Bending Beam Theory

Bending Beam Theory

Bending beam theory refers to a simplification of the theory of linear elasticity. It enables calculation of deflection and load carrying characteristics of beams. Mostly it is a case of minute deflections of beams only subjected to lateral loading (Bauchau et al., 2011). As such, bending beam theory is a specialized case of Timoshenko beam theory. Its first enunciation was in the years 1750, although it was not applicable in large scale until the engineering advances of Ferris wheel and Eiffel Tower late in the 19^th Century. Due to the success of such demonstrations, bending beam theory is regarded as an engineering cornerstone and a milestone that prompted Second Industrial Revolution. 

Evolutions

Galileo Galilei first published the theory of beam strength when bending. However, there was a recent discovery of Leonardo da Vinci’s work “Codex Madrid” in Spain National Library. Leonardo accurately appreciated strains and stress in beams subjected to bending. However, he did not provide beam strength assessment technique, material’s tensile strength tensile strength and beam dimensions. In 1638, Galileo addressed gaps in Leonardo’s work. He illustrated his theory with cantilever beam pinned to the wall (Love, 2013). His assumption was that the beam’s rotation point was at the base of its support. As such, he predicted a presence of uniform tensile stress that is equal to material’s tensile strength. 

Galileo’s assumptions regarding uniform stress were incorrect. In fact, the outcome of his experiment was three times higher than the actual result. In 1686, Mariette Edme identified flaws in Galileo’s projections and published an alternative approach. In addition to Galileo’s theory, he proposed a triangular stress distribution mechanism. Therefore, variation was evident in failure stress at the peak and base zero. Without proof, Edme proposed that the neutral axis should be at the midsection. In addition, he conceived an error for his theory which led to a doubling of the accurate value. It would be this way especially if the neutral axis is placed at the base section he assumed initially. 

Jacob Bernoulli, on the other hand, carefully investigated Cantilever beam’s deflection as a practice in infinitesimal calculus application by Leibnitz (Sylla et al., 2010). In 1694, Bernoulli published an initial discussion on the issue. A few years before his death in 1705, he authored and published a final piece version. His assumption was strikingly similar to Marriotte’s, that beam’s bottom will be the location of a neutral axis. He derived an equation based on his assumption that the plane sections were unaltered. His equation was therefore based on cross-sectional net bending and external bending moment. Bernoulli’s stiffness factor in his equation was double the actual value because he borrowed his assumption from Marriote(Teugels, 2014). The general form of his equation was still correct and applicable, as Euler used it in his investigation. Johann Bernoulli’s contribution to the field of material elastic property was dismal and less important, according to Timoshenko. Daniel (Han et al., 2013). On the other hand, Daniel Bernoulli derived a differential equation that governs lateral vibrations of prismatic bars.

Eventually, Antonie Parent created a correct formula in 1713 by assuming correctly the linear stress distribution at the top and central neutral axis to opposite and equal bottom compression. As such, he derived a precise modulus for elastic section area multiplied by section depth and all divided by six. It is unfortunate that his theory had a little impact. It took centuries into the future before actual scientific principles were applied regularly in the analysis of beam strength in bending (Wang et al., 2012).

In 1773, Coulomb reinvented bending beam theory. He discovered that wood and stone exhibited distinct traits and behaviours. Resultantly, he devised two theories as per a common approach to satisfy mechanics’ requirements. His conclusion was that Galileo’s coefficient of a half was best suited for stone. However, he failed to satisfactorily verify his coefficient of a six for timber. He also could not verify Parent’s coefficient on the same (Cowper, 2013). As the century neared its end, Ecole Polytechnique’s standard text was still stuck on Mariotte’s formula of a third for wood and Galileo’s formula for stone.

It is curious to note that though timber’s weight is approximately equal to that of a plastic material, Mariotte’s coefficient of a third should provide a close approximation of bending strength. However, given that stone is indeed a brittle material that can possibly fail during tension especially in the case of triangular stress distribution, its appropriate coefficient should be a sixth rather than a half (Park et al., 2012).

It is, therefore, unfortunate that Heyman offers no detail on Coulomb’s experiment. Still, such an anomaly can only be explained by assuming that Coulomb conducted tensile tests that yielded outcome much lower than the actual value, or he utilised back-calculation from multiple bending tests to derive tensile strength. Whatever the case, stress concentrations in his testing apparatuses affected the outcome. By the end of 19^th century, an incorrect formula by a factor of 3 on the stone’s bending strength was still being recommended in most engineering texts.

Bending beam theory focuses on slender beams with an assumption of non-alteration of plane sections. It does not account for shear deformations across the sections (Reddy, 2014). Furthermore, it is only applicable if the material yield stress is more than the maximum overall stress. Therefore, the equation for bending beam theory can be used for objects that are slender and straight. They should also be composed of linear elastic material, especially on the cross sections. Only tiny deflections can be considered.

            In light of this, bending beam theory is applicable in plastic bending. The theory’s equation is only valid in this case when the extreme fiber exhibits stress levels significantly below material’s yield stress. Notably, stress distributions turn to non-linear at higher loadings. The theory’s formula can also be used to calculate large bending deformations if cross sectional stress detectible (Civalek et al., 2011). Some of its assumptions include the flatness of considered body section after and before deformations, and normal and shear stress in such sections perpendicular to normal vector.

In summary, it is clear that the bending beam theory has evolved over centuries. Credit should be given to Eduardo da Vinci for his ideas that formed the basis of this theory. However, it is worth noticing that early its early forms were marred by inaccuracies and lack of precisions despite the efforts made by the researchers. Interestingly, it took several generations for the evident flaws to be discovered and rectified. Partly, this can be attributed to inapplicability of this theory in engineering until early 19^th century.

Bibliography

Bauchau, O.A. and Craig, J.I., 2011. Euler-Bernoulli Beam Theory. InStructural analysis (pp. 173-221). Springer Netherlands.

Civalek, Ö. and Demir, Ç., 2011. Bending Analysis of Microtubules Using Nonlocal Euler–Bernoulli Beam Theory. Applied Mathematical Modelling, 35(5), pp.2053-2067.

Cowper, G.R., 2013. The Shear Coefficient in Timoshenko’s Beam Theory.Journal Of Applied Mechanics, 33(2), pp.335-340.

Han, S.M., Benaroya, H. and Wei, T., 2013. Dynamics of Transversely Vibrating Beams Using Four Engineering Theories. Journal of Sound and Vibration, 225(5), pp.935-988.

Love, A.E.H., 2013. A Treatise on the Mathematical Theory Of Elasticity (Vol. 1). London: Cambridge University Press.

Park, S.K. and Gao, X.L., 2012. Bernoulli–Euler Beam Model Based on a Modified Couple Stress Theory. Journal of Micromechanics and Microengineering, 16(11), p.2355.

Reddy, J.N., 1914. On Locking-Free Shear Deformable Beam Finite Elements.Computer Methods in Applied Mechanics and Engineering, 149(1), pp.113-132.

Sylla, E.D., 2010. The Emergence of Mathematical Probability from the Perspective Of The Leibniz-Jacob Bernoulli Correspondence. Perspectives on Science, 6(1), pp.41-76.

Teugels, J.L., 2004. Bernoulli Family.  New York: John Wiley & Sons, Ltd.

Wang, C.M., Kitipornchai, S., Lim, C.W. and Eisenberger, M., 2012. Beam bending solutions based on nonlocal Timoshenko beam theory. Journal of Engineering Mechanics, 134(6), pp.475-481.