Applied Finite Element Analysis
for Civil and Mechanical
Engineering Applications
A Conceptual Course
Presented by
Rajendra Machavaram
Assistant Professor, AgFE, IIT Kharagpur
for
Faculty Development Program
at
Audisankara College of Engineering and Technology
Gudur, Andhra Pradesh, India.
Rajendra M. 1 AgFE, IIT KGP

Finite Element Method (Introduction & History)
The Finite Element Method (FEM) is a numerical
procedure for analyzing structures and continua.
It was first applied to stress analysis and today it is applied
to other problems, such as, heat transfer, fluid flow,
lubrication, electric and magnetic fields.
It is also used to design buildings, electric motors, heat
engines, ships, airframes and space crafts.
History:
Fig. A coarse mesh
of gear tooth.
R. Courant (1943) described a piecewise polynomial solution for the torsion problem.
The procedure was impractical at that time due to lack of digital computers.
1950s: Work in the aircraft industry introduced the FE method to practicing engineers
1960s: The name Finite Element was coined (R. W. Clough)
1963: The mathematical validity of the FE method was recognized and the method was
expanded from its structural beginnings to include heat transfer, groundwater flow,
magnetic fields and other areas
1970s: Large general purpose FE software began to appear
1980s: The software was available on microcomputers, complete with color graphics and
pre and post processors (G. E. Smith)
1990 to Present: Application of FE using various softwares on different areas is started
Rajendra M. 2 AgFE, IIT KGP

Finite Element Method (General Procedure)
Step 1: Divide the structure or continuum into finite
elements, called as mesh generation.
Step 2: Formulate the properties of each element
Step 3: Assemble the elements to obtain the finite
element model of the structure
Step 4: Apply the known loads (Initial conditions)
Step 5: Apply the support conditions (Boundary
conditions)
Step 6: Solve the simultaneous linear algebraic
equations to determine nodal DOF
(Displacements/ temperatures).
Step 7: Calculate the element properties, such as
strains, stresses, heat flux etc.
Rajendra M. 3 AgFE, IIT KGP

Finite Element Method (Terminology)
Elements: Finite elements are fragments of the structure, these
are triangular or quadrilateral areas on two dimensional
space (OR) Tetrahedron or Hexahedron on three
dimensional volumes.
Nodes: Nodes are the connectors that fasten elements together,
which appear on element boundaries.
Degrees of Freedom (DOF): Number of independent parameters
Shape Function: It is a polynomial function which defines the
element field variables in terms of field variables of the
nodes.
Stiffness Matrix: It is an element characteristic matrix in
structural mechanics. It relates nodal displacements to
nodal forces.
Conductivity Matrix: It is an element characteristic matrix in
heat conduction. It relates nodal temperatures to nodal
fluxes.
Mass Matrix: It is an element characteristic matrix in structural
dynamics. It relates nodal velocities to nodal fluxes.
Rajendra M. 4 AgFE, IIT KGP

Finite Element Method (Problem Formulation)
Finite Element Method Problem Formulation describes the procedure for determining
the element characteristic matrix. There are three important ways to derive an element
characteristic matrix.
1. Direct Method: It is based on physical reasoning. It is limited to very simple elements.
2. Variational Method: It is applicable to problems that can be stated by certain integral
expressions such as the expression for potential energy.
3. Weighted Residual Methods: These are particularly suited to problems for which
differential equations are known but no variational statement is available.
a) Collocation Method: Impulse functions are selected as weighted functions.
b) Subdomain Method: Each weighting function is selected as unity over a specific
region.
c) Galerkin’s Method: This uses the same functions for weights, that were used in the
approximating equation. This approach is the basis of the finite element method for
problems involving first-derivative terms (Potential or Kinetic Energy variation).
d) Least Squares Method: It utilizes the residual as the weighting function and obtains
a new error term. This error is minimized with respect to the unknown coefficients in
the approximate solution.
Rajendra M. 5 AgFE, IIT KGP

Finite Element Method (Problem Formulation)
Constitutive Matrix Formulation (Three Dimensional Problem)–Classical Mechanics
Consider a continuous three-dimensional (3D) elastic solid with a volume V and a surface
area S. The solid can be loaded by body forces fb and surface forces fs in any distributed
fashion in the volume of the solid.
Since cij=cji, there are
altogether 21
independent material
constants for a fully
anisotropic material
Rajendra M. 6 AgFE, IIT KGP

Finite Element Method (Problem Formulation)
Constitutive Matrix Formulation (Three Dimensional Problem)–Classical Mechanics
Isotropic Material
Isotropic Material – Plane Stress (Thin Shells) and Plane Strain (Thick Shells)
Rajendra M. 7 AgFE, IIT KGP

