Mohr Circle of Stress Calculator

Draw Mohr's circle from two normal stresses and a shear stress to find the principal stresses, maximum in-plane shear stress, and principal plane angle.

⭕ Mohr Circle of Stress Calculator
Normal stress (σx)80 MPa
MPa
-200200
Normal stress (σy)20 MPa
MPa
-200200
Shear stress (τxy)40 MPa
MPa
-100100
Principal stress (σ1)
—
Principal stress (σ2)
—
Max in-plane shear (τmax)
—
Principal plane angle (θp)
—
Step-by-step working

⭕ What is Mohr's Circle of Stress?

Mohr's circle is a graphical construction that represents a two-dimensional (plane) stress state as a circle in sigma-tau (normal stress versus shear stress) space, making it possible to read off the principal stresses, the maximum in-plane shear stress, and the normal and shear stress on any rotated plane directly from the diagram, without re-deriving the stress transformation equations for every angle. Named after the German civil engineer Christian Otto Mohr, who popularized the method in 1882, it remains one of the most widely taught graphical tools in mechanics of materials.

Structural, mechanical, and geotechnical engineers reach for Mohr's circle constantly. A structural engineer checking a beam's web near a support, where bending and shear stresses combine, uses Mohr's circle to find the maximum principal stress driving crack initiation. A pressure vessel designer checks the principal stresses at the shell-to-nozzle junction, where hoop stress, longitudinal stress, and local shear all interact. A geotechnical engineer uses the same underlying construction, extended with the Mohr-Coulomb failure envelope, to assess whether a soil element is on the verge of shear failure.

A common misconception is that Mohr's circle is a separate theory from the stress transformation equations taught alongside it. In reality, they describe exactly the same physics and produce identical numbers, Mohr's circle is purely a geometric interpretation: the circle's center is the average normal stress, its radius is the maximum in-plane shear stress, and every point on the circle corresponds to the normal and shear stress on some rotated plane through the same material point.

This calculator takes the two normal stresses sigma-x and sigma-y and the shear stress tau-xy for a 2D stress element, then returns the two principal stresses, the maximum in-plane shear stress, the principal plane angle, full step-by-step working, and a dynamic Mohr's circle diagram plotting the current stress state and both principal-stress points.

📐 Formula

σ1,2  =  σavg ± R     R  =  √(((σx − σy)/2)² + τxy²)
σavg = (σx + σy) / 2, the center of Mohr's circle on the σ axis
R = the circle's radius, also equal to the maximum in-plane shear stress τmax
σ1 = σavg + R (larger principal stress), σ2 = σavg − R (smaller principal stress)
θp = 0.5 × atan2(2τxy, σx − σy), the rotation angle from the x-face to the σ1 principal plane
Example: σx = 80 MPa, σy = 20 MPa, τxy = 40 MPa → σavg = 50 MPa, R = 50 MPa, σ1 = 100 MPa, σ2 = 0 MPa, θp ≈ 26.57°.

📖 How to Use This Calculator

Steps

1
Enter the normal stresses. Type sigma-x and sigma-y, the normal stresses acting on the x-face and y-face of the stress element.
2
Enter the shear stress. Type tau-xy, the shear stress acting on the same two faces.
3
Read the principal stresses and Mohr's circle. See sigma1, sigma2, the maximum in-plane shear stress, the principal plane angle, and the constructed Mohr's circle diagram.

💡 Example Calculations

Example 1 — Combined Bending and Shear at a Beam Web

σx = 80 MPa, σy = 20 MPa, τxy = 40 MPa

1
σavg = (80 + 20) / 2 = 50 MPa
2
R = √(((80 − 20)/2)² + 40²) = √(30² + 40²) = √2500 = 50 MPa
3
σ1 = 50 + 50 = 100 MPa, σ2 = 50 − 50 = 0 MPa, θp = 0.5 × atan2(80, 60) ≈ 26.57°
σ1 = 100 MPa, σ2 = 0 MPa, τmax = 50 MPa
Try this example →

Example 2 — Tension-Compression Element with Shear

σx = 100 MPa, σy = −40 MPa, τxy = 30 MPa

1
σavg = (100 + (−40)) / 2 = 30 MPa
2
R = √(((100 − (−40))/2)² + 30²) = √(70² + 30²) = √5800 ≈ 76.158 MPa
3
σ1 = 30 + 76.158 ≈ 106.158 MPa, σ2 = 30 − 76.158 ≈ −46.158 MPa, θp = 0.5 × atan2(60, 140) ≈ 11.60°
σ1 ≈ 106.16 MPa, σ2 ≈ −46.16 MPa, τmax ≈ 76.16 MPa, the negative σ2 shows that principal direction is in compression
Try this example →

Example 3 — Equal Biaxial Stress with Shear (45° Principal Plane)

σx = 60 MPa, σy = 60 MPa, τxy = 25 MPa

1
σavg = (60 + 60) / 2 = 60 MPa
2
R = √(((60 − 60)/2)² + 25²) = √(0² + 25²) = 25 MPa
3
σ1 = 60 + 25 = 85 MPa, σ2 = 60 − 25 = 35 MPa, θp = 0.5 × atan2(50, 0) = 45.00°
σ1 = 85 MPa, σ2 = 35 MPa, θp = 45°, exactly 45 degrees whenever σx equals σy
Try this example →

❓ Frequently Asked Questions

What is Mohr's circle used for?+
Mohr's circle is a graphical method for finding the principal stresses, maximum shear stress, and stress on any rotated plane from a known 2D stress state (sigma_x, sigma_y, tau_xy), without repeating the stress transformation equations for every angle. It is standard in mechanics of materials, structural engineering, and geotechnical engineering.
How do you calculate principal stresses from sigma-x, sigma-y, and tau-xy?+
sigma1 = sigma_avg + R and sigma2 = sigma_avg minus R, where sigma_avg = (sigma_x + sigma_y) / 2 is the circle's center and R = square root of (((sigma_x minus sigma_y)/2) squared + tau_xy squared) is the circle's radius. sigma1 is always the larger (more tensile) principal stress and sigma2 the smaller.
How do you find the maximum in-plane shear stress?+
Maximum in-plane shear stress tau_max equals the circle's radius R, the same R used to find the principal stresses: R = square root of (((sigma_x minus sigma_y)/2) squared + tau_xy squared). Equivalently, tau_max = (sigma1 minus sigma2) / 2.
What is the principal plane angle theta_p?+
Theta_p = 0.5 times the arctangent of (2 times tau_xy) divided by (sigma_x minus sigma_y), the angle you rotate the stress element by, measured from the original x-face, to reach the orientation where shear stress vanishes and only the principal stresses sigma1 and sigma2 act.
Why are the points X and Y on opposite ends of a diameter?+
Point X represents the stress state on the x-face (sigma_x, tau_xy) and point Y represents the stress state on the perpendicular y-face (sigma_y, minus tau_xy). Because the x-face and y-face are 90 degrees apart physically, they are 180 degrees apart on Mohr's circle, which doubles every physical angle, placing X and Y at opposite ends of a diameter through the center.
What does it mean if sigma2 comes out negative?+
A negative principal stress simply means that principal direction is in compression rather than tension, Mohr's circle and the underlying formulas treat tension and compression with the normal sign convention (positive for tension), so a negative sigma2 is a completely valid and common result, not an error.
How is Mohr's circle different from just using the stress transformation equations?+
Mohr's circle and the stress transformation equations describe exactly the same physics and give identical numerical answers, Mohr's circle is simply a geometric (graphical) way to visualize the transformation, making it easy to read off the principal stresses, maximum shear stress, and stress at any rotation angle from one diagram instead of re-deriving each value algebraically.
What happens to Mohr's circle when tau_xy = 0?+
When tau_xy = 0, the input stress state is already principal, meaning sigma_x and sigma_y are themselves the principal stresses (sigma1 is whichever of the two is larger), the circle still exists, centered at sigma_avg with radius R = |sigma_x minus sigma_y| / 2, but the principal plane angle theta_p comes out to 0 degrees.
Does Mohr's circle apply to 3D stress states?+
This calculator covers 2D (plane) stress, the most common case for thin plates, beam surfaces, and many structural elements. A full 3D stress state uses three interlocking Mohr's circles (one per pair of principal stresses), a more advanced construction not covered here.
Where is Mohr's circle used in real engineering practice?+
Structural engineers use it to check combined bending and shear stress at critical points in beams and plates, geotechnical engineers use an equivalent construction (Mohr-Coulomb failure criterion) to assess soil and rock shear strength, and mechanical engineers use it for pressure vessel and shaft design wherever combined normal and shear stress needs a principal-stress check.
Tags: Mohr Circle Calculator Mohr's Circle of Stress Calculator Principal Stress Calculator Maximum Shear Stress Calculator 2d Stress Transformation Calculator Calculator Formula Free Online

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What is Mohr's circle used for?

