Comparison of bending in beams

Objective: To compare the performance of three types of beams under a bending load.

Problem Statement:

To create the sketches shown below in Solidworks:

Case 1:

Case 2:

Case 3:

To extrude the sketches to a length of 400mm

Apply a load of 1500N on the smaller face(i.e. 45mm face for case 1, 40mm face for case 2 and 45 mm face for case 3)

and compare the results of stress, displacement, and strain for all three cases and suggest the best out of the three.

Theory:

The beam, or flexural member, is frequently encountered in structures and machines, and its elementary stress analysis constitutes one of the more interesting facets of mechanics of materials. A beam is a member subjected to loads applied transverse to the long dimension, causing the member to bend.

Bending stress or flexural stress commonly occurs in two instances. One is called a simply supported structural beam bending and the other is called cantilever bending. For the simply supported structural beam, the upper surface of the bending beam is in compression and the bottom surface is in tension. The neutral axis (NA) is a region of zero stress. The bending stress (σ) is defined. M is the bending moment, which is calculated by multiplying a force by the distance between that point of interest and the force. c is the distance from the NA and I is the moment of inertia. The cantilevered beam configuration has a similar formula. The formulas for M, c, and I can be complex, depending on the exact configuration and beam shape, but many are published.

Cantilever beams and simple beams have two reactions (two forces or one force
and a couple) and these reactions can be obtained from a free-body diagram of the beam
by applying the equations of equilibrium. Such beams are said to be statically
determinate since the reactions can be obtained from the equations of equilibrium.
Continuous and other beams with only transverse loads, with more than two reaction
components are called statically indeterminate since there are not enough equations of
equilibrium to determine the reactions.

Stress
The determination of stress distributions of beams in necessary for determining the
level of performance for the component. In particular, stress-based failure theories
require determination of the maximum combined stresses in which the complete stress
state must be either measured or calculated.

Strain
Although strain is not usually required for engineering evaluations (for example,
failure theories), it is used in the development of bending relations.

Deflections
Often limits must be placed on the amount of deflection a beam or shaft may undergo when it is subjected to a load. For example beams in many machines must deflect just the right amount for gears or other parts to make proper contact. Deflections of beams depend on the stiffness of the material and the dimensions of the beams, as well as the more obvious, applied loads and supports.

Study Setup:

These are the parts created with the dimensions given in the problem statement.

The following material(Alloy Steel) is applied to all the parts.

Then add the fixture to the faces as shown in the following figures.

Then add the forces on the respective faces as shown in the figure.

These mesh characteristics are used while creating the mesh for each case.

The mesh appears to be as shown for each case.

Procedure:

1. The parts are created for each case using the dimensions as mentioned in the problem statement.
2. Then using the simulation add the case study.
3. The Material type is selected by using the add material option.
4. Then the face which is to be fixed is selected in the fixtures option.
5. A force of 1500 N is added using the external loads feature to the required face as shown.
6. Then create the mesh using the characteristics as shown. The mesh and run option is to be selected to execute and plot the desired results.

Results:

These plots show the different results such as the Stress, strain, and displacement acting in the beams using the heat map which are being plotted for case 1. It can be seen from the plots the stress and the strain tend to occur at the fixed end where the maximum molecular forces act because of the bending in the beams which gradually decrease as we move away from the fixed end. And the displacement is maximum at the end because it's the area that is not bounded like the opposite so the beam is able to move freely.

These plots show the different results such as the Stress, strain, and displacement acting in the beams using the heat map which are being plotted for case 2. It can be seen from the plots the stress and the strain tend to occur at the fixed end where the maximum molecular forces act because of the bending in the beams which gradually decrease as we move away from the fixed end. And the displacement is maximum at the end because it's the area that is not bounded like the opposite so the beam is able to move freely.

These plots show the different results such as the Stress, strain, and displacement acting in the beams using the heat map which are being plotted for case 3. It can be seen from the plots the stress and the strain tend to occur at the fixed end where the maximum molecular forces act because of the bending in the beams which gradually decrease as we move away from the fixed end. And the displacement is maximum at the end because it's the area that is not bounded like the opposite so the beam is able to move freely.

Conclusion:

In case 1 we have used a rectangular beam which is completely solid, unlike the other cases where there are 'I' section and 'C' Section used in Case 2 and Case 3 respectively. So the weight of this beam would be the highest which is not ideal with the lowest stress and displacement amongst all the cases which are the only desirable aspects. Moving on to the 'I' section we find that it has a lower weight compared to Case 1 but higher than Case 3, also the stresses are slightly higher than the case 1, the strain is the lowest amongst all, with the displacement being moderate as well. Case 3 is not a good choice because it's not matching up to the expectations of a safer design in terms of the values of stress, strain, and displacement along with the plot results which doesn't instill confidence from a designer's perspective.

So, after judging all the three beams we can comprehend that Case 2 i.e 'I' section beam is the most suitable for design which covers all the bases necessary along with the safety aspects.

References:

https://www.sciencedirect.com/topics/engineering/bending-stress