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In any normally loaded beam (ie excluding torsion), the loading is either tensile or compressive.

The loading on a structural beam generally has the top being compression loaded and the bottom being stretched (try drawing the forces for clarity if needed) the "I" creates a stabilising force, preventing the top and bottom from shearing away from each other.

On any beam, the actual internal shear is generally very low, therefore (within obvious limits) the vertical web can be quite small, but the flanges need to be increased depending on the load as this is where the forces are realised.

Effectively, the I beam is strongest (per unit of mass) as the strengthening - ie material - only appears where it is required, thus eliminating unnecessary weight.

Taking this one step farther...a skeleton structure for a bridge is basically a non-solid I beam....the triangles on the joining spars being the effect of the central web.

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There are mainly two reasons, one is that the second moment of inertia for the I-Beam is more that any other shape and hence the internal shearing effect reduce drastically between the top and bottom side because of the compression and stretching respectively. The second reason is cost effectiveness. the Cross section area for the I-Beam is more than any other shape like Square , rectangular , triangular, round etc. 
One of the most importance is to mention here that in practical world the engineering is use for cost effectiveness and hence when it comes to the beam section the engineers concluded the most effective way to design a safe beam structure  is I beam

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Simply put, the center web creates a relatively large separation between the top and bottom webs which is effective in offsetting the bending moment as the beam is loaded.

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I regret to point out that ianbates1 has led us on a wild goose chase with his mathematical presentation.

1. The universal gravitational law, describing the force of attraction between two masses, is not relevant to a discussion of the bending of an I-section.

2. The resulting expression for a, the acceleration, is irrelevant as the beam bending problem is usually applied to static structures.

3. The bending deflection expression, FL^3/(3EI) is relevant for a uniform cantilever tip deflection only, but has no general applicability.

4. The expression for the area MOI, bh^3/12, is for a rectangular section as shown, which has no significance for the I-section.

5. The stress-strain relation through Young's modulus explains nothing about the I-section.

Other than that, yeah, sure .... why not?

DrD

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