Free P-Delta Analysis
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?
AutoCalcs P-Delta analysis is a bounded, linearised second-order elastic method. A first-order solve establishes the axial and membrane force state, then a second solve adds geometric stiffness to recover the applicable second-order forces and displacements. It does not update element geometry.
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
Under compressive axial load, both effects reduce effective stiffness and amplify internal forces. The linearised P-Delta formulation 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 - Uses elastic stiffness on the undeformed reference geometry. Fast and often sufficient for stiff structures.
- Second-order (P-Delta) analysis - Adds geometric stiffness derived from the first-order force state to recover linearised second-order effects.
AutoCalcs uses a bounded two-pass P-Delta solution. The first pass calculates the axial forces with a linear analysis. Those forces are then used to assemble the geometric stiffness matrix for one second-order solve.
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 two solver passes:
- Linear pass - Solve the original model and recover the member axial forces
- Geometric stiffness - Assemble the geometric stiffness matrix from those recovered axial forces
- Second-order pass - Combine the elastic and geometric stiffness terms and solve again for the amplified displacements
- Recover results - Calculate the second-order member forces and reactions from the updated displacement solution
This is not a large-displacement analysis and it does not repeatedly update the element geometry until convergence. If the model contains tension-only or compression-only members, a separate active-set process may repeat the two-pass solution until the active members stabilise. Use elastic buckling analysis and the reported stability checks to assess proximity to instability.
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 acts through the lateral displacement. This creates the additional overturning moment P × Δ. The second solver pass includes the associated geometric stiffness effect, producing amplified displacements and 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 effects arise from geometry under load, but AutoCalcs represents them with a linearised geometric-stiffness formulation rather than a fully geometric nonlinear analysis. This is different from material nonlinearity, which accounts for yielding, plasticity, or other non-elastic material behaviour.
AutoCalcs keeps materials elastic. It does not update the model through large rotations or repeated geometry updates, and it does not model yielding, plasticity, or post-buckling response.
Try P-Delta Analysis Free
AutoCalcs supports linear, P-Delta, buckling, and modal 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.