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On a modal damping identification model for non-classically damped linear building structures subjected to earthquakes
ARTICLE IN PRESS Soil Dynamics and Earthquake Engineering 29 (2009) 583–589
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Technical Note
On a modal damping identification model for non-classically damped linear building structures subjected to earthquakes George A. Papagiannopoulos a, Dimitri E. Beskos a,b, a b
Department of Civil Engineering, University of Patras, GR-26500 Patras, Greece Office of Theoretical and Applied Mechanics, Academy of Athens, 4 Soranou Efessiou street, 11527 Athens, Greece
a r t i c l e in fo
abstract
Article history: Received 24 September 2008 Accepted 29 October 2008
A simple modal damping identification model developed by the present authors for classically damped linear building frames is extended here to the non-classically damped case. The modal damping values are obtained with the aid of the frequency domain modulus of the roof-to-basement transfer function and the resonant frequencies of the structure (peaks of the transfer function) as well as the modal participation factors and mode shapes of the undamped structure. The assumption is made that the modulus of the transfer function of the non-classically damped structure matches the one of the classically damped structure in a discrete manner, i.e., at the resonant frequencies of that function modulus. This proposed approximate identification method is applied to a number of plane building frames with and without pronounced non-classical damping under different with respect to their frequency content earthquakes and its limitations and range of applicability are assessed with respect to the accuracy of both the identified damping ratios and that of the seismic structural response obtained by classical mode superposition and use of those identified modal damping ratios. & 2008 Elsevier Ltd. All rights reserved.
Keywords: Modal damping ratios Identification model Linear building frames Non-classical damping Transfer function Frequency domain Earthquakes
1. Introduction Structural identification is usually the inverse problem of structural dynamics that tries to estimate the dynamic properties of a structure (modal shapes, natural frequencies, modal damping ratios, etc.) on the basis of measurements of response to known dynamic excitations [1–3]. For linear elastic structural behaviour, energy of dissipation is usually taken into account by viscous damping in the form of modal damping ratios [4]. Among the many works on structural damping identification on the basis of measured seismic response, one can mention the works of Caravani and Thomson [5], McVerry [6], Beck and Jennings [7], Shinozuka et al. [8], Safak [9], Ruzzene et al. [10], Yin et al. [11] and Yang et al. [12]. Recently, a simple modal identification model for building structures was presented by the present authors [13] as a generalization of a model originally proposed by Hart and Vasudevan [14]. This model works in the frequency domain and provides time-invariant modal damping ratios of building structures subjected to earthquakes in terms of the modulus of the roof-to-basement transfer function, the resonant frequencies of the structure (peaks of the transfer function) as well as the modal
Corresponding author at: Department of Civil Engineering, University of Patras, GR-26500 Patras, Greece. Tel.: +30 261 099 6559; fax: +30 261 099 6578. E-mail address: [email protected] (D.E. Beskos).
0267-7261/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.soildyn.2008.10.005
participation factors and mode shapes of the undamped structure. The model was found to give excellent results for linear plane building frames on the assumption of classical viscous damping [13]. In this work, this identification model is extended to the case of non-classical damping. Classical (or proportional) damping represents an appropriate idealization if damping is distributed throughout the structure, e.g., a multi-storey building with a similar structural system and structural material over its height [4]. One well-known procedure to construct a classical damping matrix from modal damping ratios is due to Rayleigh. However, if the structural system consists of two or more parts with quite different amounts of damping, e.g., a soil-building system or a structure with special energy dissipating devices or equipment, the system damping is non-classical (non-proportional) [4]. A system with classical damping can be uncoupled by its real undamped modal shapes for the purpose of obtaining its seismic response by real modal superposition. When its damping is non-classical this can only be done by its complex damped modal shapes. Caughey and O’ Kelly [15] found the necessary and sufficient conditions that mass, stiffness and damping matrices should satisfy in order to have classical damping. Thus, one should expect that use of the simple identification model of [13] constructed on the assumption that its damping is classical, will lead to modal damping ratios in error for those cases of structures characterized by non-classical damping. In this work, the identification model of [13] is extended to also cover
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approximately the case of non-classical damping. This is accomplished by assuming that the modulus of the frequency response transfer function of the non-classically damped system matches the one of the classically damped system in a discrete manner, i.e., at the peaks (resonant frequencies) of that function modulus. Here the proposed identification model is applied to a number of plane building frames with and without pronounced non-classical damping under two different with respect to their frequency content earthquakes and its limitations and range of applicability
Thus, the roof-to-basement transfer function R(o) evaluated at o ¼ ok and on account of q€j ðoÞ ¼ o2 qj ðoÞ receives the form [13] Rðo ¼ ok Þ ¼ 1 þ
X j
frj Gj o2k ðo o2k Þ þ i ð2xj oj ok Þ 2 j
(7)
where frj is the element of the j-th eigenvector corresponding to the roof. Finally, the modulus of R(o ¼ ok) takes the real valued form [13]
jRðo ¼ ok Þj2 ¼1þ2
þ
N X
frj Gj o2k ðo2j o2k Þ
j¼1
ðo2j o2k Þ2 þ ð2 xj oj ok Þ2
N ½f2 G2 o4 ðo2 o2 Þ2 þ 4 x2 o2 o2 X rj j j k j k j k
½ðo2j o2k Þ2 þ ð2 xj oj ok Þ2 2
j¼1
þ2
N X jam;m4j
frj Gj frm Gm o4k ½ðo2j o2k Þ ðo2m o2k Þ þ 4 xj xm oj om o2k h
i (8) ðo2j o2k Þ2 þ ð2 xj oj ok Þ2 ½ðo2m o2k Þ2 þ ð2 xm om ok Þ2
are assessed both with respect to the accuracy of the identified modal damping ratios and that of the seismic structural response obtained by classical mode superposition and use of those identified modal damping ratios.
