seismic analysis of structures

César Osses Espinoza

seismic analysis of structures

César Osses Espinoza

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Abstract
AI

The paper presents an in-depth analysis of the seismic behavior of structures, focusing on the fundamental principles of plate tectonics as a precursor to understanding earthquake mechanics. It discusses the mechanisms of seismic wave propagation, interplate and intraplate earthquake occurrences, and the relevant factors influencing structural response during seismic events. Through a mix of theoretical modeling and empirical data, the research highlights the need for effective damping strategies and response analysis to improve earthquake resilience in engineered structures.

Figures (449)

Figure 1.1 Inside the earth (Source: Murty, C.V.R. “IITK-BMPTC Earthquake Tips.” Public domain, National Information Centre of Earthquake Engineering. 2005. http://nicee.org/EQTips.php - accessed April 16, 2009.) the inner core, the outer core, the mantle, and the crust, as shown in Figure 1.1. The upper-most layer, called the crust, is of varying thickness, from 5 to 40 km. The discontinuity between the crust and the next layer, the mantle, was first discovered by Mohorovi¢i¢ through observing a sharp change in the velocity of seismic waves passing from the mantle to the crust. This discontinuity is thus known as the Mohorovici¢ discontinuity (““M discontinuity”). The average seismic wave velocity (P wave) within the crust ranges from 4 to 8kms'. The oceanic crust is relatively thin (S—15 km), while the crust beneath mountains is relatively thick. This observation also demonstrates the principle of isostasy, which states that the crust is floating on the mantle. Based on this principle, the mantle is considered to consist of an upper layer that is fairly rigid, as the crust is. The upper layer along with the crust, of thickness ~120 km, is known as the lithosphere. Immediately below this is a zone called the asthenosphere, which extends for another 200 km. This zone is thought to be of molten rock and is highly plastic in character. The asthenosphere is only a small fraction of the total thickness of the mantle (~2900 km), but because of its plastic character it supports the lithosphere floating above it. Towards the bottom of the mantle (1000-2900 km), the variation of the seismic wave velocity is much less, indicating that the mass there is nearly homogeneous. The floating lithosphere does not move as a single unit but as a cluster of a number of plates of various sizes. The movement in the various plates is different both in magnitude and direction. This differential movement of the plates provides the basis of the foundation of the theory of tectonic earthquake. Below the mantle is the central core. Wichert [1] first suggested the presence of the central core. Later,

Figure 1.3. Major tectonic plates on the earth’s surface (Source: Murty, C.V.R. “IITK-BMPTC Earthquake Tips.” Public domain, National Information Centre of Earthquake Engineering. 2005. http://nicee.org/EQTips.php - accessed April 16, 2009.) The continental motions are associated with a variety of circulation patterns. As a result, the continental motion does not take place as one unit, rather it occurs through the sliding of the lithosphere in pieces, called tectonic plates. There are seven such major tectonic plates, as shown in Figure 1.3, and many smaller ones. They move in different directions and at different speeds. The tectonic plates pass each other

of continental drift. At mid-oceanic ridges, two large land masses (continents) are initially joined together. They drift apart because of the flow of hot mantle upwards to the surface of the earth at the ridges due to convective circulation of the earth’s mantle, as shown in Figure 1.2. The energy of the convective flow is derived from the radioactivity inside the earth. As the material reaches the surface and cools, it forms an additional crust on the lithosphere floating on the asthenosphere. Eventually, the newly formed crust spreads outwards because of the continuous upwelling of molten rock. The new crust sinks beneath the surface of the sea as it cools down and the outwards spreading continues. These phenomena gave rise to the concept of sea-floor spreading. The spreading continues until the lithosphere reaches a deep-sea trench where it plunges downwards into the asthenosphere (subduction). Figure 1.2. Local convective currents in the mantle (Source: Murty, C.V.R. “IITK-BMPTC Earthquake Tips.” Public domain, National Information Centre of Earthquake Engineering. 2005. http://nicee.org/EQTips.php - accessed April 16, 2009.)

Figure 1.5 Types of fault (Source: Murty, C.V.R. “IITK-BMPTC Earthquake Tips.” Public domain, National Information Centre of Earthquake Engineering. 2005. http://nicee.org/EQTips.php - accessed April 16, 2009.) The faults at the plate boundaries are the most likely locations for earthquakes to occur. These earthquakes are termed interplate earthquakes. A number of earthquakes also occur within the plate away from the faults. These earthquakes are known as intraplate earthquakes, in which a sudden release of energy takes place due to the mutual slip of the rock beds. This slip creates new faults called earthquake faults. However, faults are mainly the causes rather than the results of earthquakes. These faults, which have been undergoing deformation for the past several thousands years and will continue to do so in future, are termed active faults. At the faults (new or old), two different types of slippages are observed, namely: dip slip and strike slip. Dip slip takes place in the vertical direction while strike slip takes place in the horizontal direction, as shown in Figure 1.5.

Figure 1.4 Types of interplate boundaries (Source: Murty, C.V.R. “IITK-BMPTC Earthquake Tips.” Public domain, National Information Centre of Earthquake Engineering. 2005. http://nicee.org/EQTips.php - accessed April 16, 2009.)

Figure 1.7 Earthquake mechanism: (a) before slip; (b) rebound due to slip; (c) push and pull force; and (d) double couple

Figure 1.8 Motion caused by body and surface waves (Source: Murty, C.V.R. “IITK-BMPTC Earthquake Tips.” Public domain, National Information Centre of Earthquake Engineering. 2005S. http://nicee.org/EQTips.php - accessed April 16, 2009.)

Figure 1.10 Typical strong motion record

Figure 1.9 Reflections at the earth’s surface

Figure 1.11 Practically single shock: (a) acceleration; (b) velocity; and (c) displacement

Figure 1.12 Records with mixed frequency: (a) acceleration; (b) velocity; and (c) displacement

Figure 1.13 Records with a predominant frequency: (a) acceleration; (b) velocity; and (c) displacement

focus with a focal depth of less than 70 km. Depths of foci greater than 70 km are classified as intermediate or deep, depending on their distances. Distances from the focus and the epicenter to the point of observed ground motion are called the focal distance and epicentral distance, respectively. The limited region of the earth that is influenced by the focus of earthquake is called the focal region. The larger the earthquake, the greater is the focal region. Foreshocks are defined as those which occur before the earthquake (that is, the main shock). Similarly, aftershocks are those which occur after the main shock.

Figure 1.15 Comparison of moment magnitude scale with other magnitude scales OF wee Ee ae A AE As shown in Figure 1.15, all the above magnitude scales saturate for large earthquakes. It is apparent that M, begins to saturate at M, = 5.5 and fully saturates at 6.0. Mg does not saturate until approximately Ms =7.25 and is fully saturated at 8.0. My begins to saturate at about 6.5. Because of this saturation, it is difficult to relate one type of magnitude with another at values greater than 6. Up to a value of 6, it may be generally considered that My = My = M, = Ms = M. Beyond the value of 6, it is better to specify the type of magnitude. However, in earthquake engineering many alternation and empirical relationships are used without specific mention of the type of magnitude. The magnitude is simply denoted by M (or m). In such cases, no specific type should be attached to the magnitude. It is desirable to have a magnitude measure that does not suffer from this saturation.

Table 1.1 Frequency of occurrence of earthquakes (based on observations since 1900) 1.3.5 Energy Release Newmark and Rosenblueth [9] compared the energy released by an earthquake with that of a nuclear explosion. A nuclear explosion of 1 megaton releases 5 x 10'°J. According to Equation 1.12, an earthquake of magnitude Ms = 7.3 would release the equivalent amount of energy as a nuclear explosion of 50 megatons. A simple calculation shows that an earthquake of magnitude 7.2 produces ten times more ground motion than a magnitude 6.2 earthquake, but releases about 32 times more energy. The E of a magnitude 8.0 earthquake will be nearly 1000 times the E of a magnitude 6.0 earthquake. This explains why big earthquakes are so much more devastating than small ones. The amplitude numbers are easier to explain, and are more often used in the literature, but it is the energy that does the damage. The length of the earthquake fault L in kilometers is related to the magnitude [10] by

Table 1.2 Modified Mercalli intensity (MMI) scale (abbreviated version) Although the subjective measure of earthquake seems undesirable, subjective intensity scales have played important roles in measuring earthquakes throughout history and in areas where no strong motion instruments are installed. There have been attempts to correlate the intensity of earthquakes with instrumentally measured ground motion from observed data and the magnitude of an earthquake. An empirical relationship between the intensity and magnitude of an earthquake, as proposed by Gutenberg and Richter [6], is given as The last one, which has ten grades, 1s still used 1n some parts of Europe. The Mercalli-Cancani—Sieberg scale, developed from the Mercalli (1902) and Cancani (1904) scales, is still widely used in Western Europe. A modified Mercalli scale having 12 grades (a modification of the Mercalli-Cancani—Sieberg scale proposed by Neuman, 1931) is now widely used in most parts of the world. The 12-grade Medved— Sponheuer—Karnik (MSK) scale (1964) was an attempt to unify intensity scales internationally and with the 8-grade intensity scale used in Japan. The subjective nature of the modified Mercalli scale is depicted in Table 1.2.

[he instrument that measures the ground motion is called a seismograph and has three components, 1amely, the sensor, the recorder, and the timer. The principle of operation of the instrument is simple. Right from the earliest seismograph to the most modern one, the principle of operation is based on the »scillation of a pendulum subjected to the motion of the support. Figure 1.16 shows a pen attached to the ip of an oscillating pendulum. The support is in firm contact with the ground. Any horizontal motion of he ground will cause the same motion of the support and the drum. Because of the inertia of the bob n which the pen is attached, a relative motion of the bob with respect to the support will take place. [he relative motion of the bob can be controlled by providing damping with the aid of a magnet around the string. The trace of this relative motion can be plotted against time if the drum is rotated at a sonstant speed. Figure 1.16 Schematic diagram of an early seismograph (Source: Murty, C.V.R. “IITK-BMPTC Earthquake Tips.” Public domain, National Information Centre of Earthquake Engineering. 2005. http://nicee.org/EQTips.php - accessed April 16, 2009.)

Figure 1.17 Representation of the principle of a seismograph Amplification of the pendulum displacement can be achieved by mechanical, optical or electromag- netic means. Through optical means, amplifications of several thousand-fold and through electromagnetic means, amplifications of several millionfold are possible. Electromagnetic or fluid dampers may be used to dampen the motion of the system. The pendulum mass, string, magnet, and support together constitute

Figure 1.18 Illustration of the torsion pendulum of a Wood—Anderson seismograph COUMStAall SPCeG ty UO UIeT. CALL LY DOS OL SUISINOURT ADS HdVe Use UNECE COMIMOMEHUS. The more commonly used seismographs fall into three groups, namely, direct coupled (mechanical), moving coil, and variable reluctance. The last two use a moving coil galvanometer for greater magnification of the output from the seismographs. In the direct-coupled seismographs, the output is coupled directly to the recorder using a mechanical or optical lever arrangement. The Wood—Anderson seismograph, developed in 1925, belongs to this category. A schematic diagram of the Wood—Anderson seismograph is shown in Figure 1.18. A small copper cylinder of 2 mm diameter and 2.5 cm length having a weight of 0.7g is attached eccentrically to a taut wire. The wire is made of tungsten and is 1/50 mm in diameter. The eccentrically placed mass on the taut wire constitutes a torsion pendulum. A small plane mirror is fixed to the cylinder, which reflects the beam from a light source. By means of double reflection of the light beam from the mirror, magnification of up to 2800 can be achieved. Electromagnetic damping (0.8 of the critical) of the eddy current type is provided by placing the copper mass in a magnetic field of a permanent horseshoe magnet. The characteristics of ground motions near and far from the epicentral distances are different. In near sarthquakes, waves with a period less than 0.1s and a very large amplitude are present. For far sarthquakes, periods up to 40s may be commonly encountered with very low amplitude waves. Commonly used seismographs can record earthquake waves within a 0.5—-30s range. Strona motion seigcmocranhe (accelernoranhsc) are weed for meacnringa oronnd motions for far

Table 1.3 PGA at the site for different sources

A site is surrounded by three independent sources of earthquakes, out of which one is a line source, as shown in Figure 1.19. Locations of the sources with respect to the site are also shown in the figure. The maximum magnitudes of earthquakes that have occurred in the past for the sources are recorded as: source 1, 7.5; source 2, 6.8; source 3, 5.0. Using deterministic seismic hazard analysis compute the peak ground acceleration to be experienced at the site.

Figure 1.20 Sources of earthquake near the site (PSHA) The site under consideration is shown in Figure 1.20 with three potential earthquake source zones. Seismicity of the source zones and other characteristics are also given. Using the same predictive relationship as that considered for DSHA with o;,pga = 0.57, obtain the seismic hazard curve for the site (Table 1.4). Table 1.4 Seismicity of the source zones Example 1.2

Figure 1.21 Histogram of r for source zone 1

Figure 1.22 Histogram of r for source zone 2 For each source zone,

Figure 1.23 Histogram of r for source zone 3

Figure 1.24 Histogram of magnitude for source zone 1

Figure 1.25 Histogram of magnitude for source zone 2

Figure 1.26 Histogram of magnitude for source zone 3 Thus, the above probability is given as

Figure 1.28 A site affected by three line sources of earthquakes

Figure 1.29 A site affected by a point source and an area source of earthquake

Figure 1.30 Characteristics of the sources: (a) distribution of source to site distance for source 2; (b) distribution of the magnitude of earthquake for source 1; and (c) distribution of the magnitude of earthquake for source 2

Figure 1.31 Area to be microzoned using both DSHA and PSHA

Figure 1.32 Area to be microzoned using PSHA

The amplitudes ao, a,, and 6, are given by:

Figure 2.3. Fourier amplitude spectrum: (a) narrow band; and (b) broad band

Figure 2.5 Discrete values of time history sampled at 0.02 s

Figure 2.4 Smooth Fourier amplitude spectrum in log scale

Figure 2.6 Outputs from the FFT: (a) real part; and (b) imaginary part

Figure 2.7 Fourier spectra: (a) amplitude spectrum; and (b) phase spectrum

Figure 2.8 Power spectral density function of ground acceleration The source of the above expression is discussed in Chapter 5. A related parameter that characterizes the frequency content of ground acceleration is the predominant period or frequency. It is defined as the time period where the Fourier spectrum peaks or the frequency where the PSDF peaks.

Figure 2.9 Variation of rms value of acceleration with time for uniformly modulated non-stationary process Records of actual strong motion time history show that modeling of an earthquake process as a stationary random process is not well justified as the ensemble mean square value varies with time. The mean square value gradually increases to a peak value, remains uniform over some time, and then decreases as shown in Figure 2.9. Such type of behavior is often modeled as a uniformly modulated non- stationary process [3]. Such a process is represented by an evolutionary power spectral density function. The evolutionary spectrum is obtained by multiplying a constant spectrum with a modulating function of time ¢ and is given as:

Figure 2.10 Raw PSDF of the time history of acceleration a large number of ordinates, higher point averaging is possible. The resulting PSDF becomes smoother. If necessary, a best-fit polynomial can be drawn as shown in Figure 2.11 to obtain a very smooth spectrum. For this problem:

Figure 2.12 Response spectra of an earthquake: (a) displacement spectrum; (b) pseudo velocity spectrum; and (c) pseudo acceleration spectrum VUE TEIGALTITIUIIT AUER IelatluUil ALIVE TIGA Bat @O ALUN UI UI OL BAe RT Se Fo These three spectra, as shown in Figures 2.12(a—c), provide directly some physically meaningful quantities. The displacement spectrum is directly related to the peak deformation of the SDOF system. The pseudo velocity spectrum is directly related to the peak strain energy stored in the system during an earthquake, Equation 2.25a. The pseudo acceleration spectrum is related directly to a peak value of the earthquake force in the system, Equation 2.27. Equation 2.27 shows that spectral acceleration multiplied by the mass of the SDOF gives the maximum earthquake force induced in the system by the earthquake. Because of this reason, spectral acceleration has been defined in a way as shown by Equation 2.26 and is different to the maximum acceleration of the SDOF system. While a displacement response spectrum is a plot of the maximum displacement of the SDOF system as a function of frequency, pseudo acceleration and pseudo velocity spectra are not the plots of maximum acceleration and maximum velocity as a function of the SDOF frequency. ee Se ee ee ee a a | re ye eC ee a a Lee ee Seen Lee | ee eee |

Figure 2.14 Fourier amplitude spectrum of El Centro earthquake 2.5.3. Combined D-V-A Spectrum Each of the three spectra (displacement, pseudo velocity, and pseudo acceleration) in a way provides the same information on structural response. However, each one of them provides a physically meaningful quantity, as mentioned earlier. Therefore, all three spectra are useful in understanding the nature of an earthquake and earthquake resistant design. More importantly, all three spectra are skillfully used in defining or constructing the design response spectrum, which will be discussed later. Because of this reason, a combined plot showing all three of the spectral quantities is desirable. This integrated presentation is possible because of the relationship that exists between these three quantities. Taking the log of Equations 2.21 and 2.26:

Figure 2.13 Energy spectrum of El Centro earthquake

Figure 2.15 Pseudo acceleration spectrum of El Centro earthquake

Figure 2.16 Earthquake response spectrum (displacement, pseudo velocity, and pseudo acceleration) in tripartite plo (— for 0% and ---- for 5%)

Figure 2.17 Idealized response spectrum by a series of straight lines for El Centro earthquake

Table 2.1 Comparison of periods (T,,— T;) between Parkfield and El Centro earthquakes

Figure 2.18 Comparison between the response spectra of Parkfield and El Centro earthquakes

Figure 2.19 Construction of elastic design spectrum The values of c; and cz are determined from the recorded earthquake data. Typical values of c; and Cy may be taken as c, = 1.22 to0.92 ms"! and c) = 6. 3. On the four way log graph paper, plot the baseline showing ig max, Ug max @Nd Mg max aS Shown in Figure 2.19. Multiply tigmax, Ugmax, ANd Ugmax by the amplification factors w,, %p, and ay, respectively, to obtain the lines bc, de, and cd.

Figure 2.20 Design (pseudo acceleration) spectrum as given in IS Code

Figure 2.21 Design spectrum defined as an envelope of two design spectra Example 2.5

Figure 2.22 Design response spectrums for 50th and 84.1th percentile values

Figure 2.23. Deconvolution and convolution of ground motion to take into account local site conditions EI Site specific spectra are different from design spectra that are used for the general purpose design o: structures and are recommended in the codes of practices. Site specific spectra are those which are exclusively used for the design of structures for that site. These structures are generally specialty structures and require careful consideration of the site conditions at the time of design. Alternatively special geological, geophysical, and geotechnical conditions may exist at the site, which distinguish the past earthquake records for the site as being distinctly different to other earthquake records of the region For such sites, site specific spectra are obtained from the family of accelerograms recorded near to the site If required, they are supplemented by using recorded accelerograms of other sites having similat geological, seismological, and geophysical conditions. All accelerograms are scaled in order to adjus' them for single magnitude and single source to site distance of an earthquake. They are also scaled tc adjust for the specific site being represented. The scaling is accomplished by Fourier transforming eact accelerogram. The real and imaginary parts of the transformed results are multiplied by frequency dependent scaling factors to account for the magnitude and site condition differences. The source to site distance adjustment is done by multiplying the results by a frequency independent scaling factor. Thi: scaling factor is obtained from the attenuation relationships (expressing PGA as a function of magnitude and epicentral distance) valid for the region. After applying three scaling factors, the results are inverse Fourier transformed to obtain the full set of accelerograms compatible for the site. Response spectre obtained from the accelerograms are averaged and then smoothened (using a curve fitting technique) tc derive the desired site specific spectrum. The ahove nrocedure mav not exactly cater to the actual site effects in terms of soil conditions. The

Figure 2.25 Acceleration power spectrum (PSDF) for station 1 Figure 2.24 Recording stations and the site

Solution: Power spectrums (PSDFs) at the rock bed corresponding to those of the surface stations are shown in Figures 2.28—-2.30. They are obtained using the following relationships (Chapter 4):

Figure 2.26 Acceleration power spectrum (PSDF) for station 2

Figure 2.30 PSDF at the rock bed for station 3 The three PSDFs at the rock bed are scaled to have the same PGA of 0.38g by the following relationships:

Figure 2.31 Scaled PSDF at the rock bed below the site for station |

Figure 2.33 Scaled PSDF at the rock bed below the site for station 3 (that is, at the site) are obtained with Equation 2.35a, that is,

Figure 2.32 Scaled PSDF at the rock bed below the site for station 2

‘igure 2.35 PSDF at the site obtained from station 2 Figure 2.34 PSDF at the site obtained from station 1

Figure 2.37 Response spectrum (€ =0.05) at the site obtained from station |

Figure 2.36 PSDF at the site obtained from station 3

Figure 2.38 Response spectrum (€=0.05) at the site obtained from station 2

Figure 2.39 Response spectrum (€=0.05) at the site obtained from station 3

specified response spectrum could be a design spectrum, a site specific response spectrum, or a uniform hazard spectrum. In a similar way, the specified PSDF of ground motion may be site specific, region specific or given by an empirical formula. Many standard programs are now available to generate a response spectrum or PSDF compatible ground motions. Therefore, details of the procedure for generating such ground motions are not included here. The basic principles from which they are developed are only summarized below. 2.6.1 Response Spectrum Compatible Accelerogram

Figure 2.43 Uniform hazard spectrum for acceleration with a probability of exceedance of 0.15

Figure 2.42 Normalized response spectrum

Table 2.2 Coefficients of the attenuation relationship for PHV Table 2.3. Magnitude and distance dependence of Vinax/Qmax

Table 2.5 Comparison of PHVs obtained by different empirical equations for M=7

Table 2.4 Comparison of PHAs obtained by different empirical equations for M=7

Figure 2.44 Variation of bracketed duration with epicentral distance

Figure 2.45 Comparison between the smooth Fourier spectrum of El Centro earthquake and that obtained by empirical Equation 2.63

Figure 2.46 Comparison between design response spectrums given by Boore et al., and various codes

Different types of modulating functions have been proposed based on the observations of ground motion records. The commonly used modulating functions are box car type, trapezoidal type, and a modified form of the trapezoidal type in which the beginning and end portions of the trapezoid are non-linear. A box car type modulating function is expressed as: Figure 2.47 Comparison between PSDFs of ground acceleration (obtained by various empirical equations)

in which Apo is the scaling factor and Tp is the strong motion duration of earthquake. A box car type modulating function is shown in Figure 2.48. A 4.2.2... ...2!4.1 2. . 3..124°2. 2 fat tt ke gd 8. Figure 2.49 Modified trapezoidal type modulating function

Figure 2.50 Exponential type modulating function in which Ag is the scaling factor and f,, f2, and f3 are the transition times of the modulating function. A modified form of the trapezoidal type modulating function [29] is shown in Figure 2.49 and is given by

fable 2.6 Annual rate of exceedance (a,) of acceleration response spectrum ordinates

Table 2.7 Annual rate of exceedance (a,) of PGA

Figure 3.1 Models for a single degree of freedom (SDOF): (a) spring-mass-dashpot system; and (b) idealized single frame 3.2.2 Equation of Motion in Terms of the Absolute Motion of the Mass

a set of coupled first-order differential equations, known as the equations of motion in the state-space form. The state-space form of Equation 3.1 is The state of the system at any instant of time is represented by the vector X. Similarly, the state-space form of Equation 3.3 is

Figure 3.2 Multi-degrees of freedom system with single-support excitation

Figure 3.3. Multi-degrees of freedom system with multi-support excitations

Figure 3.4 Frame structure: (a) bracket frame; (b) shear building frame; and (c) 3D model of a shear building frame

Figure 3.5 A rigid slab supported by three columns: (a) model 1; (b) model 2; and (c) unit acceleration given to wy

Note that there is an inertia coupling between the three degrees of freedom for model | and therefore, the mass matrix is not diagonal. Effective force vectors for the two models are: Effective force vectors for the two models are: Furthermore, a mass inertia coefficient of m/2 is to be applied at c to cause the motion in the direction of u, and to equilibrate the horizontal inertia force at the center of the slab. Similarly, a mass inertia coefficient of m/2 is applied at a to equilibrate the vertical inertia force at the center of the slab. Thus, the total mass inertia coefficient corresponding to u is

Figure 3.6 Inclined portal frame subjected to the same support excitation: (a) model 1; (b) model 2; (c) unit acceleration given to u; (model 1); and (d) unit acceleration given to u2 (model 2)

Solution: For the two-bay frame shown in Figure 3.7 Find the r matrices for the two frames shown in Figures 3.7 and 3.8. All members are inextensible. EI (flexural rigidity) values are the same for all the members.

Figure 3.7 Multi-bay portal frame subjected to different support excitations For the inclined leg portal frame, shown in Figure 3.8, the stiffness matrix is written in the partitioned form with r as rotational degrees of freedom and u as translational degrees of freedom as:

A simplified model of a cable stayed bridge is shown in Figure 3.9. Obtain the r matrix corresponding to the translational DOF 1, 2, and 3. Assume the towers to be inextensible. Properties of the members are

Figure 3.8 Different support excitations: (a) kinematic DOF; (b) unit rotation given at B; (c) unit rotation given at C; (d) unit rotation given at D; (e) unit displacement given at A; and (f) unit displacement given at E

With the above assumed values, the stiffness coefficients are calculated as Solution: To solve the problem, the following values are assumed. given below. Geometric non-linearity of the pre-tensioned cable members are ignored in obtaining the stiffness matrix. Furthermore, there is no moment transfer between the deck and the pier, but the deck is vertically supported by the towers.

Figure 3.9 Simplified model of a cable stayed bridge

The resulting partitioned matrices for rotational and translational degrees of freedom are:

Equations of motion in state space can be written by extending Equations 3.4 and 3.5 foran MDOF system. The state-space forms of Equations 3.19 and 3.15 are given by 3.3.3 Equations of Motion in State Space

Absolute motion of the structure: Matrix A remains the same. Taking K,, from Example 3.4 and using tT Le iis Le ie

Figure 3.10 Idealization of earthquake excitation as a sum of a series of impulses

Substituting the expressions for x, 41 and x; 1 given by Equations 3.46 and 3.47 in Equation 3.38 results in

For constant average acceleration between two time stations, the values of 6 = 1/2 and B = 1/4 Arranging Equations 3.61, 3.64 and 3.65 in matrix form, a recursive relationship can be obtained as

Table 3.1 Comparison of displacement (m) response obtained by different methods Using the recursive Equation 3.49, q for time stations 0Q—14A¢ are given in Table 3.1.

Using the above values of the constants, A and H are determined as:

Table 3.2 Frequency components of ground acceleration, frequency response function, and frequency contents of response for a few frequency steps (Aw = 0.2093 rad s~') It is seen from the table that the responses obtained by the Duhamel integration, Newmark’s f/-method, and state-space integrations are nearly the same for time stations 0 — 14A?¢. The same parameters obtained by the frequency domain analysis are slightly different than others. However, rms and peak values of the responses obtained by all methods are nearly the same. Time histories of displacements obtained by different methods are shown in Figure 3.12(a—d). It is seen from the figure that the time history of displacement obtained by the frequency domain analysis is little different than that obtained by other methods.

Using Equation 3.73, Z for time stations 0 —14At and are shown in Table 3.1. Frequency domain analysis—

Figure 3.12 Time histories of displacements obtained by different methods: (a) frequency domain FFT analysis; (b) Newmark’s £-method; (c) Duhamel integral; (d) state-space analysis

in which is the influence coefficient matrix or vector, which can be defined for both the single-point and multi-point excitations described in Section 3.3; M, C, and K are the same as Ms, Cys, and Ks, respectively; X ek+1 iS a single time history of ground motion or is a vector consisting of several time histories of acceleration depending upon the type of excitation. Direct analysis is possible using both time domain and frequency domain analysis. In the time domain, both Duhamel integration and Newmark’s B-method can be used. However, use of Duhamel integration for direct analysis of multi-degrees of freedom systems is not popular because it is somewhat complicated. Therefore, Duhamel integration is generally used in modal time domain analysis. On the other hand, Newmark’s /-method is popular for both direct analysis and modal analysis in the time domain.

The size of the Fy matrix is 3n x 3n and that of the Hy matrix is 3n x m; in which n is the number of degrees of freedom and m is the number of different support excitations.

Referring to Example 3.4, the equation of motion for the frame can be written as:

Figure 3.13 Displacements obtained by direct integration method: (a) displacement 1; and (b) displacement wu

Figure 3.14 Time histories of displacements for the degrees of freedom 4 and 5: (a) without time delay for DOF 4; (b) without time delay for DOF 5; (c) with time delay for DOF 4; and (d) with time delay for DOF 5

A segment of a pipeline supported on soil is modeled as shown in Figure 3.15. Find the displacement responses corresponding to the dynamic degrees of freedom (1, 2 and 3) using (i) the frequency domain state-space solution and (ii) the frequency domain solution for the second-order equation of motion for El Centro earthquake. Take 3EI/mL? = 16 (rad s~!)*; spring stiffness K, = 48m, and C, = 0.6m. Consider the damping ratio to be 2%. Figure 3.15 Segment of a pipeline supported on soft soil and modeled as spring-dashpot system

Figure 3.17 Time histories of displacements for degrees of freedom | and 2 obtained by frequency domain state- space solution: (a) displacement for DOF 1; and (b) displacement for DOF 2 Figure 3.16 Time histories of displacements for degrees of freedom | and 2 obtained by frequency domain analysis: (a) displacement for DOF 1; and (b) displacement for DOF 2

As a time delay of 5s is considered between the supports, a total of 45s of excitation is considered at each support. As explained in Example 3.8, ¥; will have the first 30 s as the actual El Centro record and the last 15 s will consist of zeros. x3 will have first 10 s as zeros followed by 30s of the actual record, and the last 5s of the record will be zeros. Similarly, ¥. and X,4 may be constructed. The natural frequencv and mode shanes are: Solution: From Example 3.6, the mass matrix, the stiffness matrix corresponding to the dynamic degrees of freedom, and the r matrix are taken as: For the simplified model of a cable stayed bridge, shown in Figure 3.9, obtain the displacement responses corresponding to the dynamic degrees of freedom (1, 2, and 3) shown in the figure for the El Centro earthquake with 5 s time delay between the supports. Use modal time history (with Newmark’s /-method) and modal frequency domain (with FFT) analyses with the percentage critical damping taken as 5%.

Figure 3.18 Time histories of generalized forces normalized with modal mass: (a) first generalized force (p,1); (b) second generalized force (p,2); and (c) third generalized force (p,3)

Figure 3.19 Time histories of generalized displacements: (a) first generalized displacement (z,); (b) second generalized displacement (z2); and (c) third generalized displacement (z3)

Table 3.3 Results of a few frequency steps for frequency domain modal analysis (Aw = 0.139 rad s~')

Table 3.4 Comparison of generalized displacements of higher modes to the response are not generally significant. This is the case because more energy is required to excite higher modes that have higher frequencies and, generally, the energy contents of excitations at higher frequencies are low. Furthermore, the mode participation factors given by Equa- tion 3.114 become smaller with a higher number of modes. Therefore, only the first few equations m <n are solved to obtain the response, thereby reducing the size of the problem. The number of equations to be solved is equal to the number of modes whose contributions are considered to be significant in the analysis. However, the number of modes to be considered depends upon the nature of excitation, dynamic characteristics of the structure, and the response quantity of interest. For structures having well separated natural frequencies, the first few modes (up to 5) may be sufficient to obtain the displacement response fairly accurately. For structures having closely spaced frequencies, such as asymmetric tall buildings, suspension or cable stayed bridges, a higher number of modes may have to be considered to obtain good estimates of the responses.

For the multi-storey frame shown in Figure 3.20, obtain the shear at the base of the right-side column for the El Centro earthquake. Obtain the response quantity of interest in the time domain using: (i) the mode summation approach, (11) the mode acceleration approach; and (iii) the modal state-space analysis. Assume k/m = 100 (rads~!)* and percentage critical damping as 5%. Solution: Stiffness and mass matrices for the frame are:

The mode shapes and frequencies are: w; = 5.06 rad s-!s w2 = 12.57rads—!; w3 = 18.65 rad sl, and w4 = 23.5 rads~!.

Table 3.5 First four mode shapes Using the first two frequencies, the damping coefficients are: « = 0.3608; fB = 0.0057.

Table 3.6 Responses obtained by mode superposition method

Table 3.7 Responses obtained by mode acceleration method

Table 3.8 Responses obtained by modal state-space analysis

Figure 3.21 Time histories of shear V obtained by the three methods: (a) mode acceleration method; (b) Newmark’s B-method; and (c) state-space method

Figure 3.22 Inclined portal frame (idealized as SDOF)

Figure 3.23 Pipe embedded within non-uniform soil

Figure 3.24 Stick model for shear building frame

Figure 3.27 Four-storey shear frame Figure 3.26 Frame with a secondary system

Figure 3.28 Model of a three span cable stayed bridge earthquake with time lag between supports 4 and 5 as 2.5s, 5 and 6 as 5 ss and 6 and 7 as 2.5s. Assume the following relationships: AE//, = 12EI/s?; m, = ‘ym; 12EI/s°m = 20(rad s~ ys and the modal damping ratio as 0.05; (ED ower = 0.25(ED aeck = 0.25 El, (#4) gecr= 0.8 (= E) cable’ s = 125; /, is the cable length joining the top of the tower to the end of the bridge; 7m as the mass lumped at the top of the tower and m as the mass lumped at the centre of the bridge.

A two-storey 3D frame with rigid slab, as shown in Figure 3.30, is subjected to the El Centro earthquake. Using the mode acceleration method, find the peak values of the rotation and x displacement of the first floor and y displacement of its top floor. Lateral stiffness of the columns are the same in both directions. Total mass lumped at each floor is taken as m and the mass moment of inertia corresponding to the torsional degree of freedom is taken as m/*/6. Assume the modal damping ratio as 0.05; k/m = 80(rad s~!)°; m = 10000kg and / = 3m. Consider first three modes for the response calculation and evaluate the accuracy of the results. 2 Arigid slab is supported by three columns as shown in Figure 3.31. Columns are rigidly connected to the slab and have the same lateral stiffness k in both directions. Total mass of the slab is m and k/m = 80(rad sl)’, Find the time histories of displacements of the degree of freedom when the base of the structure is subjected to two-component ground motion as shown in the figure. Take Xx¢ as the El Centro earthquake and assume x, =! /)Xxg. Use direct time domain analysis by employing Newmark’s f-method.

Figure 3.31 Rigid slab supported on three columns

Figure 3.32 Calculation of m1, m2, and m3 : (a) DOF 1, 2, and 3; (b) unit acceleration given to DOF 1; (c) virtual displacement given to DOF 2; and (d) virtual displacement given to DOF 3

Figure 3.33 Calculation of m,, and mp: (a) DOF 1 and 2; (b) unit acceleration given to DOF 1; (c) virtual displacement (A) given to DOF 1; and (d) virtual displacement given to DOF 2

Figure 3.34 Calculation of m2»: (a) unit acceleration given to DOF 2; and (b) virtual displacement given to DOF 2

A is given to the DOF 2 keeping the DOF 1 locked. The virtual work equation provides

i) Response due to Constant Force Fo over the Time Interval t Appendix 3.B Derivation of the Response over the Time Interval At in Duhamel Integration

(ii) Response due to Triangular Variation of Force (Fs) within Time Interval to Combining Equations 3.147 and 3.148 with the homogeneous solutions given in Equations 3.29 and 3.30, it follows that The force at any time s given by

Simplifying Equation 3.154, it follows that

Table 3.9 Ground acceleration (in g unit) data for the El Centro earthquake, sampled at At = 0.02 s (read row wise)

diagram for the solution using the state-space formulation is presented, while in Figure 3.37 the solution of the second-order differential equation is presented. The out puts of the solutions are displacement, velocity, and acceleration sampled at the time interval of At. They are stored in files 1-3. Figure 3.37 Simulink diagram for the solution of the second-order equation

Figure 3.36 Simulink diagram for the solution of the state-space equation

Figure 4.1 An ensemble of time histories of stochastic process x(t) Anensemble of time histories of a stochastic process x(t) is shown in Figure 4.1. As shown in the figure x(t1), X(t2) ...,x(t,) and so on are the random variables that can assume any value across the ensemble. A stationary stochastic process (of order two) is one for which the ensemble mean square value remains invariant with the time shift t. This means that the mean square value of x(t;) across the ensemble is equal to that of x(t), in which ft) = ¢, +7 for any value of t. Thus, a stationary stochastic process can be represented by a unique mean square value, which is the ensemble mean square value. If the mean square value is obtained after deducting the ensemble mean (which also remains invariant with the time shift) from the values of the random variables, then it is called the variance. For example, if the random variable is x(t), then the variance is

Any arbitrary function x(¢) can be decomposed into a number of harmonic functions using the well-known equation: 4.3 Fourier Series and Fourier Integral

Figure 4.2 Auto correlation function of the stochastic process x(t) As with the auto correlation function, the cross correlation function is a measure of the correlation between two processes. Figure 4.3 shows the cross correlation function between two random processes. It follows from Equation 4.21 that

Figure 4.4 Shaded area at frequency w contributing to the mean square value in which r? is the mean square value of the process; E[ ] indicates the expected value, that is, the average or mean value. Equation 4.26a provides a physical meaning of the power spectral density function. The area under the curve of the power spectral density function is equal to the mean square value of the process. In other words, the power spectral density function may be defined as the frequency distribution of the mean square value. At any frequency o, the shaded area as shown in Figure 4.4 is the contribution of that frequency to the mean square value. If the square of x(t) is considered to be proportional to the energy of the process [as in E(t) =!4Kx(t)*], then the power spectral density function of a stationary random process in a way denotes the distribution of the energy of the process with frequency. For random vibration analysis of structures in the frequency domain, the power spectral density function (PSDF) forms an ideal input because of two reasons:

For a zero mean process, dy) = 0 and the mean square value becomes equal to the variance. The variance of the process is then the sum of the discrete ordinates shown in Figure 4.5a. As T — 00, 22/T — 0 and therefore, for a sufficiently large value of T, the ordinates become more packed as shown in Figure 4.5b. If the Ath ordinate is divided by dw = 27/T, then the shaded areas as shown in Figure 4.5c are obtained. The sum of such shaded areas (A = 1 to N/2) will result in the variance. A smooth curve passing through the points in Figure 4.5c would then define the PSDF of the process, according to Equations 4.26a, 4.27b and 4.27c. The concept of power spectral density function can be explained in simpler terms if the ergodicity is assumed for the random process. In that case, any sample time history of the ensemble represents the mean square value of the process. If a sample time history x(t) is considered, then use of Fourier transform and Parsavel’s theorem (Equation 4.18) gives

Figure 4.5 Generation of power spectral density function from a Fourier spectrum: (a) S; = [Xz|° at kth frequency: (b) S; at closer frequencies; and (c) Ath ordinate of the power spectral density function

Figure 4.6 A simply supported beam with two harmonic excitations Using Equations 4.26a and 4.26b, Equation 4.29 may be written as:

As the PSDF of a stochastic process is always a function of @, c within the brackets is omitted from the notation of the PSDF henceforth, for brevity. Thus, from Equation 4.31 it follows that

x(t+T), that is, The derivatives of the PSDF and cross PSDF, especially the first two derivatives, are useful in the spectral analysis of structures. Therefore, they are derived below. If x(t) is a stationary random process, the PSDFs of the derivatives of the process and cross PSDFs between the original and derivatives of the process are obtained using the simple rules of differentiation and Fourier transforms. The auto correlation of the process R,(t) is the ensemble average of x(t) and x(t+ 7), that is, 4.7 PSDFs and Cross PSDFs of the Derivatives of the Process

1.8 Single Input Single Output System (SISO) Consider the SDOF system as shown in Figure 4.7. The equation of motion for the system can be written as:

A sample time history of the stationary random process x(t) can be obtained by IFFT of x(a), that is,

Figure 4.8 Power spectral density function (PSDF) for El Centro earthquake acceleration

Figure 4.9 Variation of absolute square of frequency response function with frequency

Figure 4.11 PSDFs of displacements: (a) for u;; and (b) for uz The PSDF matrix of displacement is obtained by using Equation 4.75. The diagonal elements of the matrix provide the PSDFs of u; and uz. The plots of the PSDFs of displacements u, and up, are shown Figure 4.11 (aand b). It is seen from the figure that the peaks of the PSDFs of displacements occur near the first frequency of the structure. This means that the contribution of the first mode to the response is significantly higher than the other mode. The rms values of the displacements u, and u2 are 0.0154 and 0.0078 m, respectively. Solution: From Example 3.8, the mass, stiffness and damping matrices are reproduced below.

Figure 4.12 PSDFs and cross PSDF of responses: (a) displacement uw; (b) displacement wu; (c) real part; anc (d) imaginary part of cross PSDF between uw; and uy

Figure 4.13 PSDFs of displacement for DOF 4: (a) without time lag; and (b) with time lag Using Equation 4.36, the PSDF matrix S,, of absolute displacements may be written as:

Figure 4.14 PSDFs of displacement for DOF 5: (a) without time lag; and (b) with time lag

Figure 4.15 PSDFs of absolute displacements with time lag: (a) for DOF 4; and (b) for DOF 5 respectively. It is seen that these values are much higher than those of the relative displacements obtained in the problem in Example 4.3.

Figure 4.16 PSDFs of responses: (a) displacement for DOF 1; (b) displacement for DOF 2; and (c) bending moment at the center of the pipe

The PSDFs are calculated using Equations 4.98 and 4.99. The plots of S.1-1 and S.1-2 (both real and imaginary components) are shown in Figure 4.17(a-—c). It is seen from Figure 4.17c that the imaginary part of S-1-2 is very small compared with the real part. The PSDFs of displacements corresponding to DOF 1 and 2 are shown in Figure 4.18(a and b). The rms values of displacements obtained from the modal spectral analysis and the direct spectral analysis are compared below.

Figure 4.17 PSDF and cross PSDF of generalized displacements: (a) first generalized displacement; (b) real part; and (c) imaginary part of cross PSDF between first- and second-generalized displacement

Solution: The mass and stiffness matrices and the modal properties are given in the example problem 3.12. For convenience, A is reproduced below. Equations 4.100a and 4.100b are used to calculate the PSDFs of the displacements. The PSDFs are shown in Figure 4.19(a and b). The rms values of displacements obtained by the direct, modal and state- space spectral analyses are compared below.

Figure 4.18 PSDFs of displacements: (a) top of the left tower; and (b) center of the deck

Figure 4.19 PSDFs of displacements: (a) top storey; and (b) first storey

Figure 4.20 Suspended span of submarine pipeline

fable 4.1 Digitized values of the PSDF (m? s~? rad~!) of the El Centro earthquake acceleration record given in \ppendix 3.C at a frequency interval of 0.201 rad s~! (read row wise)

Figure 5.1 Time histories of displacement of the problem in Example 3.12: (a) top-storey displacement; (b) firs generalized displacement; and (c) second generalized displacement

Figure 5.2 Variation of correlation coefficient p;; with the modal frequency ratio «; /;; abscissa scale is logarithmic pj May be ignored as it is assumed in the SRSS combination rule. SRSS and CQC rules for the combination of peak modal responses are best derived assuming a future earthquake to be a stationary random process (Chapter 4). As both the design response spectrum and power spectral density function (PSDF) of an earthquake represent the frequency contents of ground motion, a relationship exists between the two. This relationship is investigated for the smoothed curves representing the two, which show the averaged characteristics of many ground motions. If the ground motion is assumed to be a stationary random process, then the generalized coordinate in each mode of vibration is also a random process. Therefore, there a cross correlation (represented by cross PSDF) between the generalized coordinates of any two modes is present. Because of this, it is reasonable to assume that a correlation coefficient p,; exists between the two modal peak responses. Derivation of p, (Equation 5.15) as given by Der Kiureghian [2] is based on the above concept. There have heen ceveral attemnte to ectahlich 9a relatinnchin hetween the PSDE and the recnonce

Figure 5.3. Comparison between the PSDFs obtained by Equation 5.16a and the Fourier spectrum of the El Centro earthquake: (a) unsmoothed; and (b) 21-point smoothed

Table 5.1 Comparison of peak responses obtained by different approaches

Figure 5.4 Dynamic degrees of freedom for asymmetric 3D tall buildings: (a) center of masses in one vertical axis (b) center of masses at different vertical axes

Table 5.2 Comparison of peak responses obtained by different approaches 5.4 Response Spectrum Analysis for Multi-Support Excitations

With the above quantities, the correlation matrices are calculated as follows. Solution: An acceleration response spectrum compatible PSDF of the El Centro ground motion is shown in Figure 5.3. This PSDF and the correlation function are used to calculate the correlation matrices €yy, Cuz, and ¢;. Following steps 1-8 as given above, the quantities required for calculating the expected peak value of the responses are given as below.

Using the above correlation matrices, the mean peak values of the relative displacements of u and uy are obtained as 0.078 and 0.039 m, respectively. The response spectrum method of analysis for single- support excitation for the same problem provides these responses as 0.079 and 0.041 m, respectively. The time history analysis of the same problem (Example 3.10 without the time lag) gives these values as 0.081 and 0.041 m, respectively.

Using Equation 5.26, mean peak values of the response quantities are determined. The responses are obtained as:

Figure 5.5 Cascaded analysis for a secondary system: (a) secondary system mounted on a floor on a building frame; and (b) SDOF is to be analyzed to obtain floor response spectrum Ifa secondary system is mounted on the floor of a building, as shown in Figure 5.5a, then the secondary system can be analyzed separately using the response spectrum method of analysis. The input to the

Figure 5.6 Floor displacement response spectrum

the total mass of the structure to obtain the base shear. For 1-4 storey buildings, the acceleration value is taken to be nearly lg. le nt eT ee, ie The fundamental natural period of vibration may be computed by the formula: where

Figure 5.8 Distribution of lateral force along a nine-story building frame

Figure 5.9 Variations of the coefficient § with time period T

spectrum (A/g) is drawn. The expression for A/g is given as: In addition to empirical formulae for finding the natural period of vibration of the buildings, the code provides the following formula (similar to Equation 5.49) for estimating the natural period:

Figure 5.12 Variations of C, with time period T

The variation of normalized pseudo acceleration, S,/g with T is shown in Figure 5.13 The fundamental time period of the buildings are obtained by empirical formulae given in the code. The base shear is distributed along the height of the building using a quadratic variation given by:

The seismic coefficient for base shear is obtained from the design response spectrum curve, that is, the variation of C, with Tis taken to be the same as that of S,/g with T. The variation of S,/g with Tis given by: 5.8.5 Indian Code (IS 1893-2002)

Table 5.3 Comparison of results obtained by different codes

Figure 5.14 A seven-storey building frame for analysis

Figure 5.15 Comparison between the storey displacements obtained by different codes

Figure 6.1 Non-linear restoring force (non-hysteretic type)

Figure 6.2 Non-linear restoring force (hysteretic type): (a) and (b) variation of force with displacement under cyclic loading; (c) idealized model of force-displacement curve for (b); and (d) idealized model of force-displacement curve for (a) then determined using the techniques for solving the linear differential equations. However, under certain conditions, iterations are required at each time step in order to obtain the current solution. Incremental equations of motion take the form:

Figure 6.3 Non-linear analysis of SDOF system: (a) SDOF system with non-linear spring; and (b) force- displacement behavior of the spring The value of Ax = Ax; + Ax) = eAx+ Ax, Axy = Ax, is obtained by:

Figure 6.4 Idealized shear frame for analysis with elasto-plastic behavior of columns The procedure mentioned above works well for single degree of freedom systems with elasto-plastic force-deformation behavior. It also performs well for the multi-degrees of freedom systems, such as plane frames, in which sections undergoing yielding are predefined. Such a problem is shown in Figure 6.4 for a shear frame. The yield sections and their elasto-plastic force-deformation behavior are also shown in the figure. For the solution of the incremental equations of motion (Equation 6.1), the state of the each yield section is examined at the end of each time step.

Figure 6.5 Properties of three-storey frame: (a) three-storey frame; and (b) force-displacement curve of the column

It is seen that the shear forces for the first storey columns exceed the yield values. Therefore, the displacements for this storey are scaled such that the shear forces in the columns become just equal to or

To obtain the second part of the displacement vector, Ax2, the K, matrix is modified as:

Figure 6.6 Plan of the one-storey 3D frame with a rigid slab

px and @py are the plastic eccentricities, calculated in the same way as those for elastic eccentricities.

Figure 6.7 Elasto-plastic behavior of the column elements: (a) for x-direction; and (b) for y-direction

Figure 6.8 Properties of the 3D frame: (a) 3D frame; and (b) force-displacement curve of column A

‘gk = —0.08613g. Consider the effect of bi-directional interaction on yielding columns. Solution: Initial stiffness matrix is determined using Equations 6.13a and 6.13b

To determine the stiffness matrix at t= 1.38 s, the yield condition for all column members are checked for t= 1.36s:

Thus, all the columns do not satisfy the yield criterion, and therefore AU, AV,;, and AV,; have to be recalculated by pulling back the internal forces of the members such that they satisfy the yield criterion.

Revised shear forces in the column are obtained by adding the plastic components of the shear force as calculated above with shear forces determined after pulling back to the yield surface (for satisfying th« yield condition). The resulting revised shear forces are: Considering the first guess for the incremental responses, the modification of the tangent stiffness matrix for yielding is obtained as:

Figure 6.9 Properties of the three-storey frame: (a) three-storey frame; and (b) force-displacement curve of column Figure 6.10 Time history of moment at A

Figure 6.11 Shear force-displacement plot at A

Figure 6.13. Shear force-displacement plot at A When the non-linear moment-rotation relationship, as shown in Figure 6.14, is considered instead of the elasto-plastic one, the tangent stiffness matrix for each time interval is calculated by considering the slope of the curve corresponding to the beginning of the time interval, as explained in Figure 6.1. When unloading takes place at the potential section of yielding, the section behaves elastically having a stiffness property equal to the initial tangent stiffness of the moment-rotation curve. For ease of computation, slopes of the back bone curve for different values of force or moment may be given as inputs to the program. The slope for the in between values is obtained by interpolation. If the back bone curve is bilinear, two (pre- and post-yield) slopes are given as input. The stiffness of the system is calculated using one slope or the other depending upon its state. Mee Be og 4g oye 4 - i Pa qeyecs ox a3 ze

Figure 6.12 Time history of bending moment at A

Figure 6.15 Weak beam-strong column frame with potential sections for hinge formation

Figure 6.14 Non-linear moment-rotation relationship (not idealized by straight lines) in which K, is the condensed stiffness matrix of the frame corresponding to dynamic degrees of freedom and K, is the assembled correction matrix due to the plastification of potential sections in the columns. Non-zero elements of the matrix K, are arranged in such a way that they correspond to the degrees of freedom which are affected by the plastification of column sections. Non-zero elements are computed using Equations 6.15 and 6.16. The incremental solution procedure remains the same as that described for inelastic analysis with bidirectional interaction on yielding described previously. If the 3D frame is a weak beam-strong column system, then bilinear interaction on yielding is not required as the beams undergo only one way (vertical) bending. The analysis procedure remains the same as that of the 2D frame. Note that for both 3D and 2D analyses, rotational degrees of freedom are condensed out if the mass moments of inertia for the skeletal members are ignored and point mass lumping is assumed. If rotational degrees are condensed out, then incremental rotations at the ends of the members are obtained from the incremental disnlacement usine the relationshin hetween the disnlacements and

Figure 6.16 Idealized moment-rotation relationship of elasto-plastic beam

condensed Out tO ODtaln the tangent stuimMess Matix tO be Used TOr we Next UMe step. The procedure is illustrated for a framed structure like that shown in Figure 6.15, but having only two storeys. Column and beam length are equal to /. Aj and 6, stand for top storey displacement and joint rotation. The frame is assumed to be a weak beam-strong column system. Therefore, beam ends are potential sections for yielding. The sections that undergo yielding are shown in the figure. The moment-rotation relationship for the sections is shown in Figure 6.16. The sections, other than the ones that are marked, remain elastic. Furthermore, it is assumed that masses are point lumped, that is, there is nO mass moment of inertia and all members are inextensible. Considering anti-symmetric bending, the elastic stiffness matrix corresponding to the kinematic degrees of freedom is given by:

Table 6.1 Response quantities at t= 1.36s. Yield moment for the beams is 5|0kNm

‘igure 6.17 Properties of the three-storey frame: (a) three-storey frame; and (b) force-displacement curve

of time, the stiffness matrix for the system is computed by considering hinges at sections where yieldings have taken place. After the total stiffness matrix is obtained as described above (Equation 6.28a), the stiffness matrix is condensed to only the sway degrees of freedom. The total stiffness matrix and the condensed stiffness matrix are given below:

Figure 6.18 Example frame for pushover analysis: (a) bare frame; and (b) fundamental mode shape of the frame

Figure 6.19 Properties of the seven-storey frame: (a) seven-storey frame; and (b) moment-rotation curve for beams Table 6.2 Properties of the seven-storey frame: E = 2.48 x 10’ kN m~?; mass at each floor level is calculated by lumping appropriate masses at each floor; C and B stand for columns and beams respectively

Figure 6.22 Sequence of the formation of hinges sections continue to take moment after the initial yielding, until the final rotational capacity of the section is reached. The stiffness of the structure suddenly drops leading to a drop of the load carrying capacity of the structure, when the rotational capacity of a yield section or those of a number of yield sections are reached.

Figure 6.21 Variation of base shear with top floor displacement

Table 6.3. Sequence of formation of hinges at different displacement stages 6.5 Concepts of Ductility and Inelastic Response Spectrum

Figure 6.23 Elasto-plastic system and corresponding elastic system fo is defined as the strength (or restoring force = kx) of the SDOF system if it were to remain within the elastic limit during an earthquake. A normalized strength of less than unity implies that the SDOF system will deform beyond its elastic limit. If f = 1, it denotes an elasto-plastic SDOF system with f, = fo so that the system does not deform beyond the elastic limit. A system designed with Ry = 2 means that the strength of the SDOF system is halved compared with that for which the system would have remained within an elastic limit during the earthquake. The ratio between the peak deformations x, and Xo of the elasto-plastic system and the corresponding es

Figure 6.24 Response of an SDOF system for El Centro ground motion for different values of T,: (a) normalized peak deformations for elasto-plastic system and elastic system; and (b) ratio of the peak deformations

From the ductility spectrum, it is possible to obtain the yield strength to limit the ductility demand to an allowable value for given T, and €. Peak deformation is x, = pxy and xy = fy/k = Ay/ o>. With the knowledge of the ductility spectrum for 4 = 1, it is possible to plotf, versus T, for different values of . This provides a direct idea about the strength reduction required for a particular ductility. Such a plot is shown in Figure 6.26.

Figure 6.27 Idealized variation of fy with T, T,, Ty, T., and so on are those used for the elastic design spectrum.

Figure 6.28 Construction of inelastic design spectrum: (a) illustration of the method; and (b) inelastic design spectrum for pu = 2

Figure 6.29 Force-displacement curve of the non-linear spring

Figure 6.30 Properties of the five-storey frame: (a) five-storey frame; and (b) force-displacement curve of the columns

Figure 6.31 Properties of 3D frame: (a) three-storey frame; and (b) force-displacement curve of column

Table 6.4 Properties of the five-storey frame

Figure 6.32 Properties of the five-storey frame: (a) five-storey frame; and (b) moment-rotation curve of the beams

Figure 7.1 Stresses on an infinitesimal cube Using strain displacement relationships, these three equations can be written in the following form:

Figure 7.2 Terminologies used in ground response analysis: (a) soil overlying bedrock; and (b) no soil overlyin; bedrock

Figure 7.3 One-dimensional wave propagation analysis in linear elastic soil the soil. For one-dimensional wave propagation equation of the form

Figure 7.4 Modulus of transfer function for undamped elastic layer Equation 7.23 shows that if a harmonic ground motion is prescribed at the bottom, then its amplitude will amplify as it travels upward to the surface. R(@) also denotes the transfer function for the propagating harmonic wave from rock bed to the free field. The plot of the transfer function is shown in Figure 7.4. The figure shows that amplifications at frequencies ntVs /2H(n = 1,3,5,...) approach infinity. These frequencies are the soil deposit frequencies starting from the fundamental frequency zVs /2H. Because of zero damping (considered in the analysis), amplifications approach infinity at resonance points. If soil damping is considered, the equation of wave propagation takes the form

Figure 7.5 Modulus of transfer function for damped elastic layer 7.3.1 Ground Response Analysis Using FFT With the transfer function known, the free field ground response due to a propagating S-H wave for a specific time history of bedrock motion can be easily obtained using Fourier analysis. The steps to be

Figure 7.6 Mode shapes corresponding to first three natural frequencies

Figure 7.7 Shear beam element for one-dimensional wave propagation analysis Ground response analysis can also be performed in the time domain. In this method, the soil medium is discretized as anumber of shear beam elements as shown in Figure 7.7. Appropriate soil mass is lumped at

Figure 7.8 Non-linear behavior of the soil: (a) backbone curve; (b) hysteresis loop under cyclic loading; (c) variation of shear modulus with shear strain; and (d) variation of equivalent damping ratio with shear strain in which R(u) is the non-linear force—displacement relationship of the soil under cyclic loading. This non- linear relationship is represented by a backbone curve as explained in Chapter 6. For the case of soil, the backbone curve represents the variation of G with the shear strain of the soil as shown in Figure 7.8a. The non-linear equation of motion is solved numerically to obtain the free (relative) field ground response. The method of analysis is explained in Chapter 6. The free field ground response obtained by the non-linear analysis is generally less than that obtained by the linear analysis. The reason for this is the hysteretic behavior of the non-linear soil under cyclic loading, which provides a force—deformation relationship shown in Figure 7.8b. Because of this hysteretic behavior, more seismic energy is dissipated in addition to that due to the material damping of the soil.

Figure 7.9 FEM idealization of wave propagation in soil medium: (a) 2D approximation; and (b) 3D approximation The solution of the one-dimensional wave propagation analysis can be easily extended to 2D or 31 analysis by FEM. Instead of considering one-dimensional shear beam elements, if 2D or 3D elements ar considered for the analysis (Figure 7.9), then the responses in two or three directions can be obtained. Fo 2D analysis, plane strain elements are used. Ground motions are prescribed at the rock bed. For the 3I analysis, eight node brick elements may be used. Appropriate boundaries for the soil medium are to b defined depending upon the topography. 2D or 3D analysis is generally performed for regions witl irregular topography and near the ends of a basin. In the central region of the basin having regula topography, 1D wave propagation analysis is sufficient for all practical purposes. Both linear and non linear analyses are possible for 2D and 3D wave propagation problems. The interaction effect on yieldin; and improved yield curves may be included in the non-linear analysis. Standard softwares such as ANS Y‘ or ABAQUS can be used for this purpose. Note that responses obtained by 2D or 3D wave propagatio! analysis for a single component vertically propagating ground motion will give slightly different result than those obtained from 1D wave propagation analysis because of the Poisson effect.

Figure 7.14 Time histories of absolute acceleration: (a) Vs = 80m s_'; (b) Vs = 200 ms !; and (c) Vs = 6

Figure 7.15 Pseudo acceleration response spectra for 5% damping

Figure 7.16 Time histories of ground displacement: (a) Vs = 80ms~!; (b) Vs = 200ms~!; and (c) V; =600ms~!

Figure 7.17 Time histories of absolute acceleration: (a) Vs = 80 ms~!; (b) Vs = 200 ms™!; and (c) Vs = 600 ms"

Figure 7.18 Time histories of absolute acceleration for linear and non-linear soil

Figure 7.20 Horizontally propagating shear wave in the y-direction beneath the rigid slabs Figure 7.19 Kinematic interaction: (a) vertical motion modified; (b) horizontal motion modified; (c) incoherent ground motion prevented; and (d) rocking motion introduced

Figure 7.21 Inertial effects VEIT TeU GLU LIS UGOw TUT YUU Ue LEU OFICI) Liictil Lit LI LIVI eee: diVeuldil. Inertial interaction is purely caused by the inertia forces generated in the structure due to the movemen of the masses of the structure during vibration (Figure 7.21). The inertia forces transmit dynamic forces tc the foundation. If the supporting soil is compliant, the foundation will undergo dynamic displacement tha would not occur if the soil is very stiff (as assumed for the fixed base condition). Clearly, the dynamic displacement at the foundation-soil interface is the sum total of the free field ground motion, the displacement produced due to kinematic interaction, and the displacement produced due to inertia interaction. As a portion of the soil adjacent to the foundation undergoes vibration produced by the inertia effect, some portion of the energy, imparted to the structure by the ground motion, is lost in moving the adjacent soil mass. The energy travels within the soil in the form of radiating waves and gradually dies

Figure 7.22 FEM modeling of soil-structure as bounded problem For the specified time history of the ground motion at the bed, the equation of motion can be solved either in the time domain or in the frequency domain (by Fourier transform) to obtain the relative displacement vector V. The absolute displacement or acceleration vector corresponding to the degrees of freedom may be obtained by adding the ground motion to the relative motions of the degrees of freedom. This approach For the specified time history of the ground motion at the bed, the equation of motion can be solved either

Figure 7.23 Multi-step method: (a) solution for kinematic interaction; and (b) solution for inertial interaction in which M is the mass matrix with masses corresponding to the structure and foundation degrees of freedom set to zero (that is, only soil masses are retained); and V; is the relative displacements produced at the degrees of freedom. Thus, the elements of the vector of displacements {Vi—ug}" corresponding to the structural degrees of freedom are the motions that are imparted to the structure due to the ground motion. Owing to the kinematic interaction, the ground motion as such is not imparted to the structure.

Figure 7.25 Time histories of top displacements of the stick for fixed base and flexible base (Vs = 80ms_! 7.5.3 Substructure Method of Analysis

Figure 7.24 FEM modeling of soil—structure system

Figure 7.26 Bending stresses at the base for fixed base and flexible base (Vs = 80 ms~')

Figure 7.27 Time history of the base rotation for flexible base (Vs = 80 m s})

Figure 7.29 Modeling for substructure method of analysis: (a) structure without any soil portion; and (b) structure with a portion of soil attached The superstructure is modeled invariably by finite elements. Modeling for the substructure method of analysis is shown in Figure 7.29. For some of the structures, a portion of the soil may be included in the superstructure. For such structures two interfaces are dealt with, one at the free ground surface and the other at the surface between the superstructure and foundation medium as shown in Figure 7.29b.

Figure 7.28 Elastic half space solution for soil impedance function

Figure 7.30 Substructure method of analysis for SDOF system

Unlike the SDOF system, the MDOF system with multi-support excitations, as shown in Figure 7.31, is characterized by the presence of a quasi-static component of the response in the total (absolute) response of different degrees of freedom. The quasi-static component of the response is produced due to different ground motions at different supports producing relative support motions. In order to formulate the problem of dynamic soil—structure interaction using the substructure method, the total displacement is written as the sum of three displacement components Figure 7.31 Substructure method of analysis for MDOF system with multi-support excitation

Figure 7.32 Model of the four-storey building frame The shear frame, shown in Figure 7.32, is supported by two circular footings of radius 1 m below the two columns. The footing rests on a soil with Vs = 80ms~!; p= 1/3 and p = 1600kgm~°. Using a substructure technique, find the time histories of the first- and the top-floor displacements (relative to the ground) for the El Centro earthquake acceleration applied at the base. Use a frequency independent impedance function for the soil. Assume € = 5% for the frame.

Figure 7.33 Time history of top-floor displacement relative to the ground The total displacements of different degrees of freedom are determined by Equation 7.52 in the frequency domain and taking the IFFT of them. The time histories of relative displacements with respect to ground (that is, v’, + v*) for the top floor and the first floor are shown in Figures 7.33 and 7.34 respectively. In subsequent problems, these time histories are compared with those obtained by other methods.

Assuming the superstructure to remain elastic, the dynamic displacement v%, of the structure relative to the base (foundation) may be written as a weighted summation of mode shapes of the fixed base structure as:

Taking the first three modes, the following matrices (Equation 7.67) are obtained.

Figure 7.36 Time histories of first-floor displacement relative to the ground obtained by direct substructure and modal substructure analyses Figure7.35 Time histories of top-floor displacement relative to the ground obtained by direct substructure and modal substructure analyses

Figure 7.37 Equivalent spring—dashpot analysis ALIN VALU D ere aiv eal Vail 1UL liad — oa Je In equivalent spring—dashpot analysis, the soil-foundation system is replaced by a spring and 4 lashpot, as shown in Figure 7.37. For the dynamic analysis of the soil—structure system, the stiffness and lamping coefficients of the springs and dashpots are included in the stiffness and damping matrices of he total system. The response for the equivalent spring—dashpot analysis can be carried out using the lirect integration method (Chapter 3). An approximate modal analysis can also be performed by gnoring the off-diagonal terms of the transformed damping matrix (@’ Cg). Note that the undampec node shapes and frequencies of the systems are determined with the help of the stiffness and mass natrices of the structure obtained by considering the springs at the base. Using these mode shapes. requencies, and approximate modal damping, a response spectrum analysis of the soil—structure system ‘an also be carried out.

Figure 7.38 Time histories of top-storey displacements obtained by direct substructure and equivalent spring- dashpot analyses he spring supported frame are w, = 4.31 and wz = 10.99 rads~*. The responses obtained by integrating the equation of motion in the time domain for the structure— pring dashpot system subjected to the El Centro earthquake are compared with those obtained for ixample 7.5 by the frequency domain substructure technique. The results are shown in Figures 7.38 nd 7.39. It is seen from the figures that the time histories of the two analyses differ. Further, rms and peak alues of the displacements obtained by the two methods are also not the same. The reason for the lifference between the results of the two analyses is that one is obtained in the frequency domain (which loes not require any initial conditions) and the other is obtained by time domain analysis (which requires nitial conditions to start the integration process).

Figure 7.40 Comparison of the time histories of displacements between the fixed base and the flexible base: (a) top storey; and (b) first storey

Figure 7.39 Time histories of first-storey displacements obtained by direct substructure and equivalent spring— dashpot analyses

Figure 7.41 Model of flexible base 3D building

Figure 7.43 Time histories of first-storey displacement obtained by equivalent modal damping and equivalent spring—dashpot analyses Figure 7.42 Time histories of top-storey displacement obtained by equivalent modal damping and equivalent spring—dashpot analyses

Figure 7.44 Soil—pile structure modeling using FEM: (a) with pile cap as a raft; and (b) with individual piles

Figure 7.45 Modeling for finding soil—pile impedance I a a To find the impedance function of the pile head, a finite element model of a single pile or a group of piles, along with the soil, is made as shown in Figure 7.45. The pile head is then subjected to unit complex harmonic forces in the direction of the DOF one at a time in order to obtain a complex flexibility matrix. The inverse of the flexibility matrix gives the impedance function matrix corresponding to the DOF of the pile head. This matrix has coupling terms between the translational and the rotational degrees of freedom. For each value of the frequency, the impedance function matrix is derived and is used to solve the seismic soil—pile structure interaction problem in the frequency domain (Equation 7.64) for a prescribed ground motion at the foundation level. The analysis provides the responses of the superstructure including the displacement of the pile heads or pile cap. With the help of the impedance functions for the pile cap or the pile head, equivalent spring and dash pot coefficients for the soil-pile system can be obtained. Springs and dash pots may be used to replace the soil-pile system at the pile cap level as shown in Figure 7.46.

In this analysis, soil is replaced by springs and dashpots as shown in Figure 7.46. The segment of the pile between two springs is modeled as a beam element and the stiffness matrix for the entire system corresponding to the dynamic degrees of freedom is determined following the standard procedure. Assuming Rayleigh damping, the damping matrix is obtained from the mass and stiffness matrices. Dashpot coefficients are added to the diagonal elements of the damping matrix. The total system (superstructure and the piles with springs and dashpots) is analyzed for the ground motion applied at the foundation level. Analysis may be carried out using the direct time integration method. As the method of analysis is straightforward and the number of degrees of freedom is less as compared with the direct finite element analysis, this method of analysis is preferred for most cases. Furthermore, the non-linear property of the soil can be easily incorporated into the analysis by replacing the linear springs with non-linear springs having the same force-displacement behavior as that of the soil. A non-linear dynamic analysis of the total system is performed using the procedure described in Chapter 6. Figure 7.46 Equivalent spring—dashpot model for soil—pile structure interaction

Figure 7.47 Pile founded building frame: (a) properties; (b) modeling of soil—pile frame together; and (c) modeling by replacing soil by spring—dashpot

Table 7.1 Depth of centers of segments and corresponding values of S,, and S,, for pile lengths

Figure 7.48 Comparison between the time histories of top-storey displacement obtained by the direct method and equivalent spring—dashpot analysis

Figure 7.49 Comparison between the time histories of base displacement obtained by the direct method and equivalent spring—dashpot analysis properties of the contact elements as those mentioned in Exercise 7.4 are used. A tie element is used between the frame and the pile cap. The finite element modeling of the soil—pile structure is shown in Figure 7.47b. The results are obtained by ABAQUAS (for soil, pile and structure damping ratio taken as 5%) and are shown in Figures 7.48 and 7.49.

Figure 7.50 Plane strain analysis for the cross-section of buried structures: (a) tunnel; and (b) pipeline Figure 7.50 shows cross-sections of a tunnel and a buried pipeline. For vertically propagating S waves, the cross-sections of the structures are deformed and two-dimensional stresses are developed in the cross- sections. If the cross-section is large, then seismic excitations differ substantially between the top and the bottom of the cross-sections. In this case, the time history of ground acceleration is preferably prescribed at the rock bed. For smaller cross-sections, variation in the ground motions at different nodes of the cross- section is ignored and the same time histories of excitations are applied at all nodes. In addition, if the buried depth is not large, then the free field ground motion (which is generally known and can be measured) is uniformly applied to all nodes of the structure for most practical cases.

Figure 7.52 Maximum stresses in the x-direction

The analysis replaces the soil by spring—dashpots as shown in Figure 7.55. The spring stiffness and damping coefficients of the dashpots are derived by combining Mindlin’s static analysis of the soil pressure at a depth within the soil and the soil impedance functions. The details of the derivation are given in references [12, 13]. The stiffness and damping coefficients of the matrices corresponding to the DOF are coupled unlike the condition of Winkler’s bed. However, a simplified analysis can be performed by ignoring the off-diagonal terms and by using the diagonal terms to assign stiffness and

Figure 7.55 Longitudinal pipeline model on springs and dashpots: (a) deformations of soil and pipeline; and (b) spring—dashpot model of the soil pipeline problem Figure 7.54 Time histories of the displacement at point A obtained by direct analysis and equivalent spring-dashpot analysis

een gn ne nnn en een ene en nen ee enn ne ne eee nn eee nn een ee nee enn nee ee ee OE The pipeline is also subjected to axial vibration along the length of the pipe due to P waves traveling along the pipeline. Analysis as described above can be carried out for axial vibration of the pipeline, with the soil being replaced by equivalent springs and dashpots. The stiffness and damping coefficients of the springs and dashpots are similarly obtained as for the case of transverse oscillation. For a simplified analysis, stiffness and damping coefficients of the springs and dashpots may be obtained by ignoring the off-diagonal terms of the soil stiffness and damping matrices. Figure 7.56 provides the frequency independent stiffness and damping coefficients of the equivalent springs and dashpots for transverse and axial vibrations of buried pipelines [12]. These coefficients may be used to obtain the stiffness and damping coefficients as follows: Figure 7.56 Variation of frequency independent spring and damper coefficients with d/r

Figure 7.57 Model of a segment of a buried pipeline A segment of an underground pipeline is shown in Figure 7.57. For illustrative purposes, the pipeline is divided into four elements. Determine the stiffness and damping matrices for the pipe for axial and lateral vibration. Take EJ and EA values for the pipe cross-section as 10.4 x 10° kNm*~* and 6.9 x 10° KN, respectively; r= 0.5m; embedment depth d= 8m; pipe length = 10m; € for the pipe =5%, Vs = 200ms~!; p = 1800 kg m~; and lumped masses at the nodes = 2156 kg. Example 7.11

Stiffness matrix for the structure corresponding to the DOF, shown in Figure 7.57, is obtained as: [K;], is added to the K part of the [K], matrix and then rotational degrees of freedom are condensed out. The resulting stiffness matrix is the soil—pipe stiffness matrix. For axial vibration, [K,],, is directly added to [K],,- To construct the damping matrix, [K], is condensed to eliminate 6 degrees of freedom. Assuming Rayleigh damping, [C], for the pipe is obtained using the first two frequencies of the soil—-pipe system. Note that the eigen value problem cannot be solved without adding spring stiffness to the pipe stiffness matrix. [C,], is added to [C], to obtain the damping matrix of the soil—-pipe system. In a similar way, a damping matrix for the axial vibration can be determined. Results of the computation are given below. The first two frequencies of the soil—pipe system and « and f values are:

Figure 8.1 Concept of reliability with two random variables A reliability problem is said to be time variant if the limit state function is also a function of time, that is, the limit state function is defined by g(x, y(t), in which x is a set of random variables and y(t) denotes a vector of stochastic processes. The failure event for such problems may constitute the out crossing of the vector processes, y(t), through the limit state surface g(x, y) = 0 as shown in Figure 8.2. The probability of failure is given by

loading y(t). The exact solution of the out crossing analysis is very difficult for two reasons: (i) evaluation of the conditional probability within the bracket, and (ii) determination of the joint probability density function f(x). In fact, the latter is also the reason for not being able to obtain the exact integration of Equation 8.3. Therefore, various levels of approximations and assumptions are made in the reliability analysis problems. In particular, different types of seismic reliability analysis of structures have been formulated and solved, which make reasonable approximations to arrive at good estimates of the probability of failure of structures. Before they are described and illustrated with examples, popular methods of finding the probability of failure in structural reliability are described.

Figure 8.3. Hasofer—Lind reliability index with linear performance function When the limit state function is non-linear, the computation of the minimum distance becomes an optimization problem, that is, Shinozuka [4] obtained an expression for the minimum distance as:

Figure 8.4 Hasofer—Lind reliability index with non-linear performance function limit state surface. The mean values and the standard deviations of the equivalent normal variables are given by:

Figure 8.5 Different types of uncertainties in seismic risk analysis Seismic reliability or risk analysis of structures can be performed with different degrees of complexity as outlined previously. Uncertainties that could be considered in the analysis are shown in Figure 8.5. It is practically impossible to consider all uncertainties in one analysis. Further, the estimated probability

Figure 8.6 Probable mechanism of failure for weak beam-strong column frame under lateral load Consider a multi-storey frame as shown in Figure 8.6. Three alternative designs of the frame are made with the help of three risk consistent response spectrums, which have 10, 5 and 2% probabilities of exceedance in 50 years, respectively. It is assumed that the frame collapses under the earthquake loads obtained by the response spectrums. Determine the probabilities of failure of the frame.

Figure 8.7 Structure in a region surrounded by a number of earthquake sources Seismic risk analysis of a structure located in a region surrounded by a number of faults may be obtained by integrating the concepts of reliability analysis and seismic risk analysis for the region. Consider a structure in a region as shown in Figure 8.7. There are n earthquake point sources with mean occurrence rate of yf j =1...n). For each earthquake source, the probability density function of the magnitude of earthquake is given by:

v. Threshold values for the floor displacement and the storey shear are 64 mm and 623 kN, respectively.

Table 8.1 Results of computation for source-1 (R = 20;

minimum distances for both response surfaces nearly coincide with the S; axis. in which In(S,) is the mean value of the response spectrum calculated from the attenuation law given above; In(S;) is the logarithmic value of the minor axis of the ellipse. The probability of failure is given by:

Table 8.2 Results of computation for source-2 (R = 50; y) = 2) The threshold crossing reliability analysis is the out crossing analysis in which exceedance probability of a threshold value of the response is determined. One minus the value of the threshold crossing probability is defined as the threshold crossing reliability of the structure.

Figure 8.10 A two-storey frame A two-storey frame as shown in Figure 8.10 is subjected to the El Centro earthquake. EJ, and El) are assumed to be normally distributed independent random variable with mean values as 5.617 x 10’ Nm? and 2.509 x 10’ Nm?, respectively, and coefficients of variation as 0.1. The mass of each floor, m = 12 000 kg, is assumed to be deterministic. The threshold value for the top displacement is given as 19 mm. Find the threshold crossing reliability of the frame.

Figure 8.11 A typical time history of the top displacement of the frame Table 8.3 Results of computation for threshold crossing failure

Table 8.4 Bending moment at X for different combinations of E/;, and EI,

Figure 8.12 Example frame for obtaining the damage probability matrix

In the above problem, if material uncertainty is included then the limit state function takes the form where Lo is the probability of starting below the threshold, « is the decay rate, and T is the duration of the process. At high barrier levels, Lo is practically one and the decay rate is given by the following expressions with double barrier (ap) and one sided barrier (~s) respectively [12],

Table 8.5 Results of computation of PGA, a, and So Using the computed values of So, the PSDF of the top displacement of the frame is determined using spectral analysis presented in Chapter 4. Calculation of the first passage failure probability for displacement (d) is shown in tabular form in Table 8.6 (Equations 8.50-8.58).

Table 8.6 Results of computation of first passage failure for displacement 3.5.5 Reliability Analysis of Structures Using a Damage Probability Matrix

in which a’ and b’ are related to the mean (j1,,) and standard deviation (o,,), respectively, of the peak values M(T) of the process m(t) of duration T as:

Figure 8.13 A segment of a buried pipeline \ pipe segment (of radius rp and thickness f) is modeled as shown in Figure 8.13. It is assumed that the ipe is subjected to the same (perfectly correlated) ground acceleration at all supports and is represented yy the double filter PSDF with: w, = 15.7 rads~!; Nye = Np = 0.6; wy = 0.1. The same attenuation aw used in the previous example is utilized to relate the magnitude of an earthquake with PGA. The operties of the pipe are: E=2.1 x 108 kNm~?; ro = 0.685 m; t/ro = 0.0162; & = 0.05; )y = 2500 kg m~? and L= 70m. The value of the spring stiffness K, = 8.417 x 10°kNm7—!. Find he damage probability matrix and damage indices for the displacement at the center of the pipe for UZ — 4—8. Maximum allowable displacement (dmax) is prescribed as 150mm. Moderate damage is

Table 8.7 Results of computation of damage probability matrix oq =Standard deviation of displacement at the centre. 4p =Mean peak value of the displacement at the centre. Op =Standard deviation of the peak displacement at the centre. PE= Probability of exceedance. Solution: As the same PSDF of ground motion and the attenuation law as that of the previous example problem are considered, the values of Sp for different values of M remain the same as shown in Table 8.6. The PSDF of the displacement at the center of pipe shown in Figure 8.13 is obtained for the random ground motion using the procedure given in Chapter 4. The response quantities of interest and the computation of the damage probability matrix are shown in Table 8.7. Annual damage index for major damage:

Figure 8.14 Risk consistent response spectrum may vary. These values are best obtained from the statistical analysis of collected data. For an assumed value of the PGA, the base shear for the frame shown in Figure 8.6 is obtained with the help of the risk consistent normalized response spectrum given in Figure 8.14. The fundamental time period of the frame is obtained using the mean values of the properties of the frame. Furthermore, the base shear is obtained for the 84th percentile response spectrum and f, is computed as described in (i) as above.

For the frame, shown in Figure 8.15, obtain the fragility curve showing the variation of the probability of failure of the frame with the PGA corresponding to an assumed failure mechanism (partial or complete). Mean values of EJ, EI, and m are shown in the figure. The coefficients of variation for E/,, EI, and m are assumed to be the same and taken as 0.1. The coefficient of variation of the moment capacity of the sections is assumed to be f, = 0.15. The normalized risk consistent spectrum to be used for analysis is shown in Figure 8.14. Solution: A pushover analysis is performed for the frame with mean values of the properties of the frame as shown in Figure 8.15. From the push over analysis, the section where the last hinge that is formed for the collapse of the frame is marked as X in the figure. To obtain the fragility curve, the probability of failure at X for different assumed values of PGA is calculated. A sample calculation for PGA = 200 cms ~ is shown below.

Table 8.8 Results of computation of the probability of failure

Figure 8.18 Earthquake sources for the site ii. The site specific response spectrum is given by Equation 8.38 with the constants a, b and c as:

Figure 9.1 Laminated rubber bearing system: (a) section and elements; (b) idealized force deformation behavior for low damping rubber; and (c) idealized force deformation behavior for high damping rubber The basic components of the LRB are steel and rubber plates built into alternate layers (Figure 9.1a). Generally, an LRB system exhibits high damping capacity, horizontal flexibility, and high vertical stiffness. For low damping rubber bearings, a lateral force—deformation relationship is modeled as linear, as shown in Figure 9.1b. A temperature independent model for the lateral stiffness of the bearing is given

Figure 9.2 Lead rubber bearing (New Zealand system): (a) section and elements; and (b) idealized force deformation behavior 1.2.2 New Zealand Bearing System The linear damping coefficient may be taken to be about 10%. For a high damping rubber bearing, the equivalent damping is considerably increased due to the hysteretic effect as shown in Figure 9. 1c. It could be of the order of 15-20%.

Figure 9.3 Resilient friction based isolator system: (a) section and elements; and (b) idealized force deformation behavior The friction base isolator consists of concentric layers of teflon coated plates that are in friction contact with one another, and it contains a central core of rubber (Figure 9.3a). It combines the beneficial effect of friction damping with that of resiliency of rubber. The rubber core distributes the sliding displacement and velocity along the height of the R-FBI bearing. The system provides isolation through the parallel action of friction, damping, and restoring force, and is characterized by natural frequency (wp), damping constant (€,), and coefficient of friction (uu). Recommended values of these parameters are @p = 0.5rrads7!; €, = 0.1, and 0.03 < uw < 0.05. The force deformation behavior of the isolator is shown in Figure 9.3b. 1.2.4 Pure Friction System

Figure 9.5 Elastic sliding bearing: (a) round type; (b) square type; and (c) idealized force deformation behavior This is principally composed of rubber layers, connective steel plates, sliding material, a sliding plate, and a base plate. The sliding material is set between the sliding plate and the connective plate as shown in Figure 9.5(a and b). Under the lateral force, the shearing deformation is limited to the laminated rubber

Figure 9.4 Pure friction system: (a) schematic diagram; and (b) idealized force deformation behavior

in which Wis the vertical load and R is the spherical radius of the sliding surface, and Ty is the target design period. The yield force of the curved plane sliding bearing, Q is determined from Figure 9.6 Curved plane sliding bearing (FPS): (a) section and elements; and (b) idealized force deformatior behavior

Figure 9.7 Load deformation curve of isolator under cyclic loading: (a) hysteretic curve; (b) linear isolator; and (« non-linear (bilinear or trilinear, and so on.) isolators From the maximum vertical reaction R and Tj, the effective stiffness of the isolator is

Figure 9.8 Variation of equivalent stiffness and damping with displacement: (a) stiffness variation; and (b) damping variation

Figure 9.9 Properties of the base isolated frame

Figure 9.10 Acceleration response spectrum for 5% damping

Figure 9.11 Idealized force—deformation plot for the isolator

Figure 9.12 Design dimensions of the lead core rubber bearing

Figure 9.13 Base isolation at the isolated footings: (a) isolators in position; and (b) mathematical model with effective stiffness and damping

Figure 9.14 Base isolation with a raft: (a) isolators in position; and (b) based degrees of freedom The total incremental displacements are given by:

Figure 9.15 Backbone curves for the isolators: (a) linear; (b) bilinear; (c) elastoplastic; and (d) non-linear Equation 9.20 may be split into two equations:

In forming the above matrix, itis assumed that all column bases move by the same amount at every instant of time, ¢. Therefore, there is effectively one base degree of freedom at the base. The stiffness matrix for the isolated structure is obtained by adding the total stiffness of the isolators to the last diagonal term. Note that this added stiffness varies with the displacement according to the load displacement behavior of the isolator.

The C,, matrix for the structure is obtained by assuming Rayleigh damping with coefficients (determined using the first two modes of the structure having K as K,;) as « = 0.6758; 6 = 0.00267. The C matrix of the isolated structure is obtained as:

The isolators are placed just below the columns as shown in Figure 9.17.

Figure 9.18 Comparison between controlled and uncontrolled top-floor displacements

The 3D model of the seven-storey building frame shown in Figure 9.21 is base isolated using a base slab at the base of the columns. The arrangement of the isolators is also shown in the same figure. The backbone curves of the isolators in both directions are the same and are shown in Figure 9.16. Find the time histories Figure 9.21 3D frame and its properties: (a) plan; (b) frame elevation 1-1; and (c) frame elevation 2-2

Figure 9.22 Comparison between controlled and uncontrolled displacements of the column at the top floor level: (a) x-direction; and (b) y-direction

Figure 9.23 Time histories of displacements of the base of the column: (a) x-direction; and (b) y-direction Figure 9.23(a and b) shows the time histories of displacements at the base of the column. It is seen from the figures that maximum base displacements in the x- and y-directions are 0.026 and 0.0113 m, respectively. As the building is symmetric and is supported on a symmetrically placed isolator having the same characteristics, the building vibrated along the x- and y-directions without any torsional coupling. As expected, the base movement in the y-direction is less than that in the x-direction.

Figure 9.24 Hysteretic behavior of the isolator A: (a) for x-direction; and (b) for y-direction

Figure 9.25 Use of design bi-spectrum for isolator design (first criterion) For the first criterion, the fundamental period (7) of the superstructure as a fixed base is calculated. The isolator period is then obtained as nT so that there is a good separation between the two periods. Typically nis kept around 3-4 and the isolator period is kept above 2 s (as per UBC). Once the time period of the isolator is decided and an equivalent damping of the isolator is assumed, the maximum isolator displacement and the absolute acceleration (spectral acceleration) are obtained from the bi-spectrum, as shown in Figure 9.25. An approximate base shear for the isolated building can be calculated from the spectral acceleration and the total mass of the superstructure. It is possible then to see how much reduction is achieved in the base shear of the isolated building as compared with the fixed base structure by providing the isolator. The equivalent stiffness of the isolator is obtained as Keg = 47M / (nT) in which M is the total mass of the building.

Figure 9.26 Use of design bi-spectrum for isolator design (second criterion)

Figure 9.27 Use of design bi-spectrum for isolator design (third criterion) 9.5.2 Response Spectrum Analysis of Base Isolated Structure

Table 9.1 Multiplying factors for damping (spectral ordinates to be multiplied by this factor) Solution: Total load of the structure = 1575 kN. Fixed base fundamental period of the frame is 0.75 s. Assuming a time period separation of 3, the fundamental time period of the frame is 2.25 s. Assuming a damping of 10% for the isolator, the response spectrum provides a value of The seven-storey building frame as shown in Figure 9.9 is to be base isolated using a base slab. Design the base isolation system and the base isolated structure, starting from the preliminary design to the last stage of non-linear analysis. For seismic analysis, consider 50% of the live load. Use the response spectrum as shown in Figure 9.10 with PGA = 0.36g and damping modification as given in Table 9.1.

Figure 9.28 Backbone curve for the designed isolator The back bone curve for the isolator is shown in Figure 9.28. It is seen from the figure that for a base displacement of 16cm, Ke, of the isolator is the same as that obtained above.

Figure 9.29 Bending moment (kN m) diagram for the initial design of the base isolated frame

ire 9.30 The base isolated frame with revised dimensions

Figure 9.31 Envelope of the bending moment (kN m) diagram obtained from the response spectrum analysis Using these two mode shapes, the equivalent modal damping in the first two modes, after ignoring the off-diagonal terms of the transformed modal damping matrix, are obtained as €; = 0.084; €, = 0.0167. By considering the first two modes, the response spectrum analysis is carried out using SAP 2000.The envelope of the bending moment diagram is shown in Figure 9.31. The base displacement is obtained as

The undamped mode shapes and frequencies of the isolated structure modeled by a horizontal spring at the base with stiffness equal to 3Keq are obtained. The first two undamped mode shapes of the spring supported frame are: Rayleigh damping for the frame with « = 0.676 and / = 0.0027 and the isolator damping (assuming &€ = 10%). The resulting C matrix for the structure isolator system is given below.

Figure 9.32 Response spectrum compatible simulated time history of the ground acceleration (for PGA = 0.36g) The frame along with the base slab and isolators shown in Figure 9.30 are analyzed for the response spectrum compatible time history of acceleration shown in Figure 9.32. A backbone curve of the isolator is shown in Figure 9.28. Analysis is carried out using SAP 2000. The envelope of the bending moment is shown in Figure 9.33. It is seen from the figure that the time history analysis provides less bending moments compared with the response spectrum analysis. The time history of the base displacement is shown in Figure 9.34. It is seen from the figure that the maximum base displacement is 0.096 m, little less than the maximum base displacement obtained by the response spectrum method of analysis. If the two displacements are found to be very different, then the entire procedure may have to be repeated starting from the initial design. In this case, the iteration is not needed.

Figure 9.33 The envelope of the bending moment (kN m) diagram obtained from the time history analysis

Figure 9.34 Time history of the base displacement In order to understand how the TMD works in controlling the response of the building or any structure in which it is installed, consider a two degrees of freedom undamped system subjected to the support excitation as shown in Figure 9.36. In the absence of the second system, the amplitude of the response of the first system can easily be shown to be:

Figure 9.35 A tuned mass damper as passive control device

Figure 9.37 Variation of response ratio with frequency ratio for two DOFs

Figure 9.39 Variation of the absolute normalized amplitude of response Xo2 (72) with mass ratio Figure 9.38 Variation of the absolute normalized amplitude of response Xo; (77) with mass ratio

Equations 9.72a and 9.73 can be combined to obtain the following equation of motion: The C matrix is obtained by combining the C matrix with the damping elements introduced due to TMD as shown in Equation 9.76c. The C matrix is obtained by assuming Rayleigh damping for the bare frame. Note that except for the non-zero elements shown for the last rows and last columns of K and C matrices, the other elements of those rows and columns are zero. Solution of Equation 9.76a provides the responses of both the TMD and the frame. .6.2 Direct Analysis

The mass, stiffness, and damping matrices of the frame-TMD system are:

Solution: The fundamental frequency of the bare frame is 11.42 rad s“'. For perfect tuning,

The C matrix for the bare frame is determined by assuming Rayleigh damping for the structure, obtained with the help of the first two natural frequencies w, and 2. The responses of the system are obtained by solving the state-space equation, Equation 9.77a using SIMULINK of MATLAB®. The matrix A is given by:

Figure 9.41 Controlled and uncontrolled displacements of first floor The controlled and uncontrolled responses are compared in Figures 9.41 and 9.42. It is seen from the figure that the peak displacement of the top floor of the frame is controlled by about 30%. The

Figure 9.42 Controlled and uncontrolled displacements of fifth floor

Figure 9.43 Time history of the excursion of the mass of TMD

Figure 9.45 Storage and loss moduli of viscoelastic damper

Figure 9.44 A model of Teflon coated viscoelastic damper where C|.] and DJ[.] are linear differential operators with constant coefficients. Using a Laplace transformation, Equation 9.78 provides a transfer function H(s) in the form of

Figure 9.46 Idealized building frame with viscoelastic dampers

Figure 9.47 Model of a two DOF structure with viscoelastic damper the following equation in the frequency domain may be written:

where Z is a (2n + n,) state vector, A is a (2n+n,,2n+n,) system matrix and E is a (2n +n,) excitation vector, respectively, given by: Once q(t) is obtained, the response y(t) may be determined as before by modal superposition. Note that the eigen value problem given by Equation 9.109 has to be solved using the iteration technique.

Figure 9.48 Controlled and uncontrolled displacements of fifth-floor

Figure 9.49 Time history of control force in the fifth-floor VEDs

Figure 9.50 Controlled and uncontrolled displacements of first-floor

Figure 9.51. Time history of control force in the first-floor VEDs

regulator (LQR) control, pole placement technique, instantaneous optimal control, discrete time control, sliding mode control, independent modal space control, bounded state control, predictive control, H, control, and so on [11,12]. The control mechanisms, which have been developed and used in experiments and in practice, include actuated mass damper (AMD), actuated tuned mass damper (ATMD), active tendon system (AT), and so on (Figure 9.53). Active control algorithms generate the required control signal that drives the actuators of the control device. Before describing the control algorithms, it is desirable to understand the effect of control forces on the response of a structure under ideal conditions. Consider the equation of motion of a multi-degrees of freedom system under a set of control forces u:

Figure 9.53 Active control devices: (a) active tuned mass damper; (b) actuated mass damper; and (c) active tendon response of the system may be achieved theoretically by generating control force at each instant of time ¢ that is equal and opposite to the earthquake force. However, in reality it is not realizable for several reasons, which will be discussed in the limitations of the active control scheme. Therefore, control of the response is achieved with respect to a certain norm. The choice of the norm dictates the control algorithm. As the control of the response is achieved by applying the control force that depends upon the state of the system (that is, x and x), three factors are extremely important in relation to the design of the controlled system. They are stability, controllability, and observability, that is, the system should be stable, controllable, and state observable. These three attributes are elaborated further in the following sections. To examine these factors, the controlled equations of motion, Equation 9.116, is written in state-space as:

in which e = X—X. Equation 9.144 is similar to Equation 9.136a. For complete state controllability, that is, |A—K-.C] to be a state matrix, [A—K.C] has arbitrarily desired eigen values. Thus, K, can be designed to yield the desired eigen values. The problem is similar to finding the gain matrix G for the pole placement technique (discussed later) and can be solved by Ackerman’s formula [13]. Using Ackerman’s formula, the gain matrix K, can be obtained as:

MATLAB computation to find the K, is explained with the following example. in which e = x,—Xp; Xp is the estimated state of the un-measured variables. K, as before can be obtained using Ackerman’s formula and is given by [13]: Example 9.7

There are a number of algorithms developed for finding the control force u(t). Most of the algorithms derive the control force by minimizing the norm of some responses or some appropriate performance index, and therefore are termed as optimal control algorithms. The derived control force is obtained as a linear function of the state vector, and hence they are also called linear optimal control algorithms. There are other control algorithms that are not based on any optimal criterion, but are based on a stability criterion or some other different considerations. Also, there are control algorithms that have control forces defined in terms of non-linear functions of the state vector. Here, three simple control algorithms are presented in order to demonstrate how they are developed. These control algorithms are formulated in the time domain and have been implemented for many control problems.

in which € is the damping ratio, and @, is the undamped natural frequency. The problem of absolute stability can be solved by choosing the poles of the closed loop system to lie on the left-hand side of the S-plane as shown in Figure 9.54. The further away the dominant poles are from the imaginary axis, the faster is the decay of the transient response of the system. Accordingly, poles of the closed system are selected and the gain matrix G is obtained to generate the control force. There are different methods to obtain the gain matrix G, given the desired values of the poles of the closed loop system. Note that the desired values of the poles are chosen after obtaining the poles of the uncontrolled system, that is, by finding the eigen values of the matrix A. The choice of the desired poles of the controlled systems consists in shifting the poles of the uncontrolled system to the farthest points on the left-hand side of the S-plane. Thus, the gain matrix G that is generated from the desired poles is not unique. Different G matrices can be generated by placing the poles at various locations and each of them provides a different control of the responses and different peak values of the control force. A tradeoff between the percentage control of the response and the peak control force is, therefore, an important consideration in the control procedure. There are different methods for generating the gain matrix G. Here determination of the G matrix by using Ackermann’s formula is described as MATLAB has a standard program for this. Consider the system

For a control system described by Equation 9.149, obtain the G matrix using MATLAB for the following values of A, B, and J [13]. Note that the output matrix, that is, K of MATLAB is the G matrix. As a result, both K and G are used to denote the same gain matrix. Example 9.8A

Solution: Using the mass, damping and stiffness matrices of the frame described in the Example 9.6, A and B matrices for the state-space Equation 9.149 are given as:

Figure 9.55 Controlled and uncontrolled displacements of fifth floor using pole placement technique

Figure 9.56 Controlled and uncontrolled displacements of first floor using pole placement technique

Figure 9.57 Time history of the control force using pole placement technique in which f, is the duration of the earthquake excitation; Q is a 2n x 2n semi-definite matrix, and R is an n X mpositive definite matrices. The matrices Q and R are termed weighting matrices. The relative values of the elements of the matrices are selected according to the importance attached to the different parameters of control. For example, large values of the elements of Q compared with those of R denote that response reduction is given more weightage at the cost of control force. The opposite is indicated when the elements of R are relatively large. Similarly, in the Q matrix, relatively large values of the diagonal elements corresponding to displacement response denote that the velocity response is penalized in the minimization procedure.

Solution: MATLAB program (design of quadratic optimal regulator system)-

Figure 9.58 Controlled and uncontrolled displacements of fifth floor using LQR control

Figure 9.59 Controlled and uncontrolled displacements of first floor using LQR control

Figure 9.60 Controlled and uncontrolled absolute accelerations of the fifth floor using LQR control

Figure 9.61 Time history of the control force using LQR control in which @ is 2n x 2n modal matrix. The decoupled state-space equation is written as:

Equation 9.195 is a state-space equation for a control system for which the control force can be obtained using an LOR alvorithm as: If X;(t—t) is one of the state variables, its Taylor series expansion about f is given by:

Figure 9.62 Semi-active control with SHD: (a) frame fitted with SHDs; (b) SHD in a floor; and (c) modeling of SHD

the gain matrix K (Equation 9.201b) is obtained as: Solution: Using linear quadratic control with Q and R matrices taken as:

The bracing may be assumed to be almost rigid such that the piston velocity of the damper is equal to the relative velocity between the two floors. The damper capacity is set at finax = 900 KN and the maximum damping coefficient is set at Cmax = 200kNsm!.

Figure 9.64 Time histories of controlled and uncontrolled absolute accelerations of the top floor using semi-active control Figure 9.63 Time histories of controlled and uncontrolled displacements of the top floor using semi-active control

TE The time histories of controlled and uncontrolled responses are shown in Figures 9.63 and 9.64. It is seen from Figure 9.63 that the peak displacement of the top floor is reduced by 58%. The corresponding reduction in absolute acceleration is about 47%. The time histories of the displacements of the first floor are shown in Figure 9.65. It is seen that the reduction in the peak displacement of the first floor is 56%, and hence the base shear is also controlled by the same amount. The time history of the control force generated in the SHD for the first floor is shown in Figure 9.66. It is seen from the figure that the peak control force is about 175 kN, much less than the maximum capacity of the damper. Better control may be achieved by using different trial values of Cimax, fimax, Q, and R.

Figure 9.65 Time histories of controlled and uncontrolled displacements of the first floor using semi-active control

Figure 9.66 Time history of the control force generated in the first floor SHD

Figure 9.68 Details of the frame-TMD system Figure 9.67 Properties of the frame to be base isolated

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