ENME E

Subject

ENME E

Type

course

Edition

2025-2026

Source

bulletin.columbia.edu

Corequisites: ENME E3106,ENME E3113

Subject

ENME E

Credits (min)

4

Credits (max)

4

Credit unit

points

Type

course

Edition

2026

Source

bulletin.columbia.edu

Prerequisites: (MATH UN1201) MATH V1201; ENME E3105 Corequisites: ENME E3105. Kinematics of rigid bodies; momentum and energy methods; vibrations of discrete and continuous systems; eigen-value problems, natural frequencies and modes. Basics of computer simulation of dynamics problems using MATLAB or Mathematica

Subject

ENME E

Credits (min)

3

Credits (max)

3

Credit unit

points

Type

course

Edition

2026

Source

bulletin.columbia.edu

Prerequisites: (ENME E3105) or ENME E3105; or equivalent. Stress and strain. Mechanical properties of materials. Axial load, bending, shear, and torsion. Stress transformation. Deflection of beams. Buckling of columns. Combined loadings. Thermal stresses

Subject

ENME E

Credits (min)

3

Credits (max)

3

Credit unit

points

Type

course

Edition

2026

Source

bulletin.columbia.edu

Prerequisites: (ENME E3113) ENME E3113 Material behavior and constitutive relations. Mechanical properties of metals and cement composites. Structural materials. Modern construction materials. Experimental investigation of material properties and behavior of structural elements including fracture, fatigue, bending, torsion, buckling

Subject

ENME E

Credits (min)

4

Credits (max)

4

Credit unit

points

Type

course

Edition

2026

Source

bulletin.columbia.edu

Prerequisites: (ENME E3105) and ENME E3105; and ordinary differential equations. Fluid statics. Fundamental principles and concepts of flow analysis. Differential and finite control volume approach to flow analysis. Dimensional analysis. Application of flow analysis: flow in pipes, external flow, flow in open channels

Subject

ENME E

Credits (min)

4

Credits (max)

4

Credit unit

points

Type

course

Edition

2026

Source

bulletin.columbia.edu

Prerequisites: Senior standing or instructor's permission. Recommended corequisite: differential equations. Corequisites: (Recommended): differential equations Focus on formulation and application of the finite element to engineering problems such as stress analysis, heat transfer, fluid flow, and electromagnetics. Topics include finite ele?ment formulation for one-dimensional problems, such as trusses, electrical and hydraulic systems; scalar field problems in two dimensions, such as heat transfer; and vector field problems, such as elasticity and finally usage of the commercial finite element program. Students taking ENME E3332 cannot take ENME E4332

Subject

ENME E

Credits (min)

3

Credits (max)

3

Credit unit

points

Type

course

Edition

2026

Source

bulletin.columbia.edu

Stress and deformation formulation in two-and three-dimensional solids; viscoelastic and plastic material in one and two dimensions energy methods

Subject

ENME E

Credits (min)

3

Credits (max)

3

Credit unit

points

Type

course

Edition

2026

Source

bulletin.columbia.edu

Prerequisites: Undergraduate mechanics of solids course.

Subject

ENME E

Credits (min)

3

Credits (max)

3

Credit unit

points

Type

course

Edition

2026

Source

bulletin.columbia.edu

Prerequisites: (ENME E4113) or ENME E4113; Or instructor's approval. An introduction to the constitutive modeling of composite materials: Green’s functions in heterogeneous media, Eshelby’s equivalent inclusion methods, eigenstrains, spherical and ellipsoidal inclusions, dislocations, homogenization of elastic fields, elastic, viscoelastic and elasto-plastic constitutive modeling, micromechanics-based models

Subject

ENME E

Credits (min)

3

Credits (max)

3

Credit unit

points

Type

course

Edition

2026

Source

bulletin.columbia.edu

Prerequisites: Linear and tensor algebra. Differential and integral calculus. Mechanics.

