# Poisson's ratio in analytical beam deflection equations

After looking at some of the analytical expressions for analyzing beams, I noticed that none of the equations depend on the material's Poisson ratio. Some analytical expressions can be found in https://www.linsgroup.com/MECHANICAL_DESIGN/Beam/beam_formula.htm

I believe these equations are derived from the Euler Bernoulli equations and not from 3-D Linear elasticity, and Poisson's ratio isn't included in Euler Bernoulli. What underlying assumption in Euler Bernoulli allows for the exclusion of the Poisson's ratio?

If I want to model a beam under the Euler Bernoulli assumptions in a 3-D linear elasticity code, where the Poisson's ratio is required as an input, how can this be done? Essentially, I am trying to perform a one to one comparison of the linear elasticity code I am using with the analytical beam expressions. To do this, I need to figure out what value to input for the Poisson's ratio in the code. When I model the beam with the code and vary the Poisson's ratio, the deflection changes as a function of the poisson's ratio. However, the analytical beam expressions do not depend on the Poisson's ratio, so it is not clear to me how this comparison can be performed.

• Can you please edit to clarify your second question? Are you asking which Poisson ratio to input or how to write an equation that incorporates the Poisson ratio? Mar 19, 2019 at 14:36
• @Chemomechanics I just updated the last paragraph. I am trying to compare deflection generated from a 3-D linear elasticity code (which requires Poisson's ratio as input) with the analytical expressions from beam theory. It is not clear to me how this can be done because, when I vary Poisson's ratio in the code, the deflection changes, but the beam theory equations is not a function of the poisson's ratio. So I think I either need to either modify the analytical expressions to include Poisson's ratio, or I need to make an assumption in the code so that it doesn't depend on Poisson's ratio. Mar 19, 2019 at 14:45
• Usually use Hooke's law ignoring the Poisson effect. Mar 19, 2019 at 14:50
• @AlexTrounev I don't quite understand what this means? Why does using Hooke's law ignore Poisson effect? As a side question: should the deflection with inclusion of poisson's effects be smaller than the deflection from beam theory? Mar 19, 2019 at 14:53
• Whether Poisson’s ratio is needed depends on what you are calculating. Please clarify the problem. Mar 19, 2019 at 14:54

The Poisson ratio is, of course, the lateral expansion (contraction) associated with axial compression (elongation), expressed as the negative ratio of strains: $$\nu=-\epsilon_y/\epsilon_x$$.

Intuitively, when a beam bends with positive curvature, the top (bottom) side is in a state of compression (tension), which can be expected to distort the cross section because (for a material with a positive Poisson ratio) the top and bottom regions laterally expand and contract, respectively, known as anticlastic curvature: The Euler-Bernoulli equations assume that the beam is narrow enough that these strains are unconstrained. In other words, any lateral deformation of the beam encounters negligible resistance and therefore doesn't affect the deflection. The key constitutive equation that's being applied is $$\sigma=E\epsilon$$, or simple Hooke's Law, which doesn't feature Poisson's ratio.

In contrast, if you were to try to deflect a cantilevered plate, you'd find that the stiffness is higher than can be accounted for by just scaling up the beam width. The reason is that these Poisson-ratio-mediated lateral deformations are now constrained to be zero because there's plate material in the way. Another way of saying this is that the internal stress state along the length of the beam has transitioned from plane stress to plane strain. In isotropic materials, the appropriate 3D constitutive law is generalized Hooke's Law, which incorporates Poisson's ratio.

It's not surprising that the Poisson ratio has some influence on your numerical results, since you're working with a real beam whose cross section will be distorted somewhat; in addition, the composite lamina may tend to distort to different extents, which would produce an additional stress field in the cross section. The constitutive equation is now $$\boldsymbol{\sigma}=\boldsymbol{C\epsilon}$$, where the parameters are tensors and $$\boldsymbol{C}$$ is the stiffness tensor, which definitely incorporates Poisson's ratio (actually three — $$\nu_{xy}$$, $$\nu_{xz}$$, and $$\nu_{yz}$$ — because of the anisotropy).

So I'd recommend applying at least an approximately correct Poisson ratio for your materials to improve the simulation accuracy. A Poisson ratio of zero should bring you closer to the Euler-Bernoulli solution.

• A couple of follow-ups: (1) I am modeling a 24" long, 1" thick, and 3" wide aluminum 2024 beam. The Poisson ratio is 0.33. As I decrease the Poisson ratio, I notice that the deflection increases. From beam theory, I obtain a max deflection value of ~0.0027m. The 3-D linear elasticity code gives a deflection that is only 0.001m using Poisson ratio of 0.33. When I decrease Poisson ratio to 0.2, the deflection increases to ~0.0017m. Is it actually possible to input a Poisson's ratio value such that a 3-D code's model of a beam reduces to beam theory? Mar 19, 2019 at 15:24
• (2) Is it possible to derive analytical expressions with inclusion of Poisson effects? I have never seen this done, but I'm not too well versed with structures theory. Mar 19, 2019 at 15:25

Poisson’s ratio, ν, is defined as

ν = -lateral strain/longitudinal strain.

It is applied when analyzing the lateral strain on members subjected to axial forces (compression and tension). None of the beams in the diagrams of your link involve axial loading (loads applied to the ends of the beams along the axis of the beam). Nor do they deal with strain resulting from shear forces and bending moments. Therefore the Poisson ratio would not apply.

UPDATE:

I ‘m not sure if this will help you, but the modulus of elasticity, $$E$$, is related to Poisson’s ratio, ν, by the following equation where $$G$$ is the shear modulus

$$G=\frac{E}{2(1+ν)}$$

Hope this helps

• Hmm yes I understand, but are analytical expressions for the deflection derivable if Poisson effects are included? Mar 19, 2019 at 15:14
• @David See my update Mar 19, 2019 at 15:33

One of the assumptions of Euler-Bernouli theory is that the cross sectional plane remains rigid. Consider a cantilever beam (slender) with rectangular cross section. The above assumption implies that breadth and height can NOT change even though there is extension in the longitudinal direction (unrealistic but that is the implication of the assumption). Thus poison ratio does not come into the picture. Section5 5.2 of the text book Structural analysis by O.A.Bauchau and J.I.Craig has good illustration of the implications of Euler-Bernouli assumptions.