Final Analysis Report

MIE 313 � Design of Mechanical Components

Juliana Amado

Charlene Nestor Peter Walsh

i

Table of Contents Abstract: ......................................................................................................................................... iii

Introduction:.................................................................................................................................... 4

Procedure: ....................................................................................................................................... 5

Results:............................................................................................................................................ 6

Reliability:....................................................................................................................................... 8

Redesign:......................................................................................................................................... 8

Conclusion: ..................................................................................................................................... 9

Table of Figures Figure 1: Table saw Fence Assembly ............................................................................................. 4

Figure 2: Clamp Block Assembly................................................................................................... 4

Figure 3: Critical Stress Locations.................................................................................................. 6

Figure 4: Stress Intensity for fast fracture....................................................................................... 7

Figure 5: Existing Clamp Block ..................................................................................................... 8

Figure 6: Redesigned Clamp Block ................................................................................................ 8

Tables

Table 1: Stress Analysis Results ..................................................................................................... 6

Table 2: Critical Stress Redesign vs. Original ................................................................................ 9

ii

Appendices Appendix A: Energy Analysis of Input Force .......................................................................... I

Appendix B: Force Model & Hand Calculations................................................................... III

Clamp Block: Simplified Stress Calculations ........................................................................... IV

Lifter: Simplified Hand Calculations .......................................................................................VII

Dog: Simplified Hand Calculations.........................................................................................VIII

Appendix C: FEA Loads and Constraints............................................................................... X

Clamp Block .............................................................................................................................. XI

Lifter .........................................................................................................................................XII

Dog ..........................................................................................................................................XIII

Appendix D: FEA (Existing Design) ....................................................................................... XIV

Appendix E: FEA (Redesign) ................................................................................................XVIII

Appendix F: Reliability Calculations ........................................................................................XX

Existing Clamp Block............................................................................................................. XXI

Redesigned Clamp Block ....................................................................................................... XXI

Appendix G: Engineering Drawings ......................................................................................XXIII

Clamp Block ........................................................................................................................ XXIV

Lifter ......................................................................................................................................XXV

Dog ..........................................................................................................................................XIII

iii

Abstract: The part being analyzed is the Clamp Block component of the clamping mechanism of a

table saw fence assembly. The clamping system acts as a spring and applies normal force to the

fence rails. This normal force results in friction forces that prevent the fence from sliding

horizontally along the fence rails. The Clamp Block has failed by fast fracture near an abrupt

change in geometry.

The overall objective of this project is to analyze the mode of failure of the Clamp Block

and redesign it accordingly. Finite element analysis was used in conjunction with hand

calculations to determine the cause of failure of the Clamp Block. Three critical locations were

identified in the area of fracture. The most highly stressed was at an inside corner on a horizontal

web that spans between the two sides of the Clamp Block. The magnitude of stress was not

above the yield of the material at this location, but statistical analysis shows that the failure rate

was approximately 5%. The concentration of stress at this critical point may have caused cracks

to propagate causing fast fracture on subsequent loading cycles.

To improve the reliability, changes were made in the geometry of the Clamp Block near

the critically stressed area. These included increasing the thickness of the web from 4mm to

5mm, and adding a 3mm radius at an inside corner. These changes did not interfere with the

other components in the system, and increased the reliability to a level of 99%.

4

Introduction:

The focus of this analysis and redesign project is the clamp block component of the

clamping mechanism of a table saw fence assembly. The in use operational purpose of the fence

is to set the cutting width of the saw. It moves on two pipe rails that are perpendicular to the

fence and saw blade. The user sets the cutting width, then clamps the fence to the rails by

applying force to the hand lever. Once the user has rotated the hand lever to the locked position

the system acts as a spring, storing the applied energy. This spring energy is what maintains the normal force on the rails that is necessary to keep the fence from sliding sideways.

Figure 1: Table saw Fence Assembly

The component in this system that has failed is the clamp block. It has fractured as

shown below in Figure 2.