Finite Element Method (Problem Formulation)
Shape Function Formulation (Three Dimensional Problem)
Consider an element with nd nodes at xi (i=1,2,..,nd), where xT = {x, y, z} for three
dimensional problem. There should be nd shape functions for each displacement
component for an element.
Where uh is the approximation of the displacement component, pi(x) is the basis function
of monomials in the space coordinates x, and αi is the coefficient for the monomial pi(x).
Rajendra M. 8 AgFE, IIT KGP

Finite Element Method (Problem Formulation)
Stiffness and Mass Matrix Formulation (Three Dimensional Problem)
Strain Energy
Kinetic Energy
Work Done
Dynamic Equilibrium Equation
Static Equilibrium Equation
Rajendra M. 9 AgFE, IIT KGP

Finite Element Method (Elements)
Basic Element Shapes: The shapes, sizes, number and configuration of the elements have
to be chosen carefully such that the original body or domain is simulated as closely as
possible without increasing the computational effort needed for the solution.
Mostly the choice of the type of element is dictated by the geometry of the body and the
number of independent coordinates (DOF) necessary to describe the system.
One Dimensional Elements
(a) Bar (2 DOF at 2 nodes) and (b) Beam (4 DOF at 2 nodes)
Two Dimensional Elements
(a) Triangle (b) Rectangle (c) Quadrilateral (d) Parallelogram
Three Dimensional Elements
(a) Tetrahedron (b) Rectangular Prism (c) Hexahedron
Rajendra M. 10 AgFE, IIT KGP

Finite Element Method (Bar 1D Element)
Bar Element: Stiffness Matrix Formulation: Consider a uniform prismatic elastic bar of
length ‘L’ with elastic modulus ‘E’ and cross-sectional area ‘A’. A node is located at
each end with axially directed displacements.
Direct Method: The stretch in the bar is given by
And the forces is calculated as
For displacement at node 1 due to force at node 2, and the displacement at node 2 due to
force at node 1 is given by
Rajendra M. 11 AgFE, IIT KGP

Finite Element Method (Bar 1D Element)
Bar Element: Stiffness Matrix Formulation: Consider a uniform prismatic elastic bar of
length ‘L’ with elastic modulus ‘E’ and cross-sectional area ‘A’. A node is located at
each end with axially directed displacements.
Variational Approach: Stiffness matrix of a element is given by
Where B is the strain-displacement matrix, E is the material property matrix (Constitutive
matrix) and dV is an increment of the element volume V.
Rajendra M. 12 AgFE, IIT KGP

Finite Element Method (Truss 1D Element)
The displacement within the
element varies linearly, hence it
is called linear element
Truss Element: Stiffness Matrix Formulation: A truss is one of the simplest and most
widely used structural members. It is a straight bar that is designed to take only axial
forces, therefore it deforms only in its axial direction.
It has one DOF at each node.
Rajendra M. 13 AgFE, IIT KGP

Finite Element Method (Truss 1D Element)
Truss Element: Stiffness Matrix Formulation: A truss is one of the simplest and most
widely used structural members. It is a straight bar that is designed to take only axial
forces, therefore it deforms only in its axial direction.
Rajendra M. 14 AgFE, IIT KGP

Finite Element Method (Truss 1D Element)
Truss Element: Stiffness Matrix Formulation: Considering de as element displacements
in local coordinate system and De as element displacements in global coordinate system.
If T is transformation matrix to transform the local coordinates to global coordinate
system
Rajendra M. 15 AgFE, IIT KGP

Finite Element Method (Truss 1D Element)
Truss Element: Problems
A structure is made of three planer truss members as shown in Fig.
A vertical downward force of 1000 N is applied at node 2. The
properties of each member is given in Table.
Rajendra M. 16 AgFE, IIT KGP

Finite Element Method (Truss 1D Element)
Truss Element: Problems
Rajendra M. 17 AgFE, IIT KGP

Finite Element Method (Truss 1D Element)
Truss Element: Problems
Rajendra M. 18 AgFE, IIT KGP

Finite Element Method (Truss 1D Element)
Truss Element: Problems
Rajendra M. 19 AgFE, IIT KGP

Finite Element Method (Truss 1D Element)
Truss Element: Problems
Rajendra M. 20 AgFE, IIT KGP

Finite Element Method (Truss 1D Element)
Truss Element: Problems (Exercise)
A structure is made of three planer truss members as shown in Fig.
A vertical downward force of 1000 N is applied as shown. All the
truss members are of the same material (E=69.0 GPa) and with the
same cross-sectional area of 0.01 m2.
Determine the stress and strain induced in each bar.
Rajendra M. 21 AgFE, IIT KGP

