Formulation of the element matrices and vectors for elasticity problem

<< Click to Display Table of Contents >>

Navigation:  »No topics above this level«

Formulation of the element matrices and vectors for elasticity problem

 

The finite element problem consists of calculating the individual element stiffness matrices and vectors, and assembling them into the global stiffness matrix and force vectors. The set of simultaneous equations that this produces is then solved for the nodal displacements.

 

The stress vector σ and the strain vector ε are, respectively (Figure 14):

 

 

formula_8

(5)

 

formula_9

 

 

                                   

clip0010

Fig. 14. Components of the stress.

 

 

The stress-strain relation is represented as:

 

 

formula_10

(6)

 

or as:

 

 

formula_11

(7)

 

 

 

where C is a symmetric matrix of material compliances, E is a symmetric matrix of material stiffness, and E=C-1. The components of the displacement vector u along x, y and z directions are u, v, and w respectively. The following sign convention is used: Positive y-coordinates and vertical forces are taken upward, i.e. parallel to the Y-axis. Positive x-coordinates and horizontal forces are taken to the right, i.e. parallel to the x axis. The vertical deflections are measured along the y-axis. A positive deflection therefore denotes an upward movement.

 

 

clip0009

Fig. 15.  Stresses and body forces that act on a plane differential element of constant thickness.

 

 

 

 

 

 

 

 

 

© 2020 GeoStru Software