MEG 741 Energy and Variational Methods in Mechanics I

Brendan J. O’Toole, Ph.D.Associate Professor of Mechanical Engineering

Howard R. Hughes College of EngineeringUniversity of Nevada Las Vegas

TBE B-122(702) 895 - 3885bj@me.unlv.edu

First Day Class Outline

• Review Syllabus & Course Overview• Mechanics of Materials Highlights• Discuss Vectors, Tensors, & Matrices

– Chapter 1 from text

MEG 741 Energy and Variational Methods in Mechanics I

Fall Semester 2004

Professor: Brendan J. O'Toole, Ph.D.

Office: TBE B122

Phone: 895-3885

e-Mail: bj@me.unlv.edu

Class: MW / 2:30 PM – 3:45 PM / MPE 232

Text: “Mechanics of Structures: Variational and Computational Methods”, 2nd Edition, Pilkey & Wunderlich, CRC Press 2002, ISBN 0-8493-0700-7

Website: http://www.egr.unlv.edu/~bj/

MEG 741 Course Description and Objectives

• This course provides an overview of the fundamentals used to solve many types of solid mechanics problems.

• The material presented in the course provides a solid backgroundfor other courses such as Finite Element Analysis for StructuralApplications, Advanced Strength of Materials, Theory of Plates and Shells, or Computational Solid Mechanics.

• The objective of the course is to develop a better understandingof the fundamental theories behind most solid mechanics solution methods.

• Students should be able to apply the fundamental analysis techniques to solve a variety of different truss, beam, and plate mechanics problems.

Course Outline(First Half)

Class Dates Reading Lecture Topics HW 1 2

M 8/30 W 9/1

1.1 - 1.3

Introduction: Vectors, Matrices, Tensors Definitions, Strain-Displacement Equations, Material Laws

3

M 9/6 W 9/8

1.4 – 1.6

Labor Day Recess (No Class) Equilibrium, Boundary Conditions, Governing Equations

4 5

M 9/13 W 9/15

1.7 – 1.9 2.1 - 2.2

Stress Analysis, Beam Theory & Torsion Work & Energy, Classical Variational Principles

HW 1

6 7

M 9/20 W 9/22

2.3 2.4

Generalized Variational Principles Engineering Beam Theory

8 9

M 9/27 W 9/29

2.5 App I

Differential & Integral Forms of Gov. Eqns. Fundamentals of Variational Calculus

10 11

M 10/4 W 10/6

3.1 3.1

Virtual Work: Castigliano’s Theorem Virtual Work: Unit Displacement Method

HW 2

12 13

M 10/11 W 10/13

3.2

Exam 1: Chapters 1 & 2 Complementary Virtual Work

14 15

M 10/18 W 10/20

3.3 4.1 – 4.2

Reciprocal Theorems Fundamental Relations for a Beam Element

HW 3

Course Outline(Second Half)

Class Dates Reading Lecture Topics HW 16 17

M 10/25 W 10/27

4.3 – 4.4 4.5

Element Matrices, Stiffness Matrices Transfer Matrices

18 19

M 11/1 W 11/3

5.1 - 5.3 5.3

Structural Systems, Displacement Method: Virtual Work Displacement Method: Direct Derivation, Stiffness Matrices

HW 4

20 21

M 11/8 W 11/10

5.3

Displacement Method: Trusses & Frames Exam 2: Chapters 3 & 4

22 23

M 11/15 W 11/17

5.4 6.1 – 6.3

Force Method The Finite Element Method

24 25

M 11/22 W 11/24

6.4 6.5

Plane Problems Trial Functions, Convergence, Accuracy

HW 5

26 27

M 11/29 W 12/1

6.6 – 6.7 7.1 – 7.2

Numerical Integration, Isoparametric Elements Governing Differential Equations

28 29

M 12/6 W 12/8

7.3 7.4-7.5

Residual Methods Variational & Trial Function Methods

HW 6

Final Exam Wednesday, December 15, 3:10 PM

Homework

These assignments may be modified during the semester.

