Experimental Modern

MAE 477/577

- EXPERIMENTAL TECHNIQUES

IN SOLID MECHANICS -

CLASS NOTES Version 5.0

by

John A. Gilbert, Ph.D.

Professor of Mechanical Engineering

University of Alabama in Huntsville

FALL 2013

i

Table of Contents

Table of Contents i

Introducing MAE 477/577 iv

MAE 477/577 Outline v

MAE 477 ABET Syllabus vi

MAE 577 SACS Syllabus viii

Chapter 1. Stress

1.1 Introduction 1.1

1.2 Types of Forces 1.1

1.3 Traction Vector, Body Forces/Unit Mass, Surface and Body Couples 1.1

1.4 Resolution of the Traction Vector Stress at a Point 1.3

1.5 Equilibrium Equations Conservation of Linear Momentum 1.5

1.6 Stress Symmetry Conservation of Angular Momentum 1.7

1.7 Transformation Equations Mohrs Circle 1.7

1.8 General State of Stress 1.16

1.9 Stresses in Thin Walled Pressure Vessels 1.19

1.10 Homework Problems 1.22

Chapter 2. Strain


2.2 Strain-Displacement Equations 2.1

2.3 Strain Equations of Transformation 2.2

2.4 Compatibility Equations 2.3

2.5 Constitutive Equations 2.3


Chapter 3. Linear Elasticity

3.1 Overview 3.1

3.2 Plane Stress 3.1

3.3 Plane Strain 3.2


ii

Chapter 4. Light and Electromagnetic Wave Propagation

4.1 Introduction to Light 4.1

4.2 Electromagnetic Spectrum 4.3

4.3 Light Propagation, Phase, and Retardation 4.4

4.4 Polarized Light 4.5

4.5 Optical Interference 4.6

4.6 Complex Notation 4.6

4.7 Intensity 4.7

4.8 Superposition of Wavefronts 4.8

4.9 Reflection and Refraction 4.11

4.10 Double Refraction Birefringence 4.12


Chapter 5. Photoelasticity


5.2 Plane Polariscope 5.2

5.3 Circular Polariscope 5.6

5.4 Calibration Methods for Determining the Material Fringe Value 5.10

5.5 Compensation Methods for Determining Partial Fringe Order Numbers 5.12

5.6 Calculation of the 1 Direction 5.14

5.7 Birefringent Coatings 5.15

5.8 Three-Dimensional Photoelasticity 5.17


5.10 Classroom Demonstration in Photoelasticity 5.24

Chapter 6. Photography


6.2 Single Lens Reflex Cameras 6.1

6.3 Depth of Field 6.2

6.4 Photographic Processing 6.3

6.5 Polaroid Land Film Cameras 6.5

6.6 Digital Cameras 6.9

Chapter 7. Brittle Coatings


7.2 Testing Procedures 7.2

7.3 Calibration 7.3

7.4 Measurements 7.8

7.5 Coating Selection 7.9

7.6 Application 7.11

7.7 Crack Patterns 7.11

7.8 Relaxation Techniques 7.11

iii


Chapter 8. Moir Methods


8.2 Analysis 8.2

8.3 Optical Filtering 8.4

8.4 Stress Analysis 8.5

8.5 Shadow Moir 8.6


Chapter 9. Strain Gages


9.2 Electrical Resistance Strain Gages 9.2

9.3 Length Considerations 9.4

9.4 Transverse Sensitivity Corrections 9.5

9.5 Temperature Considerations 9.5

9.6 Rosettes 9.6

9.7 Gage Selection and Series 9.8

9.8 Dummy Gages and Transducer Design 9.9

9.9 P-3500 Strain Indicator 9.11

9.10 Two-Wire and Three-Wire Strain Gage Circuits 9.13


Appendix I Review of Statics and Mechanics of Materials

A.1.1 2-D Forces in Rectangular Coordinates A1.1

A.1.2 3-D Rectangular Coordinates A1.1

A1.3 Addition of Concurrent Forces in Space/3-D Equilibrium of Particles A1.2

A1.4 Moments and Couples A1.3

A1.5 Equilibrium of Rigid Bodies A1.4

A1.6 Center of Gravity and Centroid A1.4

A1.7 Moments of Inertia A1.6

A1.8 Shear and Moment Diagrams A1.8

A1.9 Deflection of Beams A1.8

A1.10 Synopsis of Mechanics of Materials A1.10

A1.11 Homework Problems A1.13

Appendix II Basic Formulas for Mechanics of Materials A2.1

iv

Introducing MAE 477/577

EXPERIMENTAL TECHNIQUES IN SOLID MECHANICS - 3 hrs.

Required Text: Experimental Techniques in Solid Mechanics, Version 4.1 by Gilbert

Reference Text: Experimental Stress Analysis, 4th. Edition by Dally and Riley

Prerequisite: MAE/CE 370; Junior Standing

Overview:

When dealing with the majority of real engineering systems, it is not always sufficient, or

advisable, to rely on analytical results alone. The experimental determination of stress, strain,

and displacement is important in both design and testing applications. MAE 477/577,

Experimental Techniques in Solid Mechanics, presents techniques which are valuable

complements to the design and analysis process; and, in some difficult or complex situations,

provide the only practical approach to a real solution.

The bulk of the course constitutes a detailed treatment of the more conventional methods

currently used for experimental stress analysis (photoelasticity, brittle coatings, moir methods,

strain gages, etc.); however, more recent developments in the field are also introduced (hybrid

methods, speckle metrology, holographic interferometry, moir interferometry, fiber optics,

radial metrology, and STARS). In-class laboratory exercises are included so that students gain

some practical experience. The lecture and laboratory exercises are designed to provide enough

exposure for participants to secure an entry level position in the field of experimental mechanics.

Attendance:

Students are required to be present from the beginning to the end of each semester, attend all

classes, and take all examinations according to their assigned schedule. In case of absence,

students are expected to satisfy the instructor that the absence was for good reason. For

excessive cutting of classes (3 or more class periods), or for dropping the course without

following the official procedure, students may fail the course.

Homework:

Homework assignments must be done on only one side of 8 1/2" x 11" paper. Each problem

shall begin on a separate page and each page must contain the following (in the upper right hand

corner): Your name, the date, and page __ of __. All final answers must be boxed and converted

(SI to US or US to SI). Homework is due at the beginning of the class on the date prescribed.

Work must be legible and should be done in pencil. Problems shall be restated prior to solution

and free-body diagrams (FBDs) shall be drawn for problems requiring such. Loose sheets shall

be stapled together in the upper left hand corner.

Note: For further information on the course and registration procedure, call Professor Gilbert by

telephone at (256) 824-6029, or, contact him by e-mail at [email protected].

mailto:[email protected]

v

MAE 477/577 Experimental Techniques in Solid Mechanics

Fall 2013

Time/Place: Friday 8:00 a.m. - 10:40 a.m.; TH S117

Instructor: Dr. John A. Gilbert

Office: OB 301E

Telephone: (256) 824-6029 (direct); (256) 824-5117/5118 (Beth/Cindy)

Fax/E-mail: (256) 824-6758; [email protected]

Office Hours: To be announced.

Grading: 10% Attendance (100% for being in class; 0% for not being there); 30%

Homework and Labs; 30% Midterm (10/11/13); 30% Final (11/29/13).

Required Text: Gilbert, J.A., Experimental Techniques in Solid Mechanics, Volume 5.0,

University of Alabama in Huntsville, 2013.

Reference Text: Dally, J.W., Riley, W.F., Experimental Stress Analysis, 4th Edition,

College House Enterprises, 2005, ISBN 0-9762413-0-7.

Course Outline

1. Overview of Solid Mechanics - Stress, Strain, and Displacement; Equilibrium,

Transformation, Strain-Displacement, Compatibility, and Constitutive Equations.

2. Stress Analysis - Method of Attack; Stress Concentration, Failure, Design, and Examples.

3. Light and Electromagnetic Wave Propagation Amplitude, Phase, Polarization,

Coherence, Interference, Reflection, Refraction, Birefringence, and Stress Optic Law.

4. Photoelasticity Plane and Circular Polariscopes; Calibration and Compensation

Methods; Reflection Polariscope, Birefringent Coatings, and 3-D Photoelasticity.

5. Photography Cameras, Lenses, and Photographic Development; Digital Cameras,

Image Acquisition and Processing.

6. Brittle Coatings Theory, Calibration, Application, and Measurement.

7. Moir Methods Geometrical Considerations and In-Plane Displacement Measurement;

Diffraction, Optical Filtering, and Stress Analysis; Out-of-Plane Displacement

Measurement and Shadow Moir.

