Question

The effective stiffness of a cantilever beam of length L and flexural rigidity EI subjected to a transverse tip load W is 

 

• A. \(\frac{3EI}{L^3}\)
• B. \(\frac{2EI}{L^3}\)
• C. \(\frac{L^3}{2EI}\)
• D. \(\frac{L^3}{3EI}\)

Hint:

It is essential to memorize the standard beam deflection formulas for competitive exams. For a cantilever beam with a tip load, the deflection is \(\frac{WL^3}{3EI}\). For a simply supported beam with a central load, it's \(\frac{WL^3}{48EI}\). Stiffness is always Force/Deflection. Knowing these will save you a lot of time.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Stiffness (\(k\)) is a measure of an object's resistance to deformation in response to an applied force. It is defined as the ratio of the applied force (\(F\) or \(W\)) to the resulting displacement or deflection (\(\delta\)).
Step 2: Key Formula or Approach:
The formula for stiffness is: \[ k = \frac{\text{Force}}{\text{Deflection}} = \frac{W}{\delta} \] For this problem, we need the standard formula for the maximum deflection (\(\delta\)) at the free end of a cantilever beam subjected to a point load (\(W\)) at its tip. The formula is: \[ \delta = \frac{WL^3}{3EI} \] where \(E\) is the modulus of elasticity, \(I\) is the area moment of inertia, and \(EI\) is the flexural rigidity.
Step 3: Detailed Calculation:
Substitute the expression for deflection (\(\delta\)) into the stiffness formula: \[ k = \frac{W}{\delta} = \frac{W}{\left(\frac{WL^3}{3EI}\right)} \] The load term \(W\) in the numerator and denominator cancels out: \[ k = \frac{1}{\left(\frac{L^3}{3EI}\right)} \] Simplifying the expression gives the stiffness: \[ k = \frac{3EI}{L^3} \] Step 4: Final Answer:
The effective stiffness of the cantilever beam is \(\frac{3EI}{L^3}\).
Step 5: Why This is Correct:
The derivation directly applies the definition of stiffness to the standard beam deflection formula for a cantilever with a tip load. Options (C) and (D) represent compliance (deflection per unit force), which is the inverse of stiffness. Option (A) is the correct expression for stiffness.