Bending of Beams Experiment Report Essay Example

by changing the load applied to the same beam. At the end of the experiment, it is expected that the deflection of the beam is linearly proportional to the applied load as Euler-Bernoulli Beam Theory suggests. The equation of Euler-Bernoulli Beam Theory is as follows:

In this equation, w represents the deflection of the beam, E represents the modulus of elasticity of the material, I represents the second moment of area of the beam and q represents the distributed load, which can also be described as force applied per unit length.

In addition, the effect of the beam thickness on the deflection is another effect to be investigated. Depending on the beam theory, it is anticipated that the deflection of the beam is inversely proportional to the third power of the beam thickness. By this experiment, the relation between the deflection and beam thickness can also be observed and proved. Lastly, the effect of material on the deflection of the beam is to be examined, as well. Euler-Bernoulli Beam Theory suggests that deflection of the beam is inversely proportional to the modulus of elasticity. Since different materials have different elastic modulus, through this experiment, it will be proven that the materials having higher elastic modulus deform less under the same load than the ones having lower modulus of elasticity.

Theory

In construction industry and literature a beam is usually referred to a structural member which is generally horizontal and used to support generally horizontal loads such as floors, roofs, and decks. Similarly, in mechanical engineering applications and in science literature, a beam is explained as a component that is designed to

support transverse loads, that is, loads that act perpendicular to the longitudinal axis of the beam. In order to make the building and structure more stable and stronger against impacts and loads, beams are generally applied as a solution. Since there are numerous material types and numerous different shapes of materials, beams can also be found in different shapes. It can be also observable that some properties of beams can also vary from one to another due to the type difference such as ductility, stiffness and so on. Since beams are used to compensate the loads within the structures, there are different types of beam loads available. For instance; there are uniform beam loads, varied beam loads by length, single point beam loads and combination beam loads which include all types of loading.

Sometimes beams are experienced some deflections from their equilibrium states towards a completely new state due to loads applied. The direction of the deflection highly depends on the direction of load applied. There are different beam theories postulated up to now to describe the behaviour of beams under loading. The theory that is accepted and applied for this experiment is named as Euler-Bernoulli Beam Theory.

There is a requirement for this theory to be applied on a beam. Namely, the length of the beam should be at least 20 times of the thickness of it. Actually the more the ratio gets higher, the more the theory and calculations are valid.

There are also some assumptions made with respect to this theory. In fact, they are all related with normals of the beam itself. First assumption is that normals do

not bend. Secondly, normals remain unstreched. If we take into consideration both two assumptions, it can be easily obtained the conclusion that normals remains normal. If the Euler-Bernoulli Beam Theory is applied on a beam, this means that these assumptions are done and the requirement is met.

In all theories, there are also some formulas used in this theory. Some of the formulas needed to apply the Euler-Bernoulli Beam Theory are as follows;

where p is the distributed loading (force per unit length) acting in the same direction as y (and w), E is the Young's modulus of the beam, and I is the area moment of inertia of the beam's cross section.

Experimental Procedure In this experiment, there are different beams lenghts and types are used. The reason for this experiment is carried on is to learn the applicability of the Euler-Bernoulli Beam Theory on loaded beams. There are 3 different steel beam are used to see the effect of different thickness to deflection. There also 3 different types of materials are used such as brass, steel and aluminum to see the effect of modulus of elasticity to deflection.

As the first step; the steels beams are selected to be tested in an order as 3 mm, 4.5 mm and 6 mm. Before testing each steel beam, the deflection gage is set at zero (0) value. After placing the 3 mm steel beam on the experiment apparatus and the gage is set to zero value, the beam is subjected to some loading right in the middle section. Firstly, 2N load is applied and the displacement gage' s value is read.

Then 2N more is applied, and total load achieved to 4N and the displacement value is read. This procedure is repeated by 2N weights until the total load is 8N. After measuring the 3 mm steel' s displacement values, we proceeded to 4.5 mm steel beam. The same procedure is also applied for both 4.5 mm and 6 mm steel beams.

Secondly, different types of beams (different modulus of elasticities) are selected. Initially the displacement value of aluminum beam is measured with the very same procedure as in steel beams. The values in the displacement gage are read after each 2N weight addition. As final step, the brass is experienced. It is placed on the apparatus and displacement values read repeatedly.

After carrying out all these steps, we get some displacement values of same materials with different thicknesses and different materials with the same thicknesses to see the effect of different properties of materials on displacement according to the Euler-Bernoulli Beam Theory. Finally we are expected to have some graphs showing these relationships and some calculations regarding this theory.

Figure B : Max Deflections of Different Materials Under Various Loads Figure B : Max Deflections of Different Materials Under Various Loads Using the tabulated data, the following graph can be obtained to show how each sample deflected under various loads. The graph shows us that under the same applied load, 3 mm Steel deflects the most, followed by 6 mm Aluminum, 4.5 mm Steel, 6 mm Brass, 6 mm Steel respectively. As mentioned above, the theory suggests that the deflection can be expressed as PL/48EI. If this equation is used to calculate

the theoretical deflections, the following graph is obtained.

Figure C : Theoretical Deflections of Different Materials Under Various Loads.

Figure C : Theoretical Deflections of Different Materials Under Various Loads Figure C also shows a similar trend in terms of the amount of deflection of the materials under the very same load. Although Figure B and Figure C seem similar at a first glance, a closer look reveals that the slopes of the graphs are different. However, since the trends are exactly the same, some error should have been made in the experimental setup or there was a calibration mistake. The errors will be discussed thoroughly in the upcoming pages.

Figure D : Variation of Maximum Deflection With Thickness

Figure D : Variation of Maximum Deflection With Thickness The next aspect to consider is the effect of thickness in deflection. To observe this phenomenon, steel samples should be observed. If the steel data are isolated, the following graph is obtained.

Figure D clearly shows that as the thickness of the beam increases, it is harder to deflect it. This statement also agrees with the theory, which suggests that deflection is inversely proportional with the moment of inertia of the beam. The moment of the inertia of the beam increases as its thickness increases, therefore deflection should decrease.

The experiment also aims to show how the material property affects maximum deflection. To look at this effect, the thickness is standardized at 6 mm and steel, aluminum, brass data are extracted from Table A to obtain the upcoming graph.

Figure E : Variation of Maximum Deflection With Material Type.