Bike Rack Design Project

2013

The 4Racketeers Design Engineers:

Jake Wang, Daniel Tai,

Augustus Bechtold, Josh Aguilar

6/6/2013

Bike Rack Design Project

COPYRIGHT © 2013 JAKE WANG

Abstract:

The purpose of this project is to present our strength, deflection, buckling, and fatigue analysis for our bike rack design. Based on our company’s prior experience, we are implementing constraints on our design with a maximum bicycle weight of 40 pounds, a maximum dynamic load, of 3G, a design safety factor of 2.0 for strength, a design safety factor of 3.0 for resistance to buckling, total system deflection less than 1 inch, limit wide of bike and rack to maximum of 4 feet, limit the maximum overhang of the rack behind the motorcycle to a maximum of 4.5 feet, lifetime of 10,000 trips with a design safety factor of 5. With these constraints, we are basically analyzing a total force of 240 pounds (F (total) =2 X 3 X 40=240 lb.).We decided to choose our favorite mechanical design supplier at www.metalsdepot.com/products, allowing us the flexibility to compare available carbon steel types and dimensions for our updated design analysis. The preliminary justifications for using carbon steel are because of its inexpensive price, high yield strength, and broad availability. With the completion of our design analyses, we can conclude that our initial justifications were correct and design parameters our purposed bike rack would meet our company’s requirements.

Included is our table of design analysis and comparison, detailed design descriptions, detailed geometry of structural components with CAD drawings, and mechanical design calculations that justify our decision for the primary dimensions of our bike rack design.

We have picked the BMW GS (see Image A-1) motorcycle model for modeling and idealization of our bike rack design. We will connect our bike rack to the luggage rack on the BMW GS. Our bicycle rack system consists of six sections: a vertical beam mainframe, a base receiver, a horizontal beam support, two link attachments, three strap-on mechanisms, and two loop-rod bicycle secure attachments, (see CAD Drawing 1 and 2). The finalized overall geometric size for our bike rack system is the same as our primary chosen dimensions, 80 inches long and 41.40 inches wide, but it only overhangs behind the motorcycle for about 16 inches.

The estimated total weight of our design is about 86.68 pounds (although the weight for the bike rake seems heavier than the estimated bike weight, it is sturdy and has a very long lifetime, the estimated total production cost per unit of this bike rack is $216.50 ($80 cheaper compared to the only vendor, 2X2 Bicycle Rack (see Image 2A) that is commercializing the product for about

$298 plus shipping and handling), the estimated total deflection is 0.006 inch, the estimated lifetime over 10 times more than the required 10,000 trips (Table XI-5).

Our design analyses allow us to finalize a new dimension that optimize the strength and reduce in weight of our bike rack. Our candidate design meets our expectation within the constraints provided. We met our strength target, with all our Von Mises stress analyses to be less than the carbon steel’s yield strength. Our selection of material and chosen dimensioned looks ideal at this phase of the project. However further detailed analysis with fatigue performance would be required to create an optimal bicycle rack. Not only to satisfy the customer but to maintain a high quality and safety standard. We look forward to make a functional, sturdy, and inexpensive prototype for your company.

http://www.metalsdepot.com/products

















Design Engineers Background and Contributions:

We are the University of California at Davis, Mechanical Engineers. We are proud to present our bike rack design to Rack-It Ralph Engineering. Engineer Jake Wang helped organize the team, put together required paperwork, distributed the weekly work, and maintained a high quality analysis and material as a group. Jake helped divide the project up equally among the other three team members and equal work contributions have been present. Jake was in charge of all the analyses, material gathering, detailed design descriptions of the vertical beam mainframe. Fellow engineer Daniel Tai was assigned all the analyses, material gathering, and detailed design descriptions of the loop-rod support system. Fellow engineer Augustus Bechtold was assigned all the analyses, material gathering, and detailed design descriptions of the horizontal beam. Fellow engineer Josh Aguilar was assigned all the analyses, material gathering, and detailed design descriptions of the base receiver. We are all very happy to work together and making great contribution to our project. GO The4Racketeer!

Below is an abridged version of the actual report. It exemplifies the work from JakeWang.


