Earthquake response of shallow foundations with low vertical gravity
Pender, M., Wotherspoon, L. & Carr, A. (2008) Earthquake response of shallow foundations with low vertical gravity load Proc. 18th NZGS Geotechnical Symposium on Soil-Structure Interaction. Ed. CY Chin, Auckland
Earthquake response of shallow foundations with low vertical gravity load Michael Pender, Liam Wotherspoon Department of Civil and Environmental Engineering, University of Auckland, NZ. Athol Carr Department of Civil and Natural Resources Engineering, University of Canterbury, NZ. Keywords: shallow foundation, earthquake response, uplift, nonlinearity, moment and shear. ABSTRACT This paper presents the development and response of a Ruaumoko model for the representation of uplift and reattachment of a shallow foundation. The vertical load in the foundation is used to control the characteristics of moment and shear loads, as these values must reduce to zero when the entire footing has detached from the underlying soil. Through extension of the capabilities of Ruaumoko this response was able to be represented. A simple elastic two bay portal frame structure was created and shallow foundations were attached to the base of each column of the portal frame and subjected to cyclic loading. Uplift modelling was shown to have a significant effect on the shear and moment in the footings. If the point of detachment and reattachment of the foundation was at different horizontal and/or rotational displacements the result was residual horizontal and rotational displacements at the end of loading. This shift in displacements occurred in conjunction with a shift in shear and moment in the footing. 1
INTRODUCTION
There are three aspects of shallow foundation response to earthquake loading that could be regarded as “complications”. First there is nonlinear behaviour of the soil with consequent nonlinearity of the foundation-soil interaction, second the limitations imposed on combinations of vertical load, shear and moment by the bearing strength surface, and finally the possibility that part of the foundation system may uplift during the earthquake. All three of these are computationally challenging, but in this paper we will discuss the modelling of uplift and reattachment of shallow foundation elements for a simple framed structure subject to a cyclic loading (a pseudo earthquake). This will be done using the Ruaumoko software (Carr 2006), or more correctly Ruaumoko with specially developed additional features to represent shallow foundations. Ruaumoko is a non-linear dynamic structural analysis program. The point of adapting Ruaumoko to encompass foundation modelling is to promote the integrated design of structure foundation systems. It seems obvious to us that the most effective way of conducting soil structure interaction studies is to work with a single model of the whole structure foundation system. Ruaumoko and many other structural analysis systems tend to keep each degree of freedom separate. Thus, in Ruaumoko for example, there are facilities for a degree of freedom to detach from the system and reattach. However, a shallow foundation, in general, carries vertical load, horizontal shear and moment. Uplift will occur when the vertical load becomes zero and at that point the degree of freedom representing horizontal shear and moment must also detach, that is the horizontal shear and moment must become zero. Similarly, when reattachment occurs because the vertical degree of freedom
1
Pender, M., Wotherspoon, L. & Carr, A. (2008) Earthquake response of shallow foundations with low vertical gravity load
connects with the foundation again, the horizontal shear and moment degrees of freedom must also reconnect. Having developed such a facility for Ruaumoko this paper demonstrates, via the response of a simple structural model, how the uplift and reattachment operates. 2
SHALLOW FOUNDATION MODELS
The bearing strength of a shallow foundation is a complex function of the soil properties and the combinations of vertical load, horizontal shear and moment sustained by the foundation. When the bearing strength is reached all the resistance that can be provided by the shear strength of the soil has been mobilised. At loads which are small in relation to the bearing strength the soil structure interaction is expected to be elastic. Between these limits there will be a gradual transition, becoming more nonlinear, between linear elastic behaviour and bearing strength failure. A frequently used model for a shallow foundation is to assume a rigid member resting on a bed of springs. This is attractive because it allows progressive uplift of the foundation. The preferred model for the elastic behaviour of the foundation is to assume that the soil is an elastic half space. Solutions for the stiffness characteristics of this case have been developed by Gazetas and his colleagues (Gazetas et al. 1985; Gazetas and Tassoulas 1987; Hatzikonstantinou et al. 1989). With this model in mind one can set