STRENGTH OF MATERIALSSTRENGTH OF MATERIALSLABORATORY MANUALLABORATORY MANUAL

First EditionFirst Edition Revion 3, 2016Revion 3, 2016 NJIT PressNJIT Press

STRENGTH OF MATERIALSLABORATORY MANUAL

by

Cheng-Tzu Thomas Hsu, NJIT

and

Allyn C. Luke, NJIT

First Edition NJIT PressJanuary, 2005 University Heights

Newark, New JerseyRevised 2016 DRAFT USA

Copyright© 2001 by Cheng-Tzu Thomas Hsu

ALL RIGHT RESERVED

First Edition - Second Printing, January 2006Revision 3 – January 2016 DRAFT

CONTENTS

Page

Preface i

Introduction ii

Experiment 1. Tension Test of Metals 1-1

Experiment 2. Torsion Test of Metallic Materials 2-1

Experiment 3. Stresses, Strains and Deflection of Steel Beams in Pure Bending 3-1

Experiment 4. Strain Measurements using Strain Rosettes in Aluminum Beams 4-1

Experiment 5. Compression Test of Steel Columns 5-1

References 6-1

PREFACE

This laboratory manual provides the experiments for the course Strength of Materials or Mechanics of Deformable Solids, supplementary to those in the classroom textbook.

Some experiments presented here have been developed over the years at the Newark College of Engineering (NCE), the New Jersey Institute of Technology (NJIT), and the authors are indebted to our former colleagues in the Department of Civil and Environmental Engineering who have contributed to the preparations of this work. In recent years the laboratory has received funding to upgradxase and modernize the Strength of Materials Laboratory at NJIT. As a result, this laboratory manual reflects the changes and introduces several experiments using the state-of-the-art testing equipments and new instrumentation.

All experiments presented here are written in a new format, however, a few are still based on the earlier texts of Professor Matthew Ciesla, Mechanics of Materials-Experiments and Problems, and Professor Paul E. Nielsen, Strength of Materials-Laboratory and problems manual. The authors would also like to acknowledge the contribution of Professor G. F. Ramberg, who planned the test procedure and designed and constructed the testing equipment for the Beam Experiment.

The authors would like to thank our colleagues, Professors Paul Chan, Walter Konon, and John Schuring, in the Department of Civil and Environmental Engineering for many helpful suggestions concerning the preparation of this laboratory manual. Their contributions are greatly appreciated. Contributing editors are Methi Wecharatana and Geraldine Milano.

Newark, New JerseyJanuary, 2005, 2006

Revision 3 January 2016

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INTRODUCTION

PURPOSE OF THIS LABORATORY

The experiments described in this laboratory manual have the following general purposes:

(a) To study the behavior of various engineering materials through material testing.(b) To verify the principles of the Strength of Materials as described in the lectures by

means of experiments.(c) To demonstrate the use of instrumentations such as extensometers, strain gages, strain

rosettes and strain indicators and materials testing equipments, and to acquaint students with some experimental testing techniques and the data acquisition system.

LABORATORY SAFETY

Your safety and the safety of those around you are of prime importance. Efforts have been made to prevent accident and to minimize any hazard in the laboratory. Students must follow the general safety rules posted in the laboratory and as outlined in this section. If you have any questions about the safety of the experiments you are going to conduct, consult the laboratory instructor before doing any tests. The applied loads in these experiments are rather high and can be dangerous if not careful. Take your experiments seriously and conduct your testing carefully. The following are the general safety rules that must be strictly followed by every student. Failure to comply may result to failing the course.

General Safety Rules

1. Keep yourself and others safe –Be aware of your own and others’ safety2. Know emergency procedures (fire escape routes, emergency phone locations and

telephone numbers)3. Report any perceived safety hazards4. No working alone5. No eating or drinking in the lab6. Wear appropriate safety equipment7. No loose clothes and long hair around machines8. Do not work with electrical appliances in the presence of water9. Do not put obstructions in walkways –keep fire escape routes completely clear10. Clean up any spills immediately11. Know the hazards of any materials or machines with which you are working

Safety Glasses

Safety glasses must be worn at all times when work is being done in the laboratory and especially when operating the testing machines. Safety awareness training is part of our

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educational mission. Students must learn how to keep themselves and others safe in the laboratory.PREPARATION FOR EXPERIMENTS

Students must prepare for each experiment by reading the instructions for that experiment before coming to class. Quizzes can be expected to verify preparation. Apparent failure to do so will affect your grade for that experiment. Many experiments require calculations to be done before the experiment can be conducted. Students will perform these calculations and submit them as homework before the start of the experiment.

ATTENDANCE

Attendance is mandatory for all first and second level courses at NJIT. Therefore, your attendance is required at all pre-lab and lab experiment classes. Three unexcused classes will result in a failing grade. A failing grade in the lab results in a failing grade for the course.

GRADING POLICIES

Students are expected to properly maintain their registration status. If your name does not appear on the final grade sheet, it is not possible to assign you a grade and it will be necessary for you to repeat the course. Personal problems should be addressed to the Dean of Students.

The laboratory grade represents 15% of the total grade. The lab grade will be factored with your class grade to determine your final grade. You will be assigned this factored grade as your lecture and laboratory grade. You must receive a passing grade in both the lecture and the laboratory to pass the course. Failure of either requires repeating both the lecture and the laboratory. In other words, failing the laboratory means failing the course!! So, please do all of your work.

All reports should be word processed. Graphs are to be computer generated using Excel or any other analytical software.

The results of the experiment are the results you must work with. Do not make up data to produce the “expected” results. Draw your conclusions based on these results. If they are not as expected (you should have an idea of the expected results), discuss and explain the discrepancies.

Reports are also graded on your written presentation. Is the material presented in a logical way? Can all of the required results be found with ease? Are the results discussed intelligently, in a good technical language? Can all the questions that enter the reader’s mind be satisfied? Be advised that your discussions and conclusions carry more weight than production of the right answers.

All laboratories are due as instructed by your laboratory instructors. After the due date reports will be accepted for only 75% credit. After the reports have been returned to the class NO late papers will be accepted.

If you feel that your grades are not what you think they should be it is your responsibility, as it is in all your classes, to seek guidance from your instructor for the

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improvement of your performance. Poor grades usually result from a failure to answer all questions asked. Virtually, all failures occur because of insufficient submission of reports.

Strength of Materials Laboratory Reports

Written laboratory reports are required for all experiments. The reports should be word processed, spell checked and edited. To reduce the work load for everyone, report groups will consist of three or four people. The reports are considered to have three major components, theoretical, laboratory analysis, and editorial. The main body of the report will be a collaborative effort. ALL members of the group will contribute to this part of the report. But EACH member of the group will write their own Discussion and Conclusion which will be signed by that party. Each group will submit ONE report with multiple Discussions and Conclusions.

Report Writing

Report writing is often a frustrating experience for new writers. Technical writing has additional, special problems. Since there are half a dozen reports required for the Strength of Materials lab a short summary on efficient report writing will be presented. It is hoped that well before the end of the semester, students will be comfortable with the writing process and that they will be able to concentrate on the content of reports rather than the writing process. The reports present the opportunity to practice what will become one of the most important of engineering skills, the ability to write competent technical reports.

General Rules1. Use Direct Technical Language

Write simply, use as few words and as little punctuation as possible. Write in complete sentences. As much as possible use the active voice, the agent of the action should be clear especially in conclusion. Avoid using first person. Define all technical terms and acronyms.

2. Be EngagingIt is very important to draw your reader into your work. Careful consideration needs be given to the first few sentences. The conclusion is the most important part of a report. The conclusion must refer back to the introduction to bring closure or “wrap-up” to the report. Remember, your goal is to prove a point, validate the theory and properties.

3. Edit, Edit, EditThe quality of writing will be directly proportional to the amount of editing. Check for correctness and completeness. Get the opinion of other readers, knowledgeable ones when possible, for comments on how the work can be improved. Finish the report early, put it aside for at least a day then go back and edit again. Look for smooth flow of language, complete expression of idea and specificity of meaning. Use the spelling and grammar tools available in Microsoft Word.

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Basic FormatReports should include the following sections:

1. Title page with experimental objectives listed from the manual, date of the experiment, date of the submission, names and signatures of group members.

2. Abstract – In one or two paragraphs, explain what the experiment is about, a sentence or two about the procedure, and a simple statement of the results. Use specific values. For each experiment, your main goal is to prove a theory or to validate specific properties.

3. Introduction –State the objective(s) of the experiment and give background information. Begin by stating the objective of the experiment in detail. Be specific as to what values you intend to find, such as a Modulus of Elasticity. Then present some background information that may be found in your textbook, from websites, or other references. All references MUST be cited in your bibliography.

4. Theory – Explain the theory or theories under consideration. What assumptions were made developing the theory and how might the assumptions affect the experiment? What is expected of the experiment to prove the theory? What predictions are made?

5. Experimental procedure – How was the experiment conducted? Be specific about the testing apparatus. What measurements and observations need be made? How was the experiment conducted? What measurements were taken?

6. Analysis of Test Results – Use tables, graphs, and sample calculations to explain the test results. Sample calculations and sample tabulated data should be incorporated into this part of the report. You can “copy and insert” portions of your spreadsheet to explain what data was used in the calculations. Show step-by-step how your get from the test data to the final results.

7. Discussion of theory in terms of the experimental results. Did your results prove the theory? This is the most important part of the entire report. Reflect back to the objective and support your discussion with some background information. Do your results meet the guidelines of published results?

8. Conclusions – how well was the theory validated? Justify the results in detail. Why do you think the results are good? Cite published data to validate your results. Why are they suspicious? Compare your results to published data and give an explanation for discrepancies. (each group member must draw their own conclusions on each objective item.)

9. Bibliography – Cite all references used. List them alphabetically and use proper citations.

10. Appendix – Included anything related to the experiment not listed abovea. Raw data (often absorbed into spreadsheet)b. Sample handwritten calculations – show, with neat handwritten samples, how

one gets from the measured to the target values. In support of spreadsheets, show the computation in the cell. The results of the hand computations should match the spreadsheet result.

c. Any extra graphs, tables, photos not incorporated into the main body of the report

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New Jersey Institute of TechnologyStrength of Materials

Laboratory Experiment 1Tension Testing

Stress-Strain Relation Physical Properties

Objectives:1. Observe the Stress-Strain relation for Steel and/or other metallic material2. For each metal tested, determine the following properties:

Proportional Limit Yield Strength Ultimate Strength Young’s Modulus Modulus of Resilience Toughness Percent Reduction of Area Percent Elongation in 2 inches

3. Compare results with reference values

4. Compare formulas approximating E, Resilience and Toughness with values computed from the Stress-Strain Curves.

5. Assess the validity of the axial deflection equation,

6. Observe the characteristics of a tensile failure.

Safety Issues Crushing Hazard – Do not place hands or any body part into the

crush zone along the line of the piston motion Eye Hazard – Small pieces of metal might fly off failing specimens, Safety

Glasses Required

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Sharp Edges – Handle failed specimens with care to avoid cuts

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Theory

An understanding of the mechanical behavior of materials is necessary and essential for the safe engineering design of all types of structures and machines. One of the most fundamental concepts in strength of materials is the stress-strain behavior of a prismatic bar under axial tension. Most materials are assumed to be homogeneous, i.e. having the same material property throughout all parts of the bar (the same material property throughout all parts of the bar). For ductile materials such as metals, at low stress level and according to Hooke’s Law, the stress-strain curve is a straight line, of which the slope is equal to the Modulus of Elasticity or Young’s Modulus (E) of the material, which is:

(1-1)

As the stress increases, a point on the curve where the linear stress-strain relationship ends is commonly known as the “Proportional Limit” of the material. For some materials, two other significant points may also be observed in close proximity to the proportional limit, which are the Elastic Limit and the Yield Point. The former is the maximum stress that can be applied to a member, at which upon unloading the material will return to its original length without causing any permanent damage to the material. The latter is a critical stage of stress, Yield Stress (y), at which yielding starts. The Yield strength of metal is one of the critical parameters used in most engineering designs. Once loading applied beyond the elastic limit, permanent damage is induced in the material resulting to residual strain or permanent deformation upon unloading. This can be visualized from the necking of the test specimens, which directly corresponds to the reduction in the cross sectional area. Non-ductile or brittle materials do not normally exhibit a well defined Yield Point. For these materials the stress that causes 0.2% residual strain upon unloading is commonly used as the Yield strength of the material. At high stress level, the maximum stress that the material can withstand is usually referred to as the Ultimate Strength, . For brittle materials, rupture usually takes place at the ultimate load. However, in ductile materials, the Rupture Strength (f) may occur after the ultimate strength is reached as straining continues beyond the ultimate strain (ult). See the diagram shown in Figure 1-1.

