07 lecture 07 split

Cite as: Carol Livermore, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

C. Livermore: 6.777J/2.372J Spring 2007, Lecture 7 - 1

Structures

Carol Livermore

Massachusetts Institute of Technology

* With thanks to Steve Senturia, from whose lecture notes some of these materials are adapted.



Outline> Regroup

> Beam bending• Loading and supports• Bending moments and shear forces• Curvature and the beam equation• Examples: cantilevers and doubly supported beams

> A quick look at torsion and plates



Recall: Isotropic Elasticity> For a general case of loading, the constitutive relationships

between stress and elastic strain are as follows

> 6 equations, one for each normal stress and shear stress

( )[ ]

( )[ ]

( )[ ]yxzz

xzyy

zyxx

E

E

E

σσνσε

σσνσε

σσνσε

+−=

+−=

+−=

1

1

1

Shear modulus G is given by)1(2 ν+

=EG

zxzx

yzyz

xyxy

G

G

G

τγ

τγ

τγ

1

1

1

=

=

=



What we are considering today> Bending in the limit of small deflections

> For axial loading, deflections are small until something bad happens

• Nonlinearity, plastic deformation, cracking, buckling• Strains typically of order 0.1% to 1%

> For bending, small deflections are typically less than the thickness of the element (i.e. beam, plate) in question



What we are NOT considering today> Basically, anything that makes today’s theory not apply (not as

well, or not at all)

> Large deflections• Axial stretching becomes a noticeable effect

> Residual stresses• Can increase or decrease the ease of bending



Our trajectory

> What are the loads and the supports?

> What is the bending moment at point x along the beam?

> How much curvature does that bending moment create in the structure at x? (Now you have the beam equation.)

> Integrate to find deformed shape

F

M

Silicon0.5 µm

1 µm

Pull-downelectrode

Cantilever

Anchor

Adapted from Rebeiz, Gabriel M. RF MEMS: Theory, Design, andTechnology. Hoboken, NJ: John Wiley, 2003. ISBN: 9780471201694.

Image by MIT OpenCourseWare.



Outline> Regroup

> Bending• Loading and supports• Bending moments and shear forces• Curvature and the beam equation• Examples: cantilevers and doubly supported beams




Types of Loads> Three basic types of loads:

• Point force (an old friend, with its own specific point of application)

• Distributed loads (pressure) • Concentrated moment (what you get from a screwdriver, with

a specific point of application)• The forces and moments work together to make internal

bending moments – more on this shortly

F = point load

M = concentrated moment

q = distributed load



Types of supports> Four basic boundary conditions:

• Fixed: can’t translate at all, can’t rotate • Pinned: can’t translate at all, but free to rotate (like a hinge)• Pinned on rollers: can translate along the surface but not off

the surface, free to rotate• Free: unconstrained boundary condition

Image by MIT OpenCourseWare.Adapted from Figure 9.5 in: Senturia, Stephen D. Microsystem Design.Boston, MA: Kluwer Academic Publishers, 2001, p. 207. ISBN: 9780792372462.

Pinned

FreeFixed

Pinned on Rollers



Reaction Forces and Moments> Equilibrium requires that the total force on an object be zero

and that the total moment about any axis be zero

> This gives rise to reaction forces and moments

> “Can’t translate” means support can have reaction forces

> “Can’t rotate” means support can have reaction moments

FF

FLM

FLMM

R

RT

=

=

−=

:mequilibriuin zero bemust forceNet

:mequilibriuin zero bemust Moment

:supportabout moment Total

R

M R

F R

O

x

L

F

Point Load

Image by MIT OpenCourseWare.Adapted from Figure 9.7 in: Senturia, Stephen D. Microsystem Design .Boston, MA: Kluwer Academic Publishers, 2001, p. 209. ISBN: 9780792372462.

