Page 1 / 5

ME 410 MECHANICAL ENGINEERING SYSTEMS LABORATORY

EXPERIMENT 5

STRESS ANALYSIS BY USING STRAIN GAGES

OBJECTIVE

The objective of this experiment is to become familiar with the electric resistance strain gage techniques and utilize such gages for the determination of unknown quantities (such as strain and stress) at prescribed conditions of a cantilever beam and a thin walled pressure vessel.

INTRODUCTION

There are various types of experimental methods to analyze strains and stresses at a point. Strain gage methods use either electrical or mechanical means to measure strains. In these types of strain gages, electrical resistance strain gages are the most accurate and widely used ones.

This experiment consists of two parts, both utilizing electric resistance strain gages. In these experiments, gages will be used to determine the flexural rigidity of a cantilever beam, internal pressure in a pressure vessel along with principle stresses at a given point on the vessel, the Poisson’s ratio of the vessel material and the gage orientation with respect to principal directions.

PART I CANTILEVER BEAM TEST

CONCEPT

In this experiment, a cantilever beam is used as a force transducer to determine the applied force. Three axial strain gages are used in two gage locations as shown in Fig. 1. At gage location 1, the gage B on the lower surface is located precisely under the gage A which is located on the top surface. Gages A and B measure bending strains that are of equal magnitudes but of opposite signs. Any resistance change in the active gage resulting from strains of the like sign (e.g. produced by axial loads) will be canceled since the active gages are in adjacent arms of the Wheatstone bridge. The gage C on the upper surface is located 300 mm from the free end of the beam. This gage also measures bending strains.

PROCEDURE

1. Set the strain gage indicator to Half Bridge for the gage location 1 (for gages A and B).

2. Adjust the gage factor setting for the given gage factor value and balance the indicator.

ME 410 – Experiment 5: Stress Analysis by using Strain Gages

Page 2 / 5

3. Set the second indicator to Quarter Bridge (for gage C), adjust the gage factor controller and balance the indicator.

4. Set the dial gage at the free end of the beam and adjust to zero.

5. Apply the given known load to the free end of the beam.

6. Measure strains at locations 1 and 2 and measure the deflection (δD) at the free end of the beam.

7. Remove the load from the beam.

8. Apply the given unknown load P at point E.

9. Measure the strains at gage locations 1 and 2.

GIVEN QUANTITIES

Fig. 1. Schematic representation of experimental set up in cantilever beam test

REQUIRED QUANTITIES

1. Calculate the flexural rigidity, EI, of the beam.

2. Calculate the height, h, of the beam

3. Determine the distance, L (distance between the free end of the beam and the gage location 1)

4. Determine the distance, x (distance between the applied load and free end of the beam)

5. Calculate the applied unknown load P

ME 410 – Experiment 5: Stress Analysis by using Strain Gages

Page 3 / 5

PART II THIN WALLED PRESSURE VESSEL TEST

CONCEPT

Cylindrical pressure vessels, hydraulic cylinders, and pipes carrying fluid at high pressures develop both radial and tangential stresses with values which are dependent upon the radius of the element under consideration. When the wall thickness of the cylindrical pressure vessel is about one-twentieth or smaller than its radius, the radial stress which results from pressurizing the vessel is quite small compared to the tangential stress. Under these conditions the tangential stress can be assumed to be uniformly distributed across the wall thickness. When this assumption is made, the vessel is called thin walled pressure vessel.

Consider a cylindrical vessel of inner radius r and wall thickness t, containing a fluid under pressure. Because of the axi-symmetry of the vessel and its contents, it is clear that no shearing stresses are created on the element.

Fig. 2. Schematic representation of thin walled pressure vessel

The normal stresses σ1 and σ2, shown in Fig. 2 are therefore principal stresses. The stress σ1 is known as the hoop (circumferential) stress and the stress σ2 is called the longitudinal stress. Principal stresses then can be calculated as

1prt

σ = , 2pr2t

σ =

where p = internal pressure r = inner radius of the cylinder t = wall thickness of the cylinder

In this experiment, a three element rectangular rosette forming an unknown angle α with the axis of the vessel is used to determine the gage pressure in the cylindrical vessel (Fig. 3).

If you consider gage A to be x direction, and gage C in y direction as shown in Fig.3, corresponding strains become:

ME 410 – Experiment 5: Stress Analysis by using Strain Gages

Page 4 / 5

Fig. 3 Configuration of strain gages on pressure vessel

A xx C yy B xx yy xy1, , ( )2

ε = ε ε = ε ε = ε + ε + γ

Principal strains become: xx yy 2 21,2 xx yy xy

1 ( )2 2

ε + εε = + ε − ε + γ

the corresponding angle is: xy

xx yy

tan 2γ

θ =ε − ε

Thus, the principal stresses can be calculated as:

( )1 1 22

E1

σ = ε + ν ε−ν

and ( )2 2 12

E1

σ = ε + ν ε−ν

PROCEDURE Connect the strain gages to the strain indicator (use Quarter Bridge configuration) 1. Set the gage factor setting to 2.10 2. Balance the indicator 3. Load the pressure vessel 4. Read the strain values from the indicator.

GIVEN QUANTITIES

Gage factor Sg = 2.10 Outer diameter do = 112.5 mm Inner diameter di = 107.9 mm Modulus of Elasticity E = 200 GPa

REQUIRED QUANTITIES

1. Determine the Poisson’s ratio of the cylinder material. 2. Determine the unknown gage angle, α. 3. Calculate principal strains and their directions. 4. Calculate principal stresses. 5. Determine the inner pressure of the vessel.

ME 410 – Experiment 5: Stress Analysis by using Strain Gages

Page 5 / 5

ASSIGNMENT (Only for the long reports)

Make a detailed research about the _______________________________ method. (The method will be assigned during the experiment.)

REFERENCES

1. Daily, J. W. and W. F. Riley., “Experimental Stress Analysis”, McGraw-Hill, 1965.

2. Timoshenko, S.P. and Goodier, J. N., “Theory of Elasticity”, McGraw-Hill, 1982.

3. Peterson, R.E. “Stress Concentration Design Factors”, John Wiley and Sons, 1953.