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Physics of 3D Printing

Before getting to materials and settings that determine the strength of a 3D printed part, it is important to understand the physics and theory driving what aspects of a 3D printed part are important to its strength. In this article, we cover 3 concepts that lay the groundwork for strong 3D printed parts.

Beam Bending Basics

As we covered in 3D Printing Software, 3D printed parts are rarely solid. Most 3D printed parts are printed with a shell and closed-cell infill, which saves material and print time without sacrificing part strength. Does the shell thickness or infill density contribute more to part strength? The answer stems from simple beam bending theory.

The key takeaway from beam bending theory is that the top and bottom surfaces of a beam experience the most force when bent, and we can optimize the strength of a beam for its weight by only adding material at these extremes and leaving the core minimal. Say we have a simple beam, like a bridge, supported on both sides with a weight centered between the supports, as shown in the following diagram:

beam bending basics
A basic loading case of a beam suspended from two points with a weight in the center.

We can abstract this model to the 3 points of contact the beam experiences - one for the weight, and one for each of the supports. This forms a triangle, like so:

a simple way to show beam bending
We can simplify this beam to a triangle with contact points as vertices.

When the weight applies a force to this beam, imagine the forces distributing along this triangular structure. The two angled segments get compressed, and the horizontal segment gets loaded in tension:

beam bending theory in triangles
The forces distribute along the lengths of the triangle.

The magnitude of the horizontal force ultimately depends upon the thickness of the beam. As the beam gets thicker while the load remains constant, the base angles of the triangle increase, reducing the resultant horizontal force on the beam. When we increase the thickness of the beam, like so:

A larger beam decreases the forces on the extremes
The triangle gets taller as the beam thickens, changing the force distribution.

The base angle increases, so the resultant tensile force decreases. So a thicker beam means that the same weight has less of an impact on bottom face of the beam. What this boils down to is that a taller beam is more capable of handling larger loads than a thinner one, because the thickness of the beam reduces the stresses applied to its bottom face.

A related aspect of beam bending theory describes that the largest loads on a bent beam are applied at its extremes - A bending force applied to a beam resolves to tensile and compressive forces on either side of what is called the “neutral axis” - the point in a beam at which no load is experienced. In this case, material below the neutral axis is loaded in tension and above the neutral axis is loaded in compression.

beam bending force diagram
Starting from the neutral axis, tensile and compressive forces get larger as you approach a beam's edge.

This information tells that if optimizing a beam for its strength and weight, material has the biggest impact on the top and bottom surfaces of the part, whereas the middle endures relatively little load. Especially in 3D printing, tensile loads are more important and easier to optimize for than compressive loads, because each extrusion behaves like a strand (more on this in the next section).

This explains why circular and box tubing, I-beams, and T-beams are so common in construction - They save weight by only adding material where the highest stresses are in the part. A circular tube can handle loading from all sides, because no matter where you apply the load from, there are two "surfaces" at the extremes. A box can handle loads from four sides, because whichever side you apply it from, there will always be faces under tension and compression. An I beam can handle in-plane loads, for which the beam has applied forces from above or below. A T beam is designed to handle loads from one direction only, so that the horizontal section of the T takes the brunt of the load.

Different types of beams
The different types of beams cut material in different areas based on how they expect to be loaded.

So when thinking about parts in bending, remember these two things:

A thicker beam is stronger than a thinner beam because the thicker a beam gets, the less resultant force its faces experience under the same load, and a beam in bending experiences the highest loads on its faces, so adding material to the extremes of the beam is stronger than adding material near its center.

Fiber Strength

3D printed carbon fiber behaves like spaghetti
Fiber is like raw spaghetti. It behaves best in tension.

Above we have a piece of raw spaghetti loaded in various conditions. Under which type of load is it strongest? Here we bend it, and it snaps. we compress it by pushing its two endpoints together, and it also snaps. However, when we load it in tension by pulling on it, it can hold a decent load.

