Principal Stresses and Mohr's Circle

Learning Objectives

  • Describe a plane-stress state using normal and shear stress components with a consistent sign convention.
  • Transform stresses to an arbitrarily oriented plane.
  • Determine principal stresses, maximum in-plane shear stress, and their orientations.
  • Construct and interpret Mohr's Circle as a graphical representation of the same transformation equations.
  • Distinguish in-plane maximum shear from absolute maximum shear when a third principal stress is relevant.

Plane Stress

Plane stress is a two-dimensional stress state in which the out-of-plane stress components are negligible relative to the in-plane components.

Thin Plate Plane Context

The plate's small through-thickness edge and broad connected surface give physical context for idealizing a thin member as a material plane; exact out-of-plane assumptions belong to the definition and equations.

Thin steel plate connected to an orthogonal frame, with its broad material plane clearly visible.

Reading the Thin Plate Context

The broad plate surface is a qualitative material-plane cue, not a plane-stress label or a measured statement about an out-of-plane component. Apply the idealization only with the assumptions stated in the lesson.

Plane-Stress Transformation

Normal and shear stresses on a plane rotated by angle theta from the x face.

σx′=σx+σy2+σx−σy2cos⁡2θ+τxysin⁡2θ\sigma_{x'}= \frac{\sigma_x+\sigma_y}{2} + \frac{\sigma_x-\sigma_y}{2}\cos 2\theta + \tau_{xy}\sin 2\thetaτx′y′=−σx−σy2sin⁡2θ+τxycos⁡2θ\tau_{x'y'}= -\frac{\sigma_x-\sigma_y}{2}\sin 2\theta + \tau_{xy}\cos 2\theta

Why Orientation Matters

The stress state at a point is fixed, but its normal and shear components depend on the orientation of the plane used to describe it. Stress transformation resolves the same physical state onto a rotated set of faces; it does not create additional load.

Alternate Material-Plane Context

The connected steel cutaway gives a physical context for asking how the same point can be described on different plane orientations; exact stress components, angle relations, and Mohr-circle geometry remain in the equations and simulator.

Text-free cool blue-gray steel specimen with a clean oblique section cut through a connected rectangular body.

Reading the Rotated-Plane Context

Treat the oblique cut as a qualitative view of a different plane through the same material—not as a drawn principal plane or a scaled stress diagram. The exact transformed normal and shear components, angle, and sign convention come from the formulas and simulation.

Principal Stress

Principal stresses are the normal stresses acting on planes where the corresponding shear stress is zero.

Biaxial Material Region

The orthogonal frame and intact plate region provide physical context for two normal directions acting within the same material plane; use the formulas and simulator for components and principal values.

A rectangular steel plate region tied into orthogonal in-plane framing at a connected joint.

Reading the Biaxial Region

The connected plate region makes two in-plane directions visible through the surrounding geometry, but it does not encode stress signs, magnitudes, or principal values. Those quantities come from the stated stress state and transformation model.

Principal Stresses

Maximum and minimum in-plane normal stresses for a plane-stress state.

σ1,2=σx+σy2±(σx−σy2)2+τxy2\sigma_{1,2} = \frac{\sigma_x+\sigma_y}{2} \pm \sqrt{ \left(\frac{\sigma_x-\sigma_y}{2}\right)^2 + \tau_{xy}^2 }

Maximum In-Plane Shear Stress

Radius of Mohr's Circle and the maximum in-plane shear stress magnitude.

τmax⁡,in\mbox−plane=(σx−σy2)2+τxy2\tau_{\max,\mathrm{in\mbox{-}plane}} = \sqrt{ \left(\frac{\sigma_x-\sigma_y}{2}\right)^2 + \tau_{xy}^2 }
Maximum-Shear Plane Context

The alternate diagonal plane changes the viewing cut through one coherent element, providing context for orientation-dependent shear; no failure or shear magnitude is implied.

A single connected material element with a clean diagonal cut plane through its thickness.

Reading the Diagonal Plane

Treat the clean diagonal cut as an idealized observation plane through the same element, not as a crack, failure surface, or measured maximum-shear plane. Exact orientation and shear magnitude remain in the equations and simulator.

Mohr's Circle

Mohr's Circle is a graphical construction in normal-stress versus shear-stress coordinates that represents all transformed plane-stress states at a point.

Mohr's Circle Geometry

The circle center is C=(σx+σy)/2C=(\sigma_x+\sigma_y)/2 on the normal-stress axis. Its radius is the maximum in-plane shear stress. The horizontal intercepts are σ1\sigma_1 and σ2\sigma_2. A physical rotation of θ\theta corresponds to a rotation of 2θ2\theta around Mohr's Circle, with the direction governed by the adopted shear-sign convention.

Interactive Exploration

Change σx\sigma_x, σy\sigma_y, and τxy\tau_{xy}. Track the circle center, radius, principal-stress intercepts, and maximum-shear points. Use the visual to verify—not replace—the transformation equations and sign convention.