8. Electrical Resistance Strain Gages - Parametrical Studies, Transverse Sensitivity,

Rosettes, Circuitry, Installation, and Transducer Design.

9. Advanced Topics Moir Interferometry, Speckle Metrology, Holographic

Interferometry, Fiber Optics, Digital Image Processing, Hybrid Methods, Panoramic

Imaging Systems, Radial Metrology, and STARS.

vi

MAE 477 Experimental Techniques in Solid Mechanics

Responsible Department:

Mechanical and Aerospace Engineering

Catalog Description:

477 Experimental Techniques in Solid Mechanics 3 hrs.

Experimental methods to determine stress, strain, displacement, velocity, and

acceleration in various media. Theory and laboratory applications of electrical

resistance strain gages, brittle coatings, and photoelasticity. Application of

transducers and experimental analysis of engineering systems. (Same as CE 477)

Prerequisites:

MAE/CE 370 and junior standing.

Textbook:

Required Text: Gilbert, J.A., Experimental Techniques in Solid

Mechanics, Volume 5.0, University of Alabama in

Huntsville, 2013.

Reference Text: Dally, J.W., Riley, W.F., Experimental Stress Analysis,

4th Edition, College House Enterprises, 2005, ISBN 0-

9762413-0-7.

Course Objectives:

1. To present techniques that are valuable complements to the design and analysis process; and in some difficult or complex situations, provide the

only practical approach to a real solution.

2. The bulk of the course constitutes a detailed treatment of the more conventional methods currently used for experimental stress analysis

(photoelasticity, brittle coatings, moir methods, strain gages, etc.);

however, more advanced topics are introduced (hybrid methods, speckle

metrology, holographic interferometry, moir interferometry, fiber optics,

radial metrology, and STARS).

3. A portion of the course is devoted to laboratory work.

Topics Covered:

1. Overview of solid mechanics

2. Stress analysis

3. Light and electromagnetic wave propagation

4. Photoelasticity

5. Photography

6. Brittle coatings

7. Moir methods

8. Electrical resistance strain gages

9. Advanced topics

vii

Class Schedule:

Once per week; class is 2 hours 40 minutes.

Contribution of Course to Meeting the Professional Component:

Basic Mathematics & Science: 0 credits.

Engineering Science: 3 credits.

Engineering Design: 0 credits.

Relationship of Course to Program Outcomes:

In this course each undergraduate student will have to:

a) apply their knowledge of mathematics, science, and engineering;

b) apply a knowledge of calculus-based physics;

c) apply advanced mathematics through differential equations;

d) apply a knowledge of linear algebra;

e) design a system, component or process to meet desired needs;

f) work professionally in mechanical systems;

g) use the techniques, skills, and modern engineering tools necessary for

engineering practice;

h) conduct experiments and to analyze and interpret data;

i) identify, formulate and solve engineering problems;

k) communicate effectively.

Person Preparing this Description:

John A. Gilbert, Ph.D., Course Coordinator, Professor

MAE Program

28 November 2012

viii

MAE 577 Experimental Techniques in Solid Mechanics

Responsible Department:

Mechanical and Aerospace Engineering

Catalog Description:

577 Experimental Techniques in Solid Mechanics 3 hrs.

Experimental methods to determine stress, strain, displacement, velocity, and

acceleration in various media. Theory and laboratory applications of electrical

resistance strain gages, brittle coatings, and photoelasticity. Application of

transducers and experimental analysis of engineering systems. (Same as CE 577)

Prerequisites:

MAE 370 and junior standing

Textbook:

Required Text: Gilbert, J.A., Experimental Techniques in Solid

Mechanics, Volume 5.0, University of Alabama in

Huntsville, 2013.

Reference Text: Dally, J.W., Riley, W.F., Experimental Stress Analysis,

4th Edition, College House Enterprises, 2005, ISBN 0-

9762413-0-7.

Course Educational Objectives:

1. To present techniques that are valuable complements to the design and analysis process; and in some difficult or complex situations, provide the

only practical approach to a real solution.

2. The bulk of the course constitutes a detailed treatment of the more conventional methods currently used for experimental stress analysis

(photoelasticity, brittle coatings, moir methods, strain gages, etc.);

however, more advanced topics are introduced (hybrid methods, speckle

metrology, holographic interferometry, moir interferometry, fiber optics,

radial metrology, and STARS).

3. A portion of the course is devoted to laboratory work.

Topics Covered:

1. Overview of solid mechanics

2. Stress analysis

3. Light and electromagnetic wave propagation

4. Photoelasticity

5. Photography

6. Brittle coatings

7. Moir methods

8. Electrical resistance strain gages

9. Advanced topics

ix

Class Schedule:

Once per week; class is 2 hours 40 minutes.

Relationship of Course to Program Outcomes:

In this course each graduate student will have to:

a) apply mathematics to solve engineering problems;

b) successfully perform advanced analysis in a technical area;

c) communicate effectively.

Person Preparing this Description:

John A. Gilbert, Course Coordinator, Professor

MAE Program

28 November 2012

1.1

CHAPTER 1 - STRESS

1.1 Introduction

When dealing with the majority of real engineering systems, it is not always sufficient, or

advisable, to rely on analytical results alone. The experimental determination of stress, strain,

and displacement is important in both design and testing applications. Experimental Techniques

in Solid Mechanics presents techniques which are valuable complements to the design and

analysis process; and, in some difficult or complex situations, provide the only practical

approach to a real solution.

The bulk of the course constitutes a detailed treatment of the more conventional methods

currently used for experimental stress analysis (photoelasticity, brittle coatings, moir methods,

strain gages, etc.); however, more recent developments in the field are also introduced (hybrid

methods, speckle metrology, holographic interferometry, moir interferometry, fiber optics,

radial metrology, and STARS). In-class laboratory exercises are included so that students gain

some practical experience. The lecture and laboratory exercises are designed to provide enough

exposure for participants to secure an entry level position in the field of experimental mechanics.

1.2 Types of Forces

The forces which act to produce stress in a material are classified according to their manner of

application. Surface forces act on the surface of the stressed body and are usually generated

when two bodies come into contact. Body forces, on the other hand, act on each element of the

body and are generated by force fields such as gravity. In a great many cases, the body forces

are small as compared to the surface forces and can be neglected in the analysis.

1.3 Traction Vector, Body Force/Unit Mass, Surface and Body Couples

When a solid body is subjected to external loads, the material from which the body is made

resists and transmits these loads. The strength of the body is measured in terms of stress.

Stress represents force intensity and is a mathematical quantity that must be computed by

dividing an applied force by the area over which the force acts. For a given force; the smaller the

area, the higher the stress. In general, the stress is different at every point in a loaded body.

The stress distribution at a given point is relatively complex and may be represented by a second

order tensor. Each direction in space (typically defined by a unit vector perpendicular to a plane

passed through the point in question) is associated with a stress vector which can be resolved into

components normal (normal stress) and parallel (shear stress) to the plane.

1.2

(1.3-1)

(1.3-2)

(1.3-3)

Figure 1, for example, shows an arbitrary internal or external surface with a small area A

enclosing a point P. As mentioned above, the orientation of the plane is defined with respect to a

reference axes system by a unit vector, n, drawn perpendicular to the surface. In general, the

distributed forces acting on A can be reduced to a force-couple system at point P where Fn

and Mn are the resultant force and couple, respectively. In most cases, the resultant force, Fn,

is not coincident with n and the couple vector lies at an arbitrary angle with respect to the line of

action of the resultant force.

Figure 1. Forces and moments act on a small area surrounding a point.

The average stress (force intensity) acting over the area is:

The traction (or resultant stress) vector, tn, at point P is found by taking the limit of the quantity

in Equation (1.3-1) as A goes to zero:

Since the magnitude and direction of the terms on opposite sides of Equation (1.3-2) are equal,

the traction vector is aligned with Fn. Simply put, tn depends upon the point in question and the

plane defined by n.

The surface couple at point P is given by:

Figure 2, on the other hand, shows a three-dimensional body with point Q surrounded by a

volume element V having a corresponding mass, m. FB and MB are the resultant body

force and body couples acting on the volume.

1.3

(1.3-5)

(1.3-4)

Figure 2. Forces and moments act on a small volume corresponding to a given mass.

Following arguments similar to those taken in obtaining Equations (1.3-2) and (1.3-3), the

resultant body force per unit mass acting at point Q is:

whereas, the resultant body couple per unit mass is:

While formulating the governing equations, surface and body couples are almost always

neglected. As a consequence, higher order terms disappear thereby simplifying the equations.

1.4 Resolution of the Traction Vector - Stress at a Point

Figure 3 shows the traction vector resolved into components normal and tangent to the plane

under consideration. The normal component, , is called the normal stress, whereas the

tangential component, , is called the shear stress.

Figure 3. The traction vector can be resolved into normal and tangential components.

1.4

(1.4-1)

(1.4-2)

A more meaningful description is obtained when one axis of a Cartesian system is aligned with

the normal to the plane under consideration. Figure 4, for example, shows the traction vector

acting at point P on a plane having a normal n acting along the Z axis. In this case, the tangential

component of the traction vector can be broken into two in-plane components. A double

subscript notation is used to define the stress components; the first subscript denotes the normal

to the surface under consideration while the second subscript denotes the direction in which the

stress component acts. For the configuration shown in Figure 4, the shear and normal stress

components are denoted by zx, zy and zz, respectively.

Figure 4. The traction vector is resolved into three Cartesian components.

Two additional sets of components are obtained for planes having their normal along the X and

Y axes. The corresponding stress components are xx,xy,xz and yx,yy,yz , respectively. The

stress distributions on these three mutually perpendicular planes can be summarized in a stress

tensor as follows:

In general, the magnitude and direction of the traction vector at a point depend upon the plane

considered. The rows in the stress tensor, on the other hand, correspond to the scalar

components of the traction vectors which act on planes having their normal vector acting along

the X,Y,Z axes, respectively.

Even though an infinite number of traction vectors may be defined by passing different planes

through a single point, each can be characterized in terms of the 9 components of stress defined

on three mutually perpendicular planes. Mathematically, the traction vector, tn, on a plane

defined by its normal, n, may be determined from the stress tensor, , using:

.

1.5

The infinitesimal region surrounding a point can be modeled by a rectangular parallelepiped. If

the sides are aligned with a Cartesian axes system, the 9 components of stress can be graphically

represented.

Figures 5, 6, and 7, for example, show the stress components on planes with their outer normals

parallel to the X,Y,Z axes, respectively. In this case, each parallelepiped is infinitesimal and the

stress distribution on opposing faces is equal and opposite, thus maintaining equilibrium.

By convention, stress components are drawn along the positive coordinate direction when the

outer normal to the plane on which they act lies along a positive coordinate direction; otherwise,

the component is drawn along the negative coordinate direction.

Each component is labeled using the double subscript notation described above. Normal stresses

acting away from the element correspond to tensile stresses while normal stresses acting toward

the element correspond to compressive stresses.

1.5 Equilibrium Equations - Conservation of Linear Momentum

The overall distribution of forces acting on the body must be such that the stress distribution

satisfies equilibrium. Figure 8, for example, shows a small element of volume taken from a

stressed body.

Figure 8. A finite element taken from a stressed body.

Figure 5. Stresses on planes

with their normal along X.


with their normal along Y.


with their normal along Z.

1.6

(1.5-1)

(1.5-2)

(1.5-4)

(1.5-3)

(1.5-5)

(1.5-6)

As opposed to the parallelepipeds depicted in Figures 5-7, the one is Figure 8 has finite

dimensions. To simplify the analysis, only the stress components which act in the X direction

have been included on the figure along with Fx, the body force per unit mass.

Assuming that the body is in equilibrium:

.

The stresses can be converted to forces by multiplying each stress by the area of the face over

which it acts and the body force per unit volume can be converted to force by multiplying it by

the volume. When this is done and Equation (1.5-1) is applied:

Dividing by the volume,

When a similar argument is applied to analyze the stresses along Y:

and for those along Z:

Summarizing,

The expressions in Equation (1.5-6) are referred to as the equilibrium equations.

1.7

(1.6-1)

(1.7-1)

1.6 Stress Symmetry - Conservation of Angular Momentum

Neglecting surface and body couples and taking moments about an XYZ axes located at the

centroid of an element subjected to stresses:

The expressions in Equation (1.6-1) show that the stress tensor, , is symmetrical. Hence, there

are 6 independent components of stress.

1.7 Transformation Equations - Mohr's Circle

In Section 1.4, it was shown that for every point in a loaded member, the components of shear

and normal stress change as different planes are passed through the point. Even though an

infinite number of planes can be passed through the point, it is sufficient to determine only the

stresses on three mutually perpendicular planes to completely characterize the state of stress on

an arbitrary plane. There are, however, certain critical planes, the analysis of which help to

determine the structural integrity. This section considers the transformation equations required

to pinpoint the orientations of these critical planes and the corresponding stresses which act on

them.

This exercise is important in experimental mechanics where the first step is usually to identify

critical points on the loaded structure. For an interior point, the measurements must be made on

three mutually perpendicular planes. In general, the stresses obtained on these planes are not the

maximum stresses at the point under consideration and a coordinate rotation is required.

When the stress tensor is formulated for a Cartesian axes system and the axes system is rotated, a

new stress tensor, ', is obtained. This is illustrated in Figures 9 and 10.

The stress tensor is modified because the shear and normal stress components on the rotated

planes are different from those on the original planes. However, the stresses on the rotated

planes may be obtained from those on the original planes by applying the law of transformation

for a tensor of the second rank. Mathematically, the transformation takes the form,

where ij' and kl are the components of the rotated and original tensors, respectively; ik and jl

correspond to the terms contained in the matrix of direction cosines and in its transpose,

respectively.

1.8

(1.7-2)

(1.7-3)

Fortunately, most experimental measurements are taken on a free surface where the components

of traction on the plane defined by the outer normal vanish. This condition is often referred to as

a plane, 2-D, or, biaxial stress state. In this case, for a point contained in the XY plane, the stress

tensor becomes:

.

From Equation (1.7-2) it is apparent that for plane stress, there are 3 independent stress

components. Thus, from an experimental mechanics standpoint, the complete determination of

the stress distribution at the point requires that 3 independent measurements be made.

The stress transformation can be studied analytically using Equation (1.7-1) or graphically by

manipulating these expressions and using Mohr's circle. Consider, for example, the stress

element shown in Figure 11.

Equation (1.7-1) can be applied to determine the stresses on the rotated element shown in Figure

12 as follows:

Figure 10. The stress distribution referred to

a rotated X'Y'Z' axes system. Figure 9. The stress distribution referred to

an XYZ axes system.

1.9

(1.7-4)

(1.7-5)

(1.7-6)

Applying the expressions in Equation (1.7-3) to the configuration shown in Figure 12, noting

that the X and Y axes correspond to = 0o and 90

o, respectively,

The expressions in Equation (1.7-4) are referred to as the transformation equations for a point in

plane stress.

An alternate approach is to manipulate each of the expressions included in Equation (1.7-3) as

follows:

and

Adding Equations (1.7-5) and (1.7-6):

Figure 12. The stress distribution on a plane

stress element referred to a rotated X'Y' axes.

Figure 11. The stress distribution on a plane

stress element referred to an XY axes system.

1.10

(1.7-7)

(1.7-8)

(1.7-9)

(1.7-10)

Equation (1.7-7) is the equation of a circle drawn in - space. This stress circle is commonly

referred to as Mohr's circle.

In X-Y space, the expression

corresponds to a circle centered at x = a and y = b with radius c.

Comparing the terms in Equation (1.7-7) to those in Equation (1.7-8), the Mohr's circle in -

space is centered at:

with radius

As illustrated in Figure 13, the circle is drawn by plotting points X with coordinates xx and xy

and Y with coordinates yy and -xy. Normal stresses are assumed positive when they act away

from the element; positive shear stresses produce a counterclockwise rotation of the element.

Figure 13. A typical Mohr's circle for graphical transformation of stress.

1.11

(1.7-11)

(1.7-12)

(1.7-13)

(1.7-14)

Each radial line in the circle represents the orientation of the unit vector normal to one of the

infinite number of planes which can be passed through the point. The intersection of the line

with the circle defines the shear and normal stresses acting on that plane. The orientation of the

planes are usually measured with respect to the X axis and are determined by computing angles

on the circle and dividing these values in half. The resulting angle is measured in the opposite

sense (clockwise or counterclockwise) on the element.

Points A and B are of special interest, since they correspond to the principal planes on which the

principal stresses act. Note that the shear stress is zero at these points. Other points of interest

lie at D and E where the maximum in-plane shearing stress occurs. As discussed later in Section

1.8, this value is the absolute maximum shearing stress when the principal normal stresses are of

opposite sign (i.e., the Mohrs circle encompasses the origin). Note that the normal stress for

both points corresponds to the location of the center of the circle.

The values of normal and shear stress at these critical locations can be computed graphically, as

demonstrated in the example problem outlined at the end of this section, or calculated on the

basis of Equation (1.7-3) as described below.

