14 The Plane Stress Problem IFEM Ch 14 Slide 1 Introduction to - - PDF document

Introduction to FEM

14

The Plane Stress Problem

IFEM Ch 14 – Slide 1

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

Introduction to FEM

Mathematical Idealization as a Two Dimensional Problem

Ω Γ

x y

Midplane Plate

IFEM Ch 14 – Slide 3

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, set to 0 Unessential but used in this course: Plate fabricated of homogeneous material through thickness

zz xz yz

IFEM Ch 14 – Slide 4

Introduction to FEM

Notation for stresses, strains, forces, displacements

x y z x x x y

In-plane stresses Positive sign convention

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 5

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

Introduction to FEM

Plane Stress Boundary Conditions

• Γ

Γ

u t

+

u

Boundary displacements u are prescribed on Γ (figure depicts fixity condition) ^

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

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

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

xy

strains (factor of 2 in e simplifies "energy dot products") stresses

IFEM Ch 14 – Slide 9

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

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

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

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 uTˆ t dŴ

body forces boundary tractions

IFEM Ch 14 – Slide 13

Introduction to FEM

Discretization into Plane Stress Finite Elements

Ω Ω Γ Γ

(a) (b)

(c)

e e

IFEM Ch 14 – Slide 14

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

Introduction to FEM

Total Potential Energy of Plane Stress Element

e γ = Ue − W e U e = 1

2

• e

e e

h σT e = 1

2

• h eT Ee de

W e =

• h uT b de

+

• Ŵ e h uT t dŴe

Ω Γ e

e

IFEM Ch 14 – Slide 16

Introduction to FEM

Constructing a Displacement Assumed Element

ue = [ 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

ue

Node displacement vector Displacement interpolation over element n nodes, n=4 in figure N is called the shape function matrix It has order 2 x 2n

. . . . . . . . .

IFEM Ch 14 – Slide 17

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       

ue = B ue

Differentiate the displacement interpolation wrt x,y to get the strain-displacement relation B is called the strain-displacement matrix It has order 3 x 2n

IFEM Ch 14 – Slide 18

Introduction to FEM

Element Construction (cont'd)

f e =

• e h NT b d e +
• Ŵe h NT ˆ

t dŴ e e =

1 2u e T Ke ue − ue Tf e

Ke =

• e h BT EB d e

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

IFEM Ch 14 – Slide 19

Introduction to FEM

Requirements on Finite Element Shape Functions

Interpolation Condition 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