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Doubly reinforced rectangular beam

Compression and tension steel when the design moment exceeds the limit of a singly reinforced section. Reference: EN 1992-1-1 3.1.7, 5.6.3.

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# Materials
f_ck := 30 MPa
f_yk := 500 MPa
gamma_c := 1.5
gamma_s := 1.15
alpha_cc := 1.0
f_cd := alpha_cc*f_ck/gamma_c =
f_yd := f_yk/gamma_s =

# Section and action
b := 300 mm
d := 500 mm
d_2 := 50 mm
M_Ed := 480 kN*m

# Limit of the singly reinforced section
K := M_Ed/(b*d^2*f_ck) =
k_1 := 0.8*alpha_cc/gamma_c =
xi_lim := 0.45
K_lim := k_1/0.4*(1 - 0.4*xi_lim)*0.4*xi_lim =
K > K_lim =

# Steel
z_lim := d*(1 - 0.4*xi_lim) = mm
x := xi_lim*d =
d_2/x <= 0.38 =
A_s2 := (K - K_lim)*f_ck*b*d^2/(f_yd*(d - d_2)) = mm^2
A_s1 := K_lim*f_ck*b*d^2/(f_yd*z_lim) + A_s2 = mm^2

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