Internal forces and moments> Each segment of beam must also be in equilibrium> This leads to internal shear forces V(x) and bending moments



( )( ) FxV

xLFxM=

−−= )(case, For thisM(x)

Images by MIT OpenCourseWare.Adapted from Figure 9.7 in Senturia, Stephen D. Microsystem DesignBoston, MA: Kluwer Academic Publishers, 2001, p. 209. ISBN: 9780792372462.

esign and Fabrication of Microelectromechanical Devices, Spring 2007. MIT Cite as: Carol Livermore, course materials for 6.777J / 2.372J DOpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].


Some conventions

Moments:

Image by MIT OpenCourseWare.Adapted from Figure 9.8 in: Senturia, Stephen D.Microsystem Design. Boston, MA: Kluwer AcademicPublishers, 2001, p. 210. ISBN: 9780792372462.

Shears:

Positive Negative

Moments

rseWare. in: Senturia, Stephen D.ston, MA: Kluwer Academic. ISBN: 9780792372462.

Image by MIT OpenCouAdapted from Figure 9.8Microsystem Design. BoPublishers, 2001, p. 210

Positive Negative

Shears



Combining all loads> A differential beam element, subjected to point loads,

distributed loads and moments in equilibrium, must obey governing differential equations

qdxdV

VdVVqdxFT

−=⇒

−++= )(

Vdx

dM

dxqdxdxdVVMdMMM T

=⇒

−+−−+=2

)()(

Image by MIT OpenCourseWare.Adapted from Figure 9.8 in: Senturia, Stephen D. Microsystem Design. Boston, MA: Kluwer AcademicPublishers, 2001, p. 210. ISBN: 9780792372462.

All Loads:

M

q

V

dx

M+dM

V+dV



Outline> Regroup



COpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].


Pure bending> Important concept: THE NEUTRAL AXIS

> Axial stress varies with transverse position relative to the neutral axis

ite as: Carol Livermore, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT

ρ−=σ

ρ−=ε

ρ−=⇒

θρ=θ−ρ=

zE

z

dxzdxdL

ddxdzdL

x

x

)(

M O M O

?

Compression

Neutral AxisTension

d?

z

Tension

Compression

?x

H/2

H/2

zImage by MIT OpenCourseWare.Adapted from Figures 9.9 and 9.10 in: Senturia, Stephen D.Microsystem Design . Boston, MA: Kluwer AcademicPublishers, 2001, pp. 211, 213. ISBN: 9780792372462.



Locating the neutral axis> In pure bending, locate the neutral axis by imposing equilibrium

of axial forces

> The neutral axis is in the middle for a one material beam of symmetric cross-section.

> Composite beams: if the beam just has a very thin film on it, can approximate neutral axis unchanged

> Composite beams: with films of comparable thickness, change in E biases the location of the neutral axis

( )

( ) ( ) 0

0

=−

==

∫

∫

dzzzWzE

dAzN

thickness

A

ρ

σ

0

beamr rectangula material, One

=− ∫ dzzEW

thicknessρ



Curvature in pure bending

ρ

ρ

EIM

dAzI

dAzEM

A

A

−=

=

−=

∫

∫

2

2

:I inertia ofMoment

beam, material-one aFor

∫

∫

−=

−=

=

A

x

Ax

dAzzEM

zE

dAzM

2)(1

:Moment Internal

ρ

ρσ

σ

Curvature is related to the internal bending moment M.

In pure bending, the internal moment M equals the externally

applied moment M0. Then for one material; for two

or more materials, calculate an effective EI.EIM 01

−=ρ



Useful case: rectangular beam

ρEWHM

WHdzWzIH

H

⎟⎠⎞

⎜⎝⎛−=

== ∫−

3

32/

2/

2

121

121

beam,r rectangula uniform aFor

M0 M0H

W



Differential equation of beam bending> Relation between curvature and the applied load

a

2

2

1

tan

cos

dxwd

dxd

ddsdxdw

dxds

=ρ

≈θ

⇓

θρ=

θ≈θ=

≈θ

= EIq

dxwd

EIV

dxwd

EIM

dxwd

=

−=

−=

4

4

3

3

2

2

ationdifferenti successiveby nd,

( )[ ] 2/3211

bending angle-largeFor

w

w′+

′′=

ρ

Large-angle bending israre in MEMS structures

Image by MIT OpenCourseWare.Adapted from Figure 9.11 in: Senturia, Stephen D. Microsystem Design.Boston, MA: Kluwer Academic Publishers, 2001, pp. 214. ISBN: 9780792372462.

