Finite Element Method
Introduction
• The finite element modelling process allows for discretizing
the intricate geometries into small fundamental volumes called
finite elements.
• It is then possible to write the governing equations and
material properties for these elements.
• These equations are then assembled by taking proper care of
the constraints and loading, which results in a set of equations.
• These equations when solved give the results that describe the
behaviour of the original complex body being analysed.
Application of FEM
Application of FEM is not limited to mechanical systems alone
but to a range of engineering problems such as:
• Stress analysis
• Dynamic analysis
• Deformation studies
• Fluid-flow analysis
• Heat-flow analysis
• Seepage analysis
• Magnetic-flux studies
• Acoustic analysis
Important Definitions
• Mesh: Subdivided part geometry is called a mesh. The process
of subdivision is called meshing.
• Element: In finite element analysis (FEA), part geometry is
divided into small volumes to solve it easily. Each such
volume is called an element.
• Node: Each element has a set of points called nodal points or
nodes for short. Nodes are usually located at the corners or
endpoints of elements.
• Degrees of Freedom: Degrees of Freedom (DOF) specify the
state of the element.
Important Definitions
• Nodal Forces: These are the forces that are applied on the
finite element model. These are generally the concentrated
forces at the nodes.
• Boundary Conditions: These specify the current state of some
nodes in the finite element mesh.
• Dimensionality: Elements that are used in FEM can have
intrinsic dimensionality of one, two or three space dimensions.
General procedure of the FEM
Steps:
1. Discretization
2. Choosing the solution approximations.
3. Forming the element matrices and equations.
4. Assembling the matrices.
5. Finding the unknowns.
6. Interpreting the results.
General procedure of the FEM
1. Discretization : In this step the entire body to be analysed is
divided into smaller elements or finite elements. These elements
are connected to each other through nodes. The elements should
not overlap each other.
• Dividing the body into elements and nodes is called mesh
generation or meshing.
General procedure of the FEM
2. Choosing the solution approximations : The analysis is initially
restricted to a single field element by finding the values of field
variables at the nodes of these elements.
• The value of the field variables on the entire element domain
can be found out by extrapolating the values at the nodes and
approximating the solution.
3. Forming the element matrices and equations:
• The analysis of a single element is done by applying equations
of equilibrium to that element.
• These equations can then be expressed in form of a matrix
called element matrix.
General procedure of the FEM
4. Assembling the matrices : The element matrices of all the
elements are combined or assembled to form the global stiffness
matrix, which represents the entire body. The boundary condition
can then be applied to the global stiffness matrix.
5. Finding the unknown field variables : The unknown field
variables can be found out from the global stiffness matrix by
using gauss elimination approach.
General procedure of the FEM
6. Interpreting the results: Once the value of field variables is
obtained from the above analysis, the conclusions are drawn and
appropriate modifications are incorporated into the original
design in order to improve the design.
• For example, in case of stress analysis of a piston, wall
thickness can be increased at the sites where the stress
concentration is more and probability of failure is high.
Types of forces
• In finite element structural analysis problems basically three
types of forces are considered :
(A) Body force
(B) Traction force
(C) Point force
Types of forces
• Body forces are the forces which are distributed uniformly
over the body and are expressed as force per unit volume. e.g.,
the body weight.
• Traction forces are the forces acting on the surface of
the body and are expressed as force per unit area. Examples
of such forces are frictional resistance, surface drag, viscous
force, pressure etc.
Types of forces
• Point force is a force applied by some external means
and is acting at a point. It is expressed in absolute units.
Stiffness matrices
• Spring Element: A structural part is shown in Fig. with four
spring elements. Each spring element has two nodes at the
endpoints of the spring.
Isolate a spring between two nodes i and j as shown in Fig.
with the respective displacements ui and uj, and forces fi and fj.
Net deflection δ is
Stiffness matrices
• Defining the spring constant k as the force per unit deflection,
the resultant axial force in the spring is
• For equilibrium,
• Hence, it is possible to write the equations for deflection at
each of the nodes as
Stiffness matrices
These can be written in matrix form as
[K] is called the stiffness matrix of the element under
consideration
Assembling the Stiffness Matrix
• Element (spring) 1 has nodes 1 and 2, while the element 2 has
nodes 2 and 3, with the node 2 being common between the two
elements.
• Considering that for the equilibrium, the sum of the internal
forces should be equal to the external forces applied at each
node, it is possible to write the equations for deflection at each
of the nodes as
Assembling the Stiffness Matrix
• These equations can then be written in matrix form as
Assembling the Stiffness Matrix
• This can be generalised for the four-element case
Boundary Conditions
• Boundary conditions are of two types:
 Homogenous
 Non-homogenous
• Each of the nodes in the structural system has six degrees of
freedom, three translational along the three axes (x, y and z) of
the rectangular coordinate system used and three rotary axes
about each of the rectangular axes.
• In the homogenous boundary conditions, all the degrees of
freedom of the node are completely fixed, whereas in the nonhomogenous type, some displacements in specified axes will
be allowed.
• Problem: Calculate the displacement of the nodal points 2 and 3.
Solution:
• The assemblage of the equation for the three-spring system is
• Applying the homogenous boundary conditions as fixed ends
will have no displacements, and the internal forces are zero. We
get
• Eliminating the columns and rows corresponding to the
boundary conditions from the above equation, we get
• Solving the above, we get, u2 = 57.1428 mm, and u3 = 14.2857
mm. To obtain the unknown forces,
f1 = –114.286 N,
f4 = –85.716 N
• The equilibrium of the system can be checked by taking
f1 + f2 + f3 + f4 = –114.286 + 200 + 0 – 85.716 = 0
Elastic Bar Element
• Figure shows an elastic bar element that obeys Hooke’s law.
• Forces are only applied at the end, and forces are only applied
axially. The deflection, δ is given by:
where, P = applied axial load on the bar,
L = length of the bar
A = uniform area of cross-section
of the bar
Elastic Bar Element
E = Elastic modulus of the bar material
The equivalent spring constant of the elastic bar can be
written as:
Problem: A tapering round bar is fixed at one end and a tensile
load of 1000 N is applied at the other end. Take elastic modulus,
E=2×105 MPa. Find the global stiffness matrix and displacements
considering it as 4 elements.
• Solution: The tapered bar is divided equally into four elements
as shown in Fig. and its elemental properties are calculated
below.
Calculate the areas of all the four elements
• Equivalent spring representation of the beam is shown in Fig.
Calculate stiffness of all the four elements.
Equilibrium equations
• where R is the reaction at the fixed end. Substitute the
boundary conditions, and crossing out the first row and
column for the fixed condition.
Advantages of FEM
• Parts with irregular geometries are difficult to be analysed by the use
of conventional strength of material approaches. In FEA, any
complex geometry can be analysed with ease.
• Parts made from different materials can be analysed using FEA.
• It is possible to analyse parts that have complex loading patterns
with multiple types of forces acting on the geometry with large
number of supports.
• The FEA procedure provides results throughout the part (all points).
• It is easy to change the model in FEA and generate a number of
options with ‘what-if’ scenarios.
• Testing of products can be done using FEA without expensive
destructive testing.