FEM for bar structures (statics)

FEM for bar structures (statics)

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

Piotr Pluciński

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

Theory - equilibrium equations

Virtual work principle

δW int = δW ext ∀δu Disassembly into finite elements

δW int = X

e

δW _int ^e , δW ext = X

e

δW _ext ^e

Equivalent to requirement of minimum of total potential energy functional in the space of admissible displacements

l

x

, u 1

2

u

u

l

x

y

, v

1 2

v

ϕ

v

ϕ

l

x

, u y

, v

1

2

u

v

ϕ

u

v

ϕ

truss FE beam FE frame FE

Theory - equilibrium equations

Variables for frame structures

u =

 u v



- displacement vector (fundamental unknown) e =

  ₀ κ



- generalized strain vector (e = Lu) s =

 N M



- generalized stress vector p =

 p _x p y



- distributed load intensity vector

Galerkin interpolation within element

u(x ^e ) = N(x ^e )q ^e , δu = N(x ^e )δq ^e

q ^e - degrees of freedom (dofs), i.e. nodal displacements

e(x ^e ) = B(x ^e )q ^e , B = LN , δe = B(x ^e )δq ^e

s(x ^e ) = D e(x ^e ) = DB(x ^e )q ^e

Theory - equilibrium equations

Virtual work principle

δW _int ^e = Z l ^e

0

δe ^T s dx ^e

δW _ext ^e = Z l ^e

0

δu ^T p dx ^e + δq ^eT f ^e

f ^e - nodal force vector (forces acting on the considered element, which come from elements connected to it at nodes)

l ^e x ^e y ^e

f 1 ^e

f ₂ ^e f ₃ ^e

f ₄ ^e f 5 ^e

f 6 ^e

p y

p x

Theory - equilibrium equations

Substitute discretization

δW _int ^e = δq ^eT Z l ^e

0

B ^T DB dx ^e q ^e = δq ^eT k ^e q ^e k ^e - element stiffness matrix

δW _ext ^e = δq ^eT Z l ^e

0

N ^T p dx ^e + δq ^eT f ^e = δq ^eT (z ^e + f ^e ) z ^e - equivalent joint loads (substitute nodal forces)

Invoke δW int = δW ext ∀δu

Element balance equations

k ^e q ^e = z ^e + f ^e

Beam element description

Bending representation

Definitions of displacement, generalized strain and generalized stress u(x) = [v(x)], e(x) = [κ(x)], s(x) = [M (x)]

Kinematic and constitutive relations at point P (x, y, z) = P (x, 0, 0) = P (x) on beam axis

κ(x)=− d ² v(x)

dx ² → e = Lu, L =



− d ² dx ²



M (x)=EI(x) κ(x) → s = De, D = 

EI(x) 

Beam element description

Approximation of deflection

l ^e x ^e y ^e , v

1 2

v ^e ₁ ϕ ^e ₁

v ₂ ^e ϕ ^e ₂

N DOF n = 2, N DOF ^e = 4 q _w

[2×1]

= {v _w , ϕ _w } q ^e

[4×1]

= {v ^e ₁ , ϕ ^e ₁ , v ^e ₂ , ϕ ^e ₂ } u(x ^e )

[1×1]

= [v(x ^e )] = N(x ^e )

[1×4]

q ^e

[4×1]

, N = [N ₁ ^e N ₂ ^e N ₃ ^e N ₄ ^e ]

0 l ^e

1

x ^e N ₁ ^e (x ^e ) = 1 − 3

 x ^e l ^e

 2

+ 2

 x ^e l ^e

 3

0 l ^e

1

x ^e N ₃ ^e (x ^e ) = 3

 x ^e l ^e

 2

− 2 

x ^e l ^e

 3

0 l ^e x ^e

N ₂ ^e (x ^e ) = x ^e h 1 − 

x ^e l ^e

i 2

0 l ^e x ^e

N ₄ ^e (x ^e ) = x ^e

 

x ^e l ^e

 2

− 

x ^e l ^e

 

Beam element description

Approximation of curvature, bending moment and stiffness

e(x ^e )

[1×1]

= [κ(x ^e )] = LN(x ^e ) · q ^e = B(x ^e )

[1×4]

· q ^e

[4×1]

s(x ^e )

[1×1]

= [M (x ^e )] = D

[1×1] · B(x ^e )

[1×4]

· q ^e

[4×1]

k ^e

[4×4] =

l ^e

Z

0

B ^T DB dx ^e

k ^e = E ^e I ^e l ^e3









12 6l ^e −12 6l ^e 6l ^e 4l ^e2 −6l ^e 2l ^e2

−12 −6l ^e 12 −6l ^e 6l ^e 2l ^e2 −6l ^e 4l ^e2









Beam element description

Computation of substitute nodal forces for constant distributed loading

l ^e x ^e y ^e

z ₁ ^e

z ^e ₂ z ^e ₃

z ₄ ^e p _y

p y l ^e 2 p y l ^e2

12

p y l ^e 2

p y l ^e2 12

z ^e =

l ^e

Z

0

N ^T 

p _y  dx ^e

z ^e =





















 p y l ^e

2 p y l ^e2

12 p y l ^e

2

− p y l ^e2 12























Global balance

Transformation and assembly

T ^e : global → local

q ^e = T ^e Q ^e , Z ^e = T ^eT z ^e , K ^e = T ^eT k ^e T ^e K = X

e

K ^e , Q = X

e

Q ^e , Z = X

e

Z ^e , F = X

e

F ^e

Equilibrium of discretized system (of nodes)

F = KQ − Z = P + R P - external point load vector

R - support reaction vector

KQ = P + Z + R

plus essential boundary conditions

Algorithm of FE computations for a bar structure

Statics

1. Discretize (set numbers, axes, topology), prepare input data 2. Compute element matrices k ^e , K ^e , assemble global matrix K 3. Compute element vectors z ^e , Z ^e , assemble global vector Z,

set up point load vector P

4. Solve equation set KQ = P + Z + R taking into account kinematic

boundary conditions, i.e. compute unknown nodal displacements in

Q and reactions in R

Algorithm of FE computations for a bar structure

Statics (cont’d)

Divide matrices into blocks

 K ₁₁ K ₁₂ K ₂₁ K ₂₂

  Q ₁ Q ₂



=

 P ₁ + Z ₁ P ₂ + Z ₂

 +

 R ₁ R ₂



Q ₂ = ˆ Q , R ₁ = 0 , K ₁₁ Q ₁ = P ₁ + Z ₁ − K ₁₂ Q → Q ˆ ₁ → Q R = KQ − Z − P

5. Compute nodal forces in elements

Q → Q ^e → q ^e → f ^e = k ^e q ^e − z ^e

or Q → Q ^e → F ^e = K ^e Q ^e − Z ^e → f ^e

6. Plot diagrams of section forces, check equilibrium