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The effects of varying: frame to wall stiffness ratios, structural ductility, frame to wall connection strengths, structural heights, level of seismicity, and structural layouts on the magnitude of transfer and inertial forces in floor diaphragms were investigated in this section.

To determine the trends that are associated with floor diaphragm forces, a number of static and dynamic analyses were carried out. Simple modal analyses were performed to determine some fundamental trends relating to transfer forces; these results are described in Section 2.3.1.

Time history analyses were then carried out to determine the time dependent behaviour of the forces which develop in floor diaphragm. A range of analyses investigating the affects of variations of frame to wall stiffness ratios, structural ductility, flexibility of the structure, variations of floor diaphragm strength and stiffness and different heights of the structures were carried out using analytical models with the frame to wall structural system. Descriptions of the frame to wall model that were used for these analyses are provided in the below paragraphs.

Time history analyses were carried out to investigate the magnitudes of floor diaphragm forces for structures designed for low seismic regions. For this analysis a structure located in the Auckland region was designed. Details of this structure and the results from this study are provided in Section 2.3.8.

Different structural systems were investigated, using time history analysis, to determine how different structural systems affect floor diaphragm forces. Podium structures and structures with different length walls were included in this study. Descriptions of these structures and the results from this study are provided in Sections 2.3.9 and 2.3.10.

A further study was carried out to investigate whether transfer and inertial forces could be determined separately and then added together. Comparisons were made between adding these forces together separately and the forces obtained from time history analysis. A description of this analysis and the results is provided in Section 2.3.11.

The parameters that were used in the frame to wall model that was used in the analyses for Sections 2.3.1 through to Section 2.3.7 are provided in the following paragraphs.

The frame to wall stiffness was varied to investigate the influence of relative stiffness of the vertical lateral force resisting system on the magnitude of the forces which develop in the floor. The variation of stiffness between the frame and wall elements was measured by determining the relative ratio of stiffness for the frame and the wall. The stiffness of the frame and the stiffness of the wall were determined separately by carrying out pushover analyses. From these stiffness values the ratio between the stiffness of the frame and the wall components (frame to wall stiffness ratio) was determined. The frame to wall stiffness ratios that were investigated in this study were; SR1:0.85, SR1:1.23, SR1:69 and SR1:2.58. These ratios were chosen as they represent a realistic range of stiffness ratios for frame to wall structures. The member sizes for the 9 storey structures, which correlated to these different stiffness ratios, are provided in Table 2 4. The properties for the 9 storey structures have only been described here as these structures have been used for the majority of the analyses. The affects of different heights of the structure on the magnitudes of diaphragm forces was considered in Section 2.3.7, the parameters fro these structures were provided in this section with the results.

Table 2(4 Geometries of the members of the models for the different stiffness ratios Stiffness ratio Beam Column Wall

1:0.85 0.9m by 0.6m 1.2m by 1.2m 9.45m by 0.40m

1:1.23 0.9m by 0.6m 1.0m by 1.0m 10.05m by 0.40m 1:1.69 0.9m by 0.6m 0.85m by 0.85m 10.53m by 0.40m 1:2.58 0.9m by 0.6m 0.70m by 0.70m 11.10m by 0.40m

The fundamental translational period of the 9 storey analytical frame to wall structure model was 0.57s. To ensure each of the structures with different stiffness ratios had the same dynamic properties, the elastic fundamental translational period of each of these structures was kept constant. An investigation on the affects of structures with different overall flexibility was carried out, descriptions of these structures and the results from this investigation are provided in Section 2.3.4.

The earthquake scale factors that were used for the time history analysis for the 9 storey structure are provided in Table 2 5.

Table 2(5 Scale factors for the earthquake records for 9(storey structure Record Component k1 9(storey

Lucerne North 1.02 Lucerne South 1.40 Izmit North 1.84 Izmit South 1.57 La Union North 1.69 La Union South 1.98 El Centro North 0.96 El Centro South 1.40 Llolleo North 0.82 Llolleo South 0.56 Tabas North 0.49 Tabas South 0.42

2.3.1 General Concepts

To obtain a general understanding of transfer forces both modal and pushover analyses were carried out.

Reviews of the literature indicated that transfer forces can develop in floor diaphragms due to compatibility constraints, imposed by the connection with the floor diaphragm, between lateral force resisting elements which have different fundamental deformation patterns; such as frames and walls for example. This indicates that transfer forces are deformation controlled, therefore it is expected that transfer forces will be the dominant force for the lower dynamic modes of the structure and inertial forces will be dominant for the higher modes of the structures. To investigate this, modal and pushover analyses were carried out to determine the contribution that different dynamic modes had to the magnitudes of transfer forces.

The 9 storey analytical frame to wall model with a stiffness ratio of SR 1:1.23 designed for structural ductility of 3 was used for this study. This structure was chosen as it represents a typical medium rise dual structure. Only one structure was considered for this initial study to allow initial trends to be identified. The affects of variations of: stiffness, strength and height are provided in the Sections 2.3.2 to 2.3.7.

Modal analysis was performed to determine the normalised mode shapes for this structure. The mode shapes were then multiplied by the participation factors and the spectral acceleration factor for the period of the mode shape and divided by the angular frequency squared (1/ω2) to determine the contribution of each mode to the displacement response of the structure. The spectral acceleration factor was associated with the region Wellington for soil type C from the New Zealand Structural Design Actions Standard (Standards New Zealand, 2004a). This allowed the relative magnitude of each mode shape to be determined. Figure 2 12 indicates the magnitudes of displacements for the different mode shapes obtained from the modal analysis.

Elastic and inelastic pushover analyses were then performed to determine the transfer forces by applying displaced shapes equivalent to the mode shapes for the structure. The analysis was terminated when the displacement reached the maximum displacement for the particular mode shape. Comparisons of the floor diaphragm transfer forces, for mode shapes 1 4, for elastic and in elastically responding structures, are shown in Figure 2 13 and Figure 2 14 below. 0 1 2 3 4 5 6 7 8 9 10 -0.05 0 0.05 0.1 0.15 0.2 Le v e l Displacement (m)

Displacements for different dynamic modes from modal analyses Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6 Mode 7

-4000 -2000 0 2000 4000 1 2 3 4 5 6 7 8 9 Transfer force L e v e l

Transfer forces in the elastic