TERM PAPER

Strength of material

Mec 201

ON

“write a report on ultimate testing machine “

B tech (ME)

Submitted by:- Submitted to:-Anshu Mr. NagvenderB tech(mechanical) Faculty & Guide, Section : RG4901

Strength of materialRoll No. RG4901B60Reg No.: 10905792

LOVELY PROFESSIONAL UNIVERSITY, JALANDHAR

(2010-2012)

Acknowledgement

I am thankful to Mr. Nagvender for providing me the task of preparing the Term Paper on“write a report on ultimate testing machine ”.At Lovely Professional University, We believe in taking challenges and the term paper which provided me the opportunity to tackle a practical challenge in the subject of kinematics of machine. This term paper tested my patience at every step of preparation but the courage provided by my teachers helped me to swim against the tide and move against the wind.

I am also thankful to my friends and parents for providing me help at every step of preparation of the Term Paper.

Abstract of Work undertaken:-

I have done my with the help of internet with reference cites mentioned at the end. I have got an idea from various encyclopedias, books and goggle search.

Contents:-

Mechanical Testing of Materials – The Tensile Test

Summary:

In this experiment you will be determining Young’s modulus, the yield strength, tensile strength, fracture stress, elongation, and other properties of several tensile bars using a screw driven MTS load frame. After testing you will need to make graphs from your data. You will

need various measurements of sample geometry to calculate engineering stress versus engineering strain to obtain the material properties. Your samples will include an annealed steel sample and a cold worked steel specimen that have the same composition. Ideally, they were cut from the same bar stock. However, because of heat treatment, the tensile samples should have quite different material properties.

Instructions:

Before coming to lab, read through this handout so that you will know what will be expected of you in the lab. Each student should answer all the questions on the preliminary question sheet to be turned in at the beginning of the lab. Each group will write one report by answering the questions at the back of this lab manual. You will receive the lab data as an Excel file. Please return your floppy disk to Chris or you can use the ME office and turn in your group’s Lab report at the same time. Reports are due in 1 week.

Timing: This lab takes about one hour. All write-ups are to be quite short (none are to exceed 4 pages excluding the graphs) but accompanied by several graphs on the same plot.

Background:

1. Theory:

The background for this lab can be found in your ME 226 textbook, Mechanics of Materials, by Bedford and Liechti and most introductory materials science texts such as Materials Science and Engineering, by Callister.

Engineering stress is the force per unit (original) area.Engineering strain is the elongation per unit (original) length. They are represented by the

following symbols:

Engineering Stress, * = F

Aoand Engineering Strain, e =

llO

Where Ao = original cross sectional area of specimenlO = original length of the gauge sectionF = applied forcel = change in length

Hooke’s law relates these parameters,

* = E e

where E is Young's modulus. It is implicit here that only axial stresses and strains are of interest. Note, it is assumed * = 0 when e = 0 so that * = E e represents the first part of the load displacement curve, a straight line that represents the elastic region with E as the slope.

True stress and true strain differ from engineering stress and strain by referring to the instantaneous areas and gauge lengths respectively. The symbols for these values are the Greek letters (in bold here) and :

True stress,

= F/Ai and true strain, d = d li li

where li = instantaneous length of gauge sectionAi = instantaneous area.

The strain has the natural logarithm or ln dependence because it is determined from the instantaneous gauge length. For the instantaneous true strain increment d, we have

d = dll

and by integration

dO

= dll

lO

li

we have

= In lilO

Note that

In a + x = In a +xa - 1

2 xa

2 + 13

xa

3 - 14

xa

4 + -

so that when

=In lo + llO

= In 1 + e e

For strains of about 1%, the "error" is of order of 2 or 10-4. Consequently, there is no significant difference in the engineering and true strains when all measurements are of small strains. The true stress and strain are also related by the modulus E, = E ε since the modulus is established at a small strain level where Ai is approximately equal to A0 and li is approximately equal to lo.

For large strains when there is mainly plastic deformation, the volume of specimens are approximately conserved. Because of this, the instantaneous area Ai can be calculated from the

true strain.

Volume = Aolo = Aili

Or, taking the log derivative, rearranging and separating the differentials

= In Ao

Ai

= In lilo

Thus, Ai = Ao exp (-). Note that a tensile true strain followed by an equal compressive true

strain reproduces the initial length of the specimen. This is not true for engineering strain.

