A MATLAB COMPUTER PROGRAM FOR 3-DIMENSIONAL DYNAMIC ANALYSIS OF MULTISTORY BUILDINGS

OBJECTIVE To present a MTLAB program for conducting three dimensional dynamic analysis of multistory building by utilizing a simple and ‘easy to understand’ plane frame analysis program. To extend the program to do dynamic response spectrum analysis of buildings.

Why MATLAB ? MATLAB has built-in functions that performs various computational tasks. This reduces, substantially, the program size. Matrix operation in MATLAB are conducted similar to scalar operation. With MATLAB graphic capability, it possible, with a little programming effort, to obtain a graphic display of the results including structure geometry, bending and shear force diagrams, deformed shape … etc.

Shear an Moment Diagrams in a Member Deformed Shape

Structural and General Purpose FE Software: SAP2000 ETABS STAAD/pro OTHERS : ANSYS, ABAQUS, ….ETC

BUILDING MODELS FOR SEISMIC ANALYSIS Three dimensional fully detailed finite element model of the building structure  very expensive Three dimensional frame model  large DOF Three dimensional frame model with rigid diaphragm constraint  3 DOF at each floor A set of orthogonal (two-way) planar frames with rigid diaphragm constraint Plane frame structure  regular and symmetric building

THREE DIMENSIONAL FRAME WITH RIGID DIAPHRAGM CONSTRAINT ASSUMTIONS: 1. The floor diaphragm is rigid in its own plane. This Allows the horizontal displacements at a floor level to be defined in terms of floor horizontal translations and rotation about a vertical axis. 2. Building Structure Is Represented by a Set of Orthogonal planar Frames.

1. RIGID DIAPHRAGM APPROXIMATION Mass center 3 DOF at each floor + 3 DOF at each node

1. Identify Building Frames Analysis Procedure: Six planar frame Two different types 1. Identify Building Frames Typical Frame

2. Construct Global Stiffness Matrix for each Typical frame 3. Apply displacement constraint to horizontal degrees of freedom at each level  eliminate horizontal degrees of freedom but one at each floor

4. Apply static condensation to eliminate all degrees of freedom but horizontal ones  Number of degrees of freedom=Number of floor. Static Condensation

5. Construct the global stiffness matrix for the 3D model by applying rigid diaphragm constraints to each plane frame stiffness matrices using the Number of degrees of freedom=3*Number of floor. Apply to frames in x-direction Apply to frames in y-direction

5. Construct the diagonal Mass matrix

Verification Examples (ETABS USER MANUALS) Two Story Three-Dimensional Unsymmetrical building Dynamic Response Spectrum Analysis Seismic excitation:Elcentro Response Spectrum at 30 to X

Results Comparison 1 .Structural periods Program Reference 0.450 0.4146 0.396 0.3753 0.247 0.2436 0.134 0.1148 0.120 0.1103 0.087 0.0729

2. DYNAMIC RESPONSE SPECTRUM BASE SHEARS:

CONCLUSIONS Building Analysis using plane frame analysis programs with constraints gives results which are comparable to 3-dimensional analysis A plane frame representation of a three dimensional building structure is easier for visualization and preparation of data MATLAB is a powerful means that could be used for programming simple equations of design to the complicated structural analysis programs.