The orientation of the principal planes with respect to the X axis, p, are obtained by

differentiating the first expression in Equation (1.7-3) with respect to and setting the result

equal to zero. This operation produces,

The magnitudes of the principal stresses are found by substituting the values found from

Equation (1.7-11) back into the first expression in Equation (1.7-3). This operation produces:

A similar argument can be applied to the second expression in Equation (1.7-3). That is, the

planes of maximum in-plane shear stress are oriented at s with respect to the X axis, where

Substituting the values in Equation (1.7-13) into the expressions in Equation (1.7-3) produces:

It should be noted, as illustrated in the following example, that the principal planes are oriented

at an angle of 45o with respect to the planes of maximum shear.

1.12

(1)

(2)

(3)

Example: For the stress state shown in Figure 14, determine the:

(a) orientation of the principal planes.

(b) magnitude of the principal stresses.

(c) orientation of the planes of maximum in-plane shearing stress.

(d) magnitude of the maximum in-plane shearing stress.

(e) magnitude of the normal stress acting on the maximum in-plane shear planes.

Figure 14. A point in a state of plane stress.

Solution: The problem can be solved analytically. From Equation (1.7-11):

.

Thus, (a) the principal planes are oriented at:

.

From the first expression in Equation (1.7-12):

1.13

(4)

(5)

(6)

(7)

(8)

(9)

.

Thus, (b) the principal stresses are:

.

It is not apparent, however which of the stresses in Equation (4) correspond to the orientations

given in Equation (2). To determine the correspondence, it is necessary to substitute the values

in Equation (2) into the first expression in Equation (1.7-3). This operation shows that max lies

at p = - 13.3o while min corresponds to p = 76.7

o. It should be noted that, from the second

expression in Equation (1.7-12), the shear stresses on the principal planes are zero.

From Equation (1.7-13):

.

Thus, (c) the orientations of the maximum in-plane shearing stresses are:

.

From the first expression in Equation (1.7-14), (d) the maximum in-plane shearing stress is:

.

Again, it is not apparent which value of s corresponds to max. This can only be determined by

substituting the values in Equation (6) into the second expression of Equation (1.7-3). This

process reveals that max corresponds to s = 31.7o; min = - max lies at s = 121.7

o.

From the second expression in Equation (1.7-14), (e) the corresponding normal stress is:

The graphical solution is obtained by drawing the Mohr's circle shown in Figure 15. This is

accomplished by plotting the two points corresponding to the planes labeled as 1 and 2 on

Figure 14 in - space. The points are connected and a circle is drawn centered at C with radius

r. The location of the center of the circle is computed from Equation (1.7-9) as:

1.14

(11)

(12)

(10)

Figure 15. A Mohrs circle can be drawn for the point in question.

The radius is computed from Equation (1.7-10) as:

The maximum and minimum values of stress are determined by adding and subtracting the

magnitude of the radius to the value of at point C, respectively. This operation produces:

.

The orientations of the principal planes are obtained by determining the orientation of the point

labeled 3 on Figure 15 with respect to the point labeled 1. Referring to the figure:

Note that on the circle, 2 is drawn counterclockwise from point 1 to point 3. Also note that

point 3 corresponds to min. Therefore, the normal to the plane corresponding to min lies at =

13.3o measured in the clockwise direction from the normal to the plane labeled as 1 in Figure 14.

These results are summarized in Figure 16 which shows the rotated element corresponding to

the principal planes.

1.15

(13)

(14)

Referring to Figure 15, the value of at the point labeled 5 is equal to the radius while the

corresponding value of is equal to the normal stress of point C. At the point labeled 5:

.

The orientation of the maximum in-plane shearing stresses may be determined by finding the

angle labeled 2 on Figure 15 as:

.

Note that 2 is drawn from point 1 to point 5 in a clockwise direction. Thus, on the element, the

plane corresponding to the maximum in-plane shearing stress lies at = 31.7o measured

counterclockwise from the plane labeled as 1 on Figure 14. Since the shear is positive, it is

drawn such that it creates a counterclockwise rotation on the element. These results are

summarized in Figure 17.

It is apparent by comparing Figures 16 and 17 that the planes of maximum in-plane shearing

stress are oriented at 45o with respect to those corresponding to the principal stresses. It should

also be noted from Figures 14-17 that the sum of the normal stresses acting on two

perpendicular planes is constant. The sum corresponds to the trace of the stress tensor (xx + yy

+ zz) which is one of three quantities which remain invariant during coordinate rotation (in this

case, about the Z axis).

Figure 17. The maximum and minimum in-

plane shear stresses and their orientations.

Figure 16. The principal stresses and their

orientations.

1.16

(1.8-1)

1.8 General State of Stress

As discussed in Sections 1.4 and 1.6, the stress tensor consists of 9 components of stress, 6 of

which are independent. The 9 values correspond to 3 normal and 6 shear stress components

acting on three mutually perpendicular planes.

As illustrated in Figure 18, the coordinate axes can be rotated, in much the same manner as in

plane stress, to determine principal planes on which the shear stress is zero. In mathematical

terms, this is called an eigenvalue problem. One obtains a characteristic equation which has

three roots called eigenvalues; each eigenvalue is equal in magnitude to one of the principal

stresses. Eigenvectors are determined for each eigenvalue, and these vectors define the

orientation of the normal to each of the principal planes on which the principal stresses act.

Figure 19 shows one possible configuration for graphically depicting the stress distribution at a

point. In this case, the three eigenvalues are all positive. Coordinate rotations around the three

eigenvectors are pictorially represented by the three circles, and points on their circumferences

define the stress distribution on different planes on a rotated element. A more general coordinate

transformation leads to a point inside of the area between the inner circles and the outer circle.

In all cases,

Figure 19. The Mohrs circle for an arbitrary

point at which the three eigenvalues are all

positive.

Figure 18. A rotated element showing the

three principal stresses.

1.17

Figure 22. Orientation of the maximum

shear stress.


shear stress.

Two cases are of interest when evaluating Equation (1.8-1) for the case of plane stress (when one

of the eigenvalues is zero).

Figure 20 shows that when the principal stresses obtained from the bi-axial transformation (the

2-D Mohr's circle shown as the solid line in the figure) are of opposite sign, the in-plane

maximum shear stress is the absolute maximum shear stress at the point under study. As

illustrated in Figures 21 and 22, the corresponding planes are oriented at 45 degrees with respect

to the principal directions.

Figure 20. A Mohr's circle for a point at which the principal stresses are of opposite sign.

1.18


shear stress.


shear stress.

Figure 23, on the other hand, illustrates that when the in-plane principal stresses are of the same

sign (see the solid circle), the maximum shear stress (shown at D') must be calculated based on

the knowledge that the third eigenvalue is zero; i.e. equal to one-half the stress at A. The

orientations of the maximum shear stress planes are shown in Figures 24 and 25.

Figure 23. A Mohr's circle for a point at which the principal stresses are of the same sign.

1.19

(1.9-1)

1.9 Stresses in Thin Walled Pressure Vessels

A thin walled pressure vessel provides an important application of the analysis of plane stress.

The following discussion will be confined to cylindrical and spherical vessels.

The cylindrical vessel shown in Figure 26 has inside radius r and wall thickness t. The vessel

contains a fluid under pressure, p.

Figure 26. A cylindrical pressure vessel.

The hoop stress, 1, and the longitudinal stress, 2, are given by:

The Mohr's circle corresponding to the cylindrical pressure vessel is shown in Figure 27.

Figure 27. Mohrs circle for a cylindrical pressure vessel.

1.20

(1.9-2)

(1.9-3)

(1.9-4)

(1.9-5)

Note,

and

Figure 28, on the other hand shows a spherical pressure vessel of inside radius r and wall

thickness t, containing a fluid of gage pressure p.

Figure 28. A spherical pressure vessel.

In this case,

The Mohr's circle for the spherical pressure vessel is shown in Figure 29.

Note,

1.21

Figure 29. Mohrs circle for a spherical pressure vessel.

1.22

1.10 Homework Problems

1. Draw a rectangular parallelepiped whose sides are aligned with a Cartesian axis system passing through the point in question. Assign stresses on each of the six faces consistent

with convention.

Answer: Combine the stress states shown in Chapter 1, Figures 5 through 7.

2. Consider a small element of volume in a continuum. Assign the stresses and the body force/unit volume which act in the Y direction. Use this figure as a basis and derive the

corresponding equilibrium equation (1.5-4) by summing forces along Y.

Answer: See Section 1.5.

3. Consider a small element of volume in a continuum. Assign the stresses and the body force/unit volume which act in the Z direction. Use this figure as a basis and derive the

corresponding equilibrium equation (1.5-5) by summing forces along Z.

Answer: See Section 1.5.

4. The following stress distribution has been determined for a machine component:

Is equilibrium satisfied in the absence of body forces?

Answer: Expand expressions in Equation (1.5-6); yes.

