How to Find the Neutral Axis — Step-by-Step Guide
The neutral axis is the line of zero bending stress. For elastic bending it passes through the centroid of the section (the transformed section, if there is more than one material). This guide walks through the hand method with a T-beam, then the extensions engineers actually need: composite sections, holes, the plastic neutral axis, and stresses at each fiber. The free neutral axis calculator follows the same steps.
- Choose a datum and split the section into rectangles
- Tabulate areas and centroid heights
- Find the elastic neutral axis
- Add moments of inertia with the parallel-axis theorem
- Get section moduli and fiber stresses
- Transform composite sections and subtract holes
- Find the plastic neutral axis and Z
- Check the answer
Choose a datum and split the section into rectangles
Draw the section and pick a horizontal reference line, usually the bottom fiber. Break the shape into rectangles that do not overlap (or overlap only where you will subtract a hole). The example throughout is a T-beam: a 12 × 20 in web with a 36 × 4 in flange on top, 24 in deep overall.
Tabulate areas and centroid heights
| Piece | A (in²) | ȳi from bottom (in) | Aȳi (in³) |
|---|---|---|---|
| Web 12 × 20 | 240 | 10 | 2,400 |
| Flange 36 × 4 | 144 | 22 | 3,168 |
| Total | 384 | 5,568 |
Find the elastic neutral axis
The neutral axis sits closer to the heavier flange. That is the reason T-beams in positive bending have small compression zones and deep tension zones.
Add moments of inertia with the parallel-axis theorem
Web: 12(20)³/12 + 240(10 − 14.5)² = 8,000 + 4,860 = 12,860
Flange: 36(4)³/12 + 144(22 − 14.5)² = 192 + 8,100 = 8,292
I = 21,152 in⁴
Get section moduli and fiber stresses
cbot = 14.5 in → Sbot = 21,152/14.5 = 1,459 in³ (smaller, governs)
At M = 100 kip-ft = 1,200 kip-in: ftop = −1,200/2,227 = −0.539 ksi, fbot = +1,200/1,459 = +0.823 ksi
Transform composite sections and subtract holes
If pieces are different materials, multiply each piece's area by n = Epiece/Ereference before Steps 2–4 (equivalently, multiply its width by n). For a concrete slab on a steel beam with f′c = 4 ksi, n = 3,605/29,000 = 0.124. The actual stress in a transformed piece is n times the stress the transformed section gives at that height. For a hole or cutout, enter it as a piece with n = −1 so its area and inertia are subtracted.
Find the plastic neutral axis and Z
For a homogeneous section at full yield, the plastic neutral axis (PNA) splits the area into equal halves. For the T-beam, half the area is 192 in². The web alone holds 12 in² per inch of height, so the PNA is at 192/12 = 16.0 in, inside the web.
Check the answer
- The neutral axis must lie between the extreme fibers, closer to the larger flange.
- I must exceed the sum of the pieces' own b h³/12 values.
- For a symmetric section, ENA and PNA coincide at mid-depth.
- Shape factor: 1.5 for a solid rectangle, roughly 1.1–1.2 for wide-flange beams, higher for tees.
Enter pieces as width, height, bottom y and n; get ENA, PNA, I, S, Z and stresses with a scaled sketch.
Open the neutral axis calculator →Worked examples
Composite steel beam (W18×35 + slab) · RC T-beam cracked section
Frequently asked questions
Is the neutral axis always at the centroid?
For linear-elastic bending with no axial force, yes — the centroid of the (transformed) section. With axial load the zero-stress line shifts away from the centroid. After yielding, the zero-stress line moves to the plastic neutral axis, which splits the area in half. In a cracked concrete section it is the centroid of the cracked transformed section, kd below the top.
Does it matter where I put the datum?
No. Any reference line gives the same neutral axis once you measure the result from that same line. The bottom fiber is convenient because every centroid height is positive.
What is the parallel-axis theorem?
I about any axis equals I about the shape's own centroidal axis plus its area times the distance squared between the two axes: I = I₀ + A·d². It is how the moments of inertia of separate pieces are added about the common neutral axis.
How do I find the neutral axis of an I-beam?
For a doubly symmetric I-shape the neutral axis is at mid-depth by symmetry, and the elastic and plastic axes coincide. For an I-shape with unequal flanges, treat the two flanges and the web as three rectangles and follow the steps in this guide.