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SELECTION AND OPTIMIZATION OF SNAP-FIT FEATURESVIA WEB-BASED SOFTWARE

DISSERTATION

Presented in Partial Fulfillment of the Requirement for

The Degree Doctor of Philosophy in the Graduate

School of The Ohio State University

By

Tieming Ruan, M.S.

* * * *

The Ohio State University

2005

Dissertation Committee:

Approved by

Professor Anthony F. Luscher, Adviser

Professor Gary L. Kinzel

Professor Krishnaswamy Srinivasan Adviser

Professor Donald R. Houser Mechanical Engineering

Graduate Program

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ABSTRACT

When used in plastics, snap-fits can be a simple, quick and cost-effective method

of assembling two parts. When designed properly, parts with snap-fits can be assembled

and disassembled numerous times without any adverse effect on the assembly. Snap-fits

also the aid in making products environmentally friendly because of their ease of

disassembly, making components of different materials easy to recycle. Traditionally,

snap-fit design methodology has been disorganized and anecdotal in nature, relying

greatly on the skill and the experience of the individual designer. The most popular

source of designing snap-fit is the design guides from resin suppliers. However the

information disseminated by these guides is usually old, obsolete and inaccurate because

they based on assumptions of small deformation and linear material property. To

overcome these two disadvantages, the author developed a two-dimensional, plane stress,

contact finite element model (FEM) considered the nonlinear material property of

 polymer. Using design of experiments (DOE) and response surface methodology (RSM),

this research obtained a second order response surface equation to predict the retention

 performance of cantilever hook with high retention angles.

With the development of Internet, there is also a demanding for a web-based

snap-fit design tool that is independent to all operating systems, easily accessible and can

 be universally upgraded by simply updating the design tool at the server location. This

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thesis developed a web-based design tool for three different snap-fit features: cantilever

hook, post & dome, and bayonet & finger. The response surface equations obtained by

using the combination between FEM, DOE and RSM were applied. Constraint

management (CM) was used to make several functions available such as sensitivity

analysis, correction advisor.

Optimization modules such as single objective optimization and multiple

objectives optimization were applied to make the design tool more flexible and powerful.

Single objective optimization was implemented in two steps. The first step is to use

Golden Section Method to identify the search direction and the second step is to use

Broydon-Fletcher-Goldfarb-Shanno (BFGS) method to find the minimum values on this

search direction. To do multiple objectives optimization, the Weighted Sum strategy was

choose. Each objective was assigned a weighted value based on their importance, then

combine them into a single objective optimization problem.

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Dedicated to my parents, my wife and kids

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ACKNOWLEDGMENTS

I wish to thank my adviser, Anthony Luscher, for intellectual support,

encouragement, and enthusiasm which made this dissertation possible, and for his

 patience in correcting both my stylistic and scientific errors.

I thank Prof. Srinivasan and Prof. Houser for their patience and to be my

committee members. Especially I thank Prof. Gary Kinzel to be my committee member

and for providing the FORTRAN source code used in this web-based design tool.

I am grateful to Gaurav Suri for discussing with me various aspects of this

dissertation. I also want to thank my officemates who helped me to handle various

 problems, especially Jonathan Pillai and Leo Rusli.

I want to thank my parents, my wife and kids. Without your support, I can’t finish

this dissertation with a full-time job.

This research was supported by a grant from CAPCE.

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VITA

January 11, 1976 …………………………………….. Born – Zhejiang, China

1996 ………..…………………………………………B.S. Mechanical Engineering,

China Univ. of Mining & Tech.

1999 ……………………………………… ………… M.S. Mechanical Engineering,

China Univ. of Mining & Tech.

1999 – 2002 …………………………………………..Graduate FellowThe Ohio State University

2002 – present ………………………………………...Graduate Research AssistantThe Ohio State University

2003 – present ……………………………………….. Sr. Associate R&D EngineerBayer HealthCare

PUBLICATIONS

Research Publication

1. T. Ruan, Z. Wang and H. Chen, “A monitoring device for the fire of belt”, Journalof Coal Science & Engineering (China), 1998.

2. Z. Luan, Z. Wang and T. Ruan, “Balance compound gear pump (motor)”,Transaction of Huainan Institute of Technology (Chinese), 1997.

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FIELDS OF STUDY

Major field: Mechanical Engineering

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TABLE OF CONTENTS

Abstract .............................................................................................................. ii

Dedication ......................................................................................................... iv

Acknowledgments.............................................................................................. v

Vita ..................................................................................................................... vi

List of Tables .................................................................................................... xi

List of Figures ................................................................................................. xiii 

Chapter 1 Introduction ....................................................................................... 1 1.1 Introduction to Snap-Fit Features ............................................................................. 21.2 Motivation for Research ........................................................................................... 4

1.2.1 Snap-fit Design Approaches .............................................................................. 81.2.2 Web-based Snap-fit Design Tool..................................................................... 111.2.3 Constraint Management ................................................................................... 12

1.2.4 Thesis Objectives ............................................................................................. 13

Chapter 2 Literature Review ............................................................................ 15 2.1 Numerical Formulation for Snap-fit Features......................................................... 15

2.2 Materials Issues in Snap-fit Design ........................................................................ 20

2.3 Failure Models of Cantilever Hook ........................................................................ 222.4 Web-based Design Tool for Snap-fit....................................................................... 25

2.5 Constraint Management .......................................................................................... 34

Chapter 3 Finite Element Analysis of Cantilever Hook With a High RetentionAngle ................................................................................................................. 37 

3.1 Combination of FEA with DOE.............................................................................. 37

3.2 Nonlinear Structural Analysis of Snap-Fits ............................................................ 39

3.2.1 Moving Contact ............................................................................................... 393.2.2 Geometric Nonlinearities................................................................................. 40

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3.2.3 Material Nonlinearities .................................................................................... 40

3.3 Numerical Model of Cantilever Hook Feature ....................................................... 413.3.1 Modeling Techniques for Contact Problem..................................................... 42

3.3.2 Improvement of Solution ................................................................................. 44

3.3.3 Comparison between FEM Result and Experiment Result.............................. 46

Chapter 4 The Empirical Model for Performance of Cantilever Hook With aHigh Retention Angle ....................................................................................... 49 

4.1 Screening Experiment of Cantilever Hooks with High Retention Angles.............. 514.2 Central Composite Design ...................................................................................... 59

4.3 Model Improvement................................................................................................ 64

4.3.1 Transformation of Empirical Model ................................................................ 654.3.2 Remove the Influential Outlying Cases ........................................................... 66

4.4 Verification Tests..................................................................................................... 74

Chapter 5 Design Optimization and Constraint Management ...................... 78 

5.1 Design Optimization ............................................................................................... 785.1.1 Single Objective Optimization......................................................................... 815.1.2 Multiple Objectives Optimization.................................................................... 82

5.2 Constraint Management .......................................................................................... 83

5.2.1 Constraint Representation................................................................................ 85

5.2.2 Constraint Management Algorithms................................................................ 895.3 Mathematic Formulations for Snap-fits Features ................................................. 102

Chapter 6 The Web-Based Application for Snap-Fits.................................. 108 6.1 Approaches for Web-based Application................................................................ 108

6.1.1 Java ................................................................................................................ 108

6.1.2 JavaScript....................................................................................................... 1106.1.3 VBScript ........................................................................................................ 110

6.1.4 VB DHTML Application............................................................................... 1116.1.5 VB IIS Application ........................................................................................ 111

