2.2.A.sagews

from sympy import *
from sympy import N as Num

# pi is wrong. Use pie!
pie  = 2*pi

# 30 deg = 1/12 pie:
(t1, t2)   = ( pie/12, pie/8 )
# 2 theta_1/2:
(tt1, tt2) = ( 2*t1, 2*t2 )

# s30 = sin(2 theta_1)
# s45 = sin(2 theta_2)
(s30, s45) = ( sin(tt1), sin(tt2) )
(c30, c45) = ( cos(tt1), cos(tt2) )

# Variables:
s_x, s_y, t_xy = var('s_x, s_y, t_xy')

# shortcuts:
pls = s_x + s_y
mns = s_x - s_y

# given:
tau30 = 2 + 3*sqrt(3)/2
sig45 = 6
tau45 = 3

# equations:
eqns = [
    Eq(tau30 ,       - mns/2*s30 + t_xy*c30),
    Eq(sig45 , pls/2 + mns/2*c45 + t_xy*s45),
    Eq(tau45 ,       - mns/2*s45 + t_xy*c45)
    ]

pprint(eqns)

unknowns=[s_x, s_y, t_xy]
sol = solve(eqns, unknowns)
pprint(sol)
# {s_x: -1, t_xy: 4, s_y: 5}

sx=sol[s_x]
sy=sol[s_y]
txy=sol[t_xy]

E = 1000
nu = S(1)/10
G = E/2/(1+nu)
ey = S(1)/E*(sy - nu*sx)
gam = S(1)/G*txy

pprint(Num(ey,3))
pprint(Num(gam,3))

___ / s_x s_y\ ___ \/ 3 *|- --- + ---| 3*\/ 3 t_xy \ 2 2 / s_x s_y s_x s [2 + ------- = ---- + -------------------, 6 = --- + --- + t_xy, 3 = - --- + - 2 2 2 2 2 2 _y --] 2 {s_x: -1, s_y: 5, t_xy: 4} 0.00510 0.00880