Structures. Carol Livermore Massachusetts Institute of Technology

Transcription

1 C. Livermore: 6.777J/.7J Spring 007, Lecture 7 - Structures Carol Livermore Massachusetts Institute of Technolog * With thanks to Steve Senturia, from whose lecture notes some of these materials are adapted. Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT

2 C. Livermore: 6.777J/.7J Spring 007, Lecture 7 - > Regroup Outline > Beam bending Loading and supports Bending moments and shear forces Curvature and the beam equation Eamples: cantilevers and doubl supported beams > A quick look at torsion and plates Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT

3 Recall: Isotropic Elasticit C. Livermore: 6.777J/.7J Spring 007, Lecture 7 - > or 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 ε ε ε z E E E [ σ ν ( σ + σ )] ( σ + σ ) [ σ ν ] [ σ ν ( σ + σ )] z z z γ γ γ z z τ G τ G τ G z z Shear modulus G is given b G E ( + ν) Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT

4 What we are considering toda C. Livermore: 6.777J/.7J Spring 007, Lecture 7-4 > Bending in the limit of small deflections > or aial loading, deflections are small until something bad happens Nonlinearit, plastic deformation, cracking, buckling Strains tpicall of order 0.% to % > or bending, small deflections are tpicall less than the thickness of the element (i.e. beam, plate) in question Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT

5 What we are NOT considering toda C. Livermore: 6.777J/.7J Spring 007, Lecture 7-5 > Basicall, anthing that makes toda s theor not appl (not as well, or not at all) > Large deflections Aial stretching becomes a noticeable effect > Residual stresses Can increase or decrease the ease of bending Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT

6 Our trajector C. Livermore: 6.777J/.7J Spring 007, Lecture 7-6 > What are the loads and the supports? µm Cantilever Silicon 0.5 µm Pull-down electrode Anchor Image b MIT OpenCourseWare. Adapted from Rebeiz, Gabriel M. R MEMS: Theor, Design, and Technolog. Hoboken, NJ: John Wile, 00. ISBN: > What is the bending moment at point along the beam? M > How much curvature does that bending moment create in the structure at? (Now ou have the beam equation.) > Integrate to find deformed shape Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT

7 C. Livermore: 6.777J/.7J Spring 007, Lecture 7-7 > Regroup Outline > Bending Loading and supports Bending moments and shear forces Curvature and the beam equation Eamples: cantilevers and doubl supported beams > A quick look at torsion and plates Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT

8 Tpes of Loads C. Livermore: 6.777J/.7J Spring 007, Lecture 7-8 > Three basic tpes of loads: Point force (an old friend, with its own specific point of application) Distributed loads (pressure) Concentrated moment (what ou get from a screwdriver, with a specific point of application) The forces and moments work together to make internal bending moments more on this shortl point load q distributed load M concentrated moment Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT

9 Tpes of supports C. Livermore: 6.777J/.7J Spring 007, Lecture 7-9 > our basic boundar conditions: ied: 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 ree: unconstrained boundar condition ied ree Pinned Pinned on Rollers Image b MIT OpenCourseWare. Adapted from igure 9.5 in: Senturia, Stephen D. Microsstem Design. Boston, MA: Kluwer Academic Publishers, 00, p. 07. ISBN: Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT

10 Reaction orces and Moments C. Livermore: 6.777J/.7J Spring 007, Lecture 7-0 > Equilibrium requires that the total force on an object be zero and that the total moment about an ais 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 Point Load M R O R Image b MIT OpenCourseWare. Adapted from igure 9.7 in: Senturia, Stephen D. Microsstem Design. Boston, MA: Kluwer Academic Publishers, 00, p. 09. ISBN: L Total moment about support : M T M R M R L Moment must be zero in equilibrium : L Net force must be zero in equilibrium : R Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT

11 Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT C. Livermore: 6.777J/.7J Spring 007, Lecture 7 - Internal forces and moments > Each segment of beam must also be in equilibrium > This leads to internal shear forces V() and bending moments M() or this case, M V ( ) ( ) ( L ) Images b MIT OpenCourseWare. Adapted from igure 9.7 in Senturia, Stephen D. Microsstem Design Boston, MA: Kluwer Academic Publishers, 00, p. 09. ISBN:

12 Some conventions C. Livermore: 6.777J/.7J Spring 007, Lecture 7 - Moments Moments: Positive Negative Image b MIT OpenCou rseware. Adapted from igure 9.8 in: Senturia, Stephen D. Microsstem Design. Boston, MA: Kluwer Academic Publishers, 00, p. 0. ISBN: Shears Shears: Positive Negative Image b MIT OpenCourseWare. Adapted from igure 9.8 in: Senturia, Stephen D. Microsstem Design. Boston, MA: Kluwer Academic Publishers, 00, p. 0. ISBN: Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT

13 C. Livermore: 6.777J/.7J Spring 007, Lecture 7 - Combining all loads > A differential beam element, subjected to point loads, distributed loads and moments in equilibrium, must obe governing differential equations All Loads: q M V M+dM d V+dV Image b MIT OpenCourseWare. Adapted from igure 9.8 in: Senturia, Stephen D. Microsstem Design. Boston, MA: Kluwer Academic Publishers, 00, p. 0. ISBN: T qd + ( V + dv ) V M T ( M + dm ) M ( V + dv ) d qd d dv d q dm d V Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT

14 C. Livermore: 6.777J/.7J Spring 007, Lecture 7-4 > Regroup Outline > Bending Loading and supports Bending moments and shear forces Curvature and the beam equation Eamples: cantilevers and doubl supported beams > A quick look at torsion and plates Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT

15 Pure bending C. Livermore: 6.777J/.7J Spring 007, Lecture 7-5 > Important concept: THE NEUTRAL AXIS > Aial stress varies with transverse position relative to the neutral ais M O Tension? z d? Neutral Ais Compression M O Image b MIT OpenCourseWare. Adapted from igures 9.9 and 9.0 in: Senturia, Stephen D. Microsstem Design. Boston, MA: Kluwer Academic Publishers, 00, pp.,. ISBN: H/ z Tension? Compression H/ dl ( z) dθ d dl ε σ dθ z d d z ze Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT

16 C. Livermore: 6.777J/.7J Spring 007, Lecture 7-6 Locating the neutral ais > In pure bending, locate the neutral ais b imposing equilibrium of aial forces N A σ thickness ( z) E da 0 ( z) W ( z) z dz 0 One material, rectangular beam EW z dz 0 thickness > The neutral ais is in the middle for a one material beam of smmetric cross-section. > Composite beams: if the beam just has a ver thin film on it, can approimate neutral ais unchanged > Composite beams: with films of comparable thickness, change in E biases the location of the neutral ais Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT

17 Curvature in pure bending C. Livermore: 6.777J/.7J Spring 007, Lecture 7-7 Curvature is related to the internal bending moment M. or a one - material beam, Internal Moment : E M z da M zσ da σ M A A ze E( z) z da Moment of inertia I : I A z A da EI M In pure bending, the internal moment M equals the eternall M applied moment M 0. Then 0 for one material; for two EI or more materials, calculate an effective EI. Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT

18 Useful case: rectangular beam C. Livermore: 6.777J/.7J Spring 007, Lecture 7-8 W M 0 M 0 H or a I uniform rectangular beam, Wz M H H / / dz WH WH E Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT

19 C. Livermore: 6.777J/.7J Spring 007, Lecture 7-9 Differential equation of beam bending > Relation between curvature and the applied load z Image b MIT OpenCourseWare. d? d ds ds dw d dθ d cosθ ds dθ?() Adapted from igure 9. in: Senturia, Stephen D. Microsstem Design. d tan θ θ d w d d w d and, b successive differentiation d w d 4 d w q d 4 d EI Boston, MA: Kluwer Academic Publishers, 00, pp. 4. ISBN: w() Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT M EI V EI or large - angle bending w [ + ( w ) ] / Large-angle bending is rare in MEMS structures

20 C. Livermore: 6.777J/.7J Spring 007, Lecture 7-0 Anticlastic curvature > If a beam is bent, then the Poisson effect causes opposite bending in the transverse direction /v ε ε z νε Original cross-section which creates Image b MIT OpenCourseWare. Adapted from igure 9. in: Senturia, Stephen D. Microsstem Design. Boston, MA: Kluwer Academic Publishers, 00, pp. 9. ISBN: ε νz a - directed internal moment Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT

21 Eample: Cantilever with point load C. Livermore: 6.777J/.7J Spring 007, Lecture 7 - d w d EI ( L ) dw d EI + L EI + A Image b MIT OpenCourseWare. Adapted from igure 9.7 in: Senturia, Stephen D. Microsstem Design. Boston, MA: Kluwer Academic Publishers, 00, p. 09. ISBN: M d w d M EI and ( ) ( V ( ) ( ) L ) w BC : 6EI w w(0) A + L EI L EI B 0, dw d L A + 0 B Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT

22 C. Livermore: 6.777J/.7J Spring 007, Lecture 7 - Spring Constant for Cantilever > Since force is applied at tip, if we find maimum tip displacement, the ratio of displacement to force is the spring constant. k w ma L EI EI L cantilever EWH 4L or the same dimensions as the uniaiall loaded beam, k cantilever 0. N/m Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT

23 C. Livermore: 6.777J/.7J Spring 007, Lecture 7 - Stress in the Bent Cantilever > To find bending stress, we find the radius of curvature, then use the pure-bending case to find stress d w Radius of curvature : ( L ) d EI Maimum value is at support ( 0) ma L EI Maimum aial strain is at surface LH 6L εma EI H WE 6L σma H W ( z H/) Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT

24 Tabulated solutions C. Livermore: 6.777J/.7J Spring 007, Lecture 7-4 > Solutions to simple situations available in introductor mechanics books Point loads, distributed loads, applied moments Handout from Crandall, Dahl, and Lardner, An Introduction to the Mechanics of Solids, 999, p. 5. > Linearit: ou can superpose the solutions > Can save a bit of time > Solutions use nomenclature of singularit functions Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT

25 Singularit functions C. Livermore: 6.777J/.7J Spring 007, Lecture 7-5 a n 0 if a < 0 a n ( a) n if a > 0 a what is variabl called an impulse or a delta function Integrate as if it were just functions of (-a); evaluate at the end. Value: a single epression describes what s going on in different regions of the beam M ( ) a a L b Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT

26 Overconstraint C. Livermore: 6.777J/.7J Spring 007, Lecture 7-6 > A cantilever s single support provides the necessar support reactions and no more > A fied-fied beam has an additional support, so it is overconstrained > Static indeterminac: must consider deformations and reactions to determine state of the structure > Man MEMS structures are staticall indeterminate: fleures, optical MEMS, switches, > What this means for us ailure modes and important operational effects: stress stiffening, buckling Your choice of how to calculate deflections Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT

27 C. Livermore: 6.777J/.7J Spring 007, Lecture 7-7 Eample: center-loaded fied-fied beam > Option : (general) Start with beam equation in terms of q Epress load as a delta function Integrate four times our B.C. give four constants > Option : (not general) Invoke smmetr 4 d w 4 d > Option : (general) Pretend beam is a cantilever with as et unknown moment and force applied at end such that w(l) slope(l) 0 Using superposition, solve for deflection and slope everwhere Impose B.C. to determine moment and force at end Plug newl-determined moment and force into solution, and ou re done q EI Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT

28 C. Livermore: 6.777J/.7J Spring 007, Lecture 7-8 Integration using singularit functions q L L/ L/ 4 d w 4 d d w d d w d dw d w q EI EI EI EI EI EI L L L L 0 ( ) C + C4 6 L + C + C + C C + C 6 + C + C C Use boundar conditions to find constants: no displacement at supports, slope 0 at supports Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT

29 Comparing spring constants C. Livermore: 6.777J/.7J Spring 007, Lecture 7-9 > Center-loaded fied-fied beam (same dimensions as previous) 6EWH L > Tip-loaded cantilever beam, same dimensions k.8 N/m > Uniaiall loaded beam, same dimensions k EWH 4L 0. N/m k EWH L 8000 N/m Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT

30 C. Livermore: 6.777J/.7J Spring 007, Lecture 7-0 > Regroup Outline > Bending Loading and supports Bending moments and shear forces Curvature and the beam equation Eamples: cantilevers and doubl supported beams > A quick look at torsion and plates Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT

31 Torsion C. Livermore: 6.777J/.7J Spring 007, Lecture 7 - The treatment of torsion mirrors that of bending. Images removed due to copright restrictions. igures 48 and 50 in: Hornbeck, Larr J. "rom Cathode Ras to Digital Micromirrors: A Histor of Electronic Projection Displa Technolog." Teas Instruments Technical Journal 5, no. (Jul-Sept ember 998): Images removed due to copright restrictions. igure 5 on p. 9 in: Hornbeck, Larr J. "rom Cathode Ras to Digital Micromirrors: A Histor of Electronic Projection Displa Technolog." Teas Instruments Technical Journal 5, no. (Jul-Sept ember 998): Digital Light Processing Technolog: Teas Instruments Digital Light Processing is a trademark of Teas Instruments. Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT

32 Bending of plates C. Livermore: 6.777J/.7J Spring 007, Lecture 7 - > 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 tring to bend a plate But ε σ νσ ε E is constrained to be zero 0 σ ε σ E ν Plate Modulus νσ E ε Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT

33 Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT C. Livermore: 6.777J/.7J Spring 007, Lecture 7 - Plate in pure bending > Analogous to beam bending, with the limit on transverse strains > Two radii of curvature along principal aes > Stresses along principal aes w w ( ) ( ) E z E z υσ σ ε ε υσ σ ε ε ( ) ( ) + + Ez Ez ν ν σ ν ν σ

34 C. Livermore: 6.777J/.7J Spring 007, Lecture 7-4 Plate in pure bending > Relate moment per unit width of plate to curvature > Treat and equivalentl M W M W M W where zσ thickness ( z) ( ν ) + ν D + D E dz EH ν ν H z H dz fleural rigidit > Note that stiffness comes from fleural rigidit as for a beam Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT

35 Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT C. Livermore: 6.777J/.7J Spring 007, Lecture 7-5 Plate in pure bending > Recall that M is two derivatives awa from a distributed load, and that > This leads to the equation for small amplitude bending of a plate > Often solve with polnomial solutions (simple cases) or eigenfunction epansions w w ( ) P w w w D, distributed load

36 Where are we now? C. Livermore: 6.777J/.7J Spring 007, Lecture 7-6 > We can handle small deflections of beams and plates > Phsics 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 Energ Methods Cite as: Carol Livermore, course materials for 6.777J /.7J Design and abrication of Microelectromechanical Devices, Spring 007. MIT