Structural Analysis Fundamentals / Bending of beams

Combined bending and axial load

Adding two stress distributions, and the point where tension disappears.

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Why this matters

Very few members carry pure bending. Columns carry moment as well as compression, portal frame rafters carry thrust as well as bending, and any beam with an inclined tie develops both. Superposing the two is straightforward — and the result explains masonry, foundation pressure and the old rule about the middle third.

By the end of this lesson you should be able to

Superpose axial and bending stress correctly, with signs
Find the position of the neutral axis under combined loading
Determine when a section goes into tension anywhere
State and use the middle-third rule

What you should already know

Direct stress σ = P/A (Module 7)
Bending stress σ = My/I (this module)
Section modulus Z (this module)

Both effects produce a stress acting along the member, so for a linearly elastic material they simply add. Superposition applies because each is proportional to its own load and neither changes the geometry:

Combined axial and bending stress

\sigma = \frac{P}{A} \pm \frac{My}{I}

What it calculates: the total direct stress at a fibre

P: axial force, tension positive (N)
A: cross-sectional area (mm²)
M: bending moment (N·mm)
y: distance from the centroidal axis (mm)
I: second moment of area (mm⁴)

This assumes

Linearly elastic material, so the two effects superpose
Small deflections — the axial force does not add moment by moving sideways
The section is not slender enough to buckle

In plain terms: The axial term is the same everywhere on the section; the bending term varies linearly and is zero at the centroid. Adding a uniform block to a linear wedge shifts the whole distribution up or down without changing its slope.

The most useful consequence concerns the neutral axis. In pure bending it passes through the centroid. Add an axial force and it moves — because the bending term now has to cancel a non-zero uniform stress rather than zero.

Setting the total stress to zero:

P/A + My/I = 0, so y = −PI/(MA)

If that y lies outside the section, the stress never crosses zero and the whole section is in one sense — all compression, or all tension. That is exactly what you want in masonry, plain concrete and most foundations, which cannot carry tension.

For a rectangular section the condition works out very neatly. Writing the moment as an eccentric load, M = Pe, and requiring the extreme-fibre stress to stay compressive:

P/A ≥ Pe/Z, so e ≤ Z/A = h/6

The load must land within h/6 either side of the centre — the middle third of the depth. For simultaneous eccentricity about both axes of a rectangular base B × L, the condition is |e_x|/B + |e_y|/L ≤ 1/6. Checking each one-sixth limit independently is insufficient. Under a linear full-contact pressure model, q_min = (P/A)[1 − 6|e_x|/B − 6|e_y|/L]. Outside this combined kern, use a no-tension contact analysis where the material or interface cannot transmit tension.

Predict first

A column carries axial compression plus a bending moment. Compared with pure bending, where is the neutral axis?

Worked example

A column with an eccentric load

Given

Rectangular column section: b = 300 mm, h = 400 mm
Axial compression P = 600 kN, applied 80 mm from the centroid along the 400 mm depth
Elastic behaviour; the column is stocky, so second-order effects are ignored

Find

The extreme-fibre stresses, and whether any part of the section goes into tension.

Practice

A rectangular column 300 mm × 400 mm carries 600 kN of compression at an eccentricity of 80 mm about the 400 mm axis. What is the magnitude of the maximum compressive stress, in N/mm²?

Practice

For the same column, what is the stress at the opposite face, in N/mm²? Give a positive number if it is tensile.

Practice

For a rectangular section 400 mm deep, what is the largest eccentricity, in mm, at which an axial compression produces no tension anywhere?

Summary

Axial and bending stresses superpose: σ = P/A ± My/I
The axial term shifts the whole distribution; the bending term sets its slope
Axial load moves the neutral axis off the centroid, and can push it off the section entirely
No tension anywhere requires e ≤ Z/A, which for a rectangle is h/6
Superposition assumes small deflections and a stocky member