Table of Contents:

I. Overview of Design Description:

II. Visual Presentation:

1) Image XI-1. Motorcycle BWM GX Motorcycle2) Image XI-2. Motorcyle with Bike Rack

3) CAD Drawing XI-1. Assembly CAD Drawing 4)Figure XI-2. Exploded View CAD Drawing5) Table XI-5-10. List of Final

Dimensions/Weight/Cost

III: General

Assumptions: IV: Overview

of the Design:

V: Detail Analysis of the Design:

Vertical Beam Mainframe Detailed Analysis

1) Detailed Description for Design Parts

2) Table A1-2: Material Properties for Design Parts

3) Analyses Formulas

i) Strength Analysis

ii) Deflection Analysis

iii) Buckling Analysis

iv) Fatigue Analysis

4) Detailed Analysis Comparison

i) Table A-: Strength Analysis

ii) Table A-: Deflection Analysis

iii) Table A-Buckling Analysis

iv) Table A-Fatigue Analysis

VI. Recommendations:

VII. Appendices:

1) Whole System Deflection Sketch

2) Whole System Damage Accumulation Sketch

3) Shigley’s Material PropertyTable

4) A500 Steel Mechanical Property Data

5) Bolt Grades Table6) References

Image XI-1: BMW GS Motorcycle Model Rear Side with Luggage Racks


Image XI-2: BMW GS Motorcycle Model Rear Side with Bike Rack


CAD Drawing XI-1: Assembly CAD Drawing with Dimensions


CAD Drawing XI-2: Exploded View CAD Drawing


CAD Drawing XI-3: Vertical Beam Components with Final Dimensions



Table XI-1: Vertical Beam Mainframe Final Dimensions/Weight/Cost

Vertical Beam: Design ParametersMaterial: A36Sets H

[inches]W

[inches]L

[inches]t [inches]

3 3 1 1/2 80 1/4Welded Square Tube: Design Parameters

Material: A513Sets H

[inches]W

[inches]L

[inches]t [inches]

3 1 1/2 1 1/2 10 3/4Subtotal Weight [lbs.] 49.25Subtotal Cost [$] 118.88Subtotal Deflections [in]

0.018472


XII: General Assumptions:

There are several assumptions we made to help us analyze our bike rack. For the vertical beam frame, we assumed the welded square tube connection to the vertical C-Channel to act like a cantilever at a fix wall. This simplifies our analysis so we do not have to analyze the additional stress created from the welding.

We used Von Mises Stress Theory to analyze our strength analysis. Because Von Mises Stress Theory is considered as the more accurate theory compared to the more conservative Maximum Shear Stress Theory. But, we understand it is not necessarily the case for all the time. We used this assumption for quicker analysis of our strength of the materials.

We have simplified our moment and shear calculations based just on the weight of the bike itself. We did not the weight of the bike rack into our calculations. This helps us save time to work on other analysis.

Primary we chose to use carbon steel as our material choice, because it is fairly inexpensive and easily to obtained. However, when we start our fatigue analysis, the S-N curve for A500, A36, and etc., were not available from our vendor. We resort to using the 1045 steel S- N curve with a home-made scale up and scale down factor. Since the Su and Sy of 1045 steel is twice of that of our steel material. We will multiple our stress amplitude by 2, and then plug into the power log formula to obtain the life cycle. At the end we will divide our interpolated life cycle by 10E11 to get our theoretical life cycle for our steel material.


XIII: Initial Research and Designs:

The two main bicycle rack products for cars are two pillars, with strap-on, and one main pillar will an L-beam attached (see Image 1A). They generally cost around $55 for the lower end products and up to $278 for the higher end ones. There are also bicycle racks on the roof top of the car, but we would not have that luxury with the motorcycle (see Image 1B).

We found only one company called 2X2 Bicycle Rack (see Image 2A) that is commercializing the product for about $298 plus shipping and handling. 2X2 Bicycle Rack has a flat steel panel and steel parts for attaching to the backseat of the motorcycle. However, most bicycle rack on the internet, are home-made by bicycle enthusiast (see Image 2B) with duck tapes and wood strapped around the backseat. Some better home-made one have an original motorcycle backseat metal frame extension, so enthusiast can attach metal rods at the extension with nuts and bolts and attach the bicycle to the metal rods with nylon bicycle strap (see Image 2C). So far, most of the designs for bicycle rack for motorcycle have the bicycle wheels at least one or both wheels removed to avoid spinning of the wheels while cruising. The bicycle is positioned facing forward on the backseat.

Here are some preliminary designs for our client. First, there is the Fork Design, with two rods forked at one end and one adjoining rod at the other end. It will attach to the motorcycle frame with a clamp. And it will hold on to the middle rod of the bicycle, with bicycle in a vertical position, front end facing the sky and bottom end facing the ground (see Sketch 1A).

With this design, the ideal way to have the bicycle attached is to remove both bicycle wheels. Although there might be issue with the bending moment induced by the weight of the bicycle, this is the simplest design with minimal material. But we will further modify this design to reduce bending moment if this is the best design for most motorcycles.

