ME 461:
Finite Element
Analysis
Spring
2016
A Semester Report on the:
Analysis of Lock Washer Operation
Group Members:
Brandon Wertz
Justin Koscianski
Department of Mechanical and Nuclear Engineering, University Park, PA
Analysis of Lock Washer Operation
Table of Contents
Table of Contents................................................................................................................................... 2
Executive Summary .............................................................................................................................. 3
Acknowledgements .............................................................................................................................. 4
Section 1: Background and Project Plan ....................................................................................... 8
Section 2: Development and Description of the CAD Geometry ........................................... 8
Section 3: Development of Finite Element Meshes ................................................................. 16
Section 4: Development and Description of the Model Assembly and Boundary
Conditions .............................................................................................................................................. 35
Section 5: Development and Description of Model Interactions ........................................ 40
Section 6: Analysis of Finite Element Model .............................................................................. 42
Section 7: Revised Approach ........................................................................................................... 46
Section 8: Summary of Major Findings ........................................................................................ 47
Section 9: Works Cited……………………………………………………………………………………………..55
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Analysis of Lock Washer Operation
Executive Summary
Objective:
The following analysis of lock washer operation will attempt to empirically examine the validity
of a lock washer’s ability to maintain the integritity of a threaded fastener joint. The team will
perform finite element analysis on various types of industrially common lock washers such as
split, tooth, bellville and wedge lock washers subjected to tensile, compressive, vibrational, and
shear loading. The primary goal is to determine if locking washers substantially increase a
threaded connections ability to resist unthreading.
A scoring algorithm will be devised in order to objectively rank each washer’s performance for
the used consistent loading condition. The primary motivation for this topic is based upon the
National Aeronautics and Space Administration’s (NASA) 1990 publication, Fastener Design
Manual, authored by Richard T. Barrett of the Lewis Research Center located in Cleveland,
Ohio. In this report, Barrett states that typical split washers “serve as a spring while the bolt is
being tightened. However, the washer is normally flat by the time the bolt is fully torqued. At
this point, it is equivalent to a flat washer, and its locking ability is non-existent.” Barrett further
asserts that tooth and bellville washers do maintain some locking ability, but are liable to damage
mating surfaces. This analysis will directly investigate his claims as well as many similar claims
that have been made throughout the engineering community. If surface damage does occur,
results will be analyzed to determine a possible relation of stress concentations, which could lead
to crack propogation and joint failure from fracture.
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Analysis of Lock Washer Operation
Acknowledgements
Reuben Kraft – Assistant professor of mechanical engineering at The Pennsylvania State
University
Our team would like to thank Dr.Kraft for exceptional efforts in instructing us
in the use of the finite element method and his genuine desire to assist us in
persuit of knowledge and understanding.
Richard T. Barrett – Former Senior Aerospace Engineer of NASA’s Lewis Research Center
Our team would like to thank Richard T. Barret for his exemplary work in the
field of fastener technology. A recognized expert in this field, our project would
most likely no have occurred without his publication.
The Pennsylvania State Univeristy –
Our team would like to extend considerable gratitude to our alma mater for
providing the technology, resources, and instructors in order to gain an
education in such a complex field as finite element analysis.
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Analysis of Lock Washer Operation
List of Figures
Table 1: List of Figures
Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
Figure 10
Figure 11
Figure 12
Figure 13
Figure 14
Figure 15
Figure 16
Figure 17
Figure 18
Figure 19
Figure 20
Figure 21
Figure 22
Figure 23
Figure 24
Figure 25
Figure 26
Figure 27
Figure 28
Figure 29
Figure 30
Basic Configuration of Testing Geometry
Side By Side View of Various Locking
Wahsers. From Right to Left Wedge Lock,
Split Lock, Tooth-Lock, and Spring Lock.
Plate the M4 Fastener Will Thread Into and an
Interation Surface for the Locking Washers
Split Lock Washer McMaster – Carr Part
Number 92148A160
Belleville Spring Lock Washer McMaster –
Carr Part Number 91477A141
Internal Tooth Lock Washer McMaster – Carr
Part Number 93925A250
Wedge Lock Washer McMaster – Carr Part
Number 91812A215
M4 Metric Stainless Steel Cap Screw
McMaster – Carr Part Number 93635A118
Imported M4 Fastener Geometry
Partitioned M4 Fastener Geometry Isometric
View
Partitioned M4 Fastener Geometry Bolt Head
Front View
Partitioned M4 Fastener Geometry Bolt Shank
Rear View
M4 Fastener Meshed Side View
M4 Fastener Meshed Bolt Shank / Head
Interface
M4 Fastener Meshed Bolt Shank Rear View
M4 Fastener Meshed Bolt Head Front View
M4 Fastener Mesh Verify Failed Elements
Imported Plate Geometry
Meshed Plate Geometry Isometric View
Meshed Plate Geometry Top View
Split Washer Meshed Side View
Split Washe Meshed Top View
Split Washer Meshed Front View
Belleville Washer Meshed Top View
Belleville Washer Meshed Side View
Wedge Lock Washer Meshed Isometric View
Wedge Lock Washer Meshed Bottom View
Wedge Lock Washer Meshed Top View
Wedge Lock Washer Mesh Verify Results
Tooth Lock Washer Meshed Isometric View
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Analysis of Lock Washer Operation
Figure 31
Figure 32
Figure 33
Figure 34
Figure 35
Figure 36
Figure 37
Figure 38
Figure 39
Figure 40
Figure 41
Figure 42
Figure 43
Figure 44
Figure 45
Figure 46
Figure 47
Figure 48
Figure 49
Figure 50
Figure 51
Figure 52
Figure 53
Figure 54
Figure 55
Figure 56
Figure 57
Figure 58
Figure 59
Figure 60
Figure 61
Tooth Lock Washer Meshed Side View
Tooth Lock Washser Mesh Verify Results
Concentric Constraint Plate & M4 Fastener
Bottom of M4 Fastener Head Mated to Plate
Top Face
Concentric Constraint M4 Fastener & Locking
Washer
Bottom Faces of the Threaded Plate and
Locking Washer Mated
Concentric Constraint M4 fastener & Nut
M4 Nut Mated to Threaded Plate Bottom Face
With Appropriate Offset
Encaste Placed Upon Threaded Block and M4
Fastener Top Faces
Reference Point Applied to Bottom Center M4
Fastener
Rigid Body Constraint Applied M4 Nut Top
Surface & Reference Point
Boundary Condition Applied Directly to the
Reference Point
General Contact Interaction Menue
IntProp-1 Contact Property Options
Tangential Behavior of IntProp-1
Normal Behavior of IntProp-1
Meshed Assembly
Meshed split lock washer Used In Assembly
Hex Mesh and Mesh Controls for Split Lock
Washer
Hex Mesh and Mesh Controls for Threaded
Plate
View Cut Beginning of Simulation
View Cut End of Simulation
Beginning of Simulation
End of Simulation
Stress Contour Threaded Plate
Split Lock Washer Beginning of Simulation
Split Lock Washer Mid Simulation
Split Lock Washer End of Simulation
Split Lock Washer – Response Frequency Vs.
