Department of Aerospace Engineering

LABORATORY MANUAL

AER 520 / MEC 430 Experimental Methods in Stress Analysis

Fall 2005

Author: Dr. H. Ghaemi

Rev. 1.2

NO REFUNDS NO EXCHANGES

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Department of Aerospace Engineering

GENERAL SAFETY RULES AND REGULATIONS FOR LABORATORIES AND RESEARCH AREAS

The following safety rules and regulations are to be followed in all Aerospace Engineering laboratories and research facilities. These rules and regulations are to insure that all personnel working in these laboratories and research areas are protected, and that a safe working environment is maintained. 1.”Horseplay” is hazardous and will not be tolerated. 2. No student may work alone in the laboratory at any time, except to prepare operating procedures for equipment or data write-up/reduction/simulations. 3. Required personal protective equipment (PPE) will be provided by the Department for use whenever specified by the Faculty, Engineering Support or Teaching Assistant, .i.e., hearing protection, face shields, dust masks, gloves, etc. 4. Contact lenses will not be worn in the laboratory when vapours or fumes are present. 5. Safety glasses with side shields and plastic lenses will be required when operating targeted class experiments as outlined in the experimental procedures. Splash goggles or face shields will also be provided and worn also, for those experiments which have been identified as a requirement. 6. Each student must know where the location of the First Aid box, emergency equipment, eye wash station is, if required in the laboratories, shops, and storage areas. 7. All Faculty, Engineering Support and Teaching Assistants must know how to use the emergency equipment and have the knowledge to take action when an accident has occurred, .i.e., emergency telephone number, location, emergency response services. 8. All Faculty, Engineering Support and Teaching Assistants, and Research Assistants, must be familiar with all elements of fire safety: alarm, evacuation and assembly, fire containment and suppression, rescue. 9. Ungrounded wiring and two-wire extension cords are prohibited. Worn or frayed extension cords or those with broken connections or exposed wiring must not be used. All electrical devices must be grounded before they are turned on. 10. All Faculty, Engineering Support and Teaching Assistants, and Research Assistants, must be familiar with an approved emergency shutdown procedure before initiating any experiment. 11. There will be NO deviation from approved equipment operating procedures. 12. All laboratory aisles and exits must remain clear and unblocked. 13. No student may sniff, breathe, or inhale any gas or vapour used or produced in any experiment. 14. All containers must be labeled as to the content, composition, and appropriate hazard warning: flammable, explosive, toxic, etc. 15. The instructions on all warning signs must be read and obeyed in all laboratories and research facilities.

Campus Security Dial: 5001/5040 Emergency Dial: 80 Mr. Karpynczyk, Safety Officer: 6420

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16. All liquid and solid waste must be segregated for disposal according to Faculty, Engineering Support or Teaching Assistant instructions. All acidic and alkaline waste should be neutralized prior to disposal. NOTE: NO organic waste material is to be poured down the sink or floor drains. These wastes should be property placed in designed waste disposal containers, labeled and stored in the department’s flammable storage cabinet which is ventilated and secured. 17. Good housekeeping must be practiced in all teaching and research laboratories, shops, and storage areas. 18. Eating, drinking, tobacco products, gum chewing or application of makeup is strictly prohibited in the laboratories and shops. 19. Only chemicals may be placed in the “Chemicals Only” refrigerator. Only food items may be placed in the Food Only refrigerator. Ice from any refrigerator is not be used for human consumption or to cool any food or drink. 20. Glassware breakage must be disposed in the cardboard boxes marked “Glass Disposal”. Any glassware breakage and malfunctioning instruments or equipment must be reported to the Faculty, Engineering Support or Teaching Assistant present. 21. All injuries, accidents, and “near misses” must be reported to the Faculty, Engineering Support or Teaching Assistant. The Accident Report must be completed as soon as possible after the event by the Faculty, Engineering Support or Teaching Assistant and reported to the Departmental Safety Officer immediately. Any person involved in an accident must be sent or escorted to the University Health Centre. All accidents are to be REPORTED. 22. All chemical spills are to be reported to the Faculty, Engineering Support or Teaching Assistant, whose direction must be followed for containment and cleanup. Faculty, Engineering Support or Teaching Assistant will follow the prescribed instructions for cleanup and decontamination of the spill area. The Departmental Safety Officer must be notified when a major spill has been reported. 23. All students and Faculty, Engineering Support or Teaching Assistant must wash their hands before leaving targeted laboratories, research facilities or shops. 24. No tools, supplies, or any other items may be tossed from one person to another. 25. Compressed gas cylinders must be secured at all times. Proper safety procedures must be followed when moving compressed gas cylinders. Cylinders not in use must be capped. 26. Only gauges that are marked “Use no oil” are for Oxygen cylinders. Do not use an oiled gauge for any oxidizing or reactive gas. 27. Students are never to play with compressed gas hoses or lines or point their discharges at any person. 28. Do not use adapters or try to modify any gas regulator or connection. 29. There will be no open flames or heating elements used when volatile chemicals are exposed to the air. 30. Any toxic chemicals will be only be exposed to the air in a properly ventilated Fume Hood. Flammable chemicals will be exposed to the air only under a properly ventilated hood or in an area which is adequately ventilated. 31. Personal items brought into the laboratory or research facility must be limited to those things necessary for the experiment and safe operation of the equipment in the laboratories and research facilities. 32. General laboratory coats, safety footwear are not provided by the Department of Aerospace Engineering, although some targeted laboratories and research areas will be supported by a reasonable stock of protective clothing and accessories, i.e., gloves, welding aprons, dust masks, face shields, safety glasses, etc.

