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Mechanical

Normal stress on a rotated element, y′ face

Normal stress on the face perpendicular to the x′ face.

Ref: MEC-51
Rev: v1
Field: Mechanical
Result: σ_y′ [MPa]

σ_{y′} = C - \frac{σ_{x} - σ_{y}}{2} \cos 2 θ - τ_{xy} \sin 2 θ C = \frac{σ_{x} + σ_{y}}{2}

σ_y′ is the normal stress on the face perpendicular to the x′ face, so it is the mirror image of σ_x′ about the circle centre C: where one is C plus an amount the other is C minus the same amount. The two always add up to σ_x + σ_y, which makes a quick check on any rotated pair. At θ = 0 the result is σ_y, and at θ = 90° it is σ_x.

Normal stress on the face perpendicular to the x′ face. Together with the x′ face it always carries the same total, so the pair of normal stresses simply swings about the centre of Mohr’s circle.

Calculator

Solve for

σ_y′Normal stress on the y′ faceComputed

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

θRotation of the x axis towards y

−9090

Companion σ_x′ (the x′ face)69.64 MPa
Shear stress τ_x′y′ on the faces−5.981 MPa
Sum σ_x′ + σ_y′ (equals σ_x + σ_y)40 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.

Normal stress on the y′ face (MPa) current value −29.64 MPa

σ_yNormal stress in y (tension positive)Key driver

−172203

τ_xyShear stress on the x face in the y direction

−125135

σ_xNormal stress in x (tension positive)

−92.132.9

θRotation of the x axis towards y

−29.869.8

At 30° the two faces carry 69.64 and −29.64 MPa. Their sum is 40 MPa, the same as σ_x + σ_y = 40 MPa, and they sit 49.64 MPa either side of the centre.
Turn the element by 90° and the roles swap: σ_y′ takes the value σ_x had, which is a useful end-point check.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (MPa)	Local step	Change in result
σ_y Normal stress in y (tension positive)	−200 to 300 MPa	−172.1	202.9	375	+10 MPa	−25.3 %
τ_xy Shear stress on the x face in the y direction	−150 to 150 MPa	−124.9	134.9	259.8	+6 MPa	+17.5 %
σ_x Normal stress in x (tension positive)	−200 to 300 MPa	−92.14	32.86	125	+10 MPa	−8.43 %
θ Rotation of the x axis towards y	−90 to 90 °	−29.8	69.8	99.6	+3.6 °	−3.85 %

Sweep of Rotation of the x axis towards y

Rotation of the x axis towards y (°)	Normal stress on the y′ face (MPa)
−90	50
−67.5	69.5
−45	60
−22.5	27.07
0	−10
22.5	−29.5
45	−20
67.5	12.93
90	50

Sweep one variable

Vary

Normal stress on the y′ face (MPa)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

σ_y′

Normal stress on the y′ face

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_y′ by sin²θ times the change: none at θ = 0 and the full amount at θ = 90°. The shares for σ_x′ and σ_y′ add to 1, so the pair keeps its total. Raising it by 10 MPa (to 60 MPa) moves the result by −8.43 %, from −29.64 to −27.14 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 σ_y′ by cos²θ times the change: the full amount at θ = 0 and none at θ = 90°. Raising it by 10 MPa (to 0 MPa) moves the result by −25.3 %, from −29.64 to −22.14 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. Lowers σ_y′ by sin 2θ times the change in τ_xy, the opposite of σ_x′. Raising it by 10 MPa (to 50 MPa) moves the result by +29.2 %, from −29.64 to −38.3 MPa.

Rotation of the x axis towards y

° · typical −90 to 90

Angle through which the x axis is turned towards the y axis, in degrees. Positive is from x towards y, the opposite way for negative. The slider covers −90° to +90°, one half-turn, because the stresses repeat every 180°. Moves the face through the cycle of stresses, in the opposite sense to σ_x′. Raising it by 10° (to 40°) moves the result by −17 %, from −29.64 to −24.6 MPa.

Worked example with the values above

Centre and half-difference: C = 20 MPa and (σ_x − σ_y)/2 = 30 MPa, with 2θ = 60°.
Swing, the same as for σ_x′: 49.64 MPa.
σ_y′ = 20 − (49.64) = −29.64 MPa.

Assumptions and limits

Plane stress at a point, with the same sign convention for every component.
The angle is measured from x towards y.

Common mistakes

Swapping the signs of the second and third terms. The y′ face has opposite signs to the x′ face for both.
Confusing the y′ face with a face turned by θ from the y axis in the other direction.

Statistics and uncertainty

Errors in the two input normal stresses pass into σ_y′ weighted by the angle, with weights (1 − cos 2θ)/2 for σ_x and (1 + cos 2θ)/2 for σ_y. At 30° these are 0.25 and 0.75, which add to 1, so a common bias in both stresses passes straight through.

Where it is used

Completing the transformed stress picture for a weld or bond line.
Checking a rotated stress pair against the original sum.

Maths you need

Double-angle form: cos 2θ and sin 2θ
Mohr’s circle: the circle is symmetric about its centre

Related formulas

Practise this formula