# A comparative study of two models for the seismic analysis of buildings

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2 A COMPARATIVE STUDY OF TWO MODELS FOR THE SEISMIC ANALYSIS OF BUILDINGS freedom in supports m, are mass matrices, damping and stiffness are associated with degrees of freedom n and/or m respectively, is a vector of n x representing associate dynamics requirements regarding degrees of freedom n, is a vector of m x representing associate reactions (unknown) to the supports degrees of freedom "m". Considering linear systems involving orthogonality for stiffness ( ), mass ( ) and damping ( ) matrices, it is convenient to diagonalise the system of equations of motion to transform it into a normal modal coordinate system. A system having free undamped vibration, which can exist in the absence of any excitation of the supports, would give: where is a mass matrix corresponding to non-restricted degrees of freedom n, is a stiffness matrix corresponding to non-restricted degrees of freedom n, is a vector of relative displacement and is a vector of relative acceleration. The solution of equation () is defined (by Aguilar Falconi, ; García Reyes, ) as: () The value of the equivalent mechanical elements acting in free nodes (McCormac, 00) is: The mechanical element components acting on members (Tena Colunga, 00) are: where and are each bar s stiffness and displacement. Application An example of the dynamic seismic design method is presented, using two different models, considering shear deformation for an office building built with a steel frame structure. The analysis is only developed transversally. Figure shows the office building s floor-plan and elevation and Figure shows the horizontal response spectrum, representing soil movement where the building is supported. Table l shows the steel profile properties. () () where is natural vibration frequency, is a modal vector associated with ω, and t is time. The values of and are determined by resolving eigenproblems as: ( ) () The modal participation factor (Clough, Penzien, ; Bazan, Meli, ) can be expressed as: where is the transposed vector of a modal vector corresponding to the mode "n" is a pseudostatic influence vector. The normal maximum coordinates of the system for each mode ( ) are: () Poisson ratio = 0. Figure. Floor-plan and elevation regarding an office building constructed with steel frame structure. ( ) () ω where is spectral acceleration corresponding to mode "n". The vectors corresponding to maximum relative displacement vector components for each mode can be defined as: ( ) () The maximum value of the vector of relative displacements in structural system is: Figure. The horizontal response spectrum or: { ( ) } ( ) ( ) ( ) () Table. Steel frame properties Frames Total area Shear area Moment of inertia (cm ) (cm ) (cm ) W0X0.., W0X.. 0, W0X 0.00., WX. 0., WX..0,00 INGENIERÍA E INVESTIGACIÓN VOL. No., DECEMBER 0 (-)

3 LUÉVANOS The load to be considered in the analysis by level was: Weight of level =, N/m Weight of level =, N/m Weight of level =,0 N/m Weight of level =, N/m Elasticity modulus = 0,0,00 N/cm l The beams and columns were considered for analysis in this model, taking into account two lumped masses per level applied in free nodes and considering three degrees of freedom at each joint (horizontal displacement, vertical displacement and rotation). The building was only analysed in a transversal direction. Figure gives the mathematical model. Figure. Two lumped masses per building level l This model considered that the beams were rigid compared to the columns and, therefore, the beams did not influence dynamic analysis of the building. It also considered one degree of freedom per level, i.e. horizontal displacement (Luévanos Rojas, 00). The mathematical model is presented in Figure. The mass and stiffness matrices for each member were evaluated (Appendix; Luévanos Rojas et al., 00) followed by the change of local system to overall system. The mass and stiffness matrices in each member s general system were then coupled and the system s general matrix obtained. This general matrix was organised to separate the degrees of freedom in the structure (M and K) and degrees of freedom in the supports (M and K). A similar transformation was applied by exchanging row and column matrices (permutation matrix). Ignoring the effect of damping in free vibration, as in equation (), being a vector (x for model and x for model ) of relative displacements corresponding to the building s structural system degrees of freedom, then the eigenvalues and eigenvectors were obtained by solving the determinant resulting from equation (). Figure. A lumped mass per building level MATLAB software was used for solving the determinant and the roots of the polynomials. Table shows the first four modes of the sixteen for model (M) and the four modes for model (M) are presented in. Table shows the spectral accelerations of the first four modes of M and the four modes of M. Equation () gave modal participation factor. The maximum normal coordinates of the system for each mode ( ) were located by using equation (). The first four values for the M and the four values for M are shown in Table. The maximum relative displacement vector components for each mode were given by equation () and the maximum value of the structural system relative displacements vector for building was obtained by equation (). These values are shown for both models in Table. Table. Eigenvalues Circular frequency (Rad / sec) Period (Sec) M M M M Table. Spectral acceleration Frequency ω n ( Hz ) Acceleration san (cm/sec) M M M M g =. 0.0g =... 0.g = g = g = g = g = g =. Table. Participation factors the system for each mode ( ) Participation factors Ln and maximum normal coordinates of Maximum normal coordinates for the system for each mode (Yn)máx M M M/M M M M/M INGENIERÍA E INVESTIGACIÓN VOL. No., DECEMBER 0 (-)

