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The Shear-Lag Model...

Make that overlap length as long as possible…the longer the better I always say…we don’t want the bonded joint to fail…says the confident designer with an unfettered surety…hmmm...well, not quite.

Longer is not necessarily better in this case. So, for those of you interested in approximating the performance of an adhesively bonded joint “quickly” while adhering to a few basic engineering principles that will establish optimal joint parameters, 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 method was used to assess failure in adhesively bonded lap joints. Of course, the method is not without some underlying assumptions which limit its use, but I will be address those later.

Let’s begin by showing the ubiquitous Volkersen’s Shear-Lag model in Figure 1. This model is nothing more than a typical single-lap joint subjected to pure tension loading. But, there's more! The figure also shows the differential shear behavior of the adhesive during loading; importantly, this model assumes that the adherends deform in tension only (note: you can assume this because the adherends are considered elastic - not infinitely rigid). Also, the adhesive is limited to deformation via shear only and both tension and shear are constant across both adherend and adhesive thicknesses. The exaggerated adhesive shear deformation also provides us with a clue as to where the maximum shear stress location is…in this case, it’s at the ends of the adherends.

Volkersen Shear-Lag Model

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

Crucially, the shear stress concentration can be manipulated by altering either the adhesive and/or adherend properties. So the focus in this article will center around one of those properties…namely, the overlap length. The intent here is to emphasize the impact that changing the overlap length will have in rasing or lowering the shear stress concentration; Graph 1 visually illustrates the effect of varying overlap length.

Graph - Stress Concentration

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

Moreover, Graph 1 shows that the stress concentration (derived using Equation 2) is highest when the overlap length is smallest. Noticeably, as the overlap length increases from the free edge, the stress concentration quickly drops and levels-out. Examining Equation 1 closer reveals why…the overlap length is in the numerator and is squared…therefore, when increasing the overlap length by say…two inches, the F-term increases by a factor of 4 in this example! 

Stress Concentration

However, you will also note that you will gain very little in terms of lowering the stress concentration…because just over 1.0 inch, the stress concertation levels-off. So, if you want to lower the shear stress concentration in a bonding joint, extending the overlap-length doesn't necessarily help you...and in this example, it would be any length over approximately 1.0 inch. To carry this concept even further, Graph 2 shows what happens to the stress concentration when you increase the overlap length to 20 inches…well, you guessed it—nothing!

Graph 2 Stress Concentration

Graph 2: Overlap Length at 20 inches

What's the lesson learned here?! Well...avoid acting on your first instinct...which is to simply increase the joint’s overlap distance. You will not be doing yourself any favors...unless, of course, you intentionally wanted to add unnecessary weight to the joint....

Oh…by the way, referring back to Figure 1, something that was shown just doesn't seem right…“Houston, we have problem!” That's right, you guessed it! The free edges have shear stresses! Huh? Let me explain...reflect 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 located at the ends which violates the stress-free condition. The implications of this violation are outlined in the assumptions section.

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

Adhesive Thickness

Graph 3 Effect on Stress Concentration when changing in Adhesive Thickness

And there you have it…a simple and quick closed-form method that you can use to partially assess a bonded joints 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 or the shear deformations, which are potentially significant for composite adherends with a low shear and transverse moduli and strength [Source: DOT/FAA/AR-05/12].

AssumptionsMoreover, the assumptions for this method restrict its practical use considerably. For instance:

  • The lateral strain (ey) is equal to zero forcing the shear stress Yxyto 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; in turn, this will overestimate the stress at the adherend ends of the overlap and prodcue a conservative failure load

Nevertheless, the Volkersen’s method does have “some” practicality when treating the analysis in terms of an initial (back-of-the-envelop) design calculation…particularly, when trying to determine a first-article overlap length which can be critical for weight sensitive designs; which, as mention earlier, can lead to unnecessary weight gains. Furthermore, the Volkersen’s approach is acceptable If the joint’s bending is assumed trivial and the adhesive is defined as brittle. A footnote: In 1944, Goland and Reissner considered the effects of the adherend bending and the peel stress, as well as the shear stress in the adhesive layer for a single lap joint. Additionally, Oplinger offered corrections to the Goland and Reissner solution by using a layered beam theory instead of the classical instead of the classical homogeneous beam model for single lap joints. Both of these alternative solutions may be discussed in a future article.

Now once you have determined the ideal overlap length, don't stop there, move forward using 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. 

Stress Equation

Lambda Equation

This article was intended to illustrate the simplest of all the joint analyses methods available. Emphasis was placed on considering the ideal overlap length. But one should never overlook adhesive strength, adhesive properties, adherend properties or the joining procedures when finalizing a bonded joint design.

quick tip
The joint strength is maximum with minimally applied adhesive which increases the load capacity of the joint.

That's it! As is customary here at ABDMatrix.com, I briefly covered the topic of bonded joints. Hopefully, this expository has increased your awareness and perhaps one day facilitate your engineering endeavors reagrding bonded joint analysis. For other kind of joints such as tubular joints, I recommend referring to papers provided by Lubkin and Reissner (1956) and Adams and Peppiatt (1977). Thank you for visisting ABDmatrix.com and your comments are aalways welcomed. 

Sources:

  • Closed-form solutions for adhesively bonded joints, Lucas F. M. da Silva, Ricardo F. T. Lima, Rui M. S. Teixeira and A. Puga
  • Methods of Analysis and Failure Predictions for Adhesively Bonded Joints of Uniform and Variable Bondline Thickness, DOT/FAA/AR-05/12, 2005
  • Single Lap Adhesive Joint (SLAJ): A Study, Sameer Shaikh, Nitinkumar Anekar, Pravin Kanase, Ajinkya Patil and Suraj TarateSameer Shaikh*, Nitinkumar Anekar, Pravin Kanase, Ajinkya Patil and Suraj TarateDepartment of Mechanical Engineering, MITCOE, Savitribai Phule Pune University, Pune, India 2017  
2018-08-25 01:56:22
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