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

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

== <br> ALESSANDRO CALVI<sup>1</sup> ==
[[alessandro.calvi84@gmail.com]]<sup>1</sup>

== '''ABSTRACT''' ==
This paper deals with the critical issue of structural collapses and introduce a novel interpretation of the concept of Structural Robustness applied to Intact and Damaged Frames in order to quantify the critical load multiplier in each case, by a simple set of mathematical equations. 

=='''PROGRESSIVE COLLAPSES: LITERATURE REVIEW''' <!-- You can enter and format the text of this document by selecting the ‘Edit’ option in the menu at the top of this frame or next to the title of every section of the document. This will give access to the visual editor. Alternatively, you can edit the source of this document (Wiki markup format) by selecting the ‘Edit source’ option.  Most of the documents in Scipedia are written in English (write your manuscript in American or British English, but not a mixture of these). Anyhow, specific publications in other languages can be published in Scipedia. In any case, the documents published in other languages must have an abstract written in English.   2.1 Subsections  Divide your article into clearly defined and numbered sections. Subsections should be numbered 1.1, 1.2, etc. and then 1.1.1, 1.1.2, ... Use this numbering also for internal cross-referencing: do not just refer to 'the text'. Any subsection may be given a brief heading. Capitalize the first word of the headings.   2.2 General guidelines  Some general guidelines that should be followed in your manuscripts are:  *  Avoid hyphenation at the end of a line.  *  Symbols denoting vectors and matrices should be indicated in bold type. Scalar variable names should normally be expressed using italics.  *  Use decimal points (not commas); use a space for thousands (10 000 and above).  *  Follow internationally accepted rules and conventions. In particular use the international system of units (SI). If other quantities are mentioned, give their equivalent in SI.   2.3 Tables, figures, lists and equations  Please insert tables as editable text and not as images. Tables should be placed next to the relevant text in the article. Number tables consecutively in accordance with their appearance in the text and place any table notes below the table body. Be sparing in the use of tables and ensure that the data presented in them do not duplicate results described elsewhere in the article.  Graphics may be inserted directly in the document and positioned as they should appear in the final manuscript.  Number the figures according to their sequence in the text. Ensure that each illustration has a caption. A caption should comprise a brief title. Keep text in the illustrations themselves to a minimum but explain all symbols and abbreviations used. Try to keep the resolution of the figures to a minimum of 300 dpi. If a finer resolution is required, the figure can be inserted as supplementary material  For tabular summations that do not deserve to be presented as a table, lists are often used. Lists may be either numbered or bulleted. Below you see examples of both.  1. The first entry in this list  2. The second entry  2.1. A subentry  3. The last entry  * A bulleted list item  * Another one  You may choose to number equations for easy referencing. In that case they must be numbered consecutively with Arabic numerals in parentheses on the right hand side of the page. Below is an example of formulae that should be referenced as eq. (1].   2.4 Supplementary material  Supplementary material can be inserted to support and enhance your article. This includes video material, animation sequences, background datasets, computational models, sound clips and more. In order to ensure that your material is directly usable, please provide the files with a preferred maximum size of 50 MB. Please supply a concise and descriptive caption for each file. -->==

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.

== '''FRAMED STRUCTURES''' ==
This section analyzes the methods of calculating the limit load or the multiplier of the collapsing loads defining the concept of structural Robustness, 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<sub>i</sub>'' and horizontal ''Q<sub>i</sub>'' acting at the various floors; in this case the overall equilibrium equations can be written in matrix form, where the matrix of horizontal actions ''Q<sub>i</sub>'' acting on the various floors must correspond to an equivalent load ''Q<sub>eq</sub>'' 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 ''D'' of the displacements of the structure at the various floors.

The above can therefore be expressed as follows:

(1)  <math>Q{\scriptstyle\text{eq}} = \lambda ED</math>

(2)  <math>Q=\bigl(\kappa{\scriptstyle\text{t}}+\lambda E\bigr) D</math>

where ''κ<sub>t</sub>'' is the translational stiffness matrix of the considered frame. Proceeding with the search for the critical collapse multiplier ''λ<sub>i</sub>'' is obtained by setting:

(3)  <math>det\bigl(\kappa{\scriptstyle\text{t}}+\lambda E\bigr)=0</math> 

=== DAMAGED STRUCTURES ===
Now considering the case of the same frame structure seen in the previous chapter in which damage is introduced due to an unforeseen event to one of the floors (assuming for example to remove a certain number of beams and pillars).

In this way equation (3) can be transformed, including two new contributions that take into account the balancing effects ''E<sub>d</sub>'' and the displacements ''D<sub>d</sub>'' induced by the damaged structure which add to the stabilizing ''E<sub>i</sub>'' and displacement ''D<sub>i</sub>'' effects of the intact structure, as follows:

(4) <math>    Q=\bigl[\kappa{\scriptstyle\text{t}}+\lambda (E_i+E_d)] (D_i+D_d)</math>

Starting from the equation obtained (4) it is therefore possible to derive the critical collapse multiplier ''λd'':

(5) <math>det[\kappa_t+\lambda(E_i+E_d)]=0</math>

=== OVERALL RESPONSE ===

At this point, the research approach for structural strength can be identified by defining the percentage of reduction of the collapse multiplier between the intact structure and the damaged structure, or by introducing the Robustness ratio ''R'', defined as:

(6) <math>R=\frac{\lambda_d}{\lambda_i}</math>

== '''CONCLUSION''' ==
In this article, a new interpretation of the P-Δ effect aimed at finding the critical multiplier of the collapse of frame structures consisting of beams and pillars has been provided. The search for this load multiplier defines the evaluation of a resistance parameter that is increasingly useful for defining the ability of a structure to resist in the event of exceptional and sudden accidental actions, which can cause loss of the ability to support even the weight of the work itself, and which lead to human deaths, as in the cases cited in the first Paragraph in historical literature.

It is increasingly necessary that the philosophy and the calculation approach aimed at the design of buildings take into account non-ordinary phenomena. In this article it is emphasized a first introduction to the topic: how, with simple mathematical rules, it is possible to deduce the behavior of a structure subject to damage with respect to an integral one.

=='''REFERENCES'''<!--[1] Author, A. and Author, B. (Year) Title of the article. Title of the Publication. Article code. Available: http://www.scipedia.com/ucode.  [2] Author, A. and Author, B. (Year) Title of the article. Title of the Publication. Volume number, first page-last page.  [3] Author, C. (Year). Title of work: Subtitle (edition.). Volume(s). Place of publication: Publisher.  [4] Author of Part, D. (Year). Title of chapter or part. In A. Editor & B. Editor (Eds.), Title: Subtitle of book (edition, inclusive page numbers). Place of publication: Publisher.  [5] Author, E. (Year, Month date). Title of the article. In A. Editor, B. Editor, and C. Editor. Title of published proceedings. Paper presented at title of conference, Volume number, first page-last page. Place of publication.  [6] Institution or author. Title of the document. Year. [Online] (Date consulted: day, month and year). Available: http://www.scipedia.com/document.pdf.  -->==

[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.

[3]. Narayanan R., "Steel framed Structures, Stability and Strenght, and Concrete framed Structures, Stability and Strenght", Elsevier Applied Sci. Publ., London, 1986.

[4]. "Manuale di ingegneria Civile e Ambientale", Volume secondo, Zanichelli, ESAC.