When incorporating earthquake excitation into the soil-structure model, we not only have to apply effective forces that are equivalent to the incoming earthquake, but we also have to account for the non-linearity of the structure, or in other words, start the transient earthquake analysis from a static state of the structure.
The effective forces to be applied to the model are derived from a scattering analysis framework as described below, wherein we consider the difference between the ground motion with the structure and without, and this scattered motion – entirely outgoing from the structure – is absorbed by a PML boundary.
The non-linear analysis of the structure can also be incorporated in the same context, as described later.
A scattering analysis framework, developed by Bielak and co-workers as part of the effective seismic input method, is adopted as the approach for soil-structure interaction analysis in LS-DYNA. This approach considers soil-structure interaction to be caused by the scattering of the free-field ground motion by the presence of the structure, and the appropriate earthquake forces and the use of an absorbing boundary follow rationally from this viewpoint.
Consider the following two independent states as part of a thought experiment:
(a) soil is excited by an earthquake in the absence of the structure, and
(b) the structure disturbs and scatters the incoming earthquake wave.
Note that both these states cannot occur in reality: it can be either one or the other.
If we take the difference of the foundation motion in the two states, we are left with only the scattered motion, having eliminated both the earthquake source and the incoming wave. The scattered motion — because it is generated solely by the structure — propagates entirely outward from the structure.
Now the unbounded domain can be replaced by a truncated bounded domain, but without only simple boundary conditions, the outer boundary will reflect spurious waves back to the structure. This may be avoided by using an absorbing boundary to reduce the wave reflection and appropriately model the unbounded domain beyond.
We shall use perfectly matched layers as the absorbing boundary, and the scattered-wave formulation will provide the equivalent earthquake forces, termed as the effective seismic input.
We consider two possible states of the soil domain during an earthquake: one with the structure, and one without.
Consider first the structure in an earthquake, for which the equations of motion of the free body are:
For the associated soil domain, the free-body equations of motion are:
The same soil domain in the absence of the structure will be governed by the following equation:
The scattered motion in the soil domain is obtained by taking the difference between the two, on all nodes other than those on the interface with the structure:
When put together with the equation of motion for the structure, the equations for the whole system become:
wherein the right-hand side give the effective earthquake forces that are equivalent to —- these depend only on the free-field ground motion at the interface, and because of the sparsity of the mass and stiffness matrices, are confined to one layer of elements around the soil-structure interface.
In other words, using the scattered motion in the soil domain creates a discontinuity at the interface with the structure, where the total motion is used, and this discontinuity creates effective forces at the interface. The discontinuity is exactly the free-field ground motion at the interface, and thus effective forces depend solely on that free-field ground motion.
This is the effective seismic input method developed by Bielak and co-workers – it directly uses the free-field earthquake ground motions at the soil-structure and does not require their deconvolution down to depth.
Since the goal of transient soil-structure interaction analysis is to predict the non-linear behaviour of the structure, the transient analysis needs to start from a static state of the structure. Furthermore, the soil itself may behave non-linearly, and this needs to be accounted for in the analysis. However, the soil domain itself is
This conflict may be resolved as follows:
1. Assume that all the non-linearity in the soil is limited to a region near the structure, and define the generalized structure to be the physical structure itself along with this non-linear part of the soil. The rest of the soil domain is then linear and can be taken to be the soil domain for the purpose of the interaction analysis.
2. For the analysis, first calculate the static reactions at the base of the generalized structure by a static analysis, and apply those reactions at the base during the transient analysis to support the weight of the structure and non-linear soil.
Effective seismic input has been implemented in LS-DYNA, with INTERFACE_SSI cards used to identify the soil-structure interface, and LOAD_SEISMIC_SSI used to specify the ground motion on such an interface. Typically, only ground acceleration histories are required to specify the ground motion, but if ground velocity and displacement curves are also available from signal processing of the accelerograms, then the ground motion may be specified using DEFINE_GROUND_MOTION.
The variations on the INTERFACE_SSI cards (_AUX, _AUX_EMBEDDED and _STATIC) are meant for different stages in the analysis. All the INTERFACE_SSI cards (except for _AUX) create a tied-contact interface between two specified segment sets, the master surface being on the soil side and the slave on the structure side.
Soil-structure interaction analysis under earthquake excitation may then be carried out in LS-DYNA using these cards as follows:
1. Carry out a static analysis of the soil-structure system (e.g. using dynamic relaxation; see *CONTROL_DYNAMIC_RELAXATION), with the soil-structure interface identified using *INTERFACE_SSI_STATIC_ID.
Optionally, carry out a free-field analysis to record free-field motions on the future soil-structure interface, using either *INTERFACE_SSI_AUX or *INTERFACE_SSI_AUX_EMBEDDED, for surface-supported or embedded structures respectively.
2. Carry out the transient analysis as a full-deck restart job (see *RESTART), with only the structure initialized to its static stress state (see *STRESS_INITIALIZATION), and the same soil-structure interface identified using *INTERFACE_SSI_ID with the same ID as in static analysis:
NOTE: Please ignore error messages from LS-Prepost flagging the INTERFACE_SSI cards as invalid.
The implementation of effective seismic input in LS-DYNA was validated by analysing a model of the Morrow Point Dam and comparing the results against the response computed from EACD, which using the substructure method for ground motion input and a boundary-element model for the foundation rock, serves as a benchmark.
The upstream-downstream displacement amplitude at the center of the crest of the dam, as computed from LS-DYNA and from EACD, are shown below. It is seen that the LS-DYNA results closely match the semi-analytical results from EACD.
The figure above shows a model representing a building (red) upon soil — the blue part is the elastic soil domain and the green part is the PML. The figure on the right shows the soil-structure interface.
An example of a purely dynamic analysis of this system — without any gravity load — is given in the input deck ssi-dynamic.k. Transient analysis of the system following an initial static analysis is demonstrated by a pair of input decks: ssi-static.k and ssi-transient.k, to be run one after the other. An example with deconvolved ground motion input is given in planessi.k. The ground motions used in the analysis are given in elcentro-x.ath, elcentro-y.ath, and elcentro-z.ath.
NOTE: Please ignore any error messages from LS-Prepost that flag the INTERFACE_SSI cards as invalid.