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Nonlinear seismic response of a gravity dam under near-fault ground motions and equivalent pulses
Soil Dynamics and Earthquake Engineering 92 (2017) 621–632
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Nonlinear seismic response of a gravity dam under near-fault ground motions and equivalent pulses
crossmark
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Y. Yazdania, M. Alembagherib, a b
Department of Civil and Environmental Engineering, Amirkabir University of Technology, Tehran, Iran Department of Civil and Environmental Engineering, Tarbiat Modares University, Tehran, Iran
A R T I C L E I N F O
A BS T RAC T
Keywords: Gravity dams Nonlinear structural response Near-fault Forward-directivity Velocity pulse
In this paper, the nonlinear seismic response of gravity dams to forward-directivity and ordinary (non forwarddirectivity) near-fault earthquake ground motions is investigated. Considering Pine Flat dam as case study, it is numerically modeled along with its full reservoir using the finite element method. Two sources of nonlinearity are considered in the analysis: (1) the material nonlinearity of dam concrete, and (2) the geometric nonlinearity by inserting a joint at the base of the dam. Seventy-five forward-directivity and sixty ordinary near-fault ground motions are used to obtain statistically significant results. The equivalent representative pulses of the selected forward-directivity ground motions are extracted using a well-known methodology. The dam-reservoir model is analyzed under the equivalent pulses as well to identify the cases for which the equivalent pulses can capture the structural response to the actual forward-directivity ground motions. Finally, the effects of pulse period, amplitude and energy on the seismic response of the dam-reservoir system are studied.
1. Introduction High concrete gravity dams are among important infrastructures playing key role in national water and power management systems. Because of large water reservoir impounded behind a high dam, its stability and safety specifically during seismic events is of great importance. The gravity dams are typically located on rock or firm soil foundations which may contain active faults near dam sites [1]. Therefore, they may undergo near-fault earthquake ground motions; for example, Pacoima dam in the US experienced high shakings during the near-fault ground motions of San Fernando 1971, and Northridge 1994 [2,3]. The potential failure modes of gravity dams that may be triggered by an earthquake can be classified as: (1) overstressing mainly in tension, which results in cracking response of the dam body; and (2) sliding along prescribed or cracked paths specifically at the dam-foundation interface [4]. Directivity effects can considerably influence ground motions in the proximity of causative faults. The extent of near-fault region depends on earthquake magnitude. Forward-directivity results when the fault rupture propagates toward the site at a velocity nearly equal to the shear wave velocity, and the direction of fault slip is aligned with the site. This causes the wavefront to arrive as a single large pulse [5]. The forward-directivity is a dynamic phenomenon that produces no permanent ground displacement and hence two-sided velocity pulses.
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Most of the energy in forward-directivity ground motions is concentrated in a narrow frequency band. These motions are characterized by a distinct, high intensity velocity pulse at the beginning of time-history records, which are oriented in perpendicular direction relative to the fault plane [5]. However, not all of near-fault ground motions are pulse-type, and they may have no pulse-like characteristics. It has been well determined that the forward-directivity pulses are critical in the analysis and design of structures in the near-fault areas. They can result in high seismic demands tending to drive structures into the nonlinear range. Typical measures such as peak ground acceleration (PGA) or spectral acceleration at periods of natural modes of structure may no longer serve as effective intensity measures [6]. The narrow band nature of the velocity pulse implies that the forwarddirectivity ground motions can be represented using equivalent pulse models [7–12]. The main characteristics of these pulses are: (1) pulse period, (2) pulse amplitude, (3) phase parameter, and (4) number of significant pulses [7,13]. These parameters can be determined such that the representative pulse acceptably approximates the original pulse-like motion [7]. Because the parameters of the forward-directivity pulses may control the response of structures [14–22], the effects of pulse-type ground motions should be considered in structural analysis. Many researchers have studied the characteristics of forwarddirectivity ground motions [5,23,24], detecting and extracting the dominant representative pulses [7,25–27], and structural response to
Corresponding author. E-mail address: [email protected] (M. Alembagheri).
http://dx.doi.org/10.1016/j.soildyn.2016.11.003 Received 21 June 2016; Received in revised form 10 September 2016; Accepted 2 November 2016 0267-7261/ © 2016 Elsevier Ltd. All rights reserved.
Soil Dynamics and Earthquake Engineering 92 (2017) 621–632
Y. Yazdani, M. Alembagheri
Fig. 1. Acceleration, velocity and displacement time-histories along with specific energy density of the actual ground motion recorded at Pacoima Dam station and its velocity pulse: (a) 1971 San Fernando earthquake, and (b) 1994 Northridge earthquake.
The objective of this paper is to compare the nonlinear seismic response of a gravity dam model to forward-directivity (FD) and ordinary (non forward-directivity, NFD) near-fault ground motions. The dam is numerically modeled along with its reservoir using finite element method based on Eulerian-Lagrangian approach. Two sources of nonlinearity are considered in the analysis: (1) the material nonlinearity of dam concrete, and (2) the geometric nonlinearity by inserting a joint at the base of the dam. Seventy-five forward-directivity and sixty ordinary near-fault ground motions are used to obtain statistically significant results. The equivalent representative pulses of the selected FD ground motions are extracted using the methodology introduced by Baker [9]. The dam-reservoir model is analyzed under the equivalent pulses as well to identify the cases for which the equivalent pulses can capture the structural response to the actual FD ground motions. Finally, the effects of pulse properties, i.e. the pulse period and amplitude, on the seismic response of the damreservoir system are studied.
the actual pulse-type motions or their simplified representative pulses [14–19,28–30]. In a series of research, Bayraktar and co-workers [31– 33] investigated near- and far-fault ground motion effects on the nonlinear dynamic response of various dam types such as gravity, arch, concrete faced rock-fill and clay core rock-fill dams. They have used limited number of near- and far-field ground motions for the analysis which have approximately identical PGAs. They showed that near-fault ground motions have different impacts on the dam types, and there is more seismic demand on displacements and stresses when the dam is subjected to near-fault ground motion. They also concluded that the seismic behavior of the concrete gravity dams is considerably affected from the length of the reservoir. Akkose and Seismik [34] focused on nonlinear seismic response of a concrete gravity dam subjected to nearand far-fault ground motions. The elastoplastic behavior of the dam concrete was idealized using Drucker–Prager yield criterion. They compared the seismic response of the selected concrete dam subjected to both near- and far-fault ground motions. Wang et al. [35] evaluated the effects of near- and far-fault ground motions on seismic performance of a concrete gravity dam. Four different near-fault ground motion records with an apparent velocity pulse were used. They showed the effects of near-fault ground motions on seismic performance of concrete gravity dams and demonstrated the importance of considering the near-fault ground excitations. Zhang and Wang [36] studied the near- and far-fault ground motion effects on nonlinear dynamic response of a concrete gravity dam. For this purpose, 10 asrecorded earthquake records which display ground motions with an apparent velocity pulse were selected to represent the near-fault ground motion characteristics. They found that near-fault ground motions would cause more severe damage to the dam body than farfault ground motions. Huang [37] investigated the effects of near-fault ground motions on the nonlinear seismic response behavior of concrete gravity dams. In particular, he evaluated the characteristics of different aspects of near-fault ground motions and examined the significance of various near-fault ground motion parameters. However, still lack of specific study about the effects of forward-directivity pulses on the seismic demand analysis of gravity dams is felt.
