Finite Element Analysis of Basketball Backboards

Finite Element Analysis of Basketball Backboards:
Stress and Deflection Analyses based on Material
by
Ryan Ansaldo
An Engineering Report Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING IN MECHANICAL ENGINEERING
Approved:
_________________________________________
Ernesto Gutierrez, Project Adviser
Rensselaer Polytechnic Institute
Hartford, Connecticut
August, 2012
i
© Copyright 2012
by
Ryan Ansaldo
All Rights Reserved
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CONTENTS
LIST OF TABLES ............................................................................................................ iv
LIST OF FIGURES ........................................................................................................... v
Nomenclature .................................................................................................................... vi
ACKNOWLEDGMENT ................................................................................................. vii
ABSTRACT ................................................................................................................... viii
1. Introduction.................................................................................................................. 1
1.1
Background ........................................................................................................ 1
1.2
Problem Description........................................................................................... 2
1.3
Material Properties ............................................................................................. 2
1.4
Failure and Yielding of Materials ...................................................................... 2
2. Methodology ................................................................................................................ 3
2.1
Abaqus Finite Element Analysis ........................................................................ 3
2.2
Finite Element Development Model .................................................................. 4
2.2.1
Backboard and Rim Geometry ............................................................... 4
2.2.2
Rim, Backboard and Mounting Plate Interactions ................................. 5
2.2.3
Rim Face Plate and Mounting Plate Constraints.................................... 6
2.2.4
Boundary Conditions and Load ............................................................. 6
2.2.5
Part Mesh ............................................................................................... 7
3. Results and Discussion ................................................................................................ 9
3.1
FEA Results ....................................................................................................... 9
3.1.1
Tempered Glass Results ......................................................................... 9
3.1.2
Acrylic Results ..................................................................................... 10
3.1.3
Steel Results ......................................................................................... 12
4. Conclusion ................................................................................................................. 13
5. References.................................................................................................................. 14
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LIST OF TABLES
Table 1: Backboard Material Properties ............................................................................ 2
Table 2: Material Failure and Yield Properties ................................................................. 2
iv
LIST OF FIGURES
Figure 1: Geometry of backboard and rim modeled in Abaqus/CAE ............................... 3
Figure 2: Backboard sections and highlighted shell-to-solid coupling ............................. 4
Figure 3: Rim and Mounting Plate Geometry ................................................................... 5
Figure 4: Surface-to-surface contact between the rim and backboard .............................. 6
Figure 5: Rigid Body Tie Constraints on Rim and Mounting Plate .................................. 6
Figure 6: Boundary Conditions Applied to the Backboard ............................................... 7
Figure 7: Load Applied to the Rim .................................................................................... 7
Figure 8: Assembly Mesh and Contact Surface Meshes ................................................... 8
Figure 9: Von Mises Stresses on Tempered Glass Backboard Contact Surfaces .............. 9
Figure 10: Remaining Glass Clamped between Two Backboard Plates [13].................. 10
Figure 11: Contour Plot with Stresses above Tempered Glass Breaking Stress ............. 10
Figure 12: Von Mises Stresses on Acrylic Backboard Contact Surfaces ........................ 11
Figure 13: Contour Plot with Stresses above Acrylic Breaking Stress ........................... 11
Figure 14: Von Mises Stresses on Steel Backboard ........................................................ 12
Figure 15: Steel Backboard Z-displacement Contour Plot .............................................. 12
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Nomenclature
E - Young’s Modulus (psi)
ʋ - Poisson’s Ratio (-)
σy - Yield Strength (psi)
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ACKNOWLEDGMENT
I would like to thank Professor Ernesto Gutierrez-Miravete for his guidance and patience
throughout the completion of my master’s project. I would also like to thank my family,
colleagues, and friends for their assistance and encouragement throughout my entire
academic career.
