Talk:Euler–Bernoulli beam theory

I forgot to login before creating this article. --Yannick 23:42, 17 Jun 2005 (UTC)

Suggestions for Further Work

I think the cantilever glass beam is a nice picture but does not help at all, and does not add any value to the article as I can not understand it, even if I have a strong knowledge on this subject. It should be deleted or replaced by another more understandable example... — Preceding unsigned comment added by 119.77.8.4 (talk) 01:06, 2 March 2012 (UTC)

Sign Convention Erros

There are some critical sign convention errors with this article. In some places it says that the fibers on the bottom are in tension and in some places it says they are in compression. They can not be both.

Also, given the equation -EI(d^2v/d^x2)= M as given in this page, the sign convention on M is VERY confusing. In one place it says the convention is that a positive M yields fibers in tension on the bottom which means M must make a smiley face when place on both ends. But the stated equation will not yield this solution.

I have seen beam theory in several books and NEVER seen the sign conventions here. —Preceding unsigned comment added by 150.250.219.220 (talk) 18:51, 2 May 2011 (UTC)

The article assumes a right handed coordinate system and that the displacement $w$ is positive in the positive $z$-direction. Otherwise the convention is the standard mechanical engineering convention that tension leads to a positive stress. In some strength of materials books a left-handed coordinate system is used while in others $w$ is assumed positive in the negative $\mathbf {e} _{z}$ direction .

If a vector is written in a right-handed coordinate system as $\mathbf {u} =u_{x}\mathbf {e} _{x}+u_{y}\mathbf {e} _{y}+u_{z}\mathbf {e} _{z}$, the standard convention is that $u_{z}$ is positive in the positive $\mathbf {e} _{z}$ direction.

The article has also been edited by a number of people, some of whom have failed to keep it consistent. A consistency check will be greatly appreciated. Bbanerje (talk) 22:03, 2 May 2011 (UTC)

Yes, there is definitely a need for a consistency check. I don't agree with the statement "The sign of the bending moment is chosen so that a positive value leads to a tensile stress at the bottom cords." in the end of the section "Static beam equation". I think it should be the other way around. In any case, this statement is contradicted by what is said in the section "Simple or symmetrical bending". Andreasdr (talk) 13:23, 25 January 2012 (UTC)

History

1. I'm curious as to when the theory was proven by laboratory experiment.
2. The nature of Bernoulli's and Euler's collaboration could be expanded.
3. Stephen Timoshenko's History of Strength of Materials (Dover Publications, Inc., New York, 1983) would probably provide more information. Anybody got a copy?

--Yannick 23:42, 17 Jun 2005 (UTC)

For number 1, I was led to believe that it was derived using laboratory experiment. I was told by a prof that this was found by measuring displacements of actual beams using a grid. - EndingPop 22 Dec 2005

In my classes, the prof told us that Euler and Bernoulli derived the equations from the full theory of elasticity matrices, which were already known, and empirical proof came later. I still have extensive notes on the derivation.--Yannick 18:34, 3 July 2006 (UTC)

Addendum

Two more assumptions are:
Material should be homogeneous.
Material should have continuity.

I believe those are assumptions of the theory of elasticity, not of beam theory.--Yannick 04:23, 18 September 2006 (UTC)

Maximum deflection for the four elementary loading cases must be added.
1) Cantilever (one end fixed, other free)
2) Cantilever prop (one end fixed, other pinned)
3) Simply-supported (both ends pinned)
4) Fixed-Fixed (both ends fixed)

I would also like to add few diagrams. I am busy with exams these days. I would like to expand this article once I am finished with exams.
Note: We referred R. C. Hibbeler for our "Engineering Mechanics", "Strength of Materials" and "Structural Analysis" couse. Hibbeler have'nt properly credited Euler-Bernoulli for this equation. We just call it classical equation.

Stress / Deflection = Moment / Inertia = Elasticity / Curvature

Yannick: I believe our library has the book u mentioned. I will confirm it though. Thanks for increasing my knowledge regarding this equation.
H.A., third year Civil Engineering Student.

---

Just found this page. I'm a (semi) retired CE who is reviewing material learned and forgotten thirty years ago.

I am thinking of adding a section on simplified beam theory which uses these assumptions:

1) Linearly elastic, isotropic 2) Small deflections 3) Shear negligible.

