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;?
đ?‘ťđ?‘ť
đ?&#x;?đ?&#x;?
���� �� � �� ���� � [�� �� ]�� �� �� ��
���� �� � �� ���� � [�� �� ]�� �� �� ��
���� �� � �� ���� � [�� �� ]�� �� �� ��
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Technical Design Calculation Report
TimberTech s.r.l.
and where ∑ đ?‘šđ?‘šđ?‘Ľđ?‘Ľ,đ?‘—đ?‘— , ∑ đ?‘šđ?‘šđ?‘Śđ?‘Ś,đ?‘—đ?‘— and ∑ đ??źđ??źđ?‘§đ?‘§,đ?‘—đ?‘— are the total masses acting in the axis X, Y and the total rotational inertia about the axis Z of the unrestrained j-th degrees of freedom.
Period [s]
Frequency [Hz]
MX [%]
Sum MX [%]
MY [%]
Sum MY [%]
MZ [%]
Sum MZ [%]
Mode 1
0.52
1.91
1.34
1.34
75.13
75.13
4.01
4.01
Mode 2
0.38
2.66
48.90
50.24
7.05
82.18
32.60
36.61
Mode 3
0.29
3.50
45.38
95.61
1.43
Mode 4
0.22
4.51
0.12
95.74
15.97
Mode 5
0.15
6.47
2.93
98.67
0.40
Mode 6
0.10
9.66
1.33
100.00
0.01
Dynamic linear analysis
83.62
48.00
84.62
99.58
4.21
88.82
99.99
9.38
98.20
100.00
1.80
100.00
rte
The dynamic linear analysis consists of:
ch .it
Mode
the calculation of the seismic effects (the seismic action is represented by the design response spectrum), of each of the vibration modes calculated in the modal analysis;
-
combination of these effects.
be
-
The seismic effects of the structural model are obtained by the application of the following external loads:
m
đ?‘đ?‘đ?’Šđ?’Šđ?’™đ?’™ = Γđ?‘Ľđ?‘Ľđ?‘–đ?‘– đ?‘†đ?‘†đ?‘‘đ?‘‘ (đ?‘‡đ?‘‡đ?‘–đ?‘– ) đ?‘´đ?‘´ đ?š˝đ?š˝ đ?’Šđ?’Š e
đ?‘đ?‘đ?’Šđ?’Šđ?’šđ?’š = Γđ?‘Śđ?‘Śđ?‘–đ?‘– đ?‘†đ?‘†đ?‘‘đ?‘‘ (đ?‘‡đ?‘‡đ?‘–đ?‘– ) đ?‘´đ?‘´ đ?š˝đ?š˝ đ?’Šđ?’Š
w .ti
where:
đ?‘đ?‘đ?’Šđ?’Šđ?’™đ?’™ and đ?‘đ?‘đ?’Šđ?’Šđ?’šđ?’š are the external loads of the i-th vibration mode due to seismic action along X and Y đ?‘†đ?‘†đ?‘‘đ?‘‘ (đ?‘‡đ?‘‡đ?‘–đ?‘– )
is the spectrum value corresponding to the i-th period
���� ������ ����
are the participating modal factor of the i-th mode given by:
is the i-th modal shape
w w
�� ��
����
đ?‘ťđ?‘ť
đ?‘ťđ?‘ť
��� �� � �� ���� ��� �� � �� ���� �� = and �� = [�� �� ]�� �� �� �� [�� �� ]�� �� �� ��
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Technical Design Calculation Report
TimberTech s.r.l.
The effects for a given direction of acceleration (along X or Y) and for each of the vibration modes are combined with the Complete Quadratic Combination technique defined as:
đ?‘–đ?‘–
is the effect of the j-th vibration mode;
ch .it
đ?‘—đ?‘—
where: đ??¸đ??¸đ?‘—đ?‘—
1/2
đ??¸đ??¸ = �� ďż˝ đ?œŒđ?œŒđ?‘–đ?‘–đ?‘–đ?‘– ∙ đ??¸đ??¸đ?‘–đ?‘– ∙ đ??¸đ??¸đ?‘—đ?‘— ďż˝
đ?œŒđ?œŒđ?‘–đ?‘–đ?‘–đ?‘– is the correlation coefficient of the i-th mode and the j-th mode, given by:
đ?œ‰đ?œ‰
đ?œŒđ?œŒđ?‘–đ?‘–đ?‘–đ?‘– =
3/2
8 đ?œ‰đ?œ‰ 2 đ?›˝đ?›˝đ?‘–đ?‘–đ?‘–đ?‘–
(1 + đ?›˝đ?›˝đ?‘–đ?‘–đ?‘–đ?‘– )��1 − đ?›˝đ?›˝đ?‘–đ?‘–đ?‘–đ?‘– ��
is the damping ratio in the i-th and j-th modes;
rte
������ = ���� /���� ).
be
The following table provides the diaphragms properties. Height above foundation zi [m]
xG [m]
yG [m]
Mass i [kg]
1
3.20
7.71
3.75
68621
2
6.69
7.62
3.62
61793
m
Diaphragm
w w
w .ti
The following table provides, for each of the vibration modes, the corresponding period and the values of the response spectra. Period [s]
SLV spectrum value [g]
SLV spectrum value [g]
Mode 1
0.54
0.10
0.06
Mode 2
0.35
0.12
0.10
Mode 3
0.30
0.12
0.11
Mode 4
0.23
0.12
0.11
Mode 5
0.14
0.12
0.11
Mode 6
0.11
0.11
0.10
Mode
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Technical Design Calculation Report
TimberTech s.r.l.
Deformation
ch .it
Mode
w .ti
m
Mode 2
be
rte
Mode 1
w w
Mode 3
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Technical Design Calculation Report
TimberTech s.r.l.
rte
ch .it
Mode 4
w .ti
m
be
Mode 5
w w
Mode 6
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