Mechanics of Material Lab Manual Essay Example for Free

Mechanics of Material Lab Manual Essay To investigate how shear strain varies with shear stress. c. To determine the Modulus of Rigidity of the rubber block. 4. Hooke’s Law for Wires a. To determine the Youngs Modulus of Elasticity of the specimen wire. b. To verify Hookes Law by experiment. 5. Strain in Compound Wires a. To determine the modulus of elasticity of two wires and hence evaluate the equivalent Young’s Modulus of Elasticity of the combination b. To position the single applied load on the slotted link in order that both wires are subjected to common strain and hence to establish the load in each wire 6. Deflection of a simply supported beam To find the slope and deflection of a simply supported beam with point load at the center and to prove the results mathematically 7. Deflection of a cantilever beam To verify the slope and deflection of a cantilever beam experimentally and theoretically. 8. Deflection of a overhanging beam To find the central deflection of overhanging arm beam and confirm the results theoretically 9. Shear center for a channel Find Shear center for a channel section cantilever. 10. Unsymmetrical Deflections To determine the deflections for symmetrical bending of an angle section beam 1. Shear Forces and Bending Moment in Beams To measure the bending moment at a normal section of a loaded beam and to check its agreement with theory 12. Study and Application of experimental photoelasticty techniques on linear crack propagation analysis 13. Direction and magnitude of principal stresses To use the Photo-elasticity as an experimental technique for stress analysis and to understand construction and operation of transmission polariscope. 14. Calculation of stress intensity factor Interpretation of Fringe Data and calculation of stress intensity factor (k) at different loading conditions 5. Micro Hardness Testing 16. Thin Cylinder Experiment No. 1Compression of a spring 1. OBJECTIVES a) To obtain the relation among the force applied to an extension spring and its change in length. b) To determine the stiffness of the test spring (s). 2. PROCEDURE a) Setup the apparatus vertically to the wall at a convenient height. b) Add increasing loads to the load hanger recording to the corresponding deflection for each load. c) Continue loading until at least 30 mm of extension has been achieved. 3. RESULTS Tabulate the results obtained and draw a graph of load (y-axis) against extension (x-axis). Note the following data for each spring used:- a. Outside diameter, b. Effective length, c. Wire diameter, d. Number of turns. The stiffness to the spring is the force required to produce a nominal extension of 1 mm. [pic] If Kg masses are used: The force applied to the spring in Newtons = Mass in Kg x 9. 81. 4. POINTS TO PONDER a. What relationship exists between the applied force and compression? b. Did the spring (s) behave according to Hooke’s Law? c. State the stiffness value (s) obtained. d. If the graph drawn does not pass through the origin state why. Experiment No. 2 Extension of a spring . OBJECTIVES a. To obtain the relation among the force applied to a compression sping and its change in length. b. To determine the stiffness of the test spring (s) 2. PROCEDURE a. Setup the apparatus vertically to the wall at a convenient height. b. Add increasing weight to the load hanger recording to c. the corresponding deflection for each load. d. Continue loading until at least 30 mm of compression has e. been achieved. 3. RESULTS Tabulate the results obtained and draw a graph of load (y-axis) against compression (x-axis). Note the following data for each spring used :- e. Outside diameter, f. Effective length, g. Wire diameter, h. Number of turns. The stiffness to the spring is the force required to produce a nominal extension of 1 mm. [pic] If Kg masses are used: The force applied to the spring in Newtons = Mass in Kg x 9. 81. 4. POINTS TO PONDER a. What relationship exists between the applied force and compression? b. Did the spring (s) behave according to Hooke’s Law? c. State the stiffness value (s) obtained. d. If the graph drawn does not pass through the origin state why. Experiment No. 3 Rubber in Shear 1. OBJECTIVES 1. To determine the variation of deflection with applied load. . To investigate how shear strain varies with shear stress. 3. To determine the Modulus of Rigidity of the rubber block. 