TEACHING (SPRING 2025)
CE 205 : FINITE ELEMENT METHOD (3:0)
Timing: TBD Venue: TBD
Description
CE 205 is a graduate-level core course which provides an introduction to the finite element method. While the primary focus of the course is the application of FE to solid and structural mechanics, the mathematical foundations of the finite element method will also be visited. A background graduate-level solid mechanics course (CE 204 / ME 242 or equivalent) is recommended. Please email me if you have any questions.
COURSE NEWS AND UPDATES
Watch this space for course-related announcements.Outline
Background: Functionals and the Calculus of variations; Principle of virtual work and Stationary Potential Energy;
Equations of equilibrium as Euler-Lagrange equations of Π.
Structural FE: A cycle of FE using rod elements; Nodal and element quantities ; Interpolation; Shape functions and their properties; element stiffness matrices using PSPE and their properties; assembly; boundary conditions; solution; member fields and forces. Coordinate transformations; Plane truss elements; Lagrange interpolation formulae; C1 continuity and Euler-Bernoulli beam elements; consistent nodal loads; plane frame and space frame elements. Constraints as optimization problems via the Penalty method and Lagrange multipliers.
Linear elasticity: FE Formulation; 2D Continuum Elements (CST, Q4); Isoparametric elements; Gauss quadrature and reduced integration; Higher-order elements (T6, Q8, and Q9) and element selection; Element pathologies (parasitic shear, locking, hour-glassing); interelement compatibility; consistent nodal loads for distributed loads and body forces; symmetry in FE. Singularities in FE. Error estimates and convergence. h-,p-, and r- refinement. Zienkiewicz-Zhu criterion.
Overview of the foundations of FE: Weak derivatives; weak and strong forms; L2, Sobolev, and Hilbert spaces and the Riesz representation theorem. Ritz-Galerkin approximation. Galerkin orthogonality. Ritz approximate problem. Computational aspects – Solution of sparse linear systems, ill-conditioning.
TEACHING (FALL 2024)
CE 297 : PROBLEMS IN THE MATHEMATICAL THEORY OF ELASTICITY (3:0)
Timing: Tu Thu 3:30 pm Venue: Civil Annex Room 205
Description
CE 297 is a graduate-level elective course which provides an in-depth look at a variety of important problems in the theory of elasticity using powerful tools from complex analysis, boundary value methods, and potential theory. The course is aimed at a broad audience of graduate students from civil, mechanical, and aerospace engineering.
A background graduate-level solid mechanics course (CE 204 / ME 242 or equivalent) is recommended. Please email me if you have any questions.
COURSE NEWS AND UPDATES
October 28th: Click here to download homework #5.
October 5th: Click here to download homework #4.
September 18th: Click here to download homework #3.
September 18th: There will be a special lecture meeting of CE 297 at 3:30 pm on Saturday, 21st September on TEAMS.
September 17th: The first mid-term exam of CE 297 will be held at 3:30 pm on Saturday, 28th September 2024 in the usual meeting place. The exam is closed book, closed notes. The time and venue will be confirmed early next week.
September 1st: Click here to download homework #2.
August 18th: Click here to download homework #1.
August 2nd: The first lecture of CE 297: Probems in the Mathematical Theory of Elasticity will be held on Tuesday, 6th August 2024. The venue is Room 205 in the Civil Annex Building.
July 27th: Watch this space for course-related announcements.
Outline
Brief review of linear elasticity; Introduction to the analysis of functions of one complex variable; multi-functions, branch cuts; holomorphic, meromorphic, and sectionally holomorphic functions; analytic continuation; singularities; monodromy; Laurent series, contour integrals, generalized Cauchy integral formulae; Schwarz reflection formulae; Cauchy integral and its properties; Plemelj-Sokhotskii formulae; integration of linear PDEs in n-connected regions; conformal mapping. Homogeneous and non-homogeneous Riemann-Hilbert problems on arcs and contours.
Kolosov-Muskhelishvili formulation on simply-connected and n-connected regions; infinite and semi-infinite regions; complex form of stresses, displacements, tractions and forces.
Analysis of selected problems: Kelvin's problem; singular and distributed solutions for halfplanes, disks, and plates with holes; stressed plate with elliptic and hypotrochoidal holes; partially loaded and reinforced holes; first boundary value problem for a halfplane; frictional and frictionless contact of a rigid punch and halfplane; the slit infinite plane and crack problems; multivalued displacements and dislocations; inclusions.
TEACHING (SPRING 2024)
CE 205 : FINITE ELEMENT METHOD (3:0)
Timing: Tu Thu 3:30 pm Venue: Civil Annex Room 205
Description
CE 205 is a graduate-level core course which provides an introduction to the finite element method. While the primary focus of the course is the application of FE to solid and structural mechanics, the mathematical foundations of the finite element method will also be visited. A background graduate-level solid mechanics course (CE 204 / ME 242 or equivalent) is recommended. Please email me if you have any questions.