Mode Number – High Load
Split Lock Washer – Response Frequency Vs.
Mode Number – Practical Load
Bellville Lock Washer – Response Frequency
Vs. Mode Number – High Load
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Analysis of Lock Washer Operation
Figure 62
Figure 63
Figure 64
Figure 65
Figure 66
Figure 67
Figure 68
Figure 69
Figure 70
Figure 71
Bellville Lock Washer – Response Frequency
Vs. Mode Number – Practical Load
Tooth Lock Washer – Response Frequency Vs.
Mode Number – High Load
Tooth Lock Washer – Response Frequency Vs.
Mode Number – Practical Load
Wedge Lock Washer – Response Frequency
Vs. Mode Number – High Load
Wedge Lock Washer – Response Frequency
Vs. Mode Number – Practical Load
Plot of 200 Newton Bolt Tension All Washer
Plot of 160 Newton Bolt Tension All Washer
Plot of 120 Newton Bolt Tension All Washer
Plot of 80 Newton Bolt Tension All Washer
Plot of 40 Newton Bolt Tension All Washer
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Analysis of Lock Washer Operation
Section 1: Background and Project Plan
Background Information:
A lock washer generally functions as a spring and is designed to stop threaded fasteners from
loosening. They are engineered to resist vibration and utilize teeth or another physical
mechanism to prevent rotation by penetrating the mating surface that the washer is in contact
with. The washers are generally designed as a left hand helix with a tooth direction for a right
handed thread and vice versa for a left handed thread. The raised edge of the washers will bite
into the nut and mating surface when turned which resists the unwanted rotation. The usefulness
of these type of lock washers have been under scrutiny due to the idea that when the washer is
tightened flat against a substrate, the edge will not bite. Therefore, there will not be any
difference in resistance when an unthreading torque is applied compared to a regular washer.
This proposed flaw with the spring style washers is the core of the objective analysis.
General Approach:
As mentioned above, an evaluation of the effectiveness of different forms of washers in fastening
applications will be investigated. To obtain the CAD models, the team opted to utilize the
McMaster Carr database. McMaster Carr offers a wide variety of fasteners and fastener
accessories, allowing easy evaluation of more than one syle of washer. In general, three to four
features will be meshed per analysis: (1) The material that the fastener will join, (2) The fastener
itself, (3) One of the various forms of washers, and (4) the nut locking the fastener to the material
and compressing the locking washer. The same material will be utilized for the plate in each test
as to ensure maximum consistency between analyses.
During the analyses, differences in stresses on the fasteners as well as the behavior that the lock
washers exhibit when compressed will be observed. As stated in the objective, the possibility of
increasing frature failure via stress concentration is also of interest.
The lock washers’ effectiveness will be ranked based upon the fasteners’ rotational displacement
for a given loading. The loadings will also be varied to explore any possible variances they could
contribute. Vibrational studies and mesh convergernce studies will also be integral to drawing
conclusions.
Section 2: Development and Description of the CAD Geometry
As our team began research on different types of lock washers, it was noticed that McMaster-Carr
has CAD drawings and 3-D renderings of all standard washers used in industrial applications.
McMaster offers these items in various shapes and sizes, and a multitude of materials.
In an attempt to ease calculations, metric units will primarily be used. The basic test geometry is
shown below. Using an M4 fastener in conjunction with the matching washer size, a specified
clockwise torque will be applied to a nut that will be on the opposite side of a 0.01270000 meter
thick plate. Various torque loads will then be applied in attempts to remove or unthread the
fastener.
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Analysis of Lock Washer Operation
M4 Fastener
M4 Nut
Lock Washer
Threaded Plate
Figure 1: Basic Configuration of Testing
Geometry
The preliminary approach is to analyze many different types of washers. Four different styles of
washers with three different materials will be modled. A total of twelve (4 washers x 3 materials)
meshings and analyses will be conducted using Abaqus. The results will be ranked according to
the rotational displacement measured in each test compared to a control with no lock washer.
Details, including 3-D renderings and 2-D CAD drawings, are shown below. These different types
of lock washers include: split lock washers, tooth-lock washers, wedge lock washers, and spring
lock washers.
Figure 2: Side By Side View of Various Locking Wahsers. From Right to Left
Wedge Lock, Split Lock, Tooth-Lock, and Spring Lock.
The materials of each washer to be analyzed are: steel, aluminum and copper. The mechanical
properties are listed in Table 2.
