EE C245 ME C218 Introduction to MEMS Design
|
|
- Chastity Nash
- 6 years ago
- Views:
Transcription
1 EE C245 ME C218 Introduction to MEMS Design Fall 2007 Prof. Clark T.-C. Nguyen Dept. of Electrical Engineering & Computer Sciences University of California at Berkeley Berkeley, CA Lecture 16: Energy Methods I EE C245: Introduction to MEMS Design Lecture 16 C. Nguyen 10/21/08 1
2 Lecture Outline Reading: Senturia, Chpt. 9, 10 Lecture Topics: Bending of beams Cantilever beam under small deflections Combining cantilevers in series and parallel Folded suspensions Design implications of residual stress and stress gradients Energy Methods Virtual Work Energy Formulations Tapered Beam Example Doubly Clamped Beam Example Large Deflection Analysis EE C245: Introduction to MEMS Design Lecture 16 C. Nguyen 10/21/08 2
3 Deflection of Folded Flexures This equivalent to two cantilevers of length L c /2 Composite cantilever free ends attach here Half of F absorbed in other half (symmetrical) 4 sets of these pairs, each of which gets ¼ of the total force F EE C245: Introduction to MEMS Design Lecture 16 C. Nguyen 10/21/08 3
4 Constituent Cantilever Spring Constant From our previous analysis: x( y) 2 Fc Lc y Fc y y L EI z L = 2 1 = c 6EI z 2 ( y ) From which the spring constant is: Fc k c = = x L ) Inserting L c = L/2 c 3EI 3 ( c Lc z k 3EI z c = = 3 ( L / 2) 24EI L 3 z EE C245: Introduction to MEMS Design Lecture 16 C. Nguyen 10/21/08 4
5 Overall Spring Constant Rigid Truss Four pairs of clamped-guided beams In each pair, beams bend in series (Assume trusses are inflexible) ibl Force is shared by each pair F pair = F/4 Leg L F pair EE C245: Introduction to MEMS Design Lecture 16 C. Nguyen 10/21/08 5
6 Folded-Beam Stiffness Ratios Folded-beam suspension Anchor Shuttle Folding truss In the x-direction: k x = 24EI 3 L In the z-direction: Same flexure and boundary conditions k z = In the y-direction: [See Senturia, 9.2] Thus: k y k x = L 4 W 24EI 3 L z x 8EWh k y = L 2 Much stiffer in y-direction! EE C245: Introduction to MEMS Design Lecture 16 C. Nguyen 10/21/08 6
7 Folded-Beam Suspensions Permeate MEMS Accelerometer [ADXL-05, Analog Devices] Gyroscope [Draper Labs.] Micromechanical Filter [K. Wang, Univ. of Michigan] EE C245: Introduction to MEMS Design Lecture 16 C. Nguyen 10/21/08 7
8 Folded-Beam Suspensions Permeate MEMS Below: Micro-Oven Controlled Folded-Beam Resonator EE C245: Introduction to MEMS Design Lecture 16 C. Nguyen 10/21/08 8
9 Stressed Folded-Flexures Flexures EE C245: Introduction to MEMS Design Lecture 16 C. Nguyen 10/21/08 9
10 Clamped-Guided Beam Under Axial Load Important case for MEMS suspensions, since the thin films comprising them are often under residual stress Consider small deflection case: y(x) «L z y x W L Governing differential equation: (Euler Beam Equation) Axial Load Unit x=l EE C245: Introduction to MEMS Design Lecture 16 C. Nguyen 10/21/08 10
11 The Euler Beam Equation Axial Stress R Axial stresses produce no net horizontal force; but as soon as the beam is bent, there is a net downward force For equilibrium, must postulate some kind of upward load on the beam to counteract the axial stress-derived force For ease of analysis, assume the beam is bent to angle π EE C245: Introduction to MEMS Design Lecture 16 C. Nguyen 10/21/08 11
12 The Euler Beam Equation Note: Use of the full bend angle of π to establish conditions for load balance; but this returns us to case of small displacements and small angles EE C245: Introduction to MEMS Design Lecture 16 C. Nguyen 10/21/08 12
13 Clamped-Guided Beam Under Axial Load Important case for MEMS suspensions, since the thin films comprising them are often under residual stress Consider small deflection case: y(x) «L z y x W L Governing differential equation: (Euler Beam Equation) Axial Load Unit x=l EE C245: Introduction to MEMS Design Lecture 16 C. Nguyen 10/21/08 13
14 Solving the ODE Can solve the ODE using standard methods Senturia, pp : solves ODE for case of point load on a clamped-clamped l d beam (which h defines B.C. s) For solution to the clamped-guided case: see S. Timoshenko, Strength of Materials II: Advanced Theory and Problems, McGraw-Hill, New York, 3 rd Ed., 1955 Result from Timoshenko: EE C245: Introduction to MEMS Design Lecture 16 C. Nguyen 10/21/08 14
15 Design Implications Straight flexures Large tensile S means flexure behaves like a tensioned wire (for which h k -1 = L/S) Large compressive S can lead to buckling (k -1 ) Folded flexures Residual stress only ypartially released Length from truss to shuttle s s centerline differs by L s for inner and outer legs EE C245: Introduction to MEMS Design Lecture 16 C. Nguyen 10/21/08 15
16 Effect on Spring Constant Residual compression on outer legs with same magnitude of tension on inner legs: ε b = ± ε r L s L = Eε r Beam Strain: ; Stress Force: ± Wh Spring constant becomes: S Ls L Remedies: Reduce the shoulder width L s to minimize stress in legs Compliance in the truss lowers the axial compression and tension and reduces its effect on the spring constant EE C245: Introduction to MEMS Design Lecture 16 C. Nguyen 10/21/08 16
17 Energy Methods EE C245: Introduction to MEMS Design Lecture 16 C. Nguyen 10/21/08 17