2. Forward-directivity ground motions and equivalent pulses Ground motions recorded close to a ruptured fault can be significantly different from far-fault ones. The near-fault ground motions can be classified as “pulse-like” and “non-pulse-like”. The pulse-like motions can be subdivide into “forward-directivity” and “fling”. The specific nature of the pulse-like motion due to forward-directivity may be revealed in its velocity time-history. The peak ground velocity (PGV) of the near-fault ground motions is substantially higher than the farfault ground motions. Previous studies about the seismic response of structures located in near-fault regions have shown that time-domain representation of the near-fault ground motions is preferable to response spectrum representation because of concentration of record's energy in a single pulse of motion [5,11,13,25]. Baker [9] described a reasonable widely-used method for quantitatively identifying ground motions containing strong velocity pulses. He employed wavelet-based signal processing to detect and extract the largest velocity pulse from a given ground motion. He also established 622
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joint. The coupled dam-reservoir system is analyzed using the EulerianLagrangian approach. In this approach, displacement-based Lagrangian formulation is considered for the dam body through the following differential equation
some criteria for classifying a ground motion as pulse-like using the size of the extracted pulse relative to the original ground motion. The Daubechies wavelet of order 4 was utilized as mother function in continuous wavelet transform (CWT), because it approximates the shape of many velocity pulses specifically in the fault-normal components of ground motions. He employed two additional criteria to identify the pulse-like records potentially caused by directivity effects: (a) the pulse arrives at the beginning of the strong ground motion, and (b) the absolute amplitude of the velocity pulse is large relative to the remainder of the record. By extracting a wavelet from the ground motion that best matches the dominant pulse, its amplitude and pseudo-period determine the amplitude and period of the equivalent velocity pulse, respectively [9]. Two examples of forward-directivity ground motions recorded at Pacoima dam site are shown in Fig. 1 along with their representative pulses. The velocity pulse is clearly observed in velocity time-history of these records. The energy of a time signal, either the original ground motion or the pulse, can be approximated using specific energy density (SED) [9,38]:
SED (t ) =
∫0
t
V 2 (ξ ) dξ
∂σji ∂xj
+ Fi = ρc
∂ 2ui ∂t 2
(2)
where σji = σij is the Cauchy stress tensor, ui is the displacement, ρc is the dam density, and Fi is the body force per unit volume [47]. Using the Galerkin's weighted residual method, and by discretizing the displacement vector of the dam body as {u}=[N]u{u}e, in which [N]u is the matrix of shape functions and {u}e is the vector of unknown nodal displacements, the finite element formulation of the dam body is obtained as
[M ]{u}̈ + [C ]{u}̇ + {FR} = {FS} + {FH}
(3)
where [M] and [C] are the structural mass and damping matrices, respectively, {FR} is the vector of nonlinear resisting force, {FS} is the vector of external forces due to ground acceleration on the dam, {FH} is the vector of hydrodynamic forces, and ‘dot’ represents derivative with respect to time. The governing differential equation of the reservoir domain considering pressure-based Eulerian formulation, assuming that the water is linearly compressible, neglecting its internal viscosity and having small amplitude irrotational motion, can be represented as wave equation
(1)
where V(ξ) is the ground motion velocity at time ξ. This parameter is also plotted in Fig. 1 for both signals. It is observed that the energy of the equivalent pulses relative to the whole records has about 50% compatibility. In this study, seventy-five forward-directivity (FD) and sixty ordinary (non forward-directivity, NFD) near-fault ground motions from 30 earthquakes are selected for the analysis. All records are taken from stations located on rock or firm soil with shear wave velocity more than 250 m/s, which is compatible with the sites where the gravity dams are founded. The FD records are compiled from a database developed by Baker [39]. The NFD near-fault ground motions, which are classified as non-pulse-like, are chosen based on these criteria: (1) closest distance to the ruptured fault R < 20 km, (2) PGV/PGA > 0.1 s, and (3) PGV > 10 cm/s. All records are obtained from Pacific Earthquake Engineering Research Center website [40]. There are various procedures for detecting the velocity pulses of FD records, however in this study, the Baker's methodology is adopted to extract the dominant pulses and identify their amplitude, Ap, and period, Tp. The ratio of the fundamental period of the structure to the pulse period, T1/Tp, can greatly affect the structure's response [7,16]. The pulse period, which is an important parameter for structural engineers, is computed as period associated with the maximum Fourier amplitude of the largest wavelet. The acceleration, velocity and displacement response spectra of all selected records, considering 5% damping, along with their mean and mean ± one standard deviation (S), are plotted in Fig. 2. The mean PGA and PGV of the NFD records are 0.24g and 39 cm/s, respectively, and for the FD records are 0.41g and 67 cm/ s, respectively. The mean value of the ratio of the amplitude of the extracted velocity pulses to the PGV of the original FD records is 0.73.
∂ 2p 1 ∂ 2p = 2 2 2 ∂xi cw ∂t
(4)
where p is the hydrodynamic pressure in excess of hydrostatic pressure, and cw is the acoustic wave speed in the water [46]. The prescribed boundary conditions of the reservoir domain are: (1) zero-pressure boundary at the free surface, (2) transmitting boundary conditions at the truncated far-end, (3) rigid boundary conditions at the reservoir bottom, and (4) water-structure interaction at the dam-reservoir interface. The last boundary condition, considering no flow across the dam-reservoir interface, can be written as
∂p ∂ 2u ni = −ρW t i ni ∂xi ∂t
(5)
where ρW is the water density, and ni is the unit normal vector on the interface. By discretizing the pressure field of the reservoir domain as {p}=[N]p{p}e, in which {p}e is the vector of unknown nodal pressures, the finite element equation of the reservoir domain is obtained as:
[H ]{p ̈ } + [D]{p ̇ } + [E ]{p} = −ρW [Q]T {u}̈ + {FW }
(6)
where [H], [D] and [E] are the equivalent mass, damping and stiffness matrices of the reservoir domain, respectively, {FW} is the force vector due to ground shaking on the dam-reservoir interface and total acceleration on the rest of the boundaries, and [Q] is the coupling matrix which relates the water pressure and hydrodynamic forces, {FH}, on the dam-reservoir interface:
3. Gravity dam-reservoir numerical model
[Q]{p} = {FH}
Pine Flat dam is selected as case-study. It is 122 m high gravity dam, with the base width of 96.8 m and crest width of 9.8 m. It is a well-studied gravity dam in literature [41–46]. The tallest two-dimensional non-over-flow monolith of the dam is numerically modeled using the finite-element method, in plane-stress manner, along with its full reservoir as shown in Fig. 3. The four-node isoparametric tetra-lateral elements are utilized in the entire model. The reservoir length is considered to be 5 times the dam height. The rock foundation is assumed to be rigid; however the dam and the underlying rigid foundation may be separated through a base joint. Two types of nonlinearity are considered in the analysis: (1) the material nonlinearity due to cracking of the dam concrete; and (2) the geometric nonlinearity due to opening, closing and frictional sliding of the base
(7)
Eqs. (3) and (6) form the coupled equations of the dam-reservoir system. They can be solved using direct or staggered solution methods [48]. 3.1. Material nonlinearity The nonlinear behavior of mass concrete is modeled using the plastic-damage method which is a continuum homogeneous damage mechanics approach [49]. In this method, the degradation of the material's stiffness beyond its strength is modeled using the damage parameter, d, which is assumed to be function of plastic strains. In this study, only the tensile damage is considered because the gravity dams 623
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Y. Yazdani, M. Alembagheri
Fig. 2. Acceleration, velocity and displacement response spectra, considering 5% damping, for all selected ground motions: (a) the NFD records, (b) the FD records, (c) the equivalent pulses of the FD records obtained from the Baker's procedure.
[50]. For detailed description of the model, one can consult [51]. The considered constitutive behavior of the mass concrete in tension is illustrated in Fig. 4. The curve and parameters have been selected based on the experimental data [52]. The material properties are tabulated in Table 1. They are assumed the same in static and dynamic analyses. Depending on the direction of the relative displacement, the gravity
are not vulnerable to compressive damage [1]. The stiffness degradation is formulated as
E = (1 − d ) E0
(8)
where E0 is the initial (undamaged) modulus of the material. The tensile damage parameter can take values from zero, representing the undamaged material, to one, which represents total loss of strength
Fig. 3. (a) Tallest non-over-flow monolith of Pine Flat dam (all dimensions in meter), (b) finite element mesh of the dam and part of the reservoir.
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and c =0 are assumed in this study for the base joint. No contact damping is considered in this model. Energy can be dissipated through friction. The penetration of the water inside the base joint is ignored. However, the hydraulic connectivity between the base joint and the reservoir would lead to elevated pore pressures at the base and higher likelihood of failure. 3.3. Cases analyzed Two cases of the dam-reservoir model are analyzed. In the case 1 (the integrated model), the dam is modeled in the integrated state without the base joint. Therefore, just the material nonlinearity is considered and the overstressing is the only failure mode of the dam structure. In the case 2 (the base-cracked model), the base joint is added to the model and both material and geometric nonlinearities are considered. Both models have the same mesh density as shown in Fig. 3(b). The material properties of the concrete and the water are tabulated in Table 1. First, the dam is statically loaded under its selfweight and hydrostatic pressure of the full reservoir, and then the damreservoir system is dynamically analyzed under the selected near-fault strong ground motions. The modified Rayleigh damping [53,54] is considered in the seismic analyses for the dam body such that it produces about 5% critical damping in the range of first six vibration modes of the integrated dam-reservoir system. Based on the considered properties, the fundamental period of the integrated dam-reservoir system is T1 =0.38 s
Fig. 4. Constitutive behavior of mass concrete in uniaxial tension [41]. Table 1 Material properties used in this study. Material
Property
Dimension
Value
Density, ρc Undamaged Young's modulus, E0 Poisson's ratio, νc Tensile strength, σt0
3
Concrete
kg/m GPa – MPa
2400 30 0.2 2.9
Water
Density, ρw Acoustic wave speed, cw
kg/m3 m/s
1000 1440
dam may separately crack from its base or neck [41]. Hence, two damage indices can be locally defined on the cracking-susceptible areas of the dam body. These areas are depicted in Fig. 3(a) as base and neck sub-regions. The damage indices, DI, for both sub-regions can be defined as
DIi =
∑e | i ∫ (de ) dAe Ae
∑e | i ∫ dAe Ae
4. Analysis results After the static loading, the analysis cases are analyzed under the FD, the equivalent pulses, and the NFD (ordinary) near-fault records. Only the fault-normal component of each record is applied to the damreservoir system. The engineering demand parameters (EDPs) such as the peak relative displacements along the dam height, the maximum hydrodynamic pressure along the dam-reservoir interface, the defined local damage indices (DIs), and the dissipated energy through cracking (Ecracking) are monitored for both cases. The local DIs and Ecracking can describe the cracking response of the dam body. They become greater than zero when at-least one element cracks during the seismic analysis. Additional EDPs such as the base joint sliding and opening displacements, and the dissipated energy through friction (Efriction) are also recorded for the case 2.
