Stress and Strain PHYSICS

Question

A metal slab is on top of a concrete column which has a height of 6.5 meters and a diameter of 1.50 meters. If

the metal slab has a total mass of 60,500 kg, determine

(a) stress,

(b) strain, and

(c) decrease in the height of the concrete column due to the metal slab.

Assume that g = 9.8 m/s2 and Young’s modulus for concrete = 20 x 109 N/m2.

Answer 1

Let

h = 6.5 m be the height of the column,

d = 1.5 m be the diameter of the column,

m = 60,500 kg be the mass of the slab,

g = 9.8 m/s² be gravitational acceleration,

E = 20×10⁹ N/m² be Young's module.

The force, due to gravity, compressing the column is

F = mg

The horizontal cross section of the column is a circular disk with area

S = π(d/2)² = πd²/4

The stress is defined as

σ = F/S = 4mg/πd²

The strain is in linear relationship with the stress

ε = σ/E

By definition of strain, the deformation is

Δh = hε

Substituting actual values

(a) σ = 4×60500×9.8/(π×1.5²) = 335513 N/m²

(b) ε = 335513/20×10⁹ = 16.7756×10^-6

(c) Δh = hε = 6.5×16.7756×10^-6 = 109.0416×10^-6 m

That is about a tenth of a mm.