CIV-32

Beam–column interaction

U = P/Pcr + M/Mcr. A linear interaction; U < 1 is the safe side of the diagram.

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StabilityRankine–Merchant

Governing equation

U=\dfrac{P}{P_{cr}}+\dfrac{M}{M_{cr}}

$P \in \left[10, 4000\right]$

where

P: Axial load (kN)
P_{cr}: Squash / buckling (kN)
M: Moment (kN·m)
M_{cr}: Moment capacity (kN·m)
U: Utilisation (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

U=\dfrac{P}{P_{cr}}+\dfrac{M}{M_{cr}}

$P \in \left[10, 4000\right]$

$P = 10$ $U = 0.408$

Along the curve

②

Domain of the variable — blue

P_{\min} = 10,\quad P_{\max} = 4000

$P \in \left[10, 4000\right]$

$P = 10$ $P = 10$

Along the curve

③

Response curve — green

U=\dfrac{P}{P_{cr}}+\dfrac{M}{M_{cr}}

$P \in \left[10, 4000\right]$

$P = 10$ $U = 0.408$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}U}{\mathrm{d}P}

$P \in \left[10, 4000\right]$

$P = 35.09$ $U^{\prime} = 8.33\times 10^{-4}$

Along the curve

Lecture brief

Historical brief

From Euler’s 1744 elastica and Navier’s beam theory to Mohr’s circle and transformed-section RC, structural mechanics grew as a closed-form craft before finite elements. The lab keeps those governing lines for buckling, flexure, joints and influence. This sheet (CIV-32 — Beam–column interaction) is the form associated with Rankine–Merchant. Working symbols: $P$ , $P_{cr}$ , $M$ , $M_{cr}$ $\rightarrow$ $U$ . Perry–Robertson / Rankine–Merchant: axial load eats the moment capacity. Amplification (1−P/Pe)⁻¹ is a cousin.

Purpose

Purpose: compute

U

from

P

P_{cr}

M

M_{cr}

in Structural & civil via

U=\dfrac{P}{P_{cr}}+\dfrac{M}{M_{cr}}

U = P/Pcr + M/Mcr. A linear interaction; U < 1 is the safe side of the diagram. Use it when a real structural & civil question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

P = 400.000\,\mathrm{kN}

P_{cr} = 1200.000\,\mathrm{kN}

M = 80.000\,\mathrm{kN·m}

M_{cr} = 200.000\,\mathrm{kN·m}

, the governing relation

U=\dfrac{P}{P_{cr}}+\dfrac{M}{M_{cr}}

yields

U = 0.733\,\mathrm{—}

. A P–M plane, a straight cutoff, a utilisation. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Axial load P400 kNSquash / buckling P_{cr}1200 kNMoment M80 kN·mMoment capacity M_{cr}200 kN·m

Outputs

Utilisation U0.733 —

Reading speed

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Narration of this film

A P–M plane, a straight cutoff, a utilisation.

Perry–Robertson / Rankine–Merchant: axial load eats the moment capacity. Amplification (1−P/Pe)⁻¹ is a cousin.

Reading speed

Watch on YouTube