An Alternative Approach to Dynamic Earth Pressure Coefficients to Improve the Seismic Limit State Design of Earth-Retaining Structures

Equivalent static earth pressure coefficients can be derived with upper and lower bound plasticity solutions (Chen and Liu 1990; Drucker et al. 1952), i.e., the true solution is found that simultaneously satisfies equilibrium, compatibility, and the perfectly plastic failure criterion. For a level soil body whose failure can be characterized by a single, macroscopic drained friction angle (${\varphi }^{\prime }$), Rankine (1856) coefficients can be obtained by finding the limiting major (${\sigma }_1^{\prime }$) to minor (${\sigma }_3^{\prime }$) principal effective stress ratio, i.e., the largest Mohr’s circle of stress that can fit within the Mohr-Coulomb failure surface for a cohesionless soil as given in Eq. (1)

Eq. (1) is often described as being equivalent to Coulomb’s earth pressure coefficient solution for the case of a vertical, smooth wall. However, bearing the derivation in mind, it is evident that the limiting principal effective stress ratio within the soil must always be given by Eq. (1). Differing lateral earth pressure coefficients for the same material assumptions (e.g., Coulomb’s (1776) or Breslau-Müller (1906) coefficients for rough, inclined walls) cannot be interpreted as defining the limiting ratio of major to minor principal effective stress, but instead are trying to capture the ratio of horizontal to (geostatic) vertical normal effective stresses. For example, kinematic constraints or interfacial shear stresses can result in soil arching, where the vertical effective stresses adjacent to a retaining system are typically reduced with respect to the geostatic values (Michalowski and Park 2003; Paik and Salgado 2003; Khosravi et al. 2016). However, methods that recover the minor principal effective stress should then use Eq. (1) to find the major principal effective stress for soil at plastic failure.

Dynamic earth pressure coefficients can be derived from predicted dynamic forces by assuming geostatic vertical effective stress distribution for the total force, or an inverted triangular distribution for the dynamic increment (Seed and Whitman 1970). This necessarily predicts linear distributions of the limiting dynamic horizontal effective stresses. While dynamic earth pressure coefficients define the ratio of horizontal to vertical effective stress, this is identical to the principal effective stress ratio for smooth wall-backfill interfaces. MO approaches and similar that include the wall-soil friction angle imply that the limiting earth pressure coefficients during dynamic loading is different to the static case. However, the present study emphasizes that Eq. (1) must also hold during dynamic loading as seismic action will not change the major to minor principal effective stress ratio that can cause failure in the material.

In this regard, dynamic earth pressure coefficients are quite different to static earth pressure coefficients. The latter can be interpreted as a material constant while the former combines external loading with a measure of the material strength. Active dynamic earth pressure coefficients greater than the static active value could be due to the locally experienced acceleration, partial mobilization of the soil strength, or a combination of the two. However, dynamic earth pressure coefficients obtained by incorrectly assuming geostatic vertical stresses adds additional scope for confusion when defining the limiting stresses earth-retaining systems will experience. Table 1 lists examples of past studies that interpreted or used dynamic earth pressure coefficients in this way. The list of experimental studies focuses on dynamic centrifuge data for experiments featuring dry soil, simple geometries, and soil-wall interfaces that are reasonably smooth. Illustrative numerical or analytical approaches with similar modeling assumptions are also listed. The interpretation of the dynamic stress states in this way is particularly surprising for numerical studies where there is no need to assume the vertical effective stress distributions. In fact, the majority of plasticity-based soil constitutive laws enforce Eq. (1) as the limiting principal effective stress ratio for both static and dynamic analyses (returned to in §Elastic Soil Behavior, Pulses and Hypothesis).

Table 1 also indicates whether the study found dynamic stress states within or outside the pseudostatic predictions. Overall, this is used to demonstrate the significant uncertainty that surrounds the predictions obtained by conventional dynamic earth pressure coefficients that did not capture the limiting stresses. Past studies have used these observations to motivate different predictions for mobilized or limiting dynamic earth pressure coefficients. However, when applied to limiting soil stresses this can cause confusion, especially when the lateral earth pressure coefficient and limiting principal effective stress ratio must be equivalent, e.g., at smooth soil-wall interfaces. Further, it is theorized that past findings that support different limiting dynamic earth pressure coefficients did not consider that under dynamic loading both the major and minor principal effective stresses could vary. It is also noted that comparisons of the inferred or predicted earth pressure coefficients implied by the horizontal effective stresses to Eq. (1) during dynamic loading were seldom made.

