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Optimized design of foundations: an application of genetic algorithms.

1. Introduction

A foundation design procedure should address at least three basic requirements: bearing capacity, settlement limit and economy. In geotechnical design part, the three primary limit states of soil supporting an isolated foundation are bearing failure, serviceability failure and total settlement. The designed foundation must be safe also structurally against shears and flexure failure, which is ensured in structural design part. It is essential that the designed foundation should satisfy the local building code requirements. The significance of economics in geotechnical engineering designs is well recognised and is discussed in various textbooks, Lambe and Whitman (1969) and Coduto (2015). Foundation designs are usually a trial-and-error procedure, in which a trial design (foundation dimensions, depth and reinforcement) is chosen and is checked against the geotechnical and structural requirements, which is followed by revision of the trial design, if necessary. Economic considerations usually carry a lesser weight. Generally speaking, multiple designs could satisfy all design requirements for the given set of material parameters, loadings and ideally, the final design should be the one with minimum construction cost. However, if the economics are not considered explicitly in the design, there is no guarantee that the final design is optimised economically. An optimisation approach in the foundation design process may confirm the economic design.

The civil engineers frequently work with optimisation problems, such as structural design, resource allocation, transportation routing and so forth. Traditionally, most optimisation problems have been solved using operations research (OR) techniques, such as mathematical programming that requires a lot of gradient information and it is difficult to solve when the number of variables are many. In recent years, genetic-based evolution algorithms have become a popular method for solving optimisation problems. In recent years, the geotechnical engineers are also using genetic algorithms (GAs) for the computerisation of their many optimisation problems. For example, Simpson and Priest (1993) demonstrate the application of a GA for identifying the maximum discontinuity frequency in a complex rock structure for three different problem sizes. Goh (1999) incorporates a GA to search for the critical slip surface in multi-wedge stability analysis. Javadi et al. (1999) have used a GA to identify material parameters in a constitutive relationship describing the time dependency of air permeability of shotcrete tunnel lining. Deng and Lee (2001) have applied a GA tor displacement back analysis of a steep slope at the Three Gorges Project site. McCombie and Wilkinson (2002) have applied a simple GA to search for the minimum factor of safety in slope stability analysis using Bishop's simplified method. Zolfaghari, Heath, and McCombie (2005) have presented a GA to search the critical non-circular failure surface in slope stability analysis and incorporated it into the Morgenstern-Price method to find the factor of safety for a variety of slope geometries and loading conditions. Samarajiva, Macari, and Wathugala (2005) use GA to find optimal material parameters in a constitutive model as a continuation of Pal's work. While the traditional OR approach requires different algorithms in solving different mathematical programming problems, the genetic algorithm-based method can solve different optimisation problem using the same algorithm. Only the fitness and constraint functions are modified for different mathematical programming problem.

In this paper, an approach using genetic-based evolution algorithm without gradient requirement is framed as an optimisation process, in which the optimisation variables are the footing dimensions and depth, the optimisation objective is to minimise the construction cost, and the all design requirements are treated as design constraints.

2. Example problem statement

A spread footing in dry sand with a unit weight (y) = 18.5 KN/[m.sup.3] and a friction angle ([phi]')=35[degrees] is illustrated in Figure 1. The sand has a Youngs modulus (E)=50 MPa and a Poissons ratio (u)=0.3. The groundwater is at considerable depth, so the spread footing is not affected by groundwater. The footing is designed to support a square column (b = c = 0.46 m) that transfers dead load of 1400 KN and live load of 900 KN to the footing. Four key design variables for the spread footing are the depth (D), width (B), length (L) and thickness (H).

3. Geotechnical design

The geotechnical design of the foundation should address at least two basic requirements: the required factor of safety, [FS.sub.r], against bearing capacity failure and the allowable settlement, [[delta].sub.r]. In this study, [FS.sub.r] [greater than or equal to] 3.0 and [[delta].sub.r] [less than or equal to] 25 mm are used as the design criteria. In addition, the foundation depth, D, should be greater than a minimum depth, [D.sub.min] = 0.5 m (e.g. to prevent frost damage) and should be limited to a maximum depth, [D.sub.max] = 2.0 m (e.g. to minimise disturbance to adjacent structures).

