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High stiffness to weight ratio, ease and speed of handling, as well as having favorable architectural appearance cause that, double layer grids with ball joint system are widely used to cover large spans. A double-layer grid has a complex behavior due to a large number of elements and a particular type of joints; hence, structural identification of this type of structure is an important issue, which refers to the determination of natural frequencies, mode shapes, and damping ratios. These results are necessary to complete the structural health monitoring, finite element model updating and damage detection. Due to the limitations of input-output methods, modal parameters of civil engineering structures such as bridges, dams, tall buildings, and double layer grids are determined mainly by output-only modal identification. In output-only methods, the vibration parameters are determined based on the information acquired only from the structure’s output. In this work, physical model of a ball jointed double-layer grid with dimensions of 2.8 m at 2.8 m, which is supported on four steel pipes in four corners was made in the laboratory. The grid consists of 32 members connected together with 13 balls, each having ten threaded holes at different angles. each member consists of a middle pipe and connecting parts including conical piece, sleeve and high strength bolt at both ends of the pipe. The middle pipe has the nominal length, diameter and thickness of 120 cm, 7.64 cm and 0.35 cm, respectively. The horizontal center to center distance of adjacent balls in each layer of the grid is 1.414 m and the total height of the structure includes the column length (1.3 m) and the distance between the top and bottom layers (1 m), which is equal to 2.3 m in total. The approximate weight of the structure is 3532 N. All the members and the balls used in the grid are identical. After all the members of the grid have been assembled, the bolt at each joint is tightened in a series of steps by twisting the corresponding sleeve. Exciting the grid, its acceleration response was measured. The modal parameters were obtained using four output-only modal identification techniques; namely enhanced frequency decomposition (EFDD), curve-fit frequency domain decomposition (CFDD), data-driven stochastic subspace identification (SSI-DD) and covariance-driven stochastic subspace identification (SSI-Cov). Two types of excitations were used in output-only modal tests, namely direct and indirect excitations. Since the modal parameters obtained via input-output modal analysis have less uncertainty compared to the output-only modal analysis techniques, an input-output modal test was also performed and the results are considered as reference values. It deduced that parameters identified in the direct excitation, were more accurate compared to indirect excitation. The results showed that the natural frequencies and mode shapes of the double-layer grid were estimated with a high accuracy via the four methods. The greatest relative difference between the natural frequencies belonged to the second mode and equaled 2.07%. The dispersion of estimated damping was much higher compared to natural frequencies and mode shapes. The results indicated that identified damping in the direct excitation was lower than indirect one. Among the 4 methods, SSI-Cov had the least error in damping estimation of the double-layer grid. The values of estimated modal damping ratios were relatively low (fraction of 1%). The mean relative error of the identified parameters showed that the time-domain methods estimated the damping ratios with less error; While the frequency-domain methods identified natural frequencies and mode shapes with higher accuracy.

In this research, the behavior of concrete rectangular ground tanks equipped with lead-rubber base isolators has been investigated under the simultaneous excitation of transitional and rotational components of earthquakes, using nonlinear time history analysis including water-structure interaction. For this purpose, first, the rotational component of three earthquakes was generated using the wave propagation and classic elasticity theories simultaneously. Then, the tanks equipped with appropriate number of base isolators were modeled in two volumes of 500 and 1000 m3 and three states of water level. In modeling the finite elements of the isolators, a model consisting of one-dimensional elements is proposed; whose hysteresis behavior is equivalent to the behavior of the three-dimensional model of the isolators.The results of the research show that the decreasing or increasing effect of the rotational component of the earthquake on the seismic response of these tanks in the full and half-full state is less than 10%, and in general, its effect on the seismic response of the rectangular ground tanks can be ignored. Also, the presence of isolators in these types of tanks has increased the response of the tanks in all states. So that the base shear force has increased up to 4 times and the normal stress has increased between 8 and 23 times, which can be attributed to the increase in the period of ground tanks from the initial range of the response spectrum to the constant acceleration region of the spectrum due to the presence of the base isolators.

Functionally graded materials are the novel class of advanced composite structures with variable properties in one or more directions. The mechanical properties of Functionally Graded Materials (FGM) such as Poisson’s ratio, Young’s modulus of elasticity, material density and shear modulus of elasticity undergo changes gradually and continuously between two surfaces in a predetermined manner. FGM structures are often made of a combination of ceramics and metals in which the metal component provides strength and fracture resistance while ceramic component provides thermal resistance. Due to desirable properties, FGM materials are used in various fields of engineering such as optics, electronics, space vehicles, shipbuilding, mechanical, biomechanical and other engineering structures subjected to high thermal and residual stresses. Therefore, the analytical study of thermoelastodynamic problems is of great importance for the functionally graded media. The use of potential functions to analyze three-dimensional elastic problems is one of the most effective methods. This method facilitates solving three-dimensional elastic problems by uncoupling the set of governing differential equations or at least simplifying them. In the present study, the displacement potential functions for solving thermoelastodynamic problems in the transversely isotropic media with functionally graded materials are introduced. For this purpose, first, the three-dimensional kinematic and thermodynamic equations for the functionally graded materials in the transversely isotropic materials are written and then using a systematic method to separate the equations, the displacement potential functions to solve thermoelastodynamic problems are obtained, which can be used further to solve problems of beams, plates, shells, and infinite and semi-infinite media. The obtained potential functions include two scalar functions F and χ; the scalar function F satisfies the sixth-order partial differential equation while the χ-scalar function satisfies the second-order partial differential equation. In addition, in the present study, the thermal potential functions for the specific state of the isotropic functional graded media are presented.

While studying large-scale systems, non-local damping definition can beneficially model contact shear and long-range forces resulting from adjacent and non-adjacent elements in the set of interconnected dampers, damping patches and foundations which are modeled as non-local domains. If two or three dimensional systems are considered one dimensional (e.g. analyzing three dimensional beams based on Euler-Bernoulli, Rayleigh or Timoshenko Theories which simplifies the behavior of structures), the concept of non-local damping models arises to improve the accuracy of numerical results. Even though defining dissipative forces which are dependent on more parameters and quantities helpfully boost the validity of results compared to experimental cases in labs and three dimensional numerical analysis done by software, many researchers have widely employed viscous damping model to demonstrate damped behavior of the structures in the sake of simplicity. Actually viscous damping does not model accurately the dissipative behavior of real systems and practical structures often demonstrate some kind of viscoelasticity while vibration.  In the recent study, external damping force at any point in the domain is influenced by the past history of velocity and long-range interactions through convolution integral over proper kernel functions. As a consequence of applying Laplace transformation and using Galerkin method, the integro-partial differential equation of Rayleigh beam as a distributed parameter system turns to an ordinary differential equation governing a discrete system with finite degrees of freedom. Galerkin method is established based on error minimization of assumed mode method and despite Rayleigh and Rayleigh-Ritz methods can suitably analyze nonconservative systems including damped beams. Corresponding undamped mode shapes of Rayleigh beam which satisfy essential boundary conditions are chosen as the best admissible functions to expand the trial response of equation of motion. In order to get  continuous functions and finite weighted residual integral, the equation is presented in the weak form. Afterwards stiffness, damping and two types of mass matrices are determined with respect to generalized coordinates. To get scaled mode shapes the mass change method is considered to evaluate scaling factor, then the results are mass normalized. By equating dynamic stiffness matrix to zero and solving the resulting algebraic equation, complex and real eigenvalues are obtained which are respectively elastic and non-viscous modes in stable systems. It’s noteworthy to announce that contrasted with the viscously damped systems the degree of algebraic equation in the case of viscoelastic non-locally damped Rayleigh beam would generally be more than 2N (which N stands for total degrees of freedom). Accordingly eigenvectors are found based on Gaussian elimination method and null space of the dynamic stiffness matrix. Additionally, Neumann series expansion is developed to determine eigenvectors of vibrating systems effected by inertia. Finally, numerical results of Rayleigh and Euler-Bernoulli beams are compared. Very thin Rayleigh beams are verified and show great accuracy but when the thickness increases and inertial effect becomes bold, results of Rayleigh and Euler-Bernoulli beams are not matched anymore. Furthermore it’s understood from the graphs and tables that when the characteristic parameter of nonlocality and relaxation constant increase and tend to infinity, the nonlocal and non-viscous effects decline.

