The study of the rocking response and overturning of electrical equipment subjected to near-source ground motions was motivated by concern about the proximity of power plants in major urban areas to active faults. The San Andreas fault runs 10 km west of San Francisco; much of Oakland, California, is within 10 km of the Hayward fault; and a large part of Greater Los Angeles lies over buried faults capable of generating large earthquakes. During strong ground shaking the sliding or rocking of a variety of rigid structures, such as electrical transformers and heavy equipment, can result in substantial damage (fig.1).

Fig. 1. Electrical equipment of Sylmar Converter Station damaged during 1971 San Fernando earthquake.– Steinbrugge Collection, PEER, University of California, Berkeley.

The study was conducted by Nicos Makris, Associate Professor, Department of Civil Engineering, University of California, Berkeley; Yiannis Roussos, Engineer, of T.Y. Lin International, San Francisco; and by Jian Zhang, Graduate Research Assistant in the Structural Engineering Mechanics and Materials division of the Department of Civil Engineering, University of California, Berkeley. The work was sponsored by a grant from Pacific Gas and Electric Co. to the Pacific Engineering Research (PEER) Center, University of California, Berkeley.

Early studies on the dynamic response of a rigid block supported on a base undergoing horizontal motion were presented by Housner (1963). The base acceleration was represented by a rectangular or a half-sine pulse and expressions were derived for the minimum acceleration required to overturn the block. An energy approach was used in an approximate analysis of the dynamics of a rigid block subjected to white noise excitation. Housner's discovery of a scale effect explained why the larger of two geometrically similar blocks can survive excitation, whereas the smaller block may topple.

Yim et al. (1980) adopted a probabilistic approach and conducted a numerical study using artificially generated ground motions to show that the rocking response of a block is sensitive to system parameters. The white-noise-type motions used by Yim et al. (1980) do not contain any coherent component, and the overturning of a block is the result of a rapid succession of small random impulses (areas under the spikes of the artificially generated high-frequency acceleration histories).

Experimental and analytical studies on the same problem (Aslam et al. 1980) have concluded that, in general, the rocking response of blocks subjected to earthquake motion is consistent with the conclusions derived from single-pulse excitations; however, when artificially generated motions were used, the rocking response showed high sensitivity to the system parameters.

While the early work of Yim et al. (1980) used artificially generated white-noise-type motions and the work of Spanos and Koh (1984) and Hogan (1989) used long-duration harmonic motions, this article refocuses on pulse-type motions which are found to dominate the kinematic characteristics of near-source ground motions (Campillo et al. 1989, Iwan and Chen 1994). What makes these motions particularly destructive to some structures is not their peak acceleration but the area under the long-duration acceleration pulse which represents the incremental velocity that the above ground mass has to reach (Anderson and Bertero 1986). Building on Housner's (1963) pioneering work, and referring to recent California earthquakes, the present study investigates the overturning potential of a near-source ground motion and presents an approximate method to evaluate whether a rigid block will overturn. It is concluded that the toppling of smaller blocks depends not only on the incremental ground velocity (area under the acceleration pulse), but also on the duration of the pulse; whereas the toppling of larger blocks tends to depend solely on the incremental ground velocity. Accordingly, a smaller block might overturn due to the high frequency fluctuations that override the long

In this study the model considered, shown in figure 2, can oscillate about the centers of rotation *O* and *O'*when it is set to rocking.

** Fig. 2. Schematic of rocking block**
Its center of gravity coincides with the geometric center, at distance

(1)

where is the ground acceleration normalized to the acceleration of gravity and is a quantity with units in rad/s. The larger the block (larger *R*), the smaller *p,* the measure of the dynamic characteristics of the block. For an electrical transformer *p* is nearly equal to 2 rad/s, and for a household brick *p* is nearly equal to 8 rad/s. Equation (1), which is well known in the literature, is valid for arbitrary values of the block angle a.

Near-source ground motions have distinguishable long-duration pulses. In some cases, coherent pulses are distinguishable not only in the displacement and velocity histories but also in the acceleration history, in which the peak acceleration reaches usually moderate values. In other cases, high spikes that resemble the traditional random-like motions can be seen, but the velocity and displacement histories reveal a coherent long-period pulse with some high-frequency fluctuations that override this.

Figure 3 (left) shows the fault-parallel components of the acceleration, velocity, and displacement histories of the June 18, 1992, Landers earthquake, recorded at the Lucerne Valley station (Iwan and Chen 1994).

The motion resulted in a forward displacement of about 1.8 m. The velocity history is remarkable for the coherent long-duration pulse responsible for most of the displacement; the acceleration history is crowded with high-frequency spikes. Figure 3 (right) plots the acceleration, velocity, and displacement histories of a type-A cycloidal pulse given by the following (Jacobsen and Ayre 1958; Makris 1997):

(2)

(3)

(4)

In constructing figure 3 (right), the values of *T*_{p} = 7.0 s and n_{p} = 0.5 m/s were used to approximate the duration and velocity amplitude of the main pulse. The figure indicates that a simple one-sine pulse can capture some of the kinematic characteristics of the motion recorded at the Lucerne Valley station. On the other hand, the resulting acceleration amplitude, a_{p} = w_{p}n_{p} / 2 is nearly equal to 0.045*g*, is one order of magnitude smaller than the recorded peak ground acceleration.

Figure 4 (left) shows the acceleration, velocity, and displacement histories of the fault-normal motions recorded at the El Centro Station, array #5, during the October 15, 1979, Imperial Valley earthquake.

This motion resulted in a forward-and-back pulse with a 3.2 sec duration. In this case, the coherent long-period pulse is distinguishable not only in the displacement and velocity records, but also in the acceleration record. Figure 4 (right) plots the acceleration, velocity, and displacement histories of a type-B cycloidal pulse (Makris 1997).

(5)

(6)

(7)

In constructing figure 4 (right) the values of *T*_{p} = 3.2 sec and n_{p} = 0.7 m/s were used as approximate values of the pulse period and velocity amplitude of the recorded motions (fig. 4 left).

Not all near-source records are forward or forward-and-back pulses. Figure 5 (left) portrays the fault-normal component of the acceleration, velocity, and displacement time histories recorded at the Sylmar station during the January 17, 1994, Northridge earthquake.

The ground displacement consists of two main long-period cycles, the first being the largest, the subsequent decaying. These long-period pulses are also distinguishable in the ground velocity history, where the amplitude of the positive pulses is larger than the amplitude of the negative pulses. Near-fault ground motions, where the displacement history exhibits one or more long-duration cycles, are approximated with type-C pulses. An *n*-cycle ground displacement is approximated with a type-C* _{n}* pulse that is defined as

(8)

(9)

(10)

In deriving these expressions it was required that the displacement and velocity be differentiable signals. The value of the phase angle, j is determined by requiring that the ground displacement at the end of the pulse be zero. A type-C* _{n}* pulse with frequency w

(11)

Equation (11) after evaluating the integral gives

(12)

The solution of the transcendental equation (12) gives the value of the phase angle j. As an example of a type-C_{1} pulse (n = 1), j = 0.0697p; whereas for a type-C_{2} pulse (n = 2), j = 0.0410p.

In a recent study (Makris and Roussos 1998), a plethora of spectra for the minimum acceleration amplitude of type-A, type-B, and type-C* _{n}* pulses that are needed to overturn a free-standing block were derived by solving equation (1). It was concluded that the minimum overturning acceleration amplitude,

where, b = 1/6 for a type-A and type-C* _{n}* pulse, and b =1/4 for a type-B pulse.

The simple expression of equation (13) is valuable to evaluate the overturning potential of a near-source ground motion that contains a distinct long-duration pulse as well as high-frequency fluctuations that override the long-duration pulse. The question that arises with such records is whether an electrical equipment structure will overturn due to the high-frequency spike or to the low-acceleration, low-frequency pulse. This issue was investigated in depth by Makris and Roussos (1998) who proposed a simple procedure that requires only hand calculations. The procedure involves the characteristics of various pulses detected in a given record, the slenderness, a, and the frequency parameter, *p*, of the block.

1. Locate the time, of the peak ground acceleration, *a*_{p}. For instance, in the Lucerne Valley record (fig. 3), the maximum acceleration, *a*_{p}, occurs at approximately t = 11.5s.

