Evaluated methods for assessment of OOP behavior of URM walls
3.1 Evaluation of assessment methods Annex H
A validation study has been performed in which the basic Tier 1 and Tier 2 method are compared against a Tier 3 method in which a complex micro-NLTH analysis is applied [46]. In the study, three buildings are analyzed. In these buildings, the Tier 1 and the Tier 2 method predict in total 9 and 7 OOP failures, respectively. According to the Tier 3 method, however, no OOP failure occurs. Only when the analyzed PGA is approximately double the original PGA, OOP failures are observed. The cases in which a wall is predicted to fail according to the Tier 1 and the Tier 2 method, while the Tier 3 method does not indicate failure, are so-called false fail predictions.
In case a false fail prediction applies for a specific URM wall, the analysis will conclude that the wall needs strengthening, when in fact this is not necessary. For several reasons, it is undesirable to unnecessarily strengthen structures. Hence, it is desired to reduce the percentage of false fail predictions. In order to reduce this percentage, the Tier 3 method can be applied. Although the application of this method will reduce the number of walls for which strengthening is advised, it
Evaluated methods for assessment of OOP behavior of URM walls page 32
might not be the most economic choice, due to the inherent high amount of time and effort that is required for the analysis.
It can be stated that the three methods that are included in the presented validation study cover a wide range of accuracies of the assessment outcomes as a result of the wide range of complexities of the included analyses regarding the overall structure. The Tier 1 and Tier 2 method include a rela-tively simple NLPO analysis and yield conservative assessments, whereas the applied Tier 3 method includes a complex micro-NLTH analysis, thus yielding more accurate assessments. Therefore, the Tier 1 and Tier 2 method can be allocated on one side of the imaginary ‘accuracy vs. efficiency’-spectrum, as illustrated in Figure 3.1, whereas the applied Tier 3 method can be allocated on the other side of this spectrum. Hence, a big ‘gap’ can be observed in the imaginary spectrum between the first two Tiers one the one side, and the last Tier on the other side.
Figure 3.1: Illustration of ‘accuracy vs. efficiency’-spectrum.
This ‘gap’ in the imaginary ‘accuracy vs. efficiency’-spectrum of assessment methods offers possibil-ities for improvement. It would be beneficial if a research utilizes an assessment method that can be applied to assess the OOP behavior of URM walls in such a way that it yields more accurate results than the Tier 2 method, however, without requiring the complexity and high effort that correspond to the applied Tier 3 method. Such an assessment method could fill the ‘gap’ of the imaginary spectrum.
A Tier 3 method in which a relatively simple macro-NLTH analysis is applied can serve as such a desired assessment method. Due to the reduction of the incorporated complexity of the NLTH analysis, the efficiency of the method is increased significantly. Such a simplified Tier 3 method is used in a sensitivity study that has been performed in order to find a reasonable value for the dynamic amplification factor (DAF) that is used in the Tier 2 method, which has eventually been set to 2 [47]. Besides the determination of the DAF, the study gives insight in the differences in assessment outcomes concerning the OOP behavior of URM walls when applying the Tier 1 and 2 method and a relatively simple Tier 3 method. The percentages that were investigated in the sensitivity study, as well as the corresponding optimal situations, are listed in Table 3.1.
Figure 3.2 shows part of the results of the conducted research. The assessment outcomes of the simplified Tier 3 method, which utilizes a macro-NLTH analysis, serve as a benchmark. A significant improvement can be observed in the assessment results of the Tier 2 method compared to the Tier 1 method. When comparing the assessment outcomes of the Tier 2 method to the benchmark, a
3.1 Evaluation of assessment methods Annex H page 33
Table 3.1: Investigated percentages and corresponding optimal situations [47].
