Deflection at the Free End of a Cantilever Beam with UDL Covering Half Span

To determine the deflection at the free end of a cantilever beam with a uniformly distributed load (UDL) covering only half the span from the fixed end using the area moment method, follow these steps:

Step 1: Identify the Given Parameters
- Length of the beam: L
- Distance of UDL from fixed end: L/2
- Magnitude of UDL: w
- Modulus of elasticity of the material: E
- Moment of inertia of the cross-section: I

Step 2: Determine the Support Reactions
- Since the UDL covers only half the span, the reaction at the fixed end, R1, can be determined by dividing the total load by 2: R1 = (w * L/2)/2 = wL/4
- The reaction at the free end, R2, can be determined by subtracting R1 from the total load: R2 = (w * L/2) - R1 = wL/2 - wL/4 = wL/4

Step 3: Calculate the Area Moment of the Loaded Part
- The area moment of the loaded part can be calculated by multiplying the load intensity by the square of the distance from the fixed end: M = (w * x^2)/2, where x is the distance from the fixed end.
- Since the UDL covers only half the span, the maximum distance from the fixed end is L/2.
- Integrate M with respect to x from 0 to L/2 to find the area moment: A = ∫(w * x^2)/2 dx = w/2 * ∫(x^2) dx from 0 to L/2 = w/2 * [x^3/3] from 0 to L/2 = w/2 * [(L/2)^3/3 - 0^3/3] = wL^3/48

Step 4: Calculate the Deflection at the Free End
- The deflection at the free end can be determined by dividing the area moment by the product of the modulus of elasticity and the moment of inertia: δ = A/(E * I) = (wL^3/48)/(E * I) = 7wL^4/384EI

Therefore, the deflection at the free end of the cantilever beam with a UDL covering half the span from the fixed end using the area moment method is 7wL^4/384EI.