1: Normal stress

1: Normal Stress

Normal stress is the most fundamental type of stress encountered in mechanics of materials. It arises when an internal force acts perpendicularly (normal) to the cross-sectional area of a material element. This force tends to either stretch the material (tension) or shorten/squeeze it (compression). Understanding normal stress is crucial because it forms the basis for analyzing the strength and deformation of structural components like rods, columns, struts, and beams under axial loads.

The defining characteristic of normal stress is its direction: it acts perpendicular to the plane on which it is being considered. We denote normal stress using the Greek letter sigma ($\sigma$). Its magnitude is calculated as the intensity of internal force per unit area acting normal to that cross-section.

The Formula:

The average normal stress ($\sigma_{avg}$) on a cross-section is given by: $\sigma_{avg} = \frac{P}{A}$ Where:

• P is the magnitude of the resultant internal normal force acting perpendicularly through the centroid of the cross-sectional area. This force is developed within the material in response to external loads.
• A is the original cross-sectional area before deformation occurs.

Sign Convention:

• Tensile Stress (Positive, $+\sigma$): Occurs when the internal force P pulls away from the material element on the cross-section. Tensile stress tends to elongate the material. We conventionally assign a positive value to tensile normal stress.
• Compressive Stress (Negative, $-\sigma$): Occurs when the internal force P pushes into the material element on the cross-section. Compressive stress tends to shorten the material. We conventionally assign a negative value to compressive normal stress. Always pay close attention to sign convention in calculations and diagrams.

Units:

Normal stress is measured in units of force per unit area. The SI unit is the Pascal (Pa), equal to one Newton per square meter (N/m²). Due to the large magnitudes often encountered, prefixes like kiloPascal (kPa = $10^3$ Pa), MegaPascal (MPa = $10^6$ Pa), and GigaPascal (GPa = $10^9$ Pa) are common. In the US Customary system, pounds per square inch (psi) or kilopounds per square inch (ksi) are used.

Uniform Distribution Assumption:

For long, straight, prismatic (constant cross-section) members subjected to concentric axial loads (forces acting through the centroid of every cross-section), the normal stress distribution can often be assumed uniform across the entire cross-section. This means the average stress ($\sigma_{avg} = P/A$) accurately represents the stress at every point on the section. This assumption simplifies analysis significantly and is valid away from points of load application or abrupt changes in geometry (Saint-Venant's principle). If the load is eccentric or the member is not prismatic, the stress distribution becomes non-uniform and more complex analysis is required.