TME 513-Theory of Elasticity
Module 1 new:3D and 2D Elasticity
Professor O. Oluwole
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This material is a compilation of materials from
various sources and it is compiled not as an original
material but for a comprehensive aid to be used in
conjunction with our primary (recommended) and
secondary texts.
Effort has been made to reference and state
sources from where materials were obtained.
I hereby acknowledge these sources and thank
them for making their resources available for quick
use on the internet.
Professor O. Oluwole
Mechanical Engineering Department
University of Ibadan
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Course content in modules
• Module 1. Applications of TOE to 2 & 3D
problems
• Module 2. Experimental Stress Analysis
• Module 3. Electrical Resistance Strain Gauging
• Module 4. Approximate Methods
• Module 5. Finite Element Method
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Texts and Tools
• Helena .Theory of elasticity and plasticity.
• Ramadas Chennamsetti. Theory of 2D
elasticity
• Handbook on Experimental Mechanics,’’
edited by A. S. Kobayashi
• MATLAB®, Abaqus, SOLIDWORKS, FUSION.
• OTHER USEFUL TEXTS
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objectives
• Student will be able to use theory of elasticity
equations to solve 2D and 3D engineering
problems.
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What we need to know
• There are all types of materials that are available
to the engineer and they have different
properties
• The Mechanical Engineer has to consult the
Materials Engineer as to the material to choose
for desired applications
• There are monoclinic, orthotropic, isotropic,
transversely orthotropic and anisotropic
materials and they present different challenges in
service.
• Materials which Mechanical Engineers normally
deal with are isotropic e.g. Metals/ alloys
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Derivation of governing equations for
Elasticity problems
• 3-D Elasticity governing equations consist of
1. strain-displacement equations
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coord
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Spherical coord
https://en.wikipedia.org/wiki/Linear_elasticity
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Comments(contd)
2. Strain Compatibility equations
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Cylinderical –polar coord
http://maeresearch.ucsd.edu/~vlubarda/research/pdf
papers/MMS20.pdf
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Spherical polar coords
Assignment: Get the [6 ]strain compatibility equations
for spherical coordinates
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Comments(contd)
3. Equilibrium Equations
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Comments(contd)
4. Constitutive Equations or Law
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Cartesian Coord
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Cylinderical-polar coords
https://www.brown.edu/Departments/Engineering/Courses/En221/Notes/Polar_Coords/Polar_C
oords.htm
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Spherical –polar coords
https://www.brown.edu/Departments/Engineering/Courses/En221/Notes/Polar_Coords/Polar_C
oords.htm
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Comments(contd)
• With these 3-D equations, it is difficult to
solve for 15 unknowns with 15 equations
(15 equations=3 equilibrium+ 6 straindispalcement+6 constitutive)
(15 unknowns=3 displacements, 6 stress
components, 6 strain components)
What we do next to simplify by recasting the
problem for either primarily displacement
seeking or stress seeking.
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Ref. Ramadas
Comments(contd)
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Ref. Ramadas
Comments(contd)
3-D Displacement –Based
Formulation
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Ref. Ramadas
3-D Displacement –Based Formulation(contd.)
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Ref. Ramadas
3-D Displacement –Based Formulation(contd.)
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Ref. Ramadas
3-D Displacement –Based Formulation(contd.)
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Ref. Ramadas
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3-D Stress –Based Formulation
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Ref. Ramadas
3-D Stress –Based Formulation(contd.)
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Ref. Ramadas
3-D Stress –Based Formulation(contd.)
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Ref. Ramadas
3-D Stress –Based Formulation(contd.)
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Ref. Ramadas
Six Beltami-Mitchel Equations for 3D Elasto-Statics. Body forces =0
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Beltrami-Mitchel Spherical coords
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Further Problems encountered in 3-D
analysis and Solution.
