Library / Mechanical / MEC-54

Mechanical

Principal stresses in order, including the zero

The three principal stresses of a plane stress state sorted from largest to smallest, with the zero out-of-plane stress counted as one of them.

Ref: MEC-54
Rev: v1
Field: Mechanical
Result: σ₁ [MPa]

σ_{1} = \max (σ_{a}, σ_{b}, 0) σ_{3} = \min (σ_{a}, σ_{b}, 0) σ_{2} = σ_{a} + σ_{b} - σ_{1} - σ_{3}

In plane stress the out-of-plane stress σ_z is zero, and zero is a principal stress. So the full set is the two in-plane values σ_a = C + R and σ_b = C − R together with 0, sorted into σ₁ ≥ σ₂ ≥ σ₃. The largest is max(σ_a, σ_b, 0), the smallest is min(σ_a, σ_b, 0), and the middle one is what remains because the three add up to σ_a + σ_b. This is step 4 of the stress route and the input to both yield criteria. The calculator gives σ₁.

The three principal stresses of a plane stress state sorted from largest to smallest, with the zero out-of-plane stress counted as one of them. Tresca and von Mises both need this complete, ordered set.

Calculator

Solve for

σ₁Largest principal stressComputed

MPa

σ_xNormal stress in x (tension positive)

MPa

−200300

σ_yNormal stress in y (tension positive)

MPa

−200300

τ_xyShear stress on the x face in the y direction

MPa

−150150

Middle principal stress σ₂0 MPa
Smallest principal stress σ₃−30 MPa
Mohr circle radius R (in-plane)50 MPa

What drives the result

Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.

Largest principal stress (MPa) current value 70 MPa

σ_xNormal stress in x (tension positive)Key driver

0305

σ_yNormal stress in y (tension positive)

56.2306

τ_xyShear stress on the x face in the y direction

50173

The lecture state gives σ₁ = 70, σ₂ = 0, σ₃ = −30 MPa. The in-plane values have opposite signs, so the zero falls in the middle.
If both in-plane values are tensile the zero becomes the smallest. Equal biaxial tension of 100 MPa gives 100, 100, 0 MPa, so the largest shear is 50 MPa even though the in-plane Mohr circle is a point. Leaving the zero out would wrongly give no shear at all.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (MPa)	Local step	Change in result
σ_x Normal stress in x (tension positive)	−200 to 300 MPa	0	305.1	305.1	+10 MPa	+11.6 %
σ_y Normal stress in y (tension positive)	−200 to 300 MPa	56.24	306.2	250	+10 MPa	+3.1 %
τ_xy Shear stress on the x face in the y direction	−150 to 150 MPa	50	173	123	+6 MPa	+7.03 %

Sweep of Shear stress on the x face in the y direction

Shear stress on the x face in the y direction (MPa)	Largest principal stress (MPa)
−150	173
−112.5	136.4
−75	100.8
−37.5	68.02
0	50
37.5	68.02
75	100.8
112.5	136.4
150	173

Sweep one variable

Vary

Largest principal stress (MPa)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

σ₁

Largest principal stress

Result · MPa

What the equation gives you. Everything else feeds it.

σ_x

Normal stress in x (tension positive)

MPa · typical −200 to 300

Normal stress on the x face, in MPa. Tension is positive and compression negative, so the slider runs from −200 to +300 MPa. Raises σ₁ when σ_a is the largest value, as it is for the default state. Raising it by 10 MPa (to 60 MPa) moves the result by +11.6 %, from 70 to 78.15 MPa.

σ_y

Normal stress in y (tension positive)

MPa · typical −200 to 300

Normal stress on the y face, in MPa. Tension is positive and compression negative, so the slider runs from −200 to +300 MPa. Raises σ₁ when σ_a is the largest value, though by less than σ_x does. Raising it by 10 MPa (to 0 MPa) moves the result by +3.1 %, from 70 to 72.17 MPa.

τ_xy

Shear stress on the x face in the y direction

MPa · typical −150 to 150

Shear stress on the x face acting in the +y direction, in MPa. The sign matters for directions and rotated stresses, so the slider runs from −150 to +150 MPa. Raises σ₁ whichever way the shear acts, because it enters the radius squared. Raising it by 10 MPa (to 50 MPa) moves the result by +11.9 %, from 70 to 78.31 MPa.

Worked example with the values above

In-plane values: σ_a = 70 MPa and σ_b = −30 MPa. Include the zero out-of-plane stress.
Largest and smallest: σ₁ = max(70, −30, 0) = 70 MPa; σ₃ = min(70, −30, 0) = −30 MPa.
Middle: σ₂ = 70 + (−30) − 70 − (−30) = 0 MPa. σ₁ = 70 MPa.

Assumptions and limits

Plane stress: the out-of-plane stress is zero and is therefore a principal stress.
The three values are ordered by algebraic value, tension largest.

Common mistakes

Sorting only the two in-plane values and forgetting the zero, so a Tresca check sees only σ_a − σ_b.
Ordering by size instead of by algebraic value, which puts a large compressive stress first.

Statistics and uncertainty

The sorting itself adds no error, but the choice of which value is σ₁ and which is σ₃ can switch when an in-plane principal stress is close to zero. Near that point a small measurement error can change which stress is the middle one, although the Tresca and von Mises results vary smoothly across the switch.

Where it is used

Preparing the principal stresses for a Tresca or von Mises check.
Finding the absolute maximum shear stress, (σ₁ − σ₃)/2.

Maths you need

Principal stresses: the stresses on planes with no shear
Max and min: the largest and smallest of a set

Related formulas

Practise this formula