Free Modal Analysis

Modal analysis determines the natural vibration characteristics of a structure — the frequencies at which it wants to vibrate and the deformed shapes it takes at each of those frequencies. These properties depend only on the stiffness and mass distribution, not on any applied dynamic load, and they underpin every seismic design calculation, floor vibration check, and dynamic feasibility assessment in modern structural engineering.

What is Modal Analysis?

Modal analysis (also called free vibration analysis or eigenvalue modal analysis) solves the generalised eigenvalue problem [K]φ = ω²[M]φ to find the natural frequencies and mode shapes of a structure. Unlike static analysis which answers “how does the structure respond to this load?”, modal analysis answers a more fundamental question: “how does the structure want to move on its own?”

The result is a set of natural frequencies (in Hz) and corresponding mode shapes. Mode 1, the lowest-frequency mode, is almost always the most important — it represents how the structure moves under the least energy and is what governs earthquake response and wind-induced vibration. Each mode also has mass participation factors that tell you how much of the total mass is mobilised in each direction, which drives modal combination in seismic design.

Types of Modes

A structure's modes can be grouped by the kind of motion they describe:

Modal analysis produces all of these in a single eigenvalue solve. The mode shapes show you exactly how the structure moves — which storeys lead, which trail, where the twist axis lies — so you can identify soft storeys, torsional irregularities, and weak diaphragms by inspection.

Natural Frequencies: What They Mean

The fundamental frequency (Mode 1) of a building is a direct indicator of its lateral stiffness-to-mass ratio. Combined with the mass participation factors, it tells you most of what you need to know about dynamic behaviour:

• f > 10 Hz - Stiff structure. Short-period seismic response, wind and vibration typically not governing.
• f = 3 to 10 Hz - Typical low-rise buildings and industrial frames. Check floor vibration limits for occupant comfort.
• f = 1 to 3 Hz - Mid-rise buildings. Seismic design almost always governs. Wind dynamic effects become relevant for tall/slender shapes.
• f < 1 Hz - Tall or flexible structure. Full dynamic analysis (response spectrum or time history) is usually required by code.

Very low fundamental frequencies (below about 0.5 Hz on a building-scale model) can indicate a modelling issue rather than a genuinely flexible design. Check for missing lateral-load-resisting systems, disconnected parts, or incorrect restraint conditions. A well-supported structure with a reasonable bracing system should produce a credible first mode.

When Do You Need Modal Analysis?

Modal analysis is required by most seismic codes and recommended whenever dynamic behaviour could govern design. Common scenarios:

Even when not strictly required by code, modal analysis is a cheap, powerful sanity check. Running it early catches modelling errors like missing bracing, disconnected members, or unrealistic mass assumptions before they propagate into the static design results.

Mass Participation Factors

Mass participation is the fraction of the total structural mass that moves in a given direction during each mode. It tells you which modes actually matter for a given excitation and is the key metric in seismic modal combination.

Most seismic codes (ASCE 7, AS 1170.4, EC8, NBCC) require that at least 90% of the total mass be accounted for in each direction before a modal response spectrum analysis is considered complete. The number of modes needed varies with the structure — the AutoCalcs modal panel reports a running cumulative sum so you can solve for more modes if a direction hasn't crossed the threshold. AutoCalcs Pro provides up to 30 modes per run.

The AutoCalcs modal results panel reports participation per direction (X, Y, Z) for each mode alongside a running cumulative sum (ΣX, ΣY, ΣZ). Cumulative cells turn green once they cross the 90% threshold in that direction, so you can see immediately which directions are covered and whether you need to solve for more modes.

How Modal Analysis Works

The mathematical process behind modal analysis is an eigenvalue problem on the stiffness and mass matrices:

1. Assemble the stiffness matrix [K] - The same cubic-Hermite beam stiffness used in linear, P-Delta, and buckling analysis
2. Assemble the mass matrix [M] - AutoCalcs uses the consistent mass matrix derived from the element shape functions (the variationally consistent formulation), which gives accurate frequencies on coarse meshes
3. Apply h-refinement - Each physical member is automatically subdivided into 4 sub-elements before the eigensolve so the consistent mass matrix can resolve distributed-mass mode shapes on long slender members
4. Solve the eigenvalue problem - [K]φ = ω²[M]φ is solved with ARPACK using shift-invert at σ = 0 for the lowest k modes, with tight tolerance to resolve near-degenerate mode pairs
5. Filter and report - Invalid modes (zero/negative eigenvalues) and pure torsional modes (a beam-element artefact from missing warping stiffness) are filtered; the remaining natural frequencies, mode shapes, effective modal masses, and mass participation factors are reported

Mass Source: What Becomes Mass?

Before running modal analysis, you select a reference load pattern (load case or combination) in the modal setup dialog. Pynite converts the gravity-direction loads in that pattern into mass for the eigensolve. The following contributions become mass:

• Self-weight on the selected load cases (material density × element volume)
• Gravity-direction force loads - node loads, point loads, and distributed loads aligned with the gravity axis, converted to mass via F/g

Moment loads do not contribute to mass (they produce no rigid-body translation). Non-gravity-direction forces (e.g. a horizontal wind point load) are forces, not mass, and are likewise excluded. The modal setup dialog only shows load patterns that contain self-weight so you cannot accidentally pick an empty mass source.

