Literature review
2.5 Assessment of OOP behavior of URM walls
In the previous section, the four main seismic analysis methods are introduced. Using these methods, the dynamic behavior of structures due to earthquake loading can be obtained. Within structures, URM walls can be located. The output data of the seismic analysis methods concerning complete structures can be used as input data for seismic analyses of URM walls within such structures. The input data for the analysis of a URM wall can therefore concern either a time series of data or a single data point which describes the critical value. This depends on the performed seismic analysis method.
After executing the transition from the seismic demand of a complete structure to the OOP demand of a URM wall within structures, the OOP demand of the wall is compared against the OOP capacity of the wall. In case the demand does not exceed the capacity, the URM wall is considered to be adequate and does not need any strengthening. This assessment process can be performed using several methods. In order to find an effective, yet accurate method to assess the OOP behavior of URM walls due to earthquake loading, different existing assessment methods are evaluated and weighed against each other. Therefore, it is necessary to make an inventory of the existing methods first. In this section, the different existing methods are introduced.
Annex H of the NPR 9998 [21] proposes three methods to assess the OOP behavior of URM walls, with an increasing level of accuracy, as well as an increasing level of complexity and time that it takes to complete the analysis:
• Tier 1: The OOP demand is based on the secondary spectrum. The OOP capacity is deter-mined using the displacement-based NLKA method or the force-based MVA;
• Tier 2: The OOP demand is based on the building specific secondary spectrum. The OOP capacity is determined using the displacement-based NLKA method or the force-based MVA;
• Tier 3: A full NLTH analysis is performed, in which both the OOP demand and the OOP capacity are included.
These methods are the common standard for assessing the OOP behavior of URM walls in the province of Groningen. In case the application of the Tier 1 method yields a negative assessment outcome, it does not automatically mean that the wall does not suffice. A reassessment can be made using a Tier that corresponds to a higher level of accuracy.
2.5.1 Tier 1
2.5.1.1 OOP demand
When applying the Tier 1 method, the spectral OOP acceleration of the wall SEa;d is determined using the secondary spectrum. This spectrum describes the spectral acceleration of the wall for different ratios between the natural period of the wall element itself, Ta, and the effective vibration period of the complete structure, Tef f. In this way, the expected dynamic interaction between the wall and the structure is taken into account. Besides, the ratio between the height of the wall element within the structure, z, and the total height of the (contributing) structure, hn, is incorporated in the secondary spectrum. Naturally, a URM wall undergoes a higher acceleration in case the wall
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is positioned at a higher level within the structure. Furthermore, the spectral OOP acceleration of a URM wall is a function of the peak ground acceleration and the element behavior factor. The peak ground acceleration, logically, represents the expected maximum earthquake loading, whereas the element behavior factor is used to take into account the ductile behavior of the URM wall. The element behavior factor, therefore, depends on the type of wall that is analyzed. Figure 2.22 shows the secondary spectrum.
Figure 2.22: Secondary spectrum corresponding to Tier 1 method.
The Tier 1 method is relatively easy to apply, since no extensive data are needed regarding the different floors in the structure in which the wall is located. Only the (relative) heights, the natural period of the wall and the effective vibration period of the total structure are needed. The latter factor can be determined by means of a simple NLPO analysis of the equivalent SDOF system that corresponds to the total structure.
2.5.1.2 OOP capacity
Annex H of the NPR [21] describes two methods to determine the OOP capacity of URM walls:
the nonlinear kinematic analysis (NLKA) method and the method of virtual work (MVA). The first method is a displacement-based method that can be used for one-way vertically spanning walls. The method is easy to apply and yields less conservative results than linear methods. The second method is a force-based method that can be used for more complex situations in which one-way horizontally spanning walls or two-way spanning walls are addressed. In the following sections, the principles that form the basis of both the NLKA method and the MVA are introduced.
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Nonlinear kinematic analysis (NLKA) method
In the nonlinear kinematic analysis (NLKA) method, the behavior of a one-way vertically spanning URM wall is represented by a perfect rigid body (RB) mechanism. The URM wall is then modelled as two rigid blocks that undergo large displacements and rotations. The rigid blocks are separated by fully cracked sections [42], as illustrated in Figure 2.23. Hence, the NLKA method, as proposed by Annex H of the NPR [21], can be categorized as a displacement-based method, since the wall is schematized and analyzed in its deformed state.
a) pinned-pinned BC b) clamped-clamped BC
Figure 2.23: Rigid body mechanism of one-way vertically spanning URM walls [43].
A nonlinear kinematic analysis on the RB mechanism yields two parameters, namely, the maximum force on the undeformed wall Fmax and the displacement ∆i, for which the deformed system just remains in equilibrium with no external force. Figure 2.24a shows the experimental pushover curve of an OOP loaded URM wall, whereas 2.24b shows the derived bilinear model.
a) experimental pushover curve b) bilinear model Figure 2.24: Force-displacement relationship of OOP loaded URM wall [42].