Mohr's Circle for Plane Stress

Concept and model scope

Adjust the plane-stress components and inspect the circle center, radius, principal stresses, and principal-plane angle.

Controls

Normal stress σx

Normal stress on the x face. Positive values are tension in this tool.

Range: -150–150 MPa. Step: 5 MPa.

80 MPa

Normal stress σy

Normal stress on the y face. Positive values are tension in this tool.

Range: -150–150 MPa. Step: 5 MPa.

20 MPa

Shear stress τxy

In-plane shear component used in the transformation equations. This graph plots positive shear upward; keep that convention consistent when reading angles.

Range: -100–100 MPa. Step: 5 MPa.

40 MPa
στx facey faceEqual σ and τ plotting scales preserve the circle geometry.
Center C
50.0 MPa

(σx + σy)/2

Radius / τmax
50.0 MPa

Maximum in-plane shear magnitude.

Principal stresses
100.0 / 0.0 MPa

σ1 and σ2 are the horizontal circle intercepts.

Principal-plane angle
26.6°

One solution from ½ atan2(2τxy, σx−σy); the other is 90° away.

Stress Transformation Workflow

The workflow below keeps the algebraic and graphical methods synchronized and provides explicit checkpoints for sign and orientation errors.

Principal Stress and Mohr's Circle Workflow
Principal Stress and Mohr's Circle WorkflowDefine sigma_x, sigma_y, tau_xy and sign convention → Compute center C and radius R; Compute center C and radius R → Compute sigma_1 = C + R and sigma_2 = C - R; Compute sigma_1 = C + R and sigma_2 = C - R → Set maximum in-plane shear magnitude equal to R; Set maximum in-plane shear magnitude equal to R → Determine principal/shear-plane orientation using 2-theta relation; Determine principal/shear-plane orientation using 2-theta relation → Do transformed stresses and circle points satisfy the chosen convention?; Do transformed stresses and circle points satisfy the chosen convention? — Yes → Report stresses, orientations, and assumptions; Do transformed stresses and circle points satisfy the chosen convention? — No → Correct sign convention or angle direction; Correct sign convention or angle direction → Compute center C and radius R

Define sigma_x, sigma_y, tau_xy and sign convention → Compute center C and radius R; Compute center C and radius R → Compute sigma_1 = C + R and sigma_2 = C - R; Compute sigma_1 = C + R and sigma_2 = C - R → Set maximum in-plane shear magnitude equal to R; Set maximum in-plane shear magnitude equal to R → Determine principal/shear-plane orientation using 2-theta relation; Determine principal/shear-plane orientation using 2-theta relation → Do transformed stresses and circle points satisfy the chosen convention?; Do transformed stresses and circle points satisfy the chosen convention? — Yes → Report stresses, orientations, and assumptions; Do transformed stresses and circle points satisfy the chosen convention? — No → Correct sign convention or angle direction; Correct sign convention or angle direction → Compute center C and radius R

  • Define sigma_x, sigma_y, tau_xy and sign convention: terminator
  • Compute center C and radius R: process
  • Compute sigma_1 = C + R and sigma_2 = C - R: process
  • Set maximum in-plane shear magnitude equal to R: process
  • Determine principal/shear-plane orientation using 2-theta relation: process
  • Do transformed stresses and circle points satisfy the chosen convention?: decision
  • Correct sign convention or angle direction: process
  • Report stresses, orientations, and assumptions: terminator

Absolute Maximum Shear

For a three-dimensional stress state, compare all three principal stresses. The absolute maximum shear stress is half the largest separation between any pair of principal stresses. Under plane stress, the third principal stress is commonly σ3=0\sigma_3=0, which may make the absolute maximum shear different from the in-plane maximum.

Three-Dimensional Stress-State Context

A finite-thickness block joined across multiple faces makes out-of-plane interaction physically visible; use the principal-stress discussion for deciding when a third component matters.

Thick steel block integrated into orthogonal structural plates with visible depth on multiple faces.

Reading the Thick Solid Context

The visible thickness distinguishes this solid context from a thin plate, but the raster does not encode coordinate axes, principal labels, a tensor, or numerical values. Those remain in the lesson's definitions, formulas, and interactive model.

Mohr's Circle Sign Convention

Textbooks and software do not all plot positive shear in the same vertical direction. State the convention and use it consistently when mapping between physical rotation and circle rotation. Principal-stress magnitudes are convention-independent; plotted rotation direction is not.

Key Takeaways
  • Stress components change with plane orientation even though the physical stress state does not.
  • Principal planes carry zero shear stress, and their normal stresses are the extrema σ1\sigma_1 and σ2\sigma_2.
  • The Mohr-circle radius equals the maximum in-plane shear stress magnitude.
  • Physical angle and Mohr-circle angle are related by a factor of two.
  • For absolute maximum shear, include the third principal stress when it is relevant.

References