CE 113

MECHANICS OF MATERIALS LABORATORY

LABORATORY MANUAL

Department of Civil and Environmental Engineering San Jose State University

Revised December 28, 2009

TABLE OF CONTENTS A. COURSE OBJECTIVES ........................................................................................................ 1 B. PRE-LAB CALCULATIONS................................................................................................. 2 C. DATA SHEETS...................................................................................................................... 3 D. ACCURACY AND SIGNIFICANT FIGURES ..................................................................... 4 E. LABORATORY REPORT..................................................................................................... 5

E.1. Grading ........................................................................................................................... 6 E.2. Report Organization........................................................................................................ 6

E.2.1 Objectives of Experiment and Procedures.................................................................. 7 E.2.2 Theoretical Estimation of Experimental Results ........................................................ 7 E.2.3 Results of Experiment................................................................................................. 7 E.2.4 Conclusions................................................................................................................. 8 E.2.5 References................................................................................................................... 9

E.3. Elements of a Good Laboratory Report .......................................................................... 9 E.3.1 Spelling and Grammar ................................................................................................ 9 E.3.2 Tables.......................................................................................................................... 9 E.3.3 Graphs ....................................................................................................................... 10 E.3.4 Figures....................................................................................................................... 10

F. STRAIN GAUGES............................................................................................................... 11 F.1. Introduction....................................................................................................................... 11 F.2. Bonded Electrical Resistance Strain Gauge...................................................................... 11 F.3. Principal Stresses and Strains from Strain Gauge Rosettes .............................................. 12

G. STRESS-STRAIN RELATIONSHIPS................................................................................. 16 G.1. Hooke's law................................................................................................................... 16

G.1.1 Uniaxial Stress Equations ......................................................................................... 16 G.1.2 Biaxial Stress Equations ........................................................................................... 16

H. Experimental Techniques...................................................................................................... 17 H.1. Pre-loads ....................................................................................................................... 17 H.2. Gathering data............................................................................................................... 19 H.3. Teamwork ..................................................................................................................... 19

I. EXPERIMENT #1 ................................................................................................................ 20 J. EXPERIMENT #2 ................................................................................................................ 25 K. EXPERIMENT #3 ................................................................................................................ 30 L. EXPERIMENT #4 ................................................................................................................ 32 M. EXPERIMENT # 5 ............................................................................................................... 33 N. EXPERIMENT #6 ................................................................................................................ 38

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A. COURSE OBJECTIVES

To verify mechanics of materials theory on real specimens

Student learning outcomes: The student will demonstrate the ability to: a) Identify when theory applies and when theory is limited by simplifying assumptions b) Identify reasons why actual measurements will differ from theoretical calculations

To learn to run an experiment

Student learning outcomes: The student will demonstrate the ability to: a) Perform pre-laboratory calculations to estimate experimental parameters, outcomes and

limits b) Develop an organized and meaningful data sheet c) Use software tools to reduce and analyze data d) Organize a team to share responsibilities for operating equipment and collecting data

To learn to use testing equipment and measurement instrumentation

Student learning outcomes: The student will demonstrate the ability to: a) Use the laboratory equipment correctly and safely to perform all experiments

To learn to write a laboratory report

Student learning outcomes: The student will demonstrate the ability to: a) Write experimental objectives and procedures b) Present results in a organized and clear manner c) Draw graphs and figures to summarize key findings d) Put together a complete report including tables of contents, references and appendices

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B. PRE-LAB CALCULATIONS

For each experiment discussed in this Manual, a set of calculations that must be completed and questions that must be answered in order to prepare for the collection of data in the lab. Examples of the types of calculations you will make include: • Determination of the maximum allowable load on a specimen. • Estimation or prediction of experimental results. • Determination of simple relationships between independent and dependent variables. • A data sheet, as described in the next section of this manual. Your laboratory preparation work will be completed and turned in during the session that meets prior to the session when actual lab is performed. This insures that you are prepared so that accurate and complete data is collected and that a high-quality report can be written. Incomplete or inadequate laboratory work will result in a similarly inadequate report. Late laboratory preparation work will not receive credit. The pre-lab calculations may vary from semester to semester so do your own work. The instructor may require the pre-lab to be completed in class the week before the actual lab occurs. If possible, calculations should be set up as a template in a spreadsheet or other computer application so that changes can easily be made. This can save time and effort if the experimental setup is found to differ from the description in this manual.

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C. DATA SHEETS

A prepared paper or spreadsheet template data sheet is required as part of each pre-lab. A copy of you paper data sheet or a printout of the spreadsheet template must be handed in with the pre-lab. It is essential that you put some thought into how you plan to collect data before you come to the laboratory session. You must design it carefully in order to help you collect complete and accurate data. Remember that you are very unlikely to write a good report with bad or incomplete data. Items to consider: • Your data sheet should include a sketch of your test setup including gauge location and

appropriate dimensions so that you can check that the experiment is set up correctly before you start. (usually you can get this sketch from the Laboratory Manual)

• How many columns of data will you need? You will generally need to have two columns for each data variable and sometimes three so that scale factor and/or zero offset calculations can be made.

• Each column should be labeled so that you can remember where the data came from and what it relates to.

• The units of the data you are taking should be in your column labels. • How many rows will you need? The rows will need labels too. • Is there a location to put comments that relate to the test setup, procedures or the collection of

data? If you plan to use a spreadsheet and enter data directly into a computer, you need to design the spreadsheet with the same considerations as above. You may want to design different sheets for different parts of the experiment. A sample data sheet is included in the section of this manual for laboratory #1.

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D. ACCURACY AND SIGNIFICANT FIGURES

The number of significant digits displayed is an indicator of the accuracy of a number. With digital calculator displays and computer spreadsheets, it is easy to overstate the accuracy of a number. When taking a measurement, the last reliable digit relative to the decimal point determines the precision of your measurement. Data should not be recorded more precisely than warranted. For example, assume you are reading load off the analog dial where there are tick-marks every 10 pounds. If the needle is between the 50 and the 60 pound tick mark you could guess that the load is 54 pounds, but the 4 is not very reliable. Certainly you would not want to write down 54.3 pounds. It is somewhat harder to determine a reliable measure with the digital machines, and one would have to read the testing machine manual to determine that reliability for each range, but normally two or three significant digits is all that can be expected for the equipment used in this course. Finally when doing calculations, the answer cannot be more precise than the data and constants you begin with. For example, if you know that the dimensions of a rectangular specimen are 25.5 mm (3 significant figures) by 30.01 mm (4 significant figures), the area of the specimen would be 25.5 x 30.01 = 765.255 mm2 = 765 mm2 (3 significant digits) You may at times want to carry more significant digits while you are calculating to avoid round-off and truncation error, but your final answer should not be more precise than the original data. Little if any, of the data collected in this course will be accurate to 2%. Material properties are seldom known to more than 2-digit accuracy. This is consistent with most work in the "real world" of engineering design and analysis. In some areas of engineering work (earthquakes and machine vibration are good examples) the actual forces a structures are subjected to are not known to the designer with much reliability. Some of the data collected in this course will be only be two-significant figure accuracy and the calculated results should properly represent this fact. Another example: A tension specimen is measured by micrometer to be 1.000 inches by 0.623 inches in cross section. The specimen is loaded in a machine that reports applied loads to the nearest 10 pounds to a displayed load of 5930 pounds. The stress in the specimen is calculated:

The calculator displays the solution as 9518.4591. But the data (measurements of dimensions and load) is only accurate to three significant figures. So, the solution should be reported as:

