Beam Deflections: Solved Problems

Problem: A cantilever beam of length $L$ carries a concentrated load $P$ at the free end. Determine the maximum deflection and slope using the Double Integration Method.

Problem: A cantilever beam of length $L$ carries a uniformly distributed load $w$ over its entire length. Determine the maximum deflection using the Double Integration Method.

Problem: A simply supported beam of length $L$ carries a uniform load $w$ over its entire length. Determine the maximum deflection.

Problem: A simply supported beam of length $L$ carries a concentrated load $P$ at its center ($x = \frac{L}{2}$). Determine the maximum deflection using Double Integration.

Problem: A cantilever beam of length $L = 4.00 \text{ m}$ is subjected to a point load $P = 15.0 \text{ kN}$ at the free end. The flexural rigidity is $EI = 1.20 \times 10^4 \text{ kN}\cdot\text{m}^2$. Find the deflection at the free end using the Area-Moment Method.

Problem: A simply supported beam AB of length $L = 6.00 \text{ m}$ has a concentrated load $P = 24.0 \text{ kN}$ at midspan. $EI$ is constant. Find the slope at support A ($\theta_A$) using the Conjugate Beam Method.

Problem: A cantilever beam of length $L = 5.00 \text{ m}$ is subjected to a uniform load $w = 10.0 \text{ kN/m}$ over its entire length, and a concentrated point load $P = 20.0 \text{ kN}$ at the free end. The beam has a constant flexural rigidity $EI = 1.00 \times 10^4 \text{ kN}\cdot\text{m}^2$. Determine the maximum deflection at the free end.

Problem: A simply supported beam of length $L = 8.00 \text{ m}$ carries a load $P_1 = 30.0 \text{ kN}$ at $a = 2.00 \text{ m}$ from the left support, and $P_2 = 40.0 \text{ kN}$ at midspan ($4.00 \text{ m}$). Given $EI = 2.00 \times 10^4 \text{ kN}\cdot\text{m}^2$, find the deflection at midspan using superposition.

Problem: An engineer needs to design a floor beam for an office building. The beam satisfies all bending moment and shear force requirements (strength), but its calculated deflection under service loads exceeds the allowable limit of $L/360$. What are the most effective ways to solve this issue without changing the material?

Problem: A long-span steel beam is designed to support a heavy concrete floor slab. Under the dead load of the wet concrete, the beam is expected to deflect significantly. How can this be addressed during fabrication to ensure a level floor?

Problem: A continuous beam rests on three supports. Over time, the middle support settles downwards by a few millimeters due to soil consolidation. How does this affect the deflection and internal forces of the beam?

Problem: Compare the maximum deflection of a simply supported beam and a fixed-ended beam, both subjected to the same uniformly distributed load $w$ and having the same span $L$ and rigidity $EI$.