AER 304S

Aerospace Laboratory II

Photoelastic Stress Analysis

http://sps.aerospace.utoronto.ca/labs/raal

Experiment Duration: 150 min

Instructor

M. R. Emami

Aerospace Undergraduate Laboratories

University of Toronto

Winter 2013

1

1. Purpose

Some basic concepts of optical stress analysis techniques are demonstrated using a

variety of photoelastic beams. The stress distribution, stress concentration factor and

maximum shear stress are obtained by analysis of the images of the beams under stress in

polarized light. Some observations are also made for the effect of beam cross section and

geometry on stress distribution. A computer-aided data/image acquisition system is used

to collect and record images of the beams under stress.

2. Apparatus

Photoelastic stress test bed with birefringent beams.

Vishay Stresslume-10 beams: rectangular, 3/8” centre hole, 3/16” centre hole,

notched, dual section, arbitrary shape, knee, and arch.

Canon PowerShot® Pro1 digital camera (8.0 megapixel DSLR).

Measurement Specialties FC22 load cells (0-100 N).

Light box with parallel polarizing filters.

Haydon linear actuators for applying force to the beams.

Data acquisition system (DaqBoard/505™

& PCI-DIO24 PCI Boards).

Power supplies and power control unit.

Web cams and audio system.

3. Notation and Numerical Constants

δ retardation (nm)

Co stress optic coefficient

Ko the inverse of Co

I transmitted light

A maximum intensity

λ wavelength (nm)

K stress optic factor (MPa/nm)

σ stress (MPa)

F force (N)

S section area (mm²)

E modulus of elasticity (MPa)

ξ deformation (strain)

ξel elastic deformation (strain)

ξpl plastic deformation (strain)

σmax maximum stress (MPa)

σavg average stress (MPa)

σy yield stress (MPa)

σf fracture stress (MPa)

σu ultimate stress (MPa)

σpl proportional limit stress (MPa)

A(x) normal section at x

C(x) section centre at x

V(x) shear force at x (N)

Mc(x) moment at point C (Nm)

Igz quadratic moment along z (m4)

Ζxy normal shear stress (MPa)

Kt Theoretical stress concentration

Kts Geometric stress concentration

Kf reduced Kt value

q notch sensitivity (no unit)

r notch radius in mm

ρ grain size of the material (mm)

2

4. Experiment Setup

The major elements that make up the aerospace undergraduate laboratory photoelastic

stress test-bed are indicated in Fig. 1. The apparatus operates in series, with all beams

sharing the same set of hardware components, e.g., actuators, camera, etc.

Figure 1: Overview of the photoelastic stress test bed.

Top Right Load

Motor

Column Load

Motor

3

4.1 Photoelastic Beams

The experiment apparatus consists of eight photoelastic beams of different shapes each

placed on a staging frame. The frame is moved by a DC servo motor with rack-and-

pinion transmission to bring the desired beam to the loading area. The exact position of

the beam in the loading area is ensured by the servo incremental encoder. The encoder

count for each beam location has been pre-programmed into the apparatus with reference

to the home limit switch position. The incremental encoder is equipped with an index

signal which triggers once per rotation and is used for error correction on intermittent

missed counts. The resolution of the encoder enables it to position the beam within 3

micron of its target position. Dimensions for each beam are provided in Appendix A.

4.2 Data Acquisition System

The data acquisition system consists of a Pentium® IV workstation equipped with a

DaqBoard/505™

and PCI-DIO24 data acquisition (DAQ) boards from IOtech and

Measurement Computing respectively. The analog DAQ board (DaqBoard/505™

) has 24

digital I/O channels, and a 16-channel analog-to-digital (A/D) converter with 16-bit

resolution (i.e., 216

discrete voltage values over the measurement range of 0 to 10V),

which is equivalent to a voltage resolution of 0.15 mV. The digital DAQ board (PCI-

DIO24) has 24 digital I/O channels. The DAQ boards allow the computer software to

control electromechanical actuators and collect data from a variety of analog and digital

sensors.

4.3 Load Applicators

Load is applied to the beams using a computer-controlled linear force applicator that can

apply a load of up to 23 kg to the beams. The linear force applicators consist of a load

cell sensor coupled to a linear stepper-motor actuator with a resolution of 0.00396 mm

per step. Limit switches are used for homing the stepper motors.

