A steel plate is subjected to tensile stresses of 200MPa and 150 MPa at right angles to each other. Determine the normal and tangential stresses on a plane inclined at 60 to the 200 MPa stress. Also find the plane on which the resultant stress has maximum Obliquity.
To determine the normal and tangential stresses on a plane inclined at 60 to the 200 MPa stress, we first need to calculate the resolved stresses.
1. Normal stress (σn) on the inclined plane:
- The normal stress on the inclined plane can be calculated using the formula:
σn = σx * cos²θ + σy * sin²θ
where σn is the normal stress, σx is the stress acting in the x-direction (200 MPa), σy is the stress acting in the y-direction (150 MPa), and θ is the angle between the plane and the x-direction (60 degrees).
- Let's substitute the values into the formula:
σn = 200 MPa * cos²60 + 150 MPa * sin²60
σn = 200 MPa * (1/2)² + 150 MPa * (√3/2)²
σn = 200 MPa * 1/4 + 150 MPa * 3/4
σn = 50 MPa + 112.5 MPa
σn = 162.5 MPa
Therefore, the normal stress on the inclined plane is 162.5 MPa.
2. Tangential stress (τ) on the inclined plane:
- The tangential stress on the inclined plane can be calculated using the formula:
τ = -σx * sinθ * cosθ + σy * sinθ * cosθ
where τ is the tangential stress, σx is the stress acting in the x-direction (200 MPa), σy is the stress acting in the y-direction (150 MPa), and θ is the angle between the plane and the x-direction (60 degrees).
- Let's substitute the values into the formula:
τ = - 200 MPa * sin60° * cos60° + 150 MPa * sin60° * cos60°
τ = -200 MPa * (√3/2) * (1/2) + 150 MPa * (√3/2) * (1/2)
τ = -200 MPa * (√3/4) + 150 MPa * (√3/4)
τ = -100√3 MPa + 75√3 MPa
τ = -25√3 MPa
Therefore, the tangential stress on the inclined plane is -25√3 MPa.
To find the plane on which the resultant stress has maximum obliquity, we can use the concept of Mohr's circle. Mohr's circle provides a graphical representation of the state of stress on a plane.
1. Construct a Mohr's circle based on the given stresses:
- In Mohr's circle, plot the normal stresses (σx and σy) on the x-axis and the shear stress (τxy) on the y-axis.
- Plot the points (σx, τxy) and (σy, -τxy) on the circle.
- The center of the circle represents the average stress (σavg) and the radius represents the difference between the maximum and minimum normal stresses.
- Draw the circle.
2. Identify the plane with maximum obliquity:
- The plane with the maximum obliquity will be the one that corresponds to the point farthest away from the center of the Mohr's circle.
- The angle between this plane and the x-axis represents the inclination angle (θ) of the plane with respect to the x-axis.
- Measure this angle from the x-axis.
By following these steps and constructing Mohr's circle, you can determine the plane with the maximum obliquity.