What is Bending Stress?
Bending stress is the internal resistance generated within a component when an external bending moment or force is applied.
This bending moment induces curvature in the component, leading to the development of tension and compression stresses, collectively known as bending stresses.
The theory behind bending stress involves two critical assumptions the plane section assumption and the linear elastic material behavior.
The plane section assumption suggests that any cross-section of a structure remains plane after bending.
The linear elastic material behavior presupposes that the normal stress is directly proportional to the strain, a relationship known as Hooke’s Law.
For instance, the bending stress in a beam is not uniformly distributed across its cross-section. The maximum stress occurs at the outermost layers of the beam the ones furthest from the neutral axis, a line along the length of the beam where the bending stress is zero.
Understanding bending stress plays a significant role in determining the overall strength and longevity of structures.
Excessive bending stress can lead to deformation, failure, or even the collapse of structures. Therefore, it’s essential for designers and engineers to comprehend the theoretical underpinnings of bending stress and normal stress to predict and prevent structural failure.
Studying the mechanics of materials, which involves how different materials respond to various types of stress, provides a crucial foundation for understanding bending stress.
This knowledge assists in the selection of appropriate materials and designs that can effectively withstand bending stresses, ensuring structural safety and reliability.
Types Of Bending Stress
The major stresses induced due to bending are normal stresses of tension and compression.
But the state of stress within the beam includes shear stresses due to the shear force in addition to the major normal stresses due to bending although the former are generally of smaller order when compared to the latter.
Bending stress is actually a catch-all term for a few different types that are worth familiarizing yourself with so you can better understand what a beam is undergoing:
#1. Pure Bending
In this form of bending stress, an object will only experience a bending moment from the force that’s being applied, as opposed to having other stresses and forces working on it, too such as shear stress, axial stress, or torsional force.
The right conditions will need to be in place for pure bending and shear stress should be equal to zero, so while it might not be the most realistic scenario, it gives engineers a good idea of how a material will respond to this stress alone.
#2. Symmetric Bending
This happens when the cross-section of the beam and the neutral access are symmetrical.
Therefore, the force or stress is applied evenly and results in symmetric bending, which is another idealized situation like pure bending, which may not always match up with the environment this beam or structure would actually exist in.
#3. Unsymmetric Bending
Logically, with how symmetric bending works, you’ll find that unsymmetric bending happens when the cross-section and beam aren’t symmetrical. Stress is then distributed more unevenly leading to this style of bending stress.
#4. Non-Uniform Bending
When you’re thinking of day-to-day scenarios that feature slightly less perfect conditions, non-uniform bending will come up.
This type of bending stress happens when forces or loads are unevenly applied and shear stress is present, too.
This tends to be the most realistic because bridges and structures existing in everyday life will have far more forces, stresses, and tensions working on them.
How To Calculate Bending Stress?
Now that we know what bending stress is, let’s now learn how to calculate it. To do that, we need this bending stress formula:
σ = M × c /I
where:
- σ – Bending stress (in pascals or newtons per square meter) that the beam experiences at a distance c from the neutral axis;
- M – Bending moment applied to the beam in newton-meters (N.m);
- c – Perpendicular distance of a point along the cross-section of the beam from its neutral axis in meters (m); and
- I – Area moment of inertia of the beam in meters to the fourth power (m4).
The bendingmoment, M, applied to the beam can be due to other perpendicular loads on the beam that causes the beam to bend.
A downward point load, say 10 N, at the center of a 3-meter beam, will induce an equivalent of 10 N×3 m/2=15 N⋅m of bending moment on the beam.
On the other hand, to find c, we first need to find the neutral axis of a beam’s cross-section. For symmetrical beams vertically and horizontally, we can quickly identify their neutral axis to be half the height of the beam.
A cross-section’s neutral axes are located along the cross-section’s centroid. From the neutral axis, we can then measure a distance above or below it and denote that distance as yc when considering vertical loads.
We use xc instead when considering horizontal loads that cause our beam to deflect horizontally.
To find the maximum bending stress on a beam, we use the largest value for c. That means it’s a measure from the neutral axis to either the beam’s top or bottom face whichever is larger.
You can visit our section modulus calculator to find the different neutral axis formulas of the typical cross-section of steel beams.
In this tool, we focus on the vertical deflection of beams and, therefore, on yc. If you’re wondering how to determine the deflection of beams due to different load cases, you can check out our beam deflection calculator for that.
Lastly, for the area moment of inertia, I, we have lots of different formulas to use depending on the cross-section of the beam. For the typical rectangular beam, we use this formula:
I= b×d3 / 12
where:
- I – Area moment of inertia in meters to the fourth power (m4);
- b – Beam’s width or breadth in meters; and
- d – Beam’s height or depth in meters also.
You can also check our section modulus calculator for the formulas of the common beam cross-sections.
But if you want to read purely about the moment of inertia, our moment of inertia calculator is for you.
We can also express our bending stress formula in terms of M and S, as shown in the bending stress equation below, where S is the section modulus:
σ=M / S
We can do that since we can also express S in terms of I and c, as you can see in this equation:
S=I / c
Always double-check that you are using the correct units for each variable or learn how to do dimensional analysis to avoid obtaining the wrong results.
To avoid that, you can also use our bending stress calculator, which we’ll explain how to use in the next section of this text.
Examples Of Ending Stress
Beam bending is arguably one of the most common examples of bending stress analysis. A beam is a generic object with a constant cross-section along its length.
Its length is significantly larger than the dimensions of its cross-section, allowing for relatively precise approximations.
Bending stress is present in all kinds of everyday objects and structures, a few of which we list out below.
#1. Bridges.
With a large mix of pedestrians, pets, cyclists, cars, trucks, and passengers going across bridges, it’s fair to say this is one of the most obvious and clear examples of bending stress and what beams and structures undergo when they’re put to work.
#2. Airplane wings.
Planes are another great example of bending stress in action, particularly when applied to their wings.
With gravity and aerodynamic forces working against the wings, bending stress is an important factor to consider when designing them so they can move successfully, especially while these forces are at play.
#3. Buildings and architectural structures.
The beams and columns that make up skyscrapers, office buildings, and homes have to account for bending stress, too, so engineers can understand how much force and how heavy of a load they can manage.
Practical Applications of Bending Stress
Understanding bending stress finds practical applications in multiple fields. In civil engineering, for instance, it’s essential for the structural analysis and design of beams, bridges, and foundations.
In mechanical engineering, knowledge of bending stress is necessary for the design of mechanical elements subjected to bending moments such as shafts, gears, and pulleys.
Real-world applications also abound; understanding the bending stress in the human femur, for example, assists in the design of prosthetics and the prediction of fracture risks.
It’s clear that the role of bending stress in your studies and subsequent applications in practical engineering are critical for you.