The computer software can control each component of the experiment individually, such

as in the case of applying a desired load to a beam or it can control multiple components

simultaneously allowing complex dynamic experiments to be performed. The computer

control system enables the experimenter to not only collect data more accurately, but also

perform multiple tasks simultaneously.

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5. Experiment User Interface

1. Create a report template (.doc) for the experiment and save it to the Portal.

2. Create a MATLAB® file (.m) for the experiment and save it to the Portal.

3. Exit the experiment. Remember to first create a report template and MATLAB® file.

1. Change the currently selected beam configuration.

2. Select the position of the top left applicator.

3. Select the force for the top left applicator.

4. Select the position of the top right applicator.

5. Select the force for the top right applicator.

6. Select the force for the column applicator.

7. Select the position of the bottom right applicator.

8. Select the force for the bottom right applicator.

9. Select the position of the bottom left applicator.

10. Select the force for the bottom left applicator.

11. Graphic of the currently configured beam.

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12. History of previously executed commands.

13. Move the motors to the specified positions (move the motors before applying a force).

14. Capture a digital image of the beam and analysis the stress state.

15. Apply the loads (forces) to the beam.

16. Save the results of the previous image capture and analysis.

17. View the last captured and analyzed images.

18. Help information is displayed here.

1. Change the currently selected beam configuration.

1. Colour map for real stress state.

2. Raw image captured from the digital camera.

3. Colour map for the analyzed stress state.

4. Image of the analyzed stress state.

5. Stress and retardation for the currently selected pixel.

6. Save the results of the previous image capture and analysis.

7. Return to the experiment control interface.

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6. Photoelastic Theory

Photoelastic analysis is a well-established technique for determining the stress

distribution in structures. It has been used since the middle of the 20th

century, before we

had the capability of finite element analysis, to investigate whole-field stress distribution

in structures, and it is still used in research and industry to test and verify mechanical

designs.

To derive meaningful results from the experiment, one must possess some knowledge of

photoelasticity and what information can be gathered from the analysis. This section will

cover three main areas: the physical principles behind photoelasticity, the structure of the

polariscope, and the analytic capabilities of the method.

6.1 Photoelasticity

Photoelastic analysis relies on three major principles: the polarization of light,

birefringence, and the photo-stress relationship.

Light can be thought of as a transverse electromagnetic wave with the electric and

magnetic fields oscillating perpendicular to the direction of the wave’s propagation, as

can be seen in Fig. 2. Most light sources produce unpolarized light, with the fields

oscillating in all directions perpendicular to the direction of propagation. Filters can be

used to produce plane polarized light, where the field oscillates in a plane. Filters can also

be used to produce circularly polarized light, where the electric field vector rotates with a

constant angular velocity as the wave propagates. Such a filter is called a circular

polarizer, and consists of a linear polarizer, and a quarter-wave plate. Fig. 3 shows

various polarization filters.

Figure 2: Electromagnetic wave.

A material may exhibit a property called birefringence, that is, it transmits light polarized

in two perpendicular directions, resulting in separation of the waves into two

perpendicular components. This introduces a phase shift between the two polarized

components, as seen in a circular polarizer in Fig. 3.

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Figure 4: General structure of a polariscope.

Some materials (in particular many transparent plastics and epoxies) are conditionally

birefringent. They only exhibit birefringence when placed under stress, and thus are

useful as models for photoelastic analysis. The difference in propagation wavelength is

linearly related to the difference between the principal stresses in the material, and results

in retardation of one polarized beam relative to the other. The property relating the

retardation to the stress is known as the stress optic coefficient (Co) of the material, with

SI units of m2/kg. The equation relating the retardation to stress optic coefficient and the

stress in the material is:

)( 21 tCo (1)

where δ is retardation, Co the stress optic coefficient, t thickness of the material, and σ1,

σ2 are the principal stresses at the point. A more useful form of the equation is:

Figure 3: Various forms of polarized light.

8

t

Ko 21 (2)

where Ko replaces 1/Co. Usually Ko is stated in units of MPamm/nm (mm represents mm

thickness of sample, nm represents retardation).

From the above, we can see that δ is proportional to the difference of the stresses in the

material. A polariscope is an apparatus that enables us to observe the retardation of light

as interference fringes in the material.

