2023 Faculty Courses School of Environment and Society Department of Civil and Environmental Engineering Graduate major in Urban Design and Built Environment
Tensor Analysis for Building Structure
- Academic unit or major
- Graduate major in Urban Design and Built Environment
- Instructor(s)
- Tadashi Ishihara
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 5-6 Tue (J2-304(J231))
- Class
- -
- Course Code
- UDE.S406
- Number of credits
- 100
- Course offered
- 2023
- Offered quarter
- 4Q
- Syllabus updated
- Jul 8, 2025
- Language
- English
Syllabus
Course overview and goals
In this lecture, tensors, an important concept in continuum mechanics, will be explained. First, tensor algebra as a basic theory will be presented and the characteristics of symmetric and skew-symmetric tensors will be explained. Then, examples of concrete tensors such as the strain tensor and stress tensor will be given. Finally, the treatment of tensors in the curvilinear coordinate system as advanced tensor analysis will be briefly touched upon.
Since building structures are mainly composed of frames such as columns and beams, a single-axis (one-dimensional) stress-strain relationship is often sufficient, as in structural mechanics or material mechanics. However, structures are three-dimensional in nature, and it is necessary to understand tensors in order to understand local three-dimensional stresses and strains and to master the basic theory of the finite element method. I hope that the students will understand the abstract concept of tensors and feel the necessity and usefulness of tensors through examples.
Course description and aims
By the end of this course, students will be able to:
1) Understand concept of tensor variables and difference from scalar or vector variables.
2) Understand the reason why the tensor analysis is used and explain usefulness of the tensor analysis.
3) Derive strain tensors, etc.
Keywords
Tensor, Continuum mechanics, Earthquake engineering
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Students will be asked to work on exercises.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Outline of this lecture | Concept of tensor and usefulness of tensor analysis, Direct notation |
| Class 2 | Basic tensor theory (1) | Tensor algebra, Principal invariants, Tensor product, Proper vectors/numbers of tensors |
| Class 3 | Basic tensor theory (2) | Symmetric tensors, Skew-symmetric tensor, Orthogonal tensor |
| Class 4 | Basic tensor theory (3) | Polar decomposition, Tensor fields, Integral theorems |
| Class 5 | Strain tensor | Definition of Green's strain tensor and difference of it from engineering strain |
| Class 6 | Stress tensor | Definitions of stress tensors and relation to strain tensors |
| Class 7 | Tensors in earthquake engineering | Input energy tensors, Effective mass tensors |
| Class 8 | Advanced tensor analysis | Tensors in curvilinear coordinate systems |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
None
Reference books, course materials, etc.
Lecture materials will be distributed as needed.
Evaluation methods and criteria
Students' knowledge of tensor and related problems will be assessed.
Exercise problems 100%.
Related courses
- ARC.S501 ： Shell Structures
- ARC.S201 ： Fundamentals of Mechanics of Materials A
- ARC.S202 ： Fundamentals of Mechanics of Materials B
- ARC.S203 ： Structural Mechanics I
- ARC.S305 ： Structural Mechanics II
- ARC.S303 ： Structural Design III
- CVE.A202 ： Structural Mechanics I
- CVE.A301 ： Structural Mechanics II
Prerequisites
Students must have successfully completed structural mechanics or have equivalent knowledge.