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Introduction and Theory

A beam is usually considered as member known to carry load diagonally to its length. In this experiment, it is symmetrically burdened as it is shown in Fig. III-1(a), therefore, P is the load applied. A shear force V is shown at any cross section of the beam (Fig. III-1(b)) a moment M is in (Fig. III-1c). V is zero and M has its maximum constant value where central part of the beam is shown. A positive moment is noted where deflection is negative.

EFGH of the beam small slice is imagined to be cut out, this is to make it clear that moment, M, that is applied should be balanced by the stress which is EF and GH. This is shown in Fig. III-2(a), the forces experienced there can deform the beam because it is of elastic material. When the beam is deformed FH would longer and EG would shorter.

There is an assumption that a plane when cut or the section of it usually remains as plane even after it is deformed. In Fig. III-2(b) EF and GH will deform into plane because there is an intersection at O and they are parallel and plane.

An arc of distance +η is usually considered from the axis that is neutral or the distance r + η from O (Fig. III-2(b)). The length of arc is l’=(r + η) Δθ. Fig. III-2(a), shows that l was the length of the arc before it was deformed. This length is equal to rΔθ. Therefore, distance +η of the strain is shown as follows:

(III-1)

However, axial strain is relative to neutral axis distance. This is because strain is positive and η is from the tensile side of N-N in Figure. III-2(b).

The other assumption that was put in the experiment is the application of the Hooke’s Law this is because it is same modulus of Elasticity. From this equation (I-3) and (III-1),

(III-2)

If the neutral axis distance is marked by c then it is assumed that the stress is found by σm = Ec/r, and Eq. (III-2) or

(III-3)

To get the formula of beam stress, the best way is to locate the neutral axis and by relating σm to M this is shown in Fig. III-2(c). For the equilibrium static its sum due to the internal forces must be zero as shown,

In another way to show the moment arms on the neutral axis is shown by

(III-4)

If I shows the moment of inertia and define it as the second moment of area about the neutral axis,

(III-5)

Eq. (III-4) is shown as:

(III-6)

Z = I/c is considered as the section modulus because the cross-sectional beam geometry depends on it. Equation (III-6) shows the stress of the beam equation and it relates to the maximum stress which is applied to the moment. This is same as Equation (I-1), because it defines the uniaxial tension equation. This can be well understood in that σm is usually the maximum bending stress. Therefore, σm and M are the functions of x, and they both related by Equation (III-6).

The degree of deformation of the beam is the remaining issue however; it is given where you get the radius r of the curve and the moment M which is related. The function y(x) curvature from calculus is given by

Therefore, x is the beam distance and y is the deflection in Fig. III-1(a). The deflection will be small if done practically and thus the slope dy/dx identified will be very small.

But, since σm = Ec/r = Mc/I, there results the differential equation of the elastic curve:

(III-7)

To obtain the elastic curve of the beam, y(x), and the maximum deflection, ym, it is necessary to integrate Eq. (III-7) using the moment function M(x) in Fig. III-1(c). Thus, using M(x) = Px/2 for

0 ≤ x ≤ a and M(x) = Pa/2 for a ≤ x ≤ a + b, it is found that

And that the maximum deflection at x = a + b/2 is

(III-8)

In particular, for a = b = L/3,

(III-9)

Procedures

MTS testing machine are the ones to be used in this test on a 1018 steel beam (E = 30x10*6 psi). The picture in the lab manual and the position of it in the beam should be same. The I-beam cross-section is not balanced as it was noted because one beam seems to be thicker than the other. It is well placed in the testing machine so that it is made squarely resting and centered as it is facing down. Then Test Work 4 software is installed in the computer and it is prompted by login in name “306A_Lab”. Then “exp-3 4 Point Flex Mod X” is selected.

This should be done so carefully so that strain gauge lead wires are not pinched. Then the thumb wheel is used when the digital load on the screen is observed so as to lower the fixture. Then an approximate 0.3 lb. the fixture is raised so that the preload is applied and the handset is locked. In the channel the strain gauge wire is connected and the data acquisition is performed by the software.

After that then in the dial indicator a magnet is usually positioned on the beam. This is because it is not supposed to touch the strain gauge. The activation of the magnetic base usually locks the MTS frame because of the zeroed dial indicator.

However, the lab view software is started and starts the strain gauge acquisition by pressing the white arrow and then the strain is zeroed. Then the arrow indicated in green is pressed this is to load the beam up to 1000 lb. then this experiment is done again when it zeroed out.

Summary of Important Results

Figure 1: Experimental Deflection vs Load

Figure 2: Theoretical Deflection vs Load

Figure 3: Stress vs Load

There is a direct proportional between deflection vs. load, stress vs. load and the relationship between theoretical deflections vs. load as shown by the three graphs. In plotting the second graph so as to find theoretical deflection we used 1.044 in which is the beam length and 0.05122 in4 which is the area moment of inertia. In the third graph the experiment is on calculating stress versus load and it is being done by section modulus dividing the moment. Moment is equal to distance multiplied by force.

Error Analysis

In every experiment done errors are unnecessary. They can only be reduced when an experiment is being conducted. Therefore, errors that were found in this experiment may be brought about by wrong calculations, the measurement mistake and also if the machines were not set well. However, the one doing the experiment may not be blamed because the machines could be having some problems or the samples given may not be in good condition.

Discussion and Conclusion

There are many things that are noted after this experiment is finished and all data calculated about the beam. These are stress and the deflection increase when the load increases. Experimental stress can found by the use of neutral axis and the moment of inertial and the theoretical stress can be found using strain. The dial indicator is used in the experiment to find the deflection and theoretical data is found by the use of length, moment of inertial and the elasticity. A directly proportional relationship is shown on the graph between stress and the load and also the deflection and the load.

Stress vs Load

Load (lb)

Stress (psi)

Experimental Deflection vs Load

Load (lb)

Deflection (in)

Theoretical Deflection vs Load

Load (lb)

Deflection (in)

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