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ADVANCED LEARNING IN ELASTICITY THEORY BY PHOTOELASTICITY-BASED EXPERIMENTAL TECHNIQUES E. Vegueria, R. Ansola, J. Santamaria, G. Urbikain, J. Canales 1

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INDEX 2 Introduction Limitations of the analytical solution Photoelasticity Solving a real problem Finite Element Method Photoelasticity technique Examples Conclusions

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EUROPEAN HIGHER EDUCATION SPACE INTRODUCTION 3 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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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 of the practical exercises explained in class 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. INTRODUCTION 4

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“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. INTRODUCTION 5

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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. INTRODUCTION 6

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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 stresses at that point. LIMITATIONS OF THE ANALYTICAL SOLUTION 7

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The technique is used to: Identify instantaneously critical areas, overstressed or substressed regions 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 8 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.

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9 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 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 x - y = K´ * N

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10 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 time to perform the exercise is an hour and a half. 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.

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11 FINITE ELEMENT METHOD 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” Students must follow the steps on the pre-processing to obtain the solution in the post-processing. SOLVING A REAL PROBLEM

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12 PHOTOELASTICITY TECHNIQUE Experimental equipment: transmission polariscope FL200 (Gunt©) Light source Filters Loading frame Stress-Opticon device Probe of photoelastic material PSM-1 SOLVING A REAL PROBLEM

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13 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 in which the value of K´ in this case is 1.1 MPa/fringe. x - y = K´ * N SOLVING A REAL PROBLEM

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14 1. Beam with constant bending moment PP a=27 mm B=73 mm a=27 mm h1=18.5 mm P.a Mz SOLVING A REAL PROBLEM EXAMPLES

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15 Analytical result from the Finite Element Analysis ( x - y ) Experimental result from the Photoelasticity equipment ( x - y ) SOLVING A REAL PROBLEM

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16 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. SOLVING A REAL PROBLEM

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17 2. Plate with a hole supporting distributed compression D=5 mm 18 mm q Analytical result from the Finite Element Analysis ( x - y ) Experimental result from the Photoelasticity equipment ( x - y ) * The stress field obtained by FEM is a butterfly shaped figure and the fringes obtained experimentally present a similar form. * It is easy to identify areas with high stress in both figures, which is more difficult to study analytically. Conclusions: SOLVING A REAL PROBLEM

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CONCLUSIONS 18 * 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 problem- solving competences and promoting the teamwork and the cooperativity among students.

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19 ADVANCED LEARNING IN ELASTICITY THEORY BY PHOTOELASTICITY-BASED EXPERIMENTAL TECHNIQUES E. Vegueria, R. Ansola, J. Santamaria, G. Urbikain, J. Canales

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