The foundation might be separated from the supporting soil if the earthquake is big enough, which is known as base mat uplift. This paper proposed a simplified calculation model in which spring element is adopted to simulate the interaction between soil and structure. The loaddeformation curve (
There have been two ways to deal with the soilstructure interaction (SSI) effect in seismic design of structures: one is treating the soil as rigid medium and neglecting the SSI effect; the other is assuming the foundation to be fully bonded to soil so that the foundation uplift cannot occur. However, the importance of the SSI in the dynamic response of structure has been realized recently by researchers, though foundation uplift has not gotten enough attention yet, partly due to the fact that interfacial behavior between foundation and soil cannot be directly observed after earthquake. However, the phenomenon of foundation uplift can be verified indirectly in some earthquakes such as that of Chile 1960, Alaska 1964, San Fernando 1971, Kocaeli 1999, and Athens 1999 [
Housner [
Four calculation models. (a) Rigid block rocking on rigid base [
With the construction of nuclear power plant (NPP) structures in Japan and America, much attention had been paid to the impact of base mat uplift on the seismic response of structure. The SR (AwayRocking) model (as shown in Figure
Recently, Gazetas et al. [
The principle of the new idea “rocking isolation” [
From the initial study on the rigid block rocking at the rigid base to the uptodata conception of rocking isolation, one crucial problem has not been solved perfectly, that is, how to establish a suitable calculation model to compute seismic response of block/structure due to base mat/foundation uplift. It is obvious that the nonlinear property of surrounding soil cannot be considered in the traditional analysis model as shown in Figures
The development of the calculation method for seismic response of structure considering base mat uplift has experienced three stages. The first stage was focusing on block rocking on soil surface. Chopra and Yim [
The computation efficiency for contact element and joint element is usually very low due to the difficulty of convergence in nonlinear analysis. In order to overcome this deficiency, a new computational method is suggested in this paper, in which the contact or joint element is replaced by spring element. The spring elements, COMBIN39 in the FE software ANSYS, can provide pressure by setting its loaddeformation curve (
The spring element in FE software ANSYS has the feature of “no length,” which means that the two nodes at each end of the spring element will locate at the same position. It should be noticed that the nodes in Figure
Schematic diagram of the working state of spring element. (1) base mat plane, (2) static equilibrium plane, and (3) copied plane (for the establishment of spring element).
Before analysis and static equilibrium phase
Critical state
Base mat uplift phase
The base mat will touch the soil again if the peak acceleration of the seismic wave passes through. It should be noted that the shock effect between soil and base mat will not be considered here for the sake of simplicity.
In current study, a fourstory frame structure (as shown in Figure
Computation model.
Plan view of base mat.
In the FE model, the beam is simulated with BEAM188 element, and the wall and slab are represented by SHELL181 element. The total number of element and node are 2274 and 1326, respectively. The axes of
Soil is modelled by springdamping element (COMBIN39 element), which has six degrees of freedom in each node: two horizontal directions, one vertical direction, two rocking directions, and one torsion direction. The shear wave velocity, mass density, and Poisson ratio of the soil are 1100 m/s, 2350 Kg/m^{3} and 0.4, respectively.
In current research, the rigid foundation is considered to be rested on the surface of a half space. The embedded depth of foundation H is neglected for a conservative treatment.
The formulae suggested by Veletsos and Verbic [
Soil impedance functions for circular foundation on a halfspace [
Mode 




Horizontal 



Vertical 



Rocking 



Torsion 



Values for fitting parameters of
Quantity 






0.775  0.65  0.6  0.6 



0.8  0.8  0.8  0.8 

0.525  0.5  0.45  0.4 

0  0  0.023  0.027 



0.25  0.35  —  0 

1.0  0.8  —  0 

0  0  —  0.17 

0.85  0.75  —  0.85 
The recommended formulae in ASCE code (Table
Soil impedance functions recommended in ASCE code [
Motion  Equivalent spring constant  Damping coefficient 

Horizontal 


Rocking 


Vertical 


Torsion 


Calculation results comparison between two methods.
Motion 

 

Veletsos  ASCE  Veletsos  ASCE  
Horizontal/N/m 


0.63  0.58 
Rocking/N 


0.15  0.11 
Vertical/N/m 


0.83  0.72 
Torsion/N 


0.16  0.12 
It can be seen from Table
The damping in a soilstructure interaction system mainly consists of two parts: soil damping and structure damping. Soil damping can be calculated by soil impedance functions, that is, the formulae listed in Table
where
As high frequency contents will be generated in the occurrence of base mat uplift, the time step is selected to be 0.002 s, which is small enough to consider the influence of high frequency contents.
There are many factors that impact the seismic response of structure, like velocity of shear wave, type and amplitude of seismic wave, structure stiffness, the ratio of structure height to width, and so on. The following factors are considered in this papers.
Time history and Fourier spectrum curves of input waves.
ElCentro wave
Taft wave
Songpan wave
Modal analysis is firstly conducted on structure system. The first five natural frequencies and their corresponding vibration modes are given in Table
The first five natural frequencies and corresponding vibration mode.
Order  1  2  3  4  5 

