Issue No. 20 – August 2010
`
`Updated from Original February 2001 Publication
`
`Cantilever Beams Part 1 - Beam Stiffness
`
`The cantilever beam is an extremely useful model for electronic spring connectors. The equations that
`govern the behavior of a straight cantilever beam with rectangular cross section are extremely simple.
`Given certain assumptions (e.g. small deflections, no yielding), the underlying principles of this type of
`beam analysis can be extended to electrical spring contacts of most any shape and size.
`
`Electrical contacts are designed to generate a certain amount of contact force (F) for a given amount of
`deflection (d). The ratio between the force and deflection in either case is referred to as the spring rate or
`stiffness of the beam or coil. Occasionally, it is also referred to by the less technically correct term
`“springiness.” For a coil spring, the force needed to elongate the spring is directly proportional to the
`distance it elongates. (This assumes that the material does not yield.) The spring rate is then expressed as
`unit force per unit distance. For example, a coil spring with a spring rate of 2.0 pounds per inch would
`generate a force of 2.0 pounds for a 1.0 inch deflection, 4.0 pounds for 2 inches, etc.
`
`There is also a linear relationship between the force and deflection of a cantilever beam, as long as the
`deflection is small and the beam material does not yield. It is also expressed as force per unit distance. For
`a deflection at the end of the beam perpendicular to the beam axis, the force can be expressed
`IE
`3
`(cid:152)
`(cid:152)
`(cid:170)
`(cid:186)
` Here, E is the elastic modulus of the spring material, I is the area moment of inertia of
`as:
`d
`F
`(cid:152)(cid:187)(cid:188)
`(cid:171)(cid:172)
`L
`3
`the beam cross section, and L is the length of the beam. Note that the spring stiffness depends on the
`geometry of the beam as well as the material stiffness of the beam. For a straight beam with a rectangular
`cross section, the moment of inertia of the beam, which is a measure of how the cross-sectional area is
`1
`distributed around its center, is easy to calculate.
` Here, w is the strip width and t is the strip
`I
`tw
`3
`(cid:152)
`(cid:152)
`(cid:32)
`12
`thickness. Therefore, the force generated by a given deflection is
`
`.
`
`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)
`
`F
`
`(cid:32)
`
`(cid:32)
`
`
`
`FF
`
`
`
`
`
`
`
`tttt
`
`The stiffness of the beam is thus given by the bracketed term in the previous equation. Note that the overall
`stiffness is a function of the elastic modulus (material stiffness) and the dimensions of the beam (geometric
`stiffness.) Note that
`this equation is only
`valid if the stress in
`the spring does not
`exceed
`the
`elastic
`limit of the metal. If
`the material should
`start
`to yield,
`the
`elastic modulus is no
`longer a constant, and
`the
`equation will
`predict a value for
`force
`that
`is much
`greater than what it is
`in reality.
`
`
`
`
`
`
`
`LLLLLLLLLL
`
`Stiffness is an Asset –
`Except in Public
`Speaking – A discussion
`of the benefits of using a
`stiff material in spring
`design applications.
`
`(cid:131) Spring Rate
`(cid:131) Stiffness
`(cid:131) Stress
`Distribution
`
`(cid:131) Neutral Axis
`(cid:131) Bending
`Moment
`
`The next issue of
`Technical will consist
`of Cantilever Beams
`Part 2 - Analysis
`
`
`
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`
`
`wwwwww
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`
`
`
`
`
`
`FFFF
`
`
`
`dd
`
`
`
`FF
`
`
`
`dd
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`Figure 1. Cantilever Beams vs. Coil Springs
`
`©2010 Brush Wellman Inc.
`
`CORNING EXHIBIT 1021
`
`1
`
`

`
`Cantilever Beams Part 1 – Beam Stiffness (continued)
`The next step would be to solve for the stress distribution in the beam generated by the given
`deflection. In a coil spring, the stress is distributed evenly along the length of the coil. However, in a
`cantilever beam under a bending load, the stress is different at every point in the beam. When a beam is
`bent downward, the top surface of the beam elongates and is in tension. The bottom surface becomes
`compressed. Somewhere near the center of the beam, there is a plane that neither elongates nor
`compresses and thus is under no stress. This is known as the neutral axis. The stress will increase from
`zero at the neutral axis to a maximum value at the upper and lower surfaces, as shown in Figure 2.
`
`the fixed end,
`
`max
`
` and
`
`The stress will vary along the length of the beam, as well as through the thickness. The stress at any
`point depends on the bending moment (torque) present at that point. The bending moment (M) at any
`point in the beam is equal to the force applied multiplied by the distance from that point to the point of
`application. It is therefore zero at the free end of the beam, and maximum at the fixed end. This means
`that there is no stress at the free end of the beam, and a maximum stress at the fixed end. The equation
`yM
`xF
`(cid:152)
`(cid:152)
`for the stress at any point in the beam is as follows:
`. Here, F is the force
`Stress
`y
`(cid:32)
`(cid:32)(cid:86)
`(cid:152)
`(cid:32)
`I
`I
`applied, x is the distance from the point of force application, I is the moment of inertia, and y is the
`distance from the neutral axis. Since the maximum stress will occur at the upper and lower surfaces at
`t
`y
`
`LxMax (cid:32) . Therefore, the maximum stress can be expressed as
`(cid:32)
`2
`. When the moment of inertia is and the force equation is inserted, the stress equation
`
`2t
`
`(cid:152)
`
`(cid:32)(cid:86)
`Max
`
`LF
`(cid:152)
`I
`reduces to:
`
`tE
`3
`(cid:152)
`(cid:152)
`d
`(cid:86)
`(cid:32)
`(cid:152)
`Max
`L
`2
`2
`(cid:152)
`longer valid. This is because the metal’s stress-strain relationship is no longer linear beyond the elastic
`limit, and therefore a linear equation is no longer valid.
`
` Once again, if the material begins to yield, then the above equation is no
`
`
`
`
`
`
`
`
`
`
`
`
`
`ttttttt
`
`There are some interesting consequences of these equations. Notice that the width of the beam affects
`the contact force but has no effect on the stress. The contact force is most influenced by thickness and
`length, while the stress is most influenced by length. Both the stress and force are linearly proportional
`to the elastic modulus and the
`deflection. It becomes much
`easier to design a contact
`spring when the relationships
`among
`force,
`stress,
`geometry, and material are
`kept in mind. Next month’s
`edition of Technical Tidbits
`will further explore
`these
`relationships.
`
`
`
`
`
`
`
`dddd
`
`
`
`LL
`
`
`
`
`
`
`Neutral Neutral Neutral Neutral Neutral Neutral
`
`
`
`
`
`AxisAxisAxisAxisAxisAxis
`
`Please contact your local
`sales representative for
`further information on
`spring rate 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.
`
`
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`2222222
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`2222222
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`
`ttttttt
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`Figure 2. Cantilever Beam Stress Distribution
`
`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.
`
`
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`2