Simple Beam Bending: Evaluating the Euler-Bernoulli Beam Theory

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1 Simple Beam Bending: Evaluating the Euler-Bernoulli Beam Theory Tanveer Singh Chandok AE 325 Introduction Euler-Bernoulli Beam Theory is a simplification of the linear theory of elasticity which provides a means of calculating the load-carrying and deflection characteristics of beams. It is of interest to evaluate how well such a theory holds compared to experimental analysis. The scope of this report is to first evaluate what the theoretical values for deflection are, for three different cases of a simple beam in bending, then to compare these results to the experimental data obtained, drawing conclusions along the way as to how the Euler- Bernoulli Simple Beam Theory holds up. Euler-Bernoulli Beam Theory can be used to predict the theoretical values of beam deflection (among other quantities). These values of beam deflection will be used in the analysis, as they will be compared to the experimental data obtained. This report will evaluate a simply supported beam that has a downward load (P) applied at the midpoint. The downward load will cause a vertical deflection (Δ) at the midpoint of the beam. Three different cases will be considered. These cases are as follows:. Single beam, weak axis, 8 ft. length 2. Single beam, weak axis, 6 ft. length 3. Single beam, strong axis, 8 ft. length For each of these cases, Δ/P will be evaluated using simple beam theory. This value of Δ/P will be compared to the Δ/P obtained from the experiments. The experiments conducted used commercial wooden 2x4 s, whose E is.6 x 0 6 psi (kiln dried Canadian fir). The loads for the experiment varied from 50 to 90 lbs and the deflections were observed on a home-made dial gauge with a scale. The deflection was measured in units. These deflections in units will be converted to inches using a sensitivity factor. The deflection vs. load for each of the tests will then be plotted and the slope of the line (Δ/P) will be compared to the theoretical value of Δ/P obtained from simple beam theory. This will serve the purpose of evaluating whether the theory accurately predicts the deflection in the experiments. Conclusions will be drawn and any errors will be analyzed. Theory Figure shows the simply supported beam with a downward load (P) applied at the midpoint (L/2). The beam has a total length of L and is supported by a fixed pin at the left hand side (no moment and no displacement) and a roller at the right hand side (prevents axial loading). The beam has a vertical deflection (Δ) at the midpoint where the load is applied.

2 Figure : Simply supported beam with downward load applied at the midpoint In order to evaluate the deflection, the beam bending stiffness equation (about the centroid) must be defined (second order O.D.E) as: Equation : Second order beam bending differential equation The bending moment distribution can be evaluated from the figure. Figure 2 shows the reaction forces at the ends of the beam. The beam is cut at two different points to evaluate M 3 from 0 to L/2 and from L/2 to L. The moments are summed at the cuts (moment equilibrium). The moments vs. distance graph is plotted thereafter. Figure 2: Simply supported beam showing reaction forces and M 3 Equation 2: M 3 from 0 to L/2 Equation 3: M 3 from L/2 to L Graph : Moment (M 3 ) vs. Distance (x ) across the beam

3 The next step towards obtaining the relationship for deflection is to evaluate the bending stiffness of the beam (using the cross section). In the three different cases, two of the beams are aligned on the weak axis, and one is aligned on the strong axis. The figures below show the weak axis and the strong axis cross sections with i 2 aligned in the vertical direction and i 3 aligned in the horizontal direction as shown. Figure 3: Weak axis Weak Axis: b = 3.5 inches h =.5 inches Strong Axis: b =.5 inches h = 3.5 inches Next, the bending stiffness for the weak and strong axis is computed using Equation 4. Equation 4: Bending stiffness Figure 4: Strong axis Therefore, we get: Evaluating all the equations above in Mathematica gives a value for Δ: Equation 5: Theoretical Deflection

4 This equation yields: Equation 6: Theoretical Deflection/Load (Slope) Experiment The experimental setup is shown in Figure Figure 5: Experimental Setup and Gauge Dial Figure 5 is a sketch of the mechanical experiment built to measure the deflection of the beam. The mechanism consists of a small aluminum pulley system which is mounted on a shaft that is supported at its ends. The pulley can rotate freely. Wrapped around the pulley is a small monofilament (very thin) fishing wire. One end of the fishing wire has a counterweight attached to it. The other end is attached to the beam using an eye hook. The counterweight ensures that there is tension in the wire. When the beam deflects downwards from the initial position shown in Figure 5, the wire will move the pulley clockwise and the attached pointer will display a number for the deflection in units, on the homemade dial (shown in the experimental video). The relationship between the downward deflection and the rotation of the pointer must be found. This can be done by considering the diameter (and circumference) of the pulley.

