Theory - equilibrium equations

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FEM for bar structures (statics)

Jerzy Pamin

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

Piotr Pluciński

e-mail: Piotr.Plucinski@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

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^e

x^e, u 1

2

u^e₁

u^e₂

lê xê yê, v

1 2

v^e₁ ϕ^e₁

v₂^e ϕ^e₂

lê xê, u yê, v

1

2

uê₁ vê₁ ϕê₁

uê₂ v₂ê ϕê₂

truss FE beam FE frame FE

Element relations written in local (element) coordinate system

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Discretization

3

1

2

100kN

10kN/m 50kNm

20kN/m

4 2

3

1

x y

x¹

x²

x³

d6

d₄

d₁ d₃

d7

d2

d₅

d8

d9

d₁₂ d₁₁

d10

Topology (element connectivities)

TOP =

" ₁ ₂ 2 3 3 4

# _{e = 1} e = 2 e = 3

Data:

Nodal coordinates

Cross-section stiffnesses EA and EI

Boundary conditions Loading

Discretization:

NN=4, NE=3, NDOFN=3, NDOF=NN*NDOFN=12

Vector of degees of freedom (dofs):

d^e=







dê₁ dê₂ dê₃ dê₄ dê₅ dê₆







, d =







d1

d₂ d3

d₄ d5

d₆ d7

d₈ d9

d₁₀ d11

d₁₂







Boundary conditions:

d₁= d₂= d₃= d₁₀ = d₁₁ = 0

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 (s = De)

p =

p_x p_y

- distributed load intensity vector

Galerkin approximation within element

u(xê) = N(xê)dê, δu = N(xê)δdê

dê - degrees of freedom (dofs), i.e. nodal displacements e(xê) = B(xê)dê, B = LN , δe = B(xê)δdê

s(xê) = D e(xê) = DB(xê)dê

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Theory - equilibrium equations

Virtual work principle

δW_int^e = Z l^e

0

δe^Ts dx^e

δW_ext^e = Z l^e

0

δu^Tp dxê+ δdêTfê

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

lê xê yê

f₁^e

f₂^e f₃^e

f₄^e f₅^e

f₆^e p_y px

f^e =





 f₁ê f₂ê f₃ê f₄ê f₅ê f₆ê







=







−N1

Q1

−M1

N2

−Q2

M2







Theory - equilibrium equations

Substitute approximation

δW_intê = δdêT Z lê

0

B^TDB dxê dê = δdêTKêdê Kê - element stiffness matrix

δW_extê = δdêT Z lê

0

N^Tp dxê + δdêTfê = δdêT(zê + fê)

zê - equivalent joint loads (substitute nodal forces) Invoke δW_intê = δW_extê ∀δdê

Element balance equations

Kêdê = zê + fê

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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)=−d²v(x)

dx² → e = Lu, L =

− d² dx²

M (x)=EI(x) κ(x) → s = De, D =

EI(x)

Beam element description

Approximation of deflection

lê xê yê, v

1 2

v₁^e ϕ^e₁

v^e₂ ϕ^e₂

N DOF_n = 2, N DOF^e = 4 d_w

[2×1]

= {v_w, ϕ_w} d^e

[4×1]= {v₁ê, ϕê₁, v₂ê, ϕê₂} u(xê)

[1×1]

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

[1×4]

d^e

[4×1] , N = [N₁ê N₂ê N₃ê N₄ê]

0 l^e

1

xê N₁ê(xê) = 1 − 3 ^x_leê

²

+ 2 ^x_le^e

³

0 l^e

1

xê N₃ê(xê) = 3 ^x_leê

²

− 2 ^x_le^e

³

N₂ê(xê) = xê

1 − ^x_le^e

²

N₄ê(xê) = xê

h x^e l^e

²

− ^x_l_e^ei

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Beam element description

Approximation of curvature and bending moment, stiffness matrix

e(x^e)

[1×1]

= [κ(xê)] = LN(xê) · dê = B(xê)

[1×4]

· d^e

[4×1]

s(x^e)

[1×1]

= [M (x^e)] = D

[1×1]

· B(x^e)

[1×4]

· d^e

[4×1]

K^e

[4×4]

=

l^e

Z

0

B^TDB dx^e

Kê = EêIê lê3







12 6lê −12 6lê 6lê 4lê2 −6lê 2lê2

−12 −6lê 12 −6lê 6lê 2lê2 −6lê 4lê2







Beam element description

Computation of substitute nodal forces for constant distributed loading

lê xê yê

z₁^e

z₂^e z₃^e

z₄^e p_y

p_yl^e 2 p_yl^e2

12

p_yl^e 2

p_yl^e2 12

z^e =

l^e

Z

0

N^T

p_y dx^e

z^e =







p_yl^e p_y2l^e2

p12_yl^e 2

−p_yl^e2 12







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Global balance

Transformation T

^e

: global → local

Matrices referred to local element axes will be marked by overbar d¯ê = Têdê, zê = TêT¯zê, Kê = TêTK¯êTê

Element balance equations in local coordinates

¯fê = ¯Kêd¯ê − ¯zê Element balance equations in global coordinates

fê = Kêdê − zê

Assembly

K =X

e

K^e, d = X

e

d^e, z = X

e

z^e, f = X

e

f^e

f = K d − z = w + r w - external point load vector

r - support reaction vector

Computation algorithm

3

1

2

100kN

10kN/m 50kNm

20kN/m

4 2

3

1

x y

x¹

x²

x³

d₆ d₄

r3

d₇

d5

d₈

d9

r₁₁ r₂

r1 d₁₂ r10

System equilibrium

Global vectors:

d =







d₁ d2

d₃ d₄ d₅ d₆ d₇ d₈ d₉ d₁₀ d₁₁ d₁₂







, r =







r₁ r2

r₃ r₄ r₅ r₆ r₇ r₈ r₉ r₁₀ r₁₁ r₁₂







Boundary conditions:

d₁= d₂= d₃= d₁₀ = d₁₁ = 0 Hence:

r₄= r₅= r₆= r₇= r₈= r₉= r₁₂ = 0 Element equilibrium

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Algorithm of FE computations for a bar structure

Equilibrium of discretized system (of nodes) K d = w + z + r

plus essential boundary conditions

Statics

1. Discretize (set numbers, axes, topology), prepare input data 2. Compute element matrices ¯Kê, Kê, assemble global matrix K 3. Compute element vectors ¯zê, zê, assemble global vector z,

set up point load vector w

4. Solve equation set Kd = w + z + r taking into account kinematic boundary conditions, i.e. compute unknown nodal displacements in d and reactions in r

Algorithm of FE computations for a bar structure

Statics (cont’d)

Divide matrices into blocks

K₁₁ K₁₂ K₂₁ K₂₂

d₁ d₂

=

w₁+ z₁ w₂+ z₂

+

r₁ r₂

Known displacements d₂ = ˆd, hence r₁ = 0 K11d1 = w1+ z1− K₁₂d → dˆ 1 → d r = Kd − z − w

5. Compute nodal forces in elements d → dê → ¯dê → ¯fê = ¯Kêd¯ê − ¯zê or d → dê → fê = Kêdê− zê → ¯fê

6. Plot diagrams of section forces, check equilibrium