Madabhushi (2018) and Madabhushi and Haigh (2021a, b) combined dynamic centrifuge tests and numerical analyses that revealed dynamic normal vertical effective stresses generated by horizontal shaking within the dry sand retained between two rows of parallel sheet pile walls tied at their heads. In this study, the descriptor “normal” is used to distinguish the stress component from shear stresses while horizontal versus vertical is used to indicate the direction. Fig. 2 shows increases of up to 100% of the geostatic values are numerically predicted for the sinusoidal $0.35g$ PGA horizontal input motion. The general rocking mechanics that can justify these variations for rough and smooth walled systems are also illustrated. The rocking that develops either from wall friction or reduced complementary shear stresses are described as relevant for dynamic stress variations in the soil outside the dual row system and thus for retained soil more generally.

The retained and excavated sides of one half the dual row wall system can be compared to a single anchored cantilever wall. Focusing on the excavated side, the kinematics of the passive wedge gives a peak vertical acceleration of $\approx 0.1\mathrm{g}$, i.e., a vertical stress change of 10% the static value, that can partially explain the observed stresses. Fig. 2 also presents the mobilized earth pressure coefficient. Calculated as the ratio of horizontal to vertical effective stresses, it approaches the ratio of principal effective stresses even for moderate shear stresses. For example, given a shear stress, $\tau$, equal to 20% of the major principal effective stress, ${\sigma }_1^{\prime }$, and under passive conditions (e.g., ${\sigma }_3^{\prime }/{\sigma }_1^{\prime }\approx 1/3$), Eq. (2) derived from a Mohr’s circle of stress shows that the horizontal effective stress, ${\sigma }_h^{\prime }$, will be $\approx 95\%$ the major principal effective stress value

With this in mind, the time history of the predicted mobilized earth pressure coefficient in Fig. 2 is observed to be bounded by the static earth pressure coefficients but not the MO coefficients. This is further exemplified by the predicted dynamic Mohr’s circles of stress remaining within the Mohr-Coulomb failure criterion.

The mechanical consistency between the measured and predicted dynamic response of the dual row retaining walls increased the confidence in the dynamic vertical effective stresses that were numerically predicted but not measured. However, the observation that the dynamic stress state is bounded by the Mohr-Coulomb envelope was necessitated by the soil constitutive model used. Elastoplastic constitutive laws typically enforce the Mohr-Coulomb failure criteria independent of strain and hence elastic stress pulses that exceed this plastic limit cannot be predicted.

The theoretical limiting plane-strain stress states that a small volume of soil can exhibit are explored in Fig. 3. Statically, uniaxial loading of soil confined within a perfectly rigid container could realize principal effective stress ratios that exceed $K_P$ depending on the Poisson ratio, $\nu$, if the soil remains elastic and the homogeneous continuum assumption holds. However, if the boundaries are not fixed, plastic soil failure within an active zone is expected.

The behavior for the rigidly confined case experiencing dynamic loading is less obvious. Analogous to the static case, elastodynamic elastic pulses would not be bounded by $K_P$ though additional considerations include rotation of the principal stresses due to the cyclic shear stresses. Within the soil body, complementary shear stresses allow this rotation, which means the lateral earth pressure coefficient and principal effective stress ratios can diverge. However, in regions approaching smooth walls the lack of complementary shear stress could lead to the principal effective stress ratio and lateral earth pressure coefficients becoming identical. Further, following Eq. (2), this will be approximately true even for moderately rough walls that provide limited complementary shear stresses. The ability of the soil volume to transmit shear stresses that exceed the material’s frictional (plastic) strength should also be considered. This could be possible in a small, perfectly confined soil volume that behaves elastically but seems less likely in a large soil volume. In this case, nonuniform horizontal strains could develop across the soil body, i.e., larger compression or extension of soil near the walls versus the middle leading to plastic passive or active limit states locally. More straightforwardly, when the wall displacement permits sufficient soil strain (i.e., the last case in Fig. 3) the limiting stress state at the smooth wall-soil interface will match the static case.

