Bending Stress Calculator

This bending stress calculator will help you determine the maximum bending stress on a beam due to the bending moment it experiences.

In this calculator, you can investigate the bending stresses on beams of different common solid and hollow beam cross-sections like square, rectangle, circle, and even T (or tee) or channel cross-sections.

Keep on reading to learn the following:

• What bending stress is;
• How to calculate bending stress;
• Understanding the bending stress equation;
• How to use this bending stress calculator; and
• Some frequently asked questions about bending stress.

Beams carrying loads, including their own, will experience some level of bending, usually downward. During that bending process, the beam also undergoes normal stresses across the beam that causes both compressive and tensile stresses. These normal stresses are the bending stress that a beam experiences.

As a beam bends downward, its upper part experiences compressive stress, and its lower part experiences tensile stress. The upper and lower faces of the beam experience a tremendous amount of stress, especially whichever is of a more significant distance from the beam's neutral axis.

In the next section of this text, let us discuss the maximum bending stress formula before we learn how to use this bending stress calculator.

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:

where:

• $\sigma$ – Bending stress (in pascals or newtons per square meter) that the beam experiences at a distance $\small c$ from the neutral axis;
• $M$ – Bending moment applied to the beam in newton-meters ($\small\text{N}\!\cdot\!\text{m}$);
• $c$ – Perpendicular distance of a point along the cross-section of the beam from its neutral axis in meters ($\small\text{m}$); and
• $I$ – Area moment of inertia of the beam in meters to the fourth power ($\small\text{m}^4$).

The bending moment, $M$, 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 $\small 10\ \text{N} \times 3\ \text{m}/2 = 15\ \text{N}\cdot\text{m}$ of bending moment on the beam.

On the other hand, to find $\small c$, 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 $y_\text{c}$ when considering vertical loads. We use $x_\text{c}$ 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 $y_\text{c}$. 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:

where:

• $I$ – Area moment of inertia in meters to the fourth power ($\small\text{m}^4$);
• $b$ – 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 $\small M$ and $\small S$, as shown in the bending stress equation below, where $\small S$ is the section modulus:

We can do that since we can also express $\small S$ in terms of $\small I$ and $\small c$, as you can see in this equation:

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.

From our bending stress equation, we can tell that having the bending moment in the fraction numerator, the larger the bending moment is, the greater the bending stress the beam experiences.

On the other hand, since the area moment of inertia (same goes with the section modulus) is in the denominator of the bending stress equation, the larger the area moment of inertia of the beam, the smaller the bending stress the beam can experience.

Using this bending stress calculator is relatively easy. Here are the steps you can follow for your maximum bending stress calculations:

1. Select your beam's cross-section.

2. Enter the dimensions of your beam by following the corresponding illustration of your chosen cross-section. At this point, you can expect our bending stress calculator to display the values for the section modulus ($\small S$), area moment of inertia ($\small I$) (both in the x and y axes), and the largest distance from the neutral axis ($\small c$).

3. Input the applied bending moment ($\small M$) on your beam to find the maximum bending stress that your beam experiences at that distance $\small c$.

You can skip step 3 and enter a value for the bending stress if you're interested in finding the maximum bending moment your beam could support to reach that bending stress.

The bending stress formula is σ = M × c / I, where σ is the maximum bending stress at point c of the beam, M is the bending moment the beam experiences, c is the maximum distance we can get from the beam's neutral axis to the outermost face of the beam (either on top or the bottom of the beam, whichever is larger), and I is the area moment of inertia of the beam's cross-section.

To find the bending stress of a square beam, you can use the following equation: σ = 6 × M / a³. Say a square beam has a side measurement, a, of 0.10 m and experiences a 200 N·m bending moment. Substituting these values into our square beam bending stress equation, we get:

σ = 6 × M / a³
σ = 6 × 200 N⋅m / (0.10 m)³
σ = 6 × 200 N⋅m / 0.001 m³
σ = 1,200,000 N/m² = 1,200,000 Pa
σ = 1.2 MPa

A 20 cm × 30 cm (0.2 m × 0.3 m) rectangular beam experiencing 10 kN·m bending moment has a bending stress of 3.333 MPa. We get that value by:

1. First, by taking the beam's area moment of inertia, I, using this equation:
   I = b × h³ / 12
I = 0.2 m × (0.3 m)³ / 12
I = 0.2 m × 0.027 m³ / 12
I = 0.00045 meters to the fourth power

2. Then, we substitute the values of I and the bending moment M into our maximum bending stress formula below, where c is equal to or 0.15 m (half of the beam's depth), to get:
   σ = M × c / I
σ = 10,000 N⋅m × 0.15 m / 0.00045
σ = 3,333,333 Pa or 3.333 MPa

Bending stress is a normal stress acting across and through the beam's cross-section caused by the beam's bending. On the other hand, shear stress is brought about by the forces that cause vertical tearing of the beam along the beam's cross-section.