Generalised beam theory: Difference between revisions

Content deleted Content added

Inline

Latest revision as of 16:31, 2 December 2023

In structural engineering and mechanical engineering, generalised beam theory (GBT) is a one-dimensional theory used to mathematically model how beams bend and twist under various loads. It is a generalization of classical Euler–Bernoulli beam theory that approximates a beam as an assembly of thin-walled plates that are constrained to deform as a linear combination of specified deformation modes.^[1]

History

Its origin is due to Richard Schardt (1966). Since then many other authors have extended the initial (first-order elastic) GBT formulations developed by Schardt and his co-workers.^[2]^[3] Many extensions and applications of GBT have been developed by Camotim (Instituto Superior Técnico, University of Lisbon, Portugal) and collaborators, since the beginning of the 21st century. ^[4]^[5]^[6]^[7]^[8]^[9]^[10]

Description

The theory can be applied without restrictions to any prismatic thin-walled structural member exhibiting straight or curved axial axis (any loading, any cross-section geometry, any boundary conditions). GBT is in some ways analogous to the finite strip method^[1] and can be a more computationally efficient method than modeling a beam with a full 2D or 3D finite element method to predict the member structural behavior.

GBT has been widely recognized as an efficient approach to analyzing thin-walled members and structural systems. The efficiency arises mostly from its modal nature – the displacement field is expressed as a linear combination of cross-section deformation modes whose amplitudes vary continuously along the member length (x axis) - see Figures 2-3. Due to GBT assumptions inherent to a thin-walled member, only 3 non-null stress components are considered in the formulations (see Fig. 1).

Membrane displacement field (i.e., in the cross-section mid-surface):

{\begin{aligned}u(s,x)&=u_{k}(s)\zeta _{k,x}(x),\\v(s,x)&=v_{k}(s)\zeta _{k}(x),\\w(s,x)&=w_{k}(s)\zeta _{k}(x).\end{aligned}}

The GBT modal nature makes it possible to (i) acquire in-depth knowledge on the mechanics of the thin-walled member behaviour and (ii) judiciously exclude, from subsequent similar GBT analyses, those deformation modes found to play no (or negligible) role in the particular behaviour under scrutiny. Eliminating modes that play no role reduces the number of degrees of freedom involved in a GBT analysis and increases its computational efficiency. GBT has proven useful in the understanding of the structural behaviour under analysis as well as in its computational efficiency.^[1]

References

^ ^a ^b ^c de Miranda, Stefano; Gutiérrez, Alejandro; Miletta, Rosario; Ubertini, Francesco (June 2013). "A generalized beam theory with shear deformation". Thin-Walled Structures. 67: 88–100. doi:10.1016/j.tws.2013.02.012.
^ Schardt, Richard (1994). "Generalised Beam Theory – an adequate method for coupled stability problems". Thin-Walled Structures. 19 (2–4): 161–180. doi:10.1016/0263-8231(94)90027-2.
^ Davies, JM (2000). "Recent research advances in cold-formed steel structures". Journal of Constructional Steel Research. 55 (1–3): 267–288. doi:10.1016/S0143-974X(99)00089-9.
^ Silvestre, N; Camotim, D (2002). "Second-order generalised beam theory for arbitrary orthotropic materials". Thin-Walled Structures. 40 (9): 791–820. doi:10.1016/S0263-8231(02)00026-5.
^ Borges Dinis, P; Camotim, D; Silvestre, N (2006). "GBT formulation to analyse the buckling behaviour of thin-walled members with arbitrarily 'branched'open cross-sections". Thin-Walled Structures. 44 (1): 20–38. doi:10.1016/j.tws.2005.09.005.
^ Camotim, D; Basaglia, C; Silvestre, N (2010). "GBT buckling analysis of thin-walled steel frames: a state-of-the-art report". Thin-Walled Structures. 48 (10): 726–743. doi:10.1016/j.tws.2009.12.003.
^ Abambres, M; Camotim, D; Silvestre, N (2014). "GBT-based elastic–plastic post-buckling analysis of stainless steel thin-walled members" (PDF). Thin-Walled Structures. 83 (October): 85–102. doi:10.1016/j.tws.2014.01.004. S2CID 120053099.
^ Abambres, M; Camotim, D; Silvestre, N; Rasmussen, KJR (2014). "GBT-based structural analysis of elastic–plastic thin-walled members" (PDF). Computers & Structures. 136 (May): 1–23. doi:10.1016/j.compstruc.2014.01.001. S2CID 122047967.
^ Bebiano, R; Gonçalves, R; Camotim, D (2015). "A cross-section analysis procedure to rationalise and automate the performance of GBT-based structural analyses". Thin-Walled Structures. 92 (July): 29–47. doi:10.1016/j.tws.2015.02.017.
^ Gonçalves, R; Camotim, D (2016). "GBT deformation modes for curved thin-walled cross-sections based on a mid-line polygonal approximation". Thin-Walled Structures. 103 (January): 231–243. doi:10.1016/j.tws.2015.12.025.