2. PROCEDURE 1. Set-up the apparatus securely to the wall at the convenient height 2. Note the initian dial gauge reading. 3. Add increasing increments of load and recird the corresponding deflections registered on the dial gauge. 4. Tabulate the results and draw a graph of deflection (x-axis) against applied load (y-axis). Describe the relationship between the deflections and the applied load. State if this follows a linear law. 3. Observations and Calculations: Load (W) |Deflection |Shear Stress |Shear Strain | | |X |= W/A |=X/L | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | Modulus of rigidity of the rubber block = shear stress/ shear strain = slope of graph |Data |Dimensions (Metric) | Dimension of block |150*75*25 mm | |Dial Gauge |12 mm travel * . 01 mm | |Load Hanger |250 mm * 2 N | |Max. Load |160 N (16 kg) | Experiment No. 4 Hooke’s Law for Wires Objectives: 1. To determine the Young’s Modulus of Elasticity of the specimen wire. 2. To verify Hooke’s Law by experiment. 3. To establish a value for the ultimate stress of the wire. [pic] Procedure: 1. Note the length (L), diameter (d) and the material of the wire under test. 2. Add sufficient initial load to the hanger to remove the flexure of the specimen. 3. Let the scale measurement now showing be the zero position. 4. Add equal increments of load to the hanger and note the corresponding total extension (x) for each case. 5. Care should be taken to ensure that the elastic limit of the material is not exceeded. 6. Tabulate the results and draw a graph of load (W) against extension (X). 7. Continue to load the specimen until fracture occurs. Note the breaking load. Observations and Calculations: |S/No. Load (N) |Stress (N/m2) |Extension(mm) |Strain |Young’s Modulus (Y) | |BRASS | |1 | | | | | | |2 | | | | | | |3 | | | | | | |4 | | | | | | |5 | | | | | | |STEEL | |1 | | | | | | |2 | | | | | | |3 | | | | | | |4 | | | | | | |5 | | | | | | Young’s Modulus of elasticity E/xA = WL Ultimate Stres s = Total Load at fracture / area of wire General Questions 1. State Hooke’s Law. Did the extension of the wire under test confirm to Hooke’s Law? 2. Quote the values obtained for E and the ultimate stress and compare these with the normally accepted values for the material. Experiment No. 5 Strain in Compound Wires Objectives: 1. To determine the Module of Elasticity of the two wires and hence evaluate the equivalent Young’s Modulus of Elasticity of the combination. 2. To postion the single applied load on the slotted link in order that both wires are subjected to common strain and hence to :- 3. Establish the load in each wire. 4. To obtain an experimental value of the equivalent Young’s Modulus of elsticityof the combination. 5. To compare the experimental and theoretical results. Procedure: 1. Note the length and the diameter of each wire and the distance between their centers. 2. Remove the slotted link and suspend the hanger from the lower and of the slide attached to one of the wires. 3. Apply a range of increasing loads and note the corresponding extension of the wire. 4. Do not allow the wire to exceed its elastic limit. 5. Plot a graph of load against extension, and from the slope of the straight-line graph, determine the value of Young’s Modulus of Elasticity of the wire. a. Repeat this procedure for the other wire. b. Replace the slotted link and suspend the hanger from its edge placed at the center of the link. The length of one of the wires may require to be adjusted until the link is level. Small adjustment to the length of either one of the wires may be obtained by applying a supplementary load to its slide using another hanger. Place a load (W) on the central hanger and maintain a common extension in the wires (i. e. level condition) by adjusting the position of the knife-edge on the link. Note the new position of the load measured from the center of the left-hand wire. Note the magnitude of the applied load and the common extension of the wires. Repeat over a range of increasing loads. Tabulate the results and plot a graph of the load (W) against the extension (X) of the compound wire arrangement. Diagram and calculations: |S/No. Force (N) |fs (MPa) |fb (MPa) |fe (MPa) |ee * 10^-4 |Ee (Pa) * 10^10 | |1 | | | | | | | |2 | | | | | | | |3 | | | | | | | |4 | | | | | | | |5 | | | | | | | Experiment No. 6 Deflection of a simply supported beam OBJECTIVES To find the slope and deflection of a simply supported beam with point load at the centre and prove the results mathemati cally. APPARATUS 5. HST 6:1 with complete accessories 6. Vernier caliper, micrometer, meter rod, etc. [pic] PROCEDURE 1. Set up the two end supports at 1m span and insert the thick steel beam in the end and fixtures. 