Introductory strength analysis

Why this matters

A drone arm carries the motor’s thrust at its tip. Too flexible and it bends and vibrates; too weak and it snaps in the air. A quick calculation before printing rules out designs that will not survive and shows where to add thickness. The drone knowledge base’s design and integration unit and lab L04 use the drone arm as the same problem.

Cantilevers

An arm fixed at the body at one end and loaded at the tip is a cantilever. Standard results from mechanics of materials (Hibbeler, 2023) give the tip deflection $\delta =F{L}^3/(3EI)$ and the maximum bending stress at the root $\sigma =FLc/I$, where $E$ is the material modulus, $I$ is the section’s second moment of area and $c$ is the distance from the neutral axis to the surface.

The formulas say two important things. Deflection goes with $L^3$, so doubling the arm length gives eight times the deflection. And for a rectangle $I=b{h}^3/12$, so the height $h$ in the load direction matters far more than the width: turning the same section on edge makes it several times stiffer without adding mass.

Anisotropy of printed parts

FDM parts are built layer by layer, and the bond between layers is weaker than the material within a layer. The Prusament PLA datasheet gives printed specimens a yield strength of about 51 MPa and a modulus of 2.3 GPa, but interlayer adhesion of only about 17 MPa. Ahn et al. (2002), working with ABS, found that raster direction within a layer changes tensile strength a great deal, and recommended orienting parts so that tensile loads run along the strands.

E, F, L = 2300.0, 10.0, 150.0                 # MPa, N, mm
STRENGTH_ALONG, STRENGTH_ACROSS = 51.0, 17.0  # MPa per the Prusament PLA datasheet

for b, h in ((20, 10), (10, 20), (15, 15)):
    inertia = b * h ** 3 / 12
    deflection = F * L ** 3 / (3 * E * inertia)
    stress = F * L * (h / 2) / inertia
    print(f"section {b}x{h} mm (h in load direction): deflection {deflection:5.2f} mm, stress {stress:4.1f} MPa, "
          f"SF along layers {STRENGTH_ALONG / stress:5.1f}, across layers {STRENGTH_ACROSS / stress:4.1f}")

section 20x10 mm (h in load direction): deflection  2.93 mm, stress  4.5 MPa, SF along layers  11.3, across layers  3.8
section 10x20 mm (h in load direction): deflection  0.73 mm, stress  2.2 MPa, SF along layers  22.7, across layers  7.6
section 15x15 mm (h in load direction): deflection  1.16 mm, stress  2.7 MPa, SF along layers  19.1, across layers  6.4

What thickness buys

E, F, L, B, RHO = 2300.0, 10.0, 60.0, 20.0, 1.24e-3     # MPa, N, mm, mm, g/mm³

for t in (3, 4, 5, 6, 8):
    inertia = B * t ** 3 / 12
    deflection = F * L ** 3 / (3 * E * inertia)
    mass = B * t * L * RHO
    print(f"t = {t} mm: deflection {deflection:6.3f} mm, mass {mass:5.2f} g")

t = 3 mm: deflection  6.957 mm, mass  4.46 g
t = 4 mm: deflection  2.935 mm, mass  5.95 g
t = 5 mm: deflection  1.503 mm, mass  7.44 g
t = 6 mm: deflection  0.870 mm, mass  8.93 g
t = 8 mm: deflection  0.367 mm, mass 11.90 g

FEM, used wisely

Real shapes are more complex than a straight beam, so FEM (finite element method) tools such as FreeCAD’s FEM workbench, with CalculiX as the default solver, are used. But results are only as good as the inputs: material, loads and supports must match reality, the mesh should be refined until results stop changing much, and simple cases should be checked against beam formulas first. Most standard FEM treats the material as isotropic, so it does not see the weak interlayer bonds of printed parts.

Module lab

Common mistakes

Summary

• For a cantilever $\delta =F{L}^3/(3EI)$ and $\sigma =FLc/I$; deflection is very sensitive to length and section height.
• Putting the section’s tall side along the load adds stiffness without adding mass.
• FDM parts are weak across layer bonds (PLA about 17 MPa versus 51 MPa).
• FEM needs correct conditions, a mesh check and comparison with formulas for simple cases.

Check your understanding

1. An arm grows from 100 to 200 mm. By what factor does deflection change?
2. A 10×20 mm section with the 20 mm side along the load has what I?
3. What is the safety factor for 5 MPa stress in a part of 17 MPa strength?
4. Why use printed-part strength rather than filament strength?
5. What is I for a round tube of 12 mm outer and 10 mm inner diameter?