Pseudo-static method

The evaluation of the seismic response of a slope may be performed by different analysis methods in relationship with the level of complexity of the problem. The simplest approach is the pseudo-static method, and at the other extreme is complete nonlinear finite element modelling (FEM). 

A substantial divergence in practice is developing over the use of the pseudo-static method. The EC8-5 standard includes various partial factors for use with the pseudo-static method, implying acceptance of that method within the EU. But, North American practice would generally not accept the pseudo-static method because (i) there is a lack of clarity in the meaning of the factor of safety under transient loads - a FS < 1 simply means the slope may move but that movement may... 

Current methods for the earthquake design of MSWs fall into three categories (1) pseudo-static methods, (2) sliding block displacement methods, and (3) dynamic finite element or finite difference methods. 

A. Stationary (mostly pseudo-stationary) methods -+ static measurement B. Non-stationary methods -> dynamic measurement ... 

Static charge-density susceptibilities have been computed ab initio by Li et al (38). The frequency-dependent susceptibility x(r, r cd) can be calculated within density functional theory, using methods developed by Ando (39 Zang-will and Soven (40 Gross and Kohn (4I and van Gisbergen, Snijders, and Baerends (42). In ab initio work, x(r, r co) can be determined by use of time-dependent perturbation techniques, pseudo-state methods (43-49), quantum Monte Carlo calculations (50-52), or by explicit construction of the linear response function in coupled cluster theory (53). Then the imaginary-frequency susceptibility can be obtained by analytic continuation from the susceptibility at real frequencies, or by a direct replacement co ico, where possible (for example, in pseudo-state expressions). 

As to the large reservoir like Lenggu Hydro-power Project, the impact of seismic activity generally should be taken into account. For the earthquake condition simulation, the pseudo-static seismic analysis method was adopted. The moderate earthquake with exceeding probability to be 10 percent in the future 50 years was considered. Only horizontal seismic load was picked out and the input seismic coefficient was 0.167. The displacement contour and vectors after 1000 iteration steps are shown in Figure 7. Figure 8 shows... 

The principles of recycling presented in this section apply equally to continuous processes taking place in flows, and to processes carried out in static apparatus according to the pseudo-continuous method. 

The essentials of the pseudo-continuous method are that the study of the continuous process is carried out in static apparatus (autoclave) by carrying out conversion cycles consecutively, while carefully maintaining the systems of movement and combination of the flows under the flow conditions. 

If both walls are spaced far apart, the pressures on one wall are not influenced by the presence of the other. For low-frequency input motions with frequency less than half the fundamental frequency of the unrestrained backfill, VJAW (Fj is the soil shear wave velocity), the pseudo static conditions are governed (i.e., the dynamic amplification is negligible). For this range of frequencies, wall pressures with plane strain assumption can be obtained from elastic solution for the case of a uniform, constant horizontal acceleration applied throughout the soil. The dynamic earth pressures obtained from this method must be... 

Okabe method is obtained from the equilibrium of the active or passive wedges (Fig. 9a and b). In addition to the forces that exist under static conditions of the failure wedge in a dry, cohesionless backfiU, the wedge is also subjected to horizontal and vertical pseudo-static forces specified as the mass of the wedge multiplied by pseudo-static accelerations ai, = khg and av = kvg. The total active and passive thrusts on a rigid waU retaining a dry, cohesionless backfill can be presented in a form similar to what is developed for the static conditions in the Coulomb s theory ... 

MSWs are complex mechanical systems and this complexity is compounded when the structures are subjected to transient dynamic loading due to earthquake. Pseudo-static and displacement methods may be sufficient for simple structures, for stable ground conditimis and MSWs in low seismic risk areas, and/or for preliminary design. Otherwise, more sophisticated analyses may be warranted using advanced dynamic finite element model or finite difference computer programs. Today commercially available computer programs offer the user a suite of constitutive models for the component materials and in some cases allow the user to implement their own constitutive models. In order to accurately model the soil in a MSW, it may be necessary to use nonlinear cyclic stress-strain models. The equivalent-linear... 

A more rigorous and direct method for computing earthquake-induced ground deformations, with and without soil liquefaction, is a nonlinear finite element dynamic analysis (Finn et al. 1994) that overcomes the uncertainties associated with the approximation embedded in the pseudo-static stability calculation and in the Newmark deformation analysis. With PWP calculations built into effective stress soil models, the finite element method has the capability to simulate the coupled effects from dynamic site response, generation of excess pore water pressures, soil liquefaction, and post-liquefaction behavior, and this is a trend method for quantitative evaluation of earthquake-induced ground deformations. 

There are four main approaches to tackle the design for transient deformations free-held deformation methods, pseudo-static soil-structure interaction analyses, dynamic soil-structure interaction hnite-element analyses, and dynamic earth pressure methods. The merits and drawbacks of these methods are summarized in Table 2. 

Pseudo-static soil-structure interaction methods 1. Good approximation of soil-structure interaction 2. Comparatively easy to formulate 3. Reasonable accuracy in determining structure response 4. Computationally efficient 5. Sensitivity analysis can be easily performed 1. Ignores inertial effects 2. Less precision with highly variable ground 3. Shear displacement not transmitted uniformly to shallow box stmctures Most conditions except for variable soil profile, shallow stmctures... 

Solution of the Kohn-Sham equations as outlined above are done within the static limit, i.e. use of the Born-Oppenheimer approximation, which implies that the motions of the nuclei and electrons are solved separately. It should however in many cases be of interest to include the dynamics of, for example, the reaction of molecules with clusters or surfaces. A combined ab initio method for solving both the geometric and electronic problem simultaneously is the Car-Parrinello method, which is a DFT dynamics method [52]. This method uses a plane wave expansion for the density, and the inner ions are replaced by pseudo-potentials [53]. Today this method has been extensively used for studies of dynamic problems in solids, clusters, fullerenes etc [54-61]. We have recently in a co-operation project with Andreoni at IBM used this technique for studying the existence of different isomers of transition metal clusters [62,63]. 

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