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33. Equipment that has been deemed unsafe must be tagged and locked out of service by the Technical Officer in charge of the laboratory or research facility. The Departmental Safety Officer must be notified of the equipment lockout IMMEDIATELY! 34. In June 1987 both the Federal & Ontario Governments passed legislation to implement the workplace hazardous material information system or WHMIS across Canada. WHMIS was designed to give workers the right-to-know about hazardous material to which they are exposed to on the job. Any person who is required to handle any hazardous material covered by this act should first read the label and the product’s material safety data sheet (MSDS). No student is to handle any hazardous materials unless supervised by a Faculty, Engineering Support or Teaching Assistant. The laboratory Technical Officer, Faculty, Engineering Support or Teaching Assistant is responsible for ensuring that any hazardous materials are stored safely using WHMIS recommended methods and storage procedures. All MSDS must be displayed and stored in a readily accessible place known to all users in the workplace and laboratory 35. All the foregoing rules and regulations are in addition to the Occupational Health and Safety Act, 1987. 36. Casual visitors to the laboratory and research areas are to be discouraged and must have permission from the Faculty, Engineering Support or Teaching Assistant to enter. All visitors must adhere to the safety guidelines and is the responsibility of the visitor. 37. Only the Safety Officer may make changes to these policies upon confirmation of the Safety Committee and approval of the Department Chair.

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Table of Contents 1. Strain Gauge Technology __________________________________________________ 5

1.1 Introduction ________________________________________________________________________ 5 1.2 Circuit and Wiring Diagram ___________________________________________________________ 5 1.3 Two and Three Lead Wire Connections __________________________________________________ 8 1.4 Connection to Strain Indicator __________________________________________________________ 9 1.5 Development of Equations ___________________________________________________________ 11

1.5.1 Case 1: Uniaxial Stress _________________________________________________________ 12 1.5.2 Case 3: Strain Gauge Rosette _____________________________________________________ 13

1.6 Engineering Material Properties _______________________________________________________ 16 2. Cantilever Beam ________________________________________________________ 17

2.1 Purpose __________________________________________________________________________ 17 2.2 Apparatus _________________________________________________________________________ 17 2.3 Procedure _________________________________________________________________________ 17 2.4 Report ___________________________________________________________________________ 18

3. Beam in Pure Bending ___________________________________________________ 19 3.1 Purpose __________________________________________________________________________ 19 3.2 Apparatus _________________________________________________________________________ 19 3.3 Procedure _________________________________________________________________________ 19 3.4 Report ___________________________________________________________________________ 20

4. Strain Distribution on a Structural Section ____________________________________ 21 4.1 Purpose __________________________________________________________________________ 21 4.2 Apparatus _________________________________________________________________________ 21 4.3 Procedure _________________________________________________________________________ 21 4.4 Report ___________________________________________________________________________ 22

5. Strain Distribution on a Casted Component ___________________________________ 23 5.1 Purpose __________________________________________________________________________ 23 5.2 Apparatus _________________________________________________________________________ 23 5.3 Procedure _________________________________________________________________________ 23 5.4 Report ___________________________________________________________________________ 24

6. Pressure Vessel _________________________________________________________ 25 6.1 Purpose __________________________________________________________________________ 25 6.2 Apparatus _________________________________________________________________________ 25 6.4 Procedure _________________________________________________________________________ 25 6.4 Report ___________________________________________________________________________ 26

7. Theory of Photoelasticity _________________________________________________ 27 7.1 Introduction _______________________________________________________________________ 27 7.2 Polarized Light ____________________________________________________________________ 27 7.3 Stress Optic Law ___________________________________________________________________ 28 7.3 Basic Data for Photoelastic Measurements _______________________________________________ 31 7.4 Measurement at a Point ______________________________________________________________ 33

7.4.1 Tardy Compensation ___________________________________________________________ 34 7.4.2 Absolute Compensation (Null Balance) ________________________________________________ 35

8. Photoelastic Coating Calibration ___________________________________________ 36 8.1 Purpose __________________________________________________________________________ 36 8.2 Apparatus _________________________________________________________________________ 36 8.3 Procedure _________________________________________________________________________ 36 8.4 Report ___________________________________________________________________________ 37

9. References _____________________________________________________________ 38

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1. Strain Gauge Technology 1.1 Introduction The method of electric resistance strain gauge is an established method in experimental stress analysis. Strain gauge may be applied in different stage in the life of the product from the conceptual design stage to the finished product. Within the field of experimental analysis, several other techniques are used such as the method of Photoelastic Coating and Brittle Coating. Amongst these techniques, electrical strain gauge is the most practical method for testing load-carrying parts. To make strain measurements a reliable method requires selection of parameters: proper strain gauge, environmental protection, proper circuit design and instrumentation. 1.2 Circuit and Wiring Diagram An electrical strain gauge applied to the surface of a body will change its resistance in proportion to the surface strain as follows:

xxgSRR ε=

∆ Sg is strain gauge sensitivity

The quantity ∆R/R is directly measured and converted to strain. In general there are two common type of circuits used in electrical strain gauge units: Potentiometer and Wheatstone bridge. Only Wheatstone bridge is considered here. A Wheatstone bridge that operates as a reading device is shown in the figure below.