The second design is the Side Cradle Design with a cradle attached to only one side of the motorcycle (see Sketch 2A). This is an adaptation from a bicycle rack for cars (see Image 3A). This also allows the users to remove only one or none of their bicycle wheels, because the cradle with its L-shaped tire contraption would fully secured the wheels in place. The bicycle would be in a vertical position with front facing the sky. There might be stability question we need to focus on because the cradle is only on one side of the motorcycle. And also we need to focus on what material to be used and what size for the rod bridging between the bicycle and the motorcycle. This rod will definitely experience great bending stress and possible axial stress.

The third design is The Claw Design (see Sketch 3A) inspired by the aluminum box frame extension for the backseat of a motorcycle (see Image 4A). By adding a center claw to hold the bicycle fork frame, and one longer and one shorter outer claws to secure the main rod and the front rod. This design would provide great stability and durability of the bicycle rack. However, this design would require more material parts and maybe more tooling time. So this would be a more costly design, unless we find cheaper material with high yielding stress. We would be using theory for bending of a curved beam for the two outer claws for determining the maximum stress it can handle.


The bicycle rack idea for motorcycle seems to be interesting with many potential possibilities. Although we might need some more information regarding the type of motorcycle we are targeting and the type of bicycle we are handling. And also we may need the budget for this project and the desired lifetime of the product. We look forward to working on the next phase of the project.

Image References:

Image 1A: Sample Bicycle Racks for Car

Image 1B: Roof Bicycle Racks for Car

Image 2A: 2x2 Bicycle Rack for Motorcycle


Image 2B: Home-made Bicycle Rack for Motorcycle’

Image 2C: Home-made Bicycle Rack for Motorcycle

Sketch 1A: Preliminary Fork Design


Sketch 2A: Preliminary Side Cradle Design

Image 3A: Bicycle Cradle Rack for Cars

Sketch 3A: Preliminary The Claw Design


Image 4A: Motorcycle Backseat Extension



XIII: Overview of Design:

We have picked the BMW GS (see Image A1) motorcycle model for modeling and idealization of our candidate design. In response to the proposed requirements for our design, our overall design measurements did not exceeds 4 feet in width of the bicycle and rack as mounted on the motorcycle, and 4.5 feet in overhang of the rack behind the motorcycle. The following is our detailed descriptions of our candidate design and our explanations on how our candidate design works.

Our bicycle rack system consists of six sections, a base receiver, a vertical beam mainframe, a horizontal beam support, two link attachments, three strap-on mechanisms, and two loop-rod bicycle secure attachments, (see Sketch A1).

We designed a vertical beam mainframe section is the section where the rack will hold the bicycle still while cruising on the motorcycle (see Sketch A-2). This section consists of a C- section beam, welded solid beam insert, two holes for the hollow square receiver, four holes for the strap-on attachment, four holes for the loop-rod attachment, four holes for the two link attachments, two holes L-rod connection between the vertical beam and the horizontal beam. The welded solid beam insert connects to the hollow square receiver. This allows bicycler to detach or attach the rack from the motorcycle easily with a lock mechanism consists of a L-rod and a R- ring. The C-section beam provides enough space to situate the two bicycle tires. However the main function of this section is only to cradle the bicycle and provide support for the horizontal beam, but not to carry the weight of the bicycle. Most of the weight will be acting like a point force on the vertical beam, which will be discussed in the later section of the memo.

We designed a base receiver section that is attached to the motorcycle, to make the connection between the motorcycle and our main rack (see Sketch A-1: Base Receiver Assembly section). It has one metal plate with a hollow square receiver welded to the top side, two rods, six bolts, six washers, one L rod, and one R ring. It is a 1'x2' plate of 1/4" thin metal with an 18" long, 1-3/4" hollow square receiver with 1/16" thickness metal welded to the top side. To secure the metal plate to the luggage rack of the motorcycle, we bolt the plate to two rods below the luggage rack. The base receiver is connected to the bicycle rack mainframe through the welded solid beam extruding out from the vertical C-section beam. We used a 1-1/2” square tube on the rack such that it can slide into the larger receiver on the base. The hollow square receiver on the rack will be made of 1/16” steel and extended from the rack 10”. 7-1/2” from the bicycle rack is a 7/16” hole on both vertical faces at the midpoint. There will be a matching hole, 2-1/2” from the end of the receiver. The fit of the rack and receiver are dimensioned as a line fit, this cannot be if the rack is to be easily removable. When the rack is inserted in the receiver, the holes will line up. An L-rod will be inserted all the way through the receiver and rack, holding the holes aligned. An R-ring will clip on the L-rod to ensure that it does not fall out of place. The utility of this design is that by removing the L-rod, the rack can be quickly slid out of the receiver and off the motorcycle. The major stresses to be considered on this section are bending and torsion.


