Problem: A rectangular beam has $b = 300 \text{ mm}$ and $d = 500 \text{ mm}$. It is reinforced with 3-25mm bars ($A_s = 1473 \text{ mm}^2$). $f'_c = 28 \text{ MPa}$, $f_y = 420 \text{ MPa}$. Determine the design moment capacity $\phi M_n$.

Problem: Design a rectangular beam to carry a factored moment $M_u = 350 \text{ kN-m}$. Use $f'_c = 28 \text{ MPa}$, $f_y = 420 \text{ MPa}$. Limit width $b = 300 \text{ mm}$. Determine required effective depth $d$ and steel area $A_s$ using $\rho = 0.00905$.

Problem: A beam with $b=350 \text{ mm}$, $d=600 \text{ mm}$, $d'=65 \text{ mm}$ has tension steel $A_s = 4000 \text{ mm}^2$ (8-25mm) and compression steel $A'_s = 1000 \text{ mm}^2$ (2-25mm). $f'_c = 30 \text{ MPa}$, $f_y = 420 \text{ MPa}$. Calculate $\phi M_n$.

Problem: A beam section is limited to $b=300 \text{ mm}$ and $h=500 \text{ mm}$ ($d=440 \text{ mm}$, $d'=60 \text{ mm}$). It must carry $M_u = 600 \text{ kN-m}$. $f'_c = 30 \text{ MPa}$, $f_y = 420 \text{ MPa}$. Determine required reinforcement.

Problem: A T-beam has flange width $b_f = 800 \text{ mm}$, flange thickness $t_f = 100 \text{ mm}$, web width $b_w = 300 \text{ mm}$, and $d = 500 \text{ mm}$. $A_s = 4000 \text{ mm}^2$. $f'_c = 25 \text{ MPa}$, $f_y = 420 \text{ MPa}$. Find $\phi M_n$.

Problem: A site engineer noticed a beam was deflecting significantly under its self-weight before the concrete had fully cured. To "strengthen" the remaining identical beams, they doubled the tension reinforcement ($A_s$) without consulting the structural designer. Analyze the potential consequences of this action on the beam's failure mode.

Problem: A large decorative cantilever beam was designed with very light reinforcement because the calculated factored moment was very small. During a mild windstorm, the beam suddenly cracked and collapsed. Analyze why the code mandates minimum reinforcement ($A_{s,min}$).

Problem: A rectangular beam has an extreme tension steel strain of $\epsilon_t = 0.0035$ at nominal strength. The steel used is Grade 420 ($f_y = 420 \text{ MPa}$). Calculate the appropriate strength reduction factor ($\phi$) for flexure.

Problem: A singly reinforced beam has a width $b_w = 400 \text{ mm}$ and effective depth $d = 600 \text{ mm}$. Concrete strength $f'_c = 28 \text{ MPa}$ and steel strength $f_y = 414 \text{ MPa}$. Calculate the absolute minimum required tension steel area ($A_{s,min}$).