Investigation of seismic response of cantilever retaining walls: Limit analysis vs shaking table testing

The earthquake response of cantilever retaining walls is explored by means of theoretical analyses and shaking table testing conducted at University of Bristol (EERC - EQUALS). The theoretical investigations employ both limit analysis and wave-propagation methods, which take into account different aspects of the problem such as inertia, strength, kinematics and compatibility of deformations. The experimental programme encompasses different combinations of retaining wall geometries, soil configurations and input ground motions. The response analysis of the systems at hand aims at shedding light onto salient features of the problem, such as: (1) the magnitude of soil thrust and its point of application; (2) the relative sliding versus rocking of the wall base and the corresponding failure modes; (3) the importance of the interplay between soil stiffness, wall dimensions and excitation characteristics, as affecting the above; (4) the importance of wall dynamics and phase differences between peak stresses and displacements. The results of the experimental investigations are in good agreement with the theoretical models and provide a better understanding on the complex mechanics of the problem.     

Seismic pressures on rigid cantilever walls retaining elastic continuously non-homogeneous soil: An exact solution

The dynamic response of an elastic continuously nonhomogeneous soil layer over bedrock retained by a pair of rigid cantilever walls to a horizontal seismic motion and the associated seismic pressure acting on these walls are determined analytically–numerically. The soil non-homogeneity is described by a shear modulus increasing nonlinearly with depth. The problem is solved in the frequency domain under conditions of plane strain and its exact solution is obtained analytically. This is accomplished with the aid of Fourier series along the horizontal direction and solution of the resulting system of two ordinary differential equations with variable coefficients by the method of Frobenius in power series. Due to the complexity of the various analytical expressions, the final results are determined numerically. These results include seismic pressures, resultant horizontal forces and bending moments acting on the walls. The solution of the problem involving a single retaining wall can be obtained as a special case by assuming the distance between the two walls to be very large. Results are presented in terms of numerical values and graphs using suitable dimensionless quantities. The effect of soil non-homogeneity on the system response is assessed through comparisons for typical sets of the parameters involved.     

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.     

Centrifuge modelling of flexible retaining walls subjected to dynamic loading

This paper outlines the results of an experimental program carried out on centrifuge models of cantilevered and propped retaining walls embedded in saturated sand. The main aim of the paper is to investigate the dynamic response of these structures when the foundation soil is saturated by measuring the accelerations and pore pressures in the soil, displacements and bending moment of the walls. A comparison among tests with different geometrical configurations and relative density of the soil is presented. The centrifuge models were subjected to dynamic loading in the form of sinusoidal accelerations applied at the base of the models. This paper also presents data from pressure sensors used to measure total earth pressure on the walls. Furthermore, these results are compared with previous dynamic centrifuge tests on flexible retaining walls in dry sand.     

Seismic stability of retaining walls with surcharge

The use of pseudo-static methods for the computation of soil thrust acting on retaining walls under seismic condition is well established in the design of such structures. Although different methods, based on the limited displacement concept, have been developed in the last 20 years, the most common design method is still the method derived from the theory developed by Mononobe and Okabe. However, the Mononobe–Okabe method presents a basic shortcoming: the solution is based on the limit equilibrium of the soil wedge without taking into account the presence of the wall. In the paper a new solution based on the pseudo-static equilibrium of the soil–wall system is presented. The developed solution takes into account the effect of the presence of the wall and it is applied to soil–wall systems with surcharged backfills. Formulas are provided to calculate directly the yield acceleration and the inclination of the failure surface. The effect of the intensity of the surcharge and of its distance from the wall is investigated and the results are compared to those obtained in the case of soil–wall systems without surcharge.     

Seismic active thrust on cantilever walls with short heel

This note presents an analytical solution of the thrust on retaining walls with short heel, where the seismic action is simulated through the pseudo-static approach. The critical shape of the thrust wedge is assumed to be that minimizing the safety factor against wall sliding. This procedure is also congruent with the original method of Coulomb. The problem is mathematically expressed by a system of two equations, the first cubic and the second quadratic, whose solution converges quickly.     