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ABSTRACT
The purpose of this project is to perform structural analyses of a basketball backboard
assembly and investigate the type of stresses imparted on the backboard from an NBA
basketball player performing a dunk shot. Players dunking on the rim had been a
structural issue in the past as it sometimes led to the glass backboard shattering. Not only
was this a problem in delaying basketball games but it was also potentially harmful to
players and spectators alike. The structural finite element analyses (FEA) on the rim and
backboard mounts were performed using Abaqus/CAE. The focus of this project is to
perform comparative analyses between the stresses and deflections applied to
backboards using common materials. The three backboard materials of interests are
tempered glass, acrylic, and steel. As expected, the tempered glass and acrylic
backboards failed under the applied load and using a simple direct mounted rim design,
however the steel backboard did not fail or deform plastically. The results confirmed the
necessity for newer sophisticated rim and backboard designs which currently exist and
are still being modified to this day.
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1. Introduction
1.1 Background
Basketball is a popular sport played all over the world. Since its invention, the sport has
gone through many evolutions. The original “basket” was in fact an actual peach basket
fastened to the wall by a nail. In modern times, the basket has been replaced by a rim
and net with a more sophisticated means of being fastened to the backboard. The athletic
ability of basketball players has also changed over the years and since the invention of
the dunk shot, the basketball rim and backboard have been subjected to many changes
and redesigns.
Previously the only loads applied to the basketball rim were imparted by the ball which
has a weight of approximately 22 ounces [1]. When a dunk is performed the rim can
experience loads in the magnitude of 1000 lbs due to the impact and force generated by a
jumping basketball player that can impart the strength of their arms and bodyweight to
the rim [2]. The force is then transferred to the bracket which mounts the rim to the
backboard.
The stress experienced by the backboard is something that still affects the design of
modern day rims as the consequences are not only catastrophic for its function but
potentially harmful. For this reason, the basketball rim and mount are constantly
redesigned to improve the function of the rim. Since basketball has also become a
recreational outdoor sport, the materials of backboards have also changed.
Basketball is a sport that is played professionally and collegiately indoors. Originally,
backboards were made of wood but in 1917, glass backboards were introduced to
college basketball in order to provide better viewing for the spectators sitting behind the
backboard. This was first done at Indiana University at the Indiana Hoosier’s
gymnasium [3]. Professional basketball backboards in the present are made of glass and
have shattered when being overstressed.
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1.2 Problem Description
The objective of this study was to perform a structural analysis of a basketball
backboard. The load is assumed to be applied at the point on the rim furthest from the
backboard to simulate the greatest possible moment. The geometry of the assembly was
modeled and analyzed in Abaqus/CAE. The load is assumed to be 1000 lbs, which is the
load applied by an NBA player hanging and attempting to shatter the backboard [2].
1.3 Material Properties
The three backboard materials of interests are tempered glass, acrylic, and steel. The
material properties of interest for the Abaqus analyses are Young’s Modulus and
Poisson’s ratio. The basketball backboard material properties are shown in the table
below as well as the material properties used for the rim.
Material
E (psi)
Poisson’s ratio (ʋ)
Tempered Glass [4]
10,152,641.6
0.22
Acrylic [5]
450,000
0.35
Steel (backboard and rim) [6] [7]
29,000,000
0.29
Table 1: Backboard Material Properties
1.4 Failure and Yielding of Materials
Tempered glass and Acrylic are both brittle materials that will break rather than yield
under a certain stress. The steel will elastically deform until it experiences a stress
equivalent to its yield strength. Beyond its yield strength, the steel will deform
plastically. Table 2 shows the criteria used in these analyses to determine whether or not
the backboard material has failed or yielded due to the applied load.
Material
Tempered Glass [8]
Acrylic [5]
Steel [7]
σy (psi)
N/A
N/A
55,100
Failure Stress (psi)
14,000
10,500
N/A
Table 2: Material Failure and Yield Properties
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2. Methodology
2.1 Abaqus Finite Element Analysis
The geometry of the basketball rim and backboard were modeled in Abaqus/CAE using
the dimensions of standard regulation size rim and backboard. The design chosen is of a
rim with no breakaway feature and assumes the load is directly transferred from the rim
to the backboard. The mounting bracket was modeled bolted to the backboard using the
appropriate constraints. Rigid body tie constraints were used to simulate the connection
between the bolted connection and the backboard.