[deMiranda2013-1] Miranda, Stefano; Gutiérrez, Alejandro; Miletta, Rosario; Ubertini, Francesco (June 2013). "A generalized beam theory with shear deformation". Thin-Walled Structures. 67: 88–100. doi:10.1016/j.tws.2013.02.012.

[2] Schardt, Richard (1994). "Generalised Beam Theory – an adequate method for coupled stability problems". Thin-Walled Structures. 19 (2–4): 161–180. doi:10.1016/0263-8231(94)90027-2.

[3] Davies, JM (2000). "Recent research advances in cold-formed steel structures". Journal of Constructional Steel Research. 55 (1–3): 267–288. doi:10.1016/S0143-974X(99)00089-9.

[4] Silvestre, N; Camotim, D (2002). "Second-order generalised beam theory for arbitrary orthotropic materials". Thin-Walled Structures. 40 (9): 791–820. doi:10.1016/S0263-8231(02)00026-5.

[5] Borges Dinis, P; Camotim, D; Silvestre, N (2006). "GBT formulation to analyse the buckling behaviour of thin-walled members with arbitrarily 'branched'open cross-sections". Thin-Walled Structures. 44 (1): 20–38. doi:10.1016/j.tws.2005.09.005.

[6] Camotim, D; Basaglia, C; Silvestre, N (2010). "GBT buckling analysis of thin-walled steel frames: a state-of-the-art report". Thin-Walled Structures. 48 (10): 726–743. doi:10.1016/j.tws.2009.12.003.

[7] Abambres, M; Camotim, D; Silvestre, N (2014). "GBT-based elastic–plastic post-buckling analysis of stainless steel thin-walled members" (PDF). Thin-Walled Structures. 83 (October): 85–102. doi:10.1016/j.tws.2014.01.004. S2CID 120053099.

[8] Abambres, M; Camotim, D; Silvestre, N; Rasmussen, KJR (2014). "GBT-based structural analysis of elastic–plastic thin-walled members" (PDF). Computers & Structures. 136 (May): 1–23. doi:10.1016/j.compstruc.2014.01.001. S2CID 122047967.

[9] Bebiano, R; Gonçalves, R; Camotim, D (2015). "A cross-section analysis procedure to rationalise and automate the performance of GBT-based structural analyses". Thin-Walled Structures. 92 (July): 29–47. doi:10.1016/j.tws.2015.02.017.

[10] Gonçalves, R; Camotim, D (2016). "GBT deformation modes for curved thin-walled cross-sections based on a mid-line polygonal approximation". Thin-Walled Structures. 103 (January): 231–243. doi:10.1016/j.tws.2015.12.025.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

