APL104F24

APL 104: Solid Mechanics (Fall semester 2024)

Course Info

Credit: 4 units (3-1-0)

Instructors: Prof. Rajdip Nayek (rajdipn@am.iitd.ac.in)

Class timings: Tue, Wed & Fri (9:00 to 10:00 AM) at LHC 308

Tutorial Session: Wed (2:00 to 3:00 PM), Block II LT 1

An attendance list of students in tutorial sessions can be found here.

Office hours (TA): Wed 4:00-5:00 pm (in Block 4 Room B-24)

Intended audience: BTech students in Applied Mechanics and Material Science

NOTE-For all course related emails, please put APL104 in the subject line

Lecture Schedule

Module	Topics	Lecture Notes	In-class notes
Module 00	Why learn Solid Mechanics	Introduction
Module 01	Analysis of Mechanical System	Study of forces Equilibrium conditions Additional requirements for deformable bodies
Module 02	Traction	Traction vector Traction decomposition	Mod1 & 2
Module 03	Stress	State of Stress at a point Stress equilibrium equation	Mod3
Module 04	Principal Stresses and Planes	Stress eigenvalue problem Maximum Normal and shear stress	Mod4-1
Module 05	Mohr’s Circle Decomposition of Stress Tensor	2D Mohr’s circle 2D & 3D Mohr’s circle Stress Invariants, Octahedral Stresses, etc.	Mod5
Module 06	Concept of Strain	Normal and Shear Strain Strain-displacement relation Strain compatibility Similarity of properties of stress and strain tensors	Mod6
Module 07	Stress-strain-temperature relations	General stress-strain curves Isotropic linear elastic material and thermal strain	Same as previous year
Module 08	Complete equations of linear elasticity	Equations and boundary conditions for unique solution	Same as previous year
Module 09	Application to extension-torsion-inflation in cylindrical coordinates	Elasticity Equations in cylindrical coordinates Simplified equations Solving simplified equations	Same as previous year
Module 10	Uniform bending of symmetrical beams	Uniform bending Non-uniform bending of symmetrical beams Deflection of beams and buckling	Same as previous year
Module 11	Energy methods	External work and strain energy Castigliano’s 2nd theorem Principle of Virtual Work and Minimum Potential Energy	Same as previous year
Module 12	Failure Theories	Three basic failure theory	Same as previous year

Tutorial Schedule

Topics	Tutorial Questions	Tutorial Solutions
Study of forces	Tutorial 1	Solution
Compatibility equations	Tutorial 2	Solution
Traction and Stress Equilibrium	Tutorial 3	Solution
Principal Stresses and Principal Planes	Tutorial 4	Solution
Mohr’s Circle	Tutorial 5	Solution
Strain	Tutorial 6	Solution
Complete equations of elasticity	Tutorial 7	Solution
Extension-torsion-inflation	Tutorial 8	Solution
Uniform beam bending	Tutorial 9	Solution
Euler-Bernoulli beams & Energy Methods	Tutorial 10	Solution

Table of Contents

• Course Outline
• Course Layout
• Course References
• Grading
• Course Attendance
• Quiz
• Policy for Cheating

Course Outline

This is the first course where the deformation of solid bodies and the underlying concepts are introduced to undergraduate students. The course begins by building a foundation of the concepts of stress and strain in three-dimensional deformable bodies. It further uses these concepts to study the extension, torsion, and bending of beams. The one-dimensional theory of beams is also introduced. Various theories of failure that are critical for the design of machine elements in the industry will also be discussed.

Course Layout

• Fundamental Principles of Mechanics; Introduction of mechanics of deformable bodies
• Stress tensor and its representation in Cartesian coordinate system; Transformation of stress matrix; Equations of equilibrium; Symmetry of stress tensor
• State of stress in simple cases; Principal stress components and principal planes; Maximizing shear component of traction; Mohr’s circle
• Stress invariants; Octahedral Plane; Decomposition of stress tensor; Concept of strain and strain tensor
• Longitudinal, shear, and volumetric strains; Local infinitesimal rotation; Strain compatibility condition
• Linear stress-strain relation for isotropic bodies; Relation between material constants
• Stress and strain matrices in the cylindrical coordinate system; Equations of equilibrium in the cylindrical coordinate system
• Axisymmetric deformations: combined extension-torsion-inflation of a cylinder
• Bending of beams having the symmetrical and non-symmetrical cross-section
• Shear center, Shear flow in thin and open cross-section beams; Euler Bernoulli and Timoshenko beam theories; beam buckling
• Energy methods, Reciprocal relations, Castigliano’s theorem, Deflection of straight and curved beams using the energy method
• Various theories of failure and their application

Course References

This course is based on three textbooks:

• Archer, Cook, Crandall, Dahl, Lardner, McClintock, Rabinowicz, Reichenbach, “An Introduction To The Mechanics Of Solids”, Tata Mcgraw Hill, 2012
• Kumar, Ajeet, “Solid Mechanics for Undergraduates - Using Vectors and Tensors”, White Falcon Publishing, 2024.
• Boresi, Arthur, “Advanced Mechanics of Materials”

Other references

• Solid Mechanics (NPTEL) by Prof. Ajeet Kumar [video link]
• Srinath, L.S., “Advanced Solid Mechanics”, Elsevier, 2018.
• Hibbeler, R. C., “Mechanics of Materials”, Prentice Hall, 2014Timoshenko, S.P. and Goodier, J.N., “Theory of Elasticity”, McGraw Hill, 2017.
• Sadd, M.H., “Elasticity: Theory, Applications and Numerics”, Elsevier, 2005

Grading

Component	Scores	Solutions
Quiz #1	15
Minor	30	Sol
Quiz #2	15
Major	40	Sol
Total	100

Course Attendance

Students are highly encouraged to attend all classes. Students who have failed this course were found to have attended less than 60% of the total classes on an average. If any student has less than 75% tutorial attendance, he/she will get one grade less than would have been awarded. In case of unavoidable absence, such as illness, please send an appropriate email within a week before/after absence with an email subject specifying the subject code APL 104.

Please note that re-quizzes will not be offered for missed quizzes, regardless of the reason.

Retakes will be provided only for Minor and Major exams.

For a missed quiz, your quiz grade will instead be calculated based on your performance in at least TWO other assessments (Minor/Reminor, Major/Remajor, and two Quizzes). If you are only able to appear for only one assessment, you will unfortunately be awarded zero marks on the missed assessments.

Policy for Cheating

Both copiers and copyees are guilty of cheating and will receive an equal penalty. The penalty includes a zero mark on the corresponding exam. Please do not do anything you might regret.