Experiment # 4:

Beams Strain Measurements

Using Strain Rosettes in Aluminum

Beam

1) To study the strain measurements of a simply supported aluminum beam in a general case of plane stress by means of the Mohr Circle analysis.

2) To verify theoretical computations of the combined stresses at several point on a beam with the experimental results.

3) To experimentally determine the combined stresses (the actual state of stress) at several points on a beam using the Strain Rosettes.

Objectives :

Let’s assume that the State of Stress at a point like Q is known:

The most general state of stress at a point

may be represented by 6 components:

σx , σy , σz : Normal Stresses

τxy , τxz , τzy : Shear Stresses

( Note: τyx= τxy , τxz = τzx , τzy = τyz )

The element at point Q is in plane stress if : σz= τxz = τzy

0

0

0

zy

zx

z

Q Q

Plane Stress :

Stresses in xy coordinates

Wedge-shaped stress element

Forces acting on the element

Transformed stresses in x′y′

Thus, if we know the state of stress at a point, all the stresses acting on different planes at that point can be calculated. And the value of transformed stresses are:

Why the stress transformation is important?

Stress Transformation :

Moreover, for the normal transformed strains we have :

Now, if by using some strain gages the normal strains are measured in 3 directions, then we would be able to figure out the state of strain at that point :

өA өB

өC

C x

y

xA

xB xC

Strain Measurement :

When the state of strain at a point is known the corresponding stresses

can be calculated from the Constitutive Law which is the generalization of Hooke’s law :

Where :

E : modulus of elasticity

G : shear modulus of elasticity

ν : Poisson's ration

Constitutive Law :

For measuring the strain in one direction uni-axial

electrical strain gages are used.

www.omega.com

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For measuring the strain in three different

directions strain rosettes are used. Strain rosettes are three strain gages positioned in a rosette-like layout.

Therefore by measuring three linearly

independent strain in three direction, the components of the plain stress state at the point where strain rosette is located can be calculated.

How does an electrical strain gage work?

Rectangular Strain-Rosette

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Strain Gages :

Two types of Strain Rosettes are used in this experiment:

1) Delta Strain Rosette (60° Rosette) :

In this case we have :

өA = 0° , өB = 60° , өC = 120°

- Therefore by plugging this values and also the measured

εA , εB and εC in the strain transformation formulas we will

have :

)(3

2

)22(3

1

BCxy

ACBy

Ax

εεγ

εεεε

εε

xA

xB xC

Strain Rosettes :

2) Rectangular Strain Rosette (45° Rosette) :

In this case we have :

өA = 0° , өB = 45° , өC = - 45°

and for this strain rosette :

xA

xB

xC

BCxy

ACBy

Ax

Strain Rosettes :

For this experiment 6 strain rosettes have been cemented to an

aluminum beam as indicated; Gages 2, 4 and 6 are 45° rosettes and 1, 3 and 5 are 60° rosettes.

P=10,000 lbs

x

y y

z

P/2 P/2

1.5”

1.5”

1.5”

1.5”

9.0” 9.0”

L= 36.0”

6.0”

2.0”

V

x

P/2

-P/2

+ _

+

PL/4

M

x

1

2

3

4

5

6

Aluminum Beam

E = 10.3 106 psi

G = 4.1 106 psi

υ = ?

Application of Strain Rosettes in This Test:

Moreover the state of stress at any point on the beam can be calculated

from the appropriate formulas.

For example to calculate the stresses at a point like i on

the beam we have:

P=10,000 lbs

x

y

P/2 P/2

1.5”

1.5”

1.5”

1.5”

9.0” 9.0”

L= 36.0”

1

2

3

4

5

6

i

yi

a

y

z

b = 2.0”

6.0” yi

a

bI

QV

I

yM

z

i

xy

y

z

i

ix

.

.

0

.

)2

( a

ybaQ ii

i

M : Bending moment at the section

V : Shear force at the section

Qi : First moment of area about the neutral axis

Iz : Moment of inertia

b : Section width

yi : Location of point i

Calculating the State of Stress:

P =15000 lbs

x

y

P/2 P/2 L= 36.0”

1

2

3

4

5

6

Using strain rosettes the state of strain and stress is measured at points 1 to 6

The theoretical state of stress is calculated at

point 1 to 6

Experimental Mohr Circle Theoretical Mohr Circle compare

τ

theoretical experimental

σ

Comparison of Mohr Circles at point x

Experiment Scope:

Mohr Circle:

(σy, τxy)

(σx, τxy)

http://www.engin.umich.edu/students/ELRC/me211/flash3/coach_stress_03.swf

An Interesting Link:

http://www.engin.umich.edu/students/ELRC/me211/mohr.html

Thank you !

Questions?