2. Place a load hanger and clamp at mid span and set up a dial gauge to measure the deflection at the load point. 3. Check that the end supports are free top rotate as the beam deflects. 4. Read the support rotation gauge and central deflection gauge. 5. Add load by increments of 1N up to 10 N recording the dial gauge reading and then move the load by the same decrements to obtain a duplicate set of readings. 6. Plot the end rotations and central deflection against the load. Observations and Calculations: |S/NO. |LOAD (N) |Slope |Deflection |Theoretical |Theoretical slope | | | | | |deflection | | | | |central |central |Y = |? | | | | | |WL3/48EI |WL2/16EI | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | Experiment No. 7 Deflection of a cantilever beam OBJECTIVE To verify the slope and deflection of a cantilever beam experimentally and theoretically. APPARATUS 1. HST 6:1 with complete accessories 2. Vernier Caliper, micrometer, meter rod etc. [pic] PROCEDURE 1. Clamp the thicker steel strip (2. 64 mm) in the position shown in diagram so that it forms a cantileve r. 2. Fix the hanger clamp (0. 3m) from the fixed support and setup a dial guage over it. 3. Apply a load in increments of 1 /2 N up to about 5N reading the gauge at each load. 4. Plot a graph of deflection against load Observations and Calculations: |S/NO. LOAD (N) |Slope |Deflection |Theoretical |Theoretical slope | | | | | |deflection | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | RESULTS: 1. From the graph obtained, the best fit linear relationship between displacement and load the steel strip, compares the graidient with the theoretical value. 2. Comment on the accuracy of the theoretical results. Experiment No. Deflection of an overhang beam To find the deflection of overhanging arm beam and confirm the results theoretically OBJECTIVE To verify the slope and deflection of a overhang beam experimentally and theoretically. APPARATUS 3. HST 6:1 with complete accessories 4. Vernier Caliper, micrometer, meter rod etc. PROCEDURE 5. Clamp the thicker steel strip in the position shown in diagram so that it forms a overhang. 6. Fix the hanger clampahead from the roller support and setup a dial guage over it. 7. Apply a load in increments of 1 /2 N up to about 5N reading the gauge at each load. 8. Plot a graph of deflection against load Observations and Calculations: |S/NO. LOAD (N) |Slope |Deflection |Theoretical |Theoretical slope | | | | | |deflection | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | RESULTS: 3. From the graph obtained, the best fit linear relationship between displacement and load the steel strip, compares the graidient with the theoretical value. 4. Comment on the accuracy of the theoretical results. Experiment No. 9. Shear center for a channel OBJECTIVE To determine the share centre of a channel section cantilever and to draw the graph between notch distance and gauge readings. APPARATUS 7. Unsymmetrical cantilever 8. Rigid based plate, weights 9. String, pulley 10. Calibrated ring 11. Grid, two dial gauges PROCEDURE: 7. Turn the routable head, so that the cantilever section is positioned relative to the pulley. 8. Fit the share assessory to the top of the cantilever and turn the dial gauge so that they rest against the attachment. The grooves in the notched bar have the spacing of 5 mm. 9. Turn the scales of the dial gauges until they read zero. 10. Tie the string to the left hand notch. Move the pulley to the left and hang the weight hanger on the end of the string. Put a weight of 1 Kg on the hanger so that the total weight is 1,5Kg. 11. Adjust the pulley position until the string is parallel to the lines on the pulley bracket. Record the reading of the both dial gauges. 12. Move the string to the next notch. Readjust the pulley position, Record the dial gauges readings. 13. Repeat for each notch position. Results: Experimental position of Shear Center from the outside of the web. Theoretical position Channel Shear Center is h = B-2 A-2 t / IA Experiment No. 10 Unsymmetrical Bending of a Cantilever Beam |Direction of pull|Displacement |Applied load (Kg) | |(degrees) | | | | | |. 5 |1. 0 |1. 5 |2. 