Introduction to FEM

download Introduction to FEM

of 60

  • date post

    30-Sep-2015
  • Category

    Documents

  • view

    214
  • download

    0

Embed Size (px)

description

Introduction to FEM

Transcript of Introduction to FEM

  • Simulation Tool

    Response of a structure or system to the loads imposed

    Structural Analysis Tool

    Measuring critical loads or failure criteria

    Structural Design Tool

    Modifying the structure to improve performance

  • Number of Edges

    Calculated

    circumference

    2r

  • Node

    Element

    Mesh

    Load

    Boundary Condition

    Material Property

    TrueGrid

  • Compatibility

    Equilibrium

    Constitutive Law

  • Things fit together with no gaps

    Each node and element boundary matches the one beside it.

  • F1 F2

    Fk Fn

  • E

  • P

    P

    A

  • P

    P

    A

    A A Pn

    0lim nA

    P

    A

  • Ps A

    0lim sA

    P

    A

  • Megson

  • Megson

    xy yx

    yz zy

    zx xz

    , ,x y zNormal components

    , , ,

    , ,

    xy yx yz

    zy zx xz

    Shear components

    Perpendicular to

    this axis

    Parallel to this

    axis

    xy

  • x xy xz

    yx y yz

    zx zy z

  • x

    y

    z

    yz

    zx

    xy

    Careful with the

    shear terms,

    different sources

    use different order

    x

    y

    z

    yz

    zx

    xy

    Sigma

    Tau

    Epsilon

    Gamma

  • E

  • 1 0 0 01 1

    1 0 0 01 1

    1 0 0 01 1(1 )

    (1 2 )(1 )(1 2 ) 0 0 0 0 02(1 )

    (1 2 )0 0 0 0 0

    2(1 )

    (1 2 )0 0 0 0 0

    2(1 )

    x x

    y y

    z z

    yz yz

    zx zx

    xy x

    E

    E

    y

  • Continuum

    (Displacement only)

    Structural

    (Displacement and Slope)

    1D

    Bar (u)

    Truss (u,v,w)

    Beam (u,v,w,

    x,y,z)

    2D

    Plane stress

    (u,v)

    Plane Strain

    (u,v)

    Plate (u,v,w,

    x,y) Shell

    (u,v,w,

    x,y,z)

    3D

    Bricks (u,v,w)

  • Real Structure

    Model of Structure

    Discretised Model

    Different Model

    of Structure

    Discretised Model

    Ma

    gn

    itude o

    f E

    rrors

  • Remember:

    Refinement does

    not make your

    closer to REALITY.

    Refinement makes

    your results closer

    to your MODEL!

  • (a) Von mises stress (b) Fore-aft load paths showing detail

    around mast step

  • 11 12 1 1

    21 22 2 2

    k k u P

    k k u P

    P1 P2

    u1 u2

    E, A, L

  • P1 P2

    u1=0 u2=1

    N N

    Node 1 Node 2

  • What if we have multiple elements or loads?

    We can assemble multiple elements using equilibrium at the nodes (+ compatibility)

    I II III

    P1 P2 P3 P4

    1

    IP2

    IP2

    IIP3

    IIP3

    IIIP4

    IIIP

    I II III

  • Garth Pearce 2012

    At Node 1:

    At Node 2:

    1 1

    1 2 1

    2 2 2

    1 2 3 2

    0

    0

    I

    I I I I

    I I

    I II

    I I I I II II II II

    I I II II

    P P

    A E A Eu u P

    L L

    P P P

    A E A E A E A Eu u u P

    L L L L

  • Garth Pearce 2012

    In matrix form

    1 1

    2 2

    3 3

    4 4

    0 0

    0

    0

    0 0

    I I I I

    I I

    I I I I II II II II

    I I II II

    II II II II III III III III

    II II III III

    III III III III

    III III

    A E A E

    L L

    u PA E A E A E A E

    L L L L u P

    u PA E A E A E A E

    L L L L u P

    A E A E

    L L

    u1 u3 u2 u4

    I II III

  • Ku P

    K u P

  • Steel Bar, A = 100mm2, L = 1m Al Bar, A = 50mm2, L = 600mm

    F1 = 3kN F2 = 1kN

    X=0 X=1000mm X=1600mm

  • ij j iij

    EAF A u u

    L

    j iij

    ij

    u u

    L

    ij j iij

    EE u u

    L

  • 1 kN

    2 kN

    1000 1000 1000

    A = 50 mm2

    E = 70 GPa

  • P1

    P2

    u1

    u2

    Q2

    Q1

    v1

    v2

    2 2

    1 1

    2 2

    1 1

    2 2

    2 2

    2 2

    2 2

    cos

    where

    sin

    j i

    j i

    u Pl lm l lm

    v Qlm m lm mAE

    u PL l lm l lm

    v Qlm m lm m

    x xl

    L

    y ym

    L

    j iijj iij

    u uEAF l m

    v vL

  • 2 2

    2 2

    2 2

    2 2

    2 2

    2 2

    2 2 2

    [ ]

    T

    i i i j j j

    T

    i i i j j j

    ij

    j i j i j i

    ij j i j i j i

    ij ij

    u v w u v w

    P Q R P Q R

    l lm ln l lm ln

    lm m mn lm m mn

    ln mn n ln mn nEA

    L l lm ln l lm ln

    lm m mn lm m mn

    ln mn n ln mn n

    x x y y z zL x x y y z z l m n

    L L

    u

    P

    K

    ij

    j i

    ij j i

    ij

    j i

    L

    u uEA

    F l m n v vL

    w w

  • http://bendingmomentdiagram.com/

    v3 v2

    u1 u3 u2

    I II v1

    Try with truss elements

  • 3 2 3 2

    1 1

    2 21 1

    2 2

    3 2 3 2

    2 2

    2 2

    12 6 12 6

    6 4 6 2

    12 6 12 6

    6 2 6 4

    EI EI EI EI

    L L L Lv PEI EI EI EI

    ML L L L

    v PEI EI EI EI

    L L L L M

    EI EI EI EI

    L L L L

    P2 P1 v2

    M1 M2

    2

    v1

    1