Engineered cementitious composites with polyvinyl alcohol fibers (ECC-PVA) are a group of cementitious composites, which not only provide high ductility and tensile strength but also control the crack width. The strain-hardening behavior under tensile and flexural loads and the formation of multiple shallow cracks instead of deep and concentrated cracks are among the main advantages of these composites, leading to their widespread application. In this paper, eight ECC concrete beam specimens, including four specimens containing 2% and the other four containing 1.5% PVA fibers, were tested and their load-displacement and load-crack mouse opening displacement curves were obtained. To determine the crack mouth opening displacement, four LVDTs were mounted around initial crack formed in the beam. As the displacement applied using the standard testing machine with a 500kN capacity increased, which were recorded using the LVDTs. Afterward, by averaging the recorded opening displacement recorded using LVDTs, the final crack mouth opening displacement was obtained. The results indicated that the ECC specimens with 2% PVA fibers had lower load-carrying capacity, and greater displacement, crack mouth opening displacement, and ductility values compared with the ECC specimens with 1.5% PVA fibers. Moreover, increasing the PVA fiber percentage elevated the flexural capacity of the beams by 10%.

This work considers an effective analytical method based on Displacement Potential Function (DPF) for solving 3-D thick and multi-layered transversely isotropic linearly elastic cylindrical shells (non-homogeneous in radial direction) with simply-supported end boundary conditions. Axisymmetric radial loads are applied on the inner and outer faces of the cylindrical shell. Three-dimensional elasticity equations are simplified using displacement potential function result in one single linear partial differential equation of fourth order as governing differential equation in term of displacement potential function. The governing equation is solved via the separation of variable method with exact satisfaction of two end boundary conditions including stress and displacement boundary conditions, stresses on the inner and outer surfaces of the shell, and the continuity conditions of the displacement and tractions on the interfacial surfaces of the multi-layered cylindrical shell. After determining displacement potential function, all other functions such as stresses and displacements can be obtained at each point of the examined shell. Comparison of the results with existing analytical results show excellent agreement at different thickness ratios and aspect ratios of the shells. Some practical problems are solved for one-layered and three-layered cylindrical shells. For this purpose, three types of materials are defined for a one-layered cylindrical shell such as composite material (Graphite epoxy), metallic substance (e.g. Zinc), and isotropic material (Aluminum). Also two combinations of materials are considered for three-layered cylindrical shell so that the inner and outer layers of the shell are made of transversely isotropic material (Graphite epoxy), while the middle layer of the isotropic material is made of aluminum and foam. The values of the non-dimensional functions containing stress and displacement components are calculated for these problems to demonstrate the effect of thickness ratio and anisotropy of the shell on the distribution of the stresses and displacements.

Alkali-Aggregate Reaction (AAR) is a type of destructive and time-dependent chemical process in concrete that occurs between alkali ions in the cement and reactive minerals in certain types of aggregates. First recognized in 1930s, AAR is divided into two major categories of Alkali-Silica Reaction (ASR) and Alkali-Carbonate Reaction (ACR), both of which produce an expansive gel in the concrete which expands as a result of water absorption. The expansion of the AAR gel exerts significant internal pressure in the concrete, which may lead to internal and external cracks. With the occurrence of such cracks, many parameters affecting the stiffness and strength of the structure, such as compressive strength, tensile strength, and modulus of elasticity are diminished. As a result, the safety and serviceability of the structure may be seriously impacted. While advances in concrete materials science have led to means to prevent AAR in new construction, numerous existing structures worldwide, such as dams, power plants, and bridges, are affected by these reactions, the replacement of which may be impractical, or in some cases, impossible. As a result, it is crucial to simulate the behavior of such structures for reliable estimation of their safety and providing rehabilitation measures as necessary. One of the major indicators of AAR is the anisotropic expansion it generates inside the concrete member, which changes drastically based on the boundary conditions and internal and external restraint imposed on the expansions. As a result, the prediction of the anisotropic expansions is of utmost importance in successful simulation of AAR-affected reinforced concrete structures. This paper presents a practical simulation methodology for estimating the directional distribution of AAR expansions. The methodology makes use of the user subroutine capability in the finite element software Abaqus. A mathematical model is used to simulation AAR-related expansion based on the stress tensor, whereas concrete damage is simulated using the concrete damage plasticity model. The model is used to simulate a variety of AAR-affected reinforced and plain concrete cube and beam specimens for which the directional expansion data have been reported in the literature. Comparison between numerical and experimental results shows that the proposed methodology is capable of reliably simulating the AAR-induced expansions and the interaction between AAR expansions and the ensuing damage for a variety of reinforcement configurations. The model showed that the yield strength of reinforcing bars plays a major role in the directional distribution of expansion. However, changes in the mechanical properties of concrete were found to be inconsequential in the distribution of the expansions. Moreover, changes in distance between reinforcing bars and the reinforcement ratio in each direction were observed to affect the accuracy of the model. However, the model was found to be successful in reasonably capturing the trends in all case studies investigated. The results of this study are of great value to the simulation of AAR-affected reinforced concrete structures.

Today nanotechnology has become important in many fields, including industry, medicine, engineering, aerospace, national security and electronics. As the dimensions of the structures decrease, the effects of size play a crucial role in properties of the media. Mechanical properties, electrical conductivity, thermal properties and other known chemical and physical properties are some examples that differ on nanoscales. Classical continuum mechanics are impotent to cover the effects of dimensions of the constituents of the media on nanoscales. Hence, several non-classical continuum theories, including non-local elasticity theory, strain gradient theory, and non-local strain gradient theory, have been developed by researchers to explain size-dependent mechanical behavior on a nanoscale. In this research, governing equations in terms of displacement potential functions based on nonlocal strain gradient theory are introduced for elastodynamic problems in homogeneous Transversely isotropic media. To this end, the three-dimensional equations of motion of the homogeneous Transversely isotropic media are first calculated using the nonlocal strain gradient theory. Then, using a systematic method, a set of complete displacement potential functions will be presented to solve elastodynamic problems in these media. By use of potential functions, the governing equations of motion will be decoupled. The proposed potentials include two scalar functions. One of them satisfies an 8th-order partial differential equation and 4th-order PDE is governed on the other. These potential functions are obtained in the form of a combination of wave operators, non-local parameter, and characteristic length, which are functionally and physically meaningful. These potential functions are obtained in the form of a combination of wave operators, non-local parameter, and characteristic length, which are functionally and physically meaningful. In addition, potential functions for limiting cases namely strain gradient theory and Eringen nonlocal elasticity theory are presented, separately. Also, by neglecting non-local parameters and characteristic length, the solution is degenerated to the Eskandari-Ghadi solution for classical theory of elasticity. Moreover, a new set of potential functions is presented to solve the elastodynamics of nonlocal strain gradient theory for the simpler case of isotropic materials.

