Modulus of Rigidity

By Aina Parasher|Updated : November 11th, 2022

Modulus of rigidity is the term of material where we know about the rigidness of any material and its stiffness. From the name of modulus of rigidity, we can understand that when any material is in its natural state it has some resistance against the applied forces.

Modulus of Rigidity PDF

The resistance is affected by pressure, force and temperature but after all the variables cause some resistance within the material. it means it's an inert property of a material where we can know how much rigidness material is having. the term that indicates this phenomenon is known as modulus of rigidity.

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What is the Modulus of Rigidity?

Modulus of rigidity or shear modulus is denoted as G, N, or C. Modulus of rigidity is defined as the factor of shear stress to shear strain. The modulus of rigidity is directly proportional to the modulus of elasticity; which is an elastic constant, it depends on the material's quality and properties.

As the modulus of elasticity or quality of material increases, modulus or rigidity increases proportionally. This is also helpful in designing shafts. Different types of materials have different moduli of rigidity.

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Modulus of Rigidity Formula

Modulus of rigidity is defined as the ratio of shear stress taken from a body to the shear strain as deformation in angle due to external load. In this, only shear stress has a unit, so the modulus of the rigidity unit is similar to the shear stress unit.

Modulus of Rigidity is the Ratio of

Modulus of Rigidity is the ratio of shear stress and shear strain as shown below:


Modulus of Rigidity Units

The unit of modulus of rigidity in the SI unit is termed Pascal (Pa). It is the same as that of Young's Modulous. Generally, it is taken as megapascal (MPa) or Giga Pascal (GPA). The dimensional term is [ M1L-1T-2]. Hence we can understand that the modulus of rigidity is the ratio of shear stress to shear strain in the term of GPa mostly. Here GPA is considered as 109 pascals or (1e+9) Pa.

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Modulus of Rigidity of Spring

The modulus of rigidity of spring is called the spring constant. It is denoted by K.

The modulus of rigidity has different values for different materials, as we discussed earlier, but the modulus of rigidity is zero for ideal gas. The modulus of rigidity is different from the flexural rigidity. An ideal gas particle has sufficient space,, and does not have any frictional resistance due to not presenting any frictional resistance, tangential stress is also zero.

Modulus of Rigidity Example

Modulus of rigidity differs from the material property or as well as material under environmental conditions. The examples shown in the table provided below are important for the GATE exam. Here, we consider the different materials having different values of modulus of rigidity.


Material Type

Modulus of Rigidity


Pure Aluminum

26 Gpa


Alloy Aluminum

26-30 Gpa



36-41 Gpa



36-44 Gpa



40-47 Gpa


Cast Iron

32-69 Gpa


Mild Steel

75-80 Gpa



0.0002-0.0001 Gpa

Relation Between Modulus of Elasticity and Modulus of Rigidity

The modulus of elasticity relates to different parameters like rigidity or shear modulus, bulk modulus and PoisPoisson'sio. When Poisson'Poisson'saries from 0 to 0.5, the modulus of rigidity is defined in the range of 33% of modulus of elasticity to 50% of modulus of elasticity; this shows the behaviour of the material.

The kind of relation modulus of elasticity with the modulus of rigidity is-


Relation Between Modulus of Rigidity and Bulk Modulus

The kind relation of the modulus of elasticity with bulk modulus is:


The relation between modulus of elasticity, bulk modulus, shear modulus and Poisson' ratio.

Modulus of Rigidity of Steel

The modulus of rigidity of steel exists in different metals, alloys, and subordinate metals such as mild steel, stainless steel, wrought steel etc. This modulus of rigidity of steel considers temperature, pressure, and other factors to determine how the modulus of rigidity changes with the variation of these factors. Here we consider steel’s steel'sy factors as 

  • Mild steel modulus of rigidity = approx 79 GPA
  • Stainless steel modulus of rigidity = approx 77 GPA

Important GATE Topics

Proof ResilienceSuperposition Theorem
Binary Search TreeWork Done By A Constant Force
Structural AnalysisMotion Under Gravity
Basic Signal OperationsIdeal Diode


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Modulus of Rigidity FAQs

  • When any material is subjected to load, the modulus of rigidity defines material rigidness under that load; this rigidness is considered as modulus of rigidity. In other words, it is the amount of force any material can handle, known as rigidity.

  • We know elasticity is inversely proportional to strain. In construction work steel is used majorly, in comparison to steel, plastic material has a large deformation strain, while steel is having minimum strain, hence steel is more elastic than any plastic material. 

  • In general, the heating affects the modulus of rigidity as when temperature increases the shear modulus decreases due to resistance to a high-temperature decrease and fluidity is increased.

  • We know the modulus of rigidity is directly proportional to the modulus of elasticity, and the modulus of elasticity is inversely proportional to strain. Strain indicates the deformation due to applying load.

  • Generally liquid has very little or nearby zero modulus of rigidity. The ideal liquid has properties such as no frictional forces and tangential stress hence it gains a stable form so its modulus of rigidity is considered zero.

  • Modulus of rigidity is formulated as when shear stress comes in the ratio with the shear strain that factor considered as modulus of rigidity.

    Modulus of rigidity = Shear Stress/Shear Strain

  • The ratio of shear stress which is symbolized as tau (ح) to the shear strain which is symbolized by (Φ). This ratio is expressed to know the rigidness capacity of the material.

  • Most general factors that affect the modulus of elasticity are material properties as which type of material that is and the second one the applicable force over the material. These factors can redefine the value of the modulus of rigidity.

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