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Shear stress

 Shear stress

Common symbol(s): τ

SI unit:  pascalDerivations from other quantities: τ =  F  /  A

A shear stress, is applied to the top of the rectangle while the bottom is held in place.

This stress results in a strain, or deformation, changing the rectangle into a parallelogram.The area involved would be the top of the parallelogram.

A shear stress, denoted (Greek  tau!, is defined as the component of stress coplanar

with a material cross section. "hear stress arises from the force vector  component parallel 

to the cross section. #ormal stress, on the other hand, arises from the force vectorcomponent perpendicular  or antiparallel to the material cross section on which it acts.

Contents

• $ General shear stress

• % &ther forms of shear stress

o %.$ 'ure shear

o %.% eam shear

o %.) "emi*monoco+ue shear

o %. -mpact shear

o %. "hear stress in fluids

• ) easurement b0 shear stress sensorso ).$ 1iverging fringe shear stress sensor

o ).% icro*pillar shear*stress sensor

• "ee also

• 2eferences

• 3 45ternal links

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General shear stress

The formula to calculate average shear stress is6$7

where

= the shear stress8= the force applied8

= the cross*sectional area of material with area parallel to the applied force

vector.

Other forms of shear stress

Pure shear

'ure shear  stress is related to pure shear strain, denoted , b0 the following e+uation6%7

where is the shear modulus of the material, given b0

9ere is :oung;s modulus and is 'oisson;s ratio.

Beam shear

eam shear is defined as the internal shear stress of a beam caused b0 the shear forceapplied to the beam.

where

V  = total shear force at the location in +uestion8

Q = statical moment of area8t  = thickness in the material perpendicular to the shear8 I  = oment of -nertia of the entire cross sectional area.

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The beam shear formula is also knowns as <huravskii "hear "tress formula after  1mitrii

-vanovich <huravskii who derived it in $.6)767

Semi-monocoque shear

"hear stresses within a semi*monoco+ue structure ma0 be calculated b0 ideali>ing thecross*section of the structure into a set of stringers (carr0ing onl0 a5ial loads! and webs(carr0ing onl0 shear flows!. 1ividing the shear flow b0 the thickness of a given portion of

the semi*monoco+ue structure 0ields the shear stress. Thus, the ma5imum shear stress

will occur either in the web of ma5imum shear flow or minimum thickness.

Also constructions in soil can fail due to shear8 e.g., the weight of an earth*filled dam ordike ma0 cause the subsoil to collapse, like a small landslide.

Imact shear

The ma5imum shear stress created in a solid round bar sub?ect to impact is given as thee+uation

where

U  = change in kinetic energ08

G = shear modulus8

V  = volume of rod8

and

= mass moment of inertia8= angular speed.

Shear stress in flui!s

"ee also @iscosit0, ouette flow, 9agen*'oiseuille e+uation, 1epth*slope product, and

"imple shear 

An0 real fluids (li+uids and gases included! moving along solid boundar0 will incur a

shear stress on that boundar0. The no*slip condition67  dictates that the speed of the fluidat the boundar0 (relative to the boundar0! is >ero, but at some height from the boundar0

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the flow speed must e+ual that of the fluid. The region between these two points is aptl0

named the boundar0 la0er . Bor all  #ewtonian fluids in laminar flow the shear stress is

 proportional to the strain rate in the fluid where the viscosit0 is the constant of proportionalit0. 9owever for #on #ewtonian fluids, this is no longer the case as for these

fluids the viscosit0 is not constant. The shear stress is imparted onto the boundar0 as a

result of this loss of velocit0. The shear stress, for a #ewtonian fluid, at a surface element parallel to a flat plate, at the point 0, is given b0

where

is the d0namic viscosit0 of the fluid8

is the velocit0 of the fluid along the boundar08

is the height above the boundar0.

"pecificall0, the wall shear stress is defined as

-n case of wind, the shear stress at the boundar0 is called wind stress.

"easurement by shear stress sensors

Diver#in# frin#e shear stress sensor

This relationship can be e5ploited to measure the wall shear stress. -f a sensor could

directl0 measure the gradient of the velocit0 profile at the wall, then multipl0ing b0 thed0namic viscosit0 would 0ield the shear stress. "uch a sensor was demonstrated b0 A. A.

 #a+wi and C. . 2e0nolds.637 The interference pattern generated b0 sending a beam of

light through two parallel slits forms a network of linearl0 diverging fringes that seem to

originate from the plane of the two slits (see double*slit e5periment!. As a particle in afluid passes through the fringes, a receiver detects the reflection of the fringe pattern. The

signal can be processed, and knowing the fringe angle, the height and velocit0 of the

 particle can be e5trapolated. The measured value of wall velocit0 gradient is independentof the fluid properties and as a result does not re+uire calibration. 2ecent advancements

in the micro*optic fabrication technologies have made it possible to use integrated

diffractive optical element to fabricate diverging fringe shear stress sensors usable both inair and li+uid.

"icro-illar shear-stress sensor

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A further techni+ue is that of slender wall*mounted micro*pillars made of the fle5ible

 pol0mer '1", which bend in reaction to the appl0ing drag forces in the vicinit0 of the

wall. The sensor thereb0 belongs to the indirect measurement principles rel0ing on therelationship between near*wall velocit0 gradients and the local wall*shear stress.6D767