'''
BSD 3-Clause License
Copyright (c) 2019, Donald N. Bockoven III
All rights reserved.
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modification, are permitted provided that the following conditions are met:
* Redistributions of source code must retain the above copyright notice, this
list of conditions and the following disclaimer.
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and/or other materials provided with the distribution.
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this software without specific prior written permission.
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AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
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'''
from __future__ import division
import math as m
#Torsion Equitions from AISC Design Guide 9 Appendix C
#Case 1 - Concentrated End Torques with Free Ends
#T = Applied Concentrated Torsional Moment, Kip-in
#G = Shear Modulus of Elasticity, Ksi, 11200 for steel
#J = Torsinal Constant of Cross Section, in^4
class torsion_case1:
def __init__(self, T_k_in, G_ksi, J_in4):
self.T_k_in = T_k_in
self.G_ksi = G_ksi
self.J_in4 = J_in4
def theta(self,z_in):
T = self.T_k_in
G = self.G_ksi
J = self.J_in4
z = z_in
thet = (T*z) / (G*J)
return thet
def thetap(self,z_in):
theta_prime = T / (G*J)
return theta_prime
def thetapp(self,z_in):
theta_doubleprime = 0
return theta_doubleprime
#Case 2 - Concentrated End Torques with Fixed Ends
#T = Applied Concentrated Torsional Moment, Kip-in
#G = Shear Modulus of Elasticity, Ksi, 11200 for steel
#J = Torsinal Constant of Cross Section, in^4
#l = Span Lenght, in
#a = Torsional Constant
class torsion_case2:
def __init__(self, T_k_in, G_ksi, J_in4, l_in, a):
self.T_k_in = T_k_in
self.G_ksi = G_ksi
self.J_in4 = J_in4
self.l_in = l_in
self.a = a
def theta(self,z_in):
T = self.T_k_in
G = self.G_ksi
J = self.J_in4
l = self.l_in
a = self.a
z = z_in
thet = ((T*a) / (G*J))*((m.tanh(l/(2*a))*m.cosh(z/a))-(m.tanh(l/(2*a)))+(z/a)-(m.sinh(z/a)))
return thet
def thetap(self,z_in):
T = self.T_k_in
G = self.G_ksi
J = self.J_in4
l = self.l_in
a = self.a
z = z_in
theta_prime = (T - T*m.cosh(z/a) + T*m.sinh(z/a)*m.tanh(l/(2*a)))/(G*J)
return theta_prime
def thetapp(self,z_in):
T = self.T_k_in
G = self.G_ksi
J = self.J_in4
l = self.l_in
a = self.a
z = z_in
theta_doubleprime = (-(T*m.sinh(z/a)) + T*m.cosh(z/a)*m.tanh(l/(2*a)))/(a*G*J)
return theta_doubleprime
def thetappp(self,z_in):
T = self.T_k_in
G = self.G_ksi
J = self.J_in4
l = self.l_in
a = self.a
z = z_in
theta_tripleprime = (-(T*m.cosh(z/a)) + T*m.sinh(z/a)*m.tanh(l/(2*a)))/(G*J*a**2)
return theta_tripleprime