Frequency/Hz  1.9581  7.4998  10.961  12.309  14.572 
Vibration Mode  Horizontal  Horizontal  Structure torsion about vertical axis  Floor torsion about vertical axis  Vertical 
The standard case is defined as that where the structure is excited by the EL wave in horizontal direction with peak acceleration of 0.5 g and vertical direction with peak acceleration of 0.33 g simultaneously (labeled as HV case), and the velocity of shear wave is set to be 1100 m/s.
The time history of horizontal and vertical acceleration and its corresponding Fourier spectrum curves for the fourth floor (the node 103 in Figure
Horizontal acceleration time history and corresponding Fourier spectrum curves at node 103 on the 4th floor and node 18 on the 1st floor under standard case (HV).
Vertical acceleration time history and corresponding Fourier spectrum curves at node 103 on the 4th floor and node 18 on the 1st floor under standard case (HV).
Figure
Horizontal and vertical acceleration response spectrum curves of the fourth and first floor under standard case (HV).
According to the prescribed method shown in Figure
Vertical displacement at the corner of base mat under standard case (HV).
Horizontal acceleration time history and its Fourier spectrum curves at node 103 on the fourth floor under standard case (HV).
The ratio of uplift area is defined as the ratio of the maximum uplifted area to the total area of base mat. This value is a key parameter to evaluate the effect of base mat uplift on the seismic response of structure. In current study, the maximum uplifted area can be determined by the following procedures: (a) extract the time history curve of vertical displacement at node 1; (b) determine the trigger time of the base mat uplift; and (c) select the nodes whose vertical displacement is bigger than zero at this time or substep. These selected nodes are seen as the uplifted nodes and the area consisted by these nodes is then defined as the maximum uplifted area.
Figure
Schematic diagram of uplifted nodes under standard case: 52 nodes uplifted and uplift area ratio is 25.0%.
Key factors affecting the dynamic response of structure are analyzed in this section, including seismic wave (input direction, amplitude, and type), velocity of shear wave in soil, and structure related factors (stiffness and the ratio of height to width,
Figure
Horizontal acceleration time history and spectrum curves at node 103 under EL wave for H and HV case.
From Figure
Vertical acceleration time history at node 103 under EL wave for H case and HV case.
Normalized vertical acceleration Fourier spectrum curves at node 103 under EL wave for H case and HV case.
Figure
Normalized horizontal and vertical acceleration spectrum curves at node 103 under EL wave for HV and H case.
Figure
Vertical displacement time history curves at node 1 under EL wave for H case and HV case.
Figures
Schematic diagram of uplifted nodes under H case: 39 nodes uplifted and uplift area ratio is 16.7%.
Based on above analysis, it can be concluded that the seismic wave input at vertical direction has little influence on structural response at horizontal direction, while mainly having impact on the structural response at vertical direction and the maximum base mat uplift area ratio.
The maximum vertical displacements at node 1 for the HV case with amplitudes of 0.15 g, 0.5 g, and 1 g are −0.42 mm, 0.334 mm, and 2.91 mm, respectively, and its corresponding maximum uplift area ratios are determined as 0%, 25%, and 75%, respectively. It can be seen that the maximum vertical displacement at base mat is negative in the case that the amplitude of seismic wave is small (0.15 g), which means the base mat uplift does not occur. Meanwhile, for the HV case with the amplitude of 0.4 g, the maximum vertical displacement at node 1 is found to be 0.02 mm, which can be treated as the critical amplitude for the base mat uplift in current study.
It is well known that the different seismic waves have different frequency components and in turn generate different seismic responses of structure. In this section, three types of seismic waves are selected with consideration of the distance effect, that is, the EL wave (nearfield wave), Taft wave (middledistance wave), and Songpan wave (longdistance wave) which is recorded during the Wenchuan earthquake in 2008 in China. Their acceleration time history and Fourier spectrum curves are shown in Figure
Due to the difference in computational times (EL wave 30 s, Taft wave 50 s, and Songpan wave 150 s), the results from different seismic waves are shown separately. Figures
Horizontal acceleration time history and Fourier spectrum curves at node 103 under Taft wave (HV case).
Horizontal acceleration time history and Fourier spectrum curves at node 103 under Songpan wave (HV case).
The vertical displacement response at node 1 (Figure
Comparison of vertical displacement time history curves at node 1 under EL, Taft, and Songpan waves.
Figures
Vertical acceleration time history and Fourier spectrum curves at node 103 under EL wave (HV case).
Vertical acceleration time history and Fourier spectrum curves at node 103 under Taft wave.
Vertical acceleration time history and Fourier spectrum curves at node 103 under Songpan wave.
Figures
Schematic diagram of uplifted nodes under Taft wave for HV case (23 nodes uplifted and uplift area ratio is 5.6%).
Schematic diagram of uplifted nodes under Songpan wave for HV case (54 nodes uplifted and uplift area ratio is 25.6%).
From the above analysis, it can be concluded that the different seismic waves have a significant influence on the dynamic response of structure due to different frequency components from each seismic wave. When the predominant frequency of seismic wave is closer to that of the natural frequency of structure, the seismic response of structure will be excited significantly.
Figures