Step 1:
The chosen design was modeled in Abaqus/CAE using the regulation dimensions of a
basketball rim and backboard. A common bolt pattern was used for an accurate
simulation of the interaction between the rim and backboard. The design chosen for the
rim and backboard assembly utilizes a rear mounting plate at the back of the backboard
as shown in the Figure 1. The backboard is simulated to be sandwiched between the rim
and mounting plate by nuts and bolts.
Dimensions:
Backboard:
72 in length x 42 in width x ½ in thick [9]
Basketball rim:
18 in diameter [6], 5/8 in diameter rod (inside diameter is 6 inches from backboard)
Mounting plate:
4 bolt pattern (5” x 5”) (based on common bolting patterns of basketball rim mounts)
Figure 1: Geometry of backboard and rim modeled in Abaqus/CAE
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Step 2:
Once the basketball rim and backboard assembly were finalized, the appropriate material
properties, boundary conditions, mesh and loads are applied to the model. A rough
model was used as a proof of concept using a coarse mesh while exploring the most
accurate and appropriate boundary conditions and constraints.
Step 3:
After the proof of concept study was completed, a suitable mesh was chosen for the
assembly. The material properties for the first model were of tempered glass (see Table
1). After appropriate boundary conditions and constraints were applied, the load
specified in Section 1.2 was applied on the rim. Once the load case is run, stress and
deflection data in the post processing menus are used measure the stresses and
deflections experienced by the backboard. After using the tempered glass material
properties, the analysis will be repeated two more times for the materials of interest
shown in Table 1.
2.2 Finite Element Development Model
2.2.1
Backboard and Rim Geometry
The finite element model of the basketball backboard and rim assembly was developed
consisting of 3 parts. The backboard was created as a 3D deformable shell with
dimensions as specified in Section 2.1 and Figure 2.
Figure 2: Backboard sections and highlighted shell-to-solid coupling
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The rim was created as a 3D deformable solid based on the geometry shown in Figure 1.
The part consists of a circular ring with a mounting bracket with a shape similar to an L
bracket. The face plate has a 4 bolt pattern as described in Section 2.1. The rim was
created using a number of solid extrudes and cut extrudes. The mounting plate
dimensions and partitions matched the rim face plate for simplicity. The rim face plate,
backboard and mounting plate had common bolt patterns as specified in Section 2.1 (see
Figure 2 and Figure 3). The circular partitions concentric with each hole on the rim face
plate and mounting plate represented the surface area where washers would apply the
pressure from a bolt and washer.
partition
face plate / mounting plate
basketball rim
Figure 3: Rim and Mounting Plate Geometry
2.2.2
Rim, Backboard and Mounting Plate Interactions
To accurately model the interactions taking place between the rim, backboard and
mounting plate, appropriate contact was defined. Since the 3 parts are 3D deformable
bodies, surface to surface contacts were applied to the face plate and the backboard at
the surfaces which they mate (see Figure 4). Similarly, the backboard and mounting
plate were assigned surface to surface contacts at their mating surfaces.
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Figure 4: Surface-to-surface contact between the rim and backboard
2.2.3
Rim Face Plate and Mounting Plate Constraints
Reference points were created at the center of each hole diameter and depth. Rigid body
tie constraints were used to tie the surfaces (highlighted in Figure 3) to the reference
points to simulate where a nut and washer would fix the mounting bracket to the
backboard with a threaded pin.
Figure 5: Rigid Body Tie Constraints on Rim and Mounting Plate
2.2.4
Boundary Conditions and Load
The boundary conditions for the backboard and rim assembly were created to simulate
previous backboard designs in which the backboard is fixed at its edges by an aluminum
frame that mounts to the structural frame and arms of the backboard stand. The
“encastre” boundary condition was applied at all four edges of the backboard which
constrains all degrees of freedom (see Figure 6).
6
Figure 6: Boundary Conditions Applied to the Backboard
A load was applied to a reference point created at the point of the rim furthest from the
backboard. The magnitude of the load was 1000 lbf as specified in Section 1.2 [2]. The
load was applied in the negative y-direction as shown in Figure 7.
