Shear stress
Overview

A shear stress, denoted (Greek
Greek alphabet
The Greek alphabet is the script that has been used to write the Greek language since at least 730 BC . The alphabet in its classical and modern form consists of 24 letters ordered in sequence from alpha to omega...

: tau
Tau
Tau is the 19th letter of the Greek alphabet. In the system of Greek numerals it has a value of 300.The name in English is pronounced , but in modern Greek it is...

), is defined as the component of stress
Stress (physics)
In continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...

coplanar with a material cross section. Shear stress arises from the force vector component parallel
Parallel
-Mathematics and science:* Parallel , an imaginary east-west line circling a globe* Parallel circuits, as opposed to series* Parallel * Parallel evolution* Parallel transport* Parallel of declination, used in astronomy-Computing:...

to the cross section. Normal stress, on the other hand, arises from the force vector component perpendicular
Perpendicular
In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...

or antiparallel
Antiparallel (mathematics)
-Definitions:Given two lines m_1 \, and m_2 \,, lines l_1 \, and l_2 \, are anti-parallel with respect to m_1 \, and m_2 \, if \angle 1 = \angle 2 \,....

to the material cross section on which it acts.
The formula to calculate average shear stress is:

where
= the shear stress; = the force applied; = the cross-sectional area of material with area parallel to the applied force vector.
Beam shear is defined as the internal shear stress of a beam caused by the shear force applied to the beam.

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
Second moment of area
The second moment of area, also known as the area moment of inertia, moment of inertia of plane area, or second moment of inertia is a property of a cross section that can be used to predict the resistance of beams to bending and deflection, around an axis that lies in the cross-sectional plane...

of the entire cross sectional area.

This formula is also known as the Jourawski formula.
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 flow
Shear flow
The term shear flow is used in solid mechanics as well as in fluid dynamics. Loosely speaking, shear flow is defined as:* the gradient of a shear stress force through the body ;...

s).
Discussions
Encyclopedia
A shear stress, denoted (Greek
Greek alphabet
The Greek alphabet is the script that has been used to write the Greek language since at least 730 BC . The alphabet in its classical and modern form consists of 24 letters ordered in sequence from alpha to omega...

: tau
Tau
Tau is the 19th letter of the Greek alphabet. In the system of Greek numerals it has a value of 300.The name in English is pronounced , but in modern Greek it is...

), is defined as the component of stress
Stress (physics)
In continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...

coplanar with a material cross section. Shear stress arises from the force vector component parallel
Parallel
-Mathematics and science:* Parallel , an imaginary east-west line circling a globe* Parallel circuits, as opposed to series* Parallel * Parallel evolution* Parallel transport* Parallel of declination, used in astronomy-Computing:...

to the cross section. Normal stress, on the other hand, arises from the force vector component perpendicular
Perpendicular
In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...

or antiparallel
Antiparallel (mathematics)
-Definitions:Given two lines m_1 \, and m_2 \,, lines l_1 \, and l_2 \, are anti-parallel with respect to m_1 \, and m_2 \, if \angle 1 = \angle 2 \,....

to the material cross section on which it acts.

## General shear stress

The formula to calculate average shear stress is:

where
= the shear stress; = the force applied; = the cross-sectional area of material with area parallel to the applied force vector.

### Beam shear

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

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
Second moment of area
The second moment of area, also known as the area moment of inertia, moment of inertia of plane area, or second moment of inertia is a property of a cross section that can be used to predict the resistance of beams to bending and deflection, around an axis that lies in the cross-sectional plane...

of the entire cross sectional area.

This formula is also known as the Jourawski formula.

