# Mohr’s Circle Calculator + Online Solver With Free Steps

**A Mohr’s Circle Calculator **is a free tool that helps you to find different stress parameters of an object.

The **calculator **returns the mohr’s circle representation and minimum and maximum values of normal and shear stress as the output.

## What Is the Mohr’s Circle Calculator?

**The Mohr’s Circle Calculator is an online calculator that is designed to solve your problems involving plane stress using Mohr’s circle.**

The concept of stress has a vast application in the field of **physics**, **mechanics**, and **engineering**. It can be used to determine the maximum pressure in a container, the extent of a stretch of an object and pressure of a fluid, etc.

Finding stress-related parameters is a **difficult** and **hectic** task. It requires a lot of time and computation for solving such problems. But this **advanced **tool can save you from the rigorous process.

This **calculator** is always accessible in your daily use browser without any installation.

## How To Use the Mohr’s Circle Calculator?

You can use **Mohr’s Circle Calculator** by entering the parameters related to the plane stress problem in their respective boxes. The calculator’s **interface** is made simple so everyone can easily operate this tool.

The basic steps to use the calculator are given below.

### Step 1

Insert the horizontal normal stress in the **“X Direction” **box and vertical normal stress in the **“Y Direction” **box.

### Step 2

Now put the value of shear stress in the third field with the name **“Shear Stress.” **Also, insert the plane angle in its slot.

### Step 3

Press the **Submit **button to get the final answer for the problem.

### Result

The calculator’s result has multiple sections. The first section displays the **shear** stress in a new frame. The next section gives **Mohr’s circle** for the problem and also highlights the points of normal and shear stress.

The last section gives the average, maximum and minimum value of **normal stress** on the object. In addition to that, it also gives the maximum and minimum value of **shear stress**.

## How Does the Mohr’s Circle Calculator Work?

The **Mohr’s Circle Calculator **works by drawing the **mohr’s circle **for the problem using the input elements. The circle has important parameters like shear and normal stress.

To better understand the functionality of the calculator we need to review some fundamental concepts.

### Stress

**Stress** is a reactional force whenever an external force is applied to any surface area. It is equal in magnitude and opposite in direction to the applied force. The stress is represented as the force per unit area and its formula is as follows:

\[ S = \frac{F}{A} \]

The unit of stress is N/m$^\mathsf{2}$ or Pascal (Pa). There are two main types of stress which are **Shear** and **Normal **stress.

#### Normal Stress

When the force applied to an object is perpendicular to its surface area, then the resulting stress is called **normal** stress. Such stress can bring a change either in the **length** or **volume** of an object. The symbol of normal stress is ($\sigma$).

#### Shear Stress

The **shear **stress is a resultant force when an external force is applied to an object parallel to its surface area. This kind of stress can vary the **shape** of an object. The shear stress is denoted by the symbol ($\tau$).

### Plane Stress

**Plane stress** means a condition in which stress along any particular axis is considered to be zero. It means that all stress forces acting on an object will exist on a singular plane.

Any three-dimensional object can have a maximum of three kinds of stress along the axes x, y, and z. Generally, both the normal and shear stress along the **z-axis** are assumed to be zero.

### What Is the Mohr’s Circle?

**Mohr’s Circle** is a method that uses graphical representation to determine the normal and shear stress acting on an object. The graph to plot mohr’s circle has normal stress on the **horizontal** axis and shear stress on the **vertical** axis.

The **right **side of the horizontal axis is the positive normal stress and the **left** side represents the negative normal stress.

On the other hand for shear stress, the **upward** side indicates negative and the **lower** side of the vertical axis represents positive stress.

### Drawing Mohr’s Circle

**Mohr’s circle** is drawn in multiple steps on the normal-shear stress plane. The first step is to find the **center** of the circle, which is the average of two normal stress. It is written as:

\[ \sigma_{avg} = \frac{\sigma_{x} + \sigma_{y}}{2} \]

Then we plot two **points**, the first point ($\sigma_x,\, \tau_{xy}$) corresponds to stress on the x-face and the second point ($\sigma_y,\, -\tau_{xy}$). represents stress on the y-face of the object.

Now both points are joined together by a line passing through the center of the circle. This new line is the **diameter **of mohr’s circle which is used to draw the circle.

