Deflection Beam Part 2

GOAL: Design a beam shape that deflects a certain distance under a given load. Use analytical and numerical methods to aid in the design process. 

SKILLS: Castigliano's theorem, Solidworks Simulation (FEA), iterative design, material properties

Design Process

Next, I did Castigliano’s analysis for a tapered c shape, with the horizontal beams tapering symmetrically from 0.65 in to 0.25 in. This was done with the same equations as the simple c shape, but here, h varies along the length of the vertical beams (desmos calculations: https://www.desmos.com/calculator/oluc1vqe7w). This got a deflection of 0.3765 in, which is close to the FEA value of 0.3633 in. 

After some trial and error, I created a shape in which the vertical beam is not directly over the left support. For this structure, Castigliano’s on the bottom beam was slightly different (https://www.desmos.com/calculator/bxveniizls). 

While the general shape is similar, this structure is still not exactly the same as my final design, which has complex asymmetrical geometry and features such as filets, splines, and holes. Although not exactly the same, this simplified Castigliano’s model aptly serves its purpose of confirming the legitimacy of the FEA setup. Since I confirmed that the results of the FEA analysis are reasonable for the simplified structure, I can assume that they will remain reasonable as I make minor changes such as rounding edges and removing material. Meticulously using piecewise functions and parallel axis theorem to add the holes would not be functionally relevant because the model still would not be exact. 

The small discrepancy between Castigliano's analysis results and the FEA results was because the geometries still were not exactly the same. Castigliano's calculations assumed that the beams were collapsed into lines, and that the leg lengths were measured from the centers of the beams. In the FEA analysis, there was excess material past the “end” of the beam length so that the beams could be attached via pins. Not only was there excess material, but the ends also had a circular shape, which removes material and can affect deflection. The material interaction at the corners of the structure is also not considered since Castigliano’s is performed by isolating each beam; this was likely the reason why all the Castigliano’s results overshot the FEA results (material in corner junctions kind of acts like a filet). Additionally, the fixtures defined in FEA were one that was constrained in every direction but rotation, and one that was only supported vertically. In the castigliano’s calculations, I only used simple supports for both pins. Furthermore, my analytical method only dealt with bending and compression of a beam to find the deflection of the beam, and did not account for other sources of deflection such as shearing. 

Final Design

My preliminary attempts at designing a shape included c-shape, s-shape, and hanger shape structures; I ended up with a design that averages a c-shape and hanger-shape. The initial Castigliano’s showed that about .05 inches more deflection was needed for the simple c-shape, which was good because this could be achieved through removing mass. After fileting and tapering the material, the deflection was over 0.3 inches without yielding, so I decided to decrease the length of the top horizontal beam in order to decrease both mass and deflection, resulting in more of a hanger shape. Then, based on the stress plots, I added holes in regions that were under less stress to further decrease the mass. With this basic design, I then integrated off of it with small changes like adding splines and changing hole and filet sizes to get as close to the target deflection as possible. Something surprising that I noticed was that, although removing material generally increases stress, sometimes removing material by adding holes actually decreased stress concentrations. I believe this is because the distribution of stress in an area depends on the surrounding geometry, and so adding a hole may get rid of a high-stress yielding region and increase stress in a less critical area through redistribution.

Testing Results

The key results of real-life testing was an observed y-Displacement at 100lbs of 0.3217 inches and a mass of 55.2 grams. This was close to my FEA projected values of 0.2986 inches and 56.795 grams, with a percent error of 7.736% and −2.808%, respectively. The mass was the lightest out of all other groups' designs. From these results, I calculated a spring constant of k=-310.848lbs/in using Hooke’s Law. 

Analyzing the graph generated by the instron system shows there is a linear relation between the force (lbf) and displacement (in). The graph is linear because the testing remained in the linear elastic deformation range. Because the deformation is elastic, I can calculate a spring constant for the beam with Hooke’s Law (F=-kx). Spring constant is equal to restoring force over displacement, so this is essentially the negative slope of the graph. When I graph the data points, I get a linear regression equation of y=310x+0.446. Ideally, I would not have a +b value in the linear equation since deflection should start immediately as force is applied. This can be explained by the initial shifting that occurs as the structure settles into the pins. Graph A shows that the settling of the structure adds points at x=0.

There is also a very slight curvature in the rest of the data, but this is not significant enough to be attributed to anything other than small material irregularities. When I calculate the spring constant using the points (0,0) and (0.3217, 100), which is reasonable if I assume the graph is linear, I get a spring constant value of k=-310.848lbs/in. This is very close to the opposite of the slope of the regression equation. In fact, an equation with the point-derived slope (y=310.848x, shown in blue) more closely resembles the data than the regression equation:

The key results of the instron testing was an observed y-Displacement at 100lbs of 0.3217 inches and a mass of 55.2 grams. This was close to my FEA projected values of 0.2986 inches and 56.795 grams, with a percent error of 7.736% and −2.808%, respectively. From these results, I calculated a spring constant of k=-310.848lbs/in using Hooke’s Law. 

Deviations between Predicted (FEM/Castigliano) and Observed Results (Instron)

My actual deflection was significantly greater than the predicted deflection; this was also the case for almost every other group. Because of this, I believe that it likely had to do with the lot of material that I received. Real materials, unlike simulated materials, can have defects such as non-uniform thickness and other irregularities. While we expected a thickness of 0.25 inches, the real thickness was measured to be between 0.2645 and 0.2652 inches. This would not have caused the overshoot, though, since a thicker material would mean a larger moment of inertia, and consequently a decreased deflection. If the source of the overshoot was not the thickness, then I believe it was probably the Young’s Modulus. The expected Young’s Modulus of the extruded Al 6061-T6511 material was 10^7 psi, but it appears that the material we got had a lower modulus, leading to inflated real deflection.

Another source of error between FEA and Instron results could be the FEA meshing precision. As I increased mesh density, the deflection increased. Although the mesh convergence analysis showed that the mesh converges, there could still be some discrepancies due to this, which makes sense with an overshot result.