9520 psi

623.0*00.15930/ == APσ

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E. LABORATORY REPORT The format and requirements for your laboratory reports will be set forth by your laboratory section instructor and in the descriptions of each laboratory provided in this manual, but some general guidelines are included below. Objective of Lab Reports:

The primary objective of a lab report is to convey to the client (general public, other engineers, the professor, etc.) the information that the experimenter has collected. The report should be organized so that someone who is not familiar with the particular experiment or test set-up can understand:

• what assumptions were made, • what procedures were followed, • what data was collected, • what analysis was completed, and • which conclusions were made

The fundamental purpose of your report is to clearly demonstrate to the instructor that you fully understand the theories, methods, variable, accuracy, data and analysis involved in the particular lab. In CE 113, students will work in teams to complete lab reports and submit them to the instructor for grading. It is the students’ responsibility to convey their understanding of the experiment through clear text, calculations, graphs, figures, and tables. Microsoft Word© can insert equations if the equation editor is installed. If it has not been installed on your computer, search in help for “troubleshoot equation editor.” To insert an equation, click “Insert” then “Object” and select “Microsoft Equation.” You can also customize your toolbar so that an “Equation Editor” button is available for easy use. Microsoft Excel© spreadsheets can also be inserted into a document by placing the curser where you want the spreadsheet and then clicking the “Inset Microsoft Excel Worksheet” button on the standard toolbar. Microsoft Excel© charts can also be inserted into a document simply by copy and paste.

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E.1. Grading Each instructor will have individual instructions about how he/she plans to assign grades to lab reports. However, in all cases, effectiveness of communication will be a major factor in determining the report grade. Effectiveness of communication will include

report organization, including table of contents and correct page numbering spelling grammar complete samples of calculations labeling of graphs, figures and tables proper style for citing references

All material, except appendices, must be developed using a computer unless otherwise authorized by your instructor. It is recommended that sample hand calculations be written using a computer, but engineering block lettering is acceptable. E.2. Report Organization (see individual labs within this manual for summaries) A complete report should contain at least the following:

Title Page, listing experiment number and title, laboratory section instructor and laboratory section identification (time and day of the week), date of submittal, and report author(s)

Table of Contents, indicating sections, sub-sections and page numbers Main Body

Objectives of Experiment (or introduction and problem statement) Procedures Theoretical Estimation of Experimental Results (Pre-lab description) Presentation of Data Results of experiment, data analysis Conclusions (were the objectives met? was the accuracy adequate? Does the theory

work? Etc.) References (if needed, but will usually at least contain this lab manual as a reference) Tabulated Data (on clearly labeled data sheet) Graphs and Figures (properly labeled) Appendices Appendix A – Raw Data (including the data sheet used in the lab) Appendix B – Supporting Calculations (if needed) Appendix C – Data Reduction (if needed) All pages should be numbered. Often numbering starts over in appendices and numbering such as A-1, A-2, … B-1, B-2, etc. is used.

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E.2.1 Objectives of Experiment and Procedures

The Objectives of the Experiment and the Procedures may be adapted from the objectives stated in this Manual, be sure the Manual is cited in the text and listed in the References section of your report. Procedures should explain the text configuration as well as the steps to follow to complete the experiment. Use of bullets or numbers for this section is acceptable and often makes for clearer, more concise organization. Be sure and clearly describe if and how your procedure or the setup varied from that provided in this manual.

E.2.2 Theoretical Estimation of Experimental Results

Each lab has a theoretical component, usually summarized and performed as part of the Pre-lab assignment. The theory should be clearly explained, including the key components of the theory, equations used, and any assumptions that are used. You should include samples of all calculations and then a table to summarize estimates of key experimental results completed during the pre-lab assignment. For example a table such as this may be useful:

Table 1: Estimated Strain on Beam #2

Gauge Number Load (lbs.) Estimated Strain (µin/in)

1 P 0.02P 2 P 0.04P 3 P -0.03P

Equations should be written using appropriate software (such as Microsoft Equation 3.0, which is an installation option for Microsoft Word). Simple equations may be developed using a traditional keyboard. All equations should be indented and numbered. Equation numbers should occur at the same tab stop for all equations. As an example:

∫−Θ

≡π

π

α21

xdx (Eq. 1)

B = (200 ax)(10z)/386EI)2d2 (Eq. 2)

E.2.3 Results of Experiment

The experimental measurements should be summarized and explained in this section. This section usually will not include the raw data, since that will be reserved for Appendix A. Often you will want to include tables and plots to show the summarized results. Be very careful to present numerical results with appropriately accurate significant figures. In charts and graphs, you should not curve-fit a line through the data points since errors (which are always present) cause the individual data points to be offset from the actual phenomena being measured. A chart that presents a theoretical relationship may be presented by a continuous line if the theory results in a continuous equation. An example of a complete table is shown here:

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Table 2: Results From Experiment 10

Actual Load (lbs.)

(Measured)

Strain (µµµµinch/inch) (Measured)

Stress (psi)

(Calculated)

Predicted Strain (µµµµinch/inch)

0 0 0 0 1020 35 6,700 32 2200 72 13,700 69 3350 106 20,200 105

An example of a complete plot is shown here:

Figure E1: Stress-strain relationship calculated from data taken in Experiment 1A on January 1, 2000 the curve was calculated using the least squares function in

Excel 97. Also see section H.2 Note: If possible, you should present your data and predictions as plots for comparison. All

plots for comparison should be to the same scale. Present numerical comparisons in tables.

E.2.4 Conclusions

This section is for the student to discuss whether the objectives (found in this Manual) of the experiment were achieved and to make observations and draw conclusions based on the comparisons of the experimental data and the theoretical predictions. Here are a few examples of points that could be included: • a quantification of the difference between theoretical predictions and measured results, • if the data and the theoretical predictions are not the same, a discussion of reasons why this

may have happened; • whether the theoretical model is appropriate for this real world specimen; • whether the test procedure is an appropriate method of validating the theory; • difficulties that occurred in completing the experiment or data analysis.

Experim ent 1A: Tension of a Sm all S teel S trap

0

1000

2000

3000

4000

5000

6000

7000

0 100 200 300

strain (10 -6 inch/inch)

stre

ss (

psi

) E xperim enta l D ata

C alcu la ted S tress-Stra in R ela tionsh ip C alcula ted slope

E = 24.9 x 10 6 ps i

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In addition, the conclusions should include at least one technical aspect of the experiment that the author understands clearly. This technical item could be related to the theory, or to the assumptions that were made that would prevent the experiment from matching theory, or some functional aspect of the instrumentation, etc. If the report is a group report, the conclusions should contain one item that is clearly understood by each student in the group.

E.2.5 References

Any material, that is not written or derived by the lab report authors, needs to be referenced. The purpose of the citation is to provide the reader with the information necessary to find the item in the library or on the Internet. The format for citing references should be one that is generally accepted by the engineering profession. The web site http://pubs.asce.org/authors/book/generalresources/references.htm gives examples of the format ASCE requires authors to use for citing many different types of books, journal articles, reports and reference documents. E.3. Elements of a Good Laboratory Report

E.3.1 Spelling and Grammar

It is highly recommended the students use the spell checker and grammar checker that come with most word processing software. However, for reasons stated below, you need to read the laboratory report carefully yourself and use the dictionary, your knowledge, grammar textbooks, and your judgment in cases where the word processing software cannot properly correct your text. It is important to understand that the standard libraries that come with the word processing software do not contain some of the words that we typically use in engineering applications and you may have to “teach” these words to the software. Examples are the words: geotechnical, uniaxial, deflectometer, Hooke’s Law, or Poisson’s ratio. Also, the spell checker will not correct the spelling of a word if it is spelled correctly but is used incorrectly (e.g. principal stress, not principle stress). Furthermore, most grammar checkers try to correct sentences that use the passive voice (e.g. The specimen was loaded, versus I loaded the specimen). However, in engineering, the passive voice is quite acceptable and sometimes even encouraged.