Seismic behaviour of flexible retaining systems subjected to short-duration moderately strong excitation

Using finite-element modelling, this paper explores the magnitude and distribution of dynamic earth pressures on several types of flexible retaining systems: L-shaped reinforced-concrete walls, piled walls with horizontal or with strongly inclined anchors, and reinforced-soil walls. The utilized base excitation is typical of earthquake motions of either high or moderately low dominant frequencies having a peak ground acceleration (PGA) of 0.40 g and relatively short duration. Linear as well as non-linear (Mohr–Coulomb) soil behaviour is investigated, under dry conditions. The results show that, as the degree of realism in the analysis increases, we can explain the frequently observed satisfactory performance of such retaining systems during strong seismic shaking.     

Seismic stability analysis of gravity retaining walls

A new approach based on the category of upper bound theorem of limit analysis is presented in this study to consider the seismic stability of gravity retaining walls. The retaining wall and the backfill soil were taken as a whole system. For a translational failure mechanism assumed, formulas are provided to calculate directly the yield acceleration and the inclination of the failure surface. An example is shown to illustrate the method. Comparisons are made with limit equilibrium method, and the results are found consistent. Based on a limited parametric study, it is shown that the wall roughness has remarkable influence on the yield acceleration.     

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.     

Effect of structural design on fundamental frequency of reinforced-soil retaining walls

The results of a numerical study on the influence of a number of structural design parameters on the fundamental frequency of reinforced-soil retaining wall models are presented and discussed. The design parameters in the study include the wall height, backfill width, reinforcement stiffness, reinforcement length, backfill friction angle and toe restraint condition. The intensity of ground motion, characterized by peak ground acceleration, is also included in the study as an additional parameter. The study shows that the fundamental frequency of reinforced-soil wall models with sufficiently wide backfill subjected to moderately strong vibrations can be estimated with reasonable accuracy from a few available formulae based on linear elastic wave theory using the shear wave speed in the backfill and the wall height. Numerical analyses showed no significant influence of the reinforcement stiffness, reinforcement length or toe restraint condition on the fundamental frequency of wall models. The strength of the granular backfill, characterized by its friction angle, also did not show any observable effect on the fundamental frequency of the reinforced-soil retaining wall. However, the resonance frequencies of wall models were dependent on the ground motion intensity and to a lesser extent, on the width to height ratio of the backfill.     

Seismic response of multi-tiered reinforced soil retaining walls

In this study, a validated Finite Element procedure was used to investigate the similarities and differences of seismic performances between single- and multi-tiered reinforced soil walls. Three-tiered walls at a total height of 9 m were analyzed together with vertical walls at the same height. It was found from the Finite Element analyses that the resonant frequency of reinforced soil walls might increase with an increase in the tier-offset. The multi-tiered configuration could considerably reduce the residual lateral facing displacement and the average reinforcement load, and the reinforcement load distribution with height was different from that in vertical walls. With the same reinforcement length and spacing, the multi-tiered walls resulted in smaller reinforcement connection loads with the facing blocks. The study filled the gap of seismic behavior of multi-tiered reinforced soil retaining walls and revealed a few unique dynamic properties of this type of earth structures.     

Seismic lateral movement prediction for gravity retaining walls on granular soils

At present, methods based on allowable displacements are frequently used in the seismic design of earth retaining structures. However, these procedures ignore both the foundation soil deformability and the seismic amplification of the soil placed behind the retaining wall. Thus, they are not able to predict neither a rotational failure mechanism nor seismic induced lateral displacements with an acceptable degree of accuracy for the most general case. In this paper, a series of 2D finite-element analyses were carried out to study the seismic behavior of gravity retaining walls on normally consolidated granular soils. Chilean strong-motion records were applied at the bedrock level. An advanced non-linear constitutive model was used to represent both the backfill and foundation soil behavior. This elastoplastic model takes into account both the stress dependency of soil stiffness and coupling between shear and volumetric strains. In unloading–reloading cycles, the non-linear shear-modulus reduction with shear strain amplitude is considered. Interface elements were used to model soil–structure interaction. Routine-design charts were derived from the numerical analyses to predict the lateral movements at the base and top of gravity retaining walls located at sites with similar seismic characteristics to the Chilean subduction zone. Thus, wall seismic rotation can also be obtained. The developed charts consider wall dimensions, granular soil properties, bedrock depth, and seismic input motion characteristics. As shown, the proposed charts match well with available experimental data.     

