Compute and visualize principal stresses, maximum shear, and stress transformations using interactive Mohr circle construction.
A plane-stress state becomes a geometric circle after entering sigma_x, sigma_y, and tau_xy. The calculator locates the circle center, finds its radius, identifies the two principal stresses and maximum in-plane shear, and relates a rotated physical plane to transformed normal and shear components. The most common surprise is angular: rotating the physical element by theta moves the corresponding point by 2 theta on Mohr's circle. Shear signs and clockwise versus counterclockwise directions vary among engineering conventions, so read the plot's sign labels before interpreting an angle.
A concrete input and expected output from the current implementation.
Input
sigma_x = 80 MPa sigma_y = 20 MPa tau_xy = +30 MPa
Expected output
Center C = 50 MPa; radius R = 42.426 MPa; principal stresses = 92.426 MPa and 7.574 MPa; maximum in-plane shear magnitude = 42.426 MPa at a normal stress of 50 MPa.
The average normal stress is (80 + 20) / 2 = 50 MPa. The radius is sqrt(((80 - 20) / 2)^2 + 30^2) = sqrt(1800) = 42.426 MPa, so adding and subtracting it from 50 gives the principal stresses.
The center lies at C = (sigma_x + sigma_y) / 2 on the normal-stress axis. The radius is R = sqrt(((sigma_x - sigma_y) / 2)^2 + tau_xy^2). The horizontal intercepts are principal stresses C + R and C - R. The circle's extreme shear coordinates have magnitudes R and normal stress C.
Changing the physical plane angle moves its stress point around the circle by twice that angle. The point coordinates represent transformed normal and shear stress, while the diametrically opposite point represents the perpendicular face. Direction around the plot depends on the shear-sign convention, so angle signs must be interpreted using the convention displayed by the calculator.
The plane-stress transformation equations contain cos(2 theta) and sin(2 theta). Consequently, a physical rotation of theta corresponds to a central angle of 2 theta on the circle. Depending on the shear-sign convention, the plotted motion may also run opposite to the physical rotation. The material element itself does not rotate twice as far.
No. It transforms the supplied stress state and identifies its principal and maximum-shear values. A failure assessment additionally needs material properties, an appropriate yield or fracture criterion, allowable stresses, and any required safety factor. The circle also cannot account for missing loads, uncertain boundary conditions, stress concentrations, or an unsuitable plane-stress assumption.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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