Linear vs Nonlinear Structural Analysis
Choosing the right analysis type is fundamental to accurate structural design. Linear analysis is faster and simpler, but nonlinear analysis captures effects that linear methods miss entirely. Understanding when each applies will make you a better engineer.
What is Linear Analysis?
Linear (first-order) structural analysis makes two key assumptions:
- Small displacements — The structure deflects so little that we can ignore the change in geometry when calculating forces
- Linear material behaviour — Stress is proportional to strain (Hooke's Law), and the material returns to its original state when unloaded
Under these assumptions, doubling the load doubles the displacement. The principle of superposition applies—you can add load cases together. This makes analysis fast and intuitive.
For most everyday structures—residential buildings, small commercial frames, typical beams and columns—linear analysis provides accurate results. The assumptions hold well enough that the simplification is justified.
What is Nonlinear Analysis?
Nonlinear analysis relaxes one or both of these assumptions. There are two main types:
Geometric Nonlinearity
Accounts for changes in geometry as the structure deforms. Equilibrium is calculated on the deformed shape, not the original.
- P-Delta effects (P-Δ and P-δ)
- Large displacement theory
- Cable and membrane structures
Material Nonlinearity
Accounts for non-elastic material behaviour such as yielding, plasticity, cracking, or creep.
- Steel yielding and plastic hinges
- Concrete cracking and crushing
- Pushover and collapse analysis
The most common nonlinear analysis in practice is P-Delta analysis—a form of geometric nonlinearity that captures second-order effects while still assuming elastic material behaviour. This strikes a practical balance: it captures the most important nonlinear effects without the complexity of full material nonlinearity.
Side-by-Side Comparison
| Aspect | Linear (First-Order) | Nonlinear (Second-Order) |
|---|---|---|
| Equilibrium | On original geometry | On deformed geometry |
| Superposition | ||
| Solution Method | Single step (direct) | Iterative |
| Computation Speed | Fast | Slower (iterations) |
| P-Delta Effects | ||
| Stability Detection | Not captured | Divergence indicates instability |
| Results Accuracy | Conservative for stiff structures | More accurate for flexible structures |
When is Linear Analysis Sufficient?
Linear analysis works well when:
- Stiff structures — Braced frames, shear wall buildings, stocky columns
- Small displacements — Deflections are small relative to member lengths (typically <1%)
- Low axial loads — Gravity loads are modest relative to buckling capacity
- Service load analysis — Checking deflections and stresses under working loads
- Preliminary design — Quick sizing before detailed analysis
For these cases, the error from ignoring second-order effects is typically less than 5%—well within the safety margins built into design codes.
When Do You Need Nonlinear Analysis?
Nonlinear (P-Delta) analysis becomes necessary when:
- Tall buildings — Multi-storey structures where drift amplification matters
- Slender members — Columns with high slenderness ratios
- Unbraced frames — Moment frames relying on frame action for stability
- Heavy gravity loads — High axial loads combined with lateral forces
- Code requirements — Most modern codes require second-order analysis for certain structure types
- Stability-sensitive structures — Arches, domes, shells, cable structures
The 10% Rule
A practical check: run both linear and P-Delta analysis on your structure. If the second-order results differ by more than 10%, you should use the nonlinear results for design. If they're within 10%, linear analysis is typically acceptable.
Many design codes formalise this through stability coefficients (θ) or amplification factors. The underlying principle is the same: quantify how much second-order effects amplify forces, and use that to decide if they can be ignored.
A Note on Material Nonlinearity
Material nonlinearity—accounting for yielding, plasticity, and post-elastic behaviour—is a more advanced topic. It's used for:
- Pushover analysis for seismic design
- Progressive collapse assessment
- Ultimate capacity calculations
- Concrete cracking and reinforcement yielding
AutoCalcs currently focuses on elastic analysis (linear and P-Delta), which covers the vast majority of design scenarios. Material nonlinearity requires defining yield surfaces, hardening rules, and failure criteria—a level of complexity beyond typical frame analysis needs.
Summary: Choosing Your Analysis Type
Start with linear analysis for quick results. If you have any of the warning signs (tall structure, slender members, unbraced frame, heavy loads), run P-Delta analysis to check. Compare results—if they differ significantly, use the second-order values.
AutoCalcs makes this easy: switch between Linear and P-Delta with a single click and see exactly how your results change.