Stress and Strain Explained: Uniaxial Loading, Normal and Shear Stresses, and Hooke's Law

This video introduces the core ideas of stress and strain using a simple loaded bar. Two equal opposite forces create uniaxial loading, stretching the bar and provoking internal forces that balance the external load. Normal stress, the internal force per unit area, is described as sigma and calculated as F divided by cross‑section area A. The video covers tensile versus compressive stresses, the sign convention (tensile positive), and how failure occurs when stress exceeds material strength. Strain, defined as delta L over L, is introduced along with Hooke's law for small deformations, leading to Young's modulus as a key material property. The discussion then moves to shear stress, shear strain, and the concept of stress transformation on inclined planes, concluding with a note on stress state in more complex scenarios like bending.

Overview

Stress and strain are fundamental concepts in how materials respond to external loads. This video uses a simple loaded bar to illustrate the ideas. Internal forces develop to oppose applied loads, and an imaginary cut through the bar helps reveal how equilibrium is maintained across a cross section. The discussion distinguishes normal stress, which acts perpendicular to the cross section, from shear stress, which acts parallel to it. A consistent sign convention is introduced: tensile stresses are positive and compressive stresses are negative. The material’s response to loading is connected to material strength and eventual failure if the stress exceeds the material's capacity.

Uniaxial Loading and Normal Stress

Under uniaxial loading, the internal normal stresses on the cut surface are perpendicular to the axis of the bar. Normal stress is defined as sigma and is calculated as the applied force F divided by the cross‑sectional area A (sigma = F/A). The video provides a numerical example using mild steel with a strength of about 250 megapascals and a bar diameter of 20 millimeters, showing a maximum allowable external force of roughly 79 kilonewtons before failure. The tensile nature of the loading is identified as an example of tensile stress, though the same framework applies to compressive stress with the opposite sign. The discussion also notes that, in real components, stress distributions are not perfectly uniform, but assuming uniform distribution is a useful simplification for introductory analysis.

Strain and Hooke's Law

Strain is a measure of deformation, defined as delta L divided by L, a dimensionless quantity that is often expressed as a percentage. For small, reversible deformations, the relationship between stress and strain is linear and described by Hooke's law. The ratio of stress to strain in this elastic region defines Young's modulus, a fundamental material property that characterizes stiffness. The video emphasizes that Hooke's law is most accurate for small strains, while larger strains lead to nonlinear behavior and plastic deformation where deformations are not fully recoverable when the load is removed.

Elastic, Plastic, and Material Strength

The material’s strength determines when the bar will fail. By comparing the computed stress to the material’s strength, one can predict failure. When the stress stays within the elastic region, deformations are fully reversible. Beyond the elastic limit, plastic deformation occurs, and permanent changes in shape remain after unloading. The video references broader material notions such as ductility and toughness as extensions of how materials respond to loading, while noting that these topics are explored in other videos.

Shear Stress and Shear Strain

If the loading is not aligned with the axis (for example, perpendicular to it), internal forces develop as shear stresses, denoted by the Greek letter tau. Shear stress is calculated as f over a, an average value since the stresses are not distributed uniformly. The small element analysis shows the necessity of equal and opposite shear stresses on opposite faces to maintain equilibrium and rotational stability. Shear strain, denoted by gamma, is the corresponding deformation measured as a change in angle. Hooke's law also applies to shear stresses, with the shear modulus G replacing Young's modulus in the stress‑strain relationship.

Stress Transformation and Planes

At a single point inside a body, the stress state can have normal and shear components in multiple directions. The magnitudes of these components depend on the orientation of the observation plane. In the uniaxial bar, a plane perpendicular to the axis sees only normal stress and no shear stress, but an inclined plane would exhibit both normal and shear components. The video introduces the two‑dimensional and three‑dimensional stress elements used to represent the stresses acting at a point, illustrating how the full stress state is a combination of normal and shear components.

Putting It All Together

The concepts introduced in this video lay the groundwork for more advanced topics such as torsion and beam bending. The material emphasizes the interconnectedness of stress and strain and points to further videos on stress transformation and material properties. Readers are encouraged to explore how these ideas extend to real-world problems in structural design, mechanical components, and safety assessments.