16.001 | Fall 2021 | Undergraduate

Unified Engineering: Materials and Structures

Pages

Week 1

Topics

Introduction to aerospace structural mechanics

Measurable Outcomes

Describe a structure, its functions, and associated objectives and tradeoffs.

Week 2

Topics

Introduction to aerospace materials

Measurable Outcomes

Describe the basic mechanical properties of aerospace materials. Describe the general classes of materials used in aerospace and their specific applications. 

Week 3

Topics

Three great principles: equilibrium, compatibility, and constitutive material response; equilibrium of a particle, system of particles (free-body diagram)

Measurable Outcomes

Define the “three great principles” of solid mechanics: equilibrium, compatibility, and constitutive material response. 

Week 4

Topics: 

Planar force systems, equipollent forces

Measurable Outcomes

Determine the relation between applied and transmitted forces and moments, for a particle, a set of particles, and a rigid body in equilibrium. Apply the concept of equipollent force/moment to model and simplify the analysis of force systems. 

Week 5

Topics

Support reactions, free-body diagrams, static determinacy

Measurable Outcomes

Represent and use idealizations of structural supports. Draw free-body diagrams for structural systems. Classify mechanical systems according to their state of equilibrium: underdetermined, determinate, or indeterminate. Calculate reactions in determinate systems.

Week 6

Topics

Truss analysis: method of joints, method of sections

Measurable Outcomes

Analyze truss structures using the method of joints and the method of sections.

Week 7

Topics

Statically indeterminate systems

Measurable Outcomes

Define the constitutive relationship for elastic bars. Apply compatibility of deformation in a variety of structural configurations. Analyze statically indeterminate bar and truss systems using the “three great principles.”

Week 8

Topics

Stress: definition, Cartesian components, equilibrium

Measurable Outcomes

Define the concept of stress at a material point and its mathematical representation as a second-order tensor. Describe the state of stress at a point using Cartesian tensorial components, and their meaning as a measure of the local measure of loading at material points in structural systems. Explain stress equilibrium in differential form. 

Week 9

Topics

Stress transformation and Mohr’s circle, principal stresses, maximum shear stress

Measurable Outcomes

Explain the basis for transforming stress states between two different Cartesian bases. Transform two-dimensional stress states and compute principal stresses and directions. 

Week 10

Topics

Definition of strain, extensional and shear strain, strain-displacement relations

Measurable Outcomes

Define the concept of strain at a material point as the fundamental measure of the local state of deformation and its relation to the displacement field. Describe strain as a second-order tensor, its Cartesian components, and their meaning.

Week 11

Topics

Transformation of strain, Mohr’s circle for strain, principal strains, maximum shear strain

Measurable Outcomes

Explain the basis for transforming strain states between two different Cartesian bases. Transform two-dimensional strain states, and compute principal strains and directions. 

Week 12

Topics

Constitutive equations for a linear elastic material; constitutive equations: isotropic and orthotropic elastic materials

Measurable Outcomes

Describe the constitutive relationship between stress and strain for isotropic and orthotropic linear elastic materials.

Week 13

Topics

Engineering elastic constants, measurement, generalized Hooke’s law

Measurable Outcomes

Discuss engineering elastic constants, their measurement, and their relationship to the tensorial description of Hooke’s law

Week 14

Topics

Summary of equations of the theory of elasticity

Measurable Outcomes

Summarize the key equations of the theory of elasticity. Formulate and simplify problems in general elasticity, apply displacement and traction boundary conditions to problems in elasticity, and solve simple cases.