2. Identification model for calculation of modal damping values The matrix equation of motion for a N degrees-of-freedom, linear, elastic, viscously damped plane building frame subjected at ¨g ¼ u ¨ g(t) can be written in its base to an earthquake acceleration u the form ½M fug € þ ½C fug _ þ ½K fug ¼ ½M fIg u€ g
(1)
where, [M], [C] and [K] are the mass, damping and stiffness matrices, respectively, of the structure, {u} ¼ {u(t)} is the displacement vector of the structure relative to its base, overdots denote differentiation and {I} is the identity vector. On the assumption that the damping matrix is of Rayleigh type, i.e., ½C ¼ a ½M þ b ½K
(2)
a solution of Eq. (1) expressed as a model superposition, i.e., as fug ¼ ½F fqg
(3)
where [F] is the modal matrix and {q} ¼ {q(t)} is the modal amplitude vector, uncouples Eq. (1) and leads to a system of N independent modal equations of the form q€j ðtÞ þ 2 xj oj q_j ðtÞ þ o2j qj ðtÞ ¼ Gj u€ g ðtÞ
(4)
In the above, oj and xj are the undamped natural frequency and modal damping ratio of mode j and Gj is the participation factor of mode j (j ¼ 1, 2, y, N). Applying the Fourier transform with respect to time onto Eq. (4) and solving for the transformed modal amplitude, one can obtain in the frequency o domain
It should be noted that the last summation term in Eq. (8) was accidentally omitted in Eq. (11) of [13] even though it was taken into account in all computations there. Eq. (8) represents a system of nonlinear algebraic equations that can be solved iteratively in conjunction with a user defined convergence criterion to estimate modal damping values xk on the basis of known values of |R(o ¼ ok)|, frj, ok and Gj. For a linear structure, its dynamic characteristics frj, ok and Gj can be obtained analytically for known stiffness and mass matrices, while |R(o ¼ ok)| can be constructed on the basis of the known seismic acceleration and the measured roof acceleration. One should point out that this method can provide modal damping values only for the modes which appear in the transfer function [13]. When the damping in the structure is non-classical, the above identification model provides modal damping ratios which are, in general, in error. The use of this model for non-classically damped structures is accomplished by assuming that the modulus of the frequency response transfer function of the non-classically damped system matches the one of the classically damped system in a discrete manner, i.e., at the peaks (resonant frequencies) of that function modulus. This matching is restricted to those peaks, which appear in the modulus of the transfer function, since higher modes do not show up in that function modulus and contribute only statically to the response. This matching approach follows Thomson et al. [16] and Tsai [17], who first proposed it in connection with the problem of using real mode superposition for non-classically damped systems in order to determine their dynamic response. For the case of non-classical damping the exact values of modal undamped frequencies oj and modal damping ratios xj can be obtained at the j-th mode as [18] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi oj ¼ jOj j ¼ ReðOj Þ2 þ ImðOj Þ2 (9)
xj ¼
ReðOj Þ jOj j
(10)
Gj u€ g ðoÞ (5) ðo2j o2 Þ þ ið2xj oj oÞ pffiffiffiffiffiffiffi where i ¼ 1 and overbars denote transformed quantities. Using Eq. (3), one can express the absolute acceleration vector ¨ (t)} in the frequency domain as {U
where Re(Oj), Im(Oj) are the real and imaginary part, respectively, of (Oj), i.e, the complex eigenfrequency of mode j of the structure. These complex eigenfrequencies are obtained by solving the quadratic eigenproblem corresponding to Eq. (1), i.e.,
€ oÞg ¼ ½F fqð € oÞg þ fIgu€ g ðoÞ fUð
where the vector {f} represents the complex eigenvector.
q j ð oÞ ¼
(6)
ðO2 ½M þ iO½C þ ½KÞffg ¼ 0
(11)
ARTICLE IN PRESS G.A. Papagiannopoulos, D.E. Beskos / Soil Dynamics and Earthquake Engineering 29 (2009) 583–589
3. Numerical examples and discussion In this section, the seismic response of linear elastic, plane, building frames with non-classical damping is obtained on the basis of the proposed method in order to illustrate that method and assess its accuracy and range of applicability. This assessment is done by comparing the approximate results of the proposed identification model, i.e., modal damping ratios and structural response obtained by modal superposition using identified modal damping ratios, against the exact ones of Eq. (10) and time integration, respectively. The assessment involves the investigation of the interplay among three parameters, namely the natural frequencies of the frames, the damping distribution in the frames and the type of seismic motion that excites the frames. These parameters influence the response of a non-classically damped system [19] and therefore, are taken into account herein. The case of seismically excited plane building frames with uniform damping distribution over their height and regularity in geometry implying well-separated frequencies has been studied in detail in [13] and found to produce highly accurate modal damping ratios and response results under the assumption of classical damping, in agreement with [4]. In this section, the cases of plane building frames with a non-uniform distribution of damping and geometric irregularity resulting in closely spaced frequencies for which damping should be considered as non-classical are investigated. The two linear elastic, plane, building frames considered here are shown in Fig. 1. The one bay and three storey steel frames of Fig. 1a have a bay span of 4.00 m and storey height of 3.00 m and consist of IPE 240 sections for beams and HEB 280 for columns. Load on beams equals 30 kN/m for the first and second storey and 15 kN/m for the third one. System damping is assumed to be concentrated at the bottom storey or at the top storey and can take the values 7%, 10%, 20% and 40%. The two frames are subjected to the longitudinal components of the OB accelerogram from the 2001 El Salvador earthquake and of the ERZ accelerogram from the 1992 Erzincan, Turkey, earthquake. The accelerograms of these earthquakes exhibit radical difference in their
585
frequency content as shown in their Fourier transforms in Figs. 2 and 3. Table 1 provides the natural frequencies oj of Eq. (9), the resonant frequencies from |R(o)| of Eq. (8), the modal damping values of the frame as computed by using Eqs. (10) and (8) for the damping location and values mentioned above and the relative error between Eqs. (10) and (8). One can easily observe that the error between the frequencies is very small, whereas the one for damping is very large (up to about 60%). To account for the quasi-static behaviour of the modes that do not appear in the transfer function, a value of 100% is considered for them when calculating the approximate response. Figs. 4 and 5 illustrate the comparison between the approximate and the exact roof displacement response of the frame structure under the ERZ and OB accelerograms for the case of 10% damping concentrated at the lower storey. Similar response plots (base shear, absolute
Fig. 2. Fourier transform of the OB accelerogram (2001 El Salvador earthquake).
