Pseudo-static Acceleration Research Articles

Displacements and stresses around a circular tunnel in an elastic medium with pseudo-static horizontal earthquake body forces

This manuscript presents an analytical solution based on the complex variable approach for computing displacements and stresses around a single circular tunnel in a linear elastic medium in the presence of pseudo-static horizontal earthquake body forces. The solution has been obtained with the imposition of uniform as well as non-uniform radial displacement boundary conditions along the tunnel’s periphery taking the ground volume loss (GVL) as a governing input parameter. The non-uniform radial displacements’ boundary condition along the tunnel periphery has been modelled by using the Gaussian distribution function. The effects of the distribution of the tunnel’s peripheral displacements, Poisson’s ratio of the continuum and the depth to radius of the tunnel on the results have been examined. Necessary validation exercise, in the presence of horizontal earthquake acceleration, has also been carried out by comparing the obtained results with that computed on the basis of the finite element method. The results were also compared with that reported in literature. It was observed that with an increase in the pseudo static acceleration, there is (i) a significant increase in the horizontal displacements, (ii) a generation of non-symmetric displacement and stress pattern, (iii) an increase in the shear stresses, and (iv) the development of high deviatoric stress zones around the tunnel periphery.

Seismic Bearing Capacity of Eccentrically and Obliquely Loaded Strip Footings on Geosynthetic-Reinforced Soil

The paper aims to evaluate the pseudostatic seismic bearing capacity of shallow foundations on geosynthetic-reinforced granular soil subjected to inclined/eccentric combined loading. This is achieved through a comprehensive set of lower-bound finite-element limit analysis (LB FELA) simulations adopting the second-order cone programming. The pseudostatic seismic loading is incorporated into the element equilibrium equations as body forces. Both sliding and structural modes of geosynthetic failure are simulated within the FELA framework. The influences of pseudostatic acceleration, embedment depth (u), and the ultimate tensile strength (Tu) of the reinforcement layer on the failure mechanism, bearing capacity ratio, and failure envelope of the overlying obliquely/eccentrically loaded shallow foundation are thoroughly investigated and discussed. It is generally concluded that at a given reinforcement embedment depth, failure envelopes shrink substantially with the increase of the seismic horizontal acceleration and the reduction of the geosynthetic ultimate tensile strength.

Seismic Response of Embedded Retaining Wall for Excavation Support

A comparative analysis of the solutions for the seismic impact on a flexible retaining wall designed to strengthen the construction excavation for a multi-storey building in Sofia has been performed. For this purpose, the following FEM calculation models of the interaction of the wall with the surrounding soil body were compiled: (1) a 1D beam-column-spring model of the wall, including spring support in its embedded length and pseudostatic seismic loading of the wall with active earth pressure, determined by the Mononobe-Okabe method; (2) a 2D model of the wall and soil body, with pseudostatic seismic loading in the form of inertial forces applied at the finite element nodes; (3) a time history FE analysis conducted with the accelerograms of two landmark earthquakes from the Balkan Peninsula - the 1976 Vrancea (Romania) earthquake and the 1986 Kalamata (Greece) earthquake. The pseudostatic seismic loading in analyzes (1) and (2) was determined on the basis of the reference pseudostatic acceleration of 2.3 m/s2, determined for the city of Sofia according to the seismic hazard map for Bulgaria. The accelerograms in analysis (3) were scaled to a maximum value of acceleration of PGA = 2.3 m/s2. The physical modelling of the soil body uses the HSM and HSsmall advanced constitutive laws, implemented in the Plaxis software.

Determining Seismic Bearing Capacity of Footings Embedded in Cohesive Soil Slopes Using Multivariate Adaptive Regression Splines

Seismic bearing capacity of strip footings in cohesive soil slopes considering various embedded depths is investigated in this study. Novel solutions using pseudo-static method and finite element limit analysis (FELA) with upper bound (LB) and lower bound (LB) theorems are presented. The influences of footing depth, slope angle, slope height, undrained shear strength and pseudo-static acceleration on bearing capacity and failure mechanisms are examined using dimensionless parameters. With the comprehensive numerical results, the multivariate adaptive regression splines (MARS) model is then utilized to simulate the sensitivity of all dimensionless input parameters (i.e., the normalized depth of footing D/B, the normalized slope height H/B, the normalized distance from top slope to edge of the footing L/B, slope angle β, the strength ratio cu/γB, and the pseudo-static acceleration factor, kh). The degree of influence of each design parameter is produced, and an empirical equation for the dimensionless output parameter (i.e., bearing capacity factor Nc) is proposed. The study results are accessible in the design charts, tables, empirical equation for design practitioners.

