Simple bending

Introduction to bending

Bending is one of the key issues in materials strength, playing an essential role in the design and analysis of structural elements. Bending is divided into quite broad and diverse subfields such as simple bending, transverse bending and oblique bending. In three separate introductions, we will briefly describe each of these topics.

Simple bending is a type of bending that occurs when the load acts in one plane and causes the element to bend in that same plane. It is the simplest type of bending in engineering practice.

Analysis of simple bending

During the analysis of simple bending, we calculate bending moments and stresses in the cross-section of the element to ensure that the material and structure can withstand the planned loads without damage.

Normal stresses in simple bending

The equation for normal stress considering the action of the bending moment in one plane (here the load plane is vertical, the moment vector is generated in the plane perpendicular to the bending plane, i.e., horizontally):

$$\sigma(x) = \frac{M_y}{I_y} \cdot z$$

where:

• \( \sigma \) is the normal stress,
• \( M_y \) is the bending moment,
• \( z \) is the distance from the neutral axis to the point in the cross-section where we want to calculate the stresses,
• \( I_y \) is the moment of inertia of the cross-section

For the more curious - the formula we described above can be used thanks to the assumption for bending of the Bernoulli's hypothesis of flat sections.

You can find the derivation of this formula here.

Fig. 1. Normal stress diagram for the case of simple bending

Neutral axis and moment vector

Of course, the issue of assumption is the naming of the axes of the cross-section, in the example above it is y,z – these designations are related to the symbols in the stress formula. The direction of the moment vector depends on the moment diagram, specifically on whether it stretches the lower fibers or upper fibers at the point of the beam we are analyzing (we most often analyze the point where the maximum moment occurs along the entire beam).

In a beam subjected to simple bending, there is a neutral axis, which is a line in the cross-section that is neither stretched nor compressed. It always coincides with the moment vector and divides the cross-section into a stretched part and a compressed part.

Understanding simple bending is the foundation for further, more complex bending analyses and the design of durable structures.

You have access to video courses and a wide database of problems with solutions for each type of bending separately.