5. The following stress distribution has been determined for a machine component:

Is equilibrium satisfied in the absence of body forces?

Answer: Expand expressions in Equation (1.5-6); yes.

6. If the state of stress at any point in a body is given by the equations:

1.23

What equations must the body-force intensities Fx, Fy, and Fz satisfy?

Answer: Apply expressions in Equation (1.5-6); Fx = - ( a + 2 p z ), Fy = - ( m + 2 e y ),

Fz = - ( l + 2 n x + 3 i z2 ).

7. If the state of stress at any point in a body is given by the equations:

What equations must the body-force intensities Fx, Fy, and Fz satisfy?

Answer: Apply expressions in Equation (1.5-6); Fx = - 4y (3y2+x), Fy = x, Fz = 0.

8. A two-dimensional state of stress (zz = zx = zy = 0) exists at a point on the surface of a loaded member. The remaining Cartesian components of stress are:

Using Mohrs circle:

(a) Determine the principal stresses and sketch the element on which they act. (b) What is the maximum shear stress at the point in question?

Answer: (a) 1 = 95.14 MPa, 2 = - 85.14 MPa, p = - 9.72o (b) maximum = 90.14 MPa.


xx = 90 MPa yy = 40 MPa xy = 60 MPa .

Using Mohrs circle:


Answer: (a) 1 = 130 MPa, 2 = 0 MPa, p = 33.7o (b) maximum = 65 MPa.


Using Mohrs circle:

1.24

25o

8'

30"


Answer: (a) 1 = 107 MPa, 2 = 43 MPa, p = - 19.3o (b) maximum = 53.5 MPa.


Using Mohrs circle:


Answer: (a) 1 = 11 ksi, 2 = 1 ksi, p = - 63.4o (b) maximum = 5.5 ksi.

12. A pressure tank is supported by two cradles as shown; one of the cradles is designed so that it does not exert any longitudinal force on the tank. The cylindrical body of the tank is

fabricated from a 3/8" steel plate by butt welding along a helix which forms an angle of 25o

with respect to a transverse plane (the vertical direction). The end caps are spherical and

have a uniform wall thickness of 5/16". The internal gage pressure is 180 psi.

Figure 30. A pressure vessel with helical welds.

(a) Determine the normal stress and the maximum shearing stress in the spherical caps.

(b) Determine the normal and shear stresses perpendicular and parallel to the weld, respectively.

Answer: (a) = 4230 psi, maximum = 2115 psi, (b) weld = 4137 psi, weld = 1345 psi.

13. A pressure vessel of 10-in. inside diameter and 0.25-in. wall thickness is fabricated from a 4-ft section of spirally welded pipe. The vessel is pressurized to 300 psi and a centric axial

1.25

35o

10 kip

4 ft

A

B

compressive force of 10-kip is applied to the upper end through a rigid end plate.

Figure 31. A compressed pressure vessel with helical welds.

(a) Draw a stress element for a point located along the weld with horizontal and vertical faces.

(b) Determine and in directions, respectively, normal and tangent to the weld. (c) What is the maximum shear stress in the wall?

Answer: (a) weld = 3.15 ksi, (b) weld = - 1.993 ksi, (c) max = 3 ksi.

14. Square plates, each of 5/8 thickness, may be welded together in either of the two ways shown to form the cylindrical portion of a compressed air tank. Knowing that the

allowable stress normal to the weld is 9 ksi, determine the largest allowable gage pressure

for each case.

Figure 32. Two different pressure vessels constructed from square plates.

Answer: Case No. 1: 62.9 psi, Case No. 2: 83.9 ksi.

2.1

X

Y

Z

O

d_

u_

v_

w_

CHAPTER 2 - STRAIN

2.1 Introduction

In Chapter 1, stress relationships were based on equilibrium. No restrictions were made

regarding the deformation of the body or the physical properties of the material from which it

was made. Strain is a purely geometric quantity and no restrictions need be placed on the

material of the body; however, restrictions must be placed on the allowable deformations in

order to linearize the strain-displacement relations. The type of material and/or material

constraints are introduced in the constitutive equations which relate stress to strain.

2.2 Strain-Displacement Equations

Displacement is a vector quantity which characterizes the movement of an arbitrary point. As

illustrated in Figure 1, each point has a displacement vector, d, which can be resolved along

Cartesian axes X,Y,Z such that the corresponding scalar components of displacement are u,v,w,

respectively.

Figure 1. The movement of a point is described by a displacement vector, d, which has scalar

components u,v,w.

The motion of the points in the body consists of two parts. Rigid body motion includes

translations and rotations during which there is no relative motion between neighboring points.

The movement of points relative to one another is called deformation.

Strain is a geometric quantity which describes the deformation which occurs in the body. There

are two types of strain. Shear strain is a measure of the angular change which takes place

between two line segments which were originally perpendicular. Since shear strain is expressed

in non-dimensional terms, the angle is measured in radians. Normal strain, on the other hand, is

defined as the change in length of a line segment divided by the original length of the segment.

2.2

(2.2-1)

(2.3-1)

(2.3-2)

(2.3-3)

(2.3-4)

The displacement components shown in Figure 1 are related to the strain components by the

strain-displacement equations which are given as follows:

2.3 Strain Equations of Transformation

The general transformation for a second order symmetric tensor illustrated by Equation (1.7-1)

can also be applied to transform strain provided that a slight modification is made when defining

the shear strain. To this end, the following definition is introduced:

In terms of Cartesian components, Equation (2.3-1) is written:

Then,

where ij' and kl are the components of the rotated and original tensors, respectively; ik and jl

correspond to the components in the matrix of direction cosines and its transpose, respectively.

From a rudimentary standpoint, the same relations developed in Section 1.7 for stress apply to

strain when the following substitutions are made:

Mohr's circle, for example, is plotted with as the abscissa and /2 as the ordinate.

2.3

(2.4-1)

2.4 Compatibility Equations

Equation (2.2-1) demonstrates that the strain field can be obtained from a given displacement

field by differentiating the displacement. However, if the strain field is measured, it becomes

necessary to integrate to obtain the displacement field. In this case, a unique displacement field

is obtained if and only if the body is simply connected (two points on the boundary can be

connected by never leaving the body or traversing it) and the strains satisfy compatibility

equations derived from the strain-displacement relations. The compatibility equations are:

2.5 Constitutive Equations

In general, stress is related to strain through 81 material constants. The linearity of stress versus

strain reduces this number to 36 while strain energy considerations further reduce the number to

21. If the body is linearly elastic, homogeneous and isotropic, the number reduces to 2.

A linearly elastic body is one which is made of a material which has a linear stress versus strain

curve. The body is homogeneous if it is made of the same material throughout. Isotropic refers

to the fact that a material property at a given point is independent of the direction in which it is

measured. For such a body, the equations which relate the stress to the strain are written in terms

of two independent material constants and are referred to as Hooke's laws. These constitutive

equations may be expressed for strain as a function of stress as:

2.4

(2.5-1)

(2.5-2)

or expressed for stress in terms of strain as:

2.5


1. The central portion of the tensile specimen shown in Figure 2 represents a uniaxial stress

condition; P is the applied load and A is the cross-sectional area. For a point located on the

surface and in the central portion of the specimen:

Figure 2.

(a) Draw a two-dimensional stress element having horizontal and vertical faces. Show the stress distribution on this element. Are these the principal stress planes?

(b) Construct the Mohrs circle for the point and draw a rotated element showing the maximum in-plane shear stress. Be sure to label all stresses and specify the angle of

orientation with respect to, say, the horizontal direction.

(c) How does the maximum in-plane shear stress compare with the absolute maximum shear stress at the point?

(d) Develop the constitutive equation(s) for this uniaxial stress condition from the three dimensional Hookes Law (see Eqn. 2.8-2).

(e) Develop the mathematical relationships for Young's Modulus and Poisson's Ratio and describe how these quantities could be calculated by making physical measurements.

(f) Specify the six components of strain if the specimen is made of steel (E = 30 x 106 psi, = 0.3) and loaded to yield (y = 45 x 10

3 psi).

(g) What would be the angular orientation of the fracture surface with respect to a horizontal axis if the material were ductile and failed along the planes of maximum shear?

What would happen if the material were brittle?

Answer: (a) xx = 0, yy = P/A, xy = 0, yes, (b) element oriented at 45o with = = P/2A, (c)

equal, (d) yy = Eyy ..., (e) E = yy/yy, = -xx/yy = - zz/yy, measure strain and

compute stress, (f) xx = zz = - 450 , yy = 1500 , xy = xz = yz = 0 , (g) ductile

@ 45o, brittle @ 0

o.

