Shear Locking effect on Finite Element Method applied to Timoshenko Beams
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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 of Shear Locking effect on Finite Element Method applied to Timoshenko Beams
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2100
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2100
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2100
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2100
100 s
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2100
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1.0025
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1.0029
1.0031
1.0033
1.0035
0 10 20 30 40
Ver
tica
l dis
pla
cem
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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
