Solid Mechanics Lab Manual
Compilation Date: January 7, 2023
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Solid Mechanics Lab Manual, January 7, 2023
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Foreword
This manual has been compiled for use by the students enrolled in the Solid Mechanics Lab (MNG
303) at the University of Kentucky. It serves as a guidance document on how to prepare for the lab
assignments, how to collect data and what to include in lab reports. It should not be viewed as a
substitute for ASTM or other guidelines for conducting experiments in Solid Mechanics.
A lot of the information in this document comes from laboratory manuals published by Mea­
surements Group Inc. (MGI) and from notes and slides by Dr. G.T. Lineberry and Dr. Thomas
Novak, professors emeriti, at the Department of Mining Engineering at the University of Ken­
tucky. Per notice of copying license from MGI, license is granted to educational institutions for
unlimited copying of the material in the MGI manuals for distribution and use by the students in
such institutions. The MGI manuals are referenced in the respective chapters.
I would like to thank the following (former) undergraduate and graduate students at the De­
partment of Mining Engineering at the University of Kentucky who helped develop this updated
manual: Cristian Cardenas, Jesus Castillo, Caroline Gerwig, Robin Flattery, Juan Diaz Martinez
and Garett Walton.
This manual will be updated periodically. The date below reflects the latest compilation date.
January 7, 2023
Zach Agioutantis
Professor and Chair
Department of Mining Engineering
University of Kentucky
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Contents
1
Introduction to error analysis
1
1.1 Background and theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Archery: A useful analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Types of measurement error and quantification of the magnitude of error . . . . . . . 5
1.3.1 The true value of a variable . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.2 Precision error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.3 Bias error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.4 Total error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.5 Blunder error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.6 Relative error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Impact of measurement error on predicted values . . . . . . . . . . . . . . . . . . 14
1.5 Lab experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5.1 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5.2 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.6 Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.7 Measurement worksheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2 Modulus of elasticity and Poisson’s ratio
2.1 General introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Poisson’s ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Measuring strain ­ strain gauges . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Principle of operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Strain sensitivity and gauge factor . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Transverse sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.4 Correcting for transverse sensitivity . . . . . . . . . . . . . . . . . . . . .
2.3.5 Measuring strain – Wheatstone bridge . . . . . . . . . . . . . . . . . . . .
2.3.6 Measuring strain – strain indicator . . . . . . . . . . . . . . . . . . . . . .
2.4 Modulus of elasticity: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Modulus of elasticity: Equipment and supplies . . . . . . . . . . . . . . . . . . .
2.6 Modulus of elasticity: Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.2 Strain gauge selection and installation . . . . . . . . . . . . . . . . . . . .
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2.6.3 Alternate procedure with two strain gauges . . . . . . . . . . . . . . . . .
2.6.4 Data acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.5 Data analysis and presentation . . . . . . . . . . . . . . . . . . . . . . . .
2.6.6 Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Modulus of elasticity: Worksheet . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.1 Beam dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.2 Computation of maximum load for 15,000 psi stress . . . . . . . . . . . .
2.7.3 Tabulation of loads, strains, and stresses . . . . . . . . . . . . . . . . . . .
2.7.4 Computation of Young’s modulus . . . . . . . . . . . . . . . . . . . . . .
2.8 Poisson’s ratio: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9 Poisson’s Ratio: Equipment and supplies . . . . . . . . . . . . . . . . . . . . . . .
2.10 Poisson’s Ratio: Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.10.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.10.2 Strain Gauge selection and installation . . . . . . . . . . . . . . . . . . . .
2.10.3 Data acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.10.4 Data analysis and presentation . . . . . . . . . . . . . . . . . . . . . . . .
2.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.12 Poisson’s ratio: Worksheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.12.1 Strain measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.12.2 Correction of lateral strain for transverse sensitivity . . . . . . . . . . . . .
2.12.3 Calculation of Poisson’s ratio . . . . . . . . . . . . . . . . . . . . . . . .
2.12.4 Published values of Poisson’s ratio . . . . . . . . . . . . . . . . . . . . . .
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The tensile test
3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7 Formulas and other assorted useful information . . . . . . . . . . . . . . . . . . .
3.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.9 Tensile test worksheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4 Experiments in shear failure
4.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Axial and shear stresses . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Axial stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3 Shear stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Shear pin test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Punch test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.5
4.6
4.7
4.8
4.9
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Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Formulas and other assorted useful information . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Worksheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Neutral axis of a T­beam
5.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6 Formulas and other assorted useful information . . . . . . . . . . . . . . . . . . .
5.7 Neutral axis test worksheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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6 Cantilever flexure: flexure stresses in beams
6.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Equipment and supplies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5 General procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6 Detailed instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6.1 Determination and comparison of stress values . . . . . . . . . . . . . . .
6.6.2 Determination of deflection . . . . . . . . . . . . . . . . . . . . . . . . .
6.7 Strain gauge selection and installation . . . . . . . . . . . . . . . . . . . . . . . .
6.8 Data acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.9 Data analysis and presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.9.1 Shear force from differential strains . . . . . . . . . . . . . . . . . . . . .
6.9.2 Strain and moment linearity . . . . . . . . . . . . . . . . . . . . . . . . .
6.9.3 Stress at station (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.9.4 Deflection measurement . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.10 Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.11 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.12 Worksheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.12.1 Beam dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.12.2 Gauge locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.12.3 Differential strain measurements . . . . . . . . . . . . . . . . . . . . . . .
6.12.4 Computation of load . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.12.5 Individual strain measurements . . . . . . . . . . . . . . . . . . . . . . .
6.12.6 Slope of best straight line through individual strain readings . . . . . . . .
6.12.7 Stress at station (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5
7 Introduction to stress concentrations
89
7.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
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7.2
Experiment on stress and strain concentration . . . . . . . . . . . . . . . . . . . . 92
7.2.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.3.1 Equipment and supplies . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.3.2 Procedure summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.3.3 General setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.4 Strain gauge selection and installation . . . . . . . . . . . . . . . . . . . . . . . . 96
7.5 Data acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7.6 Data analysis and presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.7 Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.9 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.10 Worksheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.10.1 Strain measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.10.2 Correction of ϵ4 for gauge factor . . . . . . . . . . . . . . . . . . . . . . . 102
7.10.3 Computation of extrapolation equation coefficients . . . . . . . . . . . . . 102
7.10.4 Maximum strain at edge of hole, ϵo . . . . . . . . . . . . . . . . . . . . . 102
7.10.5 Stress concentration factor, Kt . . . . . . . . . . . . . . . . . . . . . . . . 102
8
Determination of principal strains and stresses utilizing a strain gauge rosette
103
8.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
8.1.1 Mohr’s circle of stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
8.1.2 Step­by­step procedure for constructing Mohr’s circle . . . . . . . . . . . 104
8.2 Experiment on principal strains and stresses . . . . . . . . . . . . . . . . . . . . . 108
8.2.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
8.2.2 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
8.2.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
8.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
8.4 Equipment and supplies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
8.5 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
8.5.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
8.5.2 Strain gauge selection and installation . . . . . . . . . . . . . . . . . . . . 111
8.5.3 Data acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
8.5.4 Data analysis and presentation . . . . . . . . . . . . . . . . . . . . . . . . 113
8.5.5 Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
8.6 Worksheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
8.6.1 Beam dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
8.6.2 Computation of load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
8.6.3 Strain measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
8.6.4 Computation of strain measurements . . . . . . . . . . . . . . . . . . . . . 115
8.6.5 Computation of Poisson’s ratio . . . . . . . . . . . . . . . . . . . . . . . . 116
8.6.6 Computation of principal stresses . . . . . . . . . . . . . . . . . . . . . . 116
8.6.7 Computation of maximum principal stress from flexure formula . . . . . . 116
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8.6.9
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Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Supplementary exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Determination of the modulus of elasticity for four metallic and one non­metallic
materials
119
9.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
9.2 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
9.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
9.2.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
9.3 Worksheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
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List of Figures
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Archery practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Bias error vectors in archery practice . . . . . . . . . . . . . . . . . . . . . . . . . 4
Bias and precision in archery practice . . . . . . . . . . . . . . . . . . . . . . . . 4
Histogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Under normal distributions, 99% of the values lie within 2 standard deviations
from the mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
True value and best estimate of true value . . . . . . . . . . . . . . . . . . . . . 10
The concept of total error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Linearization of the function in the neighborhood of the measured values . . . . . 15
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
The Poisson effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Principle of strain measurements . . . . . . . . . . . . . . . . . . . . . . . . . .
Correcting for transverse sensitivity . . . . . . . . . . . . . . . . . . . . . . . .
Wheatstone bridge circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Typical stress ­ strain curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Model P3 Strain Indicator and Recorder . . . . . . . . . . . . . . . . . . . . . .
Flexor ­ cantilever flexure frame . . . . . . . . . . . . . . . . . . . . . . . . . .
Beam loaded in flexure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wiring diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Stress ­ strain chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cantilever beam for measuring Poisson’s ratio . . . . . . . . . . . . . . . . . . .
Wiring diagram for Poisson’s ratio . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
3.2
The tensile test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Tensile test setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
Normal and shear stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Punch test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Shear pin test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Punch test (left) and shear pin test (right) setup . . . . . . . . . . . . . . . . . .
Cantilever beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Crane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Rivet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Loading modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bridge structure subjected to bending . . . . . . . . . . . . . . . . . . . . . . . .
xi
21
22
24
25
26
27
28
29
32
33
36
38
52
53
53
55
59
59
59
60
60
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5.2
5.3
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5.4
5.5
5.6
5.7
Cross­section of the T­beam . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Neutral axis of a rectangular beam . . . . . . . . . . . . . . . . . . . . . . . . .
(a) Distribution of normal stress across the section. (b) Bending moment act­
ing on the cross­section. (c) Two­dimensional representation of stress distri­
bution and bending moment. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Location of the neutral axis of a T­beam (calculated based on T­beam geometry) .
Location of the neutral axis of a T­beam (based upon strain measurements) . . . .
Location of the neutral axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Force vs Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
68
69
70
71
6.1
6.2
6.3
6.4
6.5
6.6
6.7
Cantilever beam ­ moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cantilever beam ­ Shear force moments . . . . . . . . . . . . . . . . . . . . . .
Bending moment distribution on cantilever beam . . . . . . . . . . . . . . . . .
Strain gauge location on beam . . . . . . . . . . . . . . . . . . . . . . . . . . .
Beam deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wiring diagram 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wiring diagram 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
74
76
78
80
84
85
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
A nonuniform stress distribution with zero resultant axial force . . . . . . . . . . 89
Two plates under uniform tensile load . . . . . . . . . . . . . . . . . . . . . . . 90
Stress distributions on various sections of the two plates . . . . . . . . . . . . . . 90
Stress components around a small hole . . . . . . . . . . . . . . . . . . . . . . . 91
Stress distribution in a plate, with a small circular hole, under tension . . . . . . . 92
Stress concentration factor for uniaxial load . . . . . . . . . . . . . . . . . . . . 93
Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Wiring diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Graph sheet ­ strain distribution, cantilever beam with hole . . . . . . . . . . . . 101
8.1
8.2
8.3
8.4
8.5
8.6
2­D stress and Mohr’s circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Determination of the principal stresses by using Mohr’s circle . . . . . . . . . . 105
Transformation of the principal stresses into the maximum shearing stresses
and the associated normal stresses . . . . . . . . . . . . . . . . . . . . . . . . . 106
Stresses on Mohr’s circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Polar plot of the normal stress and strain at a point in a uniaxial stress field . . . . 109
Strain rosettes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
9.1
9.2
Apparatus / Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Beam deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
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64
66
List of Tables
9.1
9.2
9.3
Data Set 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Data Set 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Data Set 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
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Chapter 1
Introduction to error analysis
1.1
Background and theory
1.1.1
Introduction
Consider a physical quantity such as the length of a pencil. We imagine that its length has some
exact value. You might measure it with a steel rule and observe the result 195 mm. Subsequently,
you might measure it with a pair of vernier calipers and observe 194.87 mm. The calipers are “more
sensitive” than the ruler, giving a reading to the nearest 0.01 mm versus a reading to the nearest
1 mm. You might measure the length again with the calipers and observe 194.84 mm and a friend
might measure and observe 194.81 mm. Clearly these values differ and thus they could only be
approximations of the true or exact length of the pencil. The difference between the exact length
of the pencil and the measure of that length is called error.
As engineers, we often apply physical laws derived upon theory and assumptions. Laws can
be cast in the form where some response variable, y, is defined as a relationship among predictive
variables x1 , x2 , . . . , xn . We often express the law in terms of a mathematical model, which we
would write in the form
y = f (x1 , x2 , . . . , xn )
(1.1)
The x’s are generally attributes of an object or system involving physical quantities such as
weight, length, time, and force. The model is expected to predict y over some specified range of
the x’s.
Of course, it is possible that the theory used to establish such a law is invalid. Moreover, the
assumptions may be wrong. For example, in developing the law one may have assumed certain
physical phenomena to be unimportant and neglected to include their influence in the model, but
in fact they are important. Given this uncertainty regarding most mathematical models of a phys­
ical system, experimental validation of these models is a necessary exercise prior to the model’s
application.
At first glance, experimental validation of a physical law seems straightforward. One would
set up an experiment involving particular values of the x’s and then observe the value of y. One
would also apply the mathematical model to predict the value of y as a function of the same x
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values. If the observed value of y and the predicted value of y agree, the experiment gives some
positive evidence towards validation of the model. One would repeat this procedure for different
values of the x’s (within the stated domain of application of the model), observe the different y’s,
and calculate the predicted y’s. If agreement continued for these additional experiments, we would
gain more and more confidence in the validity of the model.
However matters are not quite so straightforward. The observed y would generally be a mea­
surement of the value y. As noted earlier, measurements are only estimates of the true value y and
are prone to error. Moreover, it is likely that many of the x’s are based on measured values, not on
exact values. For both of these reasons, we cannot ever expect exact correspondence between the
observed and the predicted value of y. If the model is valid, we can only expect these two values
to be “close” to each other.
The obvious question is “how close is close enough?” The answer to this question is clearly
a function of the magnitude of the measurement errors. If these errors are tiny and the model is
valid, we expect the difference between the observed and predicted value of y to be tiny. If they
are, large, we are willing to accept a larger difference before invalidating the model. In addition, it
is also sensitive to the relationship between the predictive variables and the response variables. For
example, if the model shows that the response changes rapidly with small changes in a particular
predictive variable, then a minor error in measuring that predictive variable will result in a large
error in the prediction of the response variable.
1.1.2
Objectives
On the basis of this discussion, we are now ready to state the objectives of this laboratory. Our
first objective is to clarify the concept of measurement error and provide practical approaches that
enable quantification of error. The approaches considered here generally determine an upper bound
on the magnitude of error. Subsequently, we will provide a general method where we can translate
estimates of the error in measurement of the x’s to corresponding errors in predicting y using the
mathematical model. As will be explained below, this can be used to answer the question noted
above “how close is close enough?” Thus, we will show how to empirically validate a model in the
presence of measurement error.
1.2
Archery: A useful analogy
Consider an archer that shoots 10 arrows at a target, attempting to hit the bullseye, or the center
ring (Figure 1.1).
A relative comparison among the four cases in this figure illustrates the concepts of accuracy
and precision. Case A is neither accurate nor precise. Case B is precise but not accurate. C is
accurate but not precise. Case D is both precise and accurate. Accuracy concerns the correspon­
dence between the average value (i.e., the centroid of the cloud of holes made by the arrows) and
the target value (the point where the crosshairs intersect). Precision concerns the relative closeness
of the holes.
The perfect archer would shoot without error. All 10 arrows would pierce the target through a
single hole and that hole would be centered perfectly on the crosshairs at the center of the target.
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A
B
A
A
A
A
A A
A
A A A
AAAA
AA
A
A
A
A
C
D
A
A
A A
A
A
A
A
A
A A
A A
AAA
AA
A
A
Figure 1.1: Archery practice
In shooting the bow, these four cases illustrate that the error is manifest by both a lack of precision
and a lack of accuracy.
For this reason, we define two types of error. The first type of error is related to accuracy and is
called the bias error, β. Since our target is two­dimensional, the bias error in the archery example
is a vector and is illustrated in Figure 1.2.
For this example, we construct the bias vector as the vector between the crosshairs of the target
and the centroid of the arrow holes. Both A and B have a significant bias error and it is similar in
both magnitude in direction. In cases C and D, the bias error is negligible because the centroid is
roughly at the center of the target
The second type of error is related to precision and is given the name precision error. In our
archery example the precision error is related to the spread of the holes. To quantify it, we might
assume that if we shot many arrows at the target, the pattern would tend to be circular with more
holes in the middle of the circle and fewer towards the edge of the circle. The radius of the circle
would be a measure of the precision error. We see that A and C in figure 1.2 have similar precision
error, as does B and D. The precision error of B and D is considerably smaller than the precision
error of A and C. Figure 1.3 illustrates the measure of both the bias error and the precision error
for case A, where the bias error is shown by the solid arrow and the precision error is shown by the
dotted arrow.
You might ask why do we need two measures of error? Why does not a single measure of error
suffice? The reason for that is that the two measures of error are attributable to different types of
causes:
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A
B
A
A
A
A
A A
A AA A
A A AA
AA
A
A
A
A
C
D
A
A
A A
A
A
A
A
A
A A
A A
AAA
AA
A
A
Figure 1.2: Bias error vectors in archery practice
A
A
A
A A
A
A AA A
Figure 1.3: Bias and precision in archery practice
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• The causes of bias errors are usually few, typically identifiable, consistent in effect from case
to case, and potentially correctable.
• The causes of precision errors are numerous, often hard to identify, inconsistent in effect
from case to case, and difficult to correct.
In our archery example, the most likely cause of a bias error would be that the sights on the
bow were maladjusted. Other possibilities might be that the surface where the arrow rests against
the bow is not aligned with the axis of the bow sending arrows down and to the right. This type
of error has a consistent or systematic effect. Moreover, we can hope to identify the causes of bias
and correct for them. We could adjust the position of the sights and re­machine the arrow rest and
thereby eliminate the bias error.
The precision error in this example might be caused by factors such as the archer is inconsistent
on how far back she draws each arrow against her cheek, variations in how the string on the bow
slides across her fingers when releasing the arrow, random flitching, the consistency that release is
timed with her breathing cycle (which causes the sights to move in a systematic elliptical pattern),
variations in material properties from arrow to arrow, and the effects of wind as the arrow travels
to the target. There are probably numerous other factors that were not cited. The effect of these
factors is, moreover, inconsistent from one trial to the next. For example, sometimes the wind
blows strong, other times there is little wind.
Although the causes of precision error are difficult to correct, they can often be controlled. We
can use high quality, carefully manufactured arrows and shoot the arrows indoors where there is no
wind. Precision error is reduced by experimental control.
There is one more type of error to consider. Imagine that we were sighting in the bow. This is
a feedback process where we would shoot many arrows and make adjustments to the sights based
on the vector describing the bias error. Now suppose that when shooting one arrow in this process,
the string broke on the bow. This is an example of a blunder error. Blunder errors occur when the
experiment does not proceed according to plan or standard. When blunders occur, we must throw
out the experimental results. In our example, it would be foolish to consider the arrow when the
string broke as providing any useful information regarding the bias error and how the sights should
be adjusted. We would simply ignore the results of that particular shot at the target; it is irrelevant
to our attempt to learn about the process.
1.3
Types of measurement error and quantification of the mag­
nitude of error
1.3.1
The true value of a variable
In the field or laboratory is common to take repeated measurements of a particular variable. If
the experiment or measurement procedure involved a blunder of any sort that measure would be
discarded. The values of the remaining blunder­free measurements would then be averaged in order
to obtain a best estimate of the true value of the variable being measured. Please note that a singular
true value need not exist. If the variable was the life of a light bulb we know from experience that
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some bulbs will have a longer life and others will have a shorter life. The information that we are
seeking is the average life of the light bulb. We will consider the true value and the average value
to be equivalent. In this course, we will usually be considering measures of variables that have a
unique value or vary only tiny amounts over time.
1.3.2
Precision error
Reconsidering the situation where we have taken multiple measurements of a variable, the scatter
of these measures about the best estimate or average value is considered to be the manifestation of
precision error. More exactly, we define precision error as follows:
Precision error is the deviation of independent repeated measurements from the aver­
age value of those measurements.