6.2 Principles for VB IIS Application......................................................................... 113

6.2.1 ASP Object Model ......................................................................................... 1146.2.2 Introduction to WebClasses ........................................................................... 116

6.3 Snap-fit Web-based Application ........................................................................... 117

6.3.1 New Specification Problem ........................................................................... 1216.3.2 Respecification Problem ................................................................................ 121

6.3.3 Unspecification Problem................................................................................ 121

6.3.4 Reverse Specification Problem...................................................................... 124

6.3.5 Sensitivity Analysis ....................................................................................... 1256.3.6 Correction Advisor......................................................................................... 126

6.3.7 Single Objective Optimization....................................................................... 127

6.3.8 Multiple Objectives Optimization.................................................................. 127

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Chapter 7 Case Studies ................................................................................. 131 7.1 Methods to Evaluate the Performance of Cantilever Hook .................................. 1317.2 Integrated Polymeric Lens Housing...................................................................... 132

7.3 Cantilever Hook Application of Cabinet............................................................... 136

7.4 Cantilever Hook Samples ..................................................................................... 139

7.5 Case Study Conclusions........................................................................................ 145

Chapter 8 Conclusions and Recommendations for Future Research ....... 147 8.1 Scientific and Engineering Contribution .............................................................. 1478.2 Recommendations for Future Research ................................................................ 150

Appendix A “PROCEDURE” File for MSC.MARC and MSC.PATRAN ........ 151 

Appendix B Visual Basic Subroutines for Cantilever Hook........................ 153 

Bibliography.................................................................................................... 162 

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LIST OF TABLES

Table 4.1: The upper limit and low limit of each parameter ............................................ 52

Table 4.2: The screening array.......................................................................................... 55

Table 4.3: The result of best subset regression................................................................. 58

Table 4.4: The most significant factors for cantilever hook............................................. 61

Table 4.5: The central composite design for cantilever hook........................................... 62

Table 4.6: Regression analysis for retention force of cantilever hook ............................. 64

Table 4.7: Regression analysis of logarithm transformation of cantilever hook .............. 67

Table 4.8: Regression analysis of reciprocal transformation of cantilever hook.............. 68

Table 4.9: The influential outlying cases.......................................................................... 71

Table 4.10: Regression analysis for retention force after deleted influential outlying cases

........................................................................................................................................... 73

Table 4.11: Verification test between FEA result and equation results............................ 76

Table 4.12: The analysis of variance for Lack-of-Fit test................................................. 77

Table 5.1: Parameters for strength of materials design equations .................................... 80

Table 5.2: The occurrence matrix of cantilever hook ....................................................... 86

Table 5.3: The occurrence matrix after specified x1, x2, x3, x4, and x6............................... 93

Table 5.4: Block, input/output, level and order of cantilever hook example.................... 95

Table 5.5: The occurrence matrix of respecification problem.......................................... 96

Table 5.6: The occurrence matrix of unspecification problem......................................... 98

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Table 5.7: The response surface factors for the post & dome feature ............................ 103

Table 5.8: Design variables for the post & dome feature ............................................... 104

Table 5.9: The scaled variables for bayonet & finger feature......................................... 106

Table 5.10: The design variables for bayonet & finger feature ...................................... 106

Table 5.11: The response surface factors for bayonet & finger feature.......................... 107

Table 6.1: The advantages and disadvantages of web-based languages......................... 113

Table 7.1: The comparison of cantilever hook samples ................................................. 145

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LIST OF FIGURES

Figure 1.1: Typical experimental force curve for insertion of snap-fit features................. 4

Figure 1.2: Typical experimental force curve for insertion of snap-fit features................. 5

Figure 1.3: Common snap-fit feature topologies ................................................................ 6

Figure 1.4: Equations for dimensioning cantilevers ........................................................... 7

Figure 1.5: Force multiplier of insertion and retention....................................................... 7

Figure 1.6: Cantilever hook and mating part ...................................................................... 9

Figure 1.7: Suri’s idealized model of cantilever hook and mating part, shown in deformedconfiguration. .................................................................................................................... 10

Figure 2.1: Percent engagement of cantilever hook.......................................................... 23

Figure 2.2: The different failure modes with respect to the PE ........................................ 24

Figure 2.3: The two different mechanical failure modes for cantilever hook .................. 24

Figure 2.4: The IFP snap-fit design tool ........................................................................... 28

Figure 2.5: AlledSignal Plastics’ snap-fit design guide.................................................... 29

Figure 2.6: Cantilever snap-fit design tool from Eastman chemical company................. 30

Figure 2.7: Jeff Raquest’s snap-fit calculator ................................................................... 31

Figure 2.8: GE Plastics’ snap-fit wizard ........................................................................... 32

Figure 2.9: Engineers Edge’s snap-fit straight beam calculator ....................................... 32

Figure 2.10: Engineers Edge’s snap-fit tapered beam calculator ..................................... 33

Figure 2.11: Brock & Wright’s design tool for snap fits.................................................. 33

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Figure 3.1: The actual and bilinear stress-strain curve for plastic material...................... 42

Figure 3.2: Typical cantilever hook meshed example ...................................................... 43

Figure 3.3: The nonlinear stress-strain curve of Duraform Polyamide ............................ 46

Figure 3.4: The comparison results for cantilever 1 ......................................................... 47

Figure 3.5: The comparison results for cantilever 2 ......................................................... 47

Figure 3.6: The comparison results for cantilever 3 ......................................................... 48

Figure 4.1: The typical cantilever hook ............................................................................ 53

Figure 4.2: The typical bayonet & finger.......................................................................... 54

Figure 4.3: The relationship between beam thickness and retention force....................... 54

Figure 4.4: The combination of central composite design................................................ 60

Figure 4.5: Retention force vs. position............................................................................ 61

Figure 4.6: The residual and normality plots for retention force...................................... 65

Figure 4.7: Residual plot vs. fitted values for logarithm transformation.......................... 69

Figure 4.8: Residual plot vs. fitted values for reciprocal transformation ......................... 70

Figure 4.9: The residual plot after deleted outlying cases ................................................ 73

Figure 4.10: Main effects for retention force.................................................................... 75

Figure 5.1: A procedure for identifying disjoint blocks of equations............................... 87

Figure 5.2: Design decomposition algorithm.................................................................... 91

Figure 5.3: The new specification problem of cantilever hook ........................................ 94

Figure 5.4: The respecification problem of cantilever hook............................................. 96

Figure 5.5: Forward dependency algorithm...................................................................... 97

Figure 5.6: The unspecification problem of cantilever hook............................................ 99

Figure 5.7: Backward dependency algorithm................................................................. 101

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Figure 5.8: The reverse specification problem of cantilever hook ................................. 102

Figure 5.9: A typical post & dome feature ..................................................................... 104

Figure 6.1: A typical WebClass life cycle ...................................................................... 118

Figure 6.2: The flow diagram of snap-fit web-based design tool ................................... 119

Figure 6.3: The logon page ............................................................................................. 120

Figure 6.4: The selection page ........................................................................................ 120

Figure 6.5: The design interface of new specification.................................................... 122

Figure 6.6: The design interface of respecification ........................................................ 122

Figure 6.7: The final design interface of new specification............................................ 123

Figure 6.8: The design interface of unspecification........................................................ 123

Figure 6.9: The design interface of reverse specification ............................................... 124

Figure 6.10: The typical sensitivity analysis page .......................................................... 125

Figure 6.11: The typical correction advisor page ........................................................... 126

Figure 6.12. Single objective optimization for retention force of cantilever hook......... 128

