The sensitivity of the rigid foundation assumption was investigated. It was thought that this assumption would have a reasonable affect on the forces which develop in the floor diaphragms as additional flexibility from the foundation would affect the deformations of the structure. A variety of literature has also indicated that foundation flexibility effects should be considered when modelling any type of structure. This section describes how foundation flexibility affects the magnitudes of total floor forces which develop in a variety of different structures. Foundation models of different complexity were analysed to determine whether a simple model would be adequate to represent the flexibility of the foundation.
The foundation compliance model was developed to represent the soil conditions in the CBD area of Wellington. The geological map (number 22) of Wellington provided in Begg and Mazengarb (1996) indicates that the typical soil in the CBD area of Wellington consists of: Alluvium, Silt, Peat and Loess, which include Haywards and Kaitoke gravels and subsurface Moera gravel.
Table 3%1 Elastic fundamental periods for rigid foundation structures Structure Frame%to%wall SR Period, T1 (s)
9 storey stiff SR1:0.85, SR1:1.69 0.58 9 storey flexible SR1:0.85, SR1:1.69 1.44 6 storey stiff SR1:0.61, SR1:1.49 0.32 6 storey flexible SR1:0.61, SR1:1.49 0.72 3 storey stiff SR1:0.3, SR1:1.14 0.28 3 storey flexible SR1:0.3, SR1:1.14 0.60
The analytical models of the structures, from previous sections of this study, were used in this investigation. The floor diaphragm forces from these models represented the results for the rigid foundation case. Foundation compliance models were added to these rigid foundation models. The global flexibilities and the relative frame to wall flexibilities of the rigid foundation structures are presented in Table 3 1. The labels stiff and flexible in this table refer to the relative flexibility of the building obtained from the elastic fundamental period for the building, T1.
The design of the complex foundation and the soil model used for this analysis is based on the foundation compliance model provided in the Red Book (NZCS, 1998). The foundation model that was used in this study was a reinforced concrete pile foundation system.
The foundation beams under the frame of the structure were 1m by 0.6m in depth. The wall foundation beam was much larger than the frame foundation beam due to large moments which form in the wall; the wall foundation beam was 2.5m by 0.6m. The piles in this model were 1m in diameter and were 20m long. One pile was placed under each of the columns in the frame and two piles were place under the wall of the structure.
The foundation piles for this structure were designed to remain elastic to avoid damage occurring in the foundation. Inelastic behaviour in the surrounding soil was incorporated into the model. Figure 3 1 provides a graphical representation of the analytical foundation model.
20m
1m Floor diaphragm
Frame Wall
The stiffness of the soil in this model was represented by a series of springs. The sub grade reaction or the stiffness of the soil pile interaction in the horizontal direction was obtained from information provided by a practising senior geotechnical engineer. It was advised that the modulus of subgrade reaction for alluvium soils, in the Wellington region, ranged between 10MPa from ground level to a depth of around 10m and varies around 30MPa for depths greater than 10m. The sub grade reaction of the soil was then calculated using the Vesic Equation provided in Equation 3 1. The soil parameters used in these models and for this calculation are provided in Table 3 2.
−
=
12 2 1 41
65
.
0
ν
E
I
E
EB
B
k
p p s Equation 3%1Table 3%2 Horizontal soil stiffness values for Wellington CBD Depth of soil (m) Soil stiffness
(kN/m3) 0 5 4574 5 10 9806 10 20 15037 Other Parameters Poisson’s ratio 0.25 Mass density 1.8t/m3
Pile stiffness 30 GPa
Pile diameter 1.0 m
RUAUMOKO 2D (Carr, 1981 2009b) has the ability to model foundation elements. The foundation elements used in this program are based on the Winkler spring model. A representation of the model is provided in Figure 3 2.
The model used for this analysis was similar to the Winkler beam model available in RUAUMOKO (Carr, 1981 2009d). In the foundation models for this study springs were used to represent the stiffness of the soil pile interaction along the lengths of the piles as shown in Figure 3 1.
Non linearity of the soil elements due to the movement of the piles was modelled using the Ramberg Osgood hysteresis model (Kaldjian and Fan, 1967). Figure 3 3 shows the back bone curve for this hysteresis loop.
