Stress and Strain

This lab exercise consists of two sections. In the first section, you will test the concrete cylinders made in last weeks lab for ultimate flexural strength. In the second section, you test a rectangular "beam" of wood, namely a 2 by 4, to a) determine the nature of its response to flexural stress and, b) to determine the value of the elastic modulus of the wood.

Background: Flexural Stress

When a beam is supported at either end and force is applied between the supports, the beam "flexes" and is subjected to flexural stress. Flexural stress is actually a combination of both compressive stress and tensile stress.

Look at the diagram of a rectangular beam. A single force F is applied at the center and the beam deflects under the load. The greatest distance that any portion of the beam deflects under load is called the maximum deflection. For a force applied at the center of uniform beam, it is easy to see that the maximum deflection will be at the center. In this lab, we will apply force at the center of the beams.

The top of the beam is under compression and the bottom is under tension. The imaginary surface at the center of the beam is called the neutral plane. The amount of compression or tension increases as you move away from the neutral plane. Compression and tension are a maximum at the top and bottom surfaces. Because the amount of stress and strain vary over the cross section of the beam, the formula relating these values to the elastic modulus and the deflection of the beam are not easy to derive. Nevertheless, engineers have long understood this phenomena and we will use that knowledge to find the elastic modulus.

I. Flexural Strength of Various Concrete Mixtures

In last week's laboratory, you created two sets of concrete cylinders. You will test one of those sets today for ultimate strength. The second set will be tested near the end of the semester. The results of the tests for each set will allow you to make qualitative conclusions concerning the role that curing plays in concrete strength.

Procedure:

Place the two 3-prong extension clamps on laboratory stands using the adjustable chucks, about 40 cm above the surface of the table. Open the jaws of the clamps completely and rotate them such that the single prong of the jaw is down. Place the stands such that the prongs are 10.0 cm apart and at the same height. The jaws will provide the end supports for the concrete cylinders. Place a foam mat below the apparatus. This will minimize secondary fracturing when the cylinders fall.

Choose one set of the four cylinders of varying mixture ratios and carefully unwrapped one of the cylinders. Find the mass of the cylinder, measure the length, and measure the diameter at 5 locations along its length. Record this data along with the mixture ratio information and a brief description of the cylinder. The description should include observations about the texture and uniformity, as well as notations of any voids or other imperfections.

Place the cylinder symmetrically on the jaws and mark the center with a pencil. Place a string loop around the cylinder at the center and hang the mass hanger from the loop. Gently add masses to the hanger in 20 gm intervals until the cylinder fractures. Record the total mass (including the hanger mass) at which the cylinder fractured. Inspect the cylinder at the break. Record your observations, including information as to where along the cylinder the fractured occurred and whether or not there were voids along the fracture cross section.

Repeat the procedure with the remaining three cylinders. Keep all data in a neat table format. You must keep this information until the experiment is completed later in the semester, when the same tests will be performed on the second set of cylinders.

II. Flexural Stress in Beams and the Elastic Modulus

In this portion of the laboratory you will determine the behavior of a standard 2 by 4 beam (2x4) under flexural stress resulting from a point load at the center of the beam. A standard engineering formula relating flexural displacement to load, beam dimensions, and the elastic modulus of a beam will be used to determine the elastic modulus of the 2x4.

Procedure:

In this experiment, each student will use their own weight as a point force applied to the center of a 2x4. The 2x4 will be supported at each end. Measure the distance between the supports and measure the cross section (horizontal width and vertical height) of the beam at 5 different places along its length. (Note: The standard 2x4 is not 2" by 4". Measure the dimensions directly in SI units.)

1. Mark the center of the 2x4. Measure and record the vertical distance from the floor to the upper surface of the beam at the center. This will be the initial "beam position" from which central deflection of the beam will be measured.

2. Each student in the class will stand at the center of the beam on one foot. (Classmates should help the student to maintain balance while the distance from the floor to the beam is measured.) Record the distance from the floor to the top of the 2x4 and find the mass (in kg) of the student with the bathroom scales provided. Be sure to record both these numbers together in a table. The instructor will collect your values and make them available to the entire class.

3. Rotate the beam 90^o, in effect exchanging the values width and depth. Repeat the entire procedure as was done for the first orientation.

Report:

Your report should have the following features:

1. For each orientation, place the data in a table format listing the total load mass and corresponding position of the beam in adjacent columns. In following columns, list the point load in Newtons and the deflection in meters. The deflection (denoted Dy) is the difference between the beam position corresponding to that load and the initial position.

2. For each of the two experiments, plot deflection vs load. The plot may be made with the software on the Physics lab computers (the instructor can provide a brief tutorial, if necessary) or with any standard plotting software. You may also do the plots by hand. Regardless of the method, the plots should be at least one half a page in size and have all the features outlined in the Lab Rules section found under Grading in the course syllabus.

3. Fit the best straight line through the data points. Graphing software will provide such a "least squares" fit, but it is important that you understand the software options. (Some of the plotting options are oriented for business purposes and are not appropriate for this data. If you are in doubt, check with the lab instructor.) Comment on the linear fit to the data. Does the data seem to lie along a straight line, or would a curved fit be more appropriate?

The central deflection of a beam under a point load is a very well-understood engineering problem. Although the derivation is beyond the scope of this course, we can use the following formula, which relates deflection to the point load. The deflection (Dy) is given by

where h is the vertical height, w is the horizontal width and L is the length of the beam as shown in the diagram. F is the point load at the center of the beam and Y is the elastic modulus. The variables in the expression are the deflection and the point load. If correct, this equation clearly implies that the deflection is linear with the point load. Using this formula and the plots, for each experiment:

1. Determine the slope of the line fitted to the data.
2. Determine the elastic modulus for the beam using the value of the slope and the deflection formula. Be certain that you understand how the values of h and t change for the two cases. (You may need to review the salient features of the formula for a straight line. Ask your instructor for a review. )

In your conclusion:

1. Comment on any differences between the elastic modulus values you found for the two different beam orientations. What conclusions would you make concerning how wood beams should be cut during manufacturing?
2. Assuming that the elastic modulus is independent of orientation, such as would be the case for an amorphous material, by what factor would you expect the deflection to be reduce by using the second orientation instead of the first orientation? Use the formula to support your conclusion.
3. Find several values for the elastic modulus of wood from reference literature, such as the CRC Handbook, and/or sources on the net. Be sure to note those sources. Make a qualitative comparison of the calculated values of the elastic modulus of your wood beam to those found in the sources.