Figure 6.13. Single objective optimization result for retention force of cantilever hook128

Figure 6.14: Multiple objectives optimization for retention force of cantilever hook.... 129

Figure 6.15: Multiple objectives optimization result for retention force of cantilever hook 

......................................................................................................................................... 129

Figure 7.1: Integral polymeric lens housing ................................................................... 133

Figure 7.2: Dimensions of snap-fits used in integral polymeric lens housing................ 133

Figure 7.3: The curve of retention force of cantilever hook of lens housing.................. 135

Figure 7.4: The web-based design tool for integral polymeric lens housing.................. 135

Figure 7.5: Calculation of beam thickness at broken area for power supply’s snap-fit.. 136

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Figure 7.6: Cantilever hook of power supply for Rockwell Allen Bradly PLC controller 

......................................................................................................................................... 137

Figure 7.7: The detailed dimensions of the second case study ....................................... 137

Figure 7.8: The retention force curve of cabinet snap-fit ............................................... 138

Figure 7.9: The web-based design page for power supply’s cantilever hook................. 138

Figure 7.10: The detailed dimensions of cantilever hook sample a................................ 139

Figure 7.11: The detailed dimensions of cantilever hook sample b ............................... 140

Figure 7.12: The detailed dimensions of cantilever hook sample c................................ 140

Figure 7.13: The retention force curve of cantilever hook a........................................... 141

Figure 7.14: The retention force curve of cantilever hook b .......................................... 142

Figure 7.15: The retention force curve of cantilever hook c........................................... 142

Figure 7.16: The web-based design page for cantilever hook sample a ......................... 144

Figure 7.17: The web-based design page for cantilever hook sample b......................... 144

Figure 7.18: The web-based design page for cantilever hook sample c ......................... 145

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1

CHAPTER 1

INTRODUCTION

When used in plastics, snap-fits can be a simple, quick and cost-effective method

of assembling two parts. When designed properly, parts with snap-fits can be assembled

and disassembled numerous times without any adverse effect on the assembly. Snap-fits

also aid in making products environmentally friendly because of their ease of

disassembly, making components of different materials easy to recycle. Traditionally,

snap-fit design methodology has been disorganized and anecdotal in nature, relying

greatly on the skill and the experience of the individual design engineer. A popular

source of design knowledge for snap-fit has been the design guides from resin suppliers

such as Honeywell Plastics [1], General Electric Plastics [2] and Bayer Polymer [3].

However the information disseminated by these guides is fragmentary and often

inaccurate. There has been tremendous progress in the areas of more accurate retention

equations, more accurate finite element models, identification of critical geometric

 parameters and high-performance topologies.

In the current scenario most snap-fit designs are one-of-a-kind efforts with very

little, if any, leveraging between different product families. Design knowledge generated

in a particular project is leveraged for use in subsequent design only through the

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experience of the designer, or through project reports describing the design process and

 performance data. It will be of great value if design information for snap-fit features

 became available to the designers in an abstracted as opposed to application-specific

forum. What is needed are design equations, response surfaces, design heuristics,

evaluation metrics and the like.

It is often tedious to apply design equations for unique types of snap-fits to get

 parameters such as insertion and retention force. This is especially true if response

surface modeling equations which can be algebraically long and tedious are used. For this

reason an interface for the snap-fit design tool is advantageous. With the surge in growth

and availability of the Internet and advanced programs such as Java Applet and Visual

Basic’s Internet Information Service, it was decided that the easiest and most efficient

way for distributing and maintaining a snap-fit design tool was to switch from a

computer-based to a web-based application. With a web browser, a designer can access

the snap-fit design tool independent of computer operating systems (i.e. versions of

Windows OS, Mac OSX, Unix and Linux) from anywhere in the world. This chapter

 presents a brief introduction to snap-fit features, some commonly used terminologies, and

then explains the motivation and goal of this research.

1.1 Introduction to Snap-Fit Features

The main performance attributes of a snap-fit which will be used in this thesis are:

• Insertion Force ( i F  )  is the force that needs to be applied in the insertion

direction of a snap-fit feature to engage it. Insertion force can be expressed as a

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single maximum value or as a graph of the force versus position relative to the

snap-fit feature (Figure 1.1) [4].

• Retention Force ( r  F  )  is the force that needs to be applied in the separation

direction of a snap-fit feature to disassemble it (Figure 1.2) [4]. If designed to be a

 permanent assembly, disengagement occurs due to fracture, permanent

deformation or loss-of-engagement between the two mating parts.

• Locking Ratio  is defined to be the ratio of the maximum retention force to the

maximum insertion force of a snap-fit feature.

Snap-fit are molded into plastic parts to provide attachment functionality. Unlike

rivets, screws etc., which are discrete fasteners, snap-fit features are integral to the part.

They thus help reduce part count and assembly time. The motions require for assembly

are also usually along a single axis as other features on the parts should remove all

degrees of freedom. Snap-fits have traditionally been used in lightly loaded application

such as toys and other consumer products. However, with improvement in polymer

technology and the rapid development of composite materials for structural applications,

snap-fit features are now used in more demanding products. As an example, automotive

under-hood applications that snap-fits have found use in includes air filter housings,

throttle bodies, temperature and pressure sensors, and engine intake manifolds. These

assemblies are expected to withstand harsh environmental conditions like high

temperatures and pressures and contaminants like oil and corrosive gases. Lens housings

in high-end projection television systems, pager housings, single-use cameras, and

compact disc players are other examples of consumer products in which snap-fits have

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 been successfully applied. Quite often, snap-fits are used to provide secondary joining

functionality to give the primary fastening (adhesive, ultrasonic welding) joints time to

form. Commonly used snap-fit topologies are show in Figure 1.3 [4]. The primary cause

restricting the use of snap-fit features in a larger number of polymeric products is the lack

of confidence in their design process and performance attributes.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

Time

   I  n  s  e  r   t   i  o  n   f  o  r  c  e   (   N   )

 Figure 1.1: Typical experimental force curve for insertion of snap-fit features

1.2 Motivation for Research

The essential goal driving this research is the need for more accurate and versatile

models of the performance of snap-fit design and a web-based snap-fit design tool that is

independent of operating system platforms and can be distributed and upgraded

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universally. As mentioned earlier, the models provided by the resin supplier are usually

limited in scope and often inaccurate. An analytical model developed by Suri [4]

overcame some limitations of current design guides. However it is unable to provide

accurate prediction for high retention angles due to convergence issues. And with the

development of the Internet, there is also a demanding for a web-based snap-fit design

tool that is independent of all operating systems, easily accessible and can be universally

upgraded by simply updating the design tool at the server location. The immediate need

for an improvement in these areas forms the motivation behind the work done in this

thesis.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

Time

   R  e   t  e  n   t   i  o  n   f  o  r  c  e   (   N   )

 Figure 1.2: Typical experimental force curve for insertion of snap-fit features

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(a) Cantilever hook feature  (b) Post & dome snap-fit feature(Axisymmetric cross-section)

(c) Bayonet & finger snap-fit feature(Two-dimensional cross-section)

(d) Loop-hook snap-fit feature

(e) Trap type snap-fit feature (f) Hollow-core hook snap-fit feature(Two-dimensional cross-section)

Figure 1.3: Common snap-fit feature topologies

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Figure 1.4: Equations for dimensioning cantilevers

Figure 1.5: Force multiplier of insertion and retention

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1.2.1 Snap-fit Design Approaches

The development of equations for the detailed sizing of snap-fit features and for

 predicting their response can be accomplished using analytical, numerical or

experimental method. The first two approaches, in particular, have been popular in the

 past. Because of cost and time issues, the experimental method is usually replaced by

numerical method using finite element method to simulate experiments. In the following

several sections, the analytical and numerical formulations will be discussed in detail.