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The voltage drop across R1 is denoted as VAB and is given by the following equation:

VRR

RVAB21

1

+=

and similarly voltage drop across R4 is given by:

VRR

RVCD43

4

+=

The output voltage E from the bridge is equal to VBD and thus:

ADABBD VVVE −==

VRRRR

RRRRE))(( 2143

4231

++−

=

The output voltage E would be zero only if:

4231 RRRR = This zeroing feature permits Wheatstone bridge to be used in static strain reading. The sensitivity of the bridge must be considered from two points of views:1) the fixed voltage applied to the bridge regardless of gauge current, 2) the variable voltage whose upper limits is determined by the dissipation of the power in the arm that has strain gauge. The sensitivity of the potentiometer is the ratio of the output voltage and strain.

εESD

∆=

When the bridge supply voltage is selected so that the maximum power is dissipated, different sensitivity equation must be used. Since the gauge current is the limiting factor, the number of gauges used in the bridge and their position are important. In general there are four cases of bridge arrangement:

This bridge arrangement uses one active gauge in the position of R1. This quarter bridge is used for many dynamic and some static strain measurements where the temperature compensation is not critical.

Case 1

This arrangement utilizes an active gauge in R1 and a dummy gauge in the position of R2. This arrangement is utilized for temperature compensation. This arrangement causes the circuit efficiency to reduce to 50 percent.

Case 2

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Bridge arrangement in this case uses an active gauge in place of R1 and a dummy in the place of R4. With the gauge positioned in this configuration, the strain measurement is temperature compensated since the temperature-induced resistances are cancelled out. However, the circuit sensitivity for this arrangement is the same as Case 1. Therefore, the temperature compensation can be obtained without loss in circuit sensitivity.

Case 3

In this arrangement, four active gauges are placed in the circuit. The four active gauge arrangement is nearly twice as sensitive as in the Case 1 and 3. This is also a temperature compensated circuit arrangement.

Case 4

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Consider the following constant Wheatstone bridge circuit.

For a Wheatstone bridge, the output voltage Eo is related to input voltage by:

( )( ) io ERRRR

RRRRE4321

4231

++−

=

This equation indicates that R1 R3 = R2 R4 if Eo= 0. When this equation is satisfied the bridge is balanced. 1.3 Two and Three Lead Wire Connections When a single active gauge is connected to the bridge as shown bellow, both wires would be in series with the strain gauge. One shortcoming of this connection is that the temperature-induced resistance changes in the lead wire would appear as strain. Reducing the lead wire to the maximum practicable cross-section can minimize the error.

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When two gauges are used, the effect of temperature cancels since the temperature-induced resistances in adjacent legs are self-nullifying. The temperature effect on the lead wire can be virtually eliminated in a single active gauge circuit by using three-lead wire connection. As shown above, in this case, the center point connection of the bridge is brought out and connected to one of the gauge terminals. Resistance change in the center point lead will not affect the bridge balance. Nevertheless, in order to have effective lead wire compensation, the wires have to be the same length and should be maintained at the same temperature. This method of connection is standard for a single active gauge (Quarter Bridge) with temperature compensation. 1.4 Connection to Strain Indicator There is one type of strain indicator units being used in our laboratory, Vishay P-3500. The P-3500 is a digital unit that consists of two boxes, one is the indicator and the other is the switch & balance unit.

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The portable Vishay 1011 strain indicator is essentially a resistance bridge for measuring static strain. The measuring bridge, null amplifier and direct-coupled transducer amplifier make up the complete instrumentation system. This indicator can be used in one or two active arms so that quarter or half bridge arrangement can be obtained. The range of the strain measuring indicator is ± 5000 µε (micro strain) with ± 2 µε resolution. As it can be seen from diagram below, for single gauge, the operator must have an appreciation of the strain direction prior to selecting a correct wiring procedure. A half bridge, two gauges, where one gauge is in tension and other in compression can be set up. The schematic diagram for both connections is depicted below.

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1.5 Development of Equations For basic strain measurement in a uniaxial strain field, a uniaxial single strain gauge installation is required if the direction of the principal strain is known. There might be instances where both principal strain directions are known and only the magnitude of each principal strain is required. In these cases, a 90o rosette gauge would be installed for obtaining the required strain value. In general, when more than one strain magnitude needs to be determined, the Strain Gauge Rosettes are used. A strain gauge rosette is nothing more than two or three gauges mounted close to one another at known angles with respect to one another. There are three types of rosette gauge configurations commonly used. These three types are: • Rectangular Rosette (0o, 45o, 90o) • Delta Rosette (0o, 60o, 120o) or (120o apart) • Tee Rosette (0o, 90o) Rosette strain gauges are used primarily to determine both the orientation and magnitude of the principal strain. Measurements are gathered from each of the three/two gauges and then the data are processed either analytically or geometrically using Mohr’s Circle.

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The following equations are commonly used in the analysis. 1.5.1 Case 1: Uniaxial Stress In this case, only one non-zero principal stress σx exist.