Other names given to precision error include experimental error, repeated error, and random
error. Important general characteristics of precision error are as follows:
• It is caused by numerous factors.
• The effect of these factors is inconsistent from case to case and unpredictable. The preci­
sion error may be largely positive in one experiment, zero in another experiment, and small
negative in another. In quantification of precision error we are usually most concerned with
either its average value or its maximum possible value.
• Important general sources of precision error include:
– Inconsistent measurement procedures from trial to trial. These may be related both to
the human (e.g., the observer of liquid level in a graduated cylinder observes the cylin­
der from different vantage points from trial to trial) and to the measuring instrument
(e.g., a bearing occasionally seizes a little on a vane anemometer used to measure the
velocity of airflow).
– Inconsistent or poor experimental procedures from case to case (e.g., a cylinder holding
a soil sample is not properly cleaned between trials of an experiment).
– Changes in environmental conditions from case to case including variables such as
temperature, pressure, humidity, line voltage, electrostatic charge, etc.
– The physical sample does not conform to the ideal upon which the measure was defined
(e.g., cohesive strength would not be well defined for a soil sample containing chunks
of rock that are large relative to the sample size. Also the diameter of a cylinder is not
defined for an eccentric cylinder).
These general sources of precision error are very important. They give us a means for reducing
the magnitude of precision error. Specifically, we should define and follow well­defined measure­
ment procedures. We should use quality instruments that are well maintained. We should define
proper experimental procedures and follow them carefully. Environmental conditions that might
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Figure 1.4: Histogram
have a potential impact on the experimental results should be controlled to the degree possible.
Finally, test specimens should be carefully selected and prepared. Collectively, these practices are
called experimental control. Such control is central to the scientific method.
If numerous repeated measurements were taken, we could plot a relative frequency diagram
or histogram of these measurements, as illustrated in Figure 1.4. This diagram gives the relative
likelihood of observing a particular value.
The average value of the observation is the balance point of this diagram, in this example some­
where near 25 mV. In probability theory there is an interesting behavior called a “normal process”.
A normal process is said to occur when observing values of a variable where:
• There are many causes of variation.
• The causes act independently from one another.
• The effects of these causes are additive.
When a normal process occurs, the theory proves that histograms will be symmetric and bell­
shaped. The variable will also have a normal distribution which describes the relative likelihood
that various values will be observed. Many real world measurement processes meet these stated
assumptions of a normal process quite closely, which is not surprising given how we have defined
precision error above. This allows us to use the normal distribution as a model of precision error.
We define the standard deviation, σ, of n repeated measurements of variable z, z1 , z2 , . . . , zn ,
as follows:
sP
n
2
i=1 (zi − z̄)
σ=
(1.2)
n−1
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-2.6σ
+2.6σ
99% of the observations
within this interval
z
Figure 1.5: Under normal distributions, 99% of the values lie within 2 standard deviations from
the mean
where z̄ is the average or mean value given by:
Pn
i=1 zi
z̄ =
n
(1.3)
If we ignore the “−1” in the denominator, the term inside of the denominator is simply averaging
the deviations between each observation and the mean value after they have been squared. Squaring
makes negative deviations positive and thus prevents the deviations from canceling in the averaging
process. We see that when the observations are close to the mean value, σ will be small. When
they deviate significantly from the mean value, σ will be large. It is a property of the normal
distribution that approximately 95% of the observations of a normal process random variable will
lie within plus or minus 2 standard deviations of the mean and approximately 99% will lie within
2.6 standard deviations of the mean (Figure 1.5).
Now, this lays the groundwork for a useful way of quantifying the precision error of a measure­
ment. As already noted, the best estimate of the measure is taken as the average of our observations.
What we are interested is in our uncertainty regarding this average value not the variability of the
individual observations. Imagine that your repeat a measurement process 10 times and calculate
the average. Subsequently you repeat this whole procedure 10 times. In doing this you would
have made 100 total measurements and computed 10 averages. What would you expect to be more
similar?
• Any 10 measurements selected at random from the 10 measurements
• The 10 average values
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The answer is that the average values will tend to be much more similar, and less variable, than
the individual measurements. In a given sample of 10 measurements we are likely to see some high
values and some low values. The process of averaging will cancel these effects out. Moreover, if
we averaged 20 values instead of 10 values, we would expect the averages to be even more similar.
One can show from probability theory that the average values will have a normal distribution and
its variance will be proportional to the inverse of the square root of the number of samples. An
estimate of the standard deviation of the average value, σz̄ , is given by the equation:
sP
n
2
i=1 (zi − z̄)
σ=
(1.4)
n(n − 1)
Inspection shows that the standard deviation of the average value is equal to the standard devi­
ation of the original values divided by the square root of the number of samples. If the individual
values are highly variable, then so will the average value and our uncertainty regarding the true
value will be high.
Given that 95% of the observations of a normal random variable fall within 2 standard devia­
tions of the mean, it is common practice to estimate the maximum precision error as 2σz̄ . (Some
more conservative authors use 2.6σz̄ and more liberal authors use σz̄ , which will be an upper bound
on the true precision error 2 times out of 3).
Now imagine measuring the length of a pencil with a steel rule with the smallest demarcation
on the rule equal to 1 mm. You could repeat the measurement 50 times and each time you would
record 195 mm. It appears that there is no precision error. That is not really the case. It is just that
the measuring device is not sensitive enough in this application to observe random error. If you
used a more sensitive instrument, such as calipers, you would certainly observe variation in the
results. It is common practice to use 1/2 the minimum measurement increment as an upper bound
on the precision error. This is sometimes called the a priori error. Because we do not observe
variation in our measurement results, we know that the precision error is less than this value.
1.3.3
Bias error
Now we turn our attention to bias error. Bias error is the difference between our best estimate (the
average of repeated measurements) and the true value as shown in Figure 1.6.
The characteristics of bias errors are as follows:
• There are relatively few causes of the bias.
• These causes are constantly present and have predictable results.
• We might be able to identify causes of bias, measure the magnitude of the bias, and correct
the observed result to eliminate the bias error.
• Common causes of bias include:
– Instruments are in need of calibration and/or maintenance.
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True value of z
Best estimate of true value
(mean of observed values)
Bias
z
Figure 1.6: True value and best estimate of true value
– There is a consistent deviation in environmental conditions from the standard, e.g., mea­
surements of a material property on 200 samples were done in a room where the temper­
ature was 10 degrees higher than the prescribed temperature of 72 degrees Fahrenheit.
– There is a consistent deviation in measurement procedures from standard (e.g., in mea­
suring the neck size of customers, the sales assistant consistently held the tape a little
too low on the neck).
– There is a consistent deviation in preparation of a material sample from the standard
method (e.g., a rod was loaded into the measuring fixture in an improper fashion).
Like precision error, proper procedure and experimental control is important to reduce and
eliminate bias error. Unlike precision error, we see that there is potentially the opportunity to
identify the source of bias error and correct for that error. This certainly is not always possible.
For example, it would be difficult to do in the neck size example, although an approximate bias
correction might be possible. Moreover, we would prefer that such a correction is unnecessary.
Note that with deep enough understanding and control of the process, it may become possible
to control and correct for factors that might be classified as causes of precision error. The boundary
between these two types of error is fluid and really the difference is the depth of our knowledge and
the level of our ability to control the experiment. In manufacturing, for example, we often spend
effort to study and identify the causes of minor variation in a process and subsequently control
them.
The most important example of bias control is instrument calibration. Here instruments are
applied to the measurement of standards under highly prescribed and controlled experiment condi­
tions. Adjustment mechanisms that have been built into the instrument are set so that the observed
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value matches the known standard value. The National Institute of Standards and Technology
(NIST) is the primary source for such standards in the United States.
To quantify bias error requires a comparison between measurements made under the non­
standard conditions that resulted in the bias to measurements under standard conditions. For exam­
ple, if the end of a ruler was worn down a little, we might make several measurements with the worn
ruler and repeat those measurements with a good ruler and observe that there is a consistent differ­
ence of 2 mm. We could use that value to correct values measured by the worn ruler, adding 2 mm
to each case. Sometimes bias corrections are determined as a function of experimental conditions,
e.g., a correction of bias of a viscosity measurement as a function of temperature is experimentally
established.
Measuring instruments are made from materials and mechanisms where even the most sophisti­
cated fabrication and assembly procedures fail to produce a perfectly accurate device. For example,
a circuit design on a multimeter (multi­tester) may call for a 10 ohm resistor and the one actually
installed is 9.999 993 7 ohms. To compensate, adjustment mechanisms are built into good instru­
ments. These allow tuning of the instrument to obtain correspondence between observed values
and known standard values, the calibration process, in order to compensate for these imperfections.
Nonetheless, as the instrument is used, components may degrade and wear and the adjustments may
drift or slip. The need of periodic maintenance and re­calibration is necessary in order to minimize
the bias of the measuring instrument.
Manufacturers of precision measuring instruments have extensive experience on the magnitude
of biases that can occur and this information is valuable in practical quantification of maximum bias
error. They might gather this information, for example, by calibrating a series of instruments re­
turned to them from the field and then establishing the range of bias they see among this population
of instruments. This would give an indication of how far an instrument can become biased over a
year’s worth of normal use. Let us consider a real world example of this information and how it
might be used.
The manufacturer of a good quality multimeter (volt/ohm/capacitance meter) used in a U.K.
laboratory makes the following statement regarding accuracy of the instrument:
Accuracy is specified for a period of 1 year after calibration, at 18o C to 28o C (64o F
to 82o F) with relative humidity of 90%. Accuracy specifications are given as ± ([1 %
of reading] + [number of the least significant digits]).
The bracketed information is then given for different applications of the meter. As an example,
when using the meter in measuring ac voltage in the range of 100­400 V at 60 Hz, the accuracy
would be:
±([1 % of reading] + 4 × [value of the least significant digit])
The meter has a digital readout and gives 4 digits, e.g., 125.3 V. In this application, the value
of the least significant digit is 0.1 V.
If we inserted the probes of the multimeter into a wall outlet in a room where the temperature
was 72o F, we might read the value 121.2 V. The manufacturer is thus stating that the value is
accurate to within ±([0.01 × 121.2 V] + 4 × 0.1 V) = ±1.612 V. What we are stating is that
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1.612 V is a good estimate of the maximum possible bias in the measurement, assuming the meter
has been properly maintained and calibrated on an annual basis and used in the stated temperature
and humidity ranges given above. The same manufacturer gives the formula for determining the
error for higher and lower temperatures and the error increases as we go either above or below the
temperature range.
As noted above, such statements regarding accuracy are generally available for good quality
scientific instruments and they represent the best source of information to quantify maximum bias
error that might occur in application of the instrument itself. Moreover, you should not assume in
this example that your instrument actually has a bias of 1.612 V. We are just stating that the bias
is in the range of ±1.612 V based on the manufacturer’s experience with similar instruments that
have been routinely maintained and calibrated when applied in the stated temperature and humidity
range. If you wanted more precise knowledge of the bias of your instrument you could have it tested
against a standard and have this procedure done on a very frequent basis.
You may have been struck by the large magnitude of the bias error calculated in the multimeter
example above. The meter reads to the closest 0.1 V in this application but the bias error might be
as great as 1.612 V. You might wonder why one would attempt to measure or bother to report the
measurement to the nearest 0.1 V when its true value could be off by as much as 1.612 V. Remember
that the bias of your well­maintained, routinely­calibrated instrument is actually expected to be
some value in the interval ±1.612 V. Moreover, since this error is a bias, its magnitude is relatively
constant. Hence, the fine resolution of the voltage provided by the meter enables us to compare
different situations.
For example, you might want to use the multimeter to characterize how well the utility in your
area is regulating line voltage. At 12:00 p.m. you might measure 121.2 V and at 12:15 p.m. you
measure 119.1 V. If the actual bias of your instrument was −1.0 V, the true value of voltage was
122.2 V at 12:00 p.m. and 120.1 V at 12:15 p.m. Regardless of the exact voltage level, the dif­
ference of 2.1 V between these two measurements is a valid estimate of how the voltage changed
between 12:00 and 12:15 p.m. The fine resolution of this instrument allows us to characterize the
difference. When biases are constant, the effect of bias cancels when differencing measurements.
Note, however, that in many instruments the maximum bias is known to be proportional to the
measurement itself, e.g., the multimeter example given above. In such situations the bias will only
cancel when the differences are small. This discussion illustrates the fact that, as a practical mat­
ter, it is far easier to obtain comparative sensitivity in a measuring instrument than it is to obtain
absolute accuracy.
Finally, there is one more important point to make in comparison of bias and precision error.
We saw earlier that as we took more and more observations, we reduced the precision error. Specif­
ically, our formula showed that this error reduced proportionally to the square root of the number
of observations. In contrast, taking more and more observations does nothing to eliminate bias.
This is because the bias is consistent and present in every observation. Unlike precision error, the
effects of bias error do not average out.
1.3.4
Total error
We define the total error, ∆, as the sum of the bias error, β, and the precision error, ϵ. That is:
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∆=β+ϵ
(1.5)
Total error would apply to a given trial of the experiment and would vary from trial to trial since
the precision error varies from trial to trial. In the analysis of error, we are generally interested in
the maximum total error, ∆max , i.e., the sum of the maximum bias, βmax , and the precision error,
ϵmax as given by the following equation:
∆max = βmax + ϵmax
(1.6)
The following procedure is a reasonable practical procedure for estimating total error in repli­
cated experiments where good quality, well­maintained, and properly­calibrated instruments are
used:
• Given the results of the replicated experiments (minimum of 3), estimate ϵmax as 2.5σz̄ .
• Use the manufacturer’s statement of accuracy to compute an upper bound on bias error, βmax ,
similar to the multimeter example above (different companies will state the accuracy of their
instrument in different terms).
• Add the two components to compute the total error.
In an experimental situation where the instrument was recently re­calibrated either to eliminate
bias or the magnitude of bias was numerically determined by an accepted procedure, that value
might be used instead of the manufacturer’s statement of accuracy. This requires exceptional in­
strument maintenance and frequent re­calibration. Typically the procedure above would be the
more appropriate one to apply. Figure 1.7 illustrates the concept of total error.
We do not know the true value of z. The smallest possible true value would occur when the
estimated value of z was observed at the rightmost tail of the distribution of the average value
and the bias was at a maximum positive value. The largest possible true value would occur when
the estimated value of z was observed at the leftmost tail of the distribution and the bias was at a
maximum negative value. We know the true value is within these lower and upper bounds.
1.3.5
Blunder error
As noted above, blunder error involves a miscarriage of the experiment. It might involve a failure to
use proper experimental procedure, misuse of a measuring instrument, use of defective specimens,
etc. When blunders occur, the results should simply be thrown out; they offer no information.
1.3.6
Relative error
You measure a length and it is determines that the maximum error is 3 cm. It that error big or is
it small? Clearly the answer is dependent on the object being measured, If you are measuring the
height of a person, 3 cm would be a large error. If you are measuring the distance between the center
of Lexington, KY and the center of Tokyo, Japan, 3 cm would be a very small error. We define the
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Estimated
value of z
Lower bound on
true value of z
Δmax
βmax
Upper bound on
true value of z
Δmax
εmax
εmax
βmax
z
Figure 1.7: The concept of total error
maximum relative error, Rmax , as the ratio of the maximum total error to the best estimate of the
measure:
∆max
z̄
Rmax is thus independent of the scale of the observation.
Rmax =
1.4
(1.7)
Impact of measurement error on predicted values
As noted above, the motivation for experimentation is often to validate a mathematical model (and
hence a theory) of the process. We can measure the response variable directly in the experiment and
establish ∆max for that variable. The interval ȳ ± ∆ymax represents an interval that with high prob­
ability will contain the true value of the response variable y. Similarly, we can establish intervals
for each predictive variable xi , i = 1, 2, . . . , n as x̄i ± ∆ximax . What is missing is an estimate of
the error of the predicted (i.e., calculated value). This error is not observable. Rather it is function
of the error of the predictive variables.
This error is dependent on the functional relationship between the predictive variables and the
response variable. A response variable might be more sensitive to errors in one predictive variable
than it is to errors in another predictive variable. One simple and general way to estimate the error
in the predicted value of the response variable as a function of the error in measured value of the
predicted variable is to use a first order Taylor’s series to approximate f (x1 , x2 , . . . , xn ) near the
measured values (xo1 , xo2 , . . . , xon ):
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Figure 1.8: Linearization of the function in the neighborhood of the measured values
∂f (x1 , x2 , . . . , xn )
(x1 − xo1 )+
∂x1
o
o
o
x1 ,x2 ,...,xn
∂f (x1 , x2 , . . . , xn )
(x2 − xo2 ) + · · · +
+
∂x2
o
o
o
x1 ,x2 ,...,xn
∂f (x1 , x2 , . . . , xn )
(xn − xon )
+
∂xn
xo ,xo ,...,xon
y ≈ f (xo1 , xo2 , . . . , xon ) +
1
(1.8)
2
Recall that this estimate is simply a linearization of the function in the neighborhood of the
measured values (xo1 , xo2 , . . . , xon ) and is a good estimate for small deviations about (xo1 , xo2 , . . . , xon )
as shown in Figure 1.8. Errors in well­conducted experiments can generally be viewed as “small
deviations”.
Assume that true value of the predictive variables is (xo1 , xo2 , . . . , xon ) but is measured to be
(xo1 + ∆x1 , xo2 + ∆x2 , . . . , xon + ∆xn ) where the ∆s are the measurement errors. Further, let yo be
the true value of y and let ∆yo be the error in the predicted value. If the model is valid then:
yo + ∆yo ≈ f (xo1 + ∆x1 , xo2 + ∆x2 , . . . , xon + ∆xn )
(1.9)
For small measurement errors, we may substitute into equation 1.8 to obtain:
∂f (x1 , x2 , . . . , xn )
(xo1 + ∆x1 − xo1 )+
∂x1
xo1 ,xo2 ,...,xon
∂f (x1 , x2 , . . . , xn )
(xo2 + ∆x2 − xo2 ) + · · · +
+
∂x2
xo1 ,xo2 ,...,xon
∂f (x1 , x2 , . . . , xn )
+
(xon + ∆xn − xon )
∂xn
xo1 ,xo2 ,...,xon
(1.10)
y + ∆yo ≈ f (xo1 , xo2 , . . . , xon ) +
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Using equation 1.9, equation 1.10 simplifies to:
∂f (x1 , x2 , . . . , xn )
(∆x1 )+
∂x1
xo1 ,xo2 ,...,xon
∂f (x1 , x2 , . . . , xn )
+
(∆x2 ) + · · · +
∂x2
xo1 ,xo2 ,...,xon
∂f (x1 , x2 , . . . , xn )
(∆xn )
+
∂xn
xo ,xo ,...,xon
∆yo ≈
1
(1.11)
2
Equation 1.11 gives us a way of estimating the error of the predicted value of the response
variable as a function of the errors in measuring the values of the predictive variable. First we take
the partial derivatives of the function with respect to each predictive variable and evaluate these
partial derivatives using the average measured value of the predictive variables. Subsequently, we
simply substitute our estimates of the maximum error into this equation to calculate the maximum
error in the predicted value of y.
Note that equation 1.11 is a linear equation, with each term corresponding to a particular one
of the xi values. If the coefficient is big, we see that the error is sensitive to the error in measuring
that particular x. More accurate instruments are important for that particular measurement. If the
coefficient is small, the reverse is true.
1.5
Lab experiment
In our experiment we will measure the dimensions of a can.
1.5.1
Equipment
• 3 Cans
• Ruler
• Vernier Calipers
1.5.2
Procedure
1. Make a sketch of your assigned object of measurements (the can).
2. Using Vernier calipers, take about three (3) measurements of the height h and diameter d of
the can. We will take the priori error, 1/2 the width of the minimum increment as an estimate
of the maximum precision error for both d and h. Be sure to note the minimum increments.
3. Measure d and h using the steel rule. The teaching assistant will help you to identify the
marks to be used in making this measurement. The rule is not a precise measuring device and
therefore it is also unlikely that we will be able to observe precision error through repeated
measurements (even though there is more procedural uncertainty in these measurements).
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Take a measurement of both d and h. Again, we will take the a priori error, the width of the
minimum increment, as an estimate of the total error for both d and h.
