## Balanced verse and Unbalanced Laminate

So how do we know that we have an unbalanced laminate in the first place? Moreover, if you end up with an unbalanced layup, what are the implications? Good questions to have answers to before signing-off on that laminate design.

Firstly, a perfunctory definition is in order regarding a balanced laminate. A designer will need to ensure that for every -α ply there is a +α ply (with the same material and thickness) somewhere within stacking sequence irrespective of location. Examples of balanced laminates are: [0/30/-30/0] or [45/-45/0/0]. An unbalanced laminate is: [0/30/30/0]...notice that the negative 30-degree ply is no longer present. Now maybe you desire an unbalanced laminate or maybe it’s simply unavoidable but in a majority of design cases this layup scheme should be avoided at all costs. Why? Well, perhaps you have already surmised, this layup scheme has adverse implications, one of them significant…the often dreaded and undesirable in-plane extension-shear coupling.

The in-plane, extension-shear coupling (shown in figure above) results from both non-zero A_{16} and A_{26} laminate stiffness coefficients found in the laminate’s A-Matrix. Whew, that was a lot to take in...but really, it’s not that bad and certainly avoidable and easily detectable. For example, below are two varying laminates, one is balanced (Figure 1) and the other one unbalanced (Figure 2). The rectangles locate the A_{16} and A_{26} stiffness coefficients for each laminate. They indicate that for a balanced laminate, both the A_{16} and A_{26} terms are equal to zero; for the unbalanced laminate, the A_{16} and A_{26} terms are non-zero.

### Figure 1: Balanced Laminate

**Figure 2: Unbalanced Laminate**

Therein lies the answer to our first question concerning how one can quickly ascertain whether or not a laminate is balanced or not. So, whether you are designing or analyzing, when examining the laminate, simply compute the ABD matrix for the desired laminate to quickly check if the A_{16} and A_{26} terms are zero (Note: __NO FEA__ is necessary at this point).

Now for a little technical stuff…below, in Figure 3, the transformed reduced stiffness coefficient equations for a lamina are presented. To establish a fundamental understanding, we need only focus on the Q_{16} and Q_{26} terms. You will notice that within the Q_{16} and Q_{26} equations, there are two variables that are cubed; specifically, the m and n variables. The m term is equal to cosine(ø) and the n term is equal to sine(ø).

### Figure 3: : Reduced Transformation Stiffness Coefficients

Let us for “simplicity’s” sake, reduce the Q_{16} and Q_{26} equations down to just the m^{3}n and mn^{3} variables. If we perform a simple calculation by plugging into those terms, a 30-degree and negative 30-degree ply we will get, as expected, two real numbers. This is captured in Table 1 below, showing the coupling effect that exists when staking two positive 30-degree plies as opposed to alternating their signs.

### Table 1: Numerical Difference between a Balanced and Unbalanced Laminate

Well, as you may have already surmised...when applying the same angle using different signs, your Q terms go to zero; however, once the angle plies are oriented in the same direction, the summation of the two values equals a non-zero value (0.66714). You now have an unbalanced laminate because the lamina's Q_{16} and Q_{26} transformed reduced stiffness terms are non-zero, which when applied to the laminate's A-matrix calculation, will contain an A_{16} and A_{26} non-zero term...and (yes you guessed it) a coupling effect.

So why did I include the squared terms in this example…good question. Answer: you will notice that regardless of the angle ply’s orientation, the squared terms will always equal a non-zero number. Noticeably, one sees that the squared terms remain independent of their signs. Therefore, all Q_{ij} equations containing squared terms will be non-zero. Of course, there are exceptions ...when designing a layup to be “Specially Orthotropic”, irrespective of whether it is balanced or not, some of the Q_{ij} equations will equal zero: but that will have to wait for another article...stay tuned.

Okay, that’s it for now! Hopefully, at least from a cursory perspective, your understanding and appreciation for what it means both qualitatively and quantitatively to have a balanced verse unbalanced laminate has improved. Remember unless you desire in-plane shear-extension coupling in your laminate design (See Below), keeping your layup’s balanced is a sure-fire way to avoid undesirable preloads; or, any cure related issues that effect the composite's performance requirements.

### In what situations would you desire an unbalanced laminate?

In the application of aeroelastic tailoring; for example, the forward swept wing in the Grumman's X-29, where unbalanced laminates were exploited to produce a membrane in-plane and shear coupling of the wing skins to avoid an aerodynamic divergence. They are also used to aeroelastically tailor a wind turbine blade to induce a structural bend-twist coupling to improve locally the angle of inflow.

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