Soil-Structure Interaction of Retaining Walls under Earthquake Loads

The study is devoted to both static and earthquake response analysis of retaining structures acted upon by lateral earth pressure. Two main approaches were implemented in the analysis, namely, the Mononobe-Okabe analytical method and the numerical Finite element procedure as provided in the ready software ABAQUS with explicit dynamic method. A basic case study considered in the present work is the bridge approach retaining walls as a part of AL-Jadiriya bridge intersection to obtain the effects of the backfill and the ground water on the retaining wall response including displacement of the retaining structure in addition to the behavior of the fill material. Parametric studies were carried out to evaluate the effects of several factors such as vertical and horizontal components of the earthquake, maximum peak acceleration, angle of friction, damping ratio, height of the wall and groundwater level within the medium of fill. Three heights of retaining walls were considered for those above mentioned factors, these are (2.9m, 4.7m and6.7m). A comparison is made between the responses obtained on the basis of finite element analysis with those obtained using the Mononobe-Okabe method. It is found that the lateral wall responses obtained using the FE were larger than those calculated by the Mononobe-Okabe method for all heights of the retaining wall, it was also found that pore pressure of the ground water depends on the water flow through the backfill during the earthquake. The distribution of the dynamic earth pressure on the wall is nonlinear and depends on the earthquake ground acceleration in addition to the wall height and soil properties. Based on the numerical analysis and the results obtained from the parametric studies carried out, two expressions are proposed to evaluate the maximum lateral wall response in terms of wall height, soil properties and earthquake base excitation acceleration, and hence the dynamic earth pressure acting on the retaining structure.


INTRODUCTION
Despite advances in geotechnical engineering, it is common to find retaining walls experiencing near or complete failure during strong earthquake.Effect of earthquakes on retaining walls often include large translation and rotational displacements, buckled walls, settlement of backfill soils, and failure of structures found on the backfill.Excessive displacement cannot only induce failure of the wall itself but may also cause damage to structures nearby.
Damage to retaining walls can be great due to an incomplete understanding of the complex soilstructure interaction occurring during an earthquake.The magnitude and distribution of additional seismic lateral earth pressures are particularly in question.Seismic behavior of a retaining wall soil system is a function of a backfill soil properties, relative stiffness of the wall soil system, wall fixity conditions, foundation stability, and characteristics of the applied earthquake motions.

DYNAMIC EARTH THRUST ON RETAINING WALLS
The earliest studies of dynamic lateral earth pressure on a retaining structure were presented by Okabe (1962) and Mononobe and Matsuo (1929), Dynamic centrifuge tests have been carried out to verify the Mononobe-Okabe equation by Steedman (1984), zeng and Steedman (1988) and Anderson (1987).Seed and Whitman (1970) summarized previous experimental studies and commented that the lateral earth pressure coefficients computed for a cohessionless backfill using the Mononobe-Okabe equation are in reasonable agreement with the model test observations.They proposed a simplified Mononobe-Okabe equation as (1)

And
(2) Were and are the dynamic and total thrust acting on the wall at peak acceleration respectively, is the active earth pressure coefficient with earthquake effect, is the unit weight of the backfill soil, H is the height of the backfill and is the coefficient of horizontal acceleration.
This force was originally assumed to act at 1/3 H from the base of the wall.However various experimental shaking table tests on model retaining walls have shown the resultant force to act above the 1/3 point (Seed and Whitman, 1970).Seed (1969) has recommended that the dynamic component in the Mononobe-Okabe force to be placed at 0.6 H above the base for design of vertical walls with horizontal dry backfill (1981,1982) performed a series of shaking table tests and concluded that the point of action of the earth thrust is at approximately 0.4 H above the base.Steedman (1984) assumed a height of 0.5 H. this height is more realistic in analyzing dynamic retaining wall problems.