Figure 3%3 Ramberg%Osgood hysteresis loop for non%linear soil
It was found that this model needed to be adjusted to incorporate an alpha factor which multiplies the force ratio to adjust for the damping. This factor is proportional to the level of damping in the structure. In RUAUMOKO, the damping of the structural members is accounted for by using a global damping value applied to the entire structure. When this Ramberg Osgood model is used to represent a structural material such as steel, the alpha factor is assumed to be 1, assuming no damping for the model. In the case where soil is modelled, the damping needs to be accounted for separately in each Winkler spring. The constant damping ratio of 0.1 for soil, which was recommended by McManus and Alabaster (2004), was used in this model. Checks were made on this new Ramberg Osgood hysteresis loop, which was incorporated into RUAUMOKO, to ensure it was behaving as it should.
The book by Reese and van Impe (2001) was used to determine the ultimate and yielding strength of the soil. This book provided an equation for determining the ultimate strength of a soil pile foundation for cohesionless soil. This equation is shown in Equation 3 2. This equation is approximate due to the models that it is based on. However, it provides an idea of
the magnitude of the ultimate soil resistance which can generally not be found unless various onsite tests are performed.
Equation 3%2
Where, γ is the weight density of the soil, Ko is the at rest pressure of the soil, z is the depth of the soil,
φ
the angle of shearing resistance (otherwise known as the angle of internal friction), β = 45o+φ
/2, αs describes the density of the soil, b is the diameter of the pile and Ka is the Rankin active pressure coefficient. The parameters that were used in this equation are provided in Table 3 3.Table 3%3 Parameters used to determine the ultimate and yielding strength of the soil
Parameter Value
Weight density, γ (kN/m3) 18.0
At rest pressure, Ko 0.5
Angle of internal friction,
φ
30o Density parameter, αs 0.26Active pressure, Ka 0.33
The soil yielding values were obtained using the backbone formula of the Ramberg Osgood hysteresis rule which is provided in Figure 3 3 and fitting a bi linear approximation to the curve.
McManus and Alabaster (2004) suggested that it is important to include the inertial affects of soil pile foundation systems when developing an analytical model to represent the behaviour of soil pile structure interaction. This paper describes that it is appropriate to model the foundation using generalised mass, stiffness and damping values (m*, k*, and c*). In this model the mass of the surrounding soil was lumped with the mass of the pile at the nodes of the piles which are located at regular 1m intervals over the length of the piles. Results from a study carried out by McManus and Alabaster (2004) indicated that the mass of the foundation should be made equivalent to the mass of the pile cap plus one third of the mass of the piles and two pile diameters of the surrounding soil. This suggested method of determining the masses was used to determine the foundation mass for this study.
Shaft resistance, alternatively known as “skin friction”, provides vertical stiffness between the soil and the pile. Two sets of empirical values for determining the limiting skin friction for a type of soil, one based on Standard Penetrometer Test (SPT) and the other on Cone Penetration Test (CPT), are provided in Tomlinson and Boorman (2001). The information provided from this publication indicates that the skin friction could range between 67 80kN/m2 for the soil in the Wellington CBD region. These values indicate the variability of this sort of data. Another form of vertical resistance comes from the end bearing of the piles. Tomlinson and Boorman (2001) provides a limiting end bearing value of 2.9MN/m2 based on SPT values for soil similar to that in the Wellington region. The vertical resistance of the end bearing of the pile (1m diameter) was compared to the shaft resistance of the pile to determine the relative contributions of each component. It was found that the end bearing resistance provided a contribution of the order of 30 times greater than the shaft resistance. As a result of this the model was constructed with the base of the piles fixed in the vertical direction and the shaft resistance of the piles was ignored. A sensitivity analysis was carried out to determine the effect on floor forces of ignoring the shaft resistance and fixing the pile at the base. 0 1 2 3 4 5 6 7 8 9 10 0 2000 4000 L e v e l Force (kN)
Envelop of total floor forces for building with frame to wall SR 1:0.85
El Centro N
No Skin Friction Skin Friction
Figure 3%4 Envelope of total floor forces for building with frame%to%wall SR 1:0.85 – El Centro N
Figure 3 4 shows a comparison between the two cases where skin friction is and is not included in the analytical model. This figure clearly shows that there is negligible difference in the two methods.
