3 Credit Hours

New or special course on recent developments in some phase of civil engineering. The objective of this course is to introduce students to the basic principles of three-dimensional elasticity theory for the solution of boundary value problems in mechanics. The course provides the background necessary to gain a fundamental understanding of solid mechanics principles in civil engineering applications and introduces variational principles and numerical methods, setting the stage for advanced analysis using finite element methods. This course provides a foundation for finite element methods and advanced mechanics courses.

Prerequisite

Graduate standing required. Students will need to have taken an undergraduate class in mechanics and analysis (CE225, CE325).

Course Topics

1. Mathematical principles
   ● Vector operations (addition/subtraction dot, cross and tensor products)
   ● Tensors
   ● Tensor operations on vectors, addition, multiplication, transpose
   ● Eigenvalue problems (orthogonality, repeated eigenvalues, modal
   decomposition, Cayley-Hamilton theorem)
   ● Tensor calculus: gradients and divergence of scalar, vector and tensor fields
   ● Cartesian, polar and cylindrical coordinates
   ● The divergence theorem
2. 3D Elasticity theory
   ● Deformations: deformation gradient, 3D strain measures in large deformations, linearized strain. Change in area/volume.
   ● Stress and equilibrium: equilibrium of forces/moments at a point, Cauchy stress, first and second Piola-Kirchoff stress, tractions with reference to deformed and undeformed configurations.
   ● Constitutive equations: hyperelasticity, isotropy, linear elasticity, orthotropy. Constitutive equations for 3D elasticity in terms of Lame parameters or Young’s modulus/Poisson ratio or bulk/shear modulus. Bulk and deviatoric stresses/strains. Defining the elasticity matrix for orthotropic materials/plane stress/plane strain.
   ● Dirichlet, Neumann, mixed and Robin boundary conditions.
   ● Strong form of the governing equations: Total vs updated Lagrangian formulations. Small deformations formulation.
   ● Closed-form solutions for 1D problems.
3. Variational forms for elasticity (Energy (E), weighted residual (WR) and virtual work(VW))
   ● WR from SF – weighted residuals, acceptable function spaces.
   ● SF from WR: the fundamental theorem of the calculus of variations.
   ● VW from WR and vice-versa: divergence theorem (integration by parts in 1D).
   ● Energy methods and the VW.
   ● Essential vs natural boundary conditions.
   ● Introduction to numerical discretization (the Ritz method).
   

Learning Outcomes

By the end of this course, you will be able to do the following:

• write equations of deformation, strains, stresses and constitutive equations in 3D
• characterize material behavior under different conditions (isotropy, orthotropy, linear elasticity)
• identify different strains and stress measures
• identify different types of boundary conditions
• make connections to previously known theories such as beams (identify assumptions)
• identify the relationship among the strong, weak, and virtual work forms of the governing equations for elasticity.
• use variational principles and integration theorems to obtain virtual work statements for 1D, plane-stress and plane-strain 2D elasticity
• incorporate essential and natural boundary conditions
• use numerical discretization techniques such as the Ritz method
• solve small (1D) elasticity problems by hand
• solve medium sized problems (1D/2D) using basic Matlab programming
• assess accuracy and convergence in the numerical solution.

Course Requirements

Assignment	Percentage	Details
Homework	10%	the lowest-scoring homework will be dropped
Exams	60%	the lowest-scoring exam will be dropped
Project (final)	30%	/

Grade assignment will follow standard NCSU guidelines.

Textbook(required)

Fundamentals of Structural Mechanics, 2nd Ed., K D. Hjelmstad, 2005. ISBN-13: 978-0387233307 (~ $100.00, also available online).

Created: 09/22/2025.