Why does doubling the thickness of a steel bar make it eight times stiffer in bending?

Bending stiffness is E × I, where I is the second moment of area of the cross-section. For a rectangle I = b × h³ / 12, with h measured in the direction of the load. Width enters to the first power and depth to the third, so doubling the width doubles the stiffness and doubling the thickness multiplies it by eight.

The cube has a physical reason. In bending, the strain in a fibre is proportional to its distance from the neutral axis, so the stress is too, and that stress acts at a lever arm equal to the same distance. Material twice as far out is stretched twice as much and has twice the leverage, and a deeper section also has more material out there.

For a simply supported bar with a central load, deflection is δ = F × L³ / (48 × E × I), with E = 200,000 MPa for steel.

• 50 × 5 flat: I = 50 × 5³ / 12 = 521 mm⁴, δ = 200 × 600³ / (48 × 200,000 × 521) = 8.6 mm.
• 50 × 10 flat: I = 4167 mm⁴, δ = 1.1 mm, for twice the weight.
• 100 × 5 flat: I = 1042 mm⁴, δ = 4.3 mm, also twice the weight.
• 50 × 5 on edge: I = 5 × 50³ / 12 = 52,083 mm⁴, δ = 0.09 mm, for the same weight.

On edge the bar is 100 times stiffer, but a thin strip loaded that way tends to twist and fall over sideways, so it needs lateral restraint.

A stronger steel does not help with sag. The elastic modulus is practically the same, 200 to 210 GPa, for mild steel, high-strength steel and stainless steel. Grade changes the yield strength, which is how much load it takes to bend the bar permanently, not how much it deflects before that.

Check strength separately. The maximum moment is F × L / 4 = 200 × 600 / 4 = 30,000 N·mm, and the section modulus is Z = b × h² / 6 = 208 mm³ for the 5 mm flat, so the bending stress is 30,000 / 208 = 144 MPa. That is under the 250 MPa yield of ordinary mild steel but without much margin. Note that strength grows with h squared while stiffness grows with h cubed.

The general lesson is to move material away from the neutral axis instead of adding more of it. That is why beams are I-sections, tubes and channels. For a shelf-type part, folding a 20 mm flange along each long edge of thin sheet, or welding a small angle or box section underneath, gives more stiffness than doubling the thickness, at a fraction of the weight.

As a comparison, a 25 × 25 mm square tube with a 2 mm wall has I of roughly 16,000 mm⁴ (a little less with rounded corners), about thirty times the 50 × 5 flat, with a smaller cross-sectional area: 184 mm² against 250 mm².

Do not overlook the span and the supports, which are as powerful as the section. Deflection goes with L cubed: shortening the span from 600 mm to 480 mm, 20 percent, cuts the sag almost in half (0.8³ = 0.51). Clamping both ends rigidly instead of resting them on supports changes the formula to F × L³ / (192 × E × I), four times stiffer for the same bar.

Real clamps are never perfectly rigid, so expect something between the two cases, but a bolted or welded end connection is often the cheapest stiffness you can buy.