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7 minutes reading time (1300 words)

The Volkersen’s Method

Make that overlap length as long as possible…the longer the better I always say…we don't want that bonded joint to fail…says the confident designer with an unparalleled surety…or, is it absurdity…hmmm.

Well, for those of you interested in approximating the performance of an adhesively bonded joint "quickly" while adhering to some basic engineering principles, then perhaps you may want to take a look at one of the simplest closed-form methods available. In 1938, the Volkersen's Method, also known as the "shear-lag model", was first used for mechanical joints with fasteners. Invariably, this simple method found its way into adhesively bonded lap joints. Of course, the method is not without some underlying assumptions that limit its use...but I will be delve into those later.

Let's first start by showing the ubiquitous Volkersen's Shear-Lag model in Figure 1. Nothing much to see here except the typical single-lap joint subjected to pure tension loading. However, this illustration also demonstrates the differential shear behavior of the adhesive during loading; importantly, the model assumes that the adherends deform in tension only (Note: they are considered elastic not infinitely rigid) and the adhesive deforms in shear only. Moreover, both tension and shear are constant across their respective thicknesses. The exaggerated shear deformation also provides a clue as to where the location of the maximum shear stresses is…and in this case, it's at the ends of the adherends. 

Figure 1: The Volkersen Model showing Location of maximum shear stress in the Adhesive

The magnitude of the shear stress concentration at the ends can be manipulated by altering either the adhesive and/or adherend properties. This article will focus on one of those properties…namely, the overlap length. To illustrate the impact that changing the overlap length has on the shear stress concentration, one turns to a graph that shows the change in shear stress concentration with respect to changing overlap length. Graph 1 shows the effect that a varying overlap length has on the stress concentrations in the adhesive.

Graph 1: Stress Concentration with an Overlap Length at 4 inches

Graph 1 shows that the shear stress concentration (derived using Equation 2) is highest when the overlap length is smallest. Noticeably, the further you move away from the free edge, the stress concentration reduces quickly and then exhibits asymptotic behavior at approximately one-inch. Examining Equation 1 closer reveals why…the lap length in the numerator is squared…so, when one changes the overlap length by say…two inches, the F-term increases by a factor of 4...for this example!  

Stress Concentration Equations

However, you will also note that you gain very little in terms of lowering the stress concentration…because after about one-inch the stress concentration again levels-off. So, if you want to lower the shear stress concentration in the adhesive, extending the lap-length beyond an optimal overlap length (again in this example about one-inch) doesn't help. To carry this concept further, Graph 2 shows what happens to the stress concentration when you extend the lap-length to say…20 inches…well, you guessed it—nothing!

Graph 2: Overlap Length at 20 inches

The lesson learned here…avoid your first instinct to simply increase a joint's overlap distance and expect improvement in the joint's performance...unless of course you intentionally want to add unnecessary weight. Oh…and by the way, referring back to Figure 1, you will note that something is just not right…"Houston, we have problem!" Recall back to your simple beam theory, the shear stress is highest in the middle and not at the ends where it is zero; however, in this model it's at the ends of the plates which violates the stress-free condition. The implications of this are outlined in the assumptions section.

Now, if my stress concentrations are still too high after achieving an optimal length, what can one do then?! Good question…OK…there are several other joint parameters that can be manipulated to reduce the stress concentration without having to increase overlap length. You could change the modulus of the adherend; or, reduce the thickness of either the adherend or the adhesive. Or, you could decrease the shear modulus of the adhesive relative to the adherend modulus…effectively lowering the stiffness in the adhesive. Graph 3 demonstrates what happens when you decrease the adhesive thickness.

Graph 3: Effect on Stress Concentration when changing in Adhesive Thickness

And there you have it…a simple and quick method to partially assess a bonded joints structural efficacy. Well, not so fast…the Volkersen approach is not without its shortcomings. The Volkersen solution does not reflect the effect of the adherend bending and the shear deformations that are potentially significant for composite adherends with a low shear and transverse moduli and strength [Source: DOT/FAA/AR-05/12].

The assumptions for this method restricts its wide use considerably; consequently, the following limitations must be observed:

  • Both adherends must be loaded in pure tension only
  • The lateral strain (ey) is equal to zero forcing the shear stress Yxy to remain constant over the thickness of the adhesive
  • The bending moment due to the eccentricity produced in the single lap joint is ignored…failing to account for peel stress
  • The adhesive can only carry out-of-plane stresses while adherends will carry only in-plane stresses
  • There is uniform distribution of shear strain through the adhesive thickness
  • Adherends are same thickness
  • Adherends are considered as thin beams; ignoring the through-thickness shear deformation…Note: Adherend shear is important in shear-soft adherends like composites
  • The model violates the stress-free condition will overestimate the stress at the ends of the overlap, yielding conservative failure loads 

Nevertheless, the Volkersen's method has "some" practicality when considering an analysis in terms of an initial (back-of-the-envelop) design calculation…particularly, when trying to determine a first-article overlap length; which if too long, merely adds weight unnecessarily. Furthermore, the Volkersen's approach is sufficient IF the joint's bending is assumed trivial and the adhesive is defined as brittle. There is a footnote to this: In 1944, Goland and Reissner considered the effects of the adherend bending; the peel stress and the shear stress in the adhesive layer for a single lap joint. Also, Oplinger offered corrections to the Goland and Reissner solution by using a layered beam theory instead of the classical homogeneous beam model for single lap joints. These alternative solutions will be discussed in a future article.

Now once you have determined the ideal overlap length, you can use the following equations to calculate the shear stress in the joint. The variable c is half the length of the adhesive and the x coordinate is in the middle of the adhesive. Note that the variable lambda associates the stiffness of the adhesive with the adherends.

Equations used to calculate the shear stress in the joint
Lambda

This exposition is intended to illustrate the simplest of all the methods. Emphasis was placed on considering the ideal overlap length only. But one should also consider adhesive strength, adhesive properties, adherend properties and joining procedures when designing a bounded lap joint. Remember the joint strength is maximum with minimally applied adhesive which increases the load capacity of the joint.

As is customary here at abdmatrix.com, I only cover the salient information that is intended to increase your awareness and perhaps facilitate your engineering endeavors by introducing you to another tool that can be leveraged during an analysis. Additional reading for other kind of joints such as tubular joints, one can refer to Lubkin and Reissner (1956), Adams and Peppiatt (1977).


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