### Semi-monocoque shear

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 flow
Shear flow
The term shear flow is used in solid mechanics as well as in fluid dynamics. Loosely speaking, shear flow is defined as:* the gradient of a shear stress force through the body ;...

s). 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
Dam
A dam is a barrier that impounds water or underground streams. Dams generally serve the primary purpose of retaining water, while other structures such as floodgates or levees are used to manage or prevent water flow into specific land regions. Hydropower and pumped-storage hydroelectricity are...

or dike may cause the subsoil to collapse, like a small landslide
Landslide
A landslide or landslip is a geological phenomenon which includes a wide range of ground movement, such as rockfalls, deep failure of slopes and shallow debris flows, which can occur in offshore, coastal and onshore environments...

.

### Impact shear

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

where
U = change in kinetic energy;
G = shear modulus;
V = volume of rod;

and
= mass moment of inertia; = angular speed.

### Shear stress in fluids

Any real fluids (liquid
Liquid
Liquid is one of the three classical states of matter . Like a gas, a liquid is able to flow and take the shape of a container. Some liquids resist compression, while others can be compressed. Unlike a gas, a liquid does not disperse to fill every space of a container, and maintains a fairly...

s and gas
Gas
Gas is one of the three classical states of matter . Near absolute zero, a substance exists as a solid. As heat is added to this substance it melts into a liquid at its melting point , boils into a gas at its boiling point, and if heated high enough would enter a plasma state in which the electrons...

es included) moving along solid boundary will incur a shear stress on that boundary. The no-slip condition
No-slip condition
In fluid dynamics, the no-slip condition for viscous fluids states that at a solid boundary, the fluid will have zero velocity relative to the boundary.The fluid velocity at all fluid–solid boundaries is equal to that of the solid boundary...

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
Boundary layer
In physics and fluid mechanics, a boundary layer is that layer of fluid in the immediate vicinity of a bounding surface where effects of viscosity of the fluid are considered in detail. In the Earth's atmosphere, the planetary boundary layer is the air layer near the ground affected by diurnal...

. For all Newtonian fluid
Newtonian fluid
A Newtonian fluid is a fluid whose stress versus strain rate curve is linear and passes through the origin. The constant of proportionality is known as the viscosity.-Definition:...

s in laminar flow
Laminar flow
Laminar flow, sometimes known as streamline flow, occurs when a fluid flows in parallel layers, with no disruption between the layers. At low velocities the fluid tends to flow without lateral mixing, and adjacent layers slide past one another like playing cards. There are no cross currents...

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
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher 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:

where is the dynamic viscosity of the fluid; is the velocity of the fluid along the boundary; is the height above the boundary.

Specifically, the wall shear stress is defined as:

In case of wind
Wind
Wind is the flow of gases on a large scale. On Earth, wind consists of the bulk movement of air. In outer space, solar wind is the movement of gases or charged particles from the sun through space, while planetary wind is the outgassing of light chemical elements from a planet's atmosphere into space...

, the shear stress at the boundary is called wind stress
Wind stress
In physical oceanography and fluid dynamics, the wind stress is the shear stress exerted by the wind on the surface of large bodies of water – such as oceans, seas, estuaries and lakes. It is the force component parallel to the surface, per unit area, as applied by the wind on the water surface...

.

### Diverging fringe shear stress sensor

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. 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
Double-slit experiment
The double-slit experiment, sometimes called Young's experiment, is a demonstration that matter and energy can display characteristics of both waves and particles...

). 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

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.

• Direct shear test
Direct Shear Test
A direct shear test also known as shearbox test is a laboratory or field test used by geotechnical engineers to measure the shear strength properties of soil or rock material, or of discontinuties in soil or rock masses.- Soil :...

• Shear rate
• Shear strain
• Shear strength
Shear strength
Shear strength in engineering is a term used to describe the strength of a material or component against the type of yield or structural failure where the material or component fails in shear. A shear load is a force that tends to produce a sliding failure on a material along a plane that is...

• Shear and moment diagrams
• Tensile stress
• Triaxial shear test
• Critical resolved shear stress
Critical resolved shear stress
Critical resolved shear stress is the component of shear stress, resolved in the direction of slip, necessary to initiate slip in a grain. It is a constant for a given crystal....