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 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.     

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。     

地震作用下非饱和土主动土压力极限分析半解析法

地震是诱发边坡失稳的主要因素之一,重力式挡土墙作为一种广泛采用的岩土支挡结构,有必要对其地震稳定性问题进行深入的研究.为有效评估地震作用下非饱和填土的主动土压力,基于极限分析上限原理和拟动力法,提出一种半解析水平片分法,计算具有非线性分布特征的非饱和土重力和地震惯性力所做外功率,并构建功能平衡方程,得到非饱和填土主动土压力显示半解析解.通过与解析解对比,验证该方法的合理性,并通过算例分析,揭示吸力效应的强化机制和非饱和填土主动土压力的地震响应规律.结果表明:忽略吸力效应会高估填土的主动土压力,吸力的强化作用不仅取决于填土类型,还与地震动特性密切相关;水平和竖向地震动对土压力有较大影响, 土压力系数峰值随土剪切模量的增加略有增加并向负方向移动,随地震周期的增加略有增加并向正方向移动;填土倾角较大时,坡顶附加荷载的影响更加显著;对于倾角大于100°的填土,墙G土界面摩擦角较大时,土压力相对较高.     

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.     

Seismic passive earth resistance using modified pseudo-dynamic method

In earthquake prone areas, understanding of the seismic passive earth resistance is very important for the design of different geotechnical earth retaining structures. In this study, the limit equilibrium method is used for estimation of critical seismic passive earth resistance for an inclined wall supporting horizontal cohesionless backfill. A composite failure surface is considered in the present analysis. Seismic forces are computed assuming the backfill soil as a viscoelastic material overlying a rigid stratum and the rigid stratum is subjected to a harmonic shaking. The present method satisfies the boundary conditions. The amplification of acceleration depends on the properties of the backfill soil and on the characteristics of the input motion. The acceleration distribution along the depth of the backfill is found to be nonlinear in nature. The present study shows that the horizontal and vertical acceleration distribution in the backfill soil is not always in-phase for the critical value of the seismic passive earth pressure coefficient. The effect of different parameters on the seismic passive earth pressure is studied in detail. A comparison of the present method with other theories is also presented, which shows the merits of the present study.     

Seismic rotational displacement of gravity walls by pseudo-dynamic method: Passive case

Prediction of the seismic rotational displacements of retaining wall under passive condition is an important aspect of design in earthquake prone region. In this paper, the pseudo-dynamic method is used to compute the rotational displacements of rigid retaining wall supporting cohesionless backfill under seismic loading for the passive earth pressure condition. The proposed method considers time, phase difference and effect of amplification in shear and primary waves propagating through both the backfill and the retaining wall. The influence of ground motion characteristics on rotational displacement of the wall is evaluated. Also the effects of variation of parameters like wall friction angle, soil friction angle, amplification factor, shear wave velocity, primary wave velocity, period of lateral shaking, horizontal and vertical seismic accelerations on the rotational displacements are studied. The rotational displacement of the wall increases substantially with increase in amplification of both shear and primary waves, time of input motion, period of lateral shaking and decreases with increase in soil friction angle, wall friction angle. The rotational displacements of the wall also increase when the effect of wall inertia is taken into account. Results are provided in graphical form.     

海啸作用下滨水挡土墙抗震转动稳定性上限分析

地震诱发的海啸对沿海围护结构的破坏具有强度大的特点。滨水挡土墙作为重要的围护结构,海啸与地震的联合作用极易造成其发生绕墙踵的被动破坏。采用条分法,将土楔体分割成无数平行于破裂面的刚性土条,并建立绕墙踵转动的挡墙与刚性土条之间的速度容许场。基于极限上限理论,依据外力做功功率等于其内能耗散功率,推导了地震加速度系数的表达式。与经典极限平衡理论相比,该方法考虑了挡墙的位移模式,且无需假设地震土压力的作用位置。分析了浪高与海平面高度之比,内摩擦角φ及墙土摩擦角δ对滨水挡土墙稳定性的影响。     

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.     

Computation of sliding displacements of bridge abutments by pseudo-dynamic method

The paper focuses on seismic sliding displacement calculations of gravity wall bridge abutments when subjected to passive condition during earthquakes. Pseudo-dynamic approach has been used for the calculation of the passive seismic earth pressure. A novel element of the present investigation is the computation of seismic passive earth pressure coefficients by considering the composite curved rupture surface behind the abutment wall in the framework of limit equilibrium method. Sliding failure along the wall base is considered in the new pseudo-dynamic method. The critical seismic acceleration coefficient for sliding and sliding component of the displacement, resulting from horizontal and vertical sinusoidal ground accelerations, are computed by using Newmark's sliding block method. The effect of sliding on the response of earth structures is evaluated and comparisons are made between sliding displacements calculated using planar and composite failure mechanisms. Results of the comparative study showed that the assumption of planar failure mechanism for rough soil–wall interfaces significantly overestimates the critical seismic accelerations for sliding and underestimates the sliding displacements.     