During a tension test, it is desirable to apply forces to the specimen large enough to break it. The grip region must have a large eno+ugh area to transmit the force without significant deformation or slipping. Consequently, most specimens have a reduced gauge length and enlarged grip regions. While most material properties are supposed to be specimen geometry and grip independent, there are some weak dependencies. Consequently, there are standard specimen geometries specified by the American Society for Testing Materials (ASTM). ASTM also prescribes test methods so that data reported for design purposes is obtained in a very standardized way. The specimen geometry is usually reported as part of the test results.

Returning to our discussion of the properties, the data you will record is the load vs elongation curve. Since many materials are rate-sensitive, the rate of elongation is controlled during the tensile test by moving one of the grips at a fixed displacement rate relative to the other. Usual

testing rates correspond to engineering strain rates of about 10-3

/s where the strain rate represents how quickly the strain in the gauge length is changing with respect to time. For

example, if the specimen had a one inch gauge length, the displacement of the machine is 10-3

inches per second and the load is recorded on a chart traveling at constant speed, say 1/10 inch

per second, then it is clear that the 10-3

/s strain rate will produce 10-3

inch displacement in 1/10 inch of chart or 1% strain in one inch of chart. Chart length and strain are then parametric variables, both dependent on time. This is the simplest way of measuring the load-elongation curve and is the most common. However, the elongation determined in this way also includes the elongation of the grips, the ends of specimen, the load measuring transducer (load cell) and the deflection of everything in the test frame. Typically, the elastic compliance for most test frames, i.e. the elongation outside the gauge length is about 5 to 10 times larger than the elongation inside the gauge length!

Consequently, we cannot measure the elastic modulus from the slope of the load vs elongation curve determined in this way. To make direct measurements of engineering strain, an extensometer is installed on the specimen that measures displacement within the gauge length. This transducer is designed to produce a linear voltage output with respect to displacement. Since the initial gauge length is fixed, the output is then proportional to the engineering strain. If the load signal (voltage which is proportional to the applied force) and the extensometer signals are plotted using an X-Y plot, the initial slope is then the elastic modulus.

For material stability, the load must increase all the time. The tensile deformation is inhomogeneous and strain is no longer uniform when the load reaches a maximum. Deformation stability is achieved when the specimen hardens during deformation. The result is uniform elongation. If the hardening rate is too low, an unstable situation called necking develops. To avoid neck formation, the hardening rate must be faster than the decrease in cross sectional area:

d

- dAA

Now if the volume remains constant V =Al or dV = 0 = Adl + ldAso

d = dll

= - dAA

Substituting, we havedd

In this case dF<0 and the sample is unstable. This can be shown as follows:

= FA

or F = A

dF = Ad + dA

When the load is maximum, dF = 0

Ad + dA = 0 or dd

=

So the work hardening rate has reached the critical value. As a result the specimen may neck down and begin local deformation. This occurs at the peak load. To determine the true stress strain behavior beyond the peak load requires knowledge of the non-uniform geometry of the neck in both the calculation of strain and the stress distribution. In this region the stress is non-uniform because the A changes along the tensile bars length. In ductile materials, the true stress at fracture can be several times the engineering fracture stress.

Most data you will be exposed to are engineering stress and strain unless otherwise specified. If there is a yield point, namely, a sharp transition between elastic and plastic deformation, yield stress is defined as the stress at the yield point. If there is a yield drop, there is an upper yield point and a lower yield point. If the load vs displacement curve is smooth, the material is yielding at a stress defined at a specific amount of plastic strain. Usually 0.2% permanent strain is used to define the yield stress. Then the yield stress is so identified as 0.2% yield. The proportional limit is the stress where the flow curve first deviates from linearity. This is intrinsically difficult to measure because it is related to the sensitivity of your instruments. Try to estimate the proportional limit when you analyze your data. The ultimate tensile strength is the largest engineering stress achieved during the test to failure. This value has little or no meaning as it represents the test not a material property. The true strain at this point has some meaning.