The inspection of retaining walls using GPR

In hilly regions, retaining walls along roads, motorways and railway lines are numerous. In some cases the knowledge of the details of the construction is limited. If rehabilitation work becomes necessary, a detailed knowledge of the construction is desirable for the improved planning of maintenance and repair. This paper describes the application of Ground Penetrating Radar (GPR) for the inspection of retaining walls. The work was carried out in two steps. First, an investigation was carried out on large retaining walls at a Swiss motorway within the framework of a service contract. This included the development of an apparatus enabling high precision positioning of the antennas on the walls. Second, a pilot study was performed on a smaller wall with optimized acquisition and processing parameters. This included the use of antennas with different orientations and the fusion of the two corresponding datasets as well as true 3-D data processing. This paper describes the approaches to data acquisition and processing in the form of the two case studies. Results from different acquisition and processing strategies are compared and the benefits and limits are discussed.     

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.     

Experimental developments for studying static and seismic behavior of retaining walls with liquefiable backfills

The effects of earthquakes on cantilever retaining walls with liquefiable backfills were studied. The experimental techniques utilized in this study are discussed here. A series of centrifuge tests was conducted on aluminum, fixed-base, cantilever wall models retaining saturated, cohesionless backfills. Accelerations on the walls and in the backfill, static and excess pore pressures in the soil, and deflections and bending strains in the wall were measured. In addition, direct measurements of static and dynamic lateral earth pressures were made. In some tests, sand backfills were saturated with the substitute pore fluid metolose. Modeling of model type experiments were conducted. The experimental measurements were found internally consistent and repeatable. Both static and dynamic earth pressure measurements were determined to be reliable. It was also observed that for the test configuration adopted, a special boundary treatment such as the use of duxseal is optional. Static and seismic modeling of models were also successful, which indicated that the assumed scaling relations were essentially correct.     

Earthquake-induced displacements of gravity retaining walls and anchor-reinforced slopes

This paper examines in terms of seismic performance, the effectiveness of anchor reinforcement against gravity retaining walls used to stabilize a dry homogenous fill slope in earthquake-prone environment. Both analyzed stabilizing measures have the same design yield acceleration estimated from a limit equilibrium approach. The earthquake-induced displacements are calculated using a sliding block formulation of the equation of motion. Sliding failure along the base of the gravity retaining wall and rotational failure of the soil active wedge behind the wall, as well as rotational failure of the slide mass of the anchor-reinforced slope were considered in the present formulation. For the specific characteristics of the analyzed fill slope and input horizontal ground motion, the slope reinforced with anchors appears to experience vertical and horizontal seismic displacements at slope crest smaller by 12% and respectively, 32% than the vertical and horizontal earthquake-induced deformations estimated at the top of the active wedge behind the gravity retaining wall.     

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.     

Centrifuge modeling of seismic response of layered soft clay

Centrifuge modeling is a valuable tool used to study the response of geotechnical structures to infrequent or extreme events such as earthquakes. A series of centrifuge model tests was conducted at 80g using an electro-hydraulic earthquake simulator mounted on the C-CORE geotechnical centrifuge to study the dynamic response of soft soils and seismic soil–structure interaction (SSI). The acceleration records at different locations within the soil bed and at its surface along with the settlement records at the surface were used to analyze the soft soil seismic response. In addition, the records of acceleration at the surface of a foundation model partially embedded in the soil were used to investigate the seismic SSI. Centrifuge data was used to evaluate the variation of shear modulus and damping ratio with shear strain amplitude and confining pressure, and to assess their effects on site response. Site response analysis using the measured shear wave velocity, estimated modulus reduction and damping ratio as input parameters produced good agreement with the measured site response. A spectral analysis of the results showed that the stiffness of the soil deposits had a significant effect on the characteristics of the input motions and the overall behavior of the structure. The peak surface acceleration measured in the centrifuge was significantly amplified, especially for low amplitude base acceleration. The amplification of the earthquake shaking as well as the frequency of the response spectra decreased with increasing earthquake intensity. The results clearly demonstrate that the layering system has to be considered, and not just the average shear wave velocity, when evaluating the local site effects.     

在考虑惯性力情况下挡土墙后黏性填土主动土压力的计算

本文在已有研究成果的基础上,根据库伦土压力的计算原理,从滑动土楔处于极限平衡状态时力的平衡条件出发,考虑实际地震中对挡土墙稳定性最不利的情况,推导出了计算黏性土或无黏性土主动土压力的公式。该公式适用于均布荷载作用于挡土墙后任意位置。对地震多发区考虑水平惯性力作用下重力式挡土墙设计中土压力的计算具有一定参考价值。