Introduction of Plastic Block Method in the Upper Bound Limit Analysis of Soil Stability Problems

Abstract

In this paper, a new method, called the Plastic Block method, was introduced to improve the current rigid block solutions in framework of rigorous limit analysis method. It is a combination of rigid block limit analysis and finite-element limit analysis methods used in the upper bound limit analysis solutions of stability problems in soil mechanics. The method is similar to the first one in terms of speed of solution process and consideration of a collapse mechanism as a priority, and also is similar to the second one in terms of capability of considering plastic deformability of the blocks in calculation of internal dissipation of energy (besides that of velocity discontinuities between them). Comparison of the results obtained by the proposed method, using a few number of blocks (in case of the classical bearing capacity problem of shallow foundations selected to illustrate capability of the method), showed the better (smaller) upper bound not only in comparison with the well-known rigid block solutions, but also in comparison with some cases of finite-element limit analysis solutions obtained by the models including a large number of elements.

Appendix A

Figure 

Fig. 11

Layout of the failure mechanism and labels of the points

11 displays a proposed failure mechanism modeled with triangular blocks as well as the labels given to the points. Slip surface is labeled with numbers 1–4 and ground surface is labeled with points 1′–4'.

Vector {X} is defined as:

$$\begin{array}{l} X_{i} = x_{i} ;\quad i = 1, \ldots ,n \hfill \\ X_{{n + 2^{*} j - 1}} = x_{j} ;\quad X_{{n + 2^{*} j}} = y_{j} ; \quad j = 1, \ldots ,n \hfill \\ \end{array}$$

(12)

where n is the number of nodes, on failure surface (1–4 in Fig. 11), which is also equal to the number of nodes on ground surface (1′–4′), x_j and y_j are the first and second coordinates of node j located on failure surface, respectively, and x_i represents the first coordinate of node i located on ground surface. The second coordinate of node i can be determined through geometry (this coordinate is not included in vector {X}). Initial value of vector {X} is called {X₀}. Normalized displacement vector {S} is defined as follows to formulate displacement of the nodes:

$$\begin{gathered} S_{i} = \Delta x_{i} ;\,\,\,\,i = 1,...,n \hfill \\ S_{n + 2*j - 1} = \Delta x_{j};\,S_{n + 2*j} = \Delta y_{j};\,\,\,\,j = 1,...,n \hfill \\ \end{gathered}$$

(13)

Function E(X) (potential energy function) is defined as Eq. 8. This function must be minimized through linear programming.

Assuming E(λ) = E(X₀ + λS) as potential energy function around {X₀}, this function can be locally minimal around {X₀} by satisfying the following equation:

$$\frac{dE}{{d\lambda }}(\lambda ) = 0$$

(14)

λ^* (optimal value of λ) can be obtained by solving Eq. 14.

In any stage of optimization, when ${\overline{\mathbf{X}}}^{*}$, a vector in locality of vector{X₀} locally minimizes the function E, which can be expressed as ${\overline{\mathbf{X}}}^{*} = {\mathbf{X}}_{0} + \lambda^{*} {\mathbf{S}}$ where, ${\mathbf{S}} = - \nabla E({\mathbf{X}}_{0} )$.

If the vector S is normalized relative to its length, then λ will be between zero and one. Negative gradient is chosen as a direction for minimization. In this method, an initial trial point X_i is moved iteratively toward optimal point according to the following formula:

$${\mathbf{X}}_{i + 1} = {\mathbf{X}}_{i} + \lambda_{i}^{*} {\mathbf{S}}_{i} = {\mathbf{X}}_{i} - \lambda_{i}^{*} \nabla E({\mathbf{X}}_{i} )$$

(15)

The following criteria can be used for termination of iterative process:

$$\left| {\frac{{E({\mathbf{X}}_{i + 1} ) - E({\mathbf{X}}_{i} )}}{{E({\mathbf{X}}_{i} )}}} \right|\,\,\, \le \varepsilon$$

(16)

The above method can be accelerated by several modifications (Rao [18]), perhaps the most effective of which is the PARTAN (parallel tangents).

The minimum of function E is determined and the bearing capacity of the foundation can be calculated straightforwardly using Eq. 8 by obtaining X_N (the final value of X_i that satisfied the criteria as Eq. 16).