Fig. 1. Plane building frames considered.
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roof acceleration and roof displacement) have been constructed for the rest of the damping cases of Table 1 when the frame structure is subjected to these accelerograms, but they are not shown here due to space limitations. Table 2 gives the exact and approximate absolute maximum values of base shear and roof displacement for the frames of Fig. 1a as well as the relative error.
From Table 2 and all response plots constructed it can be concluded that (a) the error between approximate and exact responses is, practically speaking, negligible (maximum 3%) when the seismic excitation has a low frequency content (ERZ accelerogram) no matter the value and the distribution of damping in the structure; (b) the error between approximate and exact
Fig. 3. Fourier transform of the ERZ acclerogram (1992 Erzincan earthquake).
Fig. 4. Comparison of roof displacements for the case of 10% damping at lower storey (ERZ accelerogram).
Table 1 Results for the one bay and three storey plane framed structure. Damping percentage and location
Mode
Natural frequencies (Hz) Eq. (9)
Resonant frequencies (Hz) Eq. (8)
Modal damping Eq. (10) (%)
Modal damping Eq. (8) (%)
Error (%) in modal damping
7%—lower storey
1 2 3
1.75 6.32 12.11
1.74 6.30 11.94
5.25 5.71 3.37
6.90 2.87 4.35
31.43 49.74 29.08
10%—lower storey
1 2 3
1.77 6.32 12.07
1.74 6.28 11.93
7.25 8.15 3.81
9.83 4.14 6.23
35.59 49.20 63.52
20%—lower storey
1 2 3
1.84 6.31 12.06
1.72 6.28 –
17.10 19.45 4.68
19.76 8.91 –
15.55 54.19 –
40%—lower storey
1 2 3
1.88 6.29 12.06
1.66 6.23 –
35.51 42.67 5.35
40.06 25.20 –
12.81 40.94 –
7%—upper storey
1 2 3
1.76 6.32 12.10
1.76 6.30 11.96
2.88 3.64 8.95
3.16 1.64 3.61
9.72 54.95 59.66
10%—upper storey
1 2 3
1.76 6.33 12.08
1.76 6.30 11.94
4.11 5.46 12.4
4.48 2.34 5.34
9.00 57.14 57.25
20%—upper storey
1 2 3
1.75 6.36 11.98
1.74 6.30 11.90
6.25 5.98 17.78
8.92 4.72 13.56
42.72 21.07 23.73
40%—upper storey
1 2 3
1.75 6.36 11.64
1.74 6.32 –
12.95 7.87 29.58
18.82 11.25 –
45.33 42.95 –
ARTICLE IN PRESS
0.82 3.85 1.39 6.82 0.00 0.75 1.90 2.63 0.121 0.100 0.071 0.041 0.139 0.132 0.103 0.074 0.122 0.104 0.072 0.044 0.139 0.133 0.105 0.076 0.80 7.76 9.82 8.37 2.36 1.31 3.98 2.29 296 240.4 158 95.3 350.6 315.9 248.5 166.1 298.4 260.6 175.2 104 342.5 320.1 258.8 170 0.00 1.33 1.47 1.82 0.00 0.00 0.00 0.00 0.077 0.074 0.067 0.054 0.08 0.079 0.075 0.068 0.077 0.075 0.068 0.055 0.08 0.079 0.075 0.068 0.00 0.69 1.87 2.92 0.43 0.05 0.11 0.00 181 175 158 129.6 187.6 183.7 175.9 160.3 181 176.2 161 133.5 186.8 183.8 175.7 160.3 7%—lower storey 10%—lower storey 20%—lower storey 40%—lower storey 7%—upper storey 10%—upper storey 20%—upper storey 40%—upper storey
Exact maximum base shear (kN) Error (%) for roof displacement Approximate maximum roof displacement (m) Exact maximum roof displace ment (m) Error (%) for base shear Approximate maximum base shear (kN) Exact maximum base shear (kN)
ERZ accelerogram Damping percentage and location
Table 2 Approximate and exact maximum response values and their relative error for the ERZ and OB accelerograms.
responses is of some significance for base shear and displacement response, (nearly 10% and 7%, respectively), when the seismic excitation has a high frequency content (OB accelerogram) no matter the distribution of damping; (c) the error in maximum responses is higher in the case of OB accelerogram than in the case of ERZ accelerogram. The error between approximate and exact responses is attributed to the damping coupling effect (effect due to non-negligible off-diagonal damping matrix terms) associated with the first mode and a higher mode of the framed structure that cannot be handled correctly by the approximate method, especially for the case of a excitation with a high frequency content. In order to assess the accuracy of the proposed method when two or more natural frequencies of a non-classically damped system are closely spaced, the case of a plane, two bay and seven storey steel moment resisting framed structure with an appendage interacting with the structure and being located on its top floor, as shown in Fig. 1b, is considered. Each bay and each storey of the frame have 4.00 m span and 3.00 m height, respectively. All beams have IPE 400 sections, while column sections vary for every other four storeys from HEB 400 to HEB 360 going from bottom to top. Load on all beams equals 25 kN/m. The damping in the frame varies but is uniformly distributed along its height. The appendage is modelled as a beam element with a mass of 0.01 times the mass of the structure and varying stiffness and damping values. The structure is excited separately by the ERZ and OB accelerograms. Two cases are examined: (a) the first mode frequency of the appendage almost coincides with the frequency of the first mode of the structure and (b) the first mode frequency of the appendage almost coincides with the frequency of a higher mode of the structure, e.g., the third mode. Non-classicality in damping is taken into account by considering that (i) the structure has 5% damping (in the first and fifth mode) and the appendage 1%, 7%and 20% damping (in the first and second mode) and (ii) the structure has 10% damping (in the first and fifth mode) and the appendage 25% damping (in the first and second mode). Tables 3 and 4 display the modal damping values as obtained by using Eqs. (10) and (8) for the aforementioned cases (a) and (b). One can easily observe that the error between damping ratios computed by Eqs. (10) and (8) reaches nearly 73%. Eq. (8) does not provide a modal damping ratio for the first mode of the appendage because that mode had coalesced into a mode of the
OB accelerogram
Fig. 5. Comparison of roof displacements for the case of 10% damping at lower storey (OB accelerogram).
Approximate maximum base shear (kN)
Error (%) for base shear
Exact maximum roof displace ment (m)
Approximate maximum roof displacement (m)
Error (%) for roof displacement
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structure, and thus it cannot appear in the transfer function. More specifically, the first mode of the appendage in case (a) coalesces into the first mode of the structure and in case (b) coalesces into its third mode. The only exception takes place in case (b) when the structure has 5% and the appendage 1% damping and the mode of the appendage appears in the transfer function. To account for the quasi-static behaviour of the modes that do not appear in the transfer function, a value of 100% is considered for them when calculating the approximate response for the cases (a) and (b). It should be also mentioned that when the approximate response is calculated, the damping value of the mode that corresponds to the appendage is considered to be equal to the modal damping of the first mode of the structure for case (a) and to that of the third mode for case (b). This is done due to the coalescence of the two modes mentioned above. Figs. 6 and 7 illustrate the comparison between the approximate and the exact displacement response of the frame structure under the OB and ERZ accelerograms for the case (a) when the structure has 10% and the appendage 25% damping. Similar exact and approximate response plots (base shear, absolute roof acceleration and roof displacement) have been constructed and the relative error between their absolute maximum values has been calculated, but they are not shown here due to space limitations. From the comparison between approximate and exact responses for cases (a) and (b), it can be concluded that (i) the errors in all kinds of responses of the structure are much greater for the case of the OB accelerogram, while they are smaller for that of the ERZ accelerogram; (ii) the error in maximum absolute response values of the structure is about 10% for the ERZ accelerogram and 19% for the case of the OB accelerogram; (iii) the error for the response of the appendage is always large (about 40%), especially for the case of the OB accelerogram, and it is always from the nonconservative side. The errors in response calculation are attributed to the interplay among the coincidence of the structure and appendage mode(s), the damping coupling effect associated with
Table 4 Modal damping values for frame with appendage for case (a): first natural frequency of appendageEthird natural frequency of frame. Modal damping Eq. (10) (%)
Modal damping Eq. (8) (%)
Error (%) in modal damping
Damping structure— appendage
Mode
5–1%
1 2 3 4 5
5.12 3.24 5.56 2.66 4.47
5.03 3.07 3.61 1.34 5.09
1.76 5.25 35.07 49.62 13.87
5–7%
1 2 3 4 5
5.42 3.33 7.72 11.27 14.53
5.08 3.16 4.18 – 17.55
6.27 5.11 45.85 – 20.78
5–20%
1 2 3 4 5
6.03 4.18 10.35 17.96 24.93
5.18 3.22 4.20 – 22.24
14.10 22.97 59.42 – 10.79
10–25%
1 2 3 4 5
9.11 6.61 17.88 25.68 42.49
10.17 6.65 10.62 – –
11.64 0.61 40.60 – –
Table 3 Modal damping values for frame with appendage for case (a): first natural frequency of appendageEfirst natural frequency of frame. Damping structure— appendage
Mode
Modal damping Eq. (10) (%)
Modal damping Eq. (8) (%)
Error (%) in modal damping
5–1%
1 2 3 4 5
4.84 2.94 4.21 5.93 11.72
5.02 – 3.12 4.04 20.29
3.72 – 25.89 31.88 73.12
1 2 3 4 5
5.26 7.79 4.35 6.04 11.74
5.02 – 3.12 4.04 20.29
4.56 – 28.28 33.11 72.83
1 2 3 4 5
6.14 13.66 5.73 6.59 12.63
5.02 – 3.12 4.04 20.29
18.24 – 45.55 38.69 60.65
1 2 3 4 5
10.23 19.84 8.71 11.78 20.47
9.87 – 6.51 10.59 –
3.52 – 25.26 10.10 –
5–7%
5–20%
10–25%
Fig. 6. Comparison of roof displacements—OB accelerogram and case (a).
these modes and the frequency content of the seismic excitation. In general, these errors prove the inability of the proposed method to give satisfactory seismic response results for a non-classically damped structure having an appendage (equipment) with a mode frequency almost coincident with the frequency of a mode of the structure.
4. Conclusions On the basis of the preceding developments, the following conclusions can be stated: (1) An approximate modal damping identification model for nonclassically damped linear plane building frames has been
ARTICLE IN PRESS G.A. Papagiannopoulos, D.E. Beskos / Soil Dynamics and Earthquake Engineering 29 (2009) 583–589
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frequency content. The response of the appendage is always underestimated by the proposed method.
Acknowledgements The authors acknowledge with thanks the support provided to them by the European Social Fund (ESF), Operational Program for Educational and Vocational Training I (EPEAEK I), through the Greek Program PYTHAGORAS I of the Greek Ministry of Education and Religious Affairs. References
Fig. 7. Comparison of roof displacements—ERZ accelerogram and case (a).
proposed and its accuracy and range of applicability have been assessed through parametric studies involving various types of frames with respect to damping distribution and degree of frequency separation under seismic motions with different frequency content. (2) The proposed approximate method provides identified modal damping ratios and seismic response results of high accuracy for regular frames with a uniform heightwise damping distribution for which the assumption of classical damping is acceptable. (3) The proposed approximate method provides response results of enough accuracy for geometrically regular frames with non-uniform distribution of damping irrespectively of the damping distribution in the frame and the frequency content of the excitation. More specifically, the accuracy is high for the case of non-uniform distribution of structural damping and seismic excitations of low frequency content, while it is satisfactory for the case of non-uniform distribution of structural damping and seismic excitations of high frequency content. On the other hand, the error in identified modal damping ratios obtained by the proposed method is large for all the above cases. (4) The proposed approximate method as applied to geometrically irregular frames with closely spaced frequencies, as it is, e.g., the case of frames having appendages (equipment), provides response results of satisfactory accuracy for the case of seismic excitations with low frequency content and of very low accuracy for the case of seismic excitations with high
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Soil Dynamics and Earthquake Engineering journal homepage: www.elsevier.com/locate/soildyn
Technical Note
On a modal damping identification model for non-classically damped linear building structures subjected to earthquakes George A. Papagiannopoulos a, Dimitri E. Beskos a,b, a b
Department of Civil Engineering, University of Patras, GR-26500 Patras, Greece Office of Theoretical and Applied Mechanics, Academy of Athens, 4 Soranou Efessiou street, 11527 Athens, Greece
a r t i c l e in fo
abstract
Article history: Received 24 September 2008 Accepted 29 October 2008
A simple modal damping identification model developed by the present authors for classically damped linear building frames is extended here to the non-classically damped case. The modal damping values are obtained with the aid of the frequency domain modulus of the roof-to-basement transfer function and the resonant frequencies of the structure (peaks of the transfer function) as well as the modal participation factors and mode shapes of the undamped structure. The assumption is made that the modulus of the transfer function of the non-classically damped structure matches the one of the classically damped structure in a discrete manner, i.e., at the resonant frequencies of that function modulus. This proposed approximate identification method is applied to a number of plane building frames with and without pronounced non-classical damping under different with respect to their frequency content earthquakes and its limitations and range of applicability are assessed with respect to the accuracy of both the identified damping ratios and that of the seismic structural response obtained by classical mode superposition and use of those identified modal damping ratios. & 2008 Elsevier Ltd. All rights reserved.
Keywords: Modal damping ratios Identification model Linear building frames Non-classical damping Transfer function Frequency domain Earthquakes
1. Introduction Structural identification is usually the inverse problem of structural dynamics that tries to estimate the dynamic properties of a structure (modal shapes, natural frequencies, modal damping ratios, etc.) on the basis of measurements of response to known dynamic excitations [1–3]. For linear elastic structural behaviour, energy of dissipation is usually taken into account by viscous damping in the form of modal damping ratios [4]. Among the many works on structural damping identification on the basis of measured seismic response, one can mention the works of Caravani and Thomson [5], McVerry [6], Beck and Jennings [7], Shinozuka et al. [8], Safak [9], Ruzzene et al. [10], Yin et al. [11] and Yang et al. [12]. Recently, a simple modal identification model for building structures was presented by the present authors [13] as a generalization of a model originally proposed by Hart and Vasudevan [14]. This model works in the frequency domain and provides time-invariant modal damping ratios of building structures subjected to earthquakes in terms of the modulus of the roof-to-basement transfer function, the resonant frequencies of the structure (peaks of the transfer function) as well as the modal
Corresponding author at: Department of Civil Engineering, University of Patras, GR-26500 Patras, Greece. Tel.: +30 261 099 6559; fax: +30 261 099 6578. E-mail address: [email protected] (D.E. Beskos).
0267-7261/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.soildyn.2008.10.005
participation factors and mode shapes of the undamped structure. The model was found to give excellent results for linear plane building frames on the assumption of classical viscous damping [13]. In this work, this identification model is extended to the case of non-classical damping. Classical (or proportional) damping represents an appropriate idealization if damping is distributed throughout the structure, e.g., a multi-storey building with a similar structural system and structural material over its height [4]. One well-known procedure to construct a classical damping matrix from modal damping ratios is due to Rayleigh. However, if the structural system consists of two or more parts with quite different amounts of damping, e.g., a soil-building system or a structure with special energy dissipating devices or equipment, the system damping is non-classical (non-proportional) [4]. A system with classical damping can be uncoupled by its real undamped modal shapes for the purpose of obtaining its seismic response by real modal superposition. When its damping is non-classical this can only be done by its complex damped modal shapes. Caughey and O’ Kelly [15] found the necessary and sufficient conditions that mass, stiffness and damping matrices should satisfy in order to have classical damping. Thus, one should expect that use of the simple identification model of [13] constructed on the assumption that its damping is classical, will lead to modal damping ratios in error for those cases of structures characterized by non-classical damping. In this work, the identification model of [13] is extended to also cover
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approximately the case of non-classical damping. This is accomplished by assuming that the modulus of the frequency response transfer function of the non-classically damped system matches the one of the classically damped system in a discrete manner, i.e., at the peaks (resonant frequencies) of that function modulus. Here the proposed identification model is applied to a number of plane building frames with and without pronounced non-classical damping under two different with respect to their frequency content earthquakes and its limitations and range of applicability
Thus, the roof-to-basement transfer function R(o) evaluated at o ¼ ok and on account of q€j ðoÞ ¼ o2 qj ðoÞ receives the form [13] Rðo ¼ ok Þ ¼ 1 þ
X j
frj Gj o2k ðo o2k Þ þ i ð2xj oj ok Þ 2 j
(7)
where frj is the element of the j-th eigenvector corresponding to the roof. Finally, the modulus of R(o ¼ ok) takes the real valued form [13]
jRðo ¼ ok Þj2 ¼1þ2
þ
N X
frj Gj o2k ðo2j o2k Þ
j¼1
ðo2j o2k Þ2 þ ð2 xj oj ok Þ2
N ½f2 G2 o4 ðo2 o2 Þ2 þ 4 x2 o2 o2 X rj j j k j k j k
½ðo2j o2k Þ2 þ ð2 xj oj ok Þ2 2
j¼1
þ2
N X jam;m4j
frj Gj frm Gm o4k ½ðo2j o2k Þ ðo2m o2k Þ þ 4 xj xm oj om o2k h
i (8) ðo2j o2k Þ2 þ ð2 xj oj ok Þ2 ½ðo2m o2k Þ2 þ ð2 xm om ok Þ2
are assessed both with respect to the accuracy of the identified modal damping ratios and that of the seismic structural response obtained by classical mode superposition and use of those identified modal damping ratios.
2. Identification model for calculation of modal damping values The matrix equation of motion for a N degrees-of-freedom, linear, elastic, viscously damped plane building frame subjected at ¨g ¼ u ¨ g(t) can be written in its base to an earthquake acceleration u the form ½M fug € þ ½C fug _ þ ½K fug ¼ ½M fIg u€ g
(1)
where, [M], [C] and [K] are the mass, damping and stiffness matrices, respectively, of the structure, {u} ¼ {u(t)} is the displacement vector of the structure relative to its base, overdots denote differentiation and {I} is the identity vector. On the assumption that the damping matrix is of Rayleigh type, i.e., ½C ¼ a ½M þ b ½K
(2)
a solution of Eq. (1) expressed as a model superposition, i.e., as fug ¼ ½F fqg
(3)
where [F] is the modal matrix and {q} ¼ {q(t)} is the modal amplitude vector, uncouples Eq. (1) and leads to a system of N independent modal equations of the form q€j ðtÞ þ 2 xj oj q_j ðtÞ þ o2j qj ðtÞ ¼ Gj u€ g ðtÞ
(4)
In the above, oj and xj are the undamped natural frequency and modal damping ratio of mode j and Gj is the participation factor of mode j (j ¼ 1, 2, y, N). Applying the Fourier transform with respect to time onto Eq. (4) and solving for the transformed modal amplitude, one can obtain in the frequency o domain
It should be noted that the last summation term in Eq. (8) was accidentally omitted in Eq. (11) of [13] even though it was taken into account in all computations there. Eq. (8) represents a system of nonlinear algebraic equations that can be solved iteratively in conjunction with a user defined convergence criterion to estimate modal damping values xk on the basis of known values of |R(o ¼ ok)|, frj, ok and Gj. For a linear structure, its dynamic characteristics frj, ok and Gj can be obtained analytically for known stiffness and mass matrices, while |R(o ¼ ok)| can be constructed on the basis of the known seismic acceleration and the measured roof acceleration. One should point out that this method can provide modal damping values only for the modes which appear in the transfer function [13]. When the damping in the structure is non-classical, the above identification model provides modal damping ratios which are, in general, in error. The use of this model for non-classically damped structures is accomplished by assuming that the modulus of the frequency response transfer function of the non-classically damped system matches the one of the classically damped system in a discrete manner, i.e., at the peaks (resonant frequencies) of that function modulus. This matching is restricted to those peaks, which appear in the modulus of the transfer function, since higher modes do not show up in that function modulus and contribute only statically to the response. This matching approach follows Thomson et al. [16] and Tsai [17], who first proposed it in connection with the problem of using real mode superposition for non-classically damped systems in order to determine their dynamic response. For the case of non-classical damping the exact values of modal undamped frequencies oj and modal damping ratios xj can be obtained at the j-th mode as [18] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi oj ¼ jOj j ¼ ReðOj Þ2 þ ImðOj Þ2 (9)
xj ¼
ReðOj Þ jOj j
(10)
Gj u€ g ðoÞ (5) ðo2j o2 Þ þ ið2xj oj oÞ pffiffiffiffiffiffiffi where i ¼ 1 and overbars denote transformed quantities. Using Eq. (3), one can express the absolute acceleration vector ¨ (t)} in the frequency domain as {U
where Re(Oj), Im(Oj) are the real and imaginary part, respectively, of (Oj), i.e, the complex eigenfrequency of mode j of the structure. These complex eigenfrequencies are obtained by solving the quadratic eigenproblem corresponding to Eq. (1), i.e.,
€ oÞg ¼ ½F fqð € oÞg þ fIgu€ g ðoÞ fUð
where the vector {f} represents the complex eigenvector.
q j ð oÞ ¼
(6)
ðO2 ½M þ iO½C þ ½KÞffg ¼ 0
(11)
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3. Numerical examples and discussion In this section, the seismic response of linear elastic, plane, building frames with non-classical damping is obtained on the basis of the proposed method in order to illustrate that method and assess its accuracy and range of applicability. This assessment is done by comparing the approximate results of the proposed identification model, i.e., modal damping ratios and structural response obtained by modal superposition using identified modal damping ratios, against the exact ones of Eq. (10) and time integration, respectively. The assessment involves the investigation of the interplay among three parameters, namely the natural frequencies of the frames, the damping distribution in the frames and the type of seismic motion that excites the frames. These parameters influence the response of a non-classically damped system [19] and therefore, are taken into account herein. The case of seismically excited plane building frames with uniform damping distribution over their height and regularity in geometry implying well-separated frequencies has been studied in detail in [13] and found to produce highly accurate modal damping ratios and response results under the assumption of classical damping, in agreement with [4]. In this section, the cases of plane building frames with a non-uniform distribution of damping and geometric irregularity resulting in closely spaced frequencies for which damping should be considered as non-classical are investigated. The two linear elastic, plane, building frames considered here are shown in Fig. 1. The one bay and three storey steel frames of Fig. 1a have a bay span of 4.00 m and storey height of 3.00 m and consist of IPE 240 sections for beams and HEB 280 for columns. Load on beams equals 30 kN/m for the first and second storey and 15 kN/m for the third one. System damping is assumed to be concentrated at the bottom storey or at the top storey and can take the values 7%, 10%, 20% and 40%. The two frames are subjected to the longitudinal components of the OB accelerogram from the 2001 El Salvador earthquake and of the ERZ accelerogram from the 1992 Erzincan, Turkey, earthquake. The accelerograms of these earthquakes exhibit radical difference in their
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frequency content as shown in their Fourier transforms in Figs. 2 and 3. Table 1 provides the natural frequencies oj of Eq. (9), the resonant frequencies from |R(o)| of Eq. (8), the modal damping values of the frame as computed by using Eqs. (10) and (8) for the damping location and values mentioned above and the relative error between Eqs. (10) and (8). One can easily observe that the error between the frequencies is very small, whereas the one for damping is very large (up to about 60%). To account for the quasi-static behaviour of the modes that do not appear in the transfer function, a value of 100% is considered for them when calculating the approximate response. Figs. 4 and 5 illustrate the comparison between the approximate and the exact roof displacement response of the frame structure under the ERZ and OB accelerograms for the case of 10% damping concentrated at the lower storey. Similar response plots (base shear, absolute
Fig. 2. Fourier transform of the OB accelerogram (2001 El Salvador earthquake).
Fig. 1. Plane building frames considered.
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roof acceleration and roof displacement) have been constructed for the rest of the damping cases of Table 1 when the frame structure is subjected to these accelerograms, but they are not shown here due to space limitations. Table 2 gives the exact and approximate absolute maximum values of base shear and roof displacement for the frames of Fig. 1a as well as the relative error.
From Table 2 and all response plots constructed it can be concluded that (a) the error between approximate and exact responses is, practically speaking, negligible (maximum 3%) when the seismic excitation has a low frequency content (ERZ accelerogram) no matter the value and the distribution of damping in the structure; (b) the error between approximate and exact
Fig. 3. Fourier transform of the ERZ acclerogram (1992 Erzincan earthquake).
Fig. 4. Comparison of roof displacements for the case of 10% damping at lower storey (ERZ accelerogram).
Table 1 Results for the one bay and three storey plane framed structure. Damping percentage and location
Mode
Natural frequencies (Hz) Eq. (9)
Resonant frequencies (Hz) Eq. (8)
Modal damping Eq. (10) (%)
Modal damping Eq. (8) (%)
Error (%) in modal damping
7%—lower storey
1 2 3
1.75 6.32 12.11
1.74 6.30 11.94
5.25 5.71 3.37
6.90 2.87 4.35
31.43 49.74 29.08
10%—lower storey
1 2 3
1.77 6.32 12.07
1.74 6.28 11.93
7.25 8.15 3.81
9.83 4.14 6.23
35.59 49.20 63.52
20%—lower storey
1 2 3
1.84 6.31 12.06
1.72 6.28 –
17.10 19.45 4.68
19.76 8.91 –
15.55 54.19 –
40%—lower storey
1 2 3
1.88 6.29 12.06
1.66 6.23 –
35.51 42.67 5.35
40.06 25.20 –
12.81 40.94 –
7%—upper storey
1 2 3
1.76 6.32 12.10
1.76 6.30 11.96
2.88 3.64 8.95
3.16 1.64 3.61
9.72 54.95 59.66
10%—upper storey
1 2 3
1.76 6.33 12.08
1.76 6.30 11.94
4.11 5.46 12.4
4.48 2.34 5.34
9.00 57.14 57.25
20%—upper storey
1 2 3
1.75 6.36 11.98
1.74 6.30 11.90
6.25 5.98 17.78
8.92 4.72 13.56
42.72 21.07 23.73
40%—upper storey
1 2 3
1.75 6.36 11.64
1.74 6.32 –
12.95 7.87 29.58
18.82 11.25 –
45.33 42.95 –
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0.82 3.85 1.39 6.82 0.00 0.75 1.90 2.63 0.121 0.100 0.071 0.041 0.139 0.132 0.103 0.074 0.122 0.104 0.072 0.044 0.139 0.133 0.105 0.076 0.80 7.76 9.82 8.37 2.36 1.31 3.98 2.29 296 240.4 158 95.3 350.6 315.9 248.5 166.1 298.4 260.6 175.2 104 342.5 320.1 258.8 170 0.00 1.33 1.47 1.82 0.00 0.00 0.00 0.00 0.077 0.074 0.067 0.054 0.08 0.079 0.075 0.068 0.077 0.075 0.068 0.055 0.08 0.079 0.075 0.068 0.00 0.69 1.87 2.92 0.43 0.05 0.11 0.00 181 175 158 129.6 187.6 183.7 175.9 160.3 181 176.2 161 133.5 186.8 183.8 175.7 160.3 7%—lower storey 10%—lower storey 20%—lower storey 40%—lower storey 7%—upper storey 10%—upper storey 20%—upper storey 40%—upper storey
Exact maximum base shear (kN) Error (%) for roof displacement Approximate maximum roof displacement (m) Exact maximum roof displace ment (m) Error (%) for base shear Approximate maximum base shear (kN) Exact maximum base shear (kN)
ERZ accelerogram Damping percentage and location
Table 2 Approximate and exact maximum response values and their relative error for the ERZ and OB accelerograms.
responses is of some significance for base shear and displacement response, (nearly 10% and 7%, respectively), when the seismic excitation has a high frequency content (OB accelerogram) no matter the distribution of damping; (c) the error in maximum responses is higher in the case of OB accelerogram than in the case of ERZ accelerogram. The error between approximate and exact responses is attributed to the damping coupling effect (effect due to non-negligible off-diagonal damping matrix terms) associated with the first mode and a higher mode of the framed structure that cannot be handled correctly by the approximate method, especially for the case of a excitation with a high frequency content. In order to assess the accuracy of the proposed method when two or more natural frequencies of a non-classically damped system are closely spaced, the case of a plane, two bay and seven storey steel moment resisting framed structure with an appendage interacting with the structure and being located on its top floor, as shown in Fig. 1b, is considered. Each bay and each storey of the frame have 4.00 m span and 3.00 m height, respectively. All beams have IPE 400 sections, while column sections vary for every other four storeys from HEB 400 to HEB 360 going from bottom to top. Load on all beams equals 25 kN/m. The damping in the frame varies but is uniformly distributed along its height. The appendage is modelled as a beam element with a mass of 0.01 times the mass of the structure and varying stiffness and damping values. The structure is excited separately by the ERZ and OB accelerograms. Two cases are examined: (a) the first mode frequency of the appendage almost coincides with the frequency of the first mode of the structure and (b) the first mode frequency of the appendage almost coincides with the frequency of a higher mode of the structure, e.g., the third mode. Non-classicality in damping is taken into account by considering that (i) the structure has 5% damping (in the first and fifth mode) and the appendage 1%, 7%and 20% damping (in the first and second mode) and (ii) the structure has 10% damping (in the first and fifth mode) and the appendage 25% damping (in the first and second mode). Tables 3 and 4 display the modal damping values as obtained by using Eqs. (10) and (8) for the aforementioned cases (a) and (b). One can easily observe that the error between damping ratios computed by Eqs. (10) and (8) reaches nearly 73%. Eq. (8) does not provide a modal damping ratio for the first mode of the appendage because that mode had coalesced into a mode of the
OB accelerogram
Fig. 5. Comparison of roof displacements for the case of 10% damping at lower storey (OB accelerogram).
Approximate maximum base shear (kN)
Error (%) for base shear
Exact maximum roof displace ment (m)
Approximate maximum roof displacement (m)
Error (%) for roof displacement
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structure, and thus it cannot appear in the transfer function. More specifically, the first mode of the appendage in case (a) coalesces into the first mode of the structure and in case (b) coalesces into its third mode. The only exception takes place in case (b) when the structure has 5% and the appendage 1% damping and the mode of the appendage appears in the transfer function. To account for the quasi-static behaviour of the modes that do not appear in the transfer function, a value of 100% is considered for them when calculating the approximate response for the cases (a) and (b). It should be also mentioned that when the approximate response is calculated, the damping value of the mode that corresponds to the appendage is considered to be equal to the modal damping of the first mode of the structure for case (a) and to that of the third mode for case (b). This is done due to the coalescence of the two modes mentioned above. Figs. 6 and 7 illustrate the comparison between the approximate and the exact displacement response of the frame structure under the OB and ERZ accelerograms for the case (a) when the structure has 10% and the appendage 25% damping. Similar exact and approximate response plots (base shear, absolute roof acceleration and roof displacement) have been constructed and the relative error between their absolute maximum values has been calculated, but they are not shown here due to space limitations. From the comparison between approximate and exact responses for cases (a) and (b), it can be concluded that (i) the errors in all kinds of responses of the structure are much greater for the case of the OB accelerogram, while they are smaller for that of the ERZ accelerogram; (ii) the error in maximum absolute response values of the structure is about 10% for the ERZ accelerogram and 19% for the case of the OB accelerogram; (iii) the error for the response of the appendage is always large (about 40%), especially for the case of the OB accelerogram, and it is always from the nonconservative side. The errors in response calculation are attributed to the interplay among the coincidence of the structure and appendage mode(s), the damping coupling effect associated with
Table 4 Modal damping values for frame with appendage for case (a): first natural frequency of appendageEthird natural frequency of frame. Modal damping Eq. (10) (%)
Modal damping Eq. (8) (%)
Error (%) in modal damping
Damping structure— appendage
Mode
5–1%
1 2 3 4 5
5.12 3.24 5.56 2.66 4.47
5.03 3.07 3.61 1.34 5.09
1.76 5.25 35.07 49.62 13.87
5–7%
1 2 3 4 5
5.42 3.33 7.72 11.27 14.53
5.08 3.16 4.18 – 17.55
6.27 5.11 45.85 – 20.78
5–20%
1 2 3 4 5
6.03 4.18 10.35 17.96 24.93
5.18 3.22 4.20 – 22.24
14.10 22.97 59.42 – 10.79
10–25%
1 2 3 4 5
9.11 6.61 17.88 25.68 42.49
10.17 6.65 10.62 – –
11.64 0.61 40.60 – –
Table 3 Modal damping values for frame with appendage for case (a): first natural frequency of appendageEfirst natural frequency of frame. Damping structure— appendage
Mode
Modal damping Eq. (10) (%)
Modal damping Eq. (8) (%)
Error (%) in modal damping
5–1%
1 2 3 4 5
4.84 2.94 4.21 5.93 11.72
5.02 – 3.12 4.04 20.29
3.72 – 25.89 31.88 73.12
1 2 3 4 5
5.26 7.79 4.35 6.04 11.74
5.02 – 3.12 4.04 20.29
4.56 – 28.28 33.11 72.83
1 2 3 4 5
6.14 13.66 5.73 6.59 12.63
5.02 – 3.12 4.04 20.29
18.24 – 45.55 38.69 60.65
1 2 3 4 5
10.23 19.84 8.71 11.78 20.47
9.87 – 6.51 10.59 –
3.52 – 25.26 10.10 –
5–7%
5–20%
10–25%
Fig. 6. Comparison of roof displacements—OB accelerogram and case (a).
these modes and the frequency content of the seismic excitation. In general, these errors prove the inability of the proposed method to give satisfactory seismic response results for a non-classically damped structure having an appendage (equipment) with a mode frequency almost coincident with the frequency of a mode of the structure.
4. Conclusions On the basis of the preceding developments, the following conclusions can be stated: (1) An approximate modal damping identification model for nonclassically damped linear plane building frames has been
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frequency content. The response of the appendage is always underestimated by the proposed method.
Acknowledgements The authors acknowledge with thanks the support provided to them by the European Social Fund (ESF), Operational Program for Educational and Vocational Training I (EPEAEK I), through the Greek Program PYTHAGORAS I of the Greek Ministry of Education and Religious Affairs. References
Fig. 7. Comparison of roof displacements—ERZ accelerogram and case (a).
proposed and its accuracy and range of applicability have been assessed through parametric studies involving various types of frames with respect to damping distribution and degree of frequency separation under seismic motions with different frequency content. (2) The proposed approximate method provides identified modal damping ratios and seismic response results of high accuracy for regular frames with a uniform heightwise damping distribution for which the assumption of classical damping is acceptable. (3) The proposed approximate method provides response results of enough accuracy for geometrically regular frames with non-uniform distribution of damping irrespectively of the damping distribution in the frame and the frequency content of the excitation. More specifically, the accuracy is high for the case of non-uniform distribution of structural damping and seismic excitations of low frequency content, while it is satisfactory for the case of non-uniform distribution of structural damping and seismic excitations of high frequency content. On the other hand, the error in identified modal damping ratios obtained by the proposed method is large for all the above cases. (4) The proposed approximate method as applied to geometrically irregular frames with closely spaced frequencies, as it is, e.g., the case of frames having appendages (equipment), provides response results of satisfactory accuracy for the case of seismic excitations with low frequency content and of very low accuracy for the case of seismic excitations with high
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