Where: E Modulus of elasticity

σ Normal Stress ε Normal Strain ν Poisson’s Ratio τ Shear Stress γ Shear Strain G Modulus of Rigidity

Ex

xσ

ε =

Ex

yσ

νε −=

Exx εσ =

νε

σ xy E−=

0=xyγ 0=xyτ

EEyx

x

σν

σε −=

EExy

yσ

νσ

ε −=

)(1 2 yxx

E νεεν

σ +−

=

)(1 2 xyy

E νεεν

σ +−

=

Gxy

x

τγ = xyxy Gγτ =

These equations hold true only when the x-axis is in the direction of the non-zero principal stress.

Case 2 Biaxial Stress

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1.5.2 Case 3: Strain Gauge Rosette Principal stresses and strain can be determined by using strain gauge rosette for a general case where both direction and magnitude of the principal stains are unknown. Rectangular Rosette (0o, 45o, 90o)

If ε1 > ε3 φp,q = φp

If ε1 < ε3 φp,q = φq

If ε1 = ε3 and ε2 < ε1 φp,q = φp = -45o

If ε1 = ε3 and ε2 > ε1 φp,q = φp = +45o

If ε1 = ε3 = ε1 φp,q is indeterminate (equal biaxial strain)

Delta Rosette (0o, 60o, 120o) or 120o

Arbitrary orientation with respect to principal axes

232

221

21 )()(2

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εεεεεε

ε −+−±+

=pq

( )232

221

21 )()(1

212

εεεενν

εεσ −+−

+±

−+

=E

pq

( ))(

))(tan

21

31

21321

εεεεεε

φ−

− −−−=pq

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231

232

221

321, )()()(

32

3εεεεεεεεεε −+−+−±

++=qp

( )231

232

221

321, )()()(

12

13εεεεεε

ννεεεσ −+−+−

+±

−++

=E

qp

( ))()(

)(3tan21

3121

321

εεεεεεφ−+

−=

−

−pq

232

1εεε +

⟩if

232

1εε

ε+

⟨if

1232

1 2εε

εεε ⟨

+= andif

1232

1 2εε

εεε ⟩

+= andif

321 εεε ==if 0.2

Arbitrary orientation with respect to principal axes

φp,q = φp

φp,q = φq

φp,q = φp = -45o

φp,q = φp = +45o

φp,q is indeterminate ( equal biaxial strain)

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Tee Rosette (0o, 90o) In this configuration, gauge element must be aligned with principal axes.

Where in all the equations described above:

p is the principal direction 1 q is the principal direction 2 ε1 is strain indicated from gauge 1 ε2 is strain indicated from gauge 2 ε3 is strain indicated from gauge 3 εp,q is the strains in p or q direction σ p,q is the stresses in p or q direction

The above equations are provided for calculating strains and stresses based on the reading from rosette gauge. The strain gauges are numbered in a particular manner. The mistake in use of gauge numbers would result in erroneous measurements and or ambiguity in interpretation of φp,q Using the conventional numbering as shown and defined above, φp,q is the angle from gauge 1 to the nearest principal axis. If φp,q is positive, the angle is in the direction of gauge numbering and when negative, in opposite direction.

1εε =p )(1 122 νεε

νσ +

−=

Eq

)(1 212 νεε

νσ +

−=

Ep2εε =q

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1.6 Engineering Material Properties The following lists intrinsic properties of the material used in this laboratory. Maximum allowable stress: Aluminum 6061 σ max = 10 ksi (70 MPa.) Steel σ max = 20 ksi (137 MPa.) Brass σ max = 10 ksi (70 MPa.) A536-60-40-18 σ max(tensile) = 15 ksi σ max(compressive) = 17.5 ksi Modulus of Elasticity: Aluminum 6061 E = 10.4 Msi (72 GPa.) Steel E = 30.0 Msi (207 GPa.) Brass E = 16.0 Msi (110GPa.) A536-60-40-18 E = 27.9 Msi (190 GPa.) Poisson’s Ratio Aluminum 6061 ν = 0.32 Steel ν = 0.29 Brass ν = 0.31 A536-60-40-18 ν = 0.27 NOTE: Do not exceed the maximum allowable stress provided.

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2. Cantilever Beam 2.1 Purpose The purpose of this experiment is to study and to compare the state of stress between the experimentally measured and analytically calculated stresses in a cantilever beam. This is designed to familiarize the student with the basic application of single strain gauge use. 2.2 Apparatus

• Cantilever beam (see Figure 2.1) • Strain indicator • Precision dead load •

Figure 2.1 Cantilever Beam

2.3 Procedure

• Measure the beam length and cross-sectional geometry • Based on the maximum allowable stress, calculate the maximum allowable load for this

beam geometry. • Measure the location of strain gauges • Connect the strain gauge wires to the digital strain indicator • Turn on the strain indicator and balance each strain channel • Apply the load in minimum of 5 increments using precision dead loads. • At each increment, read the strain gauge indicator and record the value in table below.