COURSE NEWS AND UPDATES
April 18th: The final exam of CE 205 will be held at 3:00 pm on Monday, 29th April 2024 in GJ Hall of the Civil Main Building. The exam is closed book, closed notes.March 25th: Click here to download the project assignment.
March 11th: Click here to download homework #4.
February 12th: Click here to download homework #3.
February 3rd: The first mid-term exam of CE 205 will be held at 5:00 pm on Saturday, 17th February 2024 in GJ Hall. The exam is closed book, closed notes.
January 30th: Click here to download homework #2.
January 30th: A class TEAM has been created for assignment submission. All assignments should be submitted as a single pdf file.
January 19th: Click here to download homework #1.
January 11th: The TA for this course is Naresh C.
December 31st: The first lecture of CE 205: Finite Element Method will be held on Thursday, 4th January 2024. The venue will be announced shortly.
Outline
Principles of FEM and applications to solid and structural mechanics.
TEACHING (FALL 2023)
CE 284 : PLATES, SHELLS, AND GEOMETRIC ELASTICITY (3:0)
Timing: Tu Thu 3:30 pm Venue: STLH Lecture Room
Description
CE 284 is a graduate-level elective which aims to provide a broad and modernized introduction to the theory of plates and shells. All requisite mathematical tools will be covered in-course.
A background graduate-level solid mechanics course (CE 204 / ME 242 or equivalent) is recommended. Please email me if you have any questions.
COURSE NEWS AND UPDATES
November 26th: Click here to download homework #5, part II.
November 15th: Click here to download homework #5, part I.
November 1st: Click here to download homework #4.
October 14th: Click here to download homework #3.
September 28th: Click here to download homework #2.
August 20th: Click here to download homework #1.
ANNOUNCEMENT: The TA for the class is Naresh C.
ANNOUNCEMENT:The first lecture of CE 284 will be held on Tuesday, 8th August at 3:30 pm in the STLH lecture room.
Outline
Review of elasticity and variational principles; classical plate theories; shear plates; elements of large deflection of thin plates; applications of plate theories; differential geometry of surfaces in R^3; curvature; shell theories; engineering applications. Other topics as time permits
TEACHING (SPRING 2023)
CE 228 : INTRODUCTION TO THE THEORY OF PLASTICITY (3:0)
Timing: Tu Thu 11:30 am - 1:00 pm Venue: STLH, Structures Lab
Description
CE 228 is a graduate-level plasticity theory course designed for a broad audience of engineering students. While there are no formal prerequisites, a background grad-level course in solid mechanics (CE 204, ME 242, or an equivalent) is desirable.
COURSE NEWS AND UPDATES
April 13th: As announced in class, the final exam of CE 228 will be held at 10:00 am on Tuesday, 25th April 2023 in the STLH lecture room. The exam is closed book, closed notes. The syllabus is everything covered in class this semester.
April 13th: Reminder: The project assignment submission deadline is Friday, 21st April 2023.
March 22nd: Click here to download homework #4.
March 7th: Click here to download homework #3.
February 15th: Click here to download homework #2.
February 9th: The first mid-term exam of CE 228 will be held at 4:00 pm on Saturday, 25th February 2023 in the STLH lecture room. The exam is closed book, closed notes. The syllabus is all material from Lecture #1 up to and including Lecture #12.
January 21st: Click here to download homework #1.
ANNOUNCEMENT: The TA for the class is Anupama S.
ANNOUNCEMENT: The first class of CE 228 will be held at 11:30 am on Thursday, 5th January in the STLH lecture room.
Outline
1D plasticity and viscoplasticity; dislocations and the physical basis of plasticity; uniaxial tensile test & Bauschinger effect; structure of phenomenological plasticity theories; internal variables; yield criteria (Tresca, von Mises, Mohr-Coulomb, Drucker-Prager); geometry of yield surfaces; Levy Mises equations; flow rules; plastic/ viscoplastic potentials; consistency condition; isotropic and kinematic hardening; Drucker's postulate; Principle of maximum plastic dissipation; associativity; convexity; normality; uniqueness; selected elastic-plastic boundary value problems (tension and torsion of tubes and rods, pressurized thin and thick spherical shells); collapse; advanced hardening models; introduction to computational plasticity; integration of plasticity models; return mapping; principle of virtual work; overview of finite elements for plasticity; other topics as time permits
References
Plasticity Theory - J. Lubliner
Plasticity for Engineers - C. R. Calladine
Theory of Plasticity - J. Chakrabarty
Click here for earlier courses.