ⁽¹⁾ 0 ^{y (1)} ₈ 0 − ^{y (1)} ₈ 0 0 ^{x (1)} ₈ − ¹

2 0 − ^{x (1)}

8 0 ^{x (1)} ₈ 0 ¹ ₂ − ^{x (1)}

8 x ⁽¹⁾

8 − ¹ ₂ ^{y (1)} ₈ − ₄ ¹ − ^{x (1)} ₈ ¹ ₄ − ^{y (1)} ₈ ^{x (1)} ₈ ^{y (1)} _{8 2} ¹ − ^{x (1)} ₈ − ^{y (1)} ₈











This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

Example

Statics of a panel

3 kN/m

7.5 kN/m y (1) Discretization

1 Q1 8 = Q2 Q1 6 = Q4 2 3

Matrix K – Element 1

B ¹ (x ⁽¹⁾ , y ⁽¹⁾ ) =











y ⁽¹⁾

8 − ¹ ₄ 0 ¹ ₄ − ^y ₈ ⁽¹⁾ 0 ^{y (1)} ₈ 0 − ^{y (1)} ₈ 0 0 ^{x (1)} ₈ − ¹

2 0 − ^{x (1)}

8 0 ^{x (1)} ₈ 0 ¹ ₂ − ^{x (1)}

8 x ⁽¹⁾

8 − ¹ ₂ ^{y (1)} ₈ − ₄ ¹ − ^{x (1)} ₈ ¹ ₄ − ^{y (1)} ₈ ^{x (1)} ₈ ^{y (1)} _{8 2} ¹ − ^{x (1)} ₈ − ^{y (1)} ₈











K ¹ = Z 2

0

Z 4 0

B ^{1 T} DB ¹ h dx ⁽¹⁾ dy ⁽¹⁾ =























16 6 -1.6 -1.2 -8 -6 -6.4 1.2 6 28 1.2 10.4 -6 -14 -1.2 -24.4 -1.6 1.2 16 -6 -6.4 -1.2 -8 6 -1.2 10.4 -6 28 1.2 -24.4 6 -14 -8 -6 -6.4 1.2 16 6 -1.6 -1.2 -6 -14 -1.2 -24.4 6 28 1.2 10.4 -6.4 -1.2 -8 6 -1.6 1.2 16 -6 1.2 -24.4 6 -14 -1.2 10.4 -6 28























· 10 ⁵

3 kN/m

7.5 kN/m

4 m 2 m

2 m

E = 18 GPa ν = 0.25

h = 0.2 m x (2)

Discretization y (2)

4

i j

k 1

l 1

i j

3 k

2 2

5 Q2 5 = Q3

Q2 6 = Q4

Q2 3 = Q5

Q2 4 = Q6

Q2 1 = Q9

Q2 2 = Q10

i 2 ^k 2

N _j ² (x ⁽²⁾ , y ⁽²⁾ ) = x ⁽²⁾ 2 N ² =

 N _i ² 0 N _j ² 0 N _k ² 0 0 N _i ² 0 N _j ² 0 N _k ²



This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

Example

Statics of a panel

3 kN/m

7.5 kN/m

Discretization y (2)

1 Q2 5 = Q3 2 3

Q2 6 = Q4

Q2 3 = Q5

Matrix K – Element 2

B ² (x ⁽²⁾ , y ⁽²⁾ ) =







0 0 ¹ ₂ 0 - ¹ ₂ 0 0 - ¹ ₂ 0 0 0 ¹ ₂ - ¹ ₂ 0 0 ¹ ₂ ¹ ₂ - ¹ ₂







K ² = B ^{2 T} DB ² hA ² =















7.2 0 0 -7.2 -7.2 7.2 0 19.2 -4.8 0 4.8 -19.2 0 -4.8 19.2 0 -19.2 4.8 -7.2 0 0 7.2 7.2 -7.2 -7.2 4.8 -19.2 7.2 26.4 -12 7.2 -19.2 4.8 -7.2 -12 26.4















· 10 ⁵

3 kN/m

7.5 kN/m

4 m 2 m

2 m

E = 18 GPa ν = 0.25

h = 0.2 m x (2)

Discretization y (2)