2.6

A

B

_T

_T

C

Axis of shaft

Axis of shaft

2. Figures 3 and 4 show that when a plane is passed through a circular shaft perpendicular to its

longitudinal axis, a state of pure shear exists.

Figure 3. Figure 4.

(a) Specify the formula used to determine the shear stress and define all terms. Draw a diagram showing how the stress varies over the circular cross section.

(b) Express the magnitude of the maximum shear stress at the surface in terms of the applied torque and the radius.

(c) Assuming that the shaft is oriented so that its longitudinal axis is horizontal, draw a two-dimensional stress element having horizontal and vertical faces, and show the stress

distribution on this element. Are these the principal stress planes? (d) Construct the Mohrs circle for the point and draw a rotated element showing the

principal stresses. Be sure to label all stresses and specify the angle of orientation with

respect to, say, the horizontal direction.

(e) What would be the angular orientation of the fracture surface with respect to a horizontal axis if the material were ductile and failed along the planes of maximum shear?

What would happen if the material were brittle?

Answer: (a) = T/J, (b) = 2T/c3, (c) xx = yy = 0, xy = 2T/c

3, no, (d) element oriented at

45o with = 2T/c

3, (e) ductile @ 90

o, brittle @ 45

o.

3.1

Y

X Z

Y

O

CHAPTER 3 LINEAR ELASTICITY

3.1 Overview

At each point in a loaded body, there are 15 unknowns: 3 displacements (u,v,w), 6 strains

(xx,yy,zz,xy,xz,yz) and 6 stresses (xx,yy,zz,xy,xz,yz). A complete solution to the general

elasticity problem must satisfy 15 equations (3 equilibrium, 6 compatibility and 6 constitutive)

plus boundary conditions formulated in terms of displacement and/or traction.

In general, it is possible to obtain a closed-form solution only for problems having relatively

simple loading and geometry. In more complex situations, it is necessary to resort to an

analytical formulation using a technique such as finite element analysis. In this method, the

body is mathematically modeled by partitioning it into a finite number of elements connected

together at points called nodes. Displacements and/or tractions are specified at the boundary and

a mathematical wave front is passed over the mesh. Stresses are predicted based on either a

flexibility or stiffness approach.

An alternative solution to the problem is to experimentally measure the displacements, strains or

stresses. When displacements are measured, the strain-displacement equations are used to

determine the strains. Stresses are obtained using the constitutive equations, after which the

transformation equations may be applied to obtain the principal stresses and the maximum shear

stress. When strains are measured, the compatibility equations must be satisfied to obtain a

unique displacement field.

3.2 Plane Stress

The geometry of the body and the nature of the loading allow two important types of problems to

be defined. Figure 1 illustrates an example of plane stress in which the geometry is essentially a

flat plate with a thickness much small than the other dimensions. The loads applied to the plate

act in the plane of the plate and are uniform over the thickness.

Figure 1. An example of plane stress.

3.2

(3.2-1)

(3.2-3)

(3.3-1)

(3.2-2)

Mathematically, for a plate contained in the XY plane,

.

Although the normal stress perpendicular to the surface vanishes, the normal strain, zz, does not.

The relationship between the latter and the other two in-plane strain components is found by

setting zz = 0 in the expression given for it in Equation (2.5-2). This results in:

An alternate form of Hookes law for plane stress is found by substituting Equation (3.2-2) into

the remaining expressions contained in Equation (2.5-2) as

3.3 Plane Strain

Figure 2, on the other hand, illustrates the condition of plane strain. In this case, the body is

considered a prismatic cylinder with a length much larger than the other dimensions. The loads

applied to the cylinder are distributed uniformly with respect to the large dimension and act

perpendicular to it.

Mathematically, for a point in the XY plane,

.

3.3

Y

X

Z

O

(1)

(2)

(3)

Example: A flat plate (located in the XY plane of a Cartesian coordinate system, with

normal along Z) was designed and built as part of a compressor system using steel

(Elastic Modulus = 30 x 106 psi; Poisson's Ratio = 0.3). The allowable shear

stress of the material is 12.5 ksi but the manufacturer wants a factor of safety of

2.0 to be incorporated into the design.

Preliminary tests conducted by the manufacturer indicated that the most critical location on the

structure was located at x = 1.0, y = 2.0, z = 0.0. This point was on a traction free surface where

the in-plane displacements were found to be expressed as follows:

(a) Determine the magnitudes and directions of the principal stresses at the critical location. Sketch your results on a rotated element and label the stress distribution. Specify the angle

of orientation of the element with respect to the x direction.

(b) Does the part meet the required specifications?

Solution: The point is in plane stress and the in-plane strain components may be obtained

using the strain-displacement equations given by Equation (2.2-1) as

For plane stress, zz = 0, and

Figure 2. An example of plane strain.

3.4

(4)

(5)

(6)

The in-plane stresses are computed using Equation (2.5-2) [or Equation (3.2-3)] as

Figure 3 shows the corresponding stress element; and, Figure 4 shows the Mohrs circle for the

stress distribution at the critical point.

The circle is drawn by plotting the two points corresponding to the planes labeled as 1 and 2 on

Figure 3 in - space. The points are connected and a circle is drawn centered at C with radius r.

The location of the center of the circle is computed from Equation (1.7-9) as:

The radius is computed from Equation (1.7-10) as:

The maximum and minimum values of stress are determined by adding and subtracting the

magnitude of the radius to the value of at point C, respectively. This operation produces:

Figure 3. Stress element. Figure 4. Mohrs circle.

3.5

(7)

(8)

(9)

(10)

.

The orientations of the principal planes are obtained by determining the orientation of the point

labeled 3 on Figure 4 with respect to the point labeled 1. Referring to the figure:

Note that on the circle, 2 is drawn clockwise from point 1 to point 3. Also note that point 3

corresponds to max. Therefore, the normal to the plane corresponding to max lies at = 10.9o

measured in the counterclockwise direction from the normal to the plane labeled as 1 in Figure

3. These results are summarized in Figure 5 which shows the rotated element corresponding to

the principal planes.

The in-plane shear stress is equal in magnitude to the radius. However, since the circle does not

encompass the origin, the absolute maximum shear stress is equal to one-half of the maximum

value,

The allowable stress, on the other hand, is found by dividing the maximum shear stress that the

material can take by the factor of safety,

Figure 5. Principal planes.

3.6

xx, yy, zz, xy, xz, yz

xx, yy, zz, xy, xz, yz

u, v, w

1, 2, 3

1

2

3

4

5 6

7

8

Since the maximum stress is greater than the allowable stress, the part does not meet the design

specifications.


1. Figure 6 shows a flowchart for the solution to the general elasticity problem. Add an

appropriate description to the items numbered 1 through 8.

Answer: 1. displacements; 2. strain, 3. stress, 4. principal stress, 5. compatibility equations, 6.

strain-displacement equations, 7. constitutive equations, and 8. transformation

equations.

2. Assume that a state of plane stress exists such that zz = yz = zx = 0.

(c) Reduce the equilibrium equations given in Equation (1.5-6) for plane stress. (d) Reduce the constitutive equations given in Equation (2.5-1) for plane stress. (e) Knowing zz = 0, develop an expression for zz = f (xx, yy) for plane stress and show by

substituting this expression into Equation (2.5-2) that the constitutive equations become:

Answer: (a) xx/x + yx/y + Fx = 0 ..., (b) xx = 1/E [xx - yy] ..., (c) zz = - /(1-) [xx +

yy] ...

Figure 6. The general elasticity problem.

3.7

3. A critical point on the surface of a part fabricated using steel (E = 30 x 106 psi; = 0.3) is

under plane stress conditions with zz = yz = zx = 0. Knowing that xx = 505 x 10-6

in/in [or,

505 ], yy = 495 x 10-6

in/in [or, 495 ] and xy = 100 x 10-6

in/in [or, 100 ], determine

the

(a) strain zz. (b) principal stresses and their orientation with respect to the XYZ axes system. (c) maximum shear stress at the point.

Answer: (a) zz = - 428.6 , (b) 22.59 ksi, 20.27 ksi, 0 ksi, x1 = 42o (c) max = 11.3 ksi.

4.1

E

Hz

(4.1-1)

(4.1-2)

CHAPTER 4 LIGHT AND ELECTROMAGNETIC WAVE

PROPAGATION

4.1 Introduction to Light

Electromagnetic radiation is predicted by Maxwell's theory to be a transverse wave motion

which propagates in free space with a velocity of approximately c = 3 x 108 m/s (186,000 mi/s).

The wave consists of oscillating electric and magnetic fields which are described by electric and

magnetic vectors E and H, respectively. These vectors are in phase, perpendicular to each other,

and at right angles to the direction of propagation, z.

