Wave propagation in a seven-story reinforced ... - Semantic Scholar
Soil Dynamics and Earthquake Engineering 21 (2001) 211±223
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Wave propagation in a seven-story reinforced concrete building I. Theoretical models q M.I. Todorovska a,*, S.S. Ivanovic b, M.D. Trifunac a a
Civil Engineering Department, University of Southern California, Los Angeles, CA 90089-2531, USA b Civil Engineering Department, University of Montenegro, Podgorica 81000, Yugoslavia Accepted 9 December 2000
Abstract For transient, high frequency, and pulse like excitation of structures in the near ®eld of strong earthquakes, the classical design approach based on relative response spectrum and mode superposition may not be conservative. For such excitations, it is more natural to use wave propagation methods. In this paper (Part I), we review several two-dimensional wave propagation models of buildings and show results for theoretical dispersion curves computed for these models. We also estimate the parameters of these models that would correspond to a sevenstory reinforced concrete building in Van Nuys, California. Ambient vibration tests data for this building imply vertical shear wave velocity b z 112 m/s and anisotropy factor b x/b z 0.55 for NS vibrations, and b z 88 m/s and b x/b z 1 for EW vibrations. The velocity of shear waves propagating through the slabs is estimated to be about 2000 m/s. In the companion paper (Part II), we estimate phase velocities of vertically and horizontally propagating waves between seven pairs of recording points in the building using recorded response to four earthquakes. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: Damage detection; System identi®cation; Wave propagation in structures; Earthquake response analysis
1. Introduction The earthquake resistant design must be based on realistic and representative models of soil±foundation±structure systems, veri®ed against experimental data. For each model, its accuracy has to be determined, and the domain (range of the model parameters) in which it is valid. The experimental veri®cation is best accomplished by the means of recorded response of full-scale structures to actual earthquakes [24]. A common assumption in many models that consider the soil±structure interaction effects is that the foundation is rigid. This reduces the number of degrees-of-freedom of the model, and gives good approximation for long wavelengths relative to the foundation dimensions [6]. For short wavelengths, this assumption can result in nonconservative estimates of the relative deformations in the structure [22,23] and, in general, is expected to result in excessive estimates of scattering of the incident wave energy and in q This paper is dedicated to Professor Vlatko BrcÏic (1919±2000), our teacher and mentor, in recognition of his devotion, inspiring teaching, leadership and invaluable contributions to the University of Belgrade. His ideas, goals and quest for the highest standards live on through his grateful students Ð engineers he helped create. * Corresponding author. Tel.: 11-213-740-0616; fax: 11-213-744-1426. E-mail address: [email protected] (M.I. Todorovska).
excessive radiation damping [15±19]. The extent to which this simplifying assumption is valid depends on the stiffness of the foundation system relative to that of the soil, and also on the overall rigidity of the structure [3,10,25]. Rigid foundation models are usually combined with lumped mass discrete representations of the structure. The entire system is then described by a system of differential equations, and the solution is given in terms of the motion of different building ¯oors. A soil±rigid foundation±lumped-mass structural model is usually limited to representation of one-dimensional (1-D) models, and offers useful approximation for the lower frequency modes of relative response [2]. The other extreme is to neglect the stiffness of the foundation system, ignore the soil±structure interaction, and assume that the wave energy in the soil drives the building according to the principles of wave propagation. This approximate approach underestimates the incident wave energy scattered by the foundation and overestimates the energy transmitted into the building. The reality is somewhere between these two approximations, and can be studied in detail only by numerical methods. Much of the earthquake resistant design methodology is based directly or indirectly on the linear concepts of relative response spectrum, and mode superposition. The modal
0267-7261/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0267-726 1(01)00003-3
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Fig. 1. (a) Typical ¯oor plan; (b) foundation plan; (c) typical transverse section; and (d) log of typical soil boring.
approach has a low-pass ®ltering effect on the end result, i.e. computed peak relative displacement at each ¯oor, because the mode participation factors for the lowest frequency modes are usually the largest, and because the higher modes are usually neglected. The modal approach is not appropriate to represent `early' transient response, particularly to high frequency pulses, with duration shorter than the travel time for an incident wave to reach the top of the building (t , H=b; H and b are the building height and vertical shear wave velocity). As the modes of vibration are standing waves by nature and result from constructive interference of the incoming wave and the wave re¯ected from the top of the building, the building starts vibrating in the ®rst mode only after time t 2H=b has elapsed from the time the shaking starts. Although, in principle, the representation of the response as a linear combination of the modal responses is complete and, therefore, can be used to represent any response, short impulsive representation would require consideration of many modes (in®nitely many for a continuous model) which is impractical. The wave propagation methods are more natural for representation of the `early' transient response, and therefore should be explored further and used to solve problems where the modal approach is limited. Wave propagation models of buildings have been proposed earlier and used to study the physics of the problem, but have not been veri®ed against actual observations. Continuous, 2-D wave propagation models (homogeneous, horizontally layered and vertically layered shear
plates) were proposed to study the effects of traveling waves on the response of long buildings [13±16,20], and discrete-time 1-D wave propagation models were proposed to study the seismic response of tall buildings [11,12]. In this paper (Part I), we review and explore further our previous 2-D continuous building models, with parameters selected so that they correspond to a seven-story reinforced concrete hotel in Van Nuys, California. The purpose of considering these models is to facilitate interpretation of recorded earthquake response of this building via wave propagation methods. In the companion paper [21], we estimate `observed' phase velocities of propagating waves between several pairs of recording points inside the building from computed wavenumbers using recorded response to four earthquakes. The purpose is to ®nd out whether there is at least a qualitative agreement between various independent estimates of wave velocities in the building. In a follow up paper, we will investigate whether the damage in the building can be detected using wave propagation methods. 2. Description of the building The building analyzed in this paper is a seven-story reinforced concrete hotel structure, in the city of Van Nuys, California (Figs. 1±3). It will be referred to as VN7SH for short. It was designed in 1965, constructed in 1966 [1], and
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Fig. 1. (continued)
serves as a hotel. Its plan dimensions are 62 £ 150 feet (1 foot 0.305 m; Fig. 1a). The typical framing consists of columns spaced at 20 foot centers in the transverse direction and 19 foot centers in the longitudinal direction. Lateral forces in each direction are resisted by the interior column±slab frames and exterior column spandrel beam frames. The structure is constructed of regular weight reinforced concrete [1]. The site lies on recent alluvium. A typical boring log (Fig. 1d) shows the underlying soil to be primarily ®ne sandy silts and silty ®ne sands. The average shear-wave velocity in the top 30 m is 300 m/s. The foundation system (Fig. 1b) consists of 38 inch deep pile caps (1 inch 2.54 cm), supported by groups of two±four poured-in-place 24 inch diameter reinforced concrete friction piles. These are centered under the main building columns. All pile caps are connected by a grid of the beams. Each pile is roughly 40 feet long and has design capacity of over 100 kips (1 kip 4.4482 £ 103 N) vertical load and up to 20 kips lateral load. The February 9, 1971 San Fernando earthquake caused minor structural damage. Epoxy was used to repair the spalled concrete of the second ¯oor beam column joints on the north side and east end of the building. The nonstructural damage, however, was extensive and about 80%
of all repair cost was used to ®x the drywall partitions, bathroom tiles and plumbing ®xtures. Next, the building was severely damaged by the January 17, 1994 Northridge earthquake. The structural damage was extensive in the exterior north (D) and south (A) frames, designed to take most of the lateral load in the longitudinal direction. Severe shear cracks occurred at the middle columns of frame A, near the contact with the spandrel beam of the ®fth ¯oor (Fig. 2). Those cracks signi®cantly decreased the axial, moment and shear capacity of the columns. The shear cracks, which appeared in the north (D) frame, on the third and fourth ¯oors, and the damage of columns D2, D3 and D4 on the ®rst ¯oor caused minor to moderate changes in the capacity of these structural elements. No major damage of the interior longitudinal (B and C) frames was noticed. There was no visible damage in the slabs and around the foundation [26]. Analyses of the displacement time histories for channels 1, 2, 3 and 13 show that during the larger peaks of the relative response, the torsion within the building contributed 20±40% of the peak relative response, at the location of channel 2 (see Fig. 4 in Ref. [25]).
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Fig. 2. Schematic representation of damage: (top) frame D (north view), and (bottom) frame A (south view). The sensor locations for channels 1±8 and 13 (oriented towards north), are also shown (see also Fig. 3).
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Fig. 3. Channel locations (1±16) and coordinate system conventions.
3. Model
3.1. Equations of motion and eigenfunction representation
Continuous medium models are convenient to understand wave propagation in buildings. Such models are appropriate for wavelengths longer than the size of the constitutive elements (beams, columns, partition walls, etc.). The following four examples are considered here: (1) homogeneous isotropic shear plate; (2) piecewise homogeneous (layered) isotropic shear plate; (3) homogeneous anisotropic shear plate; and (4) piecewise homogeneous (layered) anisotropic shear plate (Fig. 4). In the layered models, there are two types of material, `hard' and `soft', representing respectively the ¯oor slabs and the interstory space (columns, shear walls, partition walls, etc.). Because the isotropic models are special cases of the anisotropic models, the equations will be presented only for the more general (anisotropic) case.
We consider anti-plane displacement v x; z; t in y-direction (see Fig. 3) only and assume plain stress condition. All models have length L and height H. Each layer, anisotropic in general, is characterized by shear wave velocities b x and b z, shear moduli m x and m z, for the x- and z-directions. Later, superscript `h' for `hard' and `s' for `soft' will be added to differentiate the constants for the `hard' and `soft' media. In each medium, v x; z; t has to satisfy the 2-D wave equation
b2x
22 v 22 v 22 v 1 b2z 2 2 2 2x 2z 2t
1
and zero stress boundary conditions
txy 0 at x 0; 0 # z # H
2a
Fig. 4. Four two-dimensional models of the VN7SH building used to study shear wave propagation resulting in anti-plane (north±south) response.
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txy 0 at x L; 0 # z # H
2b
tzy 0 at z 0; 0 # x , L
2c
The surface z H, i.e. the boundary between the soil and the building, is ¯exible in general. The displacement on this surface can be some prescribed displacement (driving motion), or the resultant motion of a more complex process, e.g. an incident wave being partly re¯ected from this surface and partly transmitted into the building, and then again incident to this surface from the inside, after re¯ections from the building stress-free boundaries. Because this surface is neither ®xed nor stress-free, there will be no eigenvalues associated with the boundary conditions associated with it. Eq. (1) is variable separable and the solution in the frequency domain can be represented as v x; z; t X xZ zeivt
3
The boundary conditions in x imply the following frequency equation np ; n 0; 1; ¼ 4 kx;n L where kx;n
v cx;n
5
is the x-wavenumber for the nth mode and cx;n is the associated phase velocity in the x-direction. The corresponding eigenfunctions in x are np x; n 0; 1; ¼ 6 X x cos L
When kz;n is imaginary, Zn z becomes a cosine hyperbolic function, with unit amplitude at the top and monotonically increasing amplitude towards the base. While the real kz;n are associated with transport of energy inside the building, the imaginary kz;n are associated with quasi-static deformations that are the largest near the base. We also note here that Zn z in Eq. (10) is associated with the frequency equation in x, and not with a frequency equation in z, which is the case for ®xed-base models. For a layered plate, Z(z) will be similar. It will be a continuous function, with piecewise continuous slope (the slope will be discontinuous at the interfaces between different layers), with zero slope at z 0, and can be represented by 8 1 > coskz;n z; 0 # z , hp1 > > > > 2 2 2 > > A 2 hp1 # z , hp2 n coskz;n z 1 Bn coskz;n z; > > > > > .. .. > < . . Zn z > i i i i p > An coskz;n z 1 Bn coskz;n z; hi21 # z , hpi > > > > > .. .. > > > . . > > > : M M M p An coskz;n z 1 B M n coskz;n z; hM21 # z # H 11 i The coef®cients A i n and Bn can be calculated by the following recursive equations " i21 i21 p i p coskz;n hi21 coskz;n hi21 A i n An
m k i21 i21 p i p 1 z;i21 z;n i sinkz;n hi21 sinkz;n hi21 mz;i kz;n "
and are the same in all layers of the building (otherwise the continuity conditions cannot hold, for the multi-layer model). The z-wavenumber for the nth mode and the corresponding phase velocity in the z-direction, cz;n , where v 7 kz;n cz;n are computed from the relation s 2 bx v np 2 2 ; kz;n bx L bz
n 0; 1; ¼
8
and are different for each layer (because b x and b z are different). Eq. (8) implies that kz;n is real only for the ®rst few modes for which v np 9 $ bx L and is imaginary for all other modes. For n 0, kz;n is always real, and for given frequency, kz;n is real up to higher n in softer materials (smaller b x). For a homogeneous plate, Z(z) corresponding to mode n is s 2 bx v np 2 2 z; n 0; 1; ¼ 10 Zn z cos bx L bz
#
i21 p i p hi21 coskz;n hi21 1 Bn i21 sinkz;n
m k i21 i21 p i i 2 z;i21 z;n i coskz;n hi21 sinkz;n hi21 mz;i kz;n
# 12a
" B i n
An i21
i21 p i p hi21 sinkz;n hi21 coskz;n
m k i21 i21 p i p 2 z;i21 z;n i sinkz;n hi21 coskz;n hi21 mz;i kz;n "
#
i21 p i p hi21 sinkz;n hi21 1 Bn i21 sinkz;n
m k i21 i21 p i p 1 z;i21 z;n i coskz;n hi21 coskz;n hi21 mz;i kz;n
# 12b
for i 2, 3, ¼, M, where M is the number of layers, and 1 with A 1 n 1 and Bn 0.
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The general representation of motion is then v x; z; t
1 X n0
Cnp cos
npx Z zeivt L n
13
where Cnp are the Fourier coef®cients to be determined for speci®c excitation. For prescribed driving harmonic motion at the base vs x; t v0 xeivt
14
with Fourier series expansion v0 x
1 X n0
Cn cos
npx L
15 m 0, Eq. (17) gives
the coef®cients Cnp in Eq. (13) are Cnp
Cn =Zn H
Fig. 5. A plan view of a building excited by the passage of a long plane SH or a Love wave. The base motion can be approximated as a sum of translation D0 eivt and rotation f0 eivt about x 0.
bz 4f0;0 H 16
19
Coef®cients Cnp for driving motion that is a horizontally propagating wave can be found in Refs. [14,20], and for driving translational and torsional motion in Ref. [16].
and for n 1 and m 0 it gives v ! u u f1;0 2 bx L t 21 2H bz f0;0
3.2. Fixed-base frequencies
3.3. Response to ground translation and rotation
It is seen from Eq. (16) that Cnp ! 1 for frequencies v for which Zn H 0. The condition on Zn H 0 implies that these are the ®xed-base frequencies of the plate. For a homogeneous plate model, the ®xed-base frequencies are those frequencies vn;m that satisfy s vn;m 2 bx np 2 1 2 p; n 0; 1; 2; ¼; H m1 bz bx L 2
For incident SH-waves and for long wavelengths, the wave passage effects may be considered approximately by representing the motion at the base as a sum of synchronous translation D0 eivt and rotation in horizontal plane f0 eivt
m 0; 1; 2; ¼ 17 For a horizontally layered plate, vn;m are found by solving the transcendental equation cotan kz;n H 2
Bn M An M
18
Solutions for different m are obtained by solving Eq. (18) for different branches of cotan kz;n H. If there is no material or radiation damping, the response of the model to driving motion at the base will be in®nite at v vn;m . If there is damping, the response at these frequencies will be ®nite but large. These frequencies can be determined experimentally by analysis of the peaks of the building transfer-function determined from recorded response to ambient noise or to earthquake ground shaking. The ®rst translational and torsional ®xed-base frequencies can be used to estimate the shear wave velocities of the homogeneous building model as follows. The ®rst translational mode corresponds to n 0 and m 0, and the ®rst torsional mode corresponds to n 1 and m 0. Let f0,0 and f1,0 be the frequencies in Hertz, respectively, for the ®rst translational and torsional modes. Then, for n 0 and
vs x; t < D0 eivt 1 f0 xeivt
20
21
This follows from the ®rst order Taylor series approximation of a monochromatic SH-wave, and is illustrated in Fig. 5. The dashed contour in this ®gure indicates the position of the building at rest and the solid contour indicates its position and deformation during the wave passage at some time t. Let the incident wave have unit amplitude, frequency v and propagate in the x-direction with phase velocity c vs x; t eiv t2x=c
22
For very long incident waves compared to the length of the building, v x/c is small, and e2i v=cx can be expanded in Taylor series about point x 0. The ®rst order Taylor series approximation is iv x e i vt 23 vs x; t 1 2 c and represents displacement that is a sum of unit amplitude translation and rotation about x 0 with amplitude 2iv /c. Comparison with Eq. (21) implies D0 1
24
and
f0 2
iv c
25
It is seen that the translation and rotation are delayed with
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Foundations of real structures are not rigid, and deform during the wave passage and as the wave energy goes in and out of the building [25]. The building response near the ®xed-base frequencies (i.e. the resonant part of the response) can be approximated by the simpli®ed rigid foundation model shown in Fig. 6. This response represents standing waves resulting from constructive interference of the incident waves with the waves re¯ected from the top of the building. For all other frequencies, and in general, assuming shear deformation of the structure only, and negligible foundation warping, solution for the building response ur x; z; t (Fig. 6) can be derived analogous to the one presented in Section 3.1 for incident SH-waves.
4. Results and analysis 4.1. Shear wave velocities for the VN7SH building models Fig. 6. Displacements of a building (as a shear beam) on a rigid base and excited by horizontal and vertical ground translations u and w and by ground rocking u0;x .
respect to each other by phase angle p/2. Synthesis of rotational accelerograms from translational accelerograms was studied by Lee and Trifunac [7,8] and the reader is referred to their work for further details on this topic. The building response to ground motion represented by Eq. (23) is the series in Eq. (13), with coef®cients Cnp as in Eq. (16), and coef®cients Cn of the Fourier series expansion of v0 x 1 2
iv x c
26
as in Eqs. (15) are [20] as follows. For n 0 C0 1 2
i vL 2 c
and for n . 0 8 > 0; < Cn 4 v L > ; : i L c np
27a
n even n odd
27b
The ®rst term of C0 is due to the ground translation, while the second term and all other coef®cients Cn, n . 0, are due to the ground torsion. Body P- and SV-waves and surface Rayleigh waves propagating in the y±z plane result in horizontal, u, vertical, w, and rocking, ux , components of ground motion (Figs. 3 and 6). A structure excited by these motions and on ¯exible soil will experience, through soil±structure interaction, additional relative motions at its base. Assuming the foundation is rigid, these additional relative motions are translations u0 and w0 and rocking u0;x (Fig. 6). The relative response of the structure, ur will depend on the resultant motion of the base, due to all of these motions.
To evaluate b z for the continuous models in Fig. 4, the equivalent shear modulus m eq and the equivalent density r eq need to be determined. It can be shown [20] that s fEc b2 bz 28 h2 rc where Ec is the Young's modulus of elasticity of concrete, f . 1 is a factor depending on the percentage of reinforcement in the columns, b is the dimension of a column in the direction of bending, h is the story height, and r c the density of concrete. For example, assuming b 50 cm, h 3.0 m, Ec 2.8 £ 10 5 kg/cm 2, r c 2500 kg/m 3 and f , 1.2 gives b z 611 m/s for ®xed-®xed columns. The effective length (h kLc) of the columns between two stories depends on their relative bending stiffness and that of the girders, and can be computed from charts published for unbraced frames, e.g. [9], in terms of c a and c b where (see Fig. 7)
c
s E c Ic =Lc s E g Ig =Lg
29
and Ic;g is the moment of inertia of a column/girder; Lc;g is the length of a column/girder, center to center from joints; and Ec;g is the modulus of elasticity of column/girder. For a typical story of the VN7SH building with a 50 cm wide strip of ¯oor slabs, centered on the columns acting as a `girder' to make a moment resistant frame (Fig. 7b), c a , c b ,32. The above values of c give k , 5, and b z 611/5 122 m/s. This value is an estimate representative of wave propagation between two adjacent ¯oors, i.e. of b zs for the layered model, but is approximately equal to b z for an equivalent homogeneous model because the slabs are very thin and do not affect much the vertical wave propagation in the building. The shear-wave velocity associated with horizontal wave
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These experimentally determined values for b z for an equivalent homogeneous model are consistent with bsz b z estimated from the characteristics of the columns. The experimentally determined ratios of horizontal to vertical shear wave velocities for the equivalent homogeneous model b x/b z # 1, even though the shear wave velocity in the slabs is very large (20 times larger than b z). This is also because the slabs are very thin compared with the interstory space and represent a very small fraction of the building height. From the above we conclude that for a representative homogeneous model for two-dimensional wave propagation in the VN7SH building in the x±z plane, we can choose bsz 112 m/s and b x/b z 0.55, and for a layered model, we can choose bsz 112 m/s, bsx =bsz 0.55, bhz 2000 m/s and bhx =bhz 1 (Fig. 4). 4.2. Theoretical dispersion curves for the VN7SH building
Fig. 7. (a) A typical column±slab group and (b) `working width' of a slab used to de®ne c a and c b in calculation of the effective length of the columns.
propagation through the ¯oor slabs is s s G E =r 2049 m=s bx rc 2 1 1 v c
30
Shear waves are propagated horizontally also through the soft layers, via deformation of the partition walls and shear walls, with a much smaller velocity. The average shear-wave velocities b x and b z for an equivalent homogeneous model can be estimated also from the experimentally determined ®rst translational and torsional frequencies from ambient vibration test data and Eqs. (19) and (20). We note that the condition in Eq. (19) is equivalent to the building height being equal to 1/4 of the wavelength of shear waves propagating vertically in the building. The ambient vibration surveys for this building [4,5], assuming a ®xed-base structure, imply f0,0 1.4 Hz for the ®rst NS mode, f0,0 1.1 Hz for the ®rst EW mode, and f1,0 1.6 Hz for the ®rst torsional mode. This gives for NS vibrations b z 112 m/s and b x/b z 0.55, and for EW vibrations b z 88 m/s and b x/b z 1, for H=L 0:5. In reality, these frequencies are higher (even for response in the linear range) because the measured frequencies are not the ®xed-base but the system frequencies of the soil±structure system.
Dispersion curves are plots of allowable phase velocities versus frequency. In this section, we present dispersion curves cx f v and cz f v for the four models in Fig. 4 drawn for parameters approximately equal to those estimated for the VN7SH building. The purpose is to understand how these curves look and vary depending on the model parameters, which will be useful for future work on damage detection. The dispersion characteristics of a building will depend on its state of damage, and changes in experimentally determined dispersion curves can indicate that damage has occurred. Therefore, understanding of the theoretical dispersion curves and their variability will be useful for development of new methods for damage detection and for selection of optimal sensor locations to detect the changes. For the homogeneous model, we chose L 40 m, H/ L 0.5, b z 100 m/s, and b x/b z 0.5 and 1. For the layered model, L 40 m, H/L 0.5, bsz 100 m/s, bsx =bsz 0.5 and 1, bhz =bsz 20, and bhx =bhz 1. The layered model has 14 layers. The thickness of the hard layers is h1 8 inches, h3 h5 h7 h9 h11 8:5 inches and h13 10 inches, and the thickness of the soft layers is h2 h4 h6 h8 h10 h12 96 inches, and h14 152 inches, where 1 inch 2.54 cm. The equations for the dispersion curves in terms of normalized phase velocities cx =b and cz =b and dimensionless frequency vL=b, where b bz for the homogeneous model and b bsz for the layered model follow from Eqs. (4), (5), (7) and (8) and are as follows. For the homogeneous plate model cx;n 1 vL ; bz np bz
n 0; 1; 2; ¼
" #21=2 cz;n np bx 2 12 ´ ; bz vL=bz bz
31a
n 0; 1; 2; ¼ 31b
Eq. (31a) implies that cx;n is always real, and Eq. (31b)
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Fig. 8. Horizontal (left) and vertical (right) phase velocities versus dimensionless frequency vL=pb for a homogeneous isotopic plate model of the VN7SH building.
implies that cz;n is real for vL=bz $ npbx =bz , is imaginary for vL=bz , npbx =bz , and cz;n ! 1 as vL=bz ! npbx =bz . For the layered plate model cx;n 1 vL ; bsz np bsz
n 0; 1; 2; ¼
" #21=2 csz;n np bsx 2 12 ´ ; bsz vL=bsz bsz " chz;n bhx h 12 bsz bz
np bhx bhz ´ ´ vL=bsz bhz bsz
32a
n 0; 1; 2; ¼ 32b
!2 #21=2 ;
n 0; 1; 2; ¼
vL=bsz
npbsx =bsz ,
32c that csz;n is real for vL=bsz , npbsx =bsz ,
$ is Eq. (32b) implies and csz;n ! 1 as imaginary for vL=bsz ! npbsx =bsz . Similarly, Eq. (32c) implies that chz;n is real for vL=bsz $ npbhx =bhz , is imaginary for vL=bsz , npbhx =bhz , and chz;n ! 1 as vL=bsz ! npbhx =bhz . Fig. 8 shows the dispersion curves for the isotropic model (b x/b z 1), and Fig. 9 shows the dispersion curves for an anisotropic model with b x/b z 0.5. The frequency in Hertz on the second scale for the x-axis corresponds to L 40 m and b z 100 m/s. It is seen from the plot on the left hand side that the dispersion curves for horizontal wave propagation are straight lines, with 908 slope for n 0 (cx;0 1) and progressively smaller slope as n increases. In the plot on
the right hand side, the real branches of the dispersion curves for vertical wave propagation are shown by solid lines, and the imaginary branches are shown by dashed lines. For given n, both branches have a common vertical asymptote at vL=bz npbx =bz . For n 0, cz bz and is always real. This implies that regardless of the frequency, there is always at least one mode with real vertical phase velocity. All the imaginary branches start from zero and end at the corresponding vertical asymptote. The real branches start from their vertical asymptote and approach unity for large frequencies (i.e. cz;n =bz ! 1 as vL=bsz ! 1) implying that for large frequency, cz;n < bz . The solid heavier lines in the same plot are dispersion curves associated with ®xed-base response and stress free boundary conditions at the top of the building (with no constraints in x). These are de®ned by cz;m 1 H vL ; bz m 1 p=2 L bz
m 0; 1; 2; ¼
33
The frequencies vn;m and associated phase velocities cz for the ®xed-base frequencies of the 2-D models can be obtained by reading the x- and y-coordinates of the intersection of the real branch of the curve for cz,n with the line for cz,m. A consequence of the asymptotic behavior of the real branches of cz,n is that for larger m, the corresponding translational and torsional modes will have very close frequencies and vertical phase velocities. This would make it dif®cult to extract these modes from experimental data.
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Fig. 9. Horizontal (left) and vertical (right) phase velocities versus dimensionless frequency vL=pbz for a homogeneous anisotropic plate model of the VN7SH building.
Fig. 10 shows dispersion curves for an isotropic layered model with b x/b z 1 and bh =bs 20, and Fig. 11 shows the dispersion curves for an anisotropic layered model, with bsx =bsz 0.5 and bhx =bhz 1 and bhz =bsz 20. The frequency in Hertz corresponds to L 40 m and bsz 100 m/s. The dispersion curves for horizontal wave propagation are the same as for the homogeneous models. The dispersion curves for vertical wave propagation in the soft layers are also the same as those for the homogeneous models, because we chose the shear wave velocities in the soft layers to be the same as those for the corresponding homogeneous model. It is seen that for the frequency range shown (0±12.5 Hz), cz =bsz in the hard layers is imaginary for all n except n 0. The dispersion curves for the homogeneous models would be useful for interpretation of building recordings at the base and at the top. Those for the soft layers of the layered models would be useful for interpretation of recordings at neighboring ¯oors, and those for the hard layers would hardly be useful for damage detection, but are included for completeness. 5. Summary and conclusions The wave propagation methods for estimation of building response are superior to the modal methods for early, high frequency impulsive excitations (here `early' refers to times shorter than it takes for a standing wave corresponding to
the ®rst mode of vibration to develop). Such excitations occur in the near-®eld of earthquake sources. The wave propagation methods also offer possibilities for development of new methods for damage detection in buildings. In this paper (Part I), we review and deliberate on several simple 2-D wave propagation building models (homogeneous anisotropic and horizontally layered anisotropic shear plates with soft and hard layers representing the inter-story space and the slabs) and analyze dispersion in these models. The ultimate aim is to interpret, via wave propagation methods, the recorded earthquake response of a seven-story reinforced concrete building, damaged by the 1994 Northridge earthquake. Ambient vibration experiments in the VN7SH building imply vertical shear-wave velocity b z 122 and 88 m/s and ratios of horizontal to vertical shear-wave velocity b x/b z 0.55 and 1, respectively, for NS and EW vibrations. These values were obtained from the requirement that the anisotropic homogeneous building model matches the frequencies of the ®rst translational and the ®rst torsional frequencies, determined from ambient vibration data based on the assumption that the apparent frequencies seen in the data approximate the ®xed-base frequencies. The shear wave velocity in the concrete slabs is estimated to be about 2000 m/s. For the layered model it is suggested that the shear wave velocities in the soft layer should be approximately the same as for the homogeneous model. The dispersion analysis showed that the horizontal phase
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Fig. 10. Horizontal (left), and vertical (center for the soft layers and right for the hard layers) phase velocities versus dimensionless frequency vL=pb s for a layered isotropic plate model of the VN7SH building.
Fig. 11. Horizontal (left), and vertical (center for the soft layers and right for the hard layers) phase velocities versus dimensionless frequency vL=pbsz for a layered isotropic plate model of the VN7SH building.
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velocities in the models are always real (this is true only for models without vertical discontinuities; [13]). The vertical phase velocities are always real for n 0 and can be real or imaginary for higher n depending on frequency. They are imaginary for frequencies up to a critical frequency and real for frequencies higher than the critical frequency. At the critical frequency, the vertical phase velocities are in®nite, which corresponds to horizontal wave propagation. This critical frequency increases with increasing order of the mode and decreases with increasing shear-wave velocity. The eigenfunctions corresponding to real phase velocities (i.e. wavenumbers) are harmonic functions and are associated with propagation of wave energy, while those corresponding to imaginary phase velocities are hyperbolic (i.e. exponential) functions and are associated with quasi-static deformations of the building. The relative contributions of these modes will depend on the excitation. Understanding of the dispersion in building models will be useful in interpretation of recorded building response and in damage detection.
References [1] Blume JA et al. In: San Fernando, California Earthquake of February 9, 1971, vol. 1, Part A, p. 359±393. Washington, DC: US Department of Commerce, National Oceanic and Atmospheric Administration, 1973. [2] Duncan WJ. A critical examination of the representation of massive and elastic bodies by systems of rigid masses elastically connected. Quart J Mech Appl Math 1952;5(1):97±108. [3] Iguchi M, Luco JE. Vibration of ¯exible plate on viscoelastic medium. J Engng Mech ASCE 1982;108(6):1103±20. [4] Ivanovic SS, Novikova EI, Gladkov AA, Todorovska MI. Ambient vibration tests of a seven story reinforced concrete building damaged by the 1994 Northridge, California earthquake. Soil Dynam Earthquake Engng 2000;19:391±411. [5] Ivanovic SS, Trifunac MD, Todorovska MI. Ambient vibration tests of structuresÐa review. Indian Society of Earthquake Technology Journal 2000;37: in press. [6] Lee VW. Investigation of three-dimensional soil±structure interaction. Report CE 79-11, Department of Civil Engineering, University of Southern California, Los Angeles, California, 1979. [7] Lee WW, Trifunac MD. Torsional accelerograms. Soil Dynam Earthquake Engng 1985;4(3):132±9. [8] Lee VW, Trifunac MD. Rocking strong earthquake accelerations. Soil Dynam Earthquake Engng 1987;6(2):75±89. [9] Leet K. Reinforced concrete design. New York: McGraw Hill, 1991.
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[10] Liou G-S, Huang PR. Effects of ¯exibility on impedance functions for circular foundations. J Engng Mech ASCE 1994;120(7):1429±46. [11] Safak E. New approach to analyzing soil±structure systems. Soil Dynam Earthquake Engng 1998;17:509±17. [12] Safak E. Wave-propagation formulation of seismic response of multistory buildings. J Struct Engng ASCE 1999;125(4):426±37. [13] Todorovska MI, Lee VW. Seismic waves in buildings with shear walls or central core. J Engng Mech ASCE 1989;115(12):2669±86. [14] Todorovska MI, Trifunac MD. Antiplane earthquake waves in long structures. J Engng Mech ASCE 1989;115(12):2687±708. [15] Todorovska MI, Trifunac MD. A note on the propagation of earthquake waves in buildings with soft ®rst ¯oor. J Engng Mech ASCE 1990;116(4):892±900. [16] Todorovska MI, Trifunac MD. A note on excitation of long structures by ground waves. J Engng Mech ASCE 1990;116(4):952±64. [17] Todorovska MI, Trifunac MD. Analytical model for in-plane building±foundation±soil interaction: incident P-, SV- and Rayleigh waves. Department of Civil Engineering, Report No. 90-01, University of Southern California, Los Angeles, California, 1990. [18] Todorovska MI, Trifunac MD. Radiation damping during two-dimensional building±soil interaction. Department of Civil Engineering, Report No. CE 91-01, University of Southern California, Los Angeles, California, 1991. [19] Todorovska MI, Trifunac MD. The effects of the wave passage on the response of base-isolated buildings on rigid embedded foundations. Department of Civil Engineering, Report No. CE 93-10, University of Southern California, Los Angeles, California, 1993. [20] Todorovska MI, Trifunac MD, Lee VW. Investigation of earthquake response of long buildings. Department of Civil Engineering, Report No. CE 88-02, University of Southern California, Los Angeles, California, 1988. [21] Todorovska MI, Ivanovic SS, Trifunac MD. Wave propagation in a seven-story reinforced concrete building Part II: Observed wave numbers. Soil Dynam Earthquake Engng 2001;21:225±36. [22] Trifunac MD. Differential earthquake motion of building foundation. J Struct Engng ASCE 1997;123(4):414±22. [23] Trifunac MD, Todorovska MI. Response spectra for differential motion of columns. Earthquake Engng Struct Dynam 1997;26(2):251±68. [24] Trifunac MD, Todorovska MI. Recording and interpreting earthquake response of full-scale structures. Proc NATO Workshop on Strong Motion Instrumentation for Civil Engineering Structures, Istanbul, June 2±5, 1999 Kluwer, 1999. [25] Trifunac MD, Ivanovic SS, Todorovska MI, Novikova EI, Gladkov AA. Experimental evidence for ¯exibility of a building foundation supported by concrete friction piles. Soil Dynam Earthquake Engng 1999;18:169±87. [26] Trifunac MD, Ivanovic SS, Todorovska MI. Seven story reinforced concrete building in Van Nuys, California: strong data recorded between 7 February 1971 and 9 December 1994, and description of damage following the Northridge 17 January 1994 earthquake. Report no. CE 99-02, Dept. of Civil Eng., University of Southern California, Los Angeles, California, 1999.
www.elsevier.com/locate/soildyn
Wave propagation in a seven-story reinforced concrete building I. Theoretical models q M.I. Todorovska a,*, S.S. Ivanovic b, M.D. Trifunac a a
Civil Engineering Department, University of Southern California, Los Angeles, CA 90089-2531, USA b Civil Engineering Department, University of Montenegro, Podgorica 81000, Yugoslavia Accepted 9 December 2000
Abstract For transient, high frequency, and pulse like excitation of structures in the near ®eld of strong earthquakes, the classical design approach based on relative response spectrum and mode superposition may not be conservative. For such excitations, it is more natural to use wave propagation methods. In this paper (Part I), we review several two-dimensional wave propagation models of buildings and show results for theoretical dispersion curves computed for these models. We also estimate the parameters of these models that would correspond to a sevenstory reinforced concrete building in Van Nuys, California. Ambient vibration tests data for this building imply vertical shear wave velocity b z 112 m/s and anisotropy factor b x/b z 0.55 for NS vibrations, and b z 88 m/s and b x/b z 1 for EW vibrations. The velocity of shear waves propagating through the slabs is estimated to be about 2000 m/s. In the companion paper (Part II), we estimate phase velocities of vertically and horizontally propagating waves between seven pairs of recording points in the building using recorded response to four earthquakes. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: Damage detection; System identi®cation; Wave propagation in structures; Earthquake response analysis
1. Introduction The earthquake resistant design must be based on realistic and representative models of soil±foundation±structure systems, veri®ed against experimental data. For each model, its accuracy has to be determined, and the domain (range of the model parameters) in which it is valid. The experimental veri®cation is best accomplished by the means of recorded response of full-scale structures to actual earthquakes [24]. A common assumption in many models that consider the soil±structure interaction effects is that the foundation is rigid. This reduces the number of degrees-of-freedom of the model, and gives good approximation for long wavelengths relative to the foundation dimensions [6]. For short wavelengths, this assumption can result in nonconservative estimates of the relative deformations in the structure [22,23] and, in general, is expected to result in excessive estimates of scattering of the incident wave energy and in q This paper is dedicated to Professor Vlatko BrcÏic (1919±2000), our teacher and mentor, in recognition of his devotion, inspiring teaching, leadership and invaluable contributions to the University of Belgrade. His ideas, goals and quest for the highest standards live on through his grateful students Ð engineers he helped create. * Corresponding author. Tel.: 11-213-740-0616; fax: 11-213-744-1426. E-mail address: [email protected] (M.I. Todorovska).
excessive radiation damping [15±19]. The extent to which this simplifying assumption is valid depends on the stiffness of the foundation system relative to that of the soil, and also on the overall rigidity of the structure [3,10,25]. Rigid foundation models are usually combined with lumped mass discrete representations of the structure. The entire system is then described by a system of differential equations, and the solution is given in terms of the motion of different building ¯oors. A soil±rigid foundation±lumped-mass structural model is usually limited to representation of one-dimensional (1-D) models, and offers useful approximation for the lower frequency modes of relative response [2]. The other extreme is to neglect the stiffness of the foundation system, ignore the soil±structure interaction, and assume that the wave energy in the soil drives the building according to the principles of wave propagation. This approximate approach underestimates the incident wave energy scattered by the foundation and overestimates the energy transmitted into the building. The reality is somewhere between these two approximations, and can be studied in detail only by numerical methods. Much of the earthquake resistant design methodology is based directly or indirectly on the linear concepts of relative response spectrum, and mode superposition. The modal
0267-7261/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0267-726 1(01)00003-3
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Fig. 1. (a) Typical ¯oor plan; (b) foundation plan; (c) typical transverse section; and (d) log of typical soil boring.
approach has a low-pass ®ltering effect on the end result, i.e. computed peak relative displacement at each ¯oor, because the mode participation factors for the lowest frequency modes are usually the largest, and because the higher modes are usually neglected. The modal approach is not appropriate to represent `early' transient response, particularly to high frequency pulses, with duration shorter than the travel time for an incident wave to reach the top of the building (t , H=b; H and b are the building height and vertical shear wave velocity). As the modes of vibration are standing waves by nature and result from constructive interference of the incoming wave and the wave re¯ected from the top of the building, the building starts vibrating in the ®rst mode only after time t 2H=b has elapsed from the time the shaking starts. Although, in principle, the representation of the response as a linear combination of the modal responses is complete and, therefore, can be used to represent any response, short impulsive representation would require consideration of many modes (in®nitely many for a continuous model) which is impractical. The wave propagation methods are more natural for representation of the `early' transient response, and therefore should be explored further and used to solve problems where the modal approach is limited. Wave propagation models of buildings have been proposed earlier and used to study the physics of the problem, but have not been veri®ed against actual observations. Continuous, 2-D wave propagation models (homogeneous, horizontally layered and vertically layered shear
plates) were proposed to study the effects of traveling waves on the response of long buildings [13±16,20], and discrete-time 1-D wave propagation models were proposed to study the seismic response of tall buildings [11,12]. In this paper (Part I), we review and explore further our previous 2-D continuous building models, with parameters selected so that they correspond to a seven-story reinforced concrete hotel in Van Nuys, California. The purpose of considering these models is to facilitate interpretation of recorded earthquake response of this building via wave propagation methods. In the companion paper [21], we estimate `observed' phase velocities of propagating waves between several pairs of recording points inside the building from computed wavenumbers using recorded response to four earthquakes. The purpose is to ®nd out whether there is at least a qualitative agreement between various independent estimates of wave velocities in the building. In a follow up paper, we will investigate whether the damage in the building can be detected using wave propagation methods. 2. Description of the building The building analyzed in this paper is a seven-story reinforced concrete hotel structure, in the city of Van Nuys, California (Figs. 1±3). It will be referred to as VN7SH for short. It was designed in 1965, constructed in 1966 [1], and
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213
Fig. 1. (continued)
serves as a hotel. Its plan dimensions are 62 £ 150 feet (1 foot 0.305 m; Fig. 1a). The typical framing consists of columns spaced at 20 foot centers in the transverse direction and 19 foot centers in the longitudinal direction. Lateral forces in each direction are resisted by the interior column±slab frames and exterior column spandrel beam frames. The structure is constructed of regular weight reinforced concrete [1]. The site lies on recent alluvium. A typical boring log (Fig. 1d) shows the underlying soil to be primarily ®ne sandy silts and silty ®ne sands. The average shear-wave velocity in the top 30 m is 300 m/s. The foundation system (Fig. 1b) consists of 38 inch deep pile caps (1 inch 2.54 cm), supported by groups of two±four poured-in-place 24 inch diameter reinforced concrete friction piles. These are centered under the main building columns. All pile caps are connected by a grid of the beams. Each pile is roughly 40 feet long and has design capacity of over 100 kips (1 kip 4.4482 £ 103 N) vertical load and up to 20 kips lateral load. The February 9, 1971 San Fernando earthquake caused minor structural damage. Epoxy was used to repair the spalled concrete of the second ¯oor beam column joints on the north side and east end of the building. The nonstructural damage, however, was extensive and about 80%
of all repair cost was used to ®x the drywall partitions, bathroom tiles and plumbing ®xtures. Next, the building was severely damaged by the January 17, 1994 Northridge earthquake. The structural damage was extensive in the exterior north (D) and south (A) frames, designed to take most of the lateral load in the longitudinal direction. Severe shear cracks occurred at the middle columns of frame A, near the contact with the spandrel beam of the ®fth ¯oor (Fig. 2). Those cracks signi®cantly decreased the axial, moment and shear capacity of the columns. The shear cracks, which appeared in the north (D) frame, on the third and fourth ¯oors, and the damage of columns D2, D3 and D4 on the ®rst ¯oor caused minor to moderate changes in the capacity of these structural elements. No major damage of the interior longitudinal (B and C) frames was noticed. There was no visible damage in the slabs and around the foundation [26]. Analyses of the displacement time histories for channels 1, 2, 3 and 13 show that during the larger peaks of the relative response, the torsion within the building contributed 20±40% of the peak relative response, at the location of channel 2 (see Fig. 4 in Ref. [25]).
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Fig. 2. Schematic representation of damage: (top) frame D (north view), and (bottom) frame A (south view). The sensor locations for channels 1±8 and 13 (oriented towards north), are also shown (see also Fig. 3).
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215
Fig. 3. Channel locations (1±16) and coordinate system conventions.
3. Model
3.1. Equations of motion and eigenfunction representation
Continuous medium models are convenient to understand wave propagation in buildings. Such models are appropriate for wavelengths longer than the size of the constitutive elements (beams, columns, partition walls, etc.). The following four examples are considered here: (1) homogeneous isotropic shear plate; (2) piecewise homogeneous (layered) isotropic shear plate; (3) homogeneous anisotropic shear plate; and (4) piecewise homogeneous (layered) anisotropic shear plate (Fig. 4). In the layered models, there are two types of material, `hard' and `soft', representing respectively the ¯oor slabs and the interstory space (columns, shear walls, partition walls, etc.). Because the isotropic models are special cases of the anisotropic models, the equations will be presented only for the more general (anisotropic) case.
We consider anti-plane displacement v x; z; t in y-direction (see Fig. 3) only and assume plain stress condition. All models have length L and height H. Each layer, anisotropic in general, is characterized by shear wave velocities b x and b z, shear moduli m x and m z, for the x- and z-directions. Later, superscript `h' for `hard' and `s' for `soft' will be added to differentiate the constants for the `hard' and `soft' media. In each medium, v x; z; t has to satisfy the 2-D wave equation
b2x
22 v 22 v 22 v 1 b2z 2 2 2 2x 2z 2t
1
and zero stress boundary conditions
txy 0 at x 0; 0 # z # H
2a
Fig. 4. Four two-dimensional models of the VN7SH building used to study shear wave propagation resulting in anti-plane (north±south) response.
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txy 0 at x L; 0 # z # H
2b
tzy 0 at z 0; 0 # x , L
2c
The surface z H, i.e. the boundary between the soil and the building, is ¯exible in general. The displacement on this surface can be some prescribed displacement (driving motion), or the resultant motion of a more complex process, e.g. an incident wave being partly re¯ected from this surface and partly transmitted into the building, and then again incident to this surface from the inside, after re¯ections from the building stress-free boundaries. Because this surface is neither ®xed nor stress-free, there will be no eigenvalues associated with the boundary conditions associated with it. Eq. (1) is variable separable and the solution in the frequency domain can be represented as v x; z; t X xZ zeivt
3
The boundary conditions in x imply the following frequency equation np ; n 0; 1; ¼ 4 kx;n L where kx;n
v cx;n
5
is the x-wavenumber for the nth mode and cx;n is the associated phase velocity in the x-direction. The corresponding eigenfunctions in x are np x; n 0; 1; ¼ 6 X x cos L
When kz;n is imaginary, Zn z becomes a cosine hyperbolic function, with unit amplitude at the top and monotonically increasing amplitude towards the base. While the real kz;n are associated with transport of energy inside the building, the imaginary kz;n are associated with quasi-static deformations that are the largest near the base. We also note here that Zn z in Eq. (10) is associated with the frequency equation in x, and not with a frequency equation in z, which is the case for ®xed-base models. For a layered plate, Z(z) will be similar. It will be a continuous function, with piecewise continuous slope (the slope will be discontinuous at the interfaces between different layers), with zero slope at z 0, and can be represented by 8 1 > coskz;n z; 0 # z , hp1 > > > > 2 2 2 > > A 2 hp1 # z , hp2 n coskz;n z 1 Bn coskz;n z; > > > > > .. .. > < . . Zn z > i i i i p > An coskz;n z 1 Bn coskz;n z; hi21 # z , hpi > > > > > .. .. > > > . . > > > : M M M p An coskz;n z 1 B M n coskz;n z; hM21 # z # H 11 i The coef®cients A i n and Bn can be calculated by the following recursive equations " i21 i21 p i p coskz;n hi21 coskz;n hi21 A i n An
m k i21 i21 p i p 1 z;i21 z;n i sinkz;n hi21 sinkz;n hi21 mz;i kz;n "
and are the same in all layers of the building (otherwise the continuity conditions cannot hold, for the multi-layer model). The z-wavenumber for the nth mode and the corresponding phase velocity in the z-direction, cz;n , where v 7 kz;n cz;n are computed from the relation s 2 bx v np 2 2 ; kz;n bx L bz
n 0; 1; ¼
8
and are different for each layer (because b x and b z are different). Eq. (8) implies that kz;n is real only for the ®rst few modes for which v np 9 $ bx L and is imaginary for all other modes. For n 0, kz;n is always real, and for given frequency, kz;n is real up to higher n in softer materials (smaller b x). For a homogeneous plate, Z(z) corresponding to mode n is s 2 bx v np 2 2 z; n 0; 1; ¼ 10 Zn z cos bx L bz
#
i21 p i p hi21 coskz;n hi21 1 Bn i21 sinkz;n
m k i21 i21 p i i 2 z;i21 z;n i coskz;n hi21 sinkz;n hi21 mz;i kz;n
# 12a
" B i n
An i21
i21 p i p hi21 sinkz;n hi21 coskz;n
m k i21 i21 p i p 2 z;i21 z;n i sinkz;n hi21 coskz;n hi21 mz;i kz;n "
#
i21 p i p hi21 sinkz;n hi21 1 Bn i21 sinkz;n
m k i21 i21 p i p 1 z;i21 z;n i coskz;n hi21 coskz;n hi21 mz;i kz;n
# 12b
for i 2, 3, ¼, M, where M is the number of layers, and 1 with A 1 n 1 and Bn 0.
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217
The general representation of motion is then v x; z; t
1 X n0
Cnp cos
npx Z zeivt L n
13
where Cnp are the Fourier coef®cients to be determined for speci®c excitation. For prescribed driving harmonic motion at the base vs x; t v0 xeivt
14
with Fourier series expansion v0 x
1 X n0
Cn cos
npx L
15 m 0, Eq. (17) gives
the coef®cients Cnp in Eq. (13) are Cnp
Cn =Zn H
Fig. 5. A plan view of a building excited by the passage of a long plane SH or a Love wave. The base motion can be approximated as a sum of translation D0 eivt and rotation f0 eivt about x 0.
bz 4f0;0 H 16
19
Coef®cients Cnp for driving motion that is a horizontally propagating wave can be found in Refs. [14,20], and for driving translational and torsional motion in Ref. [16].
and for n 1 and m 0 it gives v ! u u f1;0 2 bx L t 21 2H bz f0;0
3.2. Fixed-base frequencies
3.3. Response to ground translation and rotation
It is seen from Eq. (16) that Cnp ! 1 for frequencies v for which Zn H 0. The condition on Zn H 0 implies that these are the ®xed-base frequencies of the plate. For a homogeneous plate model, the ®xed-base frequencies are those frequencies vn;m that satisfy s vn;m 2 bx np 2 1 2 p; n 0; 1; 2; ¼; H m1 bz bx L 2
For incident SH-waves and for long wavelengths, the wave passage effects may be considered approximately by representing the motion at the base as a sum of synchronous translation D0 eivt and rotation in horizontal plane f0 eivt
m 0; 1; 2; ¼ 17 For a horizontally layered plate, vn;m are found by solving the transcendental equation cotan kz;n H 2
Bn M An M
18
Solutions for different m are obtained by solving Eq. (18) for different branches of cotan kz;n H. If there is no material or radiation damping, the response of the model to driving motion at the base will be in®nite at v vn;m . If there is damping, the response at these frequencies will be ®nite but large. These frequencies can be determined experimentally by analysis of the peaks of the building transfer-function determined from recorded response to ambient noise or to earthquake ground shaking. The ®rst translational and torsional ®xed-base frequencies can be used to estimate the shear wave velocities of the homogeneous building model as follows. The ®rst translational mode corresponds to n 0 and m 0, and the ®rst torsional mode corresponds to n 1 and m 0. Let f0,0 and f1,0 be the frequencies in Hertz, respectively, for the ®rst translational and torsional modes. Then, for n 0 and
vs x; t < D0 eivt 1 f0 xeivt
20
21
This follows from the ®rst order Taylor series approximation of a monochromatic SH-wave, and is illustrated in Fig. 5. The dashed contour in this ®gure indicates the position of the building at rest and the solid contour indicates its position and deformation during the wave passage at some time t. Let the incident wave have unit amplitude, frequency v and propagate in the x-direction with phase velocity c vs x; t eiv t2x=c
22
For very long incident waves compared to the length of the building, v x/c is small, and e2i v=cx can be expanded in Taylor series about point x 0. The ®rst order Taylor series approximation is iv x e i vt 23 vs x; t 1 2 c and represents displacement that is a sum of unit amplitude translation and rotation about x 0 with amplitude 2iv /c. Comparison with Eq. (21) implies D0 1
24
and
f0 2
iv c
25
It is seen that the translation and rotation are delayed with
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Foundations of real structures are not rigid, and deform during the wave passage and as the wave energy goes in and out of the building [25]. The building response near the ®xed-base frequencies (i.e. the resonant part of the response) can be approximated by the simpli®ed rigid foundation model shown in Fig. 6. This response represents standing waves resulting from constructive interference of the incident waves with the waves re¯ected from the top of the building. For all other frequencies, and in general, assuming shear deformation of the structure only, and negligible foundation warping, solution for the building response ur x; z; t (Fig. 6) can be derived analogous to the one presented in Section 3.1 for incident SH-waves.
4. Results and analysis 4.1. Shear wave velocities for the VN7SH building models Fig. 6. Displacements of a building (as a shear beam) on a rigid base and excited by horizontal and vertical ground translations u and w and by ground rocking u0;x .
respect to each other by phase angle p/2. Synthesis of rotational accelerograms from translational accelerograms was studied by Lee and Trifunac [7,8] and the reader is referred to their work for further details on this topic. The building response to ground motion represented by Eq. (23) is the series in Eq. (13), with coef®cients Cnp as in Eq. (16), and coef®cients Cn of the Fourier series expansion of v0 x 1 2
iv x c
26
as in Eqs. (15) are [20] as follows. For n 0 C0 1 2
i vL 2 c
and for n . 0 8 > 0; < Cn 4 v L > ; : i L c np
27a
n even n odd
27b
The ®rst term of C0 is due to the ground translation, while the second term and all other coef®cients Cn, n . 0, are due to the ground torsion. Body P- and SV-waves and surface Rayleigh waves propagating in the y±z plane result in horizontal, u, vertical, w, and rocking, ux , components of ground motion (Figs. 3 and 6). A structure excited by these motions and on ¯exible soil will experience, through soil±structure interaction, additional relative motions at its base. Assuming the foundation is rigid, these additional relative motions are translations u0 and w0 and rocking u0;x (Fig. 6). The relative response of the structure, ur will depend on the resultant motion of the base, due to all of these motions.
To evaluate b z for the continuous models in Fig. 4, the equivalent shear modulus m eq and the equivalent density r eq need to be determined. It can be shown [20] that s fEc b2 bz 28 h2 rc where Ec is the Young's modulus of elasticity of concrete, f . 1 is a factor depending on the percentage of reinforcement in the columns, b is the dimension of a column in the direction of bending, h is the story height, and r c the density of concrete. For example, assuming b 50 cm, h 3.0 m, Ec 2.8 £ 10 5 kg/cm 2, r c 2500 kg/m 3 and f , 1.2 gives b z 611 m/s for ®xed-®xed columns. The effective length (h kLc) of the columns between two stories depends on their relative bending stiffness and that of the girders, and can be computed from charts published for unbraced frames, e.g. [9], in terms of c a and c b where (see Fig. 7)
c
s E c Ic =Lc s E g Ig =Lg
29
and Ic;g is the moment of inertia of a column/girder; Lc;g is the length of a column/girder, center to center from joints; and Ec;g is the modulus of elasticity of column/girder. For a typical story of the VN7SH building with a 50 cm wide strip of ¯oor slabs, centered on the columns acting as a `girder' to make a moment resistant frame (Fig. 7b), c a , c b ,32. The above values of c give k , 5, and b z 611/5 122 m/s. This value is an estimate representative of wave propagation between two adjacent ¯oors, i.e. of b zs for the layered model, but is approximately equal to b z for an equivalent homogeneous model because the slabs are very thin and do not affect much the vertical wave propagation in the building. The shear-wave velocity associated with horizontal wave
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These experimentally determined values for b z for an equivalent homogeneous model are consistent with bsz b z estimated from the characteristics of the columns. The experimentally determined ratios of horizontal to vertical shear wave velocities for the equivalent homogeneous model b x/b z # 1, even though the shear wave velocity in the slabs is very large (20 times larger than b z). This is also because the slabs are very thin compared with the interstory space and represent a very small fraction of the building height. From the above we conclude that for a representative homogeneous model for two-dimensional wave propagation in the VN7SH building in the x±z plane, we can choose bsz 112 m/s and b x/b z 0.55, and for a layered model, we can choose bsz 112 m/s, bsx =bsz 0.55, bhz 2000 m/s and bhx =bhz 1 (Fig. 4). 4.2. Theoretical dispersion curves for the VN7SH building
Fig. 7. (a) A typical column±slab group and (b) `working width' of a slab used to de®ne c a and c b in calculation of the effective length of the columns.
propagation through the ¯oor slabs is s s G E =r 2049 m=s bx rc 2 1 1 v c
30
Shear waves are propagated horizontally also through the soft layers, via deformation of the partition walls and shear walls, with a much smaller velocity. The average shear-wave velocities b x and b z for an equivalent homogeneous model can be estimated also from the experimentally determined ®rst translational and torsional frequencies from ambient vibration test data and Eqs. (19) and (20). We note that the condition in Eq. (19) is equivalent to the building height being equal to 1/4 of the wavelength of shear waves propagating vertically in the building. The ambient vibration surveys for this building [4,5], assuming a ®xed-base structure, imply f0,0 1.4 Hz for the ®rst NS mode, f0,0 1.1 Hz for the ®rst EW mode, and f1,0 1.6 Hz for the ®rst torsional mode. This gives for NS vibrations b z 112 m/s and b x/b z 0.55, and for EW vibrations b z 88 m/s and b x/b z 1, for H=L 0:5. In reality, these frequencies are higher (even for response in the linear range) because the measured frequencies are not the ®xed-base but the system frequencies of the soil±structure system.
Dispersion curves are plots of allowable phase velocities versus frequency. In this section, we present dispersion curves cx f v and cz f v for the four models in Fig. 4 drawn for parameters approximately equal to those estimated for the VN7SH building. The purpose is to understand how these curves look and vary depending on the model parameters, which will be useful for future work on damage detection. The dispersion characteristics of a building will depend on its state of damage, and changes in experimentally determined dispersion curves can indicate that damage has occurred. Therefore, understanding of the theoretical dispersion curves and their variability will be useful for development of new methods for damage detection and for selection of optimal sensor locations to detect the changes. For the homogeneous model, we chose L 40 m, H/ L 0.5, b z 100 m/s, and b x/b z 0.5 and 1. For the layered model, L 40 m, H/L 0.5, bsz 100 m/s, bsx =bsz 0.5 and 1, bhz =bsz 20, and bhx =bhz 1. The layered model has 14 layers. The thickness of the hard layers is h1 8 inches, h3 h5 h7 h9 h11 8:5 inches and h13 10 inches, and the thickness of the soft layers is h2 h4 h6 h8 h10 h12 96 inches, and h14 152 inches, where 1 inch 2.54 cm. The equations for the dispersion curves in terms of normalized phase velocities cx =b and cz =b and dimensionless frequency vL=b, where b bz for the homogeneous model and b bsz for the layered model follow from Eqs. (4), (5), (7) and (8) and are as follows. For the homogeneous plate model cx;n 1 vL ; bz np bz
n 0; 1; 2; ¼
" #21=2 cz;n np bx 2 12 ´ ; bz vL=bz bz
31a
n 0; 1; 2; ¼ 31b
Eq. (31a) implies that cx;n is always real, and Eq. (31b)
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Fig. 8. Horizontal (left) and vertical (right) phase velocities versus dimensionless frequency vL=pb for a homogeneous isotopic plate model of the VN7SH building.
implies that cz;n is real for vL=bz $ npbx =bz , is imaginary for vL=bz , npbx =bz , and cz;n ! 1 as vL=bz ! npbx =bz . For the layered plate model cx;n 1 vL ; bsz np bsz
n 0; 1; 2; ¼
" #21=2 csz;n np bsx 2 12 ´ ; bsz vL=bsz bsz " chz;n bhx h 12 bsz bz
np bhx bhz ´ ´ vL=bsz bhz bsz
32a
n 0; 1; 2; ¼ 32b
!2 #21=2 ;
n 0; 1; 2; ¼
vL=bsz
npbsx =bsz ,
32c that csz;n is real for vL=bsz , npbsx =bsz ,
$ is Eq. (32b) implies and csz;n ! 1 as imaginary for vL=bsz ! npbsx =bsz . Similarly, Eq. (32c) implies that chz;n is real for vL=bsz $ npbhx =bhz , is imaginary for vL=bsz , npbhx =bhz , and chz;n ! 1 as vL=bsz ! npbhx =bhz . Fig. 8 shows the dispersion curves for the isotropic model (b x/b z 1), and Fig. 9 shows the dispersion curves for an anisotropic model with b x/b z 0.5. The frequency in Hertz on the second scale for the x-axis corresponds to L 40 m and b z 100 m/s. It is seen from the plot on the left hand side that the dispersion curves for horizontal wave propagation are straight lines, with 908 slope for n 0 (cx;0 1) and progressively smaller slope as n increases. In the plot on
the right hand side, the real branches of the dispersion curves for vertical wave propagation are shown by solid lines, and the imaginary branches are shown by dashed lines. For given n, both branches have a common vertical asymptote at vL=bz npbx =bz . For n 0, cz bz and is always real. This implies that regardless of the frequency, there is always at least one mode with real vertical phase velocity. All the imaginary branches start from zero and end at the corresponding vertical asymptote. The real branches start from their vertical asymptote and approach unity for large frequencies (i.e. cz;n =bz ! 1 as vL=bsz ! 1) implying that for large frequency, cz;n < bz . The solid heavier lines in the same plot are dispersion curves associated with ®xed-base response and stress free boundary conditions at the top of the building (with no constraints in x). These are de®ned by cz;m 1 H vL ; bz m 1 p=2 L bz
m 0; 1; 2; ¼
33
The frequencies vn;m and associated phase velocities cz for the ®xed-base frequencies of the 2-D models can be obtained by reading the x- and y-coordinates of the intersection of the real branch of the curve for cz,n with the line for cz,m. A consequence of the asymptotic behavior of the real branches of cz,n is that for larger m, the corresponding translational and torsional modes will have very close frequencies and vertical phase velocities. This would make it dif®cult to extract these modes from experimental data.
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Fig. 9. Horizontal (left) and vertical (right) phase velocities versus dimensionless frequency vL=pbz for a homogeneous anisotropic plate model of the VN7SH building.
Fig. 10 shows dispersion curves for an isotropic layered model with b x/b z 1 and bh =bs 20, and Fig. 11 shows the dispersion curves for an anisotropic layered model, with bsx =bsz 0.5 and bhx =bhz 1 and bhz =bsz 20. The frequency in Hertz corresponds to L 40 m and bsz 100 m/s. The dispersion curves for horizontal wave propagation are the same as for the homogeneous models. The dispersion curves for vertical wave propagation in the soft layers are also the same as those for the homogeneous models, because we chose the shear wave velocities in the soft layers to be the same as those for the corresponding homogeneous model. It is seen that for the frequency range shown (0±12.5 Hz), cz =bsz in the hard layers is imaginary for all n except n 0. The dispersion curves for the homogeneous models would be useful for interpretation of building recordings at the base and at the top. Those for the soft layers of the layered models would be useful for interpretation of recordings at neighboring ¯oors, and those for the hard layers would hardly be useful for damage detection, but are included for completeness. 5. Summary and conclusions The wave propagation methods for estimation of building response are superior to the modal methods for early, high frequency impulsive excitations (here `early' refers to times shorter than it takes for a standing wave corresponding to
the ®rst mode of vibration to develop). Such excitations occur in the near-®eld of earthquake sources. The wave propagation methods also offer possibilities for development of new methods for damage detection in buildings. In this paper (Part I), we review and deliberate on several simple 2-D wave propagation building models (homogeneous anisotropic and horizontally layered anisotropic shear plates with soft and hard layers representing the inter-story space and the slabs) and analyze dispersion in these models. The ultimate aim is to interpret, via wave propagation methods, the recorded earthquake response of a seven-story reinforced concrete building, damaged by the 1994 Northridge earthquake. Ambient vibration experiments in the VN7SH building imply vertical shear-wave velocity b z 122 and 88 m/s and ratios of horizontal to vertical shear-wave velocity b x/b z 0.55 and 1, respectively, for NS and EW vibrations. These values were obtained from the requirement that the anisotropic homogeneous building model matches the frequencies of the ®rst translational and the ®rst torsional frequencies, determined from ambient vibration data based on the assumption that the apparent frequencies seen in the data approximate the ®xed-base frequencies. The shear wave velocity in the concrete slabs is estimated to be about 2000 m/s. For the layered model it is suggested that the shear wave velocities in the soft layer should be approximately the same as for the homogeneous model. The dispersion analysis showed that the horizontal phase
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Fig. 10. Horizontal (left), and vertical (center for the soft layers and right for the hard layers) phase velocities versus dimensionless frequency vL=pb s for a layered isotropic plate model of the VN7SH building.
Fig. 11. Horizontal (left), and vertical (center for the soft layers and right for the hard layers) phase velocities versus dimensionless frequency vL=pbsz for a layered isotropic plate model of the VN7SH building.
M.I. Todorovska et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 211±223
velocities in the models are always real (this is true only for models without vertical discontinuities; [13]). The vertical phase velocities are always real for n 0 and can be real or imaginary for higher n depending on frequency. They are imaginary for frequencies up to a critical frequency and real for frequencies higher than the critical frequency. At the critical frequency, the vertical phase velocities are in®nite, which corresponds to horizontal wave propagation. This critical frequency increases with increasing order of the mode and decreases with increasing shear-wave velocity. The eigenfunctions corresponding to real phase velocities (i.e. wavenumbers) are harmonic functions and are associated with propagation of wave energy, while those corresponding to imaginary phase velocities are hyperbolic (i.e. exponential) functions and are associated with quasi-static deformations of the building. The relative contributions of these modes will depend on the excitation. Understanding of the dispersion in building models will be useful in interpretation of recorded building response and in damage detection.
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