An alternative to the Mononobe–Okabe equations for seismic earth pressures

A closed-form stress plasticity solution is presented for gravitational and earthquake-induced earth pressures on retaining walls. The proposed solution is essentially an approximate yield-line approach, based on the theory of discontinuous stress fields, and takes into account the following parameters: (1) weight and friction angle of the soil material, (2) wall inclination, (3) backfill inclination, (4) wall roughness, (5) surcharge at soil surface, and (6) horizontal and vertical seismic acceleration. Both active and passive conditions are considered by means of different inclinations of the stress characteristics in the backfill. Results are presented in the form of dimensionless graphs and charts that elucidate the salient features of the problem. Comparisons with established numerical solutions, such as those of Chen and Sokolovskii, show satisfactory agreement (maximum error for active pressures about 10%). It is shown that the solution does not perfectly satisfy equilibrium at certain points in the medium, and hence cannot be classified in the context of limit analysis theorems. Nevertheless, extensive comparisons with rigorous numerical results indicate that the solution consistently overestimates active pressures and under-predicts the passive. Accordingly, it can be viewed as an approximate lower-bound solution, than a mere predictor of soil thrust. Compared to the Coulomb and Mononobe–Okabe equations, the proposed solution is simpler, more accurate (especially for passive pressures) and safe, as it overestimates active pressures and underestimates the passive. Contrary to the aforementioned solutions, the proposed solution is symmetric, as it can be expressed by a single equation—describing both active and passive pressures—using appropriate signs for friction angle and wall roughness.     

Reliability-based load and resistance factor design approach for external seismic stability of reinforced soil walls

Load and resistance factor design (LRFD) approach for the design of reinforced soil walls is presented to produce designs with consistent and uniform levels of risk for the whole range of design applications. The evaluation of load and resistance factors for the reinforced soil walls based on reliability theory is presented. A first order reliability method (FORM) is used to determine appropriate ranges for the values of the load and resistance factors. Using pseudo-static limit equilibrium method, analysis is conducted to evaluate the external stability of reinforced soil walls subjected to earthquake loading. The potential failure mechanisms considered in the analysis are sliding failure, eccentricity failure of resultant force (or overturning failure) and bearing capacity failure. The proposed procedure includes the variability associated with reinforced backfill, retained backfill, foundation soil, horizontal seismic acceleration and surcharge load acting on the wall. Partial factors needed to maintain the stability against three modes of failure by targeting component reliability index of 3.0 are obtained for various values of coefficients of variation (COV) of friction angle of backfill and foundation soil, distributed dead load surcharge, cohesion of the foundation soil and horizontal seismic acceleration. A comparative study between LRFD and allowable stress design (ASD) is also presented with a design example.     

Seismic behavior of an inverted T-shape flexible retaining wall via dynamic centrifuge tests

In the design procedure for a retaining wall, the pseudo-static method has been widely used and dynamic earth pressure is calculated by the Mononobe–Okabe method, which is an extension of Coulomb’s earth pressure theory computed by force equilibrium. However, there is no clear empirical basis for treating the seismic force as a static force, and recent experimental research has shown that the Mononobe–Okabe method is quite conservative, and there exists a discrepancy between the assumed conditions and real seismic behavior during an earthquake. Two dynamic centrifuge tests were designed and conducted to reexamine the Mononobe–Okabe method and to evaluate the seismic lateral earth pressure on an inverted T-shape flexible retaining wall with a dry medium sand backfill. Results from two sets of dynamic centrifuge experiments show that inertial force has a significant impact on the seismic behavior on the flexible retaining wall. The dynamic earth pressure at the time of maximum moment during the earthquake was not synchronized and almost zero. The relationship between the back-calculated dynamic earth pressure coefficient at the time of maximum dynamic wall moment and the peak ground acceleration obtained from the wall base peak ground acceleration indicates that the seismic earth pressure on flexible cantilever retaining walls can be neglected at accelerations below 0.4 g. These results suggest that a wall designed with a static factor of safety should be able to resist seismic loads up to 0.3–0.4 g.     