#Case 3 - Concentrated Torque at alpha*l with Pinned Ends
#T = Applied Concentrated Torsional Moment, Kip-in
#G = Shear Modulus of Elasticity, Ksi, 11200 for steel
#J = Torsinal Constant of Cross Section, in^4
#l = Span Lenght, in
#a = Torsional Constant
#alpa = load application point/l
class torsion_case3:
def __init__(self, T_k_in, G_ksi, J_in4, l_in, a, alpha):
self.T_k_in = T_k_in
self.G_ksi = G_ksi
self.J_in4 = J_in4
self.l_in = l_in
self.a = a
self.alpha = alpha
def theta(self,z_in):
T = self.T_k_in
G = self.G_ksi
J = self.J_in4
l = self.l_in
a = self.a
alpha = self.alpha
z = z_in
if 0 <= z_in <= (alpha*l):
thet = ((T*l) / (G*J))*(((1.0-alpha)*(z/l))+(((a/l)*((m.sinh((alpha*l)/a)/m.tanh(l/a)) - m.cosh((alpha*l)/a)))*m.sinh(z/a)))
else:
thet = ((T*l) / (G*J))*(((l-z)*(alpha/l))+((a/l)*(((m.sinh((alpha*l)/a) / m.tanh(l/a))*m.sinh(z/a)) - (m.sinh((alpha*l)/a)*m.cosh(z/a)))))
return thet
def thetap(self,z_in):
T = self.T_k_in
G = self.G_ksi
J = self.J_in4
l = self.l_in
a = self.a
alpha = self.alpha
z = z_in
if 0 <= z_in <= (alpha*l):
theta_prime = ((-1.0*T)/(G*J))*(alpha - 1.0 + (m.cosh(z/a)*((m.cosh((l*alpha)/a) - (m.sinh((l*alpha)/a)/m.tanh(l/a))))))
else:
theta_prime = -1.0*((T*(alpha - m.cosh(z/a)*m.sinh((l*alpha)/a)/m.tanh(l/a) + m.sinh(z/a)*m.sinh((l*alpha)/a))/(G*J)))
return theta_prime
def thetapp(self,z_in):
T = self.T_k_in
G = self.G_ksi
J = self.J_in4
l = self.l_in
a = self.a
alpha = self.alpha
z = z_in
if 0 <= z_in <= (alpha*l):
theta_doubleprime = -((T*m.sinh(z/a)*(m.cosh((l*alpha)/a) - m.sinh((l*alpha)/a)/m.tanh(l/a)))/(a*G*J))
else:
theta_doubleprime = (T*(-1.0*m.cosh(z/a) + m.sinh(z/a)/m.tanh(l/a))*m.sinh((l*alpha)/a))/(a*G*J)
return theta_doubleprime
def thetappp(self,z_in):
T = self.T_k_in
G = self.G_ksi
J = self.J_in4
l = self.l_in
a = self.a
alpha = self.alpha
z = z_in
if 0 <= z_in <= (alpha*l):
theta_tripleprime = -((T*m.cosh(z/a)*(m.cosh((l*alpha)/a) - m.sinh((l*alpha)/a)/m.tanh(l/a)))/(G*J*a**2))
else:
theta_tripleprime = (T*(m.cosh(z/a)/m.tanh(l/a) - m.sinh(z/a))*m.sinh((l*alpha)/a))/(G*J*a**2)
return theta_tripleprime
#Case 4 - Uniformly Distributed Torque with Pinned Ends
#t = Distributed torque, Kip-in / in
#G = Shear Modulus of Elasticity, Ksi, 11200 for steel
#J = Torsinal Constant of Cross Section, in^4
#l = Span Lenght, in
#a = Torsional Constant
class torsion_case4:
def __init__(self, t_k_inpin, G_ksi, J_in4, l_in, a):
self.t_k_inpin = t_k_inpin
self.G_ksi = G_ksi
self.J_in4 = J_in4
self.l_in = l_in
self.a = a
def theta(self,z_in):
t = self.t_k_inpin
G = self.G_ksi
J = self.J_in4
l = self.l_in
a = self.a
z = z_in
thet = ((t*a**2) / (G*J))*(((l**2 / (2*a**2))*((z/l) - ((z**2)/(l**2))))+m.cosh(z/a)-(m.tanh(l/(2*a))*m.sinh(z/a))-1.0)
return thet
def thetap(self,z_in):
t = self.t_k_inpin
G = self.G_ksi
J = self.J_in4
l = self.l_in
a = self.a
z = z_in
theta_prime = (t*(l - 2*z + 2*a*m.sinh(z/a) - 2*a*m.cosh(z/a)*m.tanh(l/(2*a))))/(2*G*J)
return theta_prime
def thetapp(self,z_in):
t = self.t_k_inpin
G = self.G_ksi
J = self.J_in4
l = self.l_in
a = self.a
z = z_in
theta_doubleprime = (t*(-1 + m.cosh(z/a) - m.sinh(z/a)*m.tanh(l/(2*a))))/(G*J)
return theta_doubleprime
def thetappp(self,z_in):
t = self.t_k_inpin
G = self.G_ksi
J = self.J_in4
l = self.l_in
a = self.a
z = z_in
theta_tripleprime = (t*(m.sinh(z/a) - m.cosh(z/a)*m.tanh(l/(2*a))))/(a*G*J)
return theta_tripleprime
#Case 5 - Linearly Varying Torque with Pinned Ends - 0 at left to t at right
#t = Distributed torque, Kip-in / in
#G = Shear Modulus of Elasticity, Ksi, 11200 for steel
#J = Torsinal Constant of Cross Section, in^4
#l = Span Lenght, in
#a = Torsional Constant
class torsion_case5:
def __init__(self, t_k_inpin, G_ksi, J_in4, l_in, a):
self.t_k_inpin = t_k_inpin
self.G_ksi = G_ksi
self.J_in4 = J_in4
self.l_in = l_in
self.a = a
def theta(self,z_in):
t = self.t_k_inpin
G = self.G_ksi
J = self.J_in4
l = self.l_in
a = self.a
z = z_in
thet = ((t*l**2) / (G*J))*((z/(6*l)) + ((a**2)/(l**2)*((m.sinh(z/a)/m.sinh(l/a))-(z/l)))-((z**3)/(6*l**3)))
return thet
def thetap(self,z_in):
t = self.t_k_inpin
G = self.G_ksi
J = self.J_in4
l = self.l_in
a = self.a
z = z_in
theta_prime = (t*(-6.0*a**2 + l**2 - 3*z**2 + 6*a*l*(m.cosh(z/a)/m.sinh(l/a))))/(6*G*J*l)
return theta_prime
def thetapp(self,z_in):
t = self.t_k_inpin
G = self.G_ksi
J = self.J_in4
l = self.l_in
a = self.a
z = z_in
theta_doubleprime = (t*(-1.0*z + l*(m.sinh(z/a)/m.sinh(l/a))))/(G*J*l)
return theta_doubleprime
def thetappp(self,z_in):
t = self.t_k_inpin
G = self.G_ksi
J = self.J_in4
l = self.l_in
a = self.a
z = z_in
theta_tripleprime = (t*(-1 + (l*(m.cosh(z/a)/m.sinh(l/a)))/a))/(G*J*l)
return theta_tripleprime
#Case 6 - Concentrated Torque at alpha*l with Fixed Ends
#T = Applied Concentrated Torsional Moment, Kip-in
#G = Shear Modulus of Elasticity, Ksi, 11200 for steel
#J = Torsinal Constant of Cross Section, in^4
#l = Span Lenght, in
#a = Torsional Constant
#alpa = load application point/l
class torsion_case6:
def __init__(self, T_k_in, G_ksi, J_in4, l_in, a, alpha):
self.T_k_in = T_k_in
self.G_ksi = G_ksi
self.J_in4 = J_in4
self.l_in = l_in
self.a = a
self.alpha = alpha
self.H = (((1.0-m.cosh((alpha*l)/a)) / m.tanh(l/a)) + ((m.cosh((alpha*l)/a)-1.0) / m.sinh(l/a)) + m.sinh((alpha*l)/a) - ((alpha*l)/a)) / \
(((m.cosh(l/a)+(m.cosh((alpha*l)/a)*m.cosh(l/a))-m.cosh((alpha*l)/a)-1.0)/m.sinh(l/a))+((l/a)*(alpha-1.0))-m.sinh((alpha*l)/a))