Figure 2: Clamp Block Assembly

5

Objectives:

The objective of this project is to analyze the mode of failure of the Clamp Block and

redesign it accordingly. Issues that will be addressed are: static loading, dynamic loading,

fatigue, and reliability. To reach this end, the previously developed solid and force models will

be the basis for finite element analysis using the Pro-Mechanica software package. The FEA will

be verified using simplified hand calculations, and the results of both techniques of stress

analysis will be used as a foundation for decisions about the redesign. FEA analysis will also be

run on the Lifter, and Dog components that are in direct contact with the Clamp Block in order to

see if they are being stressed close to the limits of their capacities.

Procedure: Design Analysis:

1. Solid model representations of critical components.

• Simplify solid models if necessary.

• Develop force model based on solid geometry and conditions of use.

2. FEA critical components

• Model loads using information developed in the solid and force models.

• Constrain components against motion in a manner that simulates the actual boundary

conditions.

• Plot graphical results of the FEA.

• Validate FEA results using simplified hand calculations.

• Determine the critical locations and magnitudes of stresses in the Clamp Block

Redesign:

• Assess significance of static versus dynamic loading as factors contributing to failure.

• Assess significance of fatigue as a factor contributing to failure.

• Assess reliability of existing part using FEA and hand calculation results.

• Determine desired reliability of the redesigned part.

• Redesign to meet desired reliability parameters.

6

Results: The results of the FEA and hand calculations agree closely, and indicate that there are

three critical locations in the area of the fracture.

Figure 3: Critical Stress Locations

The most highly stressed was point C on a horizontal web that spans between the two

sides of the clamp block. This is the result of the 773 N load causing compressive stress, σyy,

along the web. The change in section of the web at point C causes a significant stress

concentration. An examination of the broken part showed small cracks originating from corners

at point C, and extending towards the area of point B. The part was broken cleanly along the red

line indicated between points A and B.

FEA σσσσe (Mpa) Hand σσσσe (Mpa) Sy (Mpa)

Clamp Block: A

(Grey Cast Iron)

0.183 � 41.7

41.6

152

Clamp Block: B 0.183 � 41.7 31.8 152

Clamp Block: C 83.4 � 125 145 152

Lifter (HR Steel) 736-883 795 910

Dog (HR Steel) 842-947 844 910

Table 1: Stress Analysis Results

7

One explanation for the failure is that the cracks originating at point C reached the area of

point B and caused fast fracture to occur along the line A-B. Using the relationships for stress

intensity factors ki and kic, the plausibility of fast fracture as a failure mode can be assessed.

Figure 4: Stress Intensity for fast fracture

Although ki is not greater that kic,, it is close enough that fast fracture is clearly possible

given that the material properties of the casting will vary from sample to sample. This issue of

reliability will be explored in the next section of this analysis.

Fatigue due to cyclic loading was considered as a possible cause of failure, but, because

this is a hand operated system, and is used relatively infrequently, it is unlikely that the Clamp

block component experienced 106 loading cycles More likely on the order of 103 cycles.

Likewise, because the input energy is applied to the system rather slowly by a human hand

dynamic loading was also dismissed as a possible cause of failure.

The stress analysis results listed in Table (1) indicate that the Dog and Lifter are also

being stressed close to their limits. These components may also need redesign in order to achieve

an adequate level of reliability for the system as a whole, but since the analysis was primarily on

the Clamp Block, the redesign of the Lifter and the Dog were not addressed in this report.

gi ck σ⋅⋅= 8.1 and yic ck σ⋅⋅= 0.2 (Juvinall-Marshek eq. 6.1 & 6.2 pg 209)

where kI is stress intensity factor and kic is the fracture toughness of the material.

When kI ≥ kic fast fracture will occur.

Assuming a maximum crack length of 7 mm, σg = max stress on section = 125Mpa, and σy =

yield strength of grey cast iron = 152 Mpa.

