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Flexural rigidity of a bonded composite beam
Effective bending stiffness of a fully bonded composite beam, adding each component’s own stiffness EI and the parallel-axis term EA(y − yn)² about the elastic neutral axis.
- Ref
- CIV-11
- Rev
- v1
- Field
- Civil
- Result
- (EI)eff [N·m²]
The effective flexural rigidity of a bonded composite beam adds the stiffness of every part about the common neutral axis: (EI)eff = ΣEi[Ii + Ai(yi − yn)²]. Each component contributes its own bending stiffness EiIi and a parallel-axis term that is usually much larger for parts far from the axis. All parts bend with the same curvature.
Effective bending stiffness of a fully bonded composite beam, adding each component’s own stiffness EI and the parallel-axis term EA(y − yn)² about the elastic neutral axis. Every component bends with the same curvature. Use the neutral axis from the previous entry. Up to three components here; leave unused ones at zero.
Calculator
- Own-stiffness term, ΣEI400,000 N·m²
- Parallel-axis term, ΣEA(y − yn)²1,000,000 N·m²
- Share of stiffness from the parallel-axis terms0.7143
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 parallel-axis terms supply 71.43 % of the stiffness, so most of it comes from material away from the neutral axis.
- A midspan load of 10 kN on a 6 m simply supported span would deflect 32.14 mm (P L³ / 48EI).
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (N·m²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| I1 Own second moment of component 1 | 0 to 0.05 m⁴ | 1,200,000 | 1e10 | 1e10 | +10% | +1.43 % |
| I2 Own second moment of component 2 | 0 to 0.05 m⁴ | 1,200,000 | 5.001e9 | 5e9 | +10% | +1.43 % |
| yn Neutral axis position | 0 to 1 m | 1,400,000 | 3.254e8 | 3.24e8 | +10% | +2.86 % |
| y1 Centroid of component 1 | 0 to 1 m | 900,000 | 1.629e8 | 1.62e8 | +10% | −6.79 % |
| y2 Centroid of component 2 | 0 to 1 m | 900,000 | 1.629e8 | 1.62e8 | +10% | +24.6 % |
| A1 Area of component 1 | 0 to 0.05 m² | 900,000 | 2.59e7 | 2.5e7 | +10% | +3.57 % |
| A2 Area of component 2 | 0 to 0.05 m² | 900,000 | 1.34e7 | 1.25e7 | +10% | +3.57 % |
| E2 Modulus of component 2 | 0 to 4e11 Pa | 700,000 | 3,500,000 | 2,800,000 | +10% | +5 % |
| E1 Modulus of component 1 | 0 to 4e11 Pa | 700,000 | 2,100,000 | 1,400,000 | +10% | +5 % |
| E3 Modulus of component 3 | 0 to 4e11 Pa | 1,400,000 | 1,400,000 | 0 | +10% | 0 % |
| I3 Own second moment of component 3 | 0 to 0.05 m⁴ | 1,400,000 | 1,400,000 | 0 | +10% | 0 % |
| A3 Area of component 3 | 0 to 0.05 m² | 1,400,000 | 1,400,000 | 0 | +10% | 0 % |
| y3 Centroid of component 3 | 0 to 1 m | 1,400,000 | 1,400,000 | 0 | +10% | 0 % |
Sweep of Neutral axis position
| Neutral axis position (m) | Effective flexural rigidity (N·m²) |
|---|---|
| 0 | 5,400,000 |
| 0.125 | 1,650,000 |
| 0.25 | 1.04e7 |
| 0.375 | 3.165e7 |
| 0.5 | 6.54e7 |
| 0.625 | 1.117e8 |
| 0.75 | 1.704e8 |
| 0.875 | 2.417e8 |
| 1 | 3.254e8 |
Sweep one variable
Variables and how each one contributes
Effective flexural rigidity
What the equation gives you. Everything else feeds it.
Modulus of component 1
The Young’s modulus of component 1, in Pa. Raises that component’s share of both terms in direct proportion. Raising it by 10 % (to 2.2×1011) moves the result by +5 %, from 1,400,000 to 1,470,000 N·m².
Modulus of component 2
The Young’s modulus of component 2, in Pa. Raises that component’s share of both terms in direct proportion. Raising it by 10 % (to 1.1×1011) moves the result by +5 %, from 1,400,000 to 1,470,000 N·m².
Modulus of component 3
The Young’s modulus of component 3, in Pa. Raises that component’s share of both terms in direct proportion. Raising it by 10 % (to 0) moves the result by +0 %, from 1,400,000 to 1,400,000 N·m².
Own second moment of component 1
The second moment of component 1 about its own centroidal axis, in m⁴. Adds its own stiffness; small for thin parts. Raising it by 10 % (to 1.1×10−6) moves the result by +1.43 %, from 1,400,000 to 1,420,000 N·m².
Own second moment of component 2
The second moment of component 2 about its own centroidal axis, in m⁴. Adds its own stiffness; small for thin parts. Raising it by 10 % (to 2.2×10−6) moves the result by +1.43 %, from 1,400,000 to 1,420,000 N·m².
Own second moment of component 3
The second moment of component 3 about its own centroidal axis, in m⁴. Adds its own stiffness; small for thin parts. Raising it by 10 % (to 0) moves the result by +0 %, from 1,400,000 to 1,400,000 N·m².
Area of component 1
The area of component 1, in m². Raises the parallel-axis term in proportion. Raising it by 10 % (to 0.0011) moves the result by +3.57 %, from 1,400,000 to 1,450,000 N·m².
Area of component 2
The area of component 2, in m². Raises the parallel-axis term in proportion. Raising it by 10 % (to 0.0022) moves the result by +3.57 %, from 1,400,000 to 1,450,000 N·m².
Area of component 3
The area of component 3, in m². Raises the parallel-axis term in proportion. Raising it by 10 % (to 0) moves the result by +0 %, from 1,400,000 to 1,400,000 N·m².
Centroid of component 1
The distance from the datum to the centroid of component 1, in m. Raises the stiffness with the square of its distance from the neutral axis. Raising it by 10 % (to 0.055) moves the result by −6.79 %, from 1,400,000 to 1,305,000 N·m².
Centroid of component 2
The distance from the datum to the centroid of component 2, in m. Raises the stiffness with the square of its distance from the neutral axis. Raising it by 10 % (to 0.165) moves the result by +24.6 %, from 1,400,000 to 1,745,000 N·m².
Centroid of component 3
The distance from the datum to the centroid of component 3, in m. Raises the stiffness with the square of its distance from the neutral axis. Raising it by 10 % (to 0) moves the result by +0 %, from 1,400,000 to 1,400,000 N·m².
Neutral axis position
The position of the elastic neutral axis, in m, from the same datum. Moving it away from the components increases the parallel-axis terms. Raising it by 10 % (to 0.11) moves the result by +2.86 %, from 1,400,000 to 1,440,000 N·m².
Worked example with the values above
- Own stiffness: ΣEI = 400,000 N·m².
- Parallel-axis term: ΣEA(y − yn)² = 1,000,000 N·m².
- (EI)eff = 400,000 + 1,000,000 = 1,400,000 N·m².
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