Investigated percentage Optimal situation
Non-failing cases in which Tier 2 < Tier 3 as low as possible (requirement: ≤ 5%)
True fail prediction as high as possible
False fail prediction as low as possible False pass prediction as low as possible Cases in which Tier 1 > Tier 2 as high as possible
false fail prediction is obtained in 8% of the analyzed cases, as shown in Figure 3.2b. It can be concluded that the differences in assessment outcome between the Tier 2 and the Tier 3 method are significantly smaller in this sensitivity study, compared to the aforementioned validation study that incorporated a more detailed Tier 3 method. When considering the ‘accuracy vs. efficiency’-spectrum, the simplified Tier 3 method can thus be allocated in the ‘gap’.
a) passes vs. failures b) true fails vs. false fails c) false passes Figure 3.2: Assessment outcomes Tier 1, Tier 2 and Tier 3 (= macro-NLTH) method [48].
In order to further investigate the assessment process of the OOP behavior of URM walls, while considering both the efficiency of the method, as well as the accuracy of the outcome, the current research evaluates three simplified Tier 3 methods. Each Tier 3 method utilizes a relatively simple macro-NLTH analysis. Hence, the Tier 3 methods could be suitable to fill the ‘gap’ in the ‘accuracy vs. efficiency’-spectrum in a balanced manner. Besides the inclusion of the three Tier 3 methods, the basic Tier 1 and Tier 2 method are included in the current research as well.
3.1 Evaluation of assessment methods Annex H page 34
3.2 Tier 1
3.2.1 OOP demand according to secondary spectrum
When applying the Tier 1 method, the OOP acceleration demand of the wall SEa;d is determined using the secondary spectrum, which can be constructed using Eq. 6.
SEa;d=
The natural period of the wall Ta can be calculated using Eq. 7:
Ta=
This equation, however, yields a less accurate Ta than Eq. 21, which is applied in the nonlinear kinematic analysis (NLKA) method for the determination of the natural period of the wall Ta, as will be described in section 3.2.2. The higher accuracy of the latter equation results from the fact that the effective thickness of the wall, as well as the wall’s boundary conditions are taken into account.
In the current research, the natural period of the wall Ta will therefore be determined using Eq. 21, rather than using Eq. 7.
Whereas the values for the design value of the peak ground acceleration ag;d, the element behavior factor qa, the height of the wall element within the structure z and the height of the structure itself hn can be easily assigned, the effective vibration period of the overall structure Tef f must be determined using an NLPO analysis.
In section 2.4.3, the basis of the NLPO analysis has been introduced. In brief, a horizontal load is dis-tributed over the different DOFs of the idealized system that corresponds to the analyzed structure.
The distribution is based on the structure’s lumped masses and its fundamental mode shape. The horizontal load is applied in steps. At each step, the resulting displacements are calculated, while taking into account the NL behavior of the structure. The obtained force-displacement relationships are used to construct the NLPO-curve of the equivalent SDOF system of the analyzed structure.
This curve is then compared to the ADRS-curve. The intersection of both curves, the so-called performance point, depicts the effective vibration period of the structure, as shown in Figure 3.3.
The ADRS is constructed using the following steps [21]:
1. The elastic acceleration response spectrum Sa(T ) is constructed using formulas as prescribed in the NPR;
2. The elastic displacement response spectrum is constructed, using the following formula:
Sd(T ) = Sa(T )[T
2π]2 (8)
3.2 Tier 1 page 35
Figure 3.3: Comparison of NLPO- and ADRS-curve [48].
3. The elastic ADRS is created, showing the spectral accelerations Sa on the vertical axis and the spectral displacements Sd on the horizontal axis;
4. The nonlinear ADRS is constructed by means of a multiplication of the elastic ADRS with the spectral reduction factor ηζ. Due to this multiplication, the energy dissipation and damping that are associated with the achievable ductility of the structure are taken into account.