Mechanics and performance of a tied-back wall under seismic loads

In this paper the stability of a tied-back wall subjected to seismic loads is analysed for a predetermined mode of failure (rotation about the top of the wall) and the analysis is compared with data from tests on this type of wall using the seismic simulator at the State University of New York at Buffalo. We carried out a pseudo-static analysis of the problem using the Mononobe-Okabe earth pressure coefficients, wherein the dynamic effects due to the seismic loading are converted into equivalent static loads. The acceleration ratio at which the wall fails by rotation about the top was obtained by considering the moments due to the various lateral earth pressure resultants and the inertial forces induced in the soil due to the seismic loading. We found that the presence of wall friction on the passive side significantly enhances the stability of the flexible retaining wall under seismic loads. Thus, flexible retaining walls supporting dry cohesionless soil can be very efficient during earthquakes. Under moderate earthquakes, an increase in the depth of embedment increases the dynamic factor of safety significantly. However, beyond a certain acceleration ratio for a soil with a particular value of ø, any increase in the depth of emdedment has no effect in impeding failure, irrespective of any change in the geometry of the system. Seismic design charts are presented to evaluate the stability of, and to design, flexible retaining walls embedded in dry cohesionless soils under seismic loading.     

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.     

考虑填土侧向变形的地震土压力计算方法

地震土压力评价是挡土墙抗震设计的关键问题之一.以往的研究结果表明,挡墙上地震土压力的大小及分布与墙体的侧向位移或者墙后填土的侧向变形密切相关.经典的物部-冈部地震土压力公式可计算填土处于主动与被动状态的极限平衡条件下的土压力,未考虑挡墙侧向位移或填土侧向变形对土压力的影响.在研究土压力系数随应变增量比变化规律的基础上,本文指出土压力系数与挡土墙位移量之间不存在唯一性关系,发现正常固结填土的土压力系数与以应变增量比表述的填土侧向应变约束条件之间具有良好的唯一性,揭示了压剪耦合效应是土压力形成的物理本质;基于上述的唯一性关系和中间土楔等概念,提出了可考虑填土侧向变形的地震土压力实用计算方法,并通过土压力模型试验结果初步验证了该方法的合理性.     

Dynamic passive pressure from c–ϕ soil backfills

This technical note presents an analytical expression for the total passive pressure on a retaining wall from the c–? soil backfill subjected to both horizontal and vertical seismic inertial forces. The developed expression has been analysed for the special cases, and the results have been found identical to those proposed by earlier researchers on the subject. A numerical example, presented to illustrate the steps for the calculation of total dynamic passive pressure using the developed general expression, shows that the design value of total dynamic passive pressure as a resistance to the retaining wall movement should be obtained with upward vertical seismic inertial force in combination with the direction of horizontal seismic force towards the backfill.     

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.     

基于拟动力法的顺层岩质边坡稳定性极限分析

基于极限分析上限法的原理,结合随时间变化的拟动力法,考虑地震、水力和坡顶超载作用下建立含张拉裂缝的顺层岩质边坡极限分析模型,并推导出边坡安全系数的表达式。结构面采用合理的水压力分布形式,且强度参数采用非等比例折减,分析边坡前缘是否堵塞,水平和竖向地震系数、张裂缝积水深度和坡顶超载对边坡安全系数的影响。分析表明:边坡前缘堵塞、水平地震系数、张裂缝积水深度和坡顶超载的增加,降低边坡的稳定性;竖向地震系数的增加提高了边坡的稳定性。随着非等比例折减相关系数的增加,边坡稳定性显著提高。因此结构面参数的变化规律需要得到关注。     

基于拟动力法的抗滑桩加固边坡地震稳定性分析

地震力作用下土质边坡动态稳定性研究对实际边坡工程有着重要的意义。采用拟动力法结合简化毕肖普法研究坡顶抗滑桩加固土质边坡在地震力作用下的动态稳定性。尽管拟静力法是目前处理地震力最为广泛的方法之一,但其局限性在于无法考虑地震力随时间变化且忽略了地震波在土体中的传播。而拟动力法采用正弦波模拟地震波在土中传播,并考虑地震波从坡脚传递到坡顶的相位差以及阻尼力对边坡稳定性的影响,通过边坡安全系数的变化揭示土质边坡在地震力作用下的稳定性变化规律。将得到的结果与拟静力法进行对比,突出了拟动力法的优势。最后,考虑水平地震加速度系数、加速度幅值放大系数以及土体内摩擦角对边坡稳定性的影响,以期对实际工程提供理论借鉴。     

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.     

Evaluation of active and passive seismic earth pressures considering internal friction and cohesion

The Log-Spiral-Rankine (LSR) model, which is a generalized formulation for assessing the active and passive seismic earth pressures considering the internal friction and cohesion of backfill soil, is reviewed and improved in this study. System inconsistencies in the LSR model are identified, which result from an inaccurate assumption on the vertical normal stress field (σ_z=γz) in a general c–ϕ soil medium, and from omitting the effect of soil cohesion when solving for the stress states along the failure surface. The remedies to the said inconsistencies are presented, and local and global iteration schemes are introduced to solve the resulting highly coupled multivariate nonlinear system of equations. The modified LSR model provides a more complete and accurate solution for earth retaining systems, including the geometry of the mobilized soil body, the stress state along the failure surface, as well as the magnitude and the point of application of the resultant earth thrust.     

Dynamic active thrust from c–ϕ soil backfills

This technical note presents an analytical derivation of the expression for the total dynamic active thrust on a retaining wall from the c–? soil backfill considering both horizontal and vertical seismic coefficients. The derivation is based on the Coulomb sliding wedge concept, and it considers tension cracks, wall adhesion, and surcharge in order to make the expression useful for practical applications. It is found that the special cases of the general expression result in the expressions for total static and dynamic active thrusts presented by earlier researchers for different field conditions of soil backfills with and without seismic loadings.