Simplified Strength of Materials Analytical Formulation

Design guides from major resin suppliers use a strength of materials to develop

simple equations (Figure 1.4 and 1.5) for the detailed sizing of snap-fits. For example the

cantilever hook is one of the most common snap-fits and its basic shape is shown in

Figure 1.6. It is typically idealized as a cantilever beam with a rigid catch at the end. The

transverse force required to deflect the end of the beam ( P ) by an amount equal to the

offset of the catch is determined using Euler-Bernoulli beam theory. This is related to the

forces acting on the catch in the engagement direction ( i F ) to determine the insertion

force for the feature (Figure 1.6). A similar approach is used to determine the retention

force. Strength of materials formulations however suffer from a number of limitations.

Consideration of the equilibrium of the catch will show that the beam is subject to both

an axial load and a moment, in addition to the transverse load  P . The deflections due to

these forces and moments cannot be found by simple superposition. The bending moment

in the beam due to the axial component depends on the deflection of the end, rather than

 being constant. This effect has not been considered in previous snap-fit analysis [5]. The

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assumption that the maximum deflection of the end of the beam during insertion equals

the offset of the catch also introduces some inaccuracy, because the end of the beam

rotates, in addition to the transverse deflection. The Euler-Bernoulli beam theory is based

on small deformation and long-beam assumptions, which are commonly violated in real

life.

Figure 1.6: Cantilever hook and mating part

Applied Mechanics Equations

Suri [4] has worked on an analytical formulation that includes a full applied

mechanics formulation and solution. It involved idealizing the catch as a rigid body

supported on a flexible structure (Figure 1.7). A set of equations that comprehensively

described the system in its deformed configuration is formulated. The equation system

was iteratively solved for several such configurations to obtain a model of insertion and

retention processes for snap-fits. The model showed excellent agreement with

experimental results for most snap-fit geometries. However, as the retention angle

approaches 90° it becomes impossible to find a solution to the set of equations.

Insertion Direction

Retention Direction

Fi

y

P

CatchCantilever Hook 

Mating

Part 

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Figure 1.7: Suri’s idealized model of cantilever hook and mating part, shown in deformedconfiguration.

The Numerical Formulation

The numerical formulation approach involves the development of response

surface equations and main effect plot using Design of Experiment (DOE) and statistical

methods. The expected significant variables to the snap-fit’s performance and response

variables are first identified. The portion of the design space that is of interest is then

determined. An array of experimental designs spanning the region of interest is then

created using standard statistical software such as SAS®

  and Minitab®

. Physical or

computational experiments at each of the points in the DOE array are conducted. Because

of the cost related to manufacturing physical test samples and the test time, the use of

computational experiments has been more popular in the literature. Typically, finite

elements models of the snap-fit are created and queried for performance attributes.

Response surface equations are fit to the values of the response variables at the

experimental point. These equations can then be used to predict the performance of the

snap-fit at points in the design space other than those in the experimental array. The

 possibility of satisfactorily extrapolating the equations to other regions of the design

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space also exists. Main effect plots depicting the sensitivity of the response variables to

the factors are also created for quick determination of the most important design

variables. Such an approach has been used for the cantilever hook [5], the bayonet &

finger [6, 7], the compressive hook [8] and the post & dome snap-fit features [9].

While the numerical design formulation is an effective and efficient method for

generating design equations for snap-fit, it has some disadvantages. Some limitations

arise from it being a purely statistical method rather than being based on a fundamental

understanding of snap-fit’s performance. The response surface equations are simply

mathematical equations fit to the response variable values. They are very useful and

effective when the analytical formulation is very difficult to model the response surface

equation and the design space has just a single active physical phenomena.

1.2.2 Web-based Snap-fit Design Tool

Using the web as a development environment is a relatively new phenomenon

called web-based application. The CAD/CAM area is also facing a transformation from

computer-based to web-based applications. Computer-based application means that the

design software packages need to be installed and operated on a standalone computer.

Famous examples in CAD/CAM are AutoCAD®

  and ProEngineer ®. Eudora

®  and

Outlook ®

 are also examples of computer-based application. Compared to computer-based

applications, web-based applications refers to applications or services that are resident on

a server that is accessible using a web browser such as Internet Explorer ® or Netscape

® 

and is therefore accessible from anywhere in the world via the Internet and control using

the Web interface that browser provides. A popular example of a Web-based application

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would be Microsoft’s Hotmail®

  that is simply a large-scale email program that runs on

Microsoft’s server computers. They’ve allowed anyone with Internet access and a

 browser to connect to their server and check email. So instead of using Eudora®

  or

Outlook ®

 to check email, the client can check email by using Hotmail® anywhere. This

approach has the following advantages:

• It allows users to log onto the system from anywhere in the world as long asthey have a computer, an Internet connection and a web browser.

• Applications are resident on the server instead of client’s computer so thatusers don’t need to worry about software distribution and updates.

• User accessibility to data is definable.

• System administration can be performed remotely.

On the other hand, there are some limitations for web-based applications. The

major concern with web-based applications is that of control and security. Although most

Web-based application servers protect against hackers with some security systems, no

computer on the Internet is truly hacker-proof even though some web-based applications

further protect their customer’s data by encrypting it in some way. Another limitation is

speed. Web-based applications usually execute quickly, but their response is noticeably

slower. In other words, clicking on a button or link doesn’t result in an instantaneous

reaction by the program. This is simply because of the time lag that it takes for data to

travel via Internet connection from client’s computer to the application server and back.

1.2.3 Constraint Management

Constraint management is a way of planning, organizing, evaluating and

controlling complex systems and it has been used in a number of fields such as

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variational geometry based Computer-Aided Drafting systems, economical management,

mechanical design, chemical process synthesis, and artificial intelligent based constraint

satisfaction procedures. Most of work done in the field of constraint management is

closely related to sparse matrix research since the underlying matrix representations of

the designs tend to be quite sparse. Kinzel and his students [57-62] developed an

interactive engineering design framework consisting of a constraint manager coupled

with a friendly graphical user interface. Constraint management was implemented into

this design tool and several algorithms – new specification, respecification,

unspecification and reverse specification – were developed.

1.2.4 Thesis Objectives

This thesis’s objective is to make significant contributions to advancing the

accuracy and effectiveness of snap-fit modeling techniques in these areas. There are two

objectives for the current research.

• Development of an improved numerical formulation for modeling the

performance of a cantilever hook feature with a high retention angle.   As

mentioned earlier, current snap-fits design guides and Suri’s analytical model are

unable to predict the performance of high-retention cantilever hooks. Nonlinear

and contact finite element analysis are able to overcome this limitation. Design of

experiment and response surface methodology will also allow us to generate an

empirical model to predict the performance of high-retention cantilever hooks.

• Creating a web-based design tool for snap-fit features. Usually empirical

models developed by statistical methods have many polynomial terms and are

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tedious to use. A web-based, platform-independent design tool of snap-fits

implemented those empirical models will be very convenient to use. Three

different snap-fits, Cantilever Hook, Post & Dome and Bayonet & Finger, will be

 built in the design tool and constraint management will be applied. Modules such

as sensitivity analysis and correction advisor will also being applied. Two

different optimizations, single objective optimization and multiple objectives

optimization, will be applied in the design tool.