We designed a horizontal beam support section to provide the attachment for the rear wheel to the rack and support a large portion of the bicycle’s load (see Sketch B1 and B2). The C-section beam is wide enough to accommodate cruiser and mountain bicycle tires comfortably and has 1/4 inch thick walls on either side of the tire for increased stability. One strap-on mechanism is used to secure the wheel against the rack and is translatable over a 3 inch span to accommodate 26 inch diameter wheels as well as 29 inch diameter wheels. The horizontal C- section beam is wide enough that the vertical C-section beam will fit inside it and the two will be connected together with a L-rod and R-ring lock system through the holes near the solid end.

Since the rear tire is resting against the horizontal C-section beam and the bicycle is positioned vertically on the rack, the horizontal C-section beam will be supporting the majority of the bicycle’s weight. This will create a bending moment in the horizontal C-section beam (see

Sketch C-1), but with given the maximum load of sixty-five pounds and the yield strength of aluminum alloy of around 68.9 giga Pascal, this section will not yield at this stage of analysis. The end of the horizontal C-section beam farthest from the vertical C-section beam is left open to allow the bicycle tire to extend slightly beyond the rack.

For additional support of the vertical C-section beam, we designed a two base link section (see Sketch D-1). The base links will hold the horizontal and vertical rack beams, the pin attachments for the base links have not yet been designed but based on max shear stress, any common pin link should be sufficient since the necessary cross-section area is extremely small. The idea is that the base link will be movable if the user wishes to fold up the rack, but rigid when holding a bicycle.

We designed two strap-on mechanisms and two loop-rod bicycle secure attachments to attach and remove the bicycle wheels from our rack (see Sketch E1, E2, and E3). We have two strap-on mechanisms for the front and rear bicycle wheel located on the vertical beam support. Because the strap only binds the front wheel to the frame at one point of contact, additional supports must be added to prevent the bicycle from twisting around the frame. So we designed a two loop-rod section that will secure the bicycle frame in place, and to prevent scenarios like motorcycle making a sharp turn. The greatest forces that the strap would need to counter would be from the forward acceleration of the motorcycle (the bicycle rack frame itself would handle the load caused by deceleration of the motorcycle). This makes it relatively easy to design the strap since the weight of the bicycle is so small.

A-3): Detailed Description for Design Parts

The vertical beam mainframe section is the section where the rack will hold the bicycle. This section consists of a C-section beam, welded square tube, two holes for the hollow square receiver, five holes for the loop-rod support secure attachment, two holes for the two link attachments, two holes L-rod connection between the vertical beam and the horizontal beam, and two hole in place for securing the horizontal beam to the fold up vertical beam when not in use of the bike rack. The welded solid beam insert connects to the hollow square receiver. This allows biker to detach the rack from the motorcycle easily. The C-section beam provides enough space to situate the bike tire. However the main function of this section is only to cradle the bike and hold on to the vertical beam, but not to carry the weight of the bicycle. Most of the weight will be like a point force on the horizontal beam.

The critical section for the vertical beam mainframe is at the connection where the welded square tube and the vertical C-section beam joins. There is a moment created by the point for on the vertical beam that translates to the connection. There is also a shear force created by the same point force as well. For the vertical beam mainframe, we have the welded square tube that is set at 10 inches long, but the cross section size are for analysis with the variation of

heights and width from ¼ to ½ and variation of thickness from 3 to 3 . The two materials


164

selected for comparison for the welded square tube are A500 and A513 steel. As explainedpreviously, some important reason for choosing these two materials to compare with is becausethey are readily available for purchase from our preferred supplier and they are relatively inexpensive.We have created our analysis with excel tables – see Table A-1 to A-10, for easy comparison. Although all our Sets and material options are well under the yield strength along with safety factory of 2, the following explain more detail of the best options from the comparison. The combination of Set 3’s dimensions and A500 steel provided the least tensile and compressive stresses. However, since we need to choose one material, we decided to choose A500 because the deflection difference compared to A513 is only 0.00597 inch, which is well under 1 inch when we combined with the deflections from other design components. The combination of Set 3’s dimensions and A513 steel provided the least deflections. There is no buckling for the vertical beam mainframe because there is no design member in compression.

We also included a bolt analysis, comparing three different grades of carbon steel bolt in our design. There are Grade 2, Grade 5, and Grade 8 that represent increasing yield strength, ultimate strength and proof load. Our analysis on Table A-2 shows that we have 3 sets of dimensions cross compared with the 3 types of grades of the bolt. The result is that all the bolt are well under the ultimate strength and all the different bolt grades have similar Von Mises stress. We selected grade 8 for its highest ultimate strength at 150 ksi to provide us a greater safety factor value.


With the three different sets of dimensions and material, we created a weight analysis.We are comparing A36 C-Channel for the C-Section beam, and the A500 and A513 for the material. The bolts weight about the same among the different grades, so they have beenassigned with the same weight. The best total weight for the vertical beam of the mainframe is estimated at 36 lbs and using A-36 C-Channel and A513 steel square tube. The material cost from this weight analysis is $49.58 and the labor cost is $69.30 (see Table A-11), with the total production cost for this vertical mainframe looking at roughly $118.88.

