Retaining walls are essential structures in civil engineering that resist lateral earth pressures and provide stability to excavations, embankments, and slopes. Understanding the forces acting on these walls is critical for designing safe and economical structures, especially in Kenya where soil conditions and rainfall patterns vary widely. This chapter focuses on determining the forces acting on retaining walls, including the calculation of the center of gravity, stability considerations, and the types of earth pressures involved. Mastery of these concepts ensures that engineers can predict wall behavior under load and prevent failures such as overturning or sliding.

5.1 Center of Gravity and Stability in Retaining Walls

The center of gravity (CG) is a fundamental concept in structural analysis that influences the stability of retaining walls. Accurate determination of the CG for both uniform and non-uniform wall sections allows engineers to assess moments and forces that affect wall equilibrium. Stability analysis ensures the wall resists overturning and sliding caused by earth pressures and other loads, a vital consideration under Kenya’s varied site conditions.

5.1.1 Determination of Center of Gravity for Uniform and Non-Uniform Sections

The center of gravity of a retaining wall is the point at which the entire weight of the wall can be considered to act. For uniform sections, the CG is usually at the geometric centroid, while for non-uniform sections, it requires decomposition into simpler shapes.

The center of gravity coordinates \( (x_{cg}, y_{cg}) \) are found using:

$$ x_{cg} = \frac{\sum (A_i x_i)}{\sum A_i}, \quad y_{cg} = \frac{\sum (A_i y_i)}{\sum A_i} $$

where \(A_i\) are the areas of individual components, and \(x_i, y_i\) are the coordinates of their centroids.

Worked Examples

Example 1: Determine the CG of a rectangular retaining wall section 3 m wide and 4 m high, uniform thickness.

Given: Width \(b = 3\,m\), Height \(h = 4\,m\)

$$ x_{cg} = \frac{3}{2} = 1.5\,m, \quad y_{cg} = \frac{4}{2} = 2\,m $$

Answer: \(x_{cg} = 1.5\,m\), \(y_{cg} = 2\,m\)

Example 2: Find the CG of an L-shaped retaining wall composed of a vertical stem 0.3 m wide and 4 m high, and a base slab 3 m wide and 0.5 m thick.

Given: Stem \(b_1 = 0.3\,m, h_1 = 4\,m\)
Base \(b_2 = 3\,m, h_2 = 0.5\,m\)

Calculate areas:

$$ A_1 = 0.3 \times 4 = 1.2\,m^2, \quad A_2 = 3 \times 0.5 = 1.5\,m^2 $$

Coordinates of centroids from left bottom corner:

$$ x_1 = 0.15\,m, \quad y_1 = 2\,m; \quad x_2 = 1.5\,m, \quad y_2 = 0.25\,m $$

Calculate \(x_{cg}\):

$$ x_{cg} = \frac{(1.2 \times 0.15) + (1.5 \times 1.5)}{1.2 + 1.5} = \frac{0.18 + 2.25}{2.7} = \frac{2.43}{2.7} = 0.9\,m $$

Calculate \(y_{cg}\):

$$ y_{cg} = \frac{(1.2 \times 2) + (1.5 \times 0.25)}{2.7} = \frac{2.4 + 0.375}{2.7} = \frac{2.775}{2.7} = 1.028\,m $$

Answer: \(x_{cg} = 0.9\,m\), \(y_{cg} = 1.028\,m\)

Example 3: Calculate the CG of a trapezoidal retaining wall with base width 4 m, top width 2 m, and height 3 m.

Given: Base \(b_1 = 4\,m\), Top \(b_2 = 2\,m\), Height \(h = 3\,m\)

Area of trapezoid:

$$ A = \frac{(b_1 + b_2)}{2} \times h = \frac{(4 + 2)}{2} \times 3 = 9\,m^2 $$

Distance of CG from base:

$$ y_{cg} = \frac{h}{3} \times \frac{2b_1 + b_2}{b_1 + b_2} = \frac{3}{3} \times \frac{2 \times 4 + 2}{4 + 2} = 1 \times \frac{10}{6} = 1.667\,m $$

\(x_{cg}\) lies at the midpoint horizontally, since symmetric:

$$ x_{cg} = \frac{b_1 + b_2}{4} = \frac{4 + 2}{4} = 1.5\,m $$

Answer: \(x_{cg} = 1.5\,m\), \(y_{cg} = 1.667\,m\)

5.1.2 Effects on Stability and Risk of Overturning

The position of the center of gravity affects the moment equilibrium of the retaining wall. Stability against overturning requires the resultant force to lie within the base width to avoid tension zones.

The factor of safety against overturning is given by:

$$ FS_{overturning} = \frac{Resisting\,Moment}{Overturning\,Moment} $$

Worked Examples

Example 1: Calculate the factor of safety against overturning for a retaining wall with weight \(W = 150\,kN\) acting at 1.5 m from toe, and active earth pressure \(P_a = 100\,kN\) acting at 0.75 m from toe.

Given: \(W=150\,kN\), distance \(d_w = 1.5\,m\), \(P_a=100\,kN\), distance \(d_p=0.75\,m\)

Calculate moments about toe:

$$ M_{resisting} = W \times d_w = 150 \times 1.5 = 225\,kN\cdot m $$

$$ M_{overturning} = P_a \times d_p = 100 \times 0.75 = 75\,kN\cdot m $$

Calculate factor of safety:

$$ FS = \frac{225}{75} = 3 $$

Answer: Factor of safety against overturning = 3

Example 2: A retaining wall weighs 200 kN with CG 1.2 m from the toe. Earth pressure of 180 kN acts at 0.9 m from the toe. Find the factor of safety.

Given: \(W=200\,kN\), \(d_w=1.2\,m\), \(P_a=180\,kN\), \(d_p=0.9\,m\)

Calculate moments:

$$ M_{resisting} = 200 \times 1.2 = 240\,kN\cdot m $$

$$ M_{overturning} = 180 \times 0.9 = 162\,kN\cdot m $$

Calculate factor of safety:

$$ FS = \frac{240}{162} = 1.48 $$

Answer: Factor of safety against overturning = 1.48

Example 3: For a wall with weight 250 kN acting at 1.1 m from toe, active pressure 220 kN acting at 0.8 m from toe, calculate FS.

Given: \(W=250\,kN\), \(d_w=1.1\,m\), \(P_a=220\,kN\), \(d_p=0.8\,m\)

Calculate moments:

$$ M_{resisting} = 250 \times 1.1 = 275\,kN\cdot m $$

$$ M_{overturning} = 220 \times 0.8 = 176\,kN\cdot m $$

Calculate factor of safety:

$$ FS = \frac{275}{176} = 1.56 $$

Answer: Factor of safety against overturning = 1.56

Chapter Summary

This chapter focused on determining forces acting on retaining walls by first examining the center of gravity and its role in stability. The process for finding the center of gravity was explained for both uniform and non-uniform wall sections, emphasizing how its position affects the wall’s resistance to overturning. The discussion then moved to the different types of pressure that act on retaining walls, including active and passive earth pressures, with methods for calculating these forces. Understanding these pressures is crucial for designing walls that can withstand soil forces safely. The chapter concluded with lateral thrust and stability analysis, detailing approaches to compute lateral forces exerted by soil. Rankine's theory was presented as a fundamental method for calculating lateral earth pressure, contributing to the overall stability assessment. Together, these topics provide a comprehensive framework for analyzing and designing retaining walls to resist various forces effectively.