Unequal Leg Angle Bending - Symmetry Parameter Bw (CSA S16-19) 3

[CivilSigma]

Hi Everyone,

I am trying to evaluate the bending capacity of an unequal length angle (L 127x89x9.5.
The CSA design equation requires you to calculate the symmetry parameter of the section and that involves the evaluation of an area integral. Please see the attached screenshot.

Does anyone know of a solved example or reference for the evaluation of this variable?

Thank you

 

[Celt83]

attached is the resource I have for an approximate solution. Link



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https://github.com/buddyd16/Structural-Engineering

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https://github.com/open-struct-engineer

 

[Celt83]

to do the actual integral convert the problem to a sum of line integrals for each segment defining the perimeter of the angle.
Green's Theorem for line integrals
you'll need to find functions for P and Q such that the difference of the partial derivates = z(w^2+z^2)

z and w can the be written in parametric form for each edge vertices as, each (z,w) pair is a transformation of the (x,y) coordinates to the principal axis angle:
w(t) = w1+t(w2-w1)
z(t) = z1+t(z2-z1)

and then back substituted into P and Q so the integral is from 0 to 1 dt instead of dz or dw.

My Personal Open Source Structural Applications:

https://github.com/buddyd16/Structural-Engineering

Open Source Structural GitHub Group:

https://github.com/open-struct-engineer

 

[Celt83]

oh you'll also need to sub in the change of variable terms for dz or dw of:

dw/dt = (w2-w1) => dw = (w2-w1)dt
dz/dt = (z2-z1) => dz = (z2-z1)dt

My Personal Open Source Structural Applications:

https://github.com/buddyd16/Structural-Engineering

Open Source Structural GitHub Group:

https://github.com/open-struct-engineer

 

[skeletron]

Thanks for the reference Celt! Where is that from anyway?

 

[Celt83]

I honestly don't remember was the result of some late night googling a few years ago, have always had it earmarked to do a spreadsheet then I got sidetracked with other stuff.

My Personal Open Source Structural Applications:

https://github.com/buddyd16/Structural-Engineering

Open Source Structural GitHub Group:

https://github.com/open-struct-engineer

 

[ClearCalcs]

The Commentary for AISC 360 also includes values of Beta_w for most typical angle sizes. You can find it for free online here - page 404 in the PDF.

-Laurent

http://www.ClearCalcs.com

 

[CivilSigma]

Amazing references! To be honest, I changed the design to use equal leg angles as it made for an easier calculation.
I will have to find time and go through the calculations in the references shown!

Much appreciated

 

[Celt83]

Sheet to compute Beta,w: Link

L8x4:

L4x3 1/2:

L2 1/2 x 1 1/2: (This one is actually sensitive to the thickness)

Equal Leg: (Numerical precision -- 1.33 E -15 is pretty much 0)

Formula for Each linear edge of the angle defined by start point (w1,z1) and end point (w2,z2):
For Green's Theorem assume P=0, so need to integrate the function with respect to w. Then perform the parametric substitutions and change of variable to dt.

The sheet models the fillets with several straight line segments.

My Personal Open Source Structural Applications:

https://github.com/buddyd16/Structural-Engineering

Open Source Structural GitHub Group:

https://github.com/open-struct-engineer