Reduction of DOF for the Dynamic Equilibrium Equation of Building Frames

Introduction

Solving a dynamic system requires several time steps and is computationally demanding. The bending moment frame (Fig. 1a), while most common for static analysis, is inefficient for dynamic analysis, particularly for seismic analysis and structural design. Guyan (1965) proposed the reduction technique as an extension to static condensation to reduce the DOF with zero mass, i.e., at rotational dofs.

(Adapted from Cimellaro and Marasco (2018))

Guyan Reduction

By ignoring the mass and damping of rotational DOF {us}, the dynamic equilibrium equation of the bending moment frame (Fig. 1a), subjected to ground acceleration of magnitude a_g, can be written as

*The unity vector [1] is the spatial vector to apply a_g to the lateral dof.

From the second row of the equation

The reduced DOF system has a dimension that is usually equal to the number of non-zero masses, i.e., 3x3 for the shear frame shown in Fig. 1b, compared with 9x9 for the bending moment frame in Fig. 1a.

It should be noted that the reduced system contains only non-zero mass DOF with the condensed stiffness matrix as obtained by the static condensation process.

Numerical Example

The building frame as shown in Fig. 1 will be analysed using MATLAB.

References:

Guyan RJ(1965), Reduction of stiffness and mass matrices. AIAA J 3(2):380

Cimellaro and Marasco (2018), Introduction to Dynamics of Structures and Earthquake Engineering, Springer