0 |2. 5 |3. | |0 |U | | | | | | | | |V | | | | | | | |22. 5 |U | | | | | | | | |V | | | | | | | |45 |U | | | | | | | | |V | | | | | | | |67. |U | | | | | | | | |V | | | | | | | |90 |U | | | | | | | | |V | | | | | | | |112. 5 |U | | | | | | | | |V | | | | | | | |135 |U | | | | | | | | |V | | | | | | | |157. |U | | | | | | | | |V | | | | | | | |180 |U | | | | | | | | |V | | | | | | | |Direction of pull|Displacement |Applied load (Kg) | |(degrees) | | | | | |. 5 |1. 0 |1. 5 |2. 0 |2. 5 |3. 0 | |0 |L | | | | | | | | |R | | | | | | | |22. |L | | | | | | | | |R | | | | | | | |45 |L | | | | | | | | |R | | | | | | | |67. 5 |L | | | | | | | | |R | | | | | | | |90 |L | | | | | | | | |R | | | | | | | |112. |L | | | | | | | | |R | | | | | | | |135 |L | | | | | | | | |R | | | | | | | |157. 5 |L | | | | | | | | |R | | | | | | | |180 |L | | | | | | | | |R | | | | | | | Experiment No. 11 Bending Moment in Beams |S. No. Load (N) |Balance Reading (N)/ Net Force (N) | | | |W1 |W2 |W3 | |1 | | | | | |2 | | | | | |3 | | | | | |4 | | | | | |S. No. |Load (N) |Balnce Moment (N. mm)/ Theoretical Val. |1 | | | | | |2 | | | | | |3 | | | | | |4 | | | | | Experiment No. 13 Study and Application of experimental photoelasticty techniques on linear crack propagation analysis. OBJECTIVES To familiarize the students with the Linear Elastic Fracture Mechanics in context with photoealsticity and orientation and understanding of operation off different types of polariscopes. THEORY The name photoelasticity reflects the nature of this experimental method: photo implies the use of light rays and optical techniques, while elasticity depicts the study of stresses and deformations in elastic bodies. Photoelastic analysis is widely used for problems in which stress or strain information is required for extended regions of the structure. Photo elastic stress analysis is a simple and powerful tool for design engineers that provide them with the experimental data required for validating analytical and computational designs. In using this method, a transparent plastic model of the structural part of the machine element under study is first made. Then the specimen was placed in the polariscope, and the simulating operating force was applied. When examined in the polarized light field provided by the instrument, colored fringe patterns are seen which reveal: †¢ A visible picture of the stress distribution over the whole area of the specimen. †¢ Stress distribution which is accurately readable at any point for both direction and magnitude. Two types of pattern can be obtained: isochromatics and isoclinics. These patterns are related to the principal-stress differences and to the principal-stress directions, respectively. Principles The method is based on the property of birefringence, which is exhibited by certain transparent materials. When polarized light passes through a stressed material, the light separates into two wave fronts travelling at different velocities, each oriented parallel to the direction of principal stresses(? 1,? 2) in the material but perpendicular to each other. Photoelastic materials exhibit the property of birefringence only on the application of stress and the magnitude of the refractive indices at each point in the material is directly related to the state of stress at that point. Thus, the first task is to develop a model made out of such materials. Isoclinics and isochromatics Isoclinics are the locus of the points in the specimen along which the principal stresses are in the same direction. Isochromatics are the locus of the points along which the difference in the first and second principal stress remains the same. Thus they are the lines which join the points with equal maximum shear stress magnitude. Interpretation of the Photoelastic Pattern: Once the fringes obtained by application of load on photoelasic specimen the most important step is interpretation of complete stress field. The photoelastic fringe pattern data offer suggestion to modify design to avoid from material failure. It is also helpful in reducing average stress on actual part. Complete stress field interpretation include principal stress directions as well as magnitude of stresses on different fringe order. [pic] Stimulated stress field pattern in white light for typical edge crack plate The photoelastic pattern appears as a colorful map of lines of equal color. Beginning at the lower level line of stress and progressing