CVG 2140Mechanics of Materials I

Lab 2: Flexural Stresses in Beams

Christopher Morgan6605934

Lab Date: March 2nd, 2012Report Date: March 16th, 2012T.A.: hanva075@uottawa.caLab Group: 62Group Members: Mahabal, Mikaelian, Najafali, Najam

Table of ContentsObjectives2Theoretical Background.2Materials and Equipment.2Experimental Procedure2Collected Data..3Experimental Results5Data Analysis....5Discussion...8 Conclusions and Recommendations....8References .9

Objectives The objective of this experiment is to determine the flexural stresses at various points on a simply supported beam subjected to two concentrated loads.

Theoretical BackgroundA beam is a structural member whose length is large compared to its cross sectional area which is loaded and supported in the direction transverse to its axis. Lateral loads acting on the beam cause the beam to bend or flex, thereby deforming the axis of the beam into a curved line.

The equation below gives the bending normal stress, and is also commonly called the flexure formula. The y term is the distance from the neutral axis. The I term is the moment of inertia about the neutral axis. M is the maximum bending moment along the beam. The maximum moment along the beam can be found from the bending moment diagram

Materials and Equipment Hollow aluminum beam (2438 mm length) with rectangular cross section. Loads (weights) 20lbs each (8 weights for a total of 80lbs x 2) Strain gauges attached at 5 specific points along the beam. Deformation indicator

Experimental Procedure1. First step is to locate and measure the location of the stress gauges. 2. Once located, two equivalent concentrated loads are applied at 1/3 of the span length from the supports. The corresponding strain was then read for each gauge. 3. The loads were then increased from 20-80 lb at each point, recording the data for each interval.4. Finally, unloading the loads at the same interval of loading, once again recording the data for each load change.

Collected Data The moment of inertia with respect to the neutral axis was calculated. Theoretically, the neutral axis is situated at the centroidal axis, marked z on figure 1. Due to the cross section of the beam being hollow, the moment of inertia of shape 2 (inner cross-section) was subtracted from the moment of inertia of shape 1 (outer cross-section) demonstrated below.

Table 1: Beam dimensions

Outer (mm)Inner (mm)

b7652.875

h2518.75

= Since the loads are applied symmetrically on the beam, the reactions at the supports are equal to each other and have the same value as the applied loads.

The maximum bending moment is equal to the area under the shear force diagram as shown in figure 2. It can be calculated using the equation below.

Figure 2: Shear force diagrams for the beam under the experimental loading

The bending normal stresses at the different strain gauges can be calculated using the flexural equation presented in the introduction and shown below.

The estimated Youngs modulus at the different strain gauges can be calculated using Hookes law as shown below.

Experimental Results Table 2: Experimental Strains

LoadStrain (x10-6)

lbNS.G.1S.G.2S.G.3S.G.4S.G.5

00-29-26-28-27-19

2088.99-132-103-525183

40177.98-237-181-72129186

80355.97-443-334-116285392

40177.98-236-179-72131186

2088.99-133-102-515284

00-29-26-28-26-19

Data AnalysisTable 3: Calculated stress corresponding to gauges position

Load (lb)Calculated Stresses (MPa)

S.G.1S.G.2S.G.3S.G.4S.G.5

00.000.000.000.000.00

207.145.371.795.377.14

4014.2710.733.5810.7314.27

8028.5421.467.1521.4628.54

4014.2710.733.5810.7314.27

207.145.371.795.377.14

00.000.000.000.000.00

Table 4: Estimated Young's modulus from the measured strains and calculated stresses

Load (lb)Estimated Youngs Modulus (GPa)

S.G.1S.G.2S.G.3S.G.4S.G.5

00.000.000.000.000.00

2058.5557.7642.6388.0676.81

4062.9462.8357.7677.2972.89

8065.9966.3267.5772.8471.08

4063.2263.5757.7676.1972.89

2058.0858.3943.6786.6475.99

00.000.000.000.000.00

Bending Moment Diagrams

Figure 3: Bending moment diagram for P=20 lb

Figure 4: Bending moment diagram for P=40 lb

Figure 5: Bending moment diagram for P=80 lb

Figure 6: Neutral axis determination

DiscussionDetermining the flexural stresses of a beam is an important technique used in real life in order to assure the safety of the beams when subjected to various loads. The results obtained from this experiment were compared to the theoretical calculations obtained from derived formulas. After analysing these results, it can be stated that the experimental data coincides with the theoretical data fairly accurately. The apply loads in this experiment were applied in a hanging manner from under the beam therefore this created a tensile stress below the neutral axis of the beam and a compressive stress above the neural axis of the beam. The neutral axis of this beam was experimentally located just above the center point along the z axis, compared to the theoretical location being in the direct center. That being said, the experimental values as well as the theoretical values prove just this by having negative stress at stress gauges 1, 2 and 3 and positive stress values at stress gauges 4 and 5. From this information, the modulus of elasticity was experimentally calculated as seen in table 4 and compared to the theoretical value of aluminum of about 70 GPa. The experimental values fall well within range of the theoretical value therefore in can be said that this beam was indeed reacting appropriately.

Conclusions and recommendationsIn conclusion, the laboratory experiment was successful and the objective of determining the flexural stresses at various points on a simply supported beam subjected to two concentrated loads was established. According to the experimental and theoretical values, a small margin of error was accounted for, however within reasonable range to conclude that the theory with respect to the stress of the beam was indeed acting according to the applied loads applied to it.

References

Hibbeler, R.C. (2008). Mechanics of Material. Pearson Prentice Hall.

Martin-Prez, B. (Winter 2012). Flexural Stresses in Beams. CVG 2140 Laboratory Manual.

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