Short Answer:

Bending stress due to eccentric load is the stress developed in a structural member when the applied load does not act through the center of the section but at some distance from it, known as eccentricity. This eccentric load produces both a direct stress and a bending moment in the member.

Because of this bending moment, additional stress is created, called bending stress. The bending stress combines with the direct stress, making one side of the member more compressed while the other side may experience tension. This condition is common in columns, beams, and brackets under eccentric loading.

Detailed Explanation:

Bending Stress due to Eccentric Load

When a load acts through the centroid of a cross-section, it produces direct or normal stress only. However, when the load is applied at a distance from the centroid, known as the eccentricity (e), the load causes not only direct stress but also a bending moment (M). The bending moment leads to additional stress known as bending stress.

Thus, in the case of eccentric loading, the total stress on the section is a combination of direct stress due to the axial load and bending stress due to the moment caused by eccentricity. This combination of stresses results in non-uniform distribution across the section — one side experiences higher compression or tension, while the other experiences lower stress or even the opposite type of stress.

Mathematically, the bending moment (M) caused by the eccentric load is given by:
M = P × e
where,

P = Applied load,
e = Eccentricity (distance from the centroidal axis).

This moment produces bending stress (σb) given by:
σb = M / Z = (P × e) / Z

Here,

Z = Section modulus of the cross-section.

At the same time, the direct or normal stress (σd) due to the axial load is:
σd = P / A
where,

A = Cross-sectional area.

Therefore, the total stress at any fiber of the section is the sum of direct and bending stresses:
σ = σd ± σb = (P / A) ± (P × e / Z)

The ‘+’ sign is used for the side where the direct and bending stresses act in the same direction (resulting in maximum stress), while the ‘–’ sign is used for the opposite side where they act in opposite directions (resulting in minimum stress).

Stress Distribution due to Eccentric Load

Under an eccentric load, the stress is no longer uniform over the cross-section. One edge of the member will have more compression, and the other edge will have less compression or even tension, depending on the magnitude of eccentricity and the load.

Maximum Stress (σmax):
Occurs at the outermost fiber where direct and bending stresses add up.
σmax = (P / A) + (P × e / Z)
Minimum Stress (σmin):
Occurs at the opposite fiber where the stresses act in opposite directions.
σmin = (P / A) – (P × e / Z)

If σmin becomes negative, it indicates that the side is under tension, which is often undesirable for materials like concrete or masonry that are weak in tension.

Example of Bending Stress due to Eccentric Load

Consider a rectangular column carrying a load P at an eccentricity e from its centroidal axis. The load causes a bending moment M = P × e, producing a bending stress.

Let:

Width of the column = b
Depth of the column = d
Area, A = b × d
Section Modulus, Z = (b × d²) / 6

Then,
Bending stress, σb = (P × e) / Z = (6P × e) / (b × d²)

And,
Direct stress, σd = P / (b × d)

The combined stresses on the extreme fibers are:
σmax = (P / A) + (P × e / Z) and σmin = (P / A) – (P × e / Z)

This example shows how eccentric loading introduces bending stress that changes the uniform stress pattern across the section.

Effects of Bending Stress due to Eccentric Load

Uneven Stress Distribution:
The combination of direct and bending stress leads to non-uniform stress across the section. One side may experience higher compression, while the other may experience tension.
Possibility of Tension:
When eccentricity is large, the minimum stress may become tensile. This is dangerous for materials like concrete or stone, which are strong in compression but weak in tension.
Increased Risk of Buckling:
Eccentric loading increases the bending moment, which can make long columns more likely to buckle under the same load compared to centrally loaded columns.
Structural Deformation:
Due to bending effects, the member may bend or deflect sideways. This must be considered in design to prevent excessive deformation.

Practical Applications

Columns and Struts:
In real structures, loads are rarely applied perfectly through the centroid. Eccentric loading is common in columns where beams or slabs transfer loads slightly off-center, creating bending stress.
Brackets and Cantilevers:
When loads act away from the support face, bending moments are created, resulting in bending stress due to eccentric loading.
Masonry Walls and Foundations:
When a load acts at an eccentricity, it causes bending in walls and uneven soil pressure in foundations.
Bolted or Riveted Joints:
In mechanical joints, if the load acts away from the centroid of bolt patterns, it produces bending and uneven stresses among bolts or rivets.

Importance of Studying Bending Stress due to Eccentric Load

Safe Design:
Understanding bending stress due to eccentric load helps in designing members that can safely handle off-center loads.
Prevention of Failure:
By analyzing bending effects, engineers can predict where failure or cracking might occur and reinforce those areas.
Accurate Stress Analysis:
It allows for more realistic calculations since in practice, perfect concentric loading is rare.
Better Material Utilization:
Knowing stress variations helps in selecting appropriate materials and cross-sections for different load conditions.

Conclusion

Bending stress due to eccentric load arises when a load acts away from the centroidal axis of a member, creating a bending moment in addition to direct stress. This bending moment causes non-uniform stress distribution, increasing compression on one side and tension on the other. Understanding this concept is essential for designing columns, beams, and other structural members to ensure stability and safety under real-life conditions where perfect central loading is rarely achieved.