Progressive Collapse of Structures: A Robustness Interpretation of Intact Vs. Damaged Frames

Progressive Collapse of Structures: How to Reduce the Risk of Disproportionate Damage Through Proper Design and Use of Mechanical Properties of Materials

ALESSANDRO CALVI¹

alessandro.calvi84@gmail.com¹

ABSTRACT

This paper deals with the critical issue of structural collapses and the importance of the correct quantification of the actions in the design phase and the choice of materials with correct characteristics and properties.

The document analyzes the main strategies to be taken into account in the design phase which contemplate the quantification of external actions and the exploitation of the mechanical properties of materials in favor of safety in order to avoid disproportionate damage to the source.

PROGRESSIVE COLLAPSES: LITERATURE REVIEW

The most historically famous case of structural collapse following exceptional action is represented by the event that occurred on 11 September 2001 at the World Trade Center in New York City, when some terrorist attacks first caused the crash of planes against the Twin Towers and subsequently their collapse.

In this case the damage spread from top to bottom causing with a domino effect the loss of bearing capacity of all the floors of the buildings up to the complete vertical collapse.

The main theory with a posteriori analysis of the damage event is probably the one represented by Prof. Z.P. Bazant [1]: according to his interpretation, the initiation and propagation of the collapse occurred due to the heat given off by the fire caused by the aircrafts' fuel, which irremediably reduced the bearing capacity of the steel columns, where the potential energy, directed downwards from the upper floors, could not be absorbed by the plastic moment of the pillars, thus transforming itself into kinetic energy.

Other examples of unfortunately famous episodes of structural collapse involved buildings such as the Ronan Point Tower (1968) and the Murrah Federal Building (1994) where, respectively due to a gas leak and a terrorist attack event, there was the collapse of important portions of the buildings causing loss of life.

The events discussed above lead to the necessary introduction of a characteristic that all buildings should possess in order to guarantee resistance to exceptional events such as terrorist attacks, explosions, impacts and collisions, fires: the structural robustness [2].

The concept of structural robustness, already introduced in various calculation codes, is a fundamental requirement of structures for their ultimate resistance in the event of damage, even minimal, without manifesting consequences or collapses disproportionate to the cause/action.

In other words, it is desirable to ensure, starting from the design stages, that the structure is able to absorb a certain amount of extra load redistributing it or dissipating it in such a way as to exploit the ductility characteristics of the components and materials as much as possible, without inducing trends with fragile behavior.

COMPRESSED AND TENSE STRUCTURES

This section analyzes the methods of calculating the limit load or the multiplier of the collapsing loads of two main types of structures with reference to the type of stress to which they are subjected: compression and traction.

In both cases, the different responses are analyzed with an approach aimed at defining the concept of structural strength, which implies the search for the variation or reduction of the ultimate load multiplier starting from the intact structure and then moving on to the structure in case of damage.

INTACT STRUCTURES

To analyze the case of structures subject to compression and evaluate their critical multiplier of axial loads, the use of the P-Δ method applied to frames with movable nodes is used in structural engineering, which constitutes a simplified method but which takes into account all the unstable effects and allows to evaluate the overall response of the structure [3].

Therefore, considering a frame made up of horizontal beams supported by vertical pillars, it is possible to apply to this structural system a set of vertical loads V_i and horizontal Q_i acting on the various floors; in this case the overall equilibrium equations can be written in matrix form, where the matrix of horizontal actions Q_i acting on the various floors must correspond to an equivalent load Q_eq which considers the action expressed by the equilibrium matrix E of the unstable effects P-Δ due to the vertical loads, cosidering the multiplier of the vertical loads λ and the contribution of the vector Δ of the displacements of the structure to the various floors.

The above can therefore be expressed as follows:

(1) $Q{\scriptstyle {\text{eq}}}=\lambda ED$

(2) $Q={\bigl (}\kappa {\scriptstyle {\text{t}}}+\lambda E{\bigr )}D$

where κ_t is the translational stiffness matrix of the considered frame. Proceeding with the search for the critical collapse multiplier, Lambda is obtained by setting:

(3)

$det{\bigl (}\kappa {\scriptstyle {\text{t}}}+\lambda E{\bigr )}=0$

REFERENCES

[1]. Z.P. Bazant and Yong Zhou, "Why Did World Trade Center Collapse? - Simple Analysis, Journal of Engineering Mechanics, 2002, Vol. 128, 1.

[2]. FEA Ltd, "Development of the concept of structural toughness", Health and Safety Executive, 2001.