shear locking effect on finite element method applied to timoshenko beams
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
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s
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P EI L GA Lv
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3
4P
PLv
EI
EI sGA
2100
1 s
EI Nm
GA N
2100
100 s
EI Nm
GA N
1
2
L m
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