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School of Engineering
- Undergraduate major in Mechanical Engineering
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- Common courses
- School of Materials and Chemical Technology
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- School of Life Science and Technology
- School of Environment and Society
- 工学院，物質理工学院，環境・社会理工学院共通科目
- Liberal arts and basic science courses
- First-Year Courses
- School of Science
- School of Engineering
- School of Materials and Chemical Technology
- School of Computing
- School of Life Science and Technology
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School of Environment and Society
- Department of Architecture and Building Engineering
- Department of Civil and Environmental Engineering
- Department of Transdisciplinary Science and Engineering
- Department of Social and Human Sciences
- Department of Innovation Science
- Graduate major in Technology and Innovation Management
- Common courses
- Liberal arts and basic science courses
- Academic unit or major
- Graduate major in Urban Design and Built Environment
- Instructor(s)
- Ishihara Tadashi
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- -
- Group
- -
- Course number
- UDE.S406
- Credits
- 1
- Academic year
- 2024
- Offered quarter
- 4Q
- Syllabus updated
- 2024/3/14
- Lecture notes updated
- -
- Language used
- English
- Access Index
-
Course description and aims
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.
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.
Student learning outcomes
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 that will be developed
| ✔ 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/Required learning
| Course schedule | Required learning | |
|---|---|---|
| Class 1 | Outline of this lecture | Concept of tensor and usefulness of tensor analysis, Direct notation, Vector algebra |
| Class 2 | Basic tensor theory (1) | Tensor algebra, Transpose, Symmetric, Skew-symmetric, Principal invariants |
| Class 3 | Basic tensor theory (2) | Tensor product |
| Class 4 | Basic tensor theory (3) | Proper vectors and proper numbers of tensors, Spectral representation |
| Class 5 | Examples of tensors(1) | Strain tensor, Stress tensor, Principal axis of earthquake ground motions, Input energy tensor |
| Class 6 | Examples of tensors(2) | Effective mass tensor, Calculation of input energy by multidimensional earthquake motions |
| Class 7 | Exercise | Exercises on tensors |
| Class 8 | Advanced tensor analysis | Tensors in curvilinear coordinate systems |
Out-of-Class Study Time (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 course material.
Textbook(s)
None
Reference books, course materials, etc.
Lecture materials will be distributed as needed.
Assessment criteria and methods
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 (i.e., required knowledge, skills, courses, etc.)
Students must have successfully completed structural mechanics or have equivalent knowledge.
