# Can we use the Flexural formula and the Beam theories for bodies made up of Composite materials?

Below shows a link for the derivation of Flexural formula.

Derivation of Flexural formula

In this above link, there are a few assumptions made to reach to its final form. They are shown below.

By looking at the first three assumptions, can we still use the Flexural formula for bodies made up of composite materials? Furthermore, these first 3 assumptions as shown are also present in the beam theories (like Euler-Bernoulli and Timoshenko). So can we use these beam theories (to determine the deflections) for beams made up of composite materials?

If no, then can the results obtained from these i.e. Flexural formula and Beam theories be approximated to be somewhat okay/acceptable for the bodies made up of composite materials? Or these will give totally illogical results?

• So how would you answer 1, 2 and 3 based on your composite material? If you don't know about your composite material then you need to find out the values etc. Commented Dec 13, 2021 at 9:44

On an appropriate scale many composite materials (even unidirectional with the appropriate layup or fabric) can be considered homogeneous and (quasi-) isotropic materials.

E.g.: in the case of beams with short randomly oriented fibres should be adequately modelled using the assumption of isotropy and homogeneity. The exception in that case, would be if the beam length is comparable (i.e. in the same order of magnitude or close) to the length of the short fibre.

The main problem with bending and composites is that (at least in Classical Laminate Theory), if there is an anisotropy then there is a coupling between the different layups.

E.g. if you have a [0,0,0,0,90,90,90,90] laminate layup, and you subject it to tension, then it will bent.

Things become even worse when you try to bend other layups, because the end effect is when e.g. bending you could have also twisting (sometimes in unexpected ways).

To my experience, --for large enough structures---the problem isn't that the theory cannot predict the bending. Rather the problem is that is difficult to obtain all the mechanical parameters for the composite material unit (because there is always the chance of manufacturing introducing inconsistensies in the properties).

• The only parameters I see in the flexural formula (used to obtain the stress at a distance y above/below the neutral axis) are M (moment) and I (second moment of inertia). Now, the moment is a factor dependent on Force and distance x, so it doesn't matter if the material is isotropic or anisotropic. I is dependent on geometry so still it doesn't matter if isotropic or anisotropic. For the beam theories (especially Bernoulli), I see an additional parameter of E (Elastic Modulus) which can be calculated for the complete laminate that beam is composed of. Commented Dec 13, 2021 at 10:33
• So I don't see any other parameters which can cause the flexural formula and beam theories to become inapplicable for composites. Furthermore, I don't understand why did these theories had an assumption in the first place that the material must be homogeous and isotropic since this property is not used explicity in their derivation. As far as the E is concerned, so that can be assumed to be the same in tension and compression both for Composite. Regarding plane section remaining plane, I cannot say anything about the composites if it holds or not. Commented Dec 13, 2021 at 10:36
• That was what I was trying to say. I'll try to rephrase it: "Many composite materials can be considered homogeneous and isotropic at a large scale." from that point on I added the asterisks. At least to my understanding, its more common to decide on a quasi-isotropic layup, which is easier to model and simulate, rather that use an optimised anisotropic.
– NMech
Commented Dec 13, 2021 at 10:39
• Well, so it asserts that we can use these beam theories and flexural formula for bodies made up of composites as well. What would you comment on the plane sections remaining plane assumption for composites? Commented Dec 13, 2021 at 12:45
• This is not easy to answer. For quasi isotropic composites and small displacements the planar assumption should hold. However if you have "funny" ABD matrices (particularly in unidirectional composites) then things can start to fall off really quickly.
– NMech
Commented Dec 13, 2021 at 13:39

Boeing 787 is made largely of composite materials. Kit composite airplanes have been available to assemble by customers from the early 80s.

Composite structural beams and joists made of wood chips compressed and laminated in resin are commonly used in buildings. They are treated very similarly to regular lumber for structural purposes, with their own young modulus, E, and flexural strength Fb and shear strength which is many times higher than say Douglas-fir lumber.

There are different types of fiber_rinforced polymers, FRP. uni-weave, cross-woven, or even a mix of layers. There are carbon fiber, kevlar, or fiberglass, each with its own pros and cons.

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In Aviation, they use different lay-in methods to build directional strength aligned with the expected stress (anisotropic media). There are extensive research and data as to the strength and limitations of using FRPs in aviation. Lack of conductivity requires alternative methods to render the plane safe against the electrical discharge of thunder. Ionization under the sun's rays is another concern. Here is the abstract of a paper on the structural properties of FRPs. Fundamentals of Aerospace Composite Materials

This chapter is dedicated to the discussion of fundamental aspect of composite materials. The basic principles and notations of anisotropic elasticity theory are reviewed in tensor notations and then converted to Voigt matrix notations. Strain–displacement and stress–strain relations, equation of motion in terms of stresses, and the equation of motion in terms of displacements are introduced. Attention is next focused on the unidirectional composite lamina: formulae for the estimation of lamina elastic properties from the properties of the constituent fiber and matrix were presented in principal axes. The elastic constants in the longitudinal (L), transverse (T), in-plane shear (LT), and transverse shear (23) directions are deduced and the corresponding stiffness and compliance matrices are presented and related to the constitutive fiber and matrix properties. The properties of the rotated unidirectional lamina are considered next, first under the plane-stress (2D) assumption, and then in the fully 3D case. The 2D and 3D rotation matrices are deduced and applied to obtain the rotated 2D and 3D compliance and stiffness matrices. The proof of some of the more intricate steps during this process is also given as separate sections.