Finite Element Method (Beam 1D Element)
Beam Element: Element Characteristic Matrix Formulation
A beam is another simple but commonly used structural
component. It is also geometrically a straight bar of an
arbitrary cross-section, but it deforms only in directions
perpendicular to its axis. Beams are subjected to
transverse loading, including transverse forces and
moments that result in transverse deformation.
The stresses on the cross-section of a beam are the
normal stress and shear stress. There are several theories
for analyzing beam deflections. These are
(i) A theory for thin beams: Euler-Bernoulli beam theory
(ii) A theory for thick beams: Temoshenko beam theory
In thin beam theory, the transverse planes of the beam
before and after bending are always perpendicular to the
beam axis. The shear stress is assumed to be negligible.
Rajendra M. 22 AgFE, IIT KGP

Finite Element Method (Beam 1D Element)
Beam Element: Element Characteristic Matrix Formulation
In planar beam elements there are 2DOF at a node in
its local coordinate system. These are translation (v)
along y-axis and rotation about z-axis (θz) in xy
plane. Therefore the beam element has a total of
4DOF.
Consider a beam element of length l = 2a with nodes 1 and 2 at each end of the element.
As there are 4DOF for a beam element, there should be four shape functions. Considering
natural coordinate system, the natural coordinate system has its origin at the centre of the
element, and the element is defined from -1 to +1.
The relation between the natural coordinate system and the local coordinate system is
The shape functions of beam element using natural coordinate system is
The third order polynomial is chosen because there are four unknowns in the polynomial,
which can be related to the four nodal DOFs in the beam element.
Rajendra M. 23 AgFE, IIT KGP

Finite Element Method (Beam 1D Element)
Beam Element: Element Characteristic Matrix Formulation
The shape functions of beam element using natural
coordinate system is
Rajendra M. 24 AgFE, IIT KGP

Finite Element Method (Beam 1D Element)
Beam Element: Element Characteristic Matrix Formulation
The shape functions of beam element using natural
coordinate system is
Rajendra M. 25 AgFE, IIT KGP

Finite Element Method (Beam 1D Element)
Beam Element: Element Characteristic Matrix Formulation
The stiffness and mass matrices of beam element are
Rajendra M. 26 AgFE, IIT KGP

Finite Element Method (Beam 1D Element)
Beam Element: Problem
A cantilever beam is fixed at one end and it has a
uniform cross-sectional area as shown in Fig. A
download load of P=1000 N applied at the free end.
The beam is made of aluminium and its properties are
given in Table.
Determine the deflection at centre of the beam using
FEA.
The second moment of area of the cross-sectional area about the z-axis is given as
Rajendra M. 27 AgFE, IIT KGP

Finite Element Method (Beam 1D Element)
Beam Element: Problem
Applying boundary conditions
Rajendra M. 28 AgFE, IIT KGP

Finite Element Method (Beam 1D Element)
Beam Element: Problem
Deflection and rotation at centre of the beam
( ) are
Solve the above problem, considering it as two element FEA model as Exercise
Rajendra M. 29 AgFE, IIT KGP

Finite Element Method (Frame 1D Element)
Frame Element: Characteristic Matrix Formulation (Three Dimensional Problem)
A frame element is formulated to model a straight bar of
an arbitrary cross-section, which can deform not only in
the axial direction but also in the directions perpendicular
to the axis of the bar. The bar is capable of carrying both
axial and transverse forces, as well as moments.
Therefore, a frame element is seen to posses the
properties of both truss and beam elements.
The frame element developed is also known in many
commercial software packages as the General beam
element or simply Beam element.
Planar frame element has 3 DOF at each node, hence it
has total 6 DOF.
Spatial frame element has 6 DOF at each node, hence it
has total 12 DOF.
Rajendra M. 30 AgFE, IIT KGP

Finite Element Method (Frame 1D Element)
Frame Element: Characteristic Matrix Formulation (Three Dimensional Problem)
Spatial frame element has 6 DOF at each node, hence it
has total 12 DOF.
Rajendra M. 31 AgFE, IIT KGP

Finite Element Method (Frame 1D Element)
Frame Element: Characteristic Matrix Formulation (Three Dimensional Problem)
Spatial frame element has 6 DOF at each node, hence it
has total 12 DOF. Based on global coordinate system
Rajendra M. 32 AgFE, IIT KGP

Finite Element Method (Triangular 2D Element)
2D Element: Characteristic Matrix Formulation (Three Dimensional Problem)
Linear Triangular element is the first type of element
developed for 2D solids. The formulation is also simplest
among all the 2D solid elements. It has been found that
the linear triangular element is less accurate compared to
linear quadrilateral elements. Triangular elements are
normally used to generate the mesh of a 2D model
involving complex geometry with acute corners.
It has 2 DOF at each node and this element has 3 nodes
and a total of 6 DOF.
Area Coordinates
Rajendra M. 33 AgFE, IIT KGP

