In the second article of the MechDevs Blog, I would like to present to you the solution to another fundamental problem that students encounter in mechanics classes – calculating and drawing diagrams of internal forces in beams.
Before you start reading, I encourage you to familiarize yourself with the first article, where I explained the basic concepts that I will be using here.
This article, like the previous one, is divided into two parts. In this part, we will discuss the following topics:
- Why do we calculate internal forces?
- What are internal forces?
- Types of internal forces – basic strength cases
Why do we calculate internal forces – Run! It will collapse!
When conducting classes for students or tutoring on a given topic, I always believe that before presenting the subject, it’s worth spending at least a moment explaining why the topic is being introduced in the first place. It seems natural that if someone knows why I am explaining something to them, they approach the topic with greater engagement or simply awareness. This isn’t always possible without some sort of introduction (as was the case with the first article), but in this case, it is, and that’s where we’ll start.
Let’s take a break from the purely academic aspect of calculations and move to the gray (more or less) reality of an engineer. When designing a structure, machine, or building, beyond just the functional aspects (what it serves and its purpose), we must also consider additional factors like the safety of using the element, its durability, and many other aspects. Safety and durability are highlighted here for a reason – both depend heavily on the stresses acting within the element. There will be a separate article about stress, its analysis, and calculations. For now, let’s simplify things by assuming that stress describes the intensity with which a local fragment of the material of an element carries the forces acting on it. In other words, the greater the forces acting on the material, the greater the stresses within it.
As you may know from materials classes, every material has its characteristic physical, mechanical, and strength properties. These include things like the yield strength and tensile or compressive strength, which, in a way, define whether a given material can be used in a project based on its safety.
To summarize – to be able to design a structure or another object, select materials, etc., we must absolutely know (or at least estimate well) the stress values that may occur in it as a result of its operation and the loads acting on it. A lack of thorough analysis will, in the best case, result in the destruction of the structure (this is where the cat meme should appear), and in the worst case… well, we’ve all heard about various disasters (YouTube is full of videos).
In the case of many technical objects, stress calculation methods are documented in standards, while for others (e.g., those with complex geometry or specific loading conditions), computer methods such as Finite Element Method (FEM) are used. Fortunately, for simple objects like beams, trusses, or frames, simple analytical methods can be applied, yielding results that closely reflect reality.
Now, let’s get to the point – answering the question posed in the title of this subsection.
Internal forces in beams, frames, and trusses are calculated primarily to determine the stresses acting in these structures.
Analytical formulas derived by prominent scientists show that stresses in beams, frames, etc.—that is, one-dimensional elements (those whose length is much greater than the cross-sectional dimensions)—are directly dependent on the magnitude of internal forces and on the cross-sectional properties (e.g., cross-sectional area and geometric moment of inertia—more on this in another article).
Additionally, internal forces directly affect the displacements of the element, which can also be critical in design. To answer another potential question—internal force diagrams are drawn to identify locations in the structure where stress values may be maximal, as it is these stresses that can limit the safety of the structure
What are internal forces – think globally, act locally
The topic of internal force analysis and calculations should start with what internal forces are and where they come from.
From the first article, you already know that for a system to be static (immobile), we must ensure force equilibrium in the system. This static equilibrium, as it turns out, must be maintained not only “globally,” i.e., for support reactions and loads, but also “locally”—for every microscopic point of the material. If equilibrium were not maintained, each material particle, according to Newton’s second law of motion, would begin to accelerate independently of other particles, which, as you can imagine, would cause quite an interesting mess.
Speaking of messes—such situations are not impossible and often occur, for example, in fluids. Unfortunately, most structural elements are made of solid materials, and uncontrolled internal movements are generally undesirable (look up “creep,” but please don’t search for images ☺).
Returning to equilibrium—if we are dealing with one-dimensional elements (as mentioned earlier), the analysis of force equilibrium in the material can be significantly simplified, more precisely reduced to analyzing what happens in the cross-section. Let’s imagine we have a beam properly supported (geometrically rigid/stable) and loaded—now we mentally split that beam at a specific point.
Or rather, let’s not just do it mentally… let’s take a saw or a large angle grinder and cut it in half, inspired by our favorite horror characters… What happens to the cut beam? Let’s wait for the answer…
Yes – in most cases, both parts of the beam will begin to move due to the imbalance of forces and will either fall to the ground or fly upward (depending on the load).
This situation can be interpreted purely mechanically—if the beam was stationary before the cut, then forces were acting at the cut point that balanced the applied loads. Moreover, if, instead of a grinder, we used an ultra-strong, ultra-thin, and ultra-fast laser and caught both parts of the beam at the speed of light—we could hold them in the same position by applying forces with the appropriate directions and magnitudes.
As you can probably guess, the forces needed to keep the beam parts stationary are exactly the internal forces acting at that point before the cut! Additionally, since we can cut the beam at any point, internal forces can also be calculated at any point along the beam—the result of such calculations is an internal force diagram.
Types of internal forces – how to fix a cut beam
Now, it’s important to consider how our “super alter ego” might act on the ends of the cut beam, as these interactions correspond to the so-called basic strength/load cases. These cases include:
tension/compression
Occurs when the direction of the forces acting on the beam is aligned with its axis and, importantly, the direction of the forces coincides with the centroid of the cross-section (in other cases, we deal with eccentric tension/compression, which I’ll explain another time…). Such forces cause normal (axial) forces to develop in the material. Why “normal” forces (and not abnormal)? In physics and mathematics, “normal” simply means perpendicular to something.
In this case, normal forces are perpendicular to the plane of the cross-section (the one left after our laser cut). Normal forces create a uniform distribution of normal stresses—uniform meaning that the stresses have a constant value across the entire cross-section.
shear
Occurs when the direction of the forces acting on the beam is perpendicular to its axis (tangential to the cross-section—so-called transverse forces) and, again importantly, the direction of the forces passes through the shear center (in other cases, we deal with additional torsion of the cross-section—more on that in a moment).
Such forces cause shear forces to develop in the material. Shear forces create a fairly complex distribution of shear stresses, though they are often averaged across the cross-section in simplified calculations. If you are more interested in this topic, I invite you to read the article on stresses.
bending
Occurs when a moment acts on the beam, whose direction is perpendicular to the transverse axis. Such a moment can also be caused by a force couple and generally occurs when transverse forces are distant from the point of analysis (the point where the beam is cut).
Such forces cause bending moments to develop in the material. Bending moments create a linearly varying distribution of normal stresses.
torsion
Occurs when the direction of the applied moment is parallel to the axis of the beam. Such loading causes torsional moments (torque) to develop in the material, which result in shear stresses with a complex distribution. Only for circular cross-sections and square tubes do the values of shear stresses change linearly from the center of the cross-section (value of 0) to the maximum at the outer edge of the cross-section.
OK. You’ve had the chance to learn a lot of theory and dry facts related to internal force analysis. You now know what the term means, where internal forces come from, and how to classify them. In the second part of the article, you will learn methods for analyzing them and commonly accepted principles for determining their signs. I invite you to read it.
