Technical Design Calculation Report
TimberTech s.r.l.
Horizontal actions Modal analysis
ch .it
The modal analysis is used to determine the vibration modes of the structure, useful to understand the seismic behaviour of the building and to proceed with the linear dynamic analysis. The modal analysis involves the solution of the generalized eigenvalue problem: [đ?‘˛đ?‘˛ − đ?œ´đ?œ´2 đ?‘´đ?‘´]đ?š˝đ?š˝ = đ?&#x;Žđ?&#x;Ž
rte
where đ?‘˛đ?‘˛ is the stiffness matrix, đ?‘´đ?‘´ the mass matrix, đ?œ´đ?œ´2 is the diagonal matrix of the eigenvalues and đ?š˝đ?š˝ is the corresponding matrix of eigenvectors or modal shapes (normalized with respect to the mass matrix); the seismic masses of the diaphragms are calculated with the following combination of vertical loads: G1 + G2 + ďż˝ Ďˆ2j â‹… Q kj j
The eigenvalue, obtained by the solution of the generalized eigenvalue problem, is the square of the circular frequency đ??Žđ??Ž related to the period, T, and to the frequency, f, by the following equations: đ?œ”đ?œ” 2đ?œ‹đ?œ‹
be
1
�� = �� and �� =
m
The participating mass ratio for the mode i-th, corresponding to an acceleration in the global axis X and Y and to a rotational acceleration around the vertical axis Z, is given by: ������ =
w .ti
������ =
w w
where:
������ =
������ = ������ = ������ =
đ?‘šđ?‘šđ?‘Ľđ?‘Ľđ?‘–đ?‘– [%] ∑ đ?‘šđ?‘šđ?‘Ľđ?‘Ľ,đ?‘—đ?‘—
đ?‘šđ?‘šđ?‘Śđ?‘Śđ?‘–đ?‘– [%] ∑ đ?‘šđ?‘šđ?‘Śđ?‘Ś,đ?‘—đ?‘— đ?‘šđ?‘šđ?‘§đ?‘§đ?‘–đ?‘– [%] ∑ đ??źđ??źđ?‘§đ?‘§,đ?‘—đ?‘— đ?‘ťđ?‘ť
đ?&#x;?đ?&#x;?
đ?‘ťđ?‘ť
đ?&#x;?đ?&#x;?
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