Sloshing in LNG membrane tanks may cause large pressures on the tank structure. To keep the cargo at the required low temperature, the tank structure is covered with an insulation, which has a much less strength than steel. The containment system is a very complex structure, which consists of different materials and requires a careful analysis with due consideration of the load process and dynamic effects in the response. The structural response of the membrane tank wall is investigated in this paper by finite element analyses. First, a modal composition of the structural response is studied. It is shown that many modes contribute to the response, which makes it difficult to establish the simplified DLF approach. The dynamic structural response to a typical sloshing impact is investigated in detail. An important observation is that, although the containment system has traditionally been modeled with a rigid support, the steel plate that supports the insulation may be flexible under the relevant load conditions. It is shown that the flexibility of the steel plate causes significant stress variation in the insulation. Different response patterns of the Mark III containment system are presented, and mechanisms that cause large stress concentrations and different response patterns in the static and dynamic cases are discussed. The scaling issue in view of the response is also investigated. Various scaling formulations may apply in post-processing sloshing experiments. While the Froude law yields conservative scaling for pressure magnitude, its conservatism for scaling the time needs to be investigated in view of the relevant dynamic response. By analyzing the structural response to the differently scaled loads, it is found that the Froude approach is conservative, but the scatter of results may be very large.