FEA Isoparametric Quadrilaterals Part 2: The Strain-Displacement Matrix

This video continues from part 1 (   • FEA Isoparametric Quadrilaterals Part 1: J...  ) which described how natural coordinates and the Jacobian matrix are used in the development of the stiffness matrix for linear 4-node isoparametric quadrilateral elements in finite element analysis (FEA). The focus of this video is on the strain displacement matrix [B] and how the Jacobian matrix is necessary to solve for the elements within [B].

0:00 Introduction
0:28 Review of the Jacobian matrix and natural coordinates
1:41 Review of the stiffness matrix equation
2:44 Development of the strain displacement matrix
5:30 Example to determine just one element in the strain displacement matrix
8:52 Intro to why Gauss Quadrature is required to integrate the stiffness matrix (see part 3!)
9:51 Reflection questions

Suggested answers to reflection questions
1.) t = thickness of the element
[B] is the strain displacement matrix (from basic strain equation where strains are equal to the derivative of displacement with respect to x or y)
[C] is the constitutive matrix (from Hooke's Law - changes if plane stress or plane strain)
|J| is the determinate of the Jacobian matrix. This is used to map between global x, y and natural xi, eta coordinates.
2.) The Jacobian matrix is necessary when solving for components within the strain displacement matrix because strains are defined as the derivative of displacement with respect to x or y (e.g., normal strain in x = derivative of displacement in x, u, with respect to x). However, the displacement functions are in terms of natural coordinates xi, eta instead of physical coordinates. For this reason, the Jacobian matrix is required to map between the two coordinate systems.