Design steps of flanged beam sections

T- and L-beams occur when a slab is cast monolithically with the beam, so the slab acts as the flange. Design steps (IS 456 cl. 23.1.2 and Annex G-2) are as follows.

1. Fix dimensions: assume web width $b_w$ (usually 230-300 mm) and depth $D\approx L/12$ to $L/10$; slab thickness $D_f$ is known from slab design; $d=D-$ cover.
2. Effective flange width $b_f$:
   • T-beam: $b_f=\dfrac{l_0}{6}+b_w+6D_f$
   • L-beam: $b_f=\dfrac{l_0}{12}+b_w+3D_f$
   • limited to the actual centre-to-centre spacing of beams. $l_0$ is the distance between points of zero moment (span for simply supported).
3. Loads: self weight of web and slab, finishes, live load; factor by 1.5.
4. Moment: $M_u=w_uL^2/8$ (or from analysis).
5. Locate neutral axis: find the moment capacity with $x_u=D_f$: $$M_{u,f}=0.36f_{ck}b_fD_f(d-0.42D_f)$$
   • If $M_u\le M_{u,f}$, NA lies in the flange: design as a rectangular beam of width $b_f$.
   • If $M_u>M_{u,f}$, NA lies in the web: go to step 6.
6. NA in the web:
   • If $D_f/d\le0.2$: $M_u=0.36f_{ck}b_wx_u(d-0.416x_u)+0.45f_{ck}(b_f-b_w)D_f(d-D_f/2)$.
   • If $D_f/d>0.2$: replace $D_f$ by $y_f=0.15x_u+0.65D_f$.
   • Solve for $x_u$ and check $x_u\le x_{u,max}$ (if not, increase section or add compression steel).
7. Tension steel: from $0.87f_yA_{st}=0.36f_{ck}b_wx_u+0.45f_{ck}(b_f-b_w)D_f$ (or $y_f$).
8. Checks: minimum steel $0.85b_wd/f_y$, maximum 4% of $b_wD$, shear, deflection ($L/d$ with modification factor and $b_w/b_f$ factor), cracking and detailing. Flange transverse steel is provided in the slab.