*2.2. Structural Requirements*

The moment and shear capacity of all components of the retaining wall must be greater than their corresponding demands. The flexural strength of each component can be computed as follows [20]:

$$M\_{\rm n} = \phi\_{\rm m} A\_{\rm s} f\_{\rm y} \Big( d - \frac{a}{2} \Big), \tag{14}$$

in which φm is known as the nominal strength coefficient (equal to 0.9 [20]); *A*s is the cross-sectional area of the steel reinforcement; *f* y is the yield strength of steel; *d* is the effective depth of the cross section; and *a* is the depth of the compressive stress block. The shear strength is estimated by [20]:

$$V\_{\mathbf{n}} = 0.17 \phi\_{\mathbf{v}} \sqrt{f\_{\mathbf{c}}} b d\_{\mathbf{r}} \tag{15}$$

where φv is the nominal strength coefficient (equal to 0.75 [20]); *f* c is the specified compressive strength of concrete; and *b* is the width of the cross section.

### 2.2.1. Flexural Moment and Shear Force Demands of Stem

As shown in Figure 2, by considering the active force acting on the unit length of the wall due to surcharge load and the weight of the backfill on the stem, the critical section for flexural moment is at the intersection of the stem with the base slab. In addition, the critical section for shear force is at a distance *d*s from the intersection of the stem with the foot slab, defined as *d*s = *X*3 − *C*C, where *C*C is the concrete cover. The flexural moment and shear force demands of the stem at its critical section are computed by the following equations:

$$M\_{\rm s} = 1.6 \left[ \frac{1}{2} \left( \left( k\_{\rm a} q H^2 \right) + \left( \frac{1}{3} k\_{\rm a} \nu\_{\rm rs} H^3 \right) \right) \right] \cos \beta\_{\prime} \tag{16}$$

$$V\_s = 1.6 \left( (k\_\text{a} \eta (H - d\_\text{s}) \, + \frac{1}{2} k\_\text{a} \gamma\_\text{rs} (H - d\_\text{s})^2 \right) \cos \beta . \tag{17}$$

### 2.2.2. Flexural Moment and Shear Force Demands of Toe Slab

The effective forces on the toe slab include the soil weight on the toe slab, the weight of the toe concrete slab, and the force caused by earth pressure under the toe slab. The foot of the front face of the stem is the critical section for flexural moment and the critical section for shear force is formed at a distance *d*t from the front face of the stem (*d*t = *X*5 − *C*C). The flexural moment and shear force demands of the toe slab at its critical section are computed by the following equations [6]:

$$M\_{\rm t} = \left[1.6\left(\frac{q\_2}{6} + \frac{q\_{\rm max}}{3}\right) - 0.9(\gamma\_{\rm c}X\_{\rm 5} + \gamma\_{\rm bs}(D - X\_{\rm 5}))\right]l\_{\rm toe}^2\tag{18}$$

$$V\_t = \left[1.6\left(\frac{q\_{dt} + q\_{\text{max}}}{2}\right) - 0.9(\gamma\_c X\_5 + \gamma\_{\text{bs}}(D - X\_5))\right] \cdot (l\_{\text{toe}} - dt),\tag{19}$$

where *q*2 is the soil pressure intensity at the foot of the front face of the stem; γc is the unit weight of concrete; *l*toe is the length of the toe slab; and *q*dt is the soil pressure at a distance *d*t from the foot of the stem front face.

### 2.2.3. Flexural Moment and Shear Force Demands of Heel Slab

The forces acting on the heel section include the soil weight of top of the heel slab, the weight of the heel concrete slab, the surcharge load, and the force resulting from earth pressure under the heel slab. The foot of the stem back face is the critical section for flexural moment and the critical section for shear force is at a distance *d*h from the foot of the stem back face (*d*h = *X*5 − *C*C). The flexural moment and shear force demands of the heel slab at its critical section are computed by the following equations [6]:

$$M\_{\rm h} = \left[ \left( \frac{1.6q + 1.2\gamma\_{\rm C}X\_{5} + 1.2\gamma\_{\rm Fs}H}{2} \right) + \frac{1.2\mathcal{W}\_{\rm bs}}{3} - \left( \frac{q\_{1} + 2q\_{\rm min}}{6} \right) \right] l\_{\rm hsel}^{2} \tag{20}$$

$$\mathbf{V\_h} = \left[1.6\boldsymbol{\eta} + 1.2\boldsymbol{\gamma}\_5 \mathbf{X\_5} + 1.2\boldsymbol{\gamma}\_{\rm fb} \boldsymbol{H} + 1.2\frac{\mathcal{W}\_{\rm bs} + \mathcal{W}\_{\rm bcdh}}{2} - 0.9\frac{q\_{\rm dh} + q\_{\rm min}}{2}\right] (l\_{\rm hecl} - d\_{\rm h})\_{\rm r} \tag{21}$$

in which *W*bs is the maximum load due to triangular backfill soil weight at the top of the heel slab; *q*1 is the soil pressure intensity at the foot of the stem back face; *l*heel is the length of retaining wall heel; *W*bsdh is the load resulting from triangular backfill soil weight; and *q*dh is the intensity of soil pressure at a distance *d*h from the stem back face.

### **3. Optimization Problem**
