A prismatic bar is carrying only an axial force. The two planes on which normal and shearing stresses are equal are inclined to the axial force at angles?

Question

Answer

Solution:

Definition:

Prismatic bar is a structural member with constant cross-sectional area along its length.

Normal Stress:

Normal stress is the component of stress that acts perpendicular to the cross-section of the bar.

Shearing Stress:

Shearing stress is the component of stress that acts parallel to the cross-section of the bar.

Plane with equal normal and shearing stresses:

When a prismatic bar is subjected to axial force, normal stresses are developed across the cross-section of the bar. Shearing stresses are also developed across the cross-section of the bar. There are two planes on which normal and shearing stresses are equal. These planes are inclined to the axial force at certain angles.

Angles of inclination:

The angles of inclination of these planes depend on the material properties of the bar. For a ductile material, the angles of inclination are 45 degrees. For a brittle material, the angles of inclination are different. The angles of inclination can be calculated using the following formulas:

θ1 = tan^-1(2μ/1-μ) and θ2 = 90 - θ1

where μ is the Poisson's ratio of the material.

Explanation:

When a prismatic bar is subjected to axial force, normal stresses are developed across the cross-section of the bar. Shearing stresses are also developed across the cross-section of the bar. There are two planes on which normal and shearing stresses are equal. These planes are inclined to the axial force at certain angles.

The angles of inclination of these planes depend on the material properties of the bar. For a ductile material, the angles of inclination are 45 degrees. For a brittle material, the angles of inclination are different. The angles of inclination can be calculated using the above-mentioned formulas.

Conclusion:

In conclusion, the two planes on which normal and shearing stresses are equal are inclined to the axial force at certain angles. These angles of inclination depend on the material properties of the bar. For a ductile material, the angles of inclination are 45 degrees. For a brittle material, the angles of inclination can be calculated using the above-mentioned formulas.

Answer

Π/4 and 3π/4

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