View of ESTIMATION OF RISK FOR PILE FOUNDATION AND WELL FOUNDATION USING SUBSET SIMULATION AND OTHER METHODS

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VOLUME: 09, Special Issue 07, (IC-RSDSHPMTM-2022) Paper id-IJIERM-IX-VII, December 2022 42

ESTIMATION OF RISK FOR PILE FOUNDATION AND WELL FOUNDATION USING SUBSET SIMULATION AND OTHER METHODS

Subodh Kumar Suman

Assistant Professor, Department of Civil Engineering, Bhagalpur College of Engineering, Bhagalpur

Abstract- Nowadays, the seismic design of pile foundations is carried out in a straightforward, deterministic method. This paper explains how to generate seismic designs for pile group foundations using a performance-based approach that takes into account all possible levels of loading and their propensity to occur in a certain location. It is impractical to construct a comprehensive, integrated method that would include everything from ground vibrations to state exceedance due to the wide variety of elements that can occur at a site. A modular strategy was created to help the research sponsors make the problem more useful. The framework enables the creation and application of a structural model that represents the foundation system simply. To ensure that stiffness and damping properties matched deformation levels, the discrete soil model was created using an identical linear format. The foundation loads calculated in these analyses were then used to compute the displacements and rotations of the pile cap as a result of applying them to a three- dimensional soil-pile group model. The calculations needed to create load and resistance factors as well as demand and capacity factors were produced using a computer programme. A designer can choose a return period for limit state exceedance and the accompanying factors to develop a design that matches to the desired limit state exceedance rate using the calculations.

Keywords: Estimation of Risk, Pile Foundation, Well Foundation, Subset Simulation, Methods.

1 INTRODUCTION

When near-surface soils are either weak or too compressible to hold the weights without causing excessive settlement or lateral displacement, pile foundations are frequently employed to support heavy loads. Pile foundations are widely used to support bridges since these structures are frequently built on weak, compressible soils. Because of the complex behaviour of pile foundations, numerous possible loading scenarios must be taken into account during design. The design of a pile foundation aims to make sure that the structure has the strength to maintain stability under all possible loading scenarios. Pile foundations may be vulnerable to overturning moments as well as lateral and vertical loads. Depending on the terrain of the site, the type of bridge, and the bridge's design, these loads may occur under static conditions. Due to traffic loading, impact loading, and natural dangers, the same components may also be used dynamically. Earthquakes are one of the most significant of these natural risks. Since numerous previous earthquakes have damaged bridges, much consideration has been given to the seismic design of bridges. Moreover, pile foundations must be built to withstand earthquake loads without deviating too far from the soil's ability to sustain them or rotating or deflecting excessively.

2 PILE FOUNDATIONS

For supporting loads that are too big to be maintained by shallow foundations without causing excessive settlement, lateral displacement, or rotation, pile foundations are frequently used. Bridge foundations, including individual piers and/or abutments, are frequently supported by piles. Although they are occasionally employed singly, they are most frequently driven in groups and connected by a single pile cap to which structural loads are then applied. In this paper, pile foundations and their design are briefly discussed.

(i) Single Pile Behavior- When erected in suitable soils, single piles can withstand significant vertical loads applied to their heads. A single pile may be subjected to axial, lateral, or even an overturning moment of force. There are several methods for determining how individual heaps react to different combinations of these loads.

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piles by the pile cap, which ultimately transfers the loads to the supporting soil. The distance between each pile within a pile group can affect how the pile group behaves. When piles are placed closely together, the cost of the pile cap is reduced, but the capacity of the pile group may be decreased due to interactions between the stressed soil zones associated with the individual piles.

(iii) Pile Cap Resistance- The deep pile groups that support bridges are frequently joined by concrete pile caps. These pile caps are frequently buried and can be quite large. These should be taken into consideration while designing since they can greatly increase the rotational and lateral resistance of the pile group system. In rare instances, the amount of lateral resistance that the pile top provides can reach 50% of the overall amount of lateral resistance (Mokwa, 1999; Beatty, 1970). The rotating capacity of the pile group is significantly impacted by the pile cap. As shown in Figure 1, when a lateral force is applied to a group of piles, vertical soil resistance along the piles and at the pile tips work to counteract the tendency of the group to rotate. Hence, the rotational resistance of the pile group can be considerably influenced by the axial resistance of the pile.

Figure 1 Schematic of rotational soil resistance due to pile cap rotation

The pile cap strongly affects the lateral capacity of the pile group. Passive earth pressure on the front of the cap, sliding resistance on the bottom and sides, and active earth pressure on the back contribute to the lateral resistance. Compared to the passive resistance, however, sliding and active resistance are usually quite small and are commonly ignored.

(iv)Dynamic Response of Pile Foundations- Moreover, dynamic loads from various sources that produce a wide range of amplitudes, frequency contents, and durations are expected to be resisted by pile foundations. Pile groups will respond dynamically with some pattern and amplitude of deformations when they are subjected to dynamic loads.