HYDRODYNAMIC PRESSURE ON RETAINING WALLS
Westergaard (1933) developed a pseudo-static approximation for the change of water pressure during an earthquake for the case of a straight dam with a vertical up-stream face.The result of Westergaard analysis is the pressures are the same as if a certain body of water were forced to move back and forth with the dam while the remainder of the reservoir is left inactive.A parabolic dynamic distribution, is proposed as Where is the height below the water table, is the unit weight of the water, H is the height of the backfill and is the coefficient of horizontal acceleration.The resultant hydrodynamic thrust is (4) Acting at an elevation equal to 0.4 H above the base of the pool.Zangar (1953) presented an approximate solution for the hydrodynamic water pressure against an inclined wall surface.Chwang (1978) developed an analytical solution that is close to Zangar's approximation as follows

With
(5) In which is a parameter related to the inclination angle and can be approximated as .Where α is the angle (in radians) between the backfill face of the wall and the horizontal base away from the backfill.When the wall is vertical α = π/2, zangar's approximation is about the same as the Westergaard's approximation between H/3 and 2H/3 above the base and is slightly smaller elsewhere.

DYNAMIC EARTH PRESSURES FROM A SATURATED BACKFILL
Ishibashi and Madi (1990) proposed three methods to analyze the dynamic thrust acting on retaining walls based on case studies: A. To use the traditional Mononobe-Okabe's dynamic lateral earth pressure B. To use modified Mononobe-Okabe's in term of the point of application of the resultant force depending upon wall movement modes and to use generalized apparent soil's permeability C. To apply dynamic liquid soil pressure against the backfill face of the wall They applied these analytical methods to study the stability of three types of retaining walls.Their case studies showed that method given (C) provided the lowest safety factors.

MONONOBE AND OKABE METHOD
Details of the Mononobe-Okabe method and suggestions regarding its application to design problems are given by (Seed and Whitman, 1970).In order to facilitate a comparison of the Mononobe-Okabe method with the work method employs the following base assumptions: • The failure in the soil is assumed to occur along a plane surface through the toe of the wall and inclined at some angle to the horizontal.• The movement of the wall is sufficient to produce minimum active pressure.• The soil is assumed to satisfy the Mohr-Coulomb failure criterion.• The wedge of soil between the wall and the failure plane is assumed to be in equilibrium at the point of incipient failure, under gravity, earthquake, and the boundary forces along the wall and failure surface.• The backfill is completely above or completely below the water table, unless the top surface is horizontal in which case the backfill can be partially saturated.• Any surcharge is uniform and covers the entire surface of the soil wedge.• Liquefaction is not a problem.
The angle of failure plane is varied to give a maximum value of the wall force due to earthquake per unit width , and under the critical condition it can be shown that: Where the sum of the static ( ) and the earthquake force ( ).
is the active earth pressure coefficient with earthquake effect.Note that the term ψ is defined as The original approach by Mononobe and Okabe was to assume that the force from eq. ( 6) acts at a distance of 1⁄3H above the base of the wall.
The M-O method can be readily extended to encompass cohesive soils considering the equilibrium of cohesive forces acting along the wall boundary and the failure surface.
For the case when the water table is above the backfill, must be divided into static and dynamic components for computing the lateral forces.Buoyant soil weight is used for computing the static component below the water table, with the hydrostatic force added and saturated soil weight is used for computing the dynamic component .

FINITE ELEMENT MODEL 1. Model formulation
Full three-dimensional geometric models were used to represent the Wall-soi1 system.The wall and soil were modeled using eight-node block elements (C3D8).Each node had three translational degrees of freedom, in.X, Y and Z coordinates.A three dimensional surface-tosurface contact element was used at the wall-soi1 interface to allow sliding, Fig. 1 show the element C3D8 and its degree of freedom, Fig. 2 depicts the wall-soi1 system considered in the analysis and showing the finite element mesh used in the analysis.For water model element (EC3D8R) an 8-node linear eulerian brick is used.

Contact
Contact simulations in Abaqus/Standard are either surface based or contact element based.Surfaces that will be involved in contact must be created on the various components in the model.Then, the pairs of surfaces that may contact each other, known as contact pairs, must be identified.Finally, the constitutive models governing the interactions between the various surfaces must be defined.These surface interaction definitions include behavior such as friction.General contact is used to contact water with the retaining wall and the soil.

