In relation to water pressures in a dynamic calculation we can distinguish between:

•  pseudo-static approach
•  full dynamic (time-history) calculation

Pseudo-static approach

A pseudo-static approach is a strongly simplified approach to an earthquake situation. In this approach, we apply a constant and uniform acceleration to all the mass in the model. This approach is similar to the way gravity is introduced in the model but we can specify the magnitude and also the direction (i.e. a_x and/or a_y). Also, see the link below for further information on this approach.

Now with regard to water loads/pressures during a pseudo-static calculation: realise that water loads coming from “free” or “open” water are transformed to surface loads acting perpendicular to (active) model boundaries. During a pseudo-static approach, we increase these surface loads with the ratio of the size of the initial gravity load vector and the resulting size of the gravity+acceleration load vector (while the load direction stays perpendicular to the model boundary). So, in the following example:

• gravity: g =  9.81 m/s²
• specified acceleration: a_x = 2.5 m/s²
• scale factor = sqrt ((g² + a_x²) / g²) = sqrt ((9.81² + 2.5²) / 9.81²) = 1.03

Note that as a result of the above method we can model to some extent the hydrodynamic overpressure in a pseudo-static analysis, however, we do not take into account the real effects of free or open water, such as:

• fluctuations in water level due to sloshing;
• hydrodynamic under/over-pressures due to mass effects

Note: “normal” point loads and surface loads are not influenced by these additional accelerations.

Similarly, the unit weight of water used to compute the pore water pressures is scaled up by an identical factor. 

Full dynamic (time-history) analysis

As mentioned above realise that water loads coming from “free” or “open” water are transformed to surface loads acting on (active) model boundaries. As such they are simply static loads during a full dynamic (time-history) analysis. No dynamic water effects are considered.

This article presents two approaches to model the hydrodynamic pressures under the scope of a full dynamics analysis:

1. Model the water as a real mass:
   1. Use the Mohr-Coulomb model
   2. Use the Fluid User Defined Soil Model (UDSM)
2. Model the effect of water using the "added mass" feature

1) Model the water as a real mass

For some situations, a strategy may be to model the free or open water using a cluster with material properties assigned to it. This way, to some extent, the hydrodynamic effects may be captured on a structure. Possible settings for such a cluster may be:

a) Use Mohr Coulomb model

• Mohr-Coulomb model
• Normal water weight
• Drainage type: Undrained C
• Strength parameters: phi = 0 and very small s_u
• Stiffness parameters: v_u=0.495 (default setting for Undrained C) and E should be selected such that the resulting K_w (bulk modulus of water) is the desired value

b) Use Fluid User Defined Soil Model (UDSM)

The Fluid model is a User Defined Soil Model. This constitutive model has been particularly developed to model situations involving fluid – structure interaction, such as the effects of free water on water-retaining structures and quay walls, among others.

Moreover, unlike case a), it allows an expeditious calibration by requiring the definition of an individual parameter which is the bulk modulus of the water, typically chosen as 2.2⋅106 kPa. The article below explains how to use it: Fluid - PLAXIS UDSM

2) Model the effect of water using the "added mass" feature 

The hydrodynamic pressures may be approximated by the Westergaard (1933) formula, which uses a parabolic approximation for the additional pressures due to earthquake motion. Fig. 1 illustrates the forces due to the total water pressures during an earthquake. Note that the hydrodynamic forces act in both directions.

For example, the water pressure is regarded as an added mass acting on the upstream surface of a dam structure, and the rest of the water is assumed to be inactive.

Moreover, added mass approach neglects the effects of the compressibility of the water and the influence of the excitation frequency (no resonance effect can be studied). The article below further describes the “added mass” feature: Westergaard's added mass for hydrodynamic pressures: a simple case
In addition, it is essential to mention that the "added mass" feature currently is a Tech Preview.

The "added mass" feature can be found on the Structures mode tab, on the "Create load" side toolbar option for PLAXIS 2D/3D. Fig 2. presents the location for PLAXIS 2D.  

Fig. 1: Hydrostatic and hydrodynamic forces during earthquake excitation

Fig. 2: Location of “Added mass” feature

Note that with the above strategies:

• still effects of real dynamic water behaviour are missed, such as fluctuations in water level due to sloshing;
• for the Mohr-Coulomb case, calculation times may increase due to the large amount of plasticity introduced in the model.