John Johnston, Ph.D., P.E. and Edward Dammel, Ph.D.

Department of Civil Engineering

California State University Sacramento

October 1999

In this exercise, we will load a simply-supported beam and measure the resulting deflection at the center. By graphing the deflections vs. the loading, we will be able to calculate the modulus of elasticity of the beam material, an important measure of the material's strength. While this measurement has some practical significance in itself, the broader usefulness of this exercise is its role as a springboard to several topics of significance in engineering practice. These include elastic behavior, factors of safety, choice of materials, importance of shape in resisting loads, and consideration of multiple kinds of loading and kinds of failure. As we go along, several homework projects -- both mathematical and physical -- will be suggested.

To understand how a beam will react when subjected to a load, it is helpful to first know something about the behavior of engineering materials. Many engineering materials, particularly metals, exhibit what is known as elastic behavior. If, for instance, you pull on the ends of a steel bar, it will stretch. (See Figure 1.) Then, when you release the force (also known as the load), the bar will return to its original shape. This is how a spring or a rubber band works, except the change in length will be only fractions of an inch or millimeter. In most elastic materials, this phenomenon is the same whether you are pulling (tension forces) or pushing (compression forces) the specimen.

When a material is acting elastically, the change in length due to a force is directly proportional to the force. Usually we express the change in length as strain (e ) -- the change in length per unit length.

e = D L / L (eqn 1)

where

e = strain (length/length, or unitless)

D L = the change in length due to the load

L = the original (unloaded) length

We also measure force as stress -- the force per cross-sectional area.

s = F / A (eqn 2)

where

s = stress (lb/inch² or N/mm² or similar units)

F = applied force (lb or N)

A = cross-sectional area of the specimen (inch² or m² or similar units)

To say that the change in length is proportional to the force is the same thing as saying that the strain is proportional to the stress. The proportionality constant is called the modulus of elasticity (E).

s = E e (eqn 3)

where

s = stress (force/area)

e = strain (length/length)

E = modulus of elasticity (force/area units)

How does this relate to bending? When a beam is bent, the top of the beam is pushed together and the bottom is pulled apart. (See Figure 2.) In other words, the top of the beam experiences compression, and the bottom tension. How far the beam will deform under these forces depends on the modulus of elasticity. So it would make sense that E would appear in any equation we might come up with to predict deflection due to a load. In fact, this is what happens.

When a beam of length L is simply supported at the ends (i.e., resting on supports that don't confine it, or prevent it from moving), and loaded in the middle with a force P, the deflection of the beam at the middle point is given by

(eqn 4)

where

y = deflection at the middle (the "-" sign means the beam deflects downward)

L = length of the beam

P = load (force)

E = modulus of elasticity

I = moment of inertia

The moment of inertia (I) is a function of the cross-sectional shape of the beam. This is all we need to know to perform the experiment. We'll revisit this topic later.

Open up a paper clip so that it is a flat wire. Hold one end of the wire down at the edge of a table so that the other end extends out into the air. With your finger, pull the free end down a little bit and let it go. It returns to its original shape, doesn't it? this is elastic behavior. Now, if you push too hard, you will exceed the what's called the yield strength of the wire and it will bend permanently. This is plastic behavior. Since we don't want beams in buildings or bridges to yield, engineers normally use a factor of safety in their calculations so that the structure is never loaded enough to cause permanent deformation. The exception to this rule is in earthquake design. Here the structure is allowed to yield, as long as it doesn't fall down.

If you rearrange equation 4, you get

You might recognize this as the equation of a line:

y = m P

where m = slope = L³/(48IE).

In this experiment, we'll generate a number of paired measurements of load (P) and deflection (y) on three round beams -- one of steel, one of aluminum, and one of brass. After collecting the data, graph load on the x-axis and deflection on the y-axis. The data should fall on a straight line. A dramatic "kink" in the line would indicate that we exceeded the elastic limit and caused the beam to yield. (Look at the beam. Did it return to its original shape?) In this case, use only the linear portion of the curve (i.e., the smaller loads). Measure the slope off the graph and calculate E, given the length of the beam and the moment of inertia. The equation for the moment of inertia for a solid round cross section is:

I_x = p r⁴/4 = p d⁴/64

The experimental apparatus will be demonstrated during the workshop. Photos will be taken and posted on Dr. Johnston's home page:

(http://www.csus.edu/indiv/j/johnstonj)

A complete materials list and suggestions for materials to test will also be posted. Please note that there is nothing magical about the apparatus. What we will demonstrate in the workshop was constructed out of what we had readily available around our lab. Any set-up you devise that allows you to measure load and deflection will work. If you want to share your design with others, email photos or notes to us for posting. Or if you post them on your school's site, please send us the address so that we can provide a link to your site.

