William McMaster
Project Engineer
AlliedSignal Inc.
Engineering Plastics
Morristown, N.J.

Chul Lee
Technical Team Leader
AlliedSignal Inc.
Engineering Plastics
Southfield, Mich.

Engineers typically rely on the cantilever-beam equation for designing snap fits on mating components, defined by:

Deflection = (PL3)/(3EI)

where P = Applied force, lb

L = Beam length, in.
E = Young’s modulus, psi
I = Area moment of inertia, in.4

This equation was originally derived by making several assumptions, including: that a flexible beam is anchored to a rigid wall and that the beam is long enough to neglect the effects of shear.

While these assumptions may be true for rigid structures such as bridges, they aren’t necessarily as accurate for engineered resins. The assumptions may not always be valid because molded plastics generally have thin walls that are close to the same thickness as the cantilevered beam and short cantilevered snaps determined by packaging constraints. This results in designs that depart from predictions calculated by the classical equation. Experience with plastic snap fits shows that assembly and disassembly cause deflection in both the snap-fit beam and the base wall of the part itself.

This deflection is critical to performance because plastic snap fits will fail or fracture if the base wall acts as a rigid structure, like stressing a diving board bolted to a concrete pillar. Because plastic housings aren’t rigid, the elasticity of the supporting wall must be factored into the cantilever-beam equation for accurate results. To compensate for the deflection, an equation has been derived that includes a deflection-magnification factor, Q. Its value depends on the location of a snap-fit beam in relation to the main body and the aspect or length-to-thickness ratio (L/t).

Five beam configurations were considered when deriving Q-factors: a snap fit on a solid wall, on the edge of the molded part continuing in the same plane, with its width parallel to the part edge, in the middle of the part, and with its thickness parallel to the part edge. Engineers solved the Q-factor curves for each of the beam configurations by conducting experiments to determine the deflection caused by each, compared to its length-to-thickness ratio.

The results were compared and verified by data from finite-element analysis of the beams, which resulted in more conservative Q values, and this data was used to produce a Q-factors chart. The new strain equation that includes the Q-factor is:

e = 1.5 (tY)/(L2Q),
where e = maximum strain at the base

t = beam thickness, in.
Y = deflection, in.
L = beam length, in.
Q = deflection-magnification number

An equation for the mating force W of plastic joints is also important when designing snap fits. It calculates the force needed to push or pull snap fits on mating plastic parts. Including the Q -factor, the mating-force equations become:

W = P(μ + tan α)/(1 – μtan α)

and

P = (bt2E ε0)/(6LQ),

where W = push-on force, lb

W ´= pull-off force, lb
P = perpendicular force, lb
μ = coefficient of friction
α = lead angle, deg
α´ = return angle, deg
b = beam width, in.
t = beam thickness, in.
E = flexural modulus, psi
ε = strain
ε0 = allowable strain
L = beam length, in.
Q = deflection-magnification factor

How it works

To illustrate how these revised cantilever-beam equations can be used, the maximum deflection of a snap before failure and the mating force values will be calculated on a plastic snap-fit design. The snap fit is molded in 30% glass-reinforced polyethylene terephthalate (PET).

The maximum snap deflection is calculated by solving for Y using the revised strain equation for cantilever beams.

Ymax = (ε0 L2Q)/(1.5t)

The Q-factor is found by solving for the aspect ratio and finding the corresponding deflection-magnitude factor on the Q-factor chart.

Aspect ratio = L/t = 0.50/0.10 = 5.0

For a beam with its width on the edge of the part (type 4), the Q-factor is 2.0.

Using this value to solve for deflection:

Ymax = (0.015)(0.5)2(2.0)/(1.5)(0.1)
= 0.050 in.

Therefore, engineers would design the snap-fit assembly with a deflection less than 0.050 in. to account for a safety factor.

The mating force for this snap-fit is calculated by first solving for the force perpendicular to the beam, given by:

P = (bt2E ε0)/(6LQ)

Using the dimensions for the snap fit,

P = (0.25)(0.1)2(1.3 3106)(0.015)/(6.0)(0.5)(2.0)
= 8.1 lb

The mating-force equation is given by

W = P(μ + tan α)/(1 – μ tan α)

Substituting in the value calculated for P yields

W = (8.1)(0.2 + tan 30°) /(1 – 0.2 (tan 30°))

= 7.1 lb

This shows that it will take a force of 7.1 lb to engage the snap fit.



Acting on a hunch

While most engineers follow well-established laws of physics when designing components, sometimes it takes ingenuity and guts to depart from the norm and gamble on a hunch. This kind of pioneering led to developing the Q-factor for snap-fit designs.

Though the basic cantilever beam equation hadn’t been questioned for centuries, one engineer, with some help from a plastic molder, felt it was time to challenge it. They both knew that a lot of book theory doesn’t always exactly translate to applied science.

The design engineer working for AlliedSignal was goaded by a molder who came to him after experiencing problems molding snap fits from ABS. The molder wondered whether nylon could be used as a drop-in replacement.

After calculating the deflection using the conventional cantilever-beam equation, the engineer determined that nylon wouldn’t work and suggested that the molder change his design. However, the molder decided to act on his hunch and mold a part using nylon anyway. After successfully molding the component, the molder urged the engineer to review his calculations. Recalculations still showed that nylon should fail.

Taking a closer look, the engineer realized that plastics don’t meet the assumption made with the conventional cantilever-beam equation (derived for long, flexible beams attached to rigid walls). Plastic snap fits don’t meet these criteria because they typically have short cantilevers attached to thin, flexible walls that are near the same thickness as the beam.

The engineer from AlliedSignal devised an experiment to prove the new hypothesis. He developed a test fixture that included a short beam extending from an inverted U-shaped structure. As a fixed weight, W, moved away from the base, along scribed marks on the beam, a gage measured deflection on the wall and beam.

The results of the experiment, along with finite-element analysis, produced a set of deflection curves for cantilever beams with varying aspect ratios (length/thickness). The curves are lower for parts with a solid-wall base and are higher for beams close to the edge of the base wall. Engineers used the curves to derive a deflection-magnification factor, Q. For each curve, Q becomes significantly greater for smaller aspect ratios.

© 2010 Penton Media, Inc.