the stiffness of the springs in a bed of springs so that the vertical stiffness of the bed of springs represents correctly the stiffness of a rigid footing resting on an elastic half space. However, there is a drawback as the rotation stiffness of a bed of springs is less than the rotational stiffness of a rigid foundation resting on an elastic half space. These ideas are illustrated in Figure 1. One method for keeping the rotational stiffness as required is to have an additional rotational spring to make up for the deficiency shown in above. In the Ruaumoko model this rotational spring was included inside the compound spring element. Modifications were made to the original compound spring element in Ruaumoko to allow the vertical stiffness to control the other stiffnesses within the compound spring. By defining interaction between the internal elements, all detach when the vertical force in the element reduces to zero. An alternative method of representing the stiffness of a shallow foundation was provided by FEMA-273 (1997). Instead of uniform stiffness across the foundation, the footing was divided into zones of different stiffness. The ends of the footing were represented by zones of relatively high stiffness over one-sixth of the footing width. The stiffness of these zones used the formulations of Gazetas et al. 10 Spring Bed Gazetas
h5 h4 h3 h2 h1 s
k
L
k
k
k
k
ss e nf f ti S l ac tri e V ot l k/2 a n oi t at o R f o oi t a R
8
6
4
2
0
0
1
2
3
Footing Width (m)
Figure 1: Rigid foundation on a bed of elastic springs
2
4
5
6
Pender, M., Wotherspoon, L. & Carr, A. (2008) Earthquake response of shallow foundations with low vertical gravity load
3
STRUCTURAL MODEL AND DYNAMIC EXCITATION
In order to compare the characteristics, a two-dimensional two bay portal frame structure was created and shallow foundations were attached to the base of each column. The only nonlinear characteristic included in these analyses was uplift modelling. Each shallow foundation was assumed to be 4.0 m square and resting on the ground surface. The soil characteristics were based on the assumption of 100 kPa undrained shear strength and a Poisson’s ratio of 0.5. The properties of the portal frame structure are summarised in Figure 2. Each bay was 12.0 m high by 8.0 m wide and the beams and columns were modelled using elastic beam elements. Equal vertical loads of 4000 kN were applied to the nodes at the top of each column to represent the vertical mass of the structure. 5% viscous damping was used in the model to account for structural damping. Each combined model was analysed with the application of a horizontal sinusoidal force (Figure 3) applied to the top of the portal frame. 8m 4000 kN
8m 4000 kN
4000 kN
12000 kN
12 m
y
x
Figure 2: Two bay portal frame structure 4
1
x 10
0.75 0.5
) N k( ec r o F d ei pl p A
0.25 0 -0.25 -0.5 -0.75 -1
0
5
10
15
20
25
30
Time (secs)
Figure 3: Sinusoidal applied force characteristics for portal frame 4
RESULTS
The axial force carried by each footing was compared in Figure 4. These characteristics show that when uplift occurred, the axial force was redistributed to both the other footings. The only time that the force in the central footing was not constant was during uplift events, when a fraction of the force that would have been carried by the detached footing was transferred to the central footing. When all footings are in contact with the ground the whole system rotates about the middle of the central footing and the vertical force remains constant. When an outer
3
Pender, M., Wotherspoon, L. & Carr, A. (2008) Earthquake response of shallow foundations with low vertical gravity load 2500
0
-2500
) N k( ec r o F l ai x A
-5000
-7500
-10000
-12500
Left Footing C entral Footing Right Footing 0
5
10
15
20
25
30
Time (secs)
Figure 4: Axial force in each footing of the portal frame model 7500
5000
) N k( ec r o F r a e h S
2500
0
-2500
-5000
-7500
Left Footing Central Footing Right Footing 0
5
10
15
20
25
30
Time (secs)
Figure 5: Shear force in each footing of the portal frame model 4
2
x 10
1.5 1
) m N k( t n e m o M
0.5 0 -0.5 Left Footing C entral Footing Right Footing
-1 -1.5
0
5
10
15
20
Time (secs)
Figure 6: Moment in each footing of the portal frame model
4
25
30
Pender, M., Wotherspoon, L. & Carr, A. (2008) Earthquake response of shallow foundations with low vertical gravity load 2500
a
d
b
) N k( ec r o F r a e h S
e
0
0
c
f
-2500
-5000
c
d
a e
-7500 -0.02
-0.01
0
0.01
0.02
0.03
Horizontal Displacement (m)