1. Choice of material

One choice always facing engineers is what material to use in a structure. There are a number of materials available -- steel, concrete, wood, iron, aluminum, and recently, plastics. Each of these materials has a different set of characteristics to consider in design. The modulus of elasticity (E) is one example. Table 1 contains information on a number of other materials.

Some materials, like steel, act generally the same in both tension and compression. Others, like concrete, act differently, depending on how they are loaded. Concrete, for instance, is very strong in compression, but very weak in tension. In other words, it is hard to crush it, but relatively easy to pull it apart. If you bend a concrete beam, cracks develop quickly in the tension section and the beam fails. To counteract this, reinforcing bars made of steel (strong in tension) are imbedded in the concrete (see Figure 3). When reinforced concrete is bent, the steel provides tensile strength so that the beam doesn't crack.

Although very important, strength characteristics are only one consideration in choosing a material for a specific application. Other things that engineers need to think about are:

• weight (in a structure, the weight of the structure itself is part of the loading)
• cost of the materials themselves
• ease of construction (sometimes using a more expensive material is worth it if you save enough money on the construction), and
• availability (steel beams must be fabricated in a steel plant).

Given a set of parameters -- load, shape, length, and maximum allowable deflection, choose an appropriate material from Table 2. Students would use the information given to solve for the required E using equation 4. Then they can choose a material with an E value equal to or greater than the required E.

Extension: Point out to students that there is rarely only one technically correct answer. Ask them to choose several materials that would work and write a short memo describing the pros and cons of each.

2. Choice of shape

So far, we have noted that the deflection depends on the moment of inertia which in turn depends on the shape. The moment of inertia is the second moment about some axis. In mathematics, it is:

where

I_x = moment of inertia with respect to the x axis

y = the distance from the x axis to the differential area dA

Without calculus and more background in mechanics, it is hard to explain too much more about I. Fortunately, equations for calculating I for a number of shapes are available (see Table 2). The choice of axis is important. With vertical loads and bending in the vertical plane, I_x should be used.

Choose a value for the load that gave a moderate amount of deflection in the round beams. Now use this load on several other shapes. For instance, try the flat stock laid "flat", then the same beam laid "on its side". The deflections are very different. This is reflected in equation 4 as a change in I. Calculate I_x for the two cases using the equations in Table 2. Try other shapes to see the effect not only of the shape, but also of rotating the same shape and loading it from another side.

Give students a set of parameters -- load, maximum allowable deflection, material, and length -- and ask them to choose shape and dimension it. They can do this by first calculating the required I_x from equation 4, and then choosing by trial and error, a shape from Table 2.

Extension: Specify dimensions of a space (e.g., 3 inches wide by 5 inches high by length L) into which their beams must fit.

Extension: Ask for the most economical shape. That would be the shape that uses the least material, which can be calculated from the cross-sectional area times the length.

3. Consideration of Multiple Cases

You probably noticed that the taller a beam is, the less it deflects from a vertical load. What if we took this to the extreme and used a very tall, very thin section like a piece of aluminum foil. If we loaded this, it wouldn't deflect, but it would fail by buckling. Excessive deflection in the vertical plane is only one mode of failure. Buckling is another. (Buckling is a pretty complex subject, and we won't try to predict it here.) Another mode of failure is excessive deflection in the horizontal plane due to horizontal loads like wind.

The point is that the engineer has to think about all the different things that might happen to his or her structure -- the various things that might be loaded onto the structure (people, cars, filing cabinets, machine tools), the effect of ice or snow or water puddling on the roof, wind loads and earthquake loads (that are horizontal rather than vertical) -- and try to predict how the structure will react (vertical deflections, horizontal deflections, buckling, twisting) for each. Many, many cases must be considered. In the old days, engineering firms would have rooms full of people doing these calculations, and they wouldn't cover all the possible cases. Today, computer models help us analyze many more cases, much faster.

Do homework problem #2 except specify a horizontal load and a maximum allowable horizontal deflection. Horizontal loads are typically much smaller than vertical loads (say 5-10%). Equation 4 can be used to calculate the deflection except that I_y should be used instead of I_x.

Table 1 (excerpts from: Beer and Johnston, Mechanics of Materials, McGraw-Hill, 1981)

Table 2 (ref: Lindeburg, M., Civil Engineering Reference Manual, Professional Publications, Inc, 1992)