Shear and torsion design, 300x400 beam

Given: $b=300$ mm, $d=400$ mm, 4-20 mm bars ($A_{st}=1256.6$ mm$^2$), M25, Fe415. Loads at service state: $V=130$ kN, $T=45$ kN·m. Assume $D=450$ mm and effective cover 50 mm.

Step 1: Factored loads (service $\times$ 1.5)

$$V_u=1.5\times130=195.0\ \text{kN},\qquad T_u=1.5\times45=67.5\ \text{kN·m}$$

Step 2: Equivalent shear and check of section (IS 456 cl. 41.3)

$$V_e=V_u+1.6\frac{T_u}{b}=195.0+1.6\times\frac{67.5}{0.3}=555.0\ \text{kN},\qquad \tau_{ve}=\frac{555.0\times10^3}{300\times400}=4.62\ \text{N/mm}^2$$

$\tau_{ve}>\tau_{c,max}=3.1$ N/mm$^2$ (M25). The 300 mm wide section is therefore not adequate; the size must be increased (IS 456 cl. 41.3.1). Increase the width to $b=400$ mm:

$$V_e=195.0+1.6\times\frac{67.5}{0.4}=465.0\ \text{kN},\qquad \tau_{ve}=\frac{465.0\times10^3}{400\times400}=2.91\ \text{N/mm}^2<3.1\ \text{(OK)}$$

Step 3: Stirrups for the revised section. $b_1=400-100=300$ mm, $d_1=450-100=350$ mm. $p_t=\dfrac{100\times1256.6}{400\times400}=0.79\%$, $\tau_c=0.58$ N/mm$^2$ (Table 19, M25); $\tau_{ve}>\tau_c$.

$$\frac{A_{sv}}{s_v}=\frac{T_u}{b_1d_1(0.87f_y)}+\frac{V_u}{2.5d_1(0.87f_y)}=1.781+0.617=2.398\ \text{mm}^2/\text{mm}$$

The code minimum $\dfrac{(\tau_{ve}-\tau_c)\,b}{0.87f_y}=2.577$ mm$^2$/mm is larger and governs, so $A_{sv}/s_v=2.577$ mm$^2$/mm.

For 12 mm 2-legged closed stirrups ($A_{sv}=226.2$ mm$^2$): $s_v=88$ mm. Limits: $x_1=325$ mm, $(x_1+y_1)/4=175$ mm, 300 mm. Provide 12 mm 2-legged closed stirrups at 85 mm c/c.

If the given loads are taken as already factored (so $V_u=130$ kN, $T_u=45$ kN·m on the original 300 mm section): $V_e=370.0$ kN, $\tau_{ve}=3.083$ N/mm$^2$ (just below 3.1, OK), $p_t=1.05\%$, $\tau_c=0.65$ N/mm$^2$, $b_1=200$ mm, $d_1=350$ mm:

$$\frac{A_{sv}}{s_v}=1.781+0.411=2.192\ \text{mm}^2/\text{mm}\ \Rightarrow\ s_v=103\ \text{mm for 12 mm 2L}$$

giving 12 mm 2-legged stirrups at 100 mm c/c.

Answer: With the loads treated as service loads, the 300 mm width fails the $\tau_{c,max}$ check; with $b=400$ mm provide 12 mm 2L closed stirrups at 85 mm c/c (longitudinal steel to be provided for $M_{e1}$ in addition).