bending test on cantilever beam experiment

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Deflection of Cantilever Beam - Lab (02)

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AOE3054 - Experiment 2 - Static Deflection of a Beam

Experiment 2 - static response of a beam.

1. Introduction

Characterizing how an engineering structure will deform under a steady load is key to the design and development of a broad range of engineering hardware. Theoretical methods for predicting deformation, such as you have already met in your coursework, obviously have an important role to play. However, the structural elements of real vehicles are often complex - consider the wing of a large transport aircraft, the rudder assembly of an oil tanker or the robotic arm of a space probe. Theoretical methods, by themselves, are not completely reliable for such "real-world" structures. Predictions for these structures inevitably involve simplifying assumptions about the structure and material, and will be based on incomplete or uncertain information about the environment (i.e. the boundary conditions) in which the structure must operate. Experimental testing of structures is thus important. Testing is used not just to prove that the final design works as intended, but also to examine whether the theoretical models or boundary information used to predict the behavior of the structure are correct and complete. Testing is subject to error as well. Thus a good experiment characterizes not just the properties of the structure in question, but also investigates and quantifies the errors in the measurements themselves. Only then can one be sure enough to make decisions, based on the test, that influence the form of the vehicle, the success of the mission, or the lives and investments that they may involve.

The objectives of Experiment 2 are: 1. to introduce you to some basic techniques used to measure the static deformation of a structure, 2. to provide you with a set up where you can use those techniques to examine validity and consistency of a simple theoretical approach (beam theory) when applied to a simple "real-world" structure, 3. to provide you with a set up where you can investigate the validity and accuracy of the measurement techniques themselves.

You will find that this manual does not prescribe a specific set of measurements you should make or conditions you should test. These will be your choices to make as a group, based upon your understanding of the theory, the set up and measurement techniques, and the specific goals provided for you. It is therefore critical that you prepare for this lab by making sure you understand and can use the theory, are familiar with the descriptions of the measurement techniques and experimental set up as described below, and are organized as a group.

The apparatus to be used in this experiment consists of a simple beam (this will be your test structure), a frame in which that beam can be mounted and its deformation measured, and a loading fixture that allows weights to be attached to the beam.

The Beam The beam, shown in Figure 1 , is constructed from aluminum alloy and has a rectangular cross-section that is nominally 1/4 by 1 1/2-inches. The corners at one end of the beam are beveled. You should measure the actual cross-sectional dimensions with the dial caliper provided, and use those actual dimensions in any calculations you perform (note that the two beams may have slightly different dimensions).

Attached to the beam are two electrical resistance strain gages, one on each side. As you learned in the online lecture, the stretching of an electrical conductor increases its resistance. The gages are designed to take advantage of this effect to measure the strain. They consist of a single conducting strip deposited on a film that is then glued to the beam. At first sight the conducting strip appears to have the form of a simple rectangular region. On closer inspection, perhaps with a magnifying glass, you will see that in fact the conducting region consists of a thin conducting path that zigzags back and forth along the long axis of the beam. Confirm in your own mind that this will make the gage primarily sensitive to normal strains along that long axis. Be careful of the strain gages and their electrical connections, they are delicate and may be important for your test.

Loading Fixture and Weights The loading fixture and one of the weights are shown in Figure 2 . The two Allen bolts in the loading fixture can be loosened so that it can be slid onto the beam. Do not slide the loading fixture over the strain gage. Completely remove the fixture (by undoing the Allen bolts using the key provided) if you need to move it from one side of the strain gage to the other. The weights are machined from aluminum bronze rod stock of 5-inch nominal diameter. They are nominally 2, 3, 5, 10, and 15 pounds, and each has a hook threaded into a tapped hole on one side for hanging them from the eye screw of the load fixture. A digital weighing machine is provided for you to measure the exact values of these weights, the loading fixture and any other items you feel are relevant.

The Frame The loading frame, built according to a design developed by Durelli et al. (1965), is shown in Figure 3 with the beam, loading fixture and weight assembled in one possible configuration.

The frame includes two vertical beam supports of the same height. These can each be placed at any one of 7 positions along the bottom of the frame. The positions are at exactly 2.5 inch intervals. The beam can also be fixed using the slot cut into the left hand side of the frame, and the clamp, see Figure 3 . The slot holds the beam at the same height as the vertical supports. The clamp is tightened using Allen bolts on the outside of the frame, and a threaded knob that passes through the top of the frame. With the with one of the beam supports in its left-most position, the clamp and slot can be used in combination to approximate a cantilever support. Note that if used, the both the bolts and threaded knob must be tightened firmly to produce a repeatable support. A second threaded knob and other threaded holes for the these knobs are provided along the top of the frame, so that it is possible to pin the beam to vertical supports at other locations along the frame, if necessary.

The top of the frame contains a series of machined holes, spaced at two inch intervals along the frame. These holes are designed to accept dial indicator gages to measure beam deflection. Figure 3 shows an indicator mounted in one such hole. Note that, for this particular beam configuration, this indicator wouldn't tell you very much, as it is located at a position where the beam deflection is ensured to be zero by the vertical support.

Note that this device is very flexible and allows for many beam configurations. It is thus ideal for testing simple theoretical predictions or measuring material properties and assessing instrumentation limits and errors through beam theory. For example, Figure 4 shows an arrangement producing a cantilevered beam, Figure 5 shows a simply supported beam, and  Figure 6 shows a simply supported beam with overhang. Note that Figure 3 shows an indeterminate beam configuration.

You have a variety of instrumentation available with each set of apparatus. Minor items include a ruler which you can use to accurately position the beam and the loads, a dial caliper to check beam dimensions and a digital camera which you can use to record your various set ups, and photograph items that you want pictured in your logbook. Your lab TA will be able to explain operation of the digital camera to you. The remaining items of instrumentation are those used to measure the beam deformation, namely the dial indicators and strain gage system.

