Discussion on assessment outcomes of OOP behavior of URM walls
5.1 Influence of application of different assessment methods
In this section, the influence of the application of different methods on the assessment of the OOP behavior of the walls on BL1 and BL2 of structure A is evaluated. Figure 5.1 shows the derived UC values for the walls on BL1 and BL2 of structure A per PGA value. Due to the fact that all depicted UC values do not approach the critical value of 1, the Tier 1 and the Tier 2 method, on the one hand, and the Tier 3 methods, on the other hand, cannot be directly compared. Only the Tier 1 and the Tier 2 method can be compared to each other, as well as the UC values of the different Tier 3 methods. As a result of the absence of observed OOP failures in the NLTH analyses for structure A, the UC values of the Tier 3 methods all represent the average of the total number of 11 UC values.
a) wall on BL1 b) wall on BL2
Figure 5.1: Unity check values for structure A per PGA value.
The influence of the application of the different assessment methods on the derived UC values is discussed in four parts. In the first part, the differences in UC values for the Tier 1 and the Tier 2 method are examined. Then, the Tier 3a method is compared to the Tier 3b method. Afterwards, the variations between the Tier 3b and the Tier 3c method are investigated. Finally, an attempt is made to describe how the outcomes of the Tier 1 and the Tier 2 method, on the one hand, and the Tier 3 methods, on the other hand, relate to each other.
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Tier 1 vs. Tier 2
In all cases, the Tier 1 method yields higher UC values than the Tier 2 method. This is in accordance with the expectations.
Tier 3a vs. Tier 3b
When comparing the Tier 3a and the Tier 3b method, a significant influence of the applied PGA value can be observed. For low PGA values, linear wall behavior applies. Hence, no major differences can be obtained: the Tier 3a and the Tier 3b method yield very similar results. This corresponds to the expectations. In case the wall springs remain in the elastic range, there is no difference in the dynamic response of the wall’s lumped mass for a SDOF wall system with a single spring and a SDOF wall system with two springs that are connected in parallel.
For high PGA values, on the other hand, significant differences can be seen in the UC values for the Tier 3a method and the Tier 3b method. The increase in the PGA value causes nonlinear behavior of the wall springs, which results in differences in the dynamic response of the SDOF wall systems that are incorporated in the Tier 3a and the Tier 3b method. The two springs that are connected in parallel in the SDOF wall system of the Tier 3b method start behaving nonlinear before the single spring in the SDOF wall system of the Tier 3a method starts behaving nonlinear. Hence, higher displacements are obtained from the Tier 3b method, yielding higher UC values. It is interesting to note that the onset of the nonlinear behavior of the wall springs for the wall on BL2 corresponds to lower PGA values than for the wall on BL1. This is in accordance to the expectations, since the wall on BL2 has a lower stiffness due to the lower overburden load that acts on the wall. Also, the wall is located on a higher floor. This wall on BL2 thus experiences a higher degree of dynamic amplification.
In Figures 5.2-5.5, the behavior of the wall on BL1 of structure A is illustrated for the time domain of 4 to 8 seconds of GM1 as a result of two different PGA values. The PGA value of 0.15 g corresponds to linear wall spring behavior, whereas the PGA value of 0.25 g corresponds to linear behavior of the spring in the SDOF wall system of the Tier 3a method and nonlinear behavior of the springs in the SDOF wall system of the Tier 3b method. The significant impact of the nonlinear behavior of the springs in the latter SDOF wall system on the eventual deformation of the wall, relative to the average displacement of the floor on the top and the bottom side of the analyzed wall, can be seen in Figure 5.5.
a) PGA = 0.15 g b) PGA = 0.25 g
Figure 5.2: Hysteresis of wall spring Tier 3a (GM1).
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a) PGA = 0.15 g b) PGA = 0.25 g Figure 5.3: Hysteresis of wall springs Tier 3b (GM1).
a) PGA = 0.15 g b) PGA = 0.25 g
Figure 5.4: Deformation of wall springs over time (GM1).
a) PGA = 0.15 g
Note: curves overlap
b) PGA = 0.25 g
Figure 5.5: Relative deformation of wall over time (GM1).
The difference in assessment outcomes between the Tier 3a and the Tier 3b method shows the influence of the simplification that is incorporated in the Tier 3a method, in which the analyzed URM wall is subjected by the average dynamic response of the floors at the top and the bottom side of the specific wall, compared to the situation in which both sides of the wall are subjected to different displacement time histories.
