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Equations of Motion

This section presents the finite element equations of motion, based on equating internal and external virtual work expressions.

By substituting for the stress and strain expressions in the virtual work statement given by Internal Virtual Work Statement (Eq.11), and equating internal and external work increments given by Internal Virtual Work Statement (Eq.11) and External Virtual Work Statement (Eq.2), the local form of the equations of motion are derived as:

        (1)

The local hybrid stiffness matrix is decomposed to:

        (2)

where is the standard beam-column linear stiffness matrix, and and are given by:

       (3)

and:

        (4)

where:

       (5)

and are respectively the initial displacement matrix and the geometric stiffness matrix. Equation (1) above can now be transformed from the local convected to the global axes resulting in the equation:

       (6)

where:

        (7)

        (8)

       (9)

and is the transformation matrix from local to global axes.