Figure 7: Shear-horizontal displacement response of the right footing footing uplifts the system no longer rotates about the central footing, and the centre of rotation moves outwards towards the footing that has not detached. This is the reason behind the variation of vertical force in the central footing. The shear and moment springs are detached during uplift, reducing the force to zero in all springs until the spring reattaches. This characteristic is demonstrated in Figure 5 and Figure 6. When one spring detaches, the fraction of force carried by that spring is transferred to the other springs in a similar fashion to the axial spring. This results in the increase in force carried by the other footings when uplift occurs. When a spring detaches, it ceases to carry force while still being free to move. When it reattaches, it may not at the same point in space as when it detached, resulting in residual shear and moment in the spring at the end of excitation. This may also be accompanied by a residual displacement at the end of excitation. Another informative representation of the uplift modelling is provided by force-displacement or hysteretic characteristics of the shear and moment springs. For this simple model the shear and moment have been defined by elastic springs, however these are still controlled by the release of stiffness when vertical force reduces to zero. To determine the characteristics of these springs, the shear-horizontal displacement response of the right horizontal spring is presented in Figure 7. The characteristics of the rotational spring are similar to the horizontal spring so have not been shown here. To explain the processes occurring during hysteresis, labels have been used for identify each characteristics step in Figure 7. The overall process can be defined as follows: • • • • •
Prior to excitation, there is no shear force applied to the footing, resulting in zero horizontal displacement. This is defined by point 0. Prior to uplift points follow the elastic slope a-a, where force and displacement are calculated using point 0 as the origin. At point b, the vertical force reduces to zero, forcing the horizontal spring to detach. During the next time step the shear force reduces to zero while the horizontal movement increases due to zero stiffness. During uplift, there is movement along line c-c and the spring carries no shear force. Once vertical force becomes compressive again, the springs reattach. The point of reattachment is defined by the line d-d, which becomes the new origin from which force-displacement characteristics are defined.
5
Pender, M., Wotherspoon, L. & Carr, A. (2008) Earthquake response of shallow foundations with low vertical gravity load
5
•
Force-displacement characteristics follow the line e-e, which has the same elastic slope as line a-a. Points will follow this line until there is another uplift event.
•
At the end of excitation the force-displacement characteristics are defined by point f. This explains the force in the spring at the end of excitation, as well as the displacement at the end of excitation. If the point of detachment and the point of reattachment of the horizontal spring were at the same position, then there would be no residual force. CONCLUSIONS
This paper has identified the inability of a vertical spring bed to represent both the vertical and rotational stiffness characteristics. Discrete vertical springs could not represent the rotational and vertical stiffness characteristics of an elastic continuum, and required additional rotational springs to bring the rotational stiffness to the desired value. Using a modification to the Ruaumoko compound spring element, uplift of a shallow foundation could be modelled such that the vertical force in the foundation controlled the stiffness in all degrees of freedom. Uplift modelling had a significant impact on the shear and moment carried by footings. If the point of detachment and reattachment of the foundation was at different horizontal and/or rotational displacements the result was residual horizontal and rotational displacements at the end of loading. This shift in displacements occurred in conjunction with a shift in shear and moment in the footing. REFERENCES Carr, A. (2006) 3D RUAUMOKO: inelastic three-dimensional dynamic analysis program, University of Canterbury - Department of Civil Engineering, Christchurch, NZ. Federal Emergency Management Agency (FEMA) (1997) NEHRP guidelines for the seismic rehabilitation of buildings, FEMA-273, Washington, D.C. Gazetas, G., Dobry, R. and Tassoulas, J. L. (1985) Vertical response of arbitrarily shaped foundations, Jnl. Geotechnical Engineering 111 (6) 750-771. Gazetas, G. and Tassoulas, J. L. (1987) Horizontal stiffness of arbitrarily shaped foundations, Jnl. Geotechnical Engineering 113 (5) 440-457. Hatzikonstantinou, E., Tassoulas, J. L., Gazetas, G., Kotsanopoulos, P. and Fotopoulou, M. (1989) Rocking stiffness of arbitrarily shaped embedded foundations, Jnl. Geotechnical Engineering 115 (4) 457-472. Pender, M.P., Wotherspoon, L. M., Ingham, J.M., Carr, A.J. (2005) Approached to design of shallow foundations for low-rise framed structures, Proceedings of 2005 NZSEE Conference, Napier. Toh, J.C.W and Pender, M.J. (2008) Earthquake performance and permanent displacements of shallow foundations, Proceedings of 2008 NZSEE Conference, Wairakei Wotherspoon, L,M. (2008) Integrated Modelling of Structure-Foundation Systems, PhD Thesis, Civil and Environmental Engineering, The University of Auckland.
