Final answer:
The maximum displacement and rotation in a cantilevered beam subject to a distributed and concentrated load can be found using formulas that incorporate the modulus of elasticity, moment of inertia, and load details. The principle of superposition is used to sum the effects of each load.
Step-by-step explanation:
The question pertains to the deflection and rotation of a cantilevered beam subject to a distributed load over half its length and a concentrated load at the free end. To find the maximum displacement and rotation due to both loads, we use the principle of superposition, where the total deflection is the sum of the deflection due to each load acting alone.
For a distributed load q acting on a length L/2 from the fixed end, the maximum displacement (δmax) at the free end of a cantilever beam can be calculated using the formula:
δmax = (q*L^4)/(8*E*I)
Where E is the modulus of elasticity and I is the moment of inertia of the beam's cross-section. The rotation (θ) at the fixed end can be calculated as:
θ = (q*L^3)/(6*E*I)
For the concentrated load P at the free end, the maximum displacement is given by:
δmax = (P*L^3)/(3*E*I)
And the rotation at the fixed end is:
θ = (P*L^2)/(2*E*I)
As given, P = qL/2, substituting P in the earlier formulas allows us to find the deflection and rotation due to P. Adding both deflections and rotations gives us the total maximum displacement and rotation under the simultaneous action of both loads. The results should be verified again given beam specifications such as material, cross-section, and load specifics.