Schematic diagram of uplifted nodes with the soil shear wave 2000 m/s under EL wave for HV case (65 nodes uplifted and uplift area ratio is 33.3%).
Schematic diagram of uplifted nodes with the soil shear wave 400 m/s under El wave for HV case (23 nodes uplifted and uplift area ratio is 5.6%).
Two cases were considered in this section. For the first case, the stiffness of the structure is increased: the thickness of side wall is increased from 0.2 m to 0.4 m, and the section of the column is enhanced from 0.6 m
Vertical displacement time history curves at node 1 on the structure base mat under two cases: Bigger Section Size case and Bigger Modulus case.
The spectrum curves of horizontal and vertical acceleration at node 103 on the fourth floor for Standard and Bigger modulus case are given in Figure
Spectrum curves of horizontal and vertical acceleration at node 103 under standard case and Bigger Modulus case.
Horizontal acceleration
Vertical acceleration
The schematic diagram of uplifted nodes under the case of Bigger Modulus is given in Figure
Schematic diagram of uplifted nodes under the Bigger Modulus case (34 nodes uplifted and the maximum uplift area ratio is 12.5%).
The ratio of structural height to width (
Vertical displacement time history curves at node 1 on the structure base mat under two cases: structure
The comparisons of the structural natural frequency and its related structural stiffness are also conducted. The first natural frequencies for
The base mat uplift is found easier to occur for small
It should be noted that the existence of the machine, furniture, or accessory equipment can be simplified as a concentrated mass on the floor in general. Theoretically, the influence of concentrated mass on the base mat uplift should be investigated in three aspects: firstly, greater inertial force will be applied on the structure, and hence the base mat uplift is easier to occur. Secondly, the concentrated mass has little influence on structural stiffness, and then the heavier structure can prevent base mat uplift. Thirdly, the structure will be assumed to be an unsymmetrical structure and then the torsion force will affect the dynamic response of structure. However, the effect of torsion force is decided by the weight of the concentrated mass. This means whether the base mat uplift can be prevented or not is determined by the weight and location of the concentrated mass. However, the influence of the location of concentrated mass is discussed in this section because the weight can be changed a lot in the practical situation.
A concentrated mass with the onetenth of the total weight of the structure is located on the middle point in the fourth and the first floor, respectively. The time history curves of vertical displacement for the two cases are shown in Figure
Vertical displacement time history curves at node 1 on the structure base mat under two cases: concentrated mass located on the fourth and first floor.
Schematic diagram of uplifted nodes under the concentrated mass located on the fourth floor (65 nodes uplifted and the maximum uplift area ratio is 33.3%).
Schematic diagram of uplifted nodes under the concentrated mass located on the first floor (49 nodes uplifted and the maximum uplift area ratio is 22.2%).
In order to overcome the computational convergence issue in traditional contact and joint element method, a new method based on the spring element is proposed in this paper to consider the impact of base mat uplift on the seismic response of structure. Some key factors, such as seismic waves, velocity of shear wave in soil, and structural characters, are analyzed and discussed. The following conclusions can be drawn from the above analysis and comparison:
(1) The vertical direction of input seismic wave has little influence on the seismic response of structure at horizontal direction, while it significantly affects the seismic response of structure in vertical direction, and the maximum uplifted area ratio of structure. This result is consistent with the earlier research results by Joe (1993) and Kennedy et al. (1976) and in turn verified the rational of the proposed method in current study.
(2) The seismic response of structure tends to be larger with the increase of the amplitude of input seismic wave.
(3) The type of seismic wave has a notable effect on structural responses. Regarding the structure and selected seismic waves in current paper, the seismic response of structure for the longdistance Songpan wave turns out to be tenser due to the fact that the dominant frequency of the seismic wave is very close to the natural frequency of structure.
(4) The velocity of shear wave in soil has certain influence on seismic response of structure. It is found that the stiffer the soil is, the easier the occurrence of the base mat uplift is.
(5) The ratio of structural height to width (
(6) The influence of auxiliary equipment on the seismic response of structure mainly depends on its weight and location.
As has been stated previously, simulation of rocking system in seismic analysis obviously involves nonlinearity of soils and the interface between structure and soil in terms of both stressstrain relationship and geometry aspects. For the sake of simplicity, only geometry nonlinearity has been considered in this paper. In case of strong earthquake where nonlinear stressstrain relationship is prominent in material response, such property will absorb certain amount of energy and thus benefit the structure in terms of earthquakeproof capability. Further analysis should be carried out in light of more accurate simulation on nonlinearity in future researches.
The authors declare that they have no conflicts of interest.
The authors are thankful for the support from the Natural Science Foundation of China (nos. 51208406, 51678465) and Scientific Research Foundation for Chinese Ministry of Education for Returned Overseas Researchers.