The Tier 3a method appears unsuitable for the assessment of the OOP behavior of URM walls.
The higher stiffness of the single spring, compared to the two springs that are connected in parallel,
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results in reduced UC values for the Tier 3a method. Moreover, the application of the average, and thus reduced, displacement time history to the single spring, results in an underestimated dynamic response of the wall, yielding an even higher reduction of the UC values for the specific assessment method. The UC values that correspond to the Tier 3a method are therefore unrealistically low. Only in situations in which the floors that enclose the analyzed wall show a similar dynamic response, the method yields a reasonable outcome. This is, however, not a realistic situation. Therefore, for general purposes, the Tier 3a method is considered unsuitable for the assessment of the OOP behavior of URM walls.
It can be stated that the Tier 3a method does not match the nature of NLTH analyses, since the input of an NLTH analysis may not contain simplifications that cause the modeled idealized systems to not show nonlinear behavior, or to show nonlinear behavior in a too late stage. As a result, the Tier 3a assessment method is disregarded in the further discussion. In case the discussion refers to the Tier 3 methods, this concerns only the Tier 3b and the Tier 3c method.
Tier 3b vs. Tier 3c
When evaluating the Tier 3b and the Tier 3c method, it can be seen that the Tier 3c method generally results in lower UC values than the Tier 3b method.
In Figures 5.6a and 5.6d, the displacement of floor 1 in structure A, relative to the ground displace-ment, is illustrated for the time domain of 4 to 8 seconds as a result of GM1 and GM3. The wall on BL1 of the specific structure is subjected to these displacements on its top side. Theoretically, the bottom side of the wall is subjected to the ground displacement. Due to the fact that the displace-ment of floor 1 already depicts the displacedisplace-ment relative to the ground level, however, no relative displacements apply to the bottom side of the wall. The dynamic response of the wall, as a result of the imposed displacement on its top side, is presented for GM1 and GM3 in Figures 5.6b and 5.6e, respectively.
As can be seen in Figures 5.6a and 5.6d, the relative displacement of floor 1 for the Tier 3b and the Tier 3c method is very similar in the first seconds, as a result of the fact that the idealized systems that are implemented in both assessment methods represent the same structure. Yet, at a certain moment in time, the recorded dynamic responses of floor 1 start to deviate. This moment in time corresponds to differences in yielding of BL1. The time intervals of yielding of BL1 are shown in Figures 5.6c and 5.6f. The comparable idealized systems start to show significantly different behavior from the moment when differences in yielding are observed. This phenomenon typifies the nature of nonlinear dynamic problems: minor changes in similar analyses may eventually yield significant differences in the obtained behavior. This can be referred to as the so-called butterfly effect [74].
Logically, this butterfly effect, as observed for the relative displacement of floor 1, is noticed to an even larger extent when evaluating the resulting deformations of the wall, as shown in the Figures 5.6b and 5.6e.
The difference in assessment outcomes between the Tier 3b and the Tier 3c method shows the influence of the application of the cascade approach. When employing the cascade approach, as is done in the Tier 3b method, an uncoupled analysis is performed. In such an analysis, first, the dynamic response of the overall structure is determined. Afterwards, the wall’s dynamic response is evaluated. Thus, the dynamic response of the URM wall does not affect the dynamic response of the
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a) relative displacement floor 1 (GM1) b) deformation wall BL1, A (GM1)
c) yielding of BL1 (GM1)
d) relative displacement floor 1 (GM3) e) deformation wall BL1, A (GM3)
f) yielding of BL1 (GM3)
Figure 5.6: Comparison Tier 3b and 3c (PGA = 0.25 g).
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overall structure. Instead, the Tier 3c method includes the wall in the initial idealized system. Using this approach, the response of the overall structure is dependent on the response of the analyzed URM wall. The displacement time history that is subjected to the wall, that is the structure’s response, is thus influenced by the response of the wall itself. It is expected that the Tier 3c method represents the actual situation best.
