Chapter 1 Introduction
3.7 Kinetic Energy in a Half-Space and Energy Radiation EstimateRadiation Estimate
3.7 Kinetic Energy in a Half-Space and Energy
the integral (N À1). Furthermore, in order to simplify the integration, the integral over the volume where only Rayleigh waves exist, will be performed in cylindrical coordinates to simplify the integral forms involved.
R P+R P+S+R
P
P+S Source
Figure 3.10 Leading displacement field wavefronts for the dif- ferent radiated waves; Compressional (P), Shear (S), and Rayleigh (R) waves.
Therefore, the energy integral can be separated into multiple integrals over the dif- ferent components. In the following formulation, the first subscript after the velocity component represents the component being integrated, the subscripts in parentheses indicate which waves are contributing to the integral for the integrated region, and the number subscript corresponds to a region shown in Figure 3.11.
EAvg = ρ 2NT
· Z N T
T=0
Z 2π
φ=0
Z VPN T
R=0
Z θP
θ=0
VR(P2 +R)(1) R2sinθ dθ dR dφ dt + Z N T
T=0
Z 2π
φ=0
Z VRNT
sinθ
R=0
Z π
2
θ=θP
VR(P2 +R)(2) R2sinθ dθ dR dφ dt + Z N T
T=0
Z 2π
φ=0
Z VPN T
R=0
Z π
2
θ=θP
VR(P2 )(2+3) R2sinθ dθ dR dφ dt − Z N T
T=0
Z 2π
φ=0
Z VRNT sinθ
R=0
Z π
2
θ=θP
VR(P2 )(2) R2sinθ dθ dR dφ dt +
1
Source
2
3
4
9 5
6 7
8 A) Radial Component, U
B) Tangential Component, U
C) Tangential Component, Uφ θ R
θP
θS
Figure 3.11 Integral partitions over the half-space for the indi- vidual displacement components. The numbers in the differ- ent regions shown are used to identify which integral in Equa- tion 3.77 corresponds to a particular region of integration for the different displacement field components. A) Radial Com- ponent (UR), B) Tangential Component (Uθ), and C) Tangen- tial Component (Uφ).
Z NT
T=0
Z 2π
φ=0
Z ∞
z=√
(VPN T)2−r2
Z VRN T
r=0
(Vrα+Vzα)2(R)(4) r dr dz dφ dt + Z N T
T=0
Z 2π
φ=0
Z VSN T
R=0
Z θS
θ=0
Vθ(S+R)(5)2 R2sinθ dθ dR dφ dt + (3.77) Z N T
T=0
Z 2π
φ=0
Z VRNT
sinθ
R=0
Z π
2
θ=θS
Vθ(S+R)(6)2 R2sinθ dθ dR dφ dt + Z N T
T=0
Z 2π
φ=0
Z VSN T
R=0
Z π
2
θ=θS
Vθ(S)(6+7)2 R2sinθ dθ dR dφ dt − Z N T
T=0
Z 2π
φ=0
Z VRNT sinθ
R=0
Z π
2
θ=θS
Vθ(S)(6)2 R2sinθ dθ dR dφ dt + Z N T
T=0
Z 2π
φ=0
Z ∞
z=√
(VSN T)2−r2
Z VRN T
r=0
(Vrβ +Vzβ)2(R)(8) r dr dz dφ dt + Z N T
T=0
Z 2π
φ=0
Z VSN T
R=0
Z π
2
θ=0
Vφ(S)(9)2 R2sinθ dθ dR dφ dt
¸
where
θP =sin−1
³VR VP
´
θS =sin−1
³VR VS
´
(3.78)
For the time integrals (over N oscillation cycles), the integrals involved are of the following form,
Z N T
0
sin2(ωt−α)dt= Z N T
0
cos2(ωt−α)dt = NT
2 (3.79)
Z N T
0
sin(ωt−α+π
4) sin(ωt−α− π
4)dt= NT
2 (3.80)
Z N T
0
sin(ωt) cos(ωt)dt= 0 (3.81)
and can be performed first to simplify the results. Similarly, the integrals with respect to azimuth (φ, the angle along the surface) can be subsequently performed by
Z 2π
0
sin2φ dφ= Z 2π
0
cos2φ dφ=π (3.82)
Substitution of the results from Equations 3.79 and 3.82 into Equation 3.77 yields EAvg = ρπ
4
· Z VPN T
R=0
Z θP
θ=0
VR(P+R)(1)2 R2sinθ dθ dR + Z VRNT
sinθ
R=0
Z π
2
θ=θP
VR(P2 +R)(2) R2sinθ dθ dR + Z VPN T
R=0
Z π
2
θ=θP
VR(P2 )(2+3) R2sinθ dθ dR − Z VRNT
sinθ
R=0
Z π
2
θ=θP
VR(P2 )(2) R2sinθ dθ dR + Z ∞
z=√
(VPN T)2−r2
Z VRN T
r=0
(Vrα +Vzα)2(R)(4) r dr dz + Z VSN T
R=0
Z θS
θ=0
Vθ(S+R)(5)2 R2sinθ dθ dR + (3.83) Z VRNT
sinθ
R=0
Z π
2
θ=θS
Vθ(S+R)(6)2 R2sinθ dθ dR + Z VSN T
R=0
Z π
2
θ=θS
Vθ(S)(6+7)2 R2sinθ dθ dR − Z VRNT
sinθ
R=0
Z π
2
θ=θS
Vθ(S)(6)2 R2sinθ dθ dR + Z ∞
z=√
(VSN T)2−r2
Z VRN T
r=0
(Vrβ +Vzβ)2(R)(8) r dr dz +
Z VSN T
R=0
Z 2
θ=0
Vφ(S)(9)2 R2sinθ dθ dR
Due to the significant number of integrals involved and the large number of terms to be integrated, instead of presenting all of the integrals, I will only show the types of distance integrals involved. The integrals overθ will not be shown, as they need to be computed numerically (Appendix E) and the integrands are long and complicated.
The integrals dependent only on body waves (VR(P2 ), Vθ(S)2 , Vφ(S)2 ) involve integrals of the form
Z b
a
1
R2 R2dR = R|ba (3.84)
Integrals involving only Rayleigh waves (VR(R)2 ,Vθ(R)2 ) include integrals of the form Z b
a
1
rrdr = r|ba (3.85)
Z b
a
e−2αzdz = −e−2αz
2α |ba (3.86)
The more complicated of the forms includes a mix of a body wave (P or S) and a Rayleigh wave. These integrals include 3 types of integrals, namely
Z b
a
1
R2 R2dR=R|ba (3.87)
Z b
a
e−2αRcosθ
R R2dR=−e−2α1Rcosθ
4α21cos2θ(2α1Rcosθ+ 1) (3.88) Z b
a
e−αRcosθcos(hR−Rk0sinθ)
R32 R2dR (3.89)
Integrals 3.88 and 3.89 behave like near-field terms, as the integrands decay expo- nentially with distance. As a result, they are only important near the source, as the integral’s contribution with distance quickly diminishes to zero, causing the integral to reach a constant value. The same is true for 3.86, except that the limit closer
to the source is at the P wavefront, and therefore this integrand never takes on any large values. I will ignore these terms, as they can be neglected in the far-field (large N) and due to their constant values, when dividing the total average kinetic energy (EAvg) by the number of cycles of oscillation, the contribution from the integral is negligible.
I calculate the moment,M, the building induces on the soil and the shearing force, P, on the soil’s surface for each of the building’s natural frequencies in section 3.6 as
PEW ≈1.66E5N PN S ≈3.84E5N MEW ≈5.37E6Nm MN S ≈1.21E7Nm
After finding the zeros of F(ζ) for the soil properties (Λ≈ 1.89) given in Tables 3.1 (model 1) and 3.2 (model 2), we can achieve an estimate of the building’s damping.
From the zeros of F(ζ), it is found that
VR = 0.9282VS
Therefore, the integrals in Equation 3.83 can be solved as shown in Appendix E.4, where the appropriate values for a half-space have been used. However, the appendix only calculates the kinetic energy, and therefore the values given there must be mul- tiplied by two to include the potential energy (Achenbach, 1993). For model 1 the integrals give that the energy radiated per cycle is
EEW1 = 0.366J EN S1 = 3.984J and for model 2
EEW2 = 0.204J EN S2 = 2.224J
As can be seen from the detailed results presented in Appendix E.4, the generated shear body waves are the largest contributor to the radiated kinetic energy, and the
Rayleigh waves have a negligible contribution. From the radiated energy per cycle, the radiated power is estimated to be
ERad(EW1) = 0.408W atts ERad(N S1) = 6.526W atts
(3.90) ERad(EW2) = 0.228W atts ERad(N S2) = 3.642W atts
From Section 3.6, we know that the building’s kinetic energy per cycle is
EKin(EW)= 45.7J EKin(N S)= 106.1J (3.91)
In order to compute the damping due to energy being radiated away from the building, we have to compare the radiated energy with the building’s kinetic energy.
Using Equation 3.72 in section 3.6 and following the derivation of the logarithmic decrement of damping derived inHousner and Hudson (1980), the building’s damping can be calculated using the kinetic energy of two successive cycles in free vibration.
For the first cycle, the maximum kinetic energy is given by EKin1 = 1
2mω2x2 (3.92)
and for the second cycle,
EKin2 = 1
2mω2(x−∆x)2 (3.93)
where x is the maximum displacement of the first cycle and ∆x is the change in maximum displacement between the two cycles. If we calculate the change in kinetic energy (∆EKin) and divide it by the total initial kinetic energy (EKin from Equation 3.91), and furthermore assume that ∆x is small with respect to x, we get
∆EKin EKin
≈2∆x
x ≈2δ (3.94)
where δ is commonly referred to as the logarithmic decrement of damping. Further- more, δ can be related to the viscous damping ratio(ξ) by
ξ ≈ δ
2π ≈ ∆EKin
4πEKin (3.95)
In order to calculate the percentage of damping attributed to the radiated energy per cycle, substitute the computed radiated kinetic energy per cycle for ∆Ekin.
∆EKin(EW1) = 0.366J ∆EKin(N S1) = 3.984J
∆EKin(EW2) = 0.204J ∆EKin(N S2) = 2.224J
Using the values in Equation 3.91, the radiated kinetic energy damping ratios are estimated to be
ξEW1 = 0.06% ξN S1 = 0.29%
(3.96) ξEW2 = 0.04% ξN S2 = 0.17%
The observed damping ratios for Millikan Library are computed by fitting a decaying exponential to the peaks of the free amplitude decay of the building displacements, which yields the following damping values
ξEW = 1.63% ξN S = 1.65% (3.97)
As can be seen, the observed damping ratios for Millikan Library are much larger than the estimated radiated kinetic energy values computed here, and are also in general agreement with the other experimental values given in table A:7 (Bradford et al. (2004)). Therefore, it may be concluded that the half-space model applied here is not a proper mathematical model, or that processes within the structure are dissipating most of the energy input into the building. The half-space model fails to account for resonances in the soil layers, and therefore an alternative should
be explored. Multi-layer cases will be examined in Chapter 5 by utilizing a Finite Element Model, and it will be shown that the damping values estimated here for the half-space model are in general agreement with those of the Finite Element Models.
Chapter 4
Surface Wave Modelling
For the experiments performed in 1998, the field GPS units provided by SCEC were utilized to obtain precise locations of the portable seismometers. Seismometer sites were chosen throughout Pasadena at locations volunteered by members of the Cal- tech community, who kindly offered a place in their residence to install a temporary seismometer. As a result, a fairly random distribution of sites was achieved, however, with a concentration in the NW part of Caltech. At the time, it was believed that wavelengths in the order of 600 to 700 meters would be observed, and that errors in the GPS measurements (in the order of few meters) would not have a large influence on the data analysis. After collecting and processing the data, it was found that the observed wavelengths were most likely between 500 and 600 meters, which made the errors from the portable GPS units slightly more important. Furthermore, the measurements did not show the anticipated surface wave like behavior.
It was determined that the possibility existed that significant errors were intro- duced by inaccuracies in the instrument locations, and as a result, a continuous GPS campaign was performed to confirm and/or correct the seismometer positions. This campaign was carried out in 1999 with the assistance of Mr. Jeff Behr at the USGS who helped determine, through an experimental process, that the necessary accuracy of tens of centimeters could be achieved by continuously recording GPS data for as little as 30 minutes, and furthermore showed me how to process the collected data.
However, after repeating the data analysis with the improved seismometer locations, the same inconsistencies in the data remained. As a result, a second set of experi-
ments (which will be referred to as Millikan II) was designed and carried out in the year 2000. The seismometer locations for this test are shown in Figure 4.1. These ex- periments were designed to explore both the radial and azimuthal radiation patterns close to the building, and all the seismometers were located within the confines of the Caltech campus. The experiments conducted prior to the Millikan II experiments will not be mentioned in the remainder of this thesis and their data will be analyzed at a later date, utilizing the tools and knowledge developed in this thesis.
Millikan II (Caltech) Station Map
Latitudinal Distance From Millikan Library (meters)
Longitudinal Distance From Millikan Library (meters)
North line
NNE line NNW
line NE line
ENE line
East line NE quar
ter cirle
SW quarter cirle Millikan Library
Figure 4.1 Seismometer locations for the Millikan II experi- ments. The seismometers were set out in different lines and semi-circles to collect data during 6 different set-ups, while the same experiments were replicated for the different seismometer set-ups. Millikan is shown as a square at the origin, while the seismometers that functioned properly during the experiments are depicted as circles.
This chapter describes the procedure and results from performing a linear fit to both the phase and amplitude data collected during the Millikan II experiments.
The linear fit performed in this chapter supposes that the building excites dominant seismic wave phases, as it was expected that the surface waves would dominate the generated wave field.