Dial Indicators A number of basic mechanical dial indicators (see Figure 3 ) and two electronic indicators (not pictured) are available to measure beam displacements. Either type may be inserted through the holes on the top of the loading frame so that the end of the plunger rests on the beam or, if it at the same location, the top of the loading fixture.

The mechanical dial indicators are manufactured by the Chicago Dial Indicator Company (Des Plaines Il), model number 2-C100 1000.  Each gage has a range of one inch with each graduation on the large dial representing one thousandth of an inch. One complete revolution of the pointer on the large dial represents a displacement of 0.100 inches. The number of revolutions are counted on the small dial. It is possible to set the large dial to read zero, for example when the beam is unloaded, by loosening the thumb screw on the top right of the dial and rotating the outer ring of the dial face. In principle it is possible to read thes indicators to four significant figures by visually estimating the location of the needle between the smallest divisions on the scale.

The electronic indicators, Mitutoyu Model 575-123, have a range of one inch and read in increments of 0.0005 inches, and are generally easier to use.  They have inch/metric conversion, zero setting to any position, +/- counting direction, and position memory (the dimension indicated always reflects the movement of the stem from the last set zero position, also known as absolute (ABS) positioning). Their accuracy is ±0.0008 inches. Note that this is greater than their resolution. When inserting the electronic indicators into the holes in the loading frame, thread the indicator spindle through one or two of the washers provided at the workbench. This will raise the body of the indicator a few mm and enable it to work when the beam is in the unloaded position, or higher.

Note that the above accuracies are under ideal conditions and you may find that other factors tend to overwhelm the level of accuracy, particularly for the mechanical indicators. One of these factors is that the spindle of the dial indicator may stick. You can usually avoid this by sliding the spindle in and out a few times to loosen the gears before the indicator is installed. A second factor that may influence the accuracy of your measurements is the force exerted by the spindle on the beam, and the variation of that force with extension of the spindle. You should be aware of this factor as you proceed with your test, so that you are prepared to quantitatively assess its influence (and perhaps correct for that influence) during your tests.

You can also use the dial indicators to check that the load is supplied symmetrically to the beam. This can be done by inserting two dial indicators into the loading frame such that their spindles contact the beam at two widthwise points on the top surface that have the same spanwise location ( Figure 8 ). By monitoring the readings of these two dial indicators to make sure they remain the same, you can establish that the beam does not twist under load, and thus that no torque is applied to the beam as it is loaded.

Strain Gage System

The electrical resistance strain gages bonded to each beam are manufactured by Micro-Measurements Division, Measurements Group, Inc., Raleigh NC. As described above they consist of a single conducting strip deposited on a film that zigzags back and forth along the long axis of the beam. The gages are therefore sensitive to strain in that direction. Technically, the gage designation number is EA-13-125BZ-350.  The "EA" of this designation indicates these are polyimide-backed constantan material foil gages, the "13" designates the self temperature compensation of the gage with respect to aluminum, the "125" designates that the active gage length is 0.125 inches, "BZ" denotes the grid and tab geometry (narrow, high resistance pattern with compact geometry), and the "350" means the resistance of the gage in ohms. The engineering data sheet for these gages is included with the experimental set-up. To set up the strain gages you will need the gage factor from this data sheet, of 2.145.

Since the strain gage is just a single conductor, there are only two connections to make, one at each end. You will notice there are three wires coming from each gage. The red wire connects to one side of the gage. The black and white wires both connect to the other side. The strain gage is operated using the Wheatstone bridge circuit that you were introduced to in class, in a quarter bridge configuration. This circuit, shown in Figure 9 is arranged in a diamond shape. Three sides of the diamond consist the simple resistors,  R2, R3 and R4. The strain gage itself is connected to the fourth side of the diamond. Voltage is applied to the top and bottom corners of the diamond, P+ and P-, and the output signal, which will be a voltage proportional to the resistance of the gage and thus the strain it experiences, is measured across the left and right corners of the diamond, S+ and S-. Note that the points S- and D350 on the diagram are actually connected together through the leads to the strain gage.

The Wheatstone bridge circuits for our strain gages are provided by a Measurements Group model SB10 Switch and Balance Unit,  Figure 10 . Strain gages are connected into its bridge circuits using the array of electrical terminals on the right hand side of the unit. Note that there are 10 sets of terminals, each labeled P+, P-, S+, S-, and D corresponding to the 5 points in the bridge circuit in Figure 9 . There are 10 rows of terminals because this unit actually contains 10 bridge circuits that could be used to operate 10 strain gages simultaneously. We, of course, will only need to use rows 1 and 2 for the two strain gages on the beam. Consistent with Figure 9 one side of the first gage (the red wire) is connected to the P+ terminal in row 1. The other side of the gage (black and white wires) are connected to the S- and D terminals of row 1. The second gage is connected in the same way to row 2.

To supply the voltage to the P+ and P- terminals of the bridge circuits, and to read the strain from the S+ and S- terminals we have a Measurements Group model P-3500 Strain Indicator, also shown in Figure 10 . Obviously this device has to be connected to the switch and balance unit containing the bridges. The connections are straightforward. Use the banana plug cables provided to connect P+ on the P3500 to P+ on the SB10, P- to P-, S+ to S+, S- to S- and D350 to D. You should also connect the two ground (GND) sockets. The black knob on the left hand side of the SB10 controls which of the bridge circuits, and thus strain gages the strain indicator is connected to.

