Simple Stresses and Strains

In strength of materials we study how solid bodies behave when different kinds of forces act on them. Whenever you load a bar, a rod or any machine part, its shape or size changes a little. If the load becomes too large, the member may finally fail. To understand and design safe components, we use the basic ideas of stress and strain.

1.1 Introduction

Imagine pulling a steel rod. Your hands apply a force, so the rod tries to stretch. The material of the rod opposes this stretching because of the internal bonding between its particles. This internal opposition is what makes the material “strong”.

Up to a certain level of load the rod comes back to its original length as soon as the force is removed. This region is called the elastic region. Within this limit the internal resistance developed in the material is in step with (proportional to) the amount of deformation.

If the load is increased beyond the elastic limit, the material can no longer fully recover its original shape. Permanent deformation starts and, on further loading, the member finally fractures.

To study all these effects in a systematic way, instead of using total force and total deformation, we work with force per unit area and deformation per unit length. These two basic quantities are called stress and strain.

1.2 Stress

When an external load tries to deform a body, the material of the body develops an internal resisting force. If this internal resistance is divided by the area over which the load acts, we get stress.

Stress = (Internal resisting force) / (Loaded area)

In design work we usually take the internal resistance to be equal in magnitude to the applied external load (for equilibrium). If the external load is denoted by P and the cross-sectional area by A, the normal stress is written as:

σ = P / A

• σ : normal stress (N/m² or Pa)
• P : external load or force (N)
• A : cross-sectional area resisting the load (m²)

So stress tells us how “intense” the loading is at a particular section. Two members may carry the same total load, but the one with smaller area will experience higher stress.

1.2.1 Types of Normal Stress (Basic idea)

• Tensile stress – when the load pulls the member and it tends to elongate.
• Compressive stress – when the load pushes the member and it tends to shorten.

(Other types such as shear stress, bending stress and torsional stress are studied later, but the basic idea “force divided by area” remains the same.)

1.2.2 Units of Stress

Stress has the same dimensions as pressure: force per unit area.

• SI base unit : Pascal (Pa) 1 Pa = 1 N/m²
• In engineering, we normally use larger units: 1 kPa = 10³ Pa, 1 MPa = 10⁶ Pa.

Because machine parts and structural members often have areas expressed in mm², it is very convenient to write stress in N/mm².

1 N/mm² = 1 N / (1 mm²) = 1 N / (10^-6 m²) = 10⁶ N/m² = 1 MPa

So the common practical units are:

• Pa (N/m²) – basic SI unit
• kPa, MPa – for higher values
• N/mm² – numerically equal to MPa

In older MKS or FPS systems you may still find stresses written as kgf/cm² or lb/in² (psi), but for modern engineering calculations SI units (Pa, MPa) are preferred.

1.3 Strain

When a body is loaded, not only does internal resistance develop, but its dimensions actually change. The bar may get slightly longer, shorter, or change its shape.

Strain is a measure of this deformation relative to the original size of the body.

Strain = (Change in dimension) / (Original dimension)

Because it is a ratio of two similar quantities (for example, length divided by length), strain has no units. It is a pure number, although in practice we often express it as micro-strain (×10^-6) or as a percentage.

1.3.1 Types of Strain

Depending on how the body is loaded and how it deforms, strain can be of the following basic types:

1. Tensile strain
2. Compressive strain
3. Volumetric strain
4. Shear strain

(a) Tensile Strain

Consider a prismatic bar of original length L subjected to a tensile load that causes it to elongate by δL. The tensile strain is:

ε_t = δL / L

Here, the bar experiences increase in length along the direction of the load.

(b) Compressive Strain

If the same bar is subjected to a compressive load and its length decreases by δL, the compressive strain is also given by δL/L, but now the length is reducing. Sometimes tensile strain is taken as positive and compressive strain as negative to indicate direction of change.

(c) Volumetric Strain

In many cases a body does not only change length but its overall volume also changes (for example, a cube under uniform pressure from all sides).

If the original volume is V and the change in volume due to loading is δV, the volumetric strain is:

ε_v = δV / V

(d) Shear Strain

When a body is subjected to shear force, one layer of the material tends to slide over the adjacent layer. A rectangular block, for instance, becomes a rhombus-like shape.

Shear strain is defined as the angular distortion between two originally perpendicular faces of the element. If the right angle changes by a small angle φ (measured in radians), then:

γ = φ   (for small angles, in radians)

Here γ is the shear strain. Again, it is dimensionless because angles measured in radians are pure ratios.

1.4 Relation Between Stress and Strain (Preview)

For many engineering materials such as steel, within the elastic limit, stress is directly proportional to strain. This linear relationship is written as:

σ ∝ ε   or   σ = E ε

where E is the Young’s modulus or modulus of elasticity. Although the full discussion of elastic constants comes later, it is helpful to remember that once you know the stress, you can estimate the strain and resulting deformation using such relations.

Conclusion

In this article we have introduced the most fundamental ideas of strength of materials:

• Stress – internal resistance per unit area, caused by external loads.
• Strain – deformation per unit original dimension, a measure of how much the body changes shape or size.
• Stress has units of N/m² (Pa or MPa), while strain is dimensionless.
• Different loading conditions give rise to different types of stress and strain: tensile, compressive, shear and volumetric.