Finite Element Method (Triangular 2D Element)
2D Element: Characteristic Matrix Formulation (Three Dimensional Problem)
Linear Triangular element
It has 2 DOF at each node and this element has 3 nodes
and a total of 6 DOF.
Eisenberg and Malvern (1973)
Rajendra M. 34 AgFE, IIT KGP

Finite Element Method (Rectangular 2D Element)
2D Element: Characteristic Matrix Formulation (Three Dimensional Problem)
Linear Rectangular element
Triangular elements are less accurate
than rectangular elements, but they are
preferred due to meshing problem for
complex geometry. Its shape function
formulation is easy compared to
triangular element.
Rectangular element has 2 DOF at each node and it has four nodes, hence the element
has a total of 8 DOF. The dimensions of the element is defined here as 2a×2b×h. A local
natural coordinate system (ξ, η) with its origin located at the centre of the rectangular
element is defined.
Rajendra M. 35 AgFE, IIT KGP

Finite Element Method (Rectangular 2D Element)
2D Element: Characteristic Matrix Formulation (Three Dimensional Problem)
Linear Rectangular element
The stiffness and mass matrices of
Rectangular element are
Gauss Integration
Rajendra M. 36 AgFE, IIT KGP

Finite Element Method (Quadrilateral 2D Element)
2D Element: Characteristic Matrix Formulation (Three Dimensional Problem)
Linear Quadrilateral Element
Quadrilateral element is more practical and
useful element than rectangular and triangular
elements because of its unparalleled edges.
The Quadrilateral element has four nodes with
2 DOF at each node and a total of 8 DOF.
However, there can be a problem for the integration of the mass and stiffness matrices
for a quadrilateral element, because of the irregular shape of the integration domain. The
Gauss integration scheme cannot be implemented directly with quadrilateral elements.
Hence, key in the development of a quadrilateral element is the coordinate mapping from
irregular shape of local coordinate system to square shape of natural coordinate system.
Rajendra M. 37 AgFE, IIT KGP

Finite Element Method (Quadrilateral 2D Element)
2D Element: Characteristic Matrix Formulation (Three Dimensional Problem)
Linear Quadrilateral Element
The stiffness and mass matrices are
J is the Jacobian matrix
Rajendra M. 38 AgFE, IIT KGP

Finite Element Method (Hexahedron 3D Element)
3D Element: Characteristic Matrix Formulation (Three Dimensional Problem)
Hexahedron Element
Hexahedron element is a 3D element with
eight nodes and six surfaces. It has 3 DOF
for each node and the element has a total
of 24 DOF.
It is again useful to define a natural
coordinate system (ξ, η, ζ) with its origin
at the centre of its transformed cube, as
this makes it easier to construct the shape
functions and to evaluate the matrix
integration. The characteristic matrix
formulation is similar to that of
Quadrilateral element.
J is the Jacobian matrix
Rajendra M. 39 AgFE, IIT KGP

Finite Element Method (Hexahedron 3D Element)
3D Element: Characteristic Matrix Formulation (Three Dimensional Problem)
Hexahedron Element
Hexahedron element is a 3D element with
eight nodes and six surfaces. It has 3 DOF
for each node and the element has a total
of 24 DOF.
J is the Jacobian matrix
Rajendra M. 40 AgFE, IIT KGP

Finite Element Method (Hexahedron 3D Element)
3D Element: Characteristic Matrix Formulation (Three Dimensional Problem)
Hexahedron Element
The stiffness and mass matrices are
Rajendra M. 41 AgFE, IIT KGP

Finite Element Method (References)
1. Robert D. Cook. (1995). Finite Element Modeling for Stress Analysis. First Edition
John Wiley & Sons, Inc.
2. Robert D. Cook, David S. Malkus and Michael E. Plesha. (1989). Concepts and
Applications of Finite Element Analysis. Third Edition. John Wiley & Sons, Inc.
3. Singiresu S. Rao. (2004). The Finite Element Method in Engineering. Fourth Edition.
Elsevier Science & Technology Books.
4. Larry J. Segerlind. (1984). Applied Finite Element Analysis. Second Edition. John
Wiley & Sons, Inc.
5. G. L. Narasaiah. (2008). Finite Element Analysis. First Edition. B. S. Publications.
6. G. R. Liu and S. S. Quek. (2003). The Finite Element Method – A Practical Course.
First Edition. Elsevier Science Ltd.
7. J. N. Reddy. (2006). An Introduction to the Finite Element Method. Third Edition.
McGraw Hill, Inc.
Thank You
Rajendra M. 42 AgFE, IIT KGP