Table 2: Mechanical Properties of Materials to be Tested
Material
Steel
Aluminum
Copper
Modulus of Elasticity (GPa)
200
68.9
117
Density (g/cm3)
7.87
2.7
8.96
Poissons Ratio
0.29
0.33
0.34
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Analysis of Lock Washer Operation
Through an internet conversion table, a torque of 200 N/MM applied to the nut was used to
determine that a clamping force of 256 N would be present. The torque of 200 N/MM was
selected as a median value in the acceptable torque loading for this size fastener.
The equation T = .2DF is a general equation to estimate applied fastener axial loading for industry
standard threads with steel materials.
T = Torque
0.2 = Roughness Approximation (Steel)
D = Fastener Major Diameter
F = Clamping Force
200 N/MM = (0.2) (3.90 MM) (F)
F = 256 N
The team expects to primarily utilize Hooke’s law in conjunction with various contact algorithms.
Deformations will be minimal, well below yielding for all materials. The team will address
frictional properties, and potential interference between all three (plate, fastener, lock washer) test
components.
Tap Thru M4 x 0.7
Steel Plate
Figure 3: Plate the M4 Fastener Will Thread Into and an Interation Surface for the Locking
Washers
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Analysis of Lock Washer Operation
Figure 4: Split Lock Washer McMaster – Carr
Part Number 92148A160
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Analysis of Lock Washer Operation
Figure 5: Belleville Spring Lock Washer
McMaster – Carr Part Number
91477A141
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Analysis of Lock Washer Operation
Figure 6: Internal Tooth Lock Washer
McMaster – Carr Part Number
93925A250
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Analysis of Lock Washer Operation
Figure 7: Wedge Lock Washer McMaster –
Carr Part Number 91812A215
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Analysis of Lock Washer Operation
Figure 8: M4 Metric Stainless Steel Cap Screw
McMaster – Carr Part Number
93635A118
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Analysis of Lock Washer Operation
Section 3: Development of Finite Element Meshes
Section 3.1: Bolt
1. The CAD geometry of McMASTER-CARR part number 93635A118 was downloaded
directly from the McMASTER-CARR site as a .step file
2. The .step file was then imported to Autodesk Inventor 3D modeling software
3. Using the Autodesk software, an extrusion was created along the shank of the bolt at the
major diameter of an M4 x 0.7 mm thread (4 mm). This was done to remove the complex
geometry of the bolt thread and replace it with a simple cylindrical surface. This analysis
is not concerned with the interactions of the bolt thread, the only area of concern is the
underside of the bolt head and its interaction with the lock washers and plate. This
simpliflies meshing and reduces computational expense.
4. The file was then exported from the Autodesk software as a .step file
5. The .step file was then imported as a part into abaqus (Result Shown Below)
Figure 9: Imported M4 Fastener Geometry With Dimensions Shown In Figure 8
6. The bolt was then partitioned as shown in the figures 10-12. Partioning allowed proper
seeding of the geometry. The shank and bolt head were also partitioned form one another.
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Analysis of Lock Washer Operation
Figure 10: Partitioned M4 Fastener Geometry Isometric View
Figure 11: Partitioned M4 Fastener Geometry Bolt Head Front View
17
Analysis of Lock Washer Operation
Figure 12: Partitioned M4 Fastener Geometry Bolt Shank Rear View
7. The circular partition on the base of the shank was extruded through the entire body of the
piece
8. Using the mesh controls, the part was a assigned a linear tet mesh with 0.8 mm global seeds
9. Mesh results using C3D4 four node elements are shown below in figures 13-17
Figure 13: M4 Fastener Meshed Side View
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Analysis of Lock Washer Operation
Figure 14: M4 Fastener Meshed Bolt Shank / Head Interface
Figure 15: M4 Fastener Meshed Bolt Shank Rear View
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Analysis of Lock Washer Operation
Figure 16: M4 Fastener Meshed Bolt Head Front View
10.
The mesh verify tool was then used to determine the mesh quality. The results are shown
below along with the failed elements in figure 17.
Part: m4 x 07 bolt for modeling_abaqus
Tet elements: 24933
Min angle on Tri Faces < 5: 0 (0%)
Average min angle on tri faces: 39.35, Worst min angle on tri faces: 16.67
Max angle on Tri faces > 170: 0 (0%)
Average max angle on tri faces: 89.31, Worst max angle on tri faces: 130.49
Aspect ratio > 10: 0 (0%)
Average aspect ratio: 1.60, Worst aspect ratio: 3.31
Shape factor < 0.0001: 0 (0%)
Average shape factor: 0.663333, Worst shape factor: 0.085937
Geometric deviation factor > 0.2: 0 (0%)
Average geometric deviation factor: 1.11e-07, Worst geometric deviation factor: 1.13e-06
Min edge length < 0.01: 0 (0%)
Average min edge length: 0.440, Shortest edge: 0.240
Max edge length > 1: 50 (0.200537%)
Average max edge length: 0.696, Longest edge: 1.16
Number of elements : 24933, Analysis errors: 0 (0%), Analysis warnings: 0 (0%)
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Analysis of Lock Washer Operation
Failed Elements
Figure 17: M4 Fastener Mesh Verify Failed Elements Using the Criteria Shown Above
11.
The mesh will be assigned material properties consistent with those in table 2.
Section 3.2: Plate
1. A rectangular extrusion was first created using Autdesk Inventor software. The extrusion
had dimensions of 12.7 mm thick by 25.4 mm long and wide.