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Earthquake response of shallow foundations with low vertical gravity load Michael Pender, Liam Wotherspoon Department of Civil and Environmental Engineering, University of Auckland, NZ. Athol Carr Department of Civil and Natural Resources Engineering, University of Canterbury, NZ. Keywords: shallow foundation, earthquake response, uplift, nonlinearity, moment and shear. ABSTRACT This paper presents the development and response of a Ruaumoko model for the representation of uplift and reattachment of a shallow foundation. The vertical load in the foundation is used to control the characteristics of moment and shear loads, as these values must reduce to zero when the entire footing has detached from the underlying soil. Through extension of the capabilities of Ruaumoko this response was able to be represented. A simple elastic two bay portal frame structure was created and shallow foundations were attached to the base of each column of the portal frame and subjected to cyclic loading. Uplift modelling was shown to have a significant effect on the shear and moment in the footings. If the point of detachment and reattachment of the foundation was at different horizontal and/or rotational displacements the result was residual horizontal and rotational displacements at the end of loading. This shift in displacements occurred in conjunction with a shift in shear and moment in the footing. 1
INTRODUCTION
There are three aspects of shallow foundation response to earthquake loading that could be regarded as “complications”. First there is nonlinear behaviour of the soil with consequent nonlinearity of the foundation-soil interaction, second the limitations imposed on combinations of vertical load, shear and moment by the bearing strength surface, and finally the possibility that part of the foundation system may uplift during the earthquake. All three of these are computationally challenging, but in this paper we will discuss the modelling of uplift and reattachment of shallow foundation elements for a simple framed structure subject to a cyclic loading (a pseudo earthquake). This will be done using the Ruaumoko software (Carr 2006), or more correctly Ruaumoko with specially developed additional features to represent shallow foundations. Ruaumoko is a non-linear dynamic structural analysis program. The point of adapting Ruaumoko to encompass foundation modelling is to promote the integrated design of structure foundation systems. It seems obvious to us that the most effective way of conducting soil structure interaction studies is to work with a single model of the whole structure foundation system. Ruaumoko and many other structural analysis systems tend to keep each degree of freedom separate. Thus, in Ruaumoko for example, there are facilities for a degree of freedom to detach from the system and reattach. However, a shallow foundation, in general, carries vertical load, horizontal shear and moment. Uplift will occur when the vertical load becomes zero and at that point the degree of freedom representing horizontal shear and moment must also detach, that is the horizontal shear and moment must become zero. Similarly, when reattachment occurs because the vertical degree of freedom
1
Pender, M., Wotherspoon, L. & Carr, A. (2008) Earthquake response of shallow foundations with low vertical gravity load
connects with the foundation again, the horizontal shear and moment degrees of freedom must also reconnect. Having developed such a facility for Ruaumoko this paper demonstrates, via the response of a simple structural model, how the uplift and reattachment operates. 2
SHALLOW FOUNDATION MODELS
The bearing strength of a shallow foundation is a complex function of the soil properties and the combinations of vertical load, horizontal shear and moment sustained by the foundation. When the bearing strength is reached all the resistance that can be provided by the shear strength of the soil has been mobilised. At loads which are small in relation to the bearing strength the soil structure interaction is expected to be elastic. Between these limits there will be a gradual transition, becoming more nonlinear, between linear elastic behaviour and bearing strength failure. A frequently used model for a shallow foundation is to assume a rigid member resting on a bed of springs. This is attractive because it allows progressive uplift of the foundation. The preferred model for the elastic behaviour of the foundation is to assume that the soil is an elastic half space. Solutions for the stiffness characteristics of this case have been developed by Gazetas and his colleagues (Gazetas et al. 1985; Gazetas and Tassoulas 1987; Hatzikonstantinou et al. 1989). With this model in mind one can set the stiffness of the springs in a bed of springs so that the vertical stiffness of the bed of springs represents correctly the stiffness of a rigid footing resting on an