ES 221: Mechanics of Solids - Fall 2024

• To learn how materials behave in engineering applications.
• To learn how small and large natural and engineered systems remain stable/fail.
• To get an idea about design and analysis of different systems such as dams, buildings, bridges, etc.

• Free body diagram, Modeling of supports, Conditions for Equilibrium.
• Friction Force-deformation relationship and geometric compatibility (for small deformations) with illustrations through simple problems on axially loaded members and thin walled pressure vessels.
• Axial force, shear force, bending moment, and twisting moment diagrams of slender members.
• Concept of stress and strain at a point, Transformation of stresses and strain at a point, Principal stresses and strains, Mohr’s circle (only for plane stress and strain case).
• Displacement field, Strain Rosette, Modeling of problem as a plane stress or plane strain problem.
• Discussion of experimental results on 1-D material behavior.
• Concepts of elasticity, plasticity, strain-hardening, failure (fracture/yielding), idealization of 1-D stress-strain curve, Concepts of isotropy, orthotropy, anisotropy.
• Generalized Hooke’s law, (without and with thermal strains).
• Torsion of circular shafts and thin-walled tubes.
• Bending of beams with symmetric cross-section (normal and shear stresses), Combined stresses, Yield criteria, Deflection due to bending.
• Integration of the moment-curvature relationship for simple boundary conditions, Super position principle.
• Concepts of strain energy and complementary strain energy for simple structural elements (those under axial load, shear force, bending moment, and torsion).
• Castigliano’s theorems for deflection analysis and indeterminate problems.
• Concept of elastic instability and a brief introduction to column buckling and Euler’s formula.

• B1: Engineering Mechanics: Statics - R.C. Hibbeler (14th edition).
• B2: Mechanics of Materials - R.C. Hibbeler (10th edition).

• Strength of Materials - S. Timoshenko.

• Good background in Mathematics.
• Curiosity to learn the fundamentals behind construction of some of the largest structures like Hoover Dam and Burj Khalifa.

• Please be considerate about everyone's time.
• In all emails pertaining to this course, please have "ES221" in the subject line.
  • (note that there is no space or hyphen or anything between ES and 221)

Cheating cases (assignments/codes/exams/project) will be dealt with in accordance with the Institute norms. It is expected that everyone will uphold the honor code.

Following will be the weightage of different components of assessment

• Tutorials will involve two types of problems:
  • Set A: to be submitted within the tutorial session and will be graded towards Tutorial weightage
  • Set B: to be practiced outside the tutorial hour and will not be graded
• Assignments will entail analysis/practical problems, reports, etc. and may be individual or group
• Expect one assignment and tutorial per week.
• All assignments and tutorials can be downloaded from this google folder.

It is important to develop the habit of self-learning. A number of reading assignments and self-exercises will be given during the course. These will not be formally graded and it will be expected that students will go through them on a regular basis on their own.

• Introduction to the subject, solids vs. fluids, rigidity vs. flexibility. Failure modes of solids.
• Fundamental vs. derived quantities, dimensional analysis.
• SI units (base and derived).
• Significant figures and scientific notation, rounding off.
• Newton's laws of motion - first, second and third.
• Force systems - coplanar, collinear, concurrent.
• Moment of force.
• Resultant force and moment. Equations of static equilibrium.
• Reading from the book (B1): Chapters 1-4.

• Review of topics from Class XI.

• Moment of a force, couple moment. Equations of static equilibrium.
• Idealization of supports - fixed, roller, hinged.
• External and internal forces.
• Examples of determining support reactions and internal forces.
• Reading from the book (B1): Chapter 5.

• Two-force members.
• Introduction to truss structures.
• Analysis of pin-jointed trusses by method of joints.
• Reading from the book (B1): Chapters 5 & 6.

• In-class submission C1
• Analysis of pin-jointed trusses by method of sections.
• Reading from the book (B1): Chapter 6.

• Classification and idealization of structural members by geometry (1D, 2D, 3D).
• Classification and idealization of structural members by resisting action (tie, strut, beam, column, shaft).
• Bending moment and shear forces in beams. Sign convention for bending moment.
• Reading from the book (B1): Chapter 7.

• Class cancelled.

• Analysis of pin-jointed trusses.

• Assignment 1 given (due on 30 Aug).
• Bending moment and shear force diagrams. Sign convention for shear force.
• Reading from the book (B1): Chapter 7.

• Bending moment and shear force diagrams.
• Reading from the book (B1): Chapter 7.

• Bending moment and shear force diagrams.

