*3.3. Design Constraints*

Two sets of constraints are considered for the optimum design of a RCRW. The first set (four constraints) is related to geotechnical requirements (wall stability). Of these, the first three are considered to provide safety factors against overturning, sliding, and bearing capacity failure modes, as shown below [21]:

$$\log\_1 = \frac{FS\_\text{O}}{FS\_\text{O,all}} - 1 \ge 0,\tag{25}$$

$$\log\_2 = \frac{F \text{S}\_{\text{S}}}{F \text{S}\_{\text{S}, \text{all}}} - 1 \ge 0,\tag{26}$$

$$\log\_3 = \frac{FS\_{\rm B}}{FS\_{\rm B,all}} - 1 \ge 0,\tag{27}$$

in which *FS*O, *FS*S, and *FS*B are the safety factor demands against overturning, sliding, and bearing capacity failure modes, respectively, and *FS*O,all, *FS*S,all, and *FS*B,all are their allowable values.

The fourth constraint of the first set is used to avoid the presence of tensile stresses on the base soil, as follows:

$$g\_4 = g\_{\min} \ge 0.\tag{28}$$

The second set of constraints is related to the structural requirements having to do with providing the required strength of wall components and reinforcement arrangements in their cross sections, in accordance with ACI318-14 [20]. To design the structural sections, the reinforcement area at each section of the retaining wall should satisfy the allowable amounts of the reinforcement area, as follows:

$$g\_{5-8} = \left(\frac{A\_{\rm s}}{A\_{\rm s,min}}\right)\_{\rm st} - 1 \ge 0,\tag{29}$$

$$g\_{\text{9-12}} = \left(\frac{A\_{\text{s,max}}}{A\_{\text{s}}}\right)\_{\text{st}} - 1 \ge 0,\tag{30}$$

in which *A*s,min and *A*s,max are the minimum and maximum allowable area of steel reinforcement in accordance with the code, and *A*s is defined as the cross-sectional area of steel reinforcement in each section. The subscript *st* refers to all sections of the RCRW including stem, heel, toe and shear key, and similarly subscripts 5 to 8 are used for the stem, toe, heel and shear key, respectively. The same is also the case for subscripts 9 to 12.

As mentioned earlier, the moment and shear capacities of all sections of the retaining wall should be greater than the corresponding demands, namely:

$$\mathcal{G}\_{13-16} = \left(\frac{M\_{\rm n}}{M\_{\rm d}}\right)\_{\rm st} - 1 \ge 0 \tag{31}$$

$$\mathcal{g}\_{17-20} = \left(\frac{V\_{\rm n}}{V\_{\rm d}}\right)\_{\rm st} - 1 \ge 0 \tag{32}$$

In the equations above, *M*n and *V*n are the moment and shear nominal capacity, and *M*d and *V*d are the moment and shear demands, respectively. *M*n and *V*n are the flexural moment and shear strength formulated before. The subscripts 13 to 16 are for stem, toe, heel, and shear key. The same is true for subscripts 17 to 20.

The following geometric constraints are also applied to avoid an impossible or impracticable shape of the wall:

$$\text{g}\_{21} = \frac{X\_1}{X\_2 + X\_3} - 1 \ge 0,\tag{33}$$

$$\text{g}\_{22} = \frac{X\_1}{X\_6 + X\_7} - 1 \ge 0. \tag{34}$$

The minimum development length of the steel reinforcement bars should be considered for all the structural components. At first, the minimum basic development length *l*db against the allowable space is checked. If the available space is not enough, a hook is added to achieve the additional development length. In this case, a minimum hook development length *l*dh and minimum hook length of 12*d*bh (*d*bh is the diameter of the hooked bar) should be satisfied. The following limitations are considered for stem, toe, heel, and shear key in the design, respectively:

$$\text{g2z} = \frac{l\_{\text{dh,stem}}}{X\_{\text{5}} - C\_{\text{C}}} - 1 \ge 0 \quad \text{or} \quad \text{g2z} = \frac{l\_{\text{dh,stem}}}{X\_{\text{5}} - C\_{\text{C}}} - 1 \ge 0,\tag{35}$$

$$\text{g}\_{24} = \frac{l\_{\text{db,tot}}}{X\_1 - X\_2 - \text{C}\_{\text{C}}} - 1 \ge 0 \quad \text{or} \quad \text{g}\_{24} = \frac{12d\_{\text{b,tot}}}{X\_5 - \text{C}\_{\text{C}}} - 1 \ge 0,\tag{36}$$

$$\mathcal{g}\_{25} = \frac{l\_{\text{db,huel}}}{X\_2 + X\_3 - \mathcal{C}\_{\text{C}}} - 1 \ge 0 \quad \text{or} \quad \mathcal{g}\_{25} = \frac{12d\_{\text{b,huel}}}{X\_5 - \mathcal{C}\_{\text{C}}} - 1 \ge 0,\tag{37}$$

$$\log\_{26} = \frac{l\_{\text{dh,key}}}{X\mathfrak{z} - \mathbb{C}\_{\mathbb{C}}} - 1 \ge 0 \quad \text{or} \quad \mathfrak{g}\_{26} = \frac{l\_{\text{dh,key}}}{X\mathfrak{z} - \mathbb{C}\_{\mathbb{C}}} - 1 \ge 0,\tag{38}$$

It should be noted that the inequalities *g*23 to *g*26 as described in Equations (35)–(38) were actually not considered as design constraints in the present study. In fact, during the optimization process, the required development lengths were simply considered and their role in steel cost was computed and added to the construction cost. The cost of shrinkage reinforcement was also added to the total construction cost. Another important point is that the other constraints on the arrangements of steel bars in the wall sections, such as the number of allowable bars, bar size, and bar spacing, were all considered in the optimum design, as can be seen in Table 2.
