- Email: [email protected]
Dynamic analysis of a ten-story reinforced concrete building using a continuum model
Compurers
w
& Srrucrures
. _
> Pergamon
0045-794!MSMKI166-2
Vol. 58. No. 3, pp. 487-498. 1996 Elswier Science Ltd Printed in Great Britain 0045.7949/96 $9.50 + 0.00
DYNAMIC ANALYSIS OF A TEN-STORY REINFORCED CONCRETE BUILDING USING A CONTINUUM MODEL M. J. Chajes, W. W. Finch Jr and J. T. Kirby Department of Civil Engineering, University of Delaware, 137 DuPont Hall, Newark, DE 19716, U.S.A. (Received 18 November 1994) Abstract-During the 1989 Loma Prieta earthquake, accelerometers maintained by the California Division of Mines and Geology Strong Motion Instrumentation Program (CSMIP) recorded the response of a IO-story, reinforced-concrete building located in San Jose, California. In this paper, a computationally efficient, approximate, dynamic analysis of the building is conducted utilizing a reduced-order continuum model. Continuum methodology makes use of the repetitive nature of lattice framing systems to generate finite element models having significantly fewer degrees of freedom than models generated using classical
discrete finite element techniques. The vibrational characteristics of the continuum model, as well as the results of dynamic analyses, are compared to information gained from the recorded response of the building. The continuum model proves to be quite accurate, both in capturing the dominant periods of vibration of the structure and in predicting the time-history response. As a result, the method shows promise as a tool for use in the analysis and design of large lattice structures subjected to earthquake loads.
INTRODUCTION
On October 17, 1989, a M, = 7.1 (surface wave magnitude) earthquake occurred along the San Andreas Fault in Northern California with its epicenter located under a mountain called Loma Prieta. The earthquake triggered a large number of strongmotion accelerometers deployed throughout the San Francisco Bay area by both the California Division of Mines and Geology [l] and the U.S. Geological Survey [2]. These instruments recorded a great deal of valuable seismic response data for a variety of structures including buildings, bridges, dams and tunnels. These data provide the opportunity to evaluate the accuracy of either existing or newly developed computer analysis techniques. Classical discrete finite element methods for predicting linear response of large structures to even simple static loading may involve the solution of thousands of linear simultaneous equations. Adding time-varying dynamic loadings, along with material and geometric nonlinearities, can lead to thousands of coupled simultaneous nonlinear equations which must be solved at very small time increments. Even with modern high-speed computers, the computational effort, storage requirements and associated cost of solving such equations can prohibit the use of full-scale dynamic analysis in everyday design. As a result, approximate yet accurate methods for conducting dynamic analyses of large structural frameworks are needed. Various researchers have investigated the use of continuum models for conducting both linear and nonlinear analyses of lattice structures [3-241. Finite element models resulting from continuum method487
ology have significantly fewer degrees of freedom than discrete finite element models, which individually model each of the beam and column elements. Hence, the use of continuum models can result in a considerable reduction incomputational effort with a corresponding savings in cost. This paper focuses on the seismic response of a lo-story, reinforced-concrete building located in San Jose, California, to the Loma Prieta earthquake. In particular, the paper investigates the use of a reducedorder continuum model to predict the dynamic response of the structure. The accuracy of the analysis is evaluated by comparing the computed results to the actual building response recorded during the earthquake. To the author’s knowledge, this is the first attempt to compare the dynamic continuum analysis of a large building to its recorded structural response during an earthquake.
CONTINUUM METHODOLOGY
During the past 20 years, continuum models of large truss and frame structures have been used to study vibrational characteristics, static and dynamic response, and buckling behavior. Among the most prominent works are those by Noor et al. [3-71, who studied the static and dynamic response of beamlike lattice structures and predicted their buckling characteristics; Abdel-Ghaffar [%-lo], who used continuum models to find frequencies and mode shapes of suspension bridges; Sun et al. [l l] and Abrate and Sun [12], who used a beam element developed by Yang [13] to conduct vibrational analyses of planar truss systems; Sun and Juang [14], who incorporated
M. J. Chajes et al.
Typical cell ~_,i_____i‘i_____ii_____i_i_____i,_~ _ _______________________-__---- -*
//~~N/N//~~///////// Lattice framework
I
Fig. 1. Typical repeated cell.
energy dissipation in the form of viscous damping to study the simple transient vibration of trusses; and most recently, Necib and Sun [ 151, who used a higher-order Timoshenko beam element to solve dynamic problems involving beamlike truss structures. The work of Necib and Sun [15] included force calculations for discrete elements of the lattice. The continuum model utilized in this work was developed by McCallen and Romstad [ 16-181. This model replaces a lattice structure with a continuous structure that yields essentially the same global static and dynamic behavior. The equivalence of the continuum is established by setting the strain energy of the continuum equal to the approximate strain energy of the discrete lattice. The original application of the McCallen-Romstad model involved finding frequencies and mode shapes of trusses and frames as well as determining static displacements. Nonlinear geometric effects were incorporated into the original model. In preliminary studies, the method gave results that compared favorably with results obtained using discrete finite element analyses. McCallen and Romstad [19], Chajes [20], and Chajes et al. [21,22] extended the continuum methodology to include nonlinear material effects on trusses and frames subjected to static and dynamic loadings. Osterkamp [23] developed a method of determining discrete member forces for single-bay frames, and Chajes [20] and Chajes et al. [24] extended the work to include multiple-bay frames. Unlike earlier papers in which continuum analysis results were compared to discrete analysis
results, this paper involves the more challenging task of comparing computed results with actual measured response. Description of the continuum model Because the details of the continuum model used in this work have been published extensively, they will only be presented briefly herein. For a more in-depth description, the reader is referred to McCallen [16], McCallen and Romstad [18] and Chajes et al. [24]. Large lattice structures are made of discrete structural members that are framed together. Often these members are connected in ways that form repeated geometric patterns or “ceils.” In a building, each cell may contain many beams and columns as shown in Fig. 1. The entire framework is often a combination of just a few different types of cells. The objective of the continuum model is to replace these repetitive cells with an “equivalent” continuum, thereby greatly reducing the number of elements needed to model the lattice. This results in a significant reduction in the number of global degrees of freedom needed to describe the structure (Fig. 2). In developing an appropriate model, assumed axial, bending and shear modes of deformation are used to represent the possible deformation patterns of the lattice. The three assumed deformation modes are shown in Fig. 3. An approximate strain energy expression for the lattice, oL, is developed in terms of the axial force (Fci) and bending moment (Mci) in the columns and
Discrete model Fig. 2. Discrete and associate continuum models including global degrees of freedom.
489
Dynamic analysis of a ten-story reinforced concrete building
Axial
Bending
Shear
Fig. 3. Modes of deformation. a generalized section shear force (I’) resulting from the assumed deformations. The approximate lattice strain energy expression is given by
strains, (2) v, and u2 are transverse displacements, (3) dv,/dx and do&lx are slopes along the centerline of the continuum element and are used to get values of curvature, and (4) r3,, e2 and & are associated with rotations of the cross-section and are used to get axial strains due to these rotations. Because deformations of the lattice are limited to combinations of the assumed modes, the lattice energy expression is only approximate and its accuracy depends on how well the assumed modes represent the actual deformations.
DETAILS OF THE IO-STORY BUILDING
where P(x, y,) is the off-axis strain, K(X) is the curvature of the reference axis, 4(x) is the slope of the reference axis due to shear deformation, and NC is the number of columns in the lattice cross-section. The equivalence of the continuum is established by incorporating standard kinematic and constitutive relationships and setting the strain energy of the continuum equal to the approximate strain energy of the lattice. The continuum strain energy, Uc, is
+(GA,)~6+~(GA,)B’
The structure being studied is a IO-story, reinforced-concrete, commercial building located in downtown San Jose, California. The building was designed in 1964 and construction was completed in 1967. The building has a rectangular plan measuring 82 ft by 192 ft and rests on a 5-foot-thick mat foundation. The superstructure rises 124 ft above ground level and consists of 10 above-grade and 1 belowgrade stories. The vertical load carrying system consists of 4.5 in thick concrete floor slabs supported by concrete pan joists and a reinforced-concrete frame. For its lateral force resisting system, the building has shear walls in the east-west direction and a momentresistant, reinforced-concrete frame in the northsouth direction. Figure 5 shows a typical elevation view of the moment-resistant frame. Included in Fig. 5 are cross-sections of the typical elements (columns ClC4 and beams Bl). Figure 6 shows a typical floor framing plan.
dx, 1
where E and G are the material’s Young’s modulus and shear modulus, A, and I, are the area and moment of inertia of the columns, A, is the effective shear area of the “repeated cell,” u and u represent longitudinal and transverse displacements, and 0 represents the cross-section rotation. Applying standard energy minimization principles to eqn (2), a beam-like finite element for the continuum is formed. Additional degrees of freedom in the continuum element, not found in the typical Timoshenko beam elements, are needed as a result of continuum strain energy terms not found in the standard Timoshenko beam theory. The derived firstorder continuum finite element with nine degrees of freedom is shown in Fig. 4 and the associated element stiffness matrix is given in the Appendix. A more detailed derivation of the continuum element stiffness matrix is given in Chajes et al. [24]. The continuum element provides C, continuity of both shear and bending displacement. With regard to the geometrical representation of the displacement degrees of freedom, note that (1) u, and u2 are longitudinal displacements and are used to get axial
MEASURED STRUCTURAL
RESPONSE
The IO-story building is located 24 miles from the epicenter of the 1989 Loma Prieta earthquake. During the earthquake, 11 of 13 accelerometers located throughout the building recorded 60 s of dynamic response [ 11.The motion sensors were installed and maintained by the California Strong Motion Instrumentation Program (CSMIP) of the California Division of Mines and Geology of the Department of Conservation. (This particular structure is designated as CSMIP station 57355.) The locations of the motion sensors are shown in Fig. 7. No data was recorded by sensors 2 and 10 during the earthquake.
ur+&&l!$+ dv, dx
dv, dx
Fig. 4. Continuum finite element degrees of freedom.
490
M. J. Chajes ef al. 24 in
I
I
36 in
Cl
24 in
c2
24 in I c3
27.5 in 26 in 27.5 in
1
24 in
I
I
I
I
24 in
c4
Bl
ment
I
I ‘27.5 ft’
5 0 27 ft
‘27.5 ,I
190 ft Fig. 5. Typical moment-resistant Preliminary analysis of response Based on the recorded information, the building experienced peak ground accelerations of 9 and 11% of gravity in the north-south and east-west direc-
tions, respectively, and peak superstructure accelerations of 26 and 38% of gravity in the north-south and east-west directions, respectively. By comparing displacements of parallel sensors on the roof (sensors 3 and 4) the building is seen to exhibit virtually no torsional response (see Fig. 8). As a result, the north-south and east-west frames can be
27.5 ft I
5 @ 27 ft
frame (north-south).
studied individually as two-dimensional systems. Because the continuum methodology has been developed for lattice framing systems, this paper will focus on the response of one of the interior north-south moment-resistant frames. The base acceleration in the north-south direction (taken to be the acceleration recorded by sensor 13 in the basement) and the north-south displacement of the roof relative to the base (sensor 5 minus 13) are shown in Fig. 9. The strong motion of the earthquake lasted for approximately 10 s during which a peak acceleration of 8.6% of gravity occurred. A peak
27.5 ft
190 ft Fig. 6. Typical floor framing plan.
I
491
Dynamic analysis of a ten-story reinforced concrete building 8.0 -‘j6.0
-10
4.0
i--
2 ;
2nd Floor
tT+,
4t
Roof Fig. 7. Location of motion sensors.
2.0
B H 0.0 S d
-2.0 -4.0 -6.0
f
-8.0 + 0 relative displacement of 1.97 in occurred at 19.5 s, after which the building response damped out. A power spectral density of the 60 s of recorded roof response (see Fig. 10) indicates that the fundamental mode has a period of vibration around 1 s and dominates the response. This period is consistent with a period of 1.1 s found using the formula T = C,hff4 given in the 1988 UBC code[25], where T is the building’s fundamental period, h, is the height of the building in feet, and C, = 0.03 for reinforced-concrete moment frames. Analysis using wavelet transforms While Fourier amplitude spectra or power spectral densities computed using recorded time-history data have been used to identify dominant periods of vibration for other structures instrumented by CSMIP [26,27], these methods of analysis give only a time-averaged frequency content. Kirby et al. [28] have shown how wavelet transforms can be used to determine continuous frequency content of seismic response with respect to time. The transform technique involves computing the convolution of the vibrational response f(t) with a scaled analyzing wavelet g*((t - b)/a), where b is the lag of the wavelet relative to the time origin, a is a scale factor,
10
20 30 40 Time (seconds)
50
60
Fig. 8. Absolute horizontal displacement of parallel roof sensors (east-west).
and * denotes the complex conjugate. The wavelet transform T(b, a) is then given by T(b.a)=~jag*(~)i(t)dr.
(3)
where the analyzing wavelet g(t) to be used here is the Morlet wavelet [29] given by s(t) = eibb,ot e-1212.
(4)
While this method, originally developed by Morlet to characterize seismic signals, has been used in applications ranging from speech analysis to the study of fluid turbulence, there is little evidence of its use in studying seismic-induced structural vibrations. From the wavelet transform comes two useful quantities, the real part of the transform and the modulus lT(b, a)/. To identify the actual response of the structure, we can use the transform of the relative roof displacement measured during the first 40 s of vibration. Figure 11 illustrates, in the form of a
2.0
0.10 11.
1.5 s 1.0 .x5 = 0.5 P3 - 0.0
05. .I? 13 -1.0 -1.5 _“.I”
I
0
io io Time (seconds)
50
-2.o~,“.:.,,.‘:,.‘.I”“:“‘.I,...1 0 10 20 30 40 Time (seconds)
Fig. 9. Base acceleration and relative roof displacement (north-south).
50
60
M. J. Chajes et
492
0
-I-
1
2 0.8 -n 7 b 8a 0.6 -m ki 3 rz 0.4 -x 2 iI E
0.2 --
n -F 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Period (seconds)
Fig. IO. Power spectral density of recorded roof response. plot, the resulting modulus and real part of the transform (frequency content is plotted on the y-axis and time is plotted on the x-axis). The contours represent the magnitude of the frequency content and can be viewed as a “fingerprint” of the structural response. Wavelet transforms of the actual response can be used to identify structural frequencies of vibration (which can change over time due to material and geometric nonlinearities) and modal damping ratios. Furthermore, comparison of the computed and actual “fingerprints” provides a rigorous test of a model’s accuracy and can be used to identify deficiencies. Figure 11 indicates that the fundamental mode has a frequency of approximately 1.0 Hz (the contours contour
al.
build up to a ridge at 1.0 Hz). During the strong motion, the response is dominated by the fundamental mode. After 25 s, the fundamental mode response appears to be all that is left. The relative significance of the various modes during the motion can be seen by examining the frequency response at a distinct time. Cutting a vertical cross-section through the contour plot at 6.2 s (see Fig. 12), the dominance of the first mode can be clearly seen. From this plot, the fundamental period appears to be just under 1 s, while a second mode appears at roughly 0.33 s. The longer period content seen in the plot was found to be a characteristic of the ground motion, and not of the structure. Utilizing the real part of the wavelet transform (Fig. 11) the response of the building at a distinct frequency can be extracted. In Fig. 13, a horizontal cross-section is cut through the real part of the transform at the frequency corresponding to the fundamental mode (1 .OHz). This figure helps to isolate the response associated with the building’s dominant frequency of vibration. From the plot one can see the building response related to 1.0 Hz vibration alone results in a peak displacement of approximately 1.75 in (as compared to 1.97 in for the building itself). This again indicates the dominance of this fundamental mode in the response of the building. Extracted single frequency response plots, like the one in Fig. 13, are also very useful for determining the damping ratios of individual modes of vibration. The percentage of critical damping [ can be found by
Time (set)
I
I
I
I
I
I
15
20
25
30
35
40
Time (set) Fig. 1I. Measured response. (a) Modulus of wavelet transform. (b) Real part of wavelet transform.
Dynamic
0.50
analysis
of a ten-story
1.oo
Fig. 12. Frequency
content
(5)
where U, is the magnitude of the peak response at one time and u, +p is the magnitude of the peak response after p cycles. The percentage of critical damping for small values of 6 is given by r = 6/2K.
(6)
The strong motion ends roughly 20 s after the earthquake begins, as can be seen from the measured base acceleration (Fig. 9). From approximately 19 to 26 s, response at the fundamental frequency (Fig. 13) exhibits what appears to be damping-induced decay. For this segment of motion, the percentage of critical damping [, found from eqns 5 and 6, is approximately 3.7%. This value of damping is utilized in the dynamic continuum analyses described in the following sections.
COMPUTED STRUCTURAL RESPONSE
Having analyzed the recorded response of the building, we are now ready to develop our math-
-2.0
to”’
1 I I Ia ;‘I1 5 10
concrete
I I ’ I I ’ ” I ” 1.50 2.00 2.50 Period (seconds)
utilizing the concept of logarithmic decrement. The logarithmic decrement 6 is defined as 6 = l/p ln(u,lu,+,),
reinforced
of measured
response
493
building
I i I ” 3.00
I I I I I I I 3.50 4.00
at 6.2 s.
ematical models and conduct a dynamic analysis. Details are discussed in the following sections. Continuum models of IO-story building
In forming continuum models of lattice structures, special elements representing fixed-base and free-end cells are needed [24]. As a result, at least three elements must be used to model the lo-story building (one fixed-base element, one interior element and one free-end element). Furthermore, because of the change in column sizes at the eighth floor, at least two interior elements must be used in this case. Continuum models consisting of 10, 6 and 4 elements were investigated. Figure 14 presents a physical depiction of the three continuum models. Table 1 provides information regarding the number of elements and degrees of freedom associated with each of the models. These values are particularly important because they are directly related to the computational effort required in the anlaysis. Notice that a discrete model of this building consists of 150 elements and 254 degrees of freedom, while the 4-element continuum model has only 24 degrees of freedom. In order to conduct a dynamic analysis using the continuum models, system stiffness [K], mass [M] and damping [C] matrices are needed. Determination of the system stiffness matrix first involves the formation of individual continuum-element stiffness matrices
I [“I’\ “‘I 1 I N”;““l 15 20 25 30 Time (seconds)
‘N’lj 35
Fig. 13. Response at the fundamental frequency extracted from measured data.
40
494
M. J. Chajes et al. Roof 10th
’ 4 I 4,
5th
c
I
I
I
I
I
lo-Story Building
I
12nd
+ 10, 6, and 4 Element Continuum Models
Fig. 14. Continuum models of IO-story building. [S], defined in Appendix I. To form these element stiffness matrices, member properties for the individual elements (beams and columns) are needed. For both beams and columns, gross moments of inertia were used. In determining member properties for the beams, the floor slab was assumed to act compositely with the beam, and a T-beam incorporating the slab was used. The effective width of the T-beam was computed according to AC1 8.10-2 [30]. Column heights were taken to be the clear distance from the floor to the column capital, while beam lengths were measured from the column centerlines. The modulus of elasticity for the concrete was taken to be 5098 ksi. This corresponds to an estimated concrete compressive strength of 8000 psi. The final formulation of the system stiffness matrix [K] involves the appropriate summation of the element stiffness matrices. In forming the mass matrix a lumped mass model was used. The masses were computed based on a dead load contribution from the framing elements and floor slabs and a nominal live load contribution of 50psf. In the case of the lo-element continuum model, total story loads were computed and lumped into a single story mass. For the 6- and 4element continuum models, the loads at the nodes that were eliminated were proportional to the nearest remaining node according to distance. With regard to the damping matrix, a Rayleigh damping model was used in which 3.7% of critical damping was assigned to the first two modes. As described earlier, this was the values of damping observed in the recorded response of the fundamental mode (Fig. 13). Finally, the effects of rotary inertia and vertical vibration were neglected. This allowed the number of
degrees of freedom associated with the three continuum models to be condensed to 10,6 and 4, respectively. Dynamic analysis
The most general representation of the equation of motion for a multiple-degree-of-freedom system subjected to a forcing function is given by F,(t) + F,(t) + F,(t) = F(t),
where F,(t) represents the inertial force vector, F,(r) is the damping force vector, F,(t) is the spring force vector and F(t) is the applied load vector. Methods for solving equations of motion for multiple-degree-of-freedom, dynamic systems fall into two categories: modal analysis and direct integration. Modal analysis is best suited for linear systems subjected to loadings of significant duration. Direct integration is the most practical way of solving nonlinear dynamic problems. Because of the relatively low base accerations, and the fact that peak roof displacement (x2.0 in) is considerably less than the allowable design displacement (A <0.004/z = 5.95 in) given in the UBC-88 code [25], the behavior of the lo-story building was believed to be preliminary linear elastic. As a result, the more efficient modal analysis procedure was used. For a problem having a base excitation, the inertial, damping and spring force terms of eqn (7) can be rewritten in terms of absolute accelerations, relative velocities and relative displacements, respectively, to give
Wl(4 + [Cl{tij+ [4(u) = -WI&j,
Table 1. Mathematical models Model
Number of elements
Total number of degrees of freedom
Continuum model-10 Continuum model-6 Continuum model-4 Discrete model
10 6 4 150
54 34 24 264
(7)
(8)
where {ii} is the system’s absolute acceleration vector, (ti) is the system’s absolute velocity vector, {u} is the system’s relative displacement vector and {z$} is the ground acceleration vector. Making use of modal orthogonality, the matrix expression given by eqn (8)
Dynamic analysis of a ten-story reinforced concrete building
495
Table 2. Periods of vibration
Mode 1 2 3
Continuum, 6 elements (s) 1.02 0.34 0.19
Continuum, 10elements (s) 1.01 0.33 0.19
Continuum, 4 elements (s) 1.06 0.25 0.17
can be uncoupled into a series of modal equations of motion given by &+2&o,g,+wfq,=
-I-&i,,
(9)
where for mode i, qi is the generalized coordinate, 3, is the modal damping, oi is the circular frequency of the mode, and Ti is the modal participation factor given by
(10)
Discrete, 150elements (s) 1.02 0.33 0.19
Simple shear building (s) 0.65 0.22 0.14
CSMIP recorded data (s) 1.0 0.33 -
discrete model, are very close to the observed value.
With regard to the second and third periods, the loand 6-element continuum models and the discrete model give nearly identical results, with the second period matching the recorded data. While the 4element continuum model has a similar first period to these other models, its second period differs by 25%, and its third period differs by 10%. Finally, the simple shear representation leads to a model which is much stiffer than the actual system, and the periods of vibration are not accurately predicted. Time-history response
where {$J}~is the ith mode shape. The system displacement, velocity and acceleration vectors are then given by
i=l
(114
Using the accelerations recorded by sensor 13 (in the basement of the structure) as the input excitation, dynamic analyses of the building modeled by each of the three continuum models were conducted. The computed roof response given by the various models, as well as the recorded response, are compared in Fig. 15. Evaluation of computed response
{ii} = i {c$}$ji. i=,
(llc)
In order to solve the modal equations of motion [eqn (9)], the mode shapes {4}i and frequencies oi are first found by solving the eigenvalue problem given by
[WI- ~*wll~~~ = v4.
(12)
Once the individual mode shapes and frequencies have been found, a variety of well-known numerical integration methods can be used to evaluate eqn (9). Computed building periods The first three periods of vibration, found by solving the eigenvalue problem [eqn (1211associated with the three continuum models, are given in Table 2. In order to make comparisons to more conventional methods, a discrete model having 150 elements and a simple shear building representation (infinitely stiff beams) were also considered. The first three periods for the two additional models are also given in Table 2. Finally, based on the analysis of the recorded data, it was observed that the buildings first two periods were approximately 1.0 and 0.33 s, respectively. From the results, one sees that the fundamental periods for the three continuum models, as well as the
Figure 15(a) shows that the lo- and 6-element continuum models yield virtually identical results. On the other hand, Fig. 15(b) shows that the 4-element continuum model predicts a somewhat different timehistory response. Finally, Fig. 15(c) shows that the 6-element continuum is quite successful in predicting the actual response of the building. The frequency of the computed response is well predicted (as expected based on the accuracy of the fundamental period). In addition, the maximum displacement from the analysis compared very well to the actual response (see Table 3). In general, the predicted behavior during the large motion (15-20 s) agrees quite well with the actual motion. The only significant discrepancies appears to occur between 8 and 14 s when the predicted response has a higher amplitude than the recorded motion. While the process of comparing computed and measured time histories is straightforward, as described earlier, additional comparisons can be made by utilizing wavelet transforms. The following two comparisons involve information found from transforms of the recorded roof response and the response predicted by the B-element continuum model. We first compared the frequency content of the roof response (Fig. 16) to the content of the measured response (Fig. 11). A qualitative comparison of these two plots indicates that the continuum model predicts the frequency content of the dynamic response very well.
M. J. Chajes ef al.
496
Time (seconds)
Continuum Model (4 Elements)
Time (seconds)
Time (seconds) Fig. IS. Comparison
of measured and computed roof response. (a) IO- and belement continuum models (b) IO- and 4-clement continuum models. (c) Recorded data and 6-element continuum model.
We then compared the measured roof response at the fundamental frequency to that given by the continuum model (see Fig. 17). Overall, the response is predicted quite well. In the strong motion portion of the response (I 5-20 s), the two curves are very similar. The major difference occurs in the response prior
Table 3. Maximum
to 15 s where the computed response has a slightly higher amplitude. Differences here may be due to higher damping at the outset of the response. It is quite possible that damping during the initial forced vibration differs from that measured during the later free vibration.
relative roof displacement
CSMIP
Continuum model,
recorded response (in)
10 elements
Continuum
(in)
(in)
(in)
1.97
2.04
2.03
I .99
model, 6 elements
Continuum model, 4 elements
497
Dynamic analysis of a ten-story reinforced concrete building
I
I
15
20 Time (set)
I
,
I
25
30
35
I 40
Fig. 16. Frequency content of computed roof response.
-2.0
I I I II 0
I 8” 5
1 I’ 10
I a / I ’ I I; ” I I / ’ I I ” I I ’ ” I I ” 15 20 25 $0 45 40 Time (seconds)
Fig. 17. Comparison of response at the fundamental frequency extracted from measured and computed response. CONCLUSIONS
An approximate dynamic analysis of a IO-story, reinforced-concrete building was conducted using a continuum model. While comparisons between continuum and discrete analyses have been made in the past, this was the first attempt to compare actual seismic response of a large concrete building to continuum analysis results. Time-dependent wavelet transforms were used to identify vibrational characteristics of both recorded and computed response. The transforms proved valuable in identifying periods of vibration, developing single frequency response plots and computing modal damping values. Research results have shown that continuum models can accurately predict both the vibrational characteristics and seismic response of a large reinforced-concrete building. Using a 6-element continuum model, both the maximum roof displacement and the important periods of vibration of the lo-story building were predicted to within 5% of values recorded during the 1989 Loma Prieta earthquake. The results indicate that continuum models can provide designers with a valuable approximate analysis tool for use in preliminary as well as final design of complex structures.
RI-A-91-94. Thanks are also extended to R. Darragh, A. Shakal and M. Huang of the California Strong Motion Instrumentation Program for providing the response data used in this study. Finally, the authors would like to thank Mr Ted F. Januszka for his work on the computer models.
REFERENCES
1. California
2.
3. 4.
5.
6. 7. 8.
Acknowledgmenfs-The first author is grateful for support received from the Engineering Foundation under grant no.
Division of Mines and Geology, CSMIP strong-motion records from the Santa Cruz mountains (Loma Prieta), California earthquake of October 17, 1989. Report OSMS 89-06, Sacramento, CA (1989). U.S. Geological Survey, Strong motion records from northern California (Loma Prieta) earthquake of October 17, 1989. U.S. Geological Survey Open File 89-568, Menlo Park, CA (1989). A. K. Noor, Thermal stress analysis of double-layered grids. AXE J. Sirucr. Erzgng 104, 251-262 (1978). A. K. Noor, M. S. Anderson and W. H. Greene, Continuum models for beam and platelike lattice structures. AIAA J. 16, 1219-1228 (1978). A. K. Noor and M. S. Anderson, Analysis of beam-like lattice trusses. Comput. Meth. appl. Mech. Engng 20, 53-70 (1979). A. K. Noor and M. P. Nemeth, Micropolar beam models for lattice grids with rigid joints. Cornput. Mefh. appl. Mech. Engng 21, 249-263 (1980). A. K. Noor and L. S. Weisstein, Stability of beam-like lattice trusses. Comput. Meth. appl. Mech. 25, 179-193 (1981). A. M. Abdel-Ghaffar, Free lateral vibrations of suspension bridges. AXE J. Srrucr. Engng 104, 503-525 (1978).
498
M. J. Chajes et al.
9. A. M. Abdel-Ghaffar, Free torsional vibrations of suspension bridges. ASCE J. Struct. Engng 105, 767-788 (1979). 10. A. M. Abdel:GhatTar, Suspension bridge vibration: continuum formulation. ASCE J. Engng Mech. Div. 108, 1215-1231 (1982). 11. C. T. Sun, B. J. Kim and J. L. Bogdanoff, On the derivation of equivalent simple models for beam and plate-like structures in dynamic analysis. In: Proc. AIAA Dynamics Specialists Conf, Atlanta, GA, pp. 523-532. AIAA (1981). 12. S. Abrate and C. T. Sun, Dynamic analysis of geometrically nonlinear truss structures. Comput. Struct. 17, 491497 (1983). 13 T. Y. Yang, Matrix displacement solution to elastica problems of beams and frames. Int. J. Solia!s Struct. 9, 829-842 (1973). 14 C. T. Sun and J. N. Juang, Modeling global structural damping in trusses using simple continuum models. AIAA J. 24, 144150 (1986). 15 B. Necib and C. T. Sun, Analysis of beams using a higher order timoshenko beam finite element. J. Sound. Vibr. 130, 149-159 (1989). 16. D. B. McCallen, A continuum model for the nonlinear analysis of beam-like lattice structures. Ph.D. Dissertation, University of California Davis, Davis, CA (1986). 17. D. B. McCallen and K. M. Romstad, Application of a continuum model in building analysis. In: Proc. ASCE Structures Congress, Orlando, FL, pp. 344-358. ASCE (1987). 18. D. B. McCallen and K. M. Romstad, A continuum model for the nonlinear analysis of beam-like lattice structures. Comput. Struct. 29, 177-197 (1988). 19. D. B. McCallen and K. M. Romstad, A continuum model for lattice structures with geometric and material nonlinearities. Comput. Struct. 37, 795-822 (1990). 20. M. J. Chajes, Nonlinear dynamic analysis of building frameworks using a continuum model. Ph.D. dissertation, University of California Davis, Davis, CA (1990). 21. M. J. Chajes, K. M. Romstad and D. B. McCallen, Inelastic frame analysis using a continuum model. In: Proc. SSRC Technical Session, St. Louis, MO, pp. 109-120. SSRC (1990). 22. M. J. Chajes, K. M. Romstad and D. B. McCallen, Nonlinear frame analysis using a continuum model. In: Proc. ASCE Structures Congress, Baltimore, MD, pp. 451452. ASCE (1990). 23. T. A. Osterkamp, Force calculations for a continuum model of lattice systems. MS. thesis, University of California Davis, Davis, CA (1988). 24. M. J. Chajes, K. M. Romstad and D. B. McCallen, Analysis of multi-bay frames using a continuum model, ASCE J. Struct. Engng 119, 522-546 (1993). 25. International Conference of Building Officials, Uniform Building Code, Whittier, CA (1988). 26. K. M. Romstad, B. Maroney and M. J. Chajes, Interpretation of Rio Dell freeway response during six recorded earthquake events. In: Proc. of the StrongMotion Instrumentation Program’s Seminar on Seismological and Engineering Implications of Recent Strong-Motion Data, Sacramento, CA, pp. 8: 1-8: 13. SMIP (1989). 37 B. Maroney, K. M. Romstad and M. J. Chajes, Interpretation of rio dell freeway response during six recorded earthquake events. In: Proc. of the Fourth U.S. National Conference on Earthquake Engineering, Palm Springs, CA, pp. 1007-1016 (1990). 28. J. T. Kirby, M. J. Chajes and J. A. Melby, Wavelet transform analysis of several transient or nonstationary phenomena in engineering mechanics. In: Proc. ASCE _I.
Engineering Mechanics Conference, College Station, TX, pp. 204-207. ASCE (1992). 29. A. Grossmann and J. Morlet, Decomposition of hardy functions into square integrable wavelets of constant shape. SIAM J. Math. Anal. IS, 723-736 (1984). 30. AC1 Committee 318, Building Code Requirements for Reinforced Concrete (AC1 3 18-89). Revised 1992, American Concrete Institute, Detroit (1992). APPENDIX The nine-by-nine continuum element stiffness matrix is given by
(Al)
where the stiffness coefficients are given by
s,,=s,= -s,,
(A24
Wb)
(A24 (A24 We)
fic
C0hIII.S 7EA,y;
s,=s,=
,=I
s,, = s,, =
ZGA,L
~ 3L
“‘r
>+
“SLY:>
~ 15
I G;;L
Wf) (A%) (A2h)
*d-S
s,, =
(
16EA,yf
1 ,=I
s,,=s,=
~
3L
>
-s,,=
+
IGA,L ~ 15
-s,,=z
s,,=
_s,,2$
s,=s,,=
--
(A2i)
(A2j)
(A2k)
IGA, L 60
GA,L 15
s,, = s, = -
(A21) (A2m)
s,, = s,, = s,, = s,, = s,, = s,* = s,, = s,,
Wn) where L is the length of the element; E and G are the modulus of elasticity and shear modulus of the lattice material; A, is the shear area of the cell spanned by the element; and i,, Ai and yi are the moment of inertia, area and distance to the neutral axis of bending of each column i in the cell.
w
& Srrucrures
. _
> Pergamon
0045-794!MSMKI166-2
Vol. 58. No. 3, pp. 487-498. 1996 Elswier Science Ltd Printed in Great Britain 0045.7949/96 $9.50 + 0.00
DYNAMIC ANALYSIS OF A TEN-STORY REINFORCED CONCRETE BUILDING USING A CONTINUUM MODEL M. J. Chajes, W. W. Finch Jr and J. T. Kirby Department of Civil Engineering, University of Delaware, 137 DuPont Hall, Newark, DE 19716, U.S.A. (Received 18 November 1994) Abstract-During the 1989 Loma Prieta earthquake, accelerometers maintained by the California Division of Mines and Geology Strong Motion Instrumentation Program (CSMIP) recorded the response of a IO-story, reinforced-concrete building located in San Jose, California. In this paper, a computationally efficient, approximate, dynamic analysis of the building is conducted utilizing a reduced-order continuum model. Continuum methodology makes use of the repetitive nature of lattice framing systems to generate finite element models having significantly fewer degrees of freedom than models generated using classical
discrete finite element techniques. The vibrational characteristics of the continuum model, as well as the results of dynamic analyses, are compared to information gained from the recorded response of the building. The continuum model proves to be quite accurate, both in capturing the dominant periods of vibration of the structure and in predicting the time-history response. As a result, the method shows promise as a tool for use in the analysis and design of large lattice structures subjected to earthquake loads.
INTRODUCTION
On October 17, 1989, a M, = 7.1 (surface wave magnitude) earthquake occurred along the San Andreas Fault in Northern California with its epicenter located under a mountain called Loma Prieta. The earthquake triggered a large number of strongmotion accelerometers deployed throughout the San Francisco Bay area by both the California Division of Mines and Geology [l] and the U.S. Geological Survey [2]. These instruments recorded a great deal of valuable seismic response data for a variety of structures including buildings, bridges, dams and tunnels. These data provide the opportunity to evaluate the accuracy of either existing or newly developed computer analysis techniques. Classical discrete finite element methods for predicting linear response of large structures to even simple static loading may involve the solution of thousands of linear simultaneous equations. Adding time-varying dynamic loadings, along with material and geometric nonlinearities, can lead to thousands of coupled simultaneous nonlinear equations which must be solved at very small time increments. Even with modern high-speed computers, the computational effort, storage requirements and associated cost of solving such equations can prohibit the use of full-scale dynamic analysis in everyday design. As a result, approximate yet accurate methods for conducting dynamic analyses of large structural frameworks are needed. Various researchers have investigated the use of continuum models for conducting both linear and nonlinear analyses of lattice structures [3-241. Finite element models resulting from continuum method487
ology have significantly fewer degrees of freedom than discrete finite element models, which individually model each of the beam and column elements. Hence, the use of continuum models can result in a considerable reduction incomputational effort with a corresponding savings in cost. This paper focuses on the seismic response of a lo-story, reinforced-concrete building located in San Jose, California, to the Loma Prieta earthquake. In particular, the paper investigates the use of a reducedorder continuum model to predict the dynamic response of the structure. The accuracy of the analysis is evaluated by comparing the computed results to the actual building response recorded during the earthquake. To the author’s knowledge, this is the first attempt to compare the dynamic continuum analysis of a large building to its recorded structural response during an earthquake.
CONTINUUM METHODOLOGY
During the past 20 years, continuum models of large truss and frame structures have been used to study vibrational characteristics, static and dynamic response, and buckling behavior. Among the most prominent works are those by Noor et al. [3-71, who studied the static and dynamic response of beamlike lattice structures and predicted their buckling characteristics; Abdel-Ghaffar [%-lo], who used continuum models to find frequencies and mode shapes of suspension bridges; Sun et al. [l l] and Abrate and Sun [12], who used a beam element developed by Yang [13] to conduct vibrational analyses of planar truss systems; Sun and Juang [14], who incorporated
M. J. Chajes et al.
Typical cell ~_,i_____i‘i_____ii_____i_i_____i,_~ _ _______________________-__---- -*
//~~N/N//~~///////// Lattice framework
I
Fig. 1. Typical repeated cell.
energy dissipation in the form of viscous damping to study the simple transient vibration of trusses; and most recently, Necib and Sun [ 151, who used a higher-order Timoshenko beam element to solve dynamic problems involving beamlike truss structures. The work of Necib and Sun [15] included force calculations for discrete elements of the lattice. The continuum model utilized in this work was developed by McCallen and Romstad [ 16-181. This model replaces a lattice structure with a continuous structure that yields essentially the same global static and dynamic behavior. The equivalence of the continuum is established by setting the strain energy of the continuum equal to the approximate strain energy of the discrete lattice. The original application of the McCallen-Romstad model involved finding frequencies and mode shapes of trusses and frames as well as determining static displacements. Nonlinear geometric effects were incorporated into the original model. In preliminary studies, the method gave results that compared favorably with results obtained using discrete finite element analyses. McCallen and Romstad [19], Chajes [20], and Chajes et al. [21,22] extended the continuum methodology to include nonlinear material effects on trusses and frames subjected to static and dynamic loadings. Osterkamp [23] developed a method of determining discrete member forces for single-bay frames, and Chajes [20] and Chajes et al. [24] extended the work to include multiple-bay frames. Unlike earlier papers in which continuum analysis results were compared to discrete analysis
results, this paper involves the more challenging task of comparing computed results with actual measured response. Description of the continuum model Because the details of the continuum model used in this work have been published extensively, they will only be presented briefly herein. For a more in-depth description, the reader is referred to McCallen [16], McCallen and Romstad [18] and Chajes et al. [24]. Large lattice structures are made of discrete structural members that are framed together. Often these members are connected in ways that form repeated geometric patterns or “ceils.” In a building, each cell may contain many beams and columns as shown in Fig. 1. The entire framework is often a combination of just a few different types of cells. The objective of the continuum model is to replace these repetitive cells with an “equivalent” continuum, thereby greatly reducing the number of elements needed to model the lattice. This results in a significant reduction in the number of global degrees of freedom needed to describe the structure (Fig. 2). In developing an appropriate model, assumed axial, bending and shear modes of deformation are used to represent the possible deformation patterns of the lattice. The three assumed deformation modes are shown in Fig. 3. An approximate strain energy expression for the lattice, oL, is developed in terms of the axial force (Fci) and bending moment (Mci) in the columns and
Discrete model Fig. 2. Discrete and associate continuum models including global degrees of freedom.
489
Dynamic analysis of a ten-story reinforced concrete building
Axial
Bending
Shear
Fig. 3. Modes of deformation. a generalized section shear force (I’) resulting from the assumed deformations. The approximate lattice strain energy expression is given by
strains, (2) v, and u2 are transverse displacements, (3) dv,/dx and do&lx are slopes along the centerline of the continuum element and are used to get values of curvature, and (4) r3,, e2 and & are associated with rotations of the cross-section and are used to get axial strains due to these rotations. Because deformations of the lattice are limited to combinations of the assumed modes, the lattice energy expression is only approximate and its accuracy depends on how well the assumed modes represent the actual deformations.
DETAILS OF THE IO-STORY BUILDING
where P(x, y,) is the off-axis strain, K(X) is the curvature of the reference axis, 4(x) is the slope of the reference axis due to shear deformation, and NC is the number of columns in the lattice cross-section. The equivalence of the continuum is established by incorporating standard kinematic and constitutive relationships and setting the strain energy of the continuum equal to the approximate strain energy of the lattice. The continuum strain energy, Uc, is
+(GA,)~6+~(GA,)B’
The structure being studied is a IO-story, reinforced-concrete, commercial building located in downtown San Jose, California. The building was designed in 1964 and construction was completed in 1967. The building has a rectangular plan measuring 82 ft by 192 ft and rests on a 5-foot-thick mat foundation. The superstructure rises 124 ft above ground level and consists of 10 above-grade and 1 belowgrade stories. The vertical load carrying system consists of 4.5 in thick concrete floor slabs supported by concrete pan joists and a reinforced-concrete frame. For its lateral force resisting system, the building has shear walls in the east-west direction and a momentresistant, reinforced-concrete frame in the northsouth direction. Figure 5 shows a typical elevation view of the moment-resistant frame. Included in Fig. 5 are cross-sections of the typical elements (columns ClC4 and beams Bl). Figure 6 shows a typical floor framing plan.
dx, 1
where E and G are the material’s Young’s modulus and shear modulus, A, and I, are the area and moment of inertia of the columns, A, is the effective shear area of the “repeated cell,” u and u represent longitudinal and transverse displacements, and 0 represents the cross-section rotation. Applying standard energy minimization principles to eqn (2), a beam-like finite element for the continuum is formed. Additional degrees of freedom in the continuum element, not found in the typical Timoshenko beam elements, are needed as a result of continuum strain energy terms not found in the standard Timoshenko beam theory. The derived firstorder continuum finite element with nine degrees of freedom is shown in Fig. 4 and the associated element stiffness matrix is given in the Appendix. A more detailed derivation of the continuum element stiffness matrix is given in Chajes et al. [24]. The continuum element provides C, continuity of both shear and bending displacement. With regard to the geometrical representation of the displacement degrees of freedom, note that (1) u, and u2 are longitudinal displacements and are used to get axial
MEASURED STRUCTURAL
RESPONSE
The IO-story building is located 24 miles from the epicenter of the 1989 Loma Prieta earthquake. During the earthquake, 11 of 13 accelerometers located throughout the building recorded 60 s of dynamic response [ 11.The motion sensors were installed and maintained by the California Strong Motion Instrumentation Program (CSMIP) of the California Division of Mines and Geology of the Department of Conservation. (This particular structure is designated as CSMIP station 57355.) The locations of the motion sensors are shown in Fig. 7. No data was recorded by sensors 2 and 10 during the earthquake.
ur+&&l!$+ dv, dx
dv, dx
Fig. 4. Continuum finite element degrees of freedom.
490
M. J. Chajes ef al. 24 in
I
I
36 in
Cl
24 in
c2
24 in I c3
27.5 in 26 in 27.5 in
1
24 in
I
I
I
I
24 in
c4
Bl
ment
I
I ‘27.5 ft’
5 0 27 ft
‘27.5 ,I
190 ft Fig. 5. Typical moment-resistant Preliminary analysis of response Based on the recorded information, the building experienced peak ground accelerations of 9 and 11% of gravity in the north-south and east-west direc-
tions, respectively, and peak superstructure accelerations of 26 and 38% of gravity in the north-south and east-west directions, respectively. By comparing displacements of parallel sensors on the roof (sensors 3 and 4) the building is seen to exhibit virtually no torsional response (see Fig. 8). As a result, the north-south and east-west frames can be
27.5 ft I
5 @ 27 ft
frame (north-south).
studied individually as two-dimensional systems. Because the continuum methodology has been developed for lattice framing systems, this paper will focus on the response of one of the interior north-south moment-resistant frames. The base acceleration in the north-south direction (taken to be the acceleration recorded by sensor 13 in the basement) and the north-south displacement of the roof relative to the base (sensor 5 minus 13) are shown in Fig. 9. The strong motion of the earthquake lasted for approximately 10 s during which a peak acceleration of 8.6% of gravity occurred. A peak
27.5 ft
190 ft Fig. 6. Typical floor framing plan.
I
491
Dynamic analysis of a ten-story reinforced concrete building 8.0 -‘j6.0
-10
4.0
i--
2 ;
2nd Floor
tT+,
4t
Roof Fig. 7. Location of motion sensors.
2.0
B H 0.0 S d
-2.0 -4.0 -6.0
f
-8.0 + 0 relative displacement of 1.97 in occurred at 19.5 s, after which the building response damped out. A power spectral density of the 60 s of recorded roof response (see Fig. 10) indicates that the fundamental mode has a period of vibration around 1 s and dominates the response. This period is consistent with a period of 1.1 s found using the formula T = C,hff4 given in the 1988 UBC code[25], where T is the building’s fundamental period, h, is the height of the building in feet, and C, = 0.03 for reinforced-concrete moment frames. Analysis using wavelet transforms While Fourier amplitude spectra or power spectral densities computed using recorded time-history data have been used to identify dominant periods of vibration for other structures instrumented by CSMIP [26,27], these methods of analysis give only a time-averaged frequency content. Kirby et al. [28] have shown how wavelet transforms can be used to determine continuous frequency content of seismic response with respect to time. The transform technique involves computing the convolution of the vibrational response f(t) with a scaled analyzing wavelet g*((t - b)/a), where b is the lag of the wavelet relative to the time origin, a is a scale factor,
10
20 30 40 Time (seconds)
50
60
Fig. 8. Absolute horizontal displacement of parallel roof sensors (east-west).
and * denotes the complex conjugate. The wavelet transform T(b, a) is then given by T(b.a)=~jag*(~)i(t)dr.
(3)
where the analyzing wavelet g(t) to be used here is the Morlet wavelet [29] given by s(t) = eibb,ot e-1212.
(4)
While this method, originally developed by Morlet to characterize seismic signals, has been used in applications ranging from speech analysis to the study of fluid turbulence, there is little evidence of its use in studying seismic-induced structural vibrations. From the wavelet transform comes two useful quantities, the real part of the transform and the modulus lT(b, a)/. To identify the actual response of the structure, we can use the transform of the relative roof displacement measured during the first 40 s of vibration. Figure 11 illustrates, in the form of a
2.0
0.10 11.
1.5 s 1.0 .x5 = 0.5 P3 - 0.0
05. .I? 13 -1.0 -1.5 _“.I”
I
0
io io Time (seconds)
50
-2.o~,“.:.,,.‘:,.‘.I”“:“‘.I,...1 0 10 20 30 40 Time (seconds)
Fig. 9. Base acceleration and relative roof displacement (north-south).
50
60
M. J. Chajes et
492
0
-I-
1
2 0.8 -n 7 b 8a 0.6 -m ki 3 rz 0.4 -x 2 iI E
0.2 --
n -F 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Period (seconds)
Fig. IO. Power spectral density of recorded roof response. plot, the resulting modulus and real part of the transform (frequency content is plotted on the y-axis and time is plotted on the x-axis). The contours represent the magnitude of the frequency content and can be viewed as a “fingerprint” of the structural response. Wavelet transforms of the actual response can be used to identify structural frequencies of vibration (which can change over time due to material and geometric nonlinearities) and modal damping ratios. Furthermore, comparison of the computed and actual “fingerprints” provides a rigorous test of a model’s accuracy and can be used to identify deficiencies. Figure 11 indicates that the fundamental mode has a frequency of approximately 1.0 Hz (the contours contour
al.
build up to a ridge at 1.0 Hz). During the strong motion, the response is dominated by the fundamental mode. After 25 s, the fundamental mode response appears to be all that is left. The relative significance of the various modes during the motion can be seen by examining the frequency response at a distinct time. Cutting a vertical cross-section through the contour plot at 6.2 s (see Fig. 12), the dominance of the first mode can be clearly seen. From this plot, the fundamental period appears to be just under 1 s, while a second mode appears at roughly 0.33 s. The longer period content seen in the plot was found to be a characteristic of the ground motion, and not of the structure. Utilizing the real part of the wavelet transform (Fig. 11) the response of the building at a distinct frequency can be extracted. In Fig. 13, a horizontal cross-section is cut through the real part of the transform at the frequency corresponding to the fundamental mode (1 .OHz). This figure helps to isolate the response associated with the building’s dominant frequency of vibration. From the plot one can see the building response related to 1.0 Hz vibration alone results in a peak displacement of approximately 1.75 in (as compared to 1.97 in for the building itself). This again indicates the dominance of this fundamental mode in the response of the building. Extracted single frequency response plots, like the one in Fig. 13, are also very useful for determining the damping ratios of individual modes of vibration. The percentage of critical damping [ can be found by
Time (set)
I
I
I
I
I
I
15
20
25
30
35
40
Time (set) Fig. 1I. Measured response. (a) Modulus of wavelet transform. (b) Real part of wavelet transform.
Dynamic
0.50
analysis
of a ten-story
1.oo
Fig. 12. Frequency
content
(5)
where U, is the magnitude of the peak response at one time and u, +p is the magnitude of the peak response after p cycles. The percentage of critical damping for small values of 6 is given by r = 6/2K.
(6)
The strong motion ends roughly 20 s after the earthquake begins, as can be seen from the measured base acceleration (Fig. 9). From approximately 19 to 26 s, response at the fundamental frequency (Fig. 13) exhibits what appears to be damping-induced decay. For this segment of motion, the percentage of critical damping [, found from eqns 5 and 6, is approximately 3.7%. This value of damping is utilized in the dynamic continuum analyses described in the following sections.
COMPUTED STRUCTURAL RESPONSE
Having analyzed the recorded response of the building, we are now ready to develop our math-
-2.0
to”’
1 I I Ia ;‘I1 5 10
concrete
I I ’ I I ’ ” I ” 1.50 2.00 2.50 Period (seconds)
utilizing the concept of logarithmic decrement. The logarithmic decrement 6 is defined as 6 = l/p ln(u,lu,+,),
reinforced
of measured
response
493
building
I i I ” 3.00
I I I I I I I 3.50 4.00
at 6.2 s.
ematical models and conduct a dynamic analysis. Details are discussed in the following sections. Continuum models of IO-story building
In forming continuum models of lattice structures, special elements representing fixed-base and free-end cells are needed [24]. As a result, at least three elements must be used to model the lo-story building (one fixed-base element, one interior element and one free-end element). Furthermore, because of the change in column sizes at the eighth floor, at least two interior elements must be used in this case. Continuum models consisting of 10, 6 and 4 elements were investigated. Figure 14 presents a physical depiction of the three continuum models. Table 1 provides information regarding the number of elements and degrees of freedom associated with each of the models. These values are particularly important because they are directly related to the computational effort required in the anlaysis. Notice that a discrete model of this building consists of 150 elements and 254 degrees of freedom, while the 4-element continuum model has only 24 degrees of freedom. In order to conduct a dynamic analysis using the continuum models, system stiffness [K], mass [M] and damping [C] matrices are needed. Determination of the system stiffness matrix first involves the formation of individual continuum-element stiffness matrices
I [“I’\ “‘I 1 I N”;““l 15 20 25 30 Time (seconds)
‘N’lj 35
Fig. 13. Response at the fundamental frequency extracted from measured data.
40
494
M. J. Chajes et al. Roof 10th
’ 4 I 4,
5th
c
I
I
I
I
I
lo-Story Building
I
12nd
+ 10, 6, and 4 Element Continuum Models
Fig. 14. Continuum models of IO-story building. [S], defined in Appendix I. To form these element stiffness matrices, member properties for the individual elements (beams and columns) are needed. For both beams and columns, gross moments of inertia were used. In determining member properties for the beams, the floor slab was assumed to act compositely with the beam, and a T-beam incorporating the slab was used. The effective width of the T-beam was computed according to AC1 8.10-2 [30]. Column heights were taken to be the clear distance from the floor to the column capital, while beam lengths were measured from the column centerlines. The modulus of elasticity for the concrete was taken to be 5098 ksi. This corresponds to an estimated concrete compressive strength of 8000 psi. The final formulation of the system stiffness matrix [K] involves the appropriate summation of the element stiffness matrices. In forming the mass matrix a lumped mass model was used. The masses were computed based on a dead load contribution from the framing elements and floor slabs and a nominal live load contribution of 50psf. In the case of the lo-element continuum model, total story loads were computed and lumped into a single story mass. For the 6- and 4element continuum models, the loads at the nodes that were eliminated were proportional to the nearest remaining node according to distance. With regard to the damping matrix, a Rayleigh damping model was used in which 3.7% of critical damping was assigned to the first two modes. As described earlier, this was the values of damping observed in the recorded response of the fundamental mode (Fig. 13). Finally, the effects of rotary inertia and vertical vibration were neglected. This allowed the number of
degrees of freedom associated with the three continuum models to be condensed to 10,6 and 4, respectively. Dynamic analysis
The most general representation of the equation of motion for a multiple-degree-of-freedom system subjected to a forcing function is given by F,(t) + F,(t) + F,(t) = F(t),
where F,(t) represents the inertial force vector, F,(r) is the damping force vector, F,(t) is the spring force vector and F(t) is the applied load vector. Methods for solving equations of motion for multiple-degree-of-freedom, dynamic systems fall into two categories: modal analysis and direct integration. Modal analysis is best suited for linear systems subjected to loadings of significant duration. Direct integration is the most practical way of solving nonlinear dynamic problems. Because of the relatively low base accerations, and the fact that peak roof displacement (x2.0 in) is considerably less than the allowable design displacement (A <0.004/z = 5.95 in) given in the UBC-88 code [25], the behavior of the lo-story building was believed to be preliminary linear elastic. As a result, the more efficient modal analysis procedure was used. For a problem having a base excitation, the inertial, damping and spring force terms of eqn (7) can be rewritten in terms of absolute accelerations, relative velocities and relative displacements, respectively, to give
Wl(4 + [Cl{tij+ [4(u) = -WI&j,
Table 1. Mathematical models Model
Number of elements
Total number of degrees of freedom
Continuum model-10 Continuum model-6 Continuum model-4 Discrete model
10 6 4 150
54 34 24 264
(7)
(8)
where {ii} is the system’s absolute acceleration vector, (ti) is the system’s absolute velocity vector, {u} is the system’s relative displacement vector and {z$} is the ground acceleration vector. Making use of modal orthogonality, the matrix expression given by eqn (8)
Dynamic analysis of a ten-story reinforced concrete building
495
Table 2. Periods of vibration
Mode 1 2 3
Continuum, 6 elements (s) 1.02 0.34 0.19
Continuum, 10elements (s) 1.01 0.33 0.19
Continuum, 4 elements (s) 1.06 0.25 0.17
can be uncoupled into a series of modal equations of motion given by &+2&o,g,+wfq,=
-I-&i,,
(9)
where for mode i, qi is the generalized coordinate, 3, is the modal damping, oi is the circular frequency of the mode, and Ti is the modal participation factor given by
(10)
Discrete, 150elements (s) 1.02 0.33 0.19
Simple shear building (s) 0.65 0.22 0.14
CSMIP recorded data (s) 1.0 0.33 -
discrete model, are very close to the observed value.
With regard to the second and third periods, the loand 6-element continuum models and the discrete model give nearly identical results, with the second period matching the recorded data. While the 4element continuum model has a similar first period to these other models, its second period differs by 25%, and its third period differs by 10%. Finally, the simple shear representation leads to a model which is much stiffer than the actual system, and the periods of vibration are not accurately predicted. Time-history response
where {$J}~is the ith mode shape. The system displacement, velocity and acceleration vectors are then given by
i=l
(114
Using the accelerations recorded by sensor 13 (in the basement of the structure) as the input excitation, dynamic analyses of the building modeled by each of the three continuum models were conducted. The computed roof response given by the various models, as well as the recorded response, are compared in Fig. 15. Evaluation of computed response
{ii} = i {c$}$ji. i=,
(llc)
In order to solve the modal equations of motion [eqn (9)], the mode shapes {4}i and frequencies oi are first found by solving the eigenvalue problem given by
[WI- ~*wll~~~ = v4.
(12)
Once the individual mode shapes and frequencies have been found, a variety of well-known numerical integration methods can be used to evaluate eqn (9). Computed building periods The first three periods of vibration, found by solving the eigenvalue problem [eqn (1211associated with the three continuum models, are given in Table 2. In order to make comparisons to more conventional methods, a discrete model having 150 elements and a simple shear building representation (infinitely stiff beams) were also considered. The first three periods for the two additional models are also given in Table 2. Finally, based on the analysis of the recorded data, it was observed that the buildings first two periods were approximately 1.0 and 0.33 s, respectively. From the results, one sees that the fundamental periods for the three continuum models, as well as the
Figure 15(a) shows that the lo- and 6-element continuum models yield virtually identical results. On the other hand, Fig. 15(b) shows that the 4-element continuum model predicts a somewhat different timehistory response. Finally, Fig. 15(c) shows that the 6-element continuum is quite successful in predicting the actual response of the building. The frequency of the computed response is well predicted (as expected based on the accuracy of the fundamental period). In addition, the maximum displacement from the analysis compared very well to the actual response (see Table 3). In general, the predicted behavior during the large motion (15-20 s) agrees quite well with the actual motion. The only significant discrepancies appears to occur between 8 and 14 s when the predicted response has a higher amplitude than the recorded motion. While the process of comparing computed and measured time histories is straightforward, as described earlier, additional comparisons can be made by utilizing wavelet transforms. The following two comparisons involve information found from transforms of the recorded roof response and the response predicted by the B-element continuum model. We first compared the frequency content of the roof response (Fig. 16) to the content of the measured response (Fig. 11). A qualitative comparison of these two plots indicates that the continuum model predicts the frequency content of the dynamic response very well.
M. J. Chajes ef al.
496
Time (seconds)
Continuum Model (4 Elements)
Time (seconds)
Time (seconds) Fig. IS. Comparison
of measured and computed roof response. (a) IO- and belement continuum models (b) IO- and 4-clement continuum models. (c) Recorded data and 6-element continuum model.
We then compared the measured roof response at the fundamental frequency to that given by the continuum model (see Fig. 17). Overall, the response is predicted quite well. In the strong motion portion of the response (I 5-20 s), the two curves are very similar. The major difference occurs in the response prior
Table 3. Maximum
to 15 s where the computed response has a slightly higher amplitude. Differences here may be due to higher damping at the outset of the response. It is quite possible that damping during the initial forced vibration differs from that measured during the later free vibration.
relative roof displacement
CSMIP
Continuum model,
recorded response (in)
10 elements
Continuum
(in)
(in)
(in)
1.97
2.04
2.03
I .99
model, 6 elements
Continuum model, 4 elements
497
Dynamic analysis of a ten-story reinforced concrete building
I
I
15
20 Time (set)
I
,
I
25
30
35
I 40
Fig. 16. Frequency content of computed roof response.
-2.0
I I I II 0
I 8” 5
1 I’ 10
I a / I ’ I I; ” I I / ’ I I ” I I ’ ” I I ” 15 20 25 $0 45 40 Time (seconds)
Fig. 17. Comparison of response at the fundamental frequency extracted from measured and computed response. CONCLUSIONS
An approximate dynamic analysis of a IO-story, reinforced-concrete building was conducted using a continuum model. While comparisons between continuum and discrete analyses have been made in the past, this was the first attempt to compare actual seismic response of a large concrete building to continuum analysis results. Time-dependent wavelet transforms were used to identify vibrational characteristics of both recorded and computed response. The transforms proved valuable in identifying periods of vibration, developing single frequency response plots and computing modal damping values. Research results have shown that continuum models can accurately predict both the vibrational characteristics and seismic response of a large reinforced-concrete building. Using a 6-element continuum model, both the maximum roof displacement and the important periods of vibration of the lo-story building were predicted to within 5% of values recorded during the 1989 Loma Prieta earthquake. The results indicate that continuum models can provide designers with a valuable approximate analysis tool for use in preliminary as well as final design of complex structures.
RI-A-91-94. Thanks are also extended to R. Darragh, A. Shakal and M. Huang of the California Strong Motion Instrumentation Program for providing the response data used in this study. Finally, the authors would like to thank Mr Ted F. Januszka for his work on the computer models.
REFERENCES
1. California
2.
3. 4.
5.
6. 7. 8.
Acknowledgmenfs-The first author is grateful for support received from the Engineering Foundation under grant no.
Division of Mines and Geology, CSMIP strong-motion records from the Santa Cruz mountains (Loma Prieta), California earthquake of October 17, 1989. Report OSMS 89-06, Sacramento, CA (1989). U.S. Geological Survey, Strong motion records from northern California (Loma Prieta) earthquake of October 17, 1989. U.S. Geological Survey Open File 89-568, Menlo Park, CA (1989). A. K. Noor, Thermal stress analysis of double-layered grids. AXE J. Sirucr. Erzgng 104, 251-262 (1978). A. K. Noor, M. S. Anderson and W. H. Greene, Continuum models for beam and platelike lattice structures. AIAA J. 16, 1219-1228 (1978). A. K. Noor and M. S. Anderson, Analysis of beam-like lattice trusses. Comput. Meth. appl. Mech. Engng 20, 53-70 (1979). A. K. Noor and M. P. Nemeth, Micropolar beam models for lattice grids with rigid joints. Cornput. Mefh. appl. Mech. Engng 21, 249-263 (1980). A. K. Noor and L. S. Weisstein, Stability of beam-like lattice trusses. Comput. Meth. appl. Mech. 25, 179-193 (1981). A. M. Abdel-Ghaffar, Free lateral vibrations of suspension bridges. AXE J. Srrucr. Engng 104, 503-525 (1978).
498
M. J. Chajes et al.
9. A. M. Abdel-Ghaffar, Free torsional vibrations of suspension bridges. ASCE J. Struct. Engng 105, 767-788 (1979). 10. A. M. Abdel:GhatTar, Suspension bridge vibration: continuum formulation. ASCE J. Engng Mech. Div. 108, 1215-1231 (1982). 11. C. T. Sun, B. J. Kim and J. L. Bogdanoff, On the derivation of equivalent simple models for beam and plate-like structures in dynamic analysis. In: Proc. AIAA Dynamics Specialists Conf, Atlanta, GA, pp. 523-532. AIAA (1981). 12. S. Abrate and C. T. Sun, Dynamic analysis of geometrically nonlinear truss structures. Comput. Struct. 17, 491497 (1983). 13 T. Y. Yang, Matrix displacement solution to elastica problems of beams and frames. Int. J. Solia!s Struct. 9, 829-842 (1973). 14 C. T. Sun and J. N. Juang, Modeling global structural damping in trusses using simple continuum models. AIAA J. 24, 144150 (1986). 15 B. Necib and C. T. Sun, Analysis of beams using a higher order timoshenko beam finite element. J. Sound. Vibr. 130, 149-159 (1989). 16. D. B. McCallen, A continuum model for the nonlinear analysis of beam-like lattice structures. Ph.D. Dissertation, University of California Davis, Davis, CA (1986). 17. D. B. McCallen and K. M. Romstad, Application of a continuum model in building analysis. In: Proc. ASCE Structures Congress, Orlando, FL, pp. 344-358. ASCE (1987). 18. D. B. McCallen and K. M. Romstad, A continuum model for the nonlinear analysis of beam-like lattice structures. Comput. Struct. 29, 177-197 (1988). 19. D. B. McCallen and K. M. Romstad, A continuum model for lattice structures with geometric and material nonlinearities. Comput. Struct. 37, 795-822 (1990). 20. M. J. Chajes, Nonlinear dynamic analysis of building frameworks using a continuum model. Ph.D. dissertation, University of California Davis, Davis, CA (1990). 21. M. J. Chajes, K. M. Romstad and D. B. McCallen, Inelastic frame analysis using a continuum model. In: Proc. SSRC Technical Session, St. Louis, MO, pp. 109-120. SSRC (1990). 22. M. J. Chajes, K. M. Romstad and D. B. McCallen, Nonlinear frame analysis using a continuum model. In: Proc. ASCE Structures Congress, Baltimore, MD, pp. 451452. ASCE (1990). 23. T. A. Osterkamp, Force calculations for a continuum model of lattice systems. MS. thesis, University of California Davis, Davis, CA (1988). 24. M. J. Chajes, K. M. Romstad and D. B. McCallen, Analysis of multi-bay frames using a continuum model, ASCE J. Struct. Engng 119, 522-546 (1993). 25. International Conference of Building Officials, Uniform Building Code, Whittier, CA (1988). 26. K. M. Romstad, B. Maroney and M. J. Chajes, Interpretation of Rio Dell freeway response during six recorded earthquake events. In: Proc. of the StrongMotion Instrumentation Program’s Seminar on Seismological and Engineering Implications of Recent Strong-Motion Data, Sacramento, CA, pp. 8: 1-8: 13. SMIP (1989). 37 B. Maroney, K. M. Romstad and M. J. Chajes, Interpretation of rio dell freeway response during six recorded earthquake events. In: Proc. of the Fourth U.S. National Conference on Earthquake Engineering, Palm Springs, CA, pp. 1007-1016 (1990). 28. J. T. Kirby, M. J. Chajes and J. A. Melby, Wavelet transform analysis of several transient or nonstationary phenomena in engineering mechanics. In: Proc. ASCE _I.
Engineering Mechanics Conference, College Station, TX, pp. 204-207. ASCE (1992). 29. A. Grossmann and J. Morlet, Decomposition of hardy functions into square integrable wavelets of constant shape. SIAM J. Math. Anal. IS, 723-736 (1984). 30. AC1 Committee 318, Building Code Requirements for Reinforced Concrete (AC1 3 18-89). Revised 1992, American Concrete Institute, Detroit (1992). APPENDIX The nine-by-nine continuum element stiffness matrix is given by
(Al)
where the stiffness coefficients are given by
s,,=s,= -s,,
(A24
Wb)
(A24 (A24 We)
fic
C0hIII.S 7EA,y;
s,=s,=
,=I
s,, = s,, =
ZGA,L
~ 3L
“‘r
>+
“SLY:>
~ 15
I G;;L
Wf) (A%) (A2h)
*d-S
s,, =
(
16EA,yf
1 ,=I
s,,=s,=
~
3L
>
-s,,=
+
IGA,L ~ 15
-s,,=z
s,,=
_s,,2$
s,=s,,=
--
(A2i)
(A2j)
(A2k)
IGA, L 60
GA,L 15
s,, = s, = -
(A21) (A2m)
s,, = s,, = s,, = s,, = s,, = s,* = s,, = s,,
Wn) where L is the length of the element; E and G are the modulus of elasticity and shear modulus of the lattice material; A, is the shear area of the cell spanned by the element; and i,, Ai and yi are the moment of inertia, area and distance to the neutral axis of bending of each column i in the cell.