CIV-30

Column slenderness λ = KL/r

λ = K L / r. K encodes fixity: 1 pinned, 0.5 fixed, 2 cantilever.

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StabilityEuler slenderness

Governing equation

\lambda=KL/r

$L \in \left[0.8, 16\right]$

where

K: Effective-length factor (—)
L: Unbraced length (m)
r: Radius of gyration (cm)
\lambda: Slenderness (—)

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

\lambda=KL/r

$L \in \left[0.8, 16\right]$

$L = 0.8$ $\lambda = 13.33$

Along the curve

②

Domain of the variable — blue

L_{\min} = 0.8,\quad L_{\max} = 16

$L \in \left[0.8, 16\right]$

$L = 0.8$ $L = 0.8$

Along the curve

③

Response curve — green

\lambda=KL/r

$L \in \left[0.8, 16\right]$

$L = 0.8$ $\lambda = 13.33$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\lambda}{\mathrm{d}L}

$L \in \left[0.8, 16\right]$

$L = 0.896$ $\lambda^{\prime} = 16.67$

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-30 — Column slenderness λ = KL/r) is the form associated with Euler slenderness. Working symbols: $K$ , $L$ , $r$ $\rightarrow$ $\lambda$ . Euler's critical stress is π² E / λ². Codes switch from elastic to inelastic at a limit λc.

Purpose

Purpose: compute

\lambda

from

K

L

r

in Structural & civil via

\lambda=KL/r

λ = K L / r. K encodes fixity: 1 pinned, 0.5 fixed, 2 cantilever. 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

K = 1.000\,\mathrm{—}

L = 4.000\,\mathrm{m}

r = 6.000\,\mathrm{cm}

, the governing relation

\lambda=KL/r

yields

\lambda = 66.7\,\mathrm{—}

. A column, an effective length, a slenderness tick. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Effective-length factor K1 —Unbraced length L4 mRadius of gyration r6 cm

Outputs

Slenderness \lambda66.7 —

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Narration of this film

A column, an effective length, a slenderness tick.

Euler's critical stress is π² E / λ². Codes switch from elastic to inelastic at a limit λc.

Reading speed

Watch on YouTube