Earthquake analysis of dams provide an added challenge beyond analysis of soil-structure systems because here there are two unbounded domains — the water and the foundation rock, interacting with each other as well as the dam — as opposed to only one unbounded domain in soil-structure interaction.
We show next how the theory of effective seismic input for soil-structure interaction is extended to the earthquake analysis of dams, and how the existing implementation in LS-DYNA is repurposed for such analysis.
As mentioned earlier, the effective seismic input method is derived by viewing soil-structure interaction as a scattering problem, wherein the presence of the dam causes the scattering of the free-field ground motion in the linear foundation domain. This is entirely analogous to acoustic scattering, wherein acoustic waves in a linear acoustic fluid are made to scatter by the presence of a solid body. In both cases, the waves in a linear background medium — the soil foundation or the acoustic fluid — are scattered by a solid structure placed in the free-field, and the solution is formulated by considering the scattered motion in the background medium, which replaces a distant excitation source with equivalent effective forces at the interface with the structure.
This shows that this scattered-motion approach to solving a scattering problem depends only on the linearity of the background medium, and not on the particular physical properties of the medium. Therefore, for dam analysis, we can consider the water domain and the foundation rock together as one free-field background domain, and the dam as the structure. However, we know the free-field ground motion in the foundation rock only, not in the water, so to determine that, we need an initial auxiliary analysis where the water domain plays the part of the structure. This results in a two-step analysis procedure, where we first analyze the auxiliary water-foundation rock to compute the free-field motion in the water, and then use this in the analysis of the whole dam-water-rock system.
Note that there is a physical significance to the auxiliary system, even though it is not a physical entity by itself. The earthquake ground motion affects the dam not only directly through its base, but also through pressure waves in the water, excited by the earthquake at the reservoir bottom. The auxiliary system brings in the effect of the far-field pressure waves in the final analysis of the whole system without needing to model a large length of the reservoir.
Dam analysis can be formulated in LS-DYNA according to two-step approach outlined earlier using the INTERFACE_SSI and LOAD_SEISMIC_SSI cards implemented for soil-structure interaction.
Two preliminary analyses are required — a static analysis, and the auxiliary analysis to capture the effect of upstream ground motion.
Static analysis
In static analysis, the foundation rock and water are constrained on the outer boundary, and they are chosen to be sufficiently large to model the static stress distribution. This is not as onerous a constraint as in dynamic analysis, where the outward propagating waves would have reflected back from a fixed outer boundary. Only the dam and the water are subjected to gravity loading, the reasoning being that the current state of the foundation is already its deformed shape due to aeons of gravity acting upon it. The following figures show the model and the “soil-structure” interface, specified using INTERFACE_SSI_STATIC to record the static reactions. In a manner similar to soil-structure interaction analysis — where the nearby non-linear part of the soil was taken as part of the generalized structure — here the near-field water, which may behave non-linearly, is taken to be part of the structure, and the far-field water and the foundation rock are together taken to be the “soil” somain. Of course, any nearby non-linear portion of the foundation rock may also be taken to be part of the structure as necessary.
Auxiliary analysis
Auxiliary analysis involves only the far-field water and the foundation rock, with the water domain acting as the structure, and the unbounded water and foundation rock truncated using PML. The earthquake excitation is applied using LOAD_SEISMIC_SSI on a segment set at the water-foundation interface away from the PML boundary, and the motions at the future “soil-structure” interface are recorded using INTERFACE_SSI_AUX for use in the subsequent transient analysis.
Transient analysis
In the transient analysis of the entire dam-water-rock system, the direct earthquake excitation at the base of the dam is applied using LOAD_SEISMIC_SSI on a segment set, and the effect of the upstream ground motion, computed in the auxiliary analysis at the “soil-structure” interface, is incorporated using INTERFACE_SSI, which also reads in the static reactions computed in the static analysis.
The corresponding example input files for these analyses are dam-static.k, dam-auxiliary.k and dam-transient.k, using the ground motion files elcentro-x.ath, elcentro-y.ath, elcentro-z.ath. Please also see the README file.
NOTE: Please ignore error messages from LS-Prepost flagging the INTERFACE_SSI cards as invalid.