The ultimate bearing capacity ([q.sub.ult]) of a concentrically vertically loaded footing in cohesionless soil for general shear failure can be calculated using Equation (1), Chen (1975)

[q.sub.ult] = 0.5B[gamma]'[N.sub.[gamma]][[xi].sub.[gamma]s][[xi].sub.[gamma]d] + q'[N.sub.q][zeta.sub.qs][[zeta].sub.qd] (1)

where q' = effective overburden stress at foundation level; [gamma]' = effective unit weight of soil below the foundation level; [N.sub.q] and [N.sub.[gamma]] = bearing capacity factors and [[zeta].sub.[gamma]s], [[zeta].sub.[gamma]d], [[zeta].sub.qs], [[zeta].sub.qd] = shape and depth factors. The allowable bearing pressure, [q.sub.a], is calculated as total working load (P)/footing area (B x L).Then, the factor of safety (FS) can be calculated as:

FS = [[q.sub.ult]/[q.sub.a]] (2)

For this example, the settlement ([delta]) is calculated using the following elastic solution, Poulos and Davis (1974)

[mathematical expression not reproducible] (3)

where [[beta].sub.2] is another shape factor for settlement approximated by the following second-order polynomial function, Whitman and Richart (1967)

[[beta].sub.z]= - 0.0017 [(L/B).sup.2] + 0.0597 (L/B) + 0.9843 (4)

4. Cost estimation

The key component in the optimisation procedure is calculating the cost. The cost estimation of spread footing can be done using unit prices of each individual item; cost of material such as concrete and reinforcement, cost of excavation and backfilling and cost of formworks. The quantities of five items are function of design parameters. The unit prices of each individual item are extracted from the schedule of rates for civil works, Public Works Department (2014) are summarised in Table 1.

Then, the total construction cost (Z) for spread footing can be estimated as follows:

Z = [C.sub.c][Q.sub.c] + [C.sub.r][Q.sub.r] + [C.sub.e][Q.sub.e] + [C.sub.f][Q.sub.f] + [C.sub.b][Q.sub.b] (5)

where Q =quantity of concrete ([m.sup.3]), Q = quantity of reinforcement (kg), [Q.sub.e] = quantity of excavation ([m.sup.3]), [Q.sub.f]=formwork ([m.sup.2]), [Q.sub.b] = quantity of compacted backfill ([m.sup.3]) and [C.sub.c], [C.sub.f], [C.sub.e], [C.sub.f], [C.sub.b]=unit prices of concrete ($/[m.sup.3]), reinforcements ($/kg), excavation ($/[m.sup.3]), formwork ($/[m.sup.2]) and backfill ($/[m.sup.3]), respectively.

5. Genetic algorithm

Holland and Goldberg (1989) stated that GA is 'search procedures based on the mechanics of natural selection and natural genetics'. Genetic algorithm applies the adaptive processes of natural system to the artificial systems inspired by Darwin's theory of natural selection. The GA used in the present study codes and decodes four input parameters (L, B, H and D) represented by four strings (chromosomes) of five binary bits (genes): 0 and 1 each to process an individual (a point in the search space). Finally, an individual contains 20 bits, yielding a task search space of [2.sup.20] possible combinations. Therefore, an appropriate population size (Pop) should be on the order of 80, Goldberg, Deb, and Clark(1991). As such, population sizes of 100, 200 and 400 are used for the conventional GA runs. However, the initial sets of parameters (populations) are created randomly. In every generation, the following common operations are carried out:

5.7. Fitness function evaluation

The fitness function (Z') is the total cost of concrete and reinforcement, excavation and backfilling and form-works as a function of design variables (B, L, H and D). Both the bearing and serviceability requirements are considered as optimisation constraints and are reflected by computed FS and S. In addition, the design variables are subjected to some other practical constraints. The objective function can be expressed as follows:

MinimizeZ' =f(B, L, D, H, [C.sub.c], [C.sub.r], [C.sub.e], [C.sub.f], [C.sub.b]) (6)

Subjected to design requirements: [FS.sub.r] [greater than or equal to]3, [[delta].sub.r] [less than or equal to] 25 mm, B and L > 0, H [greater than or equal to] 150 mm, and 0.5 [less than or equal to] D [less than or equal to] 2.0 m. The real fitness value for an individual is the difference between the fitness of the fit individual and the fitness value of the individual in question in a generation.