For the switch and balance unit to accurately output the strain the gage factor (of 2.145) must be entered. This is done by pressing the 'Gage factor' button (3rd from left) and adjusting the knobs immediately above it so that the display reads '2.145'. It is likely that the gage factor will be set equal, or close, to this value already. To use the strain gage system to make a measurement, first select the gage you are interested in (black knob SB 10) then press the 'Run' button (4 from left on the P3500). The display shows the strain in microstrain (strain times 10 -6 ). That is, a reading of 1234 means a strain = change in length / length = 0.001234. Your measurement will be the difference between this displayed value when the load is applied, and when it is not. Note that for convenience you can usually set the offset of the display to zero when there is no load by turning the 'balance' knob on the SB-10 that corresponds to the strain gage you are looking at.

If you need further help setting up or understanding this system instruction booklets are provided with both the SB10 and P3500 units.

where b is the width ( x -direction) and h (y-direction) is the height.

where y = h /2 at the bottom of the beam and y = - h /2 at the top of the beam.

D. Example displacement and moment calculation Problem: Determine expressions for the midspan displacement and axial strain for the three-point bend test pictured in Figure 5 . Take the distance between the vertical supports as L and assume that the load is applied at midspan (midway between the supports).

Consequently, the midspan displacement is

A. Getting familiar with the apparatus and instrumentation. The following procedure is designed to help you get a feel for the apparatus and instrumentation and, most importantly, its limitations. This is a group activity, but one in which it is important that you get individual hands on experience and that everybody understand and appreciate the items and issues raised below.  Discussion is key - each student must know how to use the apparatus, what the problems are, what measurements (in addition to those made by the dial indicators and strain gages) will be critical. Ensure that all the important observations (measurement/error lists, error estimates etc.), both qualitative and quantitative are properly recorded, along with any explanations, in the electronic lab book.

• Try out the weighing machine. Measure some weights. Measure the weight of the loading fixture (don't put more than 17 pounds on the scales).
• Try assembling the beam, frame and loading fixture into one of the configurations pictured in Figures 3 through 6 . Be sure to position the beam so that the strain gages are at a location where they will have something to measure.
• Insert two dial indicators at the location of the loading fixture (plungers should rest on top of the loading fixture, away from the Allen head bolts) so as to monitor both the deflection of the beam here, and to warn of any twisting. Insert a third dial indicator elsewhere (touching the beam) to monitor deflection here. Try setting the ring of the indicators to indicate zero deflection in this no-load case.
• Add a load to the beam and look closely at the configuration you have assembled. Make sure you and your lab partners understand how you could do a beam theory calculation for this configuration to predict the deflection produced at each of the dial indicator locations and the strain at each of the two gages.  Compile a list of all the measurements you would need to make accurately before you could get an actual number from a beam theory calculation of your configuration (e.g. exact position of the load, supports, dimensions of the beam, locations of gages etc.).
• Compile a list of any factors that you think might introduce error (items that are not included in the theory but have an effect on the experiment) and think about how you might minimize those factors. You will probably want to add to the list as you try the items below.
• Try deflecting/twisting the beam with your hand to get an idea of how well the indicators work. Do the dial indicators show a deflection, a twist? Do you understand how to read the dials? How big is the deflection compared to the resolution of the dial? Do the dial indicators stick? (If so try moving the spindle in and out a few times to loosen it up.)
• Try tapping the beam at different points. Do the dial indicators return reliably to their starting positions? If not, how big is the change? How does the size of that change compare to the deflections you see under (hand) load? Try adding and removing a weight - do you see any change in the start position of the indicators? Note that repeatability in the indicators can be improved somewhat by making sure that the bolts on the loading fixture and clamp (if you are using it) and the threaded knob(s) (if you are using these) are tight. However, some inconsistency in the dial indicators is inevitable, try to quantify this error. The influence of this inconsistency on your results can be minimized, for example, by basing them on averages of measurements made with different loads, and in different configurations.
• Try removing the dial indicators one by one. When you remove an indicator do the remaining indicators show any significant change in deflection (indicating that the force exerted by the plunger may be significant)? If so, what is the magnitude of the change? Think about whether it will influence your results. Think about how you could estimate the spindle force using the beam apparatus.
• Wire up the two strain gages. Dial in the gage factors, balance the bridges (see the strain gage section for help here) with zero load. Add a load to the beam (with your hand if you like). Do the changes in the indicated strain make sense given what you understand about the stretching and compression of the fibers of the beam? Do their magnitudes make sense? What are the units? Remove the load. Do the indicators return to their starting readings exactly? If not, how unrepeatable are they? Try to quantify this error. Leave the strain gages and beam assembly alone for a couple of minutes. Do the readings drift? If so, estimate the rate of drift? How can you minimize the influence of this drift on your results?

Goal 1. Design, conduct and document a sequence of tests to measure, as reliably as possible, the Young's modulus of the beam material, and to to put a number on the reliability (i.e. the likely error, a.k.a. the uncertainty). Suggestions. Such a test will obviously require some application of beam theory.  The reliability of your result, and your ability to estimate that reliability, will be enhanced if you make multiple determinations of Young's modulus. For example, it is easy to make multiple measurements for a sequence of different loads. Even better are multiple measurements obtained in situations where the errors are unlikely to be the same, i.e. using independent measurement techniques and/or using different beam configurations. Keep careful documentation of what you do, why you do it, set up, characteristics, expected results, unexpected results, analysis, photos and plots in the electronic lab book as you proceed. Analysis should include uncertainty estimates for all results.

Goal 2. Design, conduct and document a sequence of tests to examine the validity (and extent of validity) of Maxwell's reciprocal theorem. Again, you should aim to gather whatever information you need to assess how good your result is. Suggestions. Do not feel constrained to a single 'reasonable' test. How unreasonable can the test be? How far apart can your two locations be? what may be between them? is there a limit to the deflection for the theorem to be valid? can you use the strain gages (you can at least use them to get additional data points on Young's modulus, for goal 1)? what about the sources of error you identified above? Keep careful documentation of what you do, why you do it, set up characteristics, expected results, unexpected results, analysis, photos and plots in the electronic lab book as you proceed. Analysis should include uncertainty estimates for all results.