E.3.2 Tables

Tables must include a table number, a title above the table, and labels at the top of each column that completely describe the contents of the column. In some cases you may also need labels for rows or subsections of the table. All labels should include units where appropriate. Since tables must be readable, you should not use a font smaller than 10 point. Tables may be printed in portrait or landscape mode.

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E.3.3 Graphs

It is likely that you will use Excel or Matlab for drawing graphs. It is very important that you understand the assumptions being made and limitations of the software when you draw a graph. For example you should know the difference between a scatter graph and a line graph in Excel. (Note: you don’t want to use the line graphs for plots you need for this class). In general data should be shown on graphs as markers without connecting lines. Theoretical predictions, or estimates of relationships developed from analysis of the data, should be shown as lines without markers. Generally a graph should be at least one half of a page so that you can fit all of the relevant information on it and still use a font that is 10 point or larger. A full page for each graph is recommended. Do not fit curves to your data unless the curve being fitted is part of the theory being tested. If there is a theory being tested, the theory will provide an appropriate equation. Graphs must include a figure number with a caption that gives information about the data such as experiment number and date collected, a title at the top of the graph, the axis names and units, and an indication of which is the experimental data. You may use a legend or you may label points and lines. An example graph is shown in section E.3.3.

E.3.4 Figures

Figures are any types of graphics that serve to support your laboratory report. Drawings of the test apparatus or the specimens, as well as graphs developed to display data are all figures. Drawings of the test apparatus and specimens are a simple way to show dimensions, orientation, and the loading configuration of each experiment. Figures of the test set up may be copied from the laboratory manual if the manual is properly cited. Digital cameras are an excellent way to acquire images to be used as figures in your report. Figures must include a figure number and a caption below the figure. The caption should clearly explain what is in the figure. If a figure is not drawn by the author, but instead taken from another source, it must contain a citation of the reference.

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F. STRAIN GAUGES

F.1. Introduction Stress is not directly measurable with current technology. We can measure the load that we apply to a specimen, but we cannot easily measure the load per unit area, stress, at any point on the specimen. Instead, the experimental analysis of engineering stresses must be based on strains, which can be used to calculate stresses. Using measured values of strain and knowledge of the mechanical properties of the material such as the modulus of elasticity and Poisson’s ratio, stresses can be calculated from the appropriate stress-strain relationship of the material. There are many types of commercially available strain gauges used in experimental mechanics. Examples are acoustical, capacitance, inductance, mechanical, optical, piezioresistive, resistance, and semiconductor. The optical strain gauges come in two types, diffraction and interferometric. Perhaps the most versatile and widely used gauge is the bonded electrical resistance strain gauge which is the type of gauge attached to the test specimens in Experiments 1 and 2 in this course. Experiment 5 uses a Berry Strain Gauge, which is a particular type of mechanical strain gauge. F.2. Bonded Electrical Resistance Strain Gauge The bonded electrical resistance strain gauge consists of a metallic strain-sensing element encapsulated by a thin polyimide film that acts as an insulator attached to tabs for leadwire connections. The sensing element is a grid of very thin metal alloy. The entire assembly can then be bonded to the specimen so that the gauge moves in unison with the specimen. Figure F.1 shows a graphic of a uniaxial strain gauge. As the gauge is stretched, the thin wires elongate, increasing the electrical resistance of the gauge in direct proportion to the strain. The measurement of the change in electrical resistance, which is measured as a change in voltage in a bridge type circuit, is converted to strain is by dividing by a gauge factor. Equation F.1 illustrates this simple relationship.

∆R/R = (Gauge Factor)(∆L/L) (Eq. F-1) Where: R = resistance ∆R = change in resistance L = original length ∆L = change in length (∆L/L = strain)

The gauge factor is a property of the particular gauge being used and is usually between 2.00 and 2.20

Figure F1: A strain gauge with a uniaxial pattern

for measuring strain in the direction of the gridlines. Courtesy of Measurements Group, Raleigh, NC.

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There is a wide selection of grid configurations, sizes, and alloy compositions to accommodate test conditions, including temperature extremes, dynamic or cyclic loading, etc. "Rosette" configuration strain gauges have two or three sensing elements and are used to determine principal strains and principal directions for general two dimensional plane stress problems. Three element strain gauge rosettes will be used in Experiments 1 and 2. Strain gauge resistance changes are very small, on the order of 10-3 Ωto10-6 Ω, due to micro strains occurring in engineering materials when stressed. Conventional ohmmeters are not capable of measuring resistances with enough precision to detect these small changes. Instead, potentiometer circuits and Wheatstone bridge circuits are used to convert these very small resistance changes to voltage signals that can be observed and recorded. The Wheatstone bridge circuits used in the laboratory are shown below. The dummy gauge in Figure F.2b is used to compensate for strain changes due to temperature only. The instruments in the laboratory are calibrated periodically and include adjustments for the strain gauge sensitivity (resistance), bridge balance (zero) and gauge factor. The instruments' readings are in proportion to strain.

Figure F2: Different types of variable resistance bridges.

(a) Typical Wheatstone bridge, (b) Circuit configuration to compensate for temperature change. A short explanation of how the instruments in the laboratory work is as follows. In Figure F.2a, the bridge is initially "balanced" so that R1∗R3 = R2∗R4 and the signal voltage Eo is zero. The Active leg of the bridge (R1) is the strain gauge and the resistance of R1 varies as the structure is loaded. If the resistance of R1 increases (tension), the signal voltage (Eo) increases in proportion to the strain. The signal voltage is amplified and displayed. F.3. Principal Stresses and Strains from Strain Gauge Rosettes Principal stresses cannot be directly determined from a strain rosette. Principal strains can be calculated from the measurements taken from the strain rosette and then Hooke’s Law can be used to convert principal strains to principal stresses. This section refers to homogeneous, isotropic materials in the linear-elastic range: Rectangular rosette (0° - 45° - 90°) A rectangular rosette has three gauges so that measurements of normal strain can be taken along three axes the rosettes used in this course are Rectangular and have three gauges oriented 45° from each other. In the example that follows, the three axes of the rosette are numbered 1, 2, 3

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counterclockwise and measure the strains ε1, ε2, and ε3 Respectively. The whole rosette is oriented at an arbitrary angle, Φp, measured from gauge number 1 with respect to the principal axes of the strain on the specimen.

Figure F3: Labeling and orientation of gauges and axes for a rectangular rosette. The angle ΦΦΦΦp is measured counterclockwise with respect to εεεε1.

