Shear Force

Unit 5 Cohesion

Shear Force

Juan Juan
13 min read
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Shear Force is the rotational, sliding force that arises when parallel internal layers of a material slide past one another, causing angular distortion without changing volume. It is the Mid Cohesion force in Superphysics’ Solid Relationalities — the rotational, structure-forming action at the intermediate scale between the aetherspace and space.

Unlike tensile/compression (linear/displacement), shear force is rotational and distortive — it twists layers sideways, creating angular deformation that binds the material into a coherent sheared structure.

The Equation: Shear Fundamentals

Shear Stress (Force per Area)

τ=FA \tau = \frac{F}{A}

Where:

SymbolMeaningUnit
( \tau )Shear stress( \text{Pa} = \text{N}/\text{m}^2 )
( F )Applied tangential force( \text{N} )
( A )Area parallel to force( \text{m}^2 )

Shear Strain (Angular Deformation)

γ=Δxh=tan⁡θ \gamma = \frac{\Delta x}{h} = \tan\theta

Where:

SymbolMeaningUnit
( \gamma )Shear straindimensionless
( \Delta x )Lateral displacement( \text{m} )
( h )Height of material( \text{m} )
( \theta )Shear angle( \text{rad} ) or ( \degree )

Shear Modulus (Rotational Cohesion)

τ=Gγ \tau = G \gamma

Where:

SymbolMeaningUnit
( G )Shear modulus (modulus of rigidity)( \text{Pa} )
( \tau )Shear stress( \text{Pa} )
( \gamma )Shear straindimensionless

Relationship to Young’s Modulus

G=E2(1+ν) G = \frac{E}{2(1 + \nu)}

Where:

SymbolMeaningUnit
( E )Young’s modulus( \text{Pa} )
( \nu )Poisson’s ratiodimensionless

Shear Force in Beams (Internal Distribution)

V=dMdx V = \frac{dM}{dx}

Where:

SymbolMeaningUnit
( V )Shear force( \text{N} )
( M )Bending moment( \text{N} \cdot \text{m} )
( x )Position along beam( \text{m} )

Shear Energy (Stored Cohesion)

U=12Gγ2Vol U = \frac{1}{2}G \gamma^2 V_{\text{ol}}

Where ( V_{\text{ol}} ) is volume. Shear deformation stores elastic energy through rotational distortion.

Key Properties of Shear Force

PropertyExplanation
RotationalLayers slide and rotate past one another
DistortiveChanges shape without changing volume
CohesiveBinds layers into a coherent sheared structure
AngularProduces angular deformation ( \gamma )
Material-dependent( G ) varies widely (steel ~80 GPa, rubber ~0.001 GPa)
Temperature-dependentHigher ( T ) → lower ( G ) (generally)
Volume-preservingNo change in volume under pure shear

Shear Strength (Failure Criterion)

τmax=FfailA \tau_{\text{max}} = \frac{F_{\text{fail}}}{A}

Where ( \tau_{\text{max}} ) is the maximum shear stress before failure. Ductile materials fail by shearing; brittle materials fail by tension.

Mohr’s Circle (Stress Transformation)

τmax=σ1−σ22 \tau_{\text{max}} = \frac{\sigma_1 - \sigma_2}{2}

Where ( \sigma_1 ) and ( \sigma_2 ) are principal stresses. Maximum shear occurs at 45° to principal axes.

Shear Wave Propagation (Rotational Cohesion in Motion)

vs=Gρ v_s = \sqrt{\frac{G}{\rho}}

Where:

SymbolMeaningUnit
( v_s )Shear wave velocity( \text{m}/\text{s} )
( G )Shear modulus( \text{Pa} )
( \rho )Density( \text{kg}/\text{m}^3 )

Real-Life Examples

ExampleHow Shear Force Works
1. Scissors cutting paperBlades apply opposing shear forces, sliding layers apart
2. Punching a holeShear stress exceeds material strength, cutting through
3. Bolts in a jointBolts resist shear as plates try to slide past each other
4. Earthquake fault slipShear stress releases suddenly, causing seismic displacement
5. Metal cuttingCutting tools shear metal along a plane
6. Playing cardsStacked cards slide and rotate under shear
7. Guacamole mixingSpatula applies shear, distorting avocado layers
8. Wind on a buildingHorizontal wind creates shear stress on structural frames
9. Rivets in aircraftRivets resist shear between fuselage panels
10. Seismic S-wavesShear waves propagate through Earth’s interior

Summary

QuestionAnswer
What is shear force?Rotational sliding force causing angular distortion between layers
What law describes it?Shear Modulus: ( \tau = G \gamma ); Beam Shear: ( V = \frac{dM}{dx} )
What type of force?Mid Cohesion (rotational, Rule 2 + 4)
What drives it?Tangential forces on parallel layers; angular distortion
Real examples?Scissors, bolts, earthquakes, metal cutting, wind on buildings
What’s its displacement counterpart?Tensile/Compression (Mid Displacement)

Bottom line: Shear force is the rotational, angular-distorting slide of parallel layers — the Mid Cohesion force that pairs with tensile/compression’s linear axial push-pull in Superphysics’ Solid Relationalities.

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