What is middle third rule?

Short Answer:

The middle third rule is a principle used in the analysis and design of compression members such as columns, piers, and retaining walls. It states that if the line of action of the load passes within the middle one-third of the base of a section, the entire section will be under compression and no tension will develop.

However, if the load falls outside this middle third region, part of the section will experience tension. This rule is very important in structures made of materials like masonry or concrete, which are strong in compression but weak in tension.

Detailed Explanation:

Middle Third Rule

The middle third rule is a simple and important rule used in the design of compressive members like columns, footings, retaining walls, and dams. It ensures that all the stresses within a section remain compressive and that no part of the structure is subjected to tension.

When a compressive load acts on a member, the stress distribution depends on the position of the line of action of the load relative to the centroid of the section.

  • If the load passes through the centroid, the stress is uniform across the section.
  • If the load is applied with some eccentricity (not through the centroid), it causes a bending moment along with direct compressive stress.

The middle third rule defines the region within which the load must act so that no tensile stress occurs anywhere in the section.

This concept is especially important for brittle materials like concrete, brick, or stone, which cannot resist tension effectively.

Derivation of Middle Third Rule

Consider a rectangular section of width  and depth , subjected to a compressive load  acting with an eccentricity  from the centroid.

The direct compressive stress is given by:

The bending stress due to the moment caused by the eccentric load is:

For a rectangular section,

Thus,

The maximum and minimum stresses at the extreme fibers are:

 

For no tension to occur anywhere in the section, the minimum stress (σmin) must be zero or positive, i.e.,

Substitute the values:

Simplifying,

This means that if the eccentricity of the load does not exceed one-sixth of the depth, there will be no tension in the section.

Hence, the region within which the resultant load must act is called the middle third of the base.

Meaning of Middle Third

The middle third of a rectangular section is the central one-third portion of its base.

If the base has a depth (or height) h, then the middle third lies between:

This rule ensures that the compressive stress is distributed uniformly or in a linear varying manner, but never becomes tensile.

If the line of action of the load lies outside this middle third zone, a part of the section will be under tension, which can cause cracks or failure in materials weak in tension.

Application of the Middle Third Rule

  1. Rectangular Sections:
    For rectangular sections, the load must act within the central one-third of the base width or height to avoid tension.
  2. Circular Sections:
    For circular sections, the load should pass through the middle quarter of the diameter (this is similar to the middle third rule for rectangular shapes).
  3. Triangular Sections:
    For triangular sections, the core is smaller, and the load must pass within a limited region near the center to prevent tension.

Importance of the Middle Third Rule

  1. Prevents Tension in Brittle Materials:
    Concrete, brick, and stone are strong in compression but weak in tension. By applying the middle third rule, tension can be completely avoided.
  2. Ensures Structural Stability:
    Keeping the load within the middle third ensures uniform stress distribution and prevents cracks, tilting, or overturning of structures.
  3. Foundation Design:
    In foundations, if the resultant load passes through the middle third of the base, the soil pressure remains fully compressive and the foundation remains stable.
  4. Retaining Walls and Dams:
    In retaining walls and dams, the line of action of the resultant force due to earth or water pressure and the weight of the structure must lie within the middle third of the base to avoid overturning and tension at the base.
  5. Columns and Piers:
    Columns subjected to eccentric loads are designed such that the resultant load falls within the middle third zone to maintain full compression.

Example

Suppose a rectangular column has a width of 300 mm and a depth of 450 mm.
According to the middle third rule, the eccentricity (e) of the load must not exceed h/6 = 450/6 = 75 mm.

Therefore, the load must pass through a central zone of width 450 – 2×75 = 300 mm, which represents the middle third of the section.
If the load acts within this 300 mm zone, the entire section remains under compression.

Conclusion

The middle third rule is a simple but essential principle for designing compressive members. It states that if the line of action of the load passes within the middle third of the base of a rectangular section, the entire section remains under compression. No part of the section will experience tension. This rule helps engineers design safe and stable structures such as columns, footings, retaining walls, and dams, especially when using materials that are strong in compression but weak in tension like concrete and masonry.