3.Geometry and Boundary Conditions
A basic case study considered in the present work is a bridge approach retaining walls, Fig. 3 shows the retaining wall detail as a part of AL-Jadiriya bridge intersection, the retaining walls and the related soil modeling as a 3D model of 1m strip long from the whole model.Motion in the longitudinal direction of the retaining wall is assumed to be prevented, that is the behavior of the structure will be as 2D.The reason for using a 3D model since ABAQUS program is based on Euler-Lagrange approach to simulate the water soil-structure interaction and the type of element in this approach is only the 3D.The distance between the retaining walls is 14m.Three heights of the retaining walls were considered in this work.The nodes at the base of the model considered as a rigid during the static analysis and released the horizontal degree of freedom during earthquake for both concrete and soil, for water element is free through all the degree of freedom because of using Eulerian-Lagrangian approach.

Material properties
The material properties that use in this study as follow, the elastic modulus E = 124 MPa, Poisson's ratio ν = 0.3, unit weight of soil γ = 20 kN/m3 .Three values of the angle of friction were considered (30°, 35°& 40°).Other hypotheses include fully associated flow rule and adoption of the Drucker-Preiger failure criterion as the yielding function.In general, design strength or compressive strength of the reinforced concrete wall fc′ = 21 MPa.Poisson's ratio and unit weight of RC are assumed as ν = 0.15 and γ = 24 kN/m3, respectively.Three damping ratios were used in the study (5%,10%&15%) for the concrete and soil.

5.LOADING
The acceleration record of El-Centro, California is used as a horizontal input motion (just as a case study which can give noticeable response of the structure).This earthquake is of magnitude equals

Static Analysis
The first analysis was a static case study with two phases, namely without ground water and with the ground water, for two cases of wall heights as shown in Table 1.The static earth pressures are shown in Fig. 4 in the static analysis, the earth pressure is linear along the depth of the retaining wall, The ground water pressure on the retaining wall is shown in Fig. 5.To obtain the water pressure on the retaining wall using ABAQUS, the explicit analysis was used with smooth step to apply the load on the structure with one second time period so the behavior of the structure under this analysis is the same under static analysis such that one can notice the difference between the water pressure on the retaining walls in the figure..It should be noted that the ground water level considered in the analysis is 1.5 m below the ground surface.

Dynamic Analysis
The second part of the study includes analysis of the retaining wall system of the bridge approach when acted upon by the earthquake base excitation.Several nodel point displacement were considered in the analysis, these are : horizontal displacement at the top backfill surface at midspan of the approach, horizontal displacement at top of the concrete retaining wall and at the base Fig. 6.It can be noticed that the overall motion of the wall-soil system is in phase, that is, relative damping is insignificant, however, the peak wall response is found to be different from that of the soil by almost 30% which highlights the effect of relative stiffness of the wall and the soil.Moreover the displacement at top of the wall is found to be larger than that of the base by about 50% at time of maximum response.Such a tendency is encountered since the wall is a cantilever free at the top and fixed at the base as compared to the base which is almost fixed.The negative sign mean the movement of the left wall to the left (active).
Analysis are also carried out for other cases of retaining wall heights (H=4.7) and (H=2.9).For such cases, in addition to the case where the wall height H=6.7m, the maximum earth pressure on the bottom of the retaining walls were evaluated at different times due to EL-Centro earthquake base excitation of.Results are given in Fig. 7 for wall heights (H=6.7),(4.7)and (2.9) respectively.
A comparison of the maximum earth pressure acting at the retaining walls is presented in Table 2, for the static case and the dynamic case when evaluated using the M-O method and using timedomain analysis.As expected, it can be seen that the dynamic earth pressures are found to be larger than the static pressures for all cases of wall heights.Moreover, the dynamic earth pressure as obtained in the time-domain analysis are larger than these obtained using the M-O method, (angle of friction=35 and damping ratio=10%).It is also seen that the peak dynamic pressure was found to occur at different times, especially for the case of a wall of 6.7m height which is considered as relatively flexible as compared the relatively rigid 2.9m wall.The distribution of earth pressure along the depth of the retaining wall (static and dynamic using time domain analysis and M-O method) are presented in Fig. 8.
A comparison between the FE analysis results and those results obtained using M-O method is presented in Table 3.It was found that the position of the resultant of the dynamic pressure acting on the wall are approximately the same; however the magnitude of the maximum earth force as obtained using the FE method is almost 48% larger than that obtained by the M-O method.Hence the dynamic magnification factor (the maximum dynamic force divided by the maximum static force is (1.23) in case of the M-O method while it was (1.82) when using the FE method.The time-domain analysis using F.E. normally results in a more précis evaluation of the soil-structure interaction and hence behavior.

