# Chapter2-Strain and stress -strain relations¶

## Ex1-pg38¶

In [2]:
import math
#calculate radius of circle and principal stresses and distance between o and c and strain and shear strain and maxi shear stress
theta1=(math.atan(130./195.))*57.3
print'%s %.2f %s'%("pricipal stresses in degree",theta1,"");## print'%s %.2f %s'%laying result
epsilonx=510.*10**(-6.)
epsilony=120.*10**(-6.)
epsilon=(epsilonx+epsilony)/2.
print'%s %.5f %s'%("distance between O and c=",epsilon,"")

##solution a
angle=60.- theta1
print'%s %.2f %s'%("angle of ACA1 in degree = ",angle,"")## print'%s %.2f %s'%laying result
print'%s %.5f %s'%("strains in x axis= ",epsilonx1,"")## print'%s %.2f %s'%laying result
print'%s %.5f %s'%("strains in y axis= ",epsilony1,"")## print'%s %.2f %s'%laying result
print'%s %.5f %s'%("shear strain",gammaxy,"")## print'%s %.2f %s'%laying result

##solution b
print'%s %.5f %s'%("strains in x axis= ",epsilon1,"")## print'%s %.2f %s'%laying result
print'%s %.5f %s'%("strains in x axis= ",epsilon2,"")## print'%s %.2f %s'%laying result

##solution c
gammamax=-+468*10**(-6)
print'%s %.5f %s'%("maxi shear stress= ",gammamax,"")

radius of the circle in degree =  0.00023
pricipal stresses in degree 33.69
distance between O and c= 0.00031
angle of ACA1 in degree =  26.31
strains in x axis=  0.00053
strains in y axis=  0.00010
shear strain -0.00021
strains in x axis=  0.00055
strains in x axis=  0.00008
maxi shear stress=  -0.00047


## Ex2-pg46¶

In [3]:
import math
#calculate value of epsilon and gamma
epsilon0=190.*10**(-6)
epsilon60=200.*10**(-6)
epsilon120=-300.*10**(-6)
E=200.## GPa
v=0.3
epsilonx=epsilon0
print'%s %.4f %s'%("value of epsilonx is= ",epsilonx,"")

## epsilon60=((epsilonx+epsilony)/2)-((epsilonx-epsilony)/4)+(gammaxy*sqrt(3))/4 eqn 1
## epsilon120=((epsilonx+epsilony)/2)-((epsilonx-epsilony)/4)-(gammaxy*sqrt(3))/4 eqn 2

epsilony=(2.*(epsilon60+epsilon120)-epsilon0)/3.
print'%s %.4f %s'%("value of epsilony is= ",epsilony,"")
gammaxy=(2./math.sqrt(3.))*(epsilon60-epsilon120)## from eqn 1 and eqn 2
print'%s %.5f %s'%("value of gammaxy is= ",gammaxy,"")
epsilon1=((epsilonx+epsilony)/2.)+math.sqrt(((epsilonx-epsilony)/2.)**2.+(gammaxy/2.)**2.)## epsilony value is in negative so the sign changes in the eqn
print'%s %.5f %s'%("value of epsilon1 is= ",epsilon1,"")
epsilon2=((epsilonx+epsilony)/2.)-math.sqrt(((epsilonx-epsilony)/2.)**2+(gammaxy/2.)**2.)##epsilony value is in negative so the sign changes in the eqn
print'%s %.4f %s'%("value of epsilon2 is= ",epsilon2,"")

gammamax=(2.*10**-6)*math.sqrt(((epsilonx-epsilony)/2.)**2.+(gammaxy/2.)**2)
print'%s %.3e %s'%("max shear strain is= ",gammamax,"")
thetap=math.atan((577./320.)/2.)*57.3
print'%s %.2f %s'%("orientations of principal axes is= ",thetap,"") ## or
thetap1=math.atan((577./320.)*2.)*57.
print'%s %.2f %s'%("orientations of principal axes is= ",thetap1,"")
sigma1=(200.*10**9/(1.-0.09))*(epsilon1+0.3*epsilon2)
print'%s %.2f %s'%("plane stresss is Pa= ",sigma1,"")
sigma2=(200*10**3/(1-0.09))*(epsilon2+0.3*epsilon1)
print'%s %.2f %s'%("plane stresss is MPa= ",sigma2,"")

taumax=(200.*10**9./(2.*(1.+0.3)))*gammamax
print'%s %.2f %s'%("plane stresss is MPa= ",taumax,"")
#error in orientationd of principal axes because of round off error

value of epsilonx is=  0.0002
value of epsilony is=  -0.0001
value of gammaxy is=  0.00058
value of epsilon1 is=  0.00036
value of epsilon2 is=  -0.0003
max shear strain is=  6.601e-10
orientations of principal axes is=  42.04
orientations of principal axes is=  74.12
plane stresss is Pa=  59348428.75
plane stresss is MPa=  -42.21
plane stresss is MPa=  50.78