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Civil
Elastic neutral axis of a multi-material beam
Position of the elastic neutral axis of a composite section, measured from one datum, as the stiffness-weighted average of the component centroid positions.
- Ref
- CIV-10
- Rev
- v1
- Field
- Civil
- Result
- yn [m]
The neutral axis of a composite section sits where the stiffness-weighted area is balanced: yn = ΣEiAiyi / ΣEiAi. A stiffer material pulls the axis toward its own centroid, which is why a steel plate on timber moves the axis close to the steel. The datum for y can be anywhere, as long as yn is measured from the same datum.
Position of the elastic neutral axis of a composite section, measured from one datum, as the stiffness-weighted average of the component centroid positions. Each component i has modulus E, area A and centroid y. Components must be perfectly bonded and not overlap, and the section carries no net axial force. Up to three components here; leave unused ones at zero.
Calculator
- Total axial stiffness, ΣEA4×108 N
- First moment of stiffness, ΣEAy4×107 N·m
- Share of EA carried by component 10.5
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The neutral axis is at 0.1 m from the datum, 50 % of the stiffness being in component 1.
- If every component had the same modulus the axis would move to the area-weighted centroid, 0.1167 m, so the stiffness difference shifts it by −0.0167 m.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| y2 Centroid of component 2 | 0 to 1 m | 0.025 | 0.525 | 0.5 | +10% | +7.5 % |
| y1 Centroid of component 1 | 0 to 1 m | 0.075 | 0.575 | 0.5 | +10% | +2.5 % |
| A1 Area of component 1 | 0 to 0.05 m² | 0.05196 | 0.15 | 0.09804 | +10% | −2.38 % |
| A2 Area of component 2 | 0 to 0.05 m² | 0.05 | 0.1462 | 0.09615 | +10% | +2.38 % |
| E2 Modulus of component 2 | 0 to 4e11 Pa | 0.05 | 0.13 | 0.08 | +10% | +2.38 % |
| E1 Modulus of component 1 | 0 to 4e11 Pa | 0.08333 | 0.15 | 0.06667 | +10% | −2.38 % |
| E3 Modulus of component 3 | 0 to 4e11 Pa | 0.1 | 0.1 | 0 | +10% | 0 % |
| A3 Area of component 3 | 0 to 0.05 m² | 0.1 | 0.1 | 0 | +10% | 0 % |
| y3 Centroid of component 3 | 0 to 1 m | 0.1 | 0.1 | 0 | +10% | 0 % |
Sweep of Modulus of component 1
| Modulus of component 1 (Pa) | Neutral axis position (m) |
|---|---|
| 0 | 0.15 |
| 5e10 | 0.13 |
| 1e11 | 0.1167 |
| 1.5e11 | 0.1071 |
| 2e11 | 0.1 |
| 2.5e11 | 0.09444 |
| 3e11 | 0.09 |
| 3.5e11 | 0.08636 |
| 4e11 | 0.08333 |
Sweep one variable
Variables and how each one contributes
Neutral axis position
What the equation gives you. Everything else feeds it.
Modulus of component 1
The Young’s modulus of component 1, in Pa. A stiffer component pulls the neutral axis toward its own centroid. Raising it by 10 % (to 2.2×1011) moves the result by −2.38 %, from 0.1 to 0.09762 m.
Modulus of component 2
The Young’s modulus of component 2, in Pa. A stiffer component pulls the neutral axis toward its own centroid. Raising it by 10 % (to 1.1×1011) moves the result by +2.38 %, from 0.1 to 0.1024 m.
Modulus of component 3
The Young’s modulus of component 3, in Pa. A stiffer component pulls the neutral axis toward its own centroid. Raising it by 10 % (to 0) moves the result by +0 %, from 0.1 to 0.1 m.
Area of component 1
The area of component 1, in m². A larger area gives the component more weight in the average. Raising it by 10 % (to 0.0011) moves the result by −2.38 %, from 0.1 to 0.09762 m.
Area of component 2
The area of component 2, in m². A larger area gives the component more weight in the average. Raising it by 10 % (to 0.0022) moves the result by +2.38 %, from 0.1 to 0.1024 m.
Area of component 3
The area of component 3, in m². A larger area gives the component more weight in the average. Raising it by 10 % (to 0) moves the result by +0 %, from 0.1 to 0.1 m.
Centroid of component 1
The distance from the datum to the centroid of component 1, in m. Moves the neutral axis in proportion to this component’s share of EA. Raising it by 10 % (to 0.055) moves the result by +2.5 %, from 0.1 to 0.1025 m.
Centroid of component 2
The distance from the datum to the centroid of component 2, in m. Moves the neutral axis in proportion to this component’s share of EA. Raising it by 10 % (to 0.165) moves the result by +7.5 %, from 0.1 to 0.1075 m.
Centroid of component 3
The distance from the datum to the centroid of component 3, in m. Moves the neutral axis in proportion to this component’s share of EA. Raising it by 10 % (to 0) moves the result by +0 %, from 0.1 to 0.1 m.
Worked example with the values above
- Total stiffness: ΣEA = 2×108 + 2×108 = 4×108 N.
- Moment of stiffness: ΣEAy = 4×107 N·m.
- yn = 4×107 / 4×108 = 0.1 m.
Assumptions and limits
- Linear elastic material, plane sections staying plane, and perfect bond between components.
- Sharp-cornered ideal geometry with no fillets, holes or local buckling.
Common mistakes
- Mixing millimetres with metres, which changes I by a factor of 10¹² when it is raised to the fourth power.
- Using the inertia about the wrong axis (width and depth swapped).
- Forgetting that composite properties depend on the modulus of each part, not only its area.
Statistics and uncertainty
These are exact geometric results for the stated shapes, so the uncertainty is in the dimensions. Because the second moment of area scales with depth cubed, a 2 % error in depth gives about a 6 % error in stiffness. Rolled and fabricated sections vary by a few per cent from nominal, and composite sections carry extra uncertainty from the modulus ratio.
Where it is used
- Calculating stiffness, stresses and capacity of beams and built-up members.
- Spacing connectors in composite and built-up sections.
Maths you need
- Second moment of area: I = ∫y² dA about the centroidal axis
- Parallel axis theorem: I = I₀ + Ad² about a parallel axis
- Neutral axis: the line of zero bending stress in an elastic section