It's not as complete as Timoshenko beam theory, but 99.9% of structures are designed this way.

-James0011

That would be more useful than the current page. Having both would make it much more complete. - EndingPop 00:55, 13 June 2006 (UTC)

Please don't edit equations recklessly

I basically reverted the main equations in the "practical simplification" section. My main reason is that a major error was introduced on December 22, 2005, by 131.151.65.70 simply by changing '=' signs to '+' signs. However, I also reverted other ongoing degradation of the equations. I changed the load variable back to 'P' from 'F' for consistency, because F was defined earlier in the article as internal axial force. I'm guessing it was originally changed to match the diagram, but changes like that should try to make the ENTIRE article consistent, not just one section. In this case, I would prefer changing the diagram to show a tip load of 'P', but if we go with 'F', then the variable for internal axial force should change to something else for the whole article. I also reverted 69.241.225.246's recent change at the same time because although concise equations may be pleasing to mathematicians, they make it harder for the intended audience (engineers) to read, and key information was lost in the edit: that these equations give the MAXIMUM deflection and stress for any point in the beam. Also, if we are going to express these equations as proper functions of '(x)', then 'My' should also be expressed as 'My(x)'. I didn't want to do that because this is a basic example section which should minimize the potential to confuse newbies.

No doubt there is a way to formulate these equations that will satisfy everyone's concerns, and I'll try to work on it if I get a chance. But if you want to take a try at it, please look at the entire equation, and see how it fits in with the rest of the article. A small change can really mess things up.--Yannick 04:14, 18 September 2006 (UTC)

I think there is an Error in the deflection ODE solution: it should be diveded by 3 instead of multiplied by 2. Pleae check me out!

Yes, it's wrong but no, not in that way. Fixed anyway, thanks for bringing attention. -Ben pcc 05:06, 20 April 2007 (UTC)

Cantilever Image

That image of the cantilevered beam in the Boundary Considerations section seems wrong to me. In the image, the free end remains horizontal. This seems to imply that there is zero rotation, but non-zero deflection at the end of the beam. Somehow we've fixed rotations and not displacements, which isn't a possible BC presented in the section. If I'm wrong on this, could someone please explain why? - EndingPop 22:16, 3 March 2007 (UTC)

Good point. Fixed.

Ben pcc 03:36, 6 March 2007 (UTC)

Cantilever example

The cantilever with a distributed load example appears to have incorrect equations for deflection. In particular, if I assume that fixed end of the beam occurs at either x = 0 or x = L, there is a nonzero deflection, but the fixed end should have a zero deflection. Also, the equations do not match one of the quoted external links "Beam Deflection Formulae". I don't know the correct formula myself. Could someone please fix it? Thanks. Eerb (talk) 18:20, 13 June 2008 (UTC)

There are more issues. I asked the author of these examples to fix these. Crowsnest (talk) 22:04, 13 June 2008 (UTC)

The author does not react, so I added tags to the relevant "Beam examples" section. 11:11, 22 June 2008 (UTC)

The maximum deflection calculation should be /8 versus /12. —Preceding unsigned comment added by 147.160.136.10 (talk) 21:18, 30 June 2008 (UTC)

There are errors in this section. If they are not removed and the examples are not provided with reliable sources, I will remove this section. Crowsnest (talk) 07:59, 1 July 2008 (UTC)

I moved the beam examples over here, for the time being, since they contain errors and lack verifiability. They are nice, so please clean them up, add citations and thereafter put them back, if you like. Crowsnest (talk) 08:32, 1 July 2008 (UTC)

Copy of the section Beam examples

==Beam examples== {{disputed-section|date=June 2008}} {{expert-subject|Civil engineering|date=June 2008}} {{verify-section|date=June 2008}}

In the examples below, the following nomenclature applies:

• $R$ is shear
• $M$ is bending moment
• $Q$ is beam slope
• $u$ is deflection
• $E$ is Young's Modulus
• $I$ is second moment of area (or moment of inertia)

Because the Euler-Bernoulli beam equation is linear, the examples below can be superposed (added and subtracted) to model more complex situations (for example, the shears, moments and deflections for a simply supported beam with a universally distributed load can be added to those of a simply supported beam with a central point load to give the shears, moments and deflections for a simply supported beam with both a universally distributed load and a central point load).