Seismic passive resistance with vertical seepage and surcharge

Present paper focuses on the computation of the seismic passive earth pressure acting on a vertical rigid retaining wall by a soil mass subjected to vertical steady-state seepage and a uniform surcharge load. Based on the basic assumptions of Coulomb's theory and a pseudo-static method of analysis, a general solution for the passive earth pressure containing two coefficients is presented. In the solution, many parameters, such as unit weight of saturated soil, soil effective internal friction angle, soil/wall friction angle, water/soil unit weight ratio, surcharge intensity coefficient, horizontal and vertical seismic acceleration coefficients, Poisson's ratio of soil mass, hydraulic gradient, and coefficients of pore water pressure, are considered. The effects of hydraulic gradient and seismic forces on passive earth pressure coefficient and passive earth pressure distribution are investigated. The results indicate that passive earth pressure increases with increasing hydraulic gradient for downward water flow case, but decreases for upward water flow case, and that the presence of seismic forces induces a reduction in passive earth pressure.     

Pseudo-dynamic active force and pressure behind battered retaining wall supporting inclined backfill

Knowledge of seismic active earth pressure behind rigid retaining wall is very important. Commonly used Mononobe–Okabe method considers pseudo-static approach, which gives the linear distribution of seismic earth force. In this paper, the pseudo-dynamic approach, which considers the effect of primary and shear wave propagations, is adopted to calculate the seismic active force. Considering the planar rupture surface, the effect of wide range of parameters like inclination of retaining wall, inclination of backfill surface, wall friction and soil friction angle, shear wave and primary wave velocity, horizontal and vertical seismic coefficients are taken into account to evaluate the seismic active force. Results are presented in terms of seismic coefficients in tabular form and variation of pressure along the depth.     

Static and seismic limit equilibrium analysis of sliding retaining walls under different surcharge conditions

The static and seismic sliding limit equilibrium condition of retaining walls is investigated, and analytical solutions for the angle of the active slip surface, the critical acceleration coefficient and the coefficient of active earth pressure are provided for different surcharge conditions. In particular, walls retaining a horizontal backfill without surcharge, walls supporting an extended uniform surcharge applied at different distances from the wall and walls supporting a limited uniform surcharge or linear uniform surcharge parallel to the wall are considered in the analysis.The solutions have been derived in the framework of the limit equilibrium approach, considering the effect of the wall through its weight, and accounting for the shear resistance at the base of the wall and the inertia force arising in the wall under seismic conditions.For the wall without surcharge the effect of the vertical component of the seismic acceleration as well as the effects of the inclination of the wall internal face and of the soil–wall friction were also investigated.The angle of the slip plane, the critical seismic acceleration coefficient and the coefficient of active earth pressure are given as functions of dimensionless parameters and the boundary conditions for the applicability of each solution are specified. The influence of soil weight, surcharge conditions and inertia forces on the active earth pressure coefficient is analysed.     

Seismic soil pressure for building walls: An updated approach

The Mononobe–Okabe (M–O) method developed in the 1920s in Japan continues to be widely used despite many criticisms and its limitations. The method was developed for gravity walls retaining cohesionless backfill materials. In design applications, however, the M–O method, or any of its derivatives, is commonly used for below ground building walls. In this regard, the M–O method is one of the most abused methods in the geotechnical practice. Recognizing the limitation of the M–O method, a simplified method was recently developed to predict lateral seismic soil pressure for building walls. The method is focused on the building walls rather than retaining walls and specifically considers the dynamic soil properties and frequency content of the design motion in its formulation.     

