L-beam reinforcement for given moment

Given: $b_f=900$ mm, $D_f=100$ mm, $b_w=250$ mm, web depth below flange 500 mm; $M_u=615$ kN·m, $V_u=50$ kN (factored); M20, Fe500.

Assumptions: overall depth $D=500+100=600$ mm; effective cover 50 mm, so $d=550$ mm and $d'=50$ mm.

Step 1: Limiting moment of the L-section. $D_f/d=100/550=0.182\le0.2$, so the flange thickness $D_f$ is used directly. For Fe500, $x_{u,max}=0.46d=253.0$ mm $>D_f$ (neutral axis in the web).

$$\begin{aligned} M_{u,lim}&=0.36f_{ck}b_wx_{u,max}(d-0.416x_{u,max})+0.45f_{ck}(b_f-b_w)D_f\left(d-\frac{D_f}{2}\right)\\ &=202.54+292.50=495.04\ \text{kN·m} \end{aligned}$$

Since $M_u=615>M_{u,lim}=495.04$ kN·m, compression steel is required (doubly reinforced L-beam).

Step 2: Compression steel

$$M_{u2}=615-495.04=119.96\ \text{kN·m}$$ $$\varepsilon_{sc}=0.0035\left(1-\frac{50}{253.0}\right)=0.00281\ \Rightarrow\ f_{sc}=414.2\ \text{N/mm}^2\ (\text{Fe500 curve}),\quad f_{cc}=0.446\times20=8.92$$ $$A_{sc}=\frac{M_{u2}}{(f_{sc}-f_{cc})(d-d')}=\frac{119.96\times10^6}{(414.2-8.92)(550-50)}=592\ \text{mm}^2$$

Step 3: Tension steel

$$A_{st1}=\frac{0.36f_{ck}b_wx_{u,max}+0.45f_{ck}(b_f-b_w)D_f}{0.87f_y}=2392\ \text{mm}^2,\qquad A_{st2}=\frac{A_{sc}(f_{sc}-f_{cc})}{0.87f_y}=552\ \text{mm}^2$$ $$A_{st}=A_{st1}+A_{st2}=2943\ \text{mm}^2$$

Provide 6-25 mm dia ($2945$ mm$^2$, in layers) and 3-16 mm dia compression bars ($603$ mm$^2$).

Step 4: Shear (V$_u$ = 50 kN)

$$\tau_v=\frac{V_u}{b_wd}=\frac{50\times10^3}{250\times550}=0.364\ \text{N/mm}^2$$

$p_t=100A_{st}/(b_wd)=2.14\%$, $\tau_c=0.80$ N/mm$^2$ (IS 456 Table 19, M20). Since $\tau_v<\tau_c$, only minimum shear reinforcement is needed: $\dfrac{A_{sv}}{b\,s_v}\ge\dfrac{0.4}{0.87f_y}$.

For 8 mm 2-legged stirrups ($A_{sv}=100.5$ mm$^2$): $s_v\le\dfrac{0.87\times500\times100.5}{0.4\times250}=437$ mm; also $\le0.75d=412$ mm and 300 mm. Provide 8 mm 2-legged stirrups at 200 mm c/c (closer than 300 mm to support the compression bars).

Answer: $A_{st}=2943$ mm$^2$ (6-25 mm), $A_{sc}=592$ mm$^2$ (3-16 mm); shear: 8 mm 2-legged stirrups at 200 mm c/c.