Chapter 12:Deflection of Beams

Problem 12.1,page no.518

In [2]:
import math

#Given
#Variable declaration
L=6*1000        #Length in mm
W=50*1000       #Point load in N
I=78e6          #Moment of Inertia in mm^4
E=2.1e5         #Young's modulus in N/sq.mm

#Calculation
yc=round((W*L**3)/(48*E*I),3)                      #The deflection at the centre in mm
thetaB=round(math.degrees((W*L**2)/(16*E*I)),4)    #The slope at the supports

#Result
print "Deflection at the centre =",yc,"mm"
print "NOTE:The answer given for slope at the support is wrong.The correct answer is,"
print "Slope at the support =",thetaB,"degree"

Deflection at the centre = 13.736 mm
NOTE:The answer given for slope at the support is wrong.The correct answer is,
Slope at the support = 0.3935 degree

Problem 12.2,page no.518

In [2]:
import math

#Given
#Variable declaration
L=4*1000              #Length in mm

#Calculation
thetaA=round(math.radians(1),5)    #Slope at the ends in radians
yc=float(str(thetaA*(L/3))[:5])    #Deflection at the centre in mm

#Result
print "Deflection at the centre =",yc,"mm"

Deflection at the centre = 23.26 mm

Problem 12.3,page no.519

In [1]:
import math

#Given
#Variable declaration
L=3*1000              #Length in mm

#Calculation
thetaA=round(math.radians(1),5)    #Slope at the ends in radians
yc=float(str(thetaA*(L/3))[:5])    #Deflection at the centre in mm

#Result
print "Deflection at the centre =",yc,"mm"

Deflection at the centre = 17.45 mm

Problem 12.4,page no.526

In [6]:
import math

#Given
#Variable declaration
L=5*1000        #Length in mm
W=5*1000        #Point load in N
a=3*1000        #Distance between point load and left end in mm
E=2e5           #Young's modulus in N/sq.mm
I=1e8           #Moment of Inertia in mm^4

#Calculation
b=L-a           #Width in mm
#case(i):The slope at the left support
thetaA=-(W*a*b)/(6*E*I*L)*(a+2*b)
#case(iii): The deflection under the load
yc=(W*a**2*b**2)/(3*E*I*L)
#case(iii):The maximum deflection
y_max=round((W*b)/(9*math.sqrt(3)*E*I*L)*(((a**2)+(2*a*b))**(3/2)),4)

#Result
print"slope at the left support =",thetaA,"radians"
print"Deflection under the load =",yc,"mm"
print"Maximum deflection =",y_max,"mm"

slope at the left support = -0.00035 radians
Deflection under the load = 0.6 mm
Maximum deflection = 0.6173 mm

Problem 12.5,page no,529

In [5]:
from __future__ import division
import math

#Given
#Variable declaration
b=200                #Width in mm
d=300                #Depth in mm
L=5*1000             #Span in mm
L_star=5             #Span in m
w=9*1000             #Uniformly distributed load in N/m            
E=1e4                #Young's modulus in N/sq.mm

#Calculation
W=w*L_star                                 #Total load in N
I=b*d**3/12                                #Moment of Inertia in mm^4

#case(i):the slope at the support
thetaA=round(-(W*(L**2))/(24*E*I),4)

#case(ii):maximum deflection
yc=str((W*L**3)/(E*I)*(5/384))[:5]

#Result
print"Slope at the support =",-thetaA,"radians"
print"Maximum deflection =",yc,"mm"

Slope at the support = 0.0104 radians
Maximum deflection = 16.27 mm

Problem 12.6,page no.529

In [3]:
#Given
#Variable declaration
L=5*1000      #Length in mm
L_star=5      #Length in m
w=9           #Uniformly distributed load in kN/m
f=7           #Bending stress in N/sq.mm
E=1e4         #Young's modulus in N/sq.mm
yc=10         #Central deflection in mm

#Calculation
W=w*L_star*1e3                       #Total load in N               
bd3=((W*(L**3)*12*5)/(E*yc*384))     #width X depth^3 in mm^4
M=(W*L/8)                            #Maximum bending moment in Nmm 
bd2=round(M*12/(f*2),2)              #width X depth^2 in mm^3
d=round(bd3/bd2,2)                   #Depth of beam in mm
b=str(M*12/(f*2)/d**2)[:6]           #Width of beam in mm

#Result
print "Depth of beam =",d,"mm"
print "Width of beam =",b,"mm"

Depth of beam = 364.58 mm
Width of beam = 181.36 mm

Problem 12.7,page no.531

In [2]:
#Given 
#Variable declaration
L=5*1000      #Length in mm
f=8           #Bending stress in N/sq.mm
yc=10         #Central deflection in mm
E=1.2e4       #Young's modulus in N/sq.mm

#Calculation
d=round((5*L**2*(f*2*8))/(E*384*yc)*1e-1,2)  #Depth of beam in cm

#Result
print "Depth of beam =",d,"cm"

Depth of beam = 34.72 cm

Problem 12.8,page no.534

In [4]:
#Given
#Variable declaration
L=6*1000       #Length in mm
W=40*1000      #Point load in N
a=4*1000       #Distance of point load from left support in mm
I=7.33e7       #Moment of Inertia in mm^4
E=2e5          #Young's modulus in sq.mm

#Calculation
b=L-a                                 #Width of beam in mm
yc=round(-(W*a**2*b**2)/(3*E*I*L),1)  #Deflection under the load in mm

#Result
print "Deflection under the load =",yc,"mm"

Deflection under the load = -9.7 mm
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