A cantilever beam with a uniform flexural rigidity (EI = 200 × 10⁶ Nm²) is loaded with a concentrated force at its free end. The area of the bending moment diagram corresponding to the full length of the beam is 10000 Nm². The magnitude of the slope of the beam at its free end is ______ micro radian (round off to the nearest integer).

The correct answer is 50.

To find the magnitude of the slope at the free end of the cantilever beam, we can apply Mohr's First Theorem (also known as the Moment-Area Theorem).
According to this theorem, the change in slope between any two points on a beam is equal to the area of the bending moment (M) diagram between those two points divided by the flexural rigidity (EI) of the beam.

For a cantilever beam fixed at one end and free at the other, the slope at the fixed support is zero. Therefore, the magnitude of the slope at the free end is simply:

$$\mathrm{Slope}=\frac{\text{Area of the bending moment diagram}}{EI}$$

We are given the following parameters:
• Area of the bending moment diagram corresponding to the full length of the beam = 10000 Nm²
• Flexural rigidity (EI) = 200 × 10⁶ Nm²

Substituting these values into the formula:

$$\mathrm{Slope}=\frac{10000}{200\times {10}^6}$$

Calculating the value:

$$\mathrm{Slope}=50\times {10}^{-6}\text{radian}$$

Since 1 micro radian = 10^-6 radian, the slope of the beam at its free end is:
Slope = 50 micro radian.