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LAGRANGE’S EQUATION OF MOTION

In the case of dynamic analysis of structures, the direct application of the well-known Lagrange’s equation of motion can be used to develop the dynamic equilibrium of a complex structural system. Lagrange’s minimization equation, written in terms of the previously defined notation, is given by:

                                                        (1)

The node point velocity is defined as  . The most general form for the kinetic energy  stored within a three-dimensional element i of mass density  is:

               (2)

The same shape functions used to calculate the strain energy within the element allow the velocities within the element to be expressed in terms of the node point velocities. Or:

  ,   , 

Therefore, the velocity transformation equations can be written in the following form:

Using exact or numerical integration, it is now possible to write the total kinetic energy within a structure as:

The total mass matrix  is the sum of the element mass matrices . The element consistent mass matrices are calculated from:

                                                   (3)

where  is the 3 by 3 diagonal mass density matrix shown in Equation (2).

Equation (3) is very general and can be used to develop the consistent mass matrix for any displacement-based finite element. The term “consistent” is used because the same shape functions are used to develop both the stiffness and mass matrices.

Direct application of Equation (1) will yield the dynamic equilibrium equations:

THE FORCE METHOD

The traditional method of cutting a statically indeterminate structure, applying redundant forces, and solving for the redundant forces by setting the relative displacements at the cuts to zero has been the most popular method of structural analysis, if hand calculations are used. At this point in time, there appears to be no compelling reason for using the force method within a computer program for solving large structural systems. In fact, programs based on the displacement approach are simple to program and, in general, require less computer time to execute. Another significant advantage of a displacement approach is that the method is easily extended to the dynamic response of structures.

To develop the stiffness of one-dimensional elements, however, the force method should be used because the internal forces can be expressed exactly in terms of the forces at the two ends of the element. Therefore, the force method will be presented here for a single-element system.

Neglecting thermal strains, the energy function can be written as:

The internal forces can be expressed in terms of the node forces using the following equation:

For linear material   and the energy function can be written as:

Where the element flexibility matrix is:

We can now minimize the complementary energy function by requiring that:

The node displacements can now be expressed in terms of node forces by:

The element stiffness can now be numerically evaluated from:

The element stiffness can be used in the direct stiffness approach where the basic unknowns are the node displacements. One can also derive the element flexibility by applying the virtual force method.

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