COMBINED LOADING OF STRAIGHT AND CURVED BEAMS

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Nov 15, 2013 (3 years and 9 months ago)

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MECH 396/398: Combined Loading of Straight and Curved Beams

Department of Mechanical and
Materials Engineering


21

Queen’s University

Faculty of Applied Science

Department of Mechanical
and Materials
Engineering


COMBINED LOADING OF STRAIGHT AND CURVED BEAMS


Location

Jackson Hall Rm 213


Objectives

1.

To calculate analytically the stress/strain distribution for combined

loading conditions of
two curved beam specimens.

2.

To experimentally observe the stress/strain distributions in the two loaded beams by using:



strain gauges



photo
-
elastic analysis


Safety Considerations

This laboratory is conducted without the use of obv
iously dangerous equipment, but it does
require the manipulation and suspension of relatively heavy metal weights at a significant height.
Care must be taken to handle the weights in such a way that they do not fall and cause injury.


Preparation Notes

In

this lab you will need to sketch the coloured photoelastic patterns seen in loaded beams. You
should bring plain paper and a few coloured pencils, and/or a digital camera.


Background

Strain Gauges

The most common method of measuring mechanical strain
in an object is with strain gauges. The
basic principle underlying the operation of strain gauges is that the resistivity of wire changes
when it undergoes mechanical strain. If the resistance element is attached directly to the object
being strained it wi
ll undergo an equal strain and in this way, the measured change in resistance
can be correlated to the strain in the object.


Stress Distribution in a Straight Beam

When a straight beam is loaded in tension or in compression, the stress in any cross
-
sectio
n of
the beam is





A
P




where

P = load





A = cross
-
sectional area


When a straight beam is loaded in bending, the stress at any point in the beam is determined by

MECH 396/398: Combined Loading of Straight and Curved Beams

Department of Mechanical and Materials Engineering


22






zz
I
My
y

)
(



where

y = distance across the cross
-
section

M = bending moment at that location along the beam

I
zz

= moment of inertia perpendicular to the cross
-
sectional area


According to the principle of superposition, the total stress acting in a certain direction on a
beam is the sum of applied stres
ses acting in that direction [1].


Stress Distribution in a Curved Beam

For the case where the beam is loaded only in tension or compression the stress components are:




sin
2
2
3
2
2














r
b
a
r
b
a
r
N
t
P
r















r
b
a
r
b
a
r
tN
p
2
2
3
2
2
3
'



sin




where

a
b
b
a
b
a
N
ln
)
(
'
2
2
2
2





t = thickness

a, b, r = shown in Figure 1.





MECH 396/398: Combined Loading of Straight and Curved Beams

Department of Mechanical and Materials Engineering


23



Figure 1: Dimensions of general curved beam and geometry of loading


In the case where the beam is loaded in a pure bending moment the stresses are:


r
a
a
b
r
b
a
b
r
b
a
tN
M
r
ln
ln
ln
4
2
2
2
2
2






























2
2
2
2
2
2
2
ln
ln
ln
4
a
b
r
a
a
b
r
b
a
b
r
b
a
tN
M




where



2
2
2
2
2
2
ln
4









a
b
b
a
a
b
N


Photo
-
elastic Analysis

Experimental stress analysis is used not only to find the magnitude of stresses. Often it is
important to be able to interpret the entire stress distribution field. Photo
-
elastic analysis is one
method that
allows us to observe the stresses over an entire component. Photo
-
elastic analysis
can also be an important design tool since it can yield valuable information on how to optimize
the design and reduce stress concentrations.


The basic principles underlying

photo
-
elastic analysis are relatively simple. Certain materials,
notably plastics, behave isotropically when unstressed but become optically anisotropic when
stressed. The index of refraction is dependent on the stress applied. When a polarized beam of

light passes through a photo
-
elastic coating on a part subjected to stress it is split into waves
traveling at different speeds. Once the waves have excited the coating, they will be put out of
phase. This phase difference is called the retardation. Ob
serving the stressed component with a
polariscope, an interference pattern can be seen. This pattern indicates the stress distribution
over the component.


If the strains along
x

and
y

are
x


and
y


and the speed
of light in these directions is
V
x

and
V
y

then the retardation between the beams is: [2]




y
x
y
x
n
n
t
V
t
V
t
C















Brewster’s Law states that: "The relative change in index of refraction is proportional to the
difference of principal strains" and is interp
reted numerically by:






y
x
y
x
K
n
n







where

K
is called the strain
-
optical coefficient and is a physical property of the material.


MECH 396/398: Combined Loading of Straight and Curved Beams

Department of Mechanical and Materials Engineering


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Combining these expressions:




y
x
tK









y
x
tK





2


with the first expression being fo
r transmission and the second for reflection. The basic relation
for strain measurement with the reflection photo
-
elastic technique is therefore:




tK
y
x
2







At every point the retardation between the light beams is:




y
x
tK
N







2



: retardation



: wave length


N: fringe order.


Simplifying:


Nf
tK
N
y
x



2




tK
f
2




t: thickness of the coating


K: strain optical coefficient of the coating.


The difference bet
ween the principal stresses in the component is:




v
E
Nf
v
E
y
x
y
x






1
1






In order to determine the stress, the fringe order must be determined. When examined under a
polariscope, the retardation between the two beams of light increases proportionally wi
th the
stress. Every time the retardation is equal to:







N
...
3
,
2
,



MECH 396/398: Combined Loading of Straight and Curved Beams

Department of Mechanical and Materials Engineering


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a particular colour disappears and is replaced by its complementary colour. The colour sequence
is summarized in
Table 1
. The colours progressively change from the no load
condition to the
maximum load. The colour sequence is black, yellow, red, blue, yellow, red, green, yellow, red,
green... From this the fringe order, N, can be determined. Note that only the magnitude,


x
-


y

can be derived, away from boundaries. However, at boundaries one stress will be zero, and
therefore the other can be found.


