Technical
NSC 25
March 19
highly non-linear ϕi functions (see Figure 8), which in turn leads
to recommendations regarding modelling.
At large values of N/Pcr , the difference between precise
and approximate values for ϕi is significant. It is therefore
recommended that individual members are modelled by at least
3 finite elements, which reduces the N/Pcr ratio by a factor of
9, and consequently reduces the error in taking approximated
values for ϕi . The maximum value of N/Pcr is 4 (when leff = 0.5l),
so modelling the member with 3 finite elements reduces the
ratio to 0.44. As can be seen from Figure 8, the error between the
approximate and precise values of ϕi functions for N/Pcr = 0.44 is
insignificant.
Conclusions
1 Buckling problems demand the consideration of the deformed
shape of the system;
2 The concept of an effective length is used to adapt the Euler
buckling load to different boundary conditions;
3 An imperfect strut buckles before the plastic section capacity is
reached;
4 Elastic section modulus must be used to back-calculate the
initial imperfection;
5 Second order effects can be allowed for by using an
amplification factor;
6 Approximate methods for stability functions ϕi are generally
used in assessing frame stability;
7 Modelling with at least three finite elements per member
reduces the error in using approximate stability functions.
References
1 Theory of Elastic Stability
S. P. Timoshenko, J. M. Gere; McGraw-Hill, 1961;
2 Mechanics and Strength of Materials
V. D. Silva; Springer-Verlag Berlin, 2006;
3 NA BS EN 1993-1-1+A1
UK National Annex to Eurocode 3 - Eurocode 3 - Design of steel
structures - Part 1-1: General rules and rules for buildings; BSI,
2014;
4 BS EN 1993-1-1+A1
Eurocode 3 - Design of steel structures - Part 1-1: General rules
and rules for buildings; BSI, 2014;
Kt
ij = EI
l
43 62 24
-62
l l
62 62 121 -121
l l2 l l2
43 62 24 -62
l l
-62 -121 -62 121
l l2 l l2
1 = 2cotg()
2 = 2
31 - cotg()
3 = 34
2 + 14
cotg()
4 = 32
2 + 12
cotg()
= 2
N
Pcr
Figure 7 – Formulation for the exact stiffness matrix 8,9.
5 The Stability of Frames
M. R. Horne, W. Merchant; Pergamon Press, 1965;
6 Manual on Stability of Steel Structures
ECCS – European Conventional for Constructional Steelwork,
1976;
7 Design for Structural Stability
P. A. Kirby, D. A. Nethercot; Constrado Monographs; Granada
Publishing Limited, 1979;
8 Stability functions for structural frameworks
R. K. Livesley, D. B. Chandler, Manchester University Press, 1956;
9 Stability and Design of Structures (in Portuguese)
A. Reis, D. Camotim; Orion editions, 2012;
Figure 8 – Stability
functions for exact
(ϕi solid lines) and
for approximate
(ϕ'i – dashed lines)
stiffness matrix
stiffness (ϕ'i) 8,9.
/Allowing_for_the_effects_of_deformed_frame_geometry#Increased_buckling_lengths
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/Concept_design#Structural_options_for_stability
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