Week 15

Topics

Analysis of rods: uniaxial loading of slender 1D structural elements

Measurable Outcomes

Analyze the structural response of uniaxially-loaded slender elements: rods and bars 

Week 16

Topics

  1. Analysis of beams: statics, internal forces and their relation to internal stresses; bending moment, shear force and axial force diagrams, concentrated and distributed loads; differential equations of internal equilibrium, kinetic boundary conditions
  2. Euler-Bernoulli beam theory: beam deflections, moment-curvature relation, kinematic boundary conditions. Statically determinate and indeterminate beams
  3. Cross-section properties: first and second moment of area, center of area, moment of inertia

Measurable Outcomes

Analyze the structural response of transversely-loaded slender elements: beams; internal forces and beam deflections

Week 17

Topics

Analysis of Torsion of slender 1D structural elements: Shafts. Kinematic assumptions, internal torque, constitutive law for the cross-section: torque-rate-of-twist relation, equilibrium; governing equation; solution for various statically- determinate and indeterminate loading cases

Measurable Outcomes

Analyze the stability of slender structural elements subject to compressive loads: buckling loads, mode shapes, effects of imperfections, and eccentric loads

Week 18

Topics

Structural instability and buckling of slender 1D elements subject to compressive loads; analysis of buckling loads and mode shapes for various boundary conditions; effect of imperfections and eccentric loading

Measurable Outcomes

n/a

Good engineers use accepted standards in reporting work. In regard to submitted work in Unified, the following is accepted practice:

  • Each problem should be stapled individually when you turn in your pset. So, if there are five problems in a pset, you will turn in five separate, stapled problems. 
  • Write your name on all sheets you turn in.
  • Number the sheets for each problem in sequential order indicating the sequence of each sheet in that answer.
  • Clearly indicate with whom you worked on every question you submit.
  • You do not need to type up your solutions but must write legibly.
  • State your assumptions, and display graphs and equations neatly.
  • Explain your solution clearly and interpret your results.
  • Clearly identify intermediate results and their importance.
  • Units must be explicitly described and identified in all work and solutions.
  • Clearly identify (e.g. underline, box, etc.) your final answer for any part of a problem.

Communication is a key part of the ability of an engineer. Demonstrating your ability to communicate is part of demonstrating your learning level. The sample that follows is meant to illustrate the proper way to write up a problem solution. You will see the concepts relevant to this problem in the lecture.

Unified Engineering Materials and Structures Sample Problem Set

Unified Engineering Materials and Structures Problem Set 01Unified Engineering Materials and Structures Problem Set 01 Solution

Unified Engineering Materials and Structures Problem Set 02; Unified Engineering Materials and Structures Problem Set 02 Solution

Unified Engineering Materials and Structures Problem Set 03; Unified Engineering Materials and Structures Problem Set 03 Solution

Unified Engineering Materials and Structures Problem Set 04; Unified Engineering Materials and Structures Problem Set 04 Solution

Unified Engineering Materials and Structures Problem Set 05; Unified Engineering Materials and Structures Problem Set 05 Solution

Unified Engineering Materials and Structures Problem Set 06; Unified Engineering Materials and Structures Problem Set 06 Solution

Unified Engineering Materials and Structures Problem Set 07; Unified Engineering Materials and Structures Problem Set 07 Solution

Unified Engineering Materials and Structures Problem Set 08; Unified Engineering Materials and Structures Problem Set 08 Solution

Unified Engineering Materials and Structures Problem Set 09; Unified Engineering Materials and Structures Problem Set 09 Solution

Unified Engineering Materials and Structures Problem Set 10; Unified Engineering Materials and Structures Problem Set 10 Solution

Unified Engineering Materials and Structures Problem Set 11; Unified Engineering Materials and Structures Problem Set 11 Solution

Unified Engineering Materials and Structures Problem Set 12; Unified Engineering Materials and Structures Problem Set 12 Solution

Unified Engineering Materials and Structures Problem Set 13; Unified Engineering Materials and Structures Problem Set 13 Solution

The following textbooks are used in the reading assignments.

[CDL] = S. Crandall, N. Dahl, et al., An Introduction to the Mechanics of Solids (In SI Units), McGraw-Hill, 1978. ISBN: 9781259006531.