Figure 1-1 shows a typical stress-strain diagram for ductile steel in tension. The engineering (nominal) stress is calculated using the original cross-sectional area (P/A) whereas the strain is determined from the ratio of the elongation over the initial length of the member ( = L). According to Hooke’s law, i.e. = E, where E is the Modulus of Elasticity or Young’s Modulus, the elongation of the member can be calculated from:

(1-2)

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True Stress-Strain curve

E’ Ultimate stress,

D Conventional or Engineering

Yield stress, E stress-strain curve B C Fracture A Proportional limit,

E =

O

Linear Perfect Strain Necking

region plasticity hardening or yielding

Figure 1-1 Typical stress-strain diagram for ductile metal in tension

1. Four equations are under consideration:

a. normal stress from axial loading

b. normal strain from axial deformation

c. Young’s Modulus or Modulus of Elasticity

d. axial deformation

Identify each variable and indicate how it will be determined for or from the experiment. (note that equation d is derived from equations a, b, and c.)

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2. Define the properties under consideration:Proportional Limit Modulus of ResilienceYield Strength ToughnessUltimate Strength Percent Reduction of AreaYoung’s Modulus Percent Elongation in 2 inches

3. From your text find equations for the Modulus of Elasticity, Resilience and Toughness. Calculate those theoretical, approximate values. (Which is theoretical and which approximate?).

4. For a 7000-pound load range within the linear elastic region, using a published value for E, a 2-inch gauge length and the dimensions of the specimen tested, predict the axial deflections, that is, the change of the deformation. Enter into table 2 for comparison with experimental results. If using the Instron test apparatus, the gauge length will be different.

ProcedureBefore testing, review all safety issues with the lab instructors.You may be using the MTS test apparatus in the basement of the Architecture building or the Instron test apparatus in Colton Hall. Your Lab Instructor will decide.

1. Measure and record the diameter of the rod to be tested.2. Measure and record the distance between the punch marks designated as the gauge

length.3. Prepare the testing apparatus and set to zero4. Place the specimen in the grips. Make sure that both ends 5. Attach the extensometer (if applicable) to test apparatus.6. Slowly apply a load to the specimen, the objective is to reach the yielding load in

one smooth stroke. After the yield, operate until the specimen breaks.7. Remove the extensometer from the specimen then remove the specimen from the

machine.8. Measure and record the final diameter at the break.9. Measure the largest distance across the break between two punch marks indicating

the original gauge length.

Required Data

Measurement ValueInitial load lbs.Maximum load lbs.Initial diameter inchFinal diameter inchInitial gauge length inchesFinal length inches

Table 1. Required data

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Analysis

1. Compute and plot the experimental stress-strain diagrams of each test specimen.

2. Draw a second stress-strain curve of each test specimen, magnifying (detailing) the linear elastic portion (not exceeding the proportional limit) of the stress-strain curve.

3. From the test data and the experimental stress-strain diagrams, determine the following material properties of materials:

a. Proportional limit b. Yield strengthc. Ultimate strengthd. Modulus of Elasticity (or Young’s Modulus)e. Percent elongation in the 2-inch gage lengthf. Percent reduction of cross sectional areag. Modulus of resilience, and h. Toughness

4. Using the references, find the value or range of values for the properties determined.

5. Using formulas where appropriate, calculate (predict) the Modulus values.

6. Compute the percent difference between the referenced or predicted values and the experimental values.

7. Determine the elongation () of all test specimens from the theoretical equation, using the load at 50% of the yield strength and the experimental E values. Compare these predicted elongations with the experimental values from the tests.

8. Tabulate all experimental and predicted values for different materials.

PropertyExperimental Reference Calculated % error

Proportional Limit naYield Strength naUltimate Strength naModulus of ElasticityModulus of ResilienceModulus of Toughness% Reduction of Area na% Elongation of gauge length

na

δ = PL/AE (P=7000 lbs., L=2 in,use reference E, zero data)

Compare to raw data

Table 2. Summary Table of Results

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Thought Questions (These questions are meant to direct your thinking in particular directions so you can draw expected conclusions. The answers by themselves do not complete the assignment, your conclusions are required)

1. What were the percentage differences between the measured and the referenced and/or calculated values for each of the properties? Comment on differences. Is this difference acceptable? Was the material tested what it was said to be?

2. What does the ability to predict reference values say about the principles of stress and strain that underlay this experiment?

3. Compare the Modulus of Elasticity, Modulus of Resilience and the Modulus of Toughness determined from the Stress-Strain Curve to values calculated from formulas. Are they in reasonable agreement? If not, why not? Which determinations give the best results?

4. Knowing the load and final area at failure, compute the actual stress on the rod at that moment and compare this value to the final values of the engineering stress as expressed in the stress strain diagram. What accounts for this difference?

5. What is the Toughness value? What does it describe? What is the Modulus of Resilience and how does it differ from the Toughness values? Describe the behavior of this material as it responds to increasing load.

6. What was the mode of failure? What observations lead you to that conclusion?

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STRENGTH OF MATERIALS LABORATORY

EXPERIMENT 2

TORSION TEST OF METALLIC MATERIALS

OBJECTIVE

1. To study the linearly elastic behavior of metallic material under torsion and to determine the shear modulus of elasticity, G, and Poisson’s Ratio, , for metals using torsional stress-strain relationships.

2. To study the complete behavior of metallic materials under torsion and to determine qualitatively the relationship between torsional load and angle of twist for a full range of strains till failure.

3. To determine whether the metallic materials fail in tension, compression, or shear when it is subjected to pure shear.

Submitted by __________________ SPECIMENGroup _______ Section __________ Cylindrical barDate Performed _________________ Materials ___________________Date Submitted _________________ Dimensions __________________Instructor ______________________

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THEORY

Assume that the material is linearly elastic, homogeneous, and isotropic (material having the same properties in all directions), and for a solid cylindrical section twists under an externally applied torque T, shown in Figure 2-1 in accordance with the torsion Formula (found in any standard Strength of Materials text book):

(2-1)

where = torsional shear stress at a point on the surface of a cylinder T = twisting momentr = radius of the cylinderd = diameter of the cylinder

Ip = J = polar moment of inertia of the cross section about its center =

The angle of twist of a cylindrical bar is also related to the applied torque, T as given below:

(2-2)

where = angle of twist having a unit of radians per unit of lengthand the total angle of twist is :

(2-3)

where = total angle of twist having a unit of radians. L = length of the bar over which the angle of twist measured

G = shear modulus of elasticityGIP = GJ = torsional rigidity

Since the material is linearly elastic, the Hooke’s law in shear is given by:

or (2-4)

where = shear strain in radians.Thus

(2-5)

or

(2-6)

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Note that the above formulas are valid only below the proportional limit and, therefore, above the proportional limit only qualitative conclusions can be drawn, such as, the load increases, decreases, or remains constant as the angle of twist increases.

