Mathematical Model of Frame

As with one story frames, a F function must be defined. A similar logic deduction process is followed to obtain this function, which is a matrix function. This function evaluates the internal reaction forces in the building. The F function is composed by the addition of reaction forces at each one of the four frames. Therefore, the F function must be defined for an isolated frame and then added with the other three frame F functions. The march in time algorithm requires this functional relation to solve the IVP ODE problem.

F-function for a Frame

The direct stiffness method is used for the development of these mathematical models. There are too many details that need to be solved to implement all these ideas in a computer program; It is out of this research scope to present them completely and exhaustively. Therefore, only some general details are presented in this chapter, to illustrate some additional considerations required to implement the non linear dynamic analysis of multi story buildings. One of the most obvious differences is that three DOF are required per story level to define the kinematic state of the frame model. Figure 3a shows these DOF. In the following deductions, the aim is to find the F-function for the prototype frame, although the mathematical relations are developed for the equivalent system. Dissertation Plans

The mathematical relations shown in equations 5.1 are applicable. The difference is that Atii must be substituted by the corresponding Aun, which is calculated with the equation 6.2. Aun is defined as the interstory drift. Aun = un-un-i (6.2) Another important difference is that equation 5.3 must be adjusted to include the n stories. Applying DSM standard procedures, the new relation between changes in the reaction forces and the change in interstory displacements can be found. After these adjustments, the remaining procedure is essentially the same used for one story frames.

Modification of Stiffness or Strength of Frame

The two parameters introduced for one story frames are used to modify the lateral stiffness and the ultimate lateral strength of the prototype frame. They are: K = stiffness factor. fn = strength factor. The same relations 5.16 are applicable to multistory frames.

Mathematical Model of the Building

The procedure followed to elaborate the mathematical model of the building is the same used for one story buildings, but considering the additional degrees of freedom at each additional story level. The original plus the additional degrees of freedom are stated in equation 1. Taking in consideration the general details mentioned in this chapter, it is possible to develop the computer algorithm for the non linear dynamic analysis of multistory buildings. Dissertation Plans