P-Delta Analysis of a Cantilever Column — Worked Example
A cantilever column with a heavy axial load and a modest lateral load: first-order analysis, the exact second-order answer, the AISC B2 amplifier, and what a stiffness program's P-Δ option really computes. The lesson is in the last table: how many elements you put in the column decides how much of the second-order effect the program can see.
The column
| Geometry | Vertical cantilever, h = 15 ft (180 in), fixed base, free top |
| Section | I = 484 in&sup4; about the bending axis, A = 14.1 in², E = 29,000 ksi |
| Loads at the top | Axial P = 200 kip, lateral H = 5 kip |
Model "Cantilever", vertical, length 15, custom section Iz = 484, axial P = 200 (D), tip P = 5 (L), P-Δ on; vary the segments.
Open 3D frame analysis →Ignore the axial load
M1 = Hh = 5 × 15 = 75.0 kip-ft
Solve the beam-column differential equation
Δ = (H / Pk)(tan kh − kh) = 0.8498 in
Mbase = Hh + PΔ = 900 + 200 × 0.8498 = 1,070.0 kip-in = 89.16 kip-ft (amplification 1.189 on moment, 1.227 on drift)
Euler and AISC Appendix 8
AISC B2: Pe,story = RMHh/ΔH = 0.85 × 5 × 180 / 0.6925 = 1,104.7 kip → B2 = 1.221 → Δ ≈ 0.846 in
Both estimates land within 0.5% of the exact drift. RM = 0.85 is the AISC allowance for the P-δ (member curvature) part of the effect in a moment-frame column.
Geometric stiffness on the element chords, iterated
| Elements in the column | Top drift (in) | Base moment (kip-ft) | Error vs exact drift |
|---|---|---|---|
| Linear (no P-Δ) | 0.6925 | 75.00 | −18.5% |
| 1 element, P-Δ | 0.8185 | 88.64 | −3.7% |
| 2 elements | 0.8403 | 89.01 | −1.1% |
| 4 elements | 0.8473 | 89.12 | −0.3% |
| 8 elements | 0.8492 | 89.15 | −0.07% |
| 16 elements | 0.8496 | 89.16 | −0.02% |
| Exact | 0.8498 | 89.16 | — |
The geometric stiffness N/L acts only on each element's chord, so a single element sees P-Δ (the sway of the top) but not P-δ (the bow of the column between its ends). Subdividing the column turns the bow into chord rotations, and the result converges on the exact answer. The single-element result matches the closed form H/(3EI/h³ − P/h) = 0.8185 in exactly.
Stability analysis is more than P-Δ
The AISC direct analysis method (Chapter C) adds two things to a second-order analysis: notional lateral loads of 0.002 × the gravity load at each level, and reduced stiffness (0.8τbEI and 0.8EA) to account for residual stresses and partial yielding. In the frame tool, apply them by adding a W-case joint load and by entering 0.8E for the material. The tool runs the second-order part of the analysis; the design checks are up to you.
Tools used in this example
3D frame analysis with P-Δ, Euler buckling for Pe, and AISC LRFD design strength for the column check.
Frequently asked questions
What is the difference between P-Delta and P-delta?
P-Δ (large delta) is the moment from gravity load acting through the lateral displacement of the member ends, the sway of a story. P-δ (small delta) is the moment from axial load acting through the member's own curvature between its ends. Stiffness programs with geometric stiffness on each element's chord capture P-Δ directly; P-δ is captured only if the member is split into several elements.
How accurate is the AISC B2 amplifier?
For this cantilever column with P/Pe ≈ 0.19, B2 = 1.221 predicted a drift within 0.5% of the exact elastic second-order solution. The amplifier becomes less accurate as P/Pe approaches 1, and AISC requires B2 ≤ 1.5 for some simplifications. Above that, a rigorous second-order analysis is required.
How many elements should a column have in a P-Delta analysis?
One element per column captures story sway (P-Δ) exactly but underestimates second-order drift by a few percent when P-δ matters. Two to four elements per column bring the error well under 1% for typical axial loads.