ME 3901 Engineering Experimentation

Mechanical Engineering Department

Worcester Polytechnic Institute

Stress and Strain:

Measurement and Analysis

· 1D Stress Analysis

· 2D Stress Analysis

· How to Measure Strain

· Strain Gage Resistance Differentiation

· Commercial Strain Gage

· Whetstone Bridge Analysis

· Voltage Divider

· Whetstone Bridge-Continued

All machines or structural members deform, to some extent, when subjected to external loads or forces.

For Axial loading:

where

= Axial Strain (Len/Len)

L₁ = Initial (Linear Dimension)

L₂ = Final (Strained) Linear

Dimension

Frequently, one uses the term micro-strain or m-strain since the actual L is very small

A Stress – Strain relationship for a simple uniaxial load or outer fiber in a beam in bending can be expressed as:

where

E = Young’s Modulus = Uniaxial Stress

= Strain in direction of Stress

and Poisson’s Ratio is defined as:

= Poisson’s Ratio = Lateral Strain

Consider a small differential element (x₁ = y₁ = z₁ = dx and dv = dx³) subject to

Orthogonal Stresses s_x and s_y

Initially, only s_x is applied

Now apply a stress in the Y direction:

The net strains are:

and

which can be rearranged to:

If one applied a stress in the Z direction using the same sequential application:

How to measure strain?

Photoelastic Coatings Great for concentration

Brittle Coatings points

Grid Methods Requires appreciable

deformation under loads

Extensometer (mechanical and optical)

Moire’ (“More Ray”) technique (wavy fringe patterns)

Used for whole-field displacements

Electric – Resistance Strain Gages

- Most widely used method

- Electrical resistance changes with mechanical deformation

A strain measurement must be made over a finite length. (base length).

Deformation Sensitivity – minimum deformation detectable with a gage.

Strain Sensitivity – deformation sensitivity / base length

Resistance of Conductor

where r = resistivity of material

Differentiating

(Simple Product Rule)

If sample is cylindrical and axial strain only is applied:

Recall:

(Poisson’s Ratio )

Define: in Local Vicinity

Then

If the quantity then F = 1 + 2n

= 1 + 2(0.3)

= 1.6

BUT However, it IS constant over the range of interest.

Therefore, manufacturers specify the value of F (They experimentally measure F)

Frequently, F ~ 2 (for metal strain gages)

For some silicon based materials F ~ 100

The Manufacturers also specify the strain gage resistance, R

Then, the local strain can be determined via:

Since F and R are given,

Measure and then calculate e

Consider:

F = 2 and R = 120 W

Most commercial strain units can detect a strain

of 1 m-strain

What causes this level of strain?

An ohm meter will have trouble detecting this change!

R = 120.00024

National Instruments has additional material for strain gage configurations

Signal Conditioning is Required.

Whetstone Bridge Circuit

most commonly used. It is a purely resistive bridge. It provides a means for accurately measuring resistance and for detecting small changes in resistance.

where the meter is a voltmeter – negligible current flow

The bridge is really a pair of voltage divider circuits

D – A – B and D – C – B

Voltage read across A and C midpoints

Consider the situation where the Bridge is balanced

(and I_g = 0)

V = I R

I₁ = I₂ since I_g = 0

I₄ = I₃

Also with V_g = 0

I₁ R₁ = I₄ R₄and

I₃ R₃ = I₂ R₂

But with I₁ = I₂Then I₂ R₁ = I₄ R₄

and I₄ = I₃ I₂ R₂ = I₄ R₃

Therefore, Resistance Ratio of any 2 adjacent arms must equal the Resistance ratio of the other 2 arms when taken in the same sense (i.e. L/R or T/B)

Consider a Voltage Divider

Know V_i = I (R₁+R₂)

Voltmeter (V_o) draws no current

Then V_o = I R₁

But I = V_i /(R₁+R₂)

And V_o = V_i R₁/(R₁+R₂) Voltage Divider

Now return to the Whetstone Bridge

V_o = V_meter = V_c – V_a

let R₄ change by a small amount, say (₄)

Vo is Zero initially.

Reduce ( assuming R₄ = R₃) then R₁ = R₂

if << 1 then 2() << 4

and