IDENTIFICATION OF SYSTEM PARAMETERS OF AN EXISTING MULTI-STOREY SYMMETRIC-PLAN SHEAR BUILDING
6.1 INTRODUCTION
6.2.5 Stability of the Identified System
Stability of the system can be easily observed fr
identification algorithm has the capability to transform state s with the help of the transfer function as expressed by Eq. 2.
functions have been plotted on the complex plane using the polar coordinates according to Eq. 2.14. Poles within the unit circle in the
system, while poles outside the unit circle refer to the unstable dynamic system. Plots of poles based on acceleration responses subjected to two selected earthquakes are as shown in Fig. 6.8. It is observed from
marked as ‘x’ have been found to be lying within the unit circle indicating stable condition of the systems.
(a) In shorter direction on 11-02
(c) In shorter direction on 12-08
Fig. 6.8 Poles for the system based on responses corresponding to two recorded earthquakes
155 6.2.5 Stability of the Identified System
Stability of the system can be easily observed from poles obtained using Eq. 2.14 identification algorithm has the capability to transform state space form to poles and zeros
function as expressed by Eq. 2.13. The poles of the transfer functions have been plotted on the complex plane using the polar coordinates according to . Poles within the unit circle in the complex plane refer to the stable dynamic system, while poles outside the unit circle refer to the unstable dynamic system. Plots of poles based on acceleration responses subjected to two selected earthquakes are as shown in Fig. 6.8. It is observed from the Fig. 6.8 that nine numbers of complex conjugate poles marked as ‘x’ have been found to be lying within the unit circle indicating stable condition
02-2006 (b) In longer direction on 11-02-2006
08-2006 (d) In longer direction on 12-08-2006 Poles for the system based on responses corresponding to two recorded
om poles obtained using Eq. 2.14. The pace form to poles and zeros . The poles of the transfer functions have been plotted on the complex plane using the polar coordinates according to complex plane refer to the stable dynamic system, while poles outside the unit circle refer to the unstable dynamic system. Plots of poles based on acceleration responses subjected to two selected earthquakes are as shown the Fig. 6.8 that nine numbers of complex conjugate poles marked as ‘x’ have been found to be lying within the unit circle indicating stable condition
2006
2006 Poles for the system based on responses corresponding to two recorded
156 6.2.6 Structural Parameter Identification
Stiffness of a structure can be evaluated using Eq. 2.42 described in Chapter 2 under section 2.4. Mass matrix of lumped floor masses in kg of the sample building is
=
362710 0
0 0
0 0
0 0
0
0 511080
0 0
0 0
0 0
0
0 0
515570 0
0 0
0 0
0
0 0
0 515570
0 0
0 0
0
0 0
0 0
515570 0
0 0
0
0 0
0 0
0 515580
0 0
0
0 0
0 0
0 0 515580
0 0
0 0
0 0
0 0 0
515930 0
0 0
0 0
0 0 0
0 511770
M
.
The identified modal matrix is incomplete as values of modal displacement are not available at the floor levels where accelerometers have not been installed for recording acceleration histories. Hence, the approach in subsection 2.4.1 under Case-II has been utilized for the evaluation of complete modal matrix. The modal matrix from a numerically simulated model in SAP has been considered as initial values for the iterative analysis. Fig. 6.9 depicts iteratively converged frequency in shorter direction from the identification study based on February 11, 2006 earthquake responses. It can be observed that 50 iterations are sufficient for convergence. The iterative procedure depicted a similar convergence pattern for the study based on August 12, 2006 earthquake responses. The converged modal frequencies along with corresponding identified damping ratios in two directions for earthquake on 11-02-2006 are presented in Table 6.5. Similarly, converged modal frequencies along with corresponding identified damping ratios in two directions for earthquake on 12-08-2006 are presented in Table 6.6. It may be observed that the converged identified frequencies of the building as evaluated based on the measured
157
ground accelerations as well as floor accelerations on two different dates show very good agreement.
Table 6.5 Converged frequencies for February 11, 2006 earthquake Shorter Direction Longer Direction
Frequency (rad/sec) Damping (%) Frequency (rad/sec) Damping (%)
10.22 2.20 12.75 2.20
29.60 4.40 35.95 4.40
48.00 7.50 59.02 7.50
65.25 1.72 79.90 1.72
80.10 9.33 98.30 9.33
92.65 1.54 113.50 1.54
102.25 4.21 125.25 4.21
108.80 2.30 133.50 2.30
112.70 3.39 138.10 3.39
Table 6.6 Converged frequencies for August 12, 2006 earthquake Shorter Direction Longer Direction
Frequency (rad/sec) Damping (%) Frequency (rad/sec) Damping (%)
10.15 3.64 12.50 3.63
29.80 3.18 35.70 1.13
48.25 4.10 58.85 3.04
65.55 9.29 79.90 0.63
80.60 4.55 98.35 1.28
93.20 6.79 112.45 1.35
102.80 2.17 123.80 1.13
109.40 3.83 131.80 2.62
113.35 4.36 136.25 5.05
It may be observed from Fig. 6.9 that for the study with limited sensors, some of the converged identified frequencies are somewhat different than their respective initially identified frequencies. A similar trend has been noticed for the identified modal frequencies based on structural responses for both the considered earthquakes acceleration histories. Hence, it is worth noting that an iterative approach is mandatory for more accurate identification of eigen pairs for buildings with limited sensors.
