4.2 Inexact damage state description (imperfect limit-state function)
4.2.5 Alternative damage models for uncoupled damage analysis
4.2.5.2 Utilizing multidimensional fragility functions
analysis, since the components that are otherwise equal may have different dead load, necessitating usage of several fragility functions (each for each dead load value) instead of one. Therefore, the improvement of the uncoupled damage analysis here comes at some cost, and it still does not match the performance of the coupled damage analysis.
where dy0 is the yield curvature at zero axial force and Pa0 is the yield compressive axial force at zero moment (marked as PYC in Figure 4.16). Thus, the “yield” damage state is assumed to be caused by either excessive moment or excessive axial force. We shall see how this function compares to the exact one by considering a 2-D space in the same coordinates as before, M0y andPatj.
There exist two different cases with respect to relative magnitudes of the maximum curvature and the maximum compressive axial force. In a case where the maximum axial force is relatively small, the safety region for the inexact limit-state function does not change and is found exactly as in the case of one-dimensional fragility function. For the exact limit-state function, the knowledge of the maximum axial force observed during the analysis does not directly affect the shape of the safety region either.
However, it puts an upper bound on the value ofPatj, effectively changing its distribution.
It also excludes a part of the 2-D space (i.e., the part corresponding to inequalityPatj>
Pmax) out of consideration. Figure 4.30 shows how the knowledge of maximum force affects the analysis in this case. Note that the conditional PDF of Patj is not a cut and inflated part of the unconditional PDF, since the conditioning here is on the event Pati≤ Pmax, i = 0…Nts where Nts is the number of time steps in the dynamic structural analysis rather than on the event Patj≤ Pmax, also axial forces at different time steps are correlated and, therefore, can not be discarded in the conditioning. It can be seen that introduction of the axial force as a second EDP improves the uncoupled damage analysis performance.
The improvement is achieved primarily due to the down-shift of the range of most likely
values of Patj, which makes the situation similar to one presented in Figure 4.29 where the non-overlapping areas of the two safety regions partially offset each other.
Pmax
Part of the space that does not affect damage occurrence
PDF of Patj conditional on Pmax tj
Pa
dmaxK0 0
M
y Unconditional PDF of PatjFigure 4.30 Safety regions for exact (hatched) limit-state function and two-dimensional approximate (shaded) limit-state function, axial force is relatively small
M
Pmax
dmaxK0
Pa0
, max
0P
My
Yield surface
Pa
Figure 4.31 Procedure of finding safety region for two dimensional fragility function, axial force is relatively high
Next, consider a case where the maximum axial force is relatively large. Figure 4.31 shows how the safety region is found in this case. It can be seen that whether damage occurs or not is controlled by the maximum axial force, that is, if Pmax < Pa0, the yield moment at zero axial force is high enough to ensure that the inequality M0y >
0 maxK
d is satisfied also, so according to (4.11), no damage is assumed. Therefore, the safety zone is given by M0y > M0y,Pmax, where My0,Pmax is the yield moment at zero axial force that makes the yield force at zero moment equal the maximum observed axial force,
0
Pa = Pmax. Note that this safety region is smaller than for the case of the one-dimensional fragility function, since M0y,Pmaxis greater than dmaxK0.
] [Pa E
M
ydmaxK0
E [Pati]
, max
0P
Ma
Pmax
Yield surface
ti
Pa
Figure 4.32 Safety regions for exact (hatched) limit-state function and two-dimensional approximate (shaded) limit-state function, axial force relatively large
Figure 4.32 pictures the two safety zones for this case. As before, the shaded area marks the safety region for the approximate limit-state function and the hatched area depicts the safety region for the exact limit-state function. It can be seen from the figure
that discrepancy between the two safety regions is greater than in the case of one- dimensional fragility function, since the safety zone for the approximate function is moved to the right as compared to Figure 4.25 and the safety zone for the exact limit- state function is approximately the same.
This analysis shows that it is unclear whether using the two-dimensional fragility function reduces the discrepancy between exact and inexact limit-state functions. For the particular form of 2-D limit-state function considered in this section, the overall effect depends on the distribution of maximum axial forces and maximum curvatures throughout the whole structure. If the case with relatively low maximum axial force is prevailing, the damage estimates for the uncoupled damage analysis will be closer to the exact solution; for the other case they will probably be less accurate than the ones obtained through the original one-dimensional fragility. Note that developing and using two-dimensional fragility functions is much more complicated than one-dimensional ones, therefore, one have to be careful when implementing such functions, since the benefits might be questionable.
In summary, we can see that there are ways to improve the uncoupled damage analysis, but they require additional efforts that have to be measured against the benefits they provide. The merits of different ways to implement the uncoupled damage analysis have to be considered on a case by case basis. For the present study, we select the implementation that requires approximately the same effort as the coupled damage analysis. Therefore, in this way, the comparison of the two approaches is valid, since it is made on the “apple to apple” basis.
5 Damage estimation coupled with structural analysis (multiple damage states)
In this chapter, we shall compare the results obtained by the three methods of damage estimation for the case of multiple damage states. For all three methods, we use the exact limit state. Therefore, only the error caused by the double sampling of structural properties is present. As before, we use a reinforced-concrete moment frame shown in Figure 4.5 as a case study. A more advanced structural model of the frame is used. We shall describe the modifications of the structural model next.