Using an energy method of your choice, calculate the deflection at point B for the beam depicted: Mechanics of Materials Report, UOL, UK

• Part A: Energy methods
• Question 1

1. Using an energy method of your choice, calculate the deflection at point B for the beam depicted in Figure 1. The second moment of area of the beam is I, with Young’s modulus E.
2. Express the answer as a function of E, I, L, and M₀. You must use an energy method and explain your steps. (You are of course free to use any other method to check your result)

• Question 2: 

For the beam depicted in Figure 2, calculate the reactions at all three supports (second moment of area I, Young’s modulus E).

(It is obvious that this problem is statically indeterminate. You could use the double-integration method or Macauley’s singularity functions. However, an energy method is significantly quicker here. We have not discussed this method directly in the lectures and tutorials and it is suggested that you carry out a bit of research if you want to use it. It is worth it.)

• Part B: Fatigue

You are analyzing the possible failure mechanisms on the shaft of a pump.

The pump is gear-driven at a uniform load and speed. The shaft is supported by bearings mounted in the pump housing. The shaft is made of steel with ultimate tensile strength S_u, and yield strength S_y. The tangential, axial, and radial components of force applied to the gear are shown in figure 3.

The surface of the shaft fillet has been shot-peened, which is estimated to be equivalent to a laboratory mirror-polished surface.

• Question 1

Calculate the Factor of Safety for static yielding. (It is possible that your result is below 1)

Please refer to the techniques taught in the year 2 modules: “Stress Analysis”. You should relate the stresses at the most critical location to the loads. Recall that some loads can contribute to different kinds of loadings. Use a suitable static failure criterion!

• Question 2

Calculate the Factor of Safety for fatigue failure for a life of 5.10⁵. (It is possible that your result is below 1).

Consider carefully which loads are cyclic and which loads are not. This is quite a lengthy procedure; make sure you break it down into sensible steps. Make also sure that you clearly identify the approximations you are using, and that you reference the origin of the tools (charts/graphs, etc…) that you are using to obtain certain parameters.

• Question 3

Considering that the data retrieved from literature or provided by the manufacturer may not be reliable, you decided to test the specimen in the laboratory yourself. Find the relevant standards for obtaining the relevant material data for your design, provide a brief description of the specimens and procedures in your report.

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