4. Tabulate the results from step 3. Compute the best estimate of the measured deflection as
the average value. Compute the maximum precision error for each measurement using the
formula (where z refers to a generic measure):
sP
n
2
i=1 (zi − z̄)
(1.12)
ϵmax = 2.6σz̄ = 2.6
n(n − 1)
5. Determine the analytical form of the first order Taylor’s series expansion of our model as a
function of the predictive variables d and h. Substitute the measured values into the analytic
form of this model to obtain numerical values for the gradient terms. Using the maximum
precision errors of the predictive variables (which we are taking as 1/2 the minimum mea­
surement increment) and equation 1.11, determine the estimate of the corresponding error
in the calculated value of volume. Be sure to properly deal with negative gradient terms as
outlined in the background section. Also, note the particular measurements having the most
significant impact on error on the calculated value.
1.6
Report
Follow the lab report guidelines given at the first class meeting. For Lab 1, be sure to address the
following items in your report:
• Give measured values of diameter (d) and height (h) and the estimate of maximum precision
error obtained using the minimum increment (a priori) method
• Tabulate the observed values of d and h. Show the computed standard deviation of the mean
and the computed maximum precision errors of this measure.
• Discuss possible approaches and controls to reduce precision errors in measuring d and h.
• Give the analytical formula for the Taylor’s series expansion, numerical values of each coef­
ficient, and calculation of the corresponding error in the predicted diameter and height value
as a function of the precision errors in the measured values of the d and h.
• On the basis of the Taylor’s series model, identify the measures where precision and accuracy
are most important and the measures where they are least important. (Remember that one
way to reduce the precision error is to increase the number of replications. This information
can tell you for which variable a high number of replicate measurements would be valuable.)
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1.7
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Measurement worksheet
Can 1
D1 Vernier
Caliper (in)
D2 Ruler (in)
H1 Vernier
Caliper (in)
H2 Ruler (in)
D1 Vernier
Caliper (in)
D2 Ruler (in)
H1 Vernier
Caliper (in)
H2 Ruler (in)
1
2
3
4
5
6
7
8
9
10
11
12
Average (in)
Std. Dev. (in)
Can 2
1
2
3
4
5
6
7
8
9
10
11
12
Average (in)
Std. Dev. (in)
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Can 3
D1 Vernier
Caliper (in)
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D2 Ruler (in)
1
2
3
4
5
6
7
8
9
10
11
12
Average (in)
Std. Dev. (in)
19
H1 Vernier
Caliper (in)
H2 Ruler (in)
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20
Chapter 2
Modulus of elasticity and Poisson’s ratio
2.1
General introduction
In Part 1, we will use the flexure equation to determine E for aluminum. In Part 2, we will determine
Poisson’s ratio for aluminum.
2.2
Poisson’s ratio
When we apply a compressive (tensile) force to an object, there is a corresponding lateral thickening
(thinning) (Figure 2.1).
F
F
Figure 2.1: The Poisson effect
Lateral strains exhibit a constant relationship to the axial strains in the elastic region. We can
define Poisson’s ratio as
ν=
Lateral Strain
Axial Strain
(2.1)
We may also define Hooke’s law for a biaxial stress state:
σx =
σy =
E
(ϵx + νϵy )
1 − ν2
E
(ϵy + νϵx )
1 − ν2
σz = 0
21
(2.2)
(2.3)
(2.4)
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2.3
Measuring strain ­ strain gauges
2.3.1
Principle of operation
Strain is measured by making very accurate measurements of the change in resistance of the strain
gauge as it is stretched or compressed (Figure 2.2).
Gauge Length
(Primary Axis)
Solder Tabs
Leads
Figure 2.2: Principle of strain measurements
The gauge length is the active, or strain­sensitive portion of the strain gauge. Strain gauges are
used to measure strain in this direction, and this strain is called axial strain. Any strain that occurs
perpendicular to the primary axis is called transverse strain. The end loops are relatively insensitive
to strain.
Strain gauges are often constructed of a metal foil attached to a plastic backing material.
2.3.2
Strain sensitivity and gauge factor
Strain Sensitivity is the ratio of relative change in electrical resistance of a conductor to the relative
change in conductor length. From this relationship, we can define the gauge factor, GF as:
GF =
(∆R/Ro )
(∆L/Lo )
where:
∆R = Resistance change of strain gauge due to strain
Ro = Unstrained strain gauge resistance
∆L = Change in length
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Lo = Unstrained length
Note that ∆L/Lo is the definition of unit strain.
2.3.3
Transverse sensitivity
Strain gauges also have a response to transverse strains because of the grid resistance that exists in
the end loops. This causes a resistivity change in the foil grid, and the extent of this change depends
on the grid material and conductor width.
Transverse sensitivity, Kt , is defined as:
Kt =
(∆R/Ro )(90o )
(∆R/Ro )(0o )
(2.6)
where:
(∆R/Ro )(90o ) = the resistance change caused by a uniaxial strain perpendicular to the grid.
(∆R/Ro )(0o ) = the resistance change caused by the same value of strain applied parallel to the
grid.
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2.3.4
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Correcting for transverse sensitivity
The chart in Figure 2.3 can be used to correct for transverse sensitivity.
1.5
5 = εt / ε2
εt / ε2 = -5
1.4
4
1.3
3
-4
-3
2
1.2
-2
1
Correction Factor (C)
1.1
-1
0
1.0
0
-1
1
0.9
-2
2
0.8
-3
3
0.7
-4
4
0.6
-5
5
0.5
-8
-6
-4
-2
0
2
4
6
8
Kt - %
Figure 2.3: Correcting for transverse sensitivity
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2.3.5
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Measuring strain – Wheatstone bridge
A Wheatstone bridge circuit is used to measure (very accurately) small changes in the resistance
of strain gauges. In the circuit in Figure 2.4, consider that R2 is the strain gauge.
R3
R1
B
A
R2
R4
strain gauge
Figure 2.4: Wheatstone bridge circuit
R2 can be determined from the following relationship:
R2
R4
VAB = VA − VB = VS
−
(R1 + R2 ) (R3 + R4 )
2.3.6
(2.7)
Measuring strain – strain indicator
The strain indicator, which is used in the laboratory, reports strain measurements directly in units
of microstrain, µϵ.
2.4
Modulus of elasticity: Introduction
The purpose of this experiment is to measure the modulus of elasticity (Young’s modulus) of an
aluminum beam by loading the beam in cantilever bending. The modulus of elasticity, a funda­
mental constant for linear elastic materials, is an index of the stiffness of the material. Note that a
cantilever beam is one in which one end is built into a wall or other support so that the built­in end
cannot move transversely or rotate.
For many common structural materials, including aluminum alloys and steels, strain is an essen­
tially linear function of the stress over the range of stresses normally encountered by load­carrying
members. Figure 2.5 represents a typical “stress­strain” diagram for a metal under uniaxial stress
(tension, compression, or bending, for instance). By definition, the slope of the linear portion of
the diagram is the modulus of elasticity, Therefore,
E = ∆σ/∆ϵ
where:
E = modulus of elasticity, psi (N/m2 )
σ = stress, psi (N/m2 )
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(2.8)
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σ
Δσ
Δ#
#
Figure 2.5: Typical stress ­ strain curve
ϵ = strain, in/in (m/m)
There are other definitions of E for materials which have no linear region in their stress­strain
diagrams.
Stress is a defined concept, and is not directly measurable. Because of this, experimental de­
termination of the stresses in a complex structural member or mechanical part ordinarily requires
measurement of the strains and subsequent calculation of the stresses from Hooke’s law. For the
uniaxial stress state, Hooke’s law is a rearranged form of equation 2.8:
σx = Eϵ, σy = 0, σz = 0
(2.9)
For the more general biaxial stress state, however, Hooke’s law is as follows:
E
(ϵx + νϵy )
1 − ν2
E
σy =
(ϵy + νϵx )
1 − ν2
σz = 0
σx =
(2.10)
where ν = Poisson’s ratio
To determine either stress, two strain measurements are required, and two elastic constants (E
and ν) must be known. It is obvious from the form of equations 2.9 and 2.10 that the percentage
error in σ will be the same as the percentage error in E. Therefore, accurate values of the elastic
moduli of structural materials are of considerable importance for engineers.
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Figure 2.6: Model P3 Strain Indicator and Recorder
2.5
Modulus of elasticity: Equipment and supplies
• Flexor, cantilever flexure frame
• High strength aluminum alloy beam, 1/8 × 1 × 12­1/2 in (3 × 25 × 350 mm)
• Micro­Measurements temperature­compensated strain gauges (1 or 2)
• Model P3 Strain Indicator and Recorder or equivalent (Figure 2.6)
• Student Strain Gauge Application Kit, containing gauge bonding supplies, hook­up wire, etc.
• Laboratory weights for loading cantilever beam
• Micrometer
• Accurate drafting or machinists scale
The Flexor is a cantilever flexure frame that offers a simple, versatile, and portable all­in­one
solution for loading beams. Only modest forces are required to develop large strains and high
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Figure 2.7: Flexor ­ cantilever flexure frame (https://www.micro­measurements.com/educators)
stresses on specimens (Figure 2.7). Deflections are produced and measured by a micrometer, and
strains of up to 2500 µϵ can be obtained on a 0.250 in (6.35 mm) thick beam. The Flexor can also
be used with deadweights.
2.6
Modulus of elasticity: Procedure
2.6.1
General
In this experiment, the flexural stress­strain diagram for an aluminum alloy will be obtained by
loading a beam in cantilever bending as shown in Figure 2.8.
With the dimensions of the beam known, the stress as a function of the applied load can be
calculated quite accurately from the flexure formula:
σ=
6P L
Mc
=
l
bt2
(2.11)
where:
M = bending moment at the gauge centerline, in­lbs (N­m)
c = semi­thickness of the beam, in. (m)
l = moment of inertia of the beam cross­section, in4 (m4 )
P = load, lbf (N)
L = effective beam length, in. (m)
b = beam width, in. (m)
t = beam thickness, in. (m)
The surface strain at the section of interest will be measured by a strain gauge bonded at the
point. The load will be applied in increments, and the corresponding strains recorded. The stress
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Figure 2.8: Beam loaded in flexure
(calculated from equation 2.11) and strains will be plotted to produce a stress­strain diagram from
which the modulus of elasticity can be determined.
2.6.2
Strain gauge selection and installation
Micro­Measurements foil strain gauges are intrinsically temperature­compensated for use on a ma­
terial with a particular thermal expansion coefficient. Because of this, the strain gauge employed
in the experiment can be used in a “quarter­bridge” arrangement, completing the bridge circuit
with 120­ohm precision resistor built into the P3 Strain Indicator and Recorder. For quarter­bridge
operation, a “three­wire” circuit ordinarily should be used to achieve lead wire compensation by
placing equal lengths or lead wire in adjacent arms of the bridge circuit (see the wiring diagram for
this experiment, and the P3 Strain Indicator and Recorder for Flexor Instruction Manual).
A strain gauge indicates the average strain under the grid area when placed in a non­uniform
strain field. In this instance, because the strain varies linearly along the beam length, the strain
indicated by the gauge is equal to the strain at the centerline of the grid. This being true, the “gauge
length” of the strain gauge is not critical, and gauges with patterns such as 250BG or 125AD (both
120 ohms, and temperature­compensated for aluminum) can be selected.
Following the gauge installation instructions supplied with the Student Strain Gauge Applica­
tion Kit, mark the desired location and orientation of the strain gauge on the beam. The gauge
should be located near the end of the beam which will be clamped in the Flexor (but clear of the
clamp), and aligned with the longitudinal centerline of the beam, with the solder tabs toward the
free end of the beam. Bond the strain gauge in place, following the directions precisely. Allow the
adhesive to cure for the recommended time, and then carefully solder 6­inch (150­mm) lead wires
to the gauge, remove the flux, and apply a protective coating over the entire installation (including
the lead wire attachment joints).
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2.6.3
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Alternate procedure with two strain gauges
If desired, two strain gauges can be bonded to the beam, one on the top and one on the bottom, at the
same point along the length of the beam. For a rectangular cross­section beam bending, the stresses
(and strain) are theoretically equal in magnitude on the upper and lower surfaces of the beam, at
any section, but opposite in sign. If the two strain gauges are connected to the strain indicator as a
half­bridge (see P3 Strain Indicator and Recorder instruction manual), the indicated strain will be
twice the actual strain on either beam surface. The procedures are otherwise the same as for the
single­gauge experiment.
2.6.4
Data acquisition
Back the calibrated loading screw out of the way, and insert the beam into the Flexor, with the
gauged end in the clamp, and with the gauge on the top surface. Center the free end of the beam
between the sides of the Flexor and firmly clamp the beam in place with the knurled clamping
screw.
Connect the lead wire from the strain gauge to the binding posts on the Flexor as shown in the
wiring diagram (Figure 2.9). Then connect the appropriate gauge leads from the Flexor cable to
the S­, P+, and D­120 binding posts of the P3 Strain Indicator and Recorder.
Measure the distance from the centerline of the strain gauge grid to point of load application at
the free end of the beam using an accurate scale, and enter it on the worksheet on page 34. Mea­
sure the width and thickness of the beam with a micrometer. Using the flexure formula (equation
2.11), calculate the load, P, to be applied at the free end of the beam to produce a surface stress of
approximately 15,000 psi (1 × 108 N/m2 = 1 × 108 Pa = 100 MPa) on the beam at the centerline
of the gauge.
Using the P3 Strain Indicator and Recorder manual for guidance, depress the Amp Zero button
and balance the amplifier. Then depress the Gauge Factor button and set the gauge factor (as
displayed in the LCD readout) to the value given on the strain gauge package data form. Next
depress the Run button. With the beam unloaded (except by its own weight and the weight of the
loading hook), use the balance controls of the P3 to achieve a bridge balance (as indicated by a zero
in the LCD readout). Do not adjust the balance controls again for the remainder of the experiment.
Apply the calibrated load in 10 steps, or increments. At each increment, record the indicated strain
and corresponding load on the worksheet. Unload the beam in 10 decrements and again record the
load and strain at each decrement.
2.6.5
Data analysis and presentation
For each loach increment, and decrement, calculate the beam stress from equation 2.11, using the
exact loads. Record the load, stress, and strain at each load level in the table provided on the
worksheet. Lay out stress and strain scales along the coordinate axes on the accompanying sheet of
graph paper. Select the scales to span the ranges of the two variables in the worksheet table, with
the least scale unit representing convenient numbers of stress and strain units such as 0.5, 1, 2, 5,
100, etc. Plot the data from the table using a small dot inside a 1/16­in (1.5­mm) diameter circle
for each data points.
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The data points should fall rather closely in a straight line, and there should be little or no trend
to the difference between incremental and decremental points (indicative of hysteresis). If the data
do not conform as described here, the gauge may be in properly installed. This can be checked by
applying the largest load to the beam and leaving the load in place for several minutes. If there is
no detectable change in the strain indication during this period, and if the strain indication is within
a few µϵ of zero when the load is removed, the strain gauge is functioning normally.
With a straightedge, draw a straight line so that the data points are approximately equally massed
on both sides of the line. Draw the line up to, but not through, any data points which may lie in
its path. The line need not pass precisely through the origin of the graph. Measure the quantities
∆σ and ∆ϵ over as large a span of the line as possible, and calculate E from equation 2.9. As an
alternative, the “least­squares” method of statistical analysis can be used to establish objectively
the slope of the best straight line through the data points.
2.6.6
Report
Prepare a brief report, describing in your own words the purpose of the experiment, the equipment
and setup used, and the procedure followed. State the results obtained, and include the table of
experimental data and the graph of stress versus strain. Discuss the probable sources of error in
the experiment, and their relative effects on the accuracy of the modulus of elasticity you have
determined.
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Figure 2.9: Wiring diagram
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Figure 2.10: Stress ­ strain chart
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2.7
Modulus of elasticity: Worksheet
2.7.1
Beam dimensions
Length
Width
Thickness
2.7.2
in. (m) (from the point of load application to the gauge centerline)
in. (m)
in. (m)
Computation of maximum load for 15,000 psi stress
σbt2
(15, 000)(
=
P =
6L
6(
2.7.3
Step
2.7.4
)2
)(
)
=
lbf (N)
(2.12)
Tabulation of loads, strains, and stresses
Load, lbf (N)
Strain ­
increasing
load, µϵ
Stress ­
increasing
load, psi (N)
Strain ­
decreasing
load, µϵ
Computation of Young’s modulus
From straight line on graph:
∆σ =
psi (N/m2 );
E = ∆σ/∆ϵ × 106 =
∆ϵ =
µϵ
psi (N/m2 )
=
34
Stress ­
decreasing
load, psi (N)
Solid Mechanics Lab Manual, January 7, 2023
2.8
MNG 303
Poisson’s ratio: Introduction
The purpose of this experiment is to measure the Poisson’s ratio of an aluminum beam by load­
ing the beam in cantilever bending. Poisson’s ratio is one of two fundamental elastic constants
(along with the Modulus of Elasticity) relating stress to strain in a biaxial stress field. For example,
Hooke’s law for the biaxial stress state can be written as:
E
(ϵx + νϵy )
1 − ν2
E
σy =
(ϵy + νϵx )
1 − ν2
σx =
(2.13)
(2.14)
where:
σ = stress, psi (N/m2 )
ϵ = strain, in/in (m/m)
ν = Poisson’s ratio (dimensionless)
E = modulus of elasticity or Young’s modulus (N/m2 )
Both the elastic modulus, E, and the Poisson’s Ratio, ν, are required to translate measured
strains into stresses.
It is an experimentally observable fact that when a test specimen of an isotopically elastic ma­
terial is subjected to uniaxial stress, the specimen not only deforms in the direction of the stress,
but also exhibits deformation of the opposite sign in the perpendicular direction. Poisson’s Ratio
is, by definition, the absolute value of the ratio of transverse strain, ϵy , to the axial strain, ϵx , in a
uniaxially stressed member:
ϵy
(2.15)
ν=
ϵx
Thus, Poisson’s Ratio is always a positive number. To obtain the traverse strain in a uniaxial stress
field, given the axial strain and ν,
ϵy = −νϵx
(2.16)
Poisson’s Ratio can be measured readily with two strain gauges on a uniaxially stressed member.
One gauge is aligned in the direction of the applied stress, and the second gauge, perpendicular to
the first. A tensile test specimen with uniform stress field is commonly used for this purpose, and
the gauges are mounted adjacent to one another in the form of a “T”. Note that because most strain
gauges display some degree of sensitivity to strains transverse to the axis of primary, sensitivity, it
is usually necessary to correct the indication of the perpendicular gauge for transverse sensitivity
in order to obtain an accurate value of Poisson’s Ratio by this method.
Poisson’s ratio can also be measured with reasonable accuracy on a cantilever beam, even
though the strain varies linearly along the beam. In this case, the axial gauge can be mounted
longitudinally on, say, the upper surface of the beam, and the lateral gauge mounted crosswise on
the lower surface at the same section. The absolute value of the ratio of the two strains (after the
indication of the lateral gauge has been corrected for transverse sensitivity) is Poisson’s ratio. This
arrangement is shown in Figure 2.11.
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Figure 2.11: Cantilever beam for measuring Poisson’s ratio
2.9
Poisson’s Ratio: Equipment and supplies
• Flexor, cantilever flexure frame
• High­strength aluminum alloy beam, 1/4 × 1 × 12­1/2 in (6 × 25 × 320 mm)
• Micro­Measurements temperature­compensated strain gauges (2)
• Model P3 Strain Indicator and Recorder or equivalent
• Student Strain Gauge Application Kit containing gauge bonding supplies, hook­up wire, etc.
2.10
Poisson’s Ratio: Procedure
2.10.1
General
In this experiment, an aluminum beam on which two strain gauges are mounted will be used to
determine Poisson’s ratio in bending. It is assumed that under flexural loading the longitudinal
strains at corresponding points on the upper and lower surfaces of the beam are numerically equal,
differing only in sign; and the same assumption is made for the lateral strains. Based upon these
assumptions, one strain gauge will be installed longitudinally on the upper surface of the beam, and
a second gauge laterally at the corresponding point on the lower surface.
The beam will be clamped in the Flexor and loaded in bending to an arbitrary strain level.
Since the stress state in the beam is uniaxial, the lateral and longitudinal strains will be measured
for calculation of Poisson’s ratio.
