jueves, 25 de febrero de 2010

Simulation Based Reliability Assesment (SBRA)

This is the resume of the papers:
Both developed under supervision of Pavel Marek (see previous post about structural engineering conception).

The first one is very interesting since allows comparison between the actual "State of the Art" (i.e. Limit State approach to design), and the proposed explicitly probabilistic approach.
It describes a step-by-step process of analysis and design comparing both methods:
  • Loading and load combination
    • LRFD/Limit States
      • Each load is expressed by nominal value and its load factor
      • Load combinations are determined according to rules established in the Codes
    • SBRA
      • Each load is expressed by a load duration curve and its corresponding histogram
      • Load combination employs Monte Carlo sampling over some 10 million iterations of the analysis
  • Resistance and reference values
    • LRFD/Limit States
      • Resistances are combined with different resistance factors associated with failure mechanisms (yield stress, compression, flexure, shear,...) also provided by regulations
      • Design capacity=Nominal capacity x provided failure factor
    • SBRA
      • Reference value corresponds to the onset of the stress/deformation curve of the material when reaching yielding, or to a tolerable deformation
      • An histogram of yield stresses is used to feed each analysis iteration altogether with the histograms for the loads mentioned before
  • Frame analysis
    • LRFD/Limit States
      • One single iteration is made
      • Direct 2nd order analysis can be used, in order to check stabilities, but the most common approach is to do a 1st order analysis and then modificate it
    • SBRA
      • Analysis is repeated 10 million times using each time the values for loads and resistances extracted from the corresponding histograms. In these histograms, the more probable a load or resistance value, the more often will be used. An histogram is also called Probability Density Function in Monte Carlo literature.
      • Initial imperfections are explicitly taken into account
      • There is no need to check individual stability of columns (2nd order)
      • Resistance is related to the onset of yielding
  • Safety assessment
    • LRDF/Limit States
      • Once analysis is computed, for each element we check that the relationship Demand/Capacity is smaller than one
    • SBRA
      • Probability of failure Pf is compared against target probability Pd provided by the codes
      • P[(RV-LE)<0]=Pf
        • RV=yielding stress
        • LE=calculated stress
The proposal is very interesting in terms of  providing a controlled design and an explicit probabilistic model of the structure.

Nevertheless, anyone would encounter the main drawback in the need for iterating 10 million times the whole structure, specially when it comes to real-time Lagrangian dynamics, as we are aimed to.
The steps for modelling the forces and displacements remain the same, it is, the traditional stiffness matrix.
Further research shows that the main time-consuming step into the stiffness matrix procedure (also applicable to FEM) is that of the assembly of the stiffness matrix (60-80%).
This means that, once a configuration is achieved and assembled, solving the matrix with different load cases should be relatively quick. Which is a relief if we want to implement SBRA.
Nevertheless, I still feel like I should pay deeper attention to this speed subject, and obtain my own benchmarking.

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