In this course you will learn a graphical solution method to solve for the principal strains from rectangular rosette data. The following equations can also be used to determine the principal strains, εp and εq, and principal stresses, σp and σq, and the angle of orientation, Φp:

( ) ( )

( ) ( )

( ) ( )

( )strainbiaxialequal ateindetermin is ,if

;45,andif

;45,andif

;,if

;,if

tan2

1

1

2

12

E

2

1

2

q,p321

pq,p1231

pq,p1231

qq,p31

pq,p31

31

21321q,p

232

221

31q,p

232

221

31q,p

Φ=∈=∈∈

°=Φ=Φ>∈∈=∈∈

°−=Φ=Φ<∈∈=∈∈

Φ=Φ<∈∈

Φ=Φ>∈∈

∈−∈∈−∈−∈−∈

=Φ

∈−∈+∈−∈

ν+±

ν−∈+∈

=σ

∈−∈+∈−∈±∈+∈

=∈

−

(Eqs. F-2)

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Delta rosette (0° - 60° - 120°) Delta rosettes are not currently used in this course. Delta Rosettes can provide slightly better accuracy in solving for the principal stresses than Rectangular Rosettes.

( ) ( ) ( )

( ) ( ) ( )

( )( ) ( )

( )( )( )( )

strainbiaxialequalif

andif

andif

if

if

E

pqp

pqp

qqp

pqp

qp

qp

qp

,

;45,)(2/1

;45,)(2/1

;),(2/1

;),(2/1

3tan

2

1

1

2

13

3

2

3

321

,12321

,12321

,321

,321

3121

321,

231

232

221

321,

231

232

221

321,

=∈=∈∈

°==>∈∈∈+∈=∈

°−==<∈∈∈+∈=∈

=∈+∈<∈

=∈+∈>∈

∈−∈+∈−∈∈−∈

=

∈−∈+∈−∈+∈−∈

+±

−∈+∈+∈

=

∈−∈+∈−∈+∈−∈±∈+∈+∈

=∈

−

φφφφ

φφφφ

φ

ννσ

(Eqs. F-3)

and φp,q is indeterminate Tee rosette (0° - 90°) Tee Rosettes are not currently used in this course. When the directions of the principal axes are known in advance, a two-element 90-degree (or "Tee") rosette can be used. However, the gauge axes must be aligned to coincide with the principal axes. The directions of the principal axes can sometimes be determined with sufficient accuracy from the shape of the test object and the mode of loading.

Figure F4: Alignment and orientation of a tee rosette.

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These equations that can be used to determine the principal strains, εp and εq, and principal stresses, σp and σq, from Tee Rosettes:

( )

( )122

212

2

1

1

1

∈+∈

−=

∈+∈

−=

=∈∈

=∈∈

νν

σ

νν

σ

E

E

q

p

q

p

(Eqs. F-4)

References for Strain Gauge Section: Beer, F. P. and Johnston, E. R. (1992) Mechanics of Materials, Second Edition, Mc-Graw Hill, Inc., New York, New York.

Gere, J. M. and S. P. Timoshenko, Mechanics of Materials, Third Edition, PWS-Kent Publishing Company, Boston, 1990.

http://www.vishay.com/strain-gauges/knowledge-base-list, this web site has detailed information about many aspects of electrical resistance strain gauges as well as the PhotoStress method for visualizing strains.

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G. STRESS-STRAIN RELATIONSHIPS

G.1. Hooke's law For homogeneous, isotropic materials in the linear-elastic strain range:

G.1.1 Uniaxial Stress Equations

A uniaxial stress state exists where there is only one nonzero principal stress. The uniaxial stress- strain equations below apply when the x-axis is in the direction of the nonzero principal stress:

00 ==

∈−=−=∈

∈==∈

xyxy

y

xx

y

xxx

x

EE

EE

τγν

σσ

ν

σσ

G.1.2 Biaxial Stress Equations

A biaxial stress state exists where there are (only) two nonzero principal stresses and both τxy and γxy equal zero in principal directions:

Note that these equations simplify to the uniaxial forms if ∈y = 0 and ∈x is the principle stress.

Where: E = modulus of elasticity, psi [1 psi = 6894.76 pascals (Pa)] σ = normal stress, psi ∈ = normal strain, in/in [or m/m, etc.] ν = Poisson's ratio, non-dimensional τ = shear stress, psi γ = shear strain, radians G = modulus of rigidity (shear modulus), psi

( )ν+=

12:

EGNote

G

EE

EE

xyxy

xyy

yxx

τγ

σνσ

σνσ

=

−=∈

−=∈

xyxy

xyy

yxx

G

vv

E

vv

E

γτ

σ

σ

=

∈+∈−

=

∈+∈−

=

)(1

)(1

2

2

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H. Experimental Techniques

H.1. Variables Experimenters often identify variables relevant to their work by assigning them to three categories; Independent Variables, Dependent Variables and Controlled Variables. The purpose of the experiment is to investigate the relationship, if any, between the independent and dependent variables. The controlled variables may influence the relationship being investigated but are not allowed to vary during the course of the experiment. As an example, a researcher might be interested in the relationship between temperature and the tensile strength of a material. She would adjust the temperature of several samples and then test them for strength. The independent variable is the one she controls, in this case the specimen temperature. The dependent variable is the one she observes, the tensile strength. The experimenter might suspect that the age of the material influences its tensile strength so all of the samples tested are the same age. The age of the specimen is a controlled variable for this experiment.

Independent Variables are changes that occur in an experiment that are manipulated directly by the experimenter. Dependent Variables are changes that occur due to independent variable changes. Controlled Variables are anything else that might influence the dependent variables but are kept constant by the experimenter.

H.2. Pre-loads Pre-load is a term for a load applied to the specimen before the prescribed experimental procedure begins. A pre-load is applied to eliminate any non-linearities that sometimes occur at low loads. These non-linearities are generally due to mechanical problems such as miss-alignment of the specimen, supports or loading fixture. As an example, imagine a prismatic beam which is rectangular in cross section and is resting on two supports, one support near each end. One support might be a knife-edge and the other might be a roller in the form of a cylinder. If the knife-edge and the cylinder are not perfectly parallel, or if the beam has a slight twist, there will not be full contact between the beam and its supports until sufficient load is applied to the beam. The load required to settle the beam onto its supports is generally not considered in the experimental analysis. By applying a pre-load before the experiment begins the effect can be eliminated. Pre-loading may be required if a plot of your data indicates an anomalous nonlinearity at low loads. Always record any pre-load applied as part of your data in case it becomes an issue later on. Example:

18

A set of load and deflection data is presented in the table below. As can be seen from the first plot, the data does not begin at the plot origin. In the first step, the deflection data is “zeroed” to the initial reading by subtracting that initial reading from subsequent readings. In the second plot, it is apparent that the first data point is not representative of the rest of the data set which is very nearly linear. The first data point is neglected and the second data point is taken as the reference zero for the final plot.

Raw Data

0100200300400500600700800900

0.00 0.02 0.04 0.06 0.08

Deflecion (in)

Lo

ad (

lb)

Deflection Zeroed

0100200300400500600700800900

0.00 0.02 0.04 0.06 0.08

Deflection (in)

Lo

ad (

lb)

Final Plot with Pre-load Offset

0100200300400500600700800900

0.00 0.02 0.04 0.06 0.08

Deflection (in)

Lo

ad (

lb)

Load Data

(pounds)

Raw Deflection

Data (inches)

Zero Adjusted Deflection (inches)

Data offset for Pre-load and Zero

Adjusted

0 0.010 0.000 pounds inches 102 0.031 0.021 0 0.000 201 0.036 0.026 99 0.005 300 0.041 0.031 198 0.010 399 0.047 0.037 297 0.016 500 0.053 0.043 398 0.022 602 0.058 0.048 500 0.027 701 0.062 0.052 599 0.031 800 0.068 0.058 698 0.037

19

H.3. Gathering data It is practically impossible to collect too much data. If you are unsure of how many data points to record collect more than you think can possibly be required. Some of the experiments may be destructive to the specimen and lost data is not recoverable. Work slowly and record everything. H.4. Teamwork Assign a specific task to each member of the group working on an experiment. Be consistent in method and do not change personnel during the experiment procedure. Any readings that are taken should be observed by at least two members of the group and cross-checked. The written record of the data should also be cross-checked. Much of your grade in this course is based on group success, if one member of your group does not do his part, the whole group will suffer. H.5. Safety Factors A structure fails when it can not resist additional load due to instability, material failure or component failure. If the purpose of an experiment is to determine the capacity of a structure, it will be tested to failure (Labs #5 and 6 are examples of this case). If the purpose of the experiment is to determine the behavior of a structure under varying load, the structure is not usually loaded to failure (Lab #1 is an example of this case). If the structure is not to be failed, a “safe” load must be determined to prevent failure or damage. In order to determine a safe load, the failure load is estimated and then reduced by a safety factor. The safety factor must be adequate to compensate for unknowns and normal variations in materials and components. Safety factors in Civil Engineering vary from about 1 to as much as 10 but are generally about 2.

20

I. EXPERIMENT #1

INTRODUCTION TO STRAIN MEASUREMENTS TENSION AND BENDING STRAINS AND STRESSES

Objectives: (1) To verify tension and bending stress models developed in a first course in Mechanics of

Materials. (Mc/I and P/A)

(2) To determine material modulus of elasticity (E) from test measurements

(3) To become familiar with and learn how strain gauges work

(4) To get hands-on experience with laboratory equipment and instruments

Preparation for the Laboratory (see pages 3 & 4 of this manual): (1) For the tension strap, calculate the maximum allowable load based on the material properties

and the specimen dimensions provided. (2) Provide predictions of the strain readings so that you can easily verify your data during the

experiment. You can do this based on a unit load or your maximum allowable load.

The two axial gauges on the two faces of the strap will read the maximum or principal tension strain. The two gauges will be wired into a single Wheatstone bridge such that the sum of the two gauge signals will be measured. The average strain for the two gauges can be calculated by dividing the measured value by two.

(3) For the beam, draw shear and moment diagrams and calculate the maximum allowable load

based on the material properties and specimen dimensions provided.

(4) Provide predictions of the strain readings at the top and bottom of the small beam for any given load so that you can easily verify your data during the experiment. You can do this based on a unit load.

(5) Prepare data sheets for the two tests. Two data columns will be needed for the load data

(target load and actual load). Two columns (displayed strain and zero adjusted or average strain) will be required for each strain gauge. Also, provide space on your data sheet to sketch the test specimens and indicate strain gauge orientation and dimensions.

(6) List dependent variable(s), independent variable(s) and controlled variable(s). (7) Submit a copy of your data sheets for the two experiments (provide photocopies with your

pre-lab submittal and keep the originals, you will use them during the lab). Note that all force measurements will be in units of pounds, strain measurements are

dimensionless but will be recorded as micro-inches per inch (µµµµ∈∈∈∈), all stresses are in pounds

21

per square inch (psi) and the modulus of elasticity also has units of pounds per square inch (psi).

Table H.1: Material strength for use in pre-lab calculations

Allowable Bending Stress +/-

16,000 psi

Allowable Shear Stress +/-

8,000 psi

Elastic Constants for Steel to Use in Predictions: E (Young's Modulus) = 28,000,000 to 30,000,000 psi v (Poisson's Ratio) = 0.27 to 0.30 In the Laboratory: (1) Students will work with their assigned group. (2) Each group will take a turn at each specimen. (3) Select proper load scale and adjust load zero reading if required. Using SLOW speed at

all times, allow a small load to be applied to get the feel of the machine - release load. (4) Tension test:

Check all dimensions. Prepare sketches to show location of strain gauges and test configuration.

Record the zero reading for the gauges. Increase the applied load in approximately five increments to the pre-determined maximum load while recording gauge data for each increment of load. Record the zero reading for the gauges.

(5) Bending test:

Check all dimensions and record them on your data sheet. Prepare sketches to show location of strain gauges and test configuration.

22

Record strain readings for the top and bottom gauges at zero load (no pre-load is required). Increase the applied load loads in four increments to the pre-determined maximum load while recording gauge data for each increment of load. Record the zero reading for the gauges.

(6) Compare you data to the predictions using the factors calculated for the pre-lab before

leaving the laboratory. You may want to ask the instructor to look over your data. Laboratory Report: Every group is required to submit a full, formal report for experiment #1. Please see Section E of this manual. Your analysis should include the following: Note: If possible, you should present your data and predictions as plots for comparison. All

plots for comparison should be to the same scale. Present numerical comparisons in tables.

(1) Using your data obtained from the tension test specimen: (a) Prepare a plot of stress vs. strain. (see section H.2) (b) Determine E (Young's Modulus) for this specimen from the slope of your plot of stress vs. strain for the axial gauge. (2) Using the data obtained from the bending test specimen (a) Prepare 3 plots of stress versus strain. One for the top gauge, one for the bottom

gauge and one showing the average of the absolute values of the strain measured at the top and bottom gauges.

(b) Determine the E (Young’s Modulus) for this specimen from the slope of the

average stress strain plot.

23

Figure H.1: Experiment 1 Tension Strap

Figure H.2: Experiment 1 Beam Geometry

24

J. Sample Data Sheet for the beam in Experiment #1

25

EXPERIMENT #2

STRESS AND STRAIN IN AN ALUMINUM "I" BEAM

Objectives:

(1) To verify the assumed linearity of the stress/strain profile of a beam subjected to bending. (My/I where y is the distance from the neutral surface)

This is the fundamental assumption upon which beam theory is based. The accuracy of this assumption influences the accuracy of beam bending stress calculations, beam shear stress calculations, beam deflection calculations and even column bucking. ("...each cross-section, originally plane, is assumed to remain plane and normal to the longitudinal fibers of the beams." Timoshenko & Young, 5th ed.)

(2) To determine Young’s modulus (E) for aluminum alloy from laboratory measurements

(3) To determine the value of Poisson's ratio for Aluminum by measuring longitudinal and transverse bending strains in a beam.

(4) To verify the equations for shear stresses and strains in beams by means of principal strain/stress analysis (strain gauge rosette) at the neutral surface.

Preparation for the Laboratory (see pages 3 & 4 of this manual):

(1) Determine the allowable load for the two beam sections. Show the shear and moment diagrams for the beam. Group 1 uses beam 1, Group 2 uses beam 2, etc.

Allowable bending stress +18,000 psi

Allowable shear stress + 9,000 psi

(2) Predict the following strains based on a unit load or your calculated allowable load.

Gauges 1, 4, 5, 8, 9, 10, 11 & The principal strains due to biaxial stress at the rosette

NOTES:

(a) Some of the calculations are repetitive and it is best to group the gauges into three categories: Bending Strain, Poisson Strain and Strain at the rosette due to beam shear.

(b) You cannot predict the strains measured by the individual gauges of the rosette because you do not know the rosette orientation. Therefore, calculate factors for the maximum and minimum strains possible at the rosette, which is located at the neutral axis. Bear in mind that we are investigating shear stresses which result in

26

biaxial normal stress and strain at this location.

(3) Predict the coordinates of the center of Mohr's circle of strain at the rosette. In the lab, your data can be partially verified by checking the location of the center of Mohr's circle, which can be found by taking the average of the strain readings from the two perpendicular gauges in the rosette.

(4) List dependent variable(s), independent variable(s) and controlled variable(s).

