A new aquarium plans a large fish tank with a water depth of 3 meters. One of the walls will be constructed from Plexiglas and supported by vertical I-beams. The beams will be spaced every 2 meters. The water pressure will cause a distributed load on the beams. The beams can be modeled as a cantilever beams since the base is fixed and the top end is free. The minimum weight S type I-beam needs to be specified using the beams listed in the appendix.

Distributed Load on Beam

The water will cause a pressure on the wall which is a linear function of the depth. The pressure on the wall will be

     p = h g ρ

where ρ is the water density and g is the gravitational constant, 9.81 m/s². The maximum pressure will be at the bottom, h = 3 m,

     p = (3 m) (9.81 m/s²) (1 g/cm³)

        = (3 m) (9.81 kN/m³) = 29.43 kN/m²

The pressure now needs to be converted to a distributed load on a typical vertical beam. There is one beam for every 2 meters, so the maximum distributed load at the bottom is

     w_max = (29.43 kN/m² )(2 m) = 58.86 kN/m

Free Body Diagram of Beam

The maximum moment for a cantilever beam is at the base. The distributed load has an equivalent load of

     F = 0.5 (58.86 kN/m)(3 m ) = 88.29 kN

Summing moments at the base gives,

     ΣM_base = 0

     M_max - (88.29 kN)(1 m) = 0

     M_max = 88.29 kN-m

Now that the moment is known and the yield stress is given, the section modulus can be determined. There is a factor of safety of 3.0 that will reduce the allowable stress.

     S_required = M/σ = 88.29 kN-m / (250,000/3 kN/m² )

        = 1,059 × 10³ mm³

Looking at the appendix listing of all S type I-beams (note, SI units) give the lightest beam with at least 1,059 × 10³ mm³ as

     S380 x 74

This beam type has a section modulus of 1,060 × 10³ mm³ so it will just barely work.