dx

w(x)?(x)

dx

ds

d?

x x

z



Anticlastic curvature> If a beam is bent, then the Poisson effect causes

opposite bending in the transverse direction

moment internal directed-a y creates whichρν

=ε

⇓

νε−=ερ

−=ε

z

z

y

xy

x

Image by MIT OpenCourseWare.Adapted from Figure 9.12 in: Senturia, Stephen D. Microsystem Design .Boston, MA: Kluwer Academic Publishers, 2001, pp. 219. ISBN: 9780792372462.

Originalcross-section

ρ/v

ρ



Example: Cantilever with point load

⎟⎠⎞

⎜⎝⎛ −=

==

==

+++−=

++−=

−=

=

Lxx

EIFLw

BA

dxdww

BAxxEI

FLxEIFw

AxEIFLx

EIF

dxdw

xLEIF

dxwd

x

31

2

0

0 ,0)0( :BC

26

2

)(

2

0

23

2

2

2

( )

( )( ) FxV

xLFxM

EIxM

dxwd

=−−=

−=

)(and

2

2

Image by MIT OpenCourseWare.Adapted from Figure 9.7 in: Senturia, Stephen D. Microsystem Design .Boston, MA: Kluwer Academic Publishers, 2001, p. 209. ISBN: 9780792372462.


cantilever


Spring Constant for Cantilever> Since force is applied at tip, if we find maximum tip

displacement, the ratio of displacement to force is the spring constant.

3

3

3

3

max

43

3

LEWH

LEIk

FEILw

cantilever ==

⇓

⎟⎟⎠

⎞⎜⎜⎝

⎛=

N/m 2.0beam, loaded uniaxially theas dimensions same For the

=k



Stress in the Bent Cantilever> To find bending stress, we find the radius of

curvature, then use the pure-bending case to find stress

( )

FWHL

FWEHLF

EILH

EIFL

x

xLEIF

dxwd

2max

2max

max

2

2

6

62

H/2z surfaceat is strain axial Maximum

1

)0(support at is valueMaximum

)(1 :curvature of Radius

=σ

==ε

=

=ρ

=

−==ρ



Tabulated solutions> Solutions to simple situations available in introductory

mechanics books• Point loads, distributed loads, applied moments• Handout from Crandall, Dahl, and Lardner, An Introduction to

the Mechanics of Solids, 1999, p. 531.

> Linearity: you can superpose the solutions

> Can save a bit of time

> Solutions use nomenclature of singularity functions



Singularity functions

( )functiondelta a or impulse an called variably is what

0 if

0 if 0

1 =−

>−−=−

<−=−

−ax

axaxax

axaxnn

n

a bL

F

x ( ) xaFxM −−=

Integrate as if it were just functions of (x-a); evaluate at the end.

Value: a single expression describes what’s going on in different regions of the beam



Overconstraint> A cantilever’s single support provides the necessary support reactions

and no more

> A fixed-fixed beam has an additional support, so it is overconstrained

> Static indeterminacy: must consider deformations and reactions to determine state of the structure

> Many MEMS structures are statically indeterminate: flexures, optical MEMS, switches,…

> What this means for us• Failure modes and important operational effects: stress stiffening,

buckling• Your choice of how to calculate deflections



Example: center-loaded fixed-fixed beam

> Option 1: (general)• Start with beam equation in terms of q• Express load as a delta function• Integrate four times• Four B.C. give four constants