158
(a) First mode (b) Second mode
(c) Third mode (d) Fourth mode
(e) Fifth mode (f) Sixth mode
(g) Seventh mode (h) Eighth mode
(i) Ninth mode
Fig. 6.9 Converged frequency in shorter direction corresponding to different mode shapes of the building for February 11, 2006 earthquake
0 50 100 150 200
10 10.5 11 11.5
Iteration (Nos.)
Frequency(rad/sec)
0 50 100 150 200
29.5 30 30.5 31 31.5 32
Iteration (Nos.)
Frequency (rad/sec)
0 50 100 150 200
47.6 47.8 48 48.2 48.4 48.6
Iteration (Nos.)
Frequency (rad/sec)
0 50 100 150 200
58 60 62 64 66
Iteration (Nos.)
Frequency (rad/sec)
0 50 100 150 200
75 76 77 78 79 80 81
Iteration (rad/sec)
Frequency (rad/sec)
0 50 100 150 200
84 86 88 90 92 94
Iteration (Nos.)
Frequency (rad/sec)
0 50 100 150 200
97 98 99 100 101 102 103
Iteration (Nos.)
Frequency (rad/sec)
0 50 100 150 200
105 106 107 108 109
Iteration (Nos.)
Frequency (rad/sec)
0 50 100 150 200
110 112 114 116 118 120 122
Iteration (Nos.)
Frequency (rad/sec)
159
An incomplete modal matrix is initially obtained from the identification study as the response data have been measured only at some selected floor levels. However, with the introduction of iterative procedure, a complete modal matrix is obtained at the end of convergence, which is considered as the modal matrix of the building under study. The converged normalized mode shapes along both the directions have been plotted for nine modes of the nine storey shear building and are as shown in Fig. 6.10 and 6.11.
Fig. 6.10 Mode shapes along shorter direction of the building
0 1 2 3 4 5 6 7 8 9
-1.1 -0.1 0.9 First Mode
Floor Levels
0 1 2 3 4 5 6 7 8 9
-1.1 -0.1 0.9
Second Mode
0 1 2 3 4 5 6 7 8 9
-1.1 -0.1 0.9
Third Mode
0 1 2 3 4 5 6 7 8 9
-1.1 -0.1 0.9
Fourth Mode
0 1 2 3 4 5 6 7 8 9
-1.1 -0.1 0.9
Fifth Mode
0 1 2 3 4 5 6 7 8 9
-1.1 -0.1 0.9 Sixth Mode
Floor Levels
0 1 2 3 4 5 6 7 8 9
-1.1 -0.1 0.9
Seventh Mode
0 1 2 3 4 5 6 7 8 9
-1.1 -0.1 0.9
Eighth Mode
0 1 2 3 4 5 6 7 8 9
-1.1 -0.1 0.9
Ninth Mode
160
Fig. 6.11 Mode shapes along longer direction of the building
The identified storey stiffness as obtained after the convergence of the iterative procedure corresponding to both the earthquakes are presented in Table 6.7. The converged storey stiffness indicates that the stiffness of the building along the shorter direction is lesser than those along the longer direction of the building. Further, the stiffness of ground floor in
0 1 2 3 4 5 6 7 8 9
-1.1 0 1.1
First Mode
Floor Levels
0 1 2 3 4 5 6 7 8 9
-1.1 0 1.1
Second Mode
0 1 2 3 4 5 6 7 8 9
-1.1 0 1.1
Third Mode
0 1 2 3 4 5 6 7 8 9
-1.1 0 1.1
Fourth Mode
0 1 2 3 4 5 6 7 8 9
-1.1 0 1.1
Fifth Mode
0 1 2 3 4 5 6 7 8 9
-1.1 0 1.1
Sixth Mode
Floor Levels
0 1 2 3 4 5 6 7 8 9
-1.1 0 1.1
Seventh Mode
0 1 2 3 4 5 6 7 8 9
-1.1 0 1.1
Eighth Mode
0 1 2 3 4 5 6 7 8 9
-1.1 0 1.1
Ninth Mode
161
both the directions have been observed to be higher than the corresponding stiffnesses of the floors above, which indicates that though the building is vertically irregular, does not exhibit any soft storey effect in either of the directions. The reduced ground floor height has primarily compensated for the absence of infill wall in the ground floor. It has also been observed from Table 6.7 that the stiffness values of all the floors along each of the two directions are more or less similar and hence it can be inferred that the structure has not deteriorated after experiencing these two earthquakes.
Table 6.7 Identified storey wise stiffness of the sample building for earthquakes data on two different dates
Storey Stiffness in 108 N/m
February 11, 2006 August 12, 2006
Shorter Longer Shorter Longer
Ground 21.64 32.60 21.94 31.31
First 16.71 25.09 16.90 24.10
Second 16.70 25.12 16.89 24.13
Third 16.72 25.09 16.91 24.10
Fourth 16.71 25.10 16.91 24.11
Fifth 16.73 25.09 16.92 24.10
Sixth 16.70 25.10 16.90 24.10
Seventh 16.70 25.11 16.89 24.12
Eighth 16.71 25.10 16.90 24.11
6.3 NUMERICALLY SIMULATED MODEL OF THE SAMPLE MULTI-STOREY