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Stability analysis of a large span timber dome

Herrström, Markus LU and Fredriksson, Gustav (2017) VBKM01 20171
Division of Structural Engingeering
Abstract
The aim of this thesis is to study the feasibility of building a timber dome with a span of
300 metres, concerning elastic stability. The load-bearing members were modelled with the
properties of glued laminated timber GL30c with the dimensions 0.8×1.6 m2. The design
loads were 2 kN/m2 and 4 kN/m2 in symmetrical and asymmetrical load cases, respectively.
The numerical calculations were performed using the software Abaqus FEA, and compared
with analytical equations, modified using empirical data.

There are many ways to arrange the surface members in a braced dome. Common arrangements
include Ribbed, Schwedler, Lattice, Kiewitt, Geodesic and Three-way grid. The three
latter arrangements were compared in terms of global linear... (More)
The aim of this thesis is to study the feasibility of building a timber dome with a span of
300 metres, concerning elastic stability. The load-bearing members were modelled with the
properties of glued laminated timber GL30c with the dimensions 0.8×1.6 m2. The design
loads were 2 kN/m2 and 4 kN/m2 in symmetrical and asymmetrical load cases, respectively.
The numerical calculations were performed using the software Abaqus FEA, and compared
with analytical equations, modified using empirical data.

There are many ways to arrange the surface members in a braced dome. Common arrangements
include Ribbed, Schwedler, Lattice, Kiewitt, Geodesic and Three-way grid. The three
latter arrangements were compared in terms of global linear elastic stability, constructability and stiffness, in order to find the pattern most suitable to span 300 metres. It was concluded that the Geodesic geometry had the most suitable arrangement, primarily due to the slightly higher critical load in symmetrical and asymmetrical load scenarios, fewer number of unique elements lengths, and smaller deviation in the member length distribution.

The non-linear global elastic stability was studied concerning initial geometrical imperfections using linear buckling mode shapes and creep was studied by reducing the Young’s modulus for permanent loads. These two phenomena were also looked studied in combination. The effect of radial and differential settlements on elastic stability, both in combination with initial geometrical imperfections, was also studied.
It was found that the structure was highly sensitive to initial imperfections with a lower bound critical value of only 0.135 q/qcr, corresponding to a uniformly distributed load equal to 8.9 kN/m2, when the structure was loaded symmetrically. This critical load value represents an outlier, over 95 % of the critical loads were above 15 kN/m2. This was compared to the empirical formula, only applicable in the symmetrical load case, which estimated the global failure load to 19.6 kN/m2.

Creep reduced the capacity down to 0.394 q/qcr of the linear buckling load in an asymmetrical load case covering 20 % of the dome area in the xy-plane. This corresponded to a uniformly distributed load of 29.9 kN/m2. Neither radial nor differential settlements caused any decrease of the critical load. Combining creep and initial imperfections reduced the capacity further, from a lower bound value of 0.135 q/qcr to 0.081 q/qcr, the latter corresponding to 5.4 kN/m2. No synergistic effect was found. It was therefore concluded that global stability likely will not cause the dome to collapse, given that the design loads were significantly lower that the stability critical failure loads.

Material failure was also investigated in relation to initial geometrical imperfections as well as the combined effect of creep and imperfections. It was found that the stress level in the most critical beam would cause material failure if the maximum imperfection was larger than 0.8 metres, or D/375, leading to the conclusion that, perhaps, the primary cause for concern would be imperfection induced stresses, not imperfection induced global stability failure. (Less)
Abstract (Swedish)
Denna rapport avser en stabilitetsanalys för att undersöka möjligheten att bygga en kupol i trä med en spännvidd på 300 meter. Alla lastbärande delar har modellerats med limträ av klassen GL30c och med dimensionerna 0.8×1.6 m2. Lastfallen valdes till 2 kN/m2 respektive 4 kN/m2 för symmetriska och asymmetriska snölaster. De numeriska beräkningarna utfördes i finita element-mjukvaran Abaqus FEA och har jämförts med analytiska ekvationer, som modifierats med experimentell data.

