Syllabus

16.21 is a junior level course on techniques for structural analysis and design. Special emphasis is given to the learning of modern computational methods used in aerospace structural design. Equal importance is given to the learning of the role of material properties in structural design, failure, and longevity. Although the focus of the course is on aerospace structures, the topics covered are relevant to a wide variety of applications in engineering. 16.21 has a specific set of learning objectives which the student should always keep present as the semester progresses. Students graduating from 16.21 will be able to:

1. Formulate and apply appropriate mathematical and numerical models to predict the state of stress and deformation of one, two and three-dimensional aerospace structures.
   
2. Explain the limitations of the models, assess their applicability to realistic configurations and estimate the errors resulting from their application.
   
3. Design and operate a computational analysis of a structural design in collaboration with members of a team.
   
4. Apply the concepts learned in the course to assess and explain the possibility of failure in aerospace structural configurations.

• Apply energy and variational principles for the determination of deflections and internal loads in one-dimensional structural elements.
• Apply Ritz Method for the approximate calculation of deflections and stresses in one-dimensional structural elements.
• Explain the principles and implementation of the finite element method in linear static problems.
• Apply the finite element method in the stress analysis of aerospace structural components.
• Design a structural component of an aerospace structure with the aid of state-of-the-art finite element techniques. Structural design criteria will account for stiffness, strength, toughness and useful life considerations.

In the first recitation a Mathematica Tutorial will be offered.

• In class concept questions with the aid of PRS cards (a custom version of the system used in Unified). These are presented during the lecture and intended to provide an immediate assessment of the understanding of the most important concepts by the students. The answers are discussed immediately after.
• Muddy cards: This is a well established and successful mechanism of feedback for the instructor which many students are familiar with. Cards are distributed at the beginning of each lecture and the students put down their comments on the lecture: specific points on the material covered in class that was not clearly presented, comments that the student considers as appropriate feedback for the instructor, suggestions that might contribute to the improvement of the lectures. The cards are turned in at the end of the lecture. The instructors will make every effort to respond to the muddy cards at the beginning of the following lecture and/or by posting them on the virtual course forum. Muddy cards are anonymous and an essential feedback mechanism for the instructor. We strongly encourage their use.
• Hands-on collaborative computing sessions: In these sessions, each student uses a computer terminal to operate a central computer shared by the whole group. A computing problem is presented to the class in advance, e.g. a homework problem, and the solution is developed and implemented during the recitation on the central computer with the active, hands-on participation of the students and instructor. A special networking technology (VNC: http://www.realvnc.com) is utilized that allows the simultaneous action of the input devices (mouse, keyboard) from all the participant clients, as well as the output display on all the participant clients’ screens. These sessions will be scheduled during the course of the semester and will take place during recitation or lecture when appropriate.
• Web-based course forum: This tool is intended to provide an additional means of interaction among the students and the instructors. The instructors will use a folder named “Mud” to answer the questions in the muddy cards. The format of this forum is completely open. Students are encouraged to use this forum to ask their questions about anything related to the course whose answer may be of use to others. In particular, we would like to see this forum develop into a placeholder for peer discussions on the concepts covered in class, textbook, etc.

• Reading assignments: Students are expected to read assigned material from the required textbook [1] prior to its discussion in class. The reading assignments are not part of the assessment process, i.e., they do not contribute directly to the grade. They can only affect the grade indirectly through the students’ participation in class. Students are strongly encouraged to read the assigned material in order to maximize the learning experience. Reading assignments will be scheduled during the term and announced in class and on the website.
• Problem sets and computer assignments: A total of approximately ten (10) problem sets and computer assignments will be given on Wednesdays and on a weekly basis. The due date for submission of assignments is at the beginning of class the Wednesday after the assignment is given. Late submission of assignments is not accepted. In specific computer assignments, students will be asked to turn in their results in electronic form. Students are strongly encouraged to discuss homework problems in groups, since this is expected to help the learning process. However, homework assignments are also used for performance assessment and, therefore, the material that is turned in must represent the student’s own understanding of the material.

• A mid-term exam during lecture time. This exam will last one hour.
• A final exam on the date, time and location assigned by the Institute.

Both exams will have an “open book” policy. There are no restrictions to the material students may bring to the exams.

Students are strongly encouraged to discuss homework problems with each other, since this is expected to help the learning process. However, homework assignments are also used for performance assessment and, therefore, the material that is turned in must represent the student’s own understanding of the material.

Each student’s total grade on the project reports will be based on a team grade and on an individual grade. The team grade will be 20% of the project report grade and will be based on the quality of the report as a whole. The individual grade will be 80% of the project report grade and will be based on the individual student’s contribution to the team effort. The instructors will assess individual contributions to the team project through their interactions with the teams throughout the semester.

The final letter grades will be assigned according to the rules and regulations of the Faculty.

1. Reddy, J. N. Energy and Variational Methods in Applied Mechanics. John Wiley & Sons, 1984. (Required)
2. Reddy, J. N. Energy and Variational Methods in Engineering. John Wiley & Sons, 2003. (Alternative)
3. Bathe, K. J. Finite Element Procedures. Prentice Hall, 1996.
4. Reddy, J. N. Introduction to the Finite Element Method. Mc Graw-Hill, 1993.
5. Rivello. Theory and Analysis of Flight Structures. McGraw-Hill, 1969.
6. Shames, and Dym. Energy and Finite Element Methods in Structural Mechanics. Hemisphere Publishing Corporation, 1985.
7. Hughes, T. J. R. The Finite Element Method, Linear Static and Dynamic Finite Element Analysis. Dover.