Consistent vs Lumped Mass

Modal analysis can be formulated with either a consistent mass matrix or a lumped mass matrix. AutoCalcs uses consistent mass, which is the variationally correct formulation derived from the same shape functions as the stiffness matrix. It gives an upper bound on frequencies that converges monotonically to the exact answer as the mesh refines, and it properly captures rotational inertia in beam elements.

Lumped mass concentrates element mass at the endpoint nodes. It is cheaper to compute (the mass matrix is diagonal) but is an older approximation — it converges more slowly and under-represents rotational kinetic energy on higher modes. Consistent mass is what modern FEA textbooks treat as the reference formulation (Cook, Malkus & Plesha; Bathe; Clough & Penzien) and is what NAFEMS uses to generate its modal benchmark solutions. Lumped mass is only preferred for explicit time integration (crash, blast, wave propagation), which is a different class of problem.

AutoCalcs' modal solver is verified against NAFEMS textbook benchmarks (FV52 framework tests) to within 0.02–0.23% on all 11 modes — the reference accuracy target for this formulation.

Mode Shape Visualisation

Each natural frequency has a corresponding mode shape that shows how the structure deforms at that frequency. Mode 1 is almost always the most important, but higher modes are equally useful for diagnosing problems: soft storeys appear as a large change in drift between floors, torsional irregularities show up as twisting about a vertical axis, and local floor modes reveal vibration-sensitive spans.

AutoCalcs renders each mode shape on the 3D model with smooth within-member bowing curves. Because the eigensolve runs on the h-refined mesh, the mode shape is captured at quarter-point stations along each member and animated with the true peak-translation as the amplitude reference, so you can see exactly where each mode concentrates motion.

Pure torsional modes, which can appear at artificially low frequencies in any 3D frame analysis that uses a 6-DOF beam element without warping stiffness, are automatically filtered out of the results. Modes with less than 5% translational participation are skipped. Flexural-torsional interaction is assessed separately by the steel design code checks using section warping constants directly.

Modal vs Buckling vs P-Delta

These three analysis types are complementary, not interchangeable. They each answer a different question:

• Modal analysis - “How does the structure want to move on its own?” Natural frequencies and mode shapes, driven by mass and stiffness only.
• Buckling analysis - “How far are we from instability?” Critical load factors and mode shapes, driven by the destabilising effect of compression.
• P-Delta analysis - “What are the amplified forces?” Second-order static forces on the deformed geometry.

In practice, seismic design uses all three. Modal gives the fundamental period and mass participation factors that feed code-based equivalent-lateral-force calculations (ASCE 7 §12.8, AS 1170.4, EC8). Buckling confirms that the gravity system has adequate stability margin under seismic gravity loads. P-Delta captures the second-order amplification of the seismic displacements so the design forces reflect the real deformed geometry.

Known Limitations

Modal analysis uses the same 6-DOF cubic-Hermite beam element as the rest of the solver. Two limitations follow from that formulation and are worth understanding before interpreting results:

• Open sections: torsional and lateral-torsional modes are underestimated. The 6-DOF beam element has no warping degree of freedom (θ′′), so the warping torsional stiffness (π²EI_w/L²) that dominates St Venant torsion on open cross-sections (I-sections like UBs/UCs/W-shapes, channels, tees, and angles) is missing from the stiffness matrix. The fundamental lateral-torsional mode can be up to ~40% low, and higher-order torsion-coupled modes are 2–10% low. Pure horizontal sway modes in the X and Z directions (Y is vertical), which govern lateral-load and seismic design, are unaffected by the missing warping term and align closely with NAFEMS reference values. Weak-axis bending modes without coupled twist are also unaffected. Closed sections (CHS, RHS, SHS) are not affected at all — their torsional stiffness is dominated by GJ, which the element models correctly.
• Phantom torsional modes are filtered. Without I_w, open sections can produce torsional modes at artificially low frequencies. AutoCalcs filters any mode with less than 5% translational participation and shows a banner warning when filtering occurs, so you know the top of the mode list is clean. Flexural-torsional interaction is then assessed directly in the steel design code checks (AS4100 §6.3.3, AISC 360 E4, Eurocode 3 cl.6.3.1.4, CSA S16 cl.13.3.2), which use I_w explicitly.
• Tension-only and compression-only members are not supported. Modal analysis requires a constant linear stiffness matrix; non-linear members whose stiffness switches based on load direction violate this. Models containing tension-only or compression-only elements are blocked with a clear error message — convert them to regular members before running modal.
• Response Spectrum Analysis (RSA) is not yet available. Modal analysis provides the inputs (natural frequencies and mass participation factors) that RSA combines against a design spectrum. The combination step itself (CQC/SRSS against a code spectrum) is not currently exposed in AutoCalcs. For now, modal outputs are intended for code-based equivalent-lateral-force calculations, floor vibration checks, and dynamic feasibility sanity-checking.

Try Modal Analysis Free

AutoCalcs supports linear, P-Delta, buckling, and modal analysis for 3D frame structures. Run modal analysis to get natural frequencies, mode shapes, and mass participation factors, then use those results to drive seismic base shear, floor vibration checks, or dynamic feasibility assessment for AISC 360, Eurocode 3, AS4100, or CSA S16 design.