The elongation to failure is the permanent engineering strain at fracture determined at zero load. It does not include elastic strain but does include both uniform strain and the localized, necking, strain. The elongation to failure is usually stated as percent strain over a given gauge length. The reduction in area is also a measure of ductility. The true strain at fracture is determined by measuring the areas of the fractured specimen at the fracture site. Recall using the constant volume approximation that

= In AO

Ai

The area under the engineering stress-strain curve is a measure of the energy needed to fracture the specimen. It has units of work/unit volume of the gauge length and it is sometimes referred to as a measure of a material's "toughness." However, the term fracture toughness more commonly refers to the energy required to fracture pre-notched and cracked samples. Although, these two quantities may be related in some extreme instances, this relationship is still unknown to the technical and scientific community.

2. Apparatus:

In this experiment we will use an MTS machine designed to do tensile tests of specimens. The machine has a 11,200 lb. capacity (50.2 kN). It consists of a large heavy-duty test frame with a fixed beam at the bottom, a moving beam (referred to as a crosshead) and a gearbox and very large motor located in its base. The specimen is mounted between two grips, one attached to the fixed beam and the other attached to the moving crosshead. The crosshead beam contains a load cell (which works on the principle of strain gauges). It measures the applied force on the tensile specimen. The movement of the crosshead relative to the fixed beam generates the strain within the specimen and consequently the corresponding load. The gearbox below selects high and low speed ranges for movement of the crosshead.

Next to the test frame is the associated electronics console and computer that uses LabVIEW, a computer software package for controlling experiments and recording data. MTS calls the program Test Works. The program contains the main start/stop controls for testing and the adjustments for the sensitivity of the strain gauge load cell (a strain gauge bridge) as well as a "chart recorder" to read the output of the load cell bridge.

Young's Modulus is measured by adding an extensometer directly to the sample to measure the actual elongation between two given points on the sample and Test Works records a file of the load and engineering strain curve versus time for the region when the extensometer is in place.

3. Experimental procedure:

Review this general description to understand the procedures you will use in this experiment.

You will have Chris Pratt calibrate the instrument and the extensometer for you so that the data collected for this experiment is of high quality. She will help you obtain and understand the details of these adjustments. Be sure you record the gauge length of the extensometer along with the calibrated units for data file that records the extensometer displacement. Also record the selected load and displacement rate settings for the crosshead on the MTS for each individual sample. These settings may vary between samples and will be used to interpret your laboratory data later. Also make sure to avoid hysteresis effects when calibrating the extensometer, Chris will help you understand how to implement these procedures.

Once the instrument is calibrated you are ready to mount the sample and perform the actual test. Measure and record the diameter and lengths of all the samples. Install the first specimen in the grips. Be careful to follow the recommended installation procedures as given by Chris so that no damage occurs to yourself or the test equipment. Be careful to avoid placing any part of your body at a pinch point. The final coupling should be performed by trail and error by slipping the pin in by hand with the machine stopped. Move the crosshead up and down at a very slow speed until you can do this manually. Zero and calibrate the load cell once the specimen is in place. Do this in Test Works, which can adjust the load cell bridge to match the zero line on the chart. The preliminary calculation that you have done in the preparatory questions should confirm that for the steel samples we should use about 5000 lb. full-scale range for measurement. Install the calibrated extensometer on the specimen. Be sure that it is centered and straight and that it is fully closed. Re-zero the extensometer so the data on the load and displacement versus time data file doesn't require you to remove the local zero offset that was used in calibration.

Strain rates on the order of 10-3

/s are reasonable. Strengths are strain rate dependent but it is not a very strong dependence. Heat treatment and chemical variations may differ for materials so some properties will not reflect the reported textbook values. The shape of the curves, however, remains fundamentally the same. We sometimes test faster than ideal in the interest of finishing the experiment within the time available. Set parameters to the values suggested by Chris.

Observe the specimen. Do not get too close because fracture of the specimen liberates all the stored elastic energy in the specimen. Do you see bands propagating along the steel specimen? These are Luders bands indicating the multiplication and motion of dislocations. They will not be visible unless the specimen is highly polished.

Be sure to record both load and strain vs time so you can obtain load vs strain for the test. After a few percent strain just before fracture remove the extensometer and then continue the test recording the load vs. time curve until fracture. Observe the neck formation. Note that it always occurs at the maximum load for ductile tensile tests.

Do this for all of your specimens. Record the conditions for each of your samples.