4

i j

k 1

l 1

i j

3 k

2 2

5 Q2 5 = Q3

Q2 6 = Q4

Q2 3 = Q5

Q2 4 = Q6

Q2 1 = Q9

Q2 2 = Q10

B (x , y ) =

 0 - ₂ 0 0 0 ₂ - ¹ ₂ 0 0 ¹ ₂ ¹ ₂ - ¹ ₂



K ² = B ^{2 T} DB ² hA ² =















7.2 0 0 -7.2 -7.2 7.2 0 19.2 -4.8 0 4.8 -19.2 0 -4.8 19.2 0 -19.2 4.8 -7.2 0 0 7.2 7.2 -7.2 -7.2 4.8 -19.2 7.2 26.4 -12 7.2 -19.2 4.8 -7.2 -12 26.4















· 10 ⁵

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

Example

Statics of a panel

3 kN/m

7.5 kN/m y (1) Discretization

1 Q1 8 = Q2 Q1 6 = Q4 2 3

Wektor P _b – Element 1 P ¹ _b =

Z

Γ ¹ _ij

N ^1T tdΓ + Z

Γ ¹ _jk

N ^1T tdΓ+

Z

Γ ¹ _kl

N ^1T tdΓ+

Z

Γ ¹ _li

N ^1T tdΓ

3 kN/m

7.5 kN/m

4 m 2 m

2 m

E = 18 GPa ν = 0.25

h = 0.2 m x (1)

y (1) Discretization

4

i j

k 1

l 1

i j

3 k

2 2

5 Q1 7 = Q1

Q1 8 = Q2

Q1 5 = Q3 Q1 6 = Q4

Q1 1 = Q7

Q1 2 = Q8 Q1 3 = Q9

Q1 4 = Q10 jk

interelem. edge force balance along line 2-5 t ¹ _jk = −t ² _ki

kl li

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

Example

Statics of a panel

3 kN/m

7.5 kN/m y (1) Discretization

1 Q1 8 = Q2 Q1 6 = Q4 2 3

Wektor P _b – Element 1 P ¹ _b =

Z

Γ ¹ _kl

N ^1T tdΓ + Z

Γ ¹ _li

N ^1T tdΓ

3 kN/m

7.5 kN/m 6 kN/m

4 m 2 m

2 m

E = 18 GPa ν = 0.25

h = 0.2 m x (1)

y (1) Discretization

4

i j

k 1

l 1

i j

3 k

2 2

5 Q1 7 = Q1

Q1 8 = Q2

Q1 5 = Q3 Q1 6 = Q4

Q1 1 = Q7

Q1 2 = Q8 Q1 3 = Q9

Q1 4 = Q10

Z

Γ ¹ _kl

N ^1T tdΓ = Z 4

0



N ¹ (x ⁽¹⁾ , y ⁽¹⁾ = 2)  T "

0

−3 

1 − ^{x (1)} ₄ 

− 6 ^x ₄ ⁽¹⁾

# dx ⁽¹⁾

= {0 0 0 0 0 -10 0 -8}

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

Example

Statics of a panel

3 kN/m

7.5 kN/m y (1) Discretization

1 Q1 8 = Q2 Q1 6 = Q4 2 3

Wektor P _b – Element 1

P ¹ _b = {0 0 0 0 0 -10 0 -8} + Z

Γ ¹ _li

N ^1T tdΓ Z

Γ ¹ _li

N ^1T tdΓ = Z 2

0



N ¹ (x ⁽¹⁾ = 0, y ⁽¹⁾ )  T

tdy ⁽¹⁾

= {R ¹ ₁ R ¹ ₂ 0 0 0 0 R ¹ ₇ R ¹ ₈ }

3 kN/m

7.5 kN/m

4 m 2 m

2 m

E = 18 GPa ν = 0.25

h = 0.2 m x (1)

y (1) Discretization

4

i j

k 1

l 1

i j

3 k

2 2

5 Q1 7 = Q1

Q1 8 = Q2

Q1 5 = Q3 Q1 6 = Q4

Q1 1 = Q7

Q1 2 = Q8 Q1 3 = Q9

Q1 4 = Q10

P ¹ _b =





















 R ₁ R ¹ ₂ 0 0 0 -10 R ¹ ₇ R ¹ ₈ -8























This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

Example

Statics of a panel

3 kN/m

7.5 kN/m

Discretization y (2)

1 Q2 5 = Q3 2 3

Q2 6 = Q4

Q2 3 = Q5

Wektor P _b – Element 2

P ² _b = Z

Γ ² _ij

N ^2T tdΓ + Z

Γ ² _jk

N ^2T tdΓ + Z

Γ ² _ki

N ^2T tdΓ

3 kN/m

7.5 kN/m

4 m 2 m

2 m

E = 18 GPa ν = 0.25

h = 0.2 m x (2)

Discretization y (2)

