# Chapter 9:Beam Design¶

## Ex9.1:pg-446¶

In [1]:
import math
l=3 #span, in m
b=225 #wall thickness, in mm
Dm=19.2 #weight of masonry, in kN/cu m
sigma_cbc=5 #in MPa
sigma_st=230 #in MPa
fy=415 #in MPa
#area of triangle of brick masonry
A=math.sqrt(3)/4.0*l**2 #in sq m
V=A*(b/10**3) #volume of triangle of masonry, in cu m
W=V*Dm #weight of masonry, in kN
M1=W*l/6 #in kN-m
D=l*10**3.0/12 #in mm
D=300 #approximately, in mm
self_weight=25*(D/10**3)*(b/10**3) #in kN/m
M2=self_weight*l**2/8 #in kN-m
M=M1+M2 #in kN-m
#check for depth
d=math.sqrt(M*10**6/0.65/b) #in mm
d=265 #approximately, in mm
dia=10.0 #in mm
D=d+dia/2+25 #<300 mm, hence OK
D=300 #in mm
Ast=M*10**6/sigma_st/0.9/d #in sq mm
n=Ast/0.785/10**2 #no. of 10 mm dia bars required
#provide 2-10 mm dia + 1-8 mm dia bars
Ast=2*0.785*10**2+0.785*8**2 #in sq mm
pt=Ast/b/d*100 #pt=0.35, approximately
W=W+self_weight*l #in kN
V=W/2 #in kN
Tv=V*10**3/b/d #in MPa
#for M15 grade concrete and pt=0.35
Tc=0.248 #in MPa
#as Tc>Tv, no shear reinforcement required; provide nominal stirrups
#provide 6 mm dia bars
Asv=2*0.785*6**2 #in sq mm
Sv=Asv*fy/0.4/b #in mm
Sv=260 #approximately, in mm
Svmax=0.75*d #in mm
Svmax=200 #approximately, in mm
Sv=min(Sv,Svmax) #in mm
print "Summary of design\nSize of lintel beam=",(b)," x ",(D)," mm\ncover = 35 mm\nsteel = 2-10 mm dia bars + 1-8 mm dia bar\nstirrups = 6 mm dia @ ",(Sv)," mm c/c throughout"

Summary of design
Size of lintel beam= 225  x  300  mm
cover = 35 mm
steel = 2-10 mm dia bars + 1-8 mm dia bar
stirrups = 6 mm dia @  200  mm c/c throughout


## Ex9.2:pg-447¶

In [2]:
import math
l=4.2 #span, in m
b=225 #width, in mm
D=300 #depth, in mm
sigma_cbc=5 #in MPa
sigma_st=230 #in MPa
fy=415 #in MPa
m=18.66 #modular ratio
W1=25*(D/10**3)*(b/10**3) #self-weight, in kN/m
W2=6 #load on beam, in kN/m
W=W1+W2 #in kN/m
M=W*l**2.0/8 #in kN-m
dia=12 #in mm
d=D-dia/2-25 #in mm
Xc=0.29*d #in mm
Mr=0.65*b*d**2/10**6 #M>Mr, hence doubly reinforced beam
Ast1=round(Mr*10**6/sigma_st/0.9/d) #steel required for singly reinforced beam, in sq mm
M1=M-Mr #balance of moment, in kN-m
d1=25 #top cover, in mm
Ast2=round(M1*10**6/sigma_st/(d-d1)) #in sq mm
Ast=Ast1+Ast2 #in sq mm
n1=Ast/0.785/12**2 #no. of 12 mm dia bars on tension side
n1=3 #assume
Asc=m*Ast2*(d-Xc)/(1.5*m-1)/(Xc-d1) #in sq mm
n2=Asc/0.785/12**2 #no. of 12 mm dia bars on compression side
n2=3 #assume
V=W*l/2.0 #in kN
Tv=V*10**3/b/d #in MPa
pt=n1*0.785*12**2/b/d*100 #pt=0.56, approximately
#for M15 grade concrete and pt=0.56
Tc=0.302 #in MPa
#as Tc>Tv, no shear reinforcement required; provide nominal stirrups
#provide 6 mm dia bars
Asv=2*0.785*6**2 #in sq mm
Sv=Asv*fy/0.4/b #in mm
Sv=260 #approximately, in mm
Svmax=0.75*d #in mm
Svmax=200 #approximately, in mm
Sv=min(Sv,Svmax) #in mm
print "Summary of design\nSize of beam = ",(b)," x ",(D)," mm\nCover, bottom = 25 mm\nTop = 25 mm\nSteel, bottom = ",(n1),"-12 mm dia bars\nTop = ",(n2),"-12 mm dia bars\nStirrups = 6 mm dia @ ",Sv," mm c/c throughout"

Summary of design
Size of beam =  225  x  300  mm
Cover, bottom = 25 mm
Top = 25 mm
Steel, bottom =  3 -12 mm dia bars
Top =  3 -12 mm dia bars
Stirrups = 6 mm dia @  200  mm c/c throughout