def theta(self,z_in):
T = self.T_k_in
G = self.G_ksi
J = self.J_in4
l = self.l_in
a = self.a
alpha = self.alpha
H = self.H
z = z_in
if 0 <= z_in <= (alpha*l):
thet = ((T*a) / ((H+1.0)*G*J))*\
((((H*((1.0/m.sinh(l/a))+m.sinh((alpha*l)/a)-(m.cosh((alpha*l)/a)/m.tanh(l/a))))+ \
(m.sinh((alpha*l)/a) - (m.cosh((alpha*l)/a)/m.tanh(l/a)) + (1.0/m.tanh(l/a))))* \
(m.cosh(z/a)-1.0)) - \
m.sinh(z/a) + \
(z/a))
else:
thet = (T*a / ((1+(1/H))*G*J))*\
((m.cosh((alpha*l)/a) - 1.0/(H*m.sinh(l/a)) +\
(m.cosh((alpha*l)/a)-m.cosh(l/a)+((l/a)*m.sinh(l/a))) / m.sinh(l/a)) +\
m.cosh(z/a)*\
((1.0-m.cosh((alpha*l)/a))/(H*m.tanh(l/a)) +\
(1.0-(m.cosh((alpha*l)/a)*m.cosh(l/a)))/m.sinh(l/a)) +\
m.sinh(z/a)*\
(((m.cosh((alpha*l)/a)-1.0)/H) + m.cosh((alpha*l)/a)) -\
(z/a))
return thet
def thetap(self,z_in):
T = self.T_k_in
G = self.G_ksi
J = self.J_in4
l = self.l_in
a = self.a
alpha = self.alpha
H = self.H
z = z_in
if 0 <= z_in <= (alpha*l):
theta_prime = (-1.0*T*(-1.0 + m.cosh(z/a) + (-1.0 + (1.0 + H)*m.cosh((l*alpha)/a))*(m.sinh(z/a)/m.tanh(l/a)) - 1.0*H*(m.sinh(z/a)/m.sinh(l/a)) - 1.0*m.sinh(z/a)*m.sinh((l*alpha)/a) - 1.0*H*m.sinh(z/a)*m.sinh((l*alpha)/a)))/(G*(1.0 + H)*J)
else:
theta_prime = 0
return theta_prime
def thetapp(self,z_in):
T = self.T_k_in
G = self.G_ksi
J = self.J_in4
l = self.l_in
a = self.a
alpha = self.alpha
H = self.H
z = z_in
if 0 <= z_in <= (alpha*l):
theta_doubleprime = -((T*((m.cosh(z/a)*(-1.0 + (1.0 + H)*(m.cosh((l*alpha)/a))/m.tanh(l/a)))/a - (H*(m.cosh(z/a)/m.sinh(l/a)))/a + m.sinh(z/a)/a - (m.cosh(z/a)*m.sinh((l*alpha)/a))/a - (H*m.cosh(z/a)*m.sinh((l*alpha)/a))/a))/(G*(1.0 + H)*J))
else:
theta_doubleprime = 0
return theta_doubleprime
def thetappp(self,z_in):
T = self.T_k_in
G = self.G_ksi
J = self.J_in4
l = self.l_in
a = self.a
alpha = self.alpha
H = self.H
z = z_in
if 0 <= z_in <= (alpha*l):
theta_tripleprime = -((T*(m.cosh(z/a)/a**2 + ((-1.0 + (1.0 + H)*(m.cosh((l*alpha)/a))/m.tanh(l/a))*m.sinh(z/a))/a**2 - (H*(m.sinh(z/a)/m.sinh(l/a)))/a**2 - (m.sinh(z/a)*m.sinh((l*alpha)/a))/a**2 - (H*m.sinh(z/a)*m.sinh((l*alpha)/a))/a**2))/(G*(1.0 + H)*J))
else:
theta_tripleprime = 0
return theta_tripleprime
#Test Area
T = 90 #k-in
G = 11200 #ksi
J = 1.39 #in4
Cw = 2070 #in6
E = 29000 #ksi
a = ((E*Cw) / (G*J))**0.5 #Torsional Constant
l = 180 #in
alpha = 0.5
test = torsion_case6(T, G, J, l, a, alpha)
check = []
test_theta = test.theta(90)
check.append(test_theta*((G*J)/T)*(1/a))
test_thetap = test.thetap(90)
check.append(test_thetap*((G*J)/(T)))
test_thetapp = test.thetapp(90)
check.append(test_thetapp*((G*J)/T)*a)
test_thetappp = test.thetappp(90)
check.append(test_thetappp * ((G*J)/T)*a*a)
