14 The Plane Stress Problem IFEM Ch 14 Slide 1 Department of - - PDF document

Introduction to FEM

14

The Plane Stress Problem

IFEM Ch 14 – Slide 1

Department of Engineering Mechanics

• PhD. TRUONG Tich Thien

Introduction to FEM

Plate in Plane Stress

x y z

Inplane dimensions: in x,y plane Thickness dimension

• r transverse dimension

IFEM Ch 14 – Slide 2

Department of Engineering Mechanics

• PhD. TRUONG Tich Thien

Introduction to FEM

Plane Stress Physical Assumptions

Plate is flat and has a symmetry plane (the midplane) All loads and support conditions are midplane symmetric Thickness dimension is much smaller than inplane dimensions Inplane displacements, strains and stresses uniform through thickness Transverse stresses σ , σ and σ negligible Unessential but used in this course: Plate fabricated of homogeneous material through thickness

zz xz yz

IFEM Ch 14 – Slide 3

Department of Engineering Mechanics

• PhD. TRUONG Tich Thien

Introduction to FEM

Notation for stresses, strains, forces, displacements

x y z x x x y

In-plane stresses

σ σ

xx yy

σ = σ

xy yx

y

In-plane strains

h h h e e

xx yy

e = e

xy yx

y

In-plane displacements

u u

x y

h y y x x

In-plane internal forces

pxx pxx p

xy

p

xy

pyy pyy

IFEM Ch 14 – Slide 4

Department of Engineering Mechanics

• PhD. TRUONG Tich Thien

Introduction to FEM

Mathematical Idealization as a Two Dimensional Problem

Ω Γ

x y

Midplane Plate

IFEM Ch 14 – Slide 5

Department of Engineering Mechanics

• PhD. TRUONG Tich Thien

Introduction to FEM

Inplane Forces are Obtained by Stress Integration Through Thickness

h

x y y z x x y

Inplane internal forces (also called membrane forces)

h

Inplane stresses

σxx σyy σxy σ x

y

= pxx pyy pxy

IFEM Ch 14 – Slide 6

Department of Engineering Mechanics

• PhD. TRUONG Tich Thien

Introduction to FEM

Plane Stress Boundary Conditions

• Γ

Γ

u t

+

u Boundary displacements u are prescribed on Γ ^ t ^ ^ Boundary tractions t or boundary forces q are prescribed on Γ u = 0 ^ n (unit exterior normal)

t

σ σ σ

n

^

^

t t t

n

^t

nt nn

Stress BC details (decomposition of forces q would be similar)

^ t ^ IFEM Ch 14 – Slide 7

Department of Engineering Mechanics

• PhD. TRUONG Tich Thien

Introduction to FEM

The Plane Stress Problem

Given: geometry material properties wall fabrication (thickness only for homogeneous plates) applied body forces boundary conditions: prescribed boundary forces or tractions prescribed displacements Find: inplane displacements inplane strains inplane stresses and/or internal forces

IFEM Ch 14 – Slide 8

Department of Engineering Mechanics

• PhD. TRUONG Tich Thien

Introduction to FEM

Matrix Notation for Internal Fields

e(x, y) =   exx(x, y) eyy(x, y) 2exy(x, y)   σ(x, y) =   σxx(x, y) σyy(x, y) σxy(x, y)   u(x, y) = ux(x, y) uy(x, y)

• displacements

strains stresses

IFEM Ch 14 – Slide 9

Department of Engineering Mechanics

• PhD. TRUONG Tich Thien

Introduction to FEM

Governing Plane Stress Elasticity Equations in Matrix Form

  exx eyy 2exy   =   ∂/∂x ∂/∂y ∂/∂y ∂/∂x   ux uy

• 

 σxx σyy σxy   =   E11 E12 E13 E12 E22 E23 E13 E23 E33     exx eyy 2exy   ∂/∂x ∂/∂y ∂/∂y ∂/∂x   σxx σyy σxy   + bx by

• =
• e = Du

σ = Ee DT σ + b = 0

• r

IFEM Ch 14 – Slide 10

Department of Engineering Mechanics

• PhD. TRUONG Tich Thien

Introduction to FEM

Strong Form Tonti Diagram of Plane Stress Governing Equations

e = D u in Ω D + b = 0 σ in Ω u = u

^

• n Γ

u

in Ω σ = E e Ω Γ

• n Γ

t

n = t

^

T

σ n = q

^

T

p

• r

Kinematic Constitutive Displacement BCs Force BCs Equilibrium Prescribed tractions t

