Question

If two cap screws of the same material have an equivalent stiffness of threaded portion and first cap screw-threaded portion has a cross-sectional area twice of second cap screw-threaded portion. Then find the relationship between their threaded portion lengths (where k = stiffness, A = cross the sectional area, E = young's modulus, L= length, axial load = P. All for threaded portion of cap screw):
1. L₁ = 2L₂
2. L₁ = 4L₂
3. L₁ = 8L₂
4. L₁ = L₂

Answer 1

Correct Answer - Option 1 : L₁ = 2L₂

Concept:

Stiffness:

A force required to produce unit deformation is called stiffness. Denoted by K.

\(K=\frac {Force}{Δ L }\)

Where \(\Delta L=\frac {PL}{AE}\) Change in length 

Where, P = Load; L = Length; A = Area of cross-section; E = Modulus of Elasticity.

Calculation:

Given:

A₁ = 2A₂

As the material is the same, their stiffness must be the same.

∴ K₁ = K₂

\(\frac {Force}{\Delta L_1} = \frac {Force}{\Delta L_2}\)

\(\frac {P}{\frac {PL_1}{A_1E}} = \frac {P}{\frac {PL_2}{A_2E}}\)

\(\frac {A_1}{L_1} = \frac {A_2}{L_2} \implies \frac {2A_2}{L_1} = \frac {A_2}{L_2}\)

∴ L1 = 2L2