@@ Line 1: / Line 1: @@
+{{short description|Engineering theory}}
-{{AFC submission|d|reason|I am going to ask for members of several projects, [[Wikipedia:WikiProject Physics/Taskforces/Fluid dynamics]], [[Wikipedia:WikiProject Mathematics]], and [[Wikipedia:WikiProject Technology]] to take a look at the notability. In the meantime, the article needs to be rewritten for formatting and structure. Things like "please follow the website of his research group..." need to go away. If you have a citation, simply use that citation.|u=181.233.197.42|ns=118|decliner=Onel5969|declinets=20160312182920|ts=20160215230723}} <!-- Do not remove this line! -->
+In [[structural engineering]] and [[mechanical engineering]], '''generalised beam theory (GBT)''' is a one-dimensional theory used to mathematically model how beams bend and twist under various loads. It is a generalization of classical [[Euler–Bernoulli beam theory]] that approximates a beam as an assembly of thin-walled plates that are constrained to deform as a [[linear combination]] of specified deformation [[Normal mode|modes]].<ref name="deMiranda2013">{{cite journal|last1=de Miranda|first1=Stefano|last2=Gutiérrez|first2=Alejandro|last3=Miletta|first3=Rosario|last4=Ubertini|first4=Francesco|title=A generalized beam theory with shear deformation|journal=Thin-Walled Structures|date=June 2013|volume=67|pages=88–100|doi=10.1016/j.tws.2013.02.012}}</ref>
-{{AFC submission|d|nn|u=Abambres|ns=118|decliner=SwisterTwister|declinets=20160213075023|small=yes|ts=20160206121840}} <!-- Do not remove this line! -->
+== History==
-{{AFC comment|1=Not entirely certain of its notability but this will also still need any further available in-depth third-party sources overall. [[User:SwisterTwister|<font color="green">SwisterTwister</font>]] [[User talk:SwisterTwister|<font color="green">talk</font>]] 07:50, 13 February 2016 (UTC)}}
-{{AFC comment|1=In addition to notability concerns, the article does not provide enough technical detail for the target audience.  For example, what is ζ?  It's never defined, and I can't figure it out from the diagrams.  The situations to which GBT is supposed to apply are never made explicit beyond "thin-walled".  (Perhaps that's jargon that I don't know, since I don't do fluid mechanics, but even so, it should be made more explicit.)  When the article says that only three non-null stress components are necessary, I consequently have no way of knowing if that's accurate or not.</p><p>My advice to the authors is: Pretend you are writing a survey for undergraduates.  You must be able to state clearly what the problem of interest is, and you should give real-world examples of where this problem arises.  Then you must explain why the techniques you're discussing are relevant.  You should state which situations these techniques apply to and what results they provide.  All of this must be done with enough detail that the reader (even an undergraduate) can get a surface understanding of the techniques involved and is able to determine whether they're relevant to their own work. [[User:Ozob|Ozob]] ([[User talk:Ozob|talk]]) 21:19, 12 March 2016 (UTC)}}
-----
-'''Generalised beam theory (GBT)''' is the most complete and disseminated [[beam theory]] to be formulated and extensively validated. Its origin is due to Richard Schardt (1966), a German scientist from the University of Darmstadt. Since then many other authors have extended the initial (first-order elastic) GBT formulations developed by Schardt and his co-workers.<ref>{{cite journal|last1=Schardt|first1=Richard|title=Generalised Beam Theory – an adequate method for coupled stability problems|journal=Thin-Walled Strcutures|date=1994|volume=19|issue=2-4|pages=161–180|doi=10.1016/0263-8231(94)90027-2}}</ref><ref>{{cite journal|last1=Davies|first1=JM|title=Recent research advances in cold-formed steel structures|journal=Journal of Constructional Steel Research|date=2000|volume=55|issue=1-3|doi=10.1016/S0143-974X(99)00089-9|pages=267-288}}</ref> However, it is fair to say that a vast majority of the most advanced/complete (in order to predict as close as possible the real behaviour (collapse and post-collapse) of a structural member) formulations and applications of GBT have been developed by the research group headed by Dinar Camotim, a full professor from the University of Lisbon, Portugal, since the beginning of the 21st century
+Its origin is due to Richard Schardt (1966). Since then many other authors have extended the initial (first-order elastic) GBT formulations developed by Schardt and his co-workers.<ref>{{cite journal|last1=Schardt|first1=Richard|title=Generalised Beam Theory – an adequate method for coupled stability problems|journal=Thin-Walled Structures|date=1994|volume=19|issue=2–4|pages=161–180|doi=10.1016/0263-8231(94)90027-2}}</ref><ref>{{cite journal|last1=Davies|first1=JM|title=Recent research advances in cold-formed steel structures|journal=Journal of Constructional Steel Research|date=2000|volume=55|issue=1–3|doi=10.1016/S0143-974X(99)00089-9|pages=267–288}}</ref> Many extensions and applications of GBT have been developed by Camotim ([[Instituto Superior Técnico]], University of Lisbon, Portugal) and collaborators, since the beginning of the 21st century.
+<ref>{{cite journal|last1=Silvestre|first1=N|last2=Camotim|first2=D|title=Second-order generalised beam theory for arbitrary orthotropic materials|journal=Thin-Walled Structures|date=2002|volume=40|issue=9|pages=791–820|doi=10.1016/S0263-8231(02)00026-5}}</ref><ref>{{cite journal|last1=Borges Dinis|first1=P|last2=Camotim|first2=D|last3=Silvestre|first3=N|title=GBT formulation to analyse the buckling behaviour of thin-walled members with arbitrarily 'branched'open cross-sections|journal=Thin-Walled Structures|date=2006|volume=44|issue=1|pages=20–38|doi=10.1016/j.tws.2005.09.005}}</ref><ref>{{cite journal|last1=Camotim|first1=D|last2=Basaglia|first2=C|last3=Silvestre|first3=N|title=GBT buckling analysis of thin-walled steel frames: a state-of-the-art report|journal=Thin-Walled Structures|date=2010|volume=48|issue=10|pages=726–743|doi=10.1016/j.tws.2009.12.003}}</ref><ref>{{cite journal|last1=Abambres|first1=M|last2=Camotim|first2=D|last3=Silvestre|first3=N|title=GBT-based elastic–plastic post-buckling analysis of stainless steel thin-walled members|journal=Thin-Walled Structures|date=2014|volume=83|issue=October|pages=85–102|doi=10.1016/j.tws.2014.01.004|s2cid=120053099|url=https://hal.archives-ouvertes.fr/hal-02450449/file/4th%20ISSSES%202012%20Paper_Abambres%2C%20Camotim%2C%20Silvestre.pdf}}</ref><ref>{{cite journal|last1=Abambres|first1=M|last2=Camotim|first2=D|last3=Silvestre|first3=N|last4=Rasmussen|first4=KJR|title=GBT-based structural analysis of elastic–plastic thin-walled members|journal=Computers & Structures|date=2014|volume=136|issue=May|pages=1–23|doi=10.1016/j.compstruc.2014.01.001|s2cid=122047967|url=https://hal.archives-ouvertes.fr/hal-02450457/file/SSRC%202013%20Paper_Abambres%20et%20al.pdf}}</ref><ref>{{cite journal|last1=Bebiano|first1=R|last2=Gonçalves|first2=R|last3=Camotim|first3=D|title=A cross-section analysis procedure to rationalise and automate the performance of GBT-based structural analyses|journal=Thin-Walled Structures|date=2015|volume=92|issue=July|pages=29–47|doi=10.1016/j.tws.2015.02.017}}</ref><ref>{{cite journal|last1=Gonçalves|first1=R|last2=Camotim|first2=D|title=GBT deformation modes for curved thin-walled cross-sections based on a mid-line polygonal approximation|journal=Thin-Walled Structures|volume=103|date=2016|issue=January|pages=231–243|doi=10.1016/j.tws.2015.12.025}}</ref>
-<ref>{{cite journal|last1=Silvestre|first1=N|last2=Camotim|first2=D|title=Second-order generalised beam theory for arbitrary orthotropic materials|journal=Thin-Walled Strcutures|date=2002|volume=40|issue=9|pages=791-820|doi=10.1016/S0263-8231(02)00026-5}}</ref>
-<ref>{{cite journal|last1=Borges Dinis|first1=P|last2=Camotim|first2=D|last3=Silvestre|first3=N|title=GBT formulation to analyse the buckling behaviour of thin-walled members with arbitrarily ‘branched’open cross-sections|journal=Thin-Walled Strcutures|date=2006|volume=44|issue=1|pages=20-38|doi=10.1016/j.tws.2005.09.005}}</ref>
-<ref>{{cite journal|last1=Camotim|first1=D|last2=Basaglia|first2=C|last3=Silvestre|first3=N|title=GBT buckling analysis of thin-walled steel frames: a state-of-the-art report|journal=Thin-Walled Strcutures|date=2010|volume=48|issue=10|pages=726-743|doi=10.1016/j.tws.2009.12.003}}</ref>
-<ref>{{cite journal|last1=Abambres|first1=M|last2=Camotim|first2=D|last3=Silvestre|first3=N|title=GBT-based elastic–plastic post-buckling analysis of stainless steel thin-walled members|journal=Thin-Walled Strcutures|date=2014|volume=83|issue=October|pages=85-102|doi=10.1016/j.tws.2014.01.004}}</ref>
-<ref>{{cite journal|last1=Abambres|first1=M|last2=Camotim|first2=D|last3=Silvestre|first3=N|last4=Rasmussen|first4=KJR|title=GBT-based structural analysis of elastic–plastic thin-walled members|journal=Computers & Structures|date=2014|volume=136|issue=May|pages=1-23|doi=10.1016/j.compstruc.2014.01.001}}</ref>
-<ref>{{cite journal|last1=Bebiano|first1=R|last2=Gonçalves|first2=R|last3=Camotim|first3=D|title=A cross-section analysis procedure to rationalise and automate the performance of GBT-based structural analyses|journal=Thin-Walled Strcutures|date=2015|volume=92|issue=July|pages=29-47|doi=10.1016/j.tws.2015.02.017}}</ref>
-<ref>{{cite journal|last1=Gonçalves|first1=R|last2=Camotim|first2=D|title=GBT deformation modes for curved thin-walled cross-sections based on a mid-line polygonal approximation|journal=Thin-Walled Strcutures|date=2016|issue=January|doi=10.1016/j.tws.2015.12.025}}</ref>   (please follow the [http://www.civil.ist.utl.pt/gbt <u>website of his research group</u>] for information on publications and available software).
