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StructuresCarol Livermore Massachusetts Institute of Technology

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

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

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 - 2

Recall: Isotropic Elasticity> For a general case of loading, the constitutive relationshipsbetween stress and elastic strain are as follows

> 6 equations, one for each normal stress and shear stress1 x = x ( y + z ) E 1 y = y ( z + x ) E 1 z = z ( x + y ) E

[

] ]

xy yz zx

[

[

]

1 = xy G 1 = yz G 1 = zx G

E Shear modulus G is given by G = 2(1 + )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 - 3

What we are considering today> Bending in the limit of small deflections > For axial loading, deflections are small until something badhappens

Nonlinearity, plastic deformation, cracking, buckling Strains typically of order 0.1% to 1% > For bending, small deflections are typically less than thethickness of the element (i.e. beam, plate) in question

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 - 4

What we are NOT considering today> Basically, anything that makes todays theory not apply (not aswell, or not at all)

> Large deflections Axial stretching becomes a noticeable effect > Residual stresses Can increase or decrease the ease of bending

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 - 5

Our trajectory> What are the loads and the supports?1 m 0.5 m Cantilever Pull-down electrode Anchor

Silicon

F

Image by MIT OpenCourseWare. Adapted from Rebeiz, Gabriel M. RF MEMS: Theory, Design, and Technology. Hoboken, NJ: John Wiley, 2003. ISBN: 9780471201694.

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

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

> Integrate to find deformed shapeCite 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 - 6

Outline

> Regroup > 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

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

Types of Loads> Three basic types of loads: Point force (an old friend, with its own specific point ofapplication)

Distributed loads (pressure) Concentrated moment (what you get from a screwdriver, witha specific point of application)

The forces and moments work together to make internalbending moments more on this shortly

F= point load

q = distributed load M= concentrated moment

C. Livermore: 6.777J/2.372J Spring 2007, Lecture 7 - 8

Types of supports

> Four basic boundary conditions: Fixed: cant translate at all, cant rotate Pinned: cant 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 conditionFree

Fixed

Pinned

Pinned on Rollers

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.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 - 9

Reaction Forces and Moments> Equilibrium requires that the total force on an object be zeroand that the total moment about any axis be zero

> This gives rise to reaction forces and moments > Cant translate means support can have reaction forces > Cant rotate means support can have reaction momentsPoint Load F x MR O L

Total moment about support : M T = M R FL Moment must be zero in equilibrium : M R = FL Net force must be zero in equilibrium : FR = F

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

C. Livermore: 6.777J/2.372J Spring 2007, Lecture 7 - 10

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

M (x ) = F ( L x) V (x ) = F

For this case,

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

C. Livermore: 6.777J/2.372J Spring 2007, Lecture 7 - 11

Some conventionsMomentsPositive Negative

Moments:

Image by MIT OpenCou rseWare. Adapted from Figure 9.8 in: Senturia, Stephen D. Microsystem Design. Bo ston, MA: Kluwer Academic Publishers, 2001, p. 210 . ISBN: 9780792372462.

ShearsPositive Negative

Shears:

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

C. Livermore: 6.777J/2.372J Spring 2007, Lecture 7 - 12

Combining all loads> A differential beam element, subjected to point loads,distributed loads and moments in equilibrium, must obey governing differential equationsAll Loads: q V

M

M+dM V+dV dx

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

FT = qdx + (V + dV ) V dV = q dx

M

T

= ( M + dM ) M (V + dV ) dx dM =V dx

qdx dx 2

C. Livermore: 6.777J/2.372J Spring 2007, Lecture 7 - 13

Outline

> Regroup > 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

C. Livermore: 6.777J/2.372J Spring 2007, Lecture 7 - 14