(a) Shear stress in a circular bar under torsion

(b) Tensile and compressive stresses acting on a stress element oriented at 45

Figure 2-1 Solid cylindrical bar under twisting moment

By subjecting a circular cylinder to torsion, a condition of pure shear can be produced at every point in the body (excluding the part of the cylinder within or near the clamps or chucks) and shear properties are usually determined in this manner. By this method the shear stress can be calculated by the above torsion formula (see Figure 2-1), and based on the force equilibrium in a stressed element, one can obtain the following relations:

(2-7)

where and are the tensile and compressive stresses oriented at 45 to the longitudinal axis Z as indicated in Figure 2-1b.

Figure 2-2 illustrates the strains in pure shear for a solid cylindrical bar under twisting moment as shown in Figure 2-1;

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d = 2r

L

zT

T

d = 2r

L

zTzk Tzk

σmax

σmax = σmin

σmin = -

(a) Shear distortion of a stressed element orientation at = 0

(b) Distortion of a stressed element oriented at = 45

Figure 2-2 Strains in pure shear in a stressed element

The positive normal strain in max-direction (see Figure 2-2b) produced by max is

equal to

Also, the stress min produces a positive strain in max-direction equal to , therefore, the total normal strain in max-direction is

(2-8)

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where is called the Poisson’s Ratio. The max is a positive normal strain, representing elongation in max-direction. The normal strain min in min-direction is a negative normal strain of the same amount, which represents shortening. These normal strains, max and min

are consistent with the deformed shape of Figure 2-2a, which produces elongation in the 135diagonal and shortening in the 45 diagonal. From the geometry of the deformed element which relates the shear strain to the normal strain max in the 135 direction (Figure 2-2b), one has

(2-9)

Equation (2-9) has been used to calculate the shear strains, , under applied torque T.Also, Equations (2-1) and (2-9) have been used to determine the relations between the torsional shear stress and shear strain under applied torque T. From the above linear relationship or Equation (2-4), one can attain the shear modulus of elasticity G, or the Poisson’s ratio from Equation (2-8), or

thus (2-10)

Note that while Poisson’s ratio can be determined in a tension test by measuring the lateral contraction, it will require such very precise measurements that it is more practical to determine the tensile and shearing moduli of elasticity and compute Poisson’s Ratio from these. Since in this torsion test the tensile modulus of elasticity is not determined, one can refer to a text for the generally accepted value, and with that and your own value for shearing modulus of elasticity one can calculate Poisson’s Ratio using Equation (2-10).

SAFETY ISSUES

Eye Hazard – Small pieces of metal can fly off failing specimens, beware splashes of surface conditioning compounds, Safety Glasses Required.

Sharp Edges – Handle failed specimens with care to avoid cuts

Failed Specimens can be hot (why?) Beware of burns.

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TEST PROCEDURE

1. Measure and record the diameter of the test specimen.2. Make sure the solid cylindrical bar is fit securely into the torsion testing machine.3. Start the data acquisition program, which acknowledges its readiness to collect data. 4. Repeat a second test if needed.5. Construct a table to collect the data of loads and deformations during the test as shown in

data sheet.6. Find the final diameter of test cylindrical bar and sketch its failure mode.

ANALYSIS OF DATA

1. Use a spreadsheet to plot the torsional shear stress-shear strain curve from the computer file to a stress level below the proportional limit. Determine the experimental shear modulus of elasticity from this curve using Equation (2-4). Then in conjunction with Young’s modulus, or modulus of elasticity E, from a reference value, calculate Poisson’s ratio from Equation (2-10).

2. Calculate the theoretical torque and compare to the experimental results.

3. Plot the experimentally obtained torsional shear stress , and shear strain , curves on two graphs, one for the entire data set (elastic and plastic behavior) and another one for the first phase (linearly elastic behavior) of torsion tests. Use the data given to you by the instructor. Compare and discuss the results.

4. Compare the values of the shear modulus of elasticity, G, and Poisson’s ratio, , obtained from the test with the authoritative values found in the text book or other resources. Discuss the test results.

5. Describe the behavior of this material as it responds to increasing load. Pay special attention to the region above the yield where linear elastic theory no longer applies.

6. Sketch and describe the appearance of the failed bar, and discuss the mode of failure (ductile failure as compared to brittle failure). It is suggested to photograph the test specimen for inclusion in your report.

7. Discuss and conclude whether the ductile metallic materials generally fail in tension, compression, or shear when it is subjected to pure shear.

POSTLAB QUESTIONS

(These questions are meant to direct your thinking so you can draw expected conclusions. The answers by themselves do not complete the assignment, your conclusions are required.)

1. Two equations are used to calculate the torsional stresses. What are they? Explain what is experimentally necessary to prove these equations.

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2. What does your text tell you about the failure modes of ductile and brittle materials when they are subjected to pure shear? Did this material fail in tension, compression or shear? What observations bring you to that conclusion?

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STRENGTH OF MATERIALS LABORATORY

EXPERIMENT 3

STRESSES, STRAINS AND DEFLECTION OF STEEL BEAMS IN PURE BENDING

OBJECTIVE

1. To compare the theoretical strain predictions with the strain measurements obtained from both the electrical and mechanical strain gage methods.

2. To experimentally determine the location of the neutral axis in the beam cross section and compare it with the theoretical predicted value.

3. To examine the validity of the assumption made in the flexural analysis of beams that cross sections remain plane during bending.

4. To compare the theoretically predicted deflections with the measured experimental values.

Submitted by __________________ SPECIMENGroup _______ Section __________ Material Steel Beam . . Date Performed _________________ Section S8 18.4 . Date Submitted _________________ Clear Span 12 ft and 6 inches Instructor ______________________

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THEORY

Consider a simply supported beam under four-point bending which is illustrated in Figure 3-1 .If the cross-section of a beam is symmetrical about its neutral axis, the maximum tensile and compressive stresses are equal.