2. Focus closely on the velocity record near the time of the peak ground acceleration, *a*_{p}, and identify the type and period, *T*_{p}, of the local pulse that results in the peak ground acceleration. Estimate the velocity amplitude, n_{p,} of the local pulse. The estimated values of *T*_{p} and n_{p} should satisfy (the Lucerne Valley record, e.g., the local pulse is of type-*B*, and its period is approximately ).

3. Compute the minimum overturning acceleration *a* _{p0} = a_{g}[1 + b(w_{p} / p)] where b 1/6 for type-A or* *C* _{n}* pulses, and b = 1/4 for a type-B pulse.

4. Compute the ratio *a*_{p0} / a_{p}. This ratio gives the approximate level of the ground motion that will overturn a block with slenderness a and frequency parameter *p*. In the Lucerne Valley record, *a*_{p0} / *a*_{p} = 2.99.

5. In case the velocity or displacement history exhibits a distinguishable long-duration pulse, identify the velocity amplitude, n_{p}, and the duration, *T*_{p}, of this pulse. Then, compute the corresponding acceleration of this pulse as .

6. Compute the minimum overturning acceleration, *a*_{p0}, of this pulse, as in step 3.

7. Repeat step 4 using the value *a*_{p} estimated in step 5, and the value of a_{p0}, computed in step 6. If the ratio *a*_{p0} / *a*_{p} is larger than the ratio computed in step 4, the block overturns due to the short-duration pulse for the level of the ground motion computed in step 4. In contrast, if the ratio *a*_{p0} / *a*_{p} is smaller, the block overturns due to the long-duration pulse and for the level of the ground motion computed in step 7.

The Makris and Roussos study (1998) has shown that electrical transformers with approximate values of slenderness a is nearly equal to 20 degrees and frequency parameter *p* is nearly equal to 2 rad/sec will most likely overturn due to the short-duration pulse. Accordingly, only steps 1 through 4 are needed to estimate the level of the ground motion that will overturn a typical electrical transformer. Only very large objects, such as nuclear heat-exchange boilers, having a frequency parameter value of less than one (*p* less than or equal to 1), may overturn due to the presence of the long-duration pulses. Thus, steps 5 to 7 should also be included in the procedure.

The three parameters that control overturning of a free-standing electrical equipment structure under horizontal excitation are the normalized pulse acceleration, *a*_{p} / a_{g}, the frequency ratio, w_{p}, and the slenderness a. It was found that at the low-frequency limit (w_{p} / *p* < 3), the normalized overturning acceleration amplitude of the pulse, a_{p0} / a_{g}, is larger than one and increases linearly with w_{p} / *p*. For values of w_{p} > 3 the normalized overturning acceleration amplitude increases nonlinearly with w_{p} / *p*, exhibiting a stiffening effect. Accordingly, the static solution (West's formula; Milne 1885) is increasingly overconservative as w_{p} increases.

Near-source ground motions do not bear any exceptional overturning potential for electrical transformers. The toppling of smaller blocks is more sensitive to the peak ground acceleration, whereas the toppling of larger blocks tends to depend solely on the incremental ground velocity. Blocks as big as typical electrical transformers (*p* is nearly equal to 2) overturn due to short-duration, high-acceleration pulses that often override the main long-duration pulse that generates most of the ground velocity and ground displacement recorded near the source of strong ground motions. In contrast, larger objects such as nuclear heat-exchange boilers (*p* is nearly equal to 1) will overturn due to the long-duration pulse. For these larger objects, long-period ground motions may be particularly destructive.

Under realistic conditions, the rocking response of a rigid block is affected by additional factors such as the vertical component of the ground acceleration and the additional energy loss due to plastic deformations at the pivot points. The effects of these factors will be the subject of a future study. The presented approximate method, although restricted to horizontal seismic excitations, elucidates the rocking response of rigid blocks, which is found to be quite ordered and predictable.

This work is supported by the Pacific Gas and Electric Company under a grant to the Pacific Earthquake Engineering Research Center. The valuable input and comments of Dr. Norman Abrahamson, Mr. Eric Fujisaki, and Mr. Henry Ho are greatly appreciated. Professor Gregory Fenves's efforts to coordinate this task are also acknowledged.

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*by Nicos Makris, Associate Professor
Structural Engineering Mechanics and Materials
Civil & Environmental Engineering
University of California, Berkeley
Yiannis Roussos, Engineer
T. Y. Lin, International, San Francisco*