The fundamental translational periods for the stiff and flexible buildings which incorporate the complex foundation compliance model, indicated in Figure 3 1, are provided in Table 3 4.
Table 3%4 Elastic fundamental periods for complex foundation compliant structures Structure description Stiff T1 (s) Flex T1 (s)
9 Storey SR 1:0.85 1.156 1.756
9 Storey SR 1:1.69 1.202 1.821
6 storey SR1:0.61 0.909 1.090
6 storey SR1:1.49 0.915 1.083
3 storey SR1:3, 1.14 0.760 0.842
The fundamental periods, for the different stiffness ratios, were found to vary slightly for the 9 and the 6 storey structures. This is due to the frame and the wall elements providing relatively less overall stiffness for the structure due to the contribution of stiffness of the foundation piles.
The twelve time history records, described in Section 2.2.6, to represent the seismic motion in the Wellington region were used for the stiff buildings with foundation compliance, in this study. Only six of the records were used for the more flexible buildings with foundation compliance. The time history scaling coefficients are provided in Table 3 5 below.
Table 3%5 Time history scale factors for foundation compliant stiff buildings k1 3 storey k1 6%storey k1 9%storey Record Comp All SR SR1:1.0.61 SR1:1.49 SR1:0.85 SR1:1.69
Lucerne North 1.06 0.98 0.98 0.99 0.82 Lucerne South 1.48 1.46 1.46 1.25 1.55 Izmit North 1.85 1.75 1.75 0.83 1.66 Izmit South 1.68 1.66 1.66 1.53 1.50 La Union North 1.55 1.57 1.57 1.67 1.69 La Union South 2.10 2.36 2.36 1.52 2.80 El Centro North 0.86 0.85 0.85 1.66 1.01 El Centro South 1.27 1.25 1.25 2.75 1.27 Llolleo North 0.81 0.86 0.86 0.98 1.00 Llolleo South 0.54 0.56 0.56 0.61 0.62 Tabas North 0.43 0.45 0.45 0.44 0.44 Tabas South 0.41 0.45 0.45 0.52 0.52
Table 3%6 Time history scale factors for foundation compliant flexible buildings k1 3 storey k1 6%storey k1 9%storey
Record Comp All SR All SR SR1:0.85 SR1:1.69
Lucerne North 1.05 0.86 0.74 0.73 Izmit North 1.78 1.70 1.61 1.63 La Union North 1.56 1.62 1.86 1.83 El Centro North 0.86 0.94 1.17 1.16 Llolleo North 0.84 0.92 1.72 1.81 Tabas North 1.56 0.44 0.44 0.44
3.1.1 Simple Foundation Model
A simple foundation model was developed to determine if a simple model could adequately represent the deformations of the structure and therefore adequately predict the magnitudes of floor forces. A simple model would reduce the complexity required and the computational effort.
The simple model employed, for this study, was an elastic model that was based on the physical model proposed by Wolf and Meek (1994). The model coupled horizontal, vertical and rocking motions of the structure on soil for horizontal seismic excitation. The layout of this model is shown in Figure 3 5.
Rest of the structure
Figure 3%5 Simple foundation model (Wolf and Meek, 1994)
The damping model, which was described in Section 2.2.3, was used for this model. RUAUMOKO allows the damping to be described at one time for all the members in the model. Therefore, individual foundation dashpots were not incorporated into the foundation model.
The simple model was based on the assumption of a rigid plate on an elastic half space. Empirical formulas that are presented in Newmark and Rosenblueth (1971) were used to determine the stiffness parameters for the analytical model. These empirical formulae are shown in Equation 3 3 to Equation 3 5 below.
Equation 3%3
Equation 3%4
Equation 3%5
Where E is Young’s modulus for soil; A is the area of the foundation; Cs, KT and Kφ are constants obtained from Newmark and Rosenblueth (1971) and v is poisons ratio. The stiffness values used in this simple model are provided in Table 3 7.
Table 3%7 Parameters for simple foundation model
Vertical stiffness, KV 498.5x10 6
Horizontal stiffness, KH 408.2x10 6
Rotational stiffness, Kφ 4.25x106