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This paper presents an optimization algorithm for the design of tied back retaining wall which is comprised of the same three basic elements: stem, toe and heel, where the stem is hinged to the base and tied to the heel by multiple tie rods at intervals along the wall. The aim of this study is to find the values of design variables for this suggested type of tied back retaining walls which minimize the cost function subjected to constraints of the problem. The optimum design of such structure is conducted by using one of the nontraditional optimization methods, genetic algorithm (GA). The formulation of the problem is based on the elastic analysis and the ultimate strength method of design as per ACI-318-2011 code. The built-in genetic algorithm optimtool of Matlab program is utilized to optimize the cost function of the wall. The cost of concrete, reinforcing steel, tie steel, formwork, excavation, and backfilling works are included. The considered design variables are the geometric dimensions and the amounts of reinforcement for the base slab and stem slab, as well as the amount of tie steel. The developed program is utilized to perform an extensive parametric study regarding the height of wall, backfill soil properties, and materials properties including concrete, reinforcing steel, and tie steel. The backfill properties are represented by a pressure coefficient, which is a function of the unit weight and the angle of internal friction. Average expressions are calculated for the total cost and optimum dimensions as ratios of the wall height H2 which may be useful for the practical design of walls.

In design process, engineers have to take many technological and managerial decisions at several stages. The present design of economical concrete structures mainly follows rules based on the experience of structural engineers. Most procedures adopt the cross-section dimensions and material grades sanctioned by common practice. Structural optimization methods are clear alternatives to designs based on experience. Optimum design is a structural synthesis which collects all important engineering aspects to develop structural versions not only safe but also economic. The economy is achieved by minimizing a cost function and the safety is guaranteed by fulfilling the design constraints.

The constraints may be based on stability, bending moment and shear force capacities, and some of the other measures. Thus, the problem can be defined mathematically as a constrained function minimization task, which may be solved by a mathematical programming method [

The improvements in numerical methods and computer technology have given impetus to this concept of optimization. In recent years, some optimization methods that are conceptually different from the traditional mathematical programming techniques have been developed. These methods are labeled as modern or nontraditional methods of optimization. Most of these methods are based on certain characteristics and behavior of biological, molecular, swarm of insects, and neurobiological systems.

Several authors have surveyed the utilization of optimization in structural design. Al-Janabi [^{3} of reinforcement in the kerb and 60 kg/m^{3} for the overall wall was reported. Magbo et al. [

In this research work, a new type of retaining walls is considered. It is comprised of the same three basic elements: stem, toe and heel, where the stem is hinged to the base and tied to the heel by multiple tie rods at intervals along the wall. The tied back retaining wall is suggested to consist of a precast stem of high heights and the base is cast in situ.

Optimization is the act of obtaining the best result under given circumstances [

One of these modern methods is the genetic algorithm (GA) method (John Holland, 1975) which has been utilized to analyze tied back retaining walls. Genetic algorithms are stochastic search methods that mimic some of the processes of natural biological evolution [

The stages of the optimization scheme consist of consideration of structural stability, general stability, geotechnical stability, and costs. The required parameters are given subsequently.

In optimal design problem of retaining wall the aim is to minimize the construction cost of the wall under constraints. This optimization problem can be expressed as follows:

Minimize f(X) which is subject to

where X is n-dimensional vector called the design vector,

For the analysis of wall, the geometry of the tied back wall can generally be described by a set of quantities, some of which are viewed as variables during the optimization process. Some quantities are fixed during the process and called “pre-assigned parameters”. These are outlined as:

・ The properties of backfill and base soil.

・ Height of tied back retaining wall.

・ Unit weight of concrete, steel and soil.

・ The minimum cover for the reinforcement of stem and base.

・ The compressive strength of the concrete and the yield strength of the steel to be used in the design.

・ The allowable tensile stress of tie steel.

・ The cost of each concrete, reinforcing steel, tie steel, formwork, excavation and backfilling works.

