Results main research
G.1 Material model for springs that represent lateral behavior of building layers
In section 3.5.1, the construction method of the material models for the NL springs that represent the lateral behavior of the building layers BL1 and BL2 is introduced. As mentioned in this section, it is desirable to define the NL springs using experimental data on the specific structure due to the high amount of aspects that influence the lateral behavior of URM structures. In case no experimental data are available for a specific building layer, however, the BBC of the layer is based on another BBC that has been constructed using actual experimental data. This latter BBC then serves as an example. The ‘example’ BBC is transformed into the BBC for the building layer of interest by means of a multiplication with two factors:
1. The scale factor (SF), which is defined using experimental data of the ‘example’ structure. It includes both the lateral force capacity Fmax, equal to the force corresponding to the second envelope point of the BBC F 2, and the top load (TL) that acts on both BL1 and BL2, equal to m1 + m2 and m2, respectively:
SF = Fmax,BL2,example× T LBL1,example
Fmax,BL1,example× T LBL2,example
(82)
2. The ratio between the TL that applies for the building layer of interest and the TL that applies for the ‘example’ building layer.
Equation 83 describes the transformation from the ‘example’ BBC into the BBC of interest:
F i = SF × T L
T Lexample × F iexample (83)
Due to the implementation of the difference in TL, the transformation method accounts for the fact that higher axial stresses result in higher lateral capacities [29,61]. In the current research, only the lateral force capacity of a building layer is scaled in accordance to the TL-value, despite the fact that the TL may also influence the layer’s lateral displacement capacity [84, 85]. However, the influence of a reduced or increased TL on the lateral displacement capacity is only limited compared to its influence on the force capacity. Hence, the displacement coordinates of the envelope points are not scaled as a result of a reduced or increased TL.
Besides the need to define the three envelope points, the β-value must be set. As described in section 3.5.1, a β-value of 0.9 is used to describe the degradation of the unloading stiffness in the material models for the springs that represent the lateral behavior of the building layers. For all three types of typologies, structure A, B and C, as well as for both the BL1 and the BL2, the same β-value is used.
G.1 Material model for springs that represent lateral behavior of building layers page 123
G.2 Structure B
Structure B consists of one RC floor, one timber diaphragm and a pitched timber roof. The masses and the stiffness of the BL1 are based on a modified version of EUC-BUILD-6 [86]. This two-story URM structure, with a RC floor and a timber diaphragm, has large openings in the facade at BL1.
Due to these large openings, a soft-story mechanism is likely to occur. An NLPO analysis has been performed for the modified version of EUC-BUILD-6, which is used to model the stiffness of the BL1 of structure B. No significant differences in dimensions are present between the EUC-BUILD-6 and the EUC-BUILD-1 structure. Therefore, the dimensions of structure B are assumed to be equal to the dimensions of structure A.
The dataset corresponding to the NLPO analysis on the modified version of EUC-BUILD-6 includes the values of the lumped masses m1 and m2, which are given in Table G.1. Two assumptions are made concerning the model of the modified version of EUC-BUILD-6 that is used for the NLPO analysis. The first assumption implies that no clay wall is implemented on the North side of the model, i.e. the analyzed structure is not an end-unit, but a mid-unit of a terraced house. The second assumption implies that the masses of the floor slabs and the roof system in the modified version of EUC-BUILD-6 are the same as in the original EUC-BUILD-6 structure. Based on these assumptions, as well as on the lumped masses as documented in the dataset of the NLPO analysis, an estimation is made of the masses of the masonry in structure B. A summary of the structural masses is given in Table G.2.
Table G.1: Summary of lumped masses of idealized systems of structure A, B and C; units of t.
A B C
m1 24.65 22.36 16.25 = half masonry BL1 + half masonry BL2 + floor slab 1
m2 24.70 13.06 15.60 = half masonry BL2 + floor slab 2 + masonry gable walls + roof
Table G.2: Summary of structural masses of structure A, B and C; units of t.
A B C
Masonry building layer 1 CS 8.5 7.57 8.5 CL 5.6 2.77 5.6 Masonry building layer 2 CS 8.7 8.7 8.7 CL 5.9 3.28 5.9 Masonry gable walls CS 2.4 2.4 2.4
CL 1.2 - 1.2
Floor slab 1 10.3 11.2 1.9
Floor slab 2 11.0 1.9 1.9
Roof 2.8 2.77 2.8
G.2 Structure B page 124
G.2.1 Backbone curves structure B
The BBC of the BL1 of structure B is constructed based on NLPO results of the modified version of EUC-BUILD-6 [86, 87]. A symmetrical BBC is obtained by taking the average of the original pushover curve in the positive and in the negative domain. Then, the symmetrical BBC is simplified such that the curve is described by only three envelope points in both the positive and the negative domain. The construction steps are illustrated in Figure G.2.
a) symmetrical BBC b) trilinear BBC
Figure G.2: Construction of BBC for BL1 of structure B using experimental data.
