Understanding the Sum of Normal Stress in Compound Systems

In the field of mechanics of materials, a compound stress system describes a situation where stresses are applied from multiple directions or on various planes within a material. A fundamental concept when analyzing these systems is understanding how the normal stresses behave. Specifically, we look at the sum of normal stresses acting on specific planes.

Stress Transformation Principles

Let's consider a 2D stress element. We have normal stresses $\sigma_x$ acting perpendicular to the x-face and $\sigma_y$ acting perpendicular to the y-face. Additionally, there might be shear stresses, denoted as $\tau_{xy}$. When we examine a plane that is inclined at an angle $\theta$ relative to the x-axis, the normal stress ($\sigma_n$) acting on this inclined plane can be calculated using the stress transformation equations:

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

Analyzing Perpendicular Planes

To understand the property of the sum of normal stresses, we need to consider two perpendicular planes. If one plane is at an angle $\theta$, the plane perpendicular to it is at angle $\theta + 90^\circ$. The normal stress ($\sigma'_n$) on this perpendicular plane is:

$$ \sigma'_n = \frac{\sigma_x + \sigma_y}{2} + \frac{\sigma_x - \sigma_y}{2} \cos(2(\theta + 90^\circ)) + \tau_{xy} \sin(2(\theta + 90^\circ)) $$

Using the trigonometric identities $\cos(A + 180^\circ) = -\cos(A)$ and $\sin(A + 180^\circ) = -\sin(A)$, the equation becomes:

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

Now, we sum the normal stresses on both perpendicular planes ($\sigma_n + \sigma'_n$):

$$ \sigma_n + \sigma'_n = \left( \frac{\sigma_x + \sigma_y}{2} + \frac{\sigma_x - \sigma_y}{2} \cos(2\theta) + \tau_{xy} \sin(2\theta) \right) + \left( \frac{\sigma_x + \sigma_y}{2} - \frac{\sigma_x - \sigma_y}{2} \cos(2\theta) - \tau_{xy} \sin(2\theta) \right) $$

Notice that the terms containing $\cos(2\theta)$ and $\sin(2\theta)$ cancel each other out:

$$ \sigma_n + \sigma'_n = \frac{\sigma_x + \sigma_y}{2} + \frac{\sigma_x + \sigma_y}{2} $$

$$ \sigma_n + \sigma'_n = \sigma_x + \sigma_y $$

The Significance of the Sum

The equation $\sigma_n + \sigma'_n = \sigma_x + \sigma_y$ reveals a crucial property of stress systems. The sum of the normal stresses on any pair of perpendicular planes within a compound stress system is equal to the sum of the normal stresses on the original x and y reference planes ($\sigma_x + \sigma_y$).

Since $\sigma_x$ and $\sigma_y$ represent the normal stresses on the primary axes for a given stress state, their sum is a fixed value. This means the sum of normal stresses is an invariant quantity, independent of the angle $\theta$ chosen for the planes. This sum is also equivalent to the sum of the principal stresses ($\sigma_1 + \sigma_2$), which is a fundamental invariant in stress analysis.

Conclusion on Normal Stress Sum

Based on the derivation, the sum of normal stresses acting on any two perpendicular planes in a compound stress system remains constant throughout the analysis.