Thus, the flexure formula (3-1)

Becomes (3-2)

where = the maximum tensile stress = the maximum compressive stress

S = section modulus = Iz / cMz = M = Pa = bending moment about the z-axis

(with respect to the neutral axis) Iz = I = moment of inertia of the cross-section about the z-axis c = distance from the neutral axis to the extreme element in y-direction a = distance from support to the applied load

Figure 3-1 A simply supported beam under four-point bending

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From the Hooke’s law or the theory of linearly elastic analysis, one has

(3-3)

or (3-4)

where E = modulus of elasticity or Young’s modulus = the maximum tensile strain = the maximum compressive strain

EIz = EI = flexural rigidity

DEFLECTION

Deflection calculations are an important part of structural analysis and design, and the design engineers are normally required to verify that the deflections under service loads are within tolerable limits as specified by the Standard Specifications and Codes.

There are several methods available for the calculation of deflections in structures. They are double integration method, moment-area method, conjugate beam method, and matrix structural analysis method, etc., and can be found in most Strength of Materials and Structural Analysis textbooks. The Double Integration method enables us to determine not only the deflection at any location of the structures but also the deflection profile or elastic curve of the structures.

Figure 3-1 shows a simply supported beam under four-point bending. In the case of a prismatic beam with constant EI, the differential equation of the deflection curve (or the elastic curve) of a beam is as follow:

(3-5)

where Mz = M = bending moment about the z-axisE = Young’s modulus or modulus of elasticityIz = I = moment of inertia about the z-axis, andEIz = EI = flexural rigidity

From the bending-moment expression of the beam, and the boundary conditions, one can solve the above governing differential equations and attain the following deflection curve or elastic curve:

(3-6)

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(3-7)

(3-8)

(3-9)

where y = deflection in the y-direction (positive upward)

= dy/dx = slope of the deflection curve = -y (L/2) = deflection at midpoint C of the beam (positive downward)

= -ymax = maximum deflection (positive downward) = - (0) = angle of rotation at left-hand end of the beam (positive clockwise) = - (L) = angle of rotation at right-hand end of the beam (positive counterclockwise)

(a) Coordinate System

(b) Deflection curve

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(c) Beam with four-point bending

Figure 3-2 A simply supported beam under four-point bending

TEST APPARATUS

The experimental setup is shown in Figure 3-3, which consists of a beam tester and the loads weight 50 lbs each .The instrumentations include the strain indicator and the switching and balancing unit. Also, ten strain gages and eleven sets of mechanical strain gages are installed on one face of the steel beam S818.4. The locations are shown in Tables 3-1and 3-2, respectively.

TABLE 3-1 Electrical Strain Gage at TABLE 3-2 Mechanical Strain Gage near mid-span mid-span

Number Distance from Bottom Number Distance from Bottom12345678910

8.08 inches 6.69 5.93 5.12 4.50 3.77 3.00 2.18 1.40 0.40

1234567891011

8.21 inches 6.77 6.05 5.37 4.71 4.04 3.37 2.67 2.03 1.39 0.47

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Figure 3-3 Test setup for a simply supported beam S818.4 under four-point bending

The mechanical (Whittemore) strain gage is commonly used to precisely measure a change in length of less than one-ten thousandth of an inch, and must be handled carefully to prevent damage. To make a reading, the two conical points are inserted into the two gage holes with the axes of the cones perpendicular to the surface on which the measurement is being made, and the instrument is pressed gently against the surface.

The electrical strain gage operates on the principle that, when strained (elongated or contracted), the electrical resistance of a wire changes because of change in length and/or change in cross section. This change in resistance is measured by a Wheatstone Bridge in which the variable resistance has been graduated to read directly in microinches per inch of strain. Each gage consists merely of a very fine wire within a protective covering which has been cemented to the surface of the specimen.

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TEST PROCEDURE

1. Use of Measurements Group P-3500 and SB-10 for Strain Measurements

a) P-3500 Strain Indicator

-Set Amp Zero to 0-Set Gage Factor to 2.015-Set Balance to 500-Press RUN button to read strain

b) SB-10 Switch and Balance Unit

-Adjust gage to a zero reading on the P-3500 by selecting the channel of interest and adjusting the Balance knob on the SB-10 for that channel

-After loading, measure the strains by selecting the channel of interest and reading the P-3500 display.

2. Use of Demountable Mechanical (DEMMEC) Gage for Strain Measurements

-Set the gage into the 10.0000 inch invar punch bar and zero the gage. The DEMMEC now reads 0 at 10.0000 inches.

-Using the DEMMEC measure the gage lengths at the various levels, the distance between the various gaging points. Deviation +/- from 0.0000 is +/- deviations from 10.0000 inches.

-After loading record readings, The readings are +/- s. Divide by the gage length to compute the strain.

3. Use of CDI Digital Dial Indicator for Deflection Measurements

-Set the gage reading to zero

-After loading record the deflections

4. Procedure

a) Zero the electrical gages

b) Measure the gage lengths

c) Zero the deflection gage

d) Apply load to the beam by placing weights onto the loading pans in 200 lbs. increments (the weights are 50 lbs. apiece). With the level arms setup for a load factor of 10, this will produce incremental loads of 2000, 4000 and 6000 lbs. Make readings of all the electrical and mechanical strain gages, and the deflection gage between each loading and unloading increment.

3-26

ANALYSIS OF DATA

1. Based on Equation (3-4), calculate the theoretical unit strain on the top and bottom surfaces of the steel beam for each of the loads P (not P1). Note that it is sufficient to determine c and t only because only two points are needed to determine the theoretically linear strain distributions across the cross-section.

2. Draw a graph of the theoretical strains for each of the loads, using the beam depth as the ordinate (y-axis) and strain as abscissa (x-axis).

3. On the same graph, show as separate points the measured strains. Indicate which are measured with the electrical strain gage and the mechanical strain gage for each of the loads, i.e., (a) loading, (b) unloading.

4. Using linear regression analysis, the neutral axis can be obtained by the y-intercept of the regression equations.

5. From Equation (3-8), calculate the theoretical maximum deflections at mid-span of the beam for each of the loads P (not P1).

6. Collect all dial gage readings of the deflection measurements for each of the loads during loading and unloading, and compare their results with the theoretical elastic- deflection values.

7. Compare the experimental deflection measurements with the theoretical values for zero, 2000 lbs. and 4000 lbs. while undergoing loading and unloading.

