Displacement Calculations – Introduction to Analysis (1)

The fourth article on the MechaDevs blog will focus on the topic of displacement calculations in beams. These calculations give many students sleepless nights, mainly due to the large amount of mathematical work involved in solving such problems, as well as the relatively complex calculation schemes—regardless of the chosen method.

However, before we move on to displacement calculations, I strongly suggest familiarizing yourself with the articles on internal force calculations because, without knowing how to calculate (or write them in formula form), it is unfortunately impossible to calculate displacements—or at least, you would waste a lot of time…

The article is divided into three parts, and in this one, we will cover the introductory information, specifically:

• Why do we calculate displacements in beams?
• How can we calculate displacements?
• Analytical approach – differential equations

Why do we calculate displacements? Stiffness is key

Two fundamental parameters to consider when designing various types of structures are stress and displacement. Stresses in a material directly affect its strength and the possibility of damage leading to the failure of such a structure. Displacements, on the other hand, result from the stiffness of the object and the loads acting on it, usually having more functional significance.

The best examples of stiffness-related requirements are conventional or CNC machine tools or 3D printers. In such devices, greater stiffness minimizes displacements caused by forces during machining (or even the head’s movement), which significantly impacts the possible precision of the workpiece and, therefore, the quality and class of the machine.

Similar examples of the need to maintain proper stiffness (minimizing displacements) can also be found in construction—no one would intentionally design a bridge that bends by, say, a meter under a passing truck… It would mean the truck would have to climb “uphill” again after crossing the bridge—totally impractical…

As a fun fact, I can also mention that the stiffness of a structure greatly influences its natural frequencies, which are crucial when designing both bridges and skyscrapers. Perhaps the most interesting example of neglecting modal analysis (determining frequencies and their possible excitation) in design is what happened to the Tacoma Narrows Bridge—I recommend looking for videos on YouTube. And just in case you’re wondering—it wasn’t made of chewing gum ????.

Another important point in the discussion about the need for displacement analysis is statically indeterminate problems. If we recall the basics of statics, it turns out that by using only equilibrium equations, we could only solve problems with the number of unknown reactions exactly matching the number of equilibrium equations we could write. Such systems were called statically determinate. Leaving aside the case where there are too few supports (making the system geometrically unstable), we are left with the issue of calculating systems where there are more supports than necessary…

Unfortunately, analyzing such problems, for example, calculating the distribution of internal forces, is impossible based on equilibrium equations alone. However, displacement-based equations come to the rescue, allowing us to write boundary conditions, which serve as additional equations supplementing our system.

By using the system’s stiffness, we can calculate reactions in statically indeterminate problems, thus determining the strength of such systems.

To sum up: many structures must not only withstand loads, but also not experience excessive displacements under those loads (or, conversely, some components are designed to be flexible, for example, snap-fit clamps ????).

Additionally, the ability to determine displacements allows us to solve “redundant” systems: statically indeterminate ones.

Now, let’s move on to how we can calculate displacements using knowledge from mechanics and material strength.

How can we calculate displacements? – Two approaches… you can’t beat us…

Essentially, two basic approaches are used for displacement calculations:

• the approach using differential equations derived from analyzing geometric conditions, constitutive equations, and internal forces (yes, I mentioned a lot of unpleasant terms here, but I’ll explain them shortly…).
• the approach using energy relationships—more specifically, potential elastic energy—don’t worry, I’ll explain this too.

Both of these approaches are available in the EquiBeam application—and the methods based on these approaches are constantly being developed ????. Now let me explain the above-mentioned informational chaos… Let’s start with the first approach.

Analytical approach – differential equations – ugh, integrals…

The easiest way to explain this is with the simplest case of strength analysis—axial compression/tension of a bar.

In such a bar, only axial forces (parallel to the bar’s axis), called normal forces N, act. In the analysis, we assume that these forces are evenly distributed over the cross-sectional area A. The normal stress σ_x​ in such a cross-section is therefore equal to the applied force divided by the cross-sectional area:

σ_x={N \over A}

Thus, we have linked the internal forces in the bar with the cross-sectional parameters and obtained the stresses. Now, we introduce the constitutive equations, which are simply models describing the relationship between stress σ_x​ and strain ε_x ​—in this case, relative elongation (remember that strain is dimensionless—it has no units). For elastic systems, such a model is Hooke’s law, and for pure compression/tension, it can be written in the following form:

ε_x={σ_x\over E}→ε_x={N\over EA}

where: E – is the material’s Young’s modulus (longitudinal elastic modulus)

Finally, we apply the last term used in the initial description—geometric conditions—i.e., the relationship between strain and displacement. In this case, the geometric conditions are also very simple:

u'(x) = ε_x → u' (x)={N(x)\over E(x)⋅A(x)}→u(x)= ∫{N(x)\over E(x)⋅A(x)} dx

In mathematical terms, the first derivative of displacement equals strain. Thus, we have the discussed differential equation. An important note here—strains are not always constant in reality—they can be functions of the bar’s length (depending on varying normal forces, varying cross-sections, or materials). However, if the values are constant over a given segment of the bar length L, the elongation/contraction of that segment can be calculated with a nice integral:

ΔL=∫_0^Lε_x  dx=ε_x ∫_0^Ldx={N \over EA}   x|_0^L={N\over EA}

In the end, we obtain the formula for the elongation of a bar under axial compression or tension. This elongation is used to determine the displacements of specific points along the bar. Here’s a bit of reassurance—in most problems, there’s no need to derive this from scratch (directly from theory)—we usually rely on the final derived formula or the integral itself.

To summarize the first approach—to calculate displacement, we must analytically determine the relationships between internal forces, stresses, strains, and displacements. These relationships will vary for each strength case (though sometimes they’re similar) but always lead to some differential equation. In the third part of the article, I’ll revisit tension and free torsion in bars and derive equations for bending.

OK, to summarize this part of the article—now you understand why displacement calculations are important from an engineering perspective and why they are also needed in strength analyses. You have an idea of how we can calculate displacements in systems and have learned the basics of the analytical calculation method.

In the next part of the article, we will describe the second approach to displacement calculations related to strain energy—so stay tuned for part two.