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Load

Increment Strain

Gauge 1 Strain

Gauge 2 Strain

Gauge 3 Tip

Deflection

2.4 Report 1. The report should include 1.1 Title 1.2 Abstract 1.3 Purpose 1.4 Apparatus: 1.5 Experimental Procedure (specific procedure) 1.6. Experimental results 1.6.1 Applied Stress (Load)

1.6.2 Measured Strain 1.6.3 Modulus of Elasticity for the material 1.6.4 Measured Deflection

1.7 Calculation 1.7.1 Theoretical Deflection 1.7.2 Theoretical Strain 2. Graphs:

2.1 Measured and Calculated Deflection vs. Load 2.2 Stress vs. Measured and Calculated Strain

3. Discussion: The errors encountered 4. Conclusion

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3. Beam in Pure Bending 3.1 Purpose The purpose of this experiment is to compare the experimental result and analytical result of a beam in pure bending. This is designed to familiarize students with a basic application of single strain gauge. 3.2 Apparatus

• Beam (see Figure 3.1) • Strain Gauge indicator • Two dial indicators • Precision dead loads

Figure 3.1 Beam in pure bending

3.3 Procedure

• Measure the beam length and cross-sectional geometry • Based on the maximum allowable stress, calculate the maximum allowable load for this

beam geometry. • Measure the location of the strain gauges and dial indicators. • Connect the strain gauge wires to the strain indicator. • Turn on the strain indicator and balance each channel of the strain indicator. • Apply the load in minimum of 5 increments using precision dead loads. • At each increment, read the strain and deflection and record these values in the table

below.

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Load Deflection 1 Deflection 2 Strain Gauge 1 Strain Gauge 2

3.4 Report 1. The report should include 1.1 Title 1.2 Abstract 1.3 Purpose 1.4 Apparatus: 1.5 Experimental Procedure (specific procedure) 1.6 Experimental results 1.6.1 Applied stress (Load)

1.6.2 Measured Strain 1.6.3 Measured Deflection 1.6.4 Modulus of elasticity for the material

1.7 Calculation 1.7.1 Theoretical deflection 1.7.2 Theoretical strain 2. Graphs:

2.1 Stress vs. Load 2.2 Measured and Calculated Deflection vs. Load 2.3 Stress vs. Measured and Calculated Strain 2.4 1/R vs. M/EI

3. Discussion: The errors encountered 4. Conclusion

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4. Strain Distribution on a Structural Section 4.1 Purpose In this laboratory experiment, students will utilize single strain gauge to determine the strain distribution along the cross-section of an I-beam. The purpose of this experiment is to familiarize students with application of strain gauge and their use in different application. 4.2 Apparatus

• Aluminum T-section (see Figure 4.1) • Strain gauge indicator • Hydraulic loading fixture with micrometer

4.3 Procedure

• Measure the beam length and cross-sectional geometry • Based on the maximum allowable stress, calculate the maximum allowable load for this

beam geometry. • Measure the location of the strain gauges and dial indicators. • Connect the strain gauge wires to the strain indicator. • Turn on the strain indicator and balance each channel of the strain indicator. • Apply the load in minimum of 4 increments using precision dead loads. • At each increment, read the strain gauge indicator and deflection at mid span of the beam

micrometer and record these values in table below.

Figure 4.1 Aluminum T-section in bending

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Load Step

Mid Point Deflection Strain 1 Strain 2 Strain 3 Strain 4 Strain 5

4.4 Report 1. The report should include 1.1 Title 1.2 Abstract 1.3 Purpose 1.4 Apparatus: 1.5 Experimental Procedure (specific procedure) 1.6 Experimental results 1.6.1 Applied Stress (Load)

1.6.2 Measured Strain 1.6.3 Measured Deflection

1.7 Calculation 1.7.1 Theoretical Deflection 1.7.2 Theoretical Strain

2. Graphs:

2.1 Deflection vs. Load 2.2 Stress vs. Measured and Calculated Strain 2.3 Strain distribution 2.4 Bending moment and force diagram

3. Discussion

Discuss your result in terms of accuracy and error encountered. 4. Conclusion

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5. Strain Distribution on a Casted Component 5.1 Purpose The purpose of this experiment is to determine strain distribution in a casted component. Students will become familiarized with compound stress phenomenon. 5.2 Apparatus

• C-Clamp (see Figure 5.1) • Load transducer • Strain Gauge indicator

Figure 5.1 C-clamp

5.3 Procedure

• Based on the maximum allowable stress, calculate the maximum allowable load for this beam geometry.

• Measure the location of the strain gauges. • Connect the strain gauge wires to the strain indicator. • Turn on the strain indicator, set the gauge factor and balance each channel of the strain

indicator. • Place the load transducer into the jaw of clamp and tighten the clamp, and read the digital

indicator of the load transducer. • Apply the maximum allowable load in minimum of 4 increments.

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• At each increment, read the strain gauge indicator and the corresponding load and record these values in table below.

Load Strain Gauge 1 Strain Gauge 2 Strain Gauge 3 Strain Gauge 4

5.4 Report 1. The report should include 1.1 Title 1.2 Abstract 1.3 Purpose 1.4 Apparatus: 1.5 Experimental Procedure (specific procedure) 1.6 Calculation 1.6.1 Theoretical Allowable Load 1.6.2 Theoretical Strain 1.7 Experimental results 1.7.1 Applied Stress (Load)

1.7.2 Measured Strain 2. Graphs:

2.1 Stress vs. Load 2.2 Stress vs. Measured and Calculated Strain 2.3 Stress vs. Strain Gauge Location

3. Discussion Discuss your results in terms error encountered and improvements that can benefits the load carrying capacity of the C-Clamp.