Note that your grade does not depend upon how close your results agree with beam theory, Maxwell's theorem or any other pre-conception of what the answers should be . Instead it depends upon how open mindedly and objectively you assess your data, its accuracy, and what it shows, or appears to show when combined with the theory.

The group should leave few minutes at the end of the lab period for discussion and to check that everybody has everything they need. As a group go through the exit checklist .

6. Recommended Report Format

Before starting your report read carefully all the requirements in Appendix 1.

Title page As detailed in Appendix 1 .

Introduction State logical objectives that best fit how your particular investigation turned out and what you actually discovered and learned in this experiment (no points for recycling the lab manual objectives). Then summarize what was done to achieve them. Follow this with a background to the technical area of the test and/or the techniques and/or the theory (and its mathematical results). This material can be drawn from the manual (no copying), the online class or even better, other sources you have tracked down yourself.  Finish with a summary of the layout of the rest of the report.

Apparatus and Instrumentation Begin this section with a description of the beam, loading frame and related hardware, and connect these items to your goals.  For example, you might write "A frame of the type designed by Durelli et al. (1965) was used to load the beam in several configurations for the purpose of determining its Young's modulus." Labeled dimensioned diagrams of the beam(s) and strain gages are probably critical. Labeled photographs may suffice for other items.  Don't forget to objectively describe any deficiencies, irregularities, imperfections in the apparatus relevant to your goals, and given any primary uncertainties. Then describe the instrumentation used to monitor the beam, how it was used, and its accuracy and limitations. For example "The dial indicators were inserted through the top of the frame to measure beam displacement. These indicators allowed measurements to a resolution of... but their actual accuracy was reduced to about ... inches as a result of ... The indicator spindles exerted a force on the beam of about ??? which complicated measurements somewhat (see below), particularly when... Dial indicators were also used to check for twisting...". With your description of the strain gages include a circuit diagram, and properly reference the manufacturers and model numbers. If there were any uncontrolled variables in your experiment this is a good place to mention them and assess there likely impact, for example -  "For the three point bend test the location of the load was set visually at centerspan but, due to an oversight, the actual location was not measured or checked. The maximum error in the load location is estimated at 0.25 inches and the implications for this error for the results of this test are discussed with the results below" - or - "While two actual beams were used during the tests, the data were combined as though they referred to a single structure. While the differences in dimensions were small, it is not actually known that the beams were constructed from the same type of aluminum alloy, introducing a potential error into the results. The maximum size of this error is estimated through further analysis of the beam results in the following section". Such honest assessments greatly raise the quality of a report (and your grade)

Results and Discussion Before writing the results and discussion make sure all your results are analyzed and plotted, and any theoretical derivations have been completed and compared. Make sure your plots are formatted correctly - default Excel plotting format is not acceptable, see Appendix 1 . Plotting results in more than one way may make them clearer or serve more than one purpose. For example, you may have plotted beam displacement against load for each of a series of configurations for the purpose of illustrating the beam deformations used in estimating Young's modulus.  Once you have computed a value of Young's modulus for each measurement/configuration, you may then consider plotting the whole set of estimates together, say against displacement to see of the fact that the deformations are actually finite has influenced the estimates (the error would increase with load). Alternatively you might consider cross-plotting estimates obtained separately from the strain gages and dial indicators against load. That might reveal any errors associated with additional forces, from the loading fixture say. Such pictures make it much easier for you to explain and discuss the significance and implications of your results.

Since the beam configurations you used are probably key to your goals they may well be best shown and described in this section along with the results. You will really need clear dimensioned and labeled diagrams for these.

A good way of writing this section may be to tie each set of tests and results to one of your objectives stated in the introduction. (If you find it hard to do this, try changing your objectives!) For example, you might begin with "The beam was tested in two configurations for the purpose of  determining the Young's modulus of the aluminum from which it was constructed. The two configurations, a three-point bend test and a cantilever test, are illustrated in Figures 7 and 8. Loads of ??? were applied at the locations marked with the symbol 'P'. Figure 9 show the displacements measured at the dial indicator locations (marked 'A') plotted against nominal load. Figure 10 shows strains measured on the at locations 'B' on the upper and lower surfaces of the beam. Note that the results do not show exactly a straight line behavior....due to non-repeatability in the... To use these results to estimate Young's modulus, it is necessary to use technical theory to analyze the beam deformation. As a first step, we note that the moment distribution for these two beams can be written as: ... (note coordinate directions defined on figures 7 and 8)......The estimates of Young's modulus based on equation ?? and the dial indicator results are shown in...Estimates based on ... strain results..." Don't forget to describe your plots. If you have trouble finding deeper things to say about a plot, ask yourself why you are presenting it.

Make sure your results and discussion include (and justify) the conclusions you want to make. Also remember to include any uncertainty estimates in derived results. You should reference a table (copied out of your Excel file) or appendix containing the uncertainty calculation.

Conclusions Begin with a brief description of what was done. Then a sequence of single sentence numbered conclusions that express what was learned. Your conclusions should mesh with the objectives stated in the introduction  (if not, change the objectives) and should be already stated (although perhaps not as succinctly) in the Results and Discussion. 7. References

• Beer, F. P., and Johnston, E. R., Jr., 1992. Mechanics of Materials , McGraw-Hill Book Company, New York, pp. 608-610.
• Durelli, A. J., Parks, V. J., and DeMarco, M, 1965. "Multipurpose Loading Device," Final Report, Mechanics Division, The Catholic University of America, Washington D. C. 20017, September (Sponsored by the National Science Foundation, Grant No. NSF-G22968).
• Megson, T. H. G., 1990. Aircraft Structures for Engineering Students , Second Edition, Halsted Press, pp 98-102.

