Analytical Model

4.1.1 Structural System

Figure 4 1 Moment resisting frame

The analytical model represented the behaviour of a moment resisting frame structure with perimeter seismic resisting frames and an interior gravity supporting system. The stiffness of the two parallel seismic frames was combined to form a 2 dimensional model which neglected the lateral capacity of the gravity system. This is represented in Figure 4 1.

Table 4 1 Member geometries for Wellington seismic frame structures

Structure Beams Corner Columns Side Column

3 Storey Flex 0.9x0.6m 0.8x0.8m 0.8x0.5m 3 Storey Stiff 0.9x0.6m 1.0x1.0m 1.0x0.5m 6 Storey Flex 0.9x0.6m 0.8x0.8m 0.8x0.5m 6 Storey Stiff 0.9x0.6m 1.0x1.0m 1.0x0.5m 9 Storey Flex 0.9x0.6m 0.8x0.8m 0.8x0.5m 9 Storey Stiff 0.9x0.6m 1.0x1.0m 1.0x0.5m

Table 4 2 Member geometries for Auckland seismic frame structures

Structure Beams Corner Columns Side Column

3 Storey Flex 0.7x0.4m 0.6x0.6m 0.6x0.4m 3 Storey Stiff 0.7x0.4m 0.8x0.8m 0.8x0.4m 6 Storey Flex 0.7x0.4m 0.6x0.6m 0.6x0.4m 6 Storey Stiff 0.7x0.4m 0.8x0.8m 0.8x0.4m 9 Storey Flex 0.7x0.4m 0.6x0.6m 0.6x0.4m 9 Storey Stiff 0.7x0.4m 0.8x0.8m 0.8x0.4m

Structures of 3, 6 and 9 storeys, with an inter storey height of 3.6m, were included in this study to ensure a range of different dynamic effects were considered. Two different levels of flexibility of the structure for the different height categories were included to incorporate any affects that the flexibility of the structure has on the structural performance. The geometry of the members used in the different structures is provided in Table 4 1 and Table 4 2.

The analytical structural models developed for this study included the complex foundation model, which was developed in Section 3.1, as this was shown to affect the magnitude of transfer forces for dual structures. The complex foundation model was incorporated into the analytical model for the study of dual structures, described in Section 3.1. For consistency between this moment resisting frame study and the dual structures study, the complex foundation was included in the analytical model for this study.

To adequately verify the proposed static design method three different soil types, categorised by the New Zealand Loadings Standard NZS1170.5 (Standards New Zealand, 2004a), were included in the study. These three soil types were rock (type A/B), shallow soil (type C) and deep or soft soil (type D). A further soil category of very soft soil exists. This type of soil has not been included in this study as it as not as common as the other soil types. For the Wellington region, all three of these soil types could exist: the Torlesse rock outcrops could represent rock, the alluvium deposits in the Central Business District (CBD) could represent the shallow soil regions and the deposits in the Lambton Quay/water front zone of marginal marine sediments could represent the deep or soft soil sites (Begg and Mazengarb, 1996). The soil stiffness parameters used to represent the foundation behaviour for these different regions are presented in previous sections. For soil type C, the soil parameters are provided in Section 3.1 and for soil type A/B and D the soil parameters are provided in Section 4.1.1

These three soil categories are also present in the Auckland region. The Auckland region is built on various volcanic rock sites, weathered rock sites resulting in shallow soil and very soft harbour mud around the waterfront region (Kermonde, 1992). The parameters used in the foundation model to represent the waterfront soil conditions were provided by a practising senior geotechnical engineer. The thickness of the harbour mud soils was taken as 10m and was assumed to be underlain with weak sandstone. The harbour mud was assumed to have an increasing stiffness of E=20MPa to 30MPa and a poisons ratio of v=0.5. The equivalent coefficient of sub grade reaction for this foundation was calculated using the Vesic equation which is presented in the foundation flexibility study in Section 3.1. The weak sandstone layer was assumed to have a shear wave velocity of vs = 500m/s. The maximum Young’s

modulus for this soil was calculated using the empirical equation shown by Equation 4 1.

) 1 ( 2 2 max =

ρν

ν

Where ρ represents the density of the soil (taken as 2400kg/m3 as advised by practising senior geotechnical engineer), vs represents the shear wave velocity and v represents the poisons

ratio taken as 0.3 for this soil type. It was advised that generally for soil foundations, 1/3 Emax

is used in the calculation of the sub grade reaction coefficient.

Table 4 3 Fundamental periods for Wellington seismic frames Structure Soil A/B Soil C Soil D

3 Storey Flex 0.54 0.76 0.95 3 Storey Stiff 0.38 0.66 0.86 6 Storey Flex 0.95 1.20 1.41 6 Storey Stiff 0.84 1.16 1.34 9 Storey Flex 1.45 1.73 2.02 9 Storey Stiff 1.28 1.60 1.90

Table 4 4 Fundamental periods for Auckland seismic frames Structure Soil A/B Soil C Soil D

3 Storey Flex 0.78 0.91 1.19 3 Storey Stiff 0.66 0.71 1.03 6 Storey Flex 1.59 1.89 1.98 6 Storey Stiff 1.38 1.64 1.88 9 Storey Flex 2.51 2.66 2.75 9 Storey Stiff 2.28 2.44 2.56

4.1.2 General Parameters

The general analytical modelling parameters that were used for this moment resisting frame study were similar to the general parameters used for the transfer force study that were described in Section 2.2.3. The time step used for the analyses described in this section was 0.002 seconds.

4.1.3 Members

The analytical models used to represent the beams and columns, for this moment resisting frame study, were the same as the models described in Section 2.2.4 for the transfer force study. Other properties that were similar to the properties described in Section 2.2.4 include:

− The hysteresis model and parameters that were used to describe the behaviour of the beam and column element;

− The method used to determine the effective section properties;

− The method used to determine the strength of the members;

− The method used to determine the plastic hinge lengths;

− The size of the rigid end blocks.

4.1.4 Weights and Loads

The weights used in the analysis are based on the values supplied in Section 2.2.5.

Table 4 5 Weights used in analyses

Wellington Auckland

Hollow core floor 3.5kPa 2.7kPa

Thickness of topping slab 90mm 75mm

Density of concrete 23.5kN/m3 23.5kN/m3

Super imposed dead load 1.3kPa 1.3kPa

Curtains walls and glazing 0.4kPa 0.4kPa

Live Load 3kPa 3kPa

Reduction factors were used to determine the appropriate live load for the structure according to the New Zealand Structural Loadings Standard. The weight was found per area of the structure and then distributed evenly per node attached to the area. The weights of the columns were lumped at the adjacent upper and lower nodal points in the analytical model.

4.1.5 Time History Records

The time history records that were used in the analysis of the Wellington structures were the same as the records described in Section 2.2.6. The time history records used in the analysis for the Auckland structures were the same as the records used for the study on transfer forces in low seismic regions that were described in Section 2.3.8.