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Principal and max shear stresses: 10, 7, 10 MPa

Short answer

For σ_x=10 MPa, σ_y=–7 MPa, τ_xy=10 MPa, principal stresses are 14.62 and –11.62 MPa at 24.82° from x. Maximum shear is 13.12 MPa with associated normal stress 1.50 MPa on planes at 69.82° and 159.82°.

Transform the state of stress into the principal stresses and the maximum shearing stresses and the associated normal stresses. Show the results for both cases on properly oriented elements. Tensile stress is 10 MPa, Compressive stress is 7 MPa and shearing stress is 10 MPa. [Figure: element with 10 MPa10\ \text{MPa} tensile on the left and right faces, 7 MPa7\ \text{MPa} compressive on the top and bottom faces and 10 MPa10\ \text{MPa} shear stress on all faces.]

Given: σx=10\sigma_x = 10 MPa (tension), σy=−7\sigma_y = -7 MPa (compression), τxy=10\tau_{xy} = 10 MPa.

Case 1: principal stresses

σ1,2=10−72±(10+72)2+102=1.50±13.12\sigma_{1,2} = \frac{10-7}{2} \pm \sqrt{\left(\frac{10+7}{2}\right)^2 + 10^2} = 1.50 \pm 13.12 σ1=14.62 MPa,σ2=−11.62 MPa\sigma_1 = 14.62\ \text{MPa}, \qquad \sigma_2 = -11.62\ \text{MPa} tan⁡2θp=2(10)10−(−7)=1.1765⇒2θp=49.64∘,θp=24.82∘ and 114.82∘\tan 2\theta_p = \frac{2(10)}{10-(-7)} = 1.1765 \Rightarrow 2\theta_p = 49.64^\circ,\quad \theta_p = 24.82^\circ\ \text{and}\ 114.82^\circ

Case 2: maximum shear stress

τmax=σ1−σ22=13.12 MPa,σ′=σx+σy2=1.50 MPa\tau_{max} = \frac{\sigma_1-\sigma_2}{2} = 13.12\ \text{MPa}, \qquad \sigma' = \frac{\sigma_x+\sigma_y}{2} = 1.50\ \text{MPa}

on planes at θs=θp±45∘\theta_s = \theta_p \pm 45^\circ, i.e. 69.82∘69.82^\circ and −20.18∘-20.18^\circ (the same plane as 159.82∘159.82^\circ).

Elements (not to scale)

 Principal element           Max-shear element
 (rotated 24.8 deg)         (rotated 69.8 deg)

   s2=-11.62                   s'=1.50
      ^                          ^   tau=13.12
      |   s1=14.62               |  /->
   <--+--> (no shear)         s'<-+-> (shear on
      |                          |  all faces)
      v                          v
   s2=-11.62                   s'=1.50

Answer: (1) σ1=14.62\sigma_1 = 14.62 MPa, σ2=−11.62\sigma_2 = -11.62 MPa at 24.82∘24.82^\circ from the xx-axis (no shear). (2) τmax=13.12\tau_{max} = 13.12 MPa with associated normal stress 1.501.50 MPa on planes at 69.82∘69.82^\circ and 159.82∘159.82^\circ.

✓ Answer checked by Claude Opus 5.5 (high effort)

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