23

19125007.08.1−

−=⋅⋅= mNki 23

5.25152007.02−

−=⋅⋅= mNkic

8

Reliability:

Reliability is a concept closely related with the factor of safety. In industrial production,

reliability factor plays an important role; the goal is to obtain a product with a high reliability

avoiding failure.

The clamp block of the table saw was subjected to a load of 125 MPA. The yield strength

of the ASTM 20 Grey Cast Iron is 152 MPA. This means that failure will not always occur under

these loading conditions.

The interference theory of reliability prediction was used to approximate the failure

percentage that would be expected from the original design. The clamp block was found to be

95% reliable and with a factor of safety equal to 1.22. (See Appendix F: Reliability Calculations)

Redesign:

To improve its reliability the Clamp Block was redesigned. Geometric changes were

made in the area of point C as follows: The thickness of the web was increased from 4mm to

5mm, and a 3mm radius replaced the sharp inside corner at point C. The results of an FEA on the

redesigned part, and reliability analysis showed that the stress in the critical locations was

reduced, and that the component as redesigned is 99% reliable with a factor of safety of 1.4.

(See Appendix F: Reliability Calculations)

Figure 5: Existing Clamp Block Figure 6: Redesigned Clamp Block

9

Conclusion:

The results of this analysis indicate that the most critically stressed location on the Clamp

block is in the area of points C and B. An inspection of the failed part showed small cracks

emanating from point C toward the area of point B. This observation, coupled with FEA and

hand calculations, led to the conclusion that the Clamp block cracked under load near point C,

and that these cracks caused fast fracture of the section along a line from point B to A on

subsequent loadings. (See Figure 3: Critical Stress Locations)

While the levels of stress intensity at the critical locations do not exceed the yield

strength of grey cast iron, an analysis of the reliability of the Clamp block using the interference

theory of reliability prediction showed that as originally designed and under the assumed loading

conditions, the Clamp block had a failure rate of approximately 5%. This is a relatively high rate

of failure � the component was redesigned to reduce the stress at critical locations and give a

failure rate of 1%.

The stress at point C is primarily due to direct compressive stress on the web between the

two sides of the Clamp block. At point C there is a sharp corner that causes a significant stress

concentration. This group�s recommendation for redesign is that the thickness of the web be

increased from 4mm to 5mm, and that the sharp corner at point C be replaced with a 3mm radius.

These changes in geometry should not cause any interference with the other components in the

system, and an FEA run on the modified part indicates that the stress at the critical location has

been reduced enough that the 99% reliability target has been reached. Graphical plots of the

redesigned FEA are in Appendix E.

Original Design FEA σσσσe (Mpa) Sy (Mpa) %Fail S.F Clamp Block: C 83.4 � 125 152 5% 1.22

Redesign

Clamp Block: C 75.7-113 152 1% 1.40

Table 2: Critical Stress Redesign vs. Original

I

Appendix A: Energy Analysis of Input Force

II

Work Analysis of Applied Force The force being applied to the hand lever results in forces acting on the Clamp Block,

Lifter, and Dog. In order to find these forces and develop FBD�s for the Clamp Block, Lifter, and

Dog, the tension in linking rod must be established.

Figure 3. Work Analysis

At the beginning of the hand lever rotation the entire system is at rest- there is no elastic

energy stored in the system and so the initial applied force is zero. As the hand lever is rotated

the components of the system are stretched and the required input force increases until it reaches

a maximum at the locked position. Using Data from Human Factors Design Handbook, an

applied force of F=80 N has been established as the upper range of force that an average person

can easily apply when pushing down on a lever from a standing position. Assuming the force

varies linearly from 0 N to 80 N over the displacement then Fav=40 N. Using the principle of

conservation of energy, the work applied at the hand lever equals the work done on the linking

rod. Wlever = Wrod Sf θf π/4

⌠ ⌠ ⌠ ⌡F ds = T ds = ⌡ Fav cos θ r dθ ⇒ Trod (.003m) = ⌡(-40 N) (0.100m) cos θ dθ Si θi 0 ⇒ 4 sin (π/4) = Trod (0.003m) ⇒ Trod = 943 N

45°

III

Appendix B: Force Model & Hand Calculations

Clamp Block

Lifter

Dog

IV

Clamp Block: Simplified Stress Calculations

Stress at (A): Bending stress σzz at extreme fiber.