• We have been able to break down the analyses
into smaller units using displacement and stress
based formulations but we still have problems
even with the reduction in number of equations
and unknowns.
• What we do is to break down further the
problem into whether it is a plane strain or plane
stress problem. With this approach, a 3-D
problem is broken down to a 2-D problem which
is easier to solve.
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Plane strain
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Ref. Ramadas
Plane strain (contd):
Displacement based
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Ref. Ramadas
Plane strain (contd)Displacement-based formulation
Note that
=G
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Ref. Ramadas
Plane strain (contd)
Displacement –based formulation
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Ref. Ramadas
Plane strain: Stress-based formulation
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Ref. Ramadas
Plane strain(contd):
Stress –based formulation
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Ref. Ramadas
Plane strain
Stress –based formulation(contd):
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Ref. Ramadas
Plane strain
Stress –based formulation(contd):
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Ref. Ramadas
Plane stress
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Ref. Ramadas
Plane stress formulations
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Ref. Ramadas
Plane stressDisplacement-based approach
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Ref. Ramadas
Plane stressDisplacement-based approach(contd.)
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Ref. Ramadas
Plane stressstress-based approach
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Ref. Ramadas
Plane stressstress-based approach(contd)
Ref. Ramadas
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Plane stressstress-based approach(contd)
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Ref. Ramadas
Governing equations in solving
elasticity problems
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Ref. Ramadas
Reference
• Ramadas Chennamsetti. Theory of 2D
elasticity
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Principle of Superposition
See picturesque next page
Martin Saad.
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Martin Saad.
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Saint-Venant’s Principle
• The stress, strain, and displacement fields
caused by two different statically equivalent
force distributions on parts of the body far
away from the loading points are
approximately the same.
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General Solution Strategies
ANALYTICAL SOLUTION METHODS
• Displacement formulation method: useful for simple geometries that have symmetry e.g
spherical and cylinderical discs.
• Stress formulation method: 3D analytical solutions are not possible. Different stress
function formulations are used for the solutions of different problems and Potential
function formulations.
• Direct integration(This method normally encounters significant mathematical difficulties,
thus limiting its application to problems with simple geometry e.g. Stretching of Prismatic
Bar Under Its own Weight). This can better be solved with semi-inverse method.
• Inverse(Using this scheme it is sometimes difficult to construct solutions to a specific
problem of practical interest.)
• semi-inverse, starting from a known field, stresses and strains can be calculated which
must conform to equilibrium and strain compatibility equations
• exact analytical solutions -useful in 2D analysis .(Power Series Method, Fourier Method,
Integral Transform Method, Complex Variable Method-)
APPROXIMATE SOLUTION SCHEMES (Variational methods e.g.Ritz Method). Because of the
difficulty in finding proper approximating functions for problems of complex geometry,
variational techniques have made only limited contributions to the solution of general
problems.
NUMERICAL SOLUTION METHODS (Finite Difference Method, Finite Element Method,
Boundary Element Method) The FEM solution technique is probably the best in this category.
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Hadi Sadati
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Hadi Sadati
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Hadi Sadati
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Transformation equations from cartesian to
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Navier Equations
Hadi Sadati
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Kinematic equations
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Sadati
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Sadati
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Sadati
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Sadati
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Sadati
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Sadati
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Sadati
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Sadati
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Sadati
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Example question on pressurised hollow sphere
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Solution
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At r=ri, uz = 0
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Sadati
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SEMI-INVERSE METHOD
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Prismatic Bar hanging under body weight
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Example 2: cube under uniaxial loading
Calculate the strains and displacements for any
point in the linear elastic cube. Use the semiinverse method of solution. Take Body forces
negligible.
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Cube under multiaxial loading
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and circular disc/
Cylinder)
Sadati
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Direct Integration Method
• This method seeks to determine the solution
by direct integration of the field equations
(5.1.5) or equivalently the stress and/or
displacement formulations given in Figure 5-5.
