Analysis and Design of Beams (Flexure) - Worked Examples

A rectangular beam has $b=300\,\text{mm}$, $d=500\,\text{mm}$, $A_s=1473\,\text{mm}^2$, $f'_c=28\,\text{MPa}$, and $f_y=420\,\text{MPa}$. Determine $M_n$, the strain classification, $\phi$, and $\phi M_n$.

A $300\,\text{mm}$ wide beam must resist $M_u=350\,\text{kN}\cdot\text{m}$. Use $f'_c=28\,\text{MPa}$, $f_y=420\,\text{MPa}$, and a trial effective depth $d=610\,\text{mm}$. A preliminary design gives $A_s=1656\,\text{mm}^2$. Verify the resulting capacity and strain state.

A beam has $b=350\,\text{mm}$, $d=600\,\text{mm}$, $d'=65\,\text{mm}$, $A_s=4000\,\text{mm}^2$, $A'_s=1000\,\text{mm}^2$, $f'_c=30\,\text{MPa}$, $f_y=420\,\text{MPa}$, and $E_s=200000\,\text{MPa}$. Determine the internally consistent neutral axis, reinforcement stresses, $M_n$, and $\phi M_n$.

A $300\,\text{mm}\times600\,\text{mm}$ beam must resist $M_u=600\,\text{kN}\cdot\text{m}$. Use $f'_c=30\,\text{MPa}$ and $f_y=420\,\text{MPa}$. Provide 3-20 mm compression bars at $d'=60\,\text{mm}$ and 6-28 mm tension bars in two equal layers whose bar-center depths from the bottom face are $64\,\text{mm}$ and $120\,\text{mm}$. Verify the selected reinforcement.

A rectangular beam has $b_w=400\,\text{mm}$, $d=600\,\text{mm}$, $f'_c=28\,\text{MPa}$, and $f_y=414\,\text{MPa}$. Determine $A_{s,\min}$.

A section has $\epsilon_t=0.0035$. Calculate $\phi$ first for $f_y=420\,\text{MPa}$ and then for $f_y=520\,\text{MPa}$, using $E_s=200000\,\text{MPa}$.

A T-beam has $b_f=800\,\text{mm}$, $h_f=100\,\text{mm}$, $b_w=300\,\text{mm}$, $d=500\,\text{mm}$, $A_s=4000\,\text{mm}^2$, $f'_c=25\,\text{MPa}$, and $f_y=420\,\text{MPa}$. Determine whether the web enters the compression block and calculate $\phi M_n$.

A T-beam has $b_f=1000\,\text{mm}$, $h_f=100\,\text{mm}$, $b_w=300\,\text{mm}$, $d=550\,\text{mm}$, $A_s=6500\,\text{mm}^2$, $f'_c=28\,\text{MPa}$, and $f_y=420\,\text{MPa}$. Assume the tension steel yields and determine $a$ and $\phi M_n$.

An interior T-beam has span $L=8.0\,\text{m}$, web width $b_w=300\,\text{mm}$, slab thickness $h_f=100\,\text{mm}$, and center-to-center beam spacing $s=1800\,\text{mm}$. Using the lesson limits, determine the effective flange width.

Compare two $1000\,\text{mm}$ deep beams. Beam A has clear span $l_n=3500\,\text{mm}$. Beam B has clear span $l_n=6000\,\text{mm}$ and no concentrated load close enough to a support to create a deep-beam region. Explain the appropriate classification.

A rectangular section has $b=300\,\text{mm}$, $d=450\,\text{mm}$, $A_s=4000\,\text{mm}^2$, $f'_c=28\,\text{MPa}$, and $f_y=420\,\text{MPa}$. Determine whether the tension steel actually yields and calculate the consistent design strength.

A preliminary calculation requires $A_s=2500\,\text{mm}^2$. The designer proposes five 25 mm bars, each with area $\pi(25)^2/4$, but the beam width only allows three bars in the first layer and two in a second layer. Explain what must be rechecked before accepting the design.