ADRSnonlinear= ADRSelastic× ηζ (9)
For unreinforced masonry structures, the spectral reduction factor ηζ can be calculated using the following formula:
ηζ =
s 7
2 + 100ζsys
≥ 0.55 (10)
where ζsys is the equivalent viscous damping of the system, which is based on the global structural ductility µsys and the occurring inelastic mechanism in the system. The following formula may be used to determine the equivalent viscous damping ζsysfor a system:
ζsys= ζ0+ ζhys+ β0≤ 0.40 (11)
where ζ0 is the inherent viscous damping (equal to 5% for URM structures), ζhys is the hysteretic damping of the system and β0 is the foundation radiation damping effect. The latter damping effect may be assumed to be negligible for residential buildings up to 2 stories (β0= 0). The hysteretic damping ζhysdepends on the inelastic mechanism. For brittle failure
3.2 Tier 1 page 36
modes (such as diagonal tensile failure), ζhys= 0. For ductile failure modes (such as flexural and sliding shear), ζhys can be determined using the following formula:
ζhys= 0.42
1 − 0.9
õsys
− 0.1√ µsys
≤ 0.15 (12)
The global structural ductility µsysis defined as follows:
µsys= uduct,sys uy,sys
(13) where uduct,sys is the lateral NC displacement capacity of the equivalent SDOF system and uy,sysis the yield displacement of the equivalent SDOF system;
5. The spectral accelerations Sa are multiplied by the mass of the equivalent SDOF system that corresponds to the analyzed structure, resulting in an ADRS in which the vertical axis shows the seismic base shear force Fb, whereas the horizontal axis shows the spectral displacements Sd.
3.2.2 OOP capacity according to NLKA method
Due to the fact that the current research focuses on one-way vertically spanning URM walls, the NLKA method is applied to determine the OOP capacity of the analyzed walls. This displacement-based method is introduced in section 2.5.1.2. When applying the NLKA method, the URM wall is modelled as two rigid blocks that undergo large lateral displacements and rotations. First, the instability rotation and the corresponding instability displacement, A and ∆i, are determined. This rotation and displacement correspond to the situation at which the deformed wall just remains in equilibrium. The two values depend on several factors, such as the wall’s dimensions, self-weight and boundary conditions, as well as the overburden load P that acts on top of the wall and the eventual presence of openings in the wall. In case the analyzed URM wall is connected to a non-structural outer leaf, which is the case for cavity walls, the adjacent outer leaf also has a certain influence on the instability rotation and displacement. The outer leaf’s mass must be incorporated in the NLKA.
For situations in which the inner leaf and the outer leaf are adequately connected, the outer leaf’s stiffness may also be incorporated in case it has sufficient capacity, since its stiffness then contributes to the OOP stiffness, and thus to the OOP capacity, of the analyzed URM wall.
Within the scope of the current research, only walls that are located in between adjacent terraced housing units are investigated. Hence, only load-bearing single leaf URM walls that are constructed of CS masonry are subject of the research. Also, no openings are present in the analyzed walls.
First, the instability rotation and displacement are determined:
∆i= h
2sin(A) ≈ h
2A (14)
A = b
a (15)
3.2 Tier 1 page 37
b = Wbeb+ Wt(e0+ eb+ et) + P (e0+ eb+ et+ eP) − θv(Wbyb+ Wtyt) (16)
a = Wbyb+ Wt(h − yt) + P h (17)
The required parameters, as shown in Figure 3.4, are defined as follows:
• ∆i is the instability displacement of the analyzed URM wall [m];
• A is the instability rotation of the wall [°];
• h is the height of the wall [m];
• Wb and Wt represent the self-weight of the bottom and the top half of the wall, respectively.