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Notes
To avoid ambiguity in application of the term “transverse strain” (i.e., to distinguish between strain
which is transverse to the longitudinal axis of the beam from that which is transverse to the primary
sensing axis of a strain gauge), the following definitions will apply exclusively in this and the
remaining parts of this experiment:
• longitudinal strain ­ the strain along the longitudinal axis of the beam
• lateral strain ­ the strain in a direction perpendicular to the longitudinal axis of the beam
• axial strain ­ the strain along the primary sensing axis of a strain gauge
• transverse strain ­ the strain in a direction perpendicular to the primary sensing axis of a strain
gauge
2.10.2
Strain Gauge selection and installation
Strain gauges have been selected and installed
2.10.3
Data acquisition
Back the calibrated loading screw out of the way, and insert the beam into the Flexor with the
gauged end in the damp and with the longitudinal gauge on the upper surface. Center the free end
of the beam between the sides of the Flexor, and firmly clamp the beam in place with the knurled
clamping screw.
The gauges will be connected (via the Flexor cable) to the strain indicator one at a time ­first,
with the beam undeflected, and again with the beam deflected. An initial “reference” reading of
the strain indicator digital readout will be obtained for each gauge with the beam undeflected, and
a final reading with the beam deflected. The differences in these two sets of readings will give the
strain indicated by the longitudinal and lateral gauges.
Connect the strain gauge leads from the beam to the binding posts of the Flexor as shown in the
Wiring Diagram (Figure 2.12). With the loading screw clear of the beam, connect one of the two
common leads (from the Flexor) to the S­ binding post of the strain indicator and the other common
lead to the appropriate D post (D120 for 120­ohm gauges or D350 for 350­ohm gauges). Connect
the independent lead from the longitudinal gauge to the P+ binding post of the strain indicator. The
instrument is now connected to measure strain in the longitudinal gauge.
After balancing the indicator amplifier, set the gauge factor adjustment to the value given on
the strain gauge package data sheet (or on the beam, if supplied pregauged). Set the instrument to
Run. With the Flexor loading screw still clear of the beam, adjust the balance control of the P3
until the LCD digital readout indicates zero. The zero beam deflection reading of the longitudinal
gauge should be recorded on the worksheet as 0 µϵ. Do not adjust the balance control again during
the experiment.
Turn the strain indicator off, and disconnect the Flexor cable lead for the longitudinal gauge
from the P+ binding post of the strain indicator. Connect the independent lead from the lateral gauge
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Figure 2.12: Wiring diagram for Poisson’s ratio
to the P+ binding post in preparation for indicating the lateral strain. Turn the strain indicator on
(Run) and, without making any adjustment, record the indicator reading as the zero beam deflection
reading for the lateral gauge.
Now add 500µϵ to the zero beam deflection reading. Deflect the beam, by rotating the loading
screw clockwise, until the indicator readout registers the number equal to this sum. The “indi­
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cated” (uncorrected) lateral strain in the beam is now 500µϵ and there remains only to find the
corresponding longitudinal strain.
Turn off the strain indicator, disconnect the independent lateral­gauge lead from the P+ binding
post, and reconnect the longitudinal gauge to the instrument. Turn the indicator on again and record
the number registered by the indicator readout. This number is the required second reading for the
longitudinal strain gauge. Subtract the initial zero beam deflection reading from this number to
obtain the longitudinal strain. Poisson’s ratio is then 500 C/X, where C is the correction factor
for the transverse sensitivity of the lateral gauge, and X is the second reading of the longitudinal
gauge.
As a check on the stability of the system, back the Flexor loading screw away until it clears the
beam. The strain indicator readout should now read very close to 0 µϵ if the system is operating
normally. If the number is more than ±10µϵ or so, the source of the error should be located, and
the experiment performed again.
Pregauged beams supplied by the Measurements Group have been tested for gauge stability
at the time of manufacture, and should perform in a highly repeatable manner unless one or more
of the gauges had been damaged. If the zero beam deflection readings of the strain indicator fail
to repeat well, the balance control may have been inadvertently moved after its initial adjustment,
or the binding post connections may not have been snug enough to avoid small contact resistance
changes between connection and reconnection. Binding post connections should be snug enough
to allow a “wiggle test” of the lead wires without a zero balanced shift.
2.10.4
Data analysis and presentation
Before calculating Poisson’s Ratio form the indicated longitudinal and lateral strains, the indicated
lateral strain should be corrected for transverse sensitivity. Because the longitudinal strain in the
beam is several times as large as the lateral strain, the lateral gauge is subjected to a much larger
strain in a direction transverse to its primary sensing axis than along the axis. As a result of the
finite width of the grid lines in the gauges, and the presence of end loops connecting the grid lines,
strain gauges are generally sensitive not only to the strain parallel to the grid direction, but also
(to a much lesser degree) to the strain perpendicular to the grid direction. This property of strain
gauges is referred to as “transverse sensitivity”, and symbolized by Kt .
The correction of transverse sensitivity can be made easily from the attached graph. To use
the graph, two quantities are needed: (1) the ratio of the longitudinal strain to the indicated lateral
strain (which is the ratio of the transverse to the indicated axial strain for the lateral gauge), and
(2) the transverse sensitivity, Kt , of the lateral strain gauge (given on the strain gauge package data
from or on the information panel of pregauged beams).
Enter the graph on the abscissa at the value of Kt , for the gauges used in this experiment. Project
a line upward to the sloped line representing the ratio of the strain’s transverse to and parallel to the
axis of the lateral strain gauge (note that the ratio is actually negative in this instance). From the
intersection of these lines, project horizontally to the correction­factor scale on the ordinate to find
“C”. Multiply the indicated lateral strain by “C” to obtain the corrected lateral strain. The indicated
longitudinal gauge is small to begin with, and similar in magnitude to the transverse strain in the
calibrated environment used to measure the gauge factor of the strain gauge (the gauge aligned with
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the applied stress axis in a uniaxial stress field, with a Poisson’s Ratio of 0.285).
As an alternative procedure, any two strains, measured at right angles to one another, can be
corrected for the transverse sensitivity of the strain gauges with the following relationships:
ϵ1 =
1 − νo K t
(ϵˆ1 − Kt ϵˆ2 )
1 − Kt2
(2.17)
ϵ2 =
1 − νo K t
(ϵˆ2 − Kt ϵˆ1 )
1 − Kt2
(2.18)
where:
ϵˆ1 , ϵˆ2 = two observed (uncorrected) orthogonal strains
ϵ1 , ϵ2 = corrects strains
Kt = transverse sensitivity of strain gauges
νo = 0.285 (Poisson’s Ratio under which strain gauges were calibrated for gauge factor)
After correcting the lateral strain indication for transverse sensitivity, divide the result by the
indicated longitudinal strain to obtain the Poisson’s Ratio.
2.11 References
• Measurements Group Inc, Experiments in Mechanics, Strain Gage Series, E­101, Modulus
of Elasticity ­ Flexure, 1982, Measurements Group Inc., Education Division, Raleigh, NC.
• Measurements Group Inc, Experiments in Mechanics, Strain Gage Series, E­102, Poisson’s
Ratio ­ Flexure, 1982, Measurements Group Inc., Education Division, Raleigh, NC.
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2.12
Poisson’s ratio: Worksheet
2.12.1
Strain measurements
Longitudinal
0 µϵ
Initial (undeflected)
Final (undeflected)
Final minus Initial
2.12.2
Lateral
500 µϵ
Correction of lateral strain for transverse sensitivity
Note that the transverse strain sensed by the lateral strain gauge (i.e., the longitudinal strain on the
bottom of the beam) is negative. Therefore,
−(
ϵt
=
ϵa
)
500
=
(2.19)
From the gauge package data form (or information panel on pregauged forms),
Kt =
(2.20)
C=
(2.21)
From the correction chart,
Corrected lateral strain = 500C =
2.12.3
Calculation of Poisson’s ratio
ν=
2.12.4
500C
(
)
=
(2.22)
Published values of Poisson’s ratio
Material
Aluminum Alloy (specify)
Steel
Cork
Concrete
Plastic (specify)
Plastic (specify)
Rubber
Poisson’s ratio
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42
Chapter 3
The tensile test
3.1
Theory
The purpose of the tensile test is to provide basic data that will assist in developing a theory capable
of predicting mechanical properties of materials. We do know that machines have and continue to
be wonderful servants of mankind and that machines are made of many materials. For this reason,
knowledge of materials and their properties is of great importance in the scheme of things.
There is a whole group of materials characteristics which is called “Mechanical Properties”.
In our case, we are concerned about the physical properties. Knowledge of these properties is a
basic prerequisite before intelligent use may be done of the materials. Everyone uses materials to
some extent, most particularly engineers whose function is in the field of design is to know the
capabilities of available materials.
The tensile test which you will perform today is a very simple test conducted to determine
stiffness, strength, and ductility properties of engineering materials. A long slender specimen is
machined from the material of interest (6061­T6 Aluminum) and placed in a test machine that ap­
plies tension in the long direction of the specimen. Deformation over a gauge length is measured
by mechanical gauges. Since the specimen geometry (cross­sectional area and gauge length), ap­
plied load, and deformation are known, Young’s modulus can be determined. Gauges can also be
mounted to measure deformation transverse to the applied loading direction. In this case, Pois­
son’s ratio can be determined. As you can see this test allows us to determine the same mechanical
property of the material as determined in Chapter 2.
There are two major differences between these two experiments:
1. during our first test we did not exceed the elastic (proportional) limit, nor reach the material
failure, and
2. we did not use any mechanical gauges to measure strains.
The ductility of a material provides a measure of the deformation that occurs prior to failure.
Ductility can be quantified by the strain­to­failure. Some ductile materials, i.e. those that have a
large strain­to­failure, exhibit a nonuniform reduction in cross­sectional area known as necking.
Two different measures of stress and strain are commonly used. Engineering stress and strain
use the original cross­sectional area and length, respectively, for references. True stress and strain
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σult
σr
σy
Stress
PL
Offset
Strain
Figure 3.1: The tensile test
use the current cross­sectional area and length, respectively, for references. The proportional limit
(PL), yield stress (or yield strength)1 (σy ), ultimate stress (or ultimate strength) (σult ), and fracture
stress (σr ) can all be determined directly from the engineering stress­strain curve plotted from a
tensile test as shown in Figure 3.1.
3.2
Objective
To determine the physical properties (characteristics) of an aluminum (6061­T6) specimen in ten­
sion.
3.3
Equipment
• Scott testing machine
• Dial extensometer
• Micrometer or dial caliper
• Aluminum specimen
• Safety glasses
1
Yield strength is the stress needed to be applied to the specimen in order to reach the yield point. For a given
specimen, yield strength and yield stress are the same. The only difference being yield strength is a property of the
material, whereas yield stress is just the amount of stress induced.
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Caution
• SAFETY GLASSES MUST BE WORN DURING THIS EXPERIMENT!
3.4
Procedure
1. Measure the diameter of the specimen as accurately as possible, using either a micrometer
or caliper. Estimate the maximum error in this measurement. Record the nominal value and
estimated error on the data sheet provided.
2. Arrange the specimen and extensometer in the Scott machine. Slowly apply a tensile load
until a gauge pressure of 2000 psi is indicated on the pressure transducer. (Note: If the appa­
ratus is working properly, the extensometer should indicate an elongation of approximately
0.003 in.). Release the load and see if the dial returns to 0. If not, tighten the screws holding
the specimen and zero the extensometer. (This initial loading is performed to ensure that the
specimen is securely held in the machine.)
3. Apply pressure slowly in increments of 500 psi up to 4000 psi. Then proceed very slowly in
increments of 200 psi up to 5000 psi. Record the pressure and extension in the specimen at
each step.
4. Carefully remove the extensometer from the specimen.
5. Apply pressure very slowly in increments of 100 psi until the specimen fractures.
6. Remove the specimen from the machine and release the load.
7. Measure, as accurately as possible, the length of the fractured specimen.
3.5
Notes
1. Be sure to set the needle on the pressure gauge to record the fracture pressure. To check
the accuracy of the pressure gauge, check the needle reading with the pressure transducer
indicator at several pressure increments during the test.
2. At least one person not busy taking readings should attempt to observe the physical changes
in the specimen just prior to fracture.
3. The piston area of the Scott machine for tension mode is 0.223 in2 . The length of the specimen
is 2.0 inches. For the purposes of the error analysis, assume that these are exact values (i.e.,
∆Ap = 0, ∆L = 0).
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3.6
MNG 303
Exercises
1. Calculate the engineering stresses and strains corresponding to your recorded pressures and
elongations. Plot these stresses and strains to construct a stress­strain diagram. Also calculate
and plot (preferably on the same diagram, using a different line style) the true stresses and
strains. Include the calculated stresses and strains, and the stress­strain diagram(s) in the
“results” section of your report. In the discussion section, compare the true stress­strain
curve to the engineering stress­strain curve. Why would one be more useful or practical than
the other?
2. Calculate the yield strength (by the 0.2% offset method), ultimate strength, and (approximate)
fracture stress and (approximate) fracture strain. Indicate these values on your stress­strain
diagram.
3. Calculate (from the linear part of the stress­strain diagram) the modulus of elasticity (E)
of the test specimen and compare with the modulus of elasticity determined Lab Session 2.
Calculate also the maximum percent error in E. Compare your estimate of E to the nominal
value given below.
3.7
Formulas and other assorted useful information
Basic and equations and derivations are provided below (from Beer and Johnston, 1981 and other
references). Assuming that material volume remains constant, the true stress σ̄ can be estimated
as:
F Ao
F Ao
F (L)
F (Lo + δ)
F
F
σ̄ =
=
=
=
=
=
(1 + ϵ) = σ(1 + ϵ)
(3.1)
A
A Ao
Ao A
Ao Lo
Ao Lo
Ao
True strain ϵ̄ can be estimated by the rate of instantaneous increase in the instantaneous gauge
length:
Z
(Lo + δ)
dL
δ
ϵ̄ =
= ln
= ln(1 + ) = ln(1 + ϵ)
(3.2)
L
Lo
Lo
where
σ̄, ϵ̄ = true values for stress and strain,
σ, ϵ = stress and strain,
F = the applied force on specimen,
δ = the reading from the extensometer that corresponds to the change in length,
A = the cross­sectional area of the test specimen under specific load,
L = the length of the specimen under specific load,
d = the diameter of the specimen under specific load,
Ao = the initial cross­sectional area of the test specimen,
Lo = the initial length of the test specimen (2.0 in),
do = the initial length of the test specimen.
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The following equations are also used in the above derivations:
σ=
F
,
Ao
ϵ=
δ
,
Lo
d2
Ao = π o ,
4
A=π
d2
4
(3.3)
The force and stress applied to the specimen are related through:
F = pAp
(3.4)
where
p = the applied pressure on the specimen,
Ap = the area of the piston (0.233 in2 ).
The stress and strain on the specimen are related through:
σ = Eϵ
(3.5)
F /Ao
stress
=
strain
δ/Lo
(3.6)
F Lo
4F Lo
4pAp Lo
F Lo
= 1 2 =
=
2
Ao δ
πdo δ
πd2o δ
πdo δ
4
(3.7)
4pAp Lo
πd2o δ
(3.8)
where E is the modulus of elasticity.
Combining the above equations:
E=
E=
E=
For the error analysis (under the assumption that Ap and Lo are “exact” values):
∂E
∂E
∂E
∆E =
∆p +
∆d +
∆δ
∂p
∂d
∂δ
(3.9)
Typical properties of 6061­T6 Aluminum are given below. Note that measured values may vary
and that 1 ksi = 1000 psi.
• Specific weight: 0.098 lb/in3 (2710 kg/m3 )
• Ultimate strength (σult ): 43,000 psi = 43 ksi (290 MPa)
• Yield strength (σy ): 37,000 psi = 37 ksi (255 MPa)
• Modulus of elasticity (E): 10 × 106 psi (69 GPa)
• 17% Elongation: (100 × (lf –lo )/lo ) over a 2 inch test range.
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3.8
MNG 303
References
• Bauld, Mechanics of Materials. 2nd Edition, PWS Publishers, Boston, 1982, pp. 43­54, 69­
70.
• Beer and Johnston, Mechanics of Materials, McGraw­Hill, New York, 1981, pp. 31­39.
• Lardner and Archer, Mechanics of Solids: An Introduction. McGraw­Hill, New York, 1994,
pp. 20­24
• Riley and Zachary, Introduction to Mechanics of Materials, Wiley and Sons, New York, 1989,
pp. 46­52.
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Figure 3.2: Tensile test setup
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3.9
MNG 303
Tensile test worksheet
Group Members:
Date:
Specimen Material: 6061­T6 Aluminum
Specimen Diameter:
Specimen Cross­Sectional Area:
Gauge Length: 2.0 in
Pressure (psi)
Elongation (in)
Force (lb)
50
Stress (psi)
Strain (in/in)
Chapter 4
Experiments in shear failure
4.1
Theory
4.1.1
Axial and shear stresses
Normal stress (σ) represents the intensity of the force F perpendicular or normal to a section (with
a cross­sectional area equal to a) and is given by the following equation. Normal stresses can be
compressive or tensile (Figure 4.1).
F
(4.1)
σ=
a
Shear stress represents the intensity of the force F that acts tangential to a section (with a cross­
sectional area equal to a) and is given by the following equation (Figure 4.1).
τ=
4.1.2
F
a
(4.2)
Axial stresses
In our study of the tensile test, we considered a thin rod of undeformed cross­sectional area a
subjected to a variable tensile force F . It was assumed that the load was distributed uniformly over
the cross­section of the rod far from the loading grips so that the engineering stress, σ, was given
by:
F
(4.3)
σ=
ao
On the other hand, the true stress, σA , also was defined by:
σA =
F
aA
(4.4)
in which aA denotes the actual, deformed cross­sectional area of the rod due to the force F applied at
that moment. It was seen that for the aluminum specimen, the engineering stress as a function of the
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Normal stress
Shear stress
Figure 4.1: Normal and shear stresses
strain eventually reached a maximum value and subsequently the specimen failed. The maximum
engineering stress encountered in this test was called the ultimate tensile stress; and it was defined
by:
Fmax
(4.5)
σult =
ao
When the ultimate stress is reached in a tensile test, a brittle material behaves differently; it breaks
suddenly. For a ductile metal, such as aluminum, the rod appears to deform fairly uniformly until,
suddenly, at the ultimate stress it begins to neck, that is, its uniformly deformed cross­section
becomes greatly reduced over a small portion of the test section. As consequence, further elongation
is accompanied by an increase in the true stress and by a decrease in the engineering stress as the
load falls from its maximum value. In response to the sharp decrease in the local cross­sectional
area until, finally, the rod breaks.
4.1.3
Shear stresses
Materials have a similar behavior in their response to shear loading. The purpose of the present
experiment is to study the ultimate shear stress in two simple situations: a punch test and a shear
pin test.
In the punch test (Figure 4.2), the ultimate shear stress is defined by the relation:
τult =
Fmax
πDt
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(4.6)
Solid Mechanics Lab Manual, January 7, 2023
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in which t denotes the undeformed specimen thickness, D is the undeformed punch diameter and
Fmax is the maximum normal load supported by the specimen under the punch before shearing
failure occurs.
Figure 4.2: Punch test
In the shear pin test (Figure 4.3), the ultimate shear stress is defined by:
τult =
Fmax
πD 2
4
(4.7)
in which D is the undeformed pin diameter and Fmax is the maximum transverse load supported
by the pin before failure occurs.
Figure 4.3: Shear pin test
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Solid Mechanics Lab Manual, January 7, 2023
4.2
MNG 303
Objective
Determine the physical characteristics of materials in shear. In our case we will test 6061 ­ T6
Aluminum.
4.3
Equipment
• Scott testing machine
• Micrometer or dial caliper
• Material specimens (6061 – T6 Aluminum)
• Safety Glasses
Caution
• SAFETY GLASSES MUST BE WORN DURING THIS EXPERIMENT!
• DO NOT EXCEED AN APPLIED PRESSURE OF 7000 PSI!
4.4
Procedure
4.4.1
Shear pin test
1. Measure the diameter of the pin as accurately as possible, using either a micrometer or caliper.
2. Insert the pin in the shear­pin test assembly in the Scott machine. Apply pressure until the
pin breaks. Record the pressure at which the pin broke (if the “peak” button has been set on
the transducer display, this value will be frozen on the display). Release the pressure in the
cylinder and reset the transducer read­out.