18 More General Geometries Euler-Bernoulli beam theory works well for simple geometries But how can we handle more complicated ones? Example: tapered cantilever beam Objective: Find an expression for displacement as a function of location x under a point load F applied at the tip of the free end of a cantilever with tapered width W(x) W 50% taper y x EE C245: Introduction to MEMS Design Lecture 16 C. Nguyen 10/21/08 18
19 Solution: Use Principle of Virtual Work In an energy-conserving system (i.e., elastic materials), the energy stored in a body due to the quasi-static (i.e., slow) action of surface and body forces is equal to the work done by these forces Implication: if we can formulate stored energy as a function of the deformation of a mechanical object, then we can determine how an object responds to a force by determining the shape the object must take in order to minimize the difference U between the stored energy and the work done by the forces: U = Stored Energy - Work Done Key idea: we don t have to reach U = 0 to produce a very useful, approximate analytical result for load-deflection EE C245: Introduction to MEMS Design Lecture 16 C. Nguyen 10/21/08 19
20 More Visual Description EE C245: Introduction to MEMS Design Lecture 16 C. Nguyen 10/21/08 20
21 Fundamentals: Energy Density Strain energy density: [J/m 3 ] To find work done in straining material Total strain energy [J]: Integrate over all strains (normal and shear) EE C245: Introduction to MEMS Design Lecture 16 C. Nguyen 10/21/08 21
22 Bending Energy Density Neutral Axis y y(x) = transverse displacement of neutral axis x First, find the bending energy dw bend in an infinitesimal length dx: EE C245: Introduction to MEMS Design Lecture 16 C. Nguyen 10/21/08 22
23 Energy Due to Axial Load y x Strain due to axial load S contributes an energy dw stretch in length dx, since lengthening of the different element dx (to ds) results in a strain ε x EE C245: Introduction to MEMS Design Lecture 16 C. Nguyen 10/21/08 23
24 Shear Strain Energy Shear Modulus See W.C. Albert, Vibrating Quartz Crystal Beam Accelerometer, Proc. ISA Int. Instrumentation Symp., May 1982, pp EE C245: Introduction to MEMS Design Lecture 16 C. Nguyen 10/21/08 24
25 Applying the Principle of Virtual Work Basic Procedure: Guess the form of the beam deflection under the applied loads Vary the parameters in the beam deflection function in order to minimize: Sum strain energies Assumes point load U = W j j i F i u i Displacement at point load Find minima by simply setting derivatives to zero See Senturia, pg. 244, for a general expression with distrubuted surface loads and body forces EE C245: Introduction to MEMS Design Lecture 16 C. Nguyen 10/21/08 25
26 Example: Tapered Cantilever Beam Objective: Find an expression for displacement as a function of location x under a point load F applied at the tip of the free end of a cantilever with tapered width W(x) W 50% taper y x Adjustable parameters: minimize U y ( x) = c x + c x Start by guessing the solution 2 3 It should satisfy the boundary conditions The strain energy integrals shouldn t be too tedious This might not matter much these days, though, h since one could just use matlab or mathematica EE C245: Introduction to MEMS Design Lecture 16 C. Nguyen 10/21/
27 Strain Energy And Work By F (Bending Energy) (Using our guess) Tip Deflection EWh 3 EE C245: Introduction to MEMS Design Lecture 16 C. Nguyen 10/21/08 27
28 Find c 2 and c 3 That Minimize U Minimize U basically, find the c 2 and c 3 that brings U closest to zero (which is what it would be if we had guessed correctly) The c 2 and c 3 that minimize U are the ones for which the partial derivatives of U with respective to them are zero: Proceed: First, evaluate the integral to get an expression for U: EE C245: Introduction to MEMS Design Lecture 16 C. Nguyen 10/21/08 28
29 Minimize U (cont) Evaluate the derivatives and set to zero: Solve the simultaneous equations to get c 2 and c 3 : EE C245: Introduction to MEMS Design Lecture 16 C. Nguyen 10/21/08 29
30 The Virtual Work-Derived Solution And the solution: Solve for tip deflection and obtain the spring constant: Compare with previous solution for constant-width cantilever beam (using Euler theory): 13% smaller than tapered-width case EE C245: Introduction to MEMS Design Lecture 16 C. Nguyen 10/21/08 30
31 Comparison With Finite Element Simulation Below: ANSYS finite element model with L = 500 μm W base = 20 μm E = 170 GPa h = 2 μm W tip = 10 μm Result: (from static analysis) k = μn/m This matches the result from energy minimization to 3 significant figures EE C245: Introduction to MEMS Design Lecture 16 C. Nguyen 10/21/08 31
32 Need a Better Approximation? Add more terms to the polynomial Add other strain energy terms: Shear: more significant as the beam gets shorter Axial: more significant as deflections become larger Both of the above remedies make the math more complex, so encourage the use of math software, such as Mathematica, Matlab, or Maple Finite element analysis is really just energy minimization If this is the case, then why ever use energy minimization analytically (i.e., by hand)? Analytical expressions, even approximate ones, give insight into parameter dependencies that FEA cannot Can compare the importance of different terms Should use in tandem with FEA for design EE C245: Introduction to MEMS Design Lecture 16 C. Nguyen 10/21/08 32