The BBC of the BL2 of structure B is constructed by means of a transformation of the BBC of the BL2 of the structure A. This transformation method has been introduced in section G.1.
Equation 84 shows the calculation of the appropriate scale factor SF that is incorporated in the BBC transformations for the analyzed Groningen typologies. Equation 85 describes the transformation that is applied for the construction of the BBC of the BL2 of structure B. The eventual BBCs of both the BL1 and the BL2 are included in the overview of the BBCs of all three types of two-story structures in Figure G.3.
SF = Fmax,BL2,A× T LBL1,A
Fmax,BL1,A× T LBL2,A
(84)
F iBL2,B= SF × T LBL2,B T LBL2,A
× F iBL2,A (85)
G.3 Structure C
Structure C consists of two timber diaphragms and a pitched timber roof. In contrast to structure A and B, structure C is not based on an actually tested structure of which data are available. The dimensions and material properties of this type of two-story URM structure are assumed to be equal to those in structure A and B. The masses are as well based on the first two structures. Despite of the fact that structure C is formulated based on assumptions and estimations, the structure can give useful insight in the OOP behavior of URM walls that are located in more light-weight structures, in which also more flexible boundary conditions apply at the top and bottom side of the wall.
G.3 Structure C page 125
A summary of the structural masses is given in Table G.2: the masses of the masonry and the roof system are equal to structure A, whereas the mass of the timber diaphragms are set equal to the mass of the second floor slab in structure B, which also corresponds to a timber diaphragm. The values of corresponding lumped masses m1 and m2 of structure C are given in Table G.1.
G.3.1 Backbone curves structure C
The BBCs of the BL1 and the BL2 of structure C are constructed by means of a transformation of the BBCs of the BL1 and the BL2 of structure A. Again, the SF, as calculated using Eq. 84, is incorporated in the transformation method that is introduced in section G.1. Equations 86 and 87 describe the transformations that are used to construct the BBCs of both the BL1 and the BL2 of structure C.
F iBL1,C= SF × T LBL1,C
T LBL1,A
× F iBL1,A (86)
F iBL2,C= SF × T LBL2,C
T LBL2,A
× F iBL2,A (87)
The constructed BBCs of both the BL1 and the BL2 are included in the overview of the BBCs of all three types of two-story structures in Figure G.3.
Figure G.3: Overview of BBCs of BL1 and BL2 for structure A, B and C.
G.3 Structure C page 126
G.4 Material model for springs that represent OOP behavior of wall
In section 3.5.2, the construction method of the material models for the NL springs that represent the OOP behavior of URM walls is introduced. The upper part of Table G.3 lists several values that correspond to the different URM walls. The self-weight W is determined as follows: W = ρCS× g × h × tnom, where ρCS is the density of the applied masonry. In the current research, ρCS = 1835 [kg/m3], equivalent to the CS masonry that is applied in the EUC-BUILD-1 structure [69]. This value is used for all three structures A, B and C. Based on the values in the table, the parameters Fy, dyand dmaxare calculated using the method that is described in section 3.5.2. The three derived parameters are listed in the lower part of Table G.3. Figure G.4 shows the positive displacement range of the resulting capacity curves of the URM walls on BL1 and BL2 for the three structures A, B and C. It must be noted that the depicted capacity curves correspond to the single spring that is applied in the Tier 3a method. In the Tier 3b and 3c method, instead, two springs that are connected in parallel are applied. The capacity curves of the latter springs would correspond to the depicted capacity curves in Figure G.4 in case the y-coordinates of the curves would be multiplied by a factor equal to 0.5.
Table G.3: Parameters OOP capacity of analyzed URM walls.
A B C
BL1 BL2 BL1 BL2 BL1 BL2
Boundary condition type [-] BC3 BC3 BC3 BC1 BC1 BC0
Self-weight W [kN/m] 4.92 4.48 4.92 4.48 4.92 4.48
Overburden load P [kN/m] 40.76 16.77 29.15 6.81 23.90 8.00
Effective thickness tef f [mm] 98.1 100.3 99.1 101.2 99.6 101.1
Height h [m] 2.68 2.44 2.68 2.44 2.68 2.44
Mass participation factor γ [-] 1.00 1.13 1.06 1.37 1.25 1.43 Force at transition point Fy∗ [kN] 9.167 4.105 6.437 1.324 3.496 0.862 Displacement at transition point d∗y [mm] 3.37 1.07 2.29 0.69 2.55 1.09 Instability displacement d∗max [mm] 98.06 88.93 93.23 60.13 61.78 35.33
Figure G.4: Positive displacement range of OOP capacity curves of analyzed URM walls.
G.4 Material model for springs that represent OOP behavior of wall page 127
Appendix