What these analyses need from the model

One element is enough for statics and not for buckling

A modern beam element uses cubic shape functions, which represent the exact deflected shape of a prismatic member under end loads. So a single element gives the exact static answer for a simple beam — subdividing it changes nothing but the number of points you can plot.

Buckling is different, and the reason is precise. A buckling mode needs a lateral displacement at mid-height. With one element, the only free degrees of freedom are the end rotations; there is no node in the middle to move sideways. The model is therefore constrained relative to the real strut, and a constrained structure is stiffer.

The result is the classic 12EI/L² for a single-element pin-ended strut, against Euler's π²EI/L² ≈ 9.87EI/L². That is 21.6 % too high — an error in the unsafe direction, produced by a model that looks entirely reasonable.

Two elements bring it within about 1 %. More than four buys nothing measurable. The rule that comes out of this is worth stating generally:

A finite element model of a continuum is a constrained version of it, so it is always stiffer, and buckling loads always converge from above.

Vibration meshing is the reverse of static meshing

Static results are driven by stress gradients, which are steepest where the structure is interrupted: supports, columns, openings, load points, re-entrant corners. That is where a static mesh is refined.

A mode shape is a displacement field, and its curvature is greatest mid-bay. Next to a column, a floor barely moves. So a vibration mesh is refined where a static mesh would be coarsened, and the two are genuinely in tension — one model cannot serve both well without being finer than either needs.

Mass is a decision, not a property

In a static analysis, load is what you apply. In a dynamic analysis, mass is what resists — and the mass that is present during the event is not the ultimate design load.

For a footfall analysis you want the mass actually on the floor: the structure, the finishes, the services, the partitions, and a realistic fraction of the imposed load. Too little mass and the floor comes out too stiff, too high in frequency and too responsive. Too much and it comes out sluggish.

For a seismic analysis the same question has a different answer, and it is set by the relevant code. Using the ultimate imposed load is not conservative in any reliable way: more mass gives more force, which looks safe, but it also lengthens the period, which may move the structure to a lower spectral acceleration. Whether the net effect is safe depends on where you land on the spectrum, and assuming is not analysis.

The worst case for a vibration problem is often the lightest one. An empty floor has the least mass to resist excitation. That is the opposite of every static intuition and it is why a sensitivity study on mass is not optional.

What a dynamic model needs that a static one does not

• Mass, distributed correctly. Lumped at the floors is right for a building's sway and wrong for a floor's own vibration.
• Connections modelled as they behave at that strain level. A nominally pinned steel connection acts as continuous under vibration strains, because they are too small to overcome the friction in the joint.
• Non-structural stiffness that matters at small amplitude. Balustrades and cladding do nothing at ultimate and can change a floor's liveliness.
• Enough modes. Participating mass is the first number a modal result should be judged on, before any response value is read.
• A time step, if the analysis marches in time. Convergence in time is the same discipline as convergence in space: halve it until the answer stops moving.

The detail of all five belongs to the Structural Dynamics course. What belongs here is the recognition that they are modelling decisions, taken before the analysis, and that a model inherited from the static work will have made all of them by default and none of them deliberately.