REPORT

1. Construct a table which compares the theoretical and the experimental maximum tensile strains at the bottom of the beam and the maximum compressive strains at the top as shown in Table 3-3. Discuss the results and findings.

TABLE 3-3 Comparison of theoretical and experimental stresses and strains

Loads, P (lbs)

(psi) at bottom (psi) at topTheoretical Electrical

strain gage

Mechanical strain gage

Theoretical Electrical strain gage

Mechanical strain gage

Loading200040006000

Unloading40002000

0Note: the above table is the sample for the stress comparison

3-27

2. Plot graphs showing the theoretical strains and the measured strains both by electrical and mechanical strain gages for each of the loads, including loading and unloading. Use the depth of the beam as the y-axis, and the strain as the x- axis of the graphs.

3. Determine and show the locations of the neutral axis in the beam cross section on the graphs for each load. Discuss the results and findings.

4. Discuss and conclude the validity of the assumption that beam cross-sections remain plane during bending.

5. Prepare a table as shown in Table 3-4 which lists all the deflections at mid-span of the test beam determined by Equation (3-8), the theoretical elastic-analysis and by the experimental measurement using the dial gage.

TABLE 3-4 Mid-span deflection

Loads, P(lbs.)

(inches)Theoretical Experimental

Loading2,0004,0006,000

Unloading4,0002,000

0Average Value

Standard Deviation

6. Discuss and conclude the test results and findings such as:

(a) Are the dial gage measurements made while loading the steel beam the same as those made while unloading;

(b) How do your calculated deflections compare to the actual measured deflections for each of loads? Is the theoretical analysis conservative or does it understate the actual deflection?

(c) What are the limitations of the theoretical formulas, Equation (3-5) through Equation (3-9), that are being used in the structural design?

SUGGESTED QUESTIONS

1. Was there good agreement between the theoretical predictions of strain and the experimentally measured strains?

2. Which strain gaging method seems to give the best results?

3. What is your determination of the neutral axis? How did you reach this value?

4. What measurement dictates the precision of the determination of the neutral axis?

3-28

5. Was there good agreement between the theoretical predictions of deflections and the measured deflection? Do you believe the expression for deflection is conservative?

6. Were the theories under consideration proved? If so, how? And if not, why?

7. How does the experiment demonstrate that plane sections do indeed remain plane under bending?

EXPERIMENT 3

Strains and Stresses in Pure Bending of Steel Beams

Load P Electrical Strain Mechanical StrainGage Reading Gage Reading(microinches per inch) (in)

0 1. _______ 6. ________ 1. _______ 6. ________ Central 2. _______ 7. ________ 2. _______ 7. ________ Deflection 3. _______ 8. ________ 3. _______ 8. ________ (in.) 4. _______ 9. ________ 4. _______ 9. ________________ 5. _______ 10. ________ 5. _______ 10. ________

11. ________

2,000 lbs 1. _______ 6. ________ 1. _______ 6. ________ Central 2. _______ 7. ________ 2. _______ 7. ________ Deflection 3. _______ 8. ________ 3. _______ 8. ________ (in.) 4. _______ 9. ________ 4. _______ 9. ________________ 5. _______ 10. ________ 5. _______ 10. ________

11. ________

4,000 lbs 1. _______ 6. ________ 1. _______ 6. ________ Central 2. _______ 7. ________ 2. _______ 7. ________ Deflection 3. _______ 8. ________ 3. _______ 8. ________ (in.) 4. _______ 9. ________ 4. _______ 9. ________________ 5. _______ 10. ________ 5. _______ 10. ________

11. ________

6,000 lbs 1. _______ 6. ________ 1. _______ 6. ________ Central 2. _______ 7. ________ 2. _______ 7. ________ Deflection 3. _______ 8. ________ 3. _______ 8. ________ (in.) 4. _______ 9. ________ 4. _______ 9. ________________ 5. _______ 10. ________ 5. _______ 10. ________

11. ________

3-29

EXPERIMENT 3 (continued)

Load P Electrical Strain Mechanical StrainGage Reading Gage Reading(microinches per inch) (in)

4000 lbs . 1. _______ 6. ________ 1. _______ 6. ________ Central 2. _______ 7. ________ 2. _______ 7. ________ Deflection 3. _______ 8. ________ 3. _______ 8. ________ (in.) 4. _______ 9. ________ 4. _______ 9. ________________ 5. _______ 10. ________ 5. _______ 10. ________

11. ________

2000 lbs . 1. _______ 6. ________ 1. _______ 6. ________ Central 2. _______ 7. ________ 2. _______ 7. ________ Deflection 3. _______ 8. ________ 3. _______ 8. ________ (in.) 4. _______ 9. ________ 4. _______ 9. ________________ 5. _______ 10. ________ 5. _______ 10. ________

11. ________

0 lbs 1. _______ 6. ________ 1. _______ 6. ________ Central 2. _______ 7. ________ 2. _______ 7. ________ Deflection 3. _______ 8. ________ 3. _______ 8. ________ (in.) 4. _______ 9. ________ 4. _______ 9. ________________ 5. _______ 10. ________ 5. _______ 10. ________

11. ________

3-30

STRENGTH OF MATERIALS LABORATORY

EXPERIMENT 4

STRAIN MEASUREMENTS USING STRAIN ROSETTES IN ALUMINUM BEAMS

OBJECTIVE

1. To study the strain measurements of a simply supported aluminum beam in a general case of plane stress by means of the Mohr's Circle analysis.

2. To verify theoretical computations of the combined stresses at several point on a beam with the experimental results.

3. To experimentally determine the combined stresses (the actual state of stress) at several points on a beam using the Strain Rosettes

Submitted by __________________ SPECIMENGroup ________ Section ___________ Material _____________________Date Performed __________________ Section _____________________Date Submitted __________________ Clear span _____________________Instructor __________________

4-1

THEORY

Consider an element in plane stress as shown in Figure 4-1; this element is infinitesimal in size and can be sketched as a rectangular parallelpiped. x and y are designated as normal stresses acting on the x- and y-face of the element, respectively. The shear stress xy acts on the x-face in the direction of the y-axis, and yx acts on the y-face in the direction of the x-axis. They are equal, i.e. xy = yx. The positive sign conventions of these plane stresses are depicted in Figure 4-1

(a) two-dimensional view in x-y axis

(b) two-dimensional view in x-y and n-t axes

Figure 4-1 An element in case of plane stress

4-2

Any Strength of Materials textbook will show that the axial strain in n-direction is given by:

, or (4-1a)

(4-1b)

Since each strain gage measures the normal strain in only one direction, at least three strain gages are needed to determine the strains in a plane stress element, as indicated as A, B, and C in Figure 4-2.