elastic half space. However, there is a drawback as the rotation stiffness of a bed of springs is less than the rotational stiffness of a rigid foundation resting on an elastic half space. These ideas are illustrated in Figure 1. One method for keeping the rotational stiffness as required is to have an additional rotational spring to make up for the deficiency shown in above. In the Ruaumoko model this rotational spring was included inside the compound spring element. Modifications were made to the original compound spring element in Ruaumoko to allow the vertical stiffness to control the other stiffnesses within the compound spring. By defining interaction between the internal elements, all detach when the vertical force in the element reduces to zero. An alternative method of representing the stiffness of a shallow foundation was provided by FEMA-273 (1997). Instead of uniform stiffness across the foundation, the footing was divided into zones of different stiffness. The ends of the footing were represented by zones of relatively high stiffness over one-sixth of the footing width. The stiffness of these zones used the formulations of Gazetas et al. 10 Spring Bed Gazetas
h5 h4 h3 h2 h1 s
k
L
k
k
k
k
ss e nf f ti S l ac tri e V ot l k/2 a n oi t at o R f o oi t a R
8
6
4
2
0
0
1
2
3
Footing Width (m)
Figure 1: Rigid foundation on a bed of elastic springs
2
4
5
6
Pender, M., Wotherspoon, L. & Carr, A. (2008) Earthquake response of shallow foundations with low vertical gravity load
3
STRUCTURAL MODEL AND DYNAMIC EXCITATION
In order to compare the characteristics, a two-dimensional two bay portal frame structure was created and shallow foundations were attached to the base of each column. The only nonlinear characteristic included in these analyses was uplift modelling. Each shallow foundation was assumed to be 4.0 m square and resting on the ground surface. The soil characteristics were based on the assumption of 100 kPa undrained shear strength and a Poisson’s ratio of 0.5. The properties of the portal frame structure are summarised in Figure 2. Each bay was 12.0 m high by 8.0 m wide and the beams and columns were modelled using elastic beam elements. Equal vertical loads of 4000 kN were applied to the nodes at the top of each column to represent the vertical mass of the structure. 5% viscous damping was used in the model to account for structural damping. Each combined model was analysed with the application of a horizontal sinusoidal force (Figure 3) applied to the top of the portal frame. 8m 4000 kN
8m 4000 kN
4000 kN
12000 kN
12 m
y
x
Figure 2: Two bay portal frame structure 4
1
x 10
0.75 0.5
) N k( ec r o F d ei pl p A
0.25 0 -0.25 -0.5 -0.75 -1
0
5
10
15
20
25
30
Time (secs)
Figure 3: Sinusoidal applied force characteristics for portal frame 4
RESULTS
The axial force carried by each footing was compared in Figure 4. These characteristics show that when uplift occurred, the axial force was redistributed to both the other footings. The only time that the force in the central footing was not constant was during uplift events, when a fraction of the force that would have been carried by the detached footing was transferred to the central footing. When all footings are in contact with the ground the whole system rotates about the middle of the central footing and the vertical force remains constant. When an outer
3
Pender, M., Wotherspoon, L. & Carr, A. (2008) Earthquake response of shallow foundations with low vertical gravity load 2500
0
-2500
) N k( ec r o F l ai x A
-5000
-7500
-10000
-12500
Left Footing C entral Footing Right Footing 0
5
10
15
20
25
30
Time (secs)
Figure 4: Axial force in each footing of the portal frame model 7500
5000
) N k( ec r o F r a e h S
2500
0
-2500
-5000
-7500
Left Footing Central Footing Right Footing 0
5
10
15
20
25
30
Time (secs)
Figure 5: Shear force in each footing of the portal frame model 4
2
x 10
1.5 1
) m N k( t n e m o M
0.5 0 -0.5 Left Footing C entral Footing Right Footing
-1 -1.5
0
5
10
15
20
Time (secs)
Figure 6: Moment in each footing of the portal frame model
4
25
30
Pender, M., Wotherspoon, L. & Carr, A. (2008) Earthquake response of shallow foundations with low vertical gravity load 2500
a
d
b
) N k( ec r o F r a e h S
e
0
0
c
f
-2500
-5000
c
d
a e
-7500 -0.02
-0.01
0
0.01
0.02
0.03
Horizontal Displacement (m)