Discussion:

Report the following qualitative data for each of the samples if it exists:

Young's modulus0.2% yield strengthUpper and lower yield stressesUltimate strength Strain at ultimateTrue strain at fractureArea strain at fractureReduction in areaTrue fracture stress

In your discussion please address the following:

1. The cold worked steel specimen does not show an upper yield point, the annealed steel does. How does this effect uniform deformation? After plastically deforming the sample, would either of these samples show a yield point upon reloading? Why?

2. Calculate the data for your Young's modulus from the load vs strain for both samples. Test Works also find the in-line spring using load vs crosshead displacement. Which is how strain is measured after extensometer is removed. Explain how Test Works reconciles the numbers measured with an in-series spring.

3. When an automobile crashes we want the energy of impact to be expended in deforming the car rather than the occupants. What material property corresponds to energy absorption? Clearly, a very strong, brittle material would be a poor choice for the car body. What about a material with high ductility but low strength? Of the materials you tested in this experiment, which one would have the best performance as an absorber? Why? Base your answers on the load vs displacement curves you measured for these materials. How much better?

4. Can you obtain the true stress vs strain curve for the steel specimen using the load vs extensometer strain data? Plot this data for the region where this calculation is valid. Use your Excel data file. Show your equation for relating this data.

5. Plot engineering strain versus engineering stress only up to the ultimate. On the same graph, plot true stress versus true strain from your data as recorded by Chris. The first part up to the load maximum should be nearly the same. Are yours? After the maximum load the meaningless engineering graph should diverge from the true stress graph comment on why.

6. Mark in red on the strain axis on your graphs where the specimen’s cross sectional area is not the same along the entire gauge length of the bar. Is the stress the same at every cross section along the length of the bar in these strain regions? Comment.

Objective: To determine the stress-strain relationship for a standard material (aluminum, steel, or

brass) by doing a tensile test and obtain the mechanical properties (Young's modulus, yield stress,

ultimate stress and failure or breaking stress) using Qtest machine.

Apparatus: Standard material specimens, calipers, Qtest machine.

Theory: The relation between stress and strain is an important characteristic of a material. The

stress-strain diagram is obtained by conducting a tensile test on the material. Whenever a material

is loaded in tension, elongation of the material takes place. This elongation is proportional to the

applied load. As the load is increased, it reaches a certain limit where it causes the material to

break. Within the elastic limit, the strain induced in the specimen is proportional to stress

according to Hooke's law as

E where = stress (Load/Area), and = strain (Change in length/Orginal length)

and E = Young's modulus of the material.

Experimental Procedure: For a given standard material test specimen,

1. Measure the dimensions (thickness, width and gage length) of the test specimen using calipers.

2. Slide the emergency stops up or down on the machine to avoid breakdown on the machine.

3. Turn on computer and Qtest machine and use Qtest as your login ID.

4. Go to main menu and select ASTM method D638 as the test method from the method

submenu.

5. Select TEST option from main menu and title it as TEST1 for sample ID.

6. For calibrating the machine, select Calibrate from pretest menu. Choose 2000 LB load cell and

X1 as load range. (Be sure that the machine is unloaded when calibrating).

7. Under Specimen submenu, enter the dimensions of the specimen (width and thickness). The

gage length can be entered under the Inputs submenu of the pretest menu.

8. Under submenu Windows, select X-axis as strain and Y-axis as stress by pressing F2 key.

9. Load the test specimen in the grips of the Qtest machine.

10. Press F9 key to enter the zero gage extension and no load on the control panel.

11. Select Run submenu from pretest menu to begin the test. Make sure the maximum load is set

to 1600 lbs.

12. Record stress-strain diagram on screen by selecting graph from Run submenu. Record the

mechanical properties of the test under Report submenu and print out the results. Consult the

Qtest manual for additional information.

13. Compare the experimental data with theoretical results (see attached graph of the engineering

stress-strain curves for selected metals and alloys) and comment on the accuracy of your results.

Refer to any "Mechanics of Materials" text book.

ME 272 LAB #2 -- HARDNESS TEST

Objective: To determine the tensile strength of a material using the Rockwell hardness test.

Apparatus: Steel, aluminum and brass material specimens, 10 A-R 10 (RAMS) machine

(Rockwell Hardness Tester), SER # 2160.

Theory: Hardness is a measure of the resistance of a metal to permanent (plastic) deformation.