3. Repeat for two additional pins.
4.4.2
Punch test
1. Measure the thickness of the specimen as accurately as possible, using either a micrometer
or caliper. Measure the diameter of the punch as accurately as possible. Record these two
values on the data sheet.
2. Place the specimen in the Scott machine, as demonstrated by the laboratory instructor. Apply
pressure to the specimen until failure occurs. Record the pressure at which failure occurred.
Release the pressure, and reset the pressure transducer.
3. Repeat for at least three total punches.
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Figure 4.4: Punch test (left) and shear pin test (right) setup
Notes
The piston area of the Scott machine for:
• Tension mode (shear pin test) is 0.223 in2 .
• Compression mode (punch test) is 0.373 in2 .
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4.5
MNG 303
Exercises
1. For the shear pin test, determine the ultimate shear stress for each pin tested, and the average
value for the three pins. Give possible reasons for any variance in the individual results.
2. For the punch test, determine the ultimate shear stress for each punch, and the average value
for the three punches. Give possible reasons for any variance in the individual results.
3. If the punch test and the pin tests were conducted with the same materials, compare the values
for ultimate shear stress. If they are not the same, give possible reasons for the discrepancy.
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4.6
MNG 303
Formulas and other assorted useful information
F
A
(4.8)
πD2
A=
4
(4.9)
F = P × Atension
(4.10)
τ=
Shear pin test
where D is the diameter of the pin
where P is the applied pressure and Atension is the piston area in tension mode (0.223 in2 ).
Punch test
A = πDt
(4.11)
F = P × Acompression
(4.12)
where D is the diameter of the pin
where P is the applied pressure and Acompression is the piston area in compression mode (0.373 in2 ).
Typical properties of 6061 – T6 Aluminum (Measured values may vary)
• Ultimate strength (σult ): 42,000 psi (290 MPa)
• Ultimate Strength in shear (τult ): 27,000 ­ 30,000 psi (185 ­ 207 MPa)
4.7
References
• Bauld, Mechanics of Materials, 2nd Edition, PWS Publishers, Boston, 1982, pp. 43­54, 69­
70.
• Beer and Johnston, Mechanics of Materials, McGraw­Hill, New York, 1981, pp. 31­39.
• Lardner and Archer, Mechanics of Solids: An Introduction, McGraw­Hill, New York, 1994,
pp. 20­24.
• Riley and Zachary, Introduction to Mechanics of Materials, Wiley and Sons, New York, 1989,
pp. 46­52.
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4.8
MNG 303
Worksheet
Group Members:
Date:
Shear pin test
A = πD4
2
Material
Pin Diameter
(inch)
Pressure (psi)
Force (lb)
Ultimate
Strength (psi)
Average
Ultimate
Strength (psi)
Pin Diameter
(inch)
Pressure (psi)
Force (lb)
Ultimate
Strength (psi)
Average
Ultimate
Strength (psi)
Punch test
A = πDt
Material
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4.9
MNG 303
Examples
Figure 4.5: Cantilever beam
Figure 4.6: Bridge
Figure 4.7: Crane
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MNG 303
Figure 4.8: Rivet
V
V
Shear
M
Bending
T
M
T
Torsion
Figure 4.9: Loading modes
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MNG 303
Figure 4.10: Bridge structure subjected to bending (created by Wikicommons user
Matthias079 commonswiki and licensed under CC BY­SA 3.0)
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MNG 303
62
Chapter 5
Neutral axis of a T­beam
5.1
Objective
To locate the neutral axis of a T­beam using three different ways:
1. Using simple balance method,
2. Calculation based on T­beam geometry (Figure 5.1),
3. Based upon strain measurements.
5.2
Equipment
• Scott testing machine
• T­Beam specimen with strain gauges
• Model P3 Strain Indicator and Recorder or equivalent
5.3
Procedure
1. Measure (from the base of the beam) the locations of each of the four strain gauges. You
may need to remove the beam from the Scott machine to do this. Be careful not to damage
any of the strain gauges or wires in the process.
2. Once these measurements have been recorded, replace the beam in the Scott machine. Be
careful that the loading yoke does not come into contact with any of the strain gauges.
3. Make sure that the wires connecting the strain gauges to the switch and balance box are
connected tightly.
4. Set the Gauge Factor of the strain indicator to 2.09.
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Figure 5.1: Cross­section of the T­beam
5. Switch the strain indicator to “Run” mode and zero the four active channels of the switch
and balance box as closely as possible.
6. Apply pressures of 1000, 2000, 3000, and 4000 psi on the Scott machine.
7. Record the strain from each channel at each pressure increment. Release the pressure on the
Scot machine and record the reading at 0 psi.
8. Repeat this procedure at least once and use the data from the last set of readings in all calcu­
lations for the report. This is to provide an estimate of the uncertainty in the strain readings.
Notes
• The Scott machine is in tension mode for this experiment. The piston area is 0.223 in2 .
5.4
Exercises
1. Before proceeding with any calculations, subtract the strain measured at 0 psi from all other
readings from each gauge. Assuming that the reading at 0 psi provides an indication of
any potential bias in the strain gauge, this will serve as a rough calibration for the strain
measurements.
2. For each of the four load levels (1000, 2000, 3000, and 4000 psi) plot the measured strains
versus vertical location of the strain gauge. Find the equation of each four lines and solve
each equation to find the vertical location correspondingly to zero strain. Include the average
of the four values in the results section of your report.
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MNG 303
3. For each of the strain gauges, plot the measured strains versus the applied force. From these
four lines, choose any two that have opposite slopes. Calculate the slopes of these two lines.
Considering that the line with negative slope (say, m1 ) corresponds to strain measured a
distance (d1 ) above the neutral axis, while the other (m2 ) corresponds to strain measured
a distance (d2 ) below the neutral axis, linearly interpolate between these two lines at each
applied load to determine the distance (d′ ) at which the slope (m′ ) is zero (corresponding to
neutral axis). Include the average of the four values in the results section of your report.
4. Calculate the location of the centroid or neutral axis using geometry of the T­Beam (see
Figure 5.1). To be able to perform these calculations measure required dimensions of the
provided piece of the T­Beam. To determine the thickness of each element of this T­Beam
assume average of two measured values.
5. Use the balance method to determine approximate values of the centroid location.
6. In the discussion section of your report, compare the results of these two above mentioned
methods with the location of the centroid you calculated based on your gauge measurements.
Provide an explanation for any discrepancy between these results.
5.5
References
• Bauld, Mechanics of Materials, 2nd Edition, PWS Publishers, Boston, 1982, pp. 43­54, 69­
70.
• Beer and Johnston, Mechanics of Materials, McGraw­Hill, New York, 1981, pp. 31­39.
• Lardner and Archer, Mechanics of Solids: An Introduction, McGraw­Hill, New York, 1994,
pp. 20­24.
• Riley and Zachary, Introduction to Mechanics of Materials, Wiley and Sons, New York, 1989,
pp. 46­52.
5.6
Formulas and other assorted useful information
The following words of wisdom are from Mechanics of Materials, Beer and Johnston, 1981, page
156:
For a member subjected to pure bending, and as long as the stresses remain in the
elastic range, the neutral axis passes through the centroid of the section.
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y
P
z
b
y
h
L
x
h
z
P
y
b
L/2
Lz = 1/12 bh³
x
P/2
V
x
- P/2
M
PL/4
x
Y
Compressive Stress
x
z
Tensile Stress
Figure 5.2: Neutral axis of a rectangular beam
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Y
MNG 303
Compressive Stress
X
Z
(a)
Tensile
Y
M
X
Z
(b)
Y
M
X
(c)
Figure 5.3: (a) Distribution of normal stress across the section. (b) Bending moment acting on the
cross­section. (c) Two­dimensional representation of stress distribution and bending moment.
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Figure 5.4: Location of the neutral axis of a T­beam (calculated based on T­beam geometry)
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Figure 5.5: Location of the neutral axis of a T­beam (based upon strain measurements)
69
#4
500
#3
Strain (x 10-6)
250
#2
0
#1
-250
-500
-750
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750
70
-1000
-1250
-1500
1000 psi
2000 psi
3000 psi
4000 psi
-1750
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
Vertical Location (in)
Figure 5.6: Location of the neutral axis
MNG 303
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71
Figure 5.7: Force vs Strain
MNG 303
Solid Mechanics Lab Manual, January 7, 2023
5.7
MNG 303
Neutral axis test worksheet
Group Members:
Date:
Gauge Factor = 2.09
Gauge Locations:
Gauge 1:
Gauge 2:
Gauge 3:
Gauge 4:
Pressure (psi)
Strain Gauge 1
Strain Gauge 2
72
Strain Gauge 3
Strain Gauge 4
Chapter 6
Cantilever flexure: flexure stresses in beams
6.1
Theory
The cantilever beam, is a very widely used structural element. Examples include airplane wings,
supports for overhanging roofs, gear teeth, leaf springs, and the front wheel spindles of automo­
biles. As seen from these examples, the cantilever beam may vary in section along its length, the
mounting details at the end of the beam may differ greatly, and the system of applied loads can take
a variety of dorms. The designation “cantilever” is commonly applied to any beam which is built­
in and supported at only one point, and which is loaded by one or more point loads or distributed
loads acting perpendicular to the beam axis. The built­in end of a cantilever beam cannot move
transversely or rotate.
This experiment will study the cantilever beam in its simplest and purest from, that is, a parallel­
sided beam of constant cross­section, rigidly clamped at its fixed end and deflected by a single point
load on the beam centerline near the free end.
The cantilever beam is shown in the schematic diagram in Figure 6.1, along with the associated
shear force and bending moment diagrams. Static equilibrium requires that the vertical shear force
at any section, X of the beam be equal to the load, P . Thus, the shear diagram is of constant height
from the point of load application to the fixed end of the beam. The bending moment at any section,
X, is the product of the shear force times the moment arm, M = P × x (Figure 6.2), where x is
the distance the distance of section X from the loading point.
Therefore, the bending moment varies linearly from zero at the loading point to P × L at the
fixed end (Figure 6.3). The forms of the diagrams are consistent with the fact that the shear force at
each section is equal to the derivative of the bending moment (the slope of the moment diagram).
That is,
dM
(6.1)
V =
dx
where:
V = the shear force, lbs (N)
M = the bending moment, in­lbs (N × m)
x = the distance of section X from the loading point, in. (m)
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P
Beam
L
P
Static Equivalent
M = P*L
P
Shear Force, V(X)
Figure 6.1: Cantilever beam ­ moments
V(X) = P
X=L
Distance, X
P
Free-Body Diagram
M(X) = P*X
X
V(X)
Figure 6.2: Cantilever beam ­ Shear force moments
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Note that for a beam in pure bending since no load is applied in the z­direction, σz is zero
throughout the beam. However, because of loads applied in the y direction to obtain the bending
moment, σy is not zero, but it is small enough compared to σx to neglect. In addition, σx while
varying linearly in the y direction is uniformly distributed in the z direction. Therefore, a beam
under only a bending load will be in a uniaxial, albeit a non uniform, stress state. In other words
the axial normal stress, like the strain, increases linearly from zero at the neutral axis to a maximum
at the outer surfaces of the beam.
Therefore, for the cantilever beam used in this experiment the stress is uniaxial everywhere on
the beam surface except in the immediate vicinity of the loading and the demand end. The surface
stress σ(x) at any section, X, along the beam axis can be calculated from:
σ(x) =
M (x)
6P x
M (x)c
=
= 2
I
Z
bt
(6.2)
where:
σ(x) = axial normal stress on the beam surface at section X, psi (N/m2 )
c
= distance from the neutral axis to extreme fiber of beam surface, in. (m)
I
= moment of inertia of the beam cross­section, in4 (m4 )
P
= the load, lbs (N)
b
= the beam width, in. (m)
t
= the beam thickness, in. (m)
Z = the section modulus of the beam, in3 (m3 )
For the uniaxial stress state, normal strain can be expressed as follows according to Hooke’s
Law:
ϵ = σ/E
(6.3)
where:
σ = the normal stress, psi (N/m2 )
ϵ = the normal strain, in/in (m/m)
E = the modulus of elasticity, psi (N/m2 )
Substituting equation 6.2 into equation 6.3 allows the determination of the longitudinal strain
at any section, X, and therefore:
ϵ(X) =
6P X
6M (X)
M (X)
=
=
Ebt2
Ebt2
EZ
(6.4)
Equation 6.4 states that the axial strain varies linearly along the beam from zero at the loading
point to a theoretical maximum of 6P L/Ebt2 at the fixed end.
6.2
Equipment and supplies
• Flexor, cantilever flexure frame
• High­strength aluminum alloy beam, 1/4 × 1 × 12­1/2 in (6 × 25 × 320 mm)
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Bending Moment, M
M(X) = P*L
ΔM
X=L
ΔX
Distance, X
X=0
Figure 6.3: Bending moment distribution on cantilever beam
• Micro­measurements temperature­compensated strain gauges (three)
• Model P3 Strain Indicator and Recorder or equivalent
• Strain gauge applications kit, containing gauge bonding supplies, hook­up wire, etc.
• Micrometer or Vernier caliper
• Accurate drafting or machinist’s scale
6.3
Objectives
The purpose of this experiment is to:
1. determine the shear force and the load from strain measurements,
2. verify the linearity of the strain along the beam axis,
3. confirm the shear force and moment relationship by comparing two different stress determi­
nations, and
4. calculate the deflection of the beam caused by the point load source.
Generally, in this experiment, the validity of elementary beam theory applied to the cantilever
beam in its simplest and purest from will be verified. This form is a parallel­sided beam of constant
cross­section, rigidly clamped at its fixed end and deflected by a single point load on the beam
centerline near the free end.
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6.4
MNG 303
References
1. Figliola R.S. and D.E. Beasley, Theory and Design for Mechanical Measurements, John Wi­
ley and Sons, New York, 1997.
2. Lardner and Archer, Mechanics of Solids: An Introduction. McGraw­Hill, New York, 1994,
pp. 20­24
3. Measurements Group Inc., Experiments in Mechanics, Strain Gage Series, E­105, Cantilever
Flexure, 1982, Measurements Group Inc., Education Division, Raleigh, NC.
6.5
General procedure
1. Measure the width, thickness and length (from point load to fixed end) of Beam No. E­105
using the finest scale division on your measuring instrument.
2. Measure the distances (a) from the point load to strain gauge (1), (b) between strain gauges
(1) and (2), and (2) and (3) and (c) from the point load to the fixed­point.
3. Wire the strain gauge meter to the flexor for differential strain measurements according to
the diagrams in the Appendix (Figures 6.6 and 6.7).
4. Zero the differential strain gauge reading between strain gauges (1) and (2). Repeat for the
differential measurement between gauges (2) and (3). Turn back to the reading for stations
(1) and (2) and zero again. Record the differential reading for gauges (2) and (3), while
realizing that this reading should be near a zero value.
5. While staying on the station for the gauge (2) and (3) differential, apply a point load by
turning the micrometer until a strain reading until a strain reading of [(ϵ2 − ϵ3 ) + 600]µϵ is
obtained between stations (2) and (3). Record the number.
6. Read and record the differential strain measurement between gauges (1) and (2).
7. Calculate the load using the two differential strain measurements and determine the average.
8. Without releasing the load, turn the indicator off and rewire the connection with the flexor to
obtain individual strain measurements as per the diagrams in the Appendix (Figures 6.6 and
6.7).
9. Zero the strain readings for the stations corresponding each strain gauge. Record any non­
zero values while realizing these values should be near zero.
10. Release the load. Read and record the strain measurements on gauges (1), (2), and (3).
11. Use the graph to determine the slope of the strain versus length relationship.
12. Determine the load and compare the value with that calculated in step 8.
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Clamp
1"
1"
(25.4) (25.4)
3"
(76.2)
MNG 303
3"
(76.2)
4.5"
(114.3)
1"
CL
1/4"
12.5”
(320mm)
Figure 6.4: Strain gauge location on beam
13. Provide a scaled diagram showing the deflected beam using the calculated deflection values,
δ, as a function of distance from the fixed end.
14. Prepare your report in accordance with the instructions given under Report.
6.6
Detailed instructions
This experiment can be performed by using a beam with three strain gauges, installed at uniform
intervals along the axis of the beam as shown in the gauge installation diagram in Figure 6.4.
Because the strain distribution along the beam is presumed to be linear, equation 6.1 can be
rewritten as:
∆M
(6.5)
V =
∆X
Solving equation 6.4 for M and substituting into equation 6.5:
Ebt2 ∆ϵ
V =
6 ∆X
(6.6)
From equation 6.6, the shear force can be obtained from the difference in strain indications of
any pair of gauges, divided by the distance between the gauges.
V(1−2) =
Ebt2 (ϵ1 − ϵ2 )
6 (X1 − X2 )
(6.7)
Ebt2 (ϵ2 − ϵ3 )
(6.8)
6 (X2 − X3 )
Either equation 6.7 or equation 6.8 gives the shear force, and thus the load applied to the beam.
Since the answers will generally differ slightly due to experimental error, their average is the best
estimate of the load.
V(2−3) =
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The technique used here is a convenient method for directly determining the shear force in
any uniform beam subjected to point loads, since in this case the moment and strain variations are
always linear between loads and/or reactions.
The linearity of strain distribution can be verified by plotting the three individual strain indica­
tions on the included graph sheet. After drawing the best straight line through the data, the slope of
the line, ∆ϵ/∆X, can be used to confirm the previously calculated load. Since the beam is uniform
in section, the graph thus drawn represents not only the strain distribution, but with a scale change,
the bending moment diagram as well.
6.6.1
Determination and comparison of stress values
With the load known from either of the above measurements, the stress at station (1) can be calcu­
lated from the following:
Mi c
6P Xi
σi =
=
(6.9)
I
bt2
However, the stress at station (1) can also be calculated directly from the measured strain at that
point with Hooke’s Law for uniaxial stress. Thus,
σi = Eϵi
(6.10)
The stress calculated from equations 6.9 and 6.10 can be compared as verification of the fun­
damental beam relationships used in this experiment.
6.6.2
Determination of deflection
As a result of the point load applied at the end of the beam, a deflection occurs whereby the beam
bends in the direction of the applied force (Figure 6.5).
The deflection of the beam, δ at any given point along the length of the beam can be determined
by the following expression:
P X2
(3L − X)
(6.11)
δ=
6IE
where L = beam length, in. (m).
Note that the deflection is zero at the fixed end of the beam and maximum at the loaded end of
the beam.
For the experimental program, the deflection of the beam will be determined and presented in
graphical form.
6.7 Strain gauge selection and installation
The Micro­measurements foil strain gauges are intrinsically temperature­compensated for use on a
material with a particular thermal coefficient of expansion. Because of this, the three strain gauges
employed in this experiment can be used individually in a “quarter­bridge” arrangement, complet­
ing the bridge each time with the precision resistors built into the P3 Strain Indicator and Recorder.
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δ(X=0)
δ(X=L) = 0
δ(X)
Figure 6.5: Beam deflection
For quarter­bridge operation, a “three­wire” circuit is ordinarily used for each gauge in order to
obtain lead wire compensation by placing equal lengths of lead wire in adjacent arms to the bridge
circuit (see Model P3 Strain Indicator and Recorder or Flexor Instruction Manual). It is often con­
venient in minimizing the lead and connection requirements to combine the leads from one side of
each gauge into a common lead. When three­wire circuitry is used, this becomes a pair of leads
which is common to every gauge in the system.
When the strain gauges are used as pairs and connected in adjacent arms of the bridge for half­
bridge operation (see Data Acquisition), two­wire circuitry should be used since the lengths of lead
wire in both arms of the bridge will necessarily be essentially the same. In this case, if the beam
is wired with a pair of common leads, only one of these needs be used, as shown in the wiring
diagram for half­bridge operation.
A strain gauge indicates the average strain under the grid area when placed in a non­uniform
strain field. In this instance, because the strain varies linearly along the beam length, the strain
indicated by each gauge is equal to the strain at the gauge centerline. This being true, the gauge
length of the strain gauge is not critical, and gauges with patterns such as 250BG or 125AD (both
120­ohms and temperature­compensated for aluminum) can be selected.
Following the gauge installation instructions included with the applications kit, lay out the
desired locations of the three gauges along the beam axis as shown in the installation diagram.
