2019 Tensor Analysis for Building Structure

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Academic unit or major
Graduate major in Urban Design and Built Environment
Motoyui Shojiro 
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Course description and aims

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.

Student learning outcomes

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.


Tensor, Tensor analysis, Base vector, Metric tensor, Descartes coordinate system, Curvilinear coordinate system, Geometrical nonlinearity, Shell structures

Competencies that will be developed

Intercultural skills Communication skills Specialist skills Critical thinking skills Practical and/or problem-solving skills
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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/Required learning

  Course schedule Required learning
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



Reference books, course materials, etc.

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Assessment criteria and methods

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 (i.e., required knowledge, skills, courses, etc.)

Students must have successfully completed structural mechanics or have equivalent knowledge.

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