Structual design

Name of the Student
Professor’s Name
Structural Design
Date
Summary
The main objective of the project was to design a simple bicycle pedal crank taking into account several structural and economical parameters. The structural parameters were related to the yield of pedal based on the rotations after subjected to combined loading and the shear stress generated thereof due to the impact. Given the stress conditions the principle angle derived was 45 degree for both with the Aluminium material and the Steel material. Therefore, the yield subjected to remain the same irrespective of the materials used to manufacture the bicycle crank.
The Mohr’s cycle represented the likely positions of the pedal that represented the positions of maximum and minimum stress. Therefore, the force on the pedal applied to either of the design will be the same. Considering the fatigue limits, certainly the steel material had higher limits of fatigue at higher stress amplitudes. However, with the yield strength and fatigue limits taken into consideration, the two materials had fairly equal limits of 4. 7and5.1. Moreover, the limits of pressure applied do not go near the fatigue limits of either the aluminium material or the steel material.
This means the shear amplitude is far away from the fatigue limits and is nominal on the ascribed number of cycles. On the other hand, considering the cost effectiveness at various cross sections, we found that the cost of aluminium rods with the cross section of 18mm was cheaper than the 14 cm cross section based steel material.

Therefore, considering the amount of projected force on the pedal and the other structural and economic features we recommend that the pedal crank should be made of aluminium material.
Methodology
The design analysis was based on finding out the Mohr’s figures from the principle angle estimated from the particulars provided in Table 1.
Calculated for Aluminium
Inputs
Top of Form
  Normal Stress x:                                                                                                                                                                    
  Normal Stress y:     
  Shear Stress xy:     
Answers
  Maximum Normal Stress 1:  7.04 × 104  MPa                                                                                                                                                                                                                                                                
  Minimum Normal Stress 2:  -6.96 × 104  MPa   Principal Angle p:  45.0  deg                                                                                    
  Maximum Shear Stress max:  7.00 × 104  MPa   Maximum Shear Angle s1:  90.0  deg   and s2:  -0.0368  deg
 
Calculation for steel
.
Top of Form
  Normal Stress x:                                                                                                                                                                    
  Normal Stress y:     
  Shear Stress xy:     
Answers
  Maximum Normal Stress 1:  2.01 × 105  MPa                                                                                                                                                                                                                                                                
  Minimum Normal Stress 2:  -1.99 × 105  MPa   Principal Angle p:  45.0  deg                                                                                    
  Maximum Shear Stress max:  2.00 × 105  MPa   Maximum Shear Angle s1:  90.0  deg   and s2:  -0.00573  deg
ii. Finding out the limits of fatigue and reserve factors
Considering that limit of fatigue at 106 cycles for both aluminium and steel, would be nearly 300 Mpa and 200Mpa respectively. The ultimate tensile strength will also be obtained from Fig1. Dividing the limit of fatigue with the ultimate yield strength would signify what factorial of strength would be required to produce fatigue or loss of material, that would lead to a formation of cracks or breakdown of the pedal crank. For, the assessment Goodman’s relation was applied to understand the area occupied by the limit of fatigue with ultimate yield strength curve. The relation of alternating stress with limit of fatigue is given by the Goodman relation:
Sa =Sfat * (1- Smean/Stensile )
Where Sa =alternating stress
Sfat= limit of fatigue
Smean= Mean stress
Stensile =tensile strength.
By convention, the relation predicts that if the mean stress is increased with a given level of alternating stress then the decrease in fatigue life takes place. The safety loading criteria is given by the area under the curve marked by alternating stress in X axis and Mean Stress in the Y axis. The cut off points represents the limit of fatigue and tensile strength in the Y axis and X axis respectively. Larger the area under the curve, larger is the limit of safety loading. If the co-ordinates of a limit of fatigue and ultimate tensile strength are beyond the curve, then the safety factor is lost and will lead to fatigue of materials (Schutz 263-300).
The forces at various angles of 0, 90 and 180 degrees would estimate to find out the maximum force the pedal has to withstand from the equation of crank rotation
P (theta) = -24.7(theta)3+ 5(theta)2 + 234(theta) +20
The logic would be to indicate that if the forces are much lower than the limit of fatigue at 106 cycles it would be considered safe.
iii. The final determination of cost would be made from considering the density of materials and the volume required by the possible cross sectional data in Table 2. The convention is that the more the cross section of the pedal crank, it would lead to better stability for applying the force; however cost considerations needs to be incorporated before approval of the final decision of any of the chosen materials or the study. The volume of the cylinder would be
2 πr2h
Where h = length of the crank
r= cross section/2
Results
Mohr Diagram for Aluminium

Inputs
Top of Form
  Normal Stress x:                                                                                                                                                                    
  Normal Stress y:     
  Shear Stress xy:     
Answers
  Maximum Normal Stress 1:  7.04 × 104  MPa                                                                                                                                                                                                                                                                
  Minimum Normal Stress 2:  -6.96 × 104  MPa   Principal Angle p:  45.0  deg                                                                                    
  Maximum Shear Stress max:  7.00 × 104  MPa   Maximum Shear Angle s1:  90.0  deg   and s2:  -0.0368  deg
 
Bottom of Form
 
The Mohr’s circle associated with the above stress state is similar to the following figure. However, the exact loaction of the center Avg, the radius of the Mohr’s circle R, and the principal angle p may be different from what are shown in the figure.