Seismic active earth pressure of cohesive-frictional soil on retaining wall based on a slice analysis method

The M–O (Mononobe–Okabe) theory is used as a standard method to determine the seismic earth pressure. However, the M–O theory does not consider the influence of soil cohesion, and it cannot determine the nonlinear distribution of the seismic earth pressure. This paper presents a general solution for the nonlinear distribution of the seismic active earth pressure of cohesive-frictional soil using the slice analysis method. A new method is proposed to determine the critical failure angle of the backfill wedge under complex conditions, and an iterative calculation method is presented to determine the tension crack depth of the seismic active earth pressure. The considered parameters in the proposed method include the horizontal and vertical seismic coefficients, wall inclination angle, backfill inclination angle, soil friction angle, wall friction angle, soil cohesion, wall adhesion and uniform surcharge. The classical methods of the M–O and Rankine theories can be regarded as special cases of the proposed method. Furthermore, the proposed method is compared with the test results and previously existing solutions to validate the correctness of the results. Additionally, the parameters׳ effect on the critical failure angle, the resultant force, the application-point position, the tension crack depth and the nonlinear distribution of seismic active earth pressure are studied in graphical form.     

Seismic earth pressures on rigid and flexible retaining walls

While limiting-equilibrium Mononobe–Okabe type solutions are still widely used in designing rigid gravity and flexible cantilever retaining walls against earthquakes, elasticity-based solutions have been given a new impetus following the analytical work of Veletsos and Younan [23]. The present paper develops a more general finite-element method of solution, the results of which are shown to be in agreement with the available analytical results for the distribution of dynamic earth pressures on rigid and flexible walls. The method is then employed to further investigate parametrically the effects of flexural wall rigidity and the rocking base compliance. Both homogeneous and inhomogeneous retained soil is considered, while a second soil layer is introduced as the foundation of the retaining system. The results confirm the approximate convergence between Mononobe–Okabe and elasticity-based solutions for structurally or rotationally flexible walls. At the same time they show the beneficial effect of soil inhomogeneity and that wave propagation in the underlying foundation layer may have an effect that cannot be simply accounted for with an appropriate rocking spring at the base.     

Seismic active pressure distribution history behind rigid retaining walls

Evaluating the seismic active earth pressure on retaining walls is currently based on pseudo-static method in practices. In this method, however, it is not simple, choosing an appropriate value for earthquake coefficient, which should fully reflect the dynamic characteristics of both soil and loading is an important problem. On the other hand, by using only two extra dynamic parameters that are shear wave velocity of soil and predominant frequency of probable earthquake, one can benefit from another more accurate tool called pseudo-dynamic method to solve the problem of earth pressure.In this study in the framework of limit equilibrium analysis, pseudo-dynamic method has been applied into horizontal slice method of analysis to account for the effect of earthquake on lateral earth pressure history behind rigid retaining walls. The pressure history resulted from a number of analyses shows that before and after reaching the peak resultant force, different pressure distributions occur behind a wall that put more local pressure than the same at peak. This method would be a tool to control this phenomenon in wall design.     

Evaluation of pseudostatic active earth pressure coefficient of cantilever retaining walls

A stress plasticity solution is proposed for evaluating the gravitational and dynamic active earth pressures on cantilever retaining walls with long heel. The solution takes into account the friction angle of the soil, wall roughness, backfill inclination and horizontal and vertical seismic accelerations. It is validated by means of the comparison with both traditional limit equilibrium methods (e.g. Mononobe–Okabe equations) and static and pseudostatic numerical FLAC analyses. For numerical analyses the soil is modelled as an elasto-plastic non-dilatant medium obeying the Mohr–Coulomb yield criterion, while the wall is elastic. The solutions for the horizontal and vertical seismic coefficients are proposed, which allow one to determine the intensity of the active thrust and its inclination δ with respect to the horizontal. It is demonstrated that the latter also depends on the soil friction angle φ. The inclination in seismic conditions δ_E is greater than the one in static conditions, δ_S, usually adopted in both cases. As a matter of fact, since wall stability conditions improve with the increase of inclination δ, the present method gives solutions that are less onerous than traditional ones, producing less conservative wall designs. Finally pseudostatic results are compared with proper dynamic analyses (by FLAC code) performed utilising four Italian accelerometric time-histories as input ground motion.     