String, chain, wire, silk, or any kind of strand of material behaves the same way - it will buckle, deform, or break when bent or compressed under relatively small loads, yet strands can handle far higher loads in tension. In any sorts of engineering applications, these materials are used in ways that make use of their tensile properties. When we talk about Beam Bending Theory, the material below and above a beam is effectively behaving as a “strand” - the segments in tension have the highest impact on a beam’s strength.

The strength of a strand depends on the bonds a material makes with itself, and how that bond distributes stresses from applied loads. Yarn is made of shorter lengths of wool twisted together, so when the twists come undone, it loses strength. An extrusion of plastic, like the ones used in 3D printing, is as strong as the molecular bonds holding it together. Adding a “filler” material to the plastic, like chopped carbon fiber, means the stresses from a load are applied to the short, stiff fibers, but the strength overall is still defined by the plastic holding it together. Likewise, a continuous strand of a strong fiber material like carbon fiber - with the fibers joined at the atomic level - will be stronger than a strand of filament that is a mixture of chopped fibers and plastic.

composite fiberglass strands
Strands of glass fibers being woven into a sheet.

This is why fiber materials like carbon fiber, fiberglass, and Kevlar® are so valuable. Whether you are 3D printing them, winding them, or weaving them, these continuous fibers are known for their material properties and behavior in tension. Carbon fiber, for example, has one of the highest strength-to-weight ratios on the planet. The key is understanding how a given load can distribute amongst local fibers to apply a tensile force.

Sandwich Panels

Combined, these two concepts can be used to create what is called a Sandwich Panel. Like an I-beam, a sandwich panel places reinforcing material on the top and bottom planes of the beam. However, while an I-beam is comprised of a single material, a sandwich panel is comprised of two materials. This makes it a composite - a part made from two or more constituent materials with different properties that, when combined, give the part additional properties different from its constituents.

sandwich panels vs I beams
Composite sandwich panels are very similar to I beams, in that the strongest sections are at the extremes.

A traditional sandwich panel consists of a stack of 3 sheets - two “skins” of the same composition sandwiching a “core” or “matrix” material. Given what we’ve learned from Beam Bending Theory, the skins will experience most of the force when bent. So the skins of a sandwich panel are usually a strong, stiff material, while the core of a sandwich panel is often a weak but durable material meant to space the two skins apart.

In structural composites, the material with the strongest properties dominates the behavior of the panel where that property takes effect. So if your core was stronger than your skin, the skin material wouldn’t have very much impact, and would fracture or tear almost immediately under an applied load. This is why you never see composite fiber skins around metal matrices.

Because the largest forces are applied to the extremes of a beam when in bending, we want the skin of a sandwich panel to take the majority of the force, while the core experiences only minimal load. So this means the skin should be stronger than the core.

fiber sandwich panels explained
Fiber (in blue) is most effective when added to the top- and bottom-most faces of a beam in bending because those faces experience the most force.

This makes a sandwich panel an effective, lightweight structure. Often higher-strength materials are more expensive. With a strong material only as the skin, where it makes the most impact, so you save cost, material, and weight - adding your fiber material on the extremes of a beam maximizes its impact. As you thicken your skin material and it gets closer to the core, you get diminishing returns on its effectiveness. A strong material closer to the core doesn’t contribute to the strength of the part, because the material will experience much lower forces than skins further away.

carbon fiber sandwich panel
A carbon fiber sheet like this one can be used as the "skin" of a sandwich panel to make the part stiff and strong.

As we learned above, a bending force on a beam resolves to tensile and compressive forces on these panels. So the skin of a sandwich panel should be made of a material that performs well under tension — which is often easier to optimize for than compression. This makes continuous fibers a good fit for sandwich panel skins, because they are built and optimized for their strength in tension. Whether woven in sheets, formed into tubes, or 3D printed, a structure is created such that these fiber materials can act as the skin of a sandwich panel so that they can provide the most impact.