• r forces q

Stresses

σ

Body forces

b

Displacements

u

Strains

e

Prescribed displacements

u

^

IFEM Ch 14 – Slide 11

Department of Engineering Mechanics

• PhD. TRUONG Tich Thien

Introduction to FEM

TPE-Based Weak Form Diagram of Plane Stress Governing Equations

δΠ= 0 in Ω e = D u in Ω u = u

^

• n Γ

u

in Ω σ = E e Ω Γ

Kinematic Constitutive Displacement BCs Force BCs (weak) Equilibrium (weak) Prescribed tractions t

• r forces q

Stresses

σ

Body forces

b

Displacements

u

Strains

e

Prescribed displacements

u

^ t

δΠ = 0

• n Γ

IFEM Ch 14 – Slide 12

Department of Engineering Mechanics

• PhD. TRUONG Tich Thien

Introduction to FEM

Total Potential Energy of Plate in Plane Stress

= U − W U = 1

2

• h σT e

=

1 2

• h eT Ee d

d W =

• h uT b d +
• Ŵt

h uT ˆ t dŴ

IFEM Ch 14 – Slide 13

Department of Engineering Mechanics

• PhD. TRUONG Tich Thien

Introduction to FEM

Discretization into Plane Stress Finite Elements

Ω Ω Γ Γ

(a) (b) (c)

(e) (e)

IFEM Ch 14 – Slide 14

Department of Engineering Mechanics

• PhD. TRUONG Tich Thien

Introduction to FEM

Plane Stress Element Geometries and Node Configurations

1

2

3 1 2 3 4 1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 10 11 12

n = 3 n = 4 n = 6 n = 12

IFEM Ch 14 – Slide 15

Department of Engineering Mechanics

• PhD. TRUONG Tich Thien

Introduction to FEM

Total Potential Energy of Plane Stress Element

(e) = U (e) − W (e) U (e) = 1

2

• (e)

(e) (e)

h σT e = 1

2

• h eT Ee d(e)

W (e) =

• h uT b d(e) +
• Ŵ(e) h uT t dŴ(e)

Ω Γ

(e) (e)

IFEM Ch 14 – Slide 16

Department of Engineering Mechanics

• PhD. TRUONG Tich Thien

Introduction to FEM

Constructing a Displacement Assumed Element

u(e) = [ ux1 uy1 ux2 . . . uxn uyn ]T u(x, y) = ux(x, y) uy(x, y)

• =

N (e)

1

N (e)

2

. . . N (e)

n

N (e)

1

N (e)

2

. . . N (e)

n

• = N u(e)

u(e)

Node displacement vector: Displacement interpolation n nodes, n=4 in figure N is called the shape function matrix

IFEM Ch 14 – Slide 17

Department of Engineering Mechanics

• PhD. TRUONG Tich Thien

Introduction to FEM

Element Construction (cont'd)

e(x, y) =        ∂N (e)

1

∂x ∂N (e)

2

∂x . . . ∂N (e)

n

∂x ∂N (e)

1

∂y ∂N (e)

2

∂y . . . ∂N (e)

n

∂y ∂N (e)

1

∂y ∂N (e)

1

∂x ∂N (e)

2

∂y ∂N (e)

2

∂x . . . ∂N (e)

n

∂y ∂N (e)

n

∂x        u(e) =Bu(e)

Differentiate the displacement interpolation wrt x,y to get the strain-displacement relation B is called the strain-displacement matrix

IFEM Ch 14 – Slide 18

Department of Engineering Mechanics

• PhD. TRUONG Tich Thien

Introduction to FEM

Element Construction (cont'd)

f(e) =

• (e) h NT b d(e) +
• Ŵ(e) h NT ˆ

t dŴ(e) (e) = 1

2u(e)T K(e)u(e) − u(e)T f(e)

K(e) =

• (e) h BT EB d(e)

Element stiffness matrix Element total potential energy Consistent node force vector body force surface force

IFEM Ch 14 – Slide 19

Department of Engineering Mechanics

• PhD. TRUONG Tich Thien

Introduction to FEM

Requirements on Finite Element Shape Functions

Interpolation Conditions: N takes on value 1 at node i, 0 at all other nodes Continuity (intra- and inter-element) and Completeness Conditions are covered later in the course (Chs. 18-19) i

IFEM Ch 14 – Slide 20

Department of Engineering Mechanics

• PhD. TRUONG Tich Thien