Designing Earth Dams Optimally

Abstract

Designing an earth dam is an optimization problem and is verycrucial from the view points of safety and cost of construction. Followinga scientic approach, the problem is formulated as a nonlinear programwhich aims at minimizing the material cost of dam with safety factor ofthe design as the main constraint. Computation of factor of safety whichis the main task is tackled intelligently by observing the convex nature offactor of safety. The main contribution of this project is the developmentof the computer program which facilitates determining factor of safety aswell as optimizing the designs with an unlimited scope. The chapter also presents a nice and brief history of earth dams.

Problems

• 1. Consider the Pendekal example. Doing the following exercises will help in understanding the project better (you may use the Excel files “ForPendekal” and “ForChevella” placed in the web for doing these exercises).

  • a) What is the width of the dam?

  • b) Divide the width into 15 equal parts and find out the corresponding x coordinates.

  • c) For each of these x coordinates, find out the corresponding points $(x,\ y)$ on the shell, core, phreatic line, MDDL line, and on a slip circle of your choice.

  • d) For each point ($x,\ y)$, determine the zone in which the point lies.

  • e) Using the coefficients given in the table in sheet titled “Specs,” find out the weight of each slice generated by your points.

  • f) At each of the 15 points $(x,\ y)$ on the arc of your slip circle, find out the angle between the vertical line and the normal to the circle, and determine the normal and tangential components of the weight of the slice. Using the formulae described in Sect. 5 of this chapter, compute the driving forces and resistance at each of the 15 points. Also compute the Cohesion.

  • g) Compute the factor of safety with respect to your slip circle.

• 2. A slip circle is considered invalid if the arc of the circle intersects a vertical line at two distinct points inside the dam. Consider the $Chord (1, 28)$. Determine the slip circle whose center $(x,\ y)$ is outside the dam and lies on the perpendicular bisector of the chord at a distance of two units from the chord. Is this an invalid slip circle? Determine the valid slip circle associated with this chord, that is, find out the center and radius, whose center is nearest to the chord.

• 3. In Sect. 7 of this chapter, we have used a nonlinear programming algorithm to search for an optimal solution. In this approach we have changed one variable at a time. It is possible to change more than one variable at a time. In fact, we can choose a direction d and explore the design change in that direction. To fix the ideas, Let x ⁰ be the initial design and let d be a nonzero vector whose dimension is same as that of x ⁰. Consider the new design $x^0+d$. Examine the new design. Instead of exploring the problem in one direction, we may choose a set of directions, say, $d^1,d^2,{\ldots},d^k$ and shift the initial point to the best of $x^0+d^j,\ j=1,2,{\ldots},k$. Once the initial point is shifted, we can repeat the process with the new initial point.

  $$\begin{aligned} D=\left[\begin{array}{rrrrrrrrrrrr} 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 \\ 1 & 1 & 1 & -1 & -1 & -1 & 1 & 1 & 1 & -1 & -1 & -1 \\ 1 & 1 & -1 & 1 & -1 & -1 & -1 & -1 & 1 & -1 & 1 & 1 \\ 1 & 1 & -1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & -1 & 1 \\ 1 & 1 & -1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 & 1 & -1 \\ 1 & -1 & 1 & 1 & -1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 \\ 1 & -1 & 1 & -1 & 1 & -1 & -1 & -1 & 1 & 1 & 1 & -1 \\ 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & -1 & -1 & 1 & 1 \\ 1 & -1 & -1 & 1 & 1 & -1 & 1 & 1 & -1 & -1 & 1 & -1\end{array} \right]\end{aligned}$$

  One way of selecting the set of directions is to use orthogonal arrays which are extensively used in Design of Experiments[4, 5, 6]. Each column of the matrix D above is extracted from L ₁₂ orthogonal array. Let D _j denote the $j{th}$ column of D. Explore the new designs given by $x^0+\frac{1}{2}d^j,\ j=1,2,{\ldots},12$ where x ⁰ corresponds to the design given in first row of Table 7.4. Which of the new designs are good? Can you rank them?

• 4. There is a proposal to construct an ED, called the Chevella ED, near Hyderabad, Andhra Pradesh, India. One of the proposed designs is shown in the Fig. 7.20. Due to limited availability of core material, the core dimensions are also fixed as core top width as 3 m and bottom width as 23.3 m. Find the factors of safety for this design. Since the core dimensions are fixed, the objective boils down to minimizing the volume of shell material which in turn amounts to minimizing the shell area or equivalently the dam cross-sectional area. The input data and constraints for designing this dam are as follows:

  The fixed parameters are: dam height = 35.185, core height = 31.185 m, FRL = 30.185 m, foundation depth is 10 m, dam top width = 6 m, and MDDL = 20.3 m.

  The constraints are: (i) the slopes of shell slanting edges must be at least one, (ii) the dam base width must not exceed 220 m, (iii) each berm should be at least 3 m wide.

  Densities and Coefficients of Friction and Cohesion: These are given in the Excel file titled “ForChevell.” Find an optimal design with two berms on each side. Compare your design with that given in Fig. 7.20.

Fig. 7.20

A proposed design for Chevella ED