7. semester Continuum mechanics Solution Exercise 4
7. semester Continuum mechanics Solution Exercise 4
7. semester Continuum mechanics Solution Exercise 4
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<strong>7.</strong> <strong>semester</strong> <strong>Continuum</strong> <strong>mechanics</strong><br />
<strong>Solution</strong> <strong>Exercise</strong> 4<br />
Question 1<br />
⎧<br />
⎪⎨<br />
σ ij =<br />
⎪⎩<br />
Question 2<br />
σ 0 0<br />
0 2σ 0<br />
0 0 0<br />
⎫<br />
⎪⎬<br />
⎪⎭<br />
⎧<br />
⎪⎨ σ 0 0<br />
t i = σ ij n j = 0 2σ 0<br />
⎪⎩<br />
0 0 0<br />
Question 3<br />
σnormal = t i n i =<br />
⎧<br />
⎪⎨<br />
⎪⎩<br />
⎫ ⎧<br />
⎪⎬ ⎪⎨<br />
⎪⎭ ⎪⎩<br />
σ cos θ<br />
2σ sin θ<br />
0<br />
cos θ<br />
sin θ<br />
0<br />
⎫T ⎧<br />
⎪⎬ ⎪⎨ cos θ<br />
sin θ<br />
⎪⎭ ⎪ ⎩<br />
0<br />
⎫<br />
⎪⎬<br />
⎪⎭ = ⎧<br />
⎪⎨<br />
⎪ ⎩<br />
σ cos θ<br />
2σ sin θ<br />
0<br />
⎫<br />
⎪⎬<br />
⎫<br />
⎪⎬<br />
⎪⎭<br />
(1)<br />
(2)<br />
⎪⎭ = σ cos2 θ + 2σ sin 2 θ (3)<br />
τ 2 = |t i | 2 − σ 2 normal (4)<br />
= σ 2 cos 2 θ + 4σ 2 sin 2 θ − (σ cos 2 θ + 2σ sin 2 θ) 2 (5)<br />
= σ 2 ( cos 2 θ(1 − cos 2 θ) + 4 sin 2 θ(1 − sin 2 θ) − 4 cos 2 θ sin 2 θ ) (6)<br />
= σ 2 cos 2 θ sin 2 θ = 1 4 σ2 sin 2 (2θ) (7)<br />
which gives τ = ± 1 σ sin (2θ)<br />
2<br />
Alternatively τ can be determined by projection of the stress vector t i on a vector perpendicular<br />
to n i and in the 1 − 2 plane.<br />
⎧<br />
⎪⎨ − sin θ<br />
τ = t i cos θ<br />
⎪ ⎩<br />
0<br />
⎫<br />
⎪⎬<br />
⎪⎭<br />
= σ cos θ(− sin θ) + 2σ sin θ cos θ = σ sin θ cos θ = 1 sin (2θ) (9)<br />
2<br />
In this way the sign of τ is determined directly.<br />
1<br />
(8)
2<br />
Question 4<br />
The maximum value for τ is obtained for θ = π 4 + n π 2<br />
where n takes the values 1, 2, 3, 4.<br />
τmax = ± 1 2 σ (10)<br />
The value of τmax corresponds to half the dierence between the 2 principal stresses.<br />
The corresponding normal stress is given by:<br />
σnormal = σ 1 2 + 2σ 1 2 = 1 (σ + 2σ) (11)<br />
2<br />
The value of the normal stress takes the average value of the two principal stresses.<br />
Question 5<br />
Fig. 1: Plot of normal- and shear stress<br />
PS: Observe the similarity with Mohr's circle.