Problem: A rectangular beam $300 \times 600 \text{ mm}$ is prestressed with a force $P_i = 1200 \text{ kN}$ at eccentricity $e = 100 \text{ mm}$. Calculate the stress at the bottom fiber immediately after transfer. Neglect self-weight.

Problem: For the beam in Example 1, assume 20% loss of prestress. The beam carries a service moment $M_{service} = 150 \text{ kN-m}$. Calculate the final stress at the bottom fiber.

Problem: A pre-tensioned beam has $E_c = 30,000 \text{ MPa}$ and $E_s = 200,000 \text{ MPa}$. The stress in concrete at the level of steel immediately after transfer is $f_{cgp} = 8 \text{ MPa}$. Calculate the loss due to elastic shortening.

Problem: A post-tensioned concrete beam is initially stressed to $P_i = 2000 \text{ kN}$. Calculate the final effective prestress force ($P_e$) if the total estimated losses (friction, anchorage slip, elastic shortening, creep, shrinkage, and steel relaxation) amount to 18% of the initial prestress.

Problem: Determine the required eccentricity ($e$) of a straight prestressing tendon ($P = 1500 \text{ kN}$) in a rectangular beam ($b=300 \text{ mm}, h=600 \text{ mm}$) to ensure that the stress at the bottom fiber is exactly zero when subjected only to the prestress force.

Problem: During the demolition of an older parking structure, an excavator operator inadvertently cut through a post-tensioning tendon in a floor slab. The tendon violently erupted from the concrete surface, whipping across the floor and causing severe damage. Explain the mechanics of this failure and the inherent dangers of unbonded post-tensioning.

Problem: A set of long-span precast, prestressed double-tee beams were installed in a warehouse roof. Initially, they had a designed upward camber of $25 \text{ mm}$. However, two years later, the building owner reported that the roof was severely uneven, with some beams cambering upwards by over $75 \text{ mm}$, tearing the roofing membrane. Diagnose the cause of this excessive upward deflection.