The final analysis is the fatigue analysis. It analysis how many life cycles does our product last. By calculating the maximum stress at the critical section and customize for different loading maneuvers. Some maneuvers are loading the bike, starting the motorcycle, riding through bumps, stopping the motorcycle, and unload the bike. We can create a sequence of loading and plot a graph (see Figure A-1) showing different types of cycles. By observing the graph we can define our maximum strength, Smax and minimum strength, Smin to find our stress amplitude, Sa, effective stress amplitude Sa,eff, using Goodman formula, fatigue life of each type of cycles, Nif, number of cycles, Ni, damage value, D, and life cycle, N. Our design requirement desired a product that last 10,000 trips with a Safety Factor of 5. From our fatigue analysis on Table A-12, we retrieve a design life at 2.4E+6, which is great than the provided design requirement.

The vertical beam frame has satisfied all the design criteria for strength, deflection, buckling, and fatigue analysis. Although the individual cost and weight are slightly high, our product will be durable and strong that last for many years to come. The final dimensions of vertical beam A36 steel C-Channel at 3 inches by 1 ½ inches by 80 inches with ¼ inches of thickness. The final dimensions of the welded A513 square tube is at 1 ½ inches by 1 ½ inches by 10 inches with a thickness of ¾ inches.

3)Table A1-2: Material Properties for Design Parts

Table A-1: Material Properties for our Steel Square Tubing

Table A-2: Material Properties for our Bolt Options

Table A-3:SAE1045 Steel Fatigue S-N Curve


Major Design ComponentSteel Tubing

Material/Grade ASTM A500Steel*

ASTM A513 Steel*

Density (p) 0.284[lbf/in^3]

Elastic Modulus (E)

20300 [kpsi] 30000 [kpsi]*

Poisson's ratio (v) 0.292Yield Strength (Sy)

45700 [psi] 72000 [psi]

Ultimate Strength (Su)

58000 [psi] 87000 [psi]

* Assume 30000. No datafound.

Bolt ComponentMaterial/Grade Low Carbon Bolt

**Grade 2 5 8Yield Strength (Sy) 57000 [psi] 92000 [psi] 130000

[psi]Ultimate Strength (Su)

74000 [psi] 120000[psi] 150000[psi]

Proof Load 55000 [psi] 85000 [psi] 120000[psi]

(low carbon steel)

A-4)Analyses Formulas

i) Strength Analysis

Axial Stress: 𝑃𝑃𝑆𝑆𝑦𝑦𝐴𝐴𝑛𝑛=𝐴𝐴 = 𝜋𝜋(𝑟𝑟)24

Bending Compressive Stress: 𝑀𝑀𝑀𝑀𝑆𝑆𝑦𝑦𝐼𝐼𝑛𝑛 =

𝐼𝐼𝑦𝑦 = 𝐼𝐼𝑦𝑦 − 𝐼𝐼𝑦𝑦 = 12 − 12 =

𝑂𝑂𝐼𝐼 𝑏𝑏𝑂𝑂𝑏𝑏𝐼𝐼4 4 𝑏𝑏𝑂𝑂 − 𝑏𝑏𝐼𝐼

4412𝜎𝜎𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 = 𝜎𝜎𝑉𝑉𝑉𝑉𝐵𝐵 𝑀𝑀𝐵𝐵𝑀𝑀𝐵𝐵𝑀𝑀 =

𝑛𝑛𝑀𝑀𝑀𝑀𝐼��

< 𝑆𝑆𝑦𝑦Bending Tensile Stress: 𝑀𝑀𝑀𝑀𝐼𝐼𝑛𝑛 = 𝑆𝑆𝑦

𝑦𝐼𝐼𝑦𝑦 = 𝐼𝐼𝑦𝑦 − 𝐼𝐼𝑦𝑦 = 12 − 12 =

𝑂𝑂𝐼𝐼 𝑏𝑏𝑂𝑂𝑏𝑏𝐼𝐼4 4 𝑏𝑏𝑂𝑂 − 𝑏𝑏𝐼𝐼

4412𝜎𝜎𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 = 𝜎𝜎𝑉𝑉𝑉𝑉𝐵𝐵 𝑀𝑀𝐵𝐵𝑀𝑀𝐵𝐵𝑀𝑀 =

𝑛𝑛𝑀𝑀𝑀𝑀𝐼��

< 𝑆𝑆𝑦𝑦


Von Mises Stress Criteria Analysis:

𝜎𝜎𝐴𝐴𝐴𝐴𝐵𝐵𝐴𝐴𝐴𝐴 = 𝜎𝜎𝑉𝑉𝑉𝑉𝐵𝐵 𝑀𝑀𝐵𝐵𝑀𝑀𝐵𝐵𝑀𝑀 < 𝑆𝑆𝑦𝑦

ii) Deflection Analysis

Deflection formula for Cantilever Beam:

𝛿𝛿(𝑃𝑃𝑃𝑃𝑃𝑃𝑛𝑛𝑃𝑃 𝐹𝐹𝑃𝑃𝑟𝑟𝑀𝑀𝐹𝐹 𝐷𝐷𝐹𝐹𝐷𝐷𝐷𝐷𝐹𝐹𝑀𝑀𝑃𝑃𝑃𝑃𝑃𝑃𝑛𝑛) =𝐹𝐹𝐿𝐿33𝐸𝐸𝐼𝐼

𝛿𝛿(𝑀𝑀𝑃𝑃𝑀𝑀𝐹𝐹𝑛𝑛𝑃𝑃 𝐷𝐷𝐹𝐹𝐷𝐷𝐷𝐷𝐹𝐹𝑀𝑀𝑃𝑃𝑃𝑃𝑃𝑃𝑛𝑛) =𝑀𝑀𝐿𝐿22𝐸𝐸𝐼𝐼

𝜃𝜃 = 𝑀𝑀𝐿𝐿𝐸𝐸𝐼𝐼𝛿𝛿(𝐷𝐷𝐹𝐹𝐷𝐷𝐷𝐷𝐹𝐹𝑀𝑀𝑃𝑃𝑃𝑃𝑃𝑃𝑛𝑛 𝑣𝑣𝑃𝑃𝑣𝑣 𝐴𝐴𝑛𝑛𝐴𝐴𝐷𝐷𝐹𝐹 𝑃𝑃𝐷𝐷 𝑅𝑅𝑃𝑃𝑃𝑃𝑣𝑣𝑃𝑃𝑃𝑃𝑃𝑃𝑛𝑛) = sin(𝜃𝜃) ∗ 𝐿𝐿 = 𝜃𝜃 ∗ 𝐿𝐿

Deflection formula for Axial Load

𝛿𝛿(𝐴𝐴𝐴𝐴𝑃𝑃𝑣𝑣𝐷𝐷 𝐷𝐷𝐹𝐹𝐷𝐷𝐷𝐷𝐹𝐹𝑀𝑀𝑃𝑃𝑃𝑃𝑃𝑃𝑛𝑛) = 𝑃𝑃𝑃𝑃𝐸𝐸𝐴𝐴

Figure 2: Vertical Beam Deflection Analysis



iii) Buckling Analysis

There is no buckling for the vertical beam mainframe because there is no design member in compression.

iv) Fatigue Analysis

Loading Maneuvers Calculations:

First establish local reference. Pointing down as positive Y and pointing to the right is positive X. For the center of the mass, our Fy will be located on the horizontal beam, 16 inch away from the vertical beam and our Fx will be position at half the length of vertical beam, which is 40 in. Using above information you can calculate moment.

For vertical beam, on the rack, my Fy is 80lb, my Fx is 0lb see my excel analysis attached. For start my Fy is still 80lb, my Fx is 40lb to the right. (Fx=80*0.5G) where 0.5 G is our criteria of load for starting. (with 5 pulses of the 0.5, which means there will be 5 times of this type of cycle and at the end we multiple by 10 because there are 10 starts in our mission) For bumps you multiple by 3G along the vertical force. For stops, the negative Fx force will provide the stronger counter moment, creating a negative final moment.

Reordering for the sequence is the hardest. So basically, the largest peak is the first in the reordering sequence, followed by the sequence coming afterwards. For example, Bump 1 has the highest value, then we continue the sequence with On rack, Bump 2, On rack, Stop, On rack, Unload, then back to the beginning, omitting the 0, with On rack, Start, On rack, and end with Bump 1.Use this power log formula, Sa=152(N)^-0.065, to calculate your Sa. To calculate Sa,eff with Goodman, use Sa,eff=(Sa)*(Su/Su-Sm). So, on the Material section, we put down 152 for A, and -0.065 for b.

Basically at the end you can categorize 4 types of cycle. Type A for Max to min, type B for Rack to Bump, type C Rack to Stop, and type D Rack to Start. Now, you just have to find the Smax and Smin by looking at your graph, and input the value in the Damage accumulation table. The table will automatically calculate the life cycle for you. Then you have to divide this life cycle by 5, because this is the safety factor assigned to us with regard to fatigue life. If at the end it is greater than 10,000 cycles or trips, then your part will meet the mission criteria.