6.1.1 The Transmission Polariscope

The key components of a polariscope are a light source, linear polarizers, quarter wave

plates and a model made out of photoelastic material. Linear polarizers produce plane-

polarized light. The quarter-wave plates are birefringent sheets, with the retardation

between the principal axes precisely defined (called so because they are designed to

produce a retardation that is ¼ of a specified wavelength).

A diagram of the polariscope is shown in Fig. 4. Light from the source is polarized by the

first polarizing filter (polarizer), and is then turned into circularly polarized light by a

quarter wave plate with its principal axes rotated 45˚ relative to the filter. The light passes

through the specimen, and then through a quarter wave plate rotated 45˚ in the opposite

direction, which cancels the effect of the previous plate, and finally through a second

polarizing filter (the analyzer) oriented in the same axis as the first one, and to the

observer.

The birefringence of the specimen under stress introduces additional retardation into the

light beam. The second quarter wave plate and the analyzer essentially make the two

oscillation directions to interfere by “squeezing” them into the same axis of oscillation.

This introduces variations in the colour and intensity of transmitted light, which can be

related to specific values of the difference between principal stresses. For a particular

wavelength λ, the relative intensity of the transmitted light I is given by:

2cos 2AI (3)

where A is the maximum intensity, and δ is the retardation caused by the specimen. The

spectrum of transmitted white light can also be determined by evaluating the intensities

across a range of frequencies.

The polariscope can work with either monochromatic light, which produces a series of

fringes, or white light, which, due to a range of frequencies, produces a transition

between various colours.

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6.2 Photoelastic Analysis

Using the relationships from Equation (3), one can derive much useful information from

photoelastic analysis. In 2-dimensional photoelasticity, assuming a specimen of constant

thickness, one can find the difference between the principal stresses at a point using an

equation similar to Equation (2):

K max21 2 (4)

where σ1 and σ2 are the principal stresses, τmax is the maximum shear stress at the point, K

is the stress optic factor Ko/t, and δ is the relative retardation of the light waves.

There are several important points to be made about the limitations of the analysis.

1) The photoelastic method cannot give us the two principal stresses explicitly, only

the maximum shear stress in the material. However, in certain cases, we can make

assumptions about one of the stresses. An important rule is that the stress

perpendicular to a free surface is zero. Thus, for example, in a constant moment

beam, we can assume that σ2 is zero everywhere in the middle section, and

equation (4) will give us the actual tension stress in the material. See Figure 5 as

an example of constant moment load analysis.

2) One cannot determine whether the material is in tension or compression, or the

directions of the stresses1 directly from photoelastic method, only the magnitude

of the stresses is known.

3) The experiment uses an RGB (Red, Green, Blue) camera for capturing images of

the specimens, and is thus limited to relatively low retardations and stresses, as

the spectral characteristics of the camera fail to distinguish between the

retardation spectra. We are also limited by the resolution of the camera in terms of

investigating fine details of the distribution.

Keeping these limitations in mind, one can still gain much useful information from the

analysis.

1 It is worth noting that if we used a plane polariscope (same as Fig. 3 minus the quarter wave plates), we

would be able to see points where the principal stresses are aligned with the filters as pure white, thus

enabling us to determine directions of principal stresses via rotation of the filters.

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Figure 5: Example of a beam under constant moment

(note: example beam dimensions are not actual dimensions, see Appendix A).

6.2.1 Colour Mapping

The apparatus used in this experiment is a 2-dimensional white light transmission

polariscope. The stress optic coefficient of the material is Ko = 1.22x10-2

MPa·mm/nm,

all beams have a thickness of t = 5.5mm, resulting in stress optic factor K = 1.95x10-3

MPa/nm.

The colour guide relating the colour seen by the observer and the relative retardation is

given in Fig. 6. The predicted guide was determined by convolving the spectral response

of the RGB components with the expected spectrum of transmitted light.

Figure 6: Top line is the predicted gradient, bottom is the actual gradient observed.

The software includes an image analysis algorithm that will calculate the stress

distribution through the entire sample. Keep in mind that it is limited due to constraints

on both image quality and processing time, and thus is not perfect – expect inaccuracies

of ±10%. It may also give inaccurate distributions immediately surrounding the points of

application (usually noticeable as colour discontinuities), and other areas of stress

concentration. It is also constrained by the fact that it is calibrated only to 2010nm

retardation. You are expected to be able to determine the retardation and stress from the

provided colour guides if necessary.