挡土墙地震被动土压力的拟动力分析

对地震土压力的研究是地震区挡土墙安全设计的一项重要课题。地震条件下,目前的研究主要是给出了土压力的近似拟静力解析解。本文采用可考虑动力荷载下的周期和纵波及横波效应的拟动力方法,对挡土墙后的地震被动土压力进行分析。在挡土墙后平面滑裂面假设的基础上,考虑了水平和垂直向地震加速度、纵波速度、横波速度、挡土墙摩擦角、填土内摩擦角、填土坡角对地震被动土压力的影响。与Mononobe-Okabe理论的拟静力法不同的是,用本方法得出了沿墙身地震被动土压力是非线性变化的结果,这更符合地震条件下土压力的变化规律。     

Predicting seismic permanent displacement of soil walls under surcharge based on limit analysis approach

Seismic permanent displacement of the soil walls plays an important role in design of these structures. Due to the increase in growth of urban areas and the limitations in use of flat grounds, many structures are built near slopes and retaining walls. During earthquakes, these structures can apply an additional surcharge on the wall. The intensity and location of the surcharge is of considerable importance on the seismic displacements of the soil wall. In this study, by using the limit analysis and upper bound theorem, seismic permanent displacement of the soil wall under surcharge has been analyzed. Thus, a formulation is presented for calculating the yield acceleration and seismic displacement for different surcharge conditions. The effect of seismic acceleration, surcharge intensity, its location and soil properties is investigated. A parameter called the “displacement coefficient” is proposed, and is a potential modification for Newmark’s sliding-block method.     

Seismic analysis of base-isolated liquid storage tanks using the BE–FE–BE coupling technique

A three-dimensional soil–structure–liquid interaction problem is numerically simulated in order to analyze the dynamic behavior of a base-isolated liquid storage tank subjected to seismic ground motion. A dynamic analysis of a liquid storage tank is carried out using a hybrid formulation, which combines the finite shell elements for structures and the boundary elements for liquid and soil. The system is composed of three parts: the liquid–structure interaction part, the soil–foundation interaction part, and the base-isolation part. In the liquid–structure interaction part, the tank structure is modeled using the finite elements and the liquid is modeled using the internal boundary elements, which satisfy the free surface boundary condition. In the soil–foundation interaction part, the foundation is modeled using the finite elements and the half-space soil media are modeled using the external boundary elements, which satisfy the radiation condition in the infinite domain. Finally, above two parts are connected with the base-isolation system to solve the system's behavior. Numerical examples are presented to demonstrate the accuracy of the developed method, and an earthquake response analysis is carried out to demonstrate the applicability of the developed technique. The properties of a real LNG tank located in the west coast of Korea are used. The effects of the ground and the base-isolation system on the behavior of the tank are analyzed.     

Simplified transverse seismic analysis of buried structures

This paper presents a simplified method for the analysis of square cross-section buried structures (tunnel) subjected to seismic motion. Finite element analyses are performed to assess the fundamental modes of vibration of the soil layer with and without the tunnel. The influence of the tunnel on the modes of vibrations is taken into account by comparing the modal deformations in the free-field to those in the presence of the tunnel. From this comparison the zone of influence of the modal displacements due to the presence of the structure is determined. The resulting model is subjected to horizontal and vertical excitation of statistically independent accelerograms compatible with the response spectra of the Regulatory Guide 1.6 of the Nuclear Energy Commission. The free-field displacement is introduced at the boundaries of the zone of influence. The proposed simplified static analysis yields a state of stresses similar to that obtained from a full dynamic analysis of the complete soil–tunnel system. Several examples are solved to corroborate the validity of the method.     