A.5) Detailed Analysis Comparison

i) Table A-: Strength Analysis

Table A-3: Complete Table of Available Dimensions Welded Square Tube Analysis

Welded Square Tube: Design ParametersDimensions

Sets H [inches] W [inches] L [inches] t [inches]1 1 1/4 1 1/4 10 3/162 1 1/2 1 1/2 10 1/43 1 1/2 1 1/2 10 3/4

Parts Design ParametersArea Calculations

Sets A [inches^2]1 2/32 1 1/43 2 1/4

Parts Design ParametersMoment of Inertia Calculations

Sets I [inches^4]1 1/62 1/33 3/7

Parts Design ParametersDistance from the edge to Bolt

(e)Sets e [inches]

1 11 1 1/21 22 12 1 1/22 23 13 1 1/23 2


ii) Table A-: Deflection Analysis

Parts MaterialStrength Set A500 Von

Mises<Sy A513 Von

Mises<Sy Best Set

RankingTensile Stress [psi] 1 16493 14400 Yes 3709 3138 Yes 3

24/25 1/11 6/132 9038 7253 Yes 2448 1360 Yes 2

10/13 1/33 7253 7253 Yes 1360 1360 Yes 1

1/3 1/3Compressive Stress 1 -16493 - Yes -3138 -3138 Yes 3[psi] 24/25 14400 6/13 6/13(Note: A500: 2 -9038 -7253 Yes -1360 -1360 Yes 2Sy=45700 [psi]) 10/13 1/3(Note: A513: 3 -7253 -7253 Yes -1360 -1360 Yes 1Sy=72000 [psi]) 1/3 1/3Best Material Ranking

1 2

Parts MaterialDeflections Set A500 A513

Vertical Deflection [in.] 1 19/400 9/280(Note: E=20300 [kpsi]) 1 21/131 32/295

1 19/50 9/352 19/800 9/5602 21/262 16/2952 19/100 9/703 15/812 1/803 25/401 17/4033 103/697 1/10

Deflections for design part [in.]

1 19/400 9/280

2 19/800 9/5603 15/812 1/80 Deflection

DifferenceMin Deflections for design part [in.]

Set 3

15/812 1/80 2/335


iii) Table A-Buckling Analysis

iv) Table A-Fatigue Analysis

Table A-F1.Fatigue Analysis

Parts MaterialBuckling Set A500 <Sy [Yes/No] A513 <Sy [Yes/No]

Tensile Stress [psi] 1 0 0 0 02 0 0 0 03 0 0 0 0

Compressive Stress [psi] 1 0 0 0 02 0 0 0 03 0 0 0 0

(Comments: Due to no compressive force acting on the vertical beam, the beam would not buckle.)

Vertical Beam Welded Tube Fatigue Analysis:GEOMETRY MATERIAL A1045 SteelD 0.85 inch Sy 105 ksi

Su 84 ksiI 0.3385 inch^4 A 152 ksi, (for fully

reversed loading)A N/A inch^2 b -0.065c 0.7500 inchKt, notch 1.0000 (unnotched)

LOADINGManeuver Fx [lbs] Fy [lbs] P [lbs] V [lbs] M

[lb*in]Stress @ Welded Tube (Mc/I) [ksi]

Load Bike 0 0 0 0 0 0.00On rack 0 80 0 80 1360 3.01Start 40 80 40 80 2960 6.56Bump 0 240 0 240 4080 9.04Stop -60 80 -60 80 -1040 -2.30On rack 0 80 0 80 1360 3.01Unload Bike

0 0 0 0 0 0

Table A-F2.Fatigue Analysis Sequence

10

8

6

4

2

0

-2

-4

Figure A-5: Sequence Stresses Graph

Figure A-6: Re-order Sequence Stresses Graph


SEQUENCEZero On

rackStart On rack Bump

1On rack

Bump 2

On rack

Stop On rack

Unload Bike

0.00 3.01 6.56 3.01 9.04 3.01 9.04 3.01 -2.30 3.01 0.00REORDER (largest peak or valley first)

9.04 3.01 9.04 3.01 -2.30 3.01 0.00 3.01 6.56 3.01 9.04


Table A-F3.Fatigue Analysis: Damage Accumulation

FATIGUE ANALYSIS -Damage accumulationCyclesLabel S

maxS

minS a

(actual)Sa ( scaled up by 2)

S m (actual)

Sm ( scaled up by 2)

Max to Min A 9.04 0.00 4.52 9.04 4.52 9.04Rack to Bump

B 9.04 3.01 3.01 6.03 6.03 12.05

Rack to Stop C 3.01 0.00 1.51 3.01 1.51 3.01Rack to Start

D 6.56 3.01 1.77 3.54 4.79 9.57

includes notch factor

kt =1 for our analysis

Sa, notch

Sm, notch

S eff (scaled)

Nfi (scaled)

Nfi ( scaled down by 10E11)

Ni D = Ni/Nfi

4.52 4.52 10.13 1.25E+18 1.25E+07 1 8.00E-083.01 6.03 7.04 3.40E+20 3.40E+09 10 2.94E-091.51 1.51 3.13 8.99E+25 8.99E+14 10 1.11E-141.77 4.79 4.00 2.01E+24 2.01E+13 50 2.48E-12

Total sum

8.30E-08

Life 1.2E+07Safety Factor

5

Design Life

2.4E+06

Greater than 10,000trips?

