RM Bridge latest big technical development efforts were made to include hydrodynamics in the analysis with the necessary loads and elements for a simplified linearized frequency domain analysis. The dynamic model is of a modal decomposed system taking frequency-dependent hydrodynamic added mass and radiation with damping. 

RM Bridge can be used for both FD and TD; but, here we focus on the FD - Frequency Domain - because these analyses are much quicker when compared to the final TD - Time Domain analysis. As the frequency is based on the tangent stiffness and linearized values some differences are expected to be found between the two methods.

Hydrodynamics in RM Bridge:

• Hydrostatic stiffness and buoyancy;
• Hydrodynamic viscous drag and damping;
  Hydrodynamic added mass and damping, the radiation forces due to the motion of structure;
• Wave excitation, the force from incoming waves, is included as load time histories.

These new tools were specifically designed for TLP - Tension-Leg Platforms - multi-span suspension bridges preliminary studies that were in place for the proposed crossings at the extremely wide and deep fjords along the E39 highway, on the west coast of Norway. These bridge piers are supported by submerged floaters that are subject to hydrodynamic and hydrostatic loads and therefore the bridge experiences dynamic responses due to wind and wave excitations combined. The foundations are formed by a hull and several tethers and anchors at the seabed, as in the offshore industry, with one example in the picture below.

The analysis starts with a full bridge model and their eigenvalue; then, for the structural equilibrium of the dynamic equation of motion in water with Mass, Stiffness, Load, and Damping, a modal decomposition is performed.

[M+MA(ω)]u''+CA(ω)u'+Ku=F(t)

User must notice that there will be:

• Hydrodynamic radiation damping
• Aerodynamic damping
• Structural damping

A step-by-step Newmark integration scheme with optimization is performed with Shydro springs to simulate the hydrodynamic effect. The hydrodynamic implementation included fully coupled matrices for:

• Added mass A(∞)
• Frequency depended on damping  C(ω)
• VDDE - Viscous Drag Damping Element load

Coupled wind and wave loads analyses are necessary, and more specific technical information can be found in papers submitted to several publications.