What Is Longitudinal Stress? How it Works and How it Is Calculated

Longitudinal stress is a fundamental concept in engineering and materials science. It describes how materials respond to forces applied along their length. This article provides an overview of what longitudinal stress is, how it operates, and the mathematical methods used to calculate it.

What Is Longitudinal Stress?

Longitudinal stress, or axial stress, is experienced by a material when force is exerted along its longitudinal axis (the axis that corresponds to the object's length). When a material encounters such a load, it gives rise to longitudinal stress. In other words, longitudinal stress arises when applied forces cause or encourage a change in the body’s length. This stress can be categorized into two main types: tensile stress and compressive stress. Tensile stress manifests when a material is stretched or pulled along its length, whereas compressive stress emerges when the material is compressed or squashed along its length. This is illustrated in Figure 1:

Are Longitudinal Stress and Shear Stress the Same?

No. Longitudinal stress and shear stress represent two distinct forms of mechanical stress that can act on materials. Longitudinal stress occurs when forces or loads are applied in the direction of the object’s length — i.e., along its longitudinal axis. This type of stress leads to either stretching (tensile stress) or compression (compressive stress) of the material along its length. In contrast, shear stress arises when two parallel but offset forces are applied parallel to a given cross-sectional area of material. Shear stress results in a deformation where the material's layers slide or shear past each other. You can, for example, apply longitudinal stress on a sheet of paper by pulling its two ends apart. Scissors, on the other hand, impart shear stress on the paper. These two types of stress have different effects on materials, as seen in Figure 2:

To learn more, see our guide on What is Shear Stress.

Is Young's Modulus Longitudinal Stress?

No, Young’s modulus, sometimes referred to as the modulus of elasticity, is not the same as longitudinal stress. Instead, Young’s modulus is the longitudinal stress divided by the longitudinal strain. 

What Is the Importance of Longitudinal Stress?

Understanding longitudinal stress is crucial for structural analysis, material selection, and ensuring the safety and durability of various engineering components and structures. Engineers rely on a thorough understanding of longitudinal stress to design structures that can withstand external forces without failure. This ensures the safety of buildings, bridges, machinery, and other critical infrastructure. Material selection hinges on the material's ability to handle tensile and compressive stresses along the item’s length. This consideration ensures that the part can bear its anticipated loads without deforming excessively or compromising safety. Additionally, longitudinal stress analysis plays a key role in predicting material durability and service life. It helps prevent premature fatigue and degradation. Aerospace and automotive manufacturers also employ longitudinal stress testing as part of their quality control measures. 

How Does Longitudinal Stress Work?

Longitudinal stress describes how a material responds to forces along its length. Imagine a cylinder with equal and opposite forces applied to its ends. This stress is known as longitudinal stress. Calculate it using the formula: 

Longitudinal Stress (σ) = Deforming Force (F) / Area of Cross-section (A)

Longitudinal stress occurs when the deforming force acts parallel to the object’s length and both opposing force vectors run through the same axis. It leads to a change in the body's length, and depending on the material's characteristics, it may also cause a change in diameter. Longitudinal stress can be further classified into two main types based on the direction of the deforming force: tensile stress, which elongates the material, and compressive stress, which compresses it. 

How Does Longitudinal Stress Differ From Other Types of Stress?

Longitudinal stress differs from other types of stress based on the direction of applied forces and their effects on materials. Longitudinal stress is a type of normal stress that specifically relates to forces that act in an axial direction. Such forces are commonly observed in rods or bars under tension or compression. This type of stress results in either elongation or shortening of the material.

In contrast, shear stress arises when opposing forces act parallel to each other but in opposite directions. The parallel forces act on axes that are offset from one another. It causes a shearing or sliding deformation within the material. Pure shear stress alters the shape without changing the volume and is distinct from longitudinal stress due to its direction and mode of deformation.

There are other types of stress as well. Bending stress arises from bending moments and results in a gradient of both tensile and compressive stresses spread across the material's cross-section. Torsional stress occurs because of twisting or rotational forces, leading to shear stresses within the material that act in radial directions. Volumetric stress affects objects in all three dimensions, changing their volume. Additionally, fatigue stress is a phenomenon associated with cycles of loading and unloading, leading to material failure over time.

What Causes Longitudinal Stress?

Longitudinal stress is induced when two cross-sectional areas of a cylinder or other body experience equal and opposite forces. The applied force operates in the direction of the body's length, potentially causing a change in that length. Longitudinal stress, therefore, leads to either stretching (tensile stress) or compression (compressive stress) of the material along its length.

How Does Longitudinal Stress Differ From Normal Stress?

Normal stress occurs when forces are applied perpendicular to the cross-sectional area of the body. These forces can either lead to tensile stress, causing the material to elongate, or compressive stress, causing it to shorten. The type depends on whether the forces are pulling or pushing against the material's surface. Normal stress can be subdivided into two categories: longitudinal stress and bulk stress (or volumetric stress). 