For the regular URM walls that are analyzed in the current research, Wb= Wt= 0.5W [N/m];
• e0is the eccentricity of Wbrelative to the mid-hinge. In the current research, e0= 0.5tef f [m];
• eb is the eccentricity of Wb relative to the bottom hinge. Its value depends on the boundary condition at the bottom side of the wall [m];
• etis the eccentricity of Wtrelative to the mid-hinge. In the current research, et= 0.5tef f [m];
• eP is the eccentricity of Wt relative to the top hinge. Its value depends on the boundary condition at the top side of the wall [m];
• P is the overburden load that acts on top of wall [N/m];
• θv is the OOP rotation of the wall as a result of the initial interstory drift [°];
• yb and yt represent, respectively, the distance from the center of Wb to the bottom side of the wall, and the distance from the center of Wt to the top side of the wall. In the current research, yb= yt= 0.25h [m];
• tef f is the effective thickness of the wall [m]:
tef f = tnom(0.975 − 0.025P W)
In the current research, however, another formula will be used for the determination of the effective thickness tef f, as described in the paragraph below;
• tnomis the nominal thickness of the wall [m];
• W is the self-weight of the total wall [N/m]:
W = ρCS× g × h × tnom
• ρCS is the density of the applied masonry. In the current research, ρCS = 1835 [kg/m3];
• g is the acceleration of gravity [m/s2].
3.2 Tier 1 page 38
Figure 3.4: Parameters NLKA method for determination OOP capacity of URM walls [21].
According to Annex H of the NPR, the prescribed formula for the calculation of the effective thickness tef f is based on the equivalent rectangular stress block that results from the OOP rocking behavior of the URM wall. For relatively high overburden loads, the formula yields a significantly reduced effective thickness compared to its original thickness tnom, which is inconsistent with experimental data [37]. Another deficiency of the prescribed formula is related to the fact that it does not include any material properties of the analyzed wall. Hence, a more valid formula for the effective thickness is constructed by including the compressive strength of the masonry, fm, as well as the force that acts at the wall’s mid-height, P + W/2, rather than the ratio P/W . Equation 18 presents the new formula for the calculation of the effective thickness tef f which will be used in the current research.
It must be noted that the eventual Tier 1 and Tier 2 outcomes differ from the outcomes that would result from the literal application of the code.
tef f = tnom−
0.5(P +W 2 ) bfm
(18) The NLKA method is based on the Near Collapse (NC) limit state [21], which defines the maximum usable displacement ∆mof one-way vertically spanning walls as 60% of the instability displacement
∆i. This ‘safety’ percentage is included in Eq. 19, which is used to determine the OOP capacity of the analyzed URM wall as a function of the instability rotation A. The OOP capacity Rdis expressed in terms of accelerations. Hence, Eq. 19 incorporates the general transformation of displacements or rotations into accelerations by means of the factor (Ta/2π)2. Lastly, the γ-factor accounts for the transformation of the behavior according to the assumed perfect RB mechanism into the behavior of an equivalent SDOF system.
Rd= 0.3hA γ(Ta
2π)2g
(19)
3.2 Tier 1 page 39
γ = (Wbyb+ Wtyt)h
2J g (20)
Ta= 4.07J
a (21)
J = Jb0+ Jt0+1
gWb(e2b+ yb2) + Wt(e0+ eb+ et)2+ yt2 + P (e0+ eb+ et+ eP)2 + Janc (22)
Jb0= Jt0= W 2g
[t2ef f + (0.5h)2]
12 (23)
Janc = 0 (24)
The required parameters are defined as follows:
• Rd is the OOP capacity of the analyzed URM wall [g];
• γ is the mass participation factor [-];
• Ta is the natural period of the wall [s];
• J, Jb0, Jt0 and Janc are the rotational moments of inertia of the complete wall, the bottom half of the wall, the top half of the wall and the elements that are connected to the wall, respectively. The latter value is assumed to be zero [kgm2].
3.2 Tier 1 page 40
3.3 Tier 2
3.3.1 OOP demand according to building specific secondary spectrum
When applying the Tier 2 method, the OOP acceleration of a URM wall SEa;d is determined using the building specific secondary spectrum. This spectrum is the average of the floor spectra that correspond to the floors at the bottom and at the top of the analyzed URM wall.
