Determine the normal and shear stress elements, Civil Engineering

Determine the normal and shear stress elements:

The state of stress at a point is given through the stress components σx = 70 MPa, σy = 10 MPa, and τxy = - 40 MPa. Using Mohr's circle find (i) Principal stresses, and (ii) Principal planes. Also, determine the normal and shear stress elements on planes making 25o, 40o and 60o associatively along with the x plane.

838_Determine the normal and shear stress elements.png

Figure

Solution

(a)        Choose (σ, - τ) coordinate system to a suitable scale.

(b)        Mark the points X (70, - 40) and Y (10, + 40).

(c)        Draw a circle with XY as diameter. This circle cuts σ axis at A and B.

(d)        Measure the coordinates of A and B to obtain principal stresses

σ1  = 90 MPa,  σ2  = - 10 MPa

(The radius of the Mohr's circle gives τmax = 50 MPa)

(e)        Measure ∠ XOA ; Here, ∠ XOA = - 52.8o .

∴          Aspect angle Φ1 of major principal plane is - 26.4o.

(f)        Measure ∠ XOB ; Here, ∠ XOB = + 127.2o

∴          Aspect angle Φ2 of major principal plane is + 63.6o.

(g)        Draw radial lines OR, OS and OT making angles 50o, 80o and 120o with OX.

(h)        Coordinates of R give the normal and shear stress components on the plane which makes 25o with the x plane. Here,

σn  = 28 MPa, τnt  = - 48.5 MPa

(i)         Coordinates of S given the normal and shear stress components on the plane which aspect angle 40o. Here,

σn  = 5.7 MPa, τnt  = - 36.8 MPa

(j)         Coordinates of T give the normal and shear stress components on the plane with aspect angle 60o. Here,

σn  = - 9.5 MPa, τnt  = - 6.5 MPa

Posted Date: 1/30/2013 6:24:48 AM | Location : United States







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