[AJ] = M. F. Ashby and D. R. H. Jones, Engineering Materials 1: An Introduction to Their Properties and Applications, Pergamon Press, 1980. ISBN: 9780080261393.

R. Hibbeler, Engineering Mechanics: Statics (any recent edition), Pearson. ISBN: 9780132915540. [Free eBook from the Internet Archive] [Additional information and access via Open Library]

R. Hibbeler, Mechanics of Materials (any recent edition), Pearson. ISBN: 9780134319650.

Also useful:

M. F. Ashby and D. R. H. Jones, Engineering Materials 2: An Introduction to Microstructures and Processing, Pergamon Press, 1986. ISBN: 9780080325323.

M. F. Ashby, Materials Selection in Mechanical Design, Pergamon Press, 1992. ISBN: 9780750627276.

W. Bickford, Mechanics of Solids: Concepts and Applications, Irwin, 1993. ISBN: 9780256114256.

Week 01

Topics

Introduction to aerospace structural mechanics

Reading Assignments

[CDL] 1.1–1.3

Week 02

Topics

Introduction to aerospace materials

Reading Assignments

[AJ] 1–2

Week 03

Topics

Three great principles: equilibrium, compatibility, constitutive material response; equilibrium of a particle, system of particles (free-body diagram)

Reading Assignments

[AJ] 1–2

Week 04

Topics: 

Planar force systems, equipollent forces

Reading Assignments

[CDL] 1.6–1.7 

Week 05

Topics

Support reactions, free-body diagrams, static determinacy

Reading Assignments

[CDL] 1.8–1.9

Week 06

Topics

Truss analysis: method of joints, method of sections

Reading Assignments

[CDL] 1.9

Week 07

Topics

Statically indeterminate systems

Reading Assignments

[CDL] 2.1, 2.4, 1.10–1.11, and 2.7

Week 08

Topics

Stress: definition, Cartesian components, equilibrium

Reading Assignments

[CDL] 4.2–4.4 

Week 09

Topics

Stress transformation and Mohr’s circle, principal stresses, maximum shear stress

Reading Assignments

[CDL] 4.5–4.7 

Week 10

Topics

Definition of strain, extensional and shear strain, strain-displacement relations

Reading Assignments

[CDL] 4.8–4.10

Week 11

Topics

Transformation of strain, Mohr’s circle for strain, principal strains, maximum shear strain

Reading Assignments

[CDL] 4.11–4.15 

Week 12

Topics

Constitutive equations for a linear elastic material; constitutive equations: isotropic and orthotropic elastic materials

Reading Assignments

[CDL] 5.3–5.4, 5.10; [AJ] 3–7

Week 13

Topics

Engineering elastic constants, measurement, generalized Hooke’s law

Reading Assignments

[CDL] 5.6

Week 14

Topics

Summary of equations of the theory of elasticity

Reading Assignments

n/a

Week 15

Topics

Analysis of rods: uniaxial loading of slender 1D structural elements

Reading Assignments

n/a

Week 16

Topics

  1. Analysis of beams: statics, internal forces and their relation to internal stresses; bending moment, shear force and axial force diagrams, concentrated and distributed loads; differential equations of internal equilibrium, kinetic boundary conditions
  2. Euler beam theory: beam deflections, moment-curvature relation, kinematic boundary conditions. Statically determinate and indeterminate beams
  3. Cross-section properties: first and second moment of area, center of area, moment of inertia

Reading Assignments

[CDL] 3.2–3.6, 3.8, 7.1–7.5, 8.1–8.2

Week 17

Topics

Analysis of torsion of slender 1D structural elements: shafts. kinematic assumptions, internal torque, constitutive law for the cross-section: torque-rate-of-twist relation, equilibrium; governing equation; solution for various statically determinate and indeterminate loading cases