2. In the center of this extrusion, a 4 mm dimeter bore was extruded thrugh the thickness.
(Major diameter of M4 fastener)
3. The file was then exported as a .step file
4. The file was then imported into abaqus as the figure shown below.
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Analysis of Lock Washer Operation
Figure 18: Imported Plate Geometry
5. Using the mesh control feature, a hex mesh was applied to the part.
6. The global seeds were set to a value of 1.0 mm
7. Meshing of the part using C3D8R linear elements resulted in the figures 19-20
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Analysis of Lock Washer Operation
Figure 19: Meshed Plate Geometry Isometric View
Figure 20: Meshed Plate Geometry Top View
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Analysis of Lock Washer Operation
8. The mesh verify command was then used to determine the quality of the mesh. The results
Upon further inspection, default failure criteria caused no elements to fail.
Part: Plate For Modeling
Hex elements: 26061
Min angle on Quad Faces < 10: 0 (0%)
Average min angle on quad faces: 79.65, Worst min angle on quad faces: 52.66
Max angle on Quad faces > 160: 0 (0%)
Average max angle on quad faces: 101.50, Worst max angle on quad faces: 131.18
Aspect ratio > 10: 0 (0%)
Average aspect ratio: 1.35, Worst aspect ratio: 2.42
Geometric deviation factor > 0.2: 0 (0%)
Average geometric deviation factor: 0.00243, Worst geometric deviation factor: 0.0437
Min edge length < 0.01: 0 (0%)
Average min edge length: 0.580, Shortest edge: 0.308
Max edge length > 1: 0 (0%)
Average max edge length: 0.761, Longest edge: 0.993
Number of elements : 26061, Analysis errors: 0 (0%), Analysis warnings: 0 (0%)
9. For all tests, the plate will be assigned material properties consistent with those found in
table 2 for steel.
Section 3.3: Split Washer
1. The CAD geometry of McMASTER-CARR part number 92148A160 was downloaded
directly from the McMASTER-CARR site as a .step file
2. The .step file was then imported as a part into abaqus
3. Using the mesh controls, the part was a assigned a tet mesh using C3D4 four node elements
4. The split washer will be modeled with each material as shown in Table 2 with 0.4 mm
global seeds
5. Mesh results are shown in figures 21-23
Figure 21: Split Washer Meshed Side View
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Analysis of Lock Washer Operation
Figure 22: Split Washe Meshed Top View
Figure 23: Split Washer Meshed Front View
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Analysis of Lock Washer Operation
6. The mesh verify tool was then used to determine the mesh quality. The results are shown
below. No elements were found to be below default element failure criteriea.
Part instance: P92148A160-1
Tet elements: 1123
Min angle on Tri Faces < 5: 0 (0%)
Average min angle on tri faces: 25.78, Worst min angle on tri faces:
15.65
Max angle on Tri faces > 170: 0 (0%)
Average max angle on tri faces: 97.26, Worst max angle on tri faces:
132.82
Aspect ratio > 10: 0 (0%)
Average aspect ratio: 2.52, Worst aspect ratio: 4.75
Shape factor < 0.0001: 0 (0%)
Average shape factor: 0.424193, Worst shape factor: 0.077937
Geometric deviation factor > 0.2: 0 (0%)
Average geometric deviation factor: 6.67e-08, Worst geometric deviation
factor: 4.44e-07
Min edge length < 0.01: 14 (1.24666%)
Average min edge length: 0.0146, Shortest edge: 0.00875
Max edge length > 1: 0 (0%)
Average max edge length: 0.0352, Longest edge: 0.0484
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Analysis of Lock Washer Operation
Section 3.4: Belleville Washer
1. The CAD geometry of McMASTER-CARR part number 91477A141 was downloaded
directly from the McMASTER-CARR site as a .step file
2. The .step file was then imported as a part into abaqus
3. Using the mesh controls, the part was a assigned a tet mesh using C3D4 four node elements
4. The Belleville washer will be modeled with each material as shown in Table 2 with 0.4
mm global seeds
5. Mesh results are shown in figures 24-25
Figure 24: Belleville Washer Meshed Top View
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Analysis of Lock Washer Operation
Figure 25: Belleville Washer Meshed Side View
7. The mesh verify tool was then used to determine the mesh quality. The results are shown
below. No elements were found to be below default element failure criteriea.
Part: Bellville Lock Washer
Tet elements: 3596
Min angle on Tri Faces < 5: 0 (0%)
Average min angle on tri faces: 37.35, Worst min angle on tri faces: 21.25
Max angle on Tri faces > 170: 0 (0%)
Average max angle on tri faces: 89.21, Worst max angle on tri faces: 122.19
Aspect ratio > 10: 0 (0%)
Average aspect ratio: 1.70, Worst aspect ratio: 2.71
Geometric deviation factor > 0.2: 0 (0%)
Average geometric deviation factor: 1.13e-07, Worst geometric deviation factor:
4.62e-07
Min edge length < 0.01: 0 (0%)
Average min edge length: 0.390, Shortest edge: 0.260
Max edge length > 1: 0 (0%)
Average max edge length: 0.648, Longest edge: 0.924
Number of elements : 3596, Analysis errors: 0 (0%), Analysis warnings: 0 (0%)
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Analysis of Lock Washer Operation
Section 3.5: Wedge Lock Washer
1. The CAD geometry of McMASTER-CARR part number 91812A215 was downloaded
directly from the McMASTER-CARR site as a .step file
2. The .step file was then imported as a part into abaqus
3. Using the mesh controls, the part was a assigned a tet mesh using C3D4 four node elements
4. The Wedge Lock Washer will be modeled with each material as shown in Table 2 with 0.4
mm global seeds
5. Mesh results are shown in figures 26-29
Figure 26: Wedge Lock Washer Meshed Isometric View
29
Analysis of Lock Washer Operation
Figure 27: Wedge Lock Washer Meshed Bottom View
Figure 28: Wedge Lock Washer Meshed Top View
6. The mesh verifytool was then used to determine the mesh quality. The results are shown
below in figure 29 along with the failed elements.
30
Analysis of Lock Washer Operation
Figure 29: Wedge Lock Washer Mesh Verify Results
Due to significant warnings, this mesh will have to be refined
Part: Wedge Lock Washer-1
Tet elements: 10178
Min angle on Tri Faces < 5: 0 (0%)
Average min angle on tri faces: 29.32, Worst min angle on tri faces: 5.25
Max angle on Tri faces > 170: 0 (0%)
Average max angle on tri faces: 97.66, Worst max angle on tri faces:
167.93
Aspect ratio > 10: 4 (0.0393004%)
Average aspect ratio: 2.58, Worst aspect ratio: 10.93
Shape factor < 0.0001: 0 (0%)
Average shape factor: 0.475971, Worst shape factor: 0.000112
Geometric deviation factor > 0.2: 0 (0%)
Average geometric deviation factor: 5.27e-07, Worst geometric deviation
factor: 2.47e-06
Min edge length < 0.01: 6759 (66.4079%)
Average min edge length: 0.00809, Shortest edge: 0.00188
Max edge length > 1: 0 (0%)
Average max edge length: 0.0166, Longest edge: 0.0269
31
Analysis of Lock Washer Operation
Section 3.6: Tooth Washer
1. The CAD geometry of McMASTER-CARR part number 91812A215 was downloaded
directly from the McMASTER-CARR site as a .step file
2. The .step file was then imported as a part into abaqus
3. Using the mesh controls, the part was a assigned a tet mesh using C3D4 four node elements
4. The Tooth Washer will be modeled with each material as shown in Table 2 with 0.4 mm
global seeds
5. Mesh results are shown below in figure 30-32
Figure 30: Tooth Lock Washer Meshed Isometric View
32
Analysis of Lock Washer Operation
Figure 31: Tooth Lock Washer Meshed Side View
7. The mesh verify tool was then used to determine the mesh quality. The results are shown
below along with the failed elements.
Figure 32: Tooth Lock Washer Mesh Verify Results
33
Analysis of Lock Washer Operation
This mesh should also be further refined because of the number of warnings around the inner
diameter
Part instance: P93925A250-1
Tet elements: 6510
Min angle on Tri Faces < 5: 248 (3.80952%)
Average min angle on tri faces: 29.90, Worst min angle on tri faces: 0.172
Max angle on Tri faces > 170: 19 (0.291859%)
Average max angle on tri faces: 99.14, Worst max angle on tri faces: 175.87
Aspect ratio > 10: 207 (3.17972%)
Average aspect ratio: 3.16, Worst aspect ratio: 152.7
Shape factor < 0.0001: 23 (0.353303%)
Average shape factor: 0.468963, Worst shape factor: 2.19e-07
Geometric deviation factor > 0.2: 0 (0%)
Average geometric deviation factor: 5.67e-05, Worst geometric deviation
factor: 0.0381
Min edge length < 0.01: 93 (1.42857%)
Average min edge length: 0.173, Shortest edge: 0.00187
Max edge length > 1: 0 (0%)
Average max edge length: 0.352, Longest edge: 0.736
34
Analysis of Lock Washer Operation
Section 4: Development and Description of the Model Assembly and
Boundary Conditions
The model assembly was consistent with the configuration shown in figure 1 for all four washer
trials. However, due to the irregular shape of the washer geometries it became necessary to
assemble the models using 3-D CAD software and then import each assembly into Abaqus as .step
files. After condiderable effort, it was not possible to create the desired assemblies using Abaqus
geometric constraints alone. The 3-D CAD software used to accomplish this was Autodesk
Inventor. Only the split washer assembly is described below, but the process is identicle for all
washers.
Figure 33: Concentric Constraint Plate & M4 Fastener
First, the longitudinal axes of the M4 Fastener and the threaded bore of the steel plate are set
concentric to one another as shown in figure 33. The bottom face of the fastener head is then mated
against the top face of the threaded plate with no offset as shown below in figure 34.
Figure 34: Bottom of M4 Fastener Head Mated to Plate Top Face
35
Analysis of Lock Washer Operation
The axis of locking washer is then set concentric to the longitudinal axis of the M4 fastener on the
bottom face of the threaded plate consistent with figure 35. Although assembly is being shown
using a split lock washer, the procedure is identicle regardless of the washer style.
Figure 35: Concentric Constraint M4 Fastener & Locking Washer
The bottom edge of the locking washer is then mated to the bottom surface of the threaded plate
with no offset as shown below in figure 36.
Figure 36: Bottom Faces of the Threaded Plate and Locking Washer Mated
36
Analysis of Lock Washer Operation
The axis of the M4 nut is then set concentric to the longitudinal axis of the M4 fastener as shown
below in figure 37.
Figure 37: Concentric Constraint M4 fastener & Nut
Finally, the M4 nut was mated to the bottom face of the threaded plate with an appropriate offset
to just touch the locking washer. This completes assembly.
Figure 38: M4 Nut Mated to Threaded Plate Bottom Face With Appropriate Offset
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Analysis of Lock Washer Operation
Once an assembly was imported into abaqus via a .step file, it was then possible to apply boundy
conditions. Initially an encaste is placed upon the top face of the threaded block and top face of
the M4 fastener head as shown in figure 39. Ensuring that neither of these instances translate or
rotate in any way is necessary to verify the effects of each locking washer.
Figure 39: Encaste Placed Upon Threaded Block and M4 Fastener Top Faces
The second boundary condition applied to the assembly was a combined translation and rotation.
To begin, a reference point was defined at the bottom center of the M4 fastener shank. This motion
simulated the tightening of the fastener and nut while loading the locking washer.