• Boundary conditions are to be satisfied
exactly. This method normally encounters
significant mathematical difficulties, thus
limiting its application to problems with
simple geometry.
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Direct Integration Example
Stretching of Prismatic Bar Under Its Own Weight
• As an example of a simple direct integration problem, consider the
case of a uniform prismatic bar stretched by its own weight, as
shown in Figure 5-9. The body forces for
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Inverse Method
• For this technique, particular displacements or stresses
are selected that satisfy the basic field equations.
• A search is then conducted to identify a specific problem
that would be solved by this solution field. This amounts
to determine appropriate problem geometry, boundary
conditions, and body forces that would enable the
solution to satisfy all conditions on the problem.
• Using this scheme it is sometimes difficult to construct
solutions to a specific problem of practical interest
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EXAMPLE Inverse Example: Pure Beam Bending
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Analytical solution methods
1.Power Series Method (Airy’s stress function):
useful for 2D problems using stress formulation and
stress functions. The method is based on the inverse
solution concept where we assume a form of the
solution to the biharmonic equation
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EXAMPLE : Uniaxial Tension of a Beam
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2. Fourier Method
• A general scheme to solve a large variety of
elasticity problems employs the Fourier
method.
• This procedure is normally applied to the
governing partial differential equations by
using separation of variables, superposition,
and Fourier series or Fourier integral theory.
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3. Integral Transform Method
• A very useful mathematical technique to solve
partial differential equations is the use of
integral transforms
• Typical transforms that have been successfully
applied to elasticity problems include Laplace,
Fourier, and Hankel transforms
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4. Complex Variable Method
• Several classes of problems in elasticity can be
formulated in terms of functions of a complex
variable
• Useful for problems that would be intractable
by other techniques e.g. plane isotropy and
anisotropy problems
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Approximate Solution Procedures
• With the recognized difficulty in finding exact analytical
solutions, considerable work has been done to develop
approximate solutions to elasticity problems using
variational methods, which are related to energy theorems
e.g. Ritz Method
• Because of the difficulty in finding proper approximating
functions for problems of complex geometry, variational
techniques have made only limited contributions to the
solution of general problems.
• However, they have made very important applications in the
finite element method.
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Numerical Solution Procedures
• Over the past several decades numerical
methods have played a primary role in
developing solutions to elasticity problems of
complex geometry e.g. Finite Difference
Method, Finite Element Method and the
Boundary Element Method
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Numerical solutions using FDM
Find the stresses at nodes 2, 5 and 8. Plane stress conditions apply132
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2
Find the displacements at nodes 2, 5 and 8. Plane stress conditions
apply
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• At Nodes 2 and 8 there will be displacement in
the y direction according to the equation
• 𝜀𝑦 =- 𝑣𝜀𝑥
10−7
• Thus, Total 𝜀𝑥 =
; 𝑇𝑜𝑡𝑎𝑙 𝜀𝑦 = -0.3 (10−7 )
1
• 𝑉2 = -0.15(10−7 )m and 𝑉8 = 0.15(10−7 )m
• 𝑉6 = 0
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Assignment s: FDM
1
2
Find the displacements at nodes 2, 5 and 8. Plane stress conditions apply
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• Numerical solutions using FEM
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Example 2D FEM ELASTICITY
CA
(1) A rectangular shaped thin plate 2 mm thick in a component (Figure 1) is fixed on one
side and subjected to uniaxial traction on the unfixed x-axis.
Solve for unknown displacements and stresses on the plate using the Gallerkin FEM.
Compare your results with MATLAB results.
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2 A rectangular shaped thin plate 2 mm thick in a component (Figure 2) is fixed on
one side and subjected to uniaxial traction on the unfixed x-axis.
Solve for unknown displacements and stresses on the plate using the Gallerkin
FEM. Compare your results with MATLAB results
N /m2
2
10
0
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USEFUL EQUATIONS
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