i = base or neck (9)
where de is the tensile damage of element e with area of Ae. The summation is done on the entire sub-region i. It simply calculates the weighted average of the damage variables over the prescribed base or neck sub-regions. Therefore, it is a measure of the amount of damage that the dam may locally experience. If entire sub-region i is cracked during the seismic analysis, then DIi will be unit. This shows the complete base cracking for the base sub-region, and the diffused cracking of the neck sub-region. The energy dissipated through cracking process, Ecracking, can be employed as a global measure of damage imposed to the dam body.
4.1. Comparison of the response under FD and NFD near-fault ground motions The peak relative displacements along the dam height into the downstream (UDS) and the upstream (UUS) directions are separately illustrated in Fig. 5 for the FD and NFD records. Because the dam monolith is un-symmetric, it is required to assess its relative displacement separately in the opposite directions. Also shown in this figure is the maximum hydrodynamic pressure (PH) along the dam-reservoir interface. The mean and mean ± one standard deviation (S) of the obtained results are depicted in each plot. From Fig. 5, the maximum relative displacements are higher in the DS direction than the US direction in the case 1, however an opposite trend is observed for the case 2. Therefore, the presence of the base joint decreases the DSoriented relative displacements and increases the US-oriented relative displacements. It also partly decreases the maximum hydrodynamic pressure along the dam height. The mean PGA and PGV of the FD records are considerably larger than the NFD records, but the mean responses under the FD records are higher just up to 30% with respect to the NFD records. This shows that the structural response of gravity dams is not essentially controlled by the presence of the initial velocity pulse; and other properties of the near-fault records can significantly contribute in the seismic response.
3.2. Joint nonlinearity The geometric nonlinearity is considered by inserting a base joint at the dam-foundation interface. It allows the relative tangential and normal displacements of the joint surfaces through a contact definition. In normal direction, the contact is defined such that unbounded normal force can be transmitted between the joint surfaces when the contact clearance becomes zero. In tangential direction, the contact is governed by Coulomb friction model which is assumed to be rigidplastic without exhibiting any difference between the static and kinetic values of the friction coefficient. The maximum shear force that can be transmitted between the contact surfaces is limited by the sliding threshold, τm, beyond which the surfaces will slip against each other. The sliding threshold is defined as
τm = σn (tan φ) + c
(10)
where σn is the normal stress (pressure) between the two contact surfaces, φ is the friction angle between the surfaces, and c is the cohesive strength [50]. The friction coefficient is defined as μ=tan φ; it is assumed to be independent of contact pressure. The values of μ=1 625
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Forward-directivity records UUS UDS PH
Case
Non forward-directivity records UDS UUS PH
1
Dam height (m)
120
80
40
0 50
cv,max =
25
0
50
0.85
25
0
0 50
1.5
0.96
0.53
25
0
50
1.19
25
0 1.5
0.98
0
0.65
2
Dam height (m)
120
80
40
0 50
cv,max =
25
0.58
0
50
25
0
0 50
1.5
0.89
0.46
25
0.55
0
50
25
0 1.5
1.12
0
0.51
Mean ± 1S
Mean
Fig. 5. Peak relative displacements in downstream (UDS) and upstream directions (UUS), in cm, and maximum hydrodynamic pressure (PH), in MPa, along the dam height under the FD and NFD records for the cases 1 and 2.
40 cm, and the heel and toe opening up to 4 cm may be observed in the base joint of the base-cracked model (the case 2). The dissipated energy through friction is much more than the dissipated energy through cracking damage. Therefore, the near-fault ground motions can result in intense nonlinear response of both models. A representative quadratic curve is separately fitted to the obtained results of the FD and NFD records to follow the trend of the responses. The trend lines show that, unlike to the multi-story buildings, here at the various levels of the IMs, the ordinary NFD records often cause more EDP values than the FD records, specifically for higher intensities. Therefore, the strong dominant velocity pulses may not govern the seismic demands of the gravity dams subjected to the FD near-fault records. Also, using Sa(T1,5%) as IM results in close trend lines between both FD and NFD records for the two cases studied. Selection of proper IM is a critical issue in performance-based engineering. The better IM is one that well correlates with the EDPs. To study the correlation of the obtained IM-EDP pairs, R-squared (R2) values of the quadratic trend lines in Figs. 6 and 7 are separately tabulated in Table 2. However, the optimality of IM-EDP pairs can be rigorously investigated using the framework of probabilistic seismic demand model, but the R-squared is a statistical measure of how close the data are to the fitted regression line, and hence how well an IM can predict an EDP. A large R2 implies a good correlation between IM and EDP, while a small R2 implies that IM is a poor predictor of EDP. Based on Table 2, totally the low R2 values and poor correlation between various EDPs and PGV indicates that PGV is generally a poor predictor of structural response of the gravity dams under both FD and NFD ground motions. However, its contrary was observed for the multistory buildings [55]. The PGA and Sa(T1,5%) are better predictors of the seismic responses of both models under the NFD motions than for the FD motions. Nonetheless, Sa(T1,5%) results in higher R2 values with respect to PGA in almost cases. Therefore, it is a more stable IM
This matter was not observed for the multi-story buildings where the presence of the velocity pulse may greatly magnify the seismic responses [55]. The maximum coefficient of variation of the responses (cv,max) along the dam height, which is reported for each EDP in Fig. 5, reveals that the NFD records cause more scattering for all shown EDPs with respect to the FD records. So, it seems that the pulse nature of the FD records may lead to lower dispersion in the structural response of gravity dams. It is opposite to what was observed for the multi-story buildings [55]. The maximum hydrodynamic pressure shows lower dispersion with respect to the peak relative displacements. The variation of peak seismic responses as EDPs against various earthquake intensity measures (IMs) is investigated in Figs. 6 and 7 for the cases 1 and 2, respectively. In these figures, the EDPs under the FD and NFD record sets are plotted in terms of PGA, PGV, and 5% damping spectral acceleration at the first mode period of the integrated dam-reservoir system, Sa(T1,5%). The first and second rows of each figure show, respectively, the maximum absolute relative crest displacement and the maximum hydrodynamic pressure adjacent to the dam heel. These figures illustrate how the record-to-record variability of the near-fault ground motions affects the seismic response of the integrated and the base-cracked gravity dams. The variability of the EDP values could be considered as aleatory variability for the models based on the chosen IMs. This is not, however, the aleatory variability of the near-fault records. It is rather the aleatory variability of the model chosen to represent the EDP responses. The presence of the base joint generally decreases the peak crest displacement, the maximum hydrodynamic pressure, the damage (cracking) energy dissipation and the DIneck. The base cracking almost vanishes due to stress relaxation because of the presence of the base joint, so the DIbase is not shown in Fig. 7 for the case 2. However, more than 90% of dam base may crack along the dam-foundation interface of the integrated model (the case 1). The base sliding up to more than
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Fig. 6. IM-EDP curves for different seismic responses and earthquake parameters, the case 1.
for seismic assessment of the integrated and the base-cracked gravity dams under the near-fault ground motions, and hence it can be used as an IM instead of PGA and PGV.