> Option 2: (not general)• Invoke symmetry

> Option 3: (general)• Pretend beam is a cantilever with as yet unknown moment and force

applied at end such that w(L) = slope(L) = 0• Using superposition, solve for deflection and slope everywhere• Impose B.C. to determine moment and force at end• Plug newly-determined moment and force into solution, and you’re

done

EIq

dxwd=4

4

F



Integration using singularity functions

12 −−= LxFq

( ) 43

22

31

3

32

21

2

211

2

2

10

3

3

14

4

2662

222

2

2

2

CxCxCxCLxEIFxw

CxCxCLxEIF

dxdw

CxCLxEIF

dxwd

CLxEIF

dxwd

LxEIF

EIq

dxwd

++++−

=

+++−

=

++−=

+−=

−== −

F

Use boundary conditions to find constants: no displacement at supports, slope = 0 at supports

L/2L/2



Comparing spring constants> Center-loaded fixed-fixed beam (same dimensions as previous)

> Tip-loaded cantilever beam, same dimensions

> Uniaxially loaded beam, same dimensions

FN/m 8.1216

3

3

==L

EWHk

FN/m 2.0

4 3

3

==L

EWHk

F N/m 8000==L

EWHk



Outline> Regroup





Torsion

Digital Light Processing is a trademark of Texas Instruments.

The treatment of torsion mirrors that of bending.

Digital Light Processing™ Technology: Texas Instruments

Images removed due to copyright restrictions.Figure 51 on p. 39 in: Hornbeck, Larry J. "From Cathode Rays to Digital Micromirrors: A History ofElectronic Projection Display Technology." Texas Instruments Technical Journal 15, no. 3 (July-September 1998): 7-46.

Images removed due to copyright restrictions.Figures 48 and 50 in: Hornbeck, Larry J. "From Cathode Rays to Digital Micromirrors: A History ofElectronic Projection Display Technology." Texas Instruments Technical Journal 15, no. 3 (July-September 1998): 7-46.



Bending of plates> A plate is a beam that is so

wide that the transverse strains are inhibited, both the Poisson contraction and its associated anticlastic curvature

> This leads to additional stiffness when trying to bend a plate

xx

xyy

y

yxx

E

E

E

ε⎟⎠⎞

⎜⎝⎛

ν−=σ

⇓

νσ−σ=ε=⇒

ε

νσ−σ=ε

ModulusPlate

21

0

zero be todconstraine is But



Plate in pure bending> Analogous to beam bending, with the limit on transverse strains

> Two radii of curvature along principal axes

> Stresses along principal axes2

2

2

2

1

1

yw

xw

y

x

∂∂

=

∂∂

=

ρ

ρ( )

( )xyyy

y

yxxx

x

Ez

Ez

υσσερ

ε

υσσερ

ε

−=−=

−=−=

1

1

( )

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

−−=

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−−=

xyy

yxx

Ez

Ez

ρν

ρνσ

ρν

ρνσ

11

11

2

2



Plate in pure bending> Relate moment per unit width of plate to curvature

> Treat x and y equivalently

> Note that stiffness comes from flexural rigidity as for a beam

( )

( )

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=

⎟⎟⎠

⎞⎜⎜⎝

⎛+−=

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−−=

=

∫

∫

−

2

3

2

2

22

1121 where

1

11

ν

ρν

ρ

ρν

ρν

σ

EHD

DWM

dzzEWM

dzzzWM

yx

x

H

Hyx

x

thicknessx

x

flexural rigidity



Plate in pure bending> Recall that M is two derivatives away from a distributed load,

and that

> This leads to the equation for small amplitude bending of a plate

> Often solve with polynomial solutions (simple cases) or eigenfunction expansions

2

21xw

x ∂∂

=ρ 2

21yw

y ∂∂

=ρ

( )yxPyw

yxw

xwD ,2 4

4

22

4

4

4

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

∂+

∂∂

distributed load



Where are we now?> We can handle small deflections of beams and plates

> Physics intervenes for large deflections and residual stress, and our solutions are no longer correct

> Now what do we do?• Residual stress: include it as an effective load• Large deflections: use Energy Methods

07 lecture 07 split

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