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CHAPTER 2

LITERATURE REVIEW

2.1 Numerical Formulation for Snap-fit Features

The cantilever hook is the most widely utilized snap-fit feature (Figure 1.3(a)).

Trantina [10] created a model of cantilever and imposed a displacement load on the dwell

surface equal to the offset of the hook. This type of model allows the geometric and

material nonlinearity of the hook to be modeled, and, therefore, produces a reasonably

accurate prediction of the transverse stiffness of the hook. A serious limitation to this

type of model, however, is that it cannot model hooks in retention. This is especially true

for non-removable hooks with retention face angles that are close to or equal to ninety

degrees.

Hotra et al.  [11] modeled retention of a compressive finger by use of contact

elements but did not model the insertion process. The modeling approach taken was to

have the cantilever hook modeled as an elastic structure which is incrementally pushed

into a retention block which is modeled as a rigid surface.

Luscher [5, 12] applied a combination of finite element analysis and design of

experiments to the cantilever hook snap-fit feature. A four-factor, two-level, orthogonal

array was used to study insertion and a five-factor, two-level array was used to study

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retention. The amount of engagement was also defined as a factor to incorporate the

effect of warpage and shrinkage on the performance of the feature. The finite element

results were compared to experimental data. Discrepancies between the two were

explained. The sensitivity of the feature to geometric variables was discussed for

insertion and retention separately. An optimal hook design formulation was also

 presented. The design equations developed by Luscher were first order and the retention

face angles investigated were 65° and 90° degrees. With such a large spread of retention

angles, it was not possible to accurately capture the subtle variations in retention force

 between angles from 85° to 90° degrees.

The post & dome feature is a high performance snap-fit that is self-datuming and

can take some shear loading in addition to retention (Figure 1.3(b)). It provides a higher

locking ratio than traditional cantilever hooks, and its retention strength is less dependent

on friction. Nichols [9] created two design arrays to study the post & dome snap-fit

feature. The first is called the catch array; a focused array constructed to determine the

optimal preload and molded-in undercut at the interface between the post & dome

segments. Optimal performance is judged with respect to both the snap-fits maximum

retention force and maximum locking ratio. The second array, called the macro array, is a

comprehensive design of experiments array. The objective of the macro array is to

generate design data for a variety of dome geometries. Design equations are provided by

subsequent analysis of the finite element data using Response Surface Methods (RSM).

The designed equations provide estimates for the maximum insertion force, insertion

strain, and retention force applicable for an idealized post & dome snap-fit feature. To

accommodate the snap-fit’s geometric nonlinearity, the post & dome are modeled with

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three dimension 8-node brick elements via the MARC®

  finite element package to

generate design data. This three-dimensional contact modeling utilizing an updated

Lagrangian [13] approach provides more accurate displacement predictions than small

displacement beam theory.

A bayonet & finger snap-fit feature contains a bayonet, a retention finger and a

support finger (Figure 1.3(c)). Wang et al. [14] analyzed a bayonet & finger feature using

a contact type finite element model in ABAQUS. High-impact polystyrene (HIPS) was

modeled as an elastic-perfectly plastic material. Finite element analysis predicted a snap-

through failure model. The simulation was compared to results from experimental tests.

Wang and Gabriele [15] created a finite-element model of the bayonet & finger

snap-fit feature. The model is used to simulate the insertion and retention processes of the

feature as well as snap-through and buckling phenomena. A “master-slave” approach is

used for contact modeling in ABAQUS. An elastic-perfect plastic material model is used

to describe the plastic behavior of high-impact polystyrene (HIPS). Finite-element

analysis predicts a snap-through failure mode. The simulation is compared to results from

experimental tests. Reasonably accurate correlation between analysis and experiment is

observed for insertion. Results for retention are not presented.

Lewis et al. [16]  expanded upon the previous work. A design of experiments

approach with two and three dimensional finite-element methods was used to generate

approximate linear response surfaces based on feature geometry which could calculate

insertion and retention forces for the bayonet & finger feature. Five dimensions were

chosen as the significant design factors. Two levels were chosen for each factor. A

fractional factorial experiment design was used to reduce the number of experiments

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(analyses). Sixteen different trials were performed for the study. Material properties were

not considered as a factor in the experiment. Material properties for HIPS were used, with

an elastic-perfect plastic material model. Insertion was modeled using a 2-D model and

retention using a 3-D model. Sensitivity information was generated using a level average

analysis technique. Design equations for predicting the insertion force and retention

force based on part dimensions are presented in the paper. An attempt at confirmation of

the validity of the equations using a sample feature design fails. The FEA analysis results

for the feature show large discrepancies when compared to the design equation

 predictions. No experimental verification is presented.

Shen [7] first found an optimum catch geometry of bayonet using an L-9

orthogonal experimental array. Then another three-lever, five-factor experimental array

was used to obtain the response surfaces. These response surfaces approximated the

amount of over-engagement as well as the insertion and retention forces for values of the

design factors. The calculations were done through a finite element package MARC®

. A

spring was connected between the root of the bayonet and an external node to simulate

the base-part stiffness of the bayonet. The sensitivity of the values of this spring constant

to the feature performance was investigated.

The compressive hook (Figure 1.3(e)) is a common snap-fit used in thermoplastic

automotive electrical connectors. It derives its name form its retention mechanism which

uses compression to provide the locking force. Hotra et al. [17] use an approach similar

to Wang et. al. [13] to model performance of compressive hook. The analysis is

 performed for 15% glass reinforced poly-butylene terephehalate (PBT). The material data

is experimentally determined at 0.08%/sec strain rate. Bending and shearing failure

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modes are investigated separately. The simulation results are compared to experimental

data and are shown to be in good agreement. The paper emphasized the need for the use

of advanced analysis techniques to model the performance of complex snap-fit features.

Hotra et. al. [11] extended the above work. Poly-phenylene oxide (PPO) was also

considered, beside PBT. Bending and shear were again considered in separate analyses

and the lower of the two taken as the actual failure mode. Tests were performed with a

hard gage as a mating part in one case, and the actual terminal as the mating part in the

other case. This was done to study the influence of mating part flexibility on the

 performance of the feature.

Roy [18] adopted an analytical approach for predicting the performance of the

compressive hook feature. The hook was modeled as a cantilever beam with variable

cross-section. In particular, it was divided into three difference beam sections and

Castigliano’s theorem used to determine the load-deflection relationships. A graphical

solution method was also presented. Stiffness matrices for the feature were derived using

an approach similar to the finite element method. Expressions for the insertion

(assembly) force were derived based on equilibrium of the feature. For retention,

 buckling was recognized as the primary model of failure. The Rayleigh-Ritz method was

used to derive approximate expression for column buckling loads under different end

conditions. The same method was also used to estimate eigen-frequencies for longitudinal

and transverse vibration models of the feature. All the results were compared to results

from a finite element analysis conducted using ANSYS.

Other snap-fit features such as cylindrical snap-fit hinges and ball snap-fits, have

 been studied using numerical simulation (FEM) and design of experiment.

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Matuschek and Michaeli [19] used a three-level, five-factor design of experiment

to model the performance of cylindrical snap-fit hinge. Each experiment is a finite

element model of the cylindrical snap-fit feature with the appropriate dimensions. A

quadratic response surface is generated using the results of these analyses. The sensitivity

of the maximum stress in the feature and the releasing force to the geometric variables is

 presented graphically. An equidistant grid strategy is used to partition the design space of

the feature snap-fit identify the optimal set of dimensions for a set of given design

criteria.