The OOP capacity of the wall, which is defined as the out-of-plane applied load for which the assumed rigid body mechanism just remains in equilibrium, depends on several factors, such as its dimensions and self-weight, the interstory drift between the floors at the top and the bottom side of the wall, the boundary conditions (BCs) of the wall and the overburden load that is subjected
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to the wall. The higher the ratio between the overburden load P and the wall’s self-weight W , the higher the vertical pre-compression in the wall. This causes an increase in effective stiffness and an increase in OOP capacity. The OOP capacity of the wall, as determined using the NLKA method as prescribed in Annex H of the NPR, corresponds to the limit state Near Collapse (NC) [21]. This limit state defines the ultimate displacement of one-way vertically spanning walls as 60% of the instability displacement ∆i.
The NLKA method is a relatively easy method to apply, yielding less conservative results than linear methods. However, the method can only be applied to one-way vertically spanning walls.
Method of virtual work (MVA)
For one-way horizontally spanning walls and two-way spanning walls, the load that causes the walls to crack is significantly higher than the load that corresponds to the kinematic mechanism of the deformed state, in which it is assumed that the walls have cracked already. In such situations, an analysis that is based on the principle of virtual work (MVA) can be applied instead of a kinematic analysis (NLKA), so that a less conservative OOP capacity is obtained. Hence, the MVA method, as proposed by Annex H of the NPR [21], can be categorized as a force-based method.
The principle of virtual work can be applied by setting the external virtual work and the internal virtual work equal to each other. The equality of these two factors is based on the fact that the work due to external loading on a structure gets transformed into an equal amount of internal work, or strain energy in case a structure deforms [44]. This latter type of virtual work depends on the moment capacity along the diagonal and the vertical crack lines, and thus also depends on the type of vertical crack line that is expected to occur [45], either line failure or stepped failure, as shown in Figure 2.25 by arrows 2 and 3. The type of vertical crack line depends on the material properties of the brick units and the mortar. In order to obtain reliable results, it is key to use an estimation of the collapse mechanism, i.e. the crack pattern and the corresponding deflected shape, of the URM wall that is in close correspondence to the actual behavior of the wall of interest.
a) line failure b) stepped failure
Figure 2.25: Types of vertical crack lines [21].
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According to the MVA as prescribed in Annex H of the NPR [21], the OOP capacity of the wall is defined as the peak cracking resistance. This resistance can be calculated as a function of several factors: the moment capacity along the diagonal and the vertical crack lines, the material and mechanical properties of the wall, the dimensions of the wall and three coefficients that account for the expected collapse mechanism of the analyzed URM wall. Within these factors, the presence or absence of expansion joints and openings in the wall, the normal stress over the cross-section, as well as the boundary conditions of the wall are implemented. The MVA presumes that the cracks that will appear will follow the natural diagonal crack line of the masonry, in which the stones keep intact and the crack runs through the joints. In case it is expected that the cracks will run through the stones, as for URM walls that are constructed using CS brick elements [21], the yield line method needs to be applied. This method, however, is not elaborared in this literature review.
Compared to the NLKA method, the method of virtual work (MVA) is a more complex method that can be used for one-way horizontally spanning walls and two-way spanning walls. In order to obtain reliable results, it is key to use an accurate estimation of the crack pattern and the corresponding deflected shape of the URM wall.
2.5.2 Tier 2
2.5.2.1 OOP demand
When applying the Tier 2 method, the spectral OOP acceleration of a URM wall SEa;d is deter-mined using the building specific secondary spectrum, which is the average of the floor spectra that correspond to the floors at the bottom and at the top of the analyzed URM wall. The building spe-cific secondary spectrum is shown in Figure 2.26, in which the horizontal axes represent the wall’s natural period Ta and the vertical axes represent the spectral OOP acceleration of the wall SEa;d. It can be seen that a maximum acceleration is obtained in case the natural period of the wall element, Ta, is positioned in the nonlinear period range of the floor. In this period range, resonance can be expected. This significant interaction is taken into account by means of a multiplication of the peak floor acceleration, P F A, by the dynamic amplification factor, DAF , which is set equal to 2.
The Tier 2 method is a more complex method than the Tier 1 method, since a more extensive NLPO analysis of the structure is needed, in which the three factors T1;i, Tef f ;i and P F Ai need to be determined for the construction of the individual floor spectra. Despite the fact that completing the Tier 2 method requires more time, the application of this method yields more accurate results than the more simple Tier 1 method, in which less building specific characteristics are implemented.
2.5.2.2 OOP capacity
No differences apply regarding the determination of the OOP capacity of URM walls for the Tier 1 method and the Tier 2 method. Hence, also in the Tier 2 method, both the displacement-based NLKA method and the force-based MVA can be applied to determine a wall’s OOP capacity. The background of both methods has been introduced in section 2.5.1.2.
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Figure 2.26: Building specific secondary spectrum corresponding to Tier 2 method [21].
2.5.3 Tier 3
When applying the Tier 3 method, a full NLTH analysis is performed, in which both the OOP demand and the OOP capacity of the URM wall are included. Compared to the Tier 1 and Tier 2 method, a dynamic load is applied to the numerical model of the analyzed structure. Therefore, the changes in magnitude and direction of ground accelerations are included, whereas the Tier 1 and Tier 2 method are only based on the maximum value of the ground acceleration that is expected to occur at the specific location.
The Tier 3 method is the most complex method, due to the fact that an NLTH analysis must be performed. The application of this method therefore requires a high computational time and effort.
When modelled correctly, the Tier 3 method will result in the most accurate results of the three assessment methods that are prescribed in Annex H.
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