Det diskreta skalet av kupolen kan bestå av en mängd olika mönster. Namnen på några
vanliga mönster är Ribbed, Schwedler, Lattice, Kiewitt, Geodesic och Three-way grid, varav de tre sistnämnda jämfördes inom områdena elastisk stabilitet, styvhet och fördelar kring... (More)
Denna rapport avser en stabilitetsanalys för att undersöka möjligheten att bygga en kupol i trä med en spännvidd på 300 meter. Alla lastbärande delar har modellerats med limträ av klassen GL30c och med dimensionerna 0.8×1.6 m2. Lastfallen valdes till 2 kN/m2 respektive 4 kN/m2 för symmetriska och asymmetriska snölaster. De numeriska beräkningarna utfördes i finita element-mjukvaran Abaqus FEA och har jämförts med analytiska ekvationer, som modifierats med experimentell data.

Det diskreta skalet av kupolen kan bestå av en mängd olika mönster. Namnen på några
vanliga mönster är Ribbed, Schwedler, Lattice, Kiewitt, Geodesic och Three-way grid, varav de tre sistnämnda jämfördes inom områdena elastisk stabilitet, styvhet och fördelar kring tillverkning och byggnation. Det fastställdes att kupolmönstret av typ Geodesic var mest fördelaktig och valdes därför för en vidare analys. Anledningarna till valet var den högre instabilitetslasten detta mönster hade vid både symmetriskt och asymmetriskt lastfall, färre antal unika balklängder och en mindre spridning på balkarnas längder.

Den olinjära globala elastiska stabiliteten studerades genom att ansätta modformer, med olika skalfaktorer, för att modellera imperfektioner. Krypning modellerades genom att reducera elasticitetsmodulen för permanenta laster. Sättningar undersöktes, både som en vertikal differenssättning och som en radiell sättning av en eftergivlig dragring. Den elastiska stabiliteten vid krypning och sättningar undersöktes också i kombination med imperfektioner.

Analysen visade att kupolen var väldigt känslig för imperfektioner, varav den olinjära knäckningslasten visade en undre gräns på 0.135 q/qcr, vilket motsvarar en jämt utbredd symmetrisk snölast på 8.9 kN/m2. Detta antogs vara ett avvikande värde då 95 % av imperfektionerna som undersöktes hade knäckningslaster på över 15 kN/m2. Detta jämfördes med knäcklasten som beräknades analytiskt, vilket uppskattade denna till 19.6 kN/m2, för ett symmetriskt lastfall.

Krypningen i trämaterialet reducerade knäcklasten som mest till 0.394 q/qcr
i ett asymmetriskt lastfall där endast 1/5 av kupolens area var täckt. Detta motsvarade jämt utbredd snölast på 29.9 kN/m2. Varken radiell sättning eller vertikal differenssättning påverkade den elastiska stabiliteten avsevärt. Kombinationen av krypning och imperfektioner reducerade knäckningslasten ytterligare, från 0.135 q/qcr
till 0.081 q/qcr vid den undre gränsen, där den senare motsvarar en last på 5.4 kN/m2. Någon synergieffekt mellan imperfektioner och krypning kunde dock inte urskiljas. Analysen av den globala elastiska stabiliteten sammanfattades
därmed att kupolen inte utgjorde någon risk för instabilitet, då den dimensionerande brottlasten var avsevärt lägre än knäckningslasten.

