def _tausmooth(omega, rovert):
ptL1 = 9
ptR1 = 11
ptL2 = 8.7*rovert - 1
ptR2 = 8.7*rovert + 1
if omega < ptL1:
C_tau = np.sqrt(1.0 + 42.0/omega**3 - 42.0/10**3)
elif omega >= ptL1 and omega <= ptR1:
fL = np.sqrt(1.0 + 42.0/ptL1**3 - 42.0/10**3)
fR = 1.0
gL = -63.0/ptL1**4/fL
gR = 0.0
C_tau = cubic_spline_eval(ptL1, ptR1, fL, fR, gL, gR, omega)
elif omega > ptR1 and omega < ptL2:
C_tau = 1.0
elif omega >= ptL2 and omega <= ptR2:
fL = 1.0
fR = 1.0/3.0*np.sqrt(ptR2/rovert) + 1 - np.sqrt(8.7)/3
gL = 0.0
gR = 1.0/6/np.sqrt(ptR2*rovert)
C_tau = cubic_spline_eval(ptL2, ptR2, fL, fR, gL, gR, omega)
else:
C_tau = 1.0/3.0*np.sqrt(omega/rovert) + 1 - np.sqrt(8.7)/3
return C_tau
def _cxsmooth(omega, rovert):
Cxb = 6.0 # clamped-clamped
constant = 1 + 1.83/1.7 - 2.07/1.7**2
ptL1 = 1.7-0.25
ptR1 = 1.7+0.25
ptL2 = 0.5*rovert - 1.0
ptR2 = 0.5*rovert + 1.0
ptL3 = (0.5+Cxb)*rovert - 1.0
ptR3 = (0.5+Cxb)*rovert + 1.0
if omega < ptL1:
Cx = constant - 1.83/omega + 2.07/omega**2
elif omega >= ptL1 and omega <= ptR1:
fL = constant - 1.83/ptL1 + 2.07/ptL1**2
fR = 1.0
gL = 1.83/ptL1**2 - 4.14/ptL1**3
gR = 0.0
Cx = cubic_spline_eval(ptL1, ptR1, fL, fR, gL, gR, omega)
elif omega > ptR1 and omega < ptL2:
Cx = 1.0
elif omega >= ptL2 and omega <= ptR2:
fL = 1.0
fR = 1 + 0.2/Cxb*(1-2.0*ptR2/rovert)
gL = 0.0
gR = -0.4/Cxb/rovert
Cx = cubic_spline_eval(ptL2, ptR2, fL, fR, gL, gR, omega)
elif omega > ptR2 and omega < ptL3:
Cx = 1 + 0.2/Cxb*(1-2.0*omega/rovert)
elif omega >= ptL3 and omega <= ptR3:
fL = 1 + 0.2/Cxb*(1-2.0*ptL3/rovert)
fR = 0.6
gL = -0.4/Cxb/rovert
gR = 0.0
Cx = cubic_spline_eval(ptL3, ptR3, fL, fR, gL, gR, omega)
else:
Cx = 0.6
return Cx
def _sigmasmooth(omega, E, rovert):
Ctheta = 1.5 # clamped-clamped
ptL = 1.63*rovert*Ctheta - 1
ptR = 1.63*rovert*Ctheta + 1
if omega < 20.0*Ctheta:
offset = (10.0/(20*Ctheta)**2 - 5/(20*Ctheta)**3)
Cthetas = 1.5 + 10.0/omega**2 - 5/omega**3 - offset
sigma = 0.92*E*Cthetas/omega/rovert
elif omega >= 20.0*Ctheta and omega < ptL:
sigma = 0.92*E*Ctheta/omega/rovert
elif omega >= ptL and omega <= ptR:
alpha1 = 0.92/1.63 - 2.03/1.63**4
fL = 0.92*E*Ctheta/ptL/rovert
fR = E*(1.0/rovert)**2*(alpha1 + 2.03*(Ctheta/ptR*rovert)**4)
gL = -0.92*E*Ctheta/rovert/ptL**2
gR = -E*(1.0/rovert)*2.03*4*(Ctheta/ptR*rovert)**3*Ctheta/ptR**2
sigma = cubic_spline_eval(ptL, ptR, fL, fR, gL, gR, omega)
else:
alpha1 = 0.92/1.63 - 2.03/1.63**4
sigma = E*(1.0/rovert)**2*(alpha1 + 2.03*(Ctheta/omega*rovert)**4)
return sigma
def _buckling_reduction_factor(alpha, beta, eta, lambda_0, lambda_bar):
"""
Computes a buckling reduction factor used in Eurocode shell buckling formula.
"""
lambda_p = np.sqrt(alpha/(1.0-beta))
ptL = 0.9*lambda_0
ptR = 1.1*lambda_0
if (lambda_bar < ptL):
chi = 1.0
elif lambda_bar >= ptL and lambda_bar <= ptR: # cubic spline section
fracR = (ptR-lambda_0)/(lambda_p-lambda_0)
fL = 1.0
fR = 1-beta*fracR**eta
gL = 0.0
gR = -beta*eta*fracR**(eta-1)/(lambda_p-lambda_0)
chi = cubic_spline_eval(ptL, ptR, fL, fR, gL, gR, lambda_bar)
elif lambda_bar > ptR and lambda_bar < lambda_p:
chi = 1.0 - beta*((lambda_bar-lambda_0)/(lambda_p-lambda_0))**eta
else:
chi = alpha/lambda_bar**2
# if (lambda_bar <= lambda_0):
# chi = 1.0
# elif (lambda_bar >= lambda_p):
# chi = alpha/lambda_bar**2
# else:
# chi = 1.0 - beta*((lambda_bar-lambda_0)/(lambda_p-lambda_0))**eta
return chi