The hardness of the metal is measured by forcing an indenter into its surface. The indenter

material which is usually a ball, pyramid, or cone, is made of a material much harder than the

material being tested. For example, hardened steel, tungsten carbide, or diamond are commonly

used materials for indenters. For most standard hardness tests a known load is applied slowly by

pressing the indenter at 90 degrees into the metal surface being tested. After the indentation has

been made, the indenter is withdrawn from the surface as shown in the figure. An empirical

hardness number is then calculated or read off a dial (or digital display) which is based on the

cross-sectional area of depth of the impression.

The hardness of a metal depends on the ease with which it plastically deforms. Thus a

relationship between hardness and strength for a particular metal can be determined empirically.

The hardness test is much simpler than the tensile test and can be nondestructive (i.e., the small

indentation of the indenter may not be detrimental to the use of an object). For these reasons, the

hardness test is used extensively in industry for quality control.

Experimental Procedure: First, we need to calibrate the test machine and then do the hardness

test for a particular material.

(a) Calibration

1. Locate the scale pointer at the C zone as shown in Figure I for a steel material.

2. Follow figures II, III, IV, and V to install a diamond penetrator into the machine. Insert

penetrator into penetrator holder with the flat face holding the screw and tighten screw (see

Figure II). Cam handle should be in the start or forward position and the flat anvil should be

placed on the anvil screw. Place C type test block (for steel material) on anvil and turn handwheel

clockwise until test block comes in contact with penetrator. Continue turning handwheel until

pointer makes two revolutions of the dial and comes to rest at the 12 O' clock position (see Figure

III). Move cam handle in a smooth motion to the rear position. Loosen penetrator holding screw

with hex wrench as shown in Figure IV. Return cam handle to original front or start position with

a smooth steady motion. Turn handwheel counterclockwise until test block and penetrator are no

longer in contact (see Figure V).

3. With the cam handle in the start position and with the test block on the anvil, turn handwheel

clockwise to raise the anvil assembly. When contact is made between the penetrator and the test

block, the dial pointer will move in a clockwise direction. Continue turning handwheel until dial

pointer makes two revolutions of the dial, coming to rest at the 12 O' clock position. Move dial

bezel so that pointer reads zero. A minor load has now been applied (refer to Figure VI).

4. Move cam handle with a smooth steady motion to its rear position (2-3 sec). Dial pointer

should now move counterclockwise. Wait until dial pointer stops (approx 4-6 sec). Move cam

handle to the forward or start position (approx 2-3 sec) and the dial pointer will now move

clockwise (see Figure VII). The black dial number indicated by the pointer is the Rockwell

number.

5. Minor adjustment is needed if the readings are consistenly higher than the readings marked on

the test block. With cam handle in the forward or start position, turn adjustment knob clockwise

approximately 1/4 turn to lower readings 1-2 points. Turn counterclockwise to increase the

readings. Continue the calibration procedure until test is reading within the limits of the test

block.

(b) Hardness Testing

With the cam handle in the forward or start position, place specimen (steel) on anvil and turn

handwheel clockwise to raise the anvil assembly. When contact is made between the penetrator

and the steel, the dial will move in a clockwise direction. Continue turning handwheel until dial

pointer makes two revolutions of the dial and comes to rest at the 12 O' clock position. Move dial

bezel so that pointer reads absolute zero. A minor load has now been applied. Move cam handle

with a smooth steady motion to its rear position (2-3 sec). Dial pointer will now move

counterclockwise. Wait until dial pointer stops (approx 4-6 sec). Move cam hanle with a smooth

and steady motion to its forward or start position (approx 2-3 sec). Pointer will now move

clockwise. The black dial number indicated by the pointer is the Rockwell number for the steel

specimen.

Aluminum and Brass Specimens: For the aluminum and brass materials, locate the scale pointer at

the B zone. Choose 1/8 inch diameter steel sphere penetrator. Follow steps 1-5 as indicated for the

steel specimen. The red dial number indicated by the pointer is the Rockwell number for aluminum

and brass materials. Using the conversion graph to determine the tensile strength of a materia as

shown in Figure VIII.

ME 272 LAB #3 -- TORSION OF CIRCULAR BARS

Objective: To obtain the torque-twist relationship for various circular bars and to determine the

material constants (E and G) for each specimen.

