Problem: A rectangular beam ($b=300$, $h=500$) has $f'_c = 28 \text{ MPa}$. Calculate the cracking moment $M_{cr}$.

Problem: For the beam in Example 1, if the service moment $M_a = 60 \text{ kN-m}$ and the cracked moment of inertia $I_{cr} = 1.5 \times 10^9 \text{ mm}^4$, calculate the effective moment of inertia.

Problem: Calculate the long-term deflection multiplier $\lambda_{\Delta}$ for a duration of 5 years or more. Compression reinforcement ratio $\rho' = 0$.

Problem: A simply supported reinforced concrete beam ($400 \text{ mm}$ wide, $600 \text{ mm}$ effective depth) has an immediate deflection under sustained loads (dead load + 20% live load) of $\Delta_{sust} = 15 \text{ mm}$. The beam has a compression reinforcement ratio $\rho' = 0.005$. Calculate the additional long-term deflection after 5 years ($\xi = 2.0$) and the total sustained deflection.

Problem: A beam has a width $b = 300 \text{ mm}$ and overall depth $h = 500 \text{ mm}$. The tension reinforcement consists of 3-25mm bars located $50 \text{ mm}$ from the tension face (centroid to extreme fiber). Calculate the expected crack width $w$ under a service stress of $f_s = 250 \text{ MPa}$.

Problem: A long-span flat slab in an office building began to sag noticeably in the center over a 5-year period. Employees complained of sloping floors and cracking partition walls. The original design calculations showed the immediate deflection under live load was well within $L/360$. Analyze why the problem occurred.

Problem: A reinforced concrete bridge deck located near the ocean began showing severe spalling and visible red rust stains on its underside after only 10 years. Inspectors found that flexural cracks on the tension face were nearly $0.5 \text{ mm}$ wide, allowing saltwater to penetrate. Diagnose the serviceability design flaw.