Design of member for bending, shear, torsion

A member under combined bending, shear and torsion is designed in IS 456:2000 (cl. 41) by converting the torsion into an equivalent shear and an equivalent bending moment, and then designing for flexure and shear in the usual way.

Step-by-step procedure

1. Analysis and loads: find the factored $M_u$, $V_u$ and $T_u$ at the critical sections.
2. Section size: assume $b$, $D$ and cover; find $d$.
3. Equivalent shear: $$V_e=V_u+1.6\frac{T_u}{b}$$ Equivalent nominal shear stress $\tau_{ve}=\dfrac{V_e}{bd}$.
4. Check section size: $\tau_{ve}\le\tau_{c,max}$ (Table 20: 2.8 for M20, 3.1 for M25). If not, increase the section.
5. Equivalent moments: $$M_t=T_u\,\frac{1+D/b}{1.7},\qquad M_{e1}=M_u+M_t$$ $M_{e1}$ is resisted by longitudinal steel on the flexural tension face.
6. Compression face steel: if $M_t>M_u$, an extra moment $M_{e2}=M_t-M_u$ acts in the opposite sense and needs extra longitudinal steel on the compression face (designed as tension steel for $M_{e2}$).
7. Longitudinal steel from $M_{e1}$ (and $M_{e2}$) using the flexure formula: $$M_e=0.87f_yA_{st}d\left(1-\frac{A_{st}f_y}{bdf_{ck}}\right)$$ If $M_{e1}>M_{u,lim}$, the section is doubly reinforced.
8. Transverse reinforcement: find $\tau_c$ from $p_t$ (Table 19). If $\tau_{ve}>\tau_c$, provide closed stirrups: $$\frac{A_{sv}}{s_v}=\frac{T_u}{b_1d_1(0.87f_y)}+\frac{V_u}{2.5d_1(0.87f_y)}\ \ \ge\ \frac{(\tau_{ve}-\tau_c)b}{0.87f_y}$$ $b_1$, $d_1$ = centre-to-centre distances between the corner bars in the width and depth directions.
9. Spacing limits: $s_v\le x_1$, $\le(x_1+y_1)/4$ and $\le300$ mm, where $x_1$ and $y_1$ are the short and long dimensions of the stirrup.
10. Detailing (cl. 26.5.1.7): stirrups must be closed, with 135$^\circ$ hooks, and there must be at least one longitudinal bar in each corner of the stirrups. When a cross-sectional dimension exceeds 450 mm, provide additional longitudinal (side face) bars as in cl. 26.5.1.3: a total of 0.1% of the web area, divided equally between the two faces, at a spacing of not more than 300 mm or the web thickness.
11. Check deflection, cracking and development length.