YES


Table A-W1.Vertical Beam and Welded Tube Weight Analysis

Table A-C1.Vertical Beam and Welded Tube Cost Analysis

Components WeightSets Vertical Beam [lb] Welded Tube [lb] Bolts

[lb]Total Weight [lb] Selection

Ranking1 34 2 1/4 36 18/61 12 45 1/2 3 5/9 1/4 49 9/35 23 42 3/5 6 2/5 1/4 49 9/35 2

Vertical Beam and Welded Tube Cost Analysis

Material: CChannel

A-36Steel

Welded Tube

A500Steel

Components CostsSets Vertical

Beam [$]Welded Tube [$]

Bolts [$]

Material Cost [$]

Labor Cost [$]

Total Cost [$]

Selection Ranking

1 40.96 21.66 2 64.62 69.30 133.92 12 52.00 13.32 2 67.32 69.53 136.85 23 51.84 18.00 2 71.84 69.91 141.75 3

Vertical Beam and Welded Tube Cost Analysis

Material: CChannel

A-36Steel

Welded Tube

A513Steel

Components CostsSets Vertical

Beam [$]Welded Tube [$]

Bolts [$]

Material Cost [$]

Labor Cost [$]

Total Cost [$]

Selection Ranking

1 40.96 6.62 2 49.58 69.30 118.88 12 52.00 6.24 2 60.24 69.53 129.77 23 51.84 7.82 2 61.66 69.91 131.57 3


Table A-L: Vertical Beam and Welded Tube Labor Cost Analysis

Vertical Beam and Welded Tube Labor Cost AnalysisMaterial: C

ChannelA-36Steel

Welded Tube

A500 or 513 Steel

Sets 1 2 3Processes Unit Cost

[$]# of Units or Length [cm]

Subtotal Cost [$]

Unit Cost [$]

# of Units or Length [cm]

Subtotal Cost [$]

Unit Cost [$]

# of Units or Length [cm]

Subtotal Cost [$]

Vertical TubeDrilled $ 5 $ $ 5 $ $ 5 $holes <25.4 mmdia. 0.35 1.75 0.35 1.75 0.35 1.75Saw or $ 40.54 $ $ 40.54 $ $ 40.54 $tubing cuts 0.40 16.22 0.40 16.22 0.40 16.22

Welded TubeTube end $ 1 $ $ 1 $ $ 1 $preparationforwelding 0.75 0.75 0.75 0.75 0.75 0.75Weld $ 1.69 $ $ 3.18 $ $ 5.72 $

0.15 0.25 0.15 0.48 0.15 0.86Tube cut $ 35.56 $ $ 35.56 $ $ 35.56 $

0.15 5.33 0.15 5.33 0.15 5.33

AssemblyAssemble, $ 12 $ $ 12 $ $ 12 $>20 kg,Line-on-Line 3.75 45.00 3.75 45.00 3.75 45.00

Total $ Total $ Total $Labor Labor Labor

Cost 69.30 Cost 69.53 Cost 69.91Ranking (Least to

Highest Price)1 2 3


Recommendation:

This is a great project that integrated lecture concepts into practical analysis. It allows us to make sense and utilize the formulas and sharpen analytical skills toward mechanical design. One of the recommendations for this project is that if time permitted, a welding analysis can be done where the welded square tube connects with the vertical C-Channel. We have read Shigley’s Mechanical Design textbook, and find out there is formula for simply weld on a cantilever beam. The weld analysis depends on the size of the weld throat and the distance between the welding. There is a formula for computing moment area of inertia and there are available tables regarding different type of welding material and welding patterns. We thought it was very interesting and would try this analytic method to analyze if we should have a smaller or bigger welded square tube or not. Because there are couple factors that might make welding a critical aspect of a design analysis such as tear of the weld due to due to bending or torsion.

Another good addition to our bike rack is a protective coating cover, preferably with paint because we are using steel as working material (it become corrosive when wet or encounter with

water vapor. This will make our product more versatile for consumer, enabling them to use our product in any weather conditions.

XVI. Appendix:

Sketch B-1: Whole System Deflection Sketch


Sketch B-2: Whole System Damage Accumulation Assessment Sketch


Bike Rack Design Project

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