Materialbrottet undersöktes i samband med initiala imperfektioner samt kombination av imperfektioner och krypning. Det observerades att spänningsnivåerna i den mest utsatta balken skulle genomgå materialbrott om imperfektionerna skulle utgöras av en maximal förskjutning på 0.8 meter, motsvarande D/375. Slutsatsen drogs att en större vikt av analysen på kupolen bör möjligen läggas på spänningar som uppstår av imperfektioner, framför stabilitetsbrott orsakad av imperfektioner. (Less)
Please use this url to cite or link to this publication:
author
Herrström, Markus LU and Fredriksson, Gustav
supervisor
organization
alternative title
Stabilitetsanalys av en träkupol
course
VBKM01 20171
year
type
H3 - Professional qualifications (4 Years - )
subject
keywords
Timber, Glulam, Dome, Instability, Buckling, FEM, Non-Linear, Imperfection, Creep, Settlements, Geodesic, Kiewitt, Abaqus, Stability
report number
TVBK-5258
ISSN
0349-4969
language
English
additional info
Examiner: Eva Frühwald Hansson

Secondary supervisor: Silas Nørager - Senior Civil Engineer - Ove Arup & Partners Danmark A/S
id
8927749
date added to LUP
2017-10-25 09:04:58
date last changed
2017-10-25 09:04:58
@misc{8927749,
  abstract     = {The aim of this thesis is to study the feasibility of building a timber dome with a span of
300 metres, concerning elastic stability. The load-bearing members were modelled with the
properties of glued laminated timber GL30c with the dimensions 0.8×1.6 m2. The design
loads were 2 kN/m2 and 4 kN/m2 in symmetrical and asymmetrical load cases, respectively.
The numerical calculations were performed using the software Abaqus FEA, and compared
with analytical equations, modified using empirical data.

There are many ways to arrange the surface members in a braced dome. Common arrangements
include Ribbed, Schwedler, Lattice, Kiewitt, Geodesic and Three-way grid. The three
latter arrangements were compared in terms of global linear elastic stability, constructability and stiffness, in order to find the pattern most suitable to span 300 metres. It was concluded that the Geodesic geometry had the most suitable arrangement, primarily due to the slightly higher critical load in symmetrical and asymmetrical load scenarios, fewer number of unique elements lengths, and smaller deviation in the member length distribution.

The non-linear global elastic stability was studied concerning initial geometrical imperfections using linear buckling mode shapes and creep was studied by reducing the Young’s modulus for permanent loads. These two phenomena were also looked studied in combination. The effect of radial and differential settlements on elastic stability, both in combination with initial geometrical imperfections, was also studied.
It was found that the structure was highly sensitive to initial imperfections with a lower bound critical value of only 0.135 q/qcr, corresponding to a uniformly distributed load equal to 8.9 kN/m2, when the structure was loaded symmetrically. This critical load value represents an outlier, over 95 % of the critical loads were above 15 kN/m2. This was compared to the empirical formula, only applicable in the symmetrical load case, which estimated the global failure load to 19.6 kN/m2.

Creep reduced the capacity down to 0.394 q/qcr of the linear buckling load in an asymmetrical load case covering 20 % of the dome area in the xy-plane. This corresponded to a uniformly distributed load of 29.9 kN/m2. Neither radial nor differential settlements caused any decrease of the critical load. Combining creep and initial imperfections reduced the capacity further, from a lower bound value of 0.135 q/qcr to 0.081 q/qcr, the latter corresponding to 5.4 kN/m2. No synergistic effect was found. It was therefore concluded that global stability likely will not cause the dome to collapse, given that the design loads were significantly lower that the stability critical failure loads.

Material failure was also investigated in relation to initial geometrical imperfections as well as the combined effect of creep and imperfections. It was found that the stress level in the most critical beam would cause material failure if the maximum imperfection was larger than 0.8 metres, or D/375, leading to the conclusion that, perhaps, the primary cause for concern would be imperfection induced stresses, not imperfection induced global stability failure.},
  author       = {Herrström, Markus and Fredriksson, Gustav},
  issn         = {0349-4969},
  keyword      = {Timber,Glulam,Dome,Instability,Buckling,FEM,Non-Linear,Imperfection,Creep,Settlements,Geodesic,Kiewitt,Abaqus,Stability},
  language     = {eng},
  note         = {Student Paper},
  title        = {Stability analysis of a large span timber dome},
  year         = {2017},
}