EE C245 ME C218 Introduction to MEMS Design Fall 2010
EE C245 ME C218 Introduction to MEMS Design Fall 2010 Prof. Clark T.-C. Nguyen Dept. of Electrical Engineering & Computer Sciences University of California at Berkeley Berkeley, CA 94720 Lecture EE C245:
More informationEE C245 ME C218 Introduction to MEMS Design
EE C245 ME C218 Introduction to MEMS Design Fall 2007 Prof. Clark T.-C. Nguyen Dept. of Electrical Engineering & Computer Sciences University of California at Berkeley Berkeley, CA 94720 ecture 15: Beam
More informationEE C245 ME C218 Introduction to MEMS Design Fall 2007
EE C245 ME C218 Introduction to MEMS Design Fall 2007 Prof. Clark T.-C. Nguyen Dept. of Electrical Engineering & Computer Sciences University of California at Berkeley Berkeley, CA 94720 Lecture 17: Energy
More informationEE C245 ME C218 Introduction to MEMS Design Fall 2007
EE C245 ME C218 Introduction to MEMS Design Fall 2007 Prof. Clark T.-C. Nguyen Dept. of Electrical Engineering & Computer Sciences University of California at Berkeley Berkeley, CA 94720 Lecture 15: Beam
More informationEE C245 ME C218 Introduction to MEMS Design
EE C245 ME C218 Introduction to MEMS Design Fall 2007 Prof. Clark T.-C. Nguyen Dept. of Electrical Engineering & Computer Sciences University of California at Berkeley Berkeley, CA 94720 Lecture 19: Resonance
More informationEE C245 ME C218 Introduction to MEMS Design Fall 2012
EE C245 ME C218 Introduction to MEMS Design Fall 2012 Prof. Clark T.-C. Nguyen Dept. of Electrical Engineering & Computer Sciences University of California at Berkeley Berkeley, CA 94720 Lecture EE C245:
More informationEE C245 - ME C218 Introduction to MEMS Design Fall Today s Lecture
EE C45 - ME C8 Introduction to MEMS Design Fall 3 Roger Howe and Thara Srinivasan Lecture 9 Energy Methods II Today s Lecture Mechanical structures under driven harmonic motion develop analytical techniques
More informationLecture 15 Strain and stress in beams
Spring, 2019 ME 323 Mechanics of Materials Lecture 15 Strain and stress in beams Reading assignment: 6.1 6.2 News: Instructor: Prof. Marcial Gonzalez Last modified: 1/6/19 9:42:38 PM Beam theory (@ ME
More informationEE C247B / ME C218 INTRODUCTION TO MEMS DESIGN SPRING 2014 C. Nguyen PROBLEM SET #4
Issued: Wednesday, Mar. 5, 2014 PROBLEM SET #4 Due (at 9 a.m.): Tuesday Mar. 18, 2014, in the EE C247B HW box near 125 Cory. 1. Suppose you would like to fabricate the suspended cross beam structure below
More informationModule 4 : Deflection of Structures Lecture 4 : Strain Energy Method
Module 4 : Deflection of Structures Lecture 4 : Strain Energy Method Objectives In this course you will learn the following Deflection by strain energy method. Evaluation of strain energy in member under
More informationEE C247B ME C218 Introduction to MEMS Design Spring 2017
247B/M 28: Introduction to MMS Design Lecture 0m2: Mechanics of Materials CTN 2/6/7 Outline C247B M C28 Introduction to MMS Design Spring 207 Prof. Clark T.- Reading: Senturia, Chpt. 8 Lecture Topics:
More informationChapter 5 Elastic Strain, Deflection, and Stability 1. Elastic Stress-Strain Relationship
Chapter 5 Elastic Strain, Deflection, and Stability Elastic Stress-Strain Relationship A stress in the x-direction causes a strain in the x-direction by σ x also causes a strain in the y-direction & z-direction
More informationMechanical Design in Optical Engineering
OPTI Buckling Buckling and Stability: As we learned in the previous lectures, structures may fail in a variety of ways, depending on the materials, load and support conditions. We had two primary concerns:
More informationChapter 5 Structural Elements: The truss & beam elements
Institute of Structural Engineering Page 1 Chapter 5 Structural Elements: The truss & beam elements Institute of Structural Engineering Page 2 Chapter Goals Learn how to formulate the Finite Element Equations
More informationComb resonator design (2)
Lecture 6: Comb resonator design () -Intro Intro. to Mechanics of Materials School of Electrical l Engineering i and Computer Science, Seoul National University Nano/Micro Systems & Controls Laboratory
More informationEE C247B / ME C218 INTRODUCTION TO MEMS DESIGN SPRING 2016 C. NGUYEN PROBLEM SET #4
Issued: Wednesday, March 4, 2016 PROBLEM SET #4 Due: Monday, March 14, 2016, 8:00 a.m. in the EE C247B homework box near 125 Cory. 1. This problem considers bending of a simple cantilever and several methods
More informationEE C245 ME C218 Introduction to MEMS Design Fall 2007
EE C245 ME C218 Introduction to MEMS Design Fall 2007 Prof. Clark T.-C. Nguyen Dept. of Electrical Engineering & Computer Sciences University of California at Berkeley Berkeley, CA 94720 Lecture 13: Material
More informationEE C245 - ME C218. Fall 2003
EE C45 - E C8 Introduction to ES Design Fall Roger Howe and Thara Srinivasan ecture 9 Energy ethods II Today s ecture echanical structures under driven harmonic motion develop analytical techniques for
More informationMechanics of Microstructures
Mechanics of Microstructures Topics Plane Stress in MEMS Thin film Residual Stress Effects of Residual Stress Reference: Stephen D. Senturia, Microsystem Design, Kluwer Academic Publishers, January 200.