Bader and Koch [20] were the first to use a viscoelastic material model in the

finite element analysis of a ball type snap-fit feature. A “standard linear solid”

viscoelastic model consisting of springs and dashpots was used. Plasticity was modeled

through the introduction of an additional mechanical element that remains inactive below

the yield point for the material. The dependency of the spring and dashpot parameters on

strain rate was modeled using a logarithmic curve fit. Plastic damage, recovery and stress

relaxation behavior were experimentally observed and phenomenological models were fit

to the data. Finite element analysis of the insertion and retention processes was performed

using the commercial software MARC and MENTAT. The results obtained using

viscoelastic and elastic-plastic material models were compared to experimental test data.

2.2 Materials Issues in Snap-fit Design

There have been some efforts to promote the use of advanced material models for

 polymers in the field of snap-fit design. Most of these have, however, been limited to

relatively simplistic elastic-plastic material formulations, in order to model residual strain

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in the polymer. This thesis will attempt to extend the state-of-the-art in snap-fit analysis

 by demonstrating improvement in the accuracy of finite element analysis predictions. A

 brief review of past work dealing with improved material models for snap-fit design is

 presented below, to emphasize the benefit of the proposed approach.

In order to evaluate the applicability and accuracy of experimental test data,

Knapp et al. [21] compare three tensile test methods. It is well known that polymer

 behavior is dependent upon strain rate and that conventional tensile tests subject the

specimen to variable strain during the test. In this work, tests were conducted at 0.002/s

and 0.02/s strain rate. The experimental stress-strain data fit to an analytical curve and the

value of tangent modulus (Et) at each strain value was determined. This material model

was used in a finite element analysis and its results compared to experimental data. The

authors propose that designers should use true stress-strain data gathered at strain rate

values appropriate for their application. Trantina and Minichelli [22] describe a software

developed for automating snap-fit finite-element analysis. Deflection limited models are

analyzed. Some elementary results describing the effect of dimensions on stress and

strain on snap-fits are presented. The authors use a 0.1/s strain rate for determining the

material properties. The work is also summarized, in more detail, in Trantina and

Minnichelli [10, 23].

Sawyer et al. [24] investigate the applicability of commonly reported coefficient

of friction (µ) values for snap-fit design. The analysis of surface friction between

 polymers is extremely complex [25-27]. Widely different values of µ have been reported

in literature. For example, coefficients of friction (µ) ranging from 0.28 to 0.62 have been

reported for unfilled nylon 6/6. In this study, friction measurements were made directly

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from polycarbonate cantilever hook lock pair contacts, and their dependence on load,

sliding speed and contact geometry was investigated. An experimental setup for

accomplishing this is detailed. A power law dependence of µ on normal load is suggested

in the paper. Changes in the value of µ with sliding speed were found to be minimal. The

contributing mechanisms to friction, viscoelastic deformation and adhesion are discussed

 briefly. The authors contend that adhesive contributions occur at a scale smaller than that

modeled by finite elements and as such should be included in the value of µ. On the other

hand, the effect of viscoelastic deformation should be captured by the finite element

model of the snap-fit and as such need not be included in the value of µ.

2.3 Failure Models of Cantilever Hook

In the past cantilever hook has always been selected primary in terms of retention

force and the molding tooling costs. Failure modes of cantilever hook are usually ignored

even although it was already recognized and understood that it will be an important factor

to the performance of cantilever hook.

Luscher [5] noted a snap-fit mode of failure called loss-of-engagement at small

Percent Engagement (PE) values (Figure 2.1). In loss-of-engagement failure the snap-fit

slips past the mating part without any mechanical damage to either half of the feature. At

high  PE   values, the snap-fit’s failure mode switched to either a shear and/or tensile

material failure.

Luscher et al. [28] extended the previous work. They determined the exact

dividing lines between these different modes of failure with respect to the  PE   (Figure

2.2). Well-defined transitions between modes of failure were observed with the cantilever

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hook. The different failure modes also observed at different levels of  PE . Three distinct

failure modes were observed: shear failure, tensile failure and loss of engagement. Loss

of engagement was the exclusive failure mode from 20 to 66  PE  while shear failure was

the only mode of a failure from 70 to 100  PE . Finally, tensile failure occurred at all

values over 100 PE . There was only one exception, a single shear failure point at the 40

 PE  level. It was considered a statistical anomaly, most likely due to a material, molding

or alignment condition. Figure 2.3 shows an example from each of the two mechanical

failure modes in the cantilever hook. As shown in the Figure 2.3, the direct shear failure

seems to occur progressively over a certain distance. Note in this figure that the failure

seems to have a “stair-step” pattern showing that the mating part is tearing out the catch

starting at the retention face. It is not able to tear out the catch along a line parallel to the

removal direction. The tensile failure case is different. The failure surface looks like it

occurs at a single point in time.

Figure 2.1: Percent engagement of cantilever hook

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Figure 2.2: The different failure modes with respect to the PE

(a) The tensile failure mode (b) The shear failure mode

Figure 2.3: The two different mechanical failure modes for cantilever hook

It is very important to identify the transition from one failure mode to another in

order to accurately calculate the retention force provided by the cantilever hook feature.

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2.4 Web-based Design Tool for Snap-fit

The Internet is changing not only our daily lives, but also the professional field of

 product design and new product development.

Wang et al. [29] developed a fluid simulation package integrated with the

Internet, which allow the user to perform fluid simulation on the World Wide Web

(WWW), to simulate the traditional Computational Fluid Dynamics (CFD) problem -

Transient Natural Convection in a Cavity. This simulation package was developed using

two different tools: one is Microsoft®

/COM ActiveX and VBScript and the other is Java

and Java Applet.

Chung and Wright [30] proposed a web-based engineering framework for an

infrastructure, which integers various advanced design/manufacturing systems such as

CAD, FEM and CAM, provides relationships among product data, and coordinates with

manufacturing system. Designers access this framework through web browsers and start

the design with some initial parameters.

Colton and Dascanio [31] described a vision and current development in a

distributed design and manufacturing environment, and emphasized how current CAD

tools will evolve to facilitate the distributed design and fabrication process. They also

 presented the development of a set of web-based design tools for fabricating parts using a

machining process via the Internet and experiments on machining 2-1/2 D and freedom

 parts through their Java-based design tool showed the feasibility for a networked

machining service via the Internet.

Kim et al. [32] developed a web-accessible CAD tool that assists a designer in

 producing designs that are manufacturable on a 3-axis milling machine, which simplify

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the previous complex testing for manufacturability by focusing on 3-axis milling only

and by making use of new fixturing techniques using a plastic compound or a metal alloy

for reference-free part encapsulation (RFPE). They also build a Java-based front end of

this design tool that can be downloaded as an applet over the Internet.

Velάsquez et al. [33] created a system named Tool Trial System (TTS) which is

capable of collating and disseminating information relating to tool trials amongst a

variety of user groups. TTS provided a platform from which is possible to submit and

retrieve highly specific technical tooling data on the WWW can be downloaded by

remote users in the form of Java applets, through any computer with Internet connection

and using conventional Java enabled browsers without the requirements of using middle

tiers software or hardware between clients and server sides.

Rajagopalan et al. [34] built an Internet-based infrastructure to provide designers

with access to multiple layered-manufacturing services. This system contains three

 primary operatives: Design Clients, Manufacturing Services and Process Brokers. The

Design Clients allows designers to submit completed designs for algorithmic

decomposition, or alternately, to compose a design from primitives and library

components that have been primed with some process-related information.