Probabilistic assessment of an earth dam stability design using the adaptive polynomial chaos expansion

Assessment of the stability for an earth dam under seismic loadings is presented in a probabilistic framework. The geotechnical parameters measured at the dam site permit a realistic description of the soil variability for the dam and the superficial foundation. An active learning sparse polynomial chaos expansions (A-bSPCE) combined with Monte Carlo simulation is employed to quantify the uncertainty propagation. The proposed dam design is assessed within two seismic scenarios: the first one considers a constant pseudo-static acceleration while the second one accounts for the uncertainties in seismic loadings by modeling the pseudo-static acceleration as a random variable. The probabilistic assessments for variation of the amplitude and standard deviation of the seismic loading are performed. In addition, the effects of the correlation coefficient between the soil’s shear strength parameters and the soil spatial variability are discussed. It is found that these effects are more significant in the first scenario. The obtained results permit an evaluation of the preliminary dam design and to provide guidance on seismic safety enhancement for the final design phase. The present study also highlights the accuracy and efficiency of the employed reliability method. It is shown that the total computational time of a probabilistic analysis can be reduced from several days to only 20 min by using the A-bSPCE.

Seismic Bearing Capacity of Strip Footings on Cohesive Soil Slopes by Using Adaptive Finite Element Limit Analysis

By employing adaptive finite element limit analysis (AFELA), the seismic bearing capacity of strip footing on cohesive soil slopes are investigated. To consider the earthquake effects, the pseudostatic method is used. The upper and lower bounds for the seismic bearing capacity factor (Nce) are calculated, and the relative errors between them are found within 3% or better by adopting the adaptive mesh strategy. Based on the obtained results, design tables and charts are provided to facilitate engineers use, and the effects of footing position, undrained shear strength, slope angle, slope height, and pseudostatic acceleration coefficient are studied in detail. The collapse mechanisms are also discussed, including overall slope failure and foundation failure.

Reliability analysis of slope stability under seismic condition during a given exposure time

Evaluating the failure probability of a slope under the seismic condition during a given exposure time is important for performance-based assessment of slope stability. In this paper, a two-stage method is suggested to study the seismic stability of a slope during a given exposure time. In the first stage, the exceedance probability of the horizontal pseudo-static acceleration is evaluated. In the second stage, the vulnerability curve of the slope, which shows the relationship between the horizontal pseudo-static acceleration and the failure probability of the slope, is established. The failure probability of the slope during a given exposure time is then assessed by combining the exceedance probability curve of the horizontal pseudo-static acceleration and the vulnerability curve of the slope. Examples investigated show that the reliability of a slope under the seismic condition is controlled by multiple slip surfaces. A slope may have different failure probabilities during the same exposure time when it is at different locations because of different levels of ground shaking. Event at the same site, different slopes may have different failure probability because of the difference in factors like slope geometries and geological conditions. The method suggested in this paper can be used to quantify the effect of the above factors on the reliability of a slope.

Seismic bearing capacity of surficial foundations on sloping cohesive ground

Engineers often assume undrained conditions in stability calculations of shallow foundations resting on soil with fine content as this assumption yields conservative results. Calculation of undrained bearing capacity of shallow foundations on level ground is a well-defined problem and when the footing is located on or near slopes, empirical equations or design charts produced based on limit equilibrium, upper bound plasticity calculations or finite element method (FEM) analyses are used. On the other hand, available studies that consider seismic bearing capacity of foundations on cohesive soils do not consider the influences of all influential parameters. Therefore, their use in design practice is rather limited. Accordingly, this study attempts to develop design charts that consider the influences of all parameters that affect the undrained bearing capacity factor of surficial strip footings under seismic conditions (Ncse). These influences are footing width, slope angle, slope height, footing position, undrained shear strength and pseudo-static acceleration coefficient. As the number of parameters to be considered is high, a parametric study is conducted using FEM models. Obtained results are consistent with the results of available studies in literature. Proposed design charts allow the selection of problem specific Ncse values. A design procedure is defined and two design examples are presented for the calculation of the magnitude of undrained bearing capacity under seismic conditions.