Equations behind the Calculator
The formulas used in this calculator are,

Given the stress components x, y, and xy, this calculator computes the principal stresses 1, 2, the principal angle p, the maximum shear stress max and its angle s. It also draws an approximate Mohr’s circle for the given stress state.
Mohr Diagram for Steel

Inputs
Top of Form
  Normal Stress x:                                                                                                                                                                    
  Normal Stress y:     
  Shear Stress xy:     
Answers
  Maximum Normal Stress 1:  2.01 × 105  MPa                                                                                                                                                                                                                                                                
  Minimum Normal Stress 2:  -1.99 × 105  MPa   Principal Angle p:  45.0  deg                                                                                    
  Maximum Shear Stress max:  2.00 × 105  MPa   Maximum Shear Angle s1:  90.0  deg   and s2:  -0.00573  deg
 
Bottom of Form
 
The Mohr’s circle associated with the above stress state is similar to the following figure. However, the exact loaction of the center Avg, the radius of the Mohr’s circle R, and the principal angle p may be different from what are shown in the figure.

Equations behind the Calculator

The formulas used in this calculator are,

Fatigue Limits and Safe Loading Criteria
Limit of fatigue/ ultimate tensile strength
For Steel= 300/ 585 =0.51
For Aluminium= 200/427 =0.47
This means the tensile strength may be increased 0.51 times till fatigue limit is reached at 106 cycles for steel while the tensile strength may be increased 0.47 times till fatigue limit is reached at 106 cycles for aluminium.
Considering forces at the pedal at different radians 0 degree, 90 degree and 180 degree and outing it in the equation:
P (theta) = -24.7(theta)3+ 5(theta)2 + 234(theta) +20
Forces at various radians are
0 degree = -0 + 0 +0+ 20kpa
90 degree= -729kpa + 40.5kpa +21kpa +20kpa
= -647.5kpa
180 degree= -144050kPa + 162kpa + 42kpa +20kpa
= – 143826kpa
Hence, the force generated at 0 degrees is maximum and is 20 kpa
However, the limit of fatigue for 106 cycles is far beyond 20kpa for both steel and aluminium and hence both presents safe loading.
Cost-Benefit Analysis
Considering steel cross section of 14mm and Aluminium cross section data as 18mm
The radius comes out to 7mm and 9mm respectively
Putting it in the formula for cylinder volume for example 20mm length with cost per kg. Kilogram is estimated from (density * volume)
The total kilograms required for 18mm Aluminium and 14mm steel still puts a kilogram of steel required to be much more than aluminium. Multiplying the same with 3.5pound for aluminium and 2.1pound for steel, we found that pedals made from steel will cost more even compared to the 18mm cross section steel. However, the prices are comparable. Hence, from our logic we would prefer the pedal to be built with 16mm cross section and should be made of aluminium. This will be the final design considerations, and the principle angle will be 45degree.
Discussion
It is very pertinent for making a final design consider all dimensions on strength of materials and cost effectiveness of the design. The Mohr’s cycle represented the likely positions of the pedal that represented the positions of maximum and minimum stress. Therefore, the force on the pedal applied to either of the design will be the same. Considering the fatigue limits, certainly the steel material had higher limits of fatigue at higher stress amplitudes. However, with the yield strength and fatigue limits taken into consideration, the two materials had fairly equal limits of 4. 7and5.1. Moreover, the limits of pressure applied do not go near the fatigue limits of either the aluminium material or the steel material. This means the shear amplitude is far away from the fatigue limits and is nominal on the ascribed number of cycle. On the other hand considering the cost effectiveness at various cross sections we found that the cost of aluminium rods with the cross section of 18mm was cheaper than the 14 cm cross section based steel material.
Conclusion
From the dimensions and respective calculations, the total kilograms required for 18mm Aluminium and 14mm steel still puts a kilogram of steel required to be much more than aluminium. Multiplying the same with 3.5pound for aluminium and 2.1pound for steel, we found that pedals made from steel will cost more even compared to the 18mm cross section steel. However, the prices are comparable. Hence, from our logic we would prefer the pedal to be built with 16mm cross section and should be made of aluminium. This will be the final design considerations, and the principle angle will be 45degree
Works Cited
W. Schutz. A history of fatigue. Engineering Fracture Mechanics 54(1996): pp. 263-300