# Design for bending for flanged sections (concrete beam: EC2)

IF h_{f} < 0.1*d THEN treat the beam as rectangular.

h_{f} = MIN(h_{f,side1}, h_{f,side2})

where

h_{f,sidei} = the depth of the slab on side "
*i*" of the beam

Calculate the value of
*K* from;

K = M_{Ed}/(f_{ck}*b_{eff}*d^{2})

Calculate the lever arm,
*z* from;

z = MIN(0.5*d*[1 + (1 - 2*K/(η*α_{cc}/γ_{C}))^{0.5}], 0.95*d)

Calculate the depth of the rectangular stress block,
*λ*x* from;

λ*x = 2*(d-z)

` `

**IF λ*x ≤ h**
_{f}
**THEN** the rectangular compression block is wholly in the depth of the flange and the section can be designed as a rectangular section by setting b_{w} = b_{eff}.

` `

**IF λ*x > h**
_{f}
**THEN** the rectangular compression block extends into the rib of the flanged section and the following design method is to be used.

The design bending strength of the flange, M_{f} is given by;

M_{f} = f_{cd}*h_{f}*(b_{eff}-b_{w})*(d-0.5*h
_{f})

The area of reinforcement required to provide this bending strength, A_{sf,reqd} is given by;

A_{sf,reqd} = M_{f}/(f_{yd}*(d-0.5*h_{f}))

The remaining design moment, (M_{Ed}-M_{f}) is then taken by the rectangular beam section.

Calculate the value of
*K* from;

K = (M_{Ed}-M_{f})/(f_{ck}*b_{w}*d^{2})

Then calculate the limiting value of K, known as
*K'* from;

K' = (2*η*α_{cc} /γ_{C})*(1 - λ*(δ - k_{1})/(2*k_{2}))*( λ*(δ - k_{1})/(2*k_{2})) for f_{ck} ≤ 50 N/mm^{2}

K' = (2*η*α_{cc} /γ_{C})*(1 - λ*(δ - k_{3})/(2*k_{4}))*( λ*(δ - k_{3})/(2*k_{4})) for f_{ck} > 50 N/mm^{2}

` `

**IF K ≤ K' THEN** compression reinforcement is not required.

Calculate the lever arm,
*z* from;

z = MIN(0.5*d*[1 + (1 - 2*K/(η*α_{cc}/γ_{C}))^{0.5}], 0.95*d)

The area of tension reinforcement required is then given by;

A_{sr,reqd} = (M_{Ed}-M_{f})/(f_{yd}*z)

The total area of tension reinforcement required, A_{st,reqd} is then given by;

A_{st,reqd} = A_{sf,reqd}+A_{sr,reqd}

The depth to the neutral axis, x_{u} is given by;

x_{u} = 2*(d-z)/λ

` `

**IF K > K' THEN** compression reinforcement is required.

Calculate the depth to the neutral axis from;

x_{u} = d*(δ-k_{1})/k_{2} for f_{ck} ≤ 50 N/mm^{2}

x_{u} = d*(δ-k_{3})/k_{4} for f_{ck} > 50 N/mm^{2}

` `

Calculate the stress in the reinforcement from;

f_{sc} = MAX(E_{s}*ε_{cu3}*(1-(d_{2}/x_{u}), f_{yd})

where

d_{2} = the distance from the extreme fibre in compression to the c of g of the compression reinforcement

Calculate the limiting bending strength,
*M'* from;

M' =K'*f_{ck}*b_{w}*d^{2}

Calculate the lever arm from;

z = 0.5*d*[1 + (1 - 2*K'/(η*α_{cc}/γ_{C}))^{0.5}]

The area of compression reinforcement required, A_{s2,reqd} is given by;

A_{s2,reqd} = (M_{Ed}-M_{f}-M')/(f_{sc}*(d-d_{2}))

The area of tension reinforcement required, A_{sr,reqd} is given by;

A_{sr,reqd} = M'/(f_{yd}*z) + A_{s2,reqd}*f_{sc}/f_{yd}

The total area of tension reinforcement required, A_{st,reqd} is then given by;

A_{st,reqd} = A_{sf,reqd}+A_{sr,reqd}