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A shear stress, Failed to parse (Missing texvc executable; please see math/README to configure.): \tau\, is applied to the top of the square while the bottom is held in place. This stress results in a strain, or deformation, changing the square into a parallelogram. The area involved would be the top of the parallelogram.

A shear stress, denoted Failed to parse (Missing texvc executable; please see math/README to configure.): \tau\,

(Greek: tau), is defined as the component of stress coplanar with a material cross section.  Shear stress arises from the force vector component parallel to the cross section.  Normal stress, on the other hand, arises from the force vector component perpendicular or antiparallel to the material cross section on which it acts.

General shear stress[link]

The formula to calculate average shear stress is^{[citation needed]}:

Failed to parse (Missing texvc executable; please see math/README to configure.): \tau = {F \over A},

where

Failed to parse (Missing texvc executable; please see math/README to configure.): \tau

= the shear stress;

Failed to parse (Missing texvc executable; please see math/README to configure.): F

= the force applied;

Failed to parse (Missing texvc executable; please see math/README to configure.): A

= the cross-sectional area of material with area parallel to the applied force vector.

Other forms of shear stress[link]

Pure shear[link]

Pure shear stress is related to pure shear strain, denoted Failed to parse (Missing texvc executable; please see math/README to configure.): \gamma , by the following equation:^[1]

Failed to parse (Missing texvc executable; please see math/README to configure.): \tau = \gamma G\,

where Failed to parse (Missing texvc executable; please see math/README to configure.): G

is the shear modulus of the material, given by

Failed to parse (Missing texvc executable; please see math/README to configure.): G = \frac{E}{2(1+\nu)}.

Here Failed to parse (Missing texvc executable; please see math/README to configure.): E

is Young's modulus and Failed to parse (Missing texvc executable; please see math/README to configure.): \nu
is Poisson's ratio.

Beam shear[link]

Beam shear is defined as the internal shear stress of a beam caused by the shear force applied to the beam.

Failed to parse (Missing texvc executable; please see math/README to configure.): \tau = {VQ \over It},

where

V = total shear force at the location in question;

Q = statical moment of area;

t = thickness in the material perpendicular to the shear;

I = Moment of Inertia of the entire cross sectional area.

This formula is also known as the Jourawski formula.^[2]

Semi-monocoque shear[link]

Shear stresses within a semi-monocoque structure may be calculated by idealizing the cross-section of the structure into a set of stringers (carrying only axial loads) and webs (carrying only shear flows). Dividing the shear flow by the thickness of a given portion of the semi-monocoque structure yields the shear stress. Thus, the maximum shear stress will occur either in the web of maximum shear flow or minimum thickness.

Also constructions in soil can fail due to shear; e.g., the weight of an earth-filled dam or dike may cause the subsoil to collapse, like a small landslide.

Impact shear[link]

The maximum shear stress created in a solid round bar subject to impact is given as the equation:

Failed to parse (Missing texvc executable; please see math/README to configure.): \tau=2\left({UG \over V}\right)^{1 \over 2},

where

U = change in kinetic energy;

G = shear modulus;

V = volume of rod;

and

Failed to parse (Missing texvc executable; please see math/README to configure.): U = U_{rotating}+U_{applied} \,;

Failed to parse (Missing texvc executable; please see math/README to configure.): U_{rotating} = {1 \over 2}I\omega^2 \,;

Failed to parse (Missing texvc executable; please see math/README to configure.): U_{applied} = T \theta_{displaced} \,;

Failed to parse (Missing texvc executable; please see math/README to configure.): I \,

= mass moment of inertia;

Failed to parse (Missing texvc executable; please see math/README to configure.): \omega \,

= angular speed.

Shear stress in fluids[link]

Any real fluids (liquids and gases included) moving along solid boundary will incur a shear stress on that boundary. The no-slip condition^[3] dictates that the speed of the fluid at the boundary (relative to the boundary) is zero, but at some height from the boundary the flow speed must equal that of the fluid. The region between these two points is aptly named the boundary layer. For all Newtonian fluids in laminar flow the shear stress is proportional to the strain rate in the fluid where the viscosity is the constant of proportionality. However for Non Newtonian fluids, this is no longer the case as for these fluids the viscosity is not constant. The shear stress is imparted onto the boundary as a result of this loss of velocity. The shear stress, for a Newtonian fluid, at a surface element parallel to a flat plate, at the point y, is given by:

Failed to parse (Missing texvc executable; please see math/README to configure.): \tau (y) = \mu \frac{\partial u}{\partial y}

where

Failed to parse (Missing texvc executable; please see math/README to configure.): \mu

is the dynamic viscosity of the fluid;

Failed to parse (Missing texvc executable; please see math/README to configure.): u

is the velocity of the fluid along the boundary;

Failed to parse (Missing texvc executable; please see math/README to configure.): y

is the height above the boundary.

Specifically, the wall shear stress is defined as:

Failed to parse (Missing texvc executable; please see math/README to configure.): \tau_\mathrm{w} \equiv \tau(y=0)= \mu \left.\frac{\partial u}{\partial y}\right|_{y = 0}~~.

In case of wind, the shear stress at the boundary is called wind stress.

Measurement by shear stress sensors[link]

Diverging fringe shear stress sensor[link]

This relationship can be exploited to measure the wall shear stress. If a sensor could directly measure the gradient of the velocity profile at the wall, then multiplying by the dynamic viscosity would yield the shear stress. Such a sensor was demonstrated by A. A. Naqwi and W. C. Reynolds.^[4] The interference pattern generated by sending a beam of light through two parallel slits forms a network of linearly diverging fringes that seem to originate from the plane of the two slits (see double-slit experiment). As a particle in a fluid 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 velocity of the particle can be extrapolated. The measured value of wall velocity gradient is independent of the fluid properties and as a result does not require calibration. Recent 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 in air and liquid.

Micro-pillar shear-stress sensor[link]

A further technique recently proposed is that of slender wall-mounted micro-pillars made of the flexible polymer PDMS, which bend in reaction to the applying drag forces in the vicinity of the wall. The deflection of the pillar tips from a reference position is detected optically and serves as a representative of the wall-shear stress. It allows the instantaneous detection of the streamwise and spanwise wall-shear stress distribution in turbulent flow up to high Reynolds numbers.^[5][6]

See also[link]

• Direct shear test
• Shear rate
• Shear strain
• Shear strength
• Shear and moment diagrams
• Tensile stress
• Triaxial shear test
• Critical resolved shear stress

References[link]

External links[link]

• The second page of this brochure explains the concept of the diverging fringe shear stress sensor mentioned above.