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Simply supported beam: bending and shear

Design actions, bending steel, shear resistance without links and minimum links for a simply supported beam. Reference: EN 1992-1-1 6.1, 6.2, 9.2.

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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 =

# Beam and actions
L := 7 m
b := 300 mm
b_w := b =
h := 600 mm
d := 545 mm
g_k := 22 kN/m
q_k := 12 kN/m
w_Ed := 1.35*g_k + 1.5*q_k =
M_Ed := w_Ed*L^2/8 = kN*m
V_Ed := w_Ed*L/2 = kN

# Bending
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 =
z := min(d*(0.5 + sqrt(0.25 - 0.4*K/k_1)), 0.95*d) = mm
A_s := M_Ed/(f_yd*z) = mm^2

# Shear
A_sl := A_s =
k := min(1 + sqrt(200 mm/d), 2) =
rho_l := min(A_sl/(b_w*d), 0.02) =
C_Rdc := 0.18/gamma_c =
v_min := 0.035*k^1.5*sqrt(f_ck/(1 MPa))*1 MPa =
V_Rdc := max(C_Rdc*k*(100*rho_l*f_ck/(1 MPa))^(1/3)*1 MPa, v_min)*b_w*d = kN
V_Ed > V_Rdc =
rho_w.min := 0.08*sqrt(f_ck/(1 MPa))/(f_yk/(1 MPa)) =

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