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 ).

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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