Dynamic loading typically has a modest amplitude, therefore the foundation's stiffness and damping properties will regulate its deformation. There is a substantial body of research on the dynamic response of pile foundations, and charts for calculating pile impedance a quantity that takes into account both stiffness and damping characteristics are easily accessible. The piles can be substituted by springs and dashpots designed to simulate the foundation's horizontal, vertical, and rotational impedance if a pile cap is stiff (Figure 2).

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Figure 2 Schematic illustration of equivalent springs and dashpots for modeling vertical/horizontal/rotational impedance of pile group

For the purposes of soil-structure interaction analysis, the dynamic stiffness of the foundation is of primary interest. The dynamic stiffness of a single pile is usually expressed as the product of its static stiffness and a dynamic stiffness factor. Piles themselves are generally quite stiff when loaded axially, so the entire pile tends to move vertically in phase and by essentially the same amount. When loaded laterally, however, piles can be quite flexible will tend to deflect more near the point of load application than farther away. For most pile dynamics problems, the loads are applied at the head of the pile so only a portion of the pile, usually termed the “active length” deflects significantly. The active length depends on the relative flexural stiffness of the pile and soil and is usually on the order of 10-TO 20-pile diameters (Randolph, 1981; Gazetas, 1991). The dynamic response of pile groups is complicated. Piles tend to interact with each other under dynamic as well as static loading. The type of elastic solutions from which impedance functions are typically derived show that the individual piles within a group produce stress waves that emanate from their perimeters as they move laterally with respect to the surrounding soil. When linear elastic behavior is assumed, these waves can push an adjacent pile on one side and pull an adjacent pile on the other side. The result is a complex set of interactions that depend on frequency, pile spacing, number of piles, and other factors. Examples of the dynamic stiffness and damping behavior of 2x2, 3x3, and 4x4 pile groups are shown in Figure 3.

Figure 3 Dynamic stiffness and damping behavior of pile groups subjected to (a) lateral load at pile cap, and (b) overturning moment applied to pile cap 3 METHODS TO SEISMIC RESPONSE AND DESIGN

Earthquake engineers are frequently required to evaluate the seismic performance of existing structures or to design new structures to achieve some desired level of

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engineer’s ability to predict the response of soil-foundation-structure systems is much greater than in the past. Previous notions of successful performance as the avoidance of collapse are giving way to more refined measures of performance at damage levels well short of collapse. Multi-level design procedures have been used for a considerable period of time and have been formalized in different ways. For bridge design, load and resistance factor (LRFD) principles have been used to establish design requirements for different limit states serviceability, ultimate, and extreme. In recent years, more explicit consideration of performance has been encapsulated in performance-based earthquake engineering, which has the potential to provide more consistent and rational designs in areas of widely-varying seismicity.

(i) Seismic Performance- The development and implementation of performance-based design requires that earthquake professionals be able to define performance in terms that are understandable and useful to the wide range of technical and non-technical professionals who make decisions on the basis of performance predictions. The term

“performance” can mean different things to different people.

To a seismologist, spectral acceleration may be a good descriptor of the potential performance of a building subjected to earthquake shaking. To an engineer, plastic rotation would likely be a better descriptor of performance. To an estimator preparing a bid for repairs, crack width and spacing could be more useful measures of performance. Finally, to an owner, the economic loss associated with earthquake damage could be the best measure of performance. These different notions of performance lead to an intuitive way of viewing the earthquake process. As illustrated in Figure 1, an earthquake produces ground motion, which leads to dynamic response of a structure. That response can lead to physical damage, and that damage leads to losses. The prediction of losses, therefore, requires the ability to predict ground motion intensity, system response, and physical damage. If losses are the ultimate measure of, the performance evaluations in an ideal implementation of performance-based design should focus on predicting losses as accurately, consistently, and reliably as possible. The following sections describe the progression of events that lead to earthquake losses.

Figure 4 Schematic illustration of the progression that leads to earthquake losses (ii) Uncertainty in Geotechnical Design- A design procedure should ensure some adequate margin of safety against failure. This objective, which requires a balance of safety and cost, is best accomplished with careful consideration of the sources and levels of uncertainty in the design problem. Geotechnical design, in comparison with many other engineering disciplines, must deal with numerous sources of significant uncertainty and, as a result, higher margins of safety have historically been incorporated in geotechnical design.

The following sections discuss the primary sources of uncertainty in capacity and demand with respect to geotechnical engineering problems.

(iii) Performance-Based Seismic Design- A complete prediction of performance requires prediction of the response, damage, and loss associated with one or more specific levels of ground shaking. This process can be illustrated schematically as shown in Figure 5. A response model is used to predict the response of a soil- structure system to earthquake shaking. The response model can range from an empirical algebraic equation to a detailed nonlinear finite element model. A damage model is used to predict physical damage from response levels. Finally, losses are predicted from damage by a loss model. The loss model may be a relatively straightforward combination of repair quantities and unit costs, or a complex financial model that considers indirect losses, future interestrates, etc.

