def _tausmooth(omega, rovert):
ptL1 = 9
ptR1 = 11
ptL2 = 8.7 * rovert - 1
ptR2 = 8.7 * rovert + 1
if omega < ptL1:
C_tau = sqrt(1.0 + 42.0 / omega**3 - 42.0 / 10**3)
elif omega >= ptL1 and omega <= ptR1:
fL = 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 * sqrt(ptR2 / rovert) + 1 - sqrt(8.7) / 3
gL = 0.0
gR = 1.0 / 6 / sqrt(ptR2 * rovert)
C_tau = cubic_spline_eval(ptL2, ptR2, fL, fR, gL, gR, omega)
else:
C_tau = 1.0 / 3.0 * sqrt(omega / rovert) + 1 - sqrt(8.7) / 3
return C_tau
def _tausmooth(omega, rovert):
ptL1 = 9
ptR1 = 11
ptL2 = 8.7*rovert - 1
ptR2 = 8.7*rovert + 1
if omega < ptL1:
C_tau = sqrt(1.0 + 42.0/omega**3 - 42.0/10**3)
elif omega >= ptL1 and omega <= ptR1:
fL = 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*sqrt(ptR2/rovert) + 1 - sqrt(8.7)/3
gL = 0.0
gR = 1.0/6/sqrt(ptR2*rovert)
C_tau = cubic_spline_eval(ptL2, ptR2, fL, fR, gL, gR, omega)
else:
C_tau = 1.0/3.0*sqrt(omega/rovert) + 1 - 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 _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 ApiFb(E, fy, D, t):
"""Allowable axial stress in bending"""
#Make sure we can process single floats besides arrays
fyy = np.array([fy])
EE = np.array([E])
DD = np.array([D])
tt = np.array([t])
n = len(fyy)
Fb = np.zeros(n)
for i in range(n):
dt = (DD / tt).squeeze()[i]
fy2 = fyy.squeeze()[i]
E = EE.squeeze()[i]
delta = 0.2 * 10340.0 / fy2
x1 = 10340.0 / fy2
x2 = 20680.0 / fy2
if dt < x1 - delta:
Fb[i] = 0.75 * fy2
elif dt < x1 + delta:
Fb[i] = cubic_spline_eval(x1 - delta, x1 + delta, 0.75 * fy2,
(0.84 - 1.74 * fy2 * (x1 + delta) / E) *
fy2, 0.0, -1.74 * fy2 / E * fy2, dt)
elif dt < x2 - delta:
Fb[i] = (0.84 - 1.74 * fy2 * dt / E) * fy2
elif dt < x2 + delta:
Fb[i] = cubic_spline_eval(
x2 - delta, x2 + delta,
(0.84 - 1.74 * fy2 * (x2 - delta) / E) * fy2,
(0.72 - 0.58 * fy2 * (x2 + delta) / E) * fy2,
-1.74 * fy2 / E * fy2, -0.58 * fy2 / E * fy2, dt)
else:
Fb[i] = (0.72 - 0.58 * fy2 * dt / E) * fy2
## Fb=(0.72 - 0.58*fy2*dt/E)*fy2 #initialize
## idx=(DD/tt) < (x1+delta)
## Fb[idx]=(cubic_spline_eval(x1-delta, x1+delta, 0.75*fy2, (0.84 - 1.74*fy2*(x1+delta)/E)*fy2,
## 0.0, -1.74*fy2/E*fy2, dt))[idx]
## idx=(DD/tt) < (x1-delta)
## Fb[idx]=0.75*fy2[idx]
##
## idx=(DD/tt) < (x2+delta)
return Fb
def ApiFb(E,fy,D,t):
"""Allowable axial stress in bending"""
#Make sure we can process single floats besides arrays
fyy=np.array([fy])
EE=np.array([E])
DD=np.array([D])
tt=np.array([t])
n = len(fyy)
Fb = np.zeros(n)
for i in range(n):
dt = (DD/tt).squeeze()[i]
fy2 = fyy.squeeze()[i]
E = EE.squeeze()[i]
delta = 0.2*10340.0/fy2
x1 = 10340.0/fy2
x2 = 20680.0/fy2
if dt < x1-delta:
Fb[i] = 0.75*fy2
elif dt < x1+delta:
Fb[i] = cubic_spline_eval(x1-delta, x1+delta, 0.75*fy2, (0.84 - 1.74*fy2*(x1+delta)/E)*fy2,
0.0, -1.74*fy2/E*fy2, dt)
elif dt < x2-delta:
Fb[i] = (0.84 - 1.74*fy2*dt/E)*fy2
elif dt < x2+delta:
Fb[i] = cubic_spline_eval(x2-delta, x2+delta, (0.84 - 1.74*fy2*(x2-delta)/E)*fy2,
(0.72 - 0.58*fy2*(x2+delta)/E)*fy2, -1.74*fy2/E*fy2, -0.58*fy2/E*fy2, dt)
else:
Fb[i] = (0.72 - 0.58*fy2*dt/E)*fy2
## Fb=(0.72 - 0.58*fy2*dt/E)*fy2 #initialize
## idx=(DD/tt) < (x1+delta)
## Fb[idx]=(cubic_spline_eval(x1-delta, x1+delta, 0.75*fy2, (0.84 - 1.74*fy2*(x1+delta)/E)*fy2,
## 0.0, -1.74*fy2/E*fy2, dt))[idx]
## idx=(DD/tt) < (x1-delta)
## Fb[idx]=0.75*fy2[idx]
##
## idx=(DD/tt) < (x2+delta)
return Fb
def _buckling_reduction_factor(alpha, beta, eta, lambda_0, lambda_bar):
"""
Computes a buckling reduction factor used in Eurocode shell buckling formula.
"""
lambda_p = 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
def _buckling_reduction_factor(alpha, beta, eta, lambda_0, lambda_bar):
"""
Computes a buckling reduction factor used in Eurocode shell buckling formula.
"""
lambda_p = 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
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