Manufacturing Service consists of a highly automated machine that can be used to build

ceramic parts, and the associated software components for design decomposition, process

 planning and machine control. The Process Broker implements a number of supporting

services including process selection and optimal part orientation.

Ebbesmeyer et al. [35] described a web-based tool Virtual Web Plant (VWP), a

tool to integrate 3D models from various CAD plant design tools and to display them

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interactively through Internet. Using the application of object-oriented database, it is

 possible to define various views of the logical plant structure so that the user is able to

navigate easily through both the plant structure and the project documentation. The

special advantages of an objected-oriental database for the storage of the graphical data

are also shown.

Huang et al. [36] described a web site, WAPIP, which has been developed

specifically to support new product introduction activities. It provides databases for

software vendors and researchers to register their web applications with the “wapip”

search engine. It also provides facilities to support practitioners in product design and

manufacture to search rapidly for the right web applications suitable for solving their

 problems.

Many response surface equations for snap-fit feature’s performance [5, 9, 15]

have already been studied using DOE and FEM. But it is often tedious to look for design

equations for unique types of snap-fits to calculate the insertion force, the retention force

and the locking ratio. If found, these equations are usually long, complex, and difficult to

use. For this reason, an easy using front end will be very helpful during the designing

 process of snap-fit features.

Oh et al. [37] created a calculator (Figure 2.4) in which the design equations of

seven snap-fit features (annular snap, bayonet & finger, post & dome, cantilever hook,

cantilever-hole, compressive hook, L-shaped hook and U-shaped hook) were

implemented. This calculator aids in designing snap-fits to meet specific loading

requirements by allowing the designer to size the feature to obtain desired estimates for

maximum insertion and retention forces. This calculator was developed in JAVA®

 

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language that is independent of operating system platforms and can be distributed at a

company site-wide over an intranet or worldwide over the Internet. This makes it easily

accessible to a user, and universal upgrades can be achieved by simply updating the

software at the server location.

Figure 2.4: The IFP snap-fit design tool

Honeywell Plastics [1] developed an online Snap-Fit Design Workspace (Figure

2.5). It is a web-based application that serves as an engineering tool in snap-fit design.

The program provides a workspace for the designer to test different scenarios by

adjusting various input parameters and selecting different engineering material for the

snap-fit design. The following snap-types were included: five different uniform beams,

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two different tapered beams, two different U  shaped beam cases, and an L shaped beam.

The cantilever beam formulas used in conventional snap-fit design poorly estimates the

amount of strain at the beam/wall interface because they do not include the deformation

in the wall itself. To obtain a more accurate prediction of total allowable deflection and

strain for short beams, a deflection magnification factor was applied in this design

workspace.

Figure 2.5: AlledSignal Plastics’ snap-fit design guide

Eastman chemical company [38] also developed an interactive online support tool

(Figure 2.6) that receives inputs from the user to generate a technical recommendation for

cantilever snap-fit design. This design tool calculates the theoretical strain that occurs

when a cantilever is deflected. And the maximum strain occurs on the outer layer of the

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thickness, usually at the base of the latch. Strains can also be calculated for a latch,

which varies linearly in both thickness and width from the base of the tip to latch. After

the users specify the material, its brand and geometry dimensions for cantilever, this

design tool will predict the deflection force and outer fiber strain.

Figure 2.6: Cantilever snap-fit design tool from Eastman chemical company

Jeff Raquet [39] used Java Applet to create a snap calculator (Figure 2.7) to assist

in the development of correct parameters in the design of a plastic snap-fit. Designers

need to select a material and give values to several specified inputs. Then based on these

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inputs, the snap-in force, engagement force and stress at root are computed. It also

 provide an “optimize” option to minimize the snap-fit’s volume.

Figure 2.7: Jeff Raquest’s snap-fit calculator

GE Plastics [2] (Figure 2.8), Engineers Edge [40, 41] (Figure 2.9 and Figure 2.10)

and Berkeley Manufacturing Institute of University of California - Berkeley [42] (Figure

2.11) also developed similar online snap-fit calculators. All of them used conventional

design equations based on small deformation and long beam assumptions.

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Figure 2.8: GE Plastics’ snap-fit wizard

Figure 2.9: Engineers Edge’s snap-fit straight beam calculator

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Figure 2.10: Engineers Edge’s snap-fit tapered beam calculator

Figure 2.11: Brock & Wright’s design tool for snap fits

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2.5 Constraint Management

For the current research, constraint management will be an important section. So

it’s necessary to review the literature of Constraint Management (CM). Constraint

Management techniques have been used in a number of fields, such as variational

geometry based Computer-Aided Drafting systems, mechanical design, chemical process

synthesis, and artificial intelligent (AI) based constraint satisfaction procedures. Most of

the work done in the field of constraint management is closely related to sparse matrix

research since the underlying matrix representations of the designs tend to be quite

sparse.

Steward [43, 44] was one of the earliest investigators who examined the structure

of simultaneous equations. Algorithms to partition and reorder directed graphs were

shown by Christensen and Rudd [45].

Duff [46] presented an excellent survey paper on sparse matrix research that

included a survey of algorithms for matching and decomposing sparse matrices. Later,

Duff and Reid [47] developed an algorithm (Harwell MC13A) to permute a square

occurrence matrix to its block triangular form in order to determine simultaneous

equations sets. Their algorithm was based on Tarjan’s algorithm for determining the

strong components of a directed graph. In order to accomplish its purpose, MC13A

required the original matrix to be converted to a zero-free diagonal (maximum

transversal) form. They  compared various algorithms to achieve this configuration and

developed algorithm MC21A to permute a sparse, square matrix to the maximum

transversal form.

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Shacham [48] presented a new method to partition simultaneous nonlinear

algebraic equations into smaller irreducible subsets. Prasad and Kinzel [49] presented an

algorithm to perform decomposition of incidence matrices as the number of degrees of

freedoms is reduced in a mathematical model. Akin [50] refers to the occurrence matrix

as a Boolean Inference Array and presents an algorithm for the detection of solvable sets

of nonlinear equations that is quite similar to the work of Prasad and Kinzel. The work of

Pothen and Fan [51] is aimed at converting a sparse rectangular matrix to its block

triangular form. Their two stage decomposition techniques is based on the Dulmange-

Mendelsohn decomposition algorithm.

A number of researchers have studied constraint management in the field of

variational geometry. Sutherland [52] was amongst the first researchers to treat 2-

dimensional drafting as a constraint satisfaction problem. Later works of Lin and Light

[53], Gossard [54], Serrano [55] added more to the mathematical basis behind CM.

Serrano used directed acyclic graphs (DAGs) to determine dependencies among variables

and to identify sets of simultaneous equations. A constraint modeler was also shown that

allows for interactive addition/deletion of constraints. Chung and Schussel [56] compared

 parametric and variational geometry environments and described a commercial package.

Srinivasan [57, 58] presented the Design Shell: an interactive engineering design

framework consisting of a constraint manager coupled with a friendly graphical user

interface. Sridhar [59, 60]  formulated strategies to interactively handle inequality

constraints in the design shell framework. Agrawal et al. [61] identified and developed

various constraint management, numerical solution, and numerical optimization

methodologies necessary for the performance of the shell as an interactive and effective

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design tool. Thomas [62] extended the design shell constraint management strategies

towards the development of a framework for performing term designs.

