Thursday, 22 November 2018

FAQ: Utilization linearization method behind the plastic interaction formulas

The possibility of tracking back the results of the utilizations of the design checks to the actual Eurocode formulas is a very legitimate requirement from our users, and we are constantly putting great effort into each development to provide it.

On the other hand, the end result the optimization process, is of course the utilization of each structural element. Based on it, we make section changes, or structural modifications, until we reach the adequate state, below 100% utilization. The scale of our changes on the model strongly depends on the given utilization of our structure. Is a simple section change enough? Should we rethink the mechanical model of the structure?

And here where the problem comes. If we check on the plastic interaction formulas, it is clear, that the result of it is not a utilization, but a statement if the section is adequate or not. By definition, utilization shows that how far from the 100% we are, how much reserve do we have or not have. For this, the result of the formula can not be used.

To resolve this issue, ConSteel uses the Newton-Raphson iterative method to solve the nonlinear equation of the plastic interaction formulas, evaluates the load level which satisfies the equality, and compares it with the actual load level to obtain a practically useful utilization value.

Example:
Section: SHS 120x4 section
Dominant check is Plastic interaction - Biaxial bending (acc. to 6.41 formula of EC1993-1-1)
N= 565,2 kN
My= -10,2 kNm
Mz= -1,1 kNm

If we would calculate the formula with the common approach, and try to determine a utilization it seems, that we are facing with quite a big structural problem. It is clear, that the value can not be used as a utilization of the section:

So ConSteel evaluates the load level of the N-My-Mz internal forces - considering conservative one parameter loading – satisfying this formula for equality and uses it to evaluate the final utilization. 

If we make our modifications considering the 116% utilization, it turns out, that a simple section change from SHS120x4 to SHS140x is enough to go below 100%. 

This formula is being applied on all of the plastic interaction checks:
  • Major axis shear + Torsion, acc. to EC3-1-1 6.2.7
  • Minor axis shear + Torsion, acc. to EC3-1-1 6.2.7
  • Bending around major axis + Shear, acc. to EC3-1-1 6.2.8
  • Bending around minor axis + Shear, acc. to EC3-1-1 6.2.8
  • Bending around major axis + Axial force, acc. to EC3-1-1 6.2.9.1
  • Bending around minor axis + Axial force, acc. to EC3-1-1 6.2.9.1
  • Biaxial bending + Axial force, acc. to EC3-1-1 6.2.9.1