Figure 4-2 Three strain gages, A, B, and C arranged in an element

From the configuration of strain gages shown in Figure 4-2,

(4-2)

(4-3)

(4-4)

Solving the above three equations simultaneously, one can find x, y, and xy, the two normal strains and one shear strain in a plane stress element.

4-3

For the 60 strain rosettes:

A = 0 B = 60 C = 120

sin 2A = 0 sin 2B =

sin 2C = -

cos 2A = 1 cos 2B = -

cos 2C = -

Figure 4-3 60 strain rosettes

From Equations (4-2), (4-3) and (4-4); one yields:

(4-5)

(4-6)

(4-7)

Solve the above Equations (4-5), (4-6), and (4-7), one obtains:

(4-8)(4-9)

(4-10)

(i) For the 45 strain rosettes:

= 0, = 45, and = -45 or 135, and

(4-11)(4-12)(4-13)

4-4

Figure 4-4 45 strain rosettes[

In the case of biaxial stress (Figure 4-1), Hooke’s law of plane stress-strain relation or the constitutive law for a linearly elastic material is given by:

(4-14)

(4-15)

and (4-16)

where E = Young’s modulus or modulus of elasticity = Poisson’s ratio

G = Shear modulus of elasticity

Now, the stress analysis of a simply support beam as illustrated in Figure 4-5, usually begins by determining the normal and shear stresses acting at any points on the cross sections of the beam. When the Hooke’s law holds, or the beam behaves in a linearly elastic manner, the following normal and shear stresses from the flexural and shear formulas as seen in most standard Strength of Materials textbooks, can be used:

Flexure formula: (4-17)

4-5

and Shear formula: ` (4-18)

where Mz = M = bending moment about the z-axis,Iz = I = moment of inertia about the z-axis,y = distance from the z-axis,Vy = V = Shear force in y-axis,b = width of the cross-section,Q = first moment of the cross-sectional area outside of the point in the cross section where the stress is being found.

TEST APPARATUS

Six strain rosettes have been cemented to an aluminum beam as indicated in Figure 4-5; gage 2, 4 and 6 are 45 rosettes and gages 1, 3 and 5 are 60 rosettes.

Figure 4-5 Tested aluminum beam with six strain rosettes (E = 10.3 106 psi and G = 4.1 106 psi)

4-6

TEST PROCEDURE

This test is conducted using a Vishay Micro Measurements System 5000 data acquisition system. This equipment is capable of conditioning and reading signals from strain gages, thermocouples, L VDTs, high range signals like those coming from DCDTs and tiltmeters or any device providing scaled voltage signals. In this experiment the System 5000 is used to measure and reduce the data from 5 rosettes, comprising 15 strain gages, and the applied load.

1. Start the computer and load the Strain Smart program.

2. Open, arm, and start the Lab4 program.

3. Apply a 25,000 lb load to the beam.

4. Click on the Read button.

5. Release the load and turn off the testing machine.

6. Identify the data file, and using the Strain Smart program reduce the data into Excel

comma delimited format.

7. Copy the data to floppy disk.

ANALYSIS OF DATA

1. Calculate the theoretical values of the combined stresses, x and xy using the flexural and shear formulas, Equations (4-17) and (4-18), at each rosette location.

2. The data file contains the strains measured by each rosette element and the experimentally measured principal stresses at each rosette location. For extra credit the strain gage data can be reduced by hand using Equations (4-8), (4-9), (4-10) for the 60° rosettes and Equations (4-11), (4-12), (4-13) for the 45° rosettes, however the principal stresses that result should match those given by the data file. (The hand computations are not needed to fulfill the purpose of the experiment).

3. Using the theoretical values of x and xy, draw the theoretical Mohr's Circles for the six Rosette locations.

4. On top of the theoretical Mohr's Circle plot the experimental principal stresses and connect them with a circle.

5. Note that gages 1, 2 and 3 are sufficiently far from the loads on the beam, so that the theoretical and experimental values should correspond. Gage 4, being under the concentrated load, is affected by the localized response of the beam to the concentrated load.

4-7

REPORT

1. Prepare a table as shown in Table 4-3 which compares the maximum tensile stress, compressive stress and shear stress determined experimentally and theoretically for each rosette.

TABLE 4-3 Comparison of theoretical and experimental values of combined stresses at each gage.

GageNo.

Theoretical ExperimentalStress (psi) Strain Stress (psi)

x x x

y y y

xy xy xy

max max

min min

2. Discuss the test results and draw conclusions. The discussion will cover the values discovered and the correspondence between the theoretical and experimental values. The conclusion will cover how well the theory was supported by the experiment.

SUGGESTED QUESTIONS

1. How well did the theory predict the experimental measurements at each of the six

points?

2. If any of the experimental Mohr's Circles happens to be radically different from its

theoretical partner, what might be the cause of the differences?

4-8

STRENGTH OF MATERIALS LABORATORY

EXPERIMENT 5

COMPRESSION TEST OF STEEL COLUMNS

OBJECTIVE

1. To study some of the important parameters which affect column buckling, such as slenderness ratio and least radius of gyration.

2. To determine the relationship between critical stress and the slenderness ratio of steel columns.

3. To confirm the validity of Euler’s analysis of Pcr, the critical buckling load and the relationship of Pcr to column slenderness.

Submitted by ___________________ SPECIMENGroup _______ Section __________ Lengths of bars, ranging from _________Date Performed _________________ to _________Date Submitted _________________ Material _________________________Instructor ______________________ Diameter _________________________

5-1

THEORY

Structural members that support compressive loads are commonly called columns, and one of the inherent difficulties with compressive loads applied in this manner is the possibility of a geometric instability of buckling. The longer a column, the greater the likelihood of buckling for any given cross-section and modulus of elasticity. In practical structural design, it is necessary to determine what limiting load may be carried by a column before buckling occurs.

In 1744, the Swiss famous mathematician Leonard Euler (1707-1783) published the result of critical or buckling load Pcr, for a slender column, known as Euler load,

(5-1)

where Pcr = critical or buckling load,E = modulus of elasticity,I = moment of inertia about the weak axis,L = length of the columnK = effective length factor, K=1 for pinned-ended columnrg = radius of gyration (usually written without the subscript)

The critical stress, cr, can be written as

(5-2)

where L/rg = L/r = slenderness ratio, and I = Arg

2, = Ar2

The radius of gyration, rg, is usually written simply as “r” in the context of Euler’s analysis.