Tier 1 and 2 vs. Tier 3a, 3b and 3c
As described before, the Tier 1 and the Tier 2 method, on the one hand, and the Tier 3 methods, on the other hand, cannot be directly compared in case the corresponding UC values are considerably smaller than the critical value of 1. This is the result of the significantly different principle that is applied in order to determine the UC values, as can be noted from Eq. 40 and Eq. 41. Only in case the UC value of an assessment method approaches or exceeds the critical value of 1, the method may be compared to other assessment methods. Therefore, for comparison purposes only, the walls in structure A are subjected to unrealistically high PGA values that yield a UC value of around 1 for the Tier 2 method. Figure 5.7 shows the UC values for the wall on BL1, as a result of a PGA value of 1.85 g, and for the wall on BL2, as a result of a PGA value of 1.35 g. For both the Tier 3b and the Tier 3c method, several NLTH analyses observe OOP failure of the analyzed URM wall.
Hence, no UC value could be extracted for these analyses. The depicted ‘average’ UC values for the Tier 3b and the Tier 3c method therefore represent the average value of less than the total number of 11 UC values. The numbers in Figure 5.7 show the number of UC values that is included in the average UC values. Although the average UC values are still incorporated in the graph, it must be emphasized that the values must be considered invalid. The vertical lines depict the range of obtained UC values.
a) wall on BL1 (PGA = 1.85 g) b) wall on BL2 (PGA = 1.35 g)
Figure 5.7: Unity check values for structure A for PGA value that causes UC value of ca. 1 for Tier 2 method.
Due to the fact that the Tier 1 and the Tier 2 method utilize NLPO data, whereas the Tier 3 methods apply NLTH analyses, the assessment outcomes naturally differ from each other. However, the observed outcomes are in contrast to the expectations that the sequential application of the Tier 1 method, the Tier 2 method and the Tier 3 methods leads to decreasing UC values, since lower
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UC values are expected to be obtained for more complex assessment methods. From Figure 5.7, it can be noted that the Tier 3 methods do not yield lower UC values, except for the disregarded Tier 3a method. The depicted UC values for both the Tier 3b and the Tier 3c method may not be considered valid due to the presence of observed OOP failures. Hence, it must be stated from the results that the Tier 3 methods yield less accurate assessment outcomes than the Tier 2 method.
Despite the fact that the depicted UC values for the Tier 3b and the Tier 3c method may not be considered valid, it is still interesting to evaluate the UC values that correspond to the NLTH analyses that did yield results. For the determination of these depicted UC values, a safety percentage has been included by means of the definition of the OOP capacity. Namely, for structural reliability, the OOP displacement capacity for the Tier 3 methods is set equal to 60% of the wall’s instability displacement. However, theoretically, the idealized system that represents the analyzed URM wall in an NLTH model only fails after 100% of the instability displacement is exceeded. In case the OOP displacement capacity is set to 100% of the instability displacement, the UC values of the Tier 3b and the Tier 3c method become smaller than the UC values of the Tier 2 method. Nevertheless, a change in the definition of the OOP capacity does not reduce the number of observed OOP failures:
the OOP demand is normative compared to the OOP capacity. Hence, the finding that the Tier 3 methods yield less accurate assessment outcomes than the Tier 2 method remains valid.
The discrepancy between the Tier 1 and the Tier 2 method, on the one hand, and the Tier 3 methods, on the other hand, can be attributed to the difference in the determination of the OOP demand, rather than in the check with the OOP capacity. The Tier 3 methods utilize an NLTH analysis to determine the OOP demand. The observed OOP failures in the NLTH analyses are a consequence of the nature of NLTH analyses and the rigid body mechanism of the walls. Initially, only a minor, limited dynamic response of the wall is obtained. However, at a specific moment in time, the wall’s response escalates. This occurrence of major dynamic amplification in the NLTH analyses is expected to be the reason for the significant difference in assessment outcomes of the Tier 1 and the Tier 2 method, on the one hand, and the Tier 3 methods, on the other hand.
A reason for the high nonlinear wall behavior in the NLTH analyses can be the way damping is modeled. The damping that is assigned to the idealized systems that represent the analyzed walls is based on the effective stiffness of the wall in order to prevent the need for an iterative analysis.
However, the usage of the effective stiffness for the assignment of damping results in a significantly reduced damping coefficient for the time domain in which the wall shows linear behavior. For this time domain, therefore, an underestimated damping level applies for the idealized system that represents the wall. This is described in Appendix D.2.2. In case the modeling of damping is optimized, the discrepancy between the assessment outcomes of the different types of methods is expected to be reduced.
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Chapter