Shear Stress Calculation for RC Beam Support

This solution explains how to calculate the nominal shear stress at the support section of a simply supported reinforced concrete (RC) beam under a concentrated load.

Beam Details and Loading

• Beam Width, $b$ = 250 mm
• Effective Depth, $d$ = 480 mm
• Concentrated Load, $P$ = 120 kN
• Load Position: Mid-span
• Assumption: Beam's self-weight is ignored.

Calculating Maximum Shear Force ($V_{max}$)

For a simply supported beam carrying a concentrated load at its center, the maximum shear force occurs at the supports. This maximum shear force is half the total load.

Formula: $ V_{max} = \frac{P}{2} $ Calculation: $ V_{max} = \frac{120 \text{ kN}}{2} = 60 \text{ kN} $ Convert kN to N: $ V_{max} = 60 \times 10^3 \text{ N} $

Determining Nominal Shear Stress ($\tau_v$)

Nominal shear stress ($\tau_v$) is the shear force acting on the cross-section divided by the section's area (width times effective depth).

Formula: $ \tau_v = \frac{V_{max}}{b \times d} $ Calculation: $ \tau_v = \frac{60 \times 10^3 \text{ N}}{(250 \text{ mm}) \times (480 \text{ mm})} $ $ \tau_v = \frac{60000}{120000} \text{ N/mm}^2 $ $ \tau_v = 0.5 \text{ N/mm}^2 $ Since $1 \text{ N/mm}^2 = 1 \text{ MPa}$: $ \tau_v = 0.5 \text{ MPa} $

Final Result

The nominal shear stress at the support section is 0.5 MPa.