Mohrs Circle Calculator

Mohr's Circle Calculator

Calculation Results

Avg Normal Stress (σ_avg):

Circle Radius (R):

Max Principal Stress (σ₁):

Min Principal Stress (σ₂):

Max Shear Stress (τ_max):

Principal Angle (θ_p): °

function calculateMohrsCircle() { var sx = parseFloat(document.getElementById('sigmaX').value); var sy = parseFloat(document.getElementById('sigmaY').value); var txy = parseFloat(document.getElementById('tauXY').value); if (isNaN(sx) || isNaN(sy) || isNaN(txy)) { alert("Please enter valid numerical values for all stress components."); return; } // Calculation Logic var sigmaAvg = (sx + sy) / 2; var radius = Math.sqrt(Math.pow((sx – sy) / 2, 2) + Math.pow(txy, 2)); var sigma1 = sigmaAvg + radius; var sigma2 = sigmaAvg – radius; var tauMax = radius; // Theta P in radians, then converted to degrees // Using atan2 for proper quadrant handling var thetaP = 0.5 * Math.atan2(2 * txy, (sx – sy)) * (180 / Math.PI); // Display Results document.getElementById('resAvg').innerText = sigmaAvg.toFixed(2); document.getElementById('resRadius').innerText = radius.toFixed(2); document.getElementById('resSigma1').innerText = sigma1.toFixed(2); document.getElementById('resSigma2').innerText = sigma2.toFixed(2); document.getElementById('resTauMax').innerText = tauMax.toFixed(2); document.getElementById('resThetaP').innerText = thetaP.toFixed(2); document.getElementById('results-area').style.display = 'block'; }

Understanding Mohr's Circle for Plane Stress

In structural engineering and material science, Mohr's Circle is a graphical representation used to determine the transformation of stresses at a specific point in a solid body. It is an essential tool for identifying the maximum and minimum stresses a material is subjected to, which is critical for preventing mechanical failure.

Key Concepts and Formulas

To calculate the values for Mohr's Circle, we start with three primary inputs from a stress element: the normal stresses (σ_x and σ_y) and the shear stress (τ_xy). The following formulas are used:

• Average Stress (Center of the circle): σ_avg = (σ_x + σ_y) / 2
• Circle Radius (Maximum Shear Stress): R = √[ ((σ_x – σ_y)/2)² + τ_xy² ]
• Principal Stresses: σ_1,2 = σ_avg ± R
• Orientation of Principal Planes: tan(2θ_p) = 2τ_xy / (σ_x – σ_y)

Why is this important?

Materials often fail not because of the applied loads directly, but because the internal stresses on a specific plane exceed the material's strength. Mohr's Circle allows engineers to find the "Principal Stresses" (σ₁ and σ₂). These represent the maximum and minimum normal stresses a material experiences when the shear stress is zero. Similarly, the Maximum Shear Stress (τ_max) is a vital metric for ductile materials, which often fail due to shearing.

Practical Example

Imagine a steel beam subjected to a normal stress σ_x of 80 MPa, σ_y of 20 MPa, and a shear stress τ_xy of 40 MPa. By plugging these into the calculator:

1. The Average Stress would be (80 + 20) / 2 = 50 MPa.
2. The Radius would be √[ ((80 – 20)/2)² + 40² ] = √[30² + 40²] = 50 MPa.
3. The Maximum Principal Stress (σ₁) is 50 + 50 = 100 MPa.
4. The Minimum Principal Stress (σ₂) is 50 – 50 = 0 MPa.
5. The Maximum Shear Stress is 50 MPa.

This result tells the engineer that the material must be able to withstand at least 100 MPa of tension, even if the primary load was only 80 MPa, due to the interaction with shear forces.

Usage Tips

Ensure that all units (MPa, psi, kPa) are consistent across all input fields. For compressive stress, enter the value as a negative number. This calculator assumes a 2D plane stress state, which is the standard for most thin-walled pressure vessel and beam analysis applications.