Shear Properties of Materials; Shear Stress-Strain

There can be stresses on materials and the strength of materials we must define according to these stress values. So there are strong values are defined for materials that are generally used in engineering calculations. So, two of them; tensile stress-strainand compression stress-strain characteristics of materials are explained in Mechanicalland. Also, in this article, we will explain the third one called Shear Properties and Shear characteristics of materials. It is very important to know the shear stresses and shear forces.

What are Shear Forces and Stresses?

Shear stress illustration.
Shear stress illustration.

Shear stress can be illustrated as above that there are opposite-sided forces that act on stress elements from above and below. This stress causes deformation as shown above called torsion.

The formula of this stress;

Shear stress formula.

T = Shear stress(MPa)(lb/in^2)
F = Opposite force value(N)(lb)
A = Area(mm^2)(in^2)
Also, we have a strain value as;

Shear strain value.

γ= Shear strain(mm/mm)(in/in)
δ= Deflection on element(mm)(in)
b = Orthogonal distance to deflection zone(mm)(in)
Like other test methods, stress-strain also has a stress test called Torsion Test.

Torsion test.
Torsion test.

Also, in the torsion test, a torque is applied to the specimen. A shear stress-strain curve is obtained with this test.

Stress-strain curve.
Shear stress-strain curve.

If we take a look at the shear stress-strain curve above, there is an elastic region as in tensile stress-strain curves. Furthermore, after the yield point of this stress, plastic deformation occurs. So, the plastic region represents the plastic deformation of material used as a specimen in Torsion tests.

The formula that derived from Torsion tests;

Torsion test formula.

τ= Shear stress at the specimen(MPa)(lb/in^2)

T = Applied torque(

R = Radius of the specimen(mm)(in)

t = Thickness of the specimen(mm)

The shear strength can be obtained from these standardized tests. This strength is generally lesser than the tensile stresses of materials.

Also, this is the general logic of the stress-strain characteristics of materials in engineering.

How to Calculate Shear Stresses on Different Cross Sections?

Shear stresses are the result of the vertical bending forces, acting from the perpendicular direction of the part axis. And also, we want to calculate the shear stress on a cross-section that this force acts upon it. So, it has a very basic principle. 

Firstly, you need to understand the neutral plane on this cross-section. In certain cross-sections, the neutral axis or neutral plane passes through the center of gravity of the cross-sectional area. And, you want to calculate this stress that occurs at a certain point on this cross-section, because of this force acting on this cross-section. 

So, take a look at the cross-section example above. Also, there is a shear force acting upon this cross-section say it is ‘F’. 

CD line is the line that contains the point that we want to calculate this stress. Also, you can see that the neutral axis is very straightforward above. 

And also the center of gravity of the area above the CD line is shown with ‘G’. Also, the distance between this ‘G’ and the neutral axis is denoted as ‘y(prime)’. 

Furthermore, the distance between the CD line and the neutral axis is denoted as ‘y’. 

Calculate the Moment of Inertia

Firstly, you need to calculate the moment of inertia of the whole cross-section concerning the neutral axis. So, you will use this moment of inertia value in this stress calculation. 

Calculate the Area

Then you need to calculate the area above the line which contains the point that you want to calculate shear stress(which is CD in the example above). 

Calculate the Moment Area

Then you need to multiply this area by the distance(y(prime)) of the center of gravity(which is G in the example above) of this area to the neutral axis. This will give you the first moment of the area.

Put the Values into the Equation to Find Shear Stress

Shear stress equation.

If you put the values that you find, you will find this stress on that line. Also, here, ‘b’ is the thickness of the part or cross-section which is perpendicular to shear force.

Shear Stress Calculators for Different Cross Sections

In engineering calculations, these stresses are very important aspects. Furthermore, shear stresses must be calculated neatly to see whether the designed system is safe or not. In general, engineers calculate the maximum of these stresses that occur on different cross-sections. 

Maximum Shear Stress for Rectangular Cross-Sections

‘b’ is the width of the rectangular cross-sectional area and ‘d’ is the height of the rectangular cross-sectional area. Also, shear force ‘V’ is acting vertically to this cross-section, which is perpendicular to ‘b’. In this case, you can calculate the average stress by using this calculator.