(5) Prepare a data sheet for use in the lab.

Elastic Constants for Aluminum:

For pre-lab calculations, E = 10,000,000 to 10,600,000 psi

For pre-lab calculations, Poisson's ratio in ~ 0.33.

The shear modulus (if needed) can be calculated from Poisson's ratio and the modulus of elasticity.

In The Laboratory: A universal testing machine will be used to apply the load to the simply supported aluminum beam. Three sets of strain gauges are bonded to the beam in the locations shown in the attached diagram. Set No. 1 consists of twelve of individual gauges distributed over the cross-section of the beam to measure strains due to pure bending at various values of "y". Set No. 2 consists of a 45° strain gauge rosette to measure strains due to beam shear. Set No. 3 consists of two strain gauges transverse to Set No. 1 to measure Poisson's effect.

(1) Using a suitable measuring tape, check all of the dimensions and record them on your data sheet.

(2) Prepare sketches to show the test configuration as well as the location and the orientations of the strain gauges including a detailed sketch of the rosette showing gauge numbers and angle to the beam axis.

(3) Take zero readings for all strain gauges.

(4) Load the beam in at least four equal increments to the allowable load determined in the pre-lab. Record all gauge readings for each increment of load.

(5) Obtain apparent strains for each gauge by subtracting the zero strain reading from all subsequent strain readings. Strains in tension are positive and those in compression are negative.

(6) Record the zero reading for the gauges. (7) Compare your data to the strains predicted in parts 2 and 3 of the pre-lab

27

Laboratory Report: If your group will be preparing a full report on this experiment, please refer to Section E of this manual and the steps below.

(1) In order to investigate objective #1, prepare a plot or plots showing:

(a) The measured strain profile determined from gauges 1 through 14 (excluding gauges 4 and 8). This is a plot of "y" (the distance from the neutral axis of the beam) on the vertical axis and strain on the horizontal axis. Use the data from the largest applied load that provided good data.

(b) Your predicted strain profile. This can be overlaid on the measured profile plot or a separate plot to the same scale.

(2) In order to investigate objective #2, determine Young’s modulus “E” from a stress vs. strain plot. Use the stain measurements from gauges 1,2,3 and 5,6,7. Calculate the stress at these gauges from the applied load and beam geometry.

(3) Determine Poisson's ratio in order to investigate objective #3. Poisson's ratio can be calculated from the strain readings from gauges 1 through 8. Taking averages of repetitive data can increase the accuracy of your result.

(4) For objective # 4, use the data from your maximum load, solve for the principal stresses and maximum shear at the neutral surface using your data from the strain gauge rosette. The analysis of your rosette data should result in both the magnitude and direction of the principal stress which can be compared to the shear stress calculation. Note that

IbVQ /21 ==−= τσσ

(5) Compare your calculated and pre-lab values of E, v, and maximum shear stress. Be sure to compare both the magnitude and the direction of the shear stresses at the rosette.

(6) Include a corrected copy of your pre-lab.

28

Table I.1: Section Properties:

Beam No. I Major inches^4

Area inches^2

1 or 3 6.06 2.25 2 or 4 6.79 2.79

Approximate Dimensions for Calculations: Use these approximate dimensions to calculate any section properties not given above. You must use these dimensions to calculate Q.

Table I.2: Approximate Section Dimensions to be used in Calculations:

Beam No. Web Thickness

inches Flange Thickness

inches Flange Width

inches Total Depth

Inches 1 or 3 0.190 0.300 2.66 4.00 2 or 4 0.326 0.300 2.80 4.00

Note that all gauges are aligned with the long axis of the beam with the exceptions of #4, #8 and the rosette

Figure I.1: Experiment 2 Beam Sections

29

FigureI.2: Experiment 2 Beam loading geometry

30

K. EXPERIMENT #3

DEFLECTION OF A BEAM

Objectives: (1) Verification of beam deflection theory for a beam. (2) Apply numerical integration, direct integration, finite difference approximation, moment-

area method, conjugate beam, etc. to solve a beam deflection problem Preparation for the Laboratory (see pages 3 & 4 of this manual): (1) Determine the allowable load given the following allowable stresses provided by the

instructor. Construct shear and moment diagrams and check both beam bending and beam shear.

(2) Predict the deflection of the beam at four locations along the beam (at 1/8 span, 1/4 span,

3/8 span, and midspan). Since the beam and its loading are symmetric, you may model half the beam by placing a fixed support at the center and considering the end reaction as a load in addition to the applied point load. Verify your work using the provided equations

(3) List dependent variable(s), independent variable(s) and controlled variable(s). (4) Prepare a data sheet for the laboratory. Deflections will be measured at the same four locations along the beam as required for your predictions. In The Laboratory: If there are two beams set up for testing, you are only required to collect data from one beam. (1) Check location and operation of the deflection gauges and take your "zero" readings. (2) Pre-load the beam to approximately 10% of the maximum load. See Section H.1. of this

manual to learn more about pre-loads. (3) Apply your calculated allowable load in at least five increments while taking deflection

readings for each increment. (4) Record the final zero reading for the gauges. (5) Compare your data to the predictions from part 3 of the pre-lab.

31

Laboratory Report: Please refer to Section E of this manual and the steps below. Note: If possible, you should present your data and predictions as plots for comparison. All

plots for comparison should be to the same scale. Present numerical comparisons in tables.

(1) Prepare four plots comparing predicted and measured deflections at each deflection gauge

(load vs. deflection). If the deflections for your first increment of load (pre-load) are not linear, you may need to consider the first load increment data as your effective “zero” for data analysis. See Section H.1. of this manual

(2) Prepare a plot comparing the predicted and measured deflected shape of the beam at the

maximum applied load. This is a plot of X (position along the beam) vs. Y (deflection). (3) Include a copy of a corrected pre-lab.

General differential equation for beam deflection:

EI

Mk

dx

d

x

d

−===

Θ=

ρδ

δδ

12

2

Note that the total applied load in this figure is 2P

32

L. EXPERIMENT #4

DESIGN AND RUN AN EXPERIEMNT

Preparation for the Laboratory: 1. Define the objectives of the experiment. Objectives need to be specific and directly or

indirectly measurable. 2. Research any relevant theory that might predict the results of the experiment. 3. Select the controlled variable(s). 4. Select the dependent and independent variable(s) to be measured. 5. Determine the proper ranges of the dependent and independent variables. 6. Verify that the equipment and instrumentation is appropriate for measuring the dependent

and independent variables. 7. Determine an appropriate number of data points needed for each type of measurement. 8. Prepare data sheets and anything else you might need to perform the experiment. Presentation of Results: See section E of this manual.

33

M. EXPERIMENT # 5

ELASTIC AND INELASTIC BUCKLING OF COLUMNS

Objective: (1) Prediction and observation of column failure modes (elastic or inelastic). (2) Prediction and measurement of the buckling loads of columns of varying lengths and

materials. Preparation for the Laboratory (see pages 3 & 4 of this manual): (1) Calculate your predicted critical loads for the pin-supported columns to be tested in the laboratory. The test specimens are 6", 9", 12", 15", 18", and 21" in length for both steel and aluminum. All of the columns are 5/8 inch diameter round bars. (2) Prepare a data sheet for the laboratory indicating the following information in a table:

length, predicted critical load, measured critical load, and failure mode (elastic or inelastic).

