Calculate Beta w for Single Angles

Algebraic formula for AISC property, Beta w
I used the Python package sympy
to develop a closed form solution
for beta w.
Thank you to AISC.
Thank you to slideshare presentation.
AISC v15.0 Shapes Database

Bw definition

Bw diagram

# For enhanced readability and easier translation to other software, 
# exponents are indicated with the (mathematically) traditional caret symbol.
# In Python, replace all carets with a double asterisk.

# Get single angle properties 
a  = np.arctan(my_angle.TANA) 
c  = np.cos(a) 
s  = np.sin(a) 

b  = my_angle.B_LOWERCASE 
d  = my_angle.D 
g  = my_angle.X 
h  = my_angle.Y 
t  = my_angle.T 

z0 = -s*(t/2 - g) + c*(t/2 - h) 

Iw = my_angle.IW 

# B is the integral term shown above 
# Then set 
B12 = 12 * B
# To avoid fractional coefficients from the expansion of B

# Now calculate B12, a lengthy expression
# Obviously, many of the terms may be grouped further
# Such grouping is not a priority for me

# Now
B12 = t^5*(5*s^3 -5*s^2*c +5*s*c^2 -5*c^3) + 

      t^4*(-6*s^3*h -16*s^3*g -3*s^3*b -6*s*c^2*h -16*s*c^2*g 
      -3*s*c^2*b +16*s^2*c*h +6*s^2*c*g +3*s^2*c*d +16*c^3*h 
      +6*c^3*g +3*c^3*d) + 

      t^3*(6*s^3*h^2 +12*s^3*h*g +18*s^3*g^2 +12*s^3*g*b +4*s^3*g*d -2*s^3*d^2 
      -18*s^2*c*h^2 -12*s^2*c*h*g -4*s^2*c*h*b -12*s^2*c*h*d -6*s^2*c*g^2 
      +2*s^2*c*b^2 +6*s*c^2*h^2 +12*s*c^2*h*g +18*s*c^2*g^2 +12*s*c^2*g*b 
      +4*s*c^2*g*d -2*s*c^2*d^2 -18*c^3*h^2 -12*c^3*h*g -4*c^3*h*b 
      -12*c^3*h*d -6*c^3*g^2 +2*c^3*b^2) + 

      t^2*(-12*s^3*h^2*g -6*s^3*h^2*b -12*s^3*h*g*d +6*s^3*h*b^2 +6*s^3*h*d^2 
      -12*s^3*g^3 -18*s^3*g^2*b -2*s^3*b^3 -12*s*c^2*h^2*g -6*s*c^2*h^2*b 
      -12*s*c^2*h*g*d +6*s*c^2*h*b^2 +6*s*c^2*h*d^2 -12*s*c^2*g^3 
      -18*s*c^2*g^2*b -2*s*c^2*b^3 +12*s^2*c*h^3 +12*s^2*c*h*g^2 
      +12*s^2*c*h*g*b +18*s^2*c*h^2*d -6*s^2*c*g*b^2 
      -6*s^2*c*g*d^2 + 6*s^2*c*g^2*d +2*s^2*c*d^3 +12*c^3*h^3 
      +12*c^3*h*g^2 +12*c^3*h*g*b + 18*c^3*h^2*d -6*c^3*g*b^2 
      -6*c^3*g*d^2 +6*c^3*g^2*d +2*c^3*d^3) + 

      t*(12*s^3*h^2*g*b +12*s^3*h^2*g*d -6*s^3*h^2*d^2 -12*s^3*h*g*b^2 
      -18*s^3*g^2*d^2 +4*s^3*g*b^3 +12*s^3*g*d^3 +12*s^3*g^3*b +12*s^3*g^3*d 
      -3*s^3*d^4 +12*s*c^2*h^2*g*b +12*s*c^2*h^2*g*d -6*s*c^2*h^2*d^2 
      -12*s*c^2*h*g*b^2 -18*s*c^2*g^2*d^2 +4*s*c^2*g*b^3 +12*s*c^2*g*d^3 
      +12*s*c^2*g^3*b +12*s*c^2*g^3*d -3*s*c^2*d^4 -12*s^2*c*h^3*b 
      -12*s^2*c*h^3*d -12*s^2*c*h*g^2*b -12*s^2*c*h*g^2*d +12*s^2*c*h*g*d^2 
      -12*s^2*c*h*b^3 -4*s^2*c*h*d^3 +18*s^2*c*h^2*b^2 +6*s^2*c*g^2*b^2 
      +3*s^2*c*b^4 -12*c^3*h^3*b -12*c^3*h^3*d -12*c^3*h*g^2*b -12*c^3*h*g^2*d 
      +12*c^3*h*g*d^2 -12*c^3*h*b^3 -4*c^3*h*d^3 +18*c^3*h^2*b^2 
      +6*c^3*g^2*b^2 +3*c^3*b^4) 

# From definition
B = B12 / 12.0

# Therefore
Beta_w = B / Iw - 2*z0

# Using a few example shapes with corresponding data from the 
# AISC v15.0 Shapes database
Beta_w = 3.2879 for L8x6x1
Beta_w = 3.3139 for L8x6x1/2
Beta_w = 3.3246 for L8x6x7/16
Beta_w = 1.5653 for L3x2x5/16

Leave a Reply

Your email address will not be published. Required fields are marked *