1 Technical University of Cluj-Napoca, Faculty of Civil Engineering. 15 C Daicoviciu Str., 400020, Cluj-Napoca, Romania Received 25 July 2011; Accepted 1 September 2011 The Generalised Beam Theory (GBT) approach was proved to be a valid substitute of the Eurocode methods used for cold-formed steel member design. The conventional GBT is applied for the linear/nonlinear analysis of prismatic thin-walled members, circular and elliptical cylindrical shells and tubes. The authors new formulations etend the GBT for the special case of tapered members. This paper presents an overview of the GBT formulations recently developed by the author and also the software application GBTPower in which these theoretical developments are implemented. GBTPower performs buckling analyses of prismatic and tapered elastic thin-walled members with various types of cross-sections and boundary conditions. At the end a few numerical results are presented and briefly discussed. Abordările specifice Teoriei Generalizate a Grinzii (TGG) pot înlocui cu succes metodele din Eurocode în privinţa proiectării barelor cu pereţi subţiri formate la rece. TGG este utilizată în mod convenţional la analiza liniară şi neliniară a barelor prismatice cu pereţi subţiri şi a tuburilor cilindrice şi eliptice. Noile formulări ale autorului etind TGG pentru cazul special al barelor cu secţiune variabilă. Acest articol face o prezentare de ansamblu ale ultimelor formulări teoretice recent concepute de autor şi de asemenea a programului de calcul GBTPower în care acestea au fost implementate. Programul efectuează analize de tip flambaj prin bifurcare în domeniul elastic pentru bare cu pereţi subţiri cu diferite secţiuni şi rezemări. În final câteva eemple numerice sunt pe scurt prezentate şi discutate.
L NL M B M L γ s = M L ε s =
= = s ε = u ε = v + w γ = ww & M L M NL M NL s ε = zw ε = zw&& γ = zw& B B B s s B B σ µ ε M M B E B σ = Eε σ s = µ ε s B µ µ B τ s γ s n n IV τ ECikφk GDikφ k + Bikφk λ X jik W j φ k X jki W j φk + X jikw j φ k = τ ( ) ( ) k i = n W j = y z W = N W = M W = M W = B C D B X X τ φ k λ b G s tv ds D
C D B X X τ τ ( ) ( ) ( ) ( ) τ E Cikφ k G Dikφ k + Bikφk λ X jikwj φk X jkiw j φk + X jikw j φ k = L y z α g g g θ z a L α θ π z t + t u v w
M { } { c} M B M { e} = { } + { } = { } + z{ c} M M L M NL w v ε = ε + ε = ( u ) + + δ M M L M NL v& wc us δ vc w& δ v& + wc εθθ = εθθ + εθθ = + + + r r r r + r M M L M NL u& vs w w& vw c v v& γ θ = γ θ + γ θ = + v δ δ r r + + r r r χ = w w&& w s vc & χθθ = + δ r r r w& ws & c v vs χ θ = + + δ r r r r c = ( α ) s = ( α ) = = θ δ δ δ = δ = δ = δ = δ = δ = M L M L γ = ε = k i θ θθ IV σ σ + + ( + ) + ( + ) σθ τ ( D kk Bkk G kk X jkk X jkk ) φk C φ C φ C D D X φ D D X φ kk k kk k kk kk kk jkk k kk kk jkk k + + + = = n C D D G B X X X σ σθ τ j j j j. λ b σ σ τ o o o θθ θ
r < < < < L φ k Matlab R2006b
Mihai Nedelcu / Acta Technica Napocensis: Civil Engineering & Architecture Vol. 54 No.1 (2011) 38-49 Fig. 8. GBTPower input interface Fig.9. GBTPower output interface Each cross-section type and variation law is handled in a different module. GBTPower also accepts different boundary conditions: cantilevers, simple supported, double fied or fied-simplesupported members. The latest developed module deals with the buckling analysis of conical shells. The input interface for opened and constant cross-section is shown in Fig. 8. The output interface displays for 1 st order analysis the displacement and stress field, the shape for each pure deformation 44
L = mm β = L = mm β =
FEA vs. GBT results for fied members β FEA P cr GBT Pcr kn cbm λ λ P E = GPa ν = t = L = mm r = mm r = λbp λc = λb o σ = P π rtc P = π r knmm kn
λc FEM r λ c GBT r
λc FEM EUROCODE 3, EN 1993-1.1 AISI AISI AISI
AS/NZS 4600 CUFSM GBTUL Verallgemeinerte Technische Biegetheorie Journal of Thin-Walled Structures Journal of Thin-Walled Structures Journal of Thin-Walled Structures Journal of Thin-Walled Structures International Journal of Solids and Structures, Journal of Thin-Walled Structures Journal of Thin-Walled Structures International Journal of Structural Stability and Dynamics Numerical Solution of Boundary Value Problems for Ordinary Differential Equation AIAA Journal, Stability of thin-walled tubes under torsion Quarterly Journal of Applied Mathematics, Theory of Plates and Shells MathWorks Journal of Computers and Structures, ABAQUS Standard (Version 6.3) Journal of the Structural Division