Chapter 13:Deflection of Cantilevers

Problem 13.1,page no.557

In [1]:
import math

#Given
#Variable declaration
L=3*1000           #Length in mm
W=25*1000          #Point load in N
I=1e8              #Moment of Inertia in mm^4
E=2.1e5            #Young's modulus in N/sq.mm

#Calculation
#case(i):Slope of the cantilever at the free end
thetaB=round((W*(L**2))/(2*E*I),6)
#case(ii):Deflection at the free end
yB=round((W*L**3)/(E*I*3),2)

#Result
print "Slope at the free end =",thetaB,"rad"
print "Deflection at the free end =",yB,"mm"

Slope at the free end = 0.005357 rad
Deflection at the free end = 10.71 mm

Problem 13.2,page no.557

In [1]:
import math

#Given
#Variable declaration
L=3*1000           #Length in mm
W=50*1000          #Point load in N
a=2*1000           #Distance between the load and fixed end in mm
I=1e8              #Moment of Inertia in mm^4
E=2e5              #Young's modulus in N/sq.mm

#Calculation
#case(i):Slope at the free end
thetaB=(W*(a**2))/(2*E*I)
#case(ii):Deflection at the free end
yB=round(((W*a**3)/(E*I*3))+((W*(a**2))/(2*E*I)*(L-a)),2)

#Result
print "Slope at the free end =",thetaB,"rad"
print "Deflection at the free end =",yB,"mm"

Slope at the free end = 0.005 rad
Deflection at the free end = 11.67 mm

Problem 13.3,page no.559

In [2]:
import math

#Given
#Variable declaration
L=2.5*1000           #Length in mm
w=16.4               #Uniformly distributed load in kN/m 
I=7.95e7             #Moment of Inertia in mm^4
E=2e5                #Young's modulus in N/sq.mm

#Calculation
W=w*L                           #Total load in N
yB=round((W*L**3)/(E*I*8),3)    #Deflection at the free end in mm

#Result
print "Deflection at the free end =",yB,"mm"

Deflection at the free end = 5.036 mm

Problem 13.4(A),page no.560

In [3]:
#Given
#Variable declaration
b=120          #Width in mm
d=200          #Depth in mm
L_star=2.5     #Length in m
L=2.5*1000     #Length in mm
yB=5           #Deflection at free end in mm
E=2e5          #Young's modulus in N/sq.mm

#Calculation
I=(b*d**3)/12                        #Moment of Inertia in mm^4
w=(yB*8*E*I)/(L**3*L_star)/1e3       #Uniformly distributed load in N/m

#Result
print "Uniformly distributed load =",w,"kN/m"

Uniformly distributed load = 16.384 kN/m

Problem 13.5,page no.563

In [4]:
import math

#Given
#Variable declaration
L=3*1000           #Length in mm
w=10               #Uniformly distributed load in N/mm
a=2*1000           #Length of Uniformly distributed load from fixed end in mm
I=1e8              #Moment of Inertia in mm^4
E=2e5              #Young's modulus in N/sq.mm

#Calculation
#case(i):Slope at the free end
thetaB=str((w*(a**3))/(6*E*I))[:7]
#case(ii):Deflection at the free end
yB=round(((w*a**4)/(E*I*8))+((w*(a**3))/(6*E*I)*(L-a)),2)

#Result
print "Slope at the free end =",thetaB,"rad"
print "Deflection at the free end =",yB,"mm"

Slope at the free end = 0.00066 rad
Deflection at the free end = 1.67 mm

Problem 13.6,page no.563

In [5]:
import math

#Given
#Variable declaration
L=3*1000           #Length in mm
w=10               #Uniformly distributed load in N/mm
a=2*1000           #Length of Uniformly distributed load from fixed end in mm
I=1e8              #Moment of Inertia in mm^4
E=2e5              #Young's modulus in N/sq.mm

#Calculation
#case(i):Slope at the free end
thetaB=round(((w*(L**3))/(6*E*I))-((w*((L-a)**3))/(6*E*I)),6)
#case(ii):Deflection at the free end
yB=round(((w*L**4)/(E*I*8))-(((w*(L-a)**4)/(8*E*I))+((w*(L-a)**3)/(6*E*I)*a)),4)

#Result
print "Slope at the free end =",thetaB,"rad"
print "Deflection at the free end =",yB,"mm"

Slope at the free end = 0.002167 rad
Deflection at the free end = 4.8333 mm

Problem 13.10,page no.569

In [6]:
import math

#Given
#Variable declaration
L=4*1000           #Length in mm
w=50               #load at fixed end in N/mm
I=1e8              #Moment of Inertia in mm^4
E=2e5              #Young's modulus in N/sq.mm

#Calculation
#case(i):Slope at the free end
thetaB=round(-(w*(L**3))/(24*E*I),5)
#case(ii):Deflection at the free end
yB=round((w*L**4)/(E*I*30),2)

#Result
print "Slope at the free end =",-thetaB,"rad"
print "Deflection at the free end =",yB,"mm"

Slope at the free end = 0.00667 rad
Deflection at the free end = 21.33 mm
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