CHAPTER 2: Macromechanics of a Lamina

2.1 Introduction

Learning Objectives:

1) Recognize how a composite lamina mechanical behavior is different from that of an isotropic layer.

2) Review the matrix algebra concepts needed for the course from EML3041 Computational Methods course

3) Review the transformation of stresses and strains from the EML3500 Mechanics of Solids course

Videos

Appendix A: Matrix Algebra

Title: Adding Two Matrices


Title: Subtracting Two Matrices


Title: Multiplying Two Matrices


Title: A real life problem of setting up simultaneous linear equations


Title: Writing simultaneous linear equations in matrix form 


Title: Can we divide two matrices?


Title: Some statements about the inverse of matrices


Title: Inverse of matrices: Example 


Title: Using concept of inverse to solve a set of equations 


Title: Finding the inverse of a matrix: Theory 


Title: Finding the inverse of a matrix: Example 


Title: Finding the inverse of a matrix by adjoints: Theory 


Title: Finding the inverse of a matrix by adjoints: Example 

Slide Shows

Deformations of unidirectional lamina under load [PDF] [Slideshow]

Worked Out Examples

Appendix B: Transformation of Stresses and Strains: Transformation of Stresses [PDF]

Handouts

None

Flashcards

Flashcards for Section 2.1 Introduction Slideshow

Flashcards for Appendix A: Matrix Algebra Slideshow

Flashcards for Appendix B: Transformation of Stresses and Strains Slideshow

Quiz

None


2.2 Review of Definitions

Learning Objectives:

1) Review definition of stress

2) Review definition of strain

3) Review definition of elastic moduli

4) Review definition of strain energy.

Videos

None

Slide Shows

Review of definitions [PDF] [Slideshow]

Worked Out Examples

None

Handouts

None

Flashcards

Review of Definitions Slideshow

Quiz

Review of Definitions [WATU] [HTML] [PDF]

2.3 Hooke’s Law for Different Types of Materials

Learning Objectives:

1) Develop stress–strain relationships for different types of materials

Videos

None

Slide Shows

3D stiffness and compliance matrices [PDF] [Slideshow]

Worked Out Examples

None

Handouts

None

Flashcards

Hooke’s Law for Different Types of Materials Slideshow

Quiz

Hooke’s Law for Different Types of Materials [WATU] [HTML] [PDF]

2.4 Hooke’s Law for a Two Dimensional Unidirectional Lamina

Learning Objectives:

1) Develop stress–strain relationships for a unidirectional/bidirectional lamina.

Videos

None

Slide Shows

Stiffness and compliance matrices for a unidirectional lamina [PDF] [Slideshow]

Worked Out Examples

None

Handouts

None

Flashcards

Hooke’s Law for a Two Dimensional Unidirectional Lamina Slideshow

Quiz

Hooke’s Law for a Two Dimensional Unidirectional Lamina [WATU] [HTML] [PDF]

2.5 Hooke’s Law for a Two Dimensional Angle Lamina

Learning Objectives:

1) Develop stress–strain relationships for an angle lamina.

Videos

None

Slide Shows

Stress-strain relationship for an angle lamina [PDF] [Slideshow]

Worked Out Examples

None

Handouts

None

Flashcards

Hooke’s Law for a Two Dimensional Angle Lamina Slideshow

Quiz

Hooke’s Law for a Two Dimensional Angle Lamina [WATU] [HTML] [PDF]

2.6 Engineering Constants of an Angle Lamina

Learning Objectives:

1) Develop formulas for engineering constants of an angle lamina

Videos

None

Slide Shows

Engineering constants of an angle lamina [PDF] [Slideshow]

Worked Out Examples

Problem 2.27 [PDF]

Problem 2.28 [PDF]

Problem 2.29 [PDF]

Problem 2.32 [PDF]

Background to #2.32: Linear Regression With Zero Intercept: Derivation

10 degree ply problem – Finding Ex [PDF] (Question asked in a test)

10 degree ply test – Finding G12 [PDF] (Question asked in a test)

10 degree ply test – Misalignment of strain gauges [PDF] (Question asked in a test)

Handouts

None

Flashcards

Engineering Constants of an Angle Lamina Slideshow

Quiz

Engineering Constants of an Angle Lamina [WATU] [HTML] [PDF]

2.7 Invariant Forms of Stiffness and Compliance Matrices for Angle Lamina

Learning Objectives:

1) Develop invariant forms of compliance and stiffness matrix

Videos

None

Slide Shows

Invariant form of transformation matrices [PDF] [Slideshow]

Worked Out Examples

Writing a transformed reduced stiffness matrix in fewer invariants [PDF]

Handouts

None

Flashcards

Invariant Forms of Stiffness and Compliance Matrices for Angle Lamina Slideshow

Quiz

Invariant Forms of Stiffness and Compliance Matrices for Angle Lamina [WATU] [HTML] [PDF]

2.8 Strength Failure Theories of Angle Lamina

Learning Objectives:

1) Define strength ratio

2) Use four failure criteria for failure of angle laminas

Videos

None

Slide Shows

Maximum stress failure theory for isotropic bodies [PDF] [Slideshow]

Maximum stress failure theory [PDF] [Slideshow]

Maximum strain failure theory [PDF] [Slideshow]

Tsai-Hill failure theory [PDF] [Slideshow]

Tsai-Wu failure theory [PDF] [Slideshow]

Comparison of failure theories [PDF] [Slideshow]

Worked Out Examples

Off-axis shear strength using maximum stress failure theory [PDF] (A question asked in the test)

Handouts

None

Flashcards

Strength Failure Theories of Angle Lamina Slideshow

Quiz

Strength Failure Theories of Angle Lamina [WATU] [HTML] [PDF]

2.9 Hygrothermal Stresses and Strains in a Lamina

Learning Objectives:

1) Develop hygrothermal coefficients for angle lamina

Videos

None

Slide Shows

Hygrothermal stresses and strains [PDF] [Slideshow]

Worked Out Examples

Finding coefficients of moisture expansion of an angle ply [PDF] (A question asked in a test)

Handouts

None

Flashcards

Hygrothermal Stresses and Strains in a Lamina Slideshow

Quiz

Hygrothermal Stresses and Strains in a Lamina [WATU] [HTML] [PDF]