# Chapter17-Contact Stress¶

## Ex1-pg6077¶

In [1]:
#calculate the maximum principal stress ad maximum shear stress and maximum octahedral stress
## initialization of variables
import math
E1=200. ##GPa
E2=200. ##Gpa
v1=0.29
v2=0.29
R1=60. ##mm
R11=130. ##mm
R2=80. ##mm
R22=200. ##mm
th=math.pi/3.
P=4.5 ##kN
P=P*10**3
E=E1*10**3
B=1./4.*(1./R1+1./R2+1./R11+1./R22)+1./4.*((1./R1+1./R2-1./R11-1./R22)**2. - 4.*(1./R1-1./R11)*(1./R2-1./R22)*(math.sin(th)**2.))**(1/2.)
A=1./4.*(1./R1+1./R2+1./R11+1./R22)-1./4.*((1./R1+1./R2-1./R11-1./R22)**2. - 4.*(1./R1-1./R11)*(1./R2-1./R22)*(math.sin(th)**2.))**(1/2.)
Del=2*(1-v1**2)/(E*(A+B))
BAr=B/A
Cb=0.77
Cs=0.724
Ct=0.24
Cg=0.22
Cz=0.53
Cd=2.10
b=Cb*(P*Del)**(1/3.)
br=b/Del
S_max=-Cs*br
tau_max=Ct*br
tau_oct=Cg*br
Zs=Cz*b
delta=Cd*P/math.pi*((A+B)/br)
print'%s %.2f %s'%('Sigma_max = ',S_max,' MPa')
print'%s %.2f %s'%('\n tau_max = ',tau_max,' MPa')
print'%s %.2f %s'%('\n tau_oct_max = ',tau_oct,' MPa')
print'%s %.2f %s'%('\n Zs = ',Zs,' mm')
print'%s %.2f %s'%('\n delta = ',delta,' mm')
## S_max doesn't match due to round off error

Sigma_max =  -1596.76  MPa

tau_max =  529.31  MPa

tau_oct_max =  485.20  MPa

Zs =  0.51  mm

delta =  0.03  mm


## Ex2-pg609¶

In [2]:
#calculate the maximum shear stressand tensile stress and compressive stress
## initialization of variables
import math
E=200. ##GPa
v=0.29
Y=1600. ##MPa
Po=4.2 ##kN
Omega=3000. ##rpm
th=math.pi/3.
P=1.75 ##kN
R1=4.76 ##mm
R11=R1
R2=-4.86 ##mm
R22=18.24 ##mm
##part (a)
E=E*10**3.
Po=Po*10**3.
P=P*10**3.
B=1/4.*(1./R1+1./R2+1./R11+1./R22)+1/4.*((1./R1+1./R2-1./R11-1./R22)**2 - 4.*(1/R1-1./R11)*(1/R2-1/R22)*(math.sin(th)**2.))**(1/2.)
A=1/4.*(1./R1+1./R2+1./R11+1./R22)-1/4.*((1./R1+1./R2-1./R11-1./R22)**2 - 4.*(1/R1-1./R11)*(1/R2-1/R22)*(math.sin(th)**2.))**(1/2.)
Del=2.*(1.-v**2)/(E*(A+B))
BAr=B/A
Cb=0.32
k=0.075
Cs=1.00
Ct=0.3
Cg=0.27
Cz=0.78
b=Cb*(P*Del)**(1/3.)
a=b/k
br=b/Del
S_max=-Cs*br
tau_max=Ct*br
tau_oct=Cg*br
Zs=Cz*b
tauo=0.486*b/(2.*Del)
Zo=0.41*b
print'%s %.2f %s'%('b = ',b,' mm')
print'%s %.2f %s'%('\n a = ',a,' mm')
print'%s %.2f %s'%('\n b/Delta = ',br,' MPa')
print'%s %.2f %s'%('\n Sigma_max = ',S_max,' MPa')
print'%s %.2f %s'%('\n tau_max = ',tau_max,' MPa')
print'%s %.2f %s'%('\n tau_oct_max =',tau_oct,' MPa')
print'%s %.2f %s'%('\n Zs = ',Zs,' mm')
print'%s %.2f %s'%('\n Tau_0 = ',tauo,' MPa')
print'%s %.2f %s'%('\n Zo = ',Zo,' mm')

## part (b)
tau_oY=math.sqrt(2)*Y/3.
Py = 1./Del*(tau_oY/(Cg*Cb)*Del)**3
print('\n part (b)')
print'%s %.2f %s'%('\n P_Y = ',Py,' N')
SF=Py/P
print'%s %.2f %s'%('\n SF =  ',SF,'')

b =  0.16  mm

a =  2.10  mm

b/Delta =  2313.81  MPa

Sigma_max =  -2313.81  MPa

tau_max =  694.14  MPa

tau_oct_max = 624.73  MPa

Zs =  0.12  mm

Tau_0 =  562.26  MPa

Zo =  0.06  mm

part (b)

P_Y =  3079.68  N

SF =   1.76


## Ex3-pg621¶

In [2]:
#calculate the maximum stress and tesile stress and compressive stress
## initialization of variables
import math
E=200. ##GPa
v=0.29
R=40. ##mm
h=20. ##mm
P=24.1 ##kN
S_max=1445. ##MPa
tau_max=433. ##MPa
tau_octM=361. ##MPa
##calculations
E=E*10**3
P=P*10**3
Del=2.*R*(1.-v**2)/E
b=math.sqrt(2*P*Del/(h*math.pi))
br=b/Del
S_maxT=2.*b/(9.*Del)
S_maxC=-1.13*br
tau_max=0.31*br
tau_octM=0.255*br
print'%s %.2f %s'%('Sigma_max (tension) = ',S_maxT,' MPa')
print'%s %.2f %s'%('\n Sigma_max (compression) = ',S_maxC,' MPa')
print'%s %.2f %s'%('\n tau_max = ',tau_max,' MPa')
print'%s %.2f %s'%('\n tau_oct_max = ',tau_octM,' MPa')

Sigma_max (tension) =  321.56  MPa

Sigma_max (compression) =  -1635.15  MPa

tau_max =  448.58  MPa

tau_oct_max =  368.99  MPa