Reading Assignments

[CDL] 6.1–6.5

Week 18

Topics

Structural instability and buckling of slender 1D elements subject to compressive loads; analysis of buckling loads and mode shapes for various boundary conditions; effect of imperfections and eccentric loading

Reading Assignments

[CDL] 9.1–9.6

Course Meeting Times

Lectures: 1 hour / session, 5 sessions / week

Recitations: 1 hour / session, 2 sessions / week

Prerequisites

18.02 Calculus II: Multivariable Calculus and 8.01 Physics I: Classical Mechanics

Corequisites

16.002 Unified Engineering II: Signals and Systems and 18.03: Differential Equations 

Introduction

In the MIT Department of Aeronautics and Astronautics and Unified Engineering (Unified), aerospace engineering is taught within the context of the CDIO (Conceive-Design-Implement-Operate) framework. Our goal is to educate future leaders in the field who can contribute to the development of new products in a modern, team-based environment. In addition to learning the disciplinary material and acquiring the technical foundations of aerospace engineering, you will begin learning skills that will enable you to become an effective aerospace engineer.

Course Objectives

The basic objective of Unified is to give a solid understanding of the fundamental disciplines of Aerospace Engineering and to give insight into their interrelationships and applications. In the fall, these disciplines are Materials and Structures (16.001) and Signals and Systems (16.002). In the spring, the disciplines are Fluid Dynamics (16.003) and Thermodynamics (16.004). This course 16.001 is the first course in the Unified series.

The second objective of Unified is to guide you toward an understanding of the fundamental skills, knowledge, and sensitivities that are the traits of a successful engineer. These include the skills necessary to work successfully in a group (including technical and graphical communication) and those of self-education (reading, research, and experimentation). Professional engineers have the knowledge and confidence to make estimates of poorly known parameters, create conceptual models of systems, assess the applicability of various models and their resulting solutions to encountered problems, and design new solutions to meet technical challenges. Aerospace systems in particular are very unforgiving; one can do 99.5% of the job well, but it is the error in the remaining 0.5% that may cause a major accident or failure. Aerospace engineers are known for their attention to detail and, while we do not expect perfection, we do expect that all work carried out as part of Unified Engineering be done with care and as well as possible given the time constraints of the course.

  1. Separate but Unified: While the disciplines have separate course numbers and will receive separate letter grades, Unified is nonetheless taught with a significant amount of unification. This includes the structure (assignments, quizlets, exams) as well as the grading policy of the individual courses. As such, the contents of this section apply equally to 16.001–16.004. Also, we will have topics that are not specifically about a single discipline, but yet will be taught by one of the disciplinary faculty. This is, in particular, true for some of the labs which often bring together more than one discipline and/or involve some aspect of design.
  2. Teaching Methods: The Unified Engineering Team is firmly committed to helping you learn. This commitment takes many forms, including a process of continuous improvement in our teaching effectiveness. During this semester, the faculty will be exploiting teaching methods proven to increase students’ learning. These methods include presenting information to the student in the context of a compelling challenge or problem, stressing active versus passive learning in the classroom, enhancing feedback to the student, providing multiple opportunities for hands-on learning, etc.
  3. Learning and Grades: Learning is a lifelong endeavor, whereas learning for grades is often quite temporary (i.e. the material often is not remembered as time after an exam increases). Further, while pursuing learning for a lifetime ability, one tends to acquire an understanding that is demonstrated in typical assessment activities, such as exams, and thereby one achieves the grades representative of the learning level desired and as expressed in the definition of grades at MIT (see below). We strongly encourage you to view all activities within Unified as opportunities for learning with the final goal of having you prepared for your career after Unified.

Textbooks

S. Crandall, N. Dahl, et al., An Introduction to the Mechanics of Solids (In SI Units), McGraw-Hill, 1978. ISBN: 9781259006531.

M. F. Ashby and D. R. H. Jones, Engineering Materials 1: An Introduction to Their Properties and Applications, Pergamon Press, 1980. ISBN: 9780080261393.

