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# Deep Excavations: Non-Linear Analysis Method

## Non-Linear Analysis Method with Soil Springs (NL)

### Beam on Elastoplastic Foundations Method

The Non-Linear analysis method is an advanced method for the design of deep excavations. DeepEX implements a non-linear finite element code for the analysis of the mechanical behavior of flexible earth retaining structures during all the intermediate steps of an open excavation. The non-linear engine is empowered by many unique advanced features.

DeepEX offers the following elastoplastic soil models:

a) Linear elastic - perfectly plastic

b) Hyperbolic soil model

On the reloading part, every soil model has a linear reloading elasticity parameter. Such a parameter should typically range from 2 to 4 times the loading elasticity value (with average 3). In excavations, the reloading elasticity parameter typically describes the remaining soil below the excavation while the loading elasticity is mostly applicable for soil on the retained side.

### Concept of the Non-Linear Analysis

The method simulates the connection of the soil with the walls with non-linear elastoplastic Winkler springs. The soil springs receive their initial properties from the at-rest situation in the initial stage, and then any action (excavation, backfil operation, support installation, dewatering etc.) adjusts the soil springs with stages. The soil pressures, wall moments, support reactions and more are calculated from the soil springs, taking into consideration the project staging and the wall stiffness.

Fig. Concept of the Non-Linear Analysis - DeepEX Software

DeepEX Software Implements all common Analysis Methods for the design of Deep Excavations

Limit Equilibrium Analysis - Non-Linear Analysis (Soil Springs) - Finite Element Analysis

In a non-linear analysis the excavation modelis reduced to a plane problem, in which a unit wide slice of the wall is analysed, as outlined in the figure below. Therefore DeepEX is not suitable to model excavation geometries in which three-dimensional effects may play an important role.

In the modelling of the soil-wall interaction, the very simple yet popular Winkler approach is adopted. The retaining wall is modelled by means of beam elements with transversal bending stiffness EI; the soil is modelled by means of a double array of independent elastoplastic springs; at each wall grid point, two opposite springs converge at most.

According to the Winkler model, it is assumed that the behaviour of every soil spring is totally uncoupled from the behaviour of adjacent elements: the actual interaction among different soil regions is totally left to the retaining wall.

Whereas most programs based on this numerical approach rely on some subgrade reaction modulus which is usually kept constant during the analysis, DeepEX internally computes and continuously updates such parameters, based on the elastic (Young) modulus and the wall geometry. In other words, at each step, the spring stiffness K is recomputed from the active and passive wedge length formulations.

While such correlations for the estimate of spring stiffness are based on simple empirical concepts, but their efficiency has been proved by several practical applications (see Theory Manual for further details).

The real progress of an excavation process is reproduced in all the intermediate steps, by means of a STATIC INCREMENTAL analysis. Due to the elastoplastic behaviour of the soil elements, every step in general depends on the solution at the previous steps. Corresponding with every new step, the solution is obtained by means of a Newton-Raphson iterative scheme.