)1530(1230)1526(12301073)1035(773 −⋅−−⋅+⋅++⋅−=AM

mmNM A −= 31975

+⋅+

−+⋅⋅= 2

32

3

1312

5.320)1519(12

385.32AI

4322816mmI A =

Kt≅2.8 (From Juvinall-Marshek Figure 4.39, pg 133)

8.232281

1531975 ⋅⋅=zzσ

Mpaezz 6.41== σσ

V

Stress at (B): Combined bending stress σzz at extreme fiber and axial compressive stress σyy from 773 N contact force.

σσσσzz:

1912303773)335(773 ⋅−⋅+−⋅=BM

mmNM B −= 3685

⋅−

⋅=12

61312

1420 33

BI

44105mmI B = C=-7mm

4105)7(3685 −⋅=zzσ

Mpazz 31.6=σ

σσσσyy:

tyy k⋅⋅

=144

773σ

5.2≈tk (From Juvinall-Marshek Figure 4.41, pg. 135)

5.2144

773 ⋅⋅

=yyσ

Mpayy 5.34=σ

σσσσe:

[ ]21

22 5.3431.65.3431.6 ⋅−+=eσ

Mpae 8.31=σ

VI

Stress at (C): Combined bending stress σzz, axial compressive stress σyy.

σσσσzz:

19123035773 ⋅−⋅=CM

mmNM C −= 3790

44105mmIC = mmc 8=

410483790 −⋅=zzσ

Mpazz 7.3−=σ

σσσσyy:

tyy k⋅⋅

=144

773σ

10≈tk (From Juvinall-Marshek Figure 4.41, pg. 135)

1064

773 ⋅⋅

=yyσ

Mpayy 147=σ

σσσσe:

[ ]21

22 1477.31477.3 ⋅−+=eσ

Mpae 145=σ

VII

Lifter: Simplified Hand Calculations

Mx=My at the element is 53 N-m

σσσσ1: IcM x

x == σσ1 433

45712)14(2

12 mmbhI === mmc 7=

2

3434457

)7(103.28mm

Nxx ==σ

σσσσ2: IcM y

y == σσ 2 433

28812)12(2

12 mmbhI === mmc 6=

2

3590288

)6(103.28mm

Nxy ==σ

yx σσ ≠ , but both are positive so there is no need to use a Mohr circle to find combined stress,

instead will use σx and σy to calculate Von Mises stress.

VIII

Simplified Hand Calculations, Lifter contd.

σσσσe: 2122 )( yxyxe σσσσσ −+=

22

122 530)]590(434590434[mm

Ne =−+=σ

This calculation of Von Mises stress does not take into account the stress concentration due to

the bore for the pin. Using the tabulated data for stress concentration factor kt, and the geometry

of the lifter kt≈2.3

36.06.166 ==b

d From Juninall/Marshek figure 4.40-a pg 134: kt≈1.5

So, the σe taking stress concentration into account is approximately:

2max 795)5.1(530mm

Nktee === σσ

Dog: Simplified Hand Calculations

IX

4

336412

)4(1212 mm

NbhI === mmc 2=

2

384464

)2(102721mm

NxI

Mc ==== σσ

( ) ( ) ( ) ( )( )[ ] 22

12221

212

22

1 844844844844844mm

Ne =−+=−+= σσσσσ

σe max = 844 N/mm2

X

Appendix C: FEA Loads and Constraints

Clamp Block

Lifter

Dog

XI

Clamp Block

Figure 10. Loads and Constraints Applied to Clamp Block

The Clamp Block has symmetry about the Y-Z plane so a half model was used as the

basis for the FEA. Using a half model requires that the cutting plane through the part be

constrained against translation along the X, and Y axes, but free to translate along the Z axis. It is

free to rotate about the X, Y, and Z-axes. The top face of the Clamp Block is bolted to the fence

extrusion, so in the FEA model it will be constrained against motion along all axes.