Figure 7: Shear-horizontal displacement response of the right footing footing uplifts the system no longer rotates about the central footing, and the centre of rotation moves outwards towards the footing that has not detached. This is the reason behind the variation of vertical force in the central footing. The shear and moment springs are detached during uplift, reducing the force to zero in all springs until the spring reattaches. This characteristic is demonstrated in Figure 5 and Figure 6. When one spring detaches, the fraction of force carried by that spring is transferred to the other springs in a similar fashion to the axial spring. This results in the increase in force carried by the other footings when uplift occurs. When a spring detaches, it ceases to carry force while still being free to move. When it reattaches, it may not at the same point in space as when it detached, resulting in residual shear and moment in the spring at the end of excitation. This may also be accompanied by a residual displacement at the end of excitation. Another informative representation of the uplift modelling is provided by force-displacement or hysteretic characteristics of the shear and moment springs. For this simple model the shear and moment have been defined by elastic springs, however these are still controlled by the release of stiffness when vertical force reduces to zero. To determine the characteristics of these springs, the shear-horizontal displacement response of the right horizontal spring is presented in Figure 7. The characteristics of the rotational spring are similar to the horizontal spring so have not been shown here. To explain the processes occurring during hysteresis, labels have been used for identify each characteristics step in Figure 7. The overall process can be defined as follows: • • • • •
Prior to excitation, there is no shear force applied to the footing, resulting in zero horizontal displacement. This is defined by point 0. Prior to uplift points follow the elastic slope a-a, where force and displacement are calculated using point 0 as the origin. At point b, the vertical force reduces to zero, forcing the horizontal spring to detach. During the next time step the shear force reduces to zero while the horizontal movement increases due to zero stiffness. During uplift, there is movement along line c-c and the spring carries no shear force. Once vertical force becomes compressive again, the springs reattach. The point of reattachment is defined by the line d-d, which becomes the new origin from which force-displacement characteristics are defined.
5
Pender, M., Wotherspoon, L. & Carr, A. (2008) Earthquake response of shallow foundations with low vertical gravity load
5
•
Force-displacement characteristics follow the line e-e, which has the same elastic slope as line a-a. Points will follow this line until there is another uplift event.
•
At the end of excitation the force-displacement characteristics are defined by point f. This explains the force in the spring at the end of excitation, as well as the displacement at the end of excitation. If the point of detachment and the point of reattachment of the horizontal spring were at the same position, then there would be no residual force. CONCLUSIONS
This paper has identified the inability of a vertical spring bed to represent both the vertical and rotational stiffness characteristics. Discrete vertical springs could not represent the rotational and vertical stiffness characteristics of an elastic continuum, and required additional rotational springs to bring the rotational stiffness to the desired value. Using a modification to the Ruaumoko compound spring element, uplift of a shallow foundation could be modelled such that the vertical force in the foundation controlled the stiffness in all degrees of freedom. Uplift modelling had a significant impact on the shear and moment carried by footings. If the point of detachment and reattachment of the foundation was at different horizontal and/or rotational displacements the result was residual horizontal and rotational displacements at the end of loading. This shift in displacements occurred in conjunction with a shift in shear and moment in the footing. REFERENCES Carr, A. (2006) 3D RUAUMOKO: inelastic three-dimensional dynamic analysis program, University of Canterbury - Department of Civil Engineering, Christchurch, NZ. Federal Emergency Management Agency (FEMA) (1997) NEHRP guidelines for the seismic rehabilitation of buildings, FEMA-273, Washington, D.C. Gazetas, G., Dobry, R. and Tassoulas, J. L. (1985) Vertical response of arbitrarily shaped foundations, Jnl. Geotechnical Engineering 111 (6) 750-771. Gazetas, G. and Tassoulas, J. L. (1987) Horizontal stiffness of arbitrarily shaped foundations, Jnl. Geotechnical Engineering 113 (5) 440-457. Hatzikonstantinou, E., Tassoulas, J. L., Gazetas, G., Kotsanopoulos, P. and Fotopoulou, M. (1989) Rocking stiffness of arbitrarily shaped embedded foundations, Jnl. Geotechnical Engineering 115 (4) 457-472. Pender, M.P., Wotherspoon, L. M., Ingham, J.M., Carr, A.J. (2005) Approached to design of shallow foundations for low-rise framed structures, Proceedings of 2005 NZSEE Conference, Napier. Toh, J.C.W and Pender, M.J. (2008) Earthquake performance and permanent displacements of shallow foundations, Proceedings of 2008 NZSEE Conference, Wairakei Wotherspoon, L,M. (2008) Integrated Modelling of Structure-Foundation Systems, PhD Thesis, Civil and Environmental Engineering, The University of Auckland.
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