Type the stresses σx, σy and τxy of a differential element and instantly get Mohr's circle, the principal stresses, the maximum shear and the stresses on the element rotated to any angle. It also works the other way round: build the circle (from σ1, σ2 and θp, or by dragging the center and point X) and get the x–y element. Includes a detailed step-by-step solution with the derivation of the transformation equations.
It is the graphical form of the plane-stress transformation equations: every point of the circle gives the normal and shear stress on one face of the rotated element.
They are C ± R, with center C = (σx + σy)/2 and radius R = √[((σx − σy)/2)² + τxy²]; their orientation comes from tan 2θp = 2τxy/(σx − σy).
At a point of a loaded part, the stresses depend on the orientation of the plane they are measured on. Mohr's circle is a graphical way to see all those orientations at once: on which plane the normal stress is largest (principal stress), where the shear is largest and how big they are. This guide covers plane stress, the formulas behind the circle and a complete example.
In a plane stress state, the differential element carries σx, σy and τxy, and the faces normal to z are free. If the element is rotated by an angle θ (counterclockwise positive), the stresses on the new faces are
σx′ = (σx + σy)/2 + (σx − σy)/2 · cos 2θ + τxy · sin 2θ
τx′y′ = −(σx − σy)/2 · sin 2θ + τxy · cos 2θ
These two equations describe a circle in the (σ, τ) plane: rotating the element by θ moves the point representing the x face by 2θ around the circle.
C = (σx + σy)/2 R = √[((σx − σy)/2)² + τxy²]
Positive σ is tension; τxy is positive if on the +x face it points toward +y; θ is counterclockwise positive. The tool draws the τ axis positive downward, as Hibbeler does: then the point X = (σx, τxy), and rotating θ on the element is rotating 2θ on the circle in the same sense. Other books, such as Beer, plot X = (σx, −τxy) with τ upward; it is the same figure.
An element with σx = 80 MPa, σy = −40 MPa and τxy = 30 MPa (the state the tool opens with).
Even when the problem is planar, there is a third principal stress, σz = 0. If σ1 and σ2 have opposite signs, as in the example, the absolute maximum shear is the in-plane one, R. But if they have the same sign (for example, in a thin-walled pressure vessel, with σh = 2σa), the absolute maximum shear is σmax/2, larger than the in-plane value. With the "Include σz = 0" option the tool draws all three circles.
R. C. Hibbeler, Mechanics of Materials (Pearson); J. M. Gere and B. J. Goodno, Mechanics of Materials (Cengage); F. P. Beer, E. R. Johnston et al., Mechanics of Materials (McGraw-Hill). Simuveliq is not affiliated with these authors or publishers.
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