Introductory Material

Experiment manuals, instrumentation manuals, course organizer: aurelien borgoltz.

Green Mechanic

Knowledge Is Free

Deflection of Beam Lab Report (Simply Supported Beam)

Objective of  deflection of beam lab report.

1. Learn basic working of beam 2. Perform theoretical calculation for deflection of beam 3. Perform series of experiment with different material, shapes and load  for beam 4. Compare the result and Discuss the finding

Introduction to  deflection of beam

Load supporting element in any structure that is installed in horizontal direction and take load perpendicular to its length is called a beam.  

Based on the way it is installed in structure, beam can be classified in to four main types. 

First simply supported beam which have supports at its both ends, second is overhanging beam which have length extended over its supports, third is overhanging beam which have more than two supports and fourth is cantilever beam which fixed only at one end and other end is free. 

There are three basic types of supports available for beams which includes fixed ends support, pined connection support and roller end support. From these three types any two types can be used to support single beam. 

When load is applied on beam, its produce bending moment in beam by apply shear force across its cross section area. Beam reacts against this bending moment and shear force according to its moment of inertia. 

Moment of inertia of beam is its shape dependent property which shows the beam ability to resist bending moment applied on it. Another factor which help beam to resist bending is modulus of elasticity of beam. 

It’s the material related property of beam which defines the strain produce when a certain load is applied on it. Moment of inertia of beam and modulus of elasticity of beam material when combine define the beam reaction against the bending moment due to applied load. 

Another factor which affects the working of beam is its length. Greater the length of beam greater will be the bending moment produce due to external load. 

Experimental Procedure of  deflection of beam

1. Selection one material from the four different material available and after that select the shape on which experiment will be conducted 2. Note down the materials Modulus of elasticity and moment of inertia of the selected shape  3. Select the number of weights and magnitude of each weight. 4. Fix the position of application of load 5. Use all the above information to find the reaction of beam theoretically 6. Use the information available to form the moment equation of beam 7. Use the moment equation of beam in Macaulay theory to drive the equation of slope and deflection 8. Use Equation of deflection to find the deflection of beam due to load at certain positions 9. Place the selected beam on apparatus and attach weight hanger and deflection measuring instrument 10. Apply load on beam by placing weight on hangers  11. Note the deflection of beam on selected points 12. Compare the theoretical and experimental results 13. Repeat the procedure for four different beams

Results of  deflection of beam lab report

Discussion on  deflection of beam

From the experimental and theoretical calculation made above it can be concluded that theoretical values are always less than the experimental values. 

This is because theoretical values are made with ideal cases ignoring many facts of real life, like damaged apparatus, human error and human/machine limitations. 

Other facts which can be concluded, the increase in modulus of elasticity of beam decrease deflection and similar to that increase in moment of inertia of beam decreases the deflection. Different material act differently during experiment.

Conclusion on  deflection of beam

The aim of this lab work to study the deflection of beam has been completed successfully and four different experiments have been conducted on three different materials with four different shapes. 

At the end of this lab work it can be concluded that increase in modulus of elasticity and moment of inertia decreases the deflection where increase in number of loads, magnitude of load and distance of load from ends increase the deflection of beam. 

Each beam show deflection based on its modulus of elasticity and moment of inertia. It also can be concluded that experimental values of deflection are always greater than calculated values.

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• What is a Beam?
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A Complete Guide to Cantilever Beam | Deflections and Moments

• Beam Deflection: Definition, Formula, and Examples
• What is Bending Moment?
• Bending Moment Formula and Equations
• How to Calculate Bending Moment Diagrams?
• Calculating Shear Force Diagrams
• How to Determine the Reactions at the Supports?
• How to Calculate an Indeterminate Beam?
• Types of Supports in Structural Analysis
• What is a Truss? Common Types of Trusses in Structural Engineering
• Truss Tutorial 1: Analysis and Calculation using Method of Joints
• Truss Tutorial 2: Analysis and Calculation using Method of Sections
• Truss Tutorial 3: Roof Truss Design Example
• Calculating the Centroid of a Beam Section
• Calculating the Statical/First Moment of Area
• Moment of Inertia – Overview, Formula, Calculations
• Moment of Inertia of a Circle
• Moment of Inertia of a Rectangle
• Calculating Bending Stress of a Beam Section
• What is Reinforced Concrete?
• Column Buckling
• What is a Column Interaction Diagram/Curve?
• Calculate the Moment Capacity of an Reinforced Concrete Beam
• Reinforced Concrete vs Prestressed Concrete
• How to Design Reinforced Concrete Beams?
• Shear Connection
• Moment Connection
• A Complete Guide to Building Foundations: Definition, Types, and Uses
• What is a Foundation
• Types of Foundation and Their Uses
• What is the Process of Designing a Footing Foundation?
• A Brief Guide on Pile Foundation Design
• Moment of Inertia Formula and Equations
• Centroid Equations of Various Beam Sections
• Beam Deflection Formula and Equations
• Bending Moment Equations
• Modelling A Greenhouse
• Modeling a Concrete Building
• Types of Loads
• What is a Frequency Analysis?
• Tributary Area and Tributary Width Explained – with Examples
• Introduction to SkyCiv Education
• How to Upload a CSV of your Students
• Case Study: Monash University
• Courses that use SkyCiv
• How to Design a Boomilever
• How to Design a Bridge
• How to Test for Common Boomilever Failures
• SkyCiv Science Olympiad 2021 Competition App
• SkyCiv Science Olympiad 2023 Competition App
• How to Model a Truss (30mins)
• AISC Steel Bridge Competition
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• Introduction to a Design Project for Engineers
• Beam Tutorials

List of Contents

• What is a Cantilever Beam?
• Bending Moment

Cantilever Beam Design

Cantilever beam software, cantilever beam definition: what is a cantilever beam.