Water storage tanks are one of the vital and essential arteries whose immediate operational performance after an earthquake is of special importance and interruptions in their operation can indirectly increase the damage and losses caused by earthquakes. Due to the significant effect of earthquakes on structures located in the vicinity of faults and, also, the unknown effect of the simultaneous transitional and rotational components of these earthquakes, present study investigated the behavior of concrete elevated tanks with central shaft and lead rubber bearing base isolation, under simultaneous excitation of translational and rotational components of near- and far-field earthquakes using nonlinear time history analysis. Base shear force and normal stress of tanks were determined considering dynamic interaction of water and tank. The corresponding rotational component of each translational pair of accelerograms was generated using the classic formulation of the theory of elasticity and wave propagation simultaneously, taking into account the frequency-dependency of wave velocity. To model the base isolator in the present study, an equivalent simplified link and combination model, based on 3D model of reference [32], has been suggested and its performance accuracy has been evaluated. The results of dynamic analysis show that the presence of the base isolator in all the studied conditions causes a reduction of about in base shear and in normal stress of the elevated tanks. Increasing the tank volume will reduce the decreasing effect of the isolator on the normal stress of tanks, but will improve the performance of the seismic isolator in reducing the base shear. Also, increasing the angular acceleration of rotational component of near-field earthquakes weakens by about the decreasing effect of the base isolator on the seismic response of elevated-tanks. Thus, neglecting rotational component of the earthquakes in analysis of tanks located near the faults will magnify the decreasing effect of the base isolator on seismic demands and the poor design of structure. Therefore, use of the isolator to reduce the seismic response in these areas will not be economically justified.

In this research, an exact solution for free vibration analysis of thick transversely isotropic simply supported rectangular plates on the Pasternak foundation was proposed using the displacement potential function method. By means of the proposed displacement potential functions for dynamic problems by Eskandari-Ghadi, the differential governing equations in terms of displacements were converted into two linear partial differential equations of second and forth order. These differential equations were solved based on the separation of variables method and satisfying exact boundary conditions. In order to validate the results, the obtained results were compared with other available analytical works for isotropic and transversely isotropic plates, and it indicated remarkable agreement. Having no simplifying assumptions for the strain or stress distribution in the plate thickness, the obtained results in this paper were applicable to any arbitrary plate thickness with no limitation on its thickness ratio such as thin, moderately thick, and thick plates. Thus, the obtained results of the present work can be used as a benchmark solution for other analytical and numerical studies. To investigate the effect of various parameters on the vibrational plate response, the precise non-dimensional frequencies were obtained in different range of thickness ratios, aspect ratios, elastic foundation coefficients, and mechanical characteristics of plates. It was observed that with increasing thickness ratio and aspect ratio of the plate, the non-dimensional natural frequencies of plate decreased and increased, respectively. In addition, comparative results of isotropic and transversely isotropic plates showed that shear modulus in transverse direction would have significant influence on the vibrational behavior of rectangular plates. It was shown that when the value of the thickness ratio increased, the sensitivity of non-dimensional frequency response to values of foundation stiffness coefficients decreased. In addition, it can be conducted that the effect of the shearing layer of elastic foundation coefficient value increased by increasing the thickness ratios of plates.

Output-only structural identification is conducted by output data of the structure. These data usually include structural response together with some noise. Success of output-only methods in determining the vibration parameters of a structure depends on the signal to noise ratio (SNR) of the output data. In this paper, the vibration parameters (Natural frequency and Mode shape) of a contilever beam have been obtained using output data which have different signal to noise ratios. The vibration parameters of the beam were determined using modal analysis of finite element model and considered as reference parameters. Then, appropriate input was applied to the beam and the acceleration signal was obtained. To generate noisy data, noise with different powers compared to signal powers were added to acceleration signal. The modal parameters of the beam were obtained using two output-only methods, Peak Picking (PP) and Stochastic Subspace Identification (SSI). The vibration parameters having signal-to-noise ratios greater than 25 (lower noise level) for all considered modes were identified properly. At a signal-to-noise ratio of 0.25 to 25 (higher noise level), it was not possible to identify the modal parameters of the first mode of the beam, but the parameters of the higher modes were identified with good accuracy.

The present paper investigates and compares the crack propagation in concrete gravity dams using two models of linear fracture mechanics and plasticity damage concrete. The first model is based on linear concrete behavior using the extended finite element method without considering the effect of strain softening on the crack tip while the second model is based on the nonlinear concrete behavior and the strain softening in tension with damage parameter. According to two different algorithms and based on two models, several benchmark examples are reviewed and the results compared with those reported in the literature. Then, path of the crack growth in Koyna gravity dam due to a seismic excitation of Koyna earthquake in 1962 has been performed by considering the dam-reservoir interaction. The results show that due to low compressive stresses during analysis of concrete gravity dams, consideration of compressive nonlinear behavior has no effect on crack initiation and almost is the same for two models. However because of crack opening and closing with tapping the crack faces together in extended finite element model, the compressive stress will be more than the allowable stress of concrete. Crack initiation at downstream and upstream face occurred at angle of 90 and zero degrees respectively, which in both models, the numerical results are in agreement with the experimental model. The crack in the extended finite element model grows faster such that the crest block of dam in this model is separated from the dam body, earlier than the concrete plastic damage model. Also the values of dam crest displacement and hydrodynamic pressure in the reservoir in extended finite element model with linear elastic fracture mechanic are more than the other model, which can be attributed to the linear and nonlinear behavior of concrete in extended finite element and concrete plastic damage model respectively. In the extended finite element model, due to using linear fracture mechanic, the maximum principal stress in the cracked elements reaches the values greater than the maximum tensile strength, but in the concrete plastic damage model as soon as the stress reaches a tension limit value, elements are damaged and the stress is reduced. In both models, the crack located at the slope change area, propagates with the downward slope from downstream dam face and connects to the crack at upstream face which is growth horizontally. Because of laboratory sample dimension and boundary condition of dam-reservoir compared with actual manner, neither of two crack profiles covered the experimental model, accurately. But it is shown that the crack profiles in the extended finite element model are more consistent with experimental results. Finally, the results show that the crack profile are slightly different in the two models because of quasi brittle behavior of the dam concrete, which can be attributed to the small fracture process zone of the crack tip in comparison with the dimension of the concrete gravity dams such that by removing strain softness part, the error in the amount of additional computation can be neglected.

The purpose of this paper is to investigate the behavior of positive buoyancy jets of urban wastewater in stationary and non-stratified water bodies using the investigation of the laboratory data acquired by a 3D-LIF laser 3D scanning system and numerical modeling results from the simulation by FLOW-3D software. Laboratory simulation was performed for three discharge jets with Froude numbers 10, 15, and 20, and compared with the results of the numerical model and the previous numerical results. The trajectory of the jet, concentration variations, the position of the impact point, and the rate of dilution at the impact point was determined. In the numerical model, both K-Ɛ and RNG turbulence models were used to evaluate the accuracy of the numerical simulation where the K-Ɛ results were closer to laboratory results and are capable of accurately modeling the jet trajectory than the RNG model. By different Froud numbers, the results at Fr = 20 are almost the same for all experimental and numerical states and indicating that by increasing the discharge and Froud number, the accuracy of the results in the numerical model increases and present closer results to the experimental model.

In this research, the hydraulic characteristics of flow through circular culverts and over broad crested weir are experimentally investigated for 12 models. Stage discharge rating curve was presented for each model. The results show that for a specific head, the discharge of combined structure is greater than the summation of discharge of weir and culvert. Also, the discharge coefficient for the combined structure is varying from 0.25 to 0.73 and for the weir is varying from 0.25 to 0.63 and for the culvert is varying from 0.33 to 0.77. The dimensionless parameter H/D (head to diameter) has the greater influence on discharge coefficient. In addition, regression equations are presented for estimation of discharge coefficient.

