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
Breast Walls Needed to buttress uphill slopes. Figure : Breast Check Walls Small retaining structures provided in series to arrest landslides. Gabio

Define Continuity Equation Mass in fluid systems is always conserved.   Ρ 1 v 1 = ???? 2 v 2   If the fluid density is the same at both points in the flow, the continui

A bed of sand 12 m thick is underlain by a compressible stratum of normally loaded clay, 6 m thick. The water table is at a depth of 5 m below the ground level. The bulk densities

Features of a Stratigraphical Survey phase The principal features of a Stratigraphical Survey phase are: a)  Determination of depth of ground water, thickness, extent and ap

Define in biref about the Final Prestress The stress which exists after all losses of prestress have occurred substantially is known as final prestress. This may be calculated


I would like to know if anyone can help me with my const estimating project

Write some appplications of lagrange''s mean value thorem

Question What is main purpose of conducting load test for piling works ? Answer Pile load test gives information on ultimate bearing ability but not settlement be

A construction project involves excavation below grade. A 24 inch diameter dewatering well is installed from grade (EL 586) to 80 ft below grade to lower the water table. Soil bori