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6.3 Stress Analysis

Most of the structural materials have mechanical characteristics featured in the following

diagram. These materials, usually labelled as ductile, possess a relatively large plastic

region before the failure point. However, some materials, labelled as hard or brittle, such

as cast iron, do not quite follow this diagram. They have no yield point and their plastic

region is small; therefore, they will break just with a little more stress than what is at the

peak of the elastic region. Some other factors such as creep and hysteresis may also make

it difficult to define a single curve for a particular material.

Figure 7: Stress-strain characteristics .

This experiment is confined to the regime below the proportional limit in the diagram

since the Modulus of Elasticity or Young's Modulus (E), as described below, is only

defined in this region.

E (5)

Strain ε is the change in length per unit length under an applied stress.

L

L (6)

6.3.1 Three-point Load System

A three-point load system consists of three lateral forces at three points on the beam

holding it at equilibrium. This will create bending stress at each normal section.

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(7)

(8)

x

X2 X1 X3

F1

F3

x

X2 X1 X3

Mmax

X

X

Y

Position : X3

Force : F3

Position : X2

Force : F2

Position : X1

Force : F1

Part I Part II

Section A(x)

Figure 8: Three point load configuration.

From Fig. 8, the static equations at point C of the section A(x) are:

0321 FFF

0)()()( 332211 xXFxXFxXF

Equations (7) and (8) are used to verify or to calculate the force that is applied at each

point of application if unknown. At each section A(x), the variation of shear force V(x)

and bending moment Mc(x) can be illustrated by the following graphs:

V(x):

Mc(x):

C(x)

Figure 9: Shear in each section A(x) under a three-point load system.

Figure 10: Moment at point C(x) in each section A(x) under a three-point load system.

13

(9)

(10)

)()( 121233max XXFXXFM

The applied moment resolves in bending stress at each section A(x) along the beam as

shown bellow:

x

X

)(y

Y

Position : X3

Force : F3

Position : X2

Force : F2

Position : X1

Force : F1

Part I Part II

Section A(x)

The bending stress acts as tension on one half of the section and compression on the other

half. Beam strength can be measured from the maximum allowed stress.

Knowing that, σ(y=0) = 0, σmax occurs at y = h/2, and the distribution of σ(y) is linear at

each section, the following equation can be written:

yh

yh

y

maxmax 2

2/)(

C(x) Figure 11: Three point load configuration causing bending stress at section A(x).

Figure 12: Detailed view of bending stress at section A(x).

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(11)

(14)

(13)

(15)

(12)

At this stage, it is only possible to calculate σ(y) for a given y by knowing the value of

σmax.

To determine σmax at section A(x) one can write:

6

2

2)(

)(

2

max)(

2max)(

max

)(

hbM

dzdyyh

M

yh

y

ydSyM

xc

xc

S

xc

Knowing that

2

12

max

32

hy

hbdSyI

SGz

,

Equation (11) can be rewritten as:

gz

xc

I

yM max)(

max

.

This finally leads to a formulation for σ(y) for any point at section A(x):

yI

My

Gz

xc

)(

)(

For a beam with rectangular section, one can write:

yhb

My

xc

3

)(12)(

15

(16)

x

X2 X1

F

F

X4 X3

(17)

6.3.2 Constant Moment

Constant moment load consists of four lateral forces at four symmetric points on the

beam (two on each side of the beam and all forces are equal) holding it at equilibrium, as

shown below:

),( yx

Y

X

X

Position : X4

Force : F

Position : X2

Force : F

Position : X3

Force : F

Position : X1

Force : F

Part I Part II

Section A(x)

From Fig. 13, the static equations at point C of section A(x) are:

04321 FFFF

0)()()()( 44332211 xXFxXFxXFxXF

At each section A(x), the variation of shear force V(x) and bending moment Mc(x) can be

illustrated by the following graphs:

V(x):

Figure 13: Beam under a constant moment load.

C(x)

Figure 14: Shear in each section A(x) under constant moment load.

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x

X2 X1

Mmax

X4 X3

(18)

M(x):

As shown above, the shear is null between X2 and X3 and the moment load at the normal

section is constant, and this is why this load system is called “constant moment.”

The bending stress at section A(x) for a rectangular beam can be obtained from Equation

(15).