(3) List dependent variable(s), independent variable(s) and controlled variable(s). (4) Submit copies of your calculations and data sheet to your laboratory instructor at the beginning of the laboratory period. In The Laboratory: (1) Check the lengths and diameters of each specimen. (2) Select the appropriate load range on the testing machine. (3) Before placing a specimen in the machine, check for zero load reading and adjust if

necessary. (4) Carefully bring the load head into contact with the specimen. (5) For each column, apply load very slowly to buckling failure. Record the critical load and failure mode. Laboratory Report: Please refer to Section E of this manual and the steps below.

34

Note: If possible, you should present your data and predictions as plots for comparison. All plots for comparison should be to the same scale. Present numerical comparisons in tables.

1) Acquire the column test data from all of the current semester lab sections and work with

the average critical load for each column type. Data may be made available on the course web-site http://www.engr.sjsu.edu/dmerrick/113/.

2) Plot the average critical load data for each material on a chart of critical load verses

length. The column length scale of your plot should extend from zero length to 24 inches. 3) Using Euler's equation and the average critical load data for the 21 inch columns, solve

for the modulus of elasticity for each material. 4) Use Johnson's equation and the average critical load for the 6 in columns to solve for the

yield strength of each material. Iterate as required. Note that Lc is a function of both E and σy.

5) Recalculate the theoretical critical loads for each column length and material using the

method described in the pre-lab basing your calculations on the yield strength determined in #4 and the modulus of elasticity determined in #3.

6) Graphically show that Euler’s equation diverges from your data as the column length

decreases. Extend the lines for the expected Euler values to near zero length and 24 inches.

7) Graphically show that Johnson’s equation diverges from your data as the column length

increases. Extend the lines for the expected Johnson values to zero length and 24 inches. 8) Provide a critical load verses length plot for each material showing:

a) The average critical load data (data points without line) b) The theoretical critical load from #5. Extend the lines for the expected values to

zero length and 24 inches. 9) Attach a corrected copy of your corrected pre-lab.

Laboratory summary (No PI assigned)

If your group does not have a principal investigator assigned to this experiment, do the following:

(1) Acquire the data from all of the current semester sections. (2) Scale your data for engineering units if required. (3) Present all of your data in a table.

35

(4) Calculate the expected values using your corrected pre-lab. (5) In a table, compare your experimental values and theoretical values for all of the tests. This can probably be done with three columns; laboratory data, expected values and percent error. (6) Provide plots of your data and predictions. Extend the lines for the expected values to zero length and 24 inches. (7) Include a corrected copy of the pre-lab. (8) Include the raw data sheet used in the lab.

Background: In this experiment, we will consider both elastic columns (also referred to as Euler or long columns) and intermediate columns. All of the columns will have pinned ends and be loaded concentrically with a force P that is parallel to the longitudinal axis of the column. This system is stable when the load P is less than the critical load (Pcr) and unstable when the load is greater than the critical load. The critical load for an elastic column can be determined as follows:

PEI

Lcre

= π2

2 Euler’s Equation for long, elastic columns

Where: EI = the flexural rigidity for bending in the xy plane Le = the effective length of the column

(Le = L for a pinned end column) The prediction of critical load for inelastic columns is more complex than it is for elastic columns. Various methods such as the Secant Formula and Engesser’s Method have been developed and can provide reliable results. A discussion of Engesser’s method (also known as the Tangent Modulus Method) can be found in an advanced Strength of Materials textbook. Engesser’s method uses a tangent modulus theory and works well if detailed material information is available. Since Engesser’s method considers a varying modulus of elasticity as the material goes inelastic; it is useful for materials that do not have a clearly defined yield point such as aluminum. The Secant Method can also be found in many textbooks. The Secant method assumes an eccentric applied load to calculate a combined bending and axial stress, which is then compared to the proportional limit of the material. The Secant method requires a known or assumed eccentricity (crookedness of column) The Secant Method requires a trial-and-error solution.

36

Several approximate methods have been developed that are usually more appropriate for general engineering design and for this laboratory experiment. These approximate methods rely on an empirical equation or equations to predict the behavior of short and intermediate columns. The equation that we will use for inelastic columns in our predictions is referred to as Johnson’s Equation and is used in structural steel design methods. In our method, all columns are divided into two categories, elastic and inelastic. The division between these two categories is the column length for which the solution to Euler’s equation provides a critical stress equal to 1/2 the yield stress. This is dividing point is reffered to as a slenderness ratio equal to cc or as a column length of Lc. The slenderness ratio, r, is a dimensionless measure of the slenderness of a particular column. cc is a dimensionless material property. For a given column cross-section, the length that corresponds to cc will be referred to as Lc.

A

Ir

L

AI

E

Ec

rcE

rA

EIL

AL

EI

c

y

yc

cyy

c

c

y

==

===

==

=

===

==

Gyration of Radius

stress yield thehalf equalsEquation sEuler'at which Length Column

Area Inertia ofMoment

Elasticity of Modulus Stress Yield

:Where

/2

property material essdimensionl a is c that Note

22

:Lfor solving Or,

A

P

2

2

c

22

c

2

2cr

σ

σπ

σπ

σπ

πσ

Material Properties: Lookup the required properties at MatWeb http://www.matweb.com/

Steel Aluminum Type 1018 7075-T6 Diameter 5/8 5/8

The following are equations will provide adequate predictions of column critical loads for this laboratory experiment:

37

Your Predictions will result in a plot similar to the following:

Figure L.1: Example of Critical Load Prediction for Lab #5

−=

<

=

≥

2

2

2

2

11

:Equation sJohnson' use wherecolumnsFor

:Equation sEuler' use wherecolumnsFor

c

eycr

ce

e

cr

ce

LLAP

LL

L

EIP

LL

σ

π

38

N. EXPERIMENT #6

THE PROPERTIES OF WOOD

Objectives: 1) Gain an understanding of the nature of a non-isotropic engineering material.

a) Investigate the bending and/or shear strength of various wood beams. b) Investigate the behavior of wood subjected to compressive stresses in various

orientations. 2) Learn about other factors that effect the engineering properties of wood

a) Species b) Moisture c) Defects d) Load duration

3) Learn about statistical representation of engineering properties and safety factors for a variable material. (see section H.4 on page 17)

Preparation for the Laboratory (see pages 3 & 4 of this manual): 1) If possible, inspect the test specimens and identify the species of wood being investigated. If

the specimens are not available, assume dry, coast Douglas Fir. 2) For the two wood beams:

a) Check bending and shear to find the maximum allowable load (Table M.1) for beam #1. b) Check bending and shear to find the maximum allowable load (Table M.1) for beam #2. c) Check bending and shear to find the expected ultimate strength (Table M.2) for beam #1. d) Check bending and shear to find the expected ultimate strength (Table M.2) for beam #2.

3) For the two compression blocks: a) Submit calculations for the maximum allowable (Table M.1) load for each compression

block b) Submit calculations for the expected ultimate strength (Table M.2) load for each

compression block 4) Calculate the expected safety factor for each specimen. The beams will actually have two

safety factors, one for shear and one for bending. 5) Summarize your results from part 2, 3 and 4 in a table. 6) Prepare a data sheet. There must be space on your data sheet to sketch the failure of the

specimens and to write a description of the test and failure. In the Laboratory: 1) Slowly load one block in compression perpendicular to the grain until the load rate decreases significantly. Initially the piece will be fairly stiff but the stiffness will abruptly decrease when the wood cells begin to collapse. Record the magnitude of the load when the cells begin to crush. Sketch and note any observed phenomena.

