Principal Stress Calculator

Normal stress in x direction (selected stress unit): Stress acting perpendicular to the x-face. Positive usually means tension, and negative usually means compression.

Normal stress in y direction (selected stress unit): Stress acting perpendicular to the y-face, using the same sign rule as the x-direction stress.

Shear stress on xy plane (selected stress unit): In-plane shear stress. Its sign affects the reported angles, so keep the convention consistent with your class or text.

Major principal stress: The larger principal normal stress. On its plane, in-plane shear stress is zero.

Minor principal stress: The smaller principal normal stress. It acts on the principal plane perpendicular to the major one.

Principal plane angle: Orientation of one principal plane. The calculator uses a quadrant-aware angle rule so it can place the angle in the correct branch.

Other principal plane angle: The companion principal orientation, always 90 deg from the first one.

Maximum in-plane shear stress: Largest shear stress found by rotating the plane within 2D plane stress. It equals the radius of Mohr's circle for this case.

Maximum shear plane angle: A plane orientation where the in-plane shear reaches its extreme value. It is 45 deg from a principal plane.

Average normal stress: The average of the x- and y-direction normal stresses. It is the center of Mohr's circle in plane stress.

Mistake: Mixing tension-positive notes with compression-positive notes when entering Normal stress in x direction (selected stress unit) or Normal stress in y direction (selected stress unit) .
Fix: Use one sign rule throughout. Enter tension as positive and compression as negative so Major principal stress and Minor principal stress keep the right meaning.

Mistake: Entering Shear stress on xy plane (selected stress unit) with the opposite sign from your class convention.
Fix: Keep your shear sign convention consistent from start to finish. A wrong shear sign often leaves the stress magnitudes close but flips Principal plane angle and Maximum shear plane angle .

Mistake: Using MPa values in the boxes but leaving Stress unit set to psi or ksi.
Fix: Set Stress unit to the same unit used for every entered stress so Major principal stress , Minor principal stress , Maximum in-plane shear stress , and Average normal stress are displayed correctly.

Mistake: Thinking Principal plane angle is the only valid principal orientation.
Fix: Also read Other principal plane angle . Principal planes come as a pair 90 deg apart.

Mistake: Selecting fixed rounding but entering a non-integer or out-of-range value for Decimal places (digits) .
Fix: When Number display format is fixed decimals, use a whole number from 0 to 8 for Decimal places (digits) .

Mistake: Treating a zero-shear, equal-stress case as if one special angle must exist.
Fix: If Normal stress in x direction (selected stress unit) equals Normal stress in y direction (selected stress unit) and Shear stress on xy plane (selected stress unit) is 0, use the ambiguity note: every in-plane orientation is principal even if the displayed Principal plane angle is 0 deg by convention.

• This tool is for 2D plane stress only. It does not include any out-of-plane stress, so it does not return full 3D principal stresses.
• All stress inputs must use the same unit selected in Stress unit. The calculator does not mix units inside one calculation.
• Angle outputs depend on the sign convention used for Shear stress on xy plane (selected stress unit) and on the chosen Angle convention. Different textbooks may label equivalent planes differently.
• When sigma_x equals sigma_y and tau_xy equals 0, the stress state is isotropic in-plane. Every in-plane direction is principal, so any displayed angle is only a reporting convention.
• The reported Maximum in-plane shear stress is the 2D in-plane value. In a full 3D stress state, the absolute maximum shear stress could be different.
• Displayed values can differ slightly from hand calculations because of rounding, especially when Number display format or Decimal places (digits) limits precision.

Core equations

The calculator uses the standard 2D plane-stress relations. First it finds the average normal stress and the Mohr's-circle radius, then it uses those values to get the principal stresses and angles.

σ_avg = (σ_x + σ_y) / 2

R = sqrt(((σ_x - σ_y) / 2)^2 + τ_xy^2)

σ_1 = σ_avg + R

σ_2 = σ_avg - R

θ_p = 0.5 * atan2(2 * τ_xy, σ_x - σ_y)

θ_p2 = θ_p + 90 deg

τ_max = R = (σ_1 - σ_2) / 2

θ_s = θ_p + 45 deg

Why atan2 is used

Many wrong answers come from using a plain tangent formula without quadrant handling. This calculator uses atan2 so the principal-plane angle is placed in the correct quadrant, including cases where σ_x = σ_y or the shear stress is negative. That matches the common Mohr's-circle interpretation of principal directions and maximum shear directions [2].

Worked mini-example

Suppose σ_x = 12 MPa, σ_y = 8 MPa, and τ_xy = 4 MPa.

σ_avg = (12 + 8) / 2 = 10 MPa

R = sqrt(((12 - 8) / 2)^2 + 4^2) = sqrt(2^2 + 16) = sqrt(20) = 4.4721 MPa

σ_1 = 10 + 4.4721 = 14.4721 MPa

σ_2 = 10 - 4.4721 = 5.5279 MPa

θ_p = 0.5 * atan2(8, 4) = 31.717 deg

θ_p2 = 121.717 deg

τ_max = 4.4721 MPa

θ_s = 76.717 deg

Interpretation: both principal stresses are positive here, so both are tensile. The larger one is the Major principal stress. The two principal planes are 90 deg apart, and the maximum-shear plane is 45 deg from a principal plane.

Important note about examples

If you compare with the calculator's default example values, use the calculator outputs as the final check. The formulas above are the governing method, and any mismatch in a handwritten example usually comes from arithmetic rounding, not from a different equation set.

Special cases the calculator handles

If all three input stresses are zero, all stress outputs are zero and the displayed angles default to 0 deg because no unique orientation exists. If σ_x = σ_y and τ_xy = 0, then σ_1 = σ_2 and every in-plane orientation is principal, so the angle is ambiguous rather than physically unique. If σ_x = σ_y but τ_xy is not zero, the atan2 form still gives a valid angle without division-by-zero trouble.

Assumptions behind the result

The method assumes a 2D plane-stress state only, with no out-of-plane stress included. It also assumes all entered stresses use one consistent sign convention and one consistent unit. Unit conversion is handled by converting through standard stress-unit relationships such as 1 MPa = 10^6 Pa and 1 psi = 6894.757293168 Pa.