This article explains how to calculate shear strain. Shear strain is the ratio of change in dimensions due to shear force to the original dimensions of the work piece.

**The change in dimensions is in the form of angular displacement. The shear force acts parallel to the plane of cross section. The plane shifts by some amount which is the linear displacement. The angle subtended by this displacement gives the angular displacement.**

**What is shear strain?**

As discussed in above section, shear strain is the ratio of change in dimensions due to shear force to the original dimensions of the work piece. It tells us about the deformation in form of percentage value.

**The work piece deforms in the plane parallel to the cross section of the work piece. It appears as if a layer of the work piece is moving.**

**Mathematically, shear strain can be given as-**

where,

gamma is shear strain

L is the original length of the work piece.

Image credits: Bob Clemintime, Uniaxial Shear Strain, CC BY-SA 4.0

**What is engineering strain?**

Engineering strain is a dimensionless number that represents the ratio of deformation of the work piece to its original dimension. It is directionally proportional to the amount of elongation along the length.

**Formula given in above section represents engineering strain. When the type of strain is not mentioned, it is understood that engineering strain is being referred. **

**What is true strain?**

True strain is the natural log of final length of work piece to the original length of the work piece. It shows the instantaneous strain being developed in the work piece.

**The strain developed in the work piece at a specific moment is called the true strain. It gives better representation of the material will behave under stress. **

Mathematically, true strain can be given as-

*∈ = ln x (A _{0}/A_{1})*

where,

eta represents the true strain

**How to calculate shear strain?**

Shear strain is the ratio of deformation length of the work piece to the original length of the work piece.

**Following steps are followed to calculate shear strain-**

- Measure the original length of the work piece.
- Apply shear stress on the material.
- Measure the angle of deformation.
- tan of angle of deformation gives the shear strain.

**How to calculate shear strain rate?**

Strain rate can be defined as the rate of change of strain with respect to time. Hence it is a function of time. It is the first differential of shear strain with respect to time.

**Shear strain rate can be calculated by the formula given below-**

*γ = dγ/dt*

where the dot on gamma represents shear strain rate

**Torsional stress**

A cylindrical bar experiences shear stress on its surface when subjected under torsional stress. Torsional stress is the stress which tends to twist the plane of cylindrical bar by some angle.

**Torsional stress can be given by the relation given below. The torsional formula can be used to find angle of twist, shear stress, shear strain torsional moment etc if adequate data is provided.**

**How to calculate shear strain in torsion**?

The formula discussed in above section can be used to find shear strain when the work piece is subjected to torsional stress.

**The formula for shear strain is discussed in above sections.**

**From the torsional formula, shear strain comes out to be-**

*γ = Rθ/L*

where,

theta is the angle of twist

L is the length of the work piece

**How to calculate shear strain from shear stress**?

**Shear modulus of rigidity is the ratio of shear stress to shear strain. To find the shear strain from shear stress, we divide shear stress by the shear modulus. We get,**

where,

G is the modulus of rigidity

**How to find angle of shear strain**?

The angle of shear strain is the angle by which the work piece is deformed.

**The angle of shear strain can be found using simple trigonometry. Angle of shear strain is the tan of angle subtended by the displacement, X and the original length, L. The adjacent side of the triangle formed acts as the original length of the work piece and the displacement acts as the opposite side in the triangle.**

The angle of shear can be found using the relation given below

where,

Phi is the angle of shear strain

**How to find average shear strain**?

The concept of shear strain comes into play when the work piece experiences strain from multiple directions.

**Lets consider shear strain occuring at two ends making angle of shear strain alpha and beta respectively.**

The average shear strain will become-

γ = α + β

**How to find angular shear strain**?

Angular shear strain is discussed in above section. The shear strain that describes the change in angle between two lines which were initially perpendicular is called as angular shear strain.

**Mathematically, it can be shown as**

γ = tanα

where,

alpha is the angular shear strain

**Shear stress in rectangular beam**

A rectangular beam is a beam whose cross section is rectangular or square. The amount of shear stress experienced by different points of cross section are different.

**This happens because the distance of the point from the application of force is variable for different points. So different amount of force acting on different points lead to variable shear stress.**

The shear stress is maximum at the centroid and minimum at the ends of the cross section. Shear stress is parabolically distributed across the rectangular cross section.

**What is bending stress?**

As the name suggests, this stress causes bending in work piece when a load is applied.

**Bending stress is the stress which acts normally on the work piece and causes bending along the longitudinal axis of the work piece.**

**What is bending stress formula?**

Similar to torsional formula. Bending formula also exists that makes it easy for us to find the bending stress. The formula for bending stress is given below

*σ _{b} = M_{y}/I*

**What is polar moment of inertia?**

Polar moment of inertia or second moment of inertia is a quantity used to explain the resistance to deformation caused by torsional stress.

**Mathematically, it can be given as-**

where,

J is the polar moment of inertia

**What is section modulus?**

Section modulus is used while designing flexural members of a structure or beams. It can be defined as the ratio of moment of inertia of the beam’s cross section about the neutral axis to the distance of extreme fiber from the center.

**Mathematically, it can be shown as-**

*Z = I/y _{max}*

where,

I is the moment of inertia

Z is the section of modulus