Introduction Flow contact with the spillway piers causes rooster tail waves which could continue along the flood evacuation system. Therefore, the information about the height and location of these waves would be useful for the design of the spillway chute walls. On the other hand, the engineers try to design the converging chutes to reduce the excavation costs, and this convergence in many cases would increase the transverse flow depth. According to the literature, three kinds of waves could be formed in the spillway and chute system. The first wave is formed just after the spillway piers and called as rooster tail waves. The Flow passing through the two sides of a pier collide each other at a near distance after the pier and forming the first wave. The second wave is formed at the middle axis of the chute and could be considered as the result of the interaction of the first waves and the channel convergence. The third wave is formed due to the collision of the mentioned waves with the channel walls. Investigating the effect of the chute convergence on the depth and location of transverse waves is critical and necessary and could present suitable information about the three waves. In this paper, the numerical simulation of the Khair Abad dam flood evacuation system is performed for investigating the effects of chute convergence on the formation of the transverse waves along the chute. Methodology In this Research, the FLOW-3D is used for simulation of the effects chute convergence on transverse wave formation. The Khair Abad dam flood evacuation system laboratory tests results were used for verification of the numerical simulation. The present simulations considered four convergence angles of the chute, including 0, 3, 5 and 7 °. Also, the simulation was performed using three discharge rates consist of 3000, 7000 and 9000 m3 / s. The convergence angle of the Khairabad Dam spillway is approximately 5 degrees and the convergence angles were changed during the chute until the chute width reached from 66 m to 40 m. The chute width is kept constant after the convergence. Results and discussion The model validation was performed using the laboratory data of the Kheirabad Dam spillway Model developed by the Iran Water Research Institute. Since the size and number of the cells in the model affect the accuracy of the results and computational cost, this study concentrates in finding the optimum value of dimensions and number of cells for the simulation domain. This procedure could result in acceptable accuracy and suitable computational cost. The results of the verification showed that the smaller grid size and the greater number of cells, result in the higher correlation of the numerical results and the laboratory data. Hence, based on the above-mentioned method and available computational device, the best size and number of the cells were selected for simulating the effect of chute convergence on the three waves height. Simulation results show that the height of the first wave increases with increases in the flow discharge, and the spillway convergence has not a significant effect on the first wave height. Also, the second wave height is increased by the increase in flow discharge. It should be mentioned, the second wave doesn't appear when there is no convergence in the chute, but with increasing the convergence angle the wave height increases and the maximum wave height will move to upstream stations of the chute. In other words, with the increase of the angle of convergence of the chute, the location of the second wave formation will be closer to the spillway crest and its height will increase. The simulation results demonstrated that the increase in the flow discharge could lead to an increase of third waves heights. Also, the results show that the third waves could not be formed when the channel convergence angle is equal to zero, and increase of convergence angle could result in an increase of the third wave height. Also, the increase in the angle of convergence of the chute results in transport of the location of the third wave formation into upstream stations. In other words, the third waves will form closer to the spillway axis by an increase in convergence angle. Numerical simulation results show that the height of the third wave which formed beside the wall can be more than twice the average flow depth at the same point. This fact should be considerate in the design of the chute with convergence. The results also showed that increasing the flow discharge will cause to increase of the wave height, but it led to the decrease of the ratio of the third wave height to the average flow depth. This means that increasing the discharge has less effect on increasing the transverse wave height. Conclusion Simulation results show that the height of transverse waves increases by increasing the discharge. But the ratio of the maximum height of the waves to the average depth of the same station will reduce by the increase of the discharge. Also, Results show that, by increasing the chute’s convergence angle, the height of the transverse waves will increase and the location of the primary transverse waves will transmit to upstream locations of the chute.

The analysis of plates on elastic foundation has a wide range of applications in various fields of engineering, such as civil, mechanical, aerospace, nuclear and marine engineering. Due to soil behavior is dependent on numerous factors, providing full and comprehensive model for foundation is very complicated. Therefore, in order to express properties of the substrate are used the simpler models based on the elastic properties. The simplest of those models, which was presented by winkler in 1867, assumes that the soil medium containing a system of independent spring (Liew et al, 1996). To increase the accuracy of modeling two-parameter models arose that the effect of shear stresses in the substrate is also included. In these models a new parameter is proposed to establish a mechanical connection between the independent springs. An approximate solution for the bending of moderately thick rectangular plates using numerical differential quadrature method was proposed by Han and Liew (1997). Teo and Liew (2002), presented a solution for analysis of shear deformable rectangular plates on Pasternak foundations by using differential cubature method. Displacement potential functions (DPF) method is one of the effective and efficient techniques that can be used to solve elasticity problems. The most advantage of DPF method is that the system of differential equations is uncoupled or at least simplified. This method has been used for 3D analysis of plates by a number of researchers such as Nematzadeh et al (2010) and Moslemi et al (2016) for the bending and buckling solution, respectively.

Seismic evaluation of concrete dams due to some considerations is very important. The effects of fluid-structure-foundation interaction, nonlinear behavior of dam material due to crack, and earthquake loading are some of these considerations and should be considered in modeling and analysis of the system.The nonlinear seismic response of concrete gravity dams is presented when the effect of the dam-reservoir interaction is included using Lagrangian-Lagrangian approach of the finite element method. Nonlinear fracture mechanics, based on the smeared crack concepts, is used to study the cracking profile and response of the dam. In this study, a comparative study between the coaxial rotating crack model and orthogonal multi-fixed smeared crack models is carried out. Based on the presented formulation, Pine Flat concrete gravity dam is analyzed and its crest response and stresses within the dam body are founded. Bosak's time integration and corrected Newton-Raphson method are used for solving nonlinear dynamic equations.Results show that the two crack approaches have negligible difference in terms of the number of cracked gauss points and the crack profile. Tensile principal stresses based on fixed crack concept are greater than those of the rotating crack concept, but for compressive principal stresses the results are vice versa. The differences are due to unloading-reloading path and shear retention factor, but crack propagating path remains the same. Displacements of dam crest in nonlinear cases are greater than those of linear ones and show the same result in two crack models. This phenomenon depended on internal damping of system in both linear and nonlinear cases. Permanent displacement of dam crest based on fixed crack concept is greater than that of the rotating crack concept, and this phenomenon depends on cracking intensity of dam body. Results show that the fixed crack concept have better convergence than that of the rotating crack concept and the number of iterations in the time steps are low.

This paper concentrates on the presentation of a closed form solution of dam-reservoir interaction considering reservoir viscosity and bottom absorption under harmonic excitations. For this purpose, the governing equations of dam-reservoir are solved in the frequency domain and using the obtained frequency response function, the effects of different parameters are considered on dam or hydrodynamic pressure response. The results show that consideration of reservoir viscosity creates an imaginary term and the responses depend on values of viscosity, bottom absorption, exited period and dam-reservoir period.

The beam theory is used in the analysis and design of a wide range of structures, from buildings to bridges to the load-bearing bones of the human body. Beams resting on elastic foundation have wide application in many branches of engineering problems namely geotechnics, road, railroad and marine engineering and bio-mechanics. The foundation is very often a rather complex medium; e.g., a rubberlike fuel binder, snow, or granular soil. The key issue in the analysis is modelling the contact between the structural elements and the elastic bed. Since of interest here is the response of the foundation at the contact area and not the stresses or displacements inside the foundation material, In most cases the contact is presented by replacing the elastic foundation with simple models, usually spring elements. The most frequently used foundation model in the analysis of beam on elastic foundation problems is the Winkler foundation model. In the Winkler model, the elastic bed is modeled as uniformly distributed, mutually independent, and linear elastic vertical springs which produce distributed reactions in the direction of the deflection of the beam. However since the model does not take into account either continuity or cohesion of the bed, it may be considered as a rather crude representation of the elastic foundation. In order to find a physically close and mathematically simple foundation model, Pasternak proposed a so-called two-parameter foundation model with shear interactions. The first foundation parameter is the same as the Winkler foundation model and the second one is the stiffness of the shearing layer in the Pasternak foundation model.
Dynamic analysis is an important part of structural investigation and the results of free vibration analysis are useful in this context. Vibration problems of beams on elastic foundation occupy an important place in many fields of structural and foundation engineering.With the increase of thickness, existence of simplifying hypotheses in beam theories such as the ignorance of rotational inertial and transverse shear deformation in classic theory, application of determination coefficient in first-order shear theory and expression of one or few unknown functions based on other functions in higher-order shear theories is accompanied by reduction in accuracy of these theories. This represents the necessity of precise and analytical solutions for beam problems with the least number of simplifying hypotheses and for different thicknesses.
In the present study, the analytical solution for the problem of free vibration of homogeneous prismatic simply supported beam with rectangular solid sections and desired thickness resting on Pasternak elastic foundation is provided for completely isotropic behaviors under two-dimensional theory of elasticity and functions of displacement potentials. Characteristic equations of natural vibration are defined by solving one partial differential equations of fourth order through separation of variables and application of boundary conditions. The major characteristics of present study are lack of limitation of thickness and its validity for beams of low, medium and large thickness. To verify, the results of present study were compared with those of other studies. The results show that increases of foundation parameters is associated with an increased natural frequency, The intensity by increasing the ratio of thickness to length and in values larger than 0.2 and in the higher modes of vibration is reduced considerably.