Simply supported beam with central point load
Shear	$R(x)={\frac {F}{2}}$	$R_{A}=R_{C}={\frac {F}{2}}$
Moments	$M(x)={\begin{cases}{\frac {Fx}{2}},&{\mbox{for }}x=0{\mbox{ to }}x=L/2\\{\frac {F(x-L)}{2}},&{\mbox{for }}x=L/2{\mbox{ to }}x=0\end{cases}}$	$M_{B}={\frac {FL}{4}}$
Slope	$Q(x)={\begin{cases}{\frac {Fx^{2}}{4EI}}-{\frac {FL^{2}}{16EI}},&{\mbox{for }}x=0{\mbox{ to }}x=L/2\\{\frac {F(x-L)^{2}}{4EI}}-{\frac {FL^{2}}{16EI}},&{\mbox{for }}x=L/2{\mbox{ to }}x=L\end{cases}}$	$Q_{A}=Q_{C}={\frac {FL^{2}}{16EI}}$
Deflections	$u(x)={\begin{cases}{\frac {Fx^{3}}{12EI}}-{\frac {FxL^{2}}{16EI}},&{\mbox{for }}x=0{\mbox{ to }}x=L/2\\{\frac {F(x-L)^{3}}{12EI}}-{\frac {F(x-L)L^{2}}{16EI}},&{\mbox{for }}x=L/2{\mbox{ to }}x=L\end{cases}}$	$u_{B}={\frac {FL^{3}}{48EI}}$

Simply supported beam with uniformly distributed load
Shear	$R(x)=-wx+{\frac {wL}{2}}$	$R_{A}=R_{C}={\frac {wL}{2}}$
Moments	$M(x)={\frac {-wx^{2}}{2}}+{\frac {wxL}{2}}$	$M_{B}={\frac {wL^{2}}{8}}$
Slope	$Q(x)={\frac {-wx^{3}}{6EI}}+{\frac {wx^{2}L}{4EI}}-{\frac {wL^{3}}{24EI}}$	$Q_{A}=Q_{C}={\frac {wL^{3}}{24EI}}$
Deflections	$u(x)={\frac {-wx^{4}}{24EI}}+{\frac {wx^{3}L}{12EI}}-{\frac {wxL^{3}}{24EI}}$	$u_{B}={\frac {5wL^{4}}{384EI}}$

Fixed ended beam with uniformly distributed load
Shear	$R(x)={-wx}+{\frac {wL}{2}}$	$R_{A}=R_{C}={\frac {wL}{2}}$
Moments	$M(x)={\frac {-wx^{2}}{2}}+{\frac {wxL}{2}}-{\frac {wL^{2}}{12}}$	$M_{A}=M_{C}={\frac {wL^{2}}{12}}$ $M_{B}={\frac {wL^{2}}{24}}$
Slope	$Q(x)={\frac {-wx^{3}}{6EI}}+{\frac {wx^{2}L}{4EI}}-{\frac {wxL^{2}}{12EI}}$	$Q_{A}=Q_{B}=Q_{C}=0$
Deflections	$u(x)={\frac {-wx^{4}}{24EI}}+{\frac {wx^{3}L}{12EI}}-{\frac {wx^{2}L^{2}}{24EI}}$	$u_{B}={\frac {wL^{4}}{384EI}}$

Cantilever with end point load
Shear	$R(x)={-F}$	$R_{B}=-F$
Moments	$M(x)=-Fx$	$M_{B}=-FL$
Slope	$Q(x)={\frac {-Fx^{2}}{2EI}}+{\frac {FL^{2}}{2EI}}$	$Q_{A}={\frac {-FL^{2}}{2EI}}$
Deflections	$u(x)={\frac {-Fx^{3}}{6EI}}+{\frac {FxL^{2}}{2EI}}-{\frac {FL^{3}}{3EI}}$	$u_{A}={\frac {-FL^{3}}{3EI}}$