R. Hibbeler, Engineering Mechanics: Statics (any recent edition), Pearson. ISBN: 9780132915540. [Free eBook from the Internet Archive] [Additional information and access via Open Library]

R. Hibbeler, Mechanics of Materials (any recent edition), Pearson. ISBN: 9780134319650.

The following books are also useful:

M. F. Ashby and D. R. H. Jones, Engineering Materials 2: An Introduction to Microstructures and Processing, Pergamon Press, 1986. ISBN: 9780080325323.

M. F. Ashby, Materials Selection in Mechanical Design, Pergamon Press, 1992. ISBN: 9780750627276.

W. Bickford, Mechanics of Solids: Concepts and Applications, Irwin, 1993. ISBN: 9780256114256.

Assignments, Quizlets, and Exams 

  • Problem Sets: A sample homework solution is provided illustrating the proper way to write up a problem solution. Doing homework helps to engage with the concepts and material taught in class on a deeper level. To enhance the learning process, we strongly suggest that you first try to solve the problems by yourself.
  • Quizlets [not available to OCW users]: Each week there will be a quizlet on the material covered in the past week. This will be a combined 50-minute exam. The questions will cover examples, previous homework questions, and recitation questions from the past week. Please note that the order of the subjects will not be the same from week to week. The quizlet problem will be drawn from (1) problems from any past problem set for which the solutions have been posted, (2) problems worked in lecture or recitation, or (3) problems included in assigned reading for which solutions have been made available. The quizlet problems will be identical to the problems from (1), (2), and (3) except possibly for some simplifications (to allow for solving the problem in the allotted quizlet time). When taking the quizlet, you will not have any materials available to you (e.g. no “cheat sheets”). For equations, etc. that we would not expect you to know from memory, they will be provided with the quizlet. 
  • Labs: Labs will often require working on a team of two or more Unified students. The lab report details and submission procedure will be specified for each lab. 
  • Exams: There will be an in-class 2-hour midterm and a final exam for each course.

Academic Honesty

A fundamental principle of academic integrity is that you must fairly represent the source of the intellectual content of the work you submit for credit. In the context of Unified Engineering, this means that if you consult other sources (such as fellow students, TAs, faculty, literature, old problems from prior years) in the process of completing homework, system problems, and labs, or any other assignment, you must acknowledge the sources in any way that reflects true ownership of the ideas and methods you used.

Doing homework helps to engage with the concepts and material taught in class on a deeper level. To enhance the learning process, we strongly suggest that you first try to solve the problems by yourself and then discuss challenges in groups or in office hours if necessary. Discussion among students and office hours to digest the material and the homework problems or to prepare for laboratories or exams is considered useful in the educational process and copies of problems and exams of previous years will be made available by the instructors on the Unified website. Collaboration on homework is allowed unless otherwise directed as long as all references (both literature and people) used are named clearly at the end of the assignment. Word-by-word copies of someone else’s solution or parts of a solution handed in for credit will be considered cheating unless there is a reference to the source for any part of the work that was copied verbatim. Failure to cite the contribution of another student to your homework solution will be considered cheating. Official Institute policy regarding academic honesty can be found in the MIT Bulletin Course and Degrees Issue under “Academic Procedures and Institute Regulations.” The Academic Integrity Handbook, a guide for students published by the Office of the Dean for Undergraduate Education, contains additional information that may be helpful. 