In engineering optimisation problems, it is vital to satisfy the performance constraints. The constraints reflect design requirements in this optimisation problem. In other words, they limit the range of acceptable designs in the problem. As GAs are unconstrained optimisation techniques, it is necessary to transform the constrained optimisation problem to an unconstrained one. Michalewicz (1995), Coello (2002), Coello, Lamont, and Van Veldhuizen (2007), Mahdavi, Fesanghary, and Damangir (2007) and Zhou et al. (2011) proposed several methods for handling constraints by GAs. Among them rejecting strategy and the methods based on penalty approach can be considered. In the rejecting strategy, any design that violates one or more constraints is not accepted to create a new population in the GA process. In a penalty method, a constrained optimisation problem is converted to an unconstrained problem by adding a penalty for each constraint violation to the objective function, Z(x), as follows:

[mathematical expression not reproducible] (7)

where Z'(x) is the penalised objective function, r is the penalty multiplier, n is the number of constraints and [[empty set]].sub.f] (x) is the /th penalty function which can be expressed in a general form as follows:

[[empty set].sub.f](x)=[[max([G.sub.f](x), 0)].sup.m] (8)

where m is the power of penalty function and G. (x) is the value of the ith constraint. If there are no violations of constraints, the penalty factor is zero. When one or more of the constraints are violated, the solution is infeasible to some degree, and the value of the corresponding objective function is meaningless. To retain some form of useful information from infeasible solutions, the value of the penalty factor is used to quantitatively describe the degree of constraint violation and to provide a relative measure of the solutions performance.

5.2. Reproduction

The reproduction process uses a tournament selection scheme, Goldberg and Deb (1991), in order to eliminate the shortcomings of the fitness-proportional selection. The number of randomly selected individuals (parents) for tournament competition is referred to as the tournament size, which, in the present study, is two. In the tournament selection, all individuals have an equal chance of being selected for the tournament competition; therefore, this method can eliminate the shortcomings of the fitness-proportional selection: stagnation and premature convergence of the search.

5.3. Crossover

Crossover means that two members of the population exchange genes. The parents are crossed at one point to produce a child (Figure 2). There are many ways of implementing crossover, for example, having a single crossover point or many crossover points. In this study, one-point (see Figure 2(a)) and uniform crossover (see Figure 2(b)) techniques are used with the probability of crossover equals to 0.5 (Carroll 1996).

5.4. Jump and creep mutations

After crossing, the jump mutation takes one of the bits inside the chromosome of a child as a simple mutation and determines the mutation of a selected bit based on the mutation probability (see Figure 2(c)). The jump and creep mutation probabilities are 1/Pop and 4/Pop, respectively (Carroll 1996).

5.5. Elitism

The propagation of the best, or elite, individual with lowest unfitness value is ensured in the next generation causing the GA to progress faster.

5.6. Convergence check

Steps (5.1) through (5.5) are repeated until the number of generations exceeds the maximum limit. The schematic diagram of the implemented GA is shown in Figure 3.

6. Optimum design

The optimisation model is coded in FORTRAN. The performance of the GA is given in Figure 4. It is observed that after only 20 generations, the algorithm has found the optimum design. The optimised design is found with total cost of 228.19 US$, which occurs when L = 1.716 m, B = 1.553 m, H= 0.574 m and D = 1.780 m, respectively.

Table 2 given summarises the optimised design and comparison with other two designs which also satisfies the all design requirements. It is observed that the savings in construction cost could be as much as 68%.

7. Parametric study

Design requirements are paramount, and their variation can result in different designs and consequently different construction costs. Because an economically optimised design incorporates construction cost estimation, it is possible to explore the effect of design requirements on the cost.

The variation of construction cost as a function of [FS.sub.r] and [S.sub.r] is shown in Figure 5. When [S.sub.r] is relatively small (e.g. 15 mm), there is no effect of [FS.sub.r] on the cost of the final design. For [S.sub.r] = 25 mm, there is no influence of FS on the cost of the final design when its value less than 3.5, but when FS exceeds this threshold value, the foundation cost starts to increase. This threshold value of [FS.sub.R] decreases with increase of [[delta].sub.r]. When the S is very large (e.g. 75 mm) the [FS.sub.r] controls the cost of the final design. Figure 6 shows the variation of construction cost as a function of Poissons ratio. It shows that construction cost may change up to maximum 30% with the change in Poissons ratio up to 0.1.