4.2. Effectiveness of equivalent pulses To better assess how the presence of the pulse may control the seismic response of the integrated and the base-cracked gravity dams subjected to the FD ground motions, in this section, the seismic 627
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Fig. 7. IM-EDP curves for different seismic responses and earthquake parameters, the case 2.
dam, and subscripts ep and ae represent equivalent pulse and actual earthquake, respectively. It is apparent that Ψ=1 shows the same prediction of EDP between the equivalent pulse and the actual FD record. This parameter is plotted in Fig. 9 for various EDPs against T1/ Tp, Baker, where T1 is the initial fundamental period of the integrated dam-reservoir system and Tp, Baker is the pulse period from the Baker's procedure. The fundamental vibration period of the dam-reservoir coupled system is as low as T1 =0.38 s, so the ratio of T1/Tp, Baker is totally less than 1.0 because the dominant pulse period of the FD records is essentially more than 0.4 s. From Fig. 9, the equivalent pulses generally underestimate the EDPs with respect to the actual FD records. This can be attributed to the fact that the Baker's procedure only attempts to replicate the velocity pulse, so it cannot reproduce the frequency content of the actual FD ground motions beyond that associated with the velocity pulse [27]. However, the specific nature of the pulses causes Ψ > 1 for the toe opening of the case 2 many times. It is observed that the equivalent pulses estimate the seismic demands of the integrated and the base-cracked gravity dams within 20% of
responses are compared under the FD records and their equivalent pulses extracted using the Baker's procedure. The mean values of the maximum absolute relative displacement and the maximum hydrodynamic pressure along the dam height are depicted in Fig. 8. It is observed that the peak responses under the equivalent pulses are essentially lower than the responses under the actual FD records. It can be explained by comparing the mean Sa(T1,5%) and Sv(T1,5%) of the FD records and the equivalent pulses which are, respectively, 0.81g against 0.30g, and 43.5 cm/s against 12.7 cm/s. It shows that the equivalent pulses extracted using the Baker's procedure cannot generally reproduce the seismic demands of the integrated and the basecracked gravity dams. The pulse effectiveness for the other EDPs can be investigated by defining a response ratio as
Ψ (T , Tp ) =
EDPep (T , Tp ) EDPae (T , Tp )
(11)
where Tp is the dominant pulse period, T is the natural period of the 628
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Fig. 7. (continued)
governed by the strong pulse, and the equivalent pulses are capable of reproducing structural response to the actual FD ground motions. The efficiency of the proposed period range can be rigorously studied by comparing the velocity response spectra of the FD records
those caused by the actual FD records for 0.7 < T1/Tp, Baker < 1.0. This has been shown for the building systems as well [56]. This range of Ψ, i.e. 0.8 < Ψ < 1.2, may be considered sufficiently accurate for engineering calculations. Therefore, in this range of periods the response is Table 2 R2 values of the quadratic trend lines for different IM-EDP pairs. Case 1 (Integrated model) IM: Ground motion: Peak crest displacement Peak hydrodynamic pressure Damage energy dissipation DIbase DIneck Friction energy dissipation Base sliding Heel opening Toe opening
PGA FD 0.55 0.73 0.72 0.49 0.75 – – – –
NFD 0.60 0.82 0.75 0.81 0.82 – – – –
PGV FD 0.36 0.30 0.27 0.24 0.25 – – – –
Case 2 (Base-cracked model)
NFD 0.41 0.70 0.63 0.59 0.66 – – – –
Sa(T1,5%) FD 0.64 0.82 0.82 0.73 0.79 – – – –
629
NFD 0.71 0.93 0.92 0.94 0.84 – – – –
PGA FD 0.53 0.72 0.60 – 0.64 0.42 0.44 0.50 0.44
NFD 0.79 0.86 0.67 – 0.69 0.70 0.71 0.74 0.84
PGV FD 0.39 0.25 0.36 – 0.30 0.47 0.47 0.18 0.33
NFD 0.69 0.68 0.57 – 0.56 0.77 0.75 0.54 0.71
Sa(T1,5%) FD 0.65 0.81 0.61 – 0.70 0.49 0.40 0.62 0.36
NFD 0.80 0.85 0.94 – 0.90 0.80 0.81 0.90 0.78
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Case 1 (Integrated model) Peak hydrodynamic pressure
Peak relative displacement
Case 2 (Base-cracked model) Peak hydrodynamic pressure
Peak relative displacement
Dam height (m)
120
80
40
0
50
25
0
1.5
Actual FD records
0
50
25
0 1.5
0
Equivalent pulses
Fig. 8. Mean values of the maximum absolute relative displacement, in cm, and the maximum hydrodynamic pressure, in MPa, along the dam height under the actual FD records and their equivalent pulses.
and their equivalent pulses. In Fig. 10, three representative sets of response spectra are compared for T1/Tp, Baker =0.95, 0.54, and 0.13. The perfect match between the response spectra is observed for T1/Tp, Baker =0.95. By decreasing the period ratio, in Fig. 10(b) and (c), the velocity response spectra of the equivalent pulses does not closely match the response spectra of the FD ground motions. Furthermore, other peaks are observed in the response spectra of the FD ground motions in the vicinity of the natural mode periods of the damreservoir system that control its behavior. These peaks are filtered out when simplified pulses are used to represent the FD ground motions. In Fig. 10(c), the period of the equivalent pulse is too long to excite the dam-reservoir model. Therefore, below the proposed period range, the equivalent pulse cannot sufficiently reproduce the frequency content and spectral values of the actual FD record specifically in the vicinity of natural frequencies of the dam, and those frequencies which govern the dam response either override or follow after the equivalent pulse [27]. So in these ranges, the equivalent pulse parameters are not adequate IM. The effects of velocity pulse parameters on the nonlinear response of both models are investigated by plotting the energy dissipation through damage and friction under the FD records in terms of period, amplitude and specific energy density of the dominant velocity pulse, Tp, Ap and SED (Fig. 11). As it is observed, the dissipated energies are generally decreased by increasing the period of the dominant velocity pulse, but, they are increased by increasing the amplitude of the dominant velocity pulse. However, about the energy content of the records (SED) no specific conclusion can be drawn especially about the damage energy dissipation. The same trend is observed for the other response parameters such as the local DIs, the base joint sliding and opening displacements. 5. Conclusions The nonlinear seismic response of concrete gravity dams under near-fault ground motions was investigated. Considering Pine Flat dam as case study, it was numerically modeled along with its full reservoir using the finite element method base on Eulerian-Lagrangian approach. Two sources of nonlinearity were considered: the material nonlinearity of mass concrete using the plastic-damage method; and the geometric nonlinearity of a prescribed base joint. Therefore, two main potential failure modes of gravity dams were studied: the tensile cracking; and the sliding along the dam-foundation interface. Two cases of the dam-reservoir model were assessed: the integrated model without any prescribed joints; and the base-cracked model including a prescribed base joint. Seventy-five forward-directivity (FD) and sixty ordinary (non forward-directivity, NFD) near-fault records were se-
Fig. 9. Variation of the response ratio, Ψ, for various EDPs in terms of T1/Tp,Baker for (a) the case 1, (b, c) the case 2.
lected. The comparison of the nonlinear seismic responses of the analysis cases subjected to the selected near-fault ground motions shows that the structural response of gravity dams may not essentially controlled by the presence of the initial velocity pulse; and other properties of the near-fault records can significantly contribute in the response. It was shown that the NFD records cause more scattering in 630
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Fig. 10. Comparison of the velocity response spectra of the FD records and their equivalent pulses when T1/Tp,Baker is equal to (a) 0.95, (b) 0.54, and (c) 0.13.
Fig. 11. Dissipated energy through damage and friction under the FD records in terms of their dominant velocity pulse period, Tp, amplitude, Ap, and specific energy density, SED.
References
the engineering demand parameters (EDPs) than the FD records. Therefore, the pulse nature of the FD records may cause lower dispersion in the structural response of gravity dams. Also, the ordinary NFD records often cause more EDP values than the FD records at the various levels of earthquake intensity measures (IMs). PGV is generally a poor predictor of structural response of the gravity dams under both FD and NFD near-fault ground motions. PGA and Sa(T1,5%) are better predictors of the seismic response of both models for the NFD ground motions than for the FD records. Nonetheless, Sa(T1,5%) it is a more stable IM for seismic assessment of the integrated and the base-cracked gravity dams under the near-fault ground motions. The equivalent pulses of the FD records were extracted using the Baker's procedure. The comparison of the seismic response under the actual FD ground motions and their equivalent pulses shows that the equivalent pulses generally underestimate the EDPs with respect to the actual FD records. This was attributed to the fact that the Baker's procedure only attempts to replicate the velocity pulse so it cannot reproduce the frequency content of the FD ground motions. But the equivalent pulses can predict the seismic demands of the integrated and the base-cracked gravity dams within 20% of those caused by the FD records for 0.7 < T1/Tp, Baker < 1.0. Therefore, in this range of periods the response is governed by the FD pulse and the equivalent pulses are capable of reproducing structural response to the FD ground motions. Finally it was shown that the nonlinear seismic responses of both models are decreased by increasing the period of the dominant velocity pulse; however, they are increased by increasing the amplitude of the dominant velocity pulse.