Bond the strain gauges in place, following the instructions precisely. Allow the adhesive to cure for
the recommended time, then carefully solder 6­inch (150­mm) lead wires to the gauges as indicated
by the wiring diagram (Figure 6.6). After solder connections are complete, remove the flux, and
apply a protective coating over all gauge installations.
6.8
Data acquisition
Back the calibrated loading screw out of the way and insert the beam into the Flexor with the gauges
on the upper surface as shown in the wiring diagrams in the Appendix (Figures 6.6 and 6.7). Center
the free end of the beam between the sides of the Flexor, and firmly clamp the beam in place with
the knurled clamping screw. Connect the lead wires from the strain gauges to the binding posts of
the Flexor according to these wiring diagrams.
The subtraction of strains required by equations 6.7 and 6.8 in determining the shear force and
load can be accomplished automatically in the P3 Strain Indicator and Recorder. When one gauge is
connected across the “tension” terminals of the instrument, and the other across the “compression”
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terminals, the strain indicated by the instrument is equal to the algebraic difference between the
strains of the two gauges. This electrical arrangement of the gauges is known as “half­bridge”
operation.
Connect one of the common cable leads to the S­ binding post of the strain indicator, and the
independent lead from gauge (1) to the P+ binding post. Connect the independent lead from gauge
(2) to the P­ binding post. The two gauges are now connected in adjacent arms of the bridge circuit
to form the half­bridge. With this arrangement, the bridge­completion resistors (D120 and D350)
are not needed.
After balancing the indicator amplifier, set the gauge factor adjustment to the value given in
the strain gauge package data sheet (or on the beam, if supplied pregauged). Set the instrument to
Run. With the Flexor loading screw still clear of the beam, adjust the balance control of the P3
until the LCD digital display indicates 0. Turn the Flexor loading screw clockwise until the ball
foot comes into contact with the beam and the instrument readout indicates a small value of, say,
100. Continue turning until the zero in the micrometer spindle is conveniently in view. Using the
small spanner wrench supplied with the Flexor, rotate the barrel of the loading micrometer until the
index line coincides with the zero on the calibrated spindle. Note that the vertical displacement of
the loading screw is 0.025 in (0.64 mm) per revolution.
The ball foot of the loading screw is now lightly in contact with the beam, and the strain indicator
is unbalanced. Readjust the strain indicator balance control to obtain an instrument reading of
exactly 0. Do not adjust the balance control again until instructed to do so. The initial (zero beam
deflection) reading for strain gauge element (1) should now be recorded on the worksheet as 0µϵ.
(Since the deflection of the beam, as well as the strain, vary linearly with load, any small load can
represent the zero condition as long as the initial strain is measured at this condition.)
Turn the strain indicator off and disconnect the independent gauge element leads from the P+
and P­ binding posts, leaving the common lead connected to the S­ binding post. Connect the
independent cable lead from gauge element (2) to the P+ binding post and the independent cable
lead from gauge element (3) to the P­ binding post. Turn the instrument on. Without adjusting the
balance controls, note the reading appearing in the indicator display. This is the initial reading for
the gauge (2) ­ (3) combination and should be recorded on the worksheet.
After obtaining and recording the initial reading for gauges (2) ­ (3), add 600µϵ to the value of
this initial reading. With gauges (2) and (3) connected, deflect the beam, by rotating the loading
screw clockwise, until the indicator readout registers the number equal to this sum. Record this
number in the appropriate place on the worksheet.
With the loading screw unchanged, turn off the strain indicator and disconnect the independent
leads for gauges (2) ­ (3) load from the instrument. Reconnect gauges (1) ­ (2) in the same manner
as before. Turn the instrument on again and note the number registered by the indicator readout.
This number is the second reading for gauges (1) ­ (2) and should be recorded in the appropriate
location on the worksheet. Note that except for experimental inaccuracies, the difference between
the final and initial readings for gauges (1) ­ (2) should be 600µϵ as it was for gauges (2) ­ (3).
Leave the beam deflected by the loading screw until unloading is required in the following steps.
Since the subtraction was accomplished in the strain indicator, the foregoing procedure gave
the difference in strain indication between adjacent stations along the beam, but not the strains at
each station. To obtain the individual strains, the gauges must be connected to the strain indicator
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one at a time.
Turn the P3 Strain Indicator and Recorder off and disconnect the gauge (2) independent lead
from the P­ binding post, leaving gauge (1) connected to the instrument. As shown in the wiring
diagram for measuring individual strains, connect the second common lead from the Flexor to the
appropriate D binding post (D120 for 120­ohm gauges or D350 for 350­ohm gauges). Turn the
instrument on and adjust the balance control until the LCD readout indicates precisely 0. This is
the new position of the balance control, and it should not be changed during the remainder of the
experiment.
Record the initial strain indication for gauge (1) as 0µϵ on the worksheet. Turn the indicator off,
disconnect the independent lead from gauge (1) from the P+ binding post, and connect the inde­
pendent lead from gauge (2) to the P+ binding post. Tum the instrument on and without adjusting
the balance control note the initial reading for gauge (2). Record this number at the appropriate
location on the worksheet. Repeat this procedure for gauge (3) and record the initial reading on the
worksheet for gauge (3).
With gauge (3) connected to the instrument, unload the beam until the calibrated loading screw
again reads zero. Note the new reading on the indicator display. This is the final reading for
gauge (3) and should be recorded on the worksheet. Successively connect gauges (2) and (1) to
the instrument and record their final readings on the worksheet as well. The strain readings for all
three gauges will decrease from the initial readings, since the load has been decreased.
6.9
Data analysis and presentation
6.9.1
Shear force from differential strains
1. Measure the width and thickness of the beam with a micrometer and record the dimensions
on the worksheet,
2. Assume the modulus of elasticity is 10.4 × 106 psi (7.17 × 1010 N/m2 = 71,700 MPa).
3. Substitute into equations 6.7 and 6.8 to obtain two values of the shear force.
4. Record the results on the worksheet and average the two numbers for the best estimate of the
load.
6.9.2
Strain and moment linearity
1. Set up the experiment to measure the individual strain at each of the three stations.
2. Construct a plot of strain versus distance and draw a best­fit line.
3. Measure the slope from the strain vs. distance line.
4. Substitute the slope into equation 6.6 and calculate another estimate of the load.
5. Calculate the stress at Station (1) using equation 6.9.
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6. Calculate the stress at Station (1) using equation 6.10.
7. Record and compare the two stress values.
6.9.3
Stress at station (1)
Calculate the stress at station (1) from equation 6.9 using the load as computed from the average
of the differential strain indications. Assuming E = 10.4 × 106 psi (7.17 × 1010 N/m2 = 71,700
MPa), calculate the stress at station (1) directly from ϵi by equation 6.10. Record the two stresses
on the worksheet for comparison.
6.9.4
Deflection measurement
Using the measured values for load, determine the deflection along the entire length of the beam
using equation 6.11.
Construct a plot showing the variation in deflection of the length of the beam.
6.10
Report
Prepare a brief report, describing in your own words the purpose of the experiment, the equipment
and setup used, the procedure followed, and the results obtained. Include the worksheet with all
original data and computations and the graph sheet in your report. Itemize the sources of error in
this experiment and discuss their relative effects on your estimates of the load, and the stress at
station (1).
On the graph of ϵ vs. X, extend the straight line you have drawn downward to the right until it
intersects the abscissa. If the intersection occurs at other than X = 0, what reason would you give?
Generalize what you have learned from this experiment to explain how and why a cantilever beam,
with properly located and connected strain gauges, can serve as a load­measuring transducer the
output of which is independent of the point of load application (over a certain portion of the beam).
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6.11 Appendix
Figure 6.6: Wiring diagram 1
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Figure 6.7: Wiring diagram 2
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6.12
Worksheet
6.12.1
Beam dimensions
Length
Width
Thickness
MNG 303
in. (m) (from the point of load application to the gauge centerline)
in. (m)
in. (m)
6.12.2
Gauge locations
X1
X1 −X2
X2 −X3
=
=
=
6.12.3
Differential strain measurements
in. (m) (from the point of load application to station (1))
in. (m) (between gauges)
in. (m) (between gauges)
ϵ1 − ϵ2
0 µϵ
Initial (“zero” deflection)
Final (maximum deflection)
Final minus initial
6.12.4
ϵ2 − ϵ3
600 µϵ
Computation of load
P1 = V(1−2) =
=
=
P2 = V(2−3) =
=
=
Ebt2 (ϵ1 − ϵ2 )
6 (X1 − X2 )
(
) × 106 (
(6.12)
(
) × 10−6
)
(6.13)
(
) × 10−6
)
)2 (
)(
6
lbs (N)
Ebt2 (ϵ2 − ϵ3 )
6 (X2 − X3 )
(
) × 106 (
)2 (
)(
6
lbs (N)
Best estimate of load:
P̄ =
(
P1 + P2
=
2
)+(
2
86
)
=
lbs (N)
(6.14)
Solid Mechanics Lab Manual, January 7, 2023
6.12.5
Individual strain measurements
ϵ1
0 µϵ
Initial (“zero” deflection)
Final (maximum deflection)
Final minus initial
6.12.6
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ϵ2
ϵ3
Slope of best straight line through individual strain readings
1. Measured from graph:
∆ϵ
(
=
∆X
(
)
=
)
µϵ/in (µϵ/m)
(6.15)
2. Recomputation of force:
P3 =
=
=
Ebt2 (∆ϵ)
6 (∆X)
(
) × 106 (
)2 (
)(
6
(
) × 10−6
)
(6.16)
lbs (N)
3. Maximum deviation from straight line:
Measured from graph:
µϵ
6.12.7
Stress at station (1)
σ1 =
σ1 = Eϵ1 = (
6X1 P
6(
=
bt2
(
)(
)(
)
)2
=
) × 10−6 =
) × 106 × (
87
psi (N/m2 )
psi (N/m2 )
(6.17)
(6.18)
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88
Chapter 7
Introduction to stress concentrations
7.1
Theory
When a load is distributed uniformly over a material surface, the stress is equal to the resultant load
divided by the total area on which it acts. Of course, loads are not always uniformly distributed, so
this simple approach cannot be used in this case to determine the stress. In pure bending of a beam,
for example, the axial load on the beam is zero, and so the average axial stress also must be zero.
Nevertheless, the internal local axial stresses can be large, as illustrated by the stress distribution
shown in Figure 7.1. It is seen that the local stress varies, yet the resultant is zero; in short, the
stress varies from place to place within the body.
But even when the surface loads may be uniformly distributed, it does not follow that the stress
will be everywhere uniform within the body. This fact will be illustrated in this session. Therefore,
it will be necessary to appeal to our broader concept of stress at a point on an infinitesimal element
of area on which a force acts. Obviously, this contact force arises from the action of material on
one side of the area element on the material on the other side of the element. This important local
concept is covered in more detail in the deformable solids course.
Any geometrical discontinuities (holes, notches, cracks, abrupt changes in cross­section) will
introduce stress concentrations into a material. The severity of the stress concentration increases
with increasing flaw size and with a decreasing radius of curvature of the flaw tip (i.e., with in­
creasing sharpness of the discontinuity). Stress concentrations may have a serious effect on fatigue
behavior; their general effect is to reduce the fatigue life. Under repeated loading, stress concentra­
Figure 7.1: A nonuniform stress distribution with zero resultant axial force
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Figure 7.2: Two plates under uniform tensile load
Figure 7.3: Stress distributions on various sections of the two plates
tions may initiate cracks in the localized, highly stressed volume of material near the tip of a flaw.
These cracks can then propagate to make matters worse. Sometimes a small increase in load stress
due to stress concentrations will bring about failure much more quickly. The reduction in fatigue
life increases with the severity of the flaw.
Let us consider what happens when a uniform stress field, σo in a plate under tension is disturbed
by drilling a hole in the plate as shown in Figure 7.2. We ask: What is the stress on the section
AA as compared with that on the section BB? It is reasonable to expect that the stress in the solid
plate in Figure 7.2a will be uniform along both sections AA and BB, as shown in Figures 7.3a and
7.3b. And it also seems intuitively clear that the stress on BB in plate having the hole, if removed
sufficiently far from the neighborhood of the hole, called the far field, ought to be very nearly
uniform too, as shown in Figure 7.3b. It is WRONG, however, to suppose that the force on AA
with the hole is simply redistributed uniformly over the reduced area. The material surrounding the
hole deforms in a nonuniform manner, and the stress at the hole on section AA may be considerably
larger than the stress at the outer edges of the plate on BB; and far away from the hole, the stress
and deformation are unaffected. The distribution of stress, therefore, varies throughout the drilled
plate; it is expected to be largest at the hole, and decay to less than the uniform stress far from the
hole at the uniformly loaded ends.
These non­uniform deformations very frequently are the cause of structural failure; and they
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σο
y
r
σθθ
σrr
σrθ
θ
x
d = 2ro
σο
Figure 7.4: Stress components around a small hole
are very difficult to analyze. The study of the nonuniform stress and deformation, even for linearly
elastic, homogeneous and isotropic materials, requires some rather advanced mathematics in the
theory of partial differential equations. Therefore, investigation of this important problem must be
left to a graduate level course in the Theory of Elasticity. For our purpose here, however, we may
note that it turns out from elasticity theory that for a small circular hole of radius ro in a wide, flat
plate, under tension or compression (uniaxial load), the plane stress components (see Figure 7.4),
are given by the following general relations:
σrr = 0.5(σx + σy )(1 −
4r2 3r4
r2
)
+
0.5(σ
−
σ
)(1
−
+ 4 cos(2θ)
x
y
ro2
ro2
ro
(7.1)
r2
3r4
)
−
0.5(σ
−
σ
)(1
+
) cos(2θ)
x
y
ro2
ro4
(7.2)
σθθ = 0.5(σx + σy )(1 +
where σy = σo and σx = 0.
When θ = 0 the tangential stress σθθ is the same as the normal stress σrr , Thus, equation (7.2)
shows that when r >> ro , the stress far away from the hole is equal to the uniform stress σo . But
at the edge of the bole at r = ro , σθθ = 3σo , i.e., the stress is three times greater! We see also
when θ = π/2, σθθ is the same as the normal stress σx . Equation (7.2) shows that when r >> ro,
σθθ = 0; whereas, at the edge of the hole where r = ro, σθθ = σo i.e., the hole is being compressed
in the x direction. These important results are illustrated in Figure 7.5.
If the diameter d = 2ro of the hole is not small compared with the plate width w (Figure 7.6),
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y
σo
3σo
~σo
x
Figure 7.5: Stress distribution in a plate, with a small circular hole, under tension
the value of the tangential stress σθθ at θ = 0 exceeds 3σo . In this case the ratio between σθθ and
σo (i.e., Kg = σθθ /σo ) can be determined from Figure 7.6. Note that Kg approaches 3 as the ratio
d/w approaches zero.
In view of the complex nature of problems of the type considered here, it is common for en­
gineers to conduct experiments to measure stresses near holes and around any imperfections in a
structure. Our purpose in this laboratory session is to study the foregoing problem and compare the
results of experiment with the theoretical results described above.
The stress concentration factor can be determined only by strain measurements. Therefore, to
be precise, our experimental measurements of a stress concentration factor will be referred to in the
Exercise as a strain concentration factor. Nevertheless, it turns out from theoretical considerations
that at the points of interest in the drilled plate, the values of the stresses and strains there are
proportional, so for this case the experimental strain concentration factor should agree with the
value determined for the theoretical stress concentration factor.
Note that in our case, working with cantilever beam, we do not have uniaxial type of load,
therefore, concentration factor Kt cannot be compared with concentration factor shown in Figure
7.6, and the value of the concentration factor will be smaller.
7.2
Experiment on stress and strain concentration
7.2.1
Objective
The purpose of this experiment is to demonstrate the existence of stress and strain concentration
in the vicinity of a geometrical discontinuity in a cantilever beam, and to obtain an approximate
measure of the elastic (theoretical or geometric) stress concentration factor, Kt . In this instance,
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σo
d
w
Stress concentration factor Kt
3
Kg = Kt (x / (w-d))
2
1
0
0.5
1.0
Ratio d/w
Figure 7.6: Stress concentration factor for uniaxial load (Riley and Zachary, 1989, p. 124)
the discontinuity is simply a circular hole, drilled through the depth of the beam on its centerline.
Generally in this experiment the validity of elementary theory applied to the concentration of stress
and strain due to the discontinuity of the material will be verified.
7.3
Introduction
The presence of any geometric irregularity in the shape of a loaded mechanical part of structural
member impedes the orderly flow of stress trajectories, causing them to crowd together, and locally
increasing the stress above the nominal load as calculated by conventional mechanics of materials
formulas. Such an irregularity or discontinuity is referred to as a “stress­raiser”. The sketch in
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Figure 7.7: Experimental setup
Figure 7.7 shows the stress distribution at two sections of a cantilever beam, and illustrates the
presence of stress concentration.
At Section A, the stress is uniform across the width of the beam, and can be calculated from
the following relationship:
MA c
6P L
=
σA =
(7.3)
IA
bt2
where:
σ = stress, psi (N/m2 )
M = bending moment, in­lbs (mN)
I = moment of inertia of beam in cross­section, in4 (m4 )
P = load, lbf (N)
c = half­thickness of beam, in. (m)
At Section B, the nominal stress, based upon the net area of the section, is:
σB (NOM) =
6P ℓ
MB c
=
IB
(b − d)t2
(7.4)
If the location of the hole is selected so that
b−d
ℓ
=
L
b
(7.5)
then the nominal stress at Section B is the same as that at Section A. The maximum stress at Section
B, however is much greater, due to the stress concentration effect. As shown in the sketch in Figure
7.7, the maximum stress exists at the edge of the hole, on the transverse diameter, and the stress
decreases rapidly with distance from the hole. By definition, the stress concentration factor, Kt , is
the ratio of the maximum stress at the stress­raiser to the nominal stress at the same point. That is,
K=
σB (MAX)
σB (MAX)
=
6P ℓ
σB (NOM)
(b−d)t2
(7.6)
Since the nominal stress at both sections of the beam, and the peak stress at the edge of the hole,
are all uniaxial, the strain and stress are proportional, if the proportional limit of the beam material
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is not exceeded in the experiment. Thus, the stress concentration factor is equal to the ratio of the
maximum nominal strains at Section B. Therefore,
K=
7.3.1
ϵB (MAX)
ϵB (MAX)
=
ϵB (NOM)
ϵA
(7.7)
Equipment and supplies
• Flexor, cantilever flexure frame
• High­strength aluminum alloy beam, 1/4 × 1 × 12­1/2 in (6 × 25 × 320 mm)
• Model P3 Strain Indicator and Recorder or equivalent
7.3.2
Procedure summary
1. To complete this experiment each student should follow the instructions provided in this
manual.
2. Compare and discuss your calculated concentration factor with value you can get from pro­
vided graph Kt vs. ratio d/w (for uniaxial load), see Figure 7.6.
3. Prepare your report in accordance with the instructions given under Report Preparation and
Organization.
7.3.3
General setup
In this experiment, the beam will be located with the Flexor until a predetermined nominal axial
strain level of 2,000 µϵ is reached at Section B (Figure 7.7). As described in the Introduction, the
nominal strain at Section B will be measured, not at that point on the beam, but instead at Section
A where the measurement can be made more conveniently and accurately. It is important not to
exceed a nominal strain of 2,000 µϵ, since the actual strain at the edge of the hole is much higher
than the nominal, and excessive strain could produce local yielding.
The actual strains at the region of the stress concentration will be measured with three very
small strain gauges placed in Section B at varying distances from the edge of the hole, with one of
the gauges directly adjacent to the edge. The strains indicated by the three gauges will be plotted on
the graph sheet (see Figure 7.9 included in the Appendix of this experiment) at the locations of the
respective gauge centerlines. A smooth curve can be drawn through the resulting three data points
to show the strain distribution in the vicinity of the hole, it is necessary to extrapolate the data to the
edge to obtain an approximate value for technique given in the following section on Data Analysis
and Presentation.
The ratio of the maximum to the nominal strain at Section B is the strain concentration due
to the disruptive presence of the hole. If the proportional limit of the beam material has not been
exceeded during the experiment, the stresses are proportional to the strains, and the same ratio
represents the stress concentration factor, Kt .
