Evaluate the principal stresses and principal planes, Civil Engineering

Evaluate the principal stresses and principal planes:

The state of stress at a point in a loaded solid is prescribed on two faces of an element whose shape is a triangular prism as display in Figure. Evaluate the principal stresses and principal planes.

20_Evaluate the principal stresses and principal planes.png

Figure

Solution

Here, we have to first obtain the value of x and subsequent calculations will be a standard set.

Given   σy = 48 MPa

τyx = 36 MPa            ∴  τxy = - 36 MPa

σ30o = 60 MPa (aspect angle of plane BC is + 30o)

σ30° = (σxy/2) +(σxy/2) cos (2× 30°) +τxy sin (2 × 30°)

60 = (σx + 48/2) + (σx  - 48/2) cos 60o  + (- 36) sin 60o

60 = σx/2 + 24 +σx/2 cos 60° -24 cos 60° -36 sin 60°

i.e. 60 = σx/2 (1+ cos 60°) +24 -24 cos 60° -36 sin 60°

∴          σx =2/1+0.5 [60-24-(-24×0.5)-(-36)× 0.866]

= 2/1.5[79.177] = 105.57 MPa

581_Evaluate the principal stresses and principal planes1.png

∴ σ1 = 76.835 + 46.093 =122.928 MPa

σ2 = 76.835 - 46.093 =30.742 MPa

Let the aspect angles of the principal planes be Φ

tan 2Φ = 2τxyx  - σy = 2 × (- 36)/105.57 - 48 = -51.35° or 128.65°

Φ = - 25.675° or 64.325°

Substituting Φ = - 25.675o (or 2Φ = - 51.35o) in expression for σn

σn = (105.57 + 48/2 ) + (105.57 - 48/2) cos (- 51.35o ) - 36 sin (- 51.35o )

= 122.92 MPa

∴ Φ1 = - 25.675o and Φ2 = 64.325o

Posted Date: 1/30/2013 6:22:37 AM | Location : United States







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