Flitched Beam Calculator: Timber Joist with Steel Plates, Moment of Resistance by the Modular Ratio

How this calculator works

Enter the size of the timber joist, the thickness, depth and number of steel plates, the moduli of the steel and the timber and the stress allowed in the timber.

The calculator finds the modular ratio m = Es over Ew, the second moment of area of each part and the equivalent I = m Is + Iw of the composite section.

It gives the stress in the steel (m times that in the timber at the same distance), the moment of resistance M = sigma I over y, and the share carried by the steel and the timber.

Formulas used

• m = \frac{E_s}{E_w}
• I = m I_s + I_w
• \sigma_s = m\,\sigma_w\frac{y_s}{y_w}
• M = \frac{\sigma_w}{y_w}I

Worked example

With the default numbers in the calculator above:

1. Modular ratio: how many times stiffer the steel is than the timber: 20.00
2. Second moment of area of the timber about the neutral axis: 6666.67 cm4
3. Second moment of area of the steel plates about the neutral axis: 1333.33 cm4
4. Equivalent second moment of area of the composite section in timber: 33333.33 cm4 (5.00 times the timber alone)
5. Stress in the steel at its outer edge, when the timber is at its limit (the steel carries m times the stress of timber at the same distance): 140.00 MPa
6. Moment of resistance of the composite section (limited by the timber): 23.33 kN m (23333.33 N m)
7. Moment the timber could carry alone, for comparison: 4.67 kN m, so the plates raise the capacity 5.00 times
8. Share of the bending moment carried by the steel and by the timber: Steel 80.00 percent, timber 20.00 percent

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: Where are the steel plates? On the top and the bottom of the joist (one plate each); Width of each plate (plates on the top and bottom) 150 mm; Width of the timber joist 150 mm; Depth of the timber joist 200 mm; Thickness of one steel plate 12.5 mm; Elastic modulus of the steel 200000 MPa; Elastic modulus of the timber 10000 MPa; Maximum stress allowed in the timber 6 MPa.

1. Modular ratio: how many times stiffer the steel is than the timber: 20.00
2. Second moment of area of the timber about the neutral axis: 10000.00 cm4
3. Second moment of area of the two steel plates about the neutral axis: 4238.28 cm4
4. Equivalent second moment of area of the composite section in timber: 94765.63 cm4 (9.48 times the timber alone)
5. Stress in the steel where it touches the timber (same strain as the timber surface): 120.00 MPa
6. Stress in the steel at its outer edge, when the timber is at its limit (the steel carries m times the stress of timber at the same distance): 135.00 MPa
7. Moment of resistance of the composite section (limited by the timber): 56.86 kN m (56859.38 N m)
8. Moment the timber could carry alone, for comparison: 6.00 kN m, so the plates raise the capacity 9.48 times
9. Share of the bending moment carried by the steel and by the timber: Steel 89.45 percent, timber 10.55 percent

Common questions

Related calculators

• Timber Beam Calculator to AS 1720.1: Bending, Shear and Deflection of a Joist, Rafter or Purlin
• Beam Analysis Calculator with Shear Force and Bending Moment Diagrams
• Section Properties Calculator: Area, Second Moment and Section Modulus

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: AS 1720.1:2010 Timber structures; AS 4100 Steel structures; Rajput, Strength of Materials (composite beams, equivalent section).

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