Real-world structural members are rarely subjected to just one simple type of force. A concrete column might carry the weight of a floor (axial compression) while simultaneously resisting the lateral force of wind or earthquakes (bending). A crane hook experiences both tension and bending. This module covers the critical analysis of combined stresses resulting from these complex, simultaneous loads.

When a member is subjected to both an axial load ($P$) acting through its centroid and a separate bending moment ($M$), the resulting normal stresses from each type of loading can simply be combined algebraically using the principle of superposition.

Because both axial stress and bending stress are "normal" stresses (acting perpendicular to the cross-section), they act along the exact same geometric axis and can be directly added or subtracted.

The total normal stress ($\sigma$) at any specific point on the cross-section is:

Where:

An eccentric load is an axial load applied off-center from the primary centroidal axis of the cross-section. This single, off-center load produces both a uniform axial force and a bending moment simultaneously. This is incredibly common in architecture, such as when a floor beam rests on a bracket attached to the side of a column, rather than resting directly on top of it.

If an axial load $P$ is applied at an eccentricity $e$ from the neutral axis, the combined stress formula becomes:

Many traditional architectural materials like unreinforced concrete, brick masonry, stone, and soil are very strong in compression but extremely weak in tension. They will easily crack or pull apart if subjected to even minor tensile forces.

For structures made of these brittle materials (like masonry retaining walls, stone pillars, or concrete spread footings), it is a strict fundamental design requirement that no tension develops anywhere in the cross-section under eccentric loading.

To absolutely guarantee zero tension on a rectangular cross-section of width $b$ and depth $d$, the resultant downward compressive force must fall within the middle third of the section's depth. This central safe zone is known mathematically as the Kern or simply the Middle-Third Rule.

If the eccentricity $e \le d/6$ measured from the centroid (meaning the load stays within the middle $d/3$ of the entire section), the entire section safely remains in compression. If $e > d/6$, tension will develop on the opposite face, leading to cracking or overturning.

Certain materials—most notably concrete, masonry (brick/stone), and soil—are extremely strong in compression but very weak or entirely unable to resist tension. When a compressive load is applied eccentrically to such a material, the bending moment $M = Pe$ induces tensile stresses on the side opposite the load.

To prevent any part of the cross-section from experiencing tension, the resultant compressive force must fall within a specific central region of the cross-section. This region is called the core or the kern of the section.

For a rectangular section ($b \times d$), if the eccentricity $e$ is measured along the depth $d$, the maximum allowable eccentricity to avoid tension is:

This is famously known as the "Middle Third Rule." If the load is applied anywhere within the middle third of the depth (a distance of $d/6$ on either side of the centroidal axis), the entire cross-section remains under compressive stress.

For a solid circular section (diameter $D$), the maximum allowable eccentricity to avoid tension is:

This creates a circular kern with a diameter of $D/4$ centered on the cross-section, known as the "Middle Fourth Rule."