Unified Study Group Guidelines

Study groups are considered an educationally beneficial activity. However, at the end of each problem on which you collaborated with other students, you must cite the students and the interaction. The purpose of this is to acknowledge their contribution to your work. Some examples follow:

  1. You discuss concepts, approaches, and methods that could be applied to a homework problem before either of you start your written solution. This process is encouraged. You are not required to make a written acknowledgment of this type of interaction.
  2. After working on a problem independently, you compare answers with another student, which confirms your solution. You must acknowledge that the other student’s solution was used to check your own. No credit will be lost due to this comparison if the acknowledgment is made.
  3. After working on a problem independently, you compare answers with another student, which alerts you to an error in your own work. You must state at the end of the problem that you corrected your error on the basis of checking answers with the other student. No credit will be lost due to this comparison if the acknowledgment is made and if no direct copying of the correct solution is involved.
  4. You and another student work through a problem together, exchanging ideas as the solution progresses, but you each write up your own solution. Each of you must state at the end of the problem that you worked jointly. No credit will be lost due to this cooperation if the acknowledgment is made.
  5. You copy all or part of a solution from a reference such as a textbook or the homework archive available on the Unified website. You must cite the reference. Partial credit will be given since there is some educational value in reading and understanding the solution. However, this practice is strongly discouraged and should be used only when you are unable to solve the problem without assistance.
  6. You copy verbatim all or part of a solution from another student. This process is not considered academically dishonest if the acknowledgment is made. However, you will receive no credit for verbatim copying from another student as you have not made any intellectual contribution to the work you are submitting for credit.
  7. VERBATIM COPYING OR USE OF ANY MATERIAL WHICH YOU SUBMIT FOR CREDIT WITHOUT REFERENCE TO THE SOURCE IS CONSIDERED TO BE ACADEMICALLY DISHONEST and will be treated as such.

Grades

The rules of the MIT faculty define grades in terms of the degree of mastery of course material. These definitions are listed in the Bulletin and will be applied in this course. The following is the MIT policy on assigning grades to students.

Passing Grades: Undergraduate and graduate students who satisfactorily complete the work of a subject by the end of the term receive one of the following grades:

  • A: Exceptionally good performance demonstrating a superior understanding of the subject matter, a foundation of extensive knowledge, and skillful use of concepts and/or materials.
  • B: Good performance demonstrating the capacity to use the appropriate concepts, a good understanding of the subject matter, and an ability to handle the problems and materials encountered in the subject.
  • C: Adequate performance demonstrating an adequate understanding of the subject matter, an ability to handle relatively simple problems, and adequate preparation for moving on to more advanced work in the field. 
  • D: Minimally acceptable performance demonstrating at least partial familiarity with the subject matter and some capacity to deal with relatively simple problems, but also demonstrating deficiencies serious enough to make it inadvisable to proceed further in the field without additional work.

Use of letter grades on submitted work and participation: The work you submit and overall participation in Unified will receive a numeric grade based on the MIT grade point scale as described below. Specifically, you will receive grades for:

  • each individual problem on a problem set
  • each exam
  • lab assignments

The mapping from numerical to letter grades is as follows: 5 = A, 4 = B, 3 = C, 2 = D, 0 = F. +/- modifiers are set at 0.2 grade points. For example, your grade on a midterm might be a 4.4.  This grade is between an A- (4.8) and B+ (4.2). 

Weights will be assigned to the individual problems on problem sets. Most problems will be of equal unit weight value; however, particularly long and/or challenging problems could be given increased weight value (e.g. a weight of 2 would be twice the default problem value). Using these weights, a weighted overall grade point for homework will be determined.

Unified course grade: The Unified course grade will be defined based on your “Unified grade point average” with a weighting of 20% for problem sets and labs, 20% for quizlets, 25% for the midterm, and 35% for the final exam. Specifically, the final course letter grades will be assigned using this numeric Unified grade point average, as well as considering the distribution of your grades throughout the semester. At a minimum, you will receive a course letter grade that is equal to the maximum letter grade that your numeric course grade is above. For example, if your numeric course grade is 4.3, you would at least receive a B+ for your course letter grade.

Course Info

As Taught In
Fall 2021
Learning Resource Types
Lecture Notes
Problem Sets with Solutions
Exams with Solutions
Labs
Online Textbook
Readings