Maximum Shear Stress In Rectangular Cross Section

To use this calculator, enter the required values then click on the ‘Calculate!’ button. To make another calculation, click on the ‘Reset’ button then re-enter the values.

Maximum Shear Stress for Circular Cross-Sections

Consider the same force acting upon the circular cross-section. In this case, you can calculate the maximum shear stress via this calculator; 

Maximum Shear Stress On Circular Cross Section

Here, ‘r’ is the radius of the circular cross-section and the ‘V’ is the shear force. 

Maximum Shear Stress for Triangular Cross-Sections

Consider the height of the triangular cross-section is ‘H’ and the base length of the triangular cross-section is ‘B’ Then the maximum stress is; 

Maximum Shear Stress On Triangular Cross Section

Here, ‘r’ is the radius of the circular cross-section and the ‘V’ is the shear force. 

Maximum Shear Stress for Hollow Circular Cross-Sections

And also, for these tubular cross-sections, consider the thickness of the tubular cross-section as ‘t’ and the radius of the cross-section is ‘r’. So the maximum stress that occurs on this cross-section can be calculated via this calculator; 

Maximum Shear Stress On Hollow Tubular Cross Section

Maximum Shear Stress for I-Beams

You can see a typical I-beam cross-section. There are dimensional definitions of this cross-section. There can be these stresses on this cross-section because of the force application vertically on that beam cross-section.

Say ‘F’ for this force. Then you can use the calculator below to calculate the maximum shear stress on your I-Beam.

Firstly, you need to calculate the area moment of inertia of this I-cross section. After this calculation, you can use the calculator below to calculate maximum shear stress;

Contact Stress Calculator For Spur Gear Mates

To use this calculator, enter the required values inside the brackets. Then click on the ‘Calculate!’ button to calculate the maximum stress. If you want to make another calculation, click on the ‘Reset’ button then re-enter all values. 

And then, enter the ‘d’ and ‘D’ values as shown by the illustration above. ‘F’ is the shear force and ‘I’ is the area moment of inertia of the I cross-section. 

Also, please take care of the units. You need to use consistent sets of units to obtain correct results from this calculator. 

When you calculate the maximum shear stresses on your I-beam, you need to compare this result with the maximum allowable stress value of your material. Also, you need to consider the safety factor.

Maximum Shear Stress Failure Theory

So, the maximum shear stress theory compares the stress that occurs in a uniaxial tension loading condition with the maximum shear stress at the yield point of the material.

Consider a situation where a stress element undergoes stresses like this;

Shear force elements.
Uniaxial tension and shear stress in a stress element(Image Source: D. K. Singh – Strength of Materials-Springer, 2020, pg.438).

As you see above, the stress element undergoes tensile stresses both in the ‘X’ and ‘Y’ directions, and these stresses are on the positive side.

Furthermore, as you know from the principal stress calculations, principal stresses can be found with these stresses.

Principal stresses.

Maximum and minimum principal stresses can be found with this equation. Once you calculate the major and minor principal stresses, you can make your comparisons;

Principal stresses.

According to these equations of Tresca’s theory, if the safety plane is drawn,

Sign Convention of These Forces 

So, shear forces are a very important aspect of most engineering mechanics problems. Also, to show the sign convention of these forces, consider that a beam is loaded with vertical forces. 

Sign conventions of shear forces.
Illustration of the sign of shear forces on a beam(Image Source: D. K. Singh – Strength of Materials-Springer-2020).

If the right side of the shear force application on the beam is tending to move upwards, the shear force is considered positive. Otherwise, it is considered negative. So, the situation on the left side above is the positive one and on the right side is the negative one. 

Sign Convention of Bending Moments

We will also consider a piece of beam that undergoes a bending moment. We will define the sign of bending moment according to the behavior of the beam under this moment. 

Sign conventions of bending.
Sign conventions of the bending moment on a beam(Image Source: D. K. Singh – Strength of Materials-Springer-2020).

As you see above, there are two situations of bending on beams. On the left side, the beam is bent from concave to upward. In this situation, the bending moment is positive. Otherwise, which is like on the left, the bending moment is considered negative. 