Mohr's circle is a graphical method for finding the principal stresses, maximum shear stress, and stress on any rotated plane from a known 2D stress state (sigma_x, sigma_y, tau_xy), without repeating the stress transformation equations for every angle. It is standard in mechanics of materials, structural engineering, and geotechnical engineering.

How do you calculate principal stresses from sigma-x, sigma-y, and tau-xy?

sigma1 = sigma_avg + R and sigma2 = sigma_avg minus R, where sigma_avg = (sigma_x + sigma_y) / 2 is the circle's center and R = square root of (((sigma_x minus sigma_y)/2) squared + tau_xy squared) is the circle's radius. sigma1 is always the larger (more tensile) principal stress and sigma2 the smaller.

How do you find the maximum in-plane shear stress?

Maximum in-plane shear stress tau_max equals the circle's radius R, the same R used to find the principal stresses: R = square root of (((sigma_x minus sigma_y)/2) squared + tau_xy squared). Equivalently, tau_max = (sigma1 minus sigma2) / 2.

What is the principal plane angle theta_p?

Theta_p = 0.5 times the arctangent of (2 times tau_xy) divided by (sigma_x minus sigma_y), the angle you rotate the stress element by, measured from the original x-face, to reach the orientation where shear stress vanishes and only the principal stresses sigma1 and sigma2 act.

Why are the points X and Y on opposite ends of a diameter?

Point X represents the stress state on the x-face (sigma_x, tau_xy) and point Y represents the stress state on the perpendicular y-face (sigma_y, minus tau_xy). Because the x-face and y-face are 90 degrees apart physically, they are 180 degrees apart on Mohr's circle, which doubles every physical angle, placing X and Y at opposite ends of a diameter through the center.

What does it mean if sigma2 comes out negative?

A negative principal stress simply means that principal direction is in compression rather than tension, Mohr's circle and the underlying formulas treat tension and compression with the normal sign convention (positive for tension), so a negative sigma2 is a completely valid and common result, not an error.

How is Mohr's circle different from just using the stress transformation equations?

Mohr's circle and the stress transformation equations describe exactly the same physics and give identical numerical answers, Mohr's circle is simply a geometric (graphical) way to visualize the transformation, making it easy to read off the principal stresses, maximum shear stress, and stress at any rotation angle from one diagram instead of re-deriving each value algebraically.

What happens to Mohr's circle when tau_xy = 0?

When tau_xy = 0, the input stress state is already principal, meaning sigma_x and sigma_y are themselves the principal stresses (sigma1 is whichever of the two is larger), the circle still exists, centered at sigma_avg with radius R = |sigma_x minus sigma_y| / 2, but the principal plane angle theta_p comes out to 0 degrees.

Does Mohr's circle apply to 3D stress states?

This calculator covers 2D (plane) stress, the most common case for thin plates, beam surfaces, and many structural elements. A full 3D stress state uses three interlocking Mohr's circles (one per pair of principal stresses), a more advanced construction not covered here.

Where is Mohr's circle used in real engineering practice?

Structural engineers use it to check combined bending and shear stress at critical points in beams and plates, geotechnical engineers use an equivalent construction (Mohr-Coulomb failure criterion) to assess soil and rock shear strength, and mechanical engineers use it for pressure vessel and shaft design wherever combined normal and shear stress needs a principal-stress check.

📌 Quick Tips

💡The circle's center always sits at sigma_avg = (sigma_x + sigma_y) / 2 on the sigma axis, and its radius R equals the maximum in-plane shear stress, this single geometric fact is the entire content of Mohr's circle.
💡Points X (sigma_x, tau_xy) and Y (sigma_y, minus tau_xy) always sit at opposite ends of a diameter, this is why entering the same tau_xy sign convention consistently matters, flipping the sign of tau_xy only mirrors the circle about the sigma axis and does not change sigma1 or sigma2.
💡When tau_xy = 0, the input state is already a principal state, sigma1 and sigma2 equal sigma_x and sigma_y directly (in whichever order gives the larger value first), and the principal plane angle is 0 degrees.
💡When sigma_x = sigma_y (pure shear superimposed on equal biaxial stress), the principal plane angle is always exactly 45 degrees, a useful mental check for a state of pure shear (sigma_x = sigma_y = 0, tau_xy nonzero).
💡Maximum in-plane shear stress always equals half the difference between the two principal stresses, tau_max = (sigma1 minus sigma2) / 2 = R, and it occurs on planes rotated 45 degrees from the principal planes, not on the same planes as the maximum normal stress.