The importance of the seismic behavior of concrete gravity dams in their safety evaluation and stability is inevitable. Many factors affect the prediction of the behavior of concrete dams such as dam-foundation-reservoir interaction, dam and foundation cracking and also displacement due to fault movement that could causes nonlinear behavior. The aim of this study is nonlinear analysis of concrete gravity dams, including displacement caused by normal fault movement in the dam foundation. For this purpose, dam-foundation-reservoir system is modeled using Lagrangian method and analysis of system is done by finite element method. The coordinate smeared crack model based on the nonlinear fracture mechanics is used for crack modeling in the dam body and foundation. Using two separate method including split node technique and contact element, the fault movement are modeled and the position and angle of fault has been studied. To verify the results, dam crest displacement and crack profile in the body of a concrete gravity dam is presented as an example. The results show that low fault movement causes the cracks in the dam body and could be jeopardizes the stability and safety of concrete dam.

The main purpose of this paper is presentation of proper finite element formulation for dynamic analysis of concrete gravity dams under nonuniform translational and rotational components of earthquake considering dam- reservoir interaction. Time delay of travelling waves is considered to generate the nonuniform ground motion and the rocking component of earthquake has been obtained using the corresponding translational components based on classical elasticity and elastic wave propagation theories in term of wave frequency functions. The translation and rocking components of ground motion are applied to support points of dam-reservoir bottom and analyses have done using finite element method base on Lagrangian-Lagrangian approach. Results show that the rocking component and time delay of travelling translation waves can be effect considerably on linear dynamic response of concrete gravity dams in some cases.

Design of bridge deck with long-term strength، durability and permanence is a significant interest for engineers. One applicable solution to this challenge could be via using hybrid system consisting of conventional materials such as concrete and steel with FRP plates which is also known as composite deck. Since these deck are relatively new so their performance is not completely known. The present study is dedicated to composite deck consists of a steel core and GFRP plates. This composite sandwich bridge deck system is composed of wrapped hybrid core of GFRP grid and multiple steel box cells with upper and lower GFRP facings. The structural performance of deck was evaluated by nonlinear finite element method and numerical results have been compared with available experimental results where possible. After ensuring the validity of numerical modeling of composite deck، parametric studies such as change in geometry، steel core shape and mechanical properties of materials have been done. It was found that failure mode of the proposed hybrid deck was more favorable because of the yielding of the steel tube when compared with that of absolute GFRP decks. Increasing the thickness and changing the steel core geometry can improve the ultimate load capacity of the deck. The grid layer has the maximum thickness among the GFRP layers and therefore ultimate load capacity of the deck enhanced by increase the elastic modulus of grid layer.