Longitudinal stress is a type of stress that aligns with the length of an item such as a bar. When the cross-sectional area is perpendicular to the bar's length, the normal stress acting on that section is also considered longitudinal. Volumetric stress, by contrast, acts in all directions on the body at once. It can alter the body’s volume as a result. Unlike normal stress, which primarily affects the length of a material, and tangential stress, which alters the shape of the body, volume stress specifically focuses on changes in the body's overall volume.

To learn more, see our guide on Material Stress.

What Is the Formula for Longitudinal Stress?

The formula for longitudinal stress is as follows:

σ = F / A

Where:

σ = longitudinal stress (Pa or N/m²)

F = force applied along the length of the material (N)

A = cross-sectional area of the material perpendicular to the applied force (m²)

How to Calculate Longitudinal Stress?

The longitudinal stress is calculated using the formula above. First, you must measure the perpendicular cross-sectional area prior to loading. Next, assess the applied force. Insert those values into the simple formula to calculate the stress. Make sure to use consistent units (e.g., newtons and square meters). Longitudinal stress can be either tensile (positive) or compressive (negative) depending on whether the material is being stretched or compressed along its length.

What Does the Longitudinal Stress Diagram Look Like?

Figure 3 below shows the relationship between longitudinal stress and circumferential stress (hoop stress): 

These stresses occur in a cylindrical vessel with internal pressure. The hoop stress is supported by the cylinder wall’s perimeter. Conversely, the longitudinal stress operates in the axial direction, perpendicular to the hoop stress.

How Does Longitudinal Stress Affect the Quality of Materials?

Longitudinal stress can directly influence the quality of materials. When they are subjected to longitudinal stress, it can lead to deformation, either elongation (under tensile stress) or shortening (under compressive stress). Excessive or prolonged stress beyond a material's yield strength can result in permanent deformation or failure, leading to dimensional inaccuracies, structural instability, and compromised material capabilities.

Moreover, repeated or cyclic longitudinal stress, especially if it surpasses the material's fatigue limit, can eventually cause fatigue failure. This can manifest as cracks, fractures, or other forms of material damage, reducing the item’s quality and service life.

How Often Does Longitudinal Stress Occur in Steel?

Steel is frequently subjected to longitudinal stress because it’s employed in so many engineering and structural operations. It’s a very durable and versatile material so steel gets used for tensile purposes like steel cables, beams, or rods. On the other side, steel gets compressed when it’s in column and structural support form. This phenomenon is not limited to structural applications; it also extends to mechanical components and machinery where steel parts can experience longitudinal stress during operation. In construction, designing steel-framed structures like buildings and bridges requires careful consideration of how longitudinal stress will affect material behavior under different loading conditions. Therefore, longitudinal stress is a common and essential consideration when working with steel.

To learn more, see our guide on Steel Properties.

Does Longitudinal Stress Occur in Titanium?

Yes, longitudinal stress can occur in titanium (and every other structural material), just as it does in steel when the material is subjected to forces or loads applied along the part’s longitudinal axis. Titanium, like steel, is a versatile and strong metal used in engineering, aerospace, and medical applications. The occurrence of longitudinal stress in titanium depends on the specific application and the nature of the forces applied to it. 

For example, titanium components in aircraft and spacecraft experience longitudinal stress caused by gravity and impinging airflow. It gets particularly extreme during maneuvers and hard landings. Similarly, in the medical field, titanium implants used in bone or joint replacement procedures can expect to see regular longitudinal stresses. 

What Is An Example of Longitudinal Stress?

An example of longitudinal stress is the stretching of a steel cable when it is tensioned. When you apply a pulling force to a steel cable, the internal longitudinal stresses within the cable act in the same direction as its length. The cable will tend to elongate or stretch along its length in response to the applied force.

Another example is the compression of buildings’ steel support columns. In this case, the forces act in a compressive manner along the length of the column, causing it to shorten or contract. This represents another instance of longitudinal stress, this time in compressive form.

What Is the Difference Between Longitudinal Stress and Tensile Stress?

Longitudinal stress and tensile stress are related concepts — one is a subset of the other. Longitudinal stress is a broader term that encompasses both tensile stress and compressive stress. It refers to the stress that develops within a material when it is loaded along its longitudinal axis, which is aligned with its length. Longitudinal stress can either stretch the material (tensile stress) or compress it (compressive stress) along that length, depending on the direction of the applied forces.

Tensile stress is a specific type of longitudinal stress that occurs when external forces act to pull or stretch the object along its longitudinal axis. Tensile stress may cause the material to elongate as it resists the applied force. It is characterized by forces attempting to pull the material apart.

To learn more, see our guide on What is Tensile Stress.

Summary

This article presented longitudinal stress, explained it, and discussed how it works and is calculated. To learn more about longitudinal stress, contact a Xometry representative.

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