6.3.3 Principal Stress

At a given point on a surface, stress quantity depends on the orientation (unit normal) of

the surface. For a different normal vector in the same stress field, the stress quantity on

the associated surface must also be different. The stress quantity on three mutually

perpendicular planes at a point specifies the state of stress at that point in a continuum.

The components of these stress quantities form a tensor, the stress tensor.

Mathematically, the stress tensor is a second-order Cartesian tensor with nine stress

components. Three stress components that are perpendicular to the planes are called

normal stresses. Those components acting tangent to these planes are called shear

stresses. The sign of a stress component can be assumed as positive when its direction

and the normal vector of the surface, on which the component of the stress tensor is

acting, are of the same sign. Otherwise, the sign of the stress component is negative.

The component σij of the stress tensor is defined as

N

j

jjii dAdF1

,

where dFi are the components of the resultant force vector acting on a small area dA that

can be represented by a vector dAj perpendicular to the area of the element. The

component σij of the stress tensor depends on the orientation of the plane that passes

through the point of interest.

Figure 15: Moment at point C(x) in each section A(x) under a three-point load system.

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All real objects occupy three-dimensional space. However, a class of common

engineering problems, involving stresses in a thin plate or on the free surface of a

structural element, have one principal stress that is much smaller than the other two.

Examples are the surfaces of thin-walled pressure vessels under external or internal

pressure, the free surfaces of shafts in torsion, and beams under transverse load. By

assuming that this small principal stress is zero, the three-dimensional stress state can be

reduced to two dimensions. Since the remaining two principal stresses lie in a plane,

these simplified 2D problems are called plane stress problems. Assume that the negligible

principal stress is oriented in the z-direction, to reduce the 3D stress tensor to the 2D

plane stress matrix, remove all components with z subscripts to get,

Figure 16: Stress tensor for a 3D element.

Figure 17: Stress components on the 2D element.

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(19)

yyx

xyx

xy

where xy = yx for static equilibrium. The sign convention for positive stress components

in plane stress is illustrated in the above figure on the 2D element.

The coordinate directions chosen to analyze a structure are usually based on the shape of

the structure. As a result, the direct and shear stress components are associated with these

directions. For example, to analyze a beam one almost always directs one of the

coordinate directions along the beam's axis. Nonetheless, stresses in directions that do not

line up with the original coordinate set are also important. For example, the failure plane

of a brittle shaft under torsion is often at a 45° angle with respect to the shaft's axis. Stress

transformation formulas are required to analyze these stresses.

The transformation of stresses with respect to the {x,y,z} coordinates to the stresses with

respect to {x',y',z'} is performed via the equations,

2cos2sin2

2sin2cos22

2sin2cos22

xy

yx

yx

xyxxy

yxyx

y

xy

yxyx

x

(20)

where is the rotation angle between the two coordinate sets (positive in the counter

clockwise direction). This angle along with the stresses for the {x',y',z'} coordinates are

shown in figure below.

Figure 18: Coordinate transformation of the stress components.

19

(22)

(23)

The normal stresses (x' and y') and the shear stress (x'y') vary smoothly with respect to

the rotation angle , in accordance with the coordinate transformation equations. There

exist a couple of particular angles where the stresses take on special values.

First, there exists an angle P where the shear stress x'y' becomes zero. That angle is

found by setting x'y' to zero in the above shear transformation equation and solving for

(set equal to P). The result is,

yx

xy

P

22tan (21)

The angle p defines the principal directions where the only stresses are normal stresses.

These stresses are called principal stresses and are found from the original stresses

(expressed in the x,y,z directions) through the following equations:

2

2

2

2

2

1

22

22

xy

yxyx

xy

yxyx

The transformation to the principal directions can be illustrated in figure below.

Figure 19: Principal Stresses on the 2D element.

20

Another important angle, s, is where the maximum shear stress occurs. This is found by

finding the maximum of the shear stress transformation equation, and solving for . The

result is

45 2

2tan

PS

xy

yx

S

(24)

The maximum shear stress is equal to one-half the difference between the two principal

stresses,

22

212

2

max

xy

yx (25)

The transformation to the maximum shear stress direction can be illustrated in figure

below.

