Advanced Stress Analysis for Mechanical Engineering Assignment

Assignment Task

1. Beam Analysis

Two beams (ABCD and DE) are connected using a pin at Point D. The pin acts as a moment release, i.e. no moments are transferred through this pinned connection. Shear forces can be transferred through the pinned connection at point D. Beam ABCD has a pinned support at point A and a roller support at Point C. Beam DE has a roller support at Point E. A concentrated load, P, is applied to the mid span of beam DE, as shown below. Another concentrated load, 2P, is applied to Point G. The bracket BFG can be considered rigid.

To analyse this structure, you will:

a) Construct the free body diagrams for the structure shown above. When constructing your FBD’s you must make section cuts at point B, C and D. You do not need to construct the free body diagram for bracket BFG. You can represent the structure as four separate beams. Following this, construct the bending moment and shear force diagrams for each segment of the beam. Do not substitute in values for P, L etc. Keep your answer in algebraic form. Clearly label all the key features of the shear force and bending moment diagrams (include features such as maximum values shear force and bending moment, and the locations where the values change from positive to negative).

b) Both beams (ACBD and DE) have a solid square cross-section with side lengths ‘aABCD’ and ‘aDE’. Design the beams to withstand the applied loading. Clearly state your calculated value of a in mm for each possible failure mode for both beams to one decimal place. Use the parameters shown in Table 1 to calculate the beam dimensions.

2. Stress Transformation and Principal Stresses A mountain bike manufacturer is considering a new product line that will include handlebar extenders applied to the ends of the handlebars, as shown below. To validate this design change, you have been tasked with analysing the structure. You have produced an idealised structure, shown below, where you consider half the length of the handlebars with a fixed boundary condition (A) applied to the plane of symmetry. You have estimated that the maximum load that a rider can apply to one handlebar extension is 75 Newtons that acts 100 mm from the centreline of the handlebars. The handlebars have a circular cross-section with an outer diameter of 32 mm and a wall thickness of 3.25 mm.

To assess the design, you will:

a) Determine state of stress at all points (P1, P2, P3 and P4) adjacent to the fixed support (A in the image above). These points are located on the exterior surface of the handlebars. You must consider any stresses at this point due to bending, transverse shear and/or torsion. Present your results in a table and ensure that your sign convention is clearly shown (and applied consistently!).

b) You have identified Point 1 (P1) as a critical point. Use the stresses at Point P1 to calculate the maximum principal (σ1) and maximum in-plane stress (τmax) adjacent to Point A. Ensure that you sketch the resulting state of stress at this point clearly indicating the magnitude of the stresses and any angles associated with the state of stress (principal or maximum inplane shear). State your answers to a precision of two decimal points. You must also show the axes and your sign convention.

c) P1 is the critical point. If the maximum permissible tensile stress is 150 MPa and the maximum permissible in-plane shear stress is 55 MPa, what is the largest load, P, that can be applied by the rider? Justify your answer.

3. Indeterminate Beam Analysis

Two identical prismatic beams are fabricated from steel and arranged as shown in the diagram below. Beam ABC can be considered cantilevered and beam DE simply supported. Due to the arrangement of the beams, there is an initial gap, ∆, separating the beams. The gap is present before any load has been applied to the beams. The horizontal distance between point A and B is 3L/4.

You have been tasked with analysing the beam arrangement.

a) A non-uniform distributed load is applied to beam ABC as shown above. Determine the intensity of the distributed load, ω0, that closes the initial gap, ∆. This means that the deflection of point B for beam ABC will be equal to ∆. You can assume that the beams will contact and that the initial gap will be closed. Use centreline dimensions in your calculations.

b) The intensity applied to beam ABC is increased by a factor of 2. (i.e. your answer for the intensity magnitude ω0, obtained in part (a) will be doubled and applied to beam AB). For this new load case, calculate the maximum deflection of beam ABC at point B. You can assume that the contact force between the two beams can be represented as aconcentrated point load. Use centreline dimensions in your calculations. Don’t forget to include the initial gap!

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