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CHAPTER 3

FINITE ELEMENT ANALYSIS OF CANTILEVER HOOK WITH A HIGH

RETENTION ANGLE

This chapter describes the finite element model of cantilever hook with a high

retention angle. Firstly three different nonlinearities of finite element analysis were

described: Moving Contact, Geometric Nonlinearities and Material Nonlinearities. Then

finite element analysis techniques applied in this thesis were explained because in a

typical FEA simulation usually 80%-90% time is spent to find the best parameter settings.

Finally a two dimensional, contact FEA model was created in MSC.MARC, and the

results were compared to the sample testing results with the same geometry to verify the

accuracy of the FEA model.

After an accurate FEA model was obtained in this chapter, in the next chapter it

was been used for the simulation of each design point and an empirical model was

generated by using Design of Experiment (DOE) and Response Surface Methodology

(RSM).

3.1 Combination of FEA with DOE

Modern finite element analysis and design of experiments techniques can be

combined into a powerful quality-engineering tool [63]. FEA has been widely used to

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simulate the snap-fit behaviors in the recent years and proved to be a powerful tool to get

 better understanding of snap-fit performance. Usually FEA users worked in the way of

“one problem-one model” interactively. That is, for a given problem (say, a cantilever

hook with certain geometry and under certain working condition such as loading and

 boundary condition), designers usually build a fixed model, solve it and get the required

results (the insertion force, the retention force, etc). This has proved to be useful as

shown in many applications.

However, when using FEA to do DOE and design optimizations, the snap-fits

have similar geometry but the parameters of components (dimensions and material

 properties) are changeable. Generally, even a simple DOE often involves more than ten

runs. Working in one problem-one model way, we need to build many models

interactively or manually which is time consuming.

Fortunately, in MARC FEA package, besides working in interactive mode, it also

allowed to use “PROCEDURE” file (Appendix A). “PROCEDURE” file allows users to

 build the model in terms of parameters instead of fixed value, which in turn make it very

easy to change the design. Compared to the traditional one problem-one model method,

the combination between DOE and FEM has the following advantages:

• By expressing all the parameters in terms of variable the model becomes very

flexible. In order to rebuild other models, designers just need to change the

 parameters’ value. This feature is especially suitable for DOE and design

optimizations.

• Store the model in text file (generally in size of a few kb) instead of in db file (in

size of a few Mb) to save space.

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• Run batch of models overnight to make full use of FEA system.

3.2 Nonlinear Structural Analysis of Snap-Fits

The reasons that the snap-fits design guides are not accurate for cantilever hook

with a high retention angle are material and structural nonlinearities. Suri’s analytical

mathematical model has difficulties to converge for high-retention cantilever hooks.

 Nonlinear structural behavior arises from a number of causes, which can be grouped into

these principal categories:

• Moving Contact

• Geometric nonlinearities

• Material nonlinearities

For cantilever hook with a high retention angle, all these three nonlinearities exist

in the retention stage. The following paragraphs explained these three nonlinearities in

details.

3.2.1 Moving Contact

Many common structural features exhibit nonlinear behavior that is status-

dependent. Status changes might be directly related to load, or they might be determined

 by some external cause.

Situations in which contact occurs are common to many different snap-fit

applications. Contact problems are highly nonlinear and require significant computer

resource to solve. Contact problems present two significant difficulties. First, users

generally do not know the regions of contact until you’ve run the problem. Depending on

the loads, material, boundary conditions, and other factors, surfaces can come into and go

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out of contact with each other in a largely unpredictable and abrupt manner. For example,

during the retention of cantilever hook there actually exist several contact pairs

depending on contact locations. Since the deformation of local contact areas it’s very

difficult to identify the point when these contact pairs change. The second reason to cause

contact problem highly nonlinear is that most contact problems need to account for

friction. There are several friction laws and models to choose from, and all are nonlinear.

Frictional response can be chaotic, making solution convergence difficult.

3.2.2 Geometric Nonlinearities

Small deflection and small strain analyses assume that displacements are small

enough that the resulting stiffness changes are insignificant. In contrast, large strain

analyses account for the stiffness changes that result from changes in an element’s shape

and orientation. All design guides of snap-fits and Suri’s analytical model were based on

small deformation theory. However cantilever hook with a high retention angle

experiences large deformations, its changing geometric configuration can cause the

structure to respond nonlinearly.

3.2.3 Material Nonlinearities

A number of material-related factors can cause your structure’s stiffness to

change during the course of an analysis. Nonlinear stress-strain relationships of plastic,

multilinear elastic, and hyperelastic materials will cause a structure’s stiffness to change

at different load levels (and, typically, at different temperatures). Creep, viscoplasticity,

and viscoelasticity will give rise to nonlinearities that can be time-, rate-, temperature-,

and stress-related. Most polymeric materials used for cantilever hooks have a time-

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dependent behavior, in which a certain amount of deformation is recoverable over time.

For the purpose of numerical modeling, it was assumed that the material was time-

independent.

For all design guides of snap-fits none of these material nonlinearities were

included in their design guides. All of them assumed that the maximum stress always

under the yield stress that means no plastic deformation occurred during the retention. So

that in order to overcome the inaccuracy caused by the ignoring of plastic deformation,

it’s necessary to include the material nonlinearity in the FEA model. Several plastic

material options are available in most of FEA packages such as MSC. MARC and

ANSYS to describe material plasticity behavior. These options are the Bilinear

Hardening, Multilinear Kinematic Hardening, Nonlinear Kinematic Hardening, Bilinear

Isotropic Hardening, Multilinear Isotropic Hardening and Nonlinear Isotropic Hardening.

Since in this thesis only Bilinear Hardening was used to describe the material plasticity

 behavior of plastics, the Bilinear Hardening option was shown in Figure 3.1. Although

this is an approximation of the actual material’s curve it is an improvement over ignoring

 plasticity. For more details of other options, please refer to [64].

3.3 Numerical Model of Cantilever Hook Feature

In order to compute the insertion force, insertion strain, retention force and

locking ratio of cantilever hook, the author created a two-dimensional, contact and

nonlinear finite element model on Marc for each experiment. A mesh was generated

using plane stress quadrilateral elements with full integration, which can prevent

instability called hourglass mode [13] in a bending dominant problem. An example of the

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meshed cantilever hook model on Marc is illustrated in Figure 3.2. A fine mesh is

required on the insertion face, dwell face and retention face of cantilever catch where are

the crucial regions at engagement and disengagement.

Figure 3.1: The actual and bilinear stress-strain curve for plastic material

3.3.1 Modeling Techniques for Contact Problem

The modeling of the contact conditions is an advanced technique in the finite

element method. To simulate the contact condition of the finite element models, the

cantilever hook was specified as a deformable contact body and the fixed displacement

 boundary conditions were applied to the end of the cantilever hook (Figure 3.2). The

mating part was defined a rigid body and the nodes on it were tied together as a group

and fixed in the vertical direction.

The analysis of surface friction between polymers is extremely complex [24-26].

The classical "laws" of friction in which friction is proportional to applied load, and is

independent of area of contact, velocity, and time of contact are poorly held for most

 polymers. In addition the conditions of a sliding snap-fit in terms of its speed, area of

Inelastic

Region

Elastic

Region

Strain

Stress

Yield

Stress 

Inelastic

Region

Elastic

Region

Strain

Stress

Yield

Stress

Workhardening

Slope

(a) An actual stress-stain curve (b) Bilinear hardening option

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contact, and residence time are not usually replicated in resin manufacturers tests and

reported friction values. Rather than delve into this area, which is not a key focus of this

research, it will be assumed that background research will be done to determine a value

of coefficient of friction which is appropriate for the snap-fit in question. Therefore in

these