4

i j

k 1

l 1

i j

3 k

2 2

5 Q2 5 = Q3

Q2 6 = Q4

Q2 3 = Q5

Q2 4 = Q6

Q2 1 = Q9

Q2 2 = Q10

Γ ² _ij Γ ² _jk Γ ² _ki

interelem. edge force balance along line 2-5 t ¹ _jk = −t ² _ki

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

Example

Statics of a panel

3 kN/m

7.5 kN/m

Discretization y (2)

1 Q2 5 = Q3 2 3

Q2 6 = Q4

Q2 3 = Q5

Wektor P _b – Element 2

P ² _b = − Z

Γ ² _jk

N ^2T tdΓ

3 kN/m

7.5 kN/m 6 kN/m

4 m 2 m

2 m

E = 18 GPa ν = 0.25

h = 0.2 m x (2)

Discretization y (2)

4

i j

k 1

l 1

i j

3 k

2 2

5 Q2 5 = Q3

Q2 6 = Q4

Q2 3 = Q5

Q2 4 = Q6

Q2 1 = Q9

Q2 2 = Q10

Γ ² _jk

Z

Γ ² _jk

N ^2T tdΓ = Z 2

0



N ¹ (x ⁽²⁾ , y ⁽²⁾ = 2)  T "

0

−6 

1 − ^{x (2)} ₂ 

− 7.5 ^{x (2)} ₂

# dx ⁽²⁾

= {0 0 0 -7 0 -6.5}

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

Example

Statics of a panel

3 kN/m

7.5 kN/m

Discretization y (2)

1 Q2 5 = Q3 2 3

Q2 6 = Q4

Q2 3 = Q5

Wektor P _b – Element 2

P ² _b =















 0 0 0 -7 0 -6.5

















3 kN/m

7.5 kN/m

4 m 2 m

2 m

E = 18 GPa ν = 0.25

h = 0.2 m x (1)

y (1) Discretization

4

i j

k 1

l 1

i j

3 k

2 2

5 Q1 7 = Q1

Q1 8 = Q2

Q1 5 = Q3 Q1 6 = Q4

Q1 1 = Q7

Q1 2 = Q8 Q1 3 = Q9

Q1 4 = Q10

I B ¹ =























0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0





















 glob. no. ^{1 2 3 4 5 6 7 8 9 10}

2 3 4 5 6 7 8

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

Example

Statics of a panel

3 kN/m

7.5 kN/m y (1) Discretization

1 Q1 8 = Q2 Q1 6 = Q4 2 3

Assembly – Boole’s matrix I B

I B ¹ =

























0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0























 loc. no.

glob. no. ^{1 2 3 4 5 6 7 8 9 10}

1

2

3

4

5

6

7

8

3 kN/m

7.5 kN/m

4 m 2 m

2 m

E = 18 GPa ν = 0.25

h = 0.2 m x (1)

y (1) Discretization

4

i j

k 1

l 1

i j

3 k

2 2

5 Q1 7 = Q1

Q1 8 = Q2

Q1 5 = Q3 Q1 6 = Q4

Q1 1 = Q7

Q1 2 = Q8 Q1 3 = Q9

Q1 4 = Q10

I B ¹ =























0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0





















 glob. no. ^{1 2 3 4 5 6 7 8 9 10}

2 3 4 5 6 7 8

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

Example

Statics of a panel

3 kN/m

7.5 kN/m y (1) Discretization

1 Q1 8 = Q2 Q1 6 = Q4 2 3

Assembly – Boole’s matrix I B

I B ¹ =

























0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0























 loc. no.

glob. no. ^{1 2 3 4 5 6 7 8 9 10}

1

2

3

4

5

6

7

8

3 kN/m

7.5 kN/m

4 m 2 m

2 m

E = 18 GPa ν = 0.25

h = 0.2 m x (2)

Discretization y (2)

4

i j

k 1

l 1

i j

3 k

2 2

5 Q2 5 = Q3

Q2 6 = Q4

Q2 3 = Q5

Q2 4 = Q6

Q2 1 = Q9

Q2 2 = Q10

I B ² =















0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0













 glob. no. ^{1 2 3 4 5 6 7 8 9 10}

2 3 4 5 6

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

Example

Statics of a panel

3 kN/m

7.5 kN/m

Discretization y (2)

1 Q2 5 = Q3 2 3

Q2 6 = Q4

Q2 3 = Q5

Assembly – Boole’s matrix I B

I B ² =

















0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0















 loc. no.

glob. no. ^{1 2 3 4 5 6 7 8 9 10}

1

2

3

4

5

6