Reference Point
Figure 40: Reference Point Applied to Bottom Center M4 Fastener
38
Analysis of Lock Washer Operation
A rigid body constraint was then used to tie the top surface of the M4 nut to the reference point as
shown below in figure 41. This accomplished the desired rotational aspect by informing the nut
about wich axis to rotate.
Figure 41: Rigid Body Constraint Applied M4 Nut Top Surface & Reference Point
Once the top surface of the M4 nut was tied to the reference point, the actual boundary condition
was applied directly to the reference point. This corresponded to a downward displacement of
0.8mm along the global X axis while rotating 3.14 radians about the same axis.
Figure 42: Boundary Condition Applied Directly to the Reference Point
39
Analysis of Lock Washer Operation
Section 5: Development and Description of Model Interactions
The assemblies were then modeled using dynamic explicit steps in order to take advantage of
Abaqus’s general contact feature. Using this feature, Abaqus defines and propogates its own
contact pairs in conjunction with our created global property assignment defined as IntProp-1.
Figure 43: General Contact Interaction Menue
Figure 44: IntProp-1 Contact Property Options
40
Analysis of Lock Washer Operation
The IntProp-1 interaction property consisted of two mechanical specifications. The first
characteristic is tangential and the second is normal.
Figure 45: Tangential Behavior of IntProp-1
The tangential behavior took the form of a penalty method friction algorithm between solid
surfaces. The coefficient of friction was taken to be 0.5. This value was chosen due to the fact that
it represents the median coefficnet of friction possible for steel on steel contact.
Figure 46: Normal Behavior of IntProp-1
The normal interaction behavior was set to what Abaqus defines as “Hard Contact”. This prevented
the plate from penetrating the locking washers while allowing the locking washers to still penetrate
the plate. This was accomplished by properly defing the washer as the master and the plate as the
slave surface.
41
Analysis of Lock Washer Operation
Section 6: Analysis of Finite Element Model
Using the previously derived geometry and meshes, an assembly was compiled using the solid
modeling software Autodesk Inventor. The assembly was then imported as a .step file into the
Abaqus program. Using the material properties previously shown in in Section 2 of this report,
steel was assigned to all components of the model (plate, bolt, washer, and nut) In order to
develop a baseline modeling.
The result is the geometry shown below in figure 47.
Plasticity was included by stating zero plastic strain at 300 MPa of stress and 0.05 plastic strain
at 310 Mpa of stress. All entities are solid homogenous bodies.
Figure 47: Meshed Assembly
In order to develop the base model, the split washer was chosen as the first to be modled as it has
the simplest geometry.
Figure 48: Meshed split lock washer Used In
Assembly
At this point the combine faces tool in the meshing module was employed to simplify several
contours that would interface with the split lock washer. Originally the nut was not going to be
42
Analysis of Lock Washer Operation
utilized in the model, but after learning the nature of simulating a bolt load including it became
necessary.
Upon reviewing the meshes, several other modifications were made as well. Efforts were
focused on improving the contact based aspects of the heavily contact dependent model. The
split lock washer and the threaded plate were both updated from tet meshes to hex meshes. The
element sizes were also matched to increase contact algorithm accuracy.
Figure 49: Hex Mesh and Mesh Controls for Split Lock Washer
The model, using the boundary conditionts stated in section 5, was then submitted using PBS script
submission to Penn State’s lion XG computing platform. The simulation utilized eight
processessing cores in parallel and was completed within six minutes and forty seven seconds.
Figure 50: Hex Mesh and Mesh Controls for Threaded Plate
43
Analysis of Lock Washer Operation
Figure 51: View Cut Beginning of Simulation
Figure 52: View Cut End of Simulation
Figures 51 and 52 clearly show the successful downward translation and compression of the lock
washer along the global X axis. There is also no unwanted penetration between bodies.
Figure 53: Beginning of Simulation
Figure 54: End of Simulation
Figures 53 and 54 support the claim stated above, but also demonstrate the successful rotation of
the M4 Nut. Note the position of the Nut corners in figure 53 versus figure 54.
44
Analysis of Lock Washer Operation
Stress Concentrations
Figure 55 shows a contour of stresses
on the surface of the threaded plate.
An intriguing result is that stress
concentrations are present at ninety
degress from the split in the locking
washer. Additionally, take note of
the position of the locking washer in
figures 56 through 58. The washer
was actually rotated slightly from the
motion of the M4 nut.
Figure 55: Stress Contour Threaded Plate
Figure 56: Split Lock Washer Beginning of Simulation
Figure 57: Split Lock Washer Mid Simulation
Figure 58: Split Lock Washer End of Simulation
45
Analysis of Lock Washer Operation
Section 7: Revised Approach
After attempting the above simulation with the more complex washer geometries, it became
increasingly difficult to carry out the original analysis. The rotation step would generally cause
Abaqus program failures that, after numerous attempts, could not be resolves. Additionally, due to
the amount of trouble shooting needed to devise the first model described above, time constraints
were encountered in terms of running enough models to test all the materials. At this point, it was
collectively decided to alter the simulation and analysis criteria. It was decided that steel would be
the only material being analyzed.
The initial step for analyzing all forms of washers was to create a displacement that would
uniformally compress each washer to the same in-use state. After this was completed, loads
ranging from 40N to 200N in increments of 40N were applied to the top surface of the nut to
simulate varying loads. Finally, to evaluate the effectiveness of the washers, a frequency analysis
was conducted to simulate in-use vibrational response with an output of the system’s natural
frequency. These results allow for a better understanding of which lock washers are better suited
for certain applications.
It has been concluded that uniformly loading each lock washer is a more realistic test criteria than
applying a displacement. Varying washers are designed to compress in unique amounts specific to
their design.
