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Mechanical

Tresca equivalent stress

Maximum-shear-stress measure of a combined stress state, found from the largest and smallest of the three principal stresses.

Ref: MEC-31
Rev: v1
Field: Mechanical
Result: σ_T [MPa]

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

Tresca says a ductile metal starts to yield when the largest shear stress in it reaches the value it has in a simple tension test at yield, S_y/2. Since the largest shear stress is (σ₁ − σ₃)/2, the equivalent stress is σ_T = σ₁ − σ₃, the difference between the largest and smallest of the three principal stresses. Here the principal stresses come from the plane stress route: σ_a = C + R and σ_b = C − R from Mohr’s circle, together with the zero out-of-plane stress, so σ_T = max(σ_a, σ_b, 0) − min(σ_a, σ_b, 0). Yield begins at σ_T = S_y. The criterion is for pressure-insensitive ductile materials; it does not assess fatigue, brittle fracture or buckling.

Maximum-shear-stress measure of a combined stress state, found from the largest and smallest of the three principal stresses. It is a more conservative yield check than von Mises.

Calculator

Solve for

σ_TTresca equivalent 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

Maximum shear stress σ_T/250 MPa
Largest principal stress σ₁70 MPa
Smallest principal stress σ₃ (the zero counts)−30 MPa
Von Mises for comparison88.88 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.

Tresca equivalent stress (MPa) current value 100 MPa

τ_xyShear stress on the x face in the y directionKey driver

60306

σ_xNormal stress in x (tension positive)

80.3320

σ_yNormal stress in y (tension positive)

82.5306

The lecture state gives σ_T = 100 MPa against 88.882 MPa for von Mises: Tresca is 12.5 % higher. Against a 250 MPa yield strength the factor of safety is 2.5 for Tresca and 2.813 for von Mises.
Because the zero out-of-plane stress is included, equal biaxial tension of 100 MPa gives σ_T = 100 MPa, not zero, and pure shear of 50 MPa gives 100 MPa, which is 2τ. Von Mises gives 86.6 MPa for the pure shear, the √3τ value, so Tresca is up to 2/√3 = 1.155 times as severe.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (MPa)	Local step	Change in result
τ_xy Shear stress on the x face in the y direction	−150 to 150 MPa	60	305.9	245.9	+6 MPa	+9.84 %
σ_x Normal stress in x (tension positive)	−200 to 300 MPa	80.28	320.2	239.9	+10 MPa	+6.3 %
σ_y Normal stress in y (tension positive)	−200 to 300 MPa	82.53	306.2	223.7	+10 MPa	−5.66 %

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

Shear stress on the x face in the y direction (MPa)	Tresca equivalent stress (MPa)
−150	305.9
−112.5	232.9
−75	161.6
−37.5	96.05
0	60
37.5	96.05
75	161.6
112.5	232.9
150	305.9

Sweep one variable

Vary

Tresca equivalent stress (MPa)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

σ_T

Tresca equivalent 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 σ_T while σ_x is the larger normal stress. With principal stresses of opposite signs σ_T is twice the circle radius, and the radius grows with the gap σ_x − σ_y. Raising it by 10 MPa (to 60 MPa) moves the result by +6.3 %, from 100 to 106.3 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. Lowering σ_y (more compressive) raises σ_T because it widens the gap σ_x − σ_y; raising it towards σ_x narrows the gap and lowers σ_T. Raising it by 10 MPa (to 0 MPa) moves the result by −5.66 %, from 100 to 94.34 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 σ_T whichever way the shear acts, because it enters the radius squared. The sign of the shear only changes the direction of the principal stresses. Raising it by 10 MPa (to 50 MPa) moves the result by +16.6 %, from 100 to 116.6 MPa.

Worked example with the values above

Principal stresses from Mohr’s circle: σ_a = 70 MPa and σ_b = −30 MPa. With the zero out-of-plane stress, σ₁ = 70 MPa and σ₃ = −30 MPa.
Difference: σ₁ − σ₃ = 70 − (−30).
σ_T = 100 MPa, which is 50 MPa of maximum shear.

Assumptions and limits

Plane stress, so the out-of-plane principal stress is zero and must be included.
An isotropic ductile material with equal tensile and compressive yield strength, under static load.

Common mistakes

Using σ_a − σ_b alone when both in-plane principal stresses have the same sign. The zero third stress then governs.
Comparing with the ultimate rather than the yield strength.

Statistics and uncertainty

The two yield criteria differ by at most 15 % in equivalent stress (a factor of 2/√3), usually less than the scatter in measured yield strength, so the choice of safety factor matters more than the choice of criterion. The result is a difference of two principal stresses, so independent errors of ±3 MPa in each give about ±4.2 MPa in σ_T.

Where it is used

Conservative yield checks in shafts, pressure vessels and structures.
Checking finite-element results against a second criterion.

Maths you need

Principal stresses sorted: σ₁ ≥ σ₂ ≥ σ₃, with the zero included
Maximum shear stress: half the largest principal difference

Related formulas

Practise this formula