def _beami_stiffness(prop: Union[PBAR, PBARL, PBEAM, PBEAML],
mat: MAT1,
L: float,
Iy: float,
Iz: float,
k1: Optional[float] = None,
k2: Optional[float] = None):
"""gets the ith Euler-Bernoulli beam stiffness"""
E = mat.E()
G = mat.G()
A = prop.Area()
J = prop.J()
kaxial = E * A / L
ktorsion = G * J / L
L2 = L * L
if k1 is not None:
phiy = 12. * E * Iy / (k1 * G * A * L2)
if k2 is not None:
phiz = 12. * E * Iy / (k2 * G * A * L2)
phiy = 1.0
phiz = 1.0
ky = E * Iy / (L * phiy)
kz = E * Iz / (L * phiz)
K = np.zeros((12, 12), dtype='float64')
# axial
K[0, 0] = K[6, 6] = kaxial
K[6, 0] = K[0, 6] = -kaxial
# torsion
K[3, 3] = K[9, 9] = ktorsion
K[9, 3] = K[3, 9] = -ktorsion
#Fx - 0, 6
#Fy - 1, 7**
#Fz - 2, 8
#Mx - 3, 9
#My - 4, 10
#Mz - 5, 11**
# bending z (Fy/1/7, Mz/5/11)
# 1 5 7 11
# 1 [12 & 6L & -12 & 6L
# 5 [6L & 4L^2 & -6L & 2L^2
# 7 [-12 &-6L & 12 & -6L
# 11 [6L & 2L^2 & -6L & 4L^2
K[1, 1] = K[7, 7] = 12. * kz
K[1, 7] = K[1, 7] = -12. * kz
K[1, 5] = K[5, 1] = K[11, 1] = K[1, 11] = 6. * L * kz
K[5, 7] = K[7, 5] = K[7, 11] = K[11, 7] = -6. * L * kz
K[5, 11] = K[11, 5] = 2. * L2 * kz * (2 - phiz)
K[5, 5] = K[11, 11] = 4. * L2 * kz * (4 + phiz)
#Fx - 0, 6
#Fy - 1, 7
#Fz - 2, 8**
#Mx - 3, 9
#My - 4, 10**
#Mz - 5, 11
# bending y (Fz/2/8, My/4/10)
# 2 4 8 10
# 2 [12 & 6L & -12 & 6L
# 4 [6L & 4L^2 & -6L & 2L^2
# 8 [-12 &-6L & 12 & -6L
# 10 [6L & 2L^2 & -6L & 4L^2
K[2, 2] = K[8, 8] = 12. * ky
K[2, 8] = K[2, 8] = -12. * ky
K[2, 4] = K[4, 2] = K[10, 2] = K[2, 10] = 6. * L * ky
K[4, 8] = K[8, 4] = K[8, 10] = K[10, 8] = -6. * L * ky
K[4, 10] = K[10, 4] = 2. * L * L * ky * (2. - phiy)
K[4, 4] = K[10, 10] = 4. * L * L * ky * (4. + phiy)
return K