Apparatus: Standard material specimens, calipers, Torsion test machine

Theory: Torsional stresses occur in a machine member, such as a power transmission shaft, when

such a member is loaded by twisting couples acting in a plane perpendicular to the axis of the

shaft. In the process, torsional strains are induced. Tests will be carried out on steel, brass and

aluminum specimens to observe the relationship between applied torque and twist (or torsional

strains). Hooke's law gives the torsional stress-strain relationship as

G where = shear stress (applied torque/area), = shear strain, and G = shear modulus of the

material. For a circular shaft of radius r and length L, the relationship between applied torque (T),

shear stress ( ), and angle of twist () is given by

TJ

G L

r

where J is the polar moment of inertia for the shaft.

Experimental Procedure: For each specimen,

1. Measure the gage length L and the diameter d. Calculate the area of cross-section A, and the

polar moment of inertia J.

2. Mount the specimen firmly in the torsion apparatus and make sure it is in a horizontal position

by using a spirit level. Begin applying twisting load gradually by turning the handle at suitably

selected intervals (e. g. 1 degree of twist), record the angle of twist in radians ( and the

applied torque (T) and tabulate your results. Plot a graph of T verus , and the graph should

be within the linear region. Measure at least ten points for the T veruscurve. Make sure that

the specimen is not loaded beyond the elastic region (donot overload the specimen).

3. Measure the linear slope of the T verus graph, and use this value to determine the modulus

of rigidity, G. By assuming a Poission ratio of 0.3, estimate the value of E for the material

according to the formula

G E2 ( 1 )

4. Repeat the same procedure (steps 1-3) for the other materials. Comment and discuss the

accuracy of the results obtained in the experiment.

ME 272 LAB #4 -- TORSION OF PRISMATIC BARS

Objective: To determine the approximate values of the torsional rigidity, K and maximum shear

stress, for prismatic bars of various cross-sections.

Apparatus: Prismatic bar specimens of different materials, calipers, Torsion test machine.

Theory: A prismatic bar is a uniform bar whose cross-section is in the shape of a "prism."

Examples of prismatic bars include circular, triangular, square and hexagonal sections. This

experiment is a continuation of the previous experiment on the torsion of circular bars. In this

experiment, we test circular bars as well as bars with square and hexagonal cross-sections. Thus,

for each material, three bars with different cross-sections will be tested. For all cases, the angle of

twist will be graphed as a function of torque in the linear range.

You are already familiar with the torsional stress-strain relationship for a bar having a solid,

circular cross-section. If the radius of the bar is R and the torques acting at the two ends have a

magnitude T, then the expression for the maximum shear stress is given by

m a x T RJ

The angle of twist is given by

T LG J

In the above equations, where R is the radius of the prismatic bar, L is the length of the bar, T is

the applied torque, G is the shear modulus, and J is the polar moment of inertia of bar.

By knowing the value of G and maximum shear stress, we can determine the angle of twist in a

circular bar of known length L and radius R. Derivation of the expressions for the maximum

shear stress for prismatic bars with non-circular sections is beyond the scope of ME 272.

However, we can simply list the applicable results. They are given as follows:

For a prismatic bar of width w and subjected to torques T, the maximum shear stress is given by

m a x

C a 3

TK

where a = w/2, C is a constant depends on the type of cross-section for the prismatic bar. For a

square cross-section, the value of C = 2.7013 whereas for a hexagonal cross-section of width w

(measured as the distance between two opposite flat faces of the hexagonal section), the value of

C = 2.4526.

The angle of twist is found from

2 LG a 4

TK

Experimental Procedure: Select one of the following materials: aluminum, brass or steel. For each

material selected, carry out the following procedures, first for the circular bar followed by the

square bar and, finally the hexagonal bar. Make sure not to exceed 4o of twist in any specimens.

1. Fix the selected bar in the torsion machine. Make sure it is securely fastened.

2. Measure the applied torques and the corresponding angle of twist, for each 0.50 of twist.

Tabulate your results. Convert angle from degrees to radians. Graph angle of twist (in radians)

versus applied torque. Note: Tables and graphs must be given for each bar tested.

3. For the circular bar only, calculate the shear modulus G.

4. For the square bar, find the torsional rigidity K from the slope of the twist-torque graph, and

equations given above. Repeat the same for the bar with the hexagonal section.