More informationChapter 2: Deflections of Structures
Chapter 2: Deflections of Structures Fig. 4.1. (Fig. 2.1.) ASTU, Dept. of C Eng., Prepared by: Melkamu E. Page 1 (2.1) (4.1) (2.2) Fig.4.2 Fig.2.2 ASTU, Dept. of C Eng., Prepared by: Melkamu E. Page 2
More informationStructural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian
Structural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian ahmadian@iust.ac.ir Dynamic Response of MDOF Systems: Mode-Superposition Method Mode-Superposition Method:
More information2 marks Questions and Answers
1. Define the term strain energy. A: Strain Energy of the elastic body is defined as the internal work done by the external load in deforming or straining the body. 2. Define the terms: Resilience and
More informationMarch 24, Chapter 4. Deflection and Stiffness. Dr. Mohammad Suliman Abuhaiba, PE
Chapter 4 Deflection and Stiffness 1 2 Chapter Outline Spring Rates Tension, Compression, and Torsion Deflection Due to Bending Beam Deflection Methods Beam Deflections by Superposition Strain Energy Castigliano
More informationEE C245 ME C218 Introduction to MEMS Design Fall 2011
EE C245 ME C218 Introduction to MEMS Design Fall 2011 Prof. Clark T.-C. Nguyen Dept. of Electrical Engineering & Computer Sciences University of California at Berkeley Berkeley, CA 94720 Lecture EE C245:
More informationLecture 19. Measurement of Solid-Mechanical Quantities (Chapter 8) Measuring Strain Measuring Displacement Measuring Linear Velocity
MECH 373 Instrumentation and Measurements Lecture 19 Measurement of Solid-Mechanical Quantities (Chapter 8) Measuring Strain Measuring Displacement Measuring Linear Velocity Measuring Accepleration and
More informationCHAPTER THREE SYMMETRIC BENDING OF CIRCLE PLATES
CHAPTER THREE SYMMETRIC BENDING OF CIRCLE PLATES * Governing equations in beam and plate bending ** Solution by superposition 1.1 From Beam Bending to Plate Bending 1.2 Governing Equations For Symmetric
More informationSTATICALLY INDETERMINATE STRUCTURES
STATICALLY INDETERMINATE STRUCTURES INTRODUCTION Generally the trusses are supported on (i) a hinged support and (ii) a roller support. The reaction components of a hinged support are two (in horizontal
More informationComb Resonator Design (2)
Lecture 6: Comb Resonator Design () -Intro. to Mechanics of Materials Sh School of felectrical ti lengineering i and dcomputer Science, Si Seoul National University Nano/Micro Systems & Controls Laboratory
More informationDue Monday, September 14 th, 12:00 midnight
Due Monday, September 14 th, 1: midnight This homework is considering the analysis of plane and space (3D) trusses as discussed in class. A list of MatLab programs that were discussed in class is provided
More informationM.S Comprehensive Examination Analysis
UNIVERSITY OF CALIFORNIA, BERKELEY Spring Semester 2014 Dept. of Civil and Environmental Engineering Structural Engineering, Mechanics and Materials Name:......................................... M.S Comprehensive
More informationEE C245 ME C218 Introduction to MEMS Design
EE C245 ME C218 Introduction to MEMS Design Fall 2007 Prof. Clark T.-C. Nguyen Dept. of Electrical Engineering & Computer Sciences University of California at Berkeley Berkeley, CA 94720 Lecture 12: Mechanical
More informationMEMS Report for Lab #3. Use of Strain Gages to Determine the Strain in Cantilever Beams
MEMS 1041 Report for Lab #3 Use of Strain Gages to Determine the Strain in Cantilever Beams Date: February 9, 2016 Lab Instructor: Robert Carey Submitted by: Derek Nichols Objective: The objective of this
More informationENG2000 Chapter 7 Beams. ENG2000: R.I. Hornsey Beam: 1
ENG2000 Chapter 7 Beams ENG2000: R.I. Hornsey Beam: 1 Overview In this chapter, we consider the stresses and moments present in loaded beams shear stress and bending moment diagrams We will also look at
More informationMechanics of Materials II. Chapter III. A review of the fundamental formulation of stress, strain, and deflection
Mechanics of Materials II Chapter III A review of the fundamental formulation of stress, strain, and deflection Outline Introduction Assumtions and limitations Axial loading Torsion of circular shafts
More informationEE C245 ME C218 Introduction to MEMS Design
EE C245 ME C218 Introduction to MEMS Design Fall 2007 Prof. Clark T.-C. Nguyen Dept. of Electrical Engineering & Computer Sciences University of California at Berkeley Berkeley, CA 94720 Lecture 23: Electrical
More informationFINAL EXAMINATION. (CE130-2 Mechanics of Materials)
UNIVERSITY OF CLIFORNI, ERKELEY FLL SEMESTER 001 FINL EXMINTION (CE130- Mechanics of Materials) Problem 1: (15 points) pinned -bar structure is shown in Figure 1. There is an external force, W = 5000N,
More informationMechanics in Energy Resources Engineering - Chapter 5 Stresses in Beams (Basic topics)
Week 7, 14 March Mechanics in Energy Resources Engineering - Chapter 5 Stresses in Beams (Basic topics) Ki-Bok Min, PhD Assistant Professor Energy Resources Engineering i Seoul National University Shear
More informationCHAPTER 5. Beam Theory
CHPTER 5. Beam Theory SangJoon Shin School of Mechanical and erospace Engineering Seoul National University ctive eroelasticity and Rotorcraft Lab. 5. The Euler-Bernoulli assumptions One of its dimensions
More informationCHAPTER 4 DESIGN AND ANALYSIS OF CANTILEVER BEAM ELECTROSTATIC ACTUATORS
61 CHAPTER 4 DESIGN AND ANALYSIS OF CANTILEVER BEAM ELECTROSTATIC ACTUATORS 4.1 INTRODUCTION The analysis of cantilever beams of small dimensions taking into the effect of fringing fields is studied and
More informationLecture 8: Flexibility Method. Example
ecture 8: lexibility Method Example The plane frame shown at the left has fixed supports at A and C. The frame is acted upon by the vertical load P as shown. In the analysis account for both flexural and
More informationEE C245 ME C218 Introduction to MEMS Design
EE C245 ME C218 Introduction to MEMS Design Fall 2007 Prof. Clark T.-C. Nguyen Dept. of Electrical Engineering & Computer Sciences University of California at Berkeley Berkeley, CA 94720 Lecture 22: Capacitive
More informationPresented By: EAS 6939 Aerospace Structural Composites
A Beam Theory for Laminated Composites and Application to Torsion Problems Dr. BhavaniV. Sankar Presented By: Sameer Luthra EAS 6939 Aerospace Structural Composites 1 Introduction Composite beams have
More informationD : SOLID MECHANICS. Q. 1 Q. 9 carry one mark each. Q.1 Find the force (in kn) in the member BH of the truss shown.
D : SOLID MECHANICS Q. 1 Q. 9 carry one mark each. Q.1 Find the force (in kn) in the member BH of the truss shown. Q.2 Consider the forces of magnitude F acting on the sides of the regular hexagon having
More informationAdvanced Vibrations. Distributed-Parameter Systems: Exact Solutions (Lecture 10) By: H. Ahmadian
Advanced Vibrations Distributed-Parameter Systems: Exact Solutions (Lecture 10) By: H. Ahmadian ahmadian@iust.ac.ir Distributed-Parameter Systems: Exact Solutions Relation between Discrete and Distributed
More informationReview of Strain Energy Methods and Introduction to Stiffness Matrix Methods of Structural Analysis
uke University epartment of Civil and Environmental Engineering CEE 42L. Matrix Structural Analysis Henri P. Gavin Fall, 22 Review of Strain Energy Methods and Introduction to Stiffness Matrix Methods
More informationStructures. Carol Livermore Massachusetts Institute of Technology
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.
