Talk:Euler–Bernoulli beam theory: Difference between revisions

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

Suggestions for Further Work

History

I'm curious as to when the theory was proven by laboratory experiment.
The nature of Bernoulli's and Euler's collaboration could be expanded.
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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

Good point. Fixed.

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

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)[reply]

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

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)[reply]

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)[reply]

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)[reply]

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
File:Simply Supported Beam with Central Point Load.jpg
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
File:SImply supported beam with UDL.jpg
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
File:Fixed ended beam with UDL.jpg
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
File:Cantilever with Point Load.jpg
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
File:Propped Cantilever with UDL.JPG
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)[reply]

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.^{[citation needed]} Colloquially stated, they are that:^{[citation needed]}

calculus is valid and is applicable to bending beams

the stresses in the beam are distributed in a particular, mathematically simple way

the force that resists the bending depends on the amount of bending in a particular, mathematically simple way

the material behaves the same way in every direction; i.e. material is isotropic.

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:^{[citation needed]}

continuum mechanics is valid for a bending beam

the stress at a cross section varies linearly in the direction of bending, and is zero at the centroid of every cross section

the bending moment at a particular cross section varies linearly with the second derivative of the deflected shape at that location

the beam is composed of an isotropic material

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)[reply]

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)[reply]

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