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7.4
MNG 303
Strain gauge selection and installation
In measuring a stress concentration with a strain gauge, it must be kept in mind that the gauge tends
to indicate the average strain in the area covered by the grid. Since the strain in the immediate
vicinity of a stress­raiser decreases steeply with distance, it is obviously necessary to select the
smallest practical strain gauge and bond it in place as close as possible to the edge of the stress­
raiser in order to minimize the error in sensing the peak strain. Even with this technique, the strain
indicated by the gauge may be significantly lower than the peak strain.
The three gauges selected for measurements in the steeply varying strain field near the hold
should have grids no larger than 0.031 × 0.031 in (0.75 × 0.75 mm). Ideally, in fact, gauge grid
dimensions should be 10% or less of the radius represented by the stress concentration configura­
tion. The fourth gauge, which is located at Section A, remote from the hole, and out of the region
of steep stress gradient, can be considerably larger, with a gauge length of, perhaps, 0.250 in (6
mm). At a distance along the beam from the stress­raiser, the axial strain is essentially uniformly
distributed across the width of the beam, and varies linearly along the length. The average strain
indicated by the gauge at Section A is thus equal to the strain at the center of the grid. Because
of the relationship expressed by equation 7.5 in the Introduction, this is also equal to the nominal
axial strain at Section B.
Due to the considerable skill and experience required to accurately locate, bond, and make
wiring connections to very small strain gauges, it is recommended that a pregauged beam be used
for this experiment. The gauges are installed on the pregauged beam in the manner indicated in the
Wiring Diagram (see Figure 7.8 in the Appendix of this chapter), with the centerlines of the gauges
(1), (2), and (3) at 0.020, 0.060, and 0.200 in (0.5, 1.5, and 5 mm), respectively, from the edge of
the hole.
Micro­Measurements foil strain gauges employed on the pregauged beam are intrinsically tem­
perature compensated for use of the material from which the beam is made. Because of this, the
four gauges involved in this experiment can be connected to the Model P3 Strain Indicator and
Recorder individually in “quarter­bridge” arrangements, completing the bridge circuit each time
with the precision resistors built into the instrument. For quarter­bridge operation, a “three­wire”
circuit is ordinarily used with each gauge in order to obtain compensation for the temperature­
induced resistance changes in the lead wires by placing equal lengths of lead wire in adjacent arms
of the bridge circuit (see P3 or Flexor Instruction Manual). It is often convenient in minimizing the
lead and connection requirements to combine the leads from one solder tab of each strain gauge to
a common lead. When three­wire circuitry is used, this becomes a pair of leads which is common
to every gauge in the system (see Wiring Diagram in the Appendix of this chapter on page 100).
7.5
Data acquisition
Back the calibrated loading screw out of the way, and insert the beam into the Flexor with the
gauged end in the clamp, and with the gauge on the top surface. Center the free end of the beam
between the sides of the Flexor, making certain that the end of the beam is inserted into the clamp
as far as it will go, and firmly clamp the beam in place with the knurled clamping screw.
The gauges will be connected (via the Flexor cable) to the strain indicator one at a time, first
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with the beam undeflected, and again with the beam deflected. An initial “reference” reading of
the strain indicator readout will be obtained for each gauge with the beam undeflected, and a final
reading with the beam deflected. The differences in these two sets of readings will give the strains
at the respective gauge locations.
Connect the strain gauge leads from the beam to the binding posts of the Flexor as shown in the
Wiring Diagram (see Figure 7.8 in the Appendix of this chapter). With the loading screw clear of
the beam, connect one of the two common leads (in the Flexor cable) to the S binding post of the P3
Strain Indicator and Recorder, and the other common lead to the appropriate D binding post (D120
for 120­ohm gauges or D350 for 350­ohm gauges). Connect the independent lead from gauge (1)
to the P+ binding post.
After balancing the indicator amplifier, set the gauge factor adjustment to the value given on the
beam for gauges (1), (2), and (3). This gauge factor setting will also be used for gauge (4). Do not
adjust the gauge factor during the experiment, even though the gauge factor for gauge (4) differs
from gauges (1), (2), and (3). A simple correction will be made later to account for the difference
in gauge factor, if it exists. Set the instrument to Run. With the Flexor loading screw still clear of
the beam, adjust the balance control of the P3 until the LCD digital display indicates exactly 0. Do
not adjust the balance control during the experiment. The initial (zero beam deflection) reading for
strain gauge element (1) should not be recorded on the worksheet as 0 µϵ.
Turn the strain indicator off, and disconnect the independent gauge element (1) lead from the
P+ binding post, leaving the common leads connected. Connect the independent cable lead from
gauge element (2) to the P+ binding post and turn the instrument on. Without adjusting the balance
controls, note the reading appearing in the indicator display. This is the initial reading for gauge
(2), and should be recorded on the worksheet. Repeat this procedure for gauge elements (3) and
(4), remembering to leave the balance control fixed in its original position at all times.
After obtaining and recording the initial reading for gauge (4), add 2,000 µϵ to the value of
this initial reading. With gauge (4) connected, deflect the beam by rotating the loading screw
clockwise until the indicator readout registers the number equal to this sum. The strain at the gauge
(4) location (and the nominal strain at the hole centerline) is now 2,000 µϵ, except for a small gauge
factor correction which can be made later.
Turn off the strain indicator, disconnect the independent gauge (4) lead from the instrument and
reconnect the gauge (3 lead. Turn the instrument on again and note the number registered by the
indicator readout. This number is the second reading for gauge (3) and should be recorded in the
appropriate location on the worksheet. Repeat this procedure for gauges (2) and (1). While gauge
(1) is still connected to the strain indicator, a simple check on system stability can be made. Back
the loading screw away until it clears the beam. This indicator display should not read 0 µϵ since
this was the initial setting for gauge (1). If the indication is more than, say, 10 from 0, the source
of the error should be located, and the experiment repeated.
Pregauged beams supplied by the Vishay Measurements Group have been tested for gauge sta­
bility at the time of manufacture, and should perform in a highly repeatable manner unless one
or more of the gauges has been damaged. If the zero beam deflection readings of the strain in­
dicator fail to repeat well, the balance control may have been inadvertently moved after its initial
adjustment, or the binding post connections may not have been snug enough to avoid small contact
resistance changes between connection and reconnection. Binding post connections should be snug
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enough to allow a “wiggle test” of the lead wires without a zero balance shift.
7.6
Data analysis and presentation
The strain indicated by gauge (4) can be corrected for gauge factor (if different from the gauge factor
for gauges (1), (2), and (3), and the instrument gauge factor setting) by the following relationship:
gauge factor setting of instrument
(7.8)
Corrected ϵ4 = 2000
gauge factor setting for gauge (4)
The result of this calculation is the nominal strain at Sections A and B of the beam and should
be entered on the worksheet.
The strains sensed by gauges (1), (2), and (3) can be calculated by subtracting the initial reading
from the final reading in each case. These numbers should be entered in the table on the worksheet.
Even though the centerline of gauge (1) is only 0.020 in (0.5 mm) from the edge of the hole,
the average strain sensed by this gauge is considerably lower than the peak strain. A satisfactory
estimate of the peak strain can be obtained by extrapolating curvilinearly to the edge of the hole.
It is not unreasonable to assume that the strain distribution can be represented approximately
by an expression of the following form:
4
2
R
R
+C
(7.9)
ϵ=A+B
X
X
where R is the radius of the hole, X is the distance from the center of the hole to any point on the
transverse centerline, and A, B, and C are coefficients to be determined from the measured strains
at three different points along the transverse centerline. Thus,
2
4
R
R
ϵ1 = A + B
+C
(7.10)
X1
X1
4
2
R
R
+C
(7.11)
ϵ2 = A + B
X2
X2
2
4
R
R
+C
(7.12)
ϵ3 = A + B
X3
X3
Noting that R = 0.125 in (3.18mm); X1 = 0.145 in (3.68 mm); X2 = 0.185 in (4.70 mm); X3 =
0.325 in (8.26 mm) and solving equations 7.10 to 7.12 simultaneously for C, B, and A,
C = 5.86(ϵ1 − ϵ2 ) − 5.44(ϵ2 − ϵ3 )
(7.13)
B = 3.49(ϵ1 − ϵ2 ) − 1.20C
(7.14)
A = ϵ1 − 0.743B − 0.552C
(7.15)
Substituting the measured strains from the table on the worksheet into equation 7.13 gives C, which
can then be substituted onto equation 7.14 with the strains to obtain B, etc.
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MNG 303
Since R/X = 1 at the edge of the hole, the peak strain can be calculated as:
ϵo = A + B + C
(7.16)
The strain (and stress) concentration factor is then:
Kt =
ϵo
ϵ′4
(7.17)
where ϵ′4 = the strain indication at gauge (4), corrected as necessary for gauge factor (see worksheet).
Plot the strains ϵo , ϵ1 , ϵ2 , and ϵ3 versus the corresponding dimensionless distance X/R on the
accompanying graph sheet to visualize the stress distribution in the vicinity of the hole.
7.7
Report
Prepare a brief report describing in your own words the purpose of the experiment, the equipment
and setup used, and the results obtained. Discuss the probable sources of error in this experiment,
and their relative effects on the accuracy of the stress concentration factor you have obtained. Com­
pare your results with published data for the theoretical stress concentration factor, Kt , due to a hole
in a beam in bending (for the same ratios of hole diameter to beam width and to beam thickness),
and discuss likely reasons for any difference that exists.
7.8 References
1. Figliola R.S. and D.E. Beasley, Theory and Design for Mechanical Measurements, John Wi­
ley and Sons, New York, 1997.
2. Lardner T.J. and R.R. Archer, Mechanics of Solids: An Introduction. McGraw­Hill, New
York, 1994, pp. 20­24
3. Riley and Zachary: Introduction to Mechanics of Materials. John Wiley and Sons, New
York, 1989, pp. 123­127.
4. Measurements Group Inc., Experiments in Mechanics, Strain Gage Series, E­104, Stress and
Strain Concentration, 1982, Measurements Group Inc., Education Division, Raleigh, NC.
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7.9
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Appendix
Figure 7.8: Wiring diagram
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Figure 7.9: Graph sheet ­ strain distribution, cantilever beam with hole
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7.10
Worksheet
7.10.1
Strain measurements
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Gauge factors for gauges (1), (2), (3) (and instrument setting) =
Gauge
1
2
3
4
7.10.2
Initial Reading
0 µϵ
Strain
2000 µϵ
Correction of ϵ4 for gauge factor
Corrected ϵ4 = ϵ′4 = 2, 000 ×
ϵ′4 = 2, 000 ×
7.10.3
Final Reading
Gauge factor setting for instrument
Gauge factor for gauge (4)
(
(
)
=(
)
(7.18)
)µϵ
(7.19)
Computation of extrapolation equation coefficients
C = 5.86(ϵ1 − ϵ2 ) − 5.44(ϵ2 − ϵ3 )
C = 5.86[(
)−(
)] − 5.44[(
)−(
)] = (
B = 3.49(ϵ1 − ϵ2 ) − 1.20C
B = 3.49[(
)−(
)] − 1.20(
)=(
)
A = ϵ1 − 0.743B − 0.552C
A=(
) − 0.743(
) − 0.552(
)=(
)
)
(7.20)
7.10.4 Maximum strain at edge of hole, ϵo
ϵo = A + B + C
ϵo = (
)+(
)+(
)=(
)
(7.21)
7.10.5 Stress concentration factor, Kt
ϵo
ϵ′4
(
Kt =
(
Kt =
)
=(
)
102
(7.22)
)
Chapter 8
Determination of principal strains and
stresses utilizing a strain gauge rosette
8.1
Theory
8.1.1
Mohr’s circle of stress
A geometrical solution for stresses in any required direction is provided by Mohr’s circle. Consider
the stress diagram in Figure 8.1. The steps in the construction of Mohr’s circle are as follows:
1. Construct a set of orthogonal axes and label the vertical axis τ and the horizontal axis σ as
shown in Figure 8.1b. It is necessary that the scale for these two axes be equal.
2. Plot the normal stresses σx and σy , given in Figure 8.1a on the normal stress axis.
3. Plot the shear stress τxy acting on the right­hand edge of the element in Figure 8.1a directly
below or above the point representing σx on the normal axis. If the shear stress is counter­
clockwise relative to the center of the element, plot τxy below the normal stress axis and, if
the shear stress is clockwise relative to the center of the element, plot τxy above the normal
stress axis.
4. Plot the shear stress τxy , acting on the section common with σy , either above or below the
point σy , but on the opposite side of the normal stress axis as used in step 3.
5. Join the two shear points with a straight line. This line will intercept the normal stress axis
at the point 1/2(σx + σy ).
6. Draw a circle with center on the normal axis at the point 1/2(σx + σy ) and diameter equal to
the length of the line joining the two shear points.
From Figure 8.1b, it is seen that the projection of the radius of the circle on the shear axis gives
the shear stress at any angle and that the projections of the ends of the diameter on the normal stress
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Figure 8.1: 2­D stress and Mohr’s circle
axis give the normal stresses at any angle. The radius of the circle is the maximum shear stress and
is given by:
q
r = τmax = 1/2
2
(σx − σy )2 + 4τxy
(8.1)
The intersections of the circle with the normal stress axis are the principal stresses, The tangent
of the angle 2θ is 2τxy /(σx − σy ), and 2θ is twice the angle between the x axis and the direction
of the principal stress. The direction of rotation of the radius from its original constructed position
to where the circle intersects the normal stress axis is in the same angular sense as the direction of
rotation of the element for the normal stress to become the principal stress.
Notice also that:
• The shear stress is zero when the normal stresses are maximum and minimum.
• When the shear stress is maximum, the normal stresses are equal to one half the sum of the
original normal stresses.
• The center of the circle is always at the point:
σθ − σθ+90
σx + σy
=
2
2
(8.2)
• The sum of the normal stresses is an invariant with the angular rotation of the element.
8.1.2
Step­by­step procedure for constructing Mohr’s circle
The main purpose in using Mohr’s circle is to have a quick mechanical procedure for transforming
a given state of stress at a point into principal stresses or into the maximum shearing stress and the
associated normal stresses. To be of any value, the procedure must be rapid and simple. As an aid
in applications, it is outlined below.
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σy
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τ
σx
τ
σx
right hand face
τ
σ2
σ2
O
σ1
C
σ1
(σx+σy)/2
Figure 8.2: Determination of the principal stresses by using Mohr’s circle
To determine the principal stresses, follow these steps (see also Figure 8.2):
1. Make a sketch of the element for which the normal and shearing stresses are known and
indicate on it the proper sense of these stresses.
2. Set up a rectangular co­ordinate system of axes where the horizontal axis is the normal stress
axis, and the vertical axis is the shearing stress axis. Directions of positive axes are taken as
usual, upward and to the right.
3. Locate the center of the circle, which is on the horizontal axis at a distance of 1/2(σx + σy )
from the origin. Tensile stresses are positive, compressive are negative.
4. From the right­hand face of the element prepared in (1), read off the values for σx and τ and
plot the controlling point A. The co­ordinate distances to this point are measured from the
origin. The sign of σx is positive if tensile, negative if compressive; that of τ is positive if
upward, negative if downward.
5. Connect the center of the circle found in (3) with the point plotted in (4) and determine this
distance, which is the radius of the circle.
6. Draw the circle, using the radius found in (5). The two points of intersection of the circle
with the σ­axis give the magnitudes and signs of the two principal stresses. If an intercept is
found to be positive, the principal stress is tensile, and vice versa.
7. To find the direction of the principal stresses, connect the point A located in (4) with the
intercepts found in (6). The principal stress given by the particular intercept found in (6) acts
normal to the line connecting this intercept point with the point A found in (4).
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σ2
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σ=(σ1+σ2)/2=(σx+σy)/2
τmax=(σ1-σ2)/2
σ1
45
Figure 8.3: Transformation of the principal stresses into the maximum shearing stresses and the
associated normal stresses
8. The solution of the problem may then be completed by orienting an element with the sides
parallel to the lines found in (7) and indicating the principal stresses on this element.
To determine the maximum or the principal shearing stress and the associated normal stresses:
1. As above, determine the principal stresses and the planes on which they act.
2. Prepare a sketch of an element with its corners located on the principal axes. The shear di­
agonals of this element will thus coincide with the directions of the principal stresses, Figure
8.3).
3. The magnitude of the maximum (principal) shearing stresses acting on mutually perpendic­
ular planes is equal to the radius of the circle. These shearing stresses act along the faces of
the element prepared it (2) toward the shear diagonal, which coincides with the direction of
the algebraically greater normal stress.
4. The normal stresses acting on all faces of the element prepared in (2) are equal to the average
of the principal stresses, considered algebraically. The magnitude and sign of these stresses
is also given by the distance from the origin of the co­ordinate system to the center of Mohr’s
circle.
To solve the problems of stress transformation, the foregoing procedure may be applied graph­
ically. However, it is recommended that trigonometric computations of the critical values be used
in conjunction with the graphical construction. Then the work may be carried out on a crude sketch
without any necessity for scaling off any of the distances or angles and the results will be accurate.
Using the Mohr’s circle construction in this manner is equivalent to applying the basic equations
of stress transformation. A little practice in using this aid should convince the reader that it is
quicker in application than the solution of the formal equations. The foregoing procedures are
greatly shortened if only the magnitudes of the stresses are sought.
Example: Given the state of stress shown in Figure 8.4a, transform it (a) into the principal
stresses, and (b) into the principal shearing stresses and the associated normal stresses. Show the
results for both cases on properly oriented elements.
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4 ksi
4 ksi
2 ksi
(a)
σ1
τ
-4 ksi
(c)
+6 ksi
B
-4 ksi
D
O
C
26º34´
E
+6 ksi
45º
σ= 1 ksi
σ
[-2, -4]
(b)
A
18º26´
τmax = 6 ksi
(d)
Figure 8.4: Stresses on Mohr’s circle
Solution: The co­ordinate axes are set up in Figure 8.4b. The center C of the circle is at 1/2
(−2, 000+4, 000) = +1,000 psi on the σ­axis. From the right­hand face, of the element the required
values for plotting the controlling point A on the circle are σx = −2,000 psi and τ = −4,000 psi.
Thus, the distancesp
CD and DA are 3,000 psi and 4,000 psi, respectively, and the radius of the circle
is equal to CA = (CD2 + DA2 ) = 5,000 psi. Hence from the diagram, τmax = 5,000 psi, and
the associated normal stress is represented by the distance OC i.e., σ ′ = 1,000 psi. The principal
stresses are given by the intercepts E and B and they are +6,000 psi and −4,000 psi, respectively.
Remember that 1 ksi = 1,000 psi.
The angle DEA = tan−1 (AD/DE) = tan−1 (4,000/8,000 = 28o 34′ .
An element with its sides parallel to the lines AB and AE is shown in Figure 8.4c. Since the
faces of this element intersect at right angles, several other angles (not shown) may be used to
specify its orientation. The principal stress given by the E intercept acts normal to the line EA. The
principal stress given by the B intercept acts normal to the line BA.
An element oriented with its planes parallel to the maximum shearing stresses is shown in Figure
8.4d. The maximum shearing stresses act toward the positive shear diagonal, which coincides with
the direction of the algebraically larger principal stress. The associated normal stresses are also
shown in the diagram. All of these are the same, and all of them are tensile in character.
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8.2
Experiment on principal strains and stresses
8.2.1
Objective
The purpose of this experiment is to measure the strains along three different axes surrounding a
point on a cantilever beam using strain gauge rosette, calculate the principal strains and then the
principal stresses from these strains, compare the result with the stress calculated from the flexure
formula for such a beam, and use the Mohr’s Circle to present the measured data.
Generally, in this experiment the validity of a biaxial (plane) stress or strain field and its pre­
sentation using the Mohr’s Circle will be verified.
8.2.2
Procedure
1. Before setting up any of the experimental equipment, read the experiment manual thorough
from cover to cover.
2. After all measurements and calculations will be completed the whole class will work together
to learn about graphical presentation of the laboratory data using Mohr’s Circle.
3. Based on calculated principal stresses we will draw the Mohr’s Circle and knowing orienta­
tion angle (reference to the main axis) for each gauge of the strain gauge rosette, we will be
able to determine strain for each gauge and compare these values with measured values.