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Nonlinear seismic response of a gravity dam under near-fault ground motions and equivalent pulses
crossmark
⁎
Y. Yazdania, M. Alembagherib, a b
Department of Civil and Environmental Engineering, Amirkabir University of Technology, Tehran, Iran Department of Civil and Environmental Engineering, Tarbiat Modares University, Tehran, Iran
A R T I C L E I N F O
A BS T RAC T
Keywords: Gravity dams Nonlinear structural response Near-fault Forward-directivity Velocity pulse
In this paper, the nonlinear seismic response of gravity dams to forward-directivity and ordinary (non forwarddirectivity) near-fault earthquake ground motions is investigated. Considering Pine Flat dam as case study, it is numerically modeled along with its full reservoir using the finite element method. Two sources of nonlinearity are considered in the analysis: (1) the material nonlinearity of dam concrete, and (2) the geometric nonlinearity by inserting a joint at the base of the dam. Seventy-five forward-directivity and sixty ordinary near-fault ground motions are used to obtain statistically significant results. The equivalent representative pulses of the selected forward-directivity ground motions are extracted using a well-known methodology. The dam-reservoir model is analyzed under the equivalent pulses as well to identify the cases for which the equivalent pulses can capture the structural response to the actual forward-directivity ground motions. Finally, the effects of pulse period, amplitude and energy on the seismic response of the dam-reservoir system are studied.
1. Introduction High concrete gravity dams are among important infrastructures playing key role in national water and power management systems. Because of large water reservoir impounded behind a high dam, its stability and safety specifically during seismic events is of great importance. The gravity dams are typically located on rock or firm soil foundations which may contain active faults near dam sites [1]. Therefore, they may undergo near-fault earthquake ground motions; for example, Pacoima dam in the US experienced high shakings during the near-fault ground motions of San Fernando 1971, and Northridge 1994 [2,3]. The potential failure modes of gravity dams that may be triggered by an earthquake can be classified as: (1) overstressing mainly in tension, which results in cracking response of the dam body; and (2) sliding along prescribed or cracked paths specifically at the dam-foundation interface [4]. Directivity effects can considerably influence ground motions in the proximity of causative faults. The extent of near-fault region depends on earthquake magnitude. Forward-directivity results when the fault rupture propagates toward the site at a velocity nearly equal to the shear wave velocity, and the direction of fault slip is aligned with the site. This causes the wavefront to arrive as a single large pulse [5]. The forward-directivity is a dynamic phenomenon that produces no permanent ground displacement and hence two-sided velocity pulses.
⁎
Most of the energy in forward-directivity ground motions is concentrated in a narrow frequency band. These motions are characterized by a distinct, high intensity velocity pulse at the beginning of time-history records, which are oriented in perpendicular direction relative to the fault plane [5]. However, not all of near-fault ground motions are pulse-type, and they may have no pulse-like characteristics. It has been well determined that the forward-directivity pulses are critical in the analysis and design of structures in the near-fault areas. They can result in high seismic demands tending to drive structures into the nonlinear range. Typical measures such as peak ground acceleration (PGA) or spectral acceleration at periods of natural modes of structure may no longer serve as effective intensity measures [6]. The narrow band nature of the velocity pulse implies that the forwarddirectivity ground motions can be represented using equivalent pulse models [7–12]. The main characteristics of these pulses are: (1) pulse period, (2) pulse amplitude, (3) phase parameter, and (4) number of significant pulses [7,13]. These parameters can be determined such that the representative pulse acceptably approximates the original pulse-like motion [7]. Because the parameters of the forward-directivity pulses may control the response of structures [14–22], the effects of pulse-type ground motions should be considered in structural analysis. Many researchers have studied the characteristics of forwarddirectivity ground motions [5,23,24], detecting and extracting the dominant representative pulses [7,25–27], and structural response to
Corresponding author. E-mail address: [email protected] (M. Alembagheri).
http://dx.doi.org/10.1016/j.soildyn.2016.11.003 Received 21 June 2016; Received in revised form 10 September 2016; Accepted 2 November 2016 0267-7261/ © 2016 Elsevier Ltd. All rights reserved.
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Fig. 1. Acceleration, velocity and displacement time-histories along with specific energy density of the actual ground motion recorded at Pacoima Dam station and its velocity pulse: (a) 1971 San Fernando earthquake, and (b) 1994 Northridge earthquake.
The objective of this paper is to compare the nonlinear seismic response of a gravity dam model to forward-directivity (FD) and ordinary (non forward-directivity, NFD) near-fault ground motions. The dam is numerically modeled along with its reservoir using finite element method based on Eulerian-Lagrangian approach. Two sources of nonlinearity are considered in the analysis: (1) the material nonlinearity of dam concrete, and (2) the geometric nonlinearity by inserting a joint at the base of the dam. Seventy-five forward-directivity and sixty ordinary near-fault ground motions are used to obtain statistically significant results. The equivalent representative pulses of the selected FD ground motions are extracted using the methodology introduced by Baker [9]. The dam-reservoir model is analyzed under the equivalent pulses as well to identify the cases for which the equivalent pulses can capture the structural response to the actual FD ground motions. Finally, the effects of pulse properties, i.e. the pulse period and amplitude, on the seismic response of the damreservoir system are studied.
the actual pulse-type motions or their simplified representative pulses [14–19,28–30]. In a series of research, Bayraktar and co-workers [31– 33] investigated near- and far-fault ground motion effects on the nonlinear dynamic response of various dam types such as gravity, arch, concrete faced rock-fill and clay core rock-fill dams. They have used limited number of near- and far-field ground motions for the analysis which have approximately identical PGAs. They showed that near-fault ground motions have different impacts on the dam types, and there is more seismic demand on displacements and stresses when the dam is subjected to near-fault ground motion. They also concluded that the seismic behavior of the concrete gravity dams is considerably affected from the length of the reservoir. Akkose and Seismik [34] focused on nonlinear seismic response of a concrete gravity dam subjected to nearand far-fault ground motions. The elastoplastic behavior of the dam concrete was idealized using Drucker–Prager yield criterion. They compared the seismic response of the selected concrete dam subjected to both near- and far-fault ground motions. Wang et al. [35] evaluated the effects of near- and far-fault ground motions on seismic performance of a concrete gravity dam. Four different near-fault ground motion records with an apparent velocity pulse were used. They showed the effects of near-fault ground motions on seismic performance of concrete gravity dams and demonstrated the importance of considering the near-fault ground excitations. Zhang and Wang [36] studied the near- and far-fault ground motion effects on nonlinear dynamic response of a concrete gravity dam. For this purpose, 10 asrecorded earthquake records which display ground motions with an apparent velocity pulse were selected to represent the near-fault ground motion characteristics. They found that near-fault ground motions would cause more severe damage to the dam body than farfault ground motions. Huang [37] investigated the effects of near-fault ground motions on the nonlinear seismic response behavior of concrete gravity dams. In particular, he evaluated the characteristics of different aspects of near-fault ground motions and examined the significance of various near-fault ground motion parameters. However, still lack of specific study about the effects of forward-directivity pulses on the seismic demand analysis of gravity dams is felt.