5. Find the maximum shear stress for the circular bar, square cross-section bar, and for the

hexagonal bar.

Discussion: Comment on the accuracy of your results.

ME 272 LAB #5 -- BENDING OF BEAMS

Objective: To obtain the radius of curvature for a given set of loads and compare to the theoretical

values.

Apparatus: Standard aluminum specimen, calipers, Beam bending test machine

Theory: Bending stresses occur when a member (is horizontal) is loaded by transverse loads. In

the process, bending strains are induced. Tests will be carried out on aluminum specimen to

observe the relationship between applied load and bending curvature. Hooke's law gives the

bending stress-strain relationship as

E

where = bending stress, = bending strain, and E = Young's modulus of the material.

For a rectangular bar of width b and thickness h, the relationship between moment due to applied

load (M), bending stress ( ), and radius of curvature () is given by

MI E

y

where I is the moment of inertia for the beam about the neutral axis.

Fig. 1. Beam set-up

Experimental Procedure: Measure the width b and the thickness h. Calculate the area of cross-

section A, and the moment of inertia I. For the specimen (use the modulus of elasticity for

Aluminum as E = 70 G Pa).

Part 1:

1. Use the Aluminum beam and set the two supports 1.0 m apart. Make sure that the two

overhangs (d) on the left and right sides have equal lengths (see Fig. 1).

2. Place one hanger on the outside of each support such that each hanger is d = 0.2m away from

the support (see Fig. 1).

3. Place the dial gage at the middle of the beam and zero it.

4. Place respectively m = 0.5kg, m = 1kg and m = 1.4kg equally on both sides and measure the

respective displacements (gamma). Zero the dial gage each time you change masses.

5. Tabulate and graph experimental radius of curvature (Rexp.) versus the three forces (F=mg)

that were placed at the end of the beam.

6. Calculate bending moments (M=mgd) for the three loads. Calculate the radius of curvature from the theoretical formula: RTh. = EI/M. Tabulate and compare these values with those

obtained in step 5. Explain and give reasons for the differences in the conclusion section of your

report.

where RTh. = theoretical radius of curvature; E = modulus of elasticity; I = mass moment of

inertia = (1/12) (b) (h**3); M = bending moment; m = applied mass at one end of the beam; g =

acceleration due to gravity; and d = distance from support to mass hanger.

Part 2:

1. Use the same support positions as in part (1) and zero the dial gage.

2. Use the mass of m = 1.5kg on each side.

3. Place the hangers at the distance d= 0.2m and measure the deflection (gamma) at the middle of

the beam caused by the applied force (F=mg).

4. Repeat the measurement in step 3 for hangers at d = 0.16m from the supports. Remember to

zero the dial gage before taking the measurement.

5. Repeat the measurement in step 3 for hangers at d = 0.12m from the supports. Remember to

zero the dial gage before taking the measurement.

6. Calculate the experimental radius of curvature (Rexp.). Tabulate the values and graph the

curve: radius of curvature (Rexp.) versus distance (d) of the hanger from the support.

7. Calculate the theoretical radius of curvature (RTh. = EI/M) for each case and compare the

results. Explain and give reasons for the differences in the conclusion section of your report.

Report Format for ME 340 Dynamic Systems & Measurements

The report should contain the following sections: title page, abstract, results, discussion,

conclusions, appendices and references. Each section is described in detail below. A type-written

copy of the report should be submitted with each experiment.

1. TITLE PAGE: This page should contain the following information:

a. The title of the experiment

b. The author's name (Your name)

c. The names of your lab partners

d. The date of the experiment

e. The name of your lab instructor

2. ABSTRACT: The abstract is a summary of the contents of the report. It is therefore, a

concise statement of the objectives of the test, and highlights of the experimental results,

discussion and conclusions. The experimental procedure should not be included in the abstract.

Although the abstract is the first section in the report, it is best to write it last- after you have

finished plotting all the graphs, analyzed the data, and written the discussion and conclusions.

3. RESULTS: The results are best presented in Tables, Graphs or Charts. Label all graphs

completely with: main title; axes labels, including appropriate units for the variables; etc.

Identify each table or figure with a number e.g. Figure 1, Table 2A, etc. Use computer

generated graphs and tables for describing results.

