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
Determine the final stresses in the bolt: A steel bolt of diameter 12 mm and length 175 mm is used to clamp a brass sleeve of length 150 mm to a rigid base plate as in Figure.

Maintenance of bituminous roads

Hello I have a take home final needs be ordered, do you have very well skilled professionals to make it?

Q. Glazed tiles for Construction Material? Glazed tiles are only meant for dadoes of bath room, Toilet and kitchens etc as they have no strength . Normally 8'' x 4'' g

What are the assumptions made in the theory of pure bending

Dead weight Dead loads are the weight of permanent components of the structures such as the weight of the elements and spheres, casing weight, weight of the false ceiling a

What is Atterberg Limits tests? The Atterberg Limits tests are designed to determine the plasticity of finer grained soils. These tests are performed on the fraction less than

Test on specially moulded test piece The test piece shall be moulded by the manufacturer with identical compound and under identical vulcanizing conditions as used in manufact

infomation about differnt brigdes