---
title: "Sum of normal stresses equals sum of principal stresses"
subject: "Strength of Materials"
chapter: 4
key: "semester_3/strength_of_materials:sum-normal-stresses-equals-principal-sum"
source: https://www.bctengineer.com/q/semester_3/strength_of_materials/sum-normal-stresses-equals-principal-sum
site: BCT Engineering
---

# Sum of normal stresses equals sum of principal stresses

Strength of Materials · Chapter 4: Stress and Strain Analysis

- Asked in: 2066 Chaitra (old course), 6 marks
- Video: [Understanding Stress Transformation and Mohr's Circle](https://www.youtube.com/watch?v=_DH3546mSCM) — The Efficient Engineer
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/strength_of_materials/sum-normal-stresses-equals-principal-sum
- Whole chapter: https://www.bctengineer.com/md/semester_3/strength_of_materials/chapter-4

## Question

Prove $\sigma_x + \sigma_y = \sigma_1 + \sigma_2$ [Sum of normal stresses is equal to sum of principal stresses]

## Short answer

Adding the formulas for principal stresses, the radical terms cancel, giving σ1+σ2 = σ_x+σ_y. This sum, the first stress invariant, remains constant for all orientations of axes.

## Answer

**To prove:** $\sigma_x + \sigma_y = \sigma_1 + \sigma_2$.

The principal stresses for a state of plane stress $(\sigma_x, \sigma_y, \tau_{xy})$ are

$$
\sigma_1 = \frac{\sigma_x+\sigma_y}{2} + \sqrt{\left(\frac{\sigma_x-\sigma_y}{2}\right)^2 + \tau_{xy}^2}, \qquad
\sigma_2 = \frac{\sigma_x+\sigma_y}{2} - \sqrt{\left(\frac{\sigma_x-\sigma_y}{2}\right)^2 + \tau_{xy}^2}
$$

Adding the two equations, the square-root terms cancel:

$$
\sigma_1 + \sigma_2 = 2\times\frac{\sigma_x+\sigma_y}{2} = \sigma_x + \sigma_y
$$

**Proof from the transformation equations (any plane):** the normal stress on a plane at $\theta$ is

$$
\sigma_\theta = \frac{\sigma_x+\sigma_y}{2} + \frac{\sigma_x-\sigma_y}{2}\cos2\theta + \tau_{xy}\sin2\theta
$$

and on the perpendicular plane ($\theta + 90^\circ$), $\cos 2(\theta+90^\circ) = -\cos2\theta$ and $\sin 2(\theta+90^\circ) = -\sin2\theta$:

$$
\sigma_{\theta+90^\circ} = \frac{\sigma_x+\sigma_y}{2} - \frac{\sigma_x-\sigma_y}{2}\cos2\theta - \tau_{xy}\sin2\theta
$$

$$
\sigma_\theta + \sigma_{\theta+90^\circ} = \sigma_x + \sigma_y = \text{constant}
$$

The principal planes are a pair of perpendicular planes, so $\sigma_1 + \sigma_2 = \sigma_x + \sigma_y$.

**Mohr's circle view:** $\sigma_x$ and $\sigma_y$ are the abscissas of the two ends of a diameter, whose mid-point (centre) is at $\frac{\sigma_x+\sigma_y}{2}$; $\sigma_1$ and $\sigma_2$ are the ends of the horizontal diameter, with the same centre. Hence twice the centre abscissa equals both sums.

This sum is called the first stress invariant: it does not change with the orientation of the axes.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. How does adding the two principal stress expressions prove the result?
   A. The square-root terms cancel, leaving 2 x (sx+sy)/2
   B. The shear terms add to zero because txy = 0
   C. The averages cancel, leaving 2 x the root term
   D. The root terms double, leaving sx - sy
   Answer: A. sigma_1 has +root and sigma_2 has -root, so only twice the average remains. (Key point: Method one)

2. On planes at theta and theta + 90°, what happens to the cos2theta and sin2theta terms?
   A. They stay unchanged and double
   B. They change sign, so they cancel when the stresses are added
   C. They interchange with each other
   D. They become zero for both planes
   Answer: B. cos2(theta+90°) = -cos2theta and sin2(theta+90°) = -sin2theta, giving a constant sum. (Key point: Perpendicular planes)

3. What does sx + sy = s1 + s2 imply?
   A. The shear stress is always zero
   B. The principal stresses equal sx and sy
   C. The sum of normal stresses on perpendicular planes is invariant with orientation
   D. The sum changes with axis rotation
   Answer: C. It is the first stress invariant, the same for any pair of perpendicular planes. (Key point: Meaning of the result)

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