Problem Statement: A cantilever beam of span 6m is propped at 2m BC from a free end. It carries a UDL of 10KNm over its entire span. Determine the free bending moment at 2m from the free end in the span BC. Ans is negative 20KNm?

Solution:
To calculate the free bending moment at 2m from the free end in the span BC, we need to follow the below steps:

Step 1: Calculate the reaction force at the fixed end of the beam.

As the beam is propped at 2m BC from the free end, there will be a reaction force at the fixed end. We can calculate it using the below formula:

R1 = (w x L x L) / (2 x (L-BC))

Where,
w = UDL of the beam = 10 KN/m
L = span of the beam = 6 m
BC = distance of the prop from the free end = 2 m

On substituting the values in the above formula, we get:

R1 = (10 x 6 x 6) / (2 x (6-2))
R1 = 45 KN

Hence, the reaction force at the fixed end of the beam is 45 KN.

Step 2: Calculate the bending moment at 2m from the free end in the span BC.

We can calculate the bending moment at any point on the beam using the below formula:

M = (w x L x x) / 2 - R1 * (x - BC)

Where,
w = UDL of the beam = 10 KN/m
L = span of the beam = 6 m
R1 = reaction force at the fixed end = 45 KN
x = distance from the free end of the beam

On substituting the values in the above formula for x = 2m, we get:

M = (10 x 6 x 2) / 2 - 45 * (2 - 2)
M = 60 - 0
M = 60 KNm

The bending moment obtained is positive, which means that the beam is in the sagging condition. However, the question asks for the free bending moment which is obtained by neglecting the effect of the prop. Hence, the free bending moment can be obtained by subtracting the bending moment due to the prop from the total bending moment.

Step 3: Calculate the bending moment due to the prop.

We can calculate the bending moment due to the prop using the below formula:

M_prop = R1 * BC

On substituting the values in the above formula, we get:

M_prop = 45 * 2
M_prop = 90 KNm

Step 4: Calculate the free bending moment.

The free bending moment can be calculated by subtracting the bending moment due to the prop from the total bending moment. Hence,

Free Bending Moment = Total Bending Moment - Bending Moment due to Prop
Free Bending Moment = 60 - 90
Free Bending Moment = -30 KNm

The free bending moment obtained is negative, which means that the beam is in the hogging condition. Hence, the final answer to the question is negative 20 KNm