Calculate the principal stresses for the state of stress, Civil Engineering

Calculate the principal stresses for the state of stress:

Calculate the principal stresses and principal planes for the state of stress shown in Figure.

2431_Calculate the principal stresses for the state of stress.png

Figure

Solution

Given σx = 60 N/mm2

σy = 20 N/mm2

τxy = - 26 N/mm2

On substituting in Eq. (6), we get

2303_Calculate the principal stresses for the state of stress1.png

∴          σ1 = 72.8 N/mm2 and σ2 = 7.2 N/mm2

Again substituting the values σx, σy and σxy in Eq. (2).

tan 2Φ = 2τxy/σx-σy = 2×(-26)/60-20 = -1.3

Since θ is general angle, the specific angles representing the principal planes are designated as Φ1 and Φ2.

∴          2Φ = - 52.43º, 127.57º

using    2Φ = - 52.43º

σn = (60 + 20/2) + (60-20/2) cos (-52.43o)- 26 sin (- 52.43º)

=72.8 N/mm2

Hence, we recognize that Φ1 = - 52.43º/2 defines the major principal plane and therefore, Φ2 = 127.57o/2 should define the minor principal plane.

Posted Date: 1/30/2013 6:13:19 AM | Location : United States







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