Figure 7: Load Applied to the Rim
2.2.5
Part Mesh
The 3 parts were meshed with 8-node linear brick elements (see Figure 8). The brick
elements were used with incompatible modes. The purpose of using incompatible modes
is to improve the bending behavior of the elements since the expected results will have
bending of the rim, backboard, and mounting plate [10]. The mesh density used for the
backboard was an approximate global seed size of 0.5. To provide more accuracy to the
contact analysis, the meshes over the contact surfaces are created to be uniform. The rim
and mounting plate were also assigned approximate global seed sizes of 0.5 in order to
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obtain as close to a uniform mesh between the contact surfaces and also provides a faster
convergence rate [11].
Backboard Mesh at Contact Surface
Assembly Mesh
Rim Mesh at Contact Surface
Figure 8: Assembly Mesh and Contact Surface Meshes
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3. Results and Discussion
3.1 FEA Results
The following section discusses the results of the backboard analyses. The focus of the
results is how the 3 backboard materials reacted to the load applied by the NBA
basketball player performing a dunk shot on a rim with no breakaway feature. The
primary outputs of interest are the stress and deflection applied to each of the different
backboard materials.
3.1.1
Tempered Glass Results
The material used as the baseline was tempered glass. Tempered glass is currently used
for professional and collegiate basketball backboards. The contour plot shown in Figure
9 shows the Von Mises stress gradient of the backboard. The maximum stress
experienced by the backboard is 22,743.6 psi.
Front Contact Surface
Rear Contact Surface
Figure 9: Von Mises Stresses on Tempered Glass Backboard Contact Surfaces
As expected, the highest stresses were seen around the bolt holes since the holes provide
high stress concentration to the backboard [12]. The contour plot determines that the
tempered glass backboard has failed due to the stresses imparted by the NBA player
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since the breaking stress of tempered glass is 14,000 psi (see Table 2). The results were
validated by a previously performed tempered glass fracture analysis (see Figure 10)
[13]. The analysis determined that the crack initiation occurred at the lower left hole
which is consistent with the contour plot in Figure 11. The highest stresses experienced
by the tempered glass backboard were at the areas around the lower bolt holes with the
stress magnitudes being higher than the breaking stress of the material.
Figure 10: Remaining Glass Clamped between Two Backboard Plates [13]
Figure 11: Contour Plot with Stresses above Tempered Glass Breaking Stress (highlighted in gray)
Since it was determined that the applied load caused the tempered backboard to fail, the
maximum deflection results of 0.1291 inches were determined to be inaccurate. Since
glass is a brittle material, it would not deflect but rather fracture and fail almost instantly
[8].
3.1.2
Acrylic Results
Acrylic is an alternative backboard material that emulates some of the characteristics of
tempered glass. Acrylic has its advantages over tempered glass as it is lighter in weight
and a lower cost material. The disadvantages of acrylic are lower stiffness and lower
breaking stress than tempered glass. The contour plot in Figure 12 shows the Von Mises
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stress gradient of the backboard. The maximum stress experienced by the backboard is
22,141.4 psi.
Front Contact Surface
Rear Contact Surface
Figure 12: Von Mises Stresses on Acrylic Backboard Contact Surfaces
Similar to the tempered glass the stresses experienced by the acrylic backboard are
higher than the specified breaking stress of 10,500 psi. The acrylic backboard also failed
at the lower bolt holes. Deflection results show that the maximum deflection occurs near
the center of the backboard and deflects 2.5 in which was determined to be inaccurate
since the material failed under the applied load.
Figure 13: Contour Plot with Stresses above Acrylic Breaking Stress (highlighted in gray)
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3.1.3
Steel Results
Steel backboards are primarily used in outdoor basketball courts. Steel is a durable
material that can withstand harsher treatment than either glass or acrylic. The contour
plot shown in Figure 14 shows the Von Mises stress gradient of the backboard. The
maximum stress experienced by the backboard is 21,419.3 psi.
Front Contact Surface
Rear Contact Surface
Figure 14: Von Mises Stresses on Steel Backboard
The highest stress experienced does not exceed the 55,100 psi which is the σy of steel
(see Table 2). Unlike tempered glass and acrylic, the steel does not fail. Since the
stresses do not exceed the σy of steel, the backboard deforms elastically due to the load
applied. The maximum deflection in the z-direction was 0.06456 inches (see Figure 15).
