Chapter 16:T Beams

Ex16.1:pg-867

In [7]:
import math
Df=120 #in mm
bf=1100 #in mm
bw=275 #in mm
d=450 #in mm
Ast=2700 #in sq mm
fy=500 #in MPa
fck=25 #in MPa
Asf=round(0.36*fck*bf*Df/0.87/fy) #area of steel required for flange, in sq mm
 #as Ast<Asf, Xu<Df
Xu=0.87*fy*Ast/0.36/fck/bf #in mm
Mu=0.36*fck*bf*Xu*(d-0.416*Xu)/10**6 #in kN-m
print "Moment of resistance of T-beam=",Mu," kN-m"

Moment of resistance of T-beam= 470.560221818  kN-m

Ex16.2:pg-868

In [8]:
import math
Df=100 #in mm
bf=1500 #in mm
bw=300 #in mm
d=600 #in mm
Ast=4500 #in sq mm
fy=415 #in MPa
fck=20 #in MPa
Asf=round(0.36*fck*bf*Df/0.87/fy) #area of steel required for flange, in sq mm
 #as Ast>Asf, Xu>Df
Xu=(0.87*fy*Ast-0.446*fck*(bf-bw)*Df)/0.36/fck/bw #in mm
Xc=0.479*d #Xc>Xu; hence OK
a=0.43*Xu #as Df<0.43 Xu, stress in flange is uniform
Mu=(0.36*fck*bw*Xu*(d-0.416*Xu)+0.446*fck*(bf-bw)*Df*(d-Df/2))/10**6 #in kN-m
print "Moment of resistance of T-beam=",Mu," kN-m"

Moment of resistance of T-beam= 862.135878917  kN-m

Ex16.3:pg-869

In [9]:
import math
Df=100 #in mm
bf=1500 #in mm
bw=300 #in mm
d=700 #in mm
Ast=4510 #in sq mm
fy=250 #in MPa
fck=15 #in MPa
Asf=round(0.36*fck*bf*Df/0.87/fy) #area of steel required for flange, in sq mm
 #as Ast>Asf, Xu>Df
Xu=round((0.87*fy*Ast-0.446*fck*(bf-bw)*Df)/0.36/fck/bw) #in mm
Xc=0.531*d #Xc>Xu; hence OK
a=0.43*Xu #as Df>0.43 Xu, stress in flange is not uniform
yf=0.15*Xu+0.65*Df #in mm
Mu=(0.36*fck*bw*Xu*(d-0.416*Xu)+0.446*fck*(bf-bw)*yf*(d-yf/2))/10**6 #in kN-m
print "Moment of resistance of T-beam=",Mu," kN-m"

Moment of resistance of T-beam= 547.9209765  kN-m

Ex16.4:pg-870

In [10]:
import math
Df=100 #in mm
bf=1250 #in mm
bw=250 #in mm
d=650 #in mm
Ast=2800 #in sq mm
fy=415 #in MPa
fck=20 #in MPa
Asf=round(0.36*fck*bf*Df/0.87/fy) #area of steel required for flange, in sq mm
 #as Ast>Asf, Xu>Df
Xu=round((0.87*fy*Ast-0.446*fck*(bf-bw)*Df)/0.36/fck/bw) #in mm
 #but Xu<Df; this indicates that stress in the flange is not uniform, hence replace Df by yf
Xu=(0.87*fy*Ast-0.446*fck*(bf-bw)*0.65*Df)/(0.36*fck*bw+0.446*fck*(bf-bw)*0.15) #in mm
Xc=0.479*d #Xc>Xu; hence OK
a=0.43*Xu #as Df>0.43 Xu, stress in flange is not uniform
yf=0.15*Xu+0.65*Df #in mm
Mu=(0.36*fck*bw*Xu*(d-0.416*Xu)+0.446*fck*(bf-bw)*yf*(d-yf/2))/10**6 #in kN-m
print "Moment of resistance of T-beam=",Mu," kN-m"
 #answer in textbook is incorrect

Moment of resistance of T-beam= 610.289086931  kN-m

Ex16.5:pg-871

In [11]:
import math
Df=100 #in mm
bf=1250 #in mm
bw=250 #in mm
d=660 #in mm
fy=250 #in MPa
fck=15 #in MPa
Xc=0.531*d #in mm
a=0.43*Xc #Df<0.43 Xu, stress in entire flange is uniform
Mu=(0.36*fck*bw*Xc*(d-0.416*Xc)+0.446*fck*(bf-bw)*Df*(d-Df/2))/10**6 #in kN-m
Ast=(0.36*fck*bw*Xc+0.446*fck*(bf-bw)*Df)/0.87/fy #in sq mm
print "Moment of resistance of T-beam=",Mu," kN-m\nArea of steel required=",Ast," sq mm"

Moment of resistance of T-beam= 651.372905965  kN-m
Area of steel required= 5251.13103448  sq mm

Ex16.6:pg-872

In [12]:
import math
Df=100 #in mm
bf=1250 #in mm
bw=250 #in mm
d=550 #in mm
Mu=400 #in kN-m
fy=415 #in MPa
fck=15 #in MPa
Asf=0.446*fck*(bf-bw)*Df/0.87/fy #in sq mm
Muf=0.446*fck*(bf-bw)*Df*(d-Df/2)/10**6 #in kN-m
Muw=Mu-Muf #in kN-m
 #using Cu=Tu, 0.36 fck bw Xu = 0.87 fy Ast, Xu = a Asw
a=0.87*fy/0.36/fck/bw
 #Muw=0.87 fy Asw (d-0.416 Xu)
p=0.87*fy*0.416*a
q=-0.87*fy*d
r=Muw*10**6
Asw=(-q-math.sqrt(q**2-4*p*r))/2/p #in sq mm
Ast=Asw+Asf #in sq mm
print "Area of steel required=",Ast," sq mm"

Area of steel required= 2208.32488139  sq mm