6.3.4 Stress Concentration

Standard stress-analysis calculations are mostly based on the assumption that the material

is uniform without any irregularities, such as abrupt change of section, notches, holes,

cracks, etc. However, shapes and sections are often more complex in real applications

such as shoulders on a shaft, keyways, screw threads, etc. Any discontinuity in the

material affects the stress distribution in the surrounding area. Such “discontinuity” is the

cause of the increase in the local stress, and it is referred to as “stress concentration.”

Figure 20: Maximum shear stresses on the 2D element.

21

(26)

(27)

(28)

(27)

Figure 21: Stress concentration at section variation.

These stress-concentration areas are mostly the points of interest in the stress analysis as

they are the most likely locations for a fatale rupture. This study is particularly important

for brittle materials (see definition of brittle material). But, it also remains important for

ductile materials (see definition of ductile material) in cases such as calculating fatigue.

The stress concentration factor, Kt, is used to relate the actual maximum stress at the

discontinuity to the nominal stress.

StressNominal

StressMaximumK t

This ratio is the translation of the stress concentration in the material around a specific

shape, such as a shoulder on a shaft, a hole in a beam, etc.

Figure 22: Illustration of the stress factor on a plate with a central hole.

hdb

P

A

Pavg

)(

avgtK max

(a) Plate with cross(sectional plane)

(b) Half of plate with stress distribution

Stress Concentration Factor

b = height of the beam

d = diameter of the hole

h = depth of the beam

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The stress concentration factors can be obtained from materials handbooks where they

are tabled or plotted based on different materials, geometries (e.g., rectangular plate with

filet, rectangular plate with central hole, round bar with filet, round bar with groove, etc.),

and applied loads (e.g., axial load, bending load, and torsion, etc.).

7. Experiment Design

Some preparation and research will be required to understand your experiments prior to

actually performing the tests with the photoelastic setup. Each experiment should not be

viewed as an independent activity. The results of one experiment may prove useful in

defining the parameters of another test.

A MATLAB® file can be generated by the interface to compute the stress and retardation

for the analyzed images, but you need to complete collecting data for all tests before

running the file.

Digital Photoelastic Analysis: To analyze the stress distribution in each of the beams use

the controls under the “Experiment Controls” box. Click the “Move Motors” button to

move the applicators to the specified position. Click the “Apply Load” button after

moving the motors to apply the specified forces to the beam. Click the “Capture and

Analyze” button to capture a digital image of the beam and to have the software analyze

the stress state. The analysis procedure can take up to a minute to process the image.

Large stress gradients may cause anomalies in the analyzed image (these will appear as

black pixels or areas. Mouse-over the captured or analyzed image to view the stress value

and retardation. Finally, click the “Save Results” button to save the captured and analyzed

images.

7.1 Three-point Load System

The purpose of this experiment is to determine how the location of a fixed applied load

(25 N) affects the stress distribution in the rectangular beam.

Try different three-point load configurations until you find the configuration that leads to

the maximum stress in the beam. For each configuration, record the maximum bending

stress along the outside edge of the beam and the maximum shear stress along the neutral

axis of the beam. Record at least three different data sets, including the one that leads to

the absolute maximum stress distribution in the beam.

7.2 Beam Section Effect

The purpose of this experiment is to determine how the cross section affects the stress

distribution in a beam. A fixed three-point load system (10 N) on Beam 5 is available on

the setup application for this experiment.

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Beam 5 consists of two segments with different cross sections. Record the stress value at

a number of points along the outside edge of each segment (sufficient number of points to

have a clear plot of the stress versus the position for each section). Do not consider the

thick blocks where the loads are applied.

7.3 Stress Concentration

The purpose of this experiment is to determine how various section anomalies affect the

stress distribution throughout a beam. A number of preset configurations are available on

the setup application for this experiment. They are listed as follows:

- Column load, rectangular beam (Beam 1)

- Column load, 3/8” hole beam (Beam 2)

- Column load, 3/16” hole beam (Beam 3)

- Column load, notched beam |(Beam 4)

- Column load, arbitrary shape load (Beam 6)

- Constant moment, rectangular beam (Beam 1)

- Constant moment, 3/8” hole beam (Beam 2)

- Constant moment, 3/16” hole beam (Beam3)

- Constant moment, notched beam (Beam4)

It is recommended that you perform both column load and constant moment experiments

with each beam at a time. For each configuration, record the maximum stress values

around the anomaly for three distinct loads (minimum difference of 5N between any two

loads). The data will be used later to plot σmax versus σavg.

