Module 9 · Lesson 9.4
Combined bending and axial load
Adding two stress distributions, and the point where tension disappears.
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:
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. Applied about both axes it becomes the middle-third rule for a rectangular pad footing, and the reason the resultant is kept inside the kern of a masonry section.
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
This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint