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Stress Formula and Equation in Beam Calculations: Understanding Bending, Shear, Normal Stress, and Tensile Strength

September 28, 2021|9 min read

Introduction to Stress Equations in Beams

In a separate article entitled “Key Factors for Structural Beam Calculation & Analysis,” there was a brief discussion of stresses and their function in structural analysis. In this article, we will review normal, bending, and shear stress in more depth.

Understanding the Normal Stress Formula in Structural Engineering

Understanding Normal Stress: Axial Forces and Their Impact on Structural Members

A normal stress is a stress that occurs when a member is loaded by an axial force. The value of the normal force for any prismatic section is simply the force divided by the cross sectional area.


A normal stress will occur when a member is placed in tension or compression. Examples of members experiencing pure normal forces would include columns, collar ties, etc.

Practical Applications: Where Normal Stress Comes into Play in Structural Design

In the realm of structural design, understanding normal stress isn't just an academic exercise; it's a practical necessity. Whether you're dealing with the load-bearing columns of a high-rise building or the collar ties in a residential attic, normal stress is a critical factor that engineers must account for.

For instance, in columns, the axial forces are predominantly compressive. Engineers must ensure that the material and dimensions chosen for these columns can withstand the compressive forces they will be subjected to, without buckling or failing.

Similarly, collar ties in roof structures are often subject to tensile forces, pulling them apart. The material and design must be robust enough to handle these forces to prevent roof collapse or other structural failures.

In both scenarios, the formula for normal stress—force divided by cross-sectional area—becomes a go-to equation for engineers. By calculating the normal stress, they can make informed decisions about material selection, cross-sectional dimensions, and overall structural integrity.

So, the next time you come across a structural element, be it a column or a tie, remember that normal stress is likely playing a crucial role in keeping that structure standing. And for those in the engineering field, ignoring normal stress is not an option; it's a fundamental parameter that can make or break your design.

The Bending Stress Equation: A Deep Dive

The Mechanics of Bending Stress in Structural Elements


When a member is being loaded similar to that in figure one bending stress (or flexure stress) will result.  Bending stress is a more specific type of normal stress.  When a beam experiences load like that shown in figure one the top fibers of the beam undergo a normal compressive stress. The stress at the horizontal plane of the neutral is zero.  The bottom fibers of the beam undergo a normal tensile stress.  It can be concluded therefore that the value of the bending stress will vary linearly with distance from the neutral axis.


Calculating the maximum bending stress is crucial for determining the adequacy of beams, rafters, joists, etc.

Real-World Implications: How Bending Stress Influences Structural Integrity

Understanding the bending stress equation is not just a theoretical endeavor; it has direct implications for the real world, especially in the fields of construction and engineering.

When you're dealing with structural elements like beams, rafters, and joists, knowing how to calculate and interpret bending stress can be the difference between a successful project and a structural failure.

For instance, in the construction of a residential home, the rafters must be designed to withstand not just the weight of the roof, but also additional loads like snow or even the force of the wind. Here, bending stress plays a crucial role in determining the rafter's material and dimensions.

Similarly, in commercial buildings, beams must be designed to handle varying levels of stress, including shear force, which acts parallel to the surface of the material.

Shear force, in conjunction with bending stress, can significantly impact a structure's integrity. For example, in a multi-story building, the beams on the lower floors must be capable of handling greater shear forces than those on the upper floors. This is because they bear the cumulative load of all the floors above them.

By understanding the interplay between bending stress and shear force, engineers can optimize their designs for both safety and cost-effectiveness. They can select materials that offer the best balance between strength and flexibility, and design structural elements that are both robust and efficient.

So, the next time you see a beautifully designed building or bridge, remember that behind its aesthetics are complex calculations of bending stress and shear force, ensuring its structural integrity and longevity.

Breaking Down the Shear Stress Formula in Beams

Understanding the Dynamics of Shear Stress in Beams

Normal stress is a result of load applied perpendicular to a member.  Shear stress however results when a load is applied parallel to an area. Looking again at figure one, it can be seen that both bending and shear stresses will develop. Like in bending stress, shear stress will vary across the cross sectional area.