7.4 Structural Beams

The purpose of this experiment is to demonstrate the effect of geometry on the stress

distribution in beams used for practical applications.

By applying the same forces on the two different beams, observe the stress distribution in

the beams, and record the maximum stress values in each beam. Make sure to save the

stress distributions.

8. Discussion of Results

8.1 Three-point Load System

1. Verify by calculation the load balance for each load configuration, and plot the

expected shear and moment graphs along the beam. Specify the maximum values

from the theoretical graph.

2. Calculate the corresponding maximum bending stress along the outside edge of

the beam, and the shear stress along the neutral axis of the beam.

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3. Compare your calculations with the recorded data and comment on the results.

4. For the load configuration that creates maximum stress in the beam, obtain the

principal stresses and principal directions. Also, obtain the direction along which

maximum shear stress occurs.

8.2 Beam Section Effect

1. Calculate and plot the theoretical moment and the corresponding stress along each

beam segment (Do not consider the thick blocks where the loads are applied).

Calculate the ratio of the two slopes, what do you conclude?

2. Plot the stress variation along the outside edge of the beam against the position for

both segments from the recorded data, determine the ratio of the two slopes, and

compare it with the ratio of the moment of area for the two segments. Comment

on your results.

8.3 Stress Concentration

1. Determine the stress concentration factors Kt for each of the tested configurations.

Calculate the corresponding maximum stress values for each configuration. Use

figure 24 for Kt of Beams 2 and 3 under column loading, figure 25 for Kt of

Beams 2 and 3 under constant moment loading. For Beam 4, the stress

concentration factor is Kt = 1.9 under column load and Kt = 1.6 under constant

moment load (provided by the manufacturer).

2. Plot σmax versus σavg from the recorded data, and determine the stress

concentration factor Kt for each setup. Compare these plots with your data from

the previous step and comment on your results.

3. For Beam 6 interpret the stress distribution throughout the beam considering the

shape of the beam (section variation and centre hole). How would the stress

distribution vary if the diameter of the centre hole is considerably smaller (divided

by two, for example)?

4. Discuss any and all sources of error and estimate the contribution to the total

uncertainty fro each source.

5. Which portion of the system do you think introduced the most significant error?

Why? How could this be improved or compensated for?

8.4 Structural Beams

1. Beams 7 and 8 have the same outside dimensions and can fulfill the same need for

support. Plot the maximum stress versus load for each beam on the same graph.

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2. From the visualization of stress distribution, point out where each beam will

fracture first. If you were to use a knee shape beam type of support (Beam 7) for a

great load, how could you simply reduce the stress variation significantly? The

original shape must stay invariant; you are only allowed to augment the beam

with additional components. If you add other components, what kind of new

stress concentrations may occur in the overall structure?

9. References

[1] A.J. Durelli, and W.F. Riley, 1965, Introduction to Photomechanics, N.M.

Newmark, Canada

[2] Hamrock, B.J., Jacobson, B., and Schmid, S.R., 1999, Fundamentals of Machine

Elements, McGraw-Hill WCB, New York

[3] Shigley, J.E., and Mischke, C. R., Mechanical Engineering Design, 5th ed.,

McGraw-Hill, New York.

[4] Krutz, G.W., Schueller, J.K., Claar, P.W., 1999, Machine Design for Mobile and

Industrial Applications. Society of Automotive Engineers (SAE), St. Joseph, MI

[5] The Stress-Opticon Instruction Manual (Vishay)

[6] Website: http://www.cs.rit.edu/~ncs/color/ (Vishnevsky colour conversion)

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Appendix A: Beam Dimensions

All dimensions are in millimetres. All the beams have a thickness of 5.5 mm.

Beam 1: Rectangular beam.

Beam 2: 3/8” centre hole beam.

Beam 3: 3/16” centre hole beam.

Beam 4: Notched beam.

Beam 5: Dual section beam.

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Beam 6: Arbitrary shape beam.

Beam 7: Knee shape beam.

Beam 8: Arch beam.

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Appendix B: Stress Concentration Factors

Graph 1: Kt rectangular plate with filet for axial load.

Graph2: Kt rectangular plate with filet for bending load.

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Graph 3: Kt rectangular plate with central hole for axial load.

Graph 4: Kt rectangular plate with central hole for out-of-plane bending load.

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Graph 5: Kt rectangular plate with central hole for in-plate bending load.