4. Conclusion

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6. Pressure Vessel 6.1 Purpose The purpose of this experiment is to familiarize students with the application of Rosette Strain Gauge and strain transformation principals. In this experiment, the student will utilize a strain gauge rosette to measure the strain along three different axes surrounding a point on a thin wall pressure vessel. 6.2 Apparatus

• Pressure vessel (see Figure 6.1) • Strain Indicator • Hydraulic pump • Pressure gauge

Figure 6.1 Pressure vessel

NOTE: The maximum Pressure is 150 psi (lb/in2). The Maximum allowable stress is 9000 lb/in2. 6.4 Procedure

• Based on the maximum allowable stress, calculate the maximum allowable pressure • Connect the strain gauge wires to the strain indicator. • Turn on the strain indicator, set the gauge factor and balance each channel of the strain

indicator. • Turn on the hydraulic pump, set the pressure relief valve to desired value. • Apply the maximum pressure in minimum of 5 increments. • At each increment, read the strain gauge indicator and corresponding pressure for record

these values in table provided.

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6.4 Report 1. The report should include 1.1 Title 1.2 Abstract 1.3 Purpose 1.4 Apparatus: 1.5 Experimental Procedure (specific procedure) 1.6 Analysis 1.6.1 Maximum allowable stress

1.6.2 Determine the safety factor for test configuration as opposed to consumer application.

1.6.3 Determine the Poisson’s Ratio 1.6.4 Determine the Hoop and Axial stress for each 50 psi increment. 1.6.5 Determine the direction of the principal stresses and strains.

2. Graphs: 2.1 Hoop stress vs. Pressure 2.2 Axial stress vs. pressure (For both theoretical and experimental results) 2.3 Shear stress vs. Pressure 2.4 Draw a sample Mohr’s strain circle for maximum pressure.

3. Discussion 3.1 Why did we use hydraulic oil as opposed to pneumatic pressure? 3.2 Why do we consider this to be a thin wall pressure vessel? 3.3 Is the strain gauge technique a viable means of determining principal strain plane? 3.4 Discuss your results in terms error encountered

4. Conclusion

Pressure (psi) Strain A Strain B Strain C

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7. Theory of Photoelasticity 7.1 Introduction The Photoelastic method of stress analysis takes advantage of the properties of transparent non-crystalline materials. Many non-crystalline materials are optically isotropic at stress free state and become anisotropic when they are under stress. The optical anisotropic (temporary double refraction) can be represented as an ellipsoid (index ellipsoid). The semi-axes of the ellipsoid represent the indices of refraction.

Figure 7.1 The index ellipsoid

For materials that exhibit isotropy, the three principal indices of refraction are equal and ellipsoid becomes sphere. There is a relationship between stress ellipsoid and index ellipsoid and this relationship is the bases for stress-optic law. PhotoStress is a very versatile method for experimental stress analysis. PhotoStress method can be used in field or laboratory and it gives visible picture of overall surface stress distribution as well as reliable quantitative values for magnitude and direction of stress. 7.2 Polarized Light Light is an electromagnetic wave. An incandescent light source emits energy in all directions, which contains the whole spectrum of light. Human eyes can see portion of the spectrum: wavelength of 400 to 800 nm. The light oscillation is perpendicular to the direction of propagation. By introducing a polarizing filter, only one component of vibration of incandescent light will be transmitted, which is parallel to the axis of filter. Light propagates in the vacuum or air at a speed of 3x1010 cm/sec. The speed of light in other transparent bodies is lower than that in air or vacuum. The ratio of the speed in air and in transparent body, C/V, is called the index of refraction.

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Certain materials such as plastics are isotropic when unstressed and become optically Anisotropic when stressed. Therefore, the change in index of refraction is a function of applied stress. When a plane polarized beam passes through a transparent body with the thickness t, the light beam split and the two polarized beam propagate in X and Y direction. If the state of stress in X and Y direction are σx and σy , and the velocities are Vx and Vy, then the time for the beam to pass through the plate is t/V, and the relative retardation of the two polarized beams can be determined as follows:

( )yxyx

nntVt

VtC −=

−=δ

The following diagram is the schematic of plane polarizing unit, plane Polariscope.

Figure 7.2 Schematic of the Plane Polariscope 7.3 Stress Optic Law The theory relates the state of stress in the material to the change of the indices of refraction. Maxwell, in 1853, discovered that there is a linear relation between stresses and indices of refraction.

( )3221101 σσσ ++=− ccnn

( )1322102 σσσ ++=− ccnn

( )2123103 σσσ ++=− ccnn

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Where σ1, σ2, and σ3 are the principal stresses at a point no is the index of refraction of a material in the unstressed state n1, n2, n3 are the principal indices of refraction which correspond to the principal stress directions C1 and C2 are the constants known as stress optic coefficients.

These equations indicate that by establishing the direction of the three principal optical axes and measuring the three principal indices, the state of stress at a point can be determined. Since measuring all three axes are very difficult, the application of Photoelasticity is common to two-dimensional problems.

221101 ςσ ccnn +=−

122102 σσ ccnn +=− The above equations describe the changes in angle of refraction due to external load as the material shows a temporary double refraction characteristic. The relative angular phase shift ∆ produced by a doubly refracting plate is given by:

( )1222 nnh

−==∆λπδ

λπ

Thus, the relative phase shift (retardation) is linearly proportional to the difference between the two principal stresses being normal to the direction of light beam. In a two dimensional state of stress where σ3 = 0, the stress optic law can be written as:

( )212 σσλπ

−=∆hc

Considering the plane state of stress and that σ1 > σ2, the above equation simplifies to:

hNfσσσ =− 21

Where π2∆

=N

The relative retardation, in terms of cycle of retardation, counted as fringe order and material fringe value is:

cf λσ =

30

The principal stress difference can be easily determined if the relative retardation N can be measured and the material fringe value can be established. If the material behave in a perfectly elastic manner, the principal strain difference (ε1 - ε2 ) can be calculated by determining the fringe order N. Hooke’s Law gives the stress/strain relationships for a two-dimensional or plane stress state.