4. Prepare your report in accordance with the instructions given under section 8.5.5.
8.2.3
References
1. Lardner and Archer, Mechanics of Solids, McGraw­Hill, New York, 1994, pp. 483­550.
2. Bauld, Mechanics of Materials, 2nd Edition, PWS Publishers, Boston, 1982, pp. 325­352.
3. Measurements Group Inc., Experiments in Mechanics, Strain Gage Series, E­103, Principal
Strains and Stresses ­ Flexure, 1982, Measurements Group Inc., Education Division, Raleigh,
NC.
4. Measurements Group Inc., Experiments in Mechanics, Strain Gage Series, E­102, Poisson’s
Ratio, 1982, Measurements Group Inc., Education Division, Raleigh, NC.
8.3
Introduction
The purpose of this experiment is to measure the strains along three different axes surrounding a
point on a cantilever beam, calculate the principal strains and then the principal stresses from these
strains, and compare the result with the stress calculated from the flexure formula for such a beam.
In a general biaxial stress or strain field, three strains along different axes at the same point must
be measured to determine the principal strains and stresses with strain gauges. While the stress field
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Figure 8.5: Polar plot of the normal stress and strain at a point in a uniaxial stress field
Figure 8.6: Strain rosettes
on the surface of a symmetrically loaded cantilever beam is uniaxial (except near the clamped end,
and near the loading point), the stress at any point nevertheless varies with angle about that point.
The strain field (which, in this case, is biaxial because of the Poisson strain) varies similarly. Figure
8.5 shows a polar plot of the normal stress and strain at a point in a uniaxial stress field.
The three axes along which strains are to be measured can be arbitrarily oriented about the point
of interest. For computational convenience, however, it is preferable to space the measurement
axes apart by submultiples of π, such as π/3 (60o ) or π/3 (45o ). An integral array of strain gauges
intended for simultaneous multiple strain measurements about a point is known as a “rosette”.
Three­gauge strain rosettes are commercially available in two principal forms corresponding to the
above angles; These are known as the “delta” or equiangular rosette and the 45o rectangular rosette,
respectively (Figure 8.6). The two rosette configurations are shown in the accompanying figure.
The delta rosette is so­named because of the arrangement of the strain­sensitive elements in the
form of an equilateral triangle is equivalent to the configuration shown, as is an arrangement of
two gauges symmetrically disposed 60o either side of a third gauge.
For the delta rosette, the principal strains can be calculated from the three measured strains with
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the following relationship:
√
ϵ1 + ϵ2 + ϵ3
2p
±
ϵp,q =
(ϵ1 − ϵ2 )2 + (ϵ2 − ϵ3 )2 + (ϵ3 − ϵ1 )2
3
3
The corresponding relationship for the rectangular rosette is:
ϵp,q =
ϵ1 + ϵ3
1 p
±√
(ϵ1 − ϵ2 )2 + (ϵ2 − ϵ3 )2
2
2
(8.3)
(8.4)
where:
ϵp,q
= algebraically maximum and minimum principal strains, respectively, in/in (cm/cm)
ϵ1 , ϵ2 , ϵ3 = strains measured along corresponding axes of rosette elements, in/in (cm/cm)
The algebraically maximum and minimum principal strains correspond to the plus and minus
alternatives, respectively, in equations 8.3 and 8.4. The algebraically minimum principal strain
may be numerically larger than the algebraically maximum, if negative.
The principal stresses can be calculated by substituting the principal strains from in equations
8.3 and 8.4 into the expressions for Hooke’s Law:
E
(ϵp + νϵq )
1 − ν2
E
(ϵq + νϵp )
σq =
1 − ν2
σp =
(8.5)
where:
σp , σq = algebraically maximum and minimum principal stresses, respectively, psi (Pa or N/m2 )
ν
= Poisson’s ratio
E
= modulus of elasticity, psi (Pa or N/m2 )
8.4
Equipment and supplies
• Flexor, cantilever flexure frame
• High strength aluminum alloy beam, 1/8 x 1 x 12­1/2 in (3 x 25 x 320 mm)
• Micro­Measurements Strain Gauge Rosette (delta or rectangular)
• Model P3 Strain Indicator and Recorder or equivalent
• Laboratory weights for loading cantilever beam
• Micrometer or vernier caliper
• Accurate drafting or machinist’s scale
• Protractor
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8.5
Procedure
8.5.1
General
MNG 303
In this experiment, the principal strains and principal stresses in a cantilever beam first will be
determined with a strain gauge rosette. Then, from the dimensions of the beam and the magnitude of
the load, the longitudinal bending stress will be calculated directly from the flexure formula. Since,
in this case, the principal axes for stress and strain are known to coincide with the geometric axes of
the beam, the maximum principal stress obtained from the rosette data should compare closely to
the calculated longitudinal bending stress. Similarly, the minimum principal stress obtained from
the rosette data should be approximately zero, since the transverse beam stress is zero.
It is evident that for the simple beam configuration and mode of loading used in this experiment,
a strain gauge rosette is not necessary to determine the principal stresses. However, by intentionally
bonding the rosette to the beam in an orientation such that no gauge axis coincides with an axis of the
beam, the generalized problem of a biaxial stress field (normally requiring the use of a rosette) can
be simulated completely. The person who successfully performs this experiment is then prepared
to determine the principal stresses in any stress field for which strain gauges are the appropriate
measuring tool.
8.5.2
Strain gauge selection and installation
Micro­Measurements foil strain gauges are intrinsically temperature­compensated for use on a ma­
terial with a particular thermal expansion coefficient. Because of this, the three gauge elements of
the rosette employed in this experiment can be used individually in “quarter­bridge” arrangements,
completing the bridge each time with the 120­ohm and 350­ohm precision resistors built into the
P3 Strain Indicator and Recorder. For quarter­bridge operation a “three­wire” circuit is ordinarily
used for each gauge element in order to obtain lead wire compensation by placing equal lengths of
lead wire in adjacent arms of the bridge circuit (see P3 or Flexor Instruction Manual). It is often
convenient in minimizing the lead and connection to combine the leads from one side of each gauge
element into a common lead. When three­wire circuitry is used, this becomes a pair of leads which
is common to every gauge element in the rosette (see wiring diagram for this experiment).
A strain gauge indicates the average strain under the grid area when placed in a non­uniform
strain field. Because the stress analyst is usually concerned with determining the stress “at a point”,
to the degree practicable, the shortest available gauge length consistent with performance and other
requirements of the gauge installation should be selected. In this experiment, since the strain gra­
dient is not very steep (in the order of 200 µϵ/in), a rosette with a 1/8­inch gauge length will be
satisfactory.
Following the gauge installation instructions included with the Student Strain Gauge Appli­
cation Kit, lay out the desired location (near what will be the clamped end of the beam) and the
orientation of the rosette as shown in the wiring diagram. Bond the rosette in place, following
the step­by­step directions precisely. Allow the adhesive to cure for the recommended time, then
carefully solder leads to the rosette as indicated by the wiring diagram. After solder connections
are complete, remove the flux, and apply protective coating over the entire gauge installation.
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8.5.3
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Data acquisition
Back the calibrated loading screw out of the way, and insert the beam (with the rosette bonded near
one end) into the Flexor with the gauged end in the clamp, and with the gauge on the top surface.
Connect the lead wires from the rosette to the binding posts of the Flexor according to the wiring
diagram given in the Appendix.
Connect one of the common leads (in the Flexor cable) to the S­ binding post of the P3 Strain
Indicator and Recorder, and the other common lead to the appropriate D binding post (D120 for
120­ohm gauges or 350 for 350­ohm gauges). Connect the independent lead from gauge element
(1) to the P+ binding post.
Measure the distance from the centerline of the rosette to the loading point on the free end of
the beam, using an accurate scale. Measure the width and thickness of the beam with a micrometer.
Record the beam dimensions on the accompanying worksheet. Using the cantilever beam flexure
formula,
σL =
Mc
6P L
=
I
bt2
(8.6)
where:
σL = the longitudinal surface stress at rosette centerline, psi (Pa or N/m2 )
M = the bending moment at rosette centerline, lb­in (N m)
c = the semi­thickness of beam, in. (m)
I = the moment of inertia of the beam cross­section, in4 (m4 )
P = the load, lbf (N)
L = the distance from the point of load application to rosette centerline, in. (m)
b = the beam width, in. (m)
t = the beam thickness, in. (m)
Note: If beam dimensions L, b, and t are substituted into equation 8.6 in terms of millimeters
instead of meters, the stress will be expressed in MPa instead of Pa.
Calculate the load, P , to be applied at the free end of the beam in order to produce a stress
of approximately 15,000 psi (100 MPa) at the center of the rosette. After balancing the indicator
amplifier, set the gauge factor adjustment to the value given on the strain gauge package data sheet
(or on the beam, if supplied pregauged). With the beam unloaded (except by its own weight and
the weight of the loading hook), set the instrument to Run. Adjust the balance control of the P3
until the LCD digital readout indicates precisely 0. Do not adjust the balance control again during
the experiment. The initial (zero beam deflection) reading for strain gauge element (1) should now
be recorded on the worksheet as 0µϵ.
Turn the strain indicator off and disconnect the independent gauge element (1) lead from the
P+ binding post, leaving the common leads connected. Connect the cable lead from gauge element
(2) to the P+ binding post and turn the instrument on. Without adjusting the balance controls, note
the reading on the indicator display. This is the initial reading for gauge (2) and should be recorded
on the worksheet. Repeat this procedure for gauge element (3), remembering to leave the balance
control fixed in its original position at all times.
After recording the initial reading for gauge element (3), leave the gauge connected to the
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instrument and apply the previously calculated load (or approximately that amount) with weights
hanging on the free end of the beam. Record the exact weight on the worksheet and record the
indicated strain in the table.
Leaving the load on the beam, repeat the above procedure for gauge elements (1) and (2).
While the last gauge is still connected to the instrument, remove the load from the beam. The
strain indicator readout should now indicate the same, within a few µϵ, as the initial reading for
this gauge. If the readings are not closely coincident, the source of error should be located, and the
experiment repeated.
Pregauged beams supplied by the Measurements Group have been tested for gauge stability
at the time of manufacture and should perform in a highly repeatable manner unless one or more
of the gauges has been damaged. If the zero beam deflection readings of the strain indicator fail
to repeat well, the balance control may have been inadvertently moved after its initial adjustment,
or the binding post connections may not have been snug enough to avoid small contact resistance
changes between connection and reconnection. Binding post connections should be snug enough
to allow a “wiggle test” of the lead wires without a zero balance shift.
8.5.4
Data analysis and presentation
The strain sensed by each of the three gauge elements is now obtained by subtracting the initial
measuring dial reading from the final reading. If the initial reading is larger than the final reading,
a negative (compressive) strain is indicated, and the sign should be retained to signify this. The
quantities just calculated are to be entered in the third column of the table on the worksheet. The
three strains can now be substituted into equation 8.3 or 8.4 to obtain the principal strains, ϵp and
ϵq .
Because the stress field on the surface of the cantilever beam is uniaxial, with the principal axes
known to be along and perpendicular to the beam axis, the absolute value of ϵq /ϵp is the Poisson’s
ratio of the beam material:
ν=
ϵq
ϵp
(8.7)
This number should also be recorded on the worksheet and then substituted into equation 8.5,
along with ϵp and ϵq and a value for the modulus of elasticity, E (assume E = 10.4 × 106 psi, or
71,700 MPa), to calculate the principal stresses, σp and σq .
The actual lateral beam stress is, of course, zero; and the σq calculated above should be very
small. Calculate the longitudinal stress, σL by substituting the exact load and the beam dimensions
into equation 8.6. Enter all of these values in the table on the worksheet for convenient comparison.
The principal strains and stresses have been calculated above without correcting the original
strain data for the transverse sensitivity of the strain gauge elements in the rosette. As a result of
the finite width of the grid lines in the gauges, and the presence of end loops connecting the grid
lines, strain gauges are generally sensitive not only to the strain parallel to the grid direction, but
also (to a much lesser degree) to strain perpendicular to the grid direction. This property of strain
gauges is referred to as “transverse sensitivity”.
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Strain gauges are customarily calibrated for gauge factor in a uniaxial stress field, on a steel
beam with ν = 0.285, with the gauge (or gauge elements, in the case of a rosette) aligned along
the applied stress axis. When the gauge is used in an environment with any different ratio of trans­
verse to axial strain, there is an error and a correction for transverse sensitivity may be necessary
(Measurements Group Inc, Experiments in Mechanics, Strain Gage Series, E­102, Poisson’s ratio).
If the strain gauge rosette or pregauged beam used in this experiment was supplied by the Mea­
surements Group, the error in the maximum principal strain is negligible. The minimum principal
strain can be corrected for transverse sensitivity by multiplying ϵq by 1.025.
8.5.5
Report
Prepare a brief report, describing in your own words the purpose of the experiment, the equipment
and setup used, and the procedure followed. Include the worksheet in the report. Discuss the
probable sources of error in the experiment, and their relative effects on the difference between the
principal stress obtained from strain gauge measurements and from the cantilever flexure equation.
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8.6
Worksheet
8.6.1
Beam dimensions
Length
Width
Thickness
8.6.2
MNG 303
in. (m) (from the point of load application to the gauge centerline)
in. (m)
in. (m)
Computation of load
Approximate load for σ = 15,000 psi or 100 MPa.
P =
(
σbt2
=
6L
)(
6(
Actual load
8.6.3
=
lbf (N)
(8.8)
lbf (N).
Strain measurements
Gauge Element
1
2
3
8.6.4
)2
)(
)
Gauge Factor
Initial Reading
Final Reading
Strain
Computation of strain measurements
a) Delta Rosette
ϵp = A + B
ϵq = A − B
A=
ϵ1 + ϵ2 + ϵ3
(
=
3
√
B=
2p
[(
3
)+(
(8.9)
)+(
3
)
=
µϵ
√
2p
B=
(ϵ1 − ϵ2 )2 + (ϵ2 − ϵ3 )2 + (ϵ3 − ϵ1 )2
3
)−(
)−(
)]2 + [(
)]2 + [(
(8.10)
(8.11)
)−(
)]2 =
ϵp = (
)+(
)=
µϵ
µϵ
(8.12)
(8.13)
ϵq = (
)−(
)=
µϵ
(8.14)
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b) Rectangular Rosette
ϵp = A + B
ϵq = A − B
A=
ϵ1 + ϵ3
(
=
2
(8.15)
)+(
2
)
=
µϵ
(8.16)
1 p
B=√
(ϵ1 − ϵ2 )2 + (ϵ2 − ϵ3 )2
2
1 p
B=√
[(
2
8.6.5
)−(
)]2 + [(
)−(
)]2 =
ϵp = (
)+(
)=
µϵ
(8.19)
ϵq = (
)−(
)=
µϵ
(8.20)
µϵ
(8.18)
Computation of Poisson’s ratio
ν=
8.6.6
(8.17)
(
ϵq
=
ϵp
(
)
=
)
(8.21)
Computation of principal stresses
(
E
(ϵp + νϵq ) =
2
1−ν
1−(
psi (MPa)
σp =
σp =
) × 106
[(
)2
)+(
)(
)] × 10−6
(8.22)
E
(
(ϵq + νϵp ) =
2
1−ν
1−(
σq =
psi (MPa)
σq =
) × 106
[(
)2
)+(
)(
)] × 10−6
(8.23)
8.6.7
Computation of maximum principal stress from flexure formula
σL =
6P L
6(
=
2
bt
(
)(
)(
)
)2
116
=
psi (MPa)
(8.24)
Solid Mechanics Lab Manual, January 7, 2023
8.6.8
MNG 303
Summary
Rosette Analysis
ϵp =
µϵ
ϵq =
µϵ
ν=
σp =
psi (MPa)
σq =
psi (MPa)
∗θp,q =
8.6.9
Flexure Formula
(σL /E) × 106 =
ν(σL /E) × 106 =
µϵ
µϵ
σL =
psi (MPa)
0
CHECK ∗θp,q =
Supplementary exercise
Compute angles between gauge (1) axis and principal axes.
a. Delta Rosette
"√
#
3(ϵ3 − ϵ2 )
θp,q = 1/2 tan−1
2ϵ1 − ϵ2 − ϵ3
"
θp,q = 1/2 tan−1
√
3[(
)−(
)−(
2(
#
)]
)−(
(8.25)
=
)
o
and
o
(8.26)
b. Rectangular Rosette
θp,q = 1/2 tan
2ϵ2 − ϵ1 − ϵ3
ϵ1 − ϵ3
)−(
)−(
−1
−1
θp,q = 1/2 tan
2(
(
)−(
(8.27)
)
)
=
o
and
o
(8.28)
In both cases, θp,q defines two angles, 90 apart, measured counterclockwise from the gauge (1)
axis.
Measure with a protractor the counterclockwise angles between gauge (1) axis and lateral and
longitudinal beam axes. Record in right­hand column of above table.
o
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MNG 303
118
Chapter 9
Determination of the modulus of elasticity
for four metallic and one non­metallic
materials
9.1
Objective
The purpose of this experiment is to determine the modulus of elasticity for four metallic materials
and one non­metallic material. To perform these measurements, use the five provided beams. Each
of these beams is made out of a different material. Based on experiences gained during laboratory
exercises with cantilever beams, design experimental procedure.
Generally, this experiment is to verify the knowledge you gained during our course and to teach
the ability how to design experiments by yourself.
9.2
Procedure
• To complete this experiment students must develop the testing procedure and find a proper
relationship to determine modulus of elasticity. Describe this procedure in your laboratory
report.
• Before setting up any of the experimental equipment, the group should discuss how these
experiments could be performed.
• Each group must take two measurements to come up with the average value for E. Compare
your values for E with data provided in the students’ textbook [l].
• The load for Copper, Brass and Aluminum beams should be between 2.5 and 3 lb, for Steel
beam around 7 lb and for Plastic beam no more than 0.5 lb.
• Calculate the relative maximum percent error in the determined modulus of elasticity, due to
the measurement precision error in P, b, t, δ and L, for each material.
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MNG 303
• Prepare your report in accordance with the instructions given under Report Preparation and
Organization.
9.2.1
Definitions
Abbreviation
Precision (*)
P = weight of the load at point “B”
∆P = ±0.05 grams or ±0.00011 lbs.
L = length of beam to point “B”
∆L = ±0.03125 in.
x = length of beam to point “A”
∆x = ±0.03125 in.
h = height of beam
∆h = ±0.0005 in.
b = width of beam
∆b = ±0.0005 in.
δi = initial deflection of beam
∆δi = ±0.0005 in.
δf = final deflection of beam
∆δf = ±0.0005 in.
(*) Precision based on half of smallest measuring increment of instrument.
9.2.2
References
1. Lardner and Archer, Mechanics of Solids, McGraw­Hill, New York, 1994, pp. 483­550.
2. Bauld, Mechanics of Materials, 2nd Edition, PWS Publishers, Boston, 1982, pp. 325­352.
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MNG 303
L
x
clamp (used
to secure
beam)
cantilever
beam
micrometer
(A) deflection
measurement
point
b
(B) load
application
point
Figure 9.1: Apparatus / Setup
δi
P
δf
A
B
Figure 9.2: Beam deflection
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9.3
MNG 303
Worksheet
Group Members:
Date:
1 3
bh
12
P X2
Deflection − δ(X) =
(3L − X)
6EI
I=
(9.1)
(9.2)
where:
∆P = 0.005 lbs
∆L = 0.03125 in.
∆X = 0.03125 in.
∆b = 0.0005 in.
∆h = 0.0005 in.
∆δ = 0.0005 in.
Table 9.1: Data Set 1
Material
Plexiglas
Brass
Copper
Aluminum
Steel
b (in)
h (in)
I (in4 )
P (lbs)
X (in)
L (in)
Table 9.2: Data Set 2
Material
Plexiglas
Brass
Copper
Aluminum
Steel
δi (in)
∂E
=
∂P
∂E
=
∂L
∂E
=
∂X
δf (in)
E (psi)
∂E
=
∂b
∂E
=
∂h
∂E
=
∂δ
122
Eref (psi)
Error (%)
(9.3)
(9.4)
(9.5)
Solid Mechanics Lab Manual, January 7, 2023
MNG 303
Table 9.3: Data Set 3
Material
Plexiglas
Brass
Copper
Aluminum
Steel
∂E
∆P
∂P
∂E
∆b
∂b
∂E
∆L
∂L
123
∂E
∆X
∂X
∂E
∆h
∂h
∂E
∆δ
∂δ
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MNG 303
124