Evaluate the maximum compressive stress

Assignment Help Mechanical Engineering

Reference no: EM132971643

SEM302 Advanced Stress Analysis - Deakin University

Problem Solving Task 1

Question 1: Prismatic Beam Design: Variational Loading

You have been tasked with designing a beam to support a lifting crane. The applied load is offset from the centre of the carriage used to mount the crane to the overhead beam. Therefore, the loads applied to the two axels, P1 and P2, differ. The position of the crane can vary (dependent on the variable, x, shown below).

Material Properties (Beam A-B)

Symbol Value Units

Loads

P₁ = 0.75P

P₂ = P

E 69.0 GPa

v 0.28 N/A

P 7.19 Mg/m³

σallow 330.0 MPa

τallow 110.0 MPa

Your task is to analyse the structure. To do this you will:

a) Determine an expression for the vertical reaction at Point A.
Note: this will be a function of x, where 0 ≤ x ≤ 3.75m.

b) Find the position of x that results in the maximum bending moment

c) If the beam has a rectangular cross-section (200mm high X 50mm wide), what is the maximum load, P, that can be applied to lifting crane? Apply a safety factor of 2 when performing this calculation.

Question 2: Combined Stresses and Superposition

A circular pier of width, d, must be designed to withstand a concentrated load, P, applied at the edge of the section. The material used to fabricate the pier is sensitive to tensile loading, hence, the allowable tensile strength (σT) must be used to design the pier. It can safely be assumed that the pier is sufficiently short to prevent buckling. As a consultant engineer you have been tasked with evaluating the circular pier and two alternative designs namely a rectangular pier and a crossed pier. It should be noted that the width of the two alternative designs, b, is unknown and you are tasked with designing the piers and evaluating the stress state.

The geometry of the three cross-sections (left) circular (middle) rectangular (right) crossed. It should be noted that all cross-sections are symmetrical about their centroid. The red dots represent the location of the applied force, P, for each pier.

The magnitude of the applied load, P, is 150kN.

Symbol	Value	Units
E	100.0	GPa
σallow(tension)	250	MPa
τallow	150.0	MPa
ρ	4.00	g/cm³

In order to compare the three concepts, you will:

a) Design the circular pier (diameter, d) to resist the maximum applied tensile stress.

Hint: to make this task easier you can use the principal of superposition to calculate the axial and bending stresses separately and sum the critical stresses to obtain the resultant stress.

b) Based on your design calculate the maximum tensile and compressive stresses in the circular pier.

c) Design the width, b, of the rectangular pier so the maximum tensile stress is equal to the maximum tensile stress in the circular pier. The dimension d is taken from your circular pier design.

d) Calculate the width of the crossed pier, b, so the maximum tensile stress is equal to the maximum tensile stress in the circular pier. The dimension d is taken from your circular pier design. Ensure that you document the method used to determine the moment of inertia I (and/or section modulus, S)

e) Evaluate the maximum compressive stress in each of the three designs and determine which design has the largest compressive stress.

Question 3: Deflection of prismatic and non-prismatic beams

You are tasked with analysing a cantilever beam with a cross-section that tapers in both width and height. The length of the beam is, L, and the cross section at the tip (x=0) has a width of 2b and a height of b. At the fixed end (x=L) the cross-section has a width of 4b and a height of 2b. Your client is interested in understanding the maximum deflection of the beam.

Side view of the tapered beam. The cross-section on the left-hand side of the image represents the tip of the beam (x=0) with a width of 2b and a height of b.

The parameters related to the problem:

Symbol	Value	Units
b	100.0	mm
L	2.0	m
P	2.0	kN
E	100.0	GPa
σallow(tension)	250.0	MPa
τallow	150.0	MPa
ρ	4.0	g/cm³

In order to assess the design, you will:
a) Use the method of integration to determine the maximum deflection of the beam assuming that the cross-section is constant along the length of the beam (width 2b and height b). Assume that x=0 coincides with the tip of the beam (as shown in the image).

b) Use the moment area method to determine the maximum deflection of the beam assuming that the cross-section is constant along the length of the beam (width 4b and height 2b).

c) Calculate the maximum deflection of the tapered cantilever beam. Note: you must account for the changing cross-section of the beam along the length.

Attachment:- Advanced Stress Analysis.rar

Reference no: EM132971643

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