Subject

ENME E

Credits (min)

3

Credits (max)

3

Credit unit

points

Type

course

Edition

2026

Source

bulletin.columbia.edu

Prerequisites: (ENME E3105) or ENME E3105 Differentiation of vector functions. Review of kinematics. Generalized coordinates and constraint equations. Generalized forces. Lagranges equations. Impulsive forces. Collisions. Hamiltonian. Hamiltons principle

Subject

ENME E

Credits (min)

3

Credits (max)

3

Credit unit

points

Type

course

Edition

2026

Source

bulletin.columbia.edu

Prerequisites: ENME E3113 Experimental techniques, including photoelasticity, strain measurements, digital image correlation, and other optics in stress analysis, emphasizing engineering applications. Fundamental concepts in solid mechanics and measurement science. Performance of high-quality experimental studies

Subject

ENME E

Credits (min)

3

Credits (max)

3

Credit unit

points

Type

course

Edition

2026

Source

bulletin.columbia.edu

Prerequisites: (ENME E3113) ENME E3113 Static flexural response of thin, elastic, rectangular, and circular plates. Exact (series) and approximate (Ritz) solutions. Circular cylindrical shells. Axisymmetric and non-axisymmetric membrane theory. Shells of arbitrary shape

Subject

ENME E

Credits (min)

3

Credits (max)

3

Credit unit

points

Type

course

Edition

2026

Source

bulletin.columbia.edu

Frequencies and modes of discrete and continuous elastic systems. Forced vibrations-steady-state and transient motion. Effect of damping. Exact and approximate methods. Applications

Subject

ENME E

Credits (min)

3

Credits (max)

3

Credit unit

points

Type

course

Edition

2026

Source

bulletin.columbia.edu

Prerequisites: Mechanics of solids, structural analysis, elementary computer programming MATLAB is recommended, linear algebra and ordinary differential equations. Direct stiffness approach for trusses. Strong and weak forms for one-dimensional problems. Galerkin finite element formulation, shape functions, Gauss quadrature, convergence. Multidimensional scalar field problems (heat conduction), triangular and rectangular elements, Isoparametric formulation. Multidimensional vector field problems (linear elasticity). Practical FE modeling with commercial software (ABAQUS). Computer implementation of the finite element method. Advanced topics. Not open to undergraduate students

Subject

ENME E

Credits (min)

3

Credits (max)

3

Credit unit

points

Type

course

Edition

2026

Source

bulletin.columbia.edu

Prerequisites: (ENME E4332) and ENME E4332; Elementary computer programming, linear algebra. Introduction to multiscale analysis. Information-passing bridging techniques: among them, generalized mathematical homogenization theory, the heterogeneous multiscale method, variational multiscale method, the discontinuous Galerkin method and the kinetic Monte Carlo–based methods. Concurrent multiscale techniques: domain bridging, local enrichment, and multigrid-based concurrent multiscale methods. Analysis of multiscale systems

Subject

ENME E

Credits (min)

3

Credits (max)

3

Credit unit

points

Type

course

Edition

2026

Source

bulletin.columbia.edu

Prerequisites: (ENME E4215) ENME E4215 Concepts, principles, and applications of various sensors for sensing structural parameters and nondestructive evaluation techniques for subsurface inspection, data acquisition, and signal processing techniques. Lectures, demonstrations, and hands-on laboratory experiments

Subject

ENME E

Credits (min)

3

Credits (max)

3

Credit unit

points

Type

course

Edition

2026

Source

bulletin.columbia.edu

Prerequisites: (ENME E4215) and (ENME E4332) ENME E4332 AND ENME E4215 Principles of traditional and emerging sensors, data acquisition and signal processing techniques, experimental modal analysis (input-output), operational modal analysis (output-only), model-based diagnostics of structural integrity, data-based diagnostics of structural integrity, long-term monitoring and intelligent maintenance. Lectures and demonstrations, hands-on laboratory experiments

