Course Syllabus MAE 520 Stress Analysis Fall 2020 Class Time: MW 9:00-10:15am (WXLR A203 and online: via zoom) Course Objectives: To further develop the students understanding of the concept of stress, strain, stressstrain relationships, strain energy, failure theories, curved beams, unsymmetrical bending, shear center, torsion of noncircular sections, energy principles, Castigliano’s theorem, inelastic behavior. Instructors: Dr. K.N. Solanki, ERC 351, (480) 965-1869, kiran.solanki@asu.edu Office Hours: TW 1:00pm-2:00pm via zoom or by appointment Reference Textbook: Arthur P. Boresi and Richard J. Schmidt; Advanced Mechanics of Materials, 6th Edition, John Wiley & Sons, Inc Tentative Grading Scheme: Homework Midterm exam final exam (comprehensive) 50% 20% (first week of October) 30% (TBD) A: 90-100; B: 80-89.9; C: 70-79.9; D: 60-69.9; F: < 60 (subjected to change) Homework: Assigned and collected each week through the gradescope. If you wish to dispute a homework or exam grade that you received, you must inform your instructor in writing no more than one week after the assignment or exam was returned. Attendance Policy: If you anticipate not being able to attend any face to face courses due to travel restrictions or personal health concerns, you are expected to contact me at the beginning of the semester or as soon as your situation changes to notify me of this fact. Unless you notify me in advance, you will attend some class sessions in-person and be remote for other sessions to ensure we keep the room occupancy below 50%. On Mondays, students with last names beginning with A-M will meet in-person while the other half of the class participates in the live class on Zoom via ASU Sync. On Wednesdays, students with last names beginning with N-Z will meet in-person while the other half of the class participates in the live class on Zoom via ASU Sync. Excused absences related to religious observances/practices that are in accord with ACD 304–04, “Accommodation for Religious Practices” Excused absences related to university sanctioned events/activities that are in accord with ACD 304–02, “Missed Classes Due to University-Sanctioned Activities” Excused absences related to missed class due to military line-of-duty activities that are in accord with ACD 304–11, “Missed Class Due to Military Line-of-Duty Activities,” and SSM 201–18, “Accommodating Active Duty Military”. Make up assignments and exams will be given for absences that are due to religious observances/practices that are approved by university policy or due to other events or activities sanctioned by the university. In those cases, the student must notify the instructor in writing before the assignment is due (preferably at least 1 week ahead) so that arrangements can be made. If an absence is not due to a university sanctioned reason, missed quizzes or exams will not be made up unless it is due to health issues (a doctor note will be required), or a personal emergency that is appropriately justified. Arrangements need to be made within 1 week following the absence, including a meeting with the instructor to provide justification for it. If there is no attempt on the part of the student to make these arrangements in this period, the missed assignment will receive zero points. ASU Sync Courses This course uses Sync. ASU Sync is a technology-enhanced approach, designed to meet the dynamic needs of the class. During Sync classes, students learn remotely through live class lectures, discussions, study groups and/or tutoring. You can find out more information about ASU Sync for students here, https://provost.asu.edu/sync/students and https://www.asu.edu/about/fall-2020. Detailed related to how to access or attend online classes, such as zoom link, etc., will be posted on the course website. General Class Policies: A set of homework problems will be assigned weekly. Following homework collection, full solutions to all homework problems will be made available via the web. No late homework will be accepted Homework must be written and organized in a professional manner or points will be deducted. When grading your homework, I should be able to trace your thought process throughout the homework problem. Comment your solutions as you work! Please remember to put your name on top of your completed assignment. Make sure you upload your solution as a single file in the format specified in the assignment statement, e.g., word or pdf, and that your homework is neat, legible, professional and wellorganized Test /Grading Discrepancies: When tests or assignments are returned to you, you have a period of one week to bring any grading discrepancies to me. 