R‌e‌g‌a‌r‌d‌i‌n‌g t‌h‌e i‌n‌c‌r‌e‌a‌s‌i‌n‌g d‌e‌m‌a‌n‌d f‌o‌r r‌a‌p‌i‌d‌l‌y c‌o‌n‌s‌t‌r‌u‌c‌t‌e‌d b‌r‌i‌d‌g‌e‌s w‌i‌t‌h l‌o‌w‌e‌r d‌e‌a‌d l‌o‌a‌d a‌n‌d c‌o‌s‌t, v‌a‌r‌i‌o‌u‌s t‌y‌p‌e‌s o‌f s‌t‌r‌u‌c‌t‌u‌r‌a‌l s‌y‌s‌t‌e‌m h‌a‌v‌e b‌e‌e‌n p‌r‌o‌p‌o‌s‌e‌d f‌o‌r b‌r‌i‌d‌g‌e d‌e‌c‌k‌s. D‌u‌e t‌o t‌h‌e h‌i‌g‌h t‌e‌n‌s‌i‌l‌e s‌t‌r‌e‌n‌g‌t‌h o‌f s‌t‌e‌e‌l a‌n‌d t‌h‌e r‌e‌m‌a‌r‌k‌a‌b‌l‌e c‌o‌m‌p‌r‌e‌s‌s‌i‌o‌n s‌t‌r‌e‌n‌g‌t‌h o‌f c‌o‌n‌c‌r‌e‌t‌e a‌t l‌o‌w‌e‌r c‌o‌s‌t, s‌t‌e‌e‌l-c‌o‌n‌c‌r‌e‌t‌e c‌o‌m‌p‌o‌s‌i‌t‌e‌s a‌r‌e m‌o‌s‌t c‌o‌m‌m‌o‌n‌l‌y u‌s‌e‌d i‌n c‌o‌n‌s‌t‌r‌u‌c‌t‌i‌o‌n. I‌n t‌h‌e‌s‌e c‌o‌m‌p‌o‌s‌i‌t‌e‌s, t‌h‌e c‌o‌n‌c‌r‌e‌t‌e s‌e‌c‌t‌i‌o‌n c‌a‌n a‌l‌s‌o e‌n‌h‌a‌n‌c‌e t‌h‌e f‌i‌r‌e r‌e‌s‌i‌s‌t‌a‌n‌c‌e o‌f t‌h‌e s‌t‌e‌e‌l s‌e‌c‌t‌i‌o‌n. A c‌o‌n‌v‌e‌n‌i‌e‌n‌t p‌e‌r‌f‌o‌r‌m‌a‌n‌c‌e f‌o‌r s‌t‌e‌e‌l-c‌o‌n‌c‌r‌e‌t‌e c‌o‌m‌p‌o‌s‌i‌t‌e s‌t‌r‌u‌c‌t‌u‌r‌e‌s c‌a‌n b‌e a‌c‌h‌i‌e‌v‌e‌d, p‌r‌o‌v‌i‌d‌e‌d t‌h‌e r‌e‌l‌a‌t‌i‌v‌e d‌i‌s‌p‌l‌a‌c‌e‌m‌e‌n‌t o‌f c‌o‌n‌c‌r‌e‌t‌e a‌n‌d s‌t‌e‌e‌l s‌e‌c‌t‌i‌o‌n‌s a‌t t‌h‌e‌i‌r i‌n‌t‌e‌r‌f‌a‌c‌e b‌e p‌r‌e‌v‌e‌n‌t‌e‌d o‌r, a‌t l‌e‌a‌s‌t, r‌e‌d‌u‌c‌e‌d. C‌h‌e‌m‌i‌c‌a‌l a‌d‌h‌e‌s‌i‌o‌n‌s, o‌r b‌o‌n‌d‌s, i‌n‌t‌e‌r‌f‌a‌c‌e f‌r‌i‌c‌t‌i‌o‌n a‌n‌d m‌e‌c‌h‌a‌n‌i‌c‌a‌l i‌n‌t‌e‌r‌l‌o‌c‌k‌i‌n‌g, d‌o n‌o‌t p‌r‌o‌v‌i‌d‌e s‌u‌f‌f‌i‌c‌i‌e‌n‌t r‌e‌s‌i‌s‌t‌a‌n‌c‌e t‌o h‌e‌a‌v‌y s‌h‌e‌a‌r f‌o‌r‌c‌e‌s. H‌e‌n‌c‌e, t‌o a‌s‌s‌u‌r‌e t‌h‌e s‌h‌e‌a‌r t‌r‌a‌n‌s‌f‌e‌r, v‌a‌r‌i‌o‌u‌s t‌y‌p‌e‌s o‌f m‌e‌c‌h‌a‌n‌i‌c‌a‌l c‌o‌n‌n‌e‌c‌t‌o‌r b‌e‌t‌w‌e‌e‌n t‌h‌e s‌t‌e‌e‌l a‌n‌d t‌h‌e c‌o‌n‌c‌r‌e‌t‌e, s‌u‌c‌h a‌s c‌h‌a‌n‌n‌e‌l‌s, r‌e‌i‌n‌f‌o‌r‌c‌i‌n‌g s‌t‌e‌e‌l, h‌e‌a‌d‌e‌d s‌h‌e‌a‌r s‌t‌u‌d‌s a‌n‌d w‌e‌l‌d‌e‌d s‌t‌r‌u‌c‌t‌u‌r‌a‌l s‌t‌e‌e‌l e‌l‌e‌m‌e‌n‌t‌s, h‌a‌v‌e b‌e‌e‌n c‌o‌n‌s‌i‌d‌e‌r‌e‌d. E‌x‌o‌d‌e‌r‌m‌i‌c d‌e‌c‌k s‌y‌s‌t‌e‌m‌s a‌r‌e n‌e‌w‌l‌y d‌e‌v‌e‌l‌o‌p‌e‌d c‌o‌m‌p‌o‌s‌i‌t‌e s‌t‌e‌e‌l g‌r‌i‌d d‌e‌c‌k s‌y‌s‌t‌e‌m‌s t‌h‌a‌t h‌a‌v‌e b‌e‌e‌n u‌s‌e‌d i‌n v‌a‌r‌i‌o‌u‌s p‌r‌o‌j‌e‌c‌t‌s d‌u‌r‌i‌n‌g t‌h‌e p‌a‌s‌t d‌e‌c‌a‌d‌e. A‌n e‌m‌i‌n‌e‌n‌t f‌e‌a‌t‌u‌r‌e o‌f t‌h‌i‌s s‌y‌s‌t‌e‌m i‌s t‌h‌e c‌o‌n‌s‌i‌d‌e‌r‌a‌b‌l‌e r‌e‌d‌u‌c‌t‌i‌o‌n i‌n s‌t‌r‌u‌c‌t‌u‌r‌a‌l w‌e‌i‌g‌h‌t i‌n o‌r‌d‌i‌n‌a‌r‌y ‌e‌i‌n‌f‌o‌r‌c‌e‌d c‌o‌n‌c‌r‌e‌t‌e d‌e‌c‌k‌s, a‌n‌d t‌h‌e r‌e‌d‌u‌c‌t‌i‌o‌n i‌n c‌o‌n‌s‌t‌r‌u‌c‌t‌i‌o‌n t‌i‌m‌e u‌s‌i‌n‌g t‌h‌e p‌r‌e‌c‌a‌s‌t e‌x‌o‌d‌e‌r‌m‌i‌c d‌e‌c‌k‌s. T‌h‌e i‌n‌c‌r‌e‌a‌s‌e i‌n t‌h‌e u‌s‌e o‌f t‌h‌i‌s s‌y‌s‌t‌e‌m f‌o‌r d‌e‌c‌k b‌r‌i‌d‌g‌e‌s h‌a‌s m‌a‌d‌e i‌t n‌e‌c‌e‌s‌s‌a‌r‌y t‌o i‌d‌e‌n‌t‌i‌f‌y i‌t‌s b‌e‌h‌a‌v‌i‌o‌r; e‌s‌p‌e‌c‌i‌a‌l‌l‌y i‌t‌s d‌y‌n‌a‌m‌i‌c b‌e‌h‌a‌v‌i‌o‌r. I‌n t‌h‌i‌s r‌e‌s‌e‌a‌r‌c‌h, n‌a‌t‌u‌r‌a‌l f‌r‌e‌q‌u‌e‌n‌c‌i‌e‌s, d‌a‌m‌p‌i‌n‌g r‌a‌t‌i‌o a‌n‌d s‌o‌m‌e p‌h‌y‌s‌i‌c‌a‌l p‌r‌o‌p‌e‌r‌t‌i‌e‌s, s‌u‌c‌h a‌s s‌t‌i‌f‌f‌n‌e‌s‌s a‌n‌d e‌f‌f‌e‌c‌t‌i‌v‌e m‌a‌s‌s, f‌o‌r e‌x‌o‌d‌e‌r‌m‌i‌c d‌e‌c‌k b‌r‌i‌d‌g‌e‌s w‌i‌t‌h a‌l‌t‌e‌r‌n‌a‌t‌i‌v‌e‌l‌y p‌e‌r‌f‌o‌r‌a‌t‌e‌d s‌h‌e‌a‌r c‌o‌n‌n‌e‌c‌t‌o‌r‌s, w‌e‌r‌e d‌e‌t‌e‌r‌m‌i‌n‌e‌d e‌x‌p‌e‌r‌i‌m‌e‌n‌t‌a‌l‌l‌y. T‌o‌w‌a‌r‌d‌s t‌h‌i‌s a‌i‌m, b‌y e‌x‌c‌i‌t‌i‌n‌g a p‌o‌i‌n‌t o‌n t‌h‌e s‌t‌r‌u‌c‌t‌u‌r‌e a‌n‌d m‌e‌a‌s‌u‌r‌i‌n‌g i‌n‌p‌u‌t f‌o‌r‌c‌e a‌n‌d o‌u‌t‌p‌u‌t a‌c‌c‌e‌l‌e‌r‌a‌t‌i‌o‌n, a‌n‌d t‌h‌e‌n r‌e‌p‌e‌a‌t‌i‌n‌g i‌t i‌n m‌o‌r‌e s‌t‌e‌p‌s, t‌h‌e v‌a‌l‌u‌e‌s o‌f n‌a‌t‌u‌r‌a‌l f‌r‌e‌q‌u‌e‌n‌c‌i‌e‌s a‌n‌d d‌a‌m‌p‌i‌n‌g r‌a‌t‌i‌o‌s w‌e‌r‌e d‌e‌t‌e‌r‌m‌i‌n‌e‌d. T‌h‌i‌s w‌a‌s u‌n‌d‌e‌r‌t‌a‌k‌e‌n u‌s‌i‌n‌g t‌h‌e f‌r‌e‌q‌u‌e‌n‌c‌y r‌e‌s‌p‌o‌n‌s‌e f‌u‌n‌c‌t‌i‌o‌n (F‌R‌F), b‌y e‌m‌p‌l‌o‌y‌i‌n‌g p‌e‌a‌k p‌i‌c‌k‌i‌n‌g a‌n‌d N‌y‌q‌u‌i‌s‌t c‌i‌r‌c‌l‌e f‌i‌t m‌e‌t‌h‌o‌d‌s, w‌h‌i‌c‌h s‌u‌i‌t‌a‌b‌l‌y m‌a‌t‌c‌h‌e‌d t‌h‌e f‌i‌n‌i‌t‌e e‌l‌e‌m‌e‌n‌t m‌e‌t‌h‌o‌d. A‌l‌s‌o t‌h‌e e‌x‌p‌e‌r‌i‌m‌e‌n‌t‌a‌l r‌e‌s‌u‌l‌t‌s s‌h‌o‌w t‌h‌a‌t t‌h‌e f‌i‌r‌s‌t m‌o‌d‌e i‌s t‌h‌e m‌o‌s‌t e‌f‌f‌e‌c‌t‌i‌v‌e.

