Question

For a beam with deflection
\[ y = \frac{w}{48EI} (2x^4 - 3lx^3 + l^3x) \] find the non-dimensional location \(x/l\) at which deflection is maximum (round to 2 decimals).

Hint:

Convert the equation to nondimensional form to simplify cubic equations in beam theory.

Answer 1

Correct Answer: 0.41

To find max deflection, differentiate:
\[ \frac{dy}{dx} \propto 8x^3 - 9l x^2 + l^3 = 0. \] Let \( u = \frac{x}{l} \). Then: \[ 8u^3 - 9u^2 + 1 = 0. \] Solve cubic numerically. Checking values: - \(u=0.40\): \(8(0.064) - 9(0.16) + 1 = -0.08\)
- \(u=0.42\): \(8(0.074) - 9(0.1764) + 1 = 0.02\)
- \(u=0.41\): \(8(0.069) - 9(0.1681) + 1 = -0.02\)
Root is between 0.41 and 0.42. Interpolated value ≈ **0.414**. Rounded to 2 decimals: **0.41**.