Cantilever with universally distributed load
Shear	$R(x)={-wx}$	$R_{B}=-wL$
Moments	$M(x)={\frac {-wx^{2}}{2}}$	$M_{B}={\frac {-wL^{2}}{2}}$
Slope	$Q(x)={\frac {-wx^{3}}{6EI}}+{\frac {wL^{3}}{6EI}}$	$Q_{A}={\frac {-wL^{3}}{6EI}}$ $Q_{B}=0$
Deflections	$u(x)={\frac {-wx^{4}}{24EI}}+{\frac {wxL^{3}}{6EI}}-{\frac {wL^{4}}{12EI}}$	$u_{B}={\frac {wL^{4}}{12EI}}$

Propped cantilever with universally distributed load
Shear	$R(x)=wx-{\frac {3wL}{8}}$	$R_{A}={\frac {-3wL}{8}}$ $R_{C}={\frac {5wL}{8}}$
Moments	$M(x)={\frac {wx^{2}}{2}}-{\frac {3wxL}{8}}$	$M_{B}={\frac {9wL^{2}}{128}}{\mbox{ at }}x={\frac {3L}{8}}$ $M_{C}={\frac {-wL^{2}}{8}}$
Slope	$Q(x)={\frac {-wx^{3}}{6EI}}-{\frac {3wx^{2}L}{16EI}}+{\frac {wL^{3}}{48EI}}$	$Q_{A}={\frac {-wL^{3}}{48EI}}$
Deflections	$u(x)={\frac {wx^{4}}{24EI}}-{\frac {3wx^{3}L}{48EI}}+{\frac {wxL^{3}}{48EI}}$	$u_{max}={\frac {wL^{4}}{185EI}}{\mbox{ at }}x=0.4215L$

Above is the copied section "Beam examples" from the article. Please add new discussions below here. -- Crowsnest (talk) 08:42, 1 July 2008 (UTC)

The beam equation: assumptions

I removed this from the section, "The beam equation", after Michael Belisle found errors in the first "colloqial" statement, and from the "rigorous" stated assumptions it is obvious they also contain omissions: e.g. it is not stated that linear elasticity is assumed.

The Euler-Bernoulli Beam Equation is based on 5 assumptions about a bending beam. Colloquially stated, they are that:

1. calculus is valid and is applicable to bending beams
2. the stresses in the beam are distributed in a particular, mathematically simple way
3. the force that resists the bending depends on the amount of bending in a particular, mathematically simple way
4. the material behaves the same way in every direction; i.e. material is isotropic.
5. the forces on the beam only cause the beam to bend, but not twist or stretch; i.e. the case is uncoupled.

More rigorously stated, these assumptions are:

1. continuum mechanics is valid for a bending beam
2. the stress at a cross section varies linearly in the direction of bending, and is zero at the centroid of every cross section
3. the bending moment at a particular cross section varies linearly with the second derivative of the deflected shape at that location
4. the beam is composed of an isotropic material
5. the applied load is orthogonal to the beam's neutral axis and acts in a unique plane.

With these assumptions, we can derive the following equation governing the relationship between the beam's deflection and the applied load.

Feel free to re-instate correct assumptions, including reliable references. -- Crowsnest (talk) 20:59, 8 August 2008 (UTC)

I am not in complete agreement with the statement, "the bending moment at a particular cross section varies linearly with the second derivative of the deflected shape at that location". At any cross section, we have the total bending moment exerted cn the cross section which is directly dependent on the second derivative of beam deflection with respect to x at that cross section. Within the cross section, there is a differential bending moment which is linearly dependent on the distance from the neutral axis or centroid.

Orthotropic elasticity

Why should this theory be valid for isotropic elasticity only? It should also be valid for orthotropic elasticity. —Preceding unsigned comment added by DonQ1906 (talk • contribs) 03:09, 10 October 2008 (UTC)

Other sources, beam theory, translate from de.wikipedia.org instead of writing articles from 0.

de.wikipedia.org is full of quality information about 2. Order, 3. Order, Hulls, Plates, non linearities, FEM, structural dynamics...

Beam theory redirects to this page. There is more than one beam theory, this one is considered the 1º. Order Theory. There is no information about other theories (2. Order, 3. Order) at the en.wikipedia. Maybe the focus should be centered into translating articles and not to write them again from the ground up.