( )2111 νσσε −=E

( )1221 νσσε −=E

And the strain difference is given by:

( )211εε

νσ −

+=

Eh

Nf

( )21211 σσνεε −+

=−E

Or For a linear elastic photoelastic model, the determination of fringe order N is sufficient to establish the principal stress and strain difference. It has been established that the relative change in index of refraction is proportional to the difference of principal strain and the governing equation is as follows:

( ) ( )yxyx Knn εε −=− Where K is called strain-optic coefficient and is the physical property of the photoelastic material. Combining the equation for relative retardation of two beams and the above equation we obtain:

( )yxtK εεδ −= in transmission

( )yxtK εεδ −= 2 in reflection In a plane polariscope the light intensity is zero when the angle between the principal stress and analyzer is zero (β - α =0). In a circular polariscope, the light intensity is zero when δ = 0, δ = 1λ, 2λ, 3λ … or in general δ = Nλ. The N is called fringe order and expresses the size of δ. Once δ = Nλ is known, the principal strain difference is obtained by:

NftK

NtKyx ===−

22λδεε

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7.3 Basic Data for Photoelastic Measurements When a coated test sample is subjected to an external load, the plane strain developed in the specimen due to the applied load is the same in photoelastic material. At every point in the test specimen under the load, we get signal indicating the retardation between two light beams, one polarized alongεx, and the other along εy:

( )yxtKN εελδ −== 2 Where δ is the retardation (in)

λ is the wave length ( in white light 22.7x10-6 in) N is called fringe order, which is measured.

Then, the fringe order, N, coupled with fringe value, f, can be used to determine the principal stress difference in the structure as follows:

then Nfyx =− εε

( ) NfEEyxyx νν

εεσσ+

=+

−=−11

Where t is the thickness of the coating K is the sensitivity of the material or Photoelastic coefficient. Note that in a normal light incidence, the quantity determined is the difference in principal strain, which facilitate the determination of principal stress difference. In many practical applications, one of the principal stresses is zero. Thus, only one measurement is needed. In the case of biaxial stress two measurements are needed to determine the individual principal stresses. The principal strain direction is always determined from a reference line, axis or plane. Thus, the initial step to determine the direction of principal stress is to select a reference. When a plane-polarized light passes through a coated test specimen under stress, it splits into waves propagating at different speed along the direction of principal stresses. These two waves will be out of phase from each other after reflecting from the photoelastic coating. However, at the point that the direction of principal stress is parallel to the direction polarization, light beam emerging from the Photoelastic coating will be the same as incident light. Observing the stressed part through a plane polariscope, dark fringes observed are called Isoclinic. At every point only on the isoclinic, the directions of the principal stresses are parallel to the direction of polarization of “A” and “P”. With respect to a reference line, the measurement is

( )tK

Nyx 2λεε =−

tKf

2λ

=

32

performed by rotating “A” and “P” together until isoclinic line coincides with the point of interest. See Figure 7.3 below.

Figure 7.3 Fringe pattern

Figure 7.4 Isoclinic line

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A simple cantilever beam provides the means for understanding fringe identification. The beam is coated with Photoelastic plastic and clamped to the edge of the test rig. At opposite end of the beam, a notch has been made to provide a recess for the weight pan. When the sample is stressed, the retardation increases proportionally such that every time δ = 1, 2λ, 3λ, 4λ,….a particular wave or colour disappear and a complementary colour is seen. The following table explains the sequence of colour observation.

Retardation 10-6 [in] Colour Observed N

0 Black 0

12 Yellow

18 Red

22.7 1st Fringe 1

25 Blue

35 Yellow

40 Red

45.5 2nd Fringe 2

50 Green

57 Yellow

63 Red

68.1 3rd Fringe 3

73 Green

In summary, the stress distribution can easily be studied by recognizing fringes, their absolute order, and their location with respect to one another on the structure or test sample. 7.4 Measurement at a Point It has been shown that in the first step of measurement, the fringe order is assigned to the whole area, i.e. N =1, 2….

Thus: Nfyx =− εε

34

However, the point of interest normally would fall in between fringes and it becomes necessary to establish fraction of fringe. The technique used for this is called compensation. There are two basic method of compensation:

1. Tardy Compensation using a rotatable analyzer. 2. Absolute Compensation or null balance.

7.4.1 Tardy Compensation When the polarizer and the analyzers are aligned with the direction of principal stress and the quarter wave plates are at 45o, the fringe orders on the test sample can be seen. If the point of interest is in between the fringes, the analyzer will be rotated (∝) until the fringe order arrives at the selected point. The analyzer moves a fringe to a position where the fraction of an order is ∝/180o. The analyzer dial is graduated in hundreds of a fringe from 0 to 100. Thus, the fraction of a fringe order can be read directly from the analyzer (see Figure 7.4 below).

