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Hibbeler Example 9.3: principal stresses of a plane-stress element by Mohr's circle

Tool	Mohr's Circle
Source	Hibbeler, R. C. (2011). Mechanics of Materials, 8th ed. Pearson Prentice Hall, Upper Saddle River, NJ. Example 9.3.
Level	Undergraduate

Problem

The state of plane stress at a point has σx = −20 MPa, σy = 90 MPa and τxy = 60 MPa. Find the principal stresses and their orientation, the maximum in-plane shear stress and the stresses on an element rotated 30°.

Inputs to Enter

Kind: plane stress; σx = −20 MPa, σy = 90 MPa, τxy = 60 MPa (loaded by default, SI)
Rotation angle θ = 30°

Open the Mohr's Circle tool (loaded with this example)

Answers

σ1 = 116 MPa at θp = 66.3° (−23.7° for σ2 = −46.4 MPa)
τmax in-plane = 81.4 MPa at θs = 21.3° with σavg = 35 MPa
At θ = 30°: σx′ = 59.5 MPa, σy′ = 10.5 MPa, τx′y′ = 77.6 MPa

Notes

The tool draws the circle in the Beer–Johnston convention (X at (σx, −τxy)); the example numbering is 9.3 in the 8th edition.