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Curved steel bar. Image Credit: Shutterstock.com/Marynchenko Oleksandr

Bending Stress: Definition, How it Works, Calculation, Types, and Examples

Xomety X
By Team Xometry
September 20, 2023
 12 min read
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Bending stress is a fundamental concept in structural engineering and materials science. It arises when external forces or moments cause an object to bend or deform. This stress can lead to both compressive and tensile forces within the material. Understanding bending stress is essential for designing safe and durable structures and optimizing material selection.

In this article, we'll delve into the key aspects of bending stress, including its types, calculations, real-world examples, and the distinction between compressive and tensile stress. 

What Is Bending Stress?

Bending stress occurs when a material is subjected to a bending load or force. This type of stress typically arises in structures or components like beams, bridges, and columns that are subjected to loads that cause them to bend or flex. Bending stress is an expression of the distribution of internal forces and moments within the beam's cross-section as it undergoes deformation.

Bending stress specifically refers to the item’s internal resistance that counteracts a bending moment. It is distributed unevenly across the cross-section, with different parts of the beam experiencing different stress levels. This non-uniform distribution comes down mainly to how the beam’s neutral axis and geometry relate to the applied force. The neutral axis is an axis within the beam's cross-section where the stress is zero. As the beam bends, the internal stress distribution causes certain parts of the beam to experience compression (negative stress) and others to experience tension (positive stress). The neutral axis separates these regions of compression and tension, and its location depends on the beam's geometry.

What Is Another Term for Bending Stress?

Bending stress is also called flexural stress. The stress is caused by a bending moment applied on a part.

Is Bending Stress the Same as Shear Stress?

No. Bending and shear are two distinct mechanical behaviors encountered in structural elements. Bending occurs when external forces act on a structural element in a way that induces it to deform by curving or flexing. Bending stress causes some parts of the material to experience tension (stretching) and others to experience compression. Shear, on the other hand, represents a different type of internal force within structural elements. Shear forces arise when external loads cause one portion of the element to shift or slide parallel to another. The internal resistance generated by these forces is referred to as shear stress, and it acts parallel to the item’s cross-section. Shear stress causes adjacent layers of material to slide past each other horizontally without bending or curving the object. Shear behavior plays a crucial role in structural stability and must be considered in engineering designs to prevent structural failures, especially in components like bolts and rivets. 

How Does Bending Stress Work?

Bending stress arises internally when an external bending force or moment is exerted upon a component. This external bending moment causes the component’s curvature to change, causing tensile and compressive stresses to accumulate in its structure. These combined stresses are commonly referred to as bending stresses. For example, picture a balance beam. It is supported at both ends and a force is applied in the middle when you walk on it. With your weight in the middle of the beam, it experiences a bending moment. This moment causes the top surface of the beam to go into compression, trying to shrink, while the bottom surface goes into tension, attempting to stretch. The material within the balance beam resists these changes, creating internal stresses. At some point between the compression and tension sides, there's a neutral axis that sees neither compression nor tension as seen in Figure 1: 


demonstration of bending stress

Bending stress demonstration of a beam bending.

Image Credit: https://www.sciencedirect.com/topics/engineering/bending-stress

Bending stress varies across the cross-section of the beam. The highest stress values occur at the top and bottom surfaces and decrease toward the neutral axis. 

What Is the Importance of Bending Stress?

Bending stress plays a very important role in the overall durability and lifespan of structures, from beams to skyscrapers to bridges. Knowledge of bending behavior empowers engineers to make informed decisions about design, materials, and construction methods, so they may optimize load-bearing capacities and minimize the risk of failure due to excessive bending stress. It's a fundamental aspect of structural engineering that directly impacts structural integrity, ensuring that structures endure and remain safe over time.

What Is the Bending Stress Formula?

Bending stress at a given point can be determined through the bending stress formula:

M/I = σ/y = E/R

Which can also be rearranged as: 

σ = My/I = Ey/R

Where:

σ = Bending stress (Pa or N/m²)

M = Bending moment applied to the beam (N·m)

y = The perpendicular distance of a specific point on the beam's cross-section from its neutral axis (m)

I = The area moment of inertia of the beam (m⁴)

E = Modulus of Elasticity (Pa or N/m²)

R = Radius of Curvature (m)

How to Calculate the Bending Stress?

Bending stress is calculated through the bending stress equation: 

σ = My/I

In this formula, M represents the moment, which is calculated by multiplying the applied force by its distance from a reference point. The variable y represents the perpendicular distance from the neutral axis to the specific point of interest, and I signify the moment of inertia for the item’s cross-sectional shape. It's essential to emphasize that this equation is valid only if we assume the material behaves elastically and adheres to Hooke's law. Hooke’s law describes the linear relationship between stress and strain in a material within its elastic limit.

What Is the Bending Stress of Metal?

The bending stress of a metal, or any material for that matter, depends on factors such as: the material's properties, the shape of the object, and the magnitude and distribution of the applied load. Each metal has a unique set of properties, including its own Young's Modulus and yield strength. The geometry of the component plays a crucial role as well. The moment of inertia, which quantifies an object's resistance to bending, and the distance from the neutral axis to the outer fibers are vital in calculating bending stress. The bending moment (whether it's uniform or concentrated) and the boundary conditions (such as whether the component is simply supported or cantilevered) also affect how bending stress is distributed. The bending equation can be used to calculate the bending stress for any given metal. 

To learn more, see our guide on Metalloids.

What Is the Bending Stress of Steel?

The bending stress of steel depends on various factors, including: the grade of steel, its yield strength (usually denoted as "fy"), and the specific application or structural element in which it is used. In general, the bending stress experienced by a steel beam or column is calculated based on the applied bending moment, the geometry of the section, and the material properties of the steel.

The allowable bending stress for steel is typically dictated by engineering design codes and standards. These provide guidelines for safe and efficient structural design. For example, in the context of steel design, the acceptable bending stress is often expressed as a fraction of the steel's yield strength. A common assumption is that a bending stress equal to two-thirds of the yield strength (0.66fy) is acceptable for both tension (bt) and compression (bc).

To learn more, see our guide on the Components of Steel.

What Is the Bending Stress of Beams?

The bending stress in beams, no matter what they’re made of, depends on various factors related to the specific beam design and loading conditions. The bending stress formula can be used to calculate stresses within the beam. To do so, you need to know the applied bending moment, the dimensions of the beam's cross-section, and the material properties of the beam (such as its modulus of elasticity, E). Additionally, the type of loading (uniform or varying) and the boundary conditions (e.g., simply supported or cantilevered) play significant roles.

The calculated bending stress should be compared to the allowable bending stress, which is typically specified by engineering design codes and standards. All beams must be designed with a margin of safety so they can safely carry the applied loads without failure or excessive deformation.

What Is the Bending Stress of Stainless Steel?

The allowable bending stress for stainless steel, like other materials, depends on factors such as the grade or type of stainless steel, the specific application, the item’s geometry, plus safety factors built into the item. However, as a general guideline, stainless steel can typically accept more bending stress than mild steel due to its superior strength and corrosion resistance properties. The exact bending stress values can be calculated using the bending stress equation. 

To learn more, see our guide on Stainless Steel Metal.

What Are the Different Types of Bending Stress?

The different types of bending stresses are listed and discussed below:

1. Pure Bending

Pure bending is a fundamental condition of stress in structural mechanics. In this scenario, a beam experiences a bending moment (M) without the presence of additional shear, axial, or torsional forces. This type of bending is considered "pure" because it isolates the beam's response to bending loads. It's important to note that in pure bending, the shear force (V) is inherently related to the bending moment. For pure bending to occur, the shear force must be equal to zero along the length of the beam, implying a uniform bending moment distribution. Pure bending conditions are idealized and provide a foundational understanding of how materials respond to bending stresses. This simplification is invaluable in structural analysis and design, allowing engineers to predict and design for various loading scenarios.

2. Symmetric Bending

Symmetric bending takes place when the applied load or bending moment acts precisely within the plane of symmetry of the object. This scenario is also a semi-idealized one that creates a balanced distribution of stresses and deformations.

3. Unsymmetric Bending

Unsymmetric bending refers to bending situations where shear forces are also present in addition to the applied bending moment. These shear forces introduce additional complexities into the bending behavior. This leads to variations in stress and deformation along the length of the member.

4. Non-uniform Bending

Non-uniform bending represents a more complex scenario than uniform bending. In non-uniform bending, the cross-section of the beam experiences both bending and shearing forces. This condition arises when some of the external loading is not perpendicular to the beam's longitudinal axis. Loads may be distributed unevenly along the beam and may incur at unusual angles. As a result, the bending moment within the beam is not constant along its length, leading to the designation "non-uniform bending." This type of bending is encountered in real-world engineering situations where beams are subjected to combined loading, such as lateral forces and bending moments together. Non-uniform bending requires a more intricate analysis due to the varying bending moments and shear forces within the beam.

What Type of Bending Stress is Used in 3D Printing?

Items made in 3D printers experience the same sorts of bending stresses as those made via any other manufacturing method. Bending stress is a result of forces applied to the part rather than a consequence of manufacturing. To learn more, see our 3D Printer Guide.

What Are Examples of Bending Stress?

Below are some examples of bending stress in everyday life:

  1. Bridges are subject to bending stress caused by the weight of vehicles and pedestrians crossing them. The beams and girders in the bridge structure experience bending stress as they support the applied loads.
  2. Many buildings are designed so that the weight of the structure and occupants is borne by beams and columns. These structural elements experience bending stress, especially in multi-story buildings.
  3. The wings of aircraft are designed to withstand bending stress during flight. They bend due to aerodynamic forces and gravitational forces acting on the aircraft.

What Is the Relationship Between Bending Stress and Compressive Stress?

Compressive stress is a component of bending stress within a bending member. As the beam bends, the convex side is subjected to tensile stress, which tends to elongate the material, while the concave side experiences compressive stress, which squeezes the material.

This relationship is governed by the fundamental principles of equilibrium and stress distribution. The distribution of stresses across the cross-section of the beam follows a linear pattern, transitioning from maximum compressive stress at the concave side to maximum tensile stress at the convex side, with a neutral axis where stress is zero.

What Is the Difference Between Bending Stress and Tensile Stress?

Bending stress and tensile stress are two distinct types of stress that affect structural elements differently. Bending stress emerges when external forces or moments cause a component like a beam to bend. This stress combines both tension and compression, affecting different parts of the material. In contrast, pure tensile stress occurs when a material is pulled along its axis, resulting in uniform tension forces. Bending stress induces stretching and squeezing within a material simultaneously, while tensile stress exclusively elongates the material along its axis. Moreover, bending stress varies across the material's cross-section, while tensile stress remains uniform under pure tension.

For more information, see our guide on What is Tensile Stress.

Summary

This article presented bending stress, explained it, and discussed how it works and its types. To learn more about bending stress, contact a Xometry representative.

Xometry provides a wide range of manufacturing capabilities and other value-added services for all of your prototyping and production needs. Visit our website to learn more or to request a free, no-obligation quote.

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Xomety X
Team Xometry
This article was written by various Xometry contributors. Xometry is a leading resource on manufacturing with CNC machining, sheet metal fabrication, 3D printing, injection molding, urethane casting, and more.

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