to areas of higher level, the colour sequence observed will be black, yellow, red, blue, yellow, red, green, yellow, red, green etc. The colour transmission from red to blue and from red to green is sharply marked. [pic] Polariscope: Polariscope: It is an instrument which consists of two polaroid plates mounted apart. The lower plate is generally fixed and is known as the polariser, while the upper plate can be rotated and is known as the analyser. Types: 1. Reflection Polariscope Particularly it is used to photoelastically stress-analyze opaque plastic parts. The part to be analyzed is coated with a photoelastic coating, service loads are applied to the part, and coating is illuminated by polarized light from the reflection polariscope. Molded-in or residual stresses cannot be observed with this technique. Fig. 13. 1 Typical reflection periscope on tripod stand 2. Transmission Polariscope. This type is useful for stress analysis if component is of transparent or glassy material. All transparent plastics, being birefringent, lend themselves to photoelastic stress analysis. The transparent part is placed between two polarizing mediums and viewed from the opposite side of the light source. In these experiments we will be only concerned with highlighting the dependence of stress distribution on geometric features, hence we can use the transparent materials and transmission type polariscope will be used. [pic] Fig. 13. 2 Transmission Polariscope Two arrangements of transmission polariscope are possible i. e. I. Plane polariscope Plane polariscope is used for direction measurement at a point of principal stresses for a specimen. The setup consists of two linear polarizers and a light source. The light source can either emit monochromatic light or white light depending upon the experiment. First the light is passed through the first polarizer which converts the light into plane polarized light. The apparatus is set up in such a way that this plane polarized light then passes through the stressed specimen. This light then follows, at each point of the specimen, the direction of principal stress at that point. The light is then made to pass through the analyzer and we finally get the fringe pattern. The fringe pattern in a plane polariscope setup consists of both the isochromatics and the isoclinics. The isoclinics change with the orientation of the polariscope while there is no change in the isochromatics. For this purpose, set the quarter wave plates on both the analyzer and the polarizer cells at position â€Å"D† (direction) to make the polariscope â€Å"plane† as shown below in fig13. 2 (b) Figure 13. 2 (a)Plane Polariscope Arrangement Figure 13. 2 (b)Pin postion at Plane Polariscope arrangement II. Circular polariscope When examining the model for determination of the stress distribution and magnitude, the polariscope must be transformed from a â€Å"PLANE† to a â€Å"CIRCULAR† operation. This is done by first making sure the clamp â€Å"A† is in the locked position and then withdrawing pins â€Å"B† on the ? wave plate from the hole â€Å"D† (direction) and rotating them until pins engage in hole â€Å"M† (magnitude). Now quarter wave plate is at 45 degrees to the polarizer-analyzer axis thus polariscope is in circular light operation Figure13. 3Circular Polariscope Arrangement(dark field) There are four different kinds of arrangements for the circular polariscope. Each arrangement produces either a dark field arrangement or a light field arrangement. In dark field arrangement, the fringes are shown by bright lines and the background is dark. The opposite holds true for the light field arrangement. Quarter Wave-Plates Arrangement |Polarizer’s Arrangement |Polariscope Field | |Crossed |Parallel |Light | |Crossed |Crossed |Dark | |Parallel |Parallel |Light | |Parallel |Crossed |Dark | Experiment No. 14 Calculation of direction and mag nitude of principal stresses using transmission polariscope. OBJECTIVES ) Application of photoelastic techniques to measure the direction of Principal Stresses at a point b) Calculation of magnitude of principal stresses by interpreting the fringe data. Apparatus Apparatus required to achieve the stated objectives are †¢ Transmission polariscope †¢ test specimen of different shapes †¢ Load measuring dial gauge †¢ Vernier Caliper and Meter Rod Construction of Transmission Polariscope: The basic polariscope consists of †¢ Rigid base frame ready to receive all of the modular accessory items. †¢ Two cells equipped with polarizing filters. †¢ Knob ‘H’ used to synchronously rotate the polarizer and analyzer (their common motion is indicated in degrees in the graduated dial). The quarter wave plate which can be used to convert plane polariscope into circular and vice versa. Fig 13. 