# Chapter 2:Doubly Reinforced Sections¶

## Ex2.1:pg-47¶

In [1]:
import math
b=200.0 #width, in mm
D=400 #overall depth, in mm
m=18.66 #modular ratio
Ast=4*0.785*22**2 #four 22 mm dia bars at bottom, in sq mm
Asc=3*0.785*20**2 #three 20 mm dia bars at top, in sq mm
bottom_cover=30 #in mm
top_cover=25 #in mm
d=D-bottom_cover #effective depth, in mm
#to find x using b(x**2)/2 + (1.5m-1)Asc(x-d')=mAst(d-x), which becomes of the form px**2+qx+r=0
p=b/2
q=(1.5*m-1)*Asc+m*Ast
r=-(1.5*m-1)*Asc*top_cover-m*Ast*d
x=(-q+math.sqrt(q**2-4*p*r))/(2*p) #in mm
print "Depth of neutral axis=",x,"mm"

Depth of neutral axis= 159.56861799 mm


## Ex2.2:pg-48¶

In [2]:
import math
b=280.0 #width, in mm
D=540.0 #overall depth, in mm
Ast=5*0.785*22**2 #five 22 mm dia bars on tension side, in sq mm
Asc=4*0.785*20**2 #four 20 mm dia bars on compression side, in sq mm
bottom_cover=40 #in mm
top_cover=30.0 #in mm
sigma_cbc=5.0#in MPa
sigma_st=140.0 #in MPa
m=18.66 #modular ratio
d=D-bottom_cover #effective depth, in mm
#to find critical depth of neutral axis
Xc=d/(1+sigma_st/(m*sigma_cbc)) #in mm
#to find x using b(x**2)/2 + (1.5m-1)Asc(x-d')=mAst(d-x), which becomes of the form px**2+qx+r=0
p=b/2
q=(1.5*m-1)*Asc+m*Ast
r=-(1.5*m-1)*Asc*top_cover-m*Ast*d
x=(-q+math.sqrt(q**2-4*p*r))/(2*p) #in mm
#as x<Xc, beam is under-reinforced
sigma_cbc=(sigma_st/m)*x/(d-x) #in MPa
sigma_cbc_dash=sigma_cbc*(x-top_cover)/x #in MPa
sigma_sc=1.5*m*sigma_cbc_dash #in MPa
#stress in compression steel is found to be less than its permissible limit of 130 N/mm**2
Mr=b*x*sigma_cbc*(d-x/3)/2+(1.5*m-1)*Asc*sigma_cbc_dash*(d-top_cover) #in N-mm
print "Moment of resistance of the beam=",Mr/10**6,"kN-m"

Moment of resistance of the beam= 120.506005456 kN-m


## Ex2.3:pg-48¶

In [3]:
import math
b=300.0 #width, in mm
d=600.0 #effective depth, in mm
Ast=1256 #in sq mm
Asc=1256.0 #in sq mm
top_cover=30.0 #in mm
sigma_cbc=7.0 #in MPa
sigma_st=190.0 #in MPa
m=13.33 #modular ratio
#using elastic theory method
#to find critical depth of neutral axis
Xc=d/(1+sigma_st/(m*sigma_cbc)) #in mm
#to find x using b(x**2)/2 + (1.5m-1)Asc(x-d')=mAst(d-x), which becomes of the form px**2+qx+r=0
p=b/2
q=(1.5*m-1)*Asc+m*Ast
r=-(1.5*m-1)*Asc*top_cover-m*Ast*d
x=(-q+math.sqrt(q**2-4*p*r))/(2.0*p) #in mm
#as x<Xc, beam is under-reinforced
sigma_cbc=(sigma_st/m)*x/(d-x) #in MPa
sigma_cbc_dash=sigma_cbc*(x-top_cover)/x #in MPa
sigma_sc=1.5*m*sigma_cbc_dash #in MPa
#stress in compression steel is found to be less than its permissible limit of 130 N/mm**2
Mr1=b*x*sigma_cbc*(d-x/3)/2+(1.5*m-1)*Asc*sigma_cbc_dash*(d-top_cover) #in N-mm
#using steel beam theory method
Mr2=Ast*sigma_st*(d-top_cover) #in N-mm
print "Moment of resistance of the beam using elastic theory method",Mr1/10**6,"kN-m\nMoment of resistance of the beam using elastic theory method=",Mr2/10**6,"kN-m"

Moment of resistance of the beam using elastic theory method 132.70169173 kN-m
Moment of resistance of the beam using elastic theory method= 136.0248 kN-m