Problems B1, B2, B3, etc. are additional assignments that do not correspond exactly to problems in the textbook.

Grading

Research ProjectObjective:

Provide students the opportunity to thoroughly review and summarize a particular solid mechanics topic.

Scope:

Perform a detailed literature review on a particular topic. The review can contain information from textbooks, reference books,

journal articles, conference proceedings, and websites. The review must include a minimum of 15 journal articles.

Topics:

All topics must be approved by Dr. O’Toole. A list of topic ideas will be distributed during the second week of the semester.

Office Hours

Office hours and a weekly schedule for Dr. O’Toole will be posted on the following webpage by the start of week 2 of the

semester.

http://www.egr.unlv.edu/~bj/

Instructor: Dr. Brendan J. O’Toole• Education

– Ph.D. & M.S. in Mechanical Engineering, University of Delaware, Newark, DE, 1993.Dissertation: “Modelling the Effect of Heterogeneity in Composite Structures”Thesis: “A Photoelastic Investigation of Crack-Inclusion Interaction”

– B.S. in Mechanical and Aerospace Engineering, University of Delaware, 1986.

• Employment– Associate Professor - Mechanical Engineering Dept., University of Nevada Las Vegas (8/92 - present)

• Teach undergraduate and graduate courses in: – Solid mechanics, composite materials, experimental mechanics, design, and dynamics

• Conduct research in the following areas: – experimental and computational solid mechanics, analysis, design, and fabrication of components and structures (emphasis on

composites), mechanics of solid cellular foams: dynamic & static properties

– Program Manager – Soldiers Future Force Electronics Reliability and Survivability Technology ProgramUNLV / U.S. Army Research Laboratory Cooperative Agreement (4/03 – present)

– Director of Engineering – High Pressure Science and Engineering CenterUniversity of Nevada Las Vegas (1/03 - present)

– Visiting Research Associate - Composites and Lightweight Structures Branch U.S. Army Research Laboratory, Aberdeen MD (10/01 - 4/02)

• Computational simulation of composite armor systems under ballistic impact loading.

– Visiting Research Engineer - Composites and Lightweight Structures BranchU.S. Army Research Laboratory, Aberdeen MD (7/00 - 8/00)

• Developed computer model for dynamic analysis of composite plates and ceramic tiles.

Active Projects for Dr. Brendan J. O’Toole• Development of a Reconfigurable Tooling System, 2Phase Technologies, Inc.• Characterization of Reversible Epoxy Foams and Property Gradients in

Polymeric Foams, Department of Energy Stockpile Stewardship Program• Soldier FERST - Soldier’s Objective Force Electronics Reliability and

Survivability Technology Program, US Army– Ballistic Shock Propagation Through Structural Joints– Characterization of Honeycomb Shock Mitigators in an Experimental Gun Tube

• Design/Education Oriented Projects– Human Powered Vehicle Design, ASME Competition– Developing a Balloon Satellite Program, NASA Space Grant/EPSCoR Program– High School First Robotics Competition, NASA Space Grant/EPSCoR Program

• Composite Blast Containment Vessels, Sandia National Laboratories• Blast Loading on Vehicle Structures, DOD EPSCoR and UNLV• Mechanistic Understanding of High-Temperature Deformation of Alloy EP-823

for Transmutation Applications, DOE Advanced Accelerator Applications• Development of a Materials Performance Laboratory, DOE Advanced

Accelerator Applications

Overview of Mechanics of Materials

IntroductionThe prerequisite for this course is a fundamental

undergraduate course in Civil or Mechanical Engineering similar to the UNLV course:

MEG 302 Materials MechanicsSometimes it is called:

“Strength of Materials” or “Mechanics of Materials”

Typical Undergraduate Text

Instructor: Dr. Brendan J. O’Toole, Ph.D.