4. DISCUSSION: This is the most important part of your report. In engineering

experimentation, more often than not, the measured data will not be in exact agreement with the

expected theoretical results. This is because theoretical results are based on mathematical

modeling. Further, in order to make a solvable mathematical model, it is necessary to make

some assumptions. The more the actual physical system differs from a mathematical model, the

greater the experimental error. Your discussion should explain the scientific basis for any

differences between the actual experimental results and those predicted from theory. Do not

blame everything on "human error." Moreover, you should strive to find ways by which such

errors may exist and may be eliminated in future tests. This is one of the keys to being a good

experimentalist. Therefore, recommendations for the improvement of the experimental

procedure or results, or any other aspect of the experiment should be included in this section.

5. CONCLUSIONS: Consider questions such as these in discussing your conclusions: What

did you learn as a result of spending several hours in the lab? Were your original objectives met,

or did you discover something else?

6. APPENDICES: Include a separate appendix for every major item that is too distracting

to include in the main report. Anything that could interrupt the flow of thought in the main

report, but which is nevertheless important for a thorough understanding of the experiment,

should be placed in an Appendix. Give a list of equipments used and their model type & make in

the appendix.

7. REFERENCES: Citations of relevant theoretical background material and other related

work should be included in the reference list.

NOTE: Details of the experimental procedure, table of contents, description of apparatus, etc.,

are not required in the reports submitted for this Course.Science Some of the most important mechanical properties of a material can be determined by means of a simple tensile test. This practical introduces tensile testing of metals and plastics and demonstrates the meaning of often-used mechanical property specifications, giving a comparison of actual values for a variety of real materials. Practical skills The first part of the practical involves the preparation of metal tensile specimens by machining, giving valuable practical experience of some workshop technology. Overview of practical After instruction from the Students' Workshop technician, students will each fabricate some metal tensile specimens (choosing from steel, copper and 70/30 brass).1 1 The tensile testing experiments use two steel and one each of the copper and brass specimens, but it is not necessary to make all of these yourselves, as the teaching lab has stocks of test specimens (besides, it would take far too long). This part of the practical is so you can gain some experience of how simple metal components and specimens can be made.

Safety note: You must not use any equipment in the students' workshop without first attending the safety talk there. You must not use any equipment in the students' workshop without supervision from the workshop Technician or a Junior demonstrator. You must be especially careful of your own, and others', safety in the workshop. This will occupy the first day of the practical. At the end of the day, the specimens should be handed to the Practical Class Technician (PCT), so that (s)he can heat treat them for 2 hours at 500°C in a sand bath to relieve residual stresses on the morning of the second day. On the second day the specimens are tested to failure in a Hounsfield Tensometer. A slightly more sophisticated testing machine, an Instron, is used to tensile test strip specimens of various plastics which are supplied ready for testing, also on the second day. Experimental Details 1) Testing of metals Mechanical testing of metals is carried out on a Hounsfield Tensometer. Use Tensometer No. 14 chucks, with the 20 kN load cell and maximum strain magnification, Before starting the testing set the Hounsfield Reduction in Area and Elongation gauges for each sample. Test one sample of each material to failure. The results obtained from the tensometer are put out in the form of a load-extension plot on the laser printer at the end of the test. To convert this into an engineering stress-strain plot, use the following relationships:

lengthoriginalextension,straingEngineerin,areationalseccrossoriginaltheisAwhereAloadapplied,stressgEngineerin=ε−=σ There are two important mechanical properties that can be calculated from the plot obtained: Yield point = point at which plastic deformation begins, i.e. end of linear region of plot. (Engineering) Ultimate tensile strength, Aload maximumUTS= Also measure and compare the % reduction in area and % elongation values for each sample. Take second sample of mild steel. Start the test as before, but stop when a small amount of plastic deformation has taken place. Unload the sample, then reload to produce further plastic deformation. Remove the sample and place it in boiling water for 15 minutes, then reload the specimen and test to failure. How do these results differ from the previous steel sample? Caution: use tongs for handling the sample in boiling water and avoid scalding yourself.

Material Young's Modulus (Nm-2)

Yield Stress (Nm-2)

U.T.S. (Nm-2)

% elongation

% reduction

in area

70/30 brass

Copper

Mild steel

Mild steel (water treated)

.