How to Draw Shear Force Diagrams?

Shear force diagrams are generally a prerequisite for bending moment diagrams. But first of all, you need to understand the sign conventions of these forces and bending moments. 

Force Equilibrium

On simply supported beams or cantilevered beams, these forces generally occur radially(perpendicularly) to beam orientation. Firstly, you need to build the force equilibrium in the application direction of shear forces. This will give you the reaction forces are the supports or cantilevers. 

After finding all reaction forces on all supports, you can start the draw these force diagrams on that beam.

Drawing Shear Force Diagram

In general, application, drawing shear force diagrams starts from left to right. 

So, the shear force diagram starts with the reaction of the left-most side of the beam, which can be simply supported or cantilevered. Furthermore, take a look at the very basic simply supported example below.

Cantilevered beam.
Cantilevered beam with a single shear force(Image Source: D. K. Singh – Strength of Materials-Springer, 2020, pg.149)

As you see in the example above, there is a simple force application at the right edge of a cantilevered beam. And also, there must be a positive reaction on the cantilevered side which is the reaction of the left side.

So, the force diagram is drawn with this reaction for up to the shear force application placed on the right side. Also, this is the general rule of the force diagram drawing. You need to start with the left-most force or reaction to draw the shear force diagram.

Beam with force.
Image Source: D. K. Singh – Strength of Materials-Springer, 2020, pg.162

In the example above, if you calculate the reaction force at the left-side simple support. So, the force diagram started with this reaction.

What you need to do is when you face a force or reaction like above, you need to add this reaction into the diagram, and you need to go on with the new value of the force diagram.

So check the example above to see that diagram turned to the opposite side, after the shear force application point. This is another important rule.

Evenly Distributed Load

If your system has an evenly distributed load, the effect of this evenly distributed load is; 

  • Furthermore, if it has the same direction as the force diagram, it has a constant increasing effect on the diagram, which will be the inclined shape. 
  • And also, if it has an opposite direction with the force, it has a constantly decreasing effect as inclined. 
Evenly distributed load.
Image Source:D. K. Singh – Strength of Materials-Springer, 2020, pg.152

Also, you can understand the general rule of it. So the reaction force is calculated for the left cantilevered side, and the evenly distributed load has a decreasing effect on the shear force diagram. 

Uniformly Varying Load

Also, the actual effect of this kind of load is the same as the evenly distributed load. But the increment and decrement will have a second-order curve form. Check the example below. 

You can see the second-order decrement on the force diagram. 

Explanation of The Shear Strain Energy Calculations

To explain these kinds of things, we require some theoretical assumptions. 

Think about a box that undergoes force ‘P’ from one side. Because of this shear force, we calculate there will be shear stress-induced, and this stress generally upon the strain value in which the angular distortion of the box element. 

Cube stress element.
Illustration of the cube(Image Source: D. K. Singh – Strength of Materials-Springer, 2020, pg.415).

The ‘G’ is the modulus of rigidity of the material of the cube. ‘V’ is the volume of the cube. ‘τ’ is the shear stress induced on the box because of the shear force. And also the stored strain energy is calculated with the formulation below;

Shear strain energy formula.

The theory is explained over a cube because the cube shape is the most used engineering stress element. 

Shear Strain Energy Calculation with Principal Stresses

Firstly, you need to calculate the principal stress values that your body undergoes. After that, you can use the calculator below to calculate the total stored strain on your body.

Shear Strain Energy Calculator Of Triaxial Principal Stresses

The use of the calculator above is very simple. You just need to enter the required values inside the brackets then click on the ‘Calculate!’ button to see the result. If you want to make another calculation, just click on the ‘Reset’ button. 

So, the first three values are the principal stress values of your system. Enter the shear modulus of the material. Also, enter the Poisson’s ratio and the total volume of your body that undergoes the entered principal stresses. 

Furthermore, you can enter the required values inside the brackets with suggested units in the parentheses.


Finally, you can leave your comments and questions below about ‘shear stress-strain characteristics of materials.

Image source: Fundamentals of modern manufacturing

Important External Source: ScienceDirect


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