Professor: Brendan J. O'Toole, Ph.D.Office: TBE B122 Phone: 895-3885E-Mail: bj@me.unlv.eduDays/Time/Room: TR / 11:30 AM – 12:45 PM / BEH 108Text: “Mechanics of Materials”,

Beer, Johnston, & DeWolfMcGraw Hill, 2002

O’Toole Website: http://www.egr.unlv.edu/~bj/Course Website:

MEG 302 Course Objectives• Learn the Vocabulary• Improve Your Skill at Drawing Free Body Diagrams• Learn About Material Behavior• Learn How To Solve Mechanics Problems. This is the largest

part of the class. The solution procedure for most mechanics problems involves one or more of the following tasks:– A statics analysis of a component to find the internal reactions

(forces & moments)– Determine stresses and strains in a component based on

internal reactions– Find the deformation of the component– Compare calculated values of stress & deformation with

known acceptable values• Improve Your Engineering Design Skills

Vocabulary

Rigid BodyDeformable BodyLinkTrussNormal StressShear StressBearing StressUltimate StressYield StressFailure StressPrincipal StressesNormal StrainShear StrainFailure StrainYield Strain

Shear ModulusPoisson’s RatioTrue StressEngineering StressTrue StrainElastic BehaviorPlastic BehaviorThermal ExpansionTorsionTorqueAngle of TwistStatic IndeterminacyPowerPure BendingArea Moment of InertiaPolar Moment of Inertia

Shear Force DiagramBending Moment DiagramTransverse Shear Cantilever BeamSimply Supported BeamClamped BeamIsotropicAnisotropicHomogeneousPrismaticThin-Walled MemberPressure VesselCombined LoadingStress TransformationMohr’s CirclePlane Stress

SuperpositionElastic CurveColumnBucklingEuler BucklingPlane StrainDuctile BehaviorBrittle BehaviorAxial Stiffness3-point Bending4-point BendingModulus of ElasticityYoung’s ModulusModulus of RigidityPrincipal StrainsFlexural Stiffness

This is a sampling of terms that are defined in the text. We will discuss them throughout the semester. You are expected to

understand the meaning of these terms. You are also expected to know the correct units for material properties and other variables.

Free Body Diagrams

• Free Body Diagrams were first introduced in Physics and Statics courses.

• They are a powerful tool that help define the important loads, reactions, geometry, and coordinate system in a problem so that the correct equilibrium equations are defined and solved.

Axial LoadingExample from Software CD included with textbook.

Material Response to Loading

Tensile Test of EP-823 Maraging Steel

RT EP823 2054U19Stress vs Strain

0

20

40

60

80

100

120

0.000 0.050 0.100 0.150 0.200 0.250

Strain

Stre

ss (k

si)

Torsion

Beams and Bending

Pressure Vessels & Design

Buckling

Overview of MEG 741• This class is a theoretical solid mechanics course. • We will discuss general governing equations that can be

applied to many solid mechanics problems.• We will discuss several solution procedures for solving

these governing equations. • We will solve some specific types of problems:

– Truss structures– Frame structures

• We will introduce structural analysis methods that define structural elements and matrices.

• We will introduce the finite element method for solving 2-dimensional problems

Overview of MEG 741• MEG 741 is not an applied finite element analysis

course.• We will not be learning how to use commercial FEA

software programs in the class.• Most (if not all) of the problems can be solved by hand

calculations.• Programs like MathCad or Matlab may be useful for

solving some of the longer problems.• FEA programs could be used to check your hand-

calculated solutions but this is not a requirement.

Example Problem• Statically Indeterminate problems were

introduced in undergraduate mechanics courses.

• Equilibrium equations alone do not provide enough information to solve some problems.

• Usually, some geometric phenomenon was considered in order to find another equation to use with the equations of equilibrium.

Example: Axial Load and Load Sharing

Example: Axial LoadDisplacement Equations

Example: Solving Axial Load Statically Indeterminate Problems

Example Related to MEG 741

• The first part of MEG 741 discusses the general equations used to solve mechanics problems.

• These general equations provide a formal procedure for solving all mechanics problems, including statically indeterminate problems.

• The governing equations include:– Equilibrium equations (e.g.: force balance)– Kinematic Equations (relate strains to displacements)– Material Laws (Hookes Law, constitutive relations)