Shear stress (often denoted by τ, Greek: tau) is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. Normal stress, on the other hand, arises from the force vector component perpendicular to the material cross section on which it acts.
General shear stress The formula to calculate average shear stress τ or force per unit area is
τ = F A , {\displaystyle \tau ={F \over A},} where F is the force applied and A is the cross-sectional area.
Other forms
Wall shear stress Wall shear stress expresses the retarding force (per unit area) from a wall in the layers of a fluid flowing next to the wall. It is defined as: τ w := μ ∂ u ∂ y | y = 0 , {\displaystyle \tau _{w}:=\mu \left.{\frac {\partial u}{\partial y}}\right|_{y=0},} where μ is the dynamic viscosity, u is the flow velocity, and y is the distance from the wall. It is used, for example, in the description of arterial blood flow, where there is evidence that it affects the atherogenic process.
Pure Pure shear stress is related to pure shear strain, denoted γ, by the equation τ = γ G , {\displaystyle \tau =\gamma G,} where G is the shear modulus of the isotropic material, given by G = E 2 ( 1 + ν ) . {\displaystyle G={\frac {E}{2(1+\nu )}}.} Here, E is Young's modulus and ν is Poisson's ratio.
Beam shear Beam shear is defined as the internal shear stress of a beam caused by the shear force applied to the beam: τ := f Q I b , {\displaystyle \tau :={\frac {fQ}{Ib}},} where
The beam shear formula is also known as Zhuravskii shear stress formula after Dmitrii Ivanovich Zhuravskii, who derived it in 1855.
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 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. Constructions in soil can also 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 The maximum shear stress created in a solid round bar subject to impact is given by the equation τ = 2 U G V , {\displaystyle \tau =2{\sqrt {\frac {UG}{V}}},} where
Furthermore, U = Urotating + Uapplied, where
Shear stress in fluids
Any real fluids (liquids and gases included) moving along a solid boundary will incur a shear stress at that boundary. The no-slip condition dictates that the speed of the fluid at the boundary (relative to the boundary) is zero, although at some height from the boundary, the flow speed must equal that of the fluid. The region between these two points is 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. For non-Newtonian fluids, the viscosity is not constant. The shear stress is imparted onto the boundary as a result of this loss of velocity. For a Newtonian fluid, the shear stress at a surface element parallel to a flat plate at the point y is given by τ ( y ) = μ ∂ u ∂ y , {\displaystyle \tau (y)=\mu {\frac {\partial u}{\partial y}},} where
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