Fringes appear as continuous bands that end at boundaries or occur as continuous loops that do
not intersect. An area cove
red by a single uniform colour indicates that the strain is uniform over
that entire area.


Table 1:

Photo
-
elastic pattern colour sequence appearance.


No colour means “no stress”

Black





is a zero fringe

Yellow

Red


1st Fringe




is black and labelled
as 1

Blue
-
Green

Yellow

Red


2nd Fringe



is black and labelled as 2

Green

Yellow

Red


3rd Fringe




is black and labelled as 3

etc.



More Information


More information on strain gauges and photo
-
elastic technology may be found in reference
books [2] and w
ebsites [3, 4].


Apparatus

The apparatus used in this lab is as follows:


1.

two beam specimens

2.

support frame

3.

loading apparatus

4.

weights

MECH 396/398: Combined Loading of Straight and Curved Beams

Department of Mechanical and Materials Engineering


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5.

strain gauges fixed to the specimens

6.

digital strain indicator

7.

Model 031 reflection polariscope


There are two specimens; each of them is roughly a C
-
shape. The smaller specimen has a
relatively large
-
radius continuous curve, Figure 2. The larger one has two small
-
radius curves
and a straight section in the middle, Figure 3. For purposes of this l
ab, the larger one is referred
to as the straight beam. Stress distributions within the straight section may be calculated using
straight
-
beam equations, for effects of a force
P

acting with a moment arm
d

from any point in
the cross
-
section of the strai
ght section.


Figure 2: Schematic of curved beam specimen. Dimensions in inches.


Material properties are given in Tables 2 and 3.


MECH 396/398: Combined Loading of Straight and Curved Beams

Department of Mechanical and Materials Engineering


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Table 2: Specifications for specimen material


Photo
-
elastic Properties

t


0.116 ± 0.002 in

K


0.15

f


655 x 10
-
6

[estimated


may be verified by experiment]




22.7 x 10
-
6

in


Mechanical Properties for 6061 Aluminum


Tensile Strength


32,000 psi


Yield Strength


28,000 psi


Shear Strength


22,000 psi


Young’s Modulus


10,000,000 psi


Shear
Modulus


3,790,000 psi


Poisson’s Ratio


0.33




Figure 3: Schematic of beam with straight section. Dimensions in cm.



MECH 396/398: Combined Loading of Straight and Curved Beams

Department of Mechanical and Materials Engineering


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Procedure

Part A: Strain Gauge Analysis

1.

Make a sketch of the apparatus. Record all appropriate dimensions and identify the
loc
ations of the strain gauges as precisely as possible.

2.

Ensure that the strain gauges are properly connected and zeroed under no load conditions.

3.

For four loads to be specified by the TA, record the strain readings for each specimen.


Part B: Photo
-
ela
stic analysis

For
one

of the loads from Part A and both specimens:

1.

Make a sketch of the apparatus.

2.

Sketch the photo
-
elastic patterns.

3.

Identify the fringe orders (N) at the strain gauge locations.


Results

Part A: Strain Gauge Analysis

1.

From the
experimental results determine the location of the neutral axis for both beams.

2.

Plot the strain distribution across both specimens. Is this linear? why or why not?

3.

From the axial strain readings, determine the axial stress values and plot these with resp
ect
to their location on the components.


Part B: Photo
-
elastic Analysis

1.

On your sketch of the photo
-
elastic pattern indicate the whole value fringe orders (changes
from red to green), and the fringe orders at each strain gauge location.

2.

Calculate t
he maximum shear stress at each gauge location.


Before leaving the laboratory, your results must be checked and verified (signed) by the TA.
These must be included as an appendix in the final report.


Report

An individual report is due one week after com
pletion of this laboratory. For this particular
laboratory, students should ensure that answers to the following Supplemental Questio
ns are
included in appropriate sections of their reports.


Supplemental Questions

1.

Do your experimental results for the st
rain distribution agree with the theoretical values?
Why or why not?

2.

Perform an uncertainty analysis on your theory and results. Note that for the theoretical
component it is only necessary to perform the analysis on the straight beam as it is very
ted
ious for the non
-
linear case.

3.

Determine the loading that would produce failure of the components.

MECH 396/398: Combined Loading of Straight and Curved Beams

Department of Mechanical and Materials Engineering


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4.

Explain the relative advantages and disadvantages of reflection and transmission photo
-
elasticity.

5.

Briefly describe how you would select a photo
-
elastic
coating.

6.

From your sketches of the photo
-
elastic patterns, where are the areas of stress
concentrations?


References

1.

R.C. Hibbeler,
Mechanics of Materials
, 3
rd

edition, Prentice Hall, 1997.


2.

J.W. Dally, W.F. Riley,

Experimental Stress Analysis
, 3rd Editi
on, McGraw Hill, 1991.


3.

“Strain Gages and Accessories”, Vishay Measurements Group,
http://www.vishay.com/brands/measurements_group/strain_gages/mm.htm


4.

“Optical Measureme
nt and Analysis of Stresses/Strains in Test Parts and Structures”,
Vishay Measurements Group,
http://www.vishay.com/test
-
measurements/photo
-
stress
-
plus/



5.

“Introduction to Photoela
sticity”, University of Cambridge, Department of Materials
Science and Metallurgy,
http://www.msm.cam.ac.uk/doitpoms/tlplib/photoelasticity/index.php



MECH 396/398: Combined Loading of Straight and Curved Beams

Department of Mechanical and Materials Engineering


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