Course title
Strength of Materials

kariya yoshiharu
Course description
Analysis techniques (mathematical theory) to calculate strength of the structure will be learned. Especially, mathematical theory of elasticity and plasticity in three dimensions to apply for finite element method will be studied in this lecture.
Purpose of class
Understanding of stress analysis techniques (mathematical theory) to safety design of the structure in three dimensions.
・Learn how to express stress and strain in the three-dimensional state of the structure.
・Learn the calculation method of yielding and stress and strain in multiaxial stress state in three dimensional space.
・Understand the fundamentals of finite element method analysis and learn the ability to calculate stress calculation with simple structure by finite element method.
Goals and objectives
  1. Understanding of mathematical theory of elasticity and plasticity in three dimensions.
  2. Understanding of the constitutive equations of plasticity and its application to calculate plasticity of metals.
  3. Understanding of basis of reliability analysis using the finite element method analysis and its application.
Class schedule

Class schedule HW assignments (Including preparation and review of the class.) Amount of Time Required
1. Guidance and Stochastic basis of reliability engineering Review handouts 190minutes
2. Stress Components in 3 dimensions
・Tensor representation of stress
・Stress in a given plane
・Coordinate transformation of stress and strain
Review handouts
(Stress Components in 3 dimensions)
3. Principal Stresses in Three Dimensions
・What is principal stresses.
・How to calculate the principal stresses.
・Principal Stresses in Two Dimensions
(Calculation of principal stress using Mole's circle)
Review handouts
(Principal Stresses)
4. Basics of Elastic Constitutive Equations
・Hook's law in three dimensions
・Plane Strain Problem
・Plane Stress Problem
Review handouts
(Elastic Constitutive Equations)
5. Finite Element Method (FEM) Analysis 1
・Stress analysis by matrix calculation
・Case study: Stress analysis of truss using rod elements
Review handouts
(Finite Element Method (FEM) Analysis 1)
6. Finite Element Method (FEM) Analysis 2
An example of elastic beam calculation, using matrices B and D
・D Matrix (Material Property)
・B Matrix (Strain-Displacement Matrix)
・Energy Conservation Law of Finite Elements Method
・An Example of Deflection Calculation of A Cantilever Beam
Review handouts
(Finite Element Method (FEM) Analysis 2 )
7. Deviatoric Stress and Stress Invariants
・Deviatoric Stress
・Stress Invariants
・Deviatoric Invariants
Review handouts
(Deviatoric Stress and Stress Invariants)
8. Yield Criterion for Isotropic Material
・How to calculate Yielding in multi axial condition
Equivalent Stress
・Scalar representation of stress in multi axial condition
・Yielding curves
Review handouts
(Yield Criterion and Equivalent Stress)
9. Basics of Constitutive Equations
Simplification of Stress-Strain Curves
・Full plastic
・Bilinear hardening law
Hook's law in three dimensions
Review handouts
(Constitutive Equations and Simplification of Stress-Strain Curves)
10. Incremental Strain Theory and Prandtl-Reuss Reuss Constitutive Equation
・Incremental Strain Theory
・Reuss Constitutive Equation
・Equivalent plastic strain increment and Equivalent plastic strain
Expression of plastic behavior by equivalent stress and equivalent plastic strain
・How to determine the undetermined multiplier
・Constitutive equations of plane strain and plane stress state
Review handouts
(Incremental Strain Theory and Prandtl-Reuss Reuss Constitutive Equation)
11. Case study of plane strain compression of block
・Equation of equilibrium of forces acting on infinitesimal element
・Equation of yield criterion
・Pressure distribution
・Load and average pressure
Review handouts
(plane strain compression of block)
12. Case study of axisymmetric compression of cylinder
・Equation of equilibrium of forces acting on infinitesimal element
・Yield criterion
・Load and average pressure
Review handouts
(axisymmetric compression of cylinder )
13. Elasto-plastic analysis of the truss using FEM
・Analysis of elastic deformation of truss
・Analysis of the secondary deformation (Plastic deformation)
・Relationship between external force and displacement
Review handouts
(Elasto-plastic analysis of the truss)
14. Final report and Commentary Review all handouts 180minutes
Total. - - 2650minutes
Relationship between 'Goals and Objectives' and 'Course Outcomes'

Practice problem Final exercise Total.
1. 10% 20% 30%
2. 15% 20% 35%
3. 15% 20% 35%
Total. 40% 60% -
Evaluation method and criteria
Practice problems in every lecture 40 points and final report 60 points, Passes over 60 points.
Textbooks and reference materials
Handouts of this lecture.
For example, The Mathematical Theory of Plasticity
This lecture is planned as Material strength for students of the Department of Materials Engineering.
Materials mechanics A and B must be completed to take this course.
Office hours and How to contact professors for questions
  • Wednesday ,10:00ー12:00
Non-regionally-oriented course
Development of social and professional independence
  • Course that cultivates a basic self-management skills
  • Course that cultivates an ability for utilizing knowledge
Active-learning course
About half of the classes are interactive
Course by professor with work experience
Work experience Work experience and relevance to the course content if applicable
Applicable Lecture of Strength of Materials is given Based on the practical experience of reliability calculation for design and manufacture of ships and bridges.
Education related SDGs:the Sustainable Development Goals
Last modified : Sun Mar 21 16:56:23 JST 2021