The kinematics of any point in a medium is ideally expressed in terms of three translational and three rotational components. Observations of earthquake events have shown that many structural failures and damage are associated to rotational components of ground motions. Newmark [1] was perhaps the first to establish a relationship between the torsional and translational components of a ground motion based on constant velocity of wave propagation assumption. Lee and Liang [2] have used wave propagation and classical elasticity theories based on constant wave velocity to develop the algorithms for generating rotational motion from the corresponding available translational motions and Hong-Nan Li et al. [3] proposed an improved approach based on frequency dependent wave velocity to generate the rotational components. Kalani Sarokolayi et al. [4] have used this method and they have verified their results using recorded rotational components. Recently the effect of rotational component on dynamic analysis of dam- reservoir system without foundation effect is considered by authors in their previous research [5]; but the effects of rotational components have not been considered in dynamic analysis of dam-reservoir-foundation systems in previous researches. The fluid- structure interaction is also an important subject to dynamic analysis of dams. The Lagrangian approach which were proposed by Hamdi [6] and completed by Wilson and Khalvati [7], have been used by many researchers such as [7-9]. In addition the reservoir bottom absorption effects in earthquake response of concrete gravity dams have been also investigated by some researchers such as Fenves and Chopra [10]. The main purpose of this research is the evaluation of dynamic response of concrete gravity dams considering three correlated translational and rotational components of ground motion and dam-reservoir-foundation interaction using finite element method. For this purpose, the rotational component of ground motion is obtained using translational components and relation of classical elasticity between rotation and wave propagation theories considering frequency dependent wave velocity. Then, these rotational and translational components are applied in finite element model and the dynamic response of system are calculated using Newmark method and Lagrangian- Lagrangian approach based on displacement unknown in both solid and fluid domains. In addition, with the change of elasticity modulus of foundation, earthquake acceleration, water elevation and absorption coefficient of reservoir bottom, the sensitivity of response with respect to these parameters are evaluated. The horizontal displacement of dam crest for dam-reservoir (D-R) and dam-reservoir-foundation (D-R-F) systems subjected to two translational components, 2C, and two translational added by their correlated rotationalcomponents, 3C, are obtained and the ratio of response due to 3C and 2C which is named as Normalized response, are presented in Table 1. The effects of absorption coefficients on Normalized response of system are also shown in Table 2 and the effects of water elevation on dam crest response subjected to 3C are shown in Fig. 2. Results showed that the effects of rotational components of ground motion on the dynamic response of concrete gravity dams can be low or high depending on their frequency range and power spectrum. In cases which the rotational effect is high, the dam response to rotational component of earthquake will be decreased with the increase of the foundation elasticity modulus and reservoir bottom absorption coefficients. In addition with the increase of water elevation, the rotational effect will increase and the response time history also will change.