At the heart of these considerations is whether a macroscopic frictional model to limit soil’s shear strength is valid at all strains (and perhaps also over what volume of soil it can be applied). If it is not valid, elastoplastic constitutive laws that enforce the Mohr-Coulomb limit at all strains may not be appropriate. If it is, large elastic pulses unbounded by friction predicted by analytical and numerical methods are potentially misleading. Finally, Fig. 3 highlights that under the horizontal shaking, $a_h(t)$, both ${\sigma }_1^{\prime }(t)$ and ${\sigma }_3^{\prime }(t)$ could vary with time. The current study investigates and hypothesizes that

To demonstrate paths forward for seismic design for soil-retaining systems, simplified analytical predictions of the dynamic normal vertical effective stresses are first explored. Rocking of a soil body due to a lack of complementary shear stresses at the boundaries (i.e., two smooth soil-wall interfaces) is considered (Madabhushi 2018; Madabhushi and Haigh 2021b; Westcott 2023). This approach cannot directly be applied for rough retaining walls where significant stress rotation is expected at the soil-wall interface.

Fig. 8(a) proposes two, simplified analytical schemes that could be adopted for a soil mass of width $b$ and height $h$ retained between two smooth walls and experiencing a purely horizontal acceleration $a$ varying with time $t$. Referred to as the CAM-DYN analytical method, a linear redistribution of normal vertical stresses that maintains vertical equilibrium and equilibrates the inertial overturning moment can be derived. This is analogous to methods used for eccentrically loaded earth-retaining systems. Eq. (3) presents the final solution, expressed as the ratio $\eta$ that is the maximum amount of under- or over-stressing relative to the geostatic stress (${\gamma }_d{h}_i$) for the $i^{\mathrm{th}}$ horizontal slice

Eq. (3) estimates an instantaneous distribution of dynamic vertical normal stress as a function of the soil block geometry ratio and normalized input acceleration. Key assumptions in this approach include independent rigid soil slices with no net vertical acceleration, infinitesimally small rotations, negligible rotational inertia, linearly redistributed stresses across the entire soil width, and rotation about the center of the block’s base.

The extended CUB-DYN methodology, also shown in Fig. 8(a), assumes a nonlinear stress distribution [of the form in Eq. (4a) based on the previous observations], and accounts for the linear and rotational inertia $I$ about an arbitrary point of rotation. By solving for moment and geostatic vertical equilibrium, a closed form solution can be derived [Eq. (4)]. For a given center of rotation $(X_i,Y_i)$ and instantaneous acceleration $a(t)$ (that also defines the rotational acceleration by their kinematic relation), the coefficients ${\beta }_1$ and ${\beta }_2$ are solved to define the (dynamic D’Alembert) equilibrating stresses. While inelegant due to the form of the assumed stress distribution, the results are tractable and analytical stress predictions were readily made by using a symbolic system of equations solver. This approach would support any other equilibrating stress distributions assumed

Fig. 8 compares the previous numerically obtained stress predictions and proposed analytical solutions. Fig. 8(b) shows the magnitudes of three instantaneous normal vertical stress predictions with depth across horizontal slices. The acceleration time history indicates the acceleration used as an input for each instant, with the CUB-DYN method additionally specifying a center of rotation at half of each slice height and 45% the width, the latter choice being inspired by Fig. 6(a). The ability of these analytical methods to broadly capture the distribution of cyclic rocking stresses is demonstrated. Fig. 8(c) shows how inputting an acceleration time history can also produce reasonable predictions of the time-varying vertical stresses versus the finite element solutions. Four locations are shown, and the comparison close to the left and right soil boundaries at two heights shows that the oscillations at the driving frequency, antiphase relation between opposite sides, and diminishing magnitude with height are all captured by the simple analytical methodologies based only on inertial considerations. It is noted that the numerically predicted time histories at the upper gauss points were filtered to remove a physically unreasonable stress drift. This was observed at low confining pressures by multiple constitutive models used in OpenSEES (Westcott 2023). Between the CAM-DYN and CUB-DYN methods, there is a trade-off between accuracy and inputs required, but either approach is far quicker and simpler than a dynamic 2D finite element analysis. Given the assumption of rigid soil blocks, the phase lag with height cannot be captured. However, the methodology is amenable to extensions accounting for wave propagations through soils with finite stiffness in a similar way to the pseudodynamic extension of pseudostatic methods (Steedman and Zeng 1990).

