CIV-25

Virtual work (axial bar)

Δ = N n L / (A E). Real force N, virtual n from a unit dummy load.

Reading speed

EnergyVirtual work

Governing equation

\Delta=\dfrac{N\, n\, L}{AE}

$N \in \left[-800, 800\right]$

where

N: Real axial (kN)
n: Virtual n (—)
L: Length (m)
A: Area (cm^2)
E: Modulus (GPa)
\Delta: Displacement (mm)

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

\Delta=\dfrac{N\, n\, L}{AE}

$N \in \left[-800, 800\right]$

$N = -800$ $\Delta = -8$

Along the curve

②

Domain of the variable — blue

N_{\min} = -800,\quad N_{\max} = 800

$N \in \left[-800, 800\right]$

$N = -800$ $N = -800$

Along the curve

③

Response curve — green

\Delta=\dfrac{N\, n\, L}{AE}

$N \in \left[-800, 800\right]$

$N = -800$ $\Delta = -8$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\Delta}{\mathrm{d}N}

$N \in \left[-800, 800\right]$

$N = -789.9$ $\Delta^{\prime} = 0.01$

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-25 — Virtual work (axial bar)) is the form associated with Virtual work. Working symbols: $N$ , $n$ , $L$ , $A$ , $E$ $\rightarrow$ $\Delta$ . Betti–Maxwell: external virtual work equals internal. One truss bar, one displacement.

Purpose

Purpose: compute

\Delta

from

N

n

L

A

E

in Structural & civil via

\Delta=\dfrac{N\, n\, L}{AE}

Δ = N n L / (A E). Real force N, virtual n from a unit dummy load. 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

N = 200.000\,\mathrm{kN}

n = 1.000\,\mathrm{—}

L = 4.000\,\mathrm{m}

A = 20.000\,\mathrm{cm^2}

E = 200.000\,\mathrm{GPa}

, the governing relation

\Delta=\dfrac{N\, n\, L}{AE}

yields

\Delta = 2.000\,\mathrm{mm}

. A bar, a dummy load, an elongation. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Real axial N200 kNVirtual n n1 —Length L4 mArea A20 cm^2Modulus E200 GPa

Outputs

Displacement \Delta2.000 mm

Reading speed

Watch on YouTube

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Narration of this film

A bar, a dummy load, an elongation.

Betti–Maxwell: external virtual work equals internal. One truss bar, one displacement.

Reading speed

Watch on YouTube