A cantilever beam is a structural element that extends horizontally and is supported on only one end. The unsupported end is known as the cantilever, and it extends beyond the support point. Cantilever beams are often used in construction to support balconies, roofs, and other overhangs. They can also be used in bridges and other structures to extend the deck out over a waterway or other obstacle.

Cantilever beams are members that are supported from a single side only – typically with fixed support. In order to ensure the structure is static, the support must be fixed so that it is able to support all forces and moments in all directions. A cantilever beam is usually modeled like so, with the left end being the support and the right end being the cantilevered end:

Cantilever Beam Equations

There are a range of equations for how to calculate cantilever beam forces and deflections. These can be simplified into simple cantilever beam formula, based on the following:

Cantilever Beam Deflections

Taken from our  beam deflection formula and equation page. Cantilever Beam equations can be calculated from the following formula, where:

• L = Member Length
• E = Young’s Modulus
• I = the beam’s Moment of Inertia

Cantilever Beam Moments

So how do we calculate the maximum bending moment force of a cantilever beam? You can do this using the same method as shown in our how to calculate bending moment in a beam article. However, there are short hand equations you can use. For instance, the equation for the bending moment at any point x along a cantilever beam is given by:

\(M_x = -Px\)

\(M_x \) = bending moment at point x \(P \)   = load applied at the end of the cantilever \(x \) = distance from the fixed end (support point) to point of interest along the length of the beam.

For a distributed load, the equation would change to:

\(M_x = – ∫wx\) over the length (x1 to x2)

where: w = distributed load x1 and x2 are the limits of integration.

This equation is valid for a simple cantilever beam with a point load or a uniformly distributed load applied at the free end of the beam. It should be considered that cantilevers beam can have complex loading and boundary conditions, such as multiple point loads, varying distributed loads, or even inclined loads, in those cases the above equation might not be valid, and a more complex approach might be required, it’s where the FEA comes in handy.

Cantilever Beam Stress

How to calculate stress in a cantilever beam? Cantilever Stress is calculated from the bending force and is dependent on the beam’s cross-section. For instance, if a member is quite small, there is not much cross-sectional area for the force to spread across, so the stress will be quite high. Cantilever beam stress can be calculated from either our tutorial on how to calculate beam stress  or using  SkyCiv Beam Software  – which will show the stresses of your beam.

It’s useful to note that cantilever beams typically result in tension on the upper fibres of the beam. This means that in the case of a concrete cantilever beam, primary tensile reinforcement is typically required along the upper surface. This is in contrast to a conventional concrete beam supported at both ends, where primary tensile reinforcement would typically exist along the bottom surface of the beam.

Cantilever Beam Reaction Forces

Cantilevers deflect more than most types of beams since they are only supported from one end. This means there is less support for the load to be transferred. Cantilever beam deflection can be calculated in a few different ways, including using simplified cantilever beam equations or cantilever beam calculators and software (more information on both is below). The equation for the reaction at a fixed support of a cantilever beam is simply given by:

Reaction Force in Y \( = R_y = P\)

Moment Force about Z  \( = {F}_{y} = Px\)

\(F_y \) = reaction force in the Y direction at support A (the fixed support) \(M_z \) = reaction moment about Z at support A (the fixed support) \(P \) = the load applied at the end of the cantilever beam \(x \) = distance of point load from support

This equation applies when the load is a point load on a cantilever. When the load is distributed, it is the summation of all forces in the vertical direction that needs to be zero. The equation becomes:

\(∑F_x = 0\)

Where the reaction force would be the algebraic sum of all the vertical forces acting on the structure. This equation assumes that the support is a fixed support, meaning that it does not have any rotation or translation. If the support has some degrees of freedom, the equation would change and become more complex. It’s important to keep in mind that this equation is just one step in analyzing a structure, in the design process of a real structure, several considerations such as load combinations, safety factors, material properties, etc. will be taken into account before finalizing a design.

When designing a cantilever structure, several important factors should be considered:

• Loads : The cantilever must be able to support the applied loads, including the weight of the structure itself and any additional loads such as wind, snow, and seismic loads. The loads should be analyzed and distributed appropriately throughout the structure.
• Strength and stiffness : The cantilever must be strong and stiff enough to resist deflection, buckling, and other types of failure. The properties of the materials used, such as the modulus of elasticity and yield strength, will affect the strength and stiffness of the structure.
• Stress concentration : The stress concentration at the fixed end of the cantilever must be taken into account in the design to prevent failure. Stress concentration can be reduced by using larger cross-sections or by using fillets or rounded corners.
• Deflection : The deflection of the cantilever under load should be analyzed to ensure that it remains within acceptable limits, both for structural safety and also for aesthetic reasons.
• Durability : The structure should be designed to last for the intended service life with minimal maintenance required. This includes considering factors such as corrosion, fatigue, and the effects of weathering.
• Safety factors : Safety factors should be considered and included in the design to ensure that the structure can withstand unexpected loads or other unforeseen circumstances.
• Construction methods: The design must take into account the method of construction to be used, whether it be pre-fabricated, cast-in-place, etc. This will affect the type of connections and the overall layout of the structure.
• Cost : Design should consider both initial cost and long-term maintenance costs.
• Building codes and regulations : The design must be compliant with the relevant building codes and regulations in the jurisdiction where the structure will be built. For instance, if the beam is steel and based in the US, it should comply with the requirements of AISC 360 Design Checks

It’s important to keep in mind that this is not an exhaustive list, and the specific requirements and considerations for a cantilever design may vary depending on the particular structure and its intended use. A structural engineer with expertise in cantilever design would take all of these factors into account and more, to ensure that the design is safe and effective.