An application of the proposed analytical approaches is shown in Fig. 9. Following Fig. 7(d), Fig. 9(a) focuses on the distributions of numerically predicted horizontal stresses at four instants during the shaking that were shown to violate limit states defined by the dynamic earth pressure coefficients. However, these stresses were limited by the product of the Rankine coefficients and time-varying vertical stresses numerically predicted. Fig. 9(a) shows that the distribution of vertical stresses adjacent to the sidewall using the CAM-DYN and CUB-DYN analytical predictions can also be combined with the Rankine coefficients to reasonably bound the stresses at a given instant. This is despite the simplified rocking mechanisms not capturing the full complexity of the instantaneous distributions of vertical stresses with height [e.g., the turning point at $\approx 5\mathrm{m}$ seen in Fig. 7(c)]. Fig. 9(b) confirms this; the horizontal effective stress time histories at two heights are not limited by conventional approaches while the similar vertical effective stress time history predictions by the CAM-DYN and CUB-DYN approach combined with the Rankine coefficients are bounding. While conventional approaches have utilized a PGA to define a potentially incorrect and constant limit state, the proposed approach could use either the PGA or acceleration time history as shown to better capture the variation of limiting stresses and thus lead to more accurate factor of safety estimates.

The proposed approach illustrates a path forward for improved seismic limit state design of earth-retaining structures. The CUB-DYN method could be readily extended to incorporate phase lag of the accelerations if the representative soil stiffness can be estimated. However, some limitations of the current study should be noted. Firstly, further study is required to adapt this approach for rough soil-wall interfaces where the principal effective stresses are substantially rotated away from the horizontal and vertical directions. Additionally, the simplified inertial method can only predict stress time histories with the same frequency content as the acceleration time history. The 1 Hz driving frequency was chosen to avoid exciting the anticipated natural shear modes of the retained soil, i.e., to avoid resonance. More complex instantaneous distributions and time histories for the vertical effective stress could arise depending on dynamic soil-structure interaction effects on the rocking mechanism. The extent of this interaction between the input motion and dynamic characteristics of the retained soil-wall system (that also vary with soil strains) requires further consideration, particularly when additional degrees of freedom for the wall system are introduced. Finally, the proposed rocking mechanisms span the soil body retained between the two smooth walls and thus predict increases and decreases of the normal vertical effective stresses across the entire horizontal plane of retained soil. In the case of a very long back-fill (between two smooth walls or behind a single wall) a transition may be expected between the rocking region and the nonrocking free-field. Suffice to say in these cases that the lateral extent of the rocking region needs to be established and the linear redistribution of the CAM-DYN approach may imply too abrupt a transition between these zones. While a nonlinear vertical stress function potentially allows a smoother transition between the rocking and vertically geostatic zones, the first step is to better understand the altered rocking mechanisms which are revisited in the “Discussion” section.

The proposed approach uses Rankine earth pressure coefficients combined with estimates of the dynamic normal vertical effective stress to define the limiting dynamic horizontal stresses. These vertical stress oscillations necessitate local vertical accelerations, i.e., the tendency for rotation and thus stress redistribution across an individual, rotating horizontal slice is caused by opposite regions of soil accelerating upwards and downwards.

Thus, an alternative approach would be to predict a single representative or distribution of vertical accelerations along a soil-wall interface from the rocking mechanism induced by the purely horizontal shaking. These vertical accelerations could be used with pseudostatic or pseudodynamic dynamic earth pressure coefficients (i.e., calculated using both a $k_h$ and $k_v$ for a purely horizontal input) and multiplied with the geostatic vertical stresses to obtain estimates of the limiting dynamic horizontal stresses. The $k_v$ becomes a proxy for the normal vertical stress change and would allow designers to continue using dynamic earth pressure coefficients. In fact, the CAM-DYN or CUB-DYN vertical stress predictions could be divided by the mass per area to directly obtain the implied distribution of vertical accelerations, which would ultimately yield identical results to Fig. 9.

However, such repurposing could continue to obscure the importance of dynamic normal vertical stresses that have been historically missed. Further, the established nomenclature can confuse ratios that are actually limiting dynamic force coefficients rather than limiting dynamic stress/earth pressure ratios.