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Doubly reinforced beam: compression and tension steel

Design of a rectangular beam whose moment exceeds the limiting moment, with the stress in the compression steel from the Fe 415 design curve. Reference: IS 456:2000 Cl. 38.1, Annex G (G-1.2).

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# Data
f_ck := 20 MPa
f_y := 415 MPa
b := 250 mm
d := 500 mm
d_c := 50 mm
M_u := 250 kN*m

# Limiting values
x_u.max := 0.48*d =
M_u.lim := 0.36*f_ck*b*x_u.max*(d - 0.42*x_u.max) = kN*m
M_u > M_u.lim =
M_u2 := M_u - M_u.lim = kN*m

# Stress in the compression steel
epsilon_sc := 0.0035*(1 - d_c/x_u.max) =
f_sc := interp(epsilon_sc, [0, 0.00144, 0.00163, 0.00192, 0.00241, 0.00276, 0.0038, 0.1], [0, 288.7, 306.7, 324.8, 342.8, 351.8, 360.9, 360.9])*1 MPa =

# Steel areas
A_sc := M_u2/((f_sc - 0.446*f_ck)*(d - d_c)) = mm^2
A_st1 := M_u.lim/(0.87*f_y*(d - 0.42*x_u.max)) = mm^2
A_st2 := A_sc*(f_sc - 0.446*f_ck)/(0.87*f_y) = mm^2
A_st := A_st1 + A_st2 = mm^2

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