Calculating the maximum shear stress is also crucial for determining the adequacy of beams, rafters, joists, etc.

Practical Applications of Shear Stress Calculations in Structural Design

Understanding shear stress isn't just an academic exercise; it has real-world implications that can't be ignored by anyone in the structural design field. The calculations for shear stress, often used in conjunction with those for compressive stress and strain, are vital for ensuring the integrity of a structure.

These calculations become particularly important when considering the forces acting parallel to the unit area of the material, which can significantly impact its overall stability.

The concept of the 'neutral axis' is also crucial here. This is the line in a beam or column where the stress changes from compressive on one side to tensile on the other. Accurate calculations around the neutral axis can be the difference between a structure that stands the test of time and one that fails under pressure.

In practical terms, these calculations are used to determine the adequacy of materials and the safety margins for beams, rafters, joists, and other structural elements. They help engineers decide the type and amount of reinforcement needed, especially in areas where both bending and shear stresses are significant.

By understanding the interplay between shear stress, compressive stress, strain, and the neutral axis, engineers can create structures that are not only safe but also economically viable.

So, the next time you're evaluating the structural integrity of a project, remember that shear stress calculations are not just numbers on a page. They are a critical component of responsible and effective structural design.

Exploring Tensile Strength in Structural Design

Tensile Strength Introduction

In structural engineering, tensile strength isn't just a term; it's a pivotal factor that influences the integrity of buildings, bridges, and other constructions. This property quantifies the maximum stress a material can endure when being stretched or pulled before necking begins, leading to material failure.

As we navigate through the complexities of stress equations, understanding tensile strength becomes indispensable. In this segment, we'll unpack the concept of tensile strength, its significance in material selection, and its interplay with other stress types in beam calculations.

The Interplay of Tensile Strength with Other Stress Types in Structural Engineering

Building on the foundational understanding of tensile strength, it's essential to recognize that this property is not a standalone factor. It interacts closely with other stress types, such as bending and shear stress, to influence the overall stability and safety of a structure. Engineers often use tensile strength as a key parameter in their stress equation calculations to ensure that the materials chosen can withstand the various forces they will encounter.

From steel beams in skyscrapers to concrete pillars in bridges, tensile strength plays a vital role in determining how these elements will respond under load. Let's explore some practical applications and methods for calculating tensile strength in structural engineering.

How Software Simplifies Stress Equation Calculations in Beams

When doing any kind of beam design using structural design software will greatly ease the entire process of calculating stresses. There are several different engineering design software packages available for beams, columns, or foundation design. If you’re in the market, try out StruCalc with a free 30-day trial, no strings attached!

Stress Formula Frequently Asked Questions

What Are the Main Types of Stresses in Beams?

The main types of stresses in beams are normal stress, bending stress, and shear stress. Each type of stress has its own formula for calculation and plays a unique role in structural integrity.

How Is Normal Stress Calculated?

Normal stress is calculated by dividing the axial force by the cross-sectional area of the material. It's crucial for understanding the tension or compression a material experiences under axial loads.

What Factors Influence Bending Stress?

Bending stress is influenced by the type of material, its dimensions, and the external forces applied. It varies linearly from the neutral axis, with the maximum stress occurring at the outermost fibers of the beam.

Is Shear Stress the Same as Normal Stress?

No, shear stress is different from normal stress. While normal stress acts perpendicular to the surface, shear stress acts parallel to the surface of the material.

How Does Shear Stress Affect the Integrity of Beams?

Shear stress is crucial for determining the adequacy of beams, rafters, joists, etc. It acts parallel to the unit area and can significantly impact the overall stability of a structure.

What Is the Neutral Axis in Bending Stress?

The neutral axis is the line in a beam where stress changes from compressive on one side to tensile on the other. Accurate calculations around the neutral axis are essential for structural integrity.

Why Is Tensile Strength Important in Structural Design?

Tensile strength is vital for understanding how a material will behave under tension. It helps engineers decide the type and amount of reinforcement needed, especially in areas where tensile forces are significant.