Eng. 6002 Ship Structures 1
An introduction to the finite
element method
using MATLAB
Overview
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This paper outlines an efficient approach to
introducing the finite element method
This approach requires that the students have
some prior experience with MATLAB and a
fundamental understanding of solid mechanics.
Only two-dimensional beam element problems
are considered, to simplify the development.
The approach emphasizes an orderly solution
procedure and involves important finite element
concepts, such as
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the stiffness matrix,
element and global coordinates,
force equilibrium, and constraints.
Overview
The ability of MATLAB to manipulate
matrices and solve matrix equations
makes the computer solution concise and
easy to follow.
 The flexibility associated with the
computer implementation allows example
problems to be easily modified into design
projects
 and you may even feel motivated to
modify things and tackle some other
problems
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Introduction
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We will decompose a structure into a finite
number of elements.
Once each element is completely described with
appropriate geometric and material properties,
the elements are properly combined or
‘assembled’
Introduction
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Each element is modeled as an elastic member for which
the displacement is linearly related to the applied loading.
 f   k u
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where {f} is the element load vector, [k] is the element
stiffness matrix, and {u} is the element displacement
vector.
To indicate the specific element under consideration, a
superscript will be used on the quantities in equation 1.
The element or member loads can be a force, in which
case the displacement is a linear translation, or a
moment, in which case the displacement is an angular
rotation.
Introduction
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To model the structure, the individual beam
elements are properly combined so that the
entire structure is described by:
F  K U 
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where {F} is the nodal load vector, [K] is the
global stiffness matrix, and {U} is the nodal
displacement vector.
The objective of the type of problems
considered is determine the unknown nodal
displacements.
Introduction
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For the purposes of this paper, the FEM can be conveniently
divided into five (or six) steps:
1.
2.
3.
4.
5.
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construction of the element stiffness matrix in local
coordinates,
transformation of the element stiffness matrix into global
coorindinates,
assembly to the global stiffness matrix using transformed
element stiffness matrices,
application of the constraints to reduce the global stiffness
matrix,
determination of unknown nodal displacements.
An additional step that might be implemented is the postprocessing of results, i.e. the determination of unknown
forces or stresses from the calculated displacements.
The Method
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The first step in the formulation is to discretize
the domain, i.e., to select the number and type of
elements. Only two-dimensional beam elements
are considered in this study.
Each beam element has two nodes.
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Two-dimensional beam elements allow for an axial force
fx, transverse force fy, and a bending moment fθ at each
node and have three degrees of freedom (dof) per node,
axial displacement ux, transverse displacement uy, and
angular rotation uθ.
The positive sign convention is shown.
The Method
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The stiffness matrix for each element is
The Method
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The element stiffness matrix is a square
matrix whose size is given by:
[number of nodes x dof per node] x [number of
nodes x dof per node]
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Thus, the stiffness matrix for a twodimensional beam element is 6 x 6
because of the three degrees of freedom
associated with each node.
The Method
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this matrix describes the stiffness of an
elastic element that relates the load vector
to the displacement vector
The Method
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To verify that this form of the stiffness
matrix is correct and to illustrate several
FEM concepts, consider the situation of a
cantilevered beam such that
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node 1 is constrained from movement, i.e., ux1
= uy1 = uθ1 = 0.
Next, consider a downward force applied at
node 2, so that fy2 = -F and fx2 = fθ2 = 0.
The Method
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With these considerations, the system of
equations can be rewritten as:
The Method
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The solution to this system of equations yields
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ux2 = 0,
uy1 = -FL3/(3EI),
uθ2 =-FL2/(EI),
which agree with the expressions for the slope and
deflection at the tip of a cantilevered beam as given in
any mechanics of materials textbook.
Further verification can be achieved with
consideration of an applied axial load and then an
applied moment (on your own).
The Method
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The second step in the development is to
transform the element load vector {f} and the
element displacement vector {u} into the global
coordinate system
Consider a beam element at an arbitrary
orientation described by an angle θ, measured
counterclockwise from the horizontal, and a
global coordinate system indicated with a bar
The Method
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From simple trigonometry, the loads and
displacements in the global coordinate system
can be related to the loads and displacements in
the element coordinate system
 f    f and u   u 
The Method
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Substitution of equation 6
into equation 1
yields:
 f  k u 
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Then, for an orthogonal
matrix []-1 = []T, the element
equation can be written in
global coordinates as:
f  k u 
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where the element stiffness
matrix transformed to global
coordinates is defined as:
k     k  
T
The Method
The third step is to properly combine the
individual transformed element stiffness
matrices to construct the global stiffness
matrix that describes the entire structure.
 We only consider beam elements that are
connected ‘end to end’.
 We’ll look at a two-element example.
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Example
Consider the case of two beam elements
connected ‘end to end’.
 The global stiffness matrix [K] for the twoelement case is 9 x 9 because beam
elements have three degrees of freedom
per node and there are three nodes in the
structure.
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Example
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The external loads at the nodes are given
by:
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And the displacements at the nodes:
Example
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The external loads at the nodes are given
by:
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And the displacements at the nodes:
Example
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Force and moment
equilibrium at each
node requires that the
sum of the element
loads must equal the
nodal loads and
compatibility at each
node requires that the
element displacements
must equal the nodal
displacements
Example cont.
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In order to easily combine the element stiffness
matrices, each element stiffness matrix is stored
in a matrix the size of the global stiffness matrix,
with the extra spaces filled with zeros.
In this example, the element stiffness matrix for
element 1 is stored in the portion of the global
stiffness matrix that involves nodes 1 and 2,
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i.e., the upper 6 x 6 portion of the matrix. Thus, the
expanded stiffness matrix that describes element 1 is
given by:
Example cont.
Example cont.
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The element stiffness matrix for element 2 is
stored in the portion of the global stiffness matrix
that involves nodes 2 and 3, i.e., the lower 6 x 6
portion of the matrix. The expanded stiffness
matrix that describes element 2 is given by:
Example cont.
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The expanded individual stiffness matrices in
equations 16 and 17 can now be added together
so that the global stiffness matrix for the twoelement structure is given as:
Example cont.
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The fourth step is to apply the constraints and
reduce the global stiffness matrix so that the
specific problem of interest can be solved.
At the point of constraint, the displacement of
the structure is known.
Because these displacements are known, matrix
algebra allows the removal of the corresponding
rows and columns.
The resulting system of equations can be written
as:
F r  K r U r
Example cont.
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[K]r is the reduced global stiffness matrix that contains
information about the structure and the boundary
conditions.
With MATLAB, rows and columns can be easily deleted,
and a shift of the remaining elements in the matrix is
performed automatically.
The final step is simply to solve the reduced system of
equations for the unknown displacements.
MATLAB efficiently solves a system of equations with the
backslash command. The backslash command uses
Cholesky factorization and Gaussian elimination to solve a
system of equations
Next Class
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In the next class we will be developing a
matlab script to apply this methodology