SkyCiv Beam Analysis Software allows users to analyze cantilever beam structures easily and accurately. You can get a simplified analysis of your beam member, including reactions, shear force, bending moment, deflection, stresses , and indeterminate beams in a matter of seconds. Apply any combination of loads and complete a full design as per American, European, Australian, Canadian Standards, to name a few!

If you want to give it a try first, Free Online Beam Calculator is a great way to start, or simply sign up for free today!

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• Educational experiments

Experiment to measure the deflection of a cantilever beam

Introduction.

1. Brief description A plastic optical fiber is attached to a (cantilever) beam to monitor its deflection. The change in the light intensity of the optical fiber is monitored using a light-dependent resistor (LDR) and a basic voltage divider circuit. The output of the LDR is continuously measured by the ADC-16 and this simple system is able to provide real-time beam deflection monitoring with a PC.

2. What the experiment is trying to teach The experiment highlights the potential use of an optical fiber as a sensor for monitoring, in real time, the deflection of a structure. Students will also gain practical experience in building up a basic electrical circuit based on simple devices such a light-dependent resistor (LDR), reading resistor values, principle of voltage dividers and Ohm’s law.

3. Prior knowledge required Students need to have an idea of how an optical fiber transmits light through its core and the principle of total internal reflection (TIR) and the effect of excessive bending on the light transmission in these light-guiding media. Secondly, students should have some appreciation of the function of a light-dependent resistor, the principles of voltage dividers and Ohm’s law.

4. Target group This experiment is suitable for students taking Advanced Physics and serves as a simple introduction to the practical use of optical, electronic and optoelectronic devices.

Equipment required

• Data logging equipment
• Light-dependent resistor (LDR)
• Light-emitting diode (LED)
• 1 kilohm resistor
• Soldering iron
• A voltage/signal amplifier (optional)
• 2 x 1.5 V batteries (to power LED) and 2 x 9 V batteries (for divider circuit)
• Optical fiber connectors (SMA multimode type connectors)
• Optical fiber sensor
• A flexible beam (e.g. a plastic ruler)
• Fast-curing adhesive

Safety warning:  Although it is generally safe to work with most low-power LEDs, for safety reasons, do not look directly into the LED when illuminated. A soldering iron should only be used under supervision.

Experiment setup

Figure 1 shows the experimental setup for the experiment.

Figure 1: Experiment set up

Figure 1a: Voltage divider circuit (within the Basic Circuitry in Figure 1)

The experimental arrangements for both the three-point bend and tensile tests are shown in Figure 4. A standard voltage supply was used to power the light emitting diode The detector and data acquisition system consisted of a light-dependent resistor (LDR) and a low-cost commercial data acquisition system from Pico Technology which automatically records voltage changes across the LDR as the light intensity varies. The data acquisition system offers up to a 16-bit resolution analog to digital conversion (ADC) with up to 8 input channels. The resolution of the ADC system allows for the detection of voltage changes as small as 40 mV in electrical signal. A sampling rate of 2 data samples per second will be sufficient for the purpose of the experiment. The data from the optical fiber were automatically collected by the computer and displayed graphically in real time.

A schematic of the beam with the bonded optical fiber sensor is shown in Figure 2. Details will be given in the following section.

Figure 2: Schematic of beam specimen with bonded optical fibre sensor

When the experiment is properly set up, the beam should flex about its centre with a load applied at the centre of the supports, as shown in Figure 3, where W is the applied load, dctr is the central deflection.

Figure 3: Properly set up beam

Carrying out the experiment

• Attach the optical fiber sensor to the beam (e.g. plastic ruler) using superglue or other available adhesive and wait for it to dry. Ensure that the bond is secure and that the sensor does not debond when the beam specimen is flexed.
• Connect the optical connectors to both ends of the optical fiber and couple them to the LED and LDR respectively. Ensure they are securly coupled to minimize any unwanted changes in the voltage signal due to loss in the coupling and environmental noise.
• Carefully put the beam on to the supports. The beam can also be secured on one end and free the other end as in a cantilever beam setup as an alternative support system to the one shown.
• Check that there is voltage output reading by PicoLog to ensure that the setup is correct and ready for the beam deflection test to begin.
• Adjust the scale of the Y-axis of the graph in PicoLog to obtain an optimum display. If you have a signal amplifier, you can also adjust the voltage level sent to the ADC.
• Apply a small deflection on the beam slowly and observe the change in the voltage level displayed by PicoLog. Did the signal increases or decreases when the deflection was applied? Depending on the sensitivity of the fiber, you can expect a change in the voltage of at least 100 mV for a deflection of 10 mm using the setup shown.
• Now release the deflection and observe the change in the voltage level as indicated by PicoLog. What happens to the signal? Does the signal return to the initial signal value?
• Now try with different amounts of deflection and observe the changes in the voltage signal. What do you observed?
• Can you explain how the circuitry works to give a proportional signal output when the light intensity of the optical fiber sensor changes?
• What do you think will happen to the voltage signal if the beam is deflected upwards instead of downwards as previously done?

Further study

Dynamic monitoring using optical fiber sensors: An important aspect in monitoring real-life engineering structures involve performing dynamic analyses of vibrating structures. Optical fiber sensors based on the above system represent a cost effective method to monitor a dynamic system. As a further study, the beam used in the above experiment may be subjected to a continuous deflection using a cam attached to a motor, providing a constant rate of deflection to the beam while monitoring the output signal of the optical fiber. Other types of ADC supplied by Pico Technology Ltd may be more appropriate for dynamic monitoring. Contact Pico Technology for advice.