Figure 15: Steel Backboard Z-displacement Contour Plot
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4. Conclusion
The results of this study validate that the stresses applied to a basketball backboard by an
NBA player dunking on the rim have the magnitude to cause failure on tempered glass
and acrylic backboards. The previous rim designs and mounting have proven to be
outdated in preventing the shattering of professional and collegiate backboards. More
sophisticated rim designs are equipped with a breakaway feature that allows the rim to
pivot slightly using springs which snap the rim back into proper alignment and prevent
the entire load to be transferred to the backboard [14]. Since this type of configuration
cannot always be used in public basketball courts, the use of steel as an alternative
material removes the risk of having a backboard shatter. The strength of the backboard
would also ensure that the rim would become damaged prior to the steel backboard
yielding which is more easily replaceable than the backboard. The study concludes that
the constant redesigns of basketball rims to relieve the backboard of high stresses were
warranted. In addition to changing the rim design, as revealed by Sports Science, the rim
in professional quality rim and backboard configurations has the rim directly mounted to
the frame and is no longer mounted through the backboard [2]. With this new design, no
loads are transferred through the backboard from the rim making it impossible for an
NBA player to shatter the backboard with a dunk.
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5. References
1. Hiner, Jason (2005). Indiana University Basketball Encyclopedia. United States:
Sports Publishing. pp. 447
2. “Busted Guts,” Sports Science. Fox Sports Network. CSN New England,
Massachusetts. 3 May 2009
3. "Basketball (ball)" Wikipedia: The Free Encyclopedia. Wikimedia Foundation, Inc. 2
July 2012, 4 Jul. 2012, http://en.wikipedia.org/wiki/Basketball_(ball)
4. W.A. Dalgliesh, D.A. Taylor, “The Strength and Testing of Window Glass,”
Canadian Journal of Civil Engineering, Vol. 17, No.5 October 1990, pages 752-762
5. Laird Plastics, Plexiglas® G Acrylic Plastic Sheets, Accessed on 7/26/12,
lairdplastics.com/content/view/264/
6. “ARENAPRO 180 GOAL,” Spalding Specifications, dated 15 Oct 2009,
http://www.spaldingequipment.com/SpecLibrary/Basketball/Breakaway-ReflexGoals/413583Arena_180_goal.pdf
7. Matweb.com AISI 1045 Steel, as cold drawn, Accessed on 7/15/2012,
http://www.matweb.com/search/DataSheet.aspx?MatGUID=20fffdaa96f14dd98f503
2c4014b9587
8. Read, Thomas L., “Failure Analysis of a Tempered Glass Basketball Backboard,”
Read Consulting, Accessed on 7/30/12,
http://readconsulting.com/publications/whitepapers/tempered_glass_backboard.html
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9. “Rule No. 1---Court Dimensions--Equipment,” The Official Site of the National
Basketball Association. 17 Oct 2006. National Basketball Association. 22 Jul. 2012,
http://www.nba.com/analysis/rules_1.html?nav=ArticleList
10. Abaqus/CAE 6.10EF-1. “Abaqus User Manual.” Dassault Systèmes, Providence,
RI, 2010.
11. Choi, Seung-Woo, “Technical Tips for Surface-To-Surface Contact Analysis in
MES,” ALGOR, Inc., Accessed on 7/29/2012,
http://www.algor.com/news_pub/tech_white_papers/surface_contact/default.asp
12. Richard G. Budynas, J. Keith Nisbett, “Shigley’s Mechanical Engineering Design,”
Eighth Edition, McGraw-Hill, 2008
13. Read, Thomas L., “Tempered Glass Fractures,” Read Consulting, Accessed on
7/30/12, http://readconsulting.com/publications/whitepapers/failure_analysistempered_glass.html
14. Winn, Luke. “Breaking Away,” Sports Illustrated 23 Dec. 2007. Accessed on
7/31/12, http://sportsillustrated.cnn.com/si_blogs/basketball/ncaa/2007/12/breakingaway.html
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