Neutral Axis Calculator — Built-Up, Composite & Transformed Sections

What the neutral axis is

In a beam bent about a horizontal axis, plane sections stay plane, so strain varies linearly over the depth. One horizontal line has zero strain: the neutral axis. Fibers above it shorten and fibers below it lengthen under a sagging moment. While the material is elastic, equilibrium of the axial forces (no net axial load) puts the neutral axis exactly at the centroid of the section. That is why "find the neutral axis" and "find the centroid" are the same problem for elastic bending.

For a built-up section the centroid is found piece by piece. Choose a datum (the bottom fiber is convenient), list each rectangle's area A_i and the height of its own centroid ȳ_i, and take the weighted average. The moment of inertia then comes from the parallel-axis theorem: each piece contributes its own b h³/12 plus A d², where d is the distance from its centroid to the neutral axis. The A d² terms usually dominate, which is why flanges far from the axis are so efficient.

Transformed sections for two materials

When a section mixes materials (a concrete slab on a steel beam, a steel plate bolted to a timber beam, reinforcing bars in concrete), the strain is still linear, but the stress at a given strain depends on each material's modulus. The transformed-section method converts every piece to the reference material by multiplying its width (equivalently its area) by n = E_piece/E_ref. The transformed section is then analyzed as one homogeneous section. To get back to real stress, multiply the transformed-section stress at that point by the piece's n.

Concrete moduli use ACI 318-19 E_c = 57,000√f′_c (psi) for normal-weight concrete. For sustained-load (creep) composite stresses many designers use 2n or 3n for the concrete.

Elastic vs plastic neutral axis

Once the whole section yields, every fiber carries the same stress magnitude, so axial equilibrium requires equal area above and below the axis — not equal first moment. That line is the plastic neutral axis (PNA). The plastic section modulus Z is the sum of each half's area times the distance from the PNA to that half's centroid, and the plastic moment is M_p = F_y Z. The ratio Z/S is the shape factor: 1.5 for a solid rectangle, about 1.1–1.2 for wide-flange shapes, and higher for T-sections.

Worked examples

Limits of this tool

• Bending about the horizontal (x) axis only. For I_y of an unsymmetric section, rotate the shape and enter it again.
• Pieces are rectangles. Approximate fillets, tapered flanges and circular pieces with an equal-area rectangle, or use the section properties tool for standard shapes.
• The transformed section is linear-elastic and uncracked unless you remove cracked concrete yourself. For reinforced-concrete cracked sections use the RC beam tool, which solves kd directly.
• Plastic values are for homogeneous sections. Composite plastic capacity (AISC 360 Ch. I) depends on 0.85f′_c and F_y, not on n.
• Effective slab width for composite beams is a code decision (AISC 360 I3.1a: span/8 each side, half the distance to the adjacent beam, or the distance to the slab edge) — enter the width you have already determined.

Frequently asked questions

How do you find the neutral axis of a beam?

For elastic bending the neutral axis passes through the centroid of the cross-section. Split the section into simple shapes, multiply each area by the height of its own centroid, add those products, and divide by the total area: ȳ = ΣA·ȳᵢ / ΣA. For a section made of more than one material, multiply each area by its modular ratio n first (the transformed-section method).

What is the difference between the elastic and plastic neutral axis?

The elastic neutral axis (ENA) is the centroid, where stress is zero while the section is elastic. The plastic neutral axis (PNA) is the line that splits the area into two equal halves, where stress changes sign once the whole section has yielded. They coincide for doubly-symmetric sections and separate for unsymmetric ones such as T-beams and plate girders with unequal flanges.

What is a transformed section?

A way to analyze a beam made of two materials as if it were one. Each piece's width is multiplied by n = E_piece / E_reference, which keeps strains compatible. A concrete slab on a steel beam is transformed into a narrower steel slab with width b·Ec/Es. After you find I of the transformed section, the actual stress in a piece is n × M·y/I.

How do I model a hole or a hollow section?

Enter the outer rectangle with n = 1, then enter the opening as another rectangle at its true position with n = −1. Its area and moment of inertia are subtracted. A hollow box 12 × 16 in with 0.5-in walls is the outer 12 × 16 rectangle minus an 11 × 15 hole whose bottom sits at y = 0.5 in.

Why are there two section moduli?

When the neutral axis is not at mid-depth, the distance to the top fiber differs from the distance to the bottom fiber, so S_top = I/c_top and S_bot = I/c_bot differ. The smaller one governs the extreme-fiber stress. For a T-beam the bottom (stem) fiber is farther from the neutral axis and usually governs.

Does horizontal position of the pieces matter?

Not for bending about the horizontal axis. Ix depends only on each piece's width, height and vertical position, so the calculator stacks pieces on a common centerline. Two webs of a box girder can be entered as one rectangle with their combined thickness.

References: Gere, J.M. and Goodno, B.J., Mechanics of Materials, 9th ed., Cengage, ch. 5–6 (centroids, parallel-axis theorem, composite beams and the transformed-section method). AISC 360-22, Specification for Structural Steel Buildings, Ch. F and I (plastic modulus, composite members). ACI 318-19 §19.2.2 (modulus of elasticity of concrete, E_c = 57,000√f′_c psi for normal-weight concrete).