A rectangular block of material is subjected to a tensile stress of
100MPa on a plane and a tensile stress of 40MPa on the Plane at right
angles to the former, together with a shear stress of 60MPa on the same
planes. Find (i) the direction of principal planes (ii) magnitude of
principal stresses, and (iii) magnitude of the greatest shear.
To determine the direction of principal planes, magnitude of principal stresses, and magnitude of the greatest shear, we need to perform a few calculations. Let's break it down step by step:
Step 1: Identify the stress components:
- Tensile stress on Plane 1: σ1 = 100 MPa
- Tensile stress on Plane 2: σ2 = 40 MPa
- Shear stress on the same planes: τ = 60 MPa
Step 2: Calculate the average normal stress and the difference in normal stresses:
- Average normal stress: σavg = (σ1 + σ2) / 2
- Difference in normal stresses: Δσ = (σ1 - σ2) / 2
Step 3: Calculate the principal stresses:
- Principal stress 1: σ1' = σavg + sqrt(Δσ^2 + τ^2)
- Principal stress 2: σ2' = σavg - sqrt(Δσ^2 + τ^2)
Step 4: Determine the direction of principal planes:
The direction of the principal planes is perpendicular to the direction of the principal stresses. Since we have the tensile stress on Plane 1 as σ1', the principal plane 1 will be perpendicular to Plane 1. Similarly, Plane 2 will be perpendicular to Plane 2.
Step 5: Calculate the magnitude of the greatest shear:
- The greatest shear stress occurs on the planes inclined at 45 degrees to the principal planes.
- Magnitude of the greatest shear stress: τmax = (σ1' - σ2') / 2
Let's calculate the values:
Step 2:
σavg = (100 MPa + 40 MPa) / 2 = 70 MPa
Δσ = (100 MPa - 40 MPa) / 2 = 30 MPa
Step 3:
σ1' = 70 MPa + sqrt(30 MPa^2 + 60 MPa^2) = 70 MPa + sqrt(900 MPa^2 + 3600 MPa^2) = 70 MPa + sqrt(4500 MPa^2) = 150 MPa
σ2' = 70 MPa - sqrt(30 MPa^2 + 60 MPa^2) = 70 MPa - sqrt(900 MPa^2 + 3600 MPa^2) = 70 MPa - sqrt(4500 MPa^2) = -10 MPa
Step 4:
The direction of principal plane 1 is the same as Plane 1 (the plane with a tensile stress of 100 MPa).
The direction of principal plane 2 is the same as Plane 2 (the plane with a tensile stress of 40 MPa).
Step 5:
τmax = (150 MPa - (-10 MPa)) / 2 = 160 MPa / 2 = 80 MPa
So, the answers are:
(i) The direction of principal planes: Plane 1 and Plane 2
(ii) The magnitude of principal stresses: σ1' = 150 MPa, σ2' = -10 MPa
(iii) The magnitude of the greatest shear: τmax = 80 MPa