There are four loads on the Clamp Block. The force magnitudes from the force model

must be halved because this is a half model. The 1230-N load in the positive Z direction has been

modeled as a line load along the points of contact with the fence rail. The 773-N force couple is

modeled as line loads along points of contact with the Dog. The load at the pin is modeled as a

distributed load over the surface of the pin bore.

XII

Lifter

Figure 11. Loads and Constraints Applied to Lifter

The lifter was modeled as a solid. Although it is symmetrical about the Y-X plane, it was

deemed to be simple enough in shape that using a half model was unnecessary. The bore of the

pin is constrained against translation along the X, Y, and Z-axes, and against rotation about the

X, Y, and Z axes.

The lifter has three loads acting on it, a distributed load on the surface of the pin bore of

1550 N, and two line loads. The force in the �Y direction is a result of the contact with the dog.

They meet at a line; therefore, the force of 1230 N is distributed along this line of contact. The

last force is 943 N in the � X direction applied as a line load along the line of contact with the

linking rod.

XIII

Dog

Figure 12. Loads and Constraints Applied to Dog

The Dog is modeled as a solid, and is constrained against translation along the X, Y, and

Z-axes, and rotation about the X, Y, and Z-axes. The constraints are applied along two lines at

points of contact with the Clamp Block.

The dog has four loads acting on it, and all of them are line loads at points of contact with

the clamp block, lifter, and fence rail. There are two force couples; one 1230-N couple in the y-

direction and 773-N couple in the x-direction.

XIV

Appendix D: FEA (Existing Design)

Clamp Block

Lifter

Dog

XV

XVI

XVII

XVIII

Appendix E: FEA (Redesign)

Clamp Block

XIX

XX

Appendix F: Reliability Calculations

Existing Clamp Block

Clamp Block Redesigned

XXI

Clamp Block Reliability: Interference theory of reliability prediction

Figure 7, Reliability Calculations: Original Design Data: µ x = 152 MPA (Yield strength) σ x = 12.2 MPA (Standard deviation) µ y = 125 MPA (Load) σ y = 10 MPA (Standard deviation) The variance is 8 % of the mean. This is an assumed variance and more accurate variance can be obtained in a Material�s Manual.

Using: σ z = (σ y ² + σ y ²) ½ σ z = [(12.2)² + (10)²] ½ = 15.8 MPA Knowing: µ z = µ x - µ y µ z = 152 MPA � 125 MPA = 27 MPA Then: µ z - k σ z = 0 k = - 1.71 The Figure 6.20 from Juvinall on page 228 has a table which relate the failure of k (number of standard deviation) and % of reliability and % of failure.

For k = -1.71 95 % of reliability or 5 % failure.

XXII

µ y = 114 MPA

Factor of safety = Significant strength of the material = SF Corresponding significant stress

SF = 152 MPA = 1.22 125 MPA For the redesign the goal is to make a product, which is 99% reliable. Calculations: To achieve 99 % reliability the k factor should be equal to (-2.4). µ Y = ? σ z = 15.8 MPA µ z - k σ z = 0 µ z = (2.4) x (15.8) = 38 MPA µ z = µ x - µ y 38 MPA = 152 MPA - µ y

This is the maximum acceptable magnitude of stress for 99% reliability.

The geometric changes made during the redesign stage meet its required stress intensity. And the

component is 99% reliable. We tested our results by running a Finite Element Analysis of the

component.

Factor of safety = Significant strength of the material = SF Corresponding significant stress

SF = 152 MPA = 1.4 114 MPA

XXIII

Appendix G: Engineering Drawings

Clamp Block

Lifter

Dog

XXIV

XXV

XXVI