A simple representation of the electric and magnetic vectors associated with an electro-magnetic

wave at a given instant of time is illustrated in Figure 1. As described below, the wave is

assumed to have a sinusoidal form and may be characterized by either the electric or magnetic

vector. In the arguments that follow, light propagation is described in terms of the electric

vector, E.

Figure 1. An electromagnetic wave.

A mathematical representation for the magnitude of the electric vector as a function of position,

z, and time, t, can be determined by considering the one-dimensional wave equation

and setting E = u. The DeLambert solution to this partial differential equation is

where z is the position along the axis of propagation and t is time.

In Equation (4.1-2), the quantity f (z - ct) represents a wave moving at velocity c in the positive z

direction while g (z + ct) represents a wave moving at velocity c in the negative z direction.

4.2

(4.1-3)

a

E

z

E = a cos 2/ (z - ct)

ct

t = 0

t > 0

(4.1-4)

(4.1-5)

(4.1-6)

Many of the optical effects of interest in experimental mechanics and optical metrology can be

described by considering a harmonic vibration propagating along positive z such that

where a is the instantaneous amplitude, is the wavelength, and 2/ is the wave number.

Equation (4.1-3) neglects the attenuation associated with the expanding spherical wavefront and

is rigorously valid only for a plane wave.

Figure 2 shows a graphical representation of the magnitude of the light vector as a function of

position along the positive z axis, at two different times. The wavelength, , is defined as the

distance between successive peaks. The time required for the wave to propagate through a

distance equal to the wavelength is called the period, T.

Figure 2. A wave, originally at time t = 0, propagates through space.

Since the wave is propagating with velocity, c,

The angular frequency of the wave is defined as

where the frequency,

represents the number of oscillations which take place per second.

4.3

(4.1-7)

The above definitions make it possible to express E(z,t) in several ways. For example,

4.2 Electromagnetic Spectrum

Although the electromagnetic spectrum has no upper or lower limits, the radiations commonly

observed have been classified in the table shown in Figure 3.

Figure 3. The electromagnetic spectrum.

The visible range of the spectrum is centered about a wavelength of 550 nm and extends from

approximately 400 to 700 nm. Figure 4 lists the different colors observed by the human eye as a

function of wavelength.

Figure 4. The visible spectrum.

4.4

(4.3-1)

E1 = a cos 2/ (z - ct + 1)

E2 = a cos 2/ (z - ct + 2)

E = a cos 2/ (z - ct)ct

2

1

a

E

z

(4.3-2)

The characteristics of an electromagnetic wave such as color depend upon the frequency; since,

the frequency is independent of the medium through which propagation occurs. The wavelength,

, on the other hand, depends upon the medium; since the velocity of propagation changes from

medium to medium (see Section 4.9).

When the light vectors corresponding to E(z,t) are all of one frequency, the light is referred to as

monochromatic. Light consisting of several frequencies is recorded by the eye as white light.

4.3 Light Propagation, Phase, and Retardation

In Equation (4.1-3) [the first form of E(z,t) shown in Equation (4.1-7)], the term in the argument

of the cosine function represents the angular phase, , while the term in the parenthesis

corresponds to the linear phase, . It is apparent that these quantities are related through the

wave number as follows:

Figure 5, for example, shows two waves having the same wavelength, , and equal amplitudes.

In this case, the waves are plotted as a function of position, along the axis of propagation.

Figure 5. Two waves out of phase.

Assuming that each wave has different amplitude given by a1 and a2, respectively,

4.5

(4.3-3)

Ellipical helixLight vector

z

x

y

a

b

Circular helix

Light vector

z

x

y

a

where 1 (1) and 2 (2) are the initial linear (angular) phases of E1 and E2, respectively. The

linear phase difference, , is given by

Since one wave trails the other, the linear phase difference is often referred to as the retardation.

4.4 Polarized Light

Most light sources consist of a large number of randomly oriented atomic or molecular emitters.

The rays emitted in any direction from such sources will have electric fields that have no

preferred orientation. In this case, the light beam is said to be unpolarized.

If, however, a light beam is made up of rays with electric fields that show a preferred direction of

vibration, the beam is said to be polarized. Figure 6, for example, shows the most general case

of elliptical polarized light where the tip of the electric vector describes an elliptical helix as it

propagates along z.

Figure 6. Elliptically polarized light. Figure 7. Circularly polarized light.

Figure 7 shows a more restrictive case where the tip of the light vector describes a circular helix

as it propagates along z. This condition, referred to as circularly polarized light, is studied in

Section 4.8. It is important because it will be encountered in Chapter 5 when studying the

circular polariscope in conjunction with the method of photoelasticity.

The most restrictive case, illustrated in Figure 8, occurs when E(z,t) vibrates in a single known

direction. This condition, referred to as linearly polarized light, is perhaps the most important

case as far as experimental mechanics is concerned. The figures used to characterize light in the

beginning of this chapter, for example, were drawn by making the assumption that the light

vector was linearly polarized. This state of polarization can be produced by using sheet-

polarizers, a Nicol prism, or, by reflection at the Brewster angle.

4.6

Plane of polarization

z

x Tip sweeps out a sine curve

(4.6-1)

Figure 8. Linearly polarized light.

4.5 Optical Interference

The light emitted by a conventional light source, such as a tungsten-filament light bulb, consists

of numerous short pulses originating from a large number of different atoms. Each pulse

consists of a finite number of oscillations known as a wave train. Each wave train is thought to

be a few meters long with duration of approximately 10-8

s. Since the light emissions occur in

individual atoms which do not act together in a cooperative manner, the wave trains differ from

each other in the plane of vibration, frequency, amplitude, and phase. Radiation produced in this

manner is referred to as incoherent light and the wavefronts simply add as scalar quantities.

For other light sources, such as a laser, the atoms act cooperatively in emitting light and produce

coherent light. In this case, the wave trains are monochromatic, in phase, linearly polarized, and

extremely intense. When the light coming from these sources is split and later recombined, the

wavefronts superimpose as vectors to produce optical interference.

Although coherent light must be monochromatic (of a single frequency) to interfere, light

coming from two different monochromatic sources is not, in general, coherent. In most cases,

interference will not take place unless the wavefronts in question originate from the same source.

4.6 Complex Notation

A convenient way to represent both the amplitude and phase of the light wave described by

Equation (4.1-3) for calculations involving a number of optical elements is through the use of

complex or exponential notation. To this end, the Euler identity is

where i2 = -1. The first term on the right hand side is called the real part of the complex function

whereas the term associated with i is the imaginary part.

Using this notation, the sinusoidal wave characterized by Equation (4.1-3) can be represented by

4.7

(4.6-2)

(4.6-3)

(4.6-4)

(4.7-1)

(4.7-2)

(4.7-3)

When the operations performed are linear, the symbol Re is dropped and calculations are

performed using the complex function

where and are the angular and linear phase, respectively.

If the waves have an initial phase, such as those described by Equation (4.3-2), they can be

expressed as

4.7 Intensity

In many practical applications, the light vector is related to a measurable quantity called the

intensity where

Assuming that k = 1, and using the form of E(z,t) given in Equation (4.6-3),

where E(z,t)* is the complex conjugate of E(z,t) and the operation is a dot product.

E(z,t)* is obtained from E(z,t) by changing the sign of the imaginary part of the latter which

amounts to changing the sign in the exponent of the exponential function. When referring to a

single wavefront, the dot product operation becomes a simple multiplication and

Equation (4.7-3) shows that the intensity of light is proportional to the square of the

instantaneous amplitude of the light vector.

4.8

(4.8-1)

(4.8-2)

(4.8-3)

(4.8-4)

(4.8-5)

4.8 Superposition of Wavefronts

As mentioned in Section 4.5, incoherent wavefronts superimpose as if they were scalars.

Assuming that two waves are represented by vectors E1 and E2, the resulting intensity for

incoherent waves is simply

If the wavefronts are coherent, the light vectors add as vectors. In this case, E1 and E2 can be

combined into a resultant ER and the intensity is

Consider, for example the superposition of two coherent (monochromatic; same frequency),

rectilinear harmonic vibrations (plane polarized wavefronts) that are of unequal amplitude and

out of phase. In this case, the expressions in Equation (4.6-4) can be used to characterize the

wavefronts; and, Equation (4.8-2) represents the intensity.

Since the two wavefronts are co-linear, the dot product becomes a simple multiplication, and

Equation (4.8-2) can be expanded as follows:

From complex variables, the last term in brackets on the right hand side of Equation (4.8-3) is

proportional to the cosine of the argument in the exponential function and the intensity becomes

where the term in the brackets of Equation (4.8-4) represents the difference in angular phase of

the wavefronts.

