Design for bending for rectangular sections (concrete beam: EC2)
Calculate the value of K from;  
K 

M_{Ed}/(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} 
where 

k_{i} 

moment redistribution factors 
δ 

moment redistribution ratio ( = 1.0 in the current release) 
γ_{C} 

the NDP partial safety factor for concrete 
α_{cc} 

coefficient to take account of long term effects on compressive strength of concrete 
λ 

0.8 for f_{ck} ≤ 50 N/mm^{2} 

0.8  (f_{ck}50)/400 for 50 < f_{ck} ≤ 90 N/mm^{2}  
η 

1.0 for f_{ck} ≤ 50 N/mm 

1.0  (f_{ck}50)/200 for 50 < f_{ck} ≤ 90 N/mm^{2}  


For design in accordance with UK NA, Irish NA, Malaysian NA, Singapore NA, Finnish NA and Norwegian NA;  
γ_{C} 

1.5 
α_{cc} 

0.85 


For design in accordance with EC2 Recommendations and Swedish NA;  
γ_{C} 

1.5 
α_{cc} 

1.0 


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_{st,reqd} 

M_{Ed}/(f_{yd}*z) 
where  
f_{yd} 

f_{yk}/γ_{S} 
γ_{S} 

the NDP partial safety factor for reinforcement 
The depth to the neutral axis, x_{u} is given by;  
x_{u} 

2*(dz)/λ 


For design in accordance with UK NA, EC2 Recommendations, Irish NA, Malaysian NA, Singapore NA, Finnish NA, Norwegian NA and Swedish NA;  
γ_{S} 

1.15 


IF K > K' THEN compression reinforcement is required. Calculate the depth to the neutral axis from; 

x_{u} 

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

d*(δk_{3})/k_{4} for f_{ck} > 50 N/mm^{2} 
Calculate the stress in the reinforcement from;  
f_{sc} 

MIN(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_{sc}*(dd_{2})) 
The area of tension reinforcement required, A_{st,reqd} is given by;  
A_{st,reqd} 

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