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Progressive collapse, abnormal loads, and building codes

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NATIONAL RESEARCH COUNCIL O F CANADA

CONS RCHES DU CANADA

P R O G R E S S I ~ ~ ~ I ; ' E K P S E ; ' - A B ~ ; ~ ' O - ~ ~ ' Z Y ~ - ~ ~ ~ S , AND BUILDING CODES by D. E. A l l e n and W. R. S c h r l e v e r R e p r i n t e d f r o m PROCEEDINGS ASCE N a t i o n a l Meeting on S t r u c t u r a l E n g i n e e r i n g h e l d i n C l e v e l a n d , O h i o , A p r i l 1 9 7 2 p. 21

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47 R e s e a r c h P a p e r No. 578 of the D i v i s i o n of Building R e s e a r c h P r i c e 50 c e n t s NRCC 13658

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E F F O N D R E M E N T PROGRESSIF, CHARGES E X C E P T I O N N E L L E S E T CODES DU BATIMENT SO MMAIRE L ' e f f o n d r e r n e n t p r o g r e s s i f d ' u n e p a r t i e d ' u n i m m e u b l e e l e v e d ' a p p a r t e m e n t s k Ronan P o i n t e n ~ n i l e t e r r e , e n 1 9 6 8 , m o n t r e de f a f o n d r a m a t i q u e l a n 6 c e s s i t 6 de t e n i r c o m p t e d ' u n nouveau f a c t e u r d a n s l a conception de b b t i m e n t s - l a r e s i s t a n c e 'a l ' e f f o n d r e m e n t p r o g r e s s i f provoqu6 p a r d e s d o m m a g e s d e s e c t e u r , c a u s e s p a r d e s c h a r g e s e x c e p t i o n n e l l e s . L e p r e s e n t a r t i c l e e x a m i n e c e r t a i n s e f f o n d r e m e n t s qui ont e u k u d a n s l e p a s s e e t c e r t a i n s c h a r g e s e x c e p t i o n n e l l e s c a p a b l e s d e p r o v o q u e r l ' e f f o n d r e r n e n t p r o g e s s i f . On donne 6 g a l e m e n t c e r t a i n e s s t a t i s t i q u e s obtenues d ' u n e Btude

d e s e f f o n d r e m e n t s p r o g r e s s i f s r a p p o r t e s p a r l e s a g e n c e s d ' i n f o r m a t i o n . Ces r e n s e i g n e m e n t s f o u r n i s s e n t une e v a l u a t i o n a p p r o x i m a t i v e d e s s i t u a t i o n s o'u l a p o s s i b i l i t e d ' e f f o n d r e r n e n t p r o g r e s sif e x i s t e . L e s a u t e u r s p a s s e n t bri'evement e n r e v u e l e s e x i g e n c e s d u Code du b b t i m e n t r e l a t i v e m e n t 'a l ' e f f o n d r e m e n t p r o g e s s i f . On n ' e x a m i n e cependant p a s 1'6pineux probl'erne de l a conception et d e l ' a n a l y s e d e s b s t i m e n t s e n fonction d e l ' e f f o n d r e r n e n t p r o g e s s i f .

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A N A L Y Z E D

PROGRESSIVE COLLAPSE, ABNORMAL LOADS, AND BUILDING CODES

By D. E. Allen and W. R. Schriever'

INTRODUCTION

The partial collapse of the Ronan Point high-rise apartment building in England in 1968 ( 1 ) showed, most dramatically, the need for a new consideration in structural

design-resistance to progressive collapse initiated by local failure due t o abnormal loads. In thls paper examples will be discussed of progressive collapses that have occurred in the past and of abnormal loads which can initiate progressive collapse. Some statistics are also included based o n a study of news reports. This information provides a rough measure of existing situations susceptible to progressive collapse. Building Code provisions on progressive collapse will also be reviewed briefly. The difficult problem of how to design or analyze structures for progressive collapse will, however, not be discussed.

PROGRESSIVE COLLAPSE

In the past, progressive collapse, although infrequently discussed has been known under various terms such as a deck of cards situation, chain reaction, or domino effect. Briefly it can be defined as the phenomenon in which local failure is followed by collapse of adjoining members which in tun1 is followed by further collapse and so on, so that widespread collapse occurs as a result of local failure. This local failure may be initiated by explosion o r impact o r may simply be a r e s ~ ~ l t of local defective strength.