2. Forward-directivity ground motions and equivalent pulses Ground motions recorded close to a ruptured fault can be significantly different from far-fault ones. The near-fault ground motions can be classified as “pulse-like” and “non-pulse-like”. The pulse-like motions can be subdivide into “forward-directivity” and “fling”. The specific nature of the pulse-like motion due to forward-directivity may be revealed in its velocity time-history. The peak ground velocity (PGV) of the near-fault ground motions is substantially higher than the farfault ground motions. Previous studies about the seismic response of structures located in near-fault regions have shown that time-domain representation of the near-fault ground motions is preferable to response spectrum representation because of concentration of record's energy in a single pulse of motion [5,11,13,25]. Baker [9] described a reasonable widely-used method for quantitatively identifying ground motions containing strong velocity pulses. He employed wavelet-based signal processing to detect and extract the largest velocity pulse from a given ground motion. He also established 622
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joint. The coupled dam-reservoir system is analyzed using the EulerianLagrangian approach. In this approach, displacement-based Lagrangian formulation is considered for the dam body through the following differential equation
some criteria for classifying a ground motion as pulse-like using the size of the extracted pulse relative to the original ground motion. The Daubechies wavelet of order 4 was utilized as mother function in continuous wavelet transform (CWT), because it approximates the shape of many velocity pulses specifically in the fault-normal components of ground motions. He employed two additional criteria to identify the pulse-like records potentially caused by directivity effects: (a) the pulse arrives at the beginning of the strong ground motion, and (b) the absolute amplitude of the velocity pulse is large relative to the remainder of the record. By extracting a wavelet from the ground motion that best matches the dominant pulse, its amplitude and pseudo-period determine the amplitude and period of the equivalent velocity pulse, respectively [9]. Two examples of forward-directivity ground motions recorded at Pacoima dam site are shown in Fig. 1 along with their representative pulses. The velocity pulse is clearly observed in velocity time-history of these records. The energy of a time signal, either the original ground motion or the pulse, can be approximated using specific energy density (SED) [9,38]:
SED (t ) =
∫0
t
V 2 (ξ ) dξ
∂σji ∂xj
+ Fi = ρc
∂ 2ui ∂t 2
(2)
where σji = σij is the Cauchy stress tensor, ui is the displacement, ρc is the dam density, and Fi is the body force per unit volume [47]. Using the Galerkin's weighted residual method, and by discretizing the displacement vector of the dam body as {u}=[N]u{u}e, in which [N]u is the matrix of shape functions and {u}e is the vector of unknown nodal displacements, the finite element formulation of the dam body is obtained as
[M ]{u}̈ + [C ]{u}̇ + {FR} = {FS} + {FH}
(3)
where [M] and [C] are the structural mass and damping matrices, respectively, {FR} is the vector of nonlinear resisting force, {FS} is the vector of external forces due to ground acceleration on the dam, {FH} is the vector of hydrodynamic forces, and ‘dot’ represents derivative with respect to time. The governing differential equation of the reservoir domain considering pressure-based Eulerian formulation, assuming that the water is linearly compressible, neglecting its internal viscosity and having small amplitude irrotational motion, can be represented as wave equation
(1)
where V(ξ) is the ground motion velocity at time ξ. This parameter is also plotted in Fig. 1 for both signals. It is observed that the energy of the equivalent pulses relative to the whole records has about 50% compatibility. In this study, seventy-five forward-directivity (FD) and sixty ordinary (non forward-directivity, NFD) near-fault ground motions from 30 earthquakes are selected for the analysis. All records are taken from stations located on rock or firm soil with shear wave velocity more than 250 m/s, which is compatible with the sites where the gravity dams are founded. The FD records are compiled from a database developed by Baker [39]. The NFD near-fault ground motions, which are classified as non-pulse-like, are chosen based on these criteria: (1) closest distance to the ruptured fault R < 20 km, (2) PGV/PGA > 0.1 s, and (3) PGV > 10 cm/s. All records are obtained from Pacific Earthquake Engineering Research Center website [40]. There are various procedures for detecting the velocity pulses of FD records, however in this study, the Baker's methodology is adopted to extract the dominant pulses and identify their amplitude, Ap, and period, Tp. The ratio of the fundamental period of the structure to the pulse period, T1/Tp, can greatly affect the structure's response [7,16]. The pulse period, which is an important parameter for structural engineers, is computed as period associated with the maximum Fourier amplitude of the largest wavelet. The acceleration, velocity and displacement response spectra of all selected records, considering 5% damping, along with their mean and mean ± one standard deviation (S), are plotted in Fig. 2. The mean PGA and PGV of the NFD records are 0.24g and 39 cm/s, respectively, and for the FD records are 0.41g and 67 cm/ s, respectively. The mean value of the ratio of the amplitude of the extracted velocity pulses to the PGV of the original FD records is 0.73.
∂ 2p 1 ∂ 2p = 2 2 2 ∂xi cw ∂t
(4)
where p is the hydrodynamic pressure in excess of hydrostatic pressure, and cw is the acoustic wave speed in the water [46]. The prescribed boundary conditions of the reservoir domain are: (1) zero-pressure boundary at the free surface, (2) transmitting boundary conditions at the truncated far-end, (3) rigid boundary conditions at the reservoir bottom, and (4) water-structure interaction at the dam-reservoir interface. The last boundary condition, considering no flow across the dam-reservoir interface, can be written as
∂p ∂ 2u ni = −ρW t i ni ∂xi ∂t
(5)
where ρW is the water density, and ni is the unit normal vector on the interface. By discretizing the pressure field of the reservoir domain as {p}=[N]p{p}e, in which {p}e is the vector of unknown nodal pressures, the finite element equation of the reservoir domain is obtained as:
[H ]{p ̈ } + [D]{p ̇ } + [E ]{p} = −ρW [Q]T {u}̈ + {FW }
(6)
where [H], [D] and [E] are the equivalent mass, damping and stiffness matrices of the reservoir domain, respectively, {FW} is the force vector due to ground shaking on the dam-reservoir interface and total acceleration on the rest of the boundaries, and [Q] is the coupling matrix which relates the water pressure and hydrodynamic forces, {FH}, on the dam-reservoir interface:
3. Gravity dam-reservoir numerical model
[Q]{p} = {FH}
Pine Flat dam is selected as case-study. It is 122 m high gravity dam, with the base width of 96.8 m and crest width of 9.8 m. It is a well-studied gravity dam in literature [41–46]. The tallest two-dimensional non-over-flow monolith of the dam is numerically modeled using the finite-element method, in plane-stress manner, along with its full reservoir as shown in Fig. 3. The four-node isoparametric tetra-lateral elements are utilized in the entire model. The reservoir length is considered to be 5 times the dam height. The rock foundation is assumed to be rigid; however the dam and the underlying rigid foundation may be separated through a base joint. Two types of nonlinearity are considered in the analysis: (1) the material nonlinearity due to cracking of the dam concrete; and (2) the geometric nonlinearity due to opening, closing and frictional sliding of the base
(7)
Eqs. (3) and (6) form the coupled equations of the dam-reservoir system. They can be solved using direct or staggered solution methods [48]. 3.1. Material nonlinearity The nonlinear behavior of mass concrete is modeled using the plastic-damage method which is a continuum homogeneous damage mechanics approach [49]. In this method, the degradation of the material's stiffness beyond its strength is modeled using the damage parameter, d, which is assumed to be function of plastic strains. In this study, only the tensile damage is considered because the gravity dams 623
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Fig. 2. Acceleration, velocity and displacement response spectra, considering 5% damping, for all selected ground motions: (a) the NFD records, (b) the FD records, (c) the equivalent pulses of the FD records obtained from the Baker's procedure.
[50]. For detailed description of the model, one can consult [51]. The considered constitutive behavior of the mass concrete in tension is illustrated in Fig. 4. The curve and parameters have been selected based on the experimental data [52]. The material properties are tabulated in Table 1. They are assumed the same in static and dynamic analyses. Depending on the direction of the relative displacement, the gravity
are not vulnerable to compressive damage [1]. The stiffness degradation is formulated as
E = (1 − d ) E0
(8)
where E0 is the initial (undamaged) modulus of the material. The tensile damage parameter can take values from zero, representing the undamaged material, to one, which represents total loss of strength
Fig. 3. (a) Tallest non-over-flow monolith of Pine Flat dam (all dimensions in meter), (b) finite element mesh of the dam and part of the reservoir.
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and c =0 are assumed in this study for the base joint. No contact damping is considered in this model. Energy can be dissipated through friction. The penetration of the water inside the base joint is ignored. However, the hydraulic connectivity between the base joint and the reservoir would lead to elevated pore pressures at the base and higher likelihood of failure. 3.3. Cases analyzed Two cases of the dam-reservoir model are analyzed. In the case 1 (the integrated model), the dam is modeled in the integrated state without the base joint. Therefore, just the material nonlinearity is considered and the overstressing is the only failure mode of the dam structure. In the case 2 (the base-cracked model), the base joint is added to the model and both material and geometric nonlinearities are considered. Both models have the same mesh density as shown in Fig. 3(b). The material properties of the concrete and the water are tabulated in Table 1. First, the dam is statically loaded under its selfweight and hydrostatic pressure of the full reservoir, and then the damreservoir system is dynamically analyzed under the selected near-fault strong ground motions. The modified Rayleigh damping [53,54] is considered in the seismic analyses for the dam body such that it produces about 5% critical damping in the range of first six vibration modes of the integrated dam-reservoir system. Based on the considered properties, the fundamental period of the integrated dam-reservoir system is T1 =0.38 s
Fig. 4. Constitutive behavior of mass concrete in uniaxial tension [41]. Table 1 Material properties used in this study. Material
Property
Dimension
Value
Density, ρc Undamaged Young's modulus, E0 Poisson's ratio, νc Tensile strength, σt0
3
Concrete
kg/m GPa – MPa
2400 30 0.2 2.9
Water
Density, ρw Acoustic wave speed, cw
kg/m3 m/s
1000 1440
dam may separately crack from its base or neck [41]. Hence, two damage indices can be locally defined on the cracking-susceptible areas of the dam body. These areas are depicted in Fig. 3(a) as base and neck sub-regions. The damage indices, DI, for both sub-regions can be defined as
DIi =
∑e | i ∫ (de ) dAe Ae
∑e | i ∫ dAe Ae
4. Analysis results After the static loading, the analysis cases are analyzed under the FD, the equivalent pulses, and the NFD (ordinary) near-fault records. Only the fault-normal component of each record is applied to the damreservoir system. The engineering demand parameters (EDPs) such as the peak relative displacements along the dam height, the maximum hydrodynamic pressure along the dam-reservoir interface, the defined local damage indices (DIs), and the dissipated energy through cracking (Ecracking) are monitored for both cases. The local DIs and Ecracking can describe the cracking response of the dam body. They become greater than zero when at-least one element cracks during the seismic analysis. Additional EDPs such as the base joint sliding and opening displacements, and the dissipated energy through friction (Efriction) are also recorded for the case 2.
