2022 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)
- Shojiro Motoyui
- Class Format
- Lecture (Livestream)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 5-6 Tue (W922)
- Class
- -
- Course Code
- UDE.S406
- Number of credits
- 100
- Course offered
- 2022
- Offered quarter
- 4Q
- Syllabus updated
- Jul 10, 2025
- Language
- English
Syllabus
Course overview and goals
This course focuses on the fundamental concept of tensor and tensor analysis used in the analysis of continua. Topics include the usefulness of tensor analysis, the definition of base vectors, metric tensors, various definitions for strain tensors and stress tensors, constitutive equations, and numerical method applied to actual structural problems.
Curved continua like arches and domes are often applied to actual building structures for indoor sport stadiums. The specific coordinate system which was the most suitable one to the configuration of the target structure was referred to when deriving basic equations for the structure. Consequently, varied expressions of the basic equations were used corresponding to the problem. According to the concept of tensor analysis, the basic equations can be derived in the common form which is independent of the reference coordinate system. Students will realize both the usefulness and complication of tensor analysis. Conversely, students will know we have received the benefit from the Cartesian coordinate system in deriving various equations.
Course description and aims
By the end of this course, students will be able to:
1) Understand concept of tensor variables and difference from scalor or vector variables.
2) Understand the reason why the tensor analysis is used and explain usefulness of the tensor analysis.
3) Derive base vectors, metric tensors and strain tensors in an arbitrary coordinate system.
4) Explain concept of plasticity theory for multi-axial stress field.
5) Explain the geometrical nonlinearity which should be considered in designing arches or domes.
Keywords
Tensor, Tensor analysis, Base vector, Metric tensor, Descartes coordinate system, Curvilinear coordinate system, Geometrical nonlinearity, Shell structures
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
At the beginning of each class, solutions to exercise problems that were assigned during the previous class are reviewed. Towards the end of class, students are given exercise problems related to the lecture given that day to solve. Required learning should be completed outside of the classroom for review purposes.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Tensor and tensor analysis | Concept of tensor and usefulness of tensor analysis |
| Class 2 | Base vector, metric tensor | Definitions of base vector and metric tensor, application of them to curvilinear coordinate systems |
| Class 3 | Strain tensor | Definition of Green's strain tensor and difference of it from engineering strain |
| Class 4 | Stress tensor | Definitions of various stress tensors and relation to strain tensors |
| Class 5 | Constitutitive equation | Elastic constitutive equation for 3D stress field, plane stress field and uni-axial stress field |
| Class 6 | Plastic theory for multi-axial stress field | Fondamental concept of plasitic theory for multi-axial stress field |
| Class 7 | Treatment in numerical analysis method | Tensor analysis and coding of numerical program |
| Class 8 | Application | Influence of geometrical nonlinearity on arch structures |
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.
Distribute document if it is neccesary
Evaluation methods and criteria
Students' knowledge of tensor analysis, and their ability to apply them to problems will be assessed.
Two 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.