Efficient computational system reliability analysis of reinforced soil-retaining structures under seismic conditions including the effect of simulated noise

Abstract

This article presents a computational reliability analysis of reinforced soil-retaining structures (RSRS) under seismic conditions. The internal stability of RSRS is evaluated using the horizontal slice method (HSM) with modified pseudo-dynamic seismic forces. Two different failure modes of RSRS are identified and their reliability indices are computed using the first-order reliability method (FORM). The critical probabilistic failure surface is identified using a three-tier optimization scheme. Reliability index of the system is computed by considering the modes of failure to be connected in series. The tension mode is found to be the most critical mode of failure. The present study identifies that the wall height (H), shear wave velocity of the soil (V_s), and predominant frequency of the input motion (ω) govern the response of RSRS. Reliability indices depend on a parameter termed as the normalized frequency (ωH/V_s) and their values decrease with an increase in the value of ωH/V_s. Increase in the damping ratio of soil, increases the value of reliability indices, especially for ωH/V_s values, which are close to π/2. The FORM suffers from few critical shortcomings such as linear assumption of limit state surface at the most probable point of failure and its ability to consider only the statistical uncertainties excluding the effect of epistemic uncertainties. This calls for sampling-based numerical techniques such as Monte-Carlo simulation (MCS) which gives more comprehensive understanding of the problem under consideration in a probabilistic framework. Thus, a computationally efficient surrogate-assisted MCS is carried out to validate the present formulation and provide numerical insights by capturing the system dynamics over the entire design domain. Adoption of the efficient surrogate-assisted approach allowed us to quantify the epistemic uncertainty associated with the system using Gaussian white noise (GWN). Subsequently, its effects on the system reliability index and probabilistic behavior of the critical parameters are presented. The numerical results clearly indicate that it is imperative to take into account the probabilistic deviations of the critical performance parameters for RSRS to ensure adequate safety and serviceability under operational condition while quantifying the reliability of such systems.

Appendices

Appendix 1: expressions for horizontal and vertical seismic accelerations and associated terms

$$ a_{h} (z_{i} ,t) = \frac{{k_{h} g}}{{C_{s}^{2} + S_{s}^{2} }}\left[ {\left( {C_{s} C_{{sz_{i} }} + S_{s} S_{{sz_{i} }} } \right)\cos \left( {2\pi \frac{t}{T}} \right) + \left( {S_{s} C_{{sz_{i} }} - C_{s} S_{{sz_{i} }} } \right)\sin \left( {2\pi \frac{t}{T}} \right)} \right], $$

(35)

$$ C_{{sz_{i} }} = \cos \left( {{{y_{s1} z_{i} } \mathord{\left/ {\vphantom {{y_{s1} z_{i} } H}} \right. \kern-\nulldelimiterspace} H}} \right)\cosh \left( {{{y_{s2} z_{i} } \mathord{\left/ {\vphantom {{y_{s2} z_{i} } H}} \right. \kern-\nulldelimiterspace} H}} \right), $$

(36)

$$ S_{{sz_{i} }} = - \sin \left( {{{y_{s1} z_{i} } \mathord{\left/ {\vphantom {{y_{s1} z_{i} } H}} \right. \kern-\nulldelimiterspace} H}} \right)\sinh \left( {{{y_{s2} z_{i} } \mathord{\left/ {\vphantom {{y_{s2} z_{i} } H}} \right. \kern-\nulldelimiterspace} H}} \right), $$

(37)

$$ C_{s} = \cos \left( {y_{s1} } \right)\cosh \left( {y_{s2} } \right), $$

(38)

$$ S_{s} = - \sin \left( {y_{s1} } \right)\sinh \left( {y_{s2} } \right), $$

(39)

$$ y_{s1} = \frac{2\pi H}{{TV_{s} }}\sqrt {\frac{{\sqrt {1 + \xi^{2} } + 1}}{{2 + 8\xi^{2} }}} , $$

(40)

$$ y_{s2} = - \frac{2\pi H}{{TV_{s} }}\sqrt {\frac{{\sqrt {1 + \xi^{2} } - 1}}{{2 + 8\xi^{2} }}} , $$

(41)

$$ a_{v} (z_{i} ,t) = \frac{{k_{v} g}}{{C_{p}^{2} + S_{p}^{2} }}\left[ {\left( {C_{p} C_{{pz_{i} }} + S_{p} S_{{pz_{i} }} } \right)\cos \left( {2\pi \frac{t}{T}} \right) + \left( {S_{p} C_{{pz_{i} }} - C_{p} S_{{pz_{i} }} } \right)\sin \left( {2\pi \frac{t}{T}} \right)} \right], $$

