The Finite Element Method A Practical Course

1
The Finite Element Method A Practical Course
CHAPTER 5

• FEM FOR 2D SOLIDS

2
CONTENTS

• INTRODUCTION
• LINEAR TRIANGULAR ELEMENTS
• Field variable interpolation
• Shape functions construction
• Using area coordinates
• Strain matrix
• Element matrices
• LINEAR RECTANGULAR ELEMENTS
• Shape functions construction
• Strain matrix
• Element matrices
• Gauss integration
• Evaluation of me

3
CONTENTS

• LINEAR QUADRILATERAL ELEMENTS
• Coordinate mapping
• Strain matrix
• Element matrices
• Remarks
• HIGHER ORDER ELEMENTS
• COMMENTS (GAUSS INTEGRATION)
• CASE STUDY

4
INTRODUCTION

• 2D solid elements are applicable for the analysis
  of plane strain and plane stress problems.
• A 2D solid element can have a triangular,
  rectangular or quadrilateral shape with straight
  or curved edges.
• 2D solid element can deform only in the plane of
  the 2D solid.
• At any point, there are two components in x and y
  directions for the displacement as well as
  forces.

5
INTRODUCTION

• For plane strain problems, the thickness of the
  element is unit, but for plane stress problems,
  the actual thickness must be used.
• In this course, it is assumed that the element
  has a uniform thickness h.
• Formulating 2-D elements with given variation of
  thickness is also straightforward, as the
  procedure is the same as that for a uniform
  element.

6
2D solids