Hollow Steel Strut



A hollow steel strut is a structural member that is commonly used to support loads in construction. It is often pin-jointed at the ends, meaning that it is free to rotate at these points. The behavior of the strut under load can be analyzed using different formulas, such as Euler's formula and Rankine's formula.

Euler's Formula



Euler's formula is used to determine the critical load at which a long, slender strut will buckle. It is based on the assumption that the strut is perfectly straight and has a uniform cross-section. The formula is given as:

P(cr) = (π^2 × E × I) / (L^2)

Where:
- P(cr) is the critical load
- E is the modulus of elasticity of the material
- I is the moment of inertia of the cross-section
- L is the length of the strut

Rankine's Formula



Rankine's formula is used to determine the safe load that a strut can carry without experiencing excessive deformation. It takes into account the yield stress of the material. The formula is given as:

P(rankine) = (σ(yield) × A) / (1.5)

Where:
- P(rankine) is the safe load
- σ(yield) is the yield stress of the material
- A is the cross-sectional area of the strut

Crippling Load Comparison



To compare the crippling load given by Euler's and Rankine's formulae, we need to calculate the moment of inertia and the cross-sectional area of the strut.

The moment of inertia (I) for a hollow circular section can be calculated using the formula:

I = (π/64) × (D^4 - d^4)

Where:
- D is the outer diameter of the strut
- d is the inner diameter of the strut

The cross-sectional area (A) of the strut can be calculated using the formula:

A = π × (D^2 - d^2) / 4

Using the given dimensions of the hollow steel strut (outer diameter = 50 mm, thickness = 5 mm), we can calculate the moment of inertia and the cross-sectional area.

D = 50 mm
d = 50 mm - 2 × 5 mm = 40 mm

I = (π/64) × ((50 mm)^4 - (40 mm)^4)
A = π × ((50 mm)^2 - (40 mm)^2) / 4

Substituting these values into Euler's and Rankine's formulae, we can calculate the crippling load.

P(cr) = (π^2 × 200 GPa × I) / (3 m)^2
P(rankine) = (350 N/mm^2 × A) / 1.5

Minimum L/R Ratio for Euler's Formula



Euler's formula is applicable for long, slender struts. To determine the minimum L/R ratio for which Euler's formula applies, we need to consider the slenderness ratio (L/R) of the strut.

The slenderness ratio (L/R) is the ratio of the length of the strut (L) to the radius of gyration (R). The radius of gyration can be calculated using the formula:

R =