46
Analysis of Lock Washer Operation
Section 8: Summary of Major Findings
Beginning with the split washer simulation consistent with the original testing criteria, figure 55
clearly diplays stess concentations on the threaded plate surface. As stated above, the interesting
fact about these concetrations is that they occurred at 90 degress from the washers split. This
location could possibly be explained by friction, as the washer is forced to compress flat. The small
portions in contact with the plate surface may be resisiting that motion by grabbing the plate. It is
also worth while to note that the split washer itself presented a stress concentation at 180 degrees
from its split. This could potentially be explained by the moment created on each side of the washer
as it is compressed. The washer’s offset would be functioning as a moment arm. Unfortunaley,
due to the inablility to recreate this model using more complex geometry, it was not possible to
interpret any mearsure of effectiveness from said simulation. A final observation regarding this
model was that when the nut was given a rotational displacement, the washer did rotate under it
without significantly interacting with either the nut or plate surface.
Consider the vaiation in frequency response of each individual washer for the range of loading
considered.
Split Lock Washer - Response Frequency Vs. Mode Number
80
60
Frequency
40
Hz
Split Washer -3300 N
20
Split Washer - 1500 N
Split Washer - 2500 N
Split Washer - 1000 N
0
1
2
3
4
5
6
7
8
9
10
Mode
Figure 59: Split Lock Washer – Response Frequency Vs. Mode Number – High Load
Under high load, the split washer displays consistent response behavior at 1500 newtons of load
and below. At even higher loadings of 2500 newtons and 3300 newtons, the response appears to
scale while following the same general pattern. Additionally, the range of responsive frequencies
for 1500 newtons and below is significantly smaller, over 80 % smaller than 2500 newtons and
above. It is surmised that a split washer would be significantly more effective in loadings below
the 1500 newton threshold.
When examining the practical loading range for an M4 fastenr and nut shown in figure 60, the
consistency remains. The range of responsive frequency is less than 3 Hz which would suggest
split washers to be best suited to higher frequency applications. The washer itself may be acting
also be acting as a damper. Under these loads, it isn’t until mode 4 that non-zero response even
takes place.
47
Analysis of Lock Washer Operation
Slit Lock Washer - Response Frequency Vs. Mode Number
3.5
3
2.5
Split Washer - 200 N
Frequency 2
Hz
1.5
Split Washer - 160 N
Split Washer - 120 N
1
Split Washer - 80 N
0.5
Split Washer - 40 N
0
1
2
3
4
5
6
7
8
9
10
Mode
Figure 60: Split Lock Washer – Response Frequency Vs. Mode Number – Practical Load
Figure 61 displays the bellville lock washer’s frequency response under high load. This washer’s
responses are all very consistent, but display a very wide range of responsive frequencies. The
total range covers about 60 Hz. The shape of these plotted lines seems to match the 3300 newton
and 2500 newton response of the split washer. The bellville and split washers are in fact the same
style of spring washer.
Bellville Lock Washer - Response Frequency Vs. Mode
Number
80
60
Bellville Washer -3300 N
Frequency
40
Hz
Bellville Washer - 2500 N
20
Bellville Washer - 1500 N
0
1
2
3
4
5
6
7
8
9
10
Bellville Washer - 1000 N
Mode
Figure 61: Bellville Lock Washer – Response Frequency Vs. Mode Number – High Load
Figure 62 reveals the bellville lock washer’s response under practical loading. The responsive
frequency range and behavior is identical to the high loading scenarios. Given that the bellville
washer responses do not occur in any mode below 10 Hz, this style of washer would be be suited
to very low frequency applications. It should be noted that bellville washers are unique in manner
in which they may be combined to adjust spring rates and compressive stroke.
48
Analysis of Lock Washer Operation
Bellville Lock Washer - Response Frequency Vs. Mode
Number
80
70
60
50
Frequency
40
Hz
30
20
10
0
Bellville Washer - 200 N
Bellville Washer - 160 N
Bellville Washer - 120 N
Bellville Washer - 80 N
Bellville Washer - 40 N
1
2
3
4
5
6
7
8
9
10
Mode
Figure 62: Bellville Lock Washer – Response Frequency Vs. Mode Number – Practical Load
Continueing with the investigation, the tooth lock washers subjected to high loads were examined,
and the results are displayed in figure 63. This washer’s responses were unique; for the first nine
modes the responses occurred at 0 Hz. The range of response was also small, less than 7 Hz in any
mode. This plot suggests tooth washers would perform best in high frequency applications.
Tooth Lock Washer - Response Frequency Vs. Mode Number
7
6
5
Frequency 4
3
Hz
2
1
0
Tooth Washer -3300 N
Tooth Washer - 2500 N
Tooth Washer - 1500 N
Tooth Washer - 1000 N
1
3
5
7
9
Mode
Figure 63: Tooth Lock Washer – Response Frequency Vs. Mode Number – High Load
Figure 64, containing the tooth lock washer’s practical load responses, displays nearly identical
behavior at a lower frequency magnigtude.
49
Analysis of Lock Washer Operation
Tooth Lock Washer - Response Frequency Vs. Mode Number
1.4
1.2
1
Tooth Washer - 200 N
Frequency 0.8
Hz
0.6
Tooth Washer - 160 N
Tooth Washer - 120 N
0.4
Tooth Washer - 80 N
0.2
Tooth Washer - 40 N
0
1
3
5
7
9
Mode
Figure 64: Tooth Lock Washer – Response Frequency Vs. Mode Number – Practical Load
Figure 65 reveals that under high loads, like the tooth washer, the wedge lock washer has zero
response frequencies until higher mode numbers are reached. The wedge washer displays the
smallest range of responsive frequencies of all designs, less than a ten thousandth of a hertz.
Wedge Lock Washer - Response Frequency Vs. Mode Number
0.00001
0.000008
Frequency 0.000006
Hz
0.000004
Wedge Washer -3300 N
0.000002
Wedge Washer - 1500 N
0
Wedge Washer - 1000 N
Wedge Washer - 2500 N
1
2
3
4
5
6
7
8
9
10
Mode
Figure 65: Wedge Lock Washer – Response Frequency Vs. Mode Number – High Load
When comparing the high load and practical load responses of the wedge lock washer, the range
of response is nearly identicle. The practical load does seem to produce higher responses at lower
mode numbers. The trend behavior is consistent between both loading patterns as well. When
considering the tooth and wedge washers, the small frequency response may result from chatter.
The tooth washer’s response results from teeth chattering on the mating surfaces of the nut and
plate and the wedge washer’s chatter could result from the small translation possible between the
two halves.
50
Analysis of Lock Washer Operation
Wedge Lock Washer - Response Frequency Vs. Mode Number
0.00001
0.000009
0.000008
0.000007
0.000006
Frequency
0.000005
Hz
0.000004
0.000003
0.000002
0.000001
0
Wedge Washer - 200 N
Wedge Washer - 160 N
Wedge Washer - 120 N
Wedge Washer - 80 N
Wedge Washer - 40 N
1
3
5
7
9
Mode
Figure 66: Wedge Lock Washer – Response Frequency Vs. Mode Number – Practical Load
Each washer’s response is plotted together for a specific practical load. In figure 67, where 200
newtons of bolt tension was applied, clear differences can be seen and attributed to the different
design of washer. The Bellville and flat washers respond at high frequencies, while the wedge,
tooth, and split washers respond at low frequencies. This pattern is reapeated for all practical
loading figures.
200 N Tension - All Washers
80
70
60
50
Split Washer - 200 N
Frequency
40
Hz
Flat Washer - 200 N
30
Bellville Washer - 200 N
20
Tooth Washer - 200 N
10
Wedge Washer - 200 N
0
1
2
3
4
5
6
7
8
9
10
Mode
Figure 67: Plot of 200 Newton Bolt Tension All Washer
51
Analysis of Lock Washer Operation
160 N Tension - All Washers
80
70
60
50
Flat Washer - 160 N
Frequency
40
Hz
Split Washer - 160 N
30
Bellville Washer - 160 N
20
Tooth Washer - 160 N
10
Wedge Washer - 160 N
0
1
2
3
4
5
6
7
8
9
10
Mode
Figure 68: Plot of 160 Newton Bolt Tension All Washer
120 N Tension - All Washers
80
70
60
50
Flat Washer - 120 N
Frequency
40
Hz
Split Washer - 120 N
30
Bellville Washer - 120 N
20
Tooth Washer - 120 N
10
Wedge Washer - 120 N
0
1
2
3
4
5
6
7
8
9
10
Mode
Figure 69: Plot of 120 Newton Bolt Tension All Washer
52
Analysis of Lock Washer Operation
80 N Tension - All Washers
80
70
60
50
Flat Washer - 80 N
Frequency
40
Hz
Split Washer - 80 N
30
Bellville Washer - 80 N
20
Tooth Washer - 80 N
10
Wedge Washer - 80 N
0
1
2
3
4
5
6
7
8
9
10
Mode
Figure 70: Plot of 80 Newton Bolt Tension All Washer
40 N Tension - All Washers
70
60
50
Flat Washer - 40 N
Frequency 40
Hz
30
Split Washer - 40 N
Bellville Washer - 40 N
20
Tooth Washer - 40 N
10
Wedge Washer - 40 N
0
1
2
3
4
5
6
7
8
9
10
Mode
Figure 71: Plot of 40 Newton Bolt Tension All Washer
In conclusion, the analysis was able clearly demonstrate that from a vibrational perspective,
locking washers have a significant effect on system response. Additionally, the analysis was able
to determine the vibrational situations in which one design of locking washing may out perform
another. If this project were revisited in the future, it would deffinatley be beneficial to perform
this analysis on multiple materials. The anlysis itself could be improved by refining the meshes of
the tooth and wedge washers. An empirical method to compare the washers was not determined,
but what was found shows that a test of that manner may not provide any beneficial results due to
the fact that certain washers respond more at differing frequency ranges.
53
Analysis of Lock Washer Operation
Section 9: Works Cited
1. Integrated Publishing, Inc. “Section IV Operation Under Unusual Conditions” 02/04/2016.
Website
http://constructionmisc.tpub.com/TM-11-5820-1118-12P/css/TM-11-5820-1118-12P_19.htm
2. Hilti Co. “Threaded plate” 02/04/2016. Website
https://www.hilti.co.uk/installation-systems/channel-systems/r1410
3. Mcmaster Carr. “Split Lock Washers – 18-8 Stainless Steel” 02/04/2016. Website
http://www.mcmaster.com/#catalog/122/3252/=10zfc5f
4. Mcmaster Carr. “Belleville Spring Lock Washers – 18-8 Stainless Steel” 02/04/2016. Website
http://www.mcmaster.com/#catalog/122/3256/=10zfcwd
5. Mcmaster Carr. “Internal Tooth Lock Washers – 18-8 Stainless Steel” 02/04/2016. Website
http://www.mcmaster.com/#catalog/122/3255/=10zfdfz
6. Mcmaster Carr. “Wedge Lock Washers – 18-8 Stainless Steel” 02/04/2016. Website
http://www.mcmaster.com/#catalog/122/3257/=10zfdzz
7. Mcmaster Carr. “Stainless Steel Cap Screws – 18-8 Stainless Steel” 02/04/2016. Website
http://www.mcmaster.com/#catalog/122/3158/=10zfes6
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Analysis of Lock Washer Operation
55
Analysis of Lock Washer Operation
56