P-Delta Analysis: A Complete Guide to Second-Order Effects
P-Delta analysis accounts for the additional forces and moments caused by structural deformation under load. When a structure deflects, gravity loads create secondary effects that first-order analysis ignores. Understanding when these effects matter is essential for accurate structural design.
What is P-Delta Analysis?
P-Delta analysis is a type of second-order structural analysis that considers how axial loads (P) interact with lateral displacements (Delta, or Δ) to create additional moments and forces in a structure. Unlike first-order (linear) analysis, which assumes the structure remains in its original geometry, P-Delta analysis accounts for equilibrium on the deformed shape.
The name comes from the fundamental equation: the additional moment equals the axial force (P) multiplied by the displacement (Δ). This seems simple, but the effects can be significant—and ignoring them can lead to unconservative designs.
Two Types of P-Delta Effects
Engineers distinguish between two related but distinct phenomena:
P-Δ (Big Delta)
The effect of axial load acting through the relative lateral displacement between member ends. This is the "sway" effect in frames.
- Storey drift in multi-storey frames
- Lateral displacement of column tops
- Global frame instability
P-δ (Small Delta)
The effect of axial load acting through the deflection along the member length. This is the "member curvature" effect.
- Bowing of individual members
- Amplification of member moments
- Member buckling behaviour
Both effects reduce the effective stiffness of the structure and amplify internal forces. A proper P-Delta analysis captures both, though P-Δ typically dominates in sway frames while P-δ is more significant for individual slender members.
First-Order vs Second-Order Analysis
The key difference lies in where equilibrium is calculated:
- First-order (linear) analysis — Equilibrium on the original, undeformed geometry. Fast and often sufficient for stiff structures.
- Second-order (P-Delta) analysis — Equilibrium on the deformed geometry. Required when displacements significantly affect force distribution.
Second-order analysis is iterative: the structure deflects, which changes the forces, which changes the deflection, and so on until equilibrium is reached (or the structure is unstable and no equilibrium exists).
When Do You Need P-Delta Analysis?
Not every structure requires second-order analysis. Here are the key indicators:
Consider P-Delta Analysis When:
- Tall or slender structures — Multi-storey buildings, towers, masts
- High axial loads — Heavy gravity loads combined with lateral forces
- Flexible lateral systems — Moment frames without bracing
- Slender columns — High slenderness ratios (L/r)
- Code requirements — Many design codes mandate second-order analysis for certain structure types
A common rule of thumb: if second-order effects increase forces by more than 10% compared to first-order analysis, they should not be ignored. Most design codes provide specific criteria based on stability coefficients or inter-storey drift ratios.
How P-Delta Analysis Works
The analysis proceeds through an iterative process:
- Initial analysis — Perform first-order analysis to get initial displacements
- Geometric stiffness — Calculate additional stiffness terms based on axial loads and displacements
- Modified stiffness — Subtract geometric stiffness from elastic stiffness (reducing overall stiffness)
- Re-analyse — Solve with modified stiffness to get updated displacements
- Check convergence — Repeat until displacements converge (or diverge, indicating instability)
If the iteration diverges—displacements keep increasing without limit—the structure is unstable under the applied loads. This is a form of buckling detection, though not as precise as eigenvalue buckling analysis.
Practical Example: Portal Frame
Consider a simple portal frame with a horizontal load at the top. In first-order analysis, the frame deflects laterally, and we calculate moments based on the original geometry. The column moment is simply the horizontal force times the height.
With P-Delta analysis, we recognise that the vertical load on the columns now acts at an eccentricity (the lateral displacement). This creates additional overturning moment: P × Δ. The structure deflects more, which increases the eccentricity, which increases the moment... until equilibrium is reached at a larger displacement with higher internal forces.
For a typical steel portal frame, second-order effects might increase moments by 5-15%. For a slender multi-storey frame, the increase could be 20-30% or more.
P-Delta vs Material Nonlinearity
P-Delta analysis is a type of geometric nonlinearity—it accounts for the change in geometry under load. This is different from material nonlinearity, which accounts for yielding, plasticity, or other non-elastic material behaviour.
Most P-Delta analyses assume linear elastic material behaviour. The nonlinearity comes purely from the geometry. For most design scenarios, this is appropriate—we want to ensure the structure remains elastic under service loads.
Try P-Delta Analysis Free
AutoCalcs supports both linear and P-Delta analysis for 3D frame structures. Select your analysis type, apply your loads, and compare results instantly. See exactly how second-order effects change your design forces.