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







Related Discussions:- Evaluate the principal stresses and principal planes, Assignment Help, Ask Question on Evaluate the principal stresses and principal planes, Get Answer, Expert's Help, Evaluate the principal stresses and principal planes Discussions

Write discussion on Evaluate the principal stresses and principal planes
Your posts are moderated
Related Questions
Step by step rehabilitation strategies with drawing, figures, and illustrations

Question It should be necessary that raking piles of a bridge abutment be placed under an embankment ? Answer For a bridge abutment to be supported on raking piles with div


EXPLAIN THE USE OF INTERPOLATION IN CREATING CONTOUR LINE

What is WBM road. How it is constructed?


Q. Explain Gauge Conversion? All the advanced countries of the world have recognized the need for uniformity of gauge. There are a number of problems that have cropped up on th

Define the Net Positive Suction Head All values in the equations are in units of feet.  Note: Only use values related to the suction side of the pump.   NPSHR = Net Posi

Question What is sign of shear slump and collapse slump in slump tests? Answer There are three varieties of slump that might occur in slumps test, specifically, slump, shear

the jet transport B is flying north with a velocity vB = 600 km/h when a smaller aircraft A passes underneath the transport headed in the 60 degree direction shown. To passengers i