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The notes are anonymous and I take them seriously - at least most of them! Tentative Course Outline: 1. tensor algebra 2. stress and strain at a point 3. constitutive equations 4. plasticity and failure theories 5. energy methods 6. torsion 7. nonsymmetrical bending of straight beams (2) (6) (2) (4) (4) (6) (4) 8. 9. 10. 11. 12. fatigue analysis of cracked bodies: fracture mechanics shear center for straight beams composite beams elastic instability (2) (2) (2) (1) (4) MAE 520 Stress Analysis Review of Mathematical Fundamentals Learning Objectives Introduction • It is common to use vectors and tensors to define and describe physical laws such as mechanical properties. • They allow us to describe materials properties independent of the selected reference (i.e., coordinate) system. • It is assumed that most students have had some prior exposure to linear algebra • In this lecture we will review just a few of the mathematical principles that will be used in this course. Scalars, Vectors, Matrices, and Tensors • Scalars: – Represent a single magnitude at each point in space. • Vectors: – Line segments that are expressible in terms of components resolved onto a three-dimensional coordinate system. • Matrices: – Three or more components are needed to quantify. • Tensors: – Physical quantities that are associated with coordinates but which do not depend on those coordinates. – You can think of them as “special” forms of scalars, vectors, or matrices that represent physical properties. SCALAR : mass, m density, Scalars are directionless numerical quantities. Scalars are 0th order tensors VECTOR : displacement, u uxex u y e y u z e z Vectors are directional quantities. Vectors are 1st order tensors that can be represented as column or row matrices. 2nd order tensor : Matrices are rectangular arrays of numbers. This particular matrix is a 2nd order tensor. A second order tensor relates two vectors. MATRIX : stiffness matrix = C Cijkl (This matrix is a 4th order tensor. It relates two 2nd order tensors) ij Cijklkl We will address this specific relationship in more detail later. Let’s review just a few relevant mathematical relations (that you should already know)! Index Notation • Used to represent entire sets of numbers, equations, elements or components using single symbols with subscripts. • A symbol Uijk…n with n distinct subscripts (aka n distinct indices) represents 3n distinct components. Index Notation – cont’d • For the sake of consistency, let s generalize our subscripts. For matrices such as Ai or Aij:* ’ • We will always assume that subscript i represents the row entry while subscript j represents the column entry. This applies when we use matrix notation. Higher order tensors require a slightly different treatment because they will have more than two subscripts. *Assume than anything w/ a single subscript is a vector; thus the subscript corresponds to a row or column entry General rules – cont’d • Simple relations like Ai = Bi and Aij = Bij imply that: A1 = B1, A2 = B2, … ; A11 = B11, A12 = B12, …; etc. • Relations like Ai = Bj and Aij = Bkl are ambiguous. They are to be avoided because the indices on each term are not the same. • In general, distinct subscripts on all individual terms in an equation must match. • If a subscript appears twice in the same term or product of terms, summation over that subscript from one to three is implied. See below. 3 Aii Aii A11 A22 A33 i1 3 Aij B j Aij B j Ai1B1 Ai 2B2 Ai3B3 (Produces a 31 matrix) j1 3 Aij Bjk Aij Bjk Ai1B1k Ai 2B2k Ai3B3k (Produces a 99 matrix) j1 • Repeated subscripts sometimes called “dummy subscripts. ” The other subscripts are “free subscripts.” • All commutative, associative, and distributive laws are obeyed (assuming that the sizes of the matrices are such that the indicated operations can be performed). Ai Bi Bi Ai (commutative law for addition) Ai Bi Ci Ai Bi Ci Ai B jk Cl Ai B jk Cl (associative law for addition) (associative law for multiplication) Ai Bi Ci Ai Bi Ci (distributive law) AijBk Bk Aij Aij Bk Ck Aij Bk A C ij k Bk Ck Bk Ck • See your mathematical methods book for a complete listing of laws.
