Global/local probability of failure

In the load bearing capacity of masonry brickwork there are several structural elements with a well-defined load bearing function ( walls under centric or excentral vertical loading, walls under horizontal and centric or excentral vertical loading (shear walls for withstanding wind loading) and walls submitted to bending in their plane. Modern codes focus on the design of these structural elements, taking into account the three-dimensional layout and the mutual interaction of the different elements.
In a system reliability analysis, a similar analysis should be carried out, considering further more the uncertainties on the different variables used in the design methods and the uncertainties on the methods themselves. Calculating the global probability of failure is however, at least for the moment, a problem too complicated to handle and, therefore, in the first research results the structural reliability method is applied to a single load bearing structural element: specifically, a so-called shear walls. For this structural element, the local probability of failure pf or reliability index b will be calculated.

Two hypotheses are presumed in the calculation of the probability of failure:

  • failure according to hypothesis one: failure = load-bearing capacity of a structural masonry element is lower then the loadings during the life time of the structure,
  • failure according to hypothesis two: failure = load-bearing capacity in a random point of a structural masonry element is smaller than the loading at that point during the life time of the structure.
The input of a certain reference period is required for calculating time invariant probability distribution functions for the loadings.
Hypothesis two avoids the problem of calculating the global probability of failure, and is illustrated on the so-called shear-walls.
Calculation the global probability of failure according to hypothesis two, involves a joining of:
  • Stress-analysis: in which, given the loadings (S) and the distribution of the stiffness of the wall (E), the stresses in different points of the wall are calculated on a deterministic manner, using a finite element analysis.
  • a probabilistic model: in which the calculated stresses are compared with the available load- bearing capacity (R) of the masonry on that place, which is function of the distribution of the strength of the material, cavities, humidity, workmanship, summarized as the overall quality of the wall. The probabilistic model calculates the probability of failure by means of a first order reliability method.
The joining of these two steps is required to evaluate the influence of local failure on the remaining parts of the masonry. Local failure will evidently cause a stress redistribution in the structural element under consideration. These stresses have to be calculated again by means of the finite element analysis, which has to take into account the masonry elements that have failed. From the calculated probabilities of failure, a global probability of failure can be extracted using the formulae of the total chance.

This research tries to extend the reliability analysis to the calculation of the global probability of failure of a structural masonry element. Therefore, the joining of the finite element analysis with the FORM-algorithm will be established. To obtain this goal, the finite element program CALFEM (Computer Aided Learning of the Finite Element Method, developed at the Lund University, Sweden) as implemented in Matlab, will by used to realize the joining with the FORM (First Order Reliability Method), which also has been implemented in the Matlab environment.


Copyright 1996, Katholieke Universiteit Leuven
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URL: http://www.bwk.kuleuven.ac.be/bwk/materials/ls02.htm