Evaluated methods for assessment of OOP behavior of URM walls
3.5 Numerical modelling
In the current research, the building layers in the idealized systems are both represented by only a single spring and damper. Furthermore, the analyzed URM walls are modelled using either a single spring and damper or two springs and dampers. Hence, the macro-modelling approach is applied for the numerical modelling, as introduced in section 2.4.5. The application of the macro-modelling approach has several advantages, such as the computational time efficiency and numerical stability.
Yet, due to the usage of macro finite elements, less details are incorporated in the model. Therefore, some aspects cannot be taken into account using macro-modelling, such as the nonlinear interaction between flexure and shear, as well as the local failure mechanism in which a URM wall fails due to crushing at its ‘toe’ (Figure 2.9c). This disadvantage, however, is not decisive in the choice of the modelling approach within the current research, due to the fact that the research focuses on the comparison of different methods to assess the OOP behavior of URM walls. Hence, there is no need to construct a highly exact and accurate model for a specific structure.
Due to the fact that the macro-modelling approach is applied for the construction of the numerical models, only a limited number of finite elements is present in the model. Each finite element represents a component within the structure, to which an appropriate material model is assigned. In this way, the behavior of the specific component is captured in the numerical model. In sections 3.5.1 and 3.5.2, the material models are introduced that are used in the numerical models. Afterwards, section 3.5.3 presents the implementation of damping in the numerical models.
3.5.1 Material model for springs that represent lateral behavior of building layers As introduced in section 2.2.4, the nonlinear behavior of dynamically loaded URM structures depends on the occurring failure mechanism, which in turn depends on several factors, such as the applied materials, the facade configuration and the axial load that is present. As a consequence, considerable differences can be obtained in the hysteretic behavior of different building layers that are constructed of unreinforced masonry. Nevertheless, specific characteristics for the lateral behavior of URM structures can always be seen in the hysteretic loops, namely, the presence of a positive post-yield stiffness and an unloading stiffness that is significantly reduced when compared to the initial stiffness.
These characteristics also apply for RC structures, for which the hysteretic Takeda model has been constructed [53]. Particularly, the modified so-called ‘thin’ Takeda model is used for modelling
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RC column and wall elements with high axial stresses, whereas the ‘fat’ Takeda model is used for modelling RC beam and frame elements [54,55]. Both modified Takeda models are shown in Figure 3.11. Due to the similarities between the lateral behavior of URM structures and RC elements with high axial stresses, the thin Takeda model may also be used to model the behavior of URM building layers [56,57]. Furthermore, the combination of shear failure and failure in compression at a wall’s toe can be closely approximated by the thin Takeda model. This combination of failures is typical for the in-plane behavior of URM walls, as described in section 2.2.2. For the reasons listed, the current research applies the thin Takeda model for simulating the lateral behavior of the URM building layers, which has also been done before in several studies on the structures in the Groningen area [58,59].
a) ‘thin’ Takeda model b) ‘fat’ Takeda model
Figure 3.11: Modified Takeda models [55].
The material model in OpenSees that is best applied to simulate the thin Takeda model is called Hysteretic. In order to construct this hysteretic bilinear material model in OpenSees, several pa-rameters must be set, as listed in the upper part of Table 3.2. By assigning values of 1 to the pinching factors for force and deformation during reloading and by using a relatively high value for β, a Takeda response can be obtained [59]. Furthermore, the damage parameters are set to 0.
Hence, no damage driven by ductility and dissipated energy is assumed. This conservative assump-tion is in accordance with several experiments in which only minor strength degradaassump-tion is observed [60, 61]. Additionally, the hysteretic behavior is assumed to be similar in both loading directions, which results in a symmetric backbone curve. Due to the fact that the main purpose of the research is the comparison of different assessment methods to each other, it is not of great importance to model a specific URM structure in an exact and accurate manner. The simplification of assuming a symmetric backbone curve therefore has no unfavorable consequences.
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Table 3.2: Explanation OpenSees material model Hysteretic [62].