39

2) Slowly load the other compression block in compression parallel to the grain until the maximum load is achieved. Record the magnitude of the maximum load. Sketch and note any observed phenomena. 3) Slowly load the first beam until the ultimate load is achieved. Record the magnitude of the maximum load. Sketch and note any observed phenomena. 4) Slowly load the second beam until the ultimate load is achieved. Record the magnitude of the maximum load. Sketch and note any observed phenomena. Laboratory Report: Note: Reports may be submitted without: table of contents, introduction, objectives, procedures

or conclusion. 1) Report the failure load and failure mode for each specimen. 2) Carefully describe your observations of the loading, failure and the failed specimen. Describe any remarkable sights, sounds, smells, etc. 3) Use the values in Table M.1 to calculate the allowable loads.

4) Determine the factor of safety for the wood specimens tested. Provide a table showing allowable load, ultimate load and factor of safety for each specimen. For the beams, you can only provide a safety factor for the actual failure mode

(Factor of Safety) = (Ultimate load from test) / (Calculated allowable load) 5) Describe any physical factors that may have had an affect on the performance of the specimens. 6) Attach a corrected copy of your pre-lab calculations. Material Information: The specimens tested in this laboratory are western softwoods. The following are typical design allowable strength values for western softwoods that include factors of safety. These values have been adjusted for size (2x4) and for assumed load duration of one minute (1.75):

Table M.1 Allowable Stresses (psi)

Compression Species Bending Shear

Parallel Perpendicular Western Hemlock 3680 260 3940 400

Sugar Pine 3280 120 3680 425 Douglas Fir 3900 170 3400 625

40

The strength of wood for most loading conditions is time dependant. In the chart below it can be seen that wood can resist about twice the stress in impact loading as it can in permanent loading.

Figure M.1

41

The following data was taken from Chapter 4 of: Forest Products Laboratory. 1999. Wood handbook--Wood as an engineering material. Gen. Tech. Rep. FPL-GTR-113. Madison, WI: U.S. Department of Agriculture, Forest Service, Forest Products Laboratory. 463 p. Download the PDF version of this publication at: http://www.fpl.fs.fed.us/ for further information.

Table M.2 (Table 4–3b in FPL Wood Handbook)

Example Strength properties for Douglas Fir Go to the above website for other species of wood

S t a t i c b e n d i n g Douglas-fird Moisture

content Specific gravityb

Modulus of rupture (lbf/in2)

Modulus of elasticityc

(106 lbf/in2)

Work to maximum load

(in- lbf/in2)

Impact bending

(in)

Com-pression

parallel to grain

(lbf/in2)

Com-pression perpen-

dicular to grain

(lbf/in2)

Shear parallel to grain (lbf/in2)

Tension perpen-

dicular to grain

(lbf/in2)

Side hard- ness (lbf)

Coast Green 0.45 7,700 1.56 7.6 26 3,780 380 900 300 500

12% 0.48 12,400 1.95 9.9 31 7,230 800 1,130 340 710

Interior West Green 0.46 7,700 1.51 7.2 26 3,870 420 940 290 510

12% 0.50 12,600 1.83 10.6 32 7,430 760 1,290 350 660

Interior North Green 0.45 7,400 1.41 8.1 22 3,470 360 950 340 420

12% 0.48 13,100 1.79 10.5 26 6,900 770 1,400 390 600

Interior South Green 0.43 9,800 1.16 8.0 15 3,110 340 950 250 360

12% 0.46 11,900 1.49 9.0 20 6,230 740 1,510 330 510

a) Results of tests on small clear specimens in the green and air-dried conditions. Definition of properties: impact bending is height of drop that causes complete failure, using 0.71-kg (50-lb) hammer; compression parallel to grain is also called maximum crushing strength; compression perpendicular to grain is fiber stress at proportional limit; shear is maximum shearing strength; tension is maximum tensile strength; and side hardness is hardness measured when load is perpendicular to grain. b) Specific gravity is based on weight when ovendry and volume when green or at 12% moisture content. c) Modulus of elasticity measured from a simply supported, center-loaded beam, on a span depth ratio of 14/1. To correct for shear deflection, the modulus can be increased by 10%. d) Coast Douglas-fir is defined as Douglas-fir growing in Oregon and Washington State west of the Cascade Mountains summit. Interior West includes California and all counties in Oregon and Washington east of, but adjacent to, the Cascade summit; Interior North, the remainder of Oregon and Washington plus Idaho, Montana, and Wyoming; and Interior South, Utah, Colorado, Arizona, and New Mexico.

Average coefficients of variation for some mechanical properties of clear wood:

Property Coefficient of variationa (%)

Static bending Modulus of rupture 16 Modulus of elasticity 22 Work to maximum load 34

Impact bending 25 Compression parallel to grain 18 Compression perpendicular to grain 28 Shear parallel to grain, maximum shearing strength 14 Tension parallel to grain 25 Side hardness 20 Toughness 34 Specific gravity 10

a) Values based on results of tests of green wood from approximately 50 species. Values for wood adjusted to 12% moisture content may be assumed to be approximately of the same magnitude.

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Glossary of Common Properties of Wood: Mechanical properties most commonly measured and represented as “strength properties” for design include modulus of rupture in bending, maximum stress in compression parallel to grain, compressive stress perpendicular to grain, and shear strength parallel to grain. Additional measurements are often Modulus of rupture—Reflects the maximum load-carrying capacity of a member in bending and is proportional to maximum moment borne by the specimen. Modulus of rupture is an accepted criterion of strength, although it is not a true stress because the formula by which it is computed is valid only to the elastic limit. The modulus of rupture is calculated by dividing the ultimate moment by the section modulus. Work to maximum load in bending—Ability to absorb shock with some permanent deformation and more or less injury to a specimen. Work to maximum load is a measure of the combined strength and toughness of wood under bending stresses. Compressive strength parallel to grain—Maximum stress sustained by a compression parallel-to-grain specimen having a ratio of length to least dimension of less than 11. Compressive stress perpendicular to grain—Reported as stress at proportional limit. There is no clearly defined ultimate stress for this property. The published values are based on a specimen deformation of 0.04 inch. Shear strength parallel to grain—Ability to resist internal slipping of one part upon another along the grain. Values presented are average strength in radial and tangential shear planes. Impact bending—In the impact bending test, a hammer of given weight is dropped upon a beam from successively increased heights until rupture occurs or the beam deflects 152 mm (6 in.) or more. The height of the maximum drop, or the drop that causes failure, is a comparative value that represents the ability of wood to absorb shocks that cause stresses beyond the proportional limit. Tensile strength perpendicular to grain—Resistance of wood to forces acting across the grain that tend to split a member. Values presented are the average of radial and tangential observations. Hardness—Generally defined as resistance to indentation using a modified Janka hardness test, measured by the load required to embed a 11.28-mm (0.444-in.) ball to one-half its diameter. Values presented are the average of radial and tangential penetrations. Tensile strength parallel to grain—Maximum tensile stress sustained in direction parallel to grain. Relatively few data are available on the tensile strength of various species of clear wood parallel to grain. The average tensile strength values available are for a limited number of

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specimens of a few species. In the absence of sufficient tension test data, modulus of rupture values are sometimes substituted for tensile strength of small, clear, straight-grained pieces of wood. The modulus of rupture is considered to be a low or conservative estimate of tensile strength for clear specimens (this is not true for lumber). The average tensile strength for small, clear, green, straight-grained specimens of Douglas-Fir Interior North is 15,600 psi. results are about 13% higher for specimens tested at 12% moisture content.

Beam #1 Beam #2

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For additional information, see the glossary on previous pages for “Modulus of Rupture”