+== Description ==
-The theory can be applied without restrictions to any prismatic '''thin-walled''' structural member exhibiting straight or curved axial axis (any loading, any cross-section geometry, any boundary conditions), and it is a very promising alternative to the powerful Finite Element Method employing 2D or 3D FEs to predict the member structural behaviour.
+The theory can be applied without restrictions to any prismatic thin-walled structural member exhibiting straight or curved axial axis (any [[Structural load|loading]], any [[Cross section (geometry)|cross-section geometry]], any boundary conditions).  GBT is in some ways analogous to the [[finite strip method]]<ref name="deMiranda2013" /> and can be a more computationally efficient method than modeling a beam with a full 2D or 3D [[finite element method]] to predict the member structural behavior.
-GBT has already been widely recognised as a powerful, versatile, elegant and efficient approach to analyse thin-walled members and structural systems. The elegance and efficiency arise mostly from its rather unique modal nature – the displacement field is expressed as a linear combination of cross-section deformation modes whose amplitudes vary continuously along the member length (x axis)  - see Figures 2-3. Due to GBT assumptions inherent to a thin-walled member, only 3 non-null stress components are considered in the formulations (see Fig. 1).
+GBT has been widely recognized as an efficient approach to analyzing thin-walled members and structural systems. The efficiency arises mostly from its modal nature – the [[Displacement field (mechanics)|displacement field]] is expressed as a linear combination of cross-section deformation modes whose amplitudes vary continuously along the member length (x axis)  - see Figures 2-3. Due to GBT assumptions inherent to a thin-walled member, only 3 non-null stress components are considered in the formulations (see Fig. 1).
 Membrane displacement field (i.e., in the cross-section mid-surface):
@@ Line 30: / Line 23: @@
 <!-- [[File:GBT membrane displacement field.png|GBT membrane displacement field]] -->
-The GBT modal nature makes it possible to (i) acquire in-depth knowledge on the mechanics of the thin-walled member behaviour and (ii) judiciously exclude, from subsequent similar GBT analyses, those deformation modes found to play no (or negligible) role in the particular behaviour under scrutiny, thus further reducing the number of d.o.f. involved in a GBT analysis (i.e., increasing its computational efficiency). GBT has already extensively prooved its potential regarding (1) deep understanding of the structural behaviour under analysis as well as (2) computational efficiency (i.e., number of degrees of freedom involved in the analysis) - results and a more detailed insight of GBT-based inelastic analyses are available online (check out works from [https://scholar.google.pt/citations?hl=pt-PT&user=PSuaS4QAAAAJ&view_op=list_works&sortby=pubdate <u>Dinar Camotim</u>], [https://scholar.google.com.au/citations?hl=en&user=mgXXcU0AAAAJ&view_op=list_works <u>Nuno Silvestre</u>], [https://www.researchgate.net/profile/Pedro_Dinis3 <u>Pedro B. Dinis</u>], [https://scholar.google.gr/citations?user=RMTIEWcAAAAJ&hl=en <u>Rodrigo Gonçalves</u>], [https://scholar.google.com/citations?user=7qf8LjQAAAAJ&hl=en <u>Cilmar Basaglia</u>], [https://scholar.google.com/citations?hl=en&user=5YjKnVYAAAAJ&view_op=list_works <u>Rui Bebiano</u>], [https://www.researchgate.net/researcher/72215672_NMF_Silva <u>Nuno Silva</u>] and [https://scholar.google.com/citations?user=Il_7GHMAAAAJ&hl=en <u>Miguel Abambres</u>]).
+The GBT modal nature makes it possible to (i) acquire in-depth knowledge on the mechanics of the thin-walled member behaviour and (ii) judiciously exclude, from subsequent similar GBT analyses, those deformation modes found to play no (or negligible) role in the particular behaviour under scrutiny. Eliminating modes that play no role reduces the number of degrees of freedom involved in a GBT analysis and increases its computational efficiency. GBT has proven useful in the understanding of the structural behaviour under analysis as well as in its computational efficiency.<ref name="deMiranda2013" />
-The investigation, application and dissemination of GBT will certainly last during many more years.
 [[File:Gbt3.png|Gbt3]]
@@ Line 40: / Line 31: @@
 ==References==
 {{reflist}}
-<!--- After listing your sources please cite them using inline citations and place them after the information they cite. Please see https://en.wikipedia.org/wiki/Wikipedia:REFB for instructions on how to add citations. --->
+[[Category:Beam theory]]
-{{AfC postpone G13|1}}
+[[Category:Civil engineering]]
+[[Category:Structural engineering]]
+[[Category:Mechanical engineering]]
+[[Category:Aerospace engineering]]