Determining Normal and Shear Stress on an Oblique Plane

Introduction
When a body is subjected to external forces, it experiences internal stresses that act on different planes within the material. These stresses can be categorized as normal stress, which is perpendicular to the plane, and shear stress, which is parallel to the plane. In this context, we will determine the normal and shear stresses on an oblique plane inclined at 45 degrees when the body is subjected to equal and unlike normal stresses on two mutually perpendicular directions passing through that point.

Normal Stress
Normal stress, denoted by σ, is the force acting per unit area perpendicular to the plane. On an oblique plane, the normal stress can be resolved into two components: one along the direction of the plane and the other perpendicular to it.

Shear Stress
Shear stress, denoted by τ, is the force acting per unit area parallel to the plane. On an oblique plane, the shear stress can also be resolved into two components: one along the direction of the plane and the other perpendicular to it.

Determining Normal Stress on an Oblique Plane
To determine the normal stress on an oblique plane inclined at 45 degrees, we consider a small element of the material with sides dx, dy, and dz. The normal stress along the plane is given by σ = (F * cosθ) / A, where F is the force acting on the plane, θ is the angle between the force and the plane, and A is the area of the plane. In this case, the angle θ is 45 degrees.

Determining Shear Stress on an Oblique Plane
To determine the shear stress on an oblique plane inclined at 45 degrees, we consider the same small element of the material. The shear stress along the plane is given by τ = (F * sinθ) / A. In this case, the angle θ is 45 degrees.

Equal and Unlike Normal Stresses on Two Perpendicular Directions
When the body is subjected to equal and unlike normal stresses on two mutually perpendicular directions passing through a point, the normal stresses acting on the oblique plane will also be equal and unlike. However, the shear stresses acting on the plane will be zero, as the forces are acting along the normal direction.

Conclusion
In conclusion, when an oblique plane inclined at 45 degrees is subjected to equal and unlike normal stresses on two mutually perpendicular directions passing through that point, the normal stresses on the plane will be equal and unlike. The shear stresses on the plane will be zero in this case. The normal stress can be determined using the formula σ = (F * cosθ) / A, and the shear stress can be determined using the formula τ = (F * sinθ) / A, where θ is the angle between the force and the plane.