Parameters Description
eip, sip deformation and force at envelope point i in positive direction ein, sin deformation and force at envelope point i in negative direction pinchX pinching factor for deformation during reloading
pinchY pinching factor for force during reloading damage1 damage due to ductility
damage1 damage due to energy
β degradation of unloading stiffness Kt based on ductility, Kt= K0µ−β (-)F i, (-)di force and deformation at envelope point i in both directions
β degradation of unloading stiffness Kt based on ductility, Kt= K0µ−β
Due to the above mentioned assumptions, several parameters that define the original OpenSees material model become irrelevant. Only a few parameters still need to be defined to construct the material model for the NL springs that represent the lateral behavior of the building layers: the three envelope points and the β-value. These parameters are listed in the lower part of Table 3.2. Figure 3.12a shows the original OpenSees material model, whereas Figure 3.12b illustrates the simplified version of the material model.
a) OpenSees material model Hysteretic b) simplified material model Figure 3.12: NL lateral behavior of building layers.
As a consequence of the high amount of aspects that influence the behavior of URM structures, it is desirable to define the NL springs that represent the lateral behavior of the building layers using experimental data on the specific structure. In case such experimental data are available, the three envelope points, and thus the backbone curves (BBCs), that correspond to a building layer are constructed using the provided force-deformation graphs. In these graphs, the force represents the shear force within a layer, whereas the deformation represents the displacement at the top of the building layer, relative to the displacement at the bottom of the building layer. In case no
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experimental data are available for a specific structure, the BBCs are constructed by transforming BBCs that are based on experimental data. Appendix G.1 introduces such a transformation method.
Besides the need to define the three envelope points, the β-value must be set in order to construct the hysteretic material model. This β-value is used to describe the degradation of the unloading stiffness Ktbased on ductility. In Eq. 28, the determination of Ktas a function of the initial stiffness K0, the ductility µ and the β-value is described.
Kt= K0µ−β (28)
In a NAM study, a β-value of 0.75 is applied for several URM structures with 2 stories [59]. Another research concerning structures in Groningen utilizes a β-value of 0.9, which is based on calibration with experimental data on typical Dutch URM characteristics [58]. This latter β-value of 0.9 is as well applied in a NEN study, due to the fact that a high value results in less energy dissipation, yielding conservative results [48]. Hence, the current research also applies a β-value of 0.9 for the construction of the hysteretic material models.
A validation check is performed to examine whether the chosen value is acceptable by means of a comparison between experimental data and numerical results, as derived after implementation of the constructed hysteretic material models. Due to the large number of simplifications implemented in the Hysteretic material models in OpenSees no close match will occur between the experimental data and the numerical results: significant discrepancies are expected. In case both the order of magnitude and the time series ‘shape’ of the resulting dynamic response are similar, the β-value is assumed to be accurate enough.
3.5.2 Material model for springs that represent OOP behavior of wall
As introduced in section 2.5.1.2, the OOP behavior of a URM wall is often modelled as a bilinear curve with a descending branch. Such force-deformation behavior implies the presence of a perfect rigid body (RB) mechanism. No energy dissipation is incorporated, hence, the unloading curve follows the loading curve. In the current research, the analyzed URM walls are assumed to behave according to the RB mechanism.
Perfect RB mechanism
The material model that is assigned to the nonlinear springs which represent the bilinear OOP RB behavior of the analyzed URM walls in the OpenSees models is called ElasticBilin. In order to construct this elastic bilinear material model in OpenSees, several parameters must be set, as listed in the upper part of Table 3.3. These input parameters are rewritten, such that the material model is described by only three parameters, as listed in the lower part of Table 3.3: Fy, dy and dmax. The bilinear curve is assumed to be similar in both loading directions. Figure 3.13a depicts the original OpenSees material model, whereas Figure 3.13b shows the simplified version of the material model.
For illustration purposes, the value of dy is chosen significantly higher than the actual dy-value for bilinear RB behavior of URM walls. Besides the three required parameters, Figure 3.13b depicts the Fmax-value as well. This value is needed for the determination of Fy and dy, as will be described later.
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Table 3.3: Explanation OpenSees material model ElasticBilin [62].
Parameters Description
EP 1 Tangent in tension for deformations in range 0 < d ≤ epsP EP 2 Tangent in tension for deformations in range d > epsP epsP Deformation at which material changes tangent in tension EN 1 Tangent in compression for deformations in range 0 < d ≤ epsN EN 2 Tangent in compression for deformations in range d > epsN epsN Deformation at which material changes tangent in compression
Fy Horizontal force at transition point of continuous model to RB mechanism dy Displacement at transition point of continuous model to RB mechanism dmax Instability displacement of URM wall
a) OpenSees material model ElasticBilin b) simplified material model Figure 3.13: NL OOP behavior of walls.