2 show these components.. Specimen prepration: In this experi ment we are using photoelastic sheets (Polyurethane material) The photoelastic sheet was made into different specimens as stated below: a) specimen with holes drilled. b) specimen with cracks, which is manually cut c) specimens with notches Typical single edge crack specimen 2-D model is shown in fig. 14. Fig. 14. 1 PROCEDURE Measurement of Direction of Principal Stresses at a Point: To measure the direction of the principal stresses at a point in the specimen we follow the following steps: Place the specimen in the polariscope making sure that the specimen is aligned correctly within the clamps, hence avoiding any twisting of the specimen. †¢ Apply load (compressive or tensile) by turning the loading screw. †¢ Set the quarter wave plates on both the analyzer and the polarizer cells at position â€Å"D† to make the polariscope â€Å"plane† (Fig 13. 2 b). †¢ By means of knob ‘C’ rotate the analyzer until pointer â€Å"P† is positioned at 0 and 100 on the scale. †¢ Release the clamp ‘A’ if it was locked previously and by means of knob â€Å"H† rotate the whole assembly during this rotation some black and all the colored fringes will be observed to move. These black fringes which move are the isoclinics. †¢ Identify the point of measurement using a grease pencil or scriber. By means of knob â€Å"H† rotate the polarizer-analyzer assembly until a black isoclinic crosses over the marked point. At this point the axes of the polarizer and analyzer are parallel and perpendicular to the directions of the principal stresses and their directions can be seen from the scale by a pointer â€Å"V†. The rotation of the assembly may be clockwise or anti-clockwise; in order to accommodate this, sign is used with the value of this direction angle. The positive sign is used for clockwise rotation and negative is used for counter clockwise. Magnitude calculations ? The polariscope, and the digital camera are turned on ? Specimen undergoes tensile force/compressive load in Transmission Polariscope with one end fixed as in fig 13. 2 Fringes formed and photographed by digital camera ? A gradual tension was then added onto specimen and record the load reading by using dial-guage ? Print and interpret fringe pattern obtained in photographs according to the proce dure explained. Formulation for Stress Distribution: When examining the specimen for determination of the stress distribution and magnitude, the polariscope must be transformed from a â€Å"PLANE† to a â€Å"CIRCULAR† operation. This is done by first making sure the clamp â€Å"A† is in the locked position and then withdrawing pins â€Å"B† on the ? wave plate from the hole â€Å"D† (direction) and rotating them until pins engage in hole â€Å"M† (magnitude). Now quarter wave plate is at 45 degrees to the polarizer-analyzer axis thus polariscope is in circular light operation. Difference of principal stresses is given by (1 (2 = (N * C)/t Where N=fringe order at point of measurement C= stress constant of specimen material T = specimen thickness C is usually given by manufacturer. Thus the remaining number to be found is N which can be found according to color pattern as mention in the topic of interpretation of fringe pattern. CALCULATIONS AND RESULTS |S/No. |Applied load |Thickness of |Fringe Order |Direction of |Direction of |Magnitude of |Magnitude of | | |lbs/. 01 inch |specimen |‘N’ |principal stress |principal stress |principal stress |principal stress | | | |‘t’ | | |(threotcal value) | |(threotcal value) | |1 | | | | | | | | |2 | | | | | | | | |3 | | | | | | | | |4 | | | | | | | | |5 | | | | | | | | POINTS TO PONDER: 1. What will be the magnitude of shear stress at a plane of principle stress? 2. Describ e the functions of plane polriscope vs circular polriscope. 3. Describe the importance of calculation of stresses with reference to safety factor in engineering design. 4. Discuss the region of maximum stress for specimen used in experiments and explain with reasoning. 5. In case of residual stresses as a result of specimen machining which recovery method is preferable and why?