Textbooks on material selection, fracture and fatigue:

1. Ashby, M. F., and D. R. H. Jones. Engineering Materials 1: An Introduction to Their Properties and Applications. Butterworth-Heinemann, 1996.
2. Anderson, T. L. Fracture Mechanics: Fundamentals and Applications. CRC Press, 1995.

• Introduction (L1)
  
  
• Kinetics (L2, L3)
  • Stress at a Point
  • Stress Tensor and the Cauchy Formula
  • Transformation of Stress Components
  • Principal Stresses and Principal Planes
  • Equations of Motion
  • Symmetry of the Stress Tensor
    
    
• Kinematics (L3)
  • Strain at a Point
  • Transformation of Stress Components
  • Compatibility conditions
    
    
• Thermodynamic Principles (L4)
  • The First Law of Thermodynamics: Energy Equation
  • The Second Law of Thermodynamics
    
    
• Constitutive Equations (L5)
  • Generalized Hooke’s Law
  • Strain Energy Density Function
  • Elastic Symmetry
  • Thermoelastic Constitutive Equations
    
    
• Boundary-Value Problems of Elasticity (L6)
  • Summary of Equations
  • Classification of Boundary-Value Problems
  • Existence and Uniqueness of Solutions

2. Energy and Variational Principles

• Preliminary Concepts (L7, L8)
  • Introduction
  • Work and Energy
  • Strain and Complementary Strain Energy
  • Virtual Work
    
    
• Concepts of Calculus of Variations (L9, L10)
  • Concept of a Functional
  • The Variational Operator
  • The First Variation of a Functional
  • Extremum of a Functional
  • The Euler Equations
  • Natural and Essential Boundary Conditions
  • A More General Functional
  • Minimization with Linear Equality Constraints
    
    
• Virtual Work and Energy Principles (L11, L12, L13)
  • Principle of Virtual Displacements
  • Unit Dummy-Displacement Method
  • Principle of Total Potential Energy
  • Principle of Virtual Forces and Complementary Potential Energy
  • Unit Dummy-Load Method

• Energy Theorems of Structural Mechanics (L14)
  • Castigliano’s First Theorem
  • Castigliano’s Second Theorem
  • Betti’s and Maxwell’s Reciprocity Theorems

3. Variational Methods of Approximation

• Some Preliminaries (L15)
  
  
• The Ritz Method (L16, L17)
  • Description of the Method
  • Matrix form of the Ritz Equations
  • One-dimensional Examples
    
    
• Weighted-Residual Methods (L17)
  • A Brief Description of Galerkin, Least-squares and Collocation Methods

4. The Finite Element Method for Linear Analysis in Solid and Structural Mechanics

• Formulation of the Displacement-Based Finite Element Method (L18, L19, L20)
  • General Derivation of Finite Element Equilibrium Equations
  • Imposition of Displacement Boundary Conditions
  • Generalized Coordinate Models for Specific Problems
  • Lumping of Structure Properties and Loads
    
    
• Convergence of Analysis Results (L21)
  • Definition of Convergence
  • Properties of the Finite Element Solution
  • Rate of Convergence
  • Calculation of Stresses and the Assessment of Error

5. Formulation and Calculation of Isoparametric Finite Element Matrices

• Isoparametric Derivation of Bar Element Stiness Matrix (L22)
  
  
• Formulation of Continuum Elements (L23, L24, L25)
  • Quadrilateral Elements
  • Triangular Elements
  • Convergence Considerations
  • Element Matrices in Global Coordinate System
    
    
• Formulation of Structural Elements (L26, L27)
  • Beam Elements and Axisymmetric Shell Elements
  • Plate and Shell Elements
    
    
• Numerical Integration (L28)
  
• Direct Solution of Linear System of Equations (L29)

6. Structural Failure and Structural Health

• Types of Structural Failure (L30, L31)
  
  Yield Stress and Ultimate Stress. Maximum Normal Stress Theory, Tresca Condition, Hydraulic Stress, Von Mises Criterion, Distortion Energy Interpretation. Graphical Representation of Failure Regions. Extension to Orthotropic Materials, Hill Criterion, Homan Criterion. Nature of Failure Criteria, Functional Forms. General Failure Analysis Procedure. Application to Pressure Tank.
  
  
• Fracture Mechanics (L31, L32, L33, L34)
  
  Description of Phenomena and Importance. Energy approach to Crack Growth, Energy consumed by Crack Growth, Grith’s Experiment and Formula. Definition of Stress Intensity Factor. Stresses at Crack Tip, Mode I, II and III Cracks. Solutions of Linear Elastic Fracture Mechanics, Geometry Effects. Combined Loading. Material Selection example.
  
  
• Fatigue and Longevity (L34, L35, L36, L37, L38)
  
  Terminology, S-N Diagrams, Goodman Diagrams. Effects of R Value, Stress Concentrations. Ground-Air-Ground Cycle, Miner’s Rule. Micromechanical Effects. Paris’ Law. Fatigue Life Prediction. R Effects and Forman’s Law, Sequencing Effects. Approach to Design for Longevity. Material selection example.