First, it must be determined whether a building layer behaves linear or nonlinear as a result of the analyzed earthquake loading. In case Tef f,i− T1,i≤ 0.1 s, a building layer i is considered to show only linear behavior. Then, the corresponding floor spectrum is assumed to be equal to the elastic response spectrum Sa(T ). This assumption is physically incorrect. However, within the building scope, the assessment method and the hazard of interest, the application of the elastic response spectrum proved to yield normative results compared to the application of a linear floor spectrum that is more complex to derive [46]. Therefore, Annex H of the NPR proposes to set the linear floor spectrum equal to the elastic response spectrum Sa(T ), so that the application of the Tier 2 method gets simplified significantly [21]. This simplification contributes to a more efficient determination of the OOP demand of URM walls.
In case a building layer shows nonlinear behavior, the corresponding nonlinear floor spectrum, as shown in Figure 3.5, can be constructed using Eq. 25.
Figure 3.5: Nonlinear floor spectrum [21].
SEa;d;i=
P F Ai× (1 + Ta
T1,i
(DAF − 1)) and SEa;d;i≥ Sa(Ta), if 0 < Ta ≤ T1,i
P F Ai× DAF and SEa;d;i≥ Sa(Ta), if T1,i< Ta≤ Tef f,i
P F Ai× DAF (Tef f,i Ta
)2 and SEa;d;i≥ Sa(Ta), if Ta> Tef f,i
(25)
where
P F Ai= Fb,P P,eq meq
Γφi,1 (26)
3.3 Tier 2 page 41
Hence, three factors must be known in order to construct a nonlinear floor spectrum of a building layer i: the natural period of the building layer, T1,i, the effective vibration period of the building layer, Tef f,i, and the peak floor acceleration, P F Ai [49]. These factors can be derived using Eq. 26 and the following output data of an NLPO analysis: respectively, the linear range, the performance point and the yield point. Figure 3.3 indicates the three factors for an example NLPO-curve. Due to the fact that only the first mode shape has to be considered for URM structures up to 4 stories, the described procedure only has to be performed for the first mode. The influence of higher modes may be neglected [21].
3.3.2 OOP capacity according to NLKA method
The OOP capacity of a URM wall is determined in accordance to the NLKA method that is described in section 3.2.2. No differences apply for the NLKA methods that are used for the determination of the OOP capacity in the Tier 1 method and the Tier 2 method.
3.3 Tier 2 page 42
3.4 Tier 3
Three assessment methods that can be categorized as a Tier 3 method are added to the Tier 1 and the Tier 2 method in order to investigate the influence of performing a transient analysis. Hence, these methods make use of a simplified NLTH analysis for the assessment of the OOP behavior of URM walls. The methods are referred to as the Tier 3a, Tier 3b and Tier 3c method. Outside the scope of the current research, this ‘subscripting’ can theoretically continue (Tier 3d, 3e, 3f, etc.), due to the fact that much more complexity and accuracy can be incorporated in the applied NLTH analysis.
In the Tier 3a and 3b method, first, a dynamic analysis is performed concerning the structure as a whole. Afterwards, the dynamic response of the structure is used as input for the dynamic analysis of the analyzed URM wall. Therefore, in these methods, the dynamic response of the overall structure is determined without taking into account any dynamic interaction between the structure and the URM wall within the structure. In literature, this is referred to as a ‘cascade approach’ [50,51].
In order to investigate the interaction between the overall structure and the URM wall throughout the entire dynamic analysis, the Tier 3c method is added. In this method, the URM wall is modelled as an individual lumped mass. Hence, the dynamic interaction between structure and wall is incorporated
In order to investigate the interaction between the overall structure and the URM wall throughout the entire dynamic analysis, the Tier 3c method is added. In this method, the URM wall is modelled as an individual lumped mass. Hence, the dynamic interaction between structure and wall is incorporated