There is some vagueness in this definition. For instance, how widespread must the collapse be for it to be designated as progressive collapse. In the British building code collapse is considered progressive when it involves an area of 750 sq ft o r more o r 15% of the floor area, whichever is less, o r more than three stories. In estimating from news reports which failures could be classified as progressive collapse it was convenient to define it roughly by the number three; that is, if the collapse involves members that are three or more members away from the original failure o r if three or more spans collapse. Another vague point is the difference between progressive and total collapse. It can be argued that every collapse is progressive inasmuch as it must start somewhere. If a structure is completely inadequate it will collapse completely and the notion of progressive collapse is not meaningful. Incidents of total collapse were not included in the information gained from news reports.

EXAMPLES O F PROGRESSIVE COLLAPSE

Table 1 shows the number of progressive collapses from two news sources-the

Engineering Nervs-Record (from 1968 to 1972), and from newspaper clippings on

'Building Structures Section, Division of Building Research. National Research Council of Canada, Ottawa.

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Structitral Failures

TABLE 1 .-NEWS lNCIDENTS INVOLVING PROGRESSIVE COLLAPSE

Engineering

News Record Canada Collapse Designation

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During Construction Due to impact, explosion

Formwork, bracing or erection error Design error

During Service Life Due to explosion Due to impact

Design, manufacture or construction error

( 4 years) ( 10 years)

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During Demolition, Adjacent Excavation 1 6

TOTAL 22 75

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Structural Failures

collapses in Canada (1962 to 1972). Coverage of Canadian collapses is much more complete and covers and longcr period, and this accounts for the greater number shown for Canada. The progressive collapses in Table I have been categorized first according t o whether collapse occurred during construction, during service life, or during demolition or excavation nearby. The total number of progressive collapses is considerable, comprising approximately 15 to 20% of the total number of collapses.

The rcports of progressive collapse during construction are categorized in Table I according t o cause. Some guesswork was involved in assessing the cause but it is considered that the relative numbers are approximately correct. They indicate that most progressive collapses during construction are due t o errors in formwork, bracing, or erection procedure.

Figs. 1-3 show examples of progressive collapse during construction. The collapse in Fig. 1 is primarily due to inadequate construction procedure and in Fig. 2 to inadequate bracing. Fig. 3 shows a precast garage under construction in which all four stories of the interior structure collapsed progressively. This failure was attributed to inadequate bearing area for the precast beams. As a contrast Fig. 4 shows an example in which local failure of steel members during construction did not result in progressive collapse.

The reports of progressive collapse during service life are also categorized in Table 1 according to causes. Only one progressive collapse as a result of explosion, Ronan Point, was reported in this period, but there have been at least two more in recent months. A number of structures, mostly old truss bridges, collapsed due to impact. The largest number of 'collapses were simply due to inadequate design, manufacture, or erection procedure.

Examples of progressive collapse during the service life of the structure are shown in Figs. 5-8; Figs. 9-13 show examples of local failure during service life that did not result in progressive collapse. Some of these examples of collapses are discussed briefly.

A reinforced concrete building which collapsed progressively after being exposed t o fire for about eight hours is shown in Fig. 7. Because evacuation is usually possible before collapse during fire, danger t o life safety due to progressive collapse resulting from fire is usually not a problem. Exceptions are very tall buildings where evacuation is n o t possible and buildings where local fires can set off largc explosions.

Fig. 8 shows an arena (2) that collapsed primarily because of defective workmanship in t h e glue-laminating of wood truss members. This is an example of a series of parallel trusses in which failure of o n e member initiated progressive collapse of the whole roof. Progressive collapse warrants consideration in such situations, where defects are possible in one member and where sufficient warning of impending failure may not occur. In contrast, Fig. 9 shows an arena in which local failure of one frame did not result in progressive collapse.

The famous St. Louis bridge, completed in 1874, is shown in Fig. 10. Recently a sec- tion of the lower chord o f one o f the arches was knocked o u t by a tug, but there was no progressive collapse. (3) Obviously the savings in life and limb and avoidance and eco- nomic disruption due t o this capability t o resist progressive collapse can be consider- able. James B. Eads, the designer, said in his address at the inauguration of the bridge- "the peculiar construction of [ t h e ] superstructure is such that any piece in it can be easily taken o u t and examined, and replaced o r renewed, without interrupting the traffic of the bridge.

. . .

In completing the western span two of the lower tubes of the inside ribs near the middle of the span were injured during erection, and were actually uncoupled and taken o u t without any difficulty whatever, after the span was completed, and two new ones put in their place in a few hours." (4)

Fig. 1 4 shows an example of unwanted progressive collapse during demolition, and Fig. 15 shows an example of progressive collapse due t o adjacent excavation.

I

EXAMPLE O F ABNORMAL LOADS CAUSING HEAVY DAMAGE

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24 Structural Failures

F I G . I . - C O L L A P S E O F R E I N F O R C E D C O N C R E T E BUILDING UNDER CONSTRUCTION-BOSTON, 25 JANUARY 1970 (WORLD WIDE PHOTOS)

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Stiuctural Failures

FIG. 2.-COLLAPSE O F STEEL FRAME BUILDING UNDER CONSTRUCTION-

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Structural Failures

FIG. 3.-COLLAPSE O F A PRECAST GARAGE UNDER CONSTRUCTION-LAS VEGAS, MAY 1969 (ENGINEERING NEWS-RECORD)

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Structural Failures

FIG. 4.-COLLAPSE O F SEVERAL BAYS O F STEEL FRAMING-NEW YORK,

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Structural Failures

FIG. 5.-COLLAPSE O F FLATS AT RONAN POINT, 16 MAY 1968-CANNING TOWN, ENGLAND (CANADA WIDE PHOTO)

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Structural Failures 29

FIG. 6.-COLLAPSE O F CORNER O F BUILDING DUE TO CAR IMPACT-NEW YORK,

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30 Stntctural Failures

FIG. 7.-COLLAPSE OF REINFORCED CONCRETE BUILDING DURING FIRE

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Structural Failures

FIG. 8.-COLLAPSE O F ARENA UNDER SNOW LOAD-LISTOWEL, ONTARIO, 28 FEBRUARY 1959 (AIR PHOTO BY RON NELSON PHOTOGRAPHY LIMITED, LONDON, ONTARIO)

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Stnrctuml Failures

FIG. 9.-COLLAPSE OF ONE STEEL FRAME OF AN ARENA-OTTAWA, 17 MARCH 1972

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Structural Failures

FIG.lO.-VIEW O F DAMAGED EADS BRIDGE-OCTOBER 1969 (PHOTO FROM THE ARTICLE 'EADS BRIDGE REPAIRED BY JACKING' BY B. H. SPINNER IN CIVIL ENGINEERING-ASCE, JUNE, 1971)

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Srmctural Failures

FIG. I 1.-A BLOCK IN ALGERIA, DAMAGED BY EXPLOSION (COURTESY OTH NORTH AMERICA LTD.)

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Struct~tml Failures 35

FIG. 12.-BRICK BUILDING IN COPENHAGEN DAMAGED BY GAS EXPLOSION- JANUARY 1969 (POLITIKENS PRESSE FOTO)

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36 Structural Failures

FIG. 13.-APARTMENT BUILDING STRUCK BY COLLAPSING BUILDING DURING CARACAS EARTHQUAKE-29 JULY, 1967 (AMERICAN IRON AND STEEL INSTI- TUTE)

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Structural Failures 37

FIG. 14.-COLLAPSE O F BUILDING DURING DEMOLITION-KANSAS CITY, 22

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Srmcrural Failures

FIG. 15.-REAR SECTION O F FOUR STORY BUILDING COLLAPSES INTO ADJOINING EXCAVATION-MONTREAL, 18 NOVEMBER 1964 (CANADA WIDE/ MONTREAL STAR)

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Structural Failures 39

of structures because the probability of their occurrence is considered extremely slight. The inquiry into the Ronan Point Collapse ( 1 ) showed, however, that if an explosion at any o n e location initiates progressive collapse of a large building, the risk becomes significant. To help assess the risk of progressive collapse due t o different causes, it is useful t o study past examples of abnormal loads of the type that could initiate progressive collapse.

Table 2 shows the results of a s t ~ ~ d y of news reports of abnormal loads causing heavy damage. These were also obtained from Etzgineering News-Record and Olnadian Press

clippings. They have been categorized according t o type-explosion, vehicle impact, impact from large falling or flying objects, unexpected ground conditions, and miscellaneous. Progressive collapse can also be initiated by abnormally low strengths d u e t o faulty design or workmanship (Figs. 1-3, 8 ) or as a result of fire (Fig. 7). In all, there arc a large number and variety of abnormal loads or strengths that could initiate progressive collapse.

The reports of explosions causing heavy damage are arranged in Table 2 according to type. Most are due to gas leaks. Complete information on gas leak explosