DYNAMIC RESPONSE OF MULTI-DEGREE OF FREEDOM FRAMES SUBJECTED TO IDENTICAL SUPPORT EXCITATION DUE TO EARTHQUAKE

REFERENCE

APPENDIX 1: The developed MATLAB program

APPENDIX 2: Forcing frequency Ө = 0

In structural dynamics, structures may be excited by two kinds of source, external loading acting directly on the super structure and secondly the ground motion acting on the sub structure (Seyoum, 2016). In the latter case ground motion may be expressed as acceleration, velocity or displacement. In the analysis of structures under ground motion, the ground acceleration, velocity, and displacement are presumed to be known since they can be obtained directly from instrument readings. When any two of these parameters are specified, analysis could be carried out effectively. Since earthquake engineering is based on the expectation that structures undergo serious deformation, and possibly collapse during intense ground vibration caused by earthquake, the Structural Engineer, is interested in predicting the expected structural responses (Hoskuldur, 2016). The most important factors influencing earthquake ground motions are earthquake magnitude, distance from the source of energy release, local soil conditions, variation in geology and propagation velocity along the travel path, and earthquake source conditions and mechanism (fault type, slip rate, stress conditions, stress drop, etc.). Past earthquake records have been used to study some of these influences (Chopra, 2006).

Earthquake excitation of structural frames in general, is a complex process with a high variability in time and space. However, the support induced vibrations are the major causes of deformation and stress in the structural systems (Rajasekaram, 2009). The support excitations can be Identical or multiple in all supports. 

For an identical-support excitation, it is assumed that all the supports undergo an identical (uniform) ground motion. In other words, due to the same ground motion at all supports, the supports move as one rigid base. Hence, the lumped masses at each floor are excited by the ground motion (Murty et al., 2017). Tall buildings with relatively small ground coverage are usually assumed to undergo identical support motion. For multi-support excitations, the ground support motions are different at various supports. This case is mostly seen in big network of pipe lines, very long tunnels, long dams, bridges, structures on soil with non-uniform ground properties or on buildings in a large land span (Murty et al., 2017).

The present study will develop a finite element displacement model for determining the dynamic response of multi-degree of freedom frames subjected to identical support excitation due to earthquake under prescribed ground motion (ground acceleration and displacement) using the lumped mass parameter and static condensation. A MATLAB computer program will be developed for the model.

 

Given a multi-degree of freedom (MDOF) flexible frame with eight floors and three bays, it is required to carry out a dynamic analysis using the following parameters. The ground motion is defined by the ground acceleration U ̈g(t) and the maximum ground displacement XS, the lumped masses M1, M2, M3, …… M8 at the floors are excited by the ground motion to induce the effective earthquake forces P¬1(t), P¬2(t), P¬3(t), …….  P¬8(t) at the floor levels, where Pi(t) = MiU ̈g(t). The fundamental problem here is to use the idealized model to determine the dynamic response due to the induced earthquake forces. The term response is used in a general sense to include any quantity such as displacements, internal stresses, and natural frequencies of the structure. In the absence of any external force the response quantities are the natural frequencies. However, when the excitation is an external force the response quantities of interest are the displacements and the internal stresses. The relative displacement U(t) associated with deformation of the structure is the most important since the internal forces in the structure are directly related to U(t).

The process of formulating and solving the equation of motion for the finite element model shall be programmed in computer using MATLAB. The developed MATLAB program shall be used to carry out the dynamic analysis of a one bay, two story building and the results compared to the results obtained by manual approach using ANOVA for validation, after which the program will be extended to resolving a three bay, eight storey flexible frame. 

The main aim of this research is to develop a Finite Element displacement model for determining the dynamic response of a multi-degree of freedom frame subjected to identical support excitation due to earthquake under prescribed motion.

The objectives of the study are:

The scope of work here includes the determination of the dynamic response of multi-degree of freedom (MDOF) flexible frame. Responses to be considered include natural frequencies, displacements and internal stresses. The static response will be used for comparison with the dynamic response. However, stability analysis is outside the scope of this work. Frames with stiffened joints and shear frames will not be considered. A finite element displacement model synthesized in a MATLAB program shall be used as analytical tool.