Table 3.4 shows information that is included in the New Zealand’s (NZ) guideline The Seismic Assessment of Existing Buildings [63]. The NZ guideline can be used to determine the maximum force on the wall, Fmax, corresponding to the ‘onset’ of the perfect RB mechanism in which zero displacements apply. Also, the wall’s instability displacement, dmax, can be determined using the guideline. This parameters corresponds to the ‘end’ of the perfect RB mechanism in which no force applies. The two parameters depend on the boundary conditions (BCs) of the wall. In Table 3.4, only information regarding BC0, BC1 and BC3 is presented, since the occurrence of BC2, in which the hinge at the top of the wall is located at the wall’s edge, while the hinge at the bottom of the wall is located at the wall’s center, is not realistic for the building stock of Groningen [48]. The used symbols and their meaning are listed on the next page.
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• dmax is the instability displacement of the wall [m];
• tef f is the effective thickness of the wall [m]. According to the NZ guideline as well as Annex H of the NPR, the value can be determined using the following formula:
tef f = tnom(0.975 − 0.025P W)
In the current research, however, another formula is used for the determination of the effective thickness tef f, as described in section 3.2.2, namely:
tef f = tnom−0.5(P + W/2) bfm
• tnomis the nominal thickness of the wall [m];
• P is the overburden load on the wall [kN];
• W is the self-weight of the wall [kN];
• b is the width of the wall [m];
• fmis the compressive strength of the masonry [kN/m2];
• h is the height of the wall [m];
• Cmis the seismic coefficient that corresponds to the onset of the perfect RB mechanism [g];
• g is the acceleration of gravity [m/s2].
Table 3.4: Formulas for determination Fmax and dmaxof URM walls for BCs of interest [63].
Boundary condition BC0 BC1 BC3
Instability displacement dmax[m] tef f h
(2W + 3P )tef f (2W + 4P ) tef f Seismic coefficient Cm [g]
2 + 4P
W
tef f
h
4 +6P
W
tef f
h
4 +8P
W
tef f
h
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The parameter dmaxcan be directly calculated using the formulas in Table 3.4. The parameter Fmax
can be calculated using the seismic coefficient Cm and the self-weight of the wall W :
Fmax= CmW (29)
The parameters Fy and dy correspond to the transition point of the phase in which the wall is represented by a continuous model to the phase in which the wall is represented by two rigid blocks that are separated by fully cracked sections, corresponding to the RB mechanism. In reality, the branch that runs from the origin to the transition point is not perfectly straight, since the stiffness of the wall reduces for an increasing load as a result of damage. For simplicity, however, it is assumed that the wall behaves linearly up to the point that the corresponding F-d-curve intersects the descending branch that depicts to the RB mechanism. Hence, an elastic analysis is performed for the analyzed wall according to the standard beam theory in which continuous properties are assigned to the wall. In order to account for the actual reduced stiffness of the wall, the secant modulus of elasticity at 33% of the compressive strength is used to describe the wall’s continuous properties. This method of deriving the appropriate values for the parameters Fyand dyis described in Appendix C.2.
The presented procedure of determining the parameters Fy, dy and dmax is validated by means of a comparison with experimental data. The gray lines in Figure 3.14 show the experimentally obtained F-d-relationship of a single-leaf masonry specimen that is subjected to dynamic OOP loading conditions [37]. The orange lines in the same figure show the capacity curve, as determined using the previously described method, while taking into account BCs, overburden loads, self-weights and dimensions that correspond to the setup of the experiment.
Figure 3.14: Validation of construction method bilinear capacity curve of OOP wall behavior [37].