(42)

$$ C_{{pz_{i} }} = \cos \left( {{{y_{p1} z_{i} } \mathord{\left/ {\vphantom {{y_{p1} z_{i} } H}} \right. \kern-\nulldelimiterspace} H}} \right)\cosh \left( {{{y_{p2} z_{i} } \mathord{\left/ {\vphantom {{y_{p2} z_{i} } H}} \right. \kern-\nulldelimiterspace} H}} \right), $$

(43)

$$ S_{{pz_{i} }} = - \sin \left( {{{y_{p1} z_{i} } \mathord{\left/ {\vphantom {{y_{p1} z_{i} } H}} \right. \kern-\nulldelimiterspace} H}} \right)\sinh \left( {{{y_{p2} z_{i} } \mathord{\left/ {\vphantom {{y_{p2} z_{i} } H}} \right. \kern-\nulldelimiterspace} H}} \right), $$

(44)

$$ C_{p} = \cos \left( {y_{p1} } \right)\cosh \left( {y_{p2} } \right), $$

(45)

$$ S_{p} = - \sin \left( {y_{p1} } \right)\sinh \left( {y_{p2} } \right), $$

(46)

$$ y_{p1} = \frac{2\pi H}{{TV_{p} }}\sqrt {\frac{{\sqrt {1 + \xi^{2} } + 1}}{{2 + 8\xi^{2} }}} , $$

(47)

$$ y_{p2} = - \frac{2\pi H}{{TV_{p} }}\sqrt {\frac{{\sqrt {1 + \xi^{2} } - 1}}{{2 + 8\xi^{2} }}} . $$

(48)

Appendix 2: computational procedure for searching the critical probabilistic failure surface

• 1. Determine the critical deterministic failure surface by maximizing the T_j using the HSM. The critical values of θ_o and θ_h are obtained using the SQP in MATLAB.

• 2. The reliability index, β_Dt or β_Dpo, is then calculated using the FORM by minimizing the distance between the origin and the most probable failure point subjected to a limit state function g₁(X) or g₂(X) = 0. The same is done using the SQP in MATLAB. However, the reliability index (β_Dt or β_Dpo) attached to the critical deterministic failure surface may not be the minimum reliability index.

• 3. Finally, the search for critical probabilistic failure surface and the minimum reliability index, β_t or β_po associated with it, forms the third step of the optimization process. It is done by assuming the deterministic critical slip surface as the first trial surface and optimizing it again to find the β_t or β_po associated with it.

Appendix 3: procedure to perform the RS-based MCS

• 1. The sampling points are generated using the Latin hypercube sampling (LHS) and the number of sampling points to be generated are (n + 1)(n + 2)/2, where n is the number of random variables considered in the analysis. In present study, random variables are V_s, T_u, ϕ, and γ. Hence, 15 LHS points are generated.

• 2. The limit equilibrium analysis using the SQP is performed on these 15 design points to obtain the value of performance function, g₁(X).

• 3. The coefficient vector $a$ of Eq. (28) is uniquely determined by solving the set of linear equations in MATLAB. It produces a tentative RS function, g₁(X).

• 4. After the generation of the RS, it is incorporated in a MATLAB program to conduct the RS-based MCS.

• 5. The P_f in tension mode is calculated by dividing the number of times the value of RS, g₁(X), falls less than zero by the total number of realizations, N′.

• 6. The size of sample space is determined by plotting a graph of P_f versus N′ and then searching for the value of abscissa having minimum fluctuations in the ordinate.

• 7. Finally, the coefficient of variation (CoV_pof) associated with the P_f is calculated by:

  $$ {CoV_{pof}} = \frac{{\sqrt {\left( {1 - P_f} \right)} }}{{\sqrt {\left( {N^{\prime} \times P_f} \right)} }}. $$

  (49)

The value of CoV_pof should be equal to or less than 10%.

Appendix 4: steps for performing the system reliability analysis of RSRS using MARS-assisted MCS

• 1. The required input parameters to describe the geometrical configuration of the RSRS, statistical parameters of random variables, and deterministic parameters are defined.

• 2. LHS method is employed to generate the training samples for all the random variables to be used in the analysis. The limit equilibrium analysis using the SQP-assisted HSM is performed on these sample points to obtain the value of performance function, either g₁(X) or g₂(X).

• 3. The MARS model is constructed using the design points.

• 4. The MARS model is tested on random testing samples and the optimum number of training samples is determined based on the obtained value of coefficient of determination (R²).

• 5. In the next step, MARS-based MCS formulation is executed in MATLAB to determine the optimum number of simulations (N′).

• 6. Interpretation of results: The P_f of RSRS for individual modes and system failure probability.