An Integrated Methodology for Displacement-Based Seismic Design of Homogeneous Slopes

This paper presents an integrated methodology for displacement-based seismic design of homogeneous slopes under typical surcharge loading and slope inclination conditions. Charts developed over the basis of dimensionless parameters are given for both selection of the static safety factor of the slope and determination of the corresponding critical pseudo-static acceleration coefficient in the seismic case. In addition, graphs based on the well-known Newmark's sliding block method are proposed for estimation of slope displacements as a function of the critical seismic coefficient. For this purpose, a series of strong ground motion records from two major Chilean earthquakes grouped by soil conditions were utilized. In this way, a simple and rational design procedure can be followed in order to integrate the desired static safety factor of the slope, the amount of seismic displacement judged to be allowable, and the minimum stability requirements for the slope in the seismic case. The proposed methodology can be used in both preliminary calculations and rapid engineering analyses of homogeneous slopes located at sites with similar seismic characteristics to the Chilean subduction zone.

Reliability Coupled Sensitivity-Based Seismic Analysis of Gravity Retaining Wall Using Pseudostatic Approach

AbstractThe stability of geotechnical earth structures is often affected by associated uncertainties present in geotechnical parameters, if they are not properly accounted for. The present paper aims at quantifying these uncertainties and proposes a modification factor, namely probabilistic risk factor (Rf) for each geotechnical random variable. A gravity retaining wall is analyzed by a pseudostatic method of analysis against four modes of failure namely, sliding, overturning, eccentricity, and bearing. The effect of variation of properties of backfill and foundation soil on stability of the wall for various earthquake conditions is analyzed. Rf simultaneously identifies the effects of Pf of a gravity retaining wall subjected to earthquake loading and also the sensitivity of geotechnical random variables on different modes of failure. The geotechnical random variables are modified by Rf and applied in design. It is observed that, apart from the seismic horizontal and vertical pseudostatic acceleration coe...

Some remarks on the seismic behaviour of embedded cantilevered retaining walls

This paper is a numerical investigation of the physical phenomena that control the dynamic behaviour of embedded cantilevered retaining walls. Recent experimental observations obtained from centrifuge tests have shown that embedded cantilevered retaining walls experience permanent displacements even before the acceleration reaches its critical value, corresponding to full mobilisation of the soil strength. The motivation for this work stems from the need to incorporate these observations in simplified design procedures. A parametric study was carried out on a pair of embedded cantilevered walls in dry sand, subjected to real earthquakes scaled at different values of the maximum acceleration. The results of these analyses indicate that, for the geotechnical design of the wall, the equivalent acceleration to be used in pseudo-static calculations can be related to the maximum displacement that the structure can sustain, and can be larger than the maximum acceleration expected at the site. For the structural design of the wall, it is suggested that the maximum bending moments of the wall can be computed using a realistic distribution of contact stress and a conservative value of the pseudo-static acceleration, taking into account two-dimensional amplification effects near the walls.

Active static and seismic earth pressure for c–φ soils

Rankine classic earth pressure solution has been expanded to predict the seismic active earth pressure behind rigid walls supporting c–φ backfill considering both wall inclination and backfill slope. The proposed formulation is based on Rankine's conjugate stress concept, without employing any additional assumptions. The developed expressions can be used for the static and pseudo-static seismic analyses of c–φ backfill. The results based on the proposed formulations are found to be identical to those computed with the Mononobe–Okabe method for cohesionless soils, provided the same wall friction angle is employed. For c–φ soils, the formulation yields comparable results to available solutions for cases where a comparison is feasible. Design charts are presented for calculating the net active horizontal thrust behind a rigid wall for a variety of horizontal pseudo-static accelerations, values of cohesion, soil internal friction angles, wall inclinations, and backfill slope combinations. The effects of the vertical pseudo-static acceleration on the active earth pressure and the depth of tension cracks have also been explored. In addition, examples are provided to illustrate the application of the proposed method.

Rankine pseudo-static earth pressure for c–ϕ soils

This study provides an analytical solution for calculation of seismic active earth pressure on the back of a rigid wall retaining cohesional–frictional (c–φ) soil. The proposed formulation is based on the conjugate stress concept, without employing any additional assumptions, similar to Rankine's original earth pressure formulation. The effects of sloping backfill and wall inclination are considered. In addition, a closed form solution has been derived for the soil–wall friction angle as a function of inertial forces and problem geometry for any given pseudo-static acceleration, as opposed to the constant value commonly employed in practice. The net seismic active force calculated based on the proposed formulations is found to be comparable with those obtained from previously published methods.

Conjugate Stress Approach for Rankine Seismic Active Earth Pressure in Cohesionless Soils

The Rankine classic earth pressure solution has been expanded for the calculation of seismic earth pressure on rigid retaining walls supporting cohesionless backfill. The expanded solution is based on the conjugate stress concept, without employing any additional assumptions. A single formulation is used for static and pseudostatic seismic analyses of active conditions. The equation accounts for sloping backfill, as well as wall inclination, and can readily accommodate layered soil profiles and the presence of groundwater, which require tedious analysis in commonly used methods. A closed-form expression, as opposed to the constant value commonly used in practice, has been derived for the soil-wall friction angle as a function of the inertial forces and problem geometry for a given pseudostatic acceleration. Seismic lateral pressure increases with depth, and the point of application of the resultant is calculated based on stratigraphy. Comparison of the formulation with the widely used Mononobe-Okabe solution revealed a perfect match for the proposed predictions, provided the same soil-wall friction angle is used in both formulations.

Evaluating the effects of the magnitude and amplification of pseudo-static acceleration on reinforced soil slopes and walls using the limit equilibrium Horizontal Slices Method

The effects of horizontal and vertical pseudo-static forces on reinforced soil structures are investigated in the paper. In particular, the effects of the magnitude and amplification of the ground acceleration on the seismic stability of reinforced soil slopes and walls have been investigated using the Horizontal Slices Method (HSM). The HSM is a limit equilibrium method for the analysis of reinforced soil structures, which offers a number of benefits over conventional vertical slice methods. First, a parametric study using acceptable geotechnical, geometrical and design parameters was undertaken. The results of the parametric analysis are presented in dimensionless form relating to the force required to maintain stability of the slope ( K) and the required length of the reinforcements ( L c/ H). Different rotational and planar slip surfaces are shown for various slopes and walls with different geotechnical strength parameters. Second, the capability of the HSM to consider the effect of earthquake amplification on the stability analysis of reinforced soil structures was considered. It has been shown that the effect of horizontal seismic acceleration on the response of reinforced slopes and walls depends mainly on the geotechnical strength parameters. The effect of vertical seismic acceleration on the performance of reinforced slopes is not significant for low values of horizontal seismic acceleration. It has been concluded that ignoring the effect of the amplification phenomenon could result in an underestimated design.

Stability of seismically loaded slopes using limit analysis

Numerical limit analysis is used to assess the stability of slopes subjected to seismic loading. The soil is assumed to follow the Mohr–Coulomb failure criterion. The lower and upper bound theorems are formulated as linear problems to be solved using linear programming techniques. Based on finite element discretisation of the slope, the velocity field is optimised to find the lowest upper bound, and the stress field is optimised to obtain the highest lower bound. Limit equilibrium computations and log-spiral upper bound solutions were also performed for comparison purposes. Additionally, finite element analyses were done for selected cases. Results from the limit equilibrium and finite element methods are in excellent agreement with the rigorous lower and upper bounds for all cases studied. The slip surfaces obtained from both the limit equilibrium and log-spiral upper bound methods lie within the plastic zones obtained for the slopes from both finite element and numerical limit analysis. Plots are presented of the horizontal pseudo-static acceleration ratio kc = ac/g required to cause collapse of simple homogeneous slopes as a function of the slope inclination and shear strength parameters.