Subject

ENME E

Credits (min)

3

Credits (max)

3

Credit unit

points

Type

course

Edition

2026

Source

bulletin.columbia.edu

Prerequisites: (CIEN E4111) and (ENME E4215) or CIEN E4111 AND ENME E4215; or equivalent or instructor's permission Review of random variables. Random process theory: stationary and ergodic processes, correlation functions and power spectra, non-stationary, non-white and non-Gaussian processes. Uncertainty quantification and simulation of environmental excitations and material/media properties, even when subject to limited/incomplete data: joint time-frequency analysis, sparse representations and compressive sampling concepts and tools. Stochastic dynamics and reliability assessment of diverse engineering systems: complex nonlinear/hysteretic behaviors and/or fractional derivative modeling. Emphasis on solution methodologies based on Monte Carlo simulation, statistical linearization, and Wiener path integral. Examples from civil, marine, mechanical and aerospace engineering

Subject

ENME E

Credits (min)

3

Credits (max)

3

Credit unit

points

Type

course

Edition

2026

Source

bulletin.columbia.edu

Prerequisites: (ENME E3332) or instructor's permission

Subject

ENME E

Credits (min)

3

Credits (max)

3

Credit unit

points

Type

course

Edition

2026

Source

bulletin.columbia.edu

Prerequisites: (ENME E4332) ENME E4332 FE formulation for beams and plates. Generalized eigenvalue problems (vibrations and buckling). FE formulation for time-dependent parabolic and hyperbolic problems. Nonlinear problems, linearization, and solution algorithms. Geometric and material nonlinearities. Introduction to continuum mechanics. Total and updated Lagrangian formulations. Hyperelasticity and plasticity. Special topics: fracture and damage mechanics, extended finite element method

Subject

ENME E

Credits (min)

3

Credits (max)

3

Credit unit

points

Type

course

Edition

2026

Source

bulletin.columbia.edu

Prerequisites: (ENME E4332) or ENME E4332; or equivalent, elementary computer programming, linear algebra. The formulations and solution strategies for finite element analysis of nonlinear problems are developed. Topics include the sources of nonlinear behavior (geometric, constitutive, boundary condition), derivation of the governing discrete equations for nonlinear systems such as large displacement, nonlinear elasticity, rate independent and dependent plasticity and other nonlinear constitutive laws, solution strategies for nonlinear problems (e.g. incrementation, iteration), and computational procedures for large systems of nonlinear algebraic equations

Subject

ENME E

Credits (min)

3

Credits (max)

3

Credit unit

points

Type

course

Edition

2026

Source

bulletin.columbia.edu

Prerequisites: (ENME E3161) or ENME E3161 Ordinary and partial differential equations. Turbulence phenomenology; spatial and temporal scales in turbulent flows; statistical description, filtering and Reynolds decomposition, equations governing the resolved flow, fluctuations and their energetics; turbulence closure problem for RANS and LES; two equation turbulence models and second moment closures

Subject

ENME E

Credits (min)

3

Credits (max)

3

Credit unit

points

Type

course

Edition

2026

Source

bulletin.columbia.edu

Prerequisites: (MECE E6422) and (MECE E6423) MECE E6422 AND MECE E6423; Open to Ph.D. students and to M.S. students with instructor’s permission.

Subject

ENME E

Credits (min)

4

Credits (max)

4

Credit unit

points

Type

course

Edition

2026

Source

bulletin.columbia.edu

Prerequisites: (ENME E6315) or ENME E6315; or equivalent, or instructor's permission. Constitutive equations of viscoelastic and plastic bodies. Formulation and methods of solution of the boundary value, problems of viscoelasticity and plasticity

Subject

ENME E

Credits (min)

4

Credits (max)

4

Credit unit

points

Type

course

Edition

2026

Source

bulletin.columbia.edu