---
title: "Stress at a point and notation"
subject: "Mechanics of Solids"
chapter: 2
key: "semester_5/mechanics_of_solids:stress-notation-symmetry-shear"
source: https://www.bctengineer.com/q/semester_5/mechanics_of_solids/stress-notation-symmetry-shear
site: BCT Engineering
---

# Stress at a point and notation

Mechanics of Solids · Chapter 2: Stress Tensor

- Asked in: year not given, 6 marks
- Video: [Modue 2 Part 1 02 3D Stress And Sign Convention](https://www.youtube.com/watch?v=y8wnKNC_O4U) — VTU e-Shikshana Programme
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/mechanics_of_solids/stress-notation-symmetry-shear
- Whole chapter: https://www.bctengineer.com/md/semester_5/mechanics_of_solids/chapter-2

## Question

Define stress at a point. Explain the stress tensor notation and sign convention. Prove that the shear stress components are complementary, i.e. $\tau_{xy}=\tau_{yx}$.

## Short answer

Stress at a point is force per unit area on an infinitesmal plane. The stress tensor σ_ij uses first index for face normal, second for stress direction; tension is positive. Shear components are complementary (τ_xy=τ_yx) because moment equilibrium of an element requires equal magnitudes to balance couples.

## Answer

#### Stress at a point

Cut the body by a plane through point $P$. If a small area $\Delta A$ around $P$ carries a resultant force $\Delta \mathbf{F}$, the **stress vector** is

$$
\mathbf{t} = \lim_{\Delta A \to 0} \frac{\Delta \mathbf{F}}{\Delta A}
$$

It depends on the orientation of the plane. The state of stress at $P$ is fully defined by the stresses on three mutually perpendicular planes, i.e. by the **stress tensor**.

#### Notation and sign convention

$\sigma_{ij}$: first subscript $i$ = direction of the normal to the face; second subscript $j$ = direction of the stress component. Normal stresses are written $\sigma_x, \sigma_y, \sigma_z$, shear stresses $\tau_{xy}$, etc.

$$
[\sigma] = \begin{bmatrix} \sigma_x & \tau_{xy} & \tau_{xz} \\ \tau_{yx} & \sigma_y & \tau_{yz} \\ \tau_{zx} & \tau_{zy} & \sigma_z \end{bmatrix}
$$

- A face is **positive** if its outward normal points in a positive axis direction.
- A stress component is **positive** if it acts in the positive axis direction on a positive face, or in the negative direction on a negative face.
- So tension is positive and compression negative; positive shear points the same way as the axis on a positive face.

#### Proof of $\tau_{xy}=\tau_{yx}$

Take an element $dx \times dy \times dz$ in equilibrium under plane stress (no body moments).

- Right and left faces (area $dy\,dz$): shear $\tau_{xy}$ acts up on the right face and down on the left face, forming a couple $(\tau_{xy}\,dy\,dz)\,dx$ (anticlockwise).
- Top and bottom faces (area $dx\,dz$): shear $\tau_{yx}$ acts to the right on the top face and to the left on the bottom face, forming a couple $(\tau_{yx}\,dx\,dz)\,dy$ (clockwise).

Moments about the z-axis through the centre:

$$
\begin{aligned}
\sum M_z &= 0 \\
(\tau_{xy}\, dy\, dz)\, dx - (\tau_{yx}\, dx\, dz)\, dy &= 0 \\
\tau_{xy} &= \tau_{yx}
\end{aligned}
$$

In the same way $\tau_{yz}=\tau_{zy}$ and $\tau_{zx}=\tau_{xz}$. Hence the stress tensor is **symmetric** and has only six independent components, not nine.

## Check yourself

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

1. How is the stress vector at a point defined?
   A. Total force divided by total area of the body
   B. Limit of dF/dA as the area shrinks to zero, depending on plane orientation
   C. Force times area
   D. Average stress on all planes through the point
   Answer: B. Stress is the limit of force per area as the area tends to zero, and it depends on the orientation of the plane. (Key point: Stress vector definition)

2. In sigma_ij (or tau_xy), what do the subscripts mean?
   A. First is the direction of the stress, second is the face normal
   B. Both give the direction of the normal
   C. First is the face normal, second is the stress direction
   D. They give the point's coordinates
   Answer: C. The first subscript is the normal to the face and the second is the direction of the stress component. (Key point: Subscript notation)

3. What gives tau_xy = tau_yx in the proof?
   A. Moment equilibrium of the element about the z-axis
   B. Force equilibrium along x
   C. Conservation of energy
   D. Hooke's law
   Answer: A. The two shear couples must balance (sum Mz = 0), which gives tau_xy dy dz dx = tau_yx dx dz dy. (Key point: Proof of complementary shear)

4. How many independent stress components remain after shear symmetry?
   A. Nine
   B. Three
   C. Four
   D. Six
   Answer: D. The symmetric tensor has three normal and three shear independent components, so six. (Key point: Independent components)

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