Understanding Principal Planes and Shear Stress

The question asks to identify the true statement regarding shear stress on principal planes. To answer this, let's first understand what principal planes are in the context of stress analysis.

What are Principal Planes?

In a stressed body, at any given point, we can consider different planes passing through that point. On each plane, there will generally be a normal stress (perpendicular to the plane) and a shear stress (parallel to the plane). Principal planes are special planes where the shear stress is zero.

Stress on Principal Planes

On a principal plane, the stress is purely normal. This purely normal stress is called a principal stress. For a 2D stress state, there are generally two principal planes, which are mutually perpendicular. For a 3D stress state, there are three principal planes, which are mutually orthogonal.

The key characteristic of principal planes is the absence of shear stress.

Analyzing the Given Options

Let's look at the provided options:

1. Shear stress on principal planes is zero. This aligns directly with the definition of principal planes. These planes are specifically defined as having zero shear stress.
2. Shear stress on principal planes is maximum. This is incorrect. Planes of maximum shear stress exist, and they are generally oriented at 45 degrees to the principal planes (in 2D). The shear stress on principal planes is not maximum; it is zero.
3. Shear stress on principal planes is minimum. While zero is the minimum possible value for the magnitude of shear stress, stating it is "minimum" without specifying it is zero might be ambiguous, especially if considering directional shear stress which can be negative. However, the most precise and fundamental definition is that it is zero. Comparing it with option 1, option 1 is a direct definition.
4. None of these. If option 1 is true, then this option is false.

Conclusion

Based on the definition and understanding of principal planes, the shear stress acting on these planes is always zero. This is a fundamental concept in mechanics of materials and theory of elasticity.

Therefore, the statement that is true is that shear stress on principal planes is zero.

Revision Table: Stress Concepts

Additional Information: Finding Principal Planes

Principal planes and stresses can be found using various methods, such as:

• Stress Transformation Equations: By setting the equation for shear stress on an arbitrary plane to zero and solving for the angle of the plane.

  For a 2D stress state ($\sigma_x, \sigma_y, \tau_{xy}$), the shear stress ($\tau_{\theta}$) on a plane at an angle $\theta$ from the x-axis is given by:

  $\tau_{\theta} = \frac{\sigma_y - \sigma_x}{2} \sin(2\theta) + \tau_{xy} \cos(2\theta)$

  Setting $\tau_{\theta} = 0$ allows us to find the angles ($2\theta$) corresponding to the principal planes.

• Mohr's Circle: Graphically, principal stresses are represented by the points where Mohr's circle intersects the horizontal axis (the normal stress axis). On the horizontal axis, the shear stress coordinate is zero. The angles to the principal planes are also easily determined from the circle.

These methods confirm that principal planes are defined by the condition where shear stress is zero.