# Shear Stress Formula and Applications

Contents

Shear stress is a fundamental concept in physics and engineering that measures a material’s resistance to deformation under a parallel force. When a shear force is applied, it causes layers of the material to slide, and the rate of deformation is determined by the material’s shear modulus.

## Average Shear Stress Equation

Following is the formula to calculate Average Shear Stress:

τ = Force (F) / Area (A)

where,
τ = Shear stress;
F = Force applied;
A = Cross-sectional area of material with area perpendicular to the applied force vector;

## Beam Shear Stress Equation

Beam shear is the internal shear stress of a beam which is caused by the shear force that is applied to the beam.

τ = VQ / (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 equation is also known as the Jourawski formula.

## What is Semi-monocoque shear?

Shear stresses in a semi-monocoque structure are determined by analyzing the cross-section as a combination of stringers and webs, with shear stress calculated by dividing shear flow by the structure’s thickness. Maximum shear stress occurs in the web with the highest shear flow or the thinnest section. In soil structures, shear failure can happen, such as when the weight of an earth-filled dam causes subsoil collapse. Impact shear stress in a solid round bar is calculated using a specific equation which is as follows:

Jourawski Formula = 2 (UG/V)(1/2)

where,

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

and,

U = Urotating +Uapplied ;

Urotating = (1/2)Iω2 ;

Uapplied = Tθdisplaced ;

where,

I = Mass moment of inertia ;
ω = Angular speed.

## Shear Stress equation of Fluids

When real fluids (including liquids and gases) flow along a solid boundary, they create shear stress on that boundary. The “no-slip” rule states that the fluid’s speed at the boundary is zero, but at a certain distance from the boundary, the flow speed matches that of the fluid. This in-between region is known as the boundary layer. In laminar flow of Newtonian fluids, shear stress is directly related to the strain rate, with viscosity as the constant factor.

However, for Non-Newtonian fluids, viscosity varies, and this leads to the transfer of shear stress to the boundary due to changes in velocity. In the case of a Newtonian fluid, the shear stress at a surface element parallel to a flat plate, at a point y, can be expressed as:

τ(y) = μ (∂u/∂y)

where,
μ = Dynamic viscosity of the fluid;
u = Velocity of the fluid along the boundary;
y = Height above the boundary.

## Wall Shear Stress Equation

τw = τ (y=0) = μ (∂u/∂y)y=0