Computational Engineering / Second-order, buckling and dynamic methods
What these analyses need from the model
Mesh density for buckling, mass for dynamics, and the modelling decisions that a static analysis lets you get away with and these do not.
Lesson review: expert verification remains open
This lesson has received a full-context editorial review and a check of its technical explanations. Selected arithmetic has been checked separately. This is not independent engineering sign-off.
Checks needed for project use
- A computational engineering reviewer must check algorithms, convergence claims, statistical assumptions and numerical examples beyond the cases explicitly recorded.
- Confirm current adopted code edition and national provisions for stability screening; SCI reference documents are explanatory and not a substitute for the governing standard.
- Audit buckling/eigenvalue helper implementations and lab mode-classification weighting; translations and rotations must not be combined unscaled.
- Numerical mesh table and all dynamic modelling choices require specialist verification.
These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.
Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.
Read the lesson register and outstanding review notesWhy this matters
Before reusing a static model for buckling or vibration, check whether its element formulation, mesh, stiffness and mass distribution can represent the required response. The same geometry may be suitable, but a successful solver run does not establish that suitability.
By the end of this lesson you should be able to
- Explain when a single cubic beam element gives exact static nodal results and why buckling still needs a convergence check
- Choose vibration mesh refinement from the modes and responses of interest
- Distinguish material density from the mass distribution appropriate to the analysed event
- List what a dynamic model needs that a static one does not
Mesh for buckling
A cubic Euler–Bernoulli beam can represent the static displacement field of a uniform member under end actions exactly. Distributed loads, variable properties and other formulations need separate consideration. For the pin-ended buckling example, end rotations produce displacement between nodes, but one cubic element cannot exactly reproduce the sinusoidal mode. Its critical load is 12EI/L², 21.6% above Euler’s value.
Refine until the relevant eigenvalues and mode shapes meet a stated tolerance. Conforming Rayleigh–Ritz models of the same self-adjoint problem can provide upper bounds; this is not a guarantee for every element, quadrature, geometric stiffness or modelling assumption.
Mesh for vibration
Resolve the spatial variation of the modes and the response quantities needed. Refinement may be needed within spans, near stiffness changes or at connections. Static and vibration requirements differ, but are not generally opposites. Check convergence of the selected frequencies and responses.
Mass and small-amplitude stiffness
Include the mass present during the event: structural and non-structural permanent mass, and an appropriate allowance for occupancy or imposed loading. A lower mass raises frequency at fixed stiffness; it does not make the stiffness itself larger. Forced response can increase or decrease as a resonance moves, so test plausible mass ranges. Seismic mass must follow the applicable design basis.
Represent the actual mass distribution. Floor-level lumping can be useful for building sway, while local floor modes need distributed mass. Use connection, concrete and non-structural stiffness appropriate to the amplitude and condition being assessed. Nominally pinned joints and partitions may contribute stiffness, but continuity, friction and cracking should not be assumed without justification.
Check the modes relevant to the load and output location. Effective modal mass is useful for directional base excitation; it does not by itself establish convergence for a local footfall load. For time integration, check both time-step convergence and the accuracy of the input loading.
Try it
Modal mass explorer
A floor whose frequency follows the mass present. Move the imposed load and watch the response — it does not move monotonically.
What is actually on the floor when it vibrates — not the ultimate design value.
1.0 is the base design.
- Mass present
- 5.70 kN/m²
- Deflection under it
- 9.12 mm
- Frequency f ≈ 18/√δ
- 5.96 Hz
- In the plausible band?
- yes, 4–10 Hz
- Nearest walking harmonic
- 5.4 Hz
- Detuning from it
- 10.4 %
- Response factor
- 10.3
The floor is within 10 % of the 5.4 Hz harmonic and the response is elevated. Move the imposed load either way and it falls — which means neither end of the range is the worst case.
Why this breaks bracketing
- Sweep the imposed load from one end of its range to the other and the response rises, peaks, and falls again. It is not monotonic.
- So the three-run bracket of Module 14 — worst case at a corner — does not apply. Two runs at the ends would report a comfortable answer and miss the peak entirely.
- The worst case for a vibration problem is often the LIGHTEST one, because there is least mass to resist the excitation. That is the opposite of every static intuition.
What this shows: In a vibration model, mass is a load-case decision, and the worst case is often not at either end of its range.
Worked example
How many elements does this strut need?
Given
- A pin-ended strut, E = 210 × 10⁶ kN/m², I = 1 × 10⁻⁴ m⁴, L = 5 m
- Modelled first with one beam element, then with two, four and eight
Find
The buckling load reported at each mesh density, against the Euler value
Practice
A single beam element models a pin-ended strut with EI = 21 000 kN·m² and L = 5 m. Using the one-element result P = 12EI/L², what buckling load does the model report, in kN?
Practice
A floor's first natural frequency is 5.2 Hz with the design mass. The mass is then found to be 30 % too low. Frequency goes as 1/√m. What is the corrected frequency, in Hz?
Check yourself
What does cumulative effective modal mass help assess in a modal response analysis with directional base excitation?
Check yourself
What does a dynamic analysis need from the model that a static one does not?
Summary
- A single cubic beam element can be exact for restricted static problems while overestimating this first buckling load by 21.6%
- Check eigenvalue convergence; upper bounds require the relevant variational and formulation assumptions
- Refine a vibration mesh to resolve the modes and output quantities of interest
- Mass is a load-case decision; the ultimate imposed load is not the seismic mass
- The worst vibration case is often the lightest one
- A model inherited from the static work has made every dynamic decision by default