4 A COMPARATIVE STUDY OF TWO MODELS FOR THE SEISMIC ANALYSIS OF BUILDINGS Once the deformations were obtained, equation () was used to find the values of the forces in X, the forces in Y and moments; these were applied at the free joints. Such effects were equivalent to what would have occurred due to a movement in the soil where the building was located. The mechanical elements at the joints on the members of the whole building were determined by equation () and then obtained for each of the building s rigid frames. The axial forces, the shear forces and the moments for a central frame are presented in Tables, and, respectively. Table. Deformation vector Node Concept Unit Amount M M M/M Displacement X cm...0 Displacement Y cm 0.00 Rotation rad 0.00 Displacement X cm.0..0 Displacement Y cm 0.0 Rotation rad 0.00 Displacement X cm.0.. Displacement Y cm 0. Rotation rad Displacement X cm...0 Displacement Y cm 0. Rotation rad Displacement X cm...0 Displacement Y cm 0. Rotation rad Displacement X cm.0.. Displacement Y cm 0. Rotation rad Displacement X cm Displacement Y cm 0.0 Rotation rad 0.00 Displacement X cm...0 Displacement Y cm 0.00 Rotation rad 0.00 Results The values for the building s vibration mode frequencies for both models are shown in Table. It was observed that values for M were lower regarding M in terms of frequency and logically the periods were inverse. The first four modes of M are presented in this Table, even though the work involved sixteen modes resulting from the dynamic analysis. Table shows spectral acceleration. These values were obtained from the frequency of each of the structure s vibration modes and these results were found by means of the horizontal response spectrum of the soil where the building was constructed; this spectrum is presented in Figure. The participation factors and maximum normal coordinates of the system for each mode are given in Table, all values in M being higher than in M. The participation factors increased by.% in M and maximum normal coordinates increased by.% (both percentages are presented in the third mode). Table. Axial forces in the bars for a central frame (N) Bar Node Concept l l M/M 0 Force Y 0. Force Y Force Y 0 0. Force Y Force Y 0 0. Force Y Force Y.0 Force Y Force Y +.0 Force Y +.0 Force Y Force Y Force Y Force Y Force Y + 0. Force Y +0. Force X 0 +0 Force X Force X 0 +0 Force X 0 0 Force X 0 + Force X 0 Force X 0 + Force X 0 Table. Shear forces in the bars for a central frame (N) Bar Node Concept l l M/M 0 Force X.0 Force X Force X. Force X + +. Force X 00. Force X Force X 00. Force X Force X +00. Force X Force X +00. Force X Force X +.0 Force X +.0 Force X Force X +00. Force Y +.0 Force Y Force Y Force Y Force Y + 0. Force Y + 0. Force Y Force Y INGENIERÍA E INVESTIGACIÓN VOL. No., DECEMBER 0 (-)

5 LUÉVANOS Table. Moments in the bars for a central frame (N-m) Bar Node l l M/M Table gives the structural system s relative deformations; all values were greater in M (.0% increase). Only horizontal displacements were compared as M did not consider the other two deformations which would be present in any given structure. Table shows the axial forces in the structure. There was a.0 times greater increase in M than M; this only occured in the columns and axial load was not presented in beams for M. Table gives the shear forces; there were differences of up to.0% in all top columns in M compared to M and a.0% increase in the upper members of the beams in M. Such difference decreased when dealing with each lower floor and became greater in M when arriving at level. The moments acting on the bars of the structure are shown in Table. M was greater for all columns by up to.0 times than M, while shear forces behaved similarly in beams, having.% increase in M in the top bar whilst the bottom bar in M was greater. Conclusions Horizontal displacement, vertical displacement and rotation at each joint was not restricted given the results obtained in M taking into account four degrees of freedom, one for each floor (i.e. horizontal displacement at each level) and M considered twenty-four degrees of freedom. According to the above, it was noted that several degrees of freedom were ignored in M, being so reflected in the system s response. Frequency analysis revealed that M involved neglecting certain modal shapes (symmetric modes and/or anti-symmetrical) system which, in the case of soil excitation, are present and should be considered, since they often reflect relatively low frequencies. General practice considering a lumped mass for each level is thus not recommendable, whereas the model proposed in this paper seems to be the most appropriate one for seismic analysis of buildings structural systems. References Aguilar Falconi, R., Acciones para el diseño sísmico de estructuras, Limusa-Wiley,, pp. -. Appendix. Formulario de Teoría de Estructuras. Matrices de Rigidez Elementales, de Masa Congruentes, y de Rigidez Geométrica. Disponible en: atrices_de_rigidez_elementales.pdf Bazan, E., Meli, R., Diseño sísmico de edificios, Limusa-Wiley,, pp. -. García Reyes, L. E., Dinámica estructural aplicada a diseño sísmico, Universidad de los Andes, Facultad de Ingeniería, Departamento de Ingeniería Civil, Bogotá, Colombia,, pp. -. Clough, R. W., Penzien, J., Dynamics of Structures, McGraw-Hill,, pp. -. Luévanos Rojas, A., Seismic analysis of a building of four levels; making a comparison, despising and considering the deformations by Sharp, International Review of Civil Engineering (I.RE.C.E.), Vol., No., Sep., 00, pp. -. Luévanos Rojas, A., Betancourt Silva, F., Martinez Garcia, I., Luévanos Rojas R. and Luévanos Soto, I., Vibrations in systems of pipes with different excitation in its ends, International Journal of Innovative Computing, Information and Control, Vol., No., Dec., 00, pp. -0. Mc Cormac, J. C., Structural analysis: using classical and matrix methods, John Wiley & Sons, 00, pp Przemieniecki, J. S., Theory of Matrix Structural Analysis, McGraw- Hill,, pp 0- y -. Tena Colunga, A., Análisis de estructuras con métodos matriciales, Limusa-Wiley, 00, pp. -. Zárate, G., Ayala, A. G., García, O., Método sísmico estático para edificios asimétricos: revisión de enfoques. Revista de Ingeniería Sísmica No., 00, pp. -. INGENIERÍA E INVESTIGACIÓN VOL. No., DECEMBER 0 (-)

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