i = base or neck (9)
where de is the tensile damage of element e with area of Ae. The summation is done on the entire sub-region i. It simply calculates the weighted average of the damage variables over the prescribed base or neck sub-regions. Therefore, it is a measure of the amount of damage that the dam may locally experience. If entire sub-region i is cracked during the seismic analysis, then DIi will be unit. This shows the complete base cracking for the base sub-region, and the diffused cracking of the neck sub-region. The energy dissipated through cracking process, Ecracking, can be employed as a global measure of damage imposed to the dam body.
4.1. Comparison of the response under FD and NFD near-fault ground motions The peak relative displacements along the dam height into the downstream (UDS) and the upstream (UUS) directions are separately illustrated in Fig. 5 for the FD and NFD records. Because the dam monolith is un-symmetric, it is required to assess its relative displacement separately in the opposite directions. Also shown in this figure is the maximum hydrodynamic pressure (PH) along the dam-reservoir interface. The mean and mean ± one standard deviation (S) of the obtained results are depicted in each plot. From Fig. 5, the maximum relative displacements are higher in the DS direction than the US direction in the case 1, however an opposite trend is observed for the case 2. Therefore, the presence of the base joint decreases the DSoriented relative displacements and increases the US-oriented relative displacements. It also partly decreases the maximum hydrodynamic pressure along the dam height. The mean PGA and PGV of the FD records are considerably larger than the NFD records, but the mean responses under the FD records are higher just up to 30% with respect to the NFD records. This shows that the structural response of gravity dams is not essentially controlled by the presence of the initial velocity pulse; and other properties of the near-fault records can significantly contribute in the seismic response.
3.2. Joint nonlinearity The geometric nonlinearity is considered by inserting a base joint at the dam-foundation interface. It allows the relative tangential and normal displacements of the joint surfaces through a contact definition. In normal direction, the contact is defined such that unbounded normal force can be transmitted between the joint surfaces when the contact clearance becomes zero. In tangential direction, the contact is governed by Coulomb friction model which is assumed to be rigidplastic without exhibiting any difference between the static and kinetic values of the friction coefficient. The maximum shear force that can be transmitted between the contact surfaces is limited by the sliding threshold, τm, beyond which the surfaces will slip against each other. The sliding threshold is defined as
τm = σn (tan φ) + c
(10)
where σn is the normal stress (pressure) between the two contact surfaces, φ is the friction angle between the surfaces, and c is the cohesive strength [50]. The friction coefficient is defined as μ=tan φ; it is assumed to be independent of contact pressure. The values of μ=1 625
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Forward-directivity records UUS UDS PH
Case
Non forward-directivity records UDS UUS PH
1
Dam height (m)
120
80
40
0 50
cv,max =
25
0
50
0.85
25
0
0 50
1.5
0.96
0.53
25
0
50
1.19
25
0 1.5
0.98
0
0.65
2
Dam height (m)
120
80
40
0 50
cv,max =
25
0.58
0
50
25
0
0 50
1.5
0.89
0.46
25
0.55
0
50
25
0 1.5
1.12
0
0.51
Mean ± 1S
Mean
Fig. 5. Peak relative displacements in downstream (UDS) and upstream directions (UUS), in cm, and maximum hydrodynamic pressure (PH), in MPa, along the dam height under the FD and NFD records for the cases 1 and 2.
40 cm, and the heel and toe opening up to 4 cm may be observed in the base joint of the base-cracked model (the case 2). The dissipated energy through friction is much more than the dissipated energy through cracking damage. Therefore, the near-fault ground motions can result in intense nonlinear response of both models. A representative quadratic curve is separately fitted to the obtained results of the FD and NFD records to follow the trend of the responses. The trend lines show that, unlike to the multi-story buildings, here at the various levels of the IMs, the ordinary NFD records often cause more EDP values than the FD records, specifically for higher intensities. Therefore, the strong dominant velocity pulses may not govern the seismic demands of the gravity dams subjected to the FD near-fault records. Also, using Sa(T1,5%) as IM results in close trend lines between both FD and NFD records for the two cases studied. Selection of proper IM is a critical issue in performance-based engineering. The better IM is one that well correlates with the EDPs. To study the correlation of the obtained IM-EDP pairs, R-squared (R2) values of the quadratic trend lines in Figs. 6 and 7 are separately tabulated in Table 2. However, the optimality of IM-EDP pairs can be rigorously investigated using the framework of probabilistic seismic demand model, but the R-squared is a statistical measure of how close the data are to the fitted regression line, and hence how well an IM can predict an EDP. A large R2 implies a good correlation between IM and EDP, while a small R2 implies that IM is a poor predictor of EDP. Based on Table 2, totally the low R2 values and poor correlation between various EDPs and PGV indicates that PGV is generally a poor predictor of structural response of the gravity dams under both FD and NFD ground motions. However, its contrary was observed for the multistory buildings [55]. The PGA and Sa(T1,5%) are better predictors of the seismic responses of both models under the NFD motions than for the FD motions. Nonetheless, Sa(T1,5%) results in higher R2 values with respect to PGA in almost cases. Therefore, it is a more stable IM
This matter was not observed for the multi-story buildings where the presence of the velocity pulse may greatly magnify the seismic responses [55]. The maximum coefficient of variation of the responses (cv,max) along the dam height, which is reported for each EDP in Fig. 5, reveals that the NFD records cause more scattering for all shown EDPs with respect to the FD records. So, it seems that the pulse nature of the FD records may lead to lower dispersion in the structural response of gravity dams. It is opposite to what was observed for the multi-story buildings [55]. The maximum hydrodynamic pressure shows lower dispersion with respect to the peak relative displacements. The variation of peak seismic responses as EDPs against various earthquake intensity measures (IMs) is investigated in Figs. 6 and 7 for the cases 1 and 2, respectively. In these figures, the EDPs under the FD and NFD record sets are plotted in terms of PGA, PGV, and 5% damping spectral acceleration at the first mode period of the integrated dam-reservoir system, Sa(T1,5%). The first and second rows of each figure show, respectively, the maximum absolute relative crest displacement and the maximum hydrodynamic pressure adjacent to the dam heel. These figures illustrate how the record-to-record variability of the near-fault ground motions affects the seismic response of the integrated and the base-cracked gravity dams. The variability of the EDP values could be considered as aleatory variability for the models based on the chosen IMs. This is not, however, the aleatory variability of the near-fault records. It is rather the aleatory variability of the model chosen to represent the EDP responses. The presence of the base joint generally decreases the peak crest displacement, the maximum hydrodynamic pressure, the damage (cracking) energy dissipation and the DIneck. The base cracking almost vanishes due to stress relaxation because of the presence of the base joint, so the DIbase is not shown in Fig. 7 for the case 2. However, more than 90% of dam base may crack along the dam-foundation interface of the integrated model (the case 1). The base sliding up to more than
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Fig. 6. IM-EDP curves for different seismic responses and earthquake parameters, the case 1.
for seismic assessment of the integrated and the base-cracked gravity dams under the near-fault ground motions, and hence it can be used as an IM instead of PGA and PGV.
