2.4.2.F.sagews

from sympy.physics.units import *
from sympy import *
from sympy import N as Num
import matplotlib.pyplot as plt
from numpy import linspace, full_like, array, arange

f22 = S(2)/2
f32 = S(3)/2
f42 = S(4)/2

# helper function:
def fill_array(f,x,xia):
    if f.is_constant(x):
        res = full_like(xia,f)
    else:
        res = lambdify(x,f,"numpy")
        res = array(res(xia))
    return res

# symbols:
a, MB, EI = var('a, MB, EI')

print("\n--- a: --------------------------------")

x, B_v = var("x, B_v")

M = (a - x) * B_v + MB

print("\n--- b: --------------------------------")
# integrating M to get w:
C1, C2 = var('C1, C2')
unknowns = [C1, C2]
wpp = - 1/EI * M
wp = integrate(wpp,x) + 1/EI*C1
w =  integrate(wp,x) + 1/EI*C2

pprint(w)

w = expand(w)
w = collect(w,B_v)
w = w.simplify()
pprint(w*EI)

pprint(latex(w*EI))

print("\n--- c: --------------------------------")
# 1: w(0)  = 0
# 2: w'(0) = 0
bc1 = Eq(w.subs(x,0), 0)
bc2 = Eq(wp.subs(x,0), 0)

eqns = [bc1, bc2]

sol = solve(eqns, unknowns)
sC1 = sol[C1]
sC2 = sol[C2]

for s in [sC1, sC2]:
    pprint(latex(s))

w = w.subs({C1: sC1, C2: sC2})
w = w.expand()
w = collect(w,B_v)
w = w.simplify()
pprint(w*EI)
pprint(latex(w*EI))

print("\n--- d: --------------------------------")
(Bv0, Bv1, Bv2) = (-f22*MB/a, -f32*MB/a, -f42*MB/a)

w0 = w.subs(B_v, Bv0)
w1 = w.subs(B_v, Bv1)
w2 = w.subs(B_v, Bv2)

# normalized deflections:
wn0 = w0 / ( MB*a*a / EI )
wn1 = w1 / ( MB*a*a / EI )
wn2 = w2 / ( MB*a*a / EI )

for wn in [wn0,wn1,wn2]:
    wn = wn.expand()
    wn = wn.simplify()
    # pprint(latex(wn))

xi = var('xi')
f0 = wn0.expand().subs(x/a,xi)
f1 = wn1.expand().subs(x/a,xi)
f2 = wn2.expand().subs(x/a,xi)
# Make array for plotting:
xia = linspace(float(0),float(1))

f0 = fill_array(f0,xi,xia)
f1 = fill_array(f1,xi,xia)
f2 = fill_array(f2,xi,xia)

# plot functions:
tmp = plt.plot(xia, f0, "b-", label="$w^- \\, / \\, \\frac{M^{\\mathsf{B}} a^2}{EI}$")
tmp = plt.plot(xia, f1, "r--", label="$w \\, / \\, \\frac{M^{\\mathsf{B}} a^2}{EI}$")
tmp = plt.plot(xia, f2, "g-.", label="$w^+ \\, / \\, \\frac{M^{\\mathsf{B}} a^2}{EI}$")
tmp = plt.grid()
tmp = plt.axhline(0, color='black')
tmp = plt.xlabel("$x / a$")
tmp = plt.legend(loc='best')
plt.gca().invert_yaxis()

plt.savefig("2.4.2.F_w1.svg", transparent=True)
plt.show()
plt.clf()


print("\n--- e: --------------------------------")

print("\n--- f: --------------------------------")

x2 = xia**2
x3 = xia**3
wBn = 3*x2/4 - x3/4
wMn = -1*x2/2
w = wBn + wMn

# plot functions:
tmp = plt.plot(xia, wBn, "b-", label="$w_B \\, / \\, \\frac{M^{\\mathsf{B}} a^2}{EI}$")
tmp = plt.plot(xia, wMn, "g-.", label="$w_M \\, / \\, \\frac{M^{\\mathsf{B}} a^2}{EI}$")
tmp = plt.plot(xia, w, "r--", label="$w \\, / \\, \\frac{M^{\\mathsf{B}} a^2}{EI}$")
tmp = plt.grid()
tmp = plt.axhline(0, color='black')
tmp = plt.xlabel("$x / a$")
tmp = plt.legend(loc='best')
plt.gca().invert_yaxis()

plt.savefig("2.4.2.F_w2.svg", transparent=True)
plt.show()

--- a: -------------------------------- --- b: -------------------------------- 3 2 B_v*x C1*x C2 x *(-B_v*a - MB) ------ + ---- + -- + ---------------- 6*EI EI EI 2*EI 2 2 B_v*x *(3*a - x) MB*x - ---------------- + C1*x + C2 - ----- 6 2 - \frac{B_{v} x^{2}}{6} \left(3 a - x\right) + C_{1} x + C_{2} - \frac{MB x^{2 }}{2} --- c: -------------------------------- 0 0 2 -x *(B_v*(3*a - x) + 3*MB) --------------------------- 6 - \frac{x^{2}}{6} \left(B_{v} \left(3 a - x\right) + 3 MB\right) --- d: --------------------------------

--- e: -------------------------------- --- f: --------------------------------