Engineering ToolBox - Resources, Tools and Basic Information for Engineering and Design of Technical Applications!

Cantilever Beams - Moments and Deflections

Maximum reaction forces, deflections and moments - single and uniform loads., cantilever beam - single load at the end.

Maximum Reaction Force

at the fixed end can be expressed as:

R A = F                              (1a)

R A = reaction force in A (N, lb)

F = single acting force in B (N, lb)

Maximum Moment

at the fixed end can be expressed as

M max = M A

          = - F L                               (1b)

M A = maximum moment in A (Nm, Nmm, lb in)

L = length of beam (m, mm, in)

Maximum Deflection

at the end of the cantilever beam can be expressed as

δ B = F L 3 / (3 E I)                                      (1c)

δ B = maximum deflection in B (m, mm, in)

E = modulus of elasticity (N/m 2 (Pa), N/mm 2 , lb/in 2 (psi))

I = moment of Inertia (m 4 , mm 4 , in 4 )

b = length between B and C (m, mm, in)

The stress in a bending beam can be expressed as

σ = y M / I                                     (1d)            

σ = stress (Pa (N/m 2 ), N/mm 2 , psi)

y = distance to point from neutral axis (m, mm, in)

M = bending moment (Nm, lb in)

The maximum moment in a cantilever beam is at the fixed point and the maximum stress can be calculated by combining 1b and 1d to

σ max = y max F L / I                (1e)

Example - Cantilever Beam with Single Load at the End, Metric Units

The maximum moment at the fixed end of a UB 305 x 127 x 42 beam steel flange cantilever beam 5000 mm long, with moment of inertia 8196 cm 4 (81960000 mm 4 ) , modulus of elasticity 200 GPa (200000 N/mm 2 ) and with a single load 3000 N at the end can be calculated as

M max = (3000 N)  (5000 mm)

         = 1.5 10 7 Nmm

         = 1.5 10 4 Nm

The maximum deflection at the free end can be calculated as

δ B = (3000 N) (5000 mm) 3 / (3 (2 10 5 N/mm 2 ) (8.196 10 7 mm 4 )) 

    = 7.6 mm

The height of the beam is 300 mm and the distance of the extreme point to the neutral axis is 150 mm . The maximum stress in the beam can be calculated as

σ max = (150 mm) (3000 N) (5000 mm) / (8.196 10 7 mm 4 )

    = 27.4 (N/mm 2 )

    = 27.4 10 6 (N/m 2 , Pa)

    = 27.4 MPa

Maximum stress is way below the ultimate tensile strength for most steel .

Cantilever Beam - Single Load

R A = F                              (2a)

          = - F a                               (2b)

M A = maximum moment in A (N.m, N.mm, lb.in)

a = length between A and B (m, mm, in)

δ C = (F a 3 / (3 E I)) (1 + 3 b / 2 a)                                       (2c)

δ C = maximum deflection in C (m, mm, in)

at the action of the single force can be expressed as

δ B = F a 3 / (3 E I)                              (2d)

Maximum Stress

The maximum stress can be calculated by combining 1d and 2b to

σ max = y max F a / I               (2e)   

Cantilever Beam - Single Load Calculator

A generic calculator - be consistent and use metric values based on m or mm, or imperial values based on inches. Default typical values are in metric mm.

F - Load (N, lb) a - Length of beam between A and B  (m, mm, in) b - Length of beam between B and C (m, mm, in) I - Moment of Inertia (m 4 , mm 4 , in 4 ) E - Modulus of Elasticity (N/m 2 , N/mm 2 , psi) y - Distance from neutral axis (m, mm, in)

Cantilever Beam - Uniform Distributed Load

Maximum Reaction

R A = q L                                (3a)

q = uniform distributed load (N/m, N/mm, lb/in)

L = length of cantilever beam (m, mm, in)

M A = - q L 2 / 2                              (3b)

at the end can be expressed as

δ B = q L 4 / (8 E I)                               (3c)

Cantilever Beam - Uniform Load Calculator

A generic calculator - use metric values based on m or mm, or imperial values based on inches. Default typical values are in metric mm.

q - Uniform load (N/m, N/mm, lb/in) L - Length of beam (m, mm, in) I - Moment of Inertia (m 4 , mm 4 , in 4 ) E - Modulus of Elasticity (Pa, N/mm 2 , psi) y - Distance from neutral axis (m, mm, in)

More than One Point Load and/or Uniform Load acting on a Cantilever Beam

If more than one point load and/or uniform load are acting on a cantilever beam - the resulting maximum moment at the fixed end A and the resulting maximum deflection at end B can be calculated by summarizing the maximum moment in A and maximum deflection in B for each point and/or uniform load. 

Cantilever Beam - Declining Distributed Load

R A = q L / 2                              (4a)

q = declining distributed load - max value at A - zero at B (N/m, lb/ft)

          = - q L 2 / 6                               (4b)

δ B = q L 4 / (30 E I)                                      (4c)

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Related Topics

Beams and columns, related documents, area moment of inertia - typical cross sections i, beam loads - support force calculator, beams - fixed at both ends - continuous and point loads, beams - fixed at one end and supported at the other - continuous and point loads, beams - supported at both ends - continuous and point loads, center mass, continuous beams - moment and reaction support forces, drawbridge - force and moment vs. elevation, he-b steel beams, mass moment of inertia, square hollow structural sections - hss, steel angles - equal legs, steel angles - unequal legs, three-hinged arches - continuous and point loads, weight of beams - stress and strain, wood headers - max. supported weight, young's modulus, tensile strength and yield strength values for some materials.

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