Introduction:

As the force transmitting members in structures and assemblies, beams are often subjected to a multitude of forces which in turn induce bending in the beam. As engineers, one of the key factors to consider during the design process is to account for the direction that this bending takes place, as unpredictable bending can result in premature failure of a member which in turn places the entire system and user into unnecessary danger.

To analyze the bending of a simple and symmetrical beam, we define two axes along the cross section of the beam called the principal axes. These axes pass through the centroid of the member but does not necessarily line up to the geometrical axes we define, since the geometrical axes are arbitrarily chosen for analysis. Any bending that occurs along both axes as a result of a moment that is directly applied to either axis will result in pure bending along that direction. The bending of this nature can easily be predicted by the bending equation. However, in real world application moments do not, and often will not, be directly applied along the principal axes. This is especially the case when the beam in question is oddly shaped, as the principal axes cannot be easily defined. When this occurs, the moment is applied at an angle and consists of two components that are projected along both principal axes, which generates bending in both directions. This type of bending is known as unsymmetrical bending, and is analyzed in the first portion of this lab using Mohr's circle for two angled and U-sectioned elements.

We generate five moments using masses five masses ranging from 100 to 500 grams, load each at 22.5 degree increments, and record the deflection of the beam in the U and V directions. Using this data, we plot the deflection against the load for each load angle, and obtain the gradients for each graph. Then, the gradients of the bending in the U direction are plotted against those in the V direction to obtain Mohr's circle. From this plot, the second moment of area can be determined. These results are then compared to the accepted values given by the second moment of area formulas.

Experiment two explores the bending of a beam when a load is applied at a point other than the shear centre. This results in an accompanied twisting of the member, is a result of reaction forces from the beam. Only at the shear centre will the moment generated by reaction forces cancel each other out, which will cause equal deflection along both axes of the beam. This is interpreted graphically as the intersection of the plot for deflection in the x and y directions versus the eccentricity of the load, which we define as the point we apply the load. These results are compared against the theoretical shear centre, which can be computed using the provided formulas.

Procedure:

Unsymmetrical Bending:

1. The angled specimen was first chosen for analysis and secured in place using the thumbscrews.

2. The deflection indicators were zeroed to ensure a load of zero results in zero deflection.3. The chuck hand wheel was set to a loading angle of zero degrees and a mass of 100

grams was hung from the pulley. The deflections on both indicators were recorded. The values for deflection in the U and V directions were also calculated.

4. The above step was repeated for the 200g, 300g, 400g and 500g masses5. The chuck hand wheel was adjusted to set a loading angle of 22.5 degrees and the above

steps were repeated to obtain a table of U and V for loads of 100 to 500 grams.6. Using the procedure above, we obtain tables that indicate the deflections in the U and V

directions for a loading angle from 0 to 180 degrees, incrementing each time by 22.5 degrees.

7. Plots of U vs P and V vs P were obtained for each angle and the slopes of the trend line noted.

8. Mohr's circle was plotted and the second moment of areas were graphically determined.9. These values were compared to the theoretical values obtained from calculations. 10. The angled specimen was swapped out for the U-shaped specimen and the above steps

were repeated.

Shear Centre:

1. A constant load of 500g was chosen as the designated load for this experiment. 2. The U-beam from the previous experiment was kept for analysis. 3. The apparatus was fitted with the shear centre plate which allowed for a varying point of

application. 4. The load was applied at eccentricity -25mm, located to the left of the shear centre and the

deflection in both indicators recorded. 5. The eccentricity of the load was increased by 5mm to the -20mm position and the load

was again applied, and the deflections recorded. This was repeated until we obtained readings for an eccentricity of positive 25 mm.

6. The deflections in both directions were plotted against the eccentricity to find the intersection point, which we designate as the experimental shear centre.

7. The theoretical shear centre was calculated and discrepancies between the two values were analyzed.