Shear Locking effect on Finite Element Method applied to Timoshenko Beams

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DESCRIPTION

Derivation of a stiffness matrix for a Timoshenko beam using linear shape functions. Parameter study on the shear locking effect and analysis on the reduced integration method.

Transcript

  • 1 1x

    NL

    2

    xN

    L

  • 11

    2 2N

    2

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    2

    1 1 2 2 1 2

    1

    1 1 1

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    i

    x N x N x N x x x L

    11

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    1 1

    12 2

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    v

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    sK B EIB N GA N Ld

    1

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    1

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    vN N

    vB B

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    N N NL L

    1 2 1

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    L L

    1 2 1 1v v

    v dN dNB

    dx dx L L

    1 2 1 1dN dNBdx dx L L

  • N

    B

    K

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    1 1

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    1 1 1 1 1 1 1 12 8 81 1 2 1 1 2

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    L

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  • 1 1 1 11 11 1

    1 1 1 12 23 3

    vv s sGA GAK f fL L

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    s s s s

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    L L

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    22

    / / 3

    / /12

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  • 4P

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    3

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    1 1 1 1(0) 2 2

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  • v

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    vv

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  • 34P

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    2100

    1 s

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    2100

    100 s

    EI Nm

    GA N

    1

    2

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    L m

    5

    10

    L m

    L m

    1 P N

  • 3

    3P

    s

    PL PLv

    EI GA

    [m]L

    [m]Pv

    2100

    1 s

    EI Nm

    GA N

    2100

    100 s

    EI Nm

    GA N

  • 2100

    1 s

    EI Nm

    GA N

    [m]L

    [m]Pv

    1.0025

    1.0027

    1.0029

    1.0031

    1.0033

    1.0035

    0 10 20 30 40

    Ver

    tica

    l dis

    pla

    cem

    ent

    dof

    Length 1 m

    1 GP

    2 GP

    Analytical solution

    2.018

    2.02

    2.022

    2.024

    2.026

    2.028

    0 10 20 30 40

    Ver

    tica

    l dis

    pla

    cem

    ent

    dof

    Length 2 m

    1 GP

    2 GP

    Analytical solution

    5.31

    5.36

    5.41

    0 10 20 30 40

    Ver

    tica

    l dis

    pla

    cem

    ent

    dof

    Length 5 m

    1 GP

    2 GP

    Analytical solution

    12.25

    12.75

    13.25

    0 10 20 30 40

    Ver

    tica

    l dis

    pla

    cem

    ent

    dof

    Length 10 m

    1 GP

    2 GP

    Analytical solution

  • 2100

    100 s

    EI Nm

    GA N

    [m]L

    [m]Pv

    0.012

    0.0125

    0.013

    0.0135

    0 10 20 30 40

    Ver

    tica

    l dis

    pla

    cem

    ent

    dof

    Length 1 m

    1 GP

    2 GP

    Analytical solution

    0.035

    0.04

    0.045

    0.05

    0 10 20 30 40

    Ver

    tica

    l dis

    pla

    cem

    ent

    dof

    Length 2 m

    1 GP

    2 GP

    Analytical solution

    0

    0.1

    0.2

    0.3

    0.4

    0.5

    0 10 20 30 40

    Ver

    tica

    l dis

    pla

    cem

    ent

    dof

    Length 5 m

    1 GP

    2 GP

    Analytical solution

    0

    1

    2

    3

    4

    0 10 20 30 40

    Ver

    tica

    l dis

    pla

    cem

    ent

    dof

    Length 10 m

    1 GP

    2 GP

    Analytical solution

  • 2100 EI Nm 1sGA N

    1sGA N

    2100 EI Nm 100sGA N

  • timoshenko_beam_single

    modeldatainput.m

    FEtype.m

    mesh.m boundary_conditions.m

    FE_type.m

    % Spatial dimension

    ProblemData.SpaceDim = 1;

    % PDE type

    ProblemData.pde = 'TimoshenkoBeam';

    % Degrees of freedom per node

    ElementData.dof = 2;

    % Nodes per element

    ElementData.nodes = 2;

    % Number of integration points per element

    ElementData.noInt = 1;

  • % Element type

    ElementData.type = 'Bar1';

    % Change the number of the elements and the lenght to change the mesh of

    the beam

    elements = 10;

    length = 10;

    pm = zeros(elements,3);

    for (i=1:elements)

    pm(i,1) = length/elements*i

    end

    x = [

    0.0 0.0 0.0

    pm

    ]';

    noel = zeros(elements,2);

    for (i=1:elements)

    noel(i,1) = i

    noel(i,2) = i+1

    end

    Connect = [

    noel

    ]';

  • Ka f

    1

    1

    2

    2

    v

    v

    a =

    % input boundary conditions

    % node number, boundary condition type (0=Neumnann, 1=Dirichlet), dof, bc

    value

    % Neumnann= force, Dirichlet=displacement

    tol = 0.000001;

    L = abs(max(Mesh.x(1,:)));

    %The loop below can find and apply the needed BCs automatically.

    j = 1;

    for (i=1:Mesh.noNodes)

    if(abs(Mesh.x(1,i)) < tol)

    BC_data(1,j)=i;

    BC_data(2,j)=1;

    BC_data(3,j)=1;

    BC_data(4,j)=0;

    j = j+1;

    BC_data(1,j)=i;

  • BC_data(2,j)=1;

    BC_data(3,j)=2;

    BC_data(4,j)=0;

    j = j+1;

    end

    if(abs(Mesh.x(1,i)) > L-tol)

    BC_data(1,j)=i;

    BC_data(2,j)=0;

    BC_data(3,j)=2;

    BC_data(4,j)=1;

    j = j+1;

    end

    end

    FEcode.m mesh.m

    mesh.m MeshInitialise.m

    FEcode.m

    % Directory where input files are located

    input_directory = './demo/timoshenko_beam_single';

    addpath(input_directory)

    disp('-Reading problem data')

    % Problem data

    FE_type

    % Read model/material data

  • ModelDataInput

    % read mesh (change this to the name of the mesh, leaving off .m)

    %Mesh = read_mesh(input_directory, ProblemData, ElementData);

    mesh

    % Intitialise mesh

    MeshInitialise