・ Strength reduction factor (

The design variables are divided into two categories: those that prescribe the geometric dimensions of wall cross-section, and those that model the steel reinforcement. In general, there are nine design variables of the geometric dimensions to be optimized as shown in _{s}: thickness of stem, T_{b}: thickness of base, B_{t}: length of toe, B_{h}: length of heel, D_{v}: vertical distance between ties, D_{h}: horizontal distance between ties, α: ratio of height of top of outer tie to wall height (H_{1}), β: ratio of horizontal distance of outer tie to length of heel (B_{h}), S: tie spacing.

Whereas the steel reinforcement design variables are modeled as a set of discrete values and include: A_{t}_{1}, A_{t}_{2}: the area of outer and inner tie section respectively, A_{s}_{1}: the area of vertical reinforcement at the stem for negative bending moment (active face of wall), A_{s}_{2}: the area of vertical reinforcement at the stem for positive bending moment (passive face of wall), A_{s}_{3}: the area of longitudinal reinforcement at the stem in the two faces for temperature and shrinkage, A_{s}_{4}, A_{s}_{5}: the area of longitudinal reinforcement at the outer tie level in the stem at tie point and between ties respectively, A_{s}_{6}, A_{s}_{7}: the area of longitudinal reinforcement at the inner tie level in the stem at tie point and between ties respectively, A_{s}_{8}: the area of main reinforcement at the bottom of the toe, A_{s}_{9}: the area of main reinforcement for negative bending moment at the heel, A_{s}_{10}: the area of main reinforcement for positive bending moment at the heel, A_{s}_{11}: the area of longitudinal reinforcement at the base in the two faces for temperature and shrinkage, A_{s}_{12}, A_{s}_{13}: the area of longitudinal reinforcement at the outer tie in the heel at tie point and between ties respectively, A_{s}_{14}, A_{s}_{15}: the area of horizontal reinforcement at the inner tie level in the

base at tie point and between ties respectively.

The typical design philosophy for retaining structures seeks designs that provide safety and stability against failure modes and comply with concrete building code requirements. These requirements may be classified into two groups of constraints namely, the general constraints and the ultimate resistance constraints. Within the optimization procedure, if all considered constraints and cost considerations are entirely met, the design will be feasible. These requirements represent the failure modes as a function of the design variables. Failures modes are summarized in

Some constraints have maximum allowable value much larger than that of other constraints. This will badly affect the convergence rate during the minimization of cost function. Therefore, normalization is used which gives after rearranging the equations of these constraints.

The net horizontal forces must be such that the wall is prevented from sliding along its foundation. The most significant sliding force component usually comes from the lateral earth pressure acting on the active (backfill) side of the wall. Sliding failure is a result of excessive lateral earth pressures with relation to retaining wall resistance thereby causing the retaining wall system to move away (slide) from the soil it retains.

where

The stabilizing moments, due to vertical forces must be greater than the overturning moments, due to horizontal forces to prevent rotation of the wall around its toe. The stabilizing moments result mainly from the self-weight of the structure, whereas the main source of overturning moments is the active earth pressure. Overturning failure is a result of excessive lateral earth pressures with relation to retaining wall resistance thereby causing the retaining wall system to topple or rotate (overturn).

where_{a}: active moment, and

To avoid excessive high toe pressure it is desirable to keep the resultant of the various forces acting on the wall

Inequality constraints | Failure mode |
---|---|

g_{1}(x) | Sliding stability |

g_{2}(x) | Overturning stability |

g_{3}(x), g_{4}(x) | Excessive high toe pressure |

g_{5}(x) | Bearing capacity |

g_{6}(x) to g_{10}(x) | Flexural moments in stem and base respectively |

g_{11}(x) to g_{23}(x) | Minimum reinforcement area criteria |

g_{24}(x) to g_{36}(x) | Tension controlled sections |

g_{37}(x), g_{38}(x) | Shear failure in stem and base respectively |

R within the middle third of the base, so that no tension is developed at the base [

where B: base width,

The bearing capacity of the foundation must be large enough to resist the stresses acting along the base of the structure.

