Linear, Second-Order & Nonlinear Structural Analysis
Choosing the right analysis type is fundamental to accurate structural design. Linear analysis is faster and simpler; linearised second-order P-Delta captures applicable stability effects; and full nonlinear analysis is needed when geometry or materials depart more fundamentally from those assumptions.
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
P-Delta is often grouped with geometric nonlinearity. AutoCalcs provides a bounded, linearised second-order elastic P-Delta solution: a first-order force state followed by one geometric-stiffness solve. It captures applicable second-order effects, not large rotations or full material nonlinearity.
Side-by-Side Comparison
| Aspect | Linear (First-Order) | Linearised Second-Order (P-Delta) |
|---|---|---|
| Equilibrium | Elastic stiffness on the undeformed reference geometry | Elastic plus geometric stiffness from the first-order force state |
| Superposition | ||
| Solution Method | Single step (direct) | Two-pass; repeats only for T/C active-set convergence |
| Computation Speed | Fast | Slower (two passes; T/C may repeat) |
| P-Delta Effects | ||
| Stability Detection | Use a separate buckling/stability check | Warnings; confirm proximity to instability with buckling analysis |
| Results Accuracy | Suitable when second-order effects are insignificant | Captures linearised second-order effects within scope |
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 Second-Order Analysis?
Linearised second-order P-Delta analysis should be considered 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
- Outside this formulation - Cable, membrane, large-displacement, and post-buckling problems need a specialised nonlinear analysis
Compare Linear and P-Delta Results
A practical screening check is to run both linear and P-Delta analysis on the same model. If the response differs materially, determine the required stability method using the applicable design standard and its stated criteria; no single percentage is a universal acceptance threshold.
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, linearised second-order P-Delta, buckling, and modal). Full material nonlinearity requires yield surfaces, hardening rules, and failure criteria, and is outside this analysis path.
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, P-Delta, Buckling, and Modal with a single click and see exactly how your results change.