Issue No. 21 – September 2010
`
`Updated from Original March 2001 Publication
`
`Analyze This – A
`discussion of the
`ramifications of cantilever
`beam theory on spring
`design.
`
`Cantilever Beams Part 2 - Analysis
`
`The last edition of Technical Tidbits introduced some key concepts of cantilever beam stress and
`force analysis. The equations for contact force and stress as a function of deflection are repeated
`in Figure 1. Both the stress and force depend on the elastic modulus of the beam material as
`well as the beam geometry. These are linear equations and hold true as long as the stress-strain
`relationship is linear. If the stress exceeds the elastic limit, and the material begins to yield,
`these relationships will no longer hold true.
`
`
`
`FF
`
`
`
`
`
`
`
`tttt
`
`
`Maximum StressMaximum Stress
`
`E tE t
`
`33
`
`
`(cid:152)(cid:152)
`(cid:152)(cid:152)
`
`LL
`
`22
`
`22
`
`(cid:152)(cid:152)
`
`
`
`(cid:152)(cid:152)
`
`
`
`(cid:86)max (cid:32)(cid:86)max (cid:32)
`
`
`
`dd
`
`
`
`
`
`
`
`wwwwww
`
`
`
`
`
`
`
`Contact ForceContact Force
`
`
`
`(cid:152)(cid:152)
`
`
`
`dd
`
`(cid:186) (cid:188)(cid:187)
`(cid:186) (cid:188)(cid:187)
`
`
`
`33
`
`
`E w tE w t
`
`
`(cid:152)(cid:152)
`(cid:152)(cid:152)
`
`LL
`
`44
`
`33
`
`(cid:152)(cid:152)
`
`(cid:170) (cid:172)(cid:171)
`(cid:170) (cid:172)(cid:171)
`
`
`
`FF
`
`
`
`(cid:32)(cid:32)
`
`
`
`FFFF
`
`
`
`dd
`
`Figure 1. Straight cantilever beam with rectangular cross-section
`along with beam stress and force per unit deflection equations.
`
`A key question to ask about any spring design is, “How much deflection can the spring tolerate
`before it yields ?” The stress equation can be rearranged to show deflection as a function of
`stress. The deflection at the yield (allowable deflection) can be found if the yield strength is
`inserted into the equation:
`
`d
`
`yield
`
`(cid:32)
`
`2
`3
`(cid:152)
`
`L
`2
`(cid:152)
`tE
`(cid:152)
`
`(cid:86)(cid:152)
`yield
`
`The next obvious question to ask is, “How much contact force will the spring give me before it
`yields ?” The deflection at yield can be inserted into the force-deflection equation to come up
`with the contact force at yield:
`
`(cid:86)(cid:152)
`yield
`
`Lt
`
`2
`
`(cid:152)(cid:152)
`
`w
`6
`
`F
`
`yield
`
`(cid:32)
`
`(cid:188)(cid:186)
`
`(cid:86)(cid:152)(cid:187)
`yield
`
`2
`3
`(cid:152)
`
`L
`2
`(cid:152)
`tE
`(cid:152)
`
`(cid:171)(cid:172)(cid:170)
`
`(cid:188)(cid:186)
`
`(cid:152)(cid:187)
`
`3
`
`twE
`(cid:152)
`(cid:152)
`L
`4
`3
`(cid:152)
`
`(cid:171)(cid:172)(cid:170)
`
`d
`
`yield
`
`(cid:32)
`
`(cid:188)(cid:186)
`
`(cid:152)(cid:187)
`
`3
`
`twE
`(cid:152)
`(cid:152)
`L
`4
`3
`(cid:152)
`
`(cid:171)(cid:172)(cid:170)
`
`F
`
`yield
`
`(cid:32)
`
`Most springs are designed to generate as much contact force as possible, while withstanding a
`high range of deflections. Let us look at some methods of maximizing the deflection and force,
`referencing the four key equations:
`
`©2010 Brush Wellman Inc.
`
`CORNING EXHIBIT 1022
`
`
`
`
`
`
`
`LLLLLLLLLL
`
`
`
`
`
`(cid:131) Allowable
`Deflection
`(cid:131) Contact Force
`at Yield
`(cid:131) Elastic
`Resilience
`
`The next issue of
`Technical Tidbits will
`discuss elastic
`resilience.