The variation of construction cost and predicted settlement of the optimum design with Young's modulus of the foundation material is shown in Figure 7. The predicted settlement decreases with increase of the elastic modulus. Specially, the cost decreases significantly with the increase of Young's modulus up to the value of 60 MPa. The Young's modulus increases by 50%, the construction cost decreases by more than 300%. Thus, additional efforts might be warranted for improving the characterisation of elastic modulus in sub-soil investigation.

The calculated settlement and cost of the final design for [[delta].sub.r] = 25 mm as the functions of frictional angle of soil is shown in Figure 8. When frictional angle is less than 35[degrees], the calculated total settlement increases and the cost decreases with the increase of friction angle. This indicates that the [FS.sub.r] controls the final design. When frictional angle exceeds 35[degrees], the cost is almost constant. Thus, additional efforts might be warranted for improving the characterisation of frictional angle in sub-soil investigation.

8. Conclusions

The design procedure of foundations for given loading, soil properties and strengths of structural material is framed in an optimisation process using genetic algorithm, in which the optimisation variables are the footing dimensions and depth and the objective function is the total construction cost, treating the design requirements as design constraints. As the optimisation is solved, a set of design dimensions is obtained, and the geo-structure is specified or designed that not only satisfies all design requirements, but also results in the minimum construction cost. Comparison of the economically optimised design example with conventional designs shows that the savings in construction cost could be as much as 68%. A parametric study is performed to identify the key soil properties that most affect the construction cost, and to quantitatively assess the benefit of improving soil property characterisation. It is found that Young's modulus and frictional angle are the key parameters that significantly affect the design of spread footings in cohensionless soils, and, therefore, the specific attention in site investigation is required to characterise these key parameters. However, if Young's modulus is relatively large, then it is no longer a key parameter. Similarly, if frictional angle is relatively large (i.e. >35[degrees] in this design example), then it is also no longer a key parameter. The Poisson's ratio of soil has relatively less effect on construction cost of spread footings in cohensionless soils. In addition, parametric studies show that, when serviceability requirement is relatively small (15 mm), stability requirement has no influence on the cost of final design. However, when the serviceability requirement is relaxed (75 mm), the stability requirement takes the control on the cost of the final design.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes on contributors

M. Shafiqul Islam is an Assistant Professor, Dept. of Civil Engineering, Khulna University of Engineering & Technology Khulna-9203, Bangladesh. Research Interest: Numerical Modelling, Soil Bearing Capacity Analysis, Optimization of Geoteclinical Problems.

M. Rokonuzzaman is an Professor, Dept. of Civil Engineering, Khulna University of Engineering & Technology Khulna-9203, Bangladesh. Research Interest: Numerical Modelling, Model Tests, Optimization and prediction using Artificial Intellegence in Geotechnical Engineering.

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M. Shafiqul Islam and M. Rokonuzzaman

Department of Civil Engineering, Khulna University of Engineering & Technology, Khulna, Bangladesh

CONTACT M. Shafiqul Islam sumon2k8@yahoo.com

ARTICLE HISTORY

Received 4 June 2017

Accepted 22 February 2018

https://doi.org/10.1080/14488353.2018.1445437
Table 1. Unit prices of each individual item.

                                                 Equivalent United
                                                 States dollars
Items               Unit       Unit price (Tk.)  (US$)

Excavation          [m.sup.3]    67.00            0.893
Formwork            [m.sup.2]   354.00            4.72
Reinforcement       kg           61.50            0.831
Concrete            [m.sup.3]  7319.00           97.59
Compacted backfill  [m.sup.3]   114.00            1.52

Note: 1US$ = 75.00 Tk.

Table 2. Comparison of three designs.

Design     Width, B  Length,  L Thickness,  Depth,  Cost    Difference
option     (m)       (m)      H (m)          O(m)   (US$)   (%)

Optimum    1.553     1.716    0.574         1.780   228.19  --
Example 1  2.040     2.040    0.639         1.308   384.13  68.3
Example 2  2.040     1.878    0.639         1.308   354.46  55.3
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Author:Islam, M. Shafiqul; Rokonuzzaman, M.
Publication:Australian Journal of Civil Engineering
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Date:Apr 1, 2018
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