If the lower order fringe moves to the point of interest, the total reading order is:

]fraction n [ N +=

Figure 7.5 Tardy Compensation, polarizer and analyzer aligned with principal strains εx and εy (β and β 90o respectively). Rotate analyzer clockwise until fringe n (or n + 1) moves to test point. Read fraction (r) as indicated on compensator scale.

35

εx = ε1 and εy = ε2 If the higher order fringe moves to the point of interest, the total fringe order is:

fraction] 1 n [ N +=

εx = ε2 and εy = ε1 7.4.2 Absolute Compensation (Null Balance) To measure the Photoelastic signal at a point, a calibrated value equal in size but opposite in sign is entered into the light path. The Photoelastic signal at the point of interest is then cancelled to read zero. In this method recognizing fringe value or assigning fringe order is eliminated. See the Figure 7.6 below.

Figure 7.6 Null balance compensation

To measure the total fringe order N, the following procedure has to be followed:

1. Bring the isoclinic to the point of measurement and establish the direction of principal strain.

2. Transfer the Polariscope to the circular light operation and attach the compensator to the Polariscope.

3. Lock through the compensator and observe the pattern. Turn the compensator knob and observe the fringe movement. Continue turning the knob until the black fringe is at the point of interest. Then the fringe order is equal to the fringe order of compensator.

4. Read the fringe value of the counter of compensator, and from the calibration chart, read the corrected N value.

36

8. Photoelastic Coating Calibration 8.1 Purpose At the completion of the laboratory, students will calibrate and determine optical coefficient of a sample photoelastic material. A simple cantilever beam is coated with photoelastic plastic. 8.2 Apparatus

• Beam with photoelastic coating (see Figure 8.1) • Loading frame • Precision dead load • Portable polariscope

Figure 8.1 Beam with photoelastic coating

8.3 Procedure

• Measure the length and cross-sectional geometry of the beam. • Based on the maximum allowable stress, calculate maximum allowable load. • Measure both the beam and the coating thickness. • Apply the load at two pound increments until the maximum allowable load is reached. • Record the fringe order, N, for each increment for all the location of interest.

37

Load

Fringe for Location 1

Fringe for Location 2

Fringe for Location 3

Fringe for Location 4

Fringe for Location 5

Null Tardy Null Tardy Null Tardy Null Tardy Null Tardy

8.4 Report 1. The report should include 1.1 Title 1.2 Abstract 1.3 Purpose 1.4 Apparatus: 1.5 Experimental Procedure (specific procedure) 1.6 Experimental results 1.6.1 Applied Stress (Load)

1.6.2 Measured Strain 1.7 Calculation

1.7.1 Determine principal strain difference 1.7.2 Determine fringe value, f, for the plastic 1.7.3 Determine the optical coefficient, K, for the plastic 1.7.4 Modulus of elasticity for the material 2. Graphs the principal strain difference vs. corrected fringe order. 3. Discussion: The error encountered 4. Conclusion

38

9. References

1. Avril, J. Encyclopedie Vishay d’Analyse des contraintes, Malakoff, France: Vishay-Micromesures pp.89-128, 1974.

2. Blum, A. E. “The Use and Understanding of Photoelastic Coatings. “Strain. Journal of British Society for Stain Measurements 13:96-101 (July1977)

3. Dixon J. R. and Visser. W. “An Investigation of the elastic—Plastic Strain Distribution around Cracks in various Sheet materials. “In Photoelasticity (proceeding of the international Symposium held in Illinois Institute of technology. Chicago, Illinois, October, 1061), edited by M.M Frocht, pp231-250. New York: Pergamon press. 1963.

4. Gerberich, W. W. “Plastic Strains and Energy Density in Cracked Plates: Part I, Experimental Technique and Results.” Proceedings of the Society for Experimental Stress Analysis 21. No2:335-344

5. Gerberich, W. W. and Swedlow J. L. “Plastic Strains and Energy Density in Cracked Plates: Part II. Comparison with Elastic Theory.” Proceeding for the experimental Stress Analysis 21. No. 2: 345-351.

6. Hawkes I. And Holisster, G. S. “Photoelastic Techniques Applied to Rock Mechanics Problems of underground Excavations and foundations. “In stress Analysis, edited by O. C. Zienkiewicz ans G. S. Holisster. Chap.12, pp.264-292. New York: John Wiely and Sons. 1965.

7. Heywood, R. B. Photoelasticity for Designers. New York, Pergamon Press, 1969, 8. Holister, G.S. Experimental Stress Analysis Principals and Methods. London: Cambridge

University Press, 1967. 9. Kuske A. and Robertson, G. Photoelastic Stress analysis, New York: John Wiley and

Sons. 1974. 10. McIver, R. W. “Structural-test Applications Utilizing Large Continuous Photoelastic

Coating.” Experimental Mechanics 5: 19A-26A (February 1965) 11. Nickola, W. E. “Photoelastic Coating on Flat Rotating Axisymmetrical Stress

Analysis21, No. 1:99-109(1964). 12. Rerdinand. P., Beer. E., Russel Johnson. JR., Mechanics of Material, Second Edition,

1985. 13. Dally, James W., and William F. Riley, Experimental Stress Analysis, McGrawhill Inc.,

Toronto, 1991. 14. Riley, William F., Leroy D. sturges, and Don H, Morris, mechanics of Materials, John

Wiley & Sons, New York, 1999.