868 Words4 Pages

Most civil structures will be subjected to some form of lateral loading during the span of
their lifetime. Loads produced by earthquakes, wind, or blast explosions, mainly induce lateral
displacement on structures. The effect of gravity loads acting through the structure’s lateral
displacement has been called the P-∆ effect. This effect can initiate a pernicious circle against
structural systems because the influence of gravity loads increase as the lateral displacement
grows while at the same time, the lateral displacement is magnified as a consequence of gravity
loads acting on them. From this short description, the nonlinear character of the problem is
clear. Additionally, the possible incursion of structures into the realm of inelastic deformation
further increases the complexity of this already difficult problem.
1.1. Historical perspective
For many years, the P-∆ effect has been a subject of study and concern to structural engineers.
In 1934 Ruge [1] made what was probably the first examination of the effects of gravity on
the response of simple elastic structures. He was able to estimate the change in period and
deflection of a vertical cantilever beam supporting a weight. In 1968, Jennings and Husid [2],
also working with Single-Degree-Of-Freedom (SDOF) systems, were the first to report on the
effects of gravity on inelastic structural response. They conclude that gravity increases the
amount of plastic drift significantly over that found when gravity is neglected, leading in many
cases to the collapse of the system. They also determined the critical value of the post-yield
slope of the force-deformation backbone curve of the SDOF system above which collapse will
not occur.
Studies on the response ...
... middle of paper ...
... stability coefficient (θ),
does not accurately represent inelastic P-∆ effects [9]. Increasing evidence shows that the
use of elastic stiffness in determining theoretical P-∆ response of highly inelastic systems is
unconservative [9].
Several researchers [6, 10, 11, 12] believe that the stability coefficient calculated based on
the initial elastic stiffness is aimed to check the static stability of elastic or slightly inelastic
structures only. Any procedure based on θ would be unable to capture inelastic response
because the effective post-yield stiffness is underestimated [10]. For the usual design case of
structures expected to respond beyond their elastic limit, Bernal [11] and Adam et al. [10]
have suggested the use of two stability coefficients: one initial or elastic stability coefficient
(θe ) and one inelastic stability coefficient (θi ).

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