STRAIN ENERGY

The strain energy stored in an element (i) within a general structural system is the area under the stress-strain diagram integrated over the volume of the element. For linear systems, the stress-strain matrix   , including initial thermal stresses   , can be written in matrix form as:

The column matrices  and  are the stresses and strain respectively.

Therefore, the strain energy within one element is given by:

Within each element, an approximation can be made on the displacements. Or:

N The shape functions used to calculate the strain energy within the element

Hence, after the application of the strain-displacement equations, the element strains can be expressed in terms of nodal displacements. Or:

The column matrix  contains all of the node, or joint, displacements of the complete structural system.

The element deformation-displacement transformation matrix  , is a function of the geometry of the structure.

After integration over the volume of the element, the strain energy, in terms of the global node displacements, can be written as:

Therefore, the element stiffness matrix is by definition:

And the element thermal force matrix is:

The total internal strain energy is the sum of the element strain energies. Or:

The global stiffness matrix  is the sum of the element stiffness matrices   .

Or:

The summation of element stiffness matrices to form the global stiffness matrix

is termed the direct stiffness method. The global thermal load vector  is the

sum of the element thermal load matrices:

Reference

Three-Dimensional Static and Dynamic Analysis of Structures A Physical Approach With Emphasis on Earthquake Engineering Edward L. Wilson