`
`1
`
`

`
`Please contact your local
`sales representative for
`further information on
`stress and force analysis
`or other questions
`pertaining to Brush
`Wellman or our products.
`
`Health and Safety
`Handling copper beryllium in
`solid form poses no special
`health risk. Like many
`industrial materials, beryllium-
`containing materials may pose a
`health risk if recommended safe
`handling practices are not
`followed. Inhalation of airborne
`beryllium may cause a serious
`lung disorder in susceptible
`individuals. The Occupational
`Safety and Health
`Administration (OSHA) has set
`mandatory limits on
`occupational respiratory
`exposures. Read and follow the
`guidance in the Material Safety
`Data Sheet (MSDS) before
`working with this material. For
`additional information on safe
`handling practices or technical
`data on copper beryllium,
`contact Brush Wellman Inc.
`
`Cantilever Beams Part 2 – Analysis (continued)
`
`(cid:86)(cid:152)
`yield
`
`Lt
`
`2
`
`(cid:152)(cid:152)
`
`w
`6
`
`(cid:86)(cid:152)
`yield
`
`F
`
`yield
`
`(cid:32)
`
`d
`
`yield
`
`(cid:32)
`
`2
`3
`(cid:152)
`
`L
`2
`(cid:152)
`tE
`(cid:152)
`
`d
`
`(cid:188)(cid:186)
`
`(cid:152)(cid:187)
`
`3
`
`twE
`(cid:152)
`(cid:152)
`L
`4
`3
`(cid:152)
`
`(cid:171)(cid:172)(cid:170)
`
`(cid:86)
`Max
`
`(cid:32)
`
`3
`(cid:152)
`2
`
`tE
`(cid:152)
`L
`2
`(cid:152)
`
`(cid:152)
`
`d
`
`F
`
`(cid:32)
`
`Note that by increasing the width, we increase the contact force but do not affect the stress. If
`we increase the thickness of the material, we slightly increase the stress, and slightly decrease
`the allowable deflection. However, we get much larger gains in the force per unit deflection
`and the force at yield. Next, if we increase the length, we substantially reduce the force per
`unit deflection. However, we only slightly decrease the force at yield, because we reduce the
`stress per unit deflection and greatly increase the allowable deflection. If we increase the
`length and thickness by an identical percentage we get that same percentage decrease in stress
`per unit deflection, while not affecting the force per unit deflection. Additionally, the total
`allowable deflection and the total force at yield both increase by the same percentage.
`Therefore, we know that we can improve performance of the spring by increasing the length,
`width, and thickness of the beam.
`
`The current trends in component design for computers, telecommunications, and automotive
`markets are towards miniaturization. This means that the length, width, and thickness of
`connectors and their contact springs must decrease in size. This will have a net negative
`impact on the contact force per unit deflection, maximum force, and allowable deflection.
`Therefore, choosing materials that have the material property benefits to overcome these
`geometrically imposed limitations becomes more important. For example, if we choose a
`material with a lower modulus, we get lower force and stress per unit deflection. The
`allowable deflection will increase, and the maximum stress will remain the same. So a low
`modulus material would seem to be a good option. However, there is very little variation in
`the elastic moduli of copper alloys, and the low modulus materials all have low yield
`strengths, which tends to offset the gains made by the lower modulus.
`
`The obvious solution is to use a material with a high yield strength. As the yield strength
`increases, the allowable deflection and maximum force increase as well. This is the best of
`both worlds. Ultimately, the greatest advantage comes from a material with a high ratio of
`yield strength to elastic modulus. This important ratio has been given the name “elastic
`resilience”. The higher the resiliency, the better the contact will perform. This concept will
`be explored in a future edition of Technical Tidbits.
`
`Written by Mike Gedeon of Brush Wellman’s Alloy Customer Technical Service Department.
`Mr. Gedeon’s primary focus is on electronic strip for the telecommunications and computer
`markets with emphasis on Finite Element Analysis (FEA) and material selection.
`
`Brush Wellman Inc.
`6070 Parkland Blvd.
`Mayfield Heights, OH 44124
`(216) 486-4200
`(216) 383-4005 Fax
`(800) 375-4205 Technical Service
`
`©2010 Brush Wellman Inc.
`
`
`
`2