Analytical solution of seismic pseudo-static active thrust acting on fascia retaining walls

This paper presents a limit equilibrium method, based on the approach of Mononobe and Okabe, for calculating the active thrust on fascia retaining walls, where common methods cannot be used owing to the narrowness of the backfill which does not permit the development of the thrust wedge in the shape and sizes predicted by these methods. The proposed method examines three distinct failure mechanisms, called Mechanism 1, Mechanism 2 and Mechanism 3, where the thrust wedge is formed by one, two or three blocks, respectively; separated by plane slip surfaces. The seismic forces have been simulated with the pseudo-static method. For all three mechanisms, the active thrust is obtained in closed form: in particular, with a cubic equation for Mechanism 2, and with a system of two equations, one cubic and the other quartic, for Mechanism 3. Mechanisms with more than three blocks cannot have analytical solutions. The study is completed by an examination of some significant cases from which the higher attenuation of the seismic thrust, with respect to the static, emerges as the backfill width reduces.     

Slope seismic stability analysis on kinematical element method and its application

In this paper, kinematical element method (KEM) is extended to the solution of seismic slope stability with the pseudo-static approach. Analytical expressions are derived to calculate the factor of safety of slopes subjected to seismic loading and pore-water pressure. KEM is adopted to assess seismic stability of slope and some examples show that the results obtained from KEM, limit equilibrium method (LEM), variational method and strength reduction method (SRM) are generally in good agreement. And then the seismic slope stability charts are developed on the basis of KEM and pseudo-static approach, providing a rapid and reliable way to calculate the factor of safety and the location of critical slip surface without iteration. Based on the above seismic slope stability charts, a new back analysis method is presented. KEM and pseudo-static approach are applied to study the effect of blasting on the stability of open pit slope, and the approach to determine the relationship between critical explosive weight and distance is presented.     

Nonlinear response of surface soil and NTT building due to soil–structure interaction during the 1995 Hyogo-ken Nanbu (Kobe) earthquake

The 1995 Hyogo-ken Nanbu (Kobe) earthquake brought about enormous damage to structures in the Hanshin and Awaji areas. In this paper the importance of investigating the relationship between ground motion and structural damage is pointed out.

Strong seismic motion was observed at the NTT (Nippon Telegraph and Telephone) Building during this earthquake. The structural damage to this building was relatively slight. In order to evaluate the relationship between ground motion and structural damage, it is necessary to assess the effects of the soil–structure interaction. In this study, the seismic response of the building and of the surface soil were evaluated by means of a nonlinear soil–structure interaction analysis using FEM.

It was found that, the nonlinearity of surface soil near the building had a great effect on the soil–structure interaction, especially the rocking of the building.     

Pseudo-static limit analysis by discontinuity layout optimization: Application to seismic analysis of retaining walls

Discontinuity layout optimization (DLO) is a recent development in the field of computational limit analysis, and to date, the literature has examined the solution of static geotechnical stability problems only by this method. In this paper the DLO method is extended to the solution of seismic problems though the use of the pseudo-static approach. The method is first validated against the solutions of Mononobe–Okabe and Richards and Elms for the seismic stability of retaining walls, and then used to study the effect of a wider range of failure modes. This is shown to significantly affect the predicted stability. A framework for modelling water pressures in the analysis is then proposed. Finally an example application of the method is illustrated through the assessment of two quay walls subjected to the Kobe earthquake.     

桩-液化土相互作用p-y关系分析

基于多工况的桩-液化土体动力相互作用振动台试验,研究地震荷载作用下液化土层中桩土间侧向相互作用力p与桩身和土体间侧向相对位移y之间的关系。将试验得到的实际p-y曲线与采用拟静力法和以API规范为基础的折减系数法计算出的p-y曲线进行对比,结果表明:(1)液化土层中试验得到的桩真实p-y响应及由拟静力法和折减系数法得到的结果都呈非线性变化,三者极限状态有接近一致的趋势,但变化过程差异明显;(2)采用拟静力法和折减系数法都会使液化土层桩基础侧向反力迅速增长,很快达到屈服极限,远远超过实际情况,会导致相当保守的结果;(3)液化进程中控制桩p-y响应的是土体位移而非惯性力,因而拟静力法和折减系数法的原理不适合桩-液化土体动力相互作用分析,不能用于液化土层中桩基础地震响应的计算。