Note that columns will buckle in the direction of the least moment of inertia, I.

If the column is made of round bar with a diameter of d, then

r =

For a pinned-ended column, K=1, Equation (5-2) becomes

5-2

g

(5-3)

Equations (5-2) and (5-3) are valid only when the critical stress is equal to or below the proportional limit, pl. The critical slenderness ratio can be represented as,

(5-4)

When a column is of intermediate length, or called intermediate columns, the critical stress in the column will reach over the proportional limit as illustrated in Figure 5-1. In this case, the column undergoes inelastic buckling, and a theory of inelastic buckling such as tangent modulus theory, the reduced or double modulus theory, and the Shanley theory, is needed.

Figure 5-1 Critical stress versus slenderness ratio

5-3

TEST APPARATUS

The column testing device is shown in Figure 5-2 which consists of a MTS closed-loop servo controlled hydraulic testing machine, the automatic control and electronic data acquisition system, as seen in the tension test (Experiment 1).

Figure 5-2 Column test setup

5-4

SAFETY ISSUES

Crushing Hazard – Only a trained operator is allowed to operate the MTS machine. Do not place hands or any body part into the 55,000 pound crush zone along the line of the piston motion.

TEST PROCEDURE

1. Several test specimens of AISI 1020 steel must be prepared and ready for testing.

i. Use three specimens of ¼” diameter with lengths of approximately 3”, 12”, and 24”

ii. Use three specimens of ½” diameter with lengths of approximately 3”, 12”, and 24”

2. Measure and record the actual lengths and diameters of all test specimens.

3. Construct a data sheet with the following column headings, as seen in Table 5-1: Diameter of Bar, Length of Bar, Load at Failure, Stress at Failure, and Slenderness Ratio.

4. After positioning the test specimen, the load is applied to the end of the bar through a hardened steel hemisphere resting on a hardened steel surface. Thus the ends of the bars are hinged and the force is applied very nearly along the axes of the bars. Use K = 1.

5. Apply the load very slowly at a stroke rate of 0.015 in/min until the load reaches a maximum and starts to decrease, then remove the load immediately as the column starts to buckle.

TABLE 5-1 Specimen details and test results. Suggested table for your report.

Specimen Number

Diameter (in.)

Length (in.) Pcr (lb.) cr (psi) L/r

5-5

ANALYSIS OF DATA

For the experimental Euler’s curve compute the radius of gyration, rg, and the slenderness for each column. From the loads, compute the stresses at failure for each column. From the slenderness compute the Euler Pcr. Plot the experimental results as points and the Euler’s prediction as a line (it will be necessary to sort all the data by L/rg to produce this graph).

1. Plot Euler’s formula, Equation (5-2) with K=1, which is similar to the Euler’s curve shown in Figure 5-1.

2. On the same graph, plot a point for each column tested using cr as ordinate and L/r as abscissa.

3. For those experimental points which do not agree with the Euler’s analytical expression, a curve-fitting or and empirical straight-line formula may be used. For the intermediate columns, the equation of a straight line is,

(5-5)

where A and B are the constants to be determined. To determine A and B, draw the straight line which most nearly fits the plotted points and is tangent to or intersects Euler’s curve. Select any two points on the line and read the co-ordinates of these two points. By substituting these numerical values for the variables in Equation (5-5), two simultaneous equations are formed which can be solved for the constants A and B.

REPORT

1. Make a table listing all specimen details, critical load, critical stress and slenderness ratio.

2. Plot both analytical and experimental curves of critical stress versus slenderness ratio similar to Figure 5-1.

3. Discuss and conclude your test results and analyses.

QUESTIONS TO DIRECT YOUR THOUGHTS

(These questions are meant to direct your thinking in particular directions so you can draw expected conclusions. The answers by themselves do not complete the assignment.. The answers to these questions should be written into the conclusion rather than answering individually. )

1. Was the critical load correctly predicted by Euler’s analysis?

2. What was the relationship between slenderness and Pcr?

3. What was the range of intermediate columns? Can you find a model, some equation to mathematically describe column behavior in the intermediate range? What models the long columns? The short columns ?

4. Do you conclude that Euler’s analysis of columns is valid or not ?

5-6

EXPERIMENT 5

Compression Test of Steel Columns Data Sheet

Length (in.) Diameter (in.) Critical Load (N)

____________________ ____________________ ________________

____________________ ____________________ ________________

____________________ ____________________ ________________

____________________ ____________________ ________________

____________________ ____________________ ________________

____________________ ____________________ ________________

____________________ ____________________ ________________

____________________ ____________________ ________________

____________________ ____________________ ________________

____________________ ____________________ ________________

____________________ ____________________ ________________

____________________ ____________________ ________________

____________________ ____________________ ________________

____________________ ____________________ ________________

____________________ ____________________ ________________

5-7

REFERENCES

1. ASTM Specifications (American Society for Testing and Materials Specifications), 1916 Race Street, Philadelphia, PA 19103. Note that the ASTM Specifications are revised every year and consist of many volumes. Sometimes a particular standard gets moved from one volume to another.

2. Manual of Steel Construction, published by the AISC (American Institute of Steel Construction), Inc., One East Wacker Drive, Chicago, Illinois 60601, USA.

3. Aluminum Construction Manual published by the Aluminum Association, Inc., 900, 19th Street NW, Washington, D.C. 20006, USA.

4. Paul E. Nielsen, “Strength of Material-Laboratory and Problem Manual”, Eight Edition, Newark College of Engineering, Newark, New Jersey USA, 1957.

5. Mathew Ciesla, “Mechanics of Materials-Experiments and Problems”, Sixth Edition, New Jersey Institute of Technology, Newark, New Jersey, USA, 1976.

6. James M. Gere, “Mechanics of Materials”, Fifth Edition, Brooks/Cole, A division of Thomson Learing, 2001. 926 pages.

7. Ferdinand P. Beer and E. Russell Johnston, Jr., “Mechanics of Materials”, 2nd Edition, McGraw-Hill, Inc. 1992, 740 pages.

8. G. S. Holister, “Experimental stress Analysis”, Cambridge University Press, 1967, 322 pages.

6-1