Principal Stress Calculator - 2D Plane Stress Solver

A principal stress calculator returns the two normal stresses on the rotated planes of a 2D stress element where the in-plane shear stress vanishes, together with the counterclockwise angle of those planes and the maximum in-plane shear stress. Type the normal stress sigma x, the normal stress sigma y, and the shear stress tau xy, and the result panel lists sigma max, sigma min, the principal angle theta p, the maximum shear tau max, and the Mohr circle radius R from one closed-form calculation.

• • Mechanics of materials homework: Confirming sigma_max and sigma_min from a textbook problem before drawing a Mohr circle or running a failure check.
• • Pressure vessel design: Reading the hoop and axial principal stresses on a thin-walled cylinder and the orientation of the largest tension.
• • Brittle material failure check: Comparing sigma_max against the ultimate strength of a ceramic or cast iron before sizing a fillet.
• • Biaxial stress with combined shear: Resolving a biaxial normal stress state with a known shear load into the rotated plane where only normal stress remains.

Principal stress is the extreme normal stress a point can carry when the cutting plane is rotated to remove the shear component. In 2D plane stress, two such orientations exist perpendicular to each other, so the rotated element has only normal stresses.

The pair of principal stresses sets the cap on the maximum tension or compression a brittle part sees, which is why design codes for cast iron, ceramics, and concrete compare the largest principal stress against the ultimate strength.

When you already know sigma x, sigma y, and tau xy but want to draw the rotated plane on a Mohr circle diagram, the Mohr circle calculator reads the same stress state and plots the principal stresses on a 2D normal-shear graph.

The calculator reads sigma x, sigma y, and tau xy, builds the average normal stress and the Mohr circle radius from the 2D stress transformation, and returns the two principal stresses plus the rotated plane angle. The same radius is reported as the maximum in-plane shear stress, and the shear plane angles sit 45 degrees off the principal plane.

• sigma_x (sigma x): Normal stress on the x-face. Tension positive, compression negative.
• sigma_y (sigma y): Normal stress on the y-face. Same sign convention as sigma_x.
• tau_xy (tau xy): In-plane shear stress on the x-face along the y-direction. By symmetry tau_xy equals tau_yx.
• sigma_avg and R: Center and radius of Mohr's circle. R is also the maximum in-plane shear stress.
• theta_p, theta_tau_max, theta_tau_min: Angles of the principal plane and the two shear planes, in degrees counterclockwise from the x-axis.

All inputs share the same unit because the formulas are linear in stress. Swap the labels to psi, ksi, GPa, or kPa and the result panel uses the same unit with no hidden conversion.

According to Omni Calculator - Principal Stress, sigma_max = sigma_avg + R and sigma_min = sigma_avg - R, with theta_p = (1/2) atan(2 tau_xy / (sigma_x - sigma_y)), and the maximum shear angle is theta_p + 45 deg while the minimum shear angle is theta_p - 45 deg.

If a hole or notch sits in the loaded plate, the local peak stress scales by the Kt from the stress concentration factor calculator, and that peak stress can be plugged into the principal stress formula to find sigma max at the hole boundary.

Four concepts sit behind every principal stress result.

These four concepts describe why the same input element can be tension on one face and compression on the rotated face, the practical insight behind brittle-failure and fatigue-strength checks.

On a simply supported beam the bending stress from the beam bending stress calculator sets sigma x in the beam cross section, and combining that with a known tau xy from a transverse shear check feeds the principal stress formula directly.

Five steps turn sigma x, sigma y, and tau xy into the principal stresses, the principal angle, and the maximum shear. This principal stress calculator updates every output as soon as any input changes.

1. 1 Enter sigma x: Type the normal stress on the x-face of the element in MPa. Use a positive value for tension and a negative value for compression; the default is 100 MPa.
2. 2 Enter sigma y: Type the normal stress on the y-face in MPa with the same sign convention; the default is 60 MPa.
3. 3 Enter tau xy: Type the in-plane shear stress on the x-face along the y-direction in MPa; the default is 30 MPa.
4. 4 Read the principal stresses: The result panel returns sigma_max and sigma_min in MPa, the average normal stress sigma_avg, and the Mohr circle radius R as the headline outputs.
5. 5 Read the principal and shear angles: The panel also lists the principal angle theta_p in degrees, the maximum in-plane shear stress tau_max, and the angles theta_tau_max and theta_tau_min of the two shear-plane orientations, 45 degrees off the principal plane.

For a thin-walled pressure vessel shell with hoop stress 120 MPa, axial stress 40 MPa, and a small torsional shear of 25 MPa from a misalignment, type 120, 40, and 25. The result panel reads sigma_max around 132 MPa and theta_p around 7.6 degrees, the orientation of the largest tension in the wall.

For a beam under combined bending and transverse load, the shear force and bending moment calculator returns the bending moment and shear force at the section, and the section-modulus and area give sigma x and tau xy for the principal stress formula.

The calculator replaces a hand calculation, a Mohr-circle sketch, and a table lookup with one result panel that returns the rotated-plane stresses and angles at the same time.

• • Closed-form sigma max and sigma min: Returns both eigenvalues of the 2D stress tensor in MPa, ready for a maximum-normal-stress or modified-Mohr failure check without drawing a Mohr circle.
• • Principal angle in degrees: Theta p is reported in degrees, so the rotated-plane orientation can be drawn on a finite element diagram without converting units.
• • Maximum in-plane shear and Mohr circle radius: Reports tau max and R side by side. Both equal the same value by construction, which gives an instant cross-check that the radius and the shear cap agree.
• • Shear-plane angles 45 degrees off the principal plane: Returns theta_tau_max and theta_tau_min from theta_p + 45 and theta_p - 45, the orientations of the two shear planes used in Tresca checks.
• • Real-time recalculation on edit: Editing any of sigma_x, sigma_y, or tau_xy refreshes all eight outputs without pressing a recalculate button, so you can sweep one input and watch the principal angle rotate.

For ductile materials the maximum principal stress alone is not the failure metric, and a comparison against the bending stress from a section-modulus check is often the next step, which the bending stress calculator returns for rectangular and circular beams in one panel.

Four factors drive the result panel, plus two limitations when 2D plane stress stands in for a real load case.

• • The 2D plane-stress model assumes the out-of-plane stress sigma_z is zero, which holds for thin plates loaded in their plane but not for thick pressure vessels or 3D contact patches.
• • The formulas assume a homogeneous, isotropic, linear-elastic element. Near a notch, a crack tip, or a weld toe the local stress state is no longer uniform, so the principal stresses at the hot spot must come from a finite-element or stress-concentration-factor model.

According to Wikipedia - Cauchy stress tensor, The principal stresses are the three eigenvalues of the symmetric Cauchy stress tensor, and in 2D plane stress they reduce to the two normal stresses on the rotated planes where shear vanishes.

According to Wikipedia - Mohr's circle, The circle sits on the normal-stress axis with center at (sigma_x + sigma_y) / 2 and radius sqrt(((sigma_x - sigma_y) / 2)^2 + tau_xy^2), with the maximum in-plane shear stress equal to the radius.