Mohr's Circle

A stress has a size p′ and a tilt: an arrow whose length is q/p′ and whose direction is that of σ₁′. Friction caps the length at sin φ′. Drag in the C/S panel to set the tilt. All four views show the same two-dimensional stress state.

What you are seeing

• The Mohr circle. Centre p′, radius q, drawn blue inside the cap, orange at it and red beyond it. The blue lines are the Mohr–Coulomb envelope, with the allowed wedge between them shaded. H and V are the stresses on horizontal and vertical planes, the ends of one diameter. The green dot is the chosen θ plane. At the cap, dashed radii go to the points where the circle touches the lines.
• The tilt. The tilt as a point (C, S) = (q/p′)(cos 2β, sin 2β). The shaded disc is the cap; the dotted circle is q/p′ = 1. The active, passive and at-rest (K₀) states are marked on the C axis. Drag anywhere to set the tilt. The small element in the corner shows which way σ₁′ pushes, and the chosen plane in green.
• The status bar. Whether the stress is inside the cap, at it, or beyond it.
• The stress element. A square of soil with compressive normal and shear tractions on its four faces. Switch to resultants to combine them into one vector per location. Opposite faces balance. Arrow lengths represent stress (force per unit area), rather than force on a specified area. The dashed green line is the chosen θ plane.
• The stress ellipse. The locus of traction-vector tips as the face normal rotates. Its semiaxes are the eigenvalues σ₁′ and σ₃′, and its axes follow their eigenvectors. The dotted reference circle has radius p′. The green vector is the traction on the θ plane, acting on the side whose outward normal points up when θ = 0. Isotropic stress gives a circle with no preferred principal direction; σ₃′ = 0 gives a line. This is a stress diagram, not a deformed shape. Each view has its own scale.

Conventions

Compression is positive throughout. β (the direction of σ₁′ from the vertical) and θ (the plane from the horizontal) are both measured anticlockwise. τ on a horizontal plane is positive when the soil above pushes the soil below to the right, so H sits at (σ_v′, τ) and a plane at θ plots 2θ clockwise from H. These are the conventions of the course notes and data sheets.