Alternatively, a loss model can be expressed in terms of downtime or increased traffic congestion due to temporary (or permanent) loss of one or more bridges.

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Figure 5 Schematic illustration of process by which response, damage, and loss are predicted

4 INTEGRATION OF EXPANDED RBD APPROACH WITH SUBSET SIMULATION

Consider designing a drilled shaft by expanded RBD approach and performing a Subset Simulation with m+1 level of simulations to estimate p(B, D|Failure), pf, and p(Failure|B,D) in Equation (1). Since the design parameters B and D are artificially treated as uncertain parameters in the expanded RBD approach, their sample space  is divided into m+1 individual subsets {i, i = 0, 1, 2, ..., m} by the intermediate threshold values {y i, i = 1, 2, ...

, m} of Y = BD/FS min . In Subset Simulation, the intermediate threshold values {y i , i = 1, 2, ... , m} are adaptively determined to generate m+1 individual subsets {i , i = 0, 1, 2, ... , m} of B and D, and samples in different subsets are generated level by level and correspond to different conditional probabilities. According to the Theorem of Total Probability (e.g., Ang and Tang, 2007), the failure probability p f is therefore expressed as where p(Failure|i ) = the conditional failure probability given sampling in i ; p(i ) = the probability of the event

i . p(Failure|i) is estimated as the ratio of the failure sample number in i over the total sample number in i . p(i ) is calculated as00 p1)p(  ; 1i 0i0i pp)p( , 1m1,i   ; m0m p)p(  Note that i , i = 0, 1, 2, ..., m, are mutually exclusive and collectively exhaustive events, i.e., p(i∩j) = 0 for i ≠ j and 1)p(m0ii . When p(Failure|i ), p(i , and pf are obtained, the conditional probability p(i|Failure) is calculated using the Bayes’

Theorem

Then, the conditional probability p(B, D|Failure) of a specific combination of B and D is given by the Theorem of Total Probability as

where p(B,D|Failure∩i ) = the conditional failure probability of a combination of B and D given sampling in i, and it is expressed as the ratio of the number of failure samples in i with a combination of B and D over the total failure sample number in i ; p(i|Failure) is estimated from Equation (5). Note that the accuracy of the estimated p(B, D|Failure∩i ) relies on the number of failure samples in i . The expected number of failure samples in i increases as the total number of samples in each simulation level increases. By this means, the accuracy of estimated p(B, D|Failure∩i ) is improved at the expense of efficiency of the proposed approach. pf and p(B, D|Failure) are calculated from Subset Simulation, respectively.

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Figure 6 Subset simulation and RBD Add-In (a) Subset Simulation Userform Page (b) Reliability-based Design Userform Page

Subsequently, p(Failure|B,D) is obtained in accordance with Equation. Since the failure is defined as FSuls < 1 or FS sls < 1 for ULS and SLS requirements, respectively, two sets of conditional probabilities p(Failure|B,D) (i.e., the respective failure probability pfULS and p fSLS of ULS and SLS failures for given B and D combinations) are calculated for the drilled shaft design. Finally, the feasible design values of B and D are determined by comparing the p(Failure|B,D) with p T . To facilitate the design practice, this study extends the Subset Simulation Add-In developed by Au et al. (2010) to include a function of calculating conditional failure probability [i.e., p(Failure|B,D) in this paper] for different failure modes (i.e., ULS and SLS failures in this paper) using Equations.

(a) Ultimate Limit State (ULS) Failure

(b) Serviceability Limit State (SLS) Failure

Figure 7 Conditional failure probability from Subset simulation and direct MCS

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VOLUME: 09, Special Issue 07, (IC-RSDSHPMTM-2022) Paper id-IJIERM-IX-VII, December 2022 48 5 CONCLUSION

The creation and application of a performance-based framework for the determination of reliability-based design elements for piling foundations was the goal of the study covered in this paper. An expansion of the PEER PBEE framework resulted from the process of establishing this framework in a way that could be utilised for the variety of potential bridge, foundation, site, and subsurface circumstances that foundation designers must take into account. Both a theoretical and an operational approach were taken in pursuing that extension. Lastly, several pile foundation systems located in various seismic zones were subjected to the numerical execution of the methodologies described in this study. The problem of the piling group response is challenging. Five response components three displacements and two rotations each of which is influenced by five components of static and dynamic loading must be predicted in order to forecast the entire response. Depending on the nature of the bridge, the site, and specific ground motions, the various components of loading may have varying relative amplitudes and degrees of correlation. Both the loads and the reaction may be connected to the ground motion either linearly or nonlinearly in terms of the loads.

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