R‌o‌t‌a‌t‌i‌o‌n‌a‌l c‌o‌m‌p‌o‌n‌e‌n‌t‌s o‌f g‌r‌o‌u‌n‌d m‌o‌t‌i‌o‌n i‌n‌c‌l‌u‌d‌i‌n‌g r‌o‌c‌k‌i‌n‌g a‌n‌d t‌o‌r‌s‌i‌o‌n‌a‌l c‌o‌m‌p‌o‌n‌e‌n‌t‌s h‌a‌v‌e b‌e‌e‌n e‌c‌c‌e‌n‌t‌r‌i‌c‌a‌l‌l‌y i‌g‌n‌o‌r‌e‌d f‌o‌r a l‌o‌n‌g t‌i‌m‌e, f‌i‌r‌s‌t, b‌e‌c‌a‌u‌s‌e t‌h‌e r‌o‌t‌a‌t‌i‌o‌n‌a‌l e‌f‌f‌e‌c‌t‌s w‌e‌r‌e t‌h‌o‌u‌g‌h‌t t‌o b‌e s‌m‌a‌l‌l f‌o‌r s‌t‌r‌u‌c‌t‌u‌r‌e‌s a‌n‌d, s‌e‌c‌o‌n‌d, d‌u‌e t‌o t‌h‌e‌i‌r s‌m‌a‌l‌l a‌m‌p‌l‌i‌t‌u‌d‌e, t‌h‌e‌y c‌a‌n‌n‌o‌t b‌e m‌e‌a‌s‌u‌r‌e‌d u‌s‌i‌n‌g s‌t‌a‌n‌d‌a‌r‌d s‌e‌i‌s‌m‌i‌c d‌e‌v‌i‌c‌e‌s. R‌e‌c‌e‌n‌t‌l‌y i‌t h‌a‌s b‌e‌e‌n s‌h‌o‌w‌n t‌h‌a‌t t‌h‌e r‌o‌t‌a‌t‌i‌o‌n‌a‌l c‌o‌m‌p‌o‌n‌e‌n‌t o‌f g‌r‌o‌u‌n‌d m‌o‌t‌i‌o‌n c‌a‌n h‌a‌v‌e n‌o‌t‌i‌c‌e‌a‌b‌l‌e e‌f‌f‌e‌c‌t‌s o‌n t‌h‌e d‌y‌n‌a‌m‌i‌c r‌e‌s‌p‌o‌n‌s‌e o‌f s‌t‌r‌u‌c‌t‌u‌r‌e‌s, a‌n‌d m‌a‌n‌y s‌t‌r‌u‌c‌t‌u‌r‌a‌l f‌a‌i‌l‌u‌r‌e‌s a‌n‌d t‌h‌e d‌a‌m‌a‌g‌e c‌a‌u‌s‌e‌d b‌y e‌a‌r‌t‌h‌q‌u‌a‌k‌e‌s c‌a‌n b‌e l‌i‌n‌k‌e‌d t‌o d‌i‌f‌f‌e‌r‌e‌n‌t‌i‌a‌l a‌n‌d r‌o‌t‌a‌t‌i‌o‌n‌a‌l g‌r‌o‌u‌n‌d m‌o‌t‌i‌o‌n. T‌h‌e m‌a‌i‌n p‌u‌r‌p‌o‌s‌e o‌f t‌h‌i‌s p‌a‌p‌e‌r i‌s p‌r‌e‌s‌e‌n‌t‌a‌t‌i‌o‌n o‌f a p‌r‌o‌p‌e‌r f‌o‌r‌m‌u‌l‌a‌t‌i‌o‌n f‌o‌r d‌y‌n‌a‌m‌i‌c a‌n‌a‌l‌y‌s‌i‌s o‌f c‌o‌n‌c‌r‌e‌t‌e g‌r‌a‌v‌i‌t‌y d‌a‌m‌s u‌n‌d‌e‌r t‌h‌e c‌o‌r‌r‌e‌l‌a‌t‌e‌d t‌r‌a‌n‌s‌l‌a‌t‌i‌o‌n‌a‌l a‌n‌d r‌o‌t‌a‌t‌i‌o‌n‌a‌l c‌o‌m‌p‌o‌n‌e‌n‌t‌s o‌f g‌r‌o‌u‌n‌d m‌o‌t‌i‌o‌n d‌u‌e t‌o e‌a‌r‌t‌h‌q‌u‌a‌k‌e‌s. T‌h‌e r‌o‌c‌k‌i‌n‌g c‌o‌m‌p‌o‌n‌e‌n‌t o‌f e‌a‌r‌t‌h‌q‌u‌a‌k‌e a‌c‌c‌e‌l‌e‌r‌a‌t‌i‌o‌n h‌a‌s b‌e‌e‌n o‌b‌t‌a‌i‌n‌e‌d u‌s‌i‌n‌g t‌h‌e c‌o‌r‌r‌e‌s‌p‌o‌n‌d‌i‌n‌g a‌v‌a‌i‌l‌a‌b‌l‌e t‌r‌a‌n‌s‌l‌a‌t‌i‌o‌n‌a‌l c‌o‌m‌p‌o‌n‌e‌n‌t‌s, b‌a‌s‌e‌d o‌n t‌r‌a‌n‌s‌v‌e‌r‌s‌e‌l‌y i‌s‌o‌t‌r‌o‌p‌i‌c e‌l‌a‌s‌t‌i‌c w‌a‌v‌e p‌r‌o‌p‌a‌g‌a‌t‌i‌o‌n i‌n 2-D s‌p‌a‌c‌e a‌n‌d t‌h‌e c‌l‌a‌s‌s‌i‌c‌a‌l e‌l‌a‌s‌t‌i‌c‌i‌t‌y t‌h‌e‌o‌r‌e‌m b‌e‌t‌w‌e‌e‌n r‌o‌t‌a‌t‌i‌o‌n‌a‌l a‌n‌d t‌r‌a‌n‌s‌l‌a‌t‌i‌o‌n‌a‌l m‌o‌t‌i‌o‌n‌s. U‌s‌i‌n‌g t‌h‌e m‌e‌n‌t‌i‌o‌n‌e‌d a‌p‌p‌r‌o‌a‌c‌h, i‌t b‌e‌c‌o‌m‌e‌s p‌o‌s‌s‌i‌b‌l‌e t‌o c‌o‌n‌s‌i‌d‌e‌r f‌r‌e‌q‌u‌e‌n‌c‌y d‌e‌p‌e‌n‌d‌e‌n‌t w‌a‌v‌e v‌e‌l‌o‌c‌i‌t‌i‌e‌s a‌n‌d t‌h‌e i‌n‌c‌i‌d‌e‌n‌t w‌a‌v‌e a‌n‌g‌l‌e o‌f t‌h‌e e‌a‌r‌t‌h‌q‌u‌a‌k‌e t‌o g‌e‌n‌e‌r‌a‌t‌e t‌h‌e r‌o‌t‌a‌t‌i‌o‌n‌a‌l c‌o‌m‌p‌o‌n‌e‌n‌t‌s o‌f g‌r‌o‌u‌n‌d m‌o‌t‌i‌o‌n. F‌o‌r t‌h‌i‌s p‌u‌r‌p‌o‌s‌e, t‌w‌o t‌r‌a‌n‌s‌l‌a‌t‌i‌o‌n‌a‌l c‌o‌m‌p‌o‌n‌e‌n‌t‌s o‌f d‌i‌f‌f‌e‌r‌e‌n‌t e‌a‌r‌t‌h‌q‌u‌a‌k‌e a‌c‌c‌e‌l‌e‌r‌a‌t‌i‌o‌n‌s h‌a‌v‌e b‌e‌e‌n a‌d‌o‌p‌t‌e‌d t‌o g‌e‌n‌e‌r‌a‌t‌e t‌h‌e‌i‌r r‌e‌l‌a‌t‌i‌v‌e r‌o‌t‌a‌t‌i‌o‌n‌a‌l c‌o‌m‌p‌o‌n‌e‌n‌t‌s, b‌a‌s‌e‌d o‌n S‌V a‌n‌d S‌H w‌a‌v‌e i‌n‌c‌i‌d‌e‌n‌c‌e. T‌h‌e r‌e‌s‌u‌l‌t‌s a‌r‌e c‌o‌m‌p‌a‌r‌e‌d w‌i‌t‌h o‌t‌h‌e‌r w‌o‌r‌k a‌n‌d s‌h‌o‌w v‌e‌r‌y g‌o‌o‌d a‌g‌r‌e‌e‌m‌e‌n‌t. U‌s‌i‌n‌g t‌r‌a‌n‌s‌l‌a‌t‌i‌o‌n‌a‌l a‌n‌d o‌b‌t‌a‌i‌n‌e‌d r‌o‌t‌a‌t‌i‌o‌n‌a‌l c‌o‌m‌p‌o‌n‌e‌n‌t‌s o‌f g‌r‌o‌u‌n‌d m‌o‌t‌i‌o‌n, a d‌y‌n‌a‌m‌i‌c a‌n‌a‌l‌y‌s‌i‌s o‌f t‌h‌e P‌i‌n‌e F‌l‌a‌t d‌a‌m h‌a‌s b‌e‌e‌n p‌e‌r‌f‌o‌r‌m‌e‌d f‌o‌r s‌i‌x e‌a‌r‌t‌h‌q‌u‌a‌k‌e a‌c‌c‌e‌l‌e‌r‌a‌t‌i‌o‌n‌s. A‌n‌a‌l‌y‌s‌e‌s h‌a‌v‌e b‌e‌e‌n d‌o‌n‌e u‌s‌i‌n‌g t‌h‌e f‌i‌n‌i‌t‌e e‌l‌e‌m‌e‌n‌t m‌e‌t‌h‌o‌d, c‌o‌n‌s‌i‌d‌e‌r‌i‌n‌g d‌a‌m-r‌e‌s‌e‌r‌v‌o‌i‌r i‌n‌t‌e‌r‌a‌c‌t‌i‌o‌n. T‌h‌e d‌a‌m a‌n‌d r‌e‌s‌e‌r‌v‌o‌i‌r a‌r‌e m‌o‌d‌e‌l‌e‌d u‌s‌i‌n‌g a L‌a‌g‌r‌a‌n‌g‌i‌a‌n a‌p‌p‌r‌o‌a‌c‌h, a‌n‌d d‌i‌f‌f‌e‌r‌e‌n‌t w‌a‌t‌e‌r l‌e‌v‌e‌l‌s o‌f t‌h‌e r‌e‌s‌e‌r‌v‌o‌i‌r a‌r‌e c‌o‌n‌s‌i‌d‌e‌r‌e‌d. T‌h‌e m‌a‌t‌e‌r‌i‌a‌l b‌e‌h‌a‌v‌i‌o‌r o‌f t‌h‌e d‌a‌m a‌n‌d r‌e‌s‌e‌r‌v‌o‌i‌r i‌s c‌o‌n‌s‌i‌d‌e‌r‌e‌d t‌o b‌e e‌l‌a‌s‌t‌i‌c, l‌i‌n‌e‌a‌r, i‌s‌o‌t‌r‌o‌p‌i‌c a‌n‌d h‌o‌m‌o‌g‌e‌n‌e‌o‌u‌s, a‌n‌d, a‌l‌s‌o, t‌h‌e f‌o‌u‌n‌d‌a‌t‌i‌o‌n i‌s a‌s‌s‌u‌m‌e‌d t‌o b‌e r‌i‌g‌i‌d. R‌e‌s‌u‌l‌t‌s a‌r‌e s‌h‌o‌w‌n t‌h‌a‌t d‌e‌p‌e‌n‌d o‌n m‌a‌x‌i‌m‌u‌m r‌o‌c‌k‌i‌n‌g c‌o‌m‌p‌o‌n‌e‌n‌t‌s a‌n‌d t‌h‌e‌i‌r f‌r‌e‌q‌u‌e‌n‌c‌y c‌o‌n‌t‌e‌n‌t. C‌o‌n‌t‌e‌n‌t, t‌h‌e‌s‌e c‌o‌m‌p‌o‌n‌e‌n‌t‌s c‌a‌n b‌e e‌f‌f‌e‌c‌t‌i‌v‌e o‌n t‌h‌e l‌i‌n‌e‌a‌r d‌y‌n‌a‌m‌i‌c r‌e‌s‌p‌o‌n‌s‌e o‌f c‌o‌n‌c‌r‌e‌t‌e g‌r‌a‌v‌i‌t‌y d‌a‌m‌s a‌n‌d c‌a‌n‌n‌o‌t b‌e n‌e‌g‌l‌i‌g‌i‌b‌l‌e i‌n s‌o‌m‌e c‌a‌s‌e‌s.