• Assignment 2 given (due on 6 Sep).
• Concept of stress. Normal and shear stresses. General state of stress. Sign convention.
• Area as a vector. Average stresses. Examples of finding normal stress.
• Concept of strain. Normal and shear strain. General state of strain. Sign convention.
• Average strain. Examples of finding normal strain.
• Reading from the book (B2): Chapters 1 and 2.

• Material properties. Relations between stress and strain.
• Young's modulus, shear modulus, Poisson ratio.
• Hooke's law.
• Reading from the book (B2): Chapter 3.

• Tutorial shifted to Sep 06 class.

• Tutorial 4.
• Assignment 3 given (due on 13 Sep).

• Saint Venant principle.
• Axial deformations of a bar.
• Principle of superposition.
• Degree of static indeterminacy; statically indeterminate and determinate system.
• Use of geometric compatibility to solve statically indeterminate systems.
• Reading from the book (B2): Chapter 4.

• Consideration of temperature changes through coefficient of thermal expansion.
• Strain decomposition into mechanical and thermal parts.
• Stresses and strains due to changes in temperature.
• Reading from the book (B2): Chapter 4.

• Axial deformations and temperature changes of bars.

• Consideration of torsion of bars.
• Useful video: https://www.youtube.com/watch?v=1YTKedLQOa0
• Angle of twist, shear strain and stress due to torsion.
• Polar moment of inertia.
• Reading from the book (B2): Chapter 5.
• In-class submission C2

• No lecture due to Monday's time table being followed.

• Torsion and uniaxial forces in bars.

• Normal strains and stresses due to bending.
• Derivation of the flexure formula and underlying discussions.
• First and second moments of area.
• Assignment 4 given (due on 27 Sep).

• Normal stresses due to bending.
• Centroid and moment of area of different shapes.

• Normal stresses due to bending.

• Shear stresses in beams. Shear formula.
• Transverse and longitudinal shear stress.
• Reading from the book (B2): Chapter 7.

• Normal and shear stresses in bending.
• Assignment 5 given (due on 25 Oct).

• Stress at a point.
• Consideration of combined states of stress.
• Sign conventions for positive/negative planes and positive/negative stresses.
• Transformation of stresses in 2D.
• Reading from the book (B2): Chapter 9.

• Transformation of stress in 2D.
• Mohr's circle for 2D stress transformation.
• Principal stresses and maximum shear stress.
• Problems related to stress transformation.
• Reading from the book (B2): Chapter 9.

• General state of plane stress and equilibrium equations.
• General state of plane strain and strain-displacement relations.
• Mohr's circle for plane strain.
• Generalized Hooke's law for triaxial loading and thermal effects.
• Reading from the book (B2): Chapters 9 and 10.
• In-class submission C3

• Triaxial state of stress.
• Mohr's circle for triaxial state of stress.
• Absolute maximum shear stress.
• Dilatation vs. distortion. Definition of bulk modulus and its relation with Young's modulus.
• Theoretical limits for Poisson's ratio.
• Introduction to failure theories
  • Maximum shear stress theory (Tresca criteron)
  • Maximum distortion energy theory (Mises-Huber criterion)
  • Maximum normal stress theory (Mohr-Coulomb criteron)
• Reading from the book (B2): Chapter 10.

• Strain and stress transformation in 2D, triaxial state of stress.
• Assignment 6 given (due on 4 Nov).

• Failure theories
  • Maximum shear stress theory (Tresca criteron)
  • Maximum distortion energy theory (Mises-Huber criterion)
• Reading from the book (B2): Chapter 10.

• Maximum normal stress failure theory (Mohr-Coulomb criteron)
• Useful video on failure theories: https://www.youtube.com/watch?v=xkbQnBAOFEg
• Reading from the book (B2): Chapter 10.

• Strain-displacement relationship and thin-walled pressure vessels.
• Assignment 7 given (due on 11 Nov).

• Thin-walled pressure vessels.
• Moment-curvature relationship in beams. Elastic curve.
• Deflections of beams.
• Reading from the book (B2): Chapters 8 and 12.

• No class due to Amalthea.

• Deflections of beams.
• Assignment 8 given (due on 18 Nov).

• Deflections of beams.
• Macaulay functions / singularity functions.
• Reading from the book (B2): Chapter 12.

• Strain energy.
• Conservation of energy.
• Castigliano's theorems.
• Deflections of indeterminate problems.
• Reading from the book (B2): Chapter 14.

• Singularity method and Energy methods.
• Assignment 9 given (due on 22 Nov).

• Stability of equilibrium.
• Buckling of slender members.

• Review of topics.