## Ex9.3:pg-448¶

In [3]:
import math
l=7 #span, in m
sigma_cbc=5 #in MPa
sigma_st=140 #in MPa
fy=250 #in MPa
m=18.66 #modular ratio
b=300 #assume, in mm
W1=35 #imposed load on beam, in kN/m
M=W1*l**2.0/8 #in kN-m
d=(M*10**6/0.87/b)**0.5 #in mm
d=910 #approximately, in mm
D=1.1*d+50 #increase d by 10% for self-weight and cover is 50 mm
D=1050 #approximately, in mm
W2=25*(b/10**3)*(D/10**3) #self-weight, in kN/m
W=W1+W2 #in kN/m
M=W*l**2.0/8 #in kN-m
d=(M*10**6/0.87/b)**0.5 #in mm
d=1000 #approximately, in mm
dia=20 #in mm
D=d+dia/2+35 #in mm
Ast=round(M*10**6/sigma_st/0.87/d) #in sq mm
n=Ast/0.785/20**2 #no. of 20 mm dia bars
n=7 #assume
Ast=n*0.785*20**2 #in sq mm
pt=Ast/b/D*100 #pt=0.7, approximately
As=round(0.85/fy*b*d) #minimum steel, As<Ast, hence OK
Asf=0.1/100*b*d/2 #side faced steel on each face, in sq mm
#provide 6 mm dia bars
s=1000*0.785*6**2/Asf #in mm
s=188 #assume, in mm
V=W*l/2.0 #in kN
Tv=V*10**3/b/d #<Tcmax=1.6 MPa, hence OK
#for M15 grade concrete and pt=0.7
Tc=0.33 #in MPa
#as Tv>Tc, shear reinforcement required
Vs=V-Tc*b*d/10**3 #in kN
#provide 6 mm dia bars
Asv=2*0.785*6**2 #in sq mm
sigma_sv=140 #in MPa
Sv=Asv*sigma_sv*d/Vs/10**3 #in mm
Sv=155 #approximately, in mm
Svmin=Asv*fy/0.4/b #in mm
Svmin=117 #approximately, in mm
Sv=min(Sv,Svmin) #in mm
print "Summary of design\nSize of beam = ",(D)," x ",(n)," mm\nCover = 35 mm\nSteel= ",(Sv),"-20 mm dia bars\nStirrups = 6 mm dia @ ",(s)," mm c/c throughout\nSide faced steel-6 mm dia @ ",(s)," mm c/c on both vertical faces of beam"

Summary of design
Size of beam =  1045  x  7  mm
Cover = 35 mm
Steel=  117 -20 mm dia bars
Stirrups = 6 mm dia @  188  mm c/c throughout
Side faced steel-6 mm dia @  188  mm c/c on both vertical faces of beam


## Ex9.4:pg-449¶

In [4]:
import math
l=10 #span, in m
sigma_cbc=5 #in MPa
sigma_st=140 #in MPa
fy=250 #in MPa
m=18.66 #modular ratio
Df=100 #slab thickness, in mm
D=l*10**3.0/12 #in mm
D=850 #approximately, in mm
d=D-100 #cover=100 mm
bw=300 #in mm
bf=l*10**3/6+bw+6*Df #>2500 mm c/c distance of beams
bf=2500 #in mm
W1=(bw/10**3)*(d-Df)/10**3*25 #in kN/m
W2=(Df/10**3)*(bf/10**3)*25 #in kN/m
W=W1+W2+W3 #in kN/m
W=24 #approximately, in kN/m
M=W*l**2.0/8 #in kN-m
V=W*l/2.0 #in kN
Ast=round(M*10**6/sigma_st/0.87/d) #in sq mm
#provide 4-25 mm dia bars + 4-20 mm dia bars
Ast=4*0.785*25**2+4*0.785*20**2 #in sq mm
#verification of trial section
#assume x>Df
x=(m*Ast*d+bf*Df**2/2)/(bf*Df+m*Ast) #in mm
#sigma_cbc'=sigma_cbc (x-Df)/x
a=(x-Df)/x
z=d-(1+2*a)/(1+a)*Df/3 #in mm
sigma_st=M*10**6/Ast/z #<140 MPa, hence OK
sigma_cbc=sigma_st/m*x/(d-x) #<5 MPa, hence OK
Tv=V*10**3/bw/d #in MPa
pt=Ast*100/(bw*d+(2500-300)*100) #pt=0.72, approximately
#for M15 grade concrete and pt=0.72
Tc=0.33 #in MPa
#as Tv>Tc, shear reinforcement required
Vs=V-Tc*bw*d/10**3 #in kN
#provide 6 mm dia bars
Asv=2*0.785*6**2 #in sq mm
sigma_sv=140 #in MPa
Sv=Asv*sigma_sv*d/Vs/10**3 #in mm
Sv=130 #approximately, in mm
Svmin=Asv*fy/0.4/bw #in mm
Svmin=117 #approximately, in mm
Sv=min(Sv,Svmin) #in mm
print "T beam:bf=",(bf)," mm\nDf=",(Df)," mm\nd=",(d)," mm\nbw=",(bw)," mm\nCover = 50 mm\nSteel= 4-25 mm dia + 4-20 mm dia bars\nStirrups = 6 mm dia @ ",(Sv)," mm c/c throughout"

T beam:bf= 2500  mm
Df= 100  mm
d= 750  mm
bw= 300  mm
Cover = 50 mm
Steel= 4-25 mm dia + 4-20 mm dia bars
Stirrups = 6 mm dia @  117  mm c/c throughout