Transcript solving a real problem

ADVANCED
LEARNING
IN
ELASTICITY
THEORY
BY
PHOTOELASTICITY-BASED EXPERIMENTAL TECHNIQUES
E. Vegueria , R. Ansola , J. Santamaria , G. Urbikain , J. Canales
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INDEX
 Introduction
 Limitations of the analytical solution
 Photoelasticity
 Solving a real problem
 Finite Element Method
 Photoelasticity technique
 Examples
 Conclusions
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INTRODUCTION
 EUROPEAN HIGHER EDUCATION SPACE
 Student in the centre of teaching-learning process
 Independent learning  creation of the opportunities and
experiences necessary for students to become capable, self-reliant,
self-motivated and life-long learners
 Lecturer is becoming the person who provides assistance to
guide the process of construction of knowledge  making
students to reflect, to ask questions, to relate new information, etc...
 Introduction of innovations in the lectures to obtain more
dynamism, connection and involvement throughout the group of
students  promote introspective and independent learning.
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INTRODUCTION
 OBJECTIVE
 “Elasticity and Stregth of Materials”
 Experimental seminar that we have introduce in the course of the
classes to enhance learning by creating a space for reflection
 Students
 Reproduce physically some
explained in class
of
the
practical
exercises
 Learn how to handle Photoelasticity experimental equipment
 Compare theoretical and experimental results getting some
conclusions.
 Overwise the concepts given in class by an another point of
view and are driven to think about their validity.
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INTRODUCTION
 “ELASTICITY AND STRENGTH OF MATERIALS”
 This subject provides the fundamental knowledge to deal with
important fields in mechanical engineering like Structural Analisys
and Machine Design.
 Students are introduced to the behavior of deformable solids under a
general case of loads.
 The first part of the course provides a presentation of the fundamental
concepts of the Theory of Elasticity, and in the second part the program
focuses on the analysis and design of structural members subjected to
axial forces, shear, bending and torsion.
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INTRODUCTION
 Due to the educational experience obtained over the years it has
been concluded that the most troublesome part of the course,
regarding the learning for students, is the first one, Theory of
Elasticity.
 This part discusses the theoretical foundations to obtain the stress
and strain fields in an elastic solid subjected to a load system
without taking into account simplifications in geometry or loads.
 Since any type of geometry and loading can be considered, the
theoretical study becomes difficult, abstract and unintuitive.
 In order to complement and make the learning process lighter, we
introduce Photoelasticity experimental techniques as a tool for the
acquisition of knowledge.
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LIMITATIONS OF THE ANALYTICAL
SOLUTION
 Through the resolution of elasticity theoretical equations it is
possible to obtain the strain and stress fields.
 This problem becomes difficult when the geometry of the solid or
the load conditions are complex.
 In those cases there are other methods to obtain the results:
 Numerical methods: Among them  Finite Element Method
(FEM), which is based on the discretization of a problem into small
elements that can be solved in relation to each other
 Experimental methods: Among them  Photoelasticity, upon
the application of stresses, photoelastic materials exhibit the
property of birefringence and the magnitude of the refractive indices
at each point in the material is directly related to the state of
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stresses at that point.
PHOTOELASTICITY
 When the loads are applied to the solid
simultaneously it is illuminated with polarized
light from a polariscope. In the view through
the polariscope the stress field is shown in
color, revealing the full distribution in the solid
and making possible to distinguish areas with
high stresses.
 The technique is used to:
 Identify instantaneously
substressed regions
critical
areas,
overstressed
or
 Measure the stress concentration around holes, chamfers, etc
 Measure principal stresses and principal directions
 Identify and measure assembly and residual stresses
 Observe the strain redistribution in the plastic behavior of the
material
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PHOTOELASTICITY
 In this work the testing solids are made of birefringent material.
Birefringence is a property whereby the loaded solid, when is
illuminated by a light beam, is able to separate it into two orthogonal
components and to transmit them at different speeds.
 During the practice, students, with the assistance of the professor,
utilize a tool called polariscope to take advantage of the property of
birefringent material and finally observe the stress field.
 The difference in principal stresses can be calculated with the
equation
x - y = K´ * N
K´  parameter that depends on the test part material, geometry (thickness) and
the monochromatic light used.
N  fringe order, an integer that can be observed experimentally
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SOLVING A REAL PROBLEM
 Students are challenged to solve different simple problems by means
of two different ways: Analytically (FEM analysis or Theory of
Elasticity) and using Photoelasticity techniques.
 Students
must
study
different
representative exercises submitted to a state
of plane stress.
 The work is collaborative because the practice is made in small
groups of three or four members, solving each example experimentally
and analytically and discussing about the similarities, differences, etc.
 The time to perform the exercise is an hour and a half.
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SOLVING A REAL PROBLEM
 FINITE ELEMENT METHOD
Students must follow the steps on
the pre-processing to obtain the
solution in the post-processing.
 General input data for the exercises :
 Elastic properties of the birefringent material:
 Young modulus: E = 2.4 GPa
 Poisson coefficient: ν = 0.38
Meshing:
 Element type = solid quadrilateral plane element 183 (8 nodes)
 Loads:
 Example 1: Beam with constant bending moment
 Imposed displacement at the point of the application of loads (∆y = - 0.5 mm)
 Example 2: Plate with hole
 A distributed compression load “q”
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SOLVING A REAL PROBLEM
 PHOTOELASTICITY TECHNIQUE
Experimental equipment: transmission polariscope FL200 (Gunt©)
Light source
Filters
Loading frame
Probe of photoelastic material PSM-1
Stress-Opticon device
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SOLVING A REAL PROBLEM
Experimental procedure:
Students must follow a few steps to obtain the experimental field of
stresses in the part.
1.- Light sodium lamp using monochromatic light (needs heating time
to account for a unique wavelength).
2.- Circular polariscope setup to get the status of circularly polarized
light.
3.- The model is placed under the desired loads and restrictions.
4.- Count isocromatic fringes: The centres of the light beams can be
located and numbered consecutively. The dark fringes are listed as:
N = 0, 1, 2, etc.
5.- Once the order is observed at the point of study, the stress
intensity is obtained using equation
x - y = K´ * N
in which the value of K´ in this case is 1.1 MPa/fringe.
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SOLVING A REAL PROBLEM
 EXAMPLES
1. Beam with constant bending moment
P
P
h1=18.5 mm
B=73 mm
a=27 mm
a=27 mm
Mz
P.a
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SOLVING A REAL PROBLEM
Analytical result from the Finite Element Analysis (x- y)
Experimental result from the Photoelasticity equipment (x- y)
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SOLVING A REAL PROBLEM
Conclusions:
* The photography obtained by the FL200 confirms the tendency shown by
the Finite Element model. The most critical zones are found below the loads
and close to the supports, showing relative good agreement in
quantitative values.
* Students calculate the theoretical and the experimental value for the
stress measured in the upper zone of the beam in the central section.
* In this step the teacher gives time to think about the differences
between experimental and theoretical values and to discuss the different
factors that have influence on this, like the material imperfections, real value
of the applied load, conditions, precision, sensitivity and resolution of the
measurement device, hypothesis made in the calculations, etc.
* The initial reaction of the students when they obtain the value of the
errors is varied. Some of them think that the error is acceptable and others
get surprised about not getting experimentally just the theoretical value.
* This is because they are used to solve numerical exercises without
thinking if the theoretical approach of the problem matches up with the
reality. In this moment they are driven to reflect about the experimental
error sources and realized that a real problem has many inherent
errors and imperfections very difficult to control.
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SOLVING A REAL PROBLEM
2. Plate with a hole supporting distributed compression
q
Analytical result from the Finite
Element Analysis (x- y)
D=5 mm
18 mm
18 mm
Experimental result from the
Photoelasticity equipment (x- y)
Conclusions:
* The stress field obtained by FEM is a butterfly
shaped figure and the fringes
experimentally present a similar form.
obtained
* It is easy to identify areas with high stress in
both figures, which is more difficult to study
analytically.
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CONCLUSIONS
* Over the years of teaching it has been noticed that students have
more difficulties to understand and learn the Theory of Elasticity,
which is more abstract and less intuitive.
* In order to enhance the curriculum development of students and
complement the theoretical basis explained in the lectures, we will use a
Photoelasticity device which is available to the students during the
workshop practices of “Elasticity and Strength of Materials”.
* Students reproduce physically some of the problems analytically
raised and solved in class, and they can see the experimental stress
and strain fields of the solid through different colors, which is more
visual and attractive.
* Furthermore, this practice is conceived as a multidisciplinary task
since it is offered the students the opportunity to resolve the problem by
Finite Element Analysis, which is addressed in subsequent courses.
* The students are divided in different groups to solve analytically
and experimentally different case problems and they gain new skills and
abilities in the field of Mechanical Engineering, improving problemsolving competences and promoting the teamwork and the
cooperativity among students.
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ADVANCED
LEARNING
IN
ELASTICITY
THEORY
BY
PHOTOELASTICITY-BASED EXPERIMENTAL TECHNIQUES
E. Vegueria , R. Ansola , J. Santamaria , G. Urbikain , J. Canales
19