At a point in a strained material the principal stresses are 100 MPa (tensile) and 60 MPa (compressive). Determine the normal stress, shear stress and resultant stress on a plane inclined at 50° to the axis of major principal stress. Also determine the maximum shear stress at a point. what will be the solution
Question
At a point in a strained material the principal stresses are 100 MPa (tensile) and 60 MPa (compressive). Determine the normal stress, shear stress and resultant stress on a plane inclined at 50° to the axis of major principal stress. Also determine the maximum shear stress at a point.
what will be the solution
Solution
To determine the normal stress, shear stress, and resultant stress on a plane inclined at 50° to the axis of major principal stress, we can use the equations for stress transformation.
- Normal Stress (σn): The normal stress on the inclined plane can be calculated using the formula: σn = (σ1 + σ2) / 2 + (σ1 - σ2) / 2 * cos(2θ) where σ1 and σ2 are the principal stresses and θ is the angle between the inclined plane and the axis of major principal stress.
Given: σ1 = 100 MPa (tensile) σ2 = -60 MPa (compressive) θ = 50°
Substituting the values into the formula: σn = (100 + (-60)) / 2 + (100 - (-60)) / 2 * cos(2 * 50°) = 20 MPa
Therefore, the normal stress on the inclined plane is 20 MPa.
- Shear Stress (τ): The shear stress on the inclined plane can be calculated using the formula: τ = (σ1 - σ2) / 2 * sin(2θ) where σ1 and σ2 are the principal stresses and θ is the angle between the inclined plane and the axis of major principal stress.
Given: σ1 = 100 MPa (tensile) σ2 = -60 MPa (compressive) θ = 50°
Substituting the values into the formula: τ = (100 - (-60)) / 2 * sin(2 * 50°) = 80 MPa
Therefore, the shear stress on the inclined plane is 80 MPa.
- Resultant Stress (σr): The resultant stress on the inclined plane can be calculated using the formula: σr = √(σn^2 + τ^2) where σn is the normal stress and τ is the shear stress.
Given: σn = 20 MPa τ = 80 MPa
Substituting the values into the formula: σr = √(20^2 + 80^2) = √(400 + 6400) = √6800 ≈ 82.46 MPa
Therefore, the resultant stress on the inclined plane is approximately 82.46 MPa.
- Maximum Shear Stress (τmax): The maximum shear stress at a point can be calculated using the formula: τmax = (σ1 - σ2) / 2 where σ1 and σ2 are the principal stresses.
Given: σ1 = 100 MPa (tensile) σ2 = -60 MPa (compressive)
Substituting the values into the formula: τmax = (100 - (-60)) / 2 = 80 MPa
Therefore, the maximum shear stress at the point is 80 MPa.
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