4.2. Effectiveness of equivalent pulses To better assess how the presence of the pulse may control the seismic response of the integrated and the base-cracked gravity dams subjected to the FD ground motions, in this section, the seismic 627
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Fig. 7. IM-EDP curves for different seismic responses and earthquake parameters, the case 2.
dam, and subscripts ep and ae represent equivalent pulse and actual earthquake, respectively. It is apparent that Ψ=1 shows the same prediction of EDP between the equivalent pulse and the actual FD record. This parameter is plotted in Fig. 9 for various EDPs against T1/ Tp, Baker, where T1 is the initial fundamental period of the integrated dam-reservoir system and Tp, Baker is the pulse period from the Baker's procedure. The fundamental vibration period of the dam-reservoir coupled system is as low as T1 =0.38 s, so the ratio of T1/Tp, Baker is totally less than 1.0 because the dominant pulse period of the FD records is essentially more than 0.4 s. From Fig. 9, the equivalent pulses generally underestimate the EDPs with respect to the actual FD records. This can be attributed to the fact that the Baker's procedure only attempts to replicate the velocity pulse, so it cannot reproduce the frequency content of the actual FD ground motions beyond that associated with the velocity pulse [27]. However, the specific nature of the pulses causes Ψ > 1 for the toe opening of the case 2 many times. It is observed that the equivalent pulses estimate the seismic demands of the integrated and the base-cracked gravity dams within 20% of
responses are compared under the FD records and their equivalent pulses extracted using the Baker's procedure. The mean values of the maximum absolute relative displacement and the maximum hydrodynamic pressure along the dam height are depicted in Fig. 8. It is observed that the peak responses under the equivalent pulses are essentially lower than the responses under the actual FD records. It can be explained by comparing the mean Sa(T1,5%) and Sv(T1,5%) of the FD records and the equivalent pulses which are, respectively, 0.81g against 0.30g, and 43.5 cm/s against 12.7 cm/s. It shows that the equivalent pulses extracted using the Baker's procedure cannot generally reproduce the seismic demands of the integrated and the basecracked gravity dams. The pulse effectiveness for the other EDPs can be investigated by defining a response ratio as
Ψ (T , Tp ) =
EDPep (T , Tp ) EDPae (T , Tp )
(11)
where Tp is the dominant pulse period, T is the natural period of the 628
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Fig. 7. (continued)
governed by the strong pulse, and the equivalent pulses are capable of reproducing structural response to the actual FD ground motions. The efficiency of the proposed period range can be rigorously studied by comparing the velocity response spectra of the FD records
those caused by the actual FD records for 0.7 < T1/Tp, Baker < 1.0. This has been shown for the building systems as well [56]. This range of Ψ, i.e. 0.8 < Ψ < 1.2, may be considered sufficiently accurate for engineering calculations. Therefore, in this range of periods the response is Table 2 R2 values of the quadratic trend lines for different IM-EDP pairs. Case 1 (Integrated model) IM: Ground motion: Peak crest displacement Peak hydrodynamic pressure Damage energy dissipation DIbase DIneck Friction energy dissipation Base sliding Heel opening Toe opening
PGA FD 0.55 0.73 0.72 0.49 0.75 – – – –
NFD 0.60 0.82 0.75 0.81 0.82 – – – –
PGV FD 0.36 0.30 0.27 0.24 0.25 – – – –
Case 2 (Base-cracked model)
NFD 0.41 0.70 0.63 0.59 0.66 – – – –
Sa(T1,5%) FD 0.64 0.82 0.82 0.73 0.79 – – – –
629
NFD 0.71 0.93 0.92 0.94 0.84 – – – –
PGA FD 0.53 0.72 0.60 – 0.64 0.42 0.44 0.50 0.44
NFD 0.79 0.86 0.67 – 0.69 0.70 0.71 0.74 0.84
PGV FD 0.39 0.25 0.36 – 0.30 0.47 0.47 0.18 0.33
NFD 0.69 0.68 0.57 – 0.56 0.77 0.75 0.54 0.71
Sa(T1,5%) FD 0.65 0.81 0.61 – 0.70 0.49 0.40 0.62 0.36
NFD 0.80 0.85 0.94 – 0.90 0.80 0.81 0.90 0.78
Soil Dynamics and Earthquake Engineering 92 (2017) 621–632
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Case 1 (Integrated model) Peak hydrodynamic pressure
Peak relative displacement
Case 2 (Base-cracked model) Peak hydrodynamic pressure
Peak relative displacement
Dam height (m)
120
80
40
0
50
25
0
1.5
Actual FD records
0
50
25
0 1.5
0
Equivalent pulses
Fig. 8. Mean values of the maximum absolute relative displacement, in cm, and the maximum hydrodynamic pressure, in MPa, along the dam height under the actual FD records and their equivalent pulses.
and their equivalent pulses. In Fig. 10, three representative sets of response spectra are compared for T1/Tp, Baker =0.95, 0.54, and 0.13. The perfect match between the response spectra is observed for T1/Tp, Baker =0.95. By decreasing the period ratio, in Fig. 10(b) and (c), the velocity response spectra of the equivalent pulses does not closely match the response spectra of the FD ground motions. Furthermore, other peaks are observed in the response spectra of the FD ground motions in the vicinity of the natural mode periods of the damreservoir system that control its behavior. These peaks are filtered out when simplified pulses are used to represent the FD ground motions. In Fig. 10(c), the period of the equivalent pulse is too long to excite the dam-reservoir model. Therefore, below the proposed period range, the equivalent pulse cannot sufficiently reproduce the frequency content and spectral values of the actual FD record specifically in the vicinity of natural frequencies of the dam, and those frequencies which govern the dam response either override or follow after the equivalent pulse [27]. So in these ranges, the equivalent pulse parameters are not adequate IM. The effects of velocity pulse parameters on the nonlinear response of both models are investigated by plotting the energy dissipation through damage and friction under the FD records in terms of period, amplitude and specific energy density of the dominant velocity pulse, Tp, Ap and SED (Fig. 11). As it is observed, the dissipated energies are generally decreased by increasing the period of the dominant velocity pulse, but, they are increased by increasing the amplitude of the dominant velocity pulse. However, about the energy content of the records (SED) no specific conclusion can be drawn especially about the damage energy dissipation. The same trend is observed for the other response parameters such as the local DIs, the base joint sliding and opening displacements. 5. Conclusions The nonlinear seismic response of concrete gravity dams under near-fault ground motions was investigated. Considering Pine Flat dam as case study, it was numerically modeled along with its full reservoir using the finite element method base on Eulerian-Lagrangian approach. Two sources of nonlinearity were considered: the material nonlinearity of mass concrete using the plastic-damage method; and the geometric nonlinearity of a prescribed base joint. Therefore, two main potential failure modes of gravity dams were studied: the tensile cracking; and the sliding along the dam-foundation interface. Two cases of the dam-reservoir model were assessed: the integrated model without any prescribed joints; and the base-cracked model including a prescribed base joint. Seventy-five forward-directivity (FD) and sixty ordinary (non forward-directivity, NFD) near-fault records were se-
Fig. 9. Variation of the response ratio, Ψ, for various EDPs in terms of T1/Tp,Baker for (a) the case 1, (b, c) the case 2.
lected. The comparison of the nonlinear seismic responses of the analysis cases subjected to the selected near-fault ground motions shows that the structural response of gravity dams may not essentially controlled by the presence of the initial velocity pulse; and other properties of the near-fault records can significantly contribute in the response. It was shown that the NFD records cause more scattering in 630
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Y. Yazdani, M. Alembagheri
Fig. 10. Comparison of the velocity response spectra of the FD records and their equivalent pulses when T1/Tp,Baker is equal to (a) 0.95, (b) 0.54, and (c) 0.13.
Fig. 11. Dissipated energy through damage and friction under the FD records in terms of their dominant velocity pulse period, Tp, amplitude, Ap, and specific energy density, SED.
References
the engineering demand parameters (EDPs) than the FD records. Therefore, the pulse nature of the FD records may cause lower dispersion in the structural response of gravity dams. Also, the ordinary NFD records often cause more EDP values than the FD records at the various levels of earthquake intensity measures (IMs). PGV is generally a poor predictor of structural response of the gravity dams under both FD and NFD near-fault ground motions. PGA and Sa(T1,5%) are better predictors of the seismic response of both models for the NFD ground motions than for the FD records. Nonetheless, Sa(T1,5%) it is a more stable IM for seismic assessment of the integrated and the base-cracked gravity dams under the near-fault ground motions. The equivalent pulses of the FD records were extracted using the Baker's procedure. The comparison of the seismic response under the actual FD ground motions and their equivalent pulses shows that the equivalent pulses generally underestimate the EDPs with respect to the actual FD records. This was attributed to the fact that the Baker's procedure only attempts to replicate the velocity pulse so it cannot reproduce the frequency content of the FD ground motions. But the equivalent pulses can predict the seismic demands of the integrated and the base-cracked gravity dams within 20% of those caused by the FD records for 0.7 < T1/Tp, Baker < 1.0. Therefore, in this range of periods the response is governed by the FD pulse and the equivalent pulses are capable of reproducing structural response to the FD ground motions. Finally it was shown that the nonlinear seismic responses of both models are decreased by increasing the period of the dominant velocity pulse; however, they are increased by increasing the amplitude of the dominant velocity pulse.
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