Determining forces on retaining walls is essential for ensuring their structural stability and safety in building construction. Retaining walls resist lateral earth pressures and other loads, and understanding the distribution of these forces requires knowledge of the wall’s center of gravity and stability conditions. This chapter provides mathematical methods to locate the center of gravity for various retaining wall sections, a critical step in analyzing overturning, sliding, and bearing capacity. Accurate calculations of these parameters help building technologists design safe retaining structures in Kenyan construction projects.

5.1 Center of Gravity and Stability in Retaining Walls

Retaining walls must be stable against overturning, sliding, and excessive bearing pressure to prevent failure. The center of gravity (CG) of the wall and the resultant force from soil pressures are fundamental in assessing these stability criteria. This section covers how to calculate the CG for uniform and non-uniform wall sections and use it to evaluate stability.

5.1.1 Concept of Center of Gravity in Retaining Walls

The center of gravity is the point where the entire weight of the retaining wall can be considered to act. It is crucial for determining the moment arms for overturning analysis.

• The CG is the balance point of the wall section.
• It helps locate the line of action of the wall’s self-weight.
• The CG position influences the magnitude of stabilizing moments.
• It is used to calculate eccentricity in bearing pressure checks.
• For composite walls, the CG is found by combining the CGs of individual parts.

The formula for the coordinate of the CG, \( \bar{x} \), along a horizontal axis is:

$$ \bar{x} = \frac{\sum (A_i x_i)}{\sum A_i} $$

where \(A_i\) is the area of part \(i\), and \(x_i\) is the distance of the centroid of part \(i\) from the reference axis.

Worked Examples

Example 1: Calculate the center of gravity (CG) of a rectangular retaining wall section 2.5 m wide and 4 m high.

Given: \(b = 2.5\, m\), \(h = 4\, m\)

$$ \bar{x} = \frac{2.5}{2} = 1.25\, m $$$$ \bar{y} = \frac{4}{2} = 2\, m $$ Answer: CG is at 1.25 m from one side horizontally and 2 m from base vertically

Example 2: For a composite wall consisting of two rectangles: base 3 m wide by 0.5 m thick (\(A_1 = 1.5\, m^2\), centroid \(y_1 = 0.25\, m\)), and stem 0.5 m wide by 3 m high (\(A_2 = 1.5\, m^2\), centroid \(y_2 = 0.5 + 1.5 = 2\, m\)). Find the vertical CG from base.

Total area:$$ A = 1.5 + 1.5 = 3\, m^2 $$ Vertical CG:$$ \bar{y} = \frac{(1.5 \times 0.25) + (1.5 \times 2)}{3} = \frac{0.375 + 3}{3} = \frac{3.375}{3} = 1.125\, m $$ Answer: CG is 1.125 m above the base

5.1.2 Stability Criteria for Retaining Walls

Stability involves ensuring that the resultant force from all acting loads passes through a safe zone within the base to prevent overturning or sliding.

• Overturning stability requires the resisting moment to be greater than the overturning moment.
• Sliding stability requires the frictional resistance at the base to exceed lateral forces.
• Bearing capacity checks ensure soil pressure under the base does not exceed allowable limits.
• The factor of safety (FoS) is used to quantify stability margins.
• The CG position is fundamental in calculating moments about the toe or heel of the wall.

5.1.3 Determining the Resultant Force and Its Line of Action

The resultant force acting on the retaining wall combines the wall’s weight and the lateral earth pressure.

• Calculate the weight force using the volume and density of wall materials.
• Calculate lateral earth pressure using Rankine or Coulomb earth pressure theories.
• The resultant force line of action is found by summing moments about a reference point.
• The position of the resultant affects eccentricity and thus soil pressure distribution.
• Accurate force location helps prevent eccentric loading that can cause failure.

5.1.4 Effect of Wall Geometry on Stability

The shape and dimensions of the retaining wall affect the CG location and stability.

• Uniform rectangular sections have CG at mid-width.
• T-shaped and L-shaped walls have CG shifted towards heavier parts.
• Non-uniform bases change moment arms and eccentricity.
• Stepped or counterfort walls require composite CG calculations.
• Proper design accounts for geometry to optimize stability and material use.

Chapter Summary

This chapter covers the fundamental principles of determining forces on retaining walls, beginning with the concept of center of gravity and its role in ensuring wall stability. It explains how to calculate the center of gravity for both uniform and non-uniform wall sections, which is crucial for assessing stability and the risk of overturning. Various types of pressures acting on retaining walls are discussed, including active, passive, and at-rest earth pressures, along with methods for calculating hydrostatic and earth pressures. The chapter then explores lateral thrust and stability analysis, presenting different methods for calculating lateral forces with a focus on Rankine's and Coulomb's theories. Finally, it addresses stability checks and design considerations tailored to gravity and cantilever retaining walls, emphasizing the importance of proper analysis to prevent failure and ensure structural integrity.