Beam Column Moment⚠ unverified
Mechanical / Columns · Compute the amplified moment in a beam-column from the P-delta effect
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| M0 | M0 | N*m | 1.0 | Primary (first-order) moment |
| P | P | N | 1.0 | Axial compressive load |
| delta | δ | m | 1.0 | Lateral deflection at the section |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | M | N*m | Total amplified moment, in newton-metres (N*m) |
The science & history
Understanding the Parameters
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Primary moment $M_0$ — the moment computed from the transverse loads alone, as if the member did not deflect axially (ordinary beam analysis). It is the "first-order" demand.
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Axial load $P$ — the compression that turns a deflection into extra moment. In tension $P$ would reduce the moment (a stiffening "P-delta" of opposite sign); in compression it amplifies, which is why only compression members are P-delta-critical.
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Lateral deflection $\delta$ — the sideways movement of the section. It is itself a function of the moment (more moment → more deflection), so the true relationship is coupled; this formula evaluates one step of it given a known $\delta$.
The product $P\delta$ has units of force × length = moment, and directly measures the destabilising contribution of axial load.
Derivation (Approaching a Proof)
Consider a member with a first-order bending-moment distribution $M_0(x)$ from transverse loads. Under axial compression $P$, the member also deflects laterally by $v(x)$. Taking moments about the deflected section, the total internal moment must balance both the transverse loading and the axial load acting through the lateral offset:
$$M(x) = M_0(x) + P\,v(x).$$
At the critical section, writing the local deflection as $\delta = v(x)$ gives the stated form $M = M_0 + P\delta$. This is the exact statement of moment equilibrium on the deformed geometry — the defining feature of second-order (geometrically nonlinear) analysis.
Because $v$ depends on $M$ through the beam equation $EI\,v'' = -M$, substituting closes the loop and yields the amplified solution. For a sinusoidal first-order moment the exact result is the amplification factor
$$M_{\max} = \frac{M_0}{1 - P/P_{cr}},$$
where $P_{cr}$ is the Euler load (Euler Buckling Load column). The simple $M = M_0 + P\delta$ is the single-increment (or known-$\delta$) form of this amplification; iterating it, or using the closed factor, recovers the full second-order moment. As $P \to P_{cr}$ the amplification diverges — the same buckling limit seen in the Secant Formula Stress.
Dimensional check. $[P\delta] = \text{N}\cdot\text{m} = \text{N}\cdot\text{m} = [M_0]$. ✓
History and Development
Beam-column and P-delta theory was developed by Timoshenko and others in the early 20th century and became essential with tall steel frames, where sway amplification governs design. Modern structural codes (AISC Direct Analysis Method, Eurocode 3) require second-order (P-delta and P-little-delta) effects to be included, either by rigorous analysis or by amplification factors $B_1$, $B_2$ that formalise exactly the $M_0 \to M_0/(1-P/P_{cr})$ magnification.
Related Concepts: Secant Formula Stress, Euler Buckling Load column, Interaction Formula, Beam Bending Stress, Beam Moment At X
Notes: Single-step P-delta form given a known deflection $\delta$. For the full amplified moment use $M_0/(1-P/P_{cr})$ or an iterative/second-order analysis. Amplification applies to compression; axial tension reduces the moment.