Technical
24 NSC
March 19
The Eurocode introduces an initial plateau (limited by λ0 in
Figure 5) for the design of imperfect struts. According to clause
6.3.1.3 of EN 1993-1-1, the plateau is determined by λ = 0.20,
where
= Ay /Pcr (the Eurocode terms are
= Afy /Ncr
). This
plateau makes an allowance for strain hardening in short
columns6. For values above the specified slenderness for the
plateau, second-order P-δ effects are always relevant for members.
The differential equation for the “perfect” struts in Figure 2
can be adapted to consider an initial bow imperfection. If the
formulation for a “perfect” problem is rather complex, including
an initial imperfection would certainly be more so. However,
to demonstrate the concept of the effects of an initial bow
imperfection, a simplified model can be adopted, where the
system from Figure 2 is replaced by an idealized problem having a
joint with a spring stiffness as shown in Figure 6 2,6.
Assuming that the upper and lower bars have an initial rotation
“θ0”, with zero rotation of the spring, and an axial load is applied,
the rotation increases to θ, and the moment on the spring
becomes Mspring = k·2(θ - θ0), where k is the (elastic) spring stiffness.
The equilibrium in the deformed shape leads to the following
expression: Pθ l ⁄ 2 = Mspring. From the two previous expressions, it
can be shown that P = 4k
l
- 0
( ). The critical buckling load Pcr is
for a perfectly straight member, i.e. θ0 = 0. In this case, Pcr = 4k ⁄ l.
Therefore, P = Pcr
- 0
( ). If θ0 ≠ 0, θ would need to be infinite for P
to be equal to Pcr . This means that the imperfect column will never
reach the Euler load (this is consistent with the line OCGAB from
Figure 4). The equation can be re-written as
1
1- ( ) , where
= 0
μ = Pcr ⁄ P. This is the so-called amplification factor. This factor
allows the consideration of second order effects by amplifying the
first order effects. EN 1993-1-1 section 5.2.2 introduces this factor
for frame stability in the form of
1
1-1/cr
which leads to αcr = Pcr ⁄ P,
where P is the applied load and Pcr is the elastic critical load (for a
strut, this will be Euler load). From a rigorous calculation, it can be
justified that the simplified formulation provides reasonable
results for P ≤ 0.5Pcr (αcr ≥ 2)7. EN 1993-1-1 clause 5.2.2 limits the
method for frame applications where αcr ≥ 3.
The global P-Δ effects, according to clause 5.2.1 of EN 1993-1-1
need to be considered for the cases where the value of αcr ≤ 10
for an elastic global analysis, and αcr ≤ 15 for a plastic global
analysis. Global imperfections for frames are defined according
to EN 1993-1-1 section 5.3.2. Basically, an initial frame rotation
ϕ = h/200 (where h is the height of the frame/structure) is
recommended (Figure 1), although the value can be reduced
based on the number of columns and height of the frame. If
the applied horizontal loads in the frame are more than 15% of
the vertical loads, clause 5.3.2 of EN 1993-1-1 allows the global
imperfections to be neglected. In this circumstance, the effects
of global imperfections are small compared to that of the applied
horizontal loads.
To assess global instability in a structure, the problem is
often addressed using the Finite Element Method. In simple
terms, the stiffness of a beam element is reduced based on the
level of axial force. The method leads to a stiffness matrix Kt
for the total structure, where the critical factor αcr is obtained
by solving the determinant |Kt| = 0. Different buckling modes
can be found (eigenvalues). For global stability, local modes
(related to individual members) are ignored. The exact answer
for the problem is complex, leading to the implementation of
simplified approaches. The exact answer for a simple beam with
no axial or shear deformation is presented in Figure 7. The terms
in the matrix depend on the stability functions ϕi. By necessity,
simplification generally involves making approximation to the
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Figure 6 – Idealized system with a joint with a spring stiffness 2.
/Allowing_for_the_effects_of_deformed_frame_geometry#Member_bow_imperfections
/Allowing_for_the_effects_of_deformed_frame_geometry#Application_of_amplifier
/Allowing_for_the_effects_of_deformed_frame_geometry#Second_order_effects