Each **point **on the circle represents normal and shear stress for different positions of the object. The radius of the circle is the maximum **shear** stress. It can be calculated as:

\[ R = \sqrt{\left(\frac{\sigma_{x} – \sigma_{y} }{2} \right)^2 + \tau_{xy}^2 } \]

Figure 1 shows the general form of mohr’s circle.

The shear stress will be zero at the points where the circle crosses the horizontal axis, at these points, we have maximum normal stress which is known as **principal** stress. To calculate them, the following formula is used.

\[ \sigma_{1,2} = \frac{\sigma_{x} + \sigma_{y}}{2} \pm \sqrt{ \left(\frac{\sigma_{x} – \sigma_{y} }{2}\right)^2 + \tau_{xy}^2 } \]

The angle between the stress element and the principal planes can also be determined by using the formula given below:

\[ \tan 2\theta_p = \frac{\tau_{xy}}{(\sigma_{x}-\sigma_{y}) \, / \, 2} \]

## Solved Examples

Some of the problems solved using the calculator are explained below.

### Example 1

Consider a stress element with the following characteristics:

\[ \sigma_{x} = -8 \text{ MPa}, \, \sigma_{y} = 12 \text{ MPa}, \, \tau_{xy} = 6 \text{ MPa} \]

Determine the principal and shear stresses using Mohr’s circle.

### Solution

The answer provided by the calculator is given as:

#### Shear Stress

It gives the value of shear stress in the new frame.

\[ \text{Shear stress} = 6 \text{ MPa} = 870.2 \text{ psi} = 6 \times 10^{6} \text{ Pa} \]

#### Schematic

Mohr’s circle representation is given in figure 2.

#### Mohr’s Circle Parameter

The fundamental parameters of mohr’s circle are:

\[ \text{Average Normal Stress} = 10 \text{ MPa},\: 1450 \text{ psi},\: 1 \times 10^{7} \text{ Pa} \]

\[ \text{Maximum Normal Stress} = 35.71 \text{ MPa},\: 5179 \text{ psi},\: 3.571 \times 10^{7} \text{ Pa} \]

\[ \text{Minimum Normal Stress} = -15.71 \text{ MPa},\: -2279 \text{ psi},\: -1.571 \times 10^{7} \text{ Pa} \]

\[ \text{Maximum Shear Stress} = 25.71 \text{ MPa},\: 3729 \text{ psi},\: 2.571 \times 10^{7} \text{ Pa} \]

\[ \text{Minimum Shear Stress} = -25.71 \text{ MPa},\: -3729 \text{ psi},\: -2.571 \times 10^{7} \text{ Pa} \]

### Example 2

A stress element has the following forces acting on it.

\[ \sigma_{x} = 16 \text{ MPa}, \, \sigma_{y} = 4 \text{ MPa}, \, \tau_{xy} = 25 \text{ MPa} \]

Draw Mohr’s circle for the element with angle $\theta_{p} = 30^{\circ}$.

### Solution

#### Shear Stress

\[ \text{Shear stress} = 7.304 \text{ MPa} = 1059 \text{ psi} = 7.304 \times 10^{6} \text{ Pa} \]

#### Schematic

#### Mohr’s Circle Parameter

\[ \text{Average Normal Stress} = 2 \text{ MPa},\: 290.1 \text{ psi},\: 2 \times 10^{6} \text{ Pa} \]

\[ \text{Maximum Normal Stress} = 13.66 \text{ MPa},\: 1981 \text{ psi},\: 1.366 \times 10^{7} \text{ Pa} \]

\[ \text{Minimum Normal Stress} = -9.66 \text{ MPa}, \:-1401 \text{ psi},\: -9.66 \times 10^{6} \text{ Pa} \]

\[ \text{Maximum Shear Stress} = 11.66 \text{ MPa},\: 1691 \text{ psi},\: 1.166 \times 10^{7} \text{ Pa} \]

\[ \text{Minimum Shear Stress} = -11.66 \text{ MPa},\: -1691 \text{ psi},\: -1.166 \times 10^{7} \text{ Pa} \]

*All the Mathematical Images/Graphs are created using GeoGebra.*