Two cases are of special interest. In the case in where the angular phase difference is a multiple

of , Equation (4.8-4) becomes,

If a1 = a2, then I = 0. In this case, there is no measurable intensity; implying that light plus light

results in darkness. Destructive interference is said to take place.

4.9

(4.8-6)

Ex = ax cos 2/ (z - ct + x)Ex

Ey

ax

ay

Ey = ay cos 2/ (z - ct + y)

(4.8-7)

(4.8-8)

If, on the other hand, the angular phase difference is either zero or a multiple of 2, then

Equation (4.8-4) becomes

If a1 = a2, then I = 4a2, thereby resulting in constructive interference.

Thus, the intensity observed when two rectilinear coherent wavefronts are superimposed depends

upon the amplitudes of the wavefronts and the relative difference in phase between them. In the

case considered above, the electric vector is restricted to a single plane and the superposition

results in a plane polarized, or, linearly polarized light. Interference effects have important

applications in optical metrology; for example, in photoelasticity, moir, holography, and

speckle metrology.

Two other important forms of polarized light, previously discussed in Section 4.4, arise as a

result of the superposition of two linearly polarized light waves having the same frequency but

mutually perpendicular planes of vibration. This condition is depicted in Figure 9.

Figure 9. Superposition of two wavefronts having perpendicular planes of vibration.

In this case, at a fixed position along the axis of propagation, the magnitude of the light vector

can be expressed as

where is the angular phase and is the angular velocity at which the electric vector rotates. To

achieve a better understanding of this case, the vector can be resolved into components along the

x- and y-axes, and the wavefronts shown in Figure 9 represented by

4.10

(4.8-9)

(4.8-10)

(4.8-11)

(4.8-12)

Ellipical helix

Light vector E

z

x

y

where x and y are the phase angles associated with waves in the xz and yz planes, and ax and ay

are the amplitudes of waves in the xz and yz planes.

The magnitude of the resulting light vector is given by vector addition as

Considerable insight into the nature of the light resulting from the superposition of two mutually

perpendicular waves is provided by studying the trace of the tip of the resulting electric vector on

a plane perpendicular to the axis of propagation at points along the axis of propagation. An

expression for this trace can be obtained by eliminating time from Equation (4.8-8). Pursuing

this argument in terms of angular phase results in the relation

Or, since

Equation (4.8-10) becomes

Equation (4.8-12) is the equation of an ellipse, and light exhibiting this behavior is known as

elliptically polarized light.

As illustrated in Figure 10, tips of the electric vectors at different positions along the z axis form

an elliptical helix. As mentioned previously, the electric vector rotates with an angular velocity,

. During an interval of time t, the helix will translate a distance z = ct in the positive direction.

As a result, the electric vector at position z will rotate in a counter-clockwise direction as the

translating helix is observed in the positive z direction.

Figure 10. Elliptically polarized light.

4.11

(4.8-13)

(4.9-1)

(4.9-2)

A special case of elliptically polarized light occurs when the amplitudes of the two waves are

equal [ax = ay = a] and = /2 [or, = /4]. In this case, Equations (4.8-10) and (4.8-12)

reduce to the equation of a circle given by

As mentioned previously in Section 4.4, light exhibiting this behavior is known as circularly

polarized light, and the tips of the light vectors form a circular helix along the z axis. For = /2

[ = /4], the helix is a left circular helix, and the light vector rotates counter-clockwise with time

when viewed from a distant position along the z axis. For = - /2 [ = - /4], the helix is a right

circular helix, and the electric vector rotates clockwise.

4.9 Reflection and Refraction

The discussions so far have been limited to light propagating in free space. Most optical effects

of interest, however, occur as a result of the interaction between a beam of light and some

physical material.

As mentioned in the beginning of the chapter, light propagates in free space with a velocity c

equal to approximately 3 x 108 m/s (186,000 mi/s). In any other medium, the velocity is less

than the velocity in free space.

The ratio of the velocity in free space to the velocity in a medium, v, is called the index of

refraction, n, of the medium. The latter is defined as

The index of refraction for most gases is only slightly greater than unity (for air, n = 1.0003).

Values for liquids range from 1.3 to 1.5 (for water, n = 1.33) and for solids range from 1.4 to 1.8

(for glass, n = 1.5). The index of refraction for a material is not constant but varies slightly with

wavelength of the light being transmitted. This dependence of index of refraction on wavelength

is referred to as dispersion.

Figure 11 illustrates that when a beam of light strikes a surface between two transparent

materials with different indices of refraction, it is divided into a reflected ray and a refracted ray.

The reflected and refracted rays lie in the plane formed by the incident ray and the normal to the

surface, known as the plane of incidence. The angle of incidence, , the angle of reflection , and

the angle of refraction are related. For reflection, light bounces off the surface with

Refraction is governed by Snells law and can be quantified in terms of the velocity of

propagation and/or index of refraction of the media as follows:

4.12

(4.9-3)

(4.10-1)

(4.10-2)

(4.10-3)

Reflected rayIncident ray

Wavefront

Material 1

Material 2

Refracted ray

Figure 10. Reflection and refraction.

4.10 Double Refraction - Birefringence

Light traveling through an isotropic medium propagates with the same velocity in all directions;

however, when light enters an anisotropic or crystalline medium along an optical axis it is

divided or refracted into two plane polarized components which are orthogonal and are

propagating at different velocities. This phenomenon is known as double refraction or

birefringence, and occurs in photoelastic models subjected to stress.

'The angular phase difference that occurs between these two wavefronts when light passes

through a plate of thickness d is

where is the wavelength and n is the index of refraction.

The corresponding linear phase difference, , is given by Equation (4.3-1); and,

The fringe order, R, to be defined later in conjunction with the method of photoelasticity,

provides a measure of the relative retardation as follows:

4.13

(4.10-4)

(4.10-5)

(4.10-6)

The difference in indices of refraction is proportional to the difference in principal stress; in

particular,

where C is the stress optic coefficient.

A more useful description of this phenomenon is given by the relation

where c is the velocity of propagation outside the model (usually air, n = l), d is the thickness of

the model, R is the fringe order, F = f/c is the model fringe value, and f = /c is the material

fringe value.

Equation (4.10-5) is used to determine the stress state in a loaded photoelastic model from the

analysis of the isochromatic fringe pattern. As illustrated in Chapter 5, the most commonly used

form of this equation is

where d is found by direct measurement, R is determined from the photoelastic fringe pattern,

and f is determined by performing a calibration test on a sample of the photoelastic material in

question.

4.14

E1 = a cos 2/ (z - ct + 1)

E2 = a cos 2/ (z - ct + 2)

E = a cos 2/ (z - ct)ct

2

1

a

E

z


1. Figure 12 shows two coherent (monochromatic; same frequency), rectilinear harmonic

vibrations (plane polarized wavefronts) which have unequal amplitude and are out of phase.

(a) Express the magnitudes of these light vectors in complex notation using the linear phase descriptions labeled in the figure.

(b) Using complex notation, derive the intensity distribution resulting from their superposition.

Hint: See Equation (4.6-4); apply Equation (4.8-2), and follow the arguments used to derive

Equation (4.8-4).

2. Analyze the intensity distribution derived in Problem 1 and specify the linear phase and any

other conditions that must occur to produce destructive and constructive interference.

Quantify the magnitude of the intensity for each of these conditions.

Hint: See the arguments made in Section 4.8. The results can be checked by applying Equation

(4.3-1).

Figure 12.

5.1

CHAPTER 5 - PHOTOELASTICITY

5.1 Introduction

Photoelasticity has long been a widely used experimental technique for determining stresses and

strains. The technique is applicable to any state of stress, but historically it has been for the most

part limited to the determination of static two-dimensional stress distributions. Later

developments led to its application to three dimensional stress problems, two dimensional

dynamic problems including both cyclic and transient, and some plasticity and viscoelastic

problems.

The technique is limited to a class of materials which are birefringent; however, the technique

can be applied to other materials and actual prototypes by cementing films or thin sheets of

photo-elastic material directly to the surface of the specimen. These techniques are covered in

this chapter.

The photoelastic technique is based upon the stress-optic law (see Section 4.10) that applies to

many transparent materials when they are stressed. Depending on the material, the phenomena

may be due to stress or strain, or both. If the material is linear and elastic the effect can be

referred to either stress or strain. If the material is viscoelastic both the strain-optic and the

stress-optic relationships must be known.

The materials in question exhibit a temporary birefringence when they are under a temporary

load. The principal optical axes coincide when the directions of the principal stresses and strains

are linearly related to the refractive indices along their two respective principal axes. The result

is that if a plane-polarized light wave enters a birefringent material under load, it will be resolved

into two plane-polarized waves that propagate through the stressed material at different

velocities. When the two waves emerge from

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