 # Shearing Stress

## What is Shearing Stress?

You may have tried to break a thick wooden stick, but failed in the attempt to do so. You may have tried to break it by stepping on it really hard. Here, what made it break is its shear stress.

Shear stress is the deforming force acting per unit area and in the direction perpendicular to the axle of the member. The impact of your load when you step in a wooden stick causes two types of stresses, these are:

• Bending Stress, which is parallel to the axle of the member also called flexural stress.

• Shear Stress, which acts in a direction perpendicular to the axle of the member.

### Do you know?

Shear means ‘to cut off’. When force is applied over the surface area of a rigid body (force acting in a direction parallel to the surface) then this force tries to cut off one part of the body from the other. As a result of this the body gets deformed and hence strain is produced (shear strain- the angular deflection of the body from its original position). Due to the rigidity of the body, it resists the deformation caused and a restoring force (equal and opposite to the applied force) is developed along the surface of the body as per Newton's 3rd law of motion). This restoring force of the body tends to oppose the shearing effect of the applied force. Thus shear stress is just an effect of shear strain.

### Shear Force Definition

It is a force that acts on a plane which passes through the body. The shear forces are unaligned and separate the structure into two different parts in opposite directions. The shear force acts in a perpendicular direction to the larger part of the body.

Shear force, in a beam, acts perpendicular to the longitudinal (x) axis. The beam's ability to resist shear force is much more important as compared to its ability to resist axial force. Axial force acts parallel to the longitudinal axis of the beam.

The given below figure represents a simple-supported beam of length L under a uniform load q:

This beam has the following support reactions:

Where,

Rl = is the reaction at the left end of the beam

Rr = is the reaction at the right end of the beam.

The shear forces at the end points of the beam are equivalent to the vertical forces of the support reactions. The shear force F(x) at any other point x, apart from the end points on the beam is calculated by using the shear force formula. This formula is:

F(x) = Rl – qx = qL/2 – qx = q(L/2 – x)

Where,

x = distance of the point from the left end of the beam.

Q = first moment of area in m^3

The shear stress acts in a direction parallel to that of the surface. Shear stress causes one object to slip over the other. It deforms the original shape of the object, like converting a rectangular shaped object into a parallelogram. It is the ratio of the applied force (F) to the cross-sectional area (A) of the structure/beam. Shear stress acts in a direction which is perpendicular to the normal stress. The shear stress is denoted by ‘Ԏ’.

Shear Stress Formula: Ԏ = F/A

Where,

F = force acting on the structure,

A = area of cross-section of the body.

Beams are also acted upon by transverse forces, which accounts for both bending moment M(x) and shear forces V(x)

Expression of distribution of shear stress in a body

Ԏxy = VQ/It

Where,

V = shear force in the cross section,

Q = First moment of area

I = moment of inertia of the area,

t = width of the section.

### Shear Stress Units

The units of shear stress are similar to that of any other type of stress. The unit for shear stress is N/m^2 or Pa (Pascal) in the SI system and lbf/ft^2 in the English system.

1. What is the difference between shear stress and shear strain?

The major difference between shear stress and shear strain is mentioned below:

• Shear stress Ԏ is the ratio of shear force to that of the area of the surface. It is tangential and is applied along the surface.

• Shear strain γ is the ratio of the length of transverse deformation occurring in the object to that of the perpendicular length in the plane of applied force. It is the tangent of the angle by which two perpendicular lines have changed with the applied shear force in a plane which is perpendicular to one of the lines.

2. Give some examples of Shear Stress.

When you cut something, you are actually applying shear stress on it. In case of stress exerted on a pipeline by a flowing fluid, the shear stress is exerted by a normal load to the soil from the top.

• While designing foundations and other structures, the Geo-technical engineer measures the shear stress of the soil in order to avoid failure under shear.

• The shear stress makes a sand castle collapse when someone just steps on it.

• Scissors exert lateral loads in an object, which causes shear stress and cuts it.

• Cutting of hair and paper involves shear stress.

• The opening of the screwed cap of a bottle or container requires shear stress.

• Painting, brushing teeth, soap, applying face cream, ointments, applying solutions on bodies etc involves shear stress.

• Walking and running also requires shear stress.

3. Explain shear stress in fluids

The movement of a fluid along a solid boundary exerts shear stress at that boundary. The no-slip condition states that the speed of the fluid with respect to that boundary is zero. At some height from the boundary, the speed of flow must be equal to that of the fluid. The region between these two points is called the boundary layer.

For Newtonian fluids, under laminar flow, the shear stress is proportional to the shear strain in the fluid where the viscosity of the fluid is a constant of proportionality. Viscosity is not constant for Non-Newtonian fluids. The shear stress is exerted into the boundary as a result of loss of velocity of fluid.

For a Newtonian fluid, the shear stress at any element of a surface, parallel to the flat plate at the point y is given by:

Ԏ(y) = μ du/dy

Where,

μ = dynamic viscosity of the flow;

u = flow velocity along the boundary;

y = height above the boundary.