where^{2}),^{2}),

The maximum bending moments at the critical sections should be less than the resistance moments for both stem and base.

where

i = index indicating the critical sections of toe, heel and stem,

The reinforcement area for bending moment at any section where tensile reinforcement is required should not be less than (

where (

Sections are tension-controlled if the net tensile strain in the extreme tensile steel (

where

The available effective depth must be greater than that required for wide beam shear requirement [

where S: distance between ties,

The objective function is a function of design variables the value of which provides the basis for choice between alternate acceptable designs. The objective of design may be minimization of weight, cost or stress concentration factor. In structural designs the objective function is usually weight or cost minimization.

In the present study, the objective function is defined as the total cost of tied back retaining wall (material and labor) for a spacing length (S) of the wall. This includes the cost of concrete

where^{3}),

The applications in this section involve solving many numerical examples in order to illustrate the effects of various design variables and different parameters on the optimal design. The minimum cost of the tied back retaining wall for a distance equal to the spacing between ties (S) is given. These examples are concerned with the following points:

・ The effect of the total height of the tied back retaining wall.

・ The effect of pressure coefficient, K, which is a function of backfill properties, i.e.

・ The effect of materials properties (concrete, reinforcing steel, and tie steel).

The program consists of two main stages. In the first stage the necessary data are specified. The second stage performs the calculations of the optimum design.

To investigate the effect of the total height of the tied back retaining wall, height equal to (6, 7, 8, 9 and 10) m is used. The basic values of required parameters are taken as pressure coefficient K = 6 kN/m^{3}, yield stress of steel f_{y} = 276 MPa, concrete cylinder compressive strength_{all} = 155 MPa, unit mass of steel ρ_{s} = 7.85 ton/m^{3}, unit weight of reinforced concrete γ_{c} = 24 kN/m^{3}, unit weight of base soil γ_{b} = 19 kN/m^{3}, cohesion of base soil c_{b} = 40 kN/m^{2}, angle of internal friction for base soil = 35˚, cost of steel P_{s} = 1,000,000 ID/ton, cost of concrete P_{c} = 150,000 ID/m^{3}, cost of formwork P_{fw} = 7500 ID/m^{2}, cost of excavation P_{ex} = 6000 ID/m^{3}, cost of backfilling P_{fi} = 10,000 ID/m^{3}.

The optimal solution has been summarized in _{2}.

It can be noticed from this table and figure that the total cost of wall increases as the total height H_{2} increases. The data from this table and figure also shows nonlinear relationship between the total cost of wall and the total height H_{2}. This relation usually depends on many factors like material properties and the different unit prices as will be discussed in following sections. The total height of tied back retaining wall has affected on the optimum stem thickness T_{s} which increases with the increase of total height H_{2}. Also the thickness of the base T_{b} increases as the wall height increases.

In order to examine the effect of backfill properties ^{3} that was used in Section 4.1) equal to 4, 4.5, 5, and 5.5 kN/m^{3} are used in order to illustrate the effect of its variation on the optimum results with different total height that were considered in previous section. The other data remains unchanged as in the previous section. The results have been shown in Figures 4-10.

It is clear from ^{3} and the effect appears after that. It can be seen that the effect of height vanishes for wall heights 9 m and 10 m where the curves coincide. It can be realized from ^{3} and after that the influence of pressure coefficient is increasing obviously. It is clear from _{1}, α, is inversely proportional to wall height and it seems not to alter as the pressure coefficient varies. It is clear from _{h}, β, it seems to be constant, about (0.92 to 1) times B_{h}.

The properties of concrete, reinforcing steel, and tie steel play an important role on the optimum design. Their effects have arisen not only in unit price of each material but also in their specifications that influence the optimal design as will be shown in this section.

In order to examine the effect of concrete compressive strength on the optimum results different grades of concrete are considered. Grades (25, 30, 40, and 50 MPa), which indicate the cylinder compressive strength, are