Direct integration

The direct integration methods seek the history of the dynamic response in time. The response is not obtained in continuous form but rather for a predetermined series of points in time t i .

One of the most popular direct integration methods is the Newmark method. Starting-point is the governing equation of a structure with various degrees of freedom:

For time t =t i , this equation takes on the following form:

The velocity and the acceleration are expressed as follows:

Substituting these expressions into the equation of motion, we obtain

where,

Usually the initial conditions are that the structure posesses neither displacement nor velocity.