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
Micro-organism ratio: Determine the F/M ratio (in units of lbs BOD5/lb MLSS-day) using data provided in the above example problem. Solution By definition, F/M = Q S

Sometimes side of concrete bridges is observed to turn black in colour . What is the main reason being for the phenomenon ?


Q. Explain about Best hydraulic section? The best hydraulic section of an open channel is considered by provision of maximum discharge with a given cross sectional area. Chann

Q. Minimum and maximum distance between bars? Minimum distance between bars and maximum distance between bars In some codes, a minimum distance amid bars is specified to

Q. Illustrate the Design discharge for foundation? To provide for an adequate margin of safety, the foundation and protective works of a bridge should be designed for a flood d


The process of installation of elastomeric bearing differs in case of precast vis-a-vis cast-in-situ girders. For precast girders, the bearing may be fixed to the underside of the

Necessity for Road Drainage: Poor drainage can cause damage to the various components of a road in the following manner: (a) Water can enter the various pavement layers th