5 Deflection (units) Equation 7: Relationship between deflection (in) and deflection (units) The diameter of the pulley is d = in. and s is the sensitivity factor. Therefore, from looking at the units of deflection on the pointer, the vertical deflection can be computed. Results of Experiments: P (lbs) Zero Owen Nick Jacob Josh Length Test Notes (ft.) Length (in.) Bending Stiffness (H 33 ) [lb in 2 ] Weak Axis - 8 ft Δ Weak Axis - 6 ft (Units) Strong Axis - 8 ft Table : Data from experiment Table shows the deflection of the test beam (for all 3 cases) in units. It also shows the length of the beam and the bending stiffness s. The graph below represents the data in Table Deflection (units) vs. Load (lbs) Load (lbs) Test Test 2 Test 3 Graph 2: Deflection (units) vs. Load (lbs) The following Table displays the deflection in inches (computed using the sensitivity factor), the experimental and theoretical slope, and the error percentage between the slopes.

6 Deflection (inches) P (lbs) Zero Owen Nick Jacob Josh Test Slope (Exp) Slope (Theory) [in/lb] [in/lb] Error (%) % 2 Δ (Inches) % % Table 2: Deflection in inches, slopes (experimental and theoretical) and error % The graph below represents the data in Table 2. Deflection (inches) vs. Load (lbs) Load (lbs) Test Test 2 Test 3 Graph 3: Deflection (inches) vs. Load (lbs) From the table and graph above, it is noticed that the plots are linear for beam deflection with increasing loads. This aspect holds true with the Euler-Bernoulli Simple Beam Theory. Looking at the Error column in Table 2, it is seen that Test and Test 2 were quite accurate, whereas Test 3 was not. These errors will be discussed in the next section. The following table summarizes all the properties of the beam and the experiment. Weak Axis Strong Axis Properties E psi b 3.5 inches h.5 inches b.5 inches h 3.5 inches

7 Sensitivity factor s in/unit Diameter of Pulley d inches Table 3: Properties of the beam and the experiment The next section will introduce the analysis/evaluation aspect of the report. Evaluation Using the theoretical data discussed, and the experimental data obtained, it is seen that the Euler- Bernoulli Beam Theory correlates well with experimental data for two of the cases. The error percentage for the first two cases was less than % and less than 2% respectively. This small error can be neglected since there are various factors (atmospheric conditions, significant figures, etc.) that could skew the data (since it is so sensitive). Further, since the experiment was homemade, there was some inherent error in the system (±0.2 in reading the scale, tension in the wire, etc.). However, Test 3 has a blatant error of 7%. This is quite large and cannot be ignored. The main cause for such a discrepancy can be attributed to the fact that only data for one load was collected for Test 3 (P = 90 lbs). The only two data points for the experimental test were (0,0) and (90, 0.48), the first of which is a simple initial condition. More data would be needed to accurately describe the experimental bending of an 8 ft. single beam on its strong axis. If only the single data point (90, 0.48) was to be evaluated, it would compare moderately with the Euler-Bernoulli Beam Theory: This theoretical value of in is moderately close to the experimental value of 0.48 in. However, it is not satisfactory. As mentioned above, the only way to accurately determine the experimental deflection would be to conduct tests over a larger band of loads and try to ensure that the experiment is conducted without grave human and mechanical errors (using a laboratory environment). Conclusion From the above experiments and theoretical data it can be concluded that the Euler-Bernoulli Beam Theory is a good estimate of beam deflection and can be used provided that enough data is collected to perform a slope analysis and that the experiments are conducted with minimal human and mechanical error. With a 66% (2/3 tests) success rate in this report, the Euler- Bernoulli Theory can be trusted to provide accurate theoretical predictions.