## Ex2.4:pg-49¶

In [4]:
import math
b=250.0 #width, in mm
D=550.0 #overall depth, in mm
Ast=4*0.785*25**2 #four 25 mm dia bars on tension side, in sq mm
Asc=3*0.785*22**2 #three 22 mm dia bars on compression side, in sq mm
bottom_cover=50.0 #in mm
top_cover=30 #in mm
d=D-bottom_cover #effective depth, in mm
sigma_cbc=5 #in MPa
sigma_st=140.0 #in MPa
sigma_sc=130 #in MPa
m=18.66 #modular ratio
#to find critical depth of neutral axis
Xc=d/(1+sigma_st/(m*sigma_cbc)) #in mm
#to find x using b(x**2)/2 + (1.5m-1)Asc(x-d')=mAst(d-x), which becomes of the form px**2+qx+r=0
p=b/2
q=(1.5*m-1)*Asc+m*Ast
r=-(1.5*m-1)*Asc*top_cover-m*Ast*d
x=(-q+math.sqrt(q**2-4*p*r))/(2*p) #in mm
#as x>Xc, beam is over-reinforced
sigma_cbc_dash=sigma_cbc*(x-top_cover)/x #in MPa
sigma_sc=1.5*m*sigma_cbc_dash #< 130 MPa, hence OK
#stress in compression steel is found to be less than its permissible limit of 130 N/mm**2
Mr=b*x*sigma_cbc*(d-x/3)/2+(1.5*m-1)*Asc*sigma_cbc_dash*(d-top_cover) #in N-mm
print "Moment of resistance of the beam=",Mr/10**6," kN-m"

Moment of resistance of the beam= 117.411305827  kN-m


## Ex2.5:pg-50¶

In [6]:
import math
b=250.0 #width, in mm
d=450 #effective depth, in mm
Ast=4*0.785*22**2 #four 22 mm dia bars on tension side, in sq mm
Asc=Ast
top_cover=30 #in mm
sigma_cbc=7 #in MPa
sigma_st=140.0 #in MPa
sigma_sc=130 #in MPa
m=13.33 #modular ratio
l=5.7 #effective span, in m
#to find critical depth of neutral axis
Xc=d/(1+sigma_st/(m*sigma_cbc)) #in mm
#to find x using b(x**2)/2 + (1.5m-1)Asc(x-d')=mAst(d-x), which becomes of the form px**2+qx+r=0
p=b/2
q=(1.5*m-1)*Asc+m*Ast
r=-(1.5*m-1)*Asc*top_cover-m*Ast*d
x=(-q+math.sqrt(q**2-4*p*r))/(2*p) #in mm
#as x<Xc, beam is under-reinforced
sigma_cbc=(sigma_st/m)*x/(d-x) #in MPa
sigma_cbc_dash=sigma_cbc*(x-top_cover)/x #in MPa
sigma_sc=1.5*m*sigma_cbc_dash #in MPa
#stress in compression steel is found to be less than its permissible limit of 130 N/mm**2
Mr=b*x*sigma_cbc*(d-x/3)/2+(1.5*m-1)*Asc*sigma_cbc_dash*(d-top_cover) #in N-mm
W=(Mr/10.0**6)*8.0/l**2 #in kN/m
print "Uniformly distributed load the beam can carry (including self-weight)=",W," kN/m"

Uniformly distributed load the beam can carry (including self-weight)= 21.5549707684  kN/m


## Ex2.6:pg-51¶

In [7]:
import math
b=200.0 #width, in mm
D=480.0 #overall depth, in mm
Ast=4*0.785*25**2 #four 25 mm dia bars on tension side, in sq mm
Asc=3*0.785*22**2 #three 22 mm dia bars on compression side, in sq mm
bottom_cover=30 #in mm
top_cover=30 #in mm
d=D-bottom_cover #effective depth, in mm
m=18.66 #modular ratio
M=100*10**6 #in N-mm
#to find x using b(x**2)/2 + (1.5m-1)Asc(x-d')=mAst(d-x), which becomes of the form px**2+qx+r=0
p=b/2
q=(1.5*m-1)*Asc+m*Ast
r=-(1.5*m-1)*Asc*top_cover-m*Ast*d
x=(-q+math.sqrt(q**2-4*p*r))/(2*p) #in mm
#sigma_cbc_dash=sigma_cbc*(x-d')/x=a*sigma_cbc
a=(x-top_cover)/x
sigma_cbc=M/(b*x*(d-x/3.0)/2+(1.5*m-1)*Asc*a*(d-top_cover)) #in MPa
sigma_st=m*sigma_cbc*(d-x)/x #in MPa
sigma_cbc_dash=a*sigma_cbc #in MPa
sigma_sc=1.5*m*sigma_cbc_dash #in MPa
print "Stress in concrete=",sigma_cbc," N/mm**2\nStress in tension steel=",sigma_st," N/mm**2\nStress in compression steel=",sigma_sc," N/mm**2"

Stress in concrete= 5.37043483737  N/mm**2
Stress in tension steel= 126.05367156  N/mm**2
Stress in compression steel= 127.691872535  N/mm**2


## Ex2.7:pg-52¶

In [8]:
import math
b=300.0 #width, in mm
d=500.0 #effective depth, in mm
Ast=4*0.785*20**2 #four 20 mm dia bars on tension and compression side, in sq mm
Asc=Ast
top_cover=25.0 #in mm
m=13.33 #modular ratio
M=120*10.0**6 #in N-mm
#to find x using b(x**2)/2 + (1.5m-1)Asc(x-d')=mAst(d-x), which becomes of the form px**2+qx+r=0
p=b/2
q=(1.5*m-1)*Asc+m*Ast
r=-(1.5*m-1)*Asc*top_cover-m*Ast*d
x=(-q+math.sqrt(q**2-4*p*r))/(2*p) #in mm
#sigma_cbc_dash=sigma_cbc*(x-d')/x=a*sigma_cbc
a=(x-top_cover)/x
sigma_cbc=M/(b*x*(d-x/3)/2+(1.5*m-1)*Asc*a*(d-top_cover)) #in MPa
sigma_st=m*sigma_cbc*(d-x)/x #in MPa
sigma_cbc_dash=a*sigma_cbc #in MPa
sigma_sc=1.5*m*sigma_cbc_dash #in MPa
print "Stress in concrete=",sigma_cbc," N/mm**2\nStress in tension steel=",sigma_st," N/mm**2\nStress in compression steel=",sigma_sc," N/mm**2"

Stress in concrete= 6.27031395641  N/mm**2
Stress in tension steel= 206.375703553  N/mm**2
Stress in compression steel= 103.628003414  N/mm**2


## Ex2.8:pg-53¶

In [9]:
import math
b=250 #width, in mm
D=600 #overall depth, in mm
bottom_cover=50 #in mm
top_cover=50 #in mm
d=D-bottom_cover #effective depth, in mm
sigma_cbc=5 #in MPa
sigma_st=140 #in MPa
m=18.66 #modular ratio
M=95*10**6 #in N-mm
#to find critical depth of neutral axis
Xc=d/(1+sigma_st/(m*sigma_cbc)) #in mm
#to find Ast1
Ast1=b*Xc*sigma_cbc/(2*sigma_st) #in sq mm
Ast1=982 #round-off, in sq mm
Mr=b*Xc*sigma_cbc/2*(d-Xc/3) #moment of resistance of singly reinforced beam, in N-mm
M1=M-Mr #remaining bending moment, in N-mm
#to find Ast2
Ast2=M1/(sigma_st*(d-top_cover)) #in sq mm
Ast2=421 #round-off, in sq mm
Ast=Ast1+Ast2 #in sq mm
#to find Asc
Asc=m*Ast2*(d-Xc)/((1.5*m-1)*(Xc-top_cover)) #in sq mm
Asc=565 #round-off, in sq mm
print "Tensile steel required=",Ast," mm**2\nCompression steel required=",Asc," mm**2"

Tensile steel required= 1403  mm**2
Compression steel required= 565  mm**2


## Ex2.9:pg-54¶

In [10]:
import math
b=360 #width, in mm
d=750.0 #effective depth, in mm
top_cover=50 #in mm
sigma_cbc=7 #in MPa
sigma_st=190 #in MPa
m=13.33 #modular ratio
M=300.0*10**6 #in N-mm
#to find critical depth of neutral axis
Xc=d/(1+sigma_st/(m*sigma_cbc)) #in mm
#to find Ast1
Ast1=b*Xc*sigma_cbc/(2*sigma_st) #in sq mm
Ast1=1638.0 #round-off, in sq mm
Mr=b*Xc*sigma_cbc/2*(d-Xc/3) #moment of resistance of singly reinforced beam, in N-mm
M1=M-Mr #remaining bending moment, in N-mm
#to find Ast2
Ast2=M1/(sigma_st*(d-top_cover)) #in sq mm
Ast=Ast1+Ast2 #in sq mm
#to find Asc
Asc=m*Ast2*(d-Xc)/((1.5*m-1)*(Xc-top_cover)) #in sq mm
print "Tensile steel required=",Ast," mm**2\nCompression steel required=",Asc," mm**2"

Tensile steel required= 2331.20260317  mm**2
Compression steel required= 1241.93655457  mm**2