Parametric Study
a parametric study was carried out to investigate the effect of earthquakes on the maximum horizontal displacement on the top of the wall and the maximum dynamic earth pressure.The following equations are developed for correlating the effect of different variables: Figures from Fig. 9 to Fig. 12 show the effect of wall height, angle of the friction, damping ratio and the maximum peak acceleration on the maximum earth pressure.Solid lines in the figures represent the values calculated from eq. ( 10).The time history of horizontal displacement at the top of the wall and the dynamic earth force and the location of the resultant are summarized in Table 4 for the cases of wall system of 4.7m height.It is clear that the dynamic earth pressure increases with the increase in the height of wall, with the decrease in angle of internal friction, decrease in damping ratio and with the increase in the maximum peak acceleration of the earthquake excitation.

LIQUEFACTION ANALYSIS
The most common type of analysis to determine the liquefaction potential is to use the standard Penetration test (SPT).The analysis is based on the simplified.This is the most commonly used method to evaluate the liquefaction potential of a site.The steps are as follows: 1. Appropriate soil type: first step is to determine if the soil has the ability to liquefy during an earthquake.The soil must meet the requirements 2. Groundwater table : The soil must be below the groundwater table.The liquefaction analysis could also be performed if it is anticipated that the groundwater table will rise in the future, and thus the soil will eventually be below the groundwater table.
3. Cyclic stress ratio (CSR) induced by earthquake: If the soil meets the above two requirements, then the simplified procedure can be performed.The first step in the simplified procedure is to determine the cyclic stress ratio (CSR) that will be induced by the earthquake.A major unknown in the calculation of the CSR induced by the earthquake is the peak horizontal base acceleration a (max) that should be used in the analysis.A liquefaction analysis would typically not be needed for those sites having peak ground acceleration a (max) less than 0.10g or a local magnitude ML less than 5. 4. CRR from standard penetration test: By using the standard penetration test, the cyclic resistance ratio (CRR) of the in situ soil is then determined If the CSR induced by the earthquake is greater than the CRR determined from the standard penetration test, and then it is likely that liquefaction will occur during the earthquake, and vice versa.The final step in the liquefaction analysis is to calculate the factor of safety against liquefaction.If the cyclic stress ratio caused by the anticipated earthquake eq.( 11) is greater than the cyclic resistance ratio of the in situ soil, then liquefaction could occur during the earthquake, and vice versa.The factor of safety against liquefaction (FS) is defined as follows: (12) The higher the factor of safety, the more resistant the soil is to liquefaction.However, soil that has a factor of safety slightly greater than 1.0 may still liquefy during an earthquake.Fig. 13 and Fig. 14 show the Factor of safety against liquefaction for two cases of ground water.

CONCLUSIONS
Based on the numerical case studies considered, the following conclusions are drown: • The finite element analysis results in larger dynamic earth pressures acting on the walls than the M-O theory for all wall heights.The finite element method can precisely predict the actual relative motions of the wall-soil and hence resulting in larger magnitude of dynamic pressure on the wall.• The distribution of the dynamic earth pressure is found to be nonlinear and depends on the earthquake ground acceleration.It is found to increase exponentially with the increase of ground acceleration.• Ground water pressure is primarily Influenced by the water flow through the soil and by the height of the wall and the contact properties with the wall.
• The value of factor of safety against liquefaction decreases with increasing the peak acceleration of the earthquake and the ground water level below the soil surface.
a maximum acceleration of 0.35g which hit on may, 18, 1940.
at the top of wall (m) H height of wall ϕ angle of friction of the soil ξ damping ratio A peak acceleration in the ground motion (as ratio of g) dynamic earth pressure (kN\m2) 5. Factor of safety (FS):The final step is to determine the factor of safety against liquefaction as FS = CRR/CSR.(11)WhereCSR cyclic stress ratio (dimensionless), also commonly referred to as seismic stress ratio Maximum horizontal acceleration at ground surface that is induced by the earthquake, Total vertical stress at a particular depth where the liquefaction analysis is being performed, Vertical effective stress at that same depth in soil deposit where was calculated factor, also known as stress reduction coefficient