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Equivalent SDOF system with a single spring
The established capacity curves correspond to the behavior of URM walls that experience a perfect RB mechanism. In the current research, however, the analyzed URM walls in the OpenSees models are modelled as an idealized system. Hence, the capacity curve must be adapted such that it represents the behavior of the equivalent SDOF system of the analyzed wall. This is done using two adjustments to the previously described method:
• The lumped mass that idealizes the wall is not exactly equal to the wall’s total mass, due to the fact that the mass of the wall only partly contributes to the inertia force that corresponds to the wall’s lumped mass. A certain percentage of the mass of the wall more significantly cooperates to the lumped masses that correspond to the floors that enclose the analyzed wall than it does to the wall’s lumped mass itself. This can be taken into account by using the equivalent self-weight of the wall W∗, rather than the wall’s total self-weight W , for the determination of the parameter Fmax. For simply supported walls with a uniformly distributed mass, the equivalent self-weight is equal to 75% of the wall’s total self-weight [64]:
Fmax= CmW∗= Cm(0.75W ) (30)
As a consequence of the implementation of the equivalent self-weight of the wall W∗, rather than the total self-weight of the wall W , the descending branch that depicts the RB mechanism of a wall changes. Hence, the parameters Fy and dy that correspond to the equivalent SDOF system are determined using the intersection point of the original linear ascending branch and the updated descending branch;
• The established capacity curves correspond to the presence of a perfect RB mechanism. Hence, in order to correctly model the bilinear OOP behavior of an idealized system, the behavior of the perfect RB mechanism must be transformed into the behavior of an equivalent SDOF system. For this, the mass participation factor γ is used. The NZ guideline [63] defines this γ-factor in the following manner:
“Participation factor for rocking system relating the deflection at the mid height hinge to that obtained from the spectrum for a simple oscillator of the same effective period and damping”
All three parameters Fy, dy and dmax are divided by the mass participation factor γ.
γ = (Wbyb+ Wtyt)h
2J g (31)
Using the presented method, the parameters Fy, dy and dmax can be defined. As a consequence, the OOP capacity curves of URM walls can be constructed. These capacity curves correspond to a single spring that includes the total OOP capacity of the wall. Therefore, this capacity curve can only be directly used in the Tier 3a method in which only one spring is incorporated in the SDOF wall system, as introduced in section 3.4.1. In the Tier 3b and Tier 3c method, the stiffness of the URM wall is modelled using two springs. The next section describes the transformation from the material model that corresponds to one spring into a material model that corresponds to two springs.
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Equivalent SDOF system with two springs
The two SDOF systems that can be used to represent the analyzed URM wall are illustrated in Figure 3.15. An illustration of the eigenmode of both systems is depicted in Figure 3.16. In this case, the floors at the top and the bottom side of the URM wall are set to zero, and the normalized displacement φ1, corresponding to the first and only eigenmode of the SDOF systems, is assigned to the wall’s lumped mass.
a) SDOF system with single spring b) SDOF system with two springs Figure 3.15: Two types of SDOF mass-spring-damper systems for representation of wall.
Note: depicted situation corresponds to wall on BL2
a) SDOF system with single spring b) SDOF system with two springs Figure 3.16: Eigenmode of the two types of SDOF mass-spring-damper systems.
Based on these configurations, several characteristics of the SDOF systems can be found, as listed in Table 3.5, with which the relationship between the stiffness of the single spring, kw, and the stiffnesses of the two springs, kw1and kw2, can be established.
Table 3.5: Characteristics of the two types of SDOF mass-spring-damper systems.
SDOF system with single spring with two springs
Spring - 1 2
Deformation of spring φ1 φ1 −φ1
Spring stiffness kw kw1 kw2
Force in spring kwφ1 kw1φ1 kw2− φ1
Total spring force on lumped mass mw kwφ1 kw1φ1+ kw2φ1
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By equating the total spring force on the lumped mass mw of both SDOF systems, the relationship between the stiffness of the single spring, kw, and the stiffnesses of the two springs, kw1 and kw2 is found, as described in Eq. 32. It is assumed that kw1 equals kw2.
kwφ1= kw1φ1+ kw2φ1 → kw= kw1+ kw2 → kw1= kw2=kw
2 (32)
This relationship corresponds to springs that are connected in parallel. However, springs that are connected in parallel generally show exactly equal deformations, whereas the springs in the SDOF system that is shown in Figure 3.16b show opposite deformations, although of the same magnitude.
As a matter of fact, the connection between the two springs can therefore be more accurately categorized as anti-parallel [65]. Due to the fact that for both parallel and anti-parallel springs the
As a matter of fact, the connection between the two springs can therefore be more accurately categorized as anti-parallel [65]. Due to the fact that for both parallel and anti-parallel springs the