Also there is no info in this page about what assumptions are being made by the euler-bernoulli beam theory (and there are a lot). For example equilibrium is calculated at the non deformed system, but in real life, forces are applied into a deformed system (2. Order, 3. Order). —Preceding unsigned comment added by 84.63.17.180 (talk • contribs) 22:38, 19 February 2009

Article name change

The name of this article should be changed to Euler–Bernoulli beam theory as it is more than only the Euler–Bernoulli beam equation. This would make it consistent with the naming in the first paragraph as well. 145.53.180.22 (talk) 16:08, 22 January 2012 (UTC)

Indeed. -- Mecanismo | Talk 21:06, 22 January 2012 (UTC)

• I've took the liberty to rename the article, in order to correct its name. -- Mecanismo | Talk 20:53, 27 January 2012 (UTC)

I disagree with the title chosen. The first words in the title should be the primary subject. The primary subject is beam theory. The primary subject is not Euler-Bernoulli. Thus the title should be "Beam Theory, Euler-Bernoulli". RHB100 (talk) 17:41, 19 June 2013 (UTC)

Sign convention corrections

I have made changes to correct the sign convention inconsistency pointed out above. Two inconsistent figures were removed.

In the History section and in the Static beam equation section a sign convention was adopted in which the positive z direction is up and the positive x direction is to the right. Since this sign conventions is the first used in the article, I followed it and added the direction of y as being into the figure as required by the right hand rule. A problem was created when someone added figures with y positive up and z positive out of the paper. Removing these figures restores consistency of the sign convention in "Static beam equation" and "Derivation of bending moment equation" as I see it. I don't know why someone added these figure with an inconsistent sign convention. I hope he or she doesn't cause trouble by putting up a fight to have their figures retained. RHB100 (talk) 20:21, 8 February 2017 (UTC)

The sign convention now seems to be consistent with the figures in the subsections of the "Stress" section and in fact throughout the article. RHB100 (talk) 21:44, 8 February 2017 (UTC)

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Derivation of final moderate rotation (small strain) beam equation

Can someone walk through the derivation of the final boxed equation for the moderate rotation (small strain) beam equation, assuming uniform cross section and no axial load? I cannot reach the same conclusion. Here are my steps: Assumptions in article state "no axial load". Axial load is previously defined as $f(x)$. Thus, this means that the internal axial force $N(x)$ is constant (but possibly non-zero!). In addition, the extensional-bending coupling term $B_{xx}$ is zero if the coordinate system is a centroidal coordinate system. The governing ODE then becomes

$EI{\frac {d^{4}w}{dx^{4}}}-EA{\frac {d^{2}w}{dx^{2}}}\left({\frac {du}{dx}}-{\frac {1}{2}}\left({\frac {dw}{dx}}\right)^{2}\right)=q(x)$

I don't see how the ${\frac {3}{2}}$ term or any other simplification in the final boxed equation was derived. Can anyone comment on the derivation?

Rchensix (talk) 16:11, 19 April 2019 (UTC)

Inadequate definition, introduction, or citing of basic concepts used to derive the equation

I have a fairly technical background, but not in engineering or mechanics. The section on the derivation of this equation begins by discussing something called a "fiber" without defining it explicitly, providing a mathematical definition, or providing common-sense intuition for the concept. There are no links to further resources that could provide the necessary background to continue. This should be remedied. i.e.: What is a fiber? Likewise, terms/concepts such as "dθ" are introduced without adequate context or definition. As-is, this section is unreadable as an introductory/didactic text for the layperson. 2A00:23C7:85A7:7801:7568:32E:9ACA:7732 (talk) 12:03, 8 October 2022 (UTC)

1. The first paragraph says: "It covers the case corresponding to small deflections of a beam that is subjected to lateral loads only." This is misleading and ignores the fact that beams are often, by convention, discussed & described in terms of vertical loads, in which case horizontal loads from the side seem to be what is meant by 'lateral loads'. Suggest 'orthogonal loads' or 'loads acting at right-angles to the beam axis'. 2. The "neutral axis" is shown as a line instead of a plane in the second diagram. This is misleading. This plane is usually at right-angles to the direction of the load. The plane is referred to as an axis only because the theory of how the beam resists applied force is based on the turning moments of infinitesimal laminae about that axis, which is a line. Projected along the beam, this line becomes a plane. It is never a single longitudinal line running along the beam. ClarkoEye (talk) 03:44, 18 February 2023 (UTC)