More informationAnalysis III for D-BAUG, Fall 2017 Lecture 11
Analysis III for D-BAUG, Fall 2017 Lecture 11 Lecturer: Alex Sisto (sisto@math.ethz.ch) 1 Beams (Balken) Beams are basic mechanical systems that appear over and over in civil engineering. We consider a
More informationSymmetric Bending of Beams
Symmetric Bending of Beams beam is any long structural member on which loads act perpendicular to the longitudinal axis. Learning objectives Understand the theory, its limitations and its applications
More informationLecture Slides. Chapter 4. Deflection and Stiffness. The McGraw-Hill Companies 2012
Lecture Slides Chapter 4 Deflection and Stiffness The McGraw-Hill Companies 2012 Chapter Outline Force vs Deflection Elasticity property of a material that enables it to regain its original configuration
More informationFinite Element Analysis of Piezoelectric Cantilever
Finite Element Analysis of Piezoelectric Cantilever Nitin N More Department of Mechanical Engineering K.L.E S College of Engineering and Technology, Belgaum, Karnataka, India. Abstract- Energy (or power)
More informationBasic Energy Principles in Stiffness Analysis
Basic Energy Principles in Stiffness Analysis Stress-Strain Relations The application of any theory requires knowledge of the physical properties of the material(s) comprising the structure. We are limiting
More informationChapter 4 Deflection and Stiffness
Chapter 4 Deflection and Stiffness Asst. Prof. Dr. Supakit Rooppakhun Chapter Outline Deflection and Stiffness 4-1 Spring Rates 4-2 Tension, Compression, and Torsion 4-3 Deflection Due to Bending 4-4 Beam
More informationEE C247B ME C218 Introduction to MEMS Design Spring 2016
EE C47B ME C18 Introduction to MEMS Design Spring 016 Prof. Clark T.-C. Nguyen Dept. of Electrical Engineering & Computer Sciences University of California at Berkeley Berkeley, CA 9470 Lecture EE C45:
More informationMechanics of Inflatable Fabric Beams
Copyright c 2008 ICCES ICCES, vol.5, no.2, pp.93-98 Mechanics of Inflatable Fabric Beams C. Wielgosz 1,J.C.Thomas 1,A.LeVan 1 Summary In this paper we present a summary of the behaviour of inflatable fabric
More informationPh.D. Preliminary Examination Analysis
UNIVERSITY OF CALIFORNIA, BERKELEY Spring Semester 2014 Dept. of Civil and Environmental Engineering Structural Engineering, Mechanics and Materials Name:......................................... Ph.D.
More informationε t increases from the compressioncontrolled Figure 9.15: Adjusted interaction diagram
CHAPTER NINE COLUMNS 4 b. The modified axial strength in compression is reduced to account for accidental eccentricity. The magnitude of axial force evaluated in step (a) is multiplied by 0.80 in case
More informationSlender Structures Load carrying principles
Slender Structures Load carrying principles Basic cases: Extension, Shear, Torsion, Cable Bending (Euler) v017-1 Hans Welleman 1 Content (preliminary schedule) Basic cases Extension, shear, torsion, cable
More informationLearning Objectives By the end of this section, the student should be able to:
Mechanics of Materials stress and strain flexural l beam bending analysis under simple loading conditions EECE 300 011 1 Learning Objectives By the end of this section, the student should be able to: describe
More informationCorrection of local-linear elasticity for nonlocal residuals: Application to Euler-Bernoulli beams
Correction of local-linear elasticity for nonlocal residuals: Application to Euler-Bernoulli beams Mohamed Shaat* Engineering and Manufacturing Technologies Department, DACC, New Mexico State University,
More informationLecture M1 Slender (one dimensional) Structures Reading: Crandall, Dahl and Lardner 3.1, 7.2
Lecture M1 Slender (one dimensional) Structures Reading: Crandall, Dahl and Lardner 3.1, 7.2 This semester we are going to utilize the principles we learnt last semester (i.e the 3 great principles and
More informationNatural vibration frequency of classic MEMS structures
Natural vibration frequency of classic MEMS structures Zacarias E. Fabrim PGCIMAT, UFRGS, Porto Alegre, RS, Brazil Wang Chong, Manoel Martín Pérez Reimbold DeTec, UNIJUI, Ijuí, RS, Brazil Abstract This
More informationNow we are going to use our free body analysis to look at Beam Bending (W3L1) Problems 17, F2002Q1, F2003Q1c
Now we are going to use our free body analysis to look at Beam Bending (WL1) Problems 17, F00Q1, F00Q1c One of the most useful applications of the free body analysis method is to be able to derive equations
More informationD : SOLID MECHANICS. Q. 1 Q. 9 carry one mark each.
GTE 2016 Q. 1 Q. 9 carry one mark each. D : SOLID MECHNICS Q.1 single degree of freedom vibrating system has mass of 5 kg, stiffness of 500 N/m and damping coefficient of 100 N-s/m. To make the system
More informationSeismic Pushover Analysis Using AASHTO Guide Specifications for LRFD Seismic Bridge Design
Seismic Pushover Analysis Using AASHTO Guide Specifications for LRFD Seismic Bridge Design Elmer E. Marx, Alaska Department of Transportation and Public Facilities Michael Keever, California Department
More informationExternal Pressure... Thermal Expansion in un-restrained pipeline... The critical (buckling) pressure is calculated as follows:
External Pressure... The critical (buckling) pressure is calculated as follows: P C = E. t s ³ / 4 (1 - ν ha.ν ah ) R E ³ P C = Critical buckling pressure, kn/m² E = Hoop modulus in flexure, kn/m² t s
More informationEE C245 ME C218 Introduction to MEMS Design
EE C245 ME C218 Introduction to MEMS Design Fall 2007 Prof. Clark T.-C. Nguyen Dept. of Electrical Engineering & Computer Sciences University of California at Berkeley Berkeley, CA 94720 Lecture 20: Equivalent
More informationME Final Exam. PROBLEM NO. 4 Part A (2 points max.) M (x) y. z (neutral axis) beam cross-sec+on. 20 kip ft. 0.2 ft. 10 ft. 0.1 ft.
ME 323 - Final Exam Name December 15, 2015 Instructor (circle) PROEM NO. 4 Part A (2 points max.) Krousgrill 11:30AM-12:20PM Ghosh 2:30-3:20PM Gonzalez 12:30-1:20PM Zhao 4:30-5:20PM M (x) y 20 kip ft 0.2
More informationConsider an elastic spring as shown in the Fig.2.4. When the spring is slowly
.3 Strain Energy Consider an elastic spring as shown in the Fig..4. When the spring is slowly pulled, it deflects by a small amount u 1. When the load is removed from the spring, it goes back to the original
More informationApplication of Finite Element Method to Create Animated Simulation of Beam Analysis for the Course of Mechanics of Materials
International Conference on Engineering Education and Research "Progress Through Partnership" 4 VSB-TUO, Ostrava, ISSN 156-35 Application of Finite Element Method to Create Animated Simulation of Beam
More informationDiscontinuous Distributions in Mechanics of Materials
Discontinuous Distributions in Mechanics of Materials J.E. Akin, Rice University 1. Introduction The study of the mechanics of materials continues to change slowly. The student needs to learn about software
More informationME 475 Modal Analysis of a Tapered Beam
ME 475 Modal Analysis of a Tapered Beam Objectives: 1. To find the natural frequencies and mode shapes of a tapered beam using FEA.. To compare the FE solution to analytical solutions of the vibratory
More informationM5 Simple Beam Theory (continued)
M5 Simple Beam Theory (continued) Reading: Crandall, Dahl and Lardner 7.-7.6 In the previous lecture we had reached the point of obtaining 5 equations, 5 unknowns by application of equations of elasticity
More informationModeling and simulation of multiport RF switch
Journal of Physics: Conference Series Modeling and simulation of multiport RF switch To cite this article: J Vijay et al 006 J. Phys.: Conf. Ser. 4 715 View the article online for updates and enhancements.
More informationDecember 1999 FINAL TECHNICAL REPORT 1 Mar Mar 98
REPORT DOCUMENTATION PAGE AFRL-SR- BL_TR " Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instruct the collection
More informationEMA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 2 Stress & Strain - Axial Loading
MA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 2 Stress & Strain - Axial Loading MA 3702 Mechanics & Materials Science Zhe Cheng (2018) 2 Stress & Strain - Axial Loading Statics
More informationPart D: Frames and Plates
Part D: Frames and Plates Plane Frames and Thin Plates A Beam with General Boundary Conditions The Stiffness Method Thin Plates Initial Imperfections The Ritz and Finite Element Approaches A Beam with
More informationLaboratory 4 Topic: Buckling
Laboratory 4 Topic: Buckling Objectives: To record the load-deflection response of a clamped-clamped column. To identify, from the recorded response, the collapse load of the column. Introduction: Buckling
More informationLATERAL STABILITY OF BEAMS WITH ELASTIC END RESTRAINTS
LATERAL STABILITY OF BEAMS WITH ELASTIC END RESTRAINTS By John J. Zahn, 1 M. ASCE ABSTRACT: In the analysis of the lateral buckling of simply supported beams, the ends are assumed to be rigidly restrained
More informationNumerical and experimental analysis of a cantilever beam: A laboratory project to introduce geometric nonlinearity in Mechanics of Materials
Numerical and experimental analysis of a cantilever beam: A laboratory project to introduce geometric nonlinearity in Mechanics of Materials Tarsicio Beléndez (1) and Augusto Beléndez (2) (1) Departamento
More informationAvailable online at ScienceDirect. Procedia IUTAM 13 (2015 ) 82 89
Available online at www.sciencedirect.com ScienceDirect Procedia IUTAM 13 (215 ) 82 89 IUTAM Symposium on Dynamical Analysis of Multibody Systems with Design Uncertainties The importance of imperfections
More informationThe University of Melbourne Engineering Mechanics
The University of Melbourne 436-291 Engineering Mechanics Tutorial Four Poisson s Ratio and Axial Loading Part A (Introductory) 1. (Problem 9-22 from Hibbeler - Statics and Mechanics of Materials) A short
More informationEE C245 ME C218 Introduction to MEMS Design
EE C45 ME C8 Introduction to MEMS Design Fall 007 Prof. Clark T.-C. Nguyen Dept. of Electrical Engineering & Computer Sciences University of California at Berkeley Berkeley, CA 9470 Lecture 5: Output t
More informationUsing the finite element method of structural analysis, determine displacements at nodes 1 and 2.
Question 1 A pin-jointed plane frame, shown in Figure Q1, is fixed to rigid supports at nodes and 4 to prevent their nodal displacements. The frame is loaded at nodes 1 and by a horizontal and a vertical
More informationToward a novel approach for damage identification and health monitoring of bridge structures
Toward a novel approach for damage identification and health monitoring of bridge structures Paolo Martino Calvi 1, Paolo Venini 1 1 Department of Structural Mechanics, University of Pavia, Italy E-mail:
More informationPune, Maharashtra, India
Volume 6, Issue 6, May 17, ISSN: 78 7798 STATIC FLEXURAL ANALYSIS OF THICK BEAM BY HYPERBOLIC SHEAR DEFORMATION THEORY Darakh P. G. 1, Dr. Bajad M. N. 1 P.G. Student, Dept. Of Civil Engineering, Sinhgad
More informationPURE BENDING. If a simply supported beam carries two point loads of 10 kn as shown in the following figure, pure bending occurs at segment BC.
BENDING STRESS The effect of a bending moment applied to a cross-section of a beam is to induce a state of stress across that section. These stresses are known as bending stresses and they act normally
More informationMembers Subjected to Torsional Loads
Members Subjected to Torsional Loads Torsion of circular shafts Definition of Torsion: Consider a shaft rigidly clamped at one end and twisted at the other end by a torque T = F.d applied in a plane perpendicular
More informationNiiranen, Jarkko; Khakalo, Sergei; Balobanov, Viacheslav Strain gradient elasticity theories in lattice structure modelling
Powered by TCPDF (www.tcpdf.org) This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Niiranen, Jarkko; Khakalo, Sergei;
More informationINTRODUCTION (Cont..)
INTRODUCTION Name : Mohamad Redhwan Abd Aziz Post : Lecturer @ DEAN CENTER OF HND STUDIES Subject : Solid Mechanics Code : BME 2033 Room : CENTER OF HND STUDIES OFFICE H/P No. : 019-2579663 W/SITE : Http://tatiuc.edu.my/redhwan
More informationA new cantilever beam-rigid-body MEMS gyroscope: mathematical model and linear dynamics
Proceedings of the International Conference on Mechanical Engineering and Mechatronics Toronto, Ontario, Canada, August 8-10 2013 Paper No. XXX (The number assigned by the OpenConf System) A new cantilever
More informationLecture 20. Measuring Pressure and Temperature (Chapter 9) Measuring Pressure Measuring Temperature MECH 373. Instrumentation and Measurements
MECH 373 Instrumentation and Measurements Lecture 20 Measuring Pressure and Temperature (Chapter 9) Measuring Pressure Measuring Temperature 1 Measuring Acceleration and Vibration Accelerometers using
More informationStress Analysis Lecture 3 ME 276 Spring Dr./ Ahmed Mohamed Nagib Elmekawy
Stress Analysis Lecture 3 ME 276 Spring 2017-2018 Dr./ Ahmed Mohamed Nagib Elmekawy Axial Stress 2 Beam under the action of two tensile forces 3 Beam under the action of two tensile forces 4 Shear Stress
More informationKINGS COLLEGE OF ENGINEERING DEPARTMENT OF MECHANICAL ENGINEERING QUESTION BANK. Subject code/name: ME2254/STRENGTH OF MATERIALS Year/Sem:II / IV
KINGS COLLEGE OF ENGINEERING DEPARTMENT OF MECHANICAL ENGINEERING QUESTION BANK Subject code/name: ME2254/STRENGTH OF MATERIALS Year/Sem:II / IV UNIT I STRESS, STRAIN DEFORMATION OF SOLIDS PART A (2 MARKS)
More information[8] Bending and Shear Loading of Beams
[8] Bending and Shear Loading of Beams Page 1 of 28 [8] Bending and Shear Loading of Beams [8.1] Bending of Beams (will not be covered in class) [8.2] Bending Strain and Stress [8.3] Shear in Straight
More informationL13 Structural Engineering Laboratory
LABORATORY PLANNING GUIDE L13 Structural Engineering Laboratory Content Covered subjects according to the curriculum of Structural Engineering... 2 Main concept... 4 Initial training provided for laboratory
More informationMECHANICS LAB AM 317 EXP 3 BENDING STRESS IN A BEAM
MECHANICS LAB AM 37 EXP 3 BENDING STRESS IN A BEAM I. OBJECTIVES I. To compare the experimentally determined stresses in a beam with those predicted from the simple beam theory (a.k.a. Euler-Bernoull beam
More informationIraq Ref. & Air. Cond. Dept/ Technical College / Kirkuk
International Journal of Scientific & Engineering Research, Volume 6, Issue 4, April-015 1678 Study the Increasing of the Cantilever Plate Stiffness by Using s Jawdat Ali Yakoob Iesam Jondi Hasan Ass.
More informationDeflection of Flexural Members - Macaulay s Method 3rd Year Structural Engineering
Deflection of Flexural Members - Macaulay s Method 3rd Year Structural Engineering 008/9 Dr. Colin Caprani 1 Contents 1. Introduction... 3 1.1 General... 3 1. Background... 4 1.3 Discontinuity Functions...
More informationIntroduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams.
Outline of Continuous Systems. Introduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams. Vibrations of Flexible Strings. Torsional Vibration of Rods. Bernoulli-Euler Beams.
More informationDeflection of Flexural Members - Macaulay s Method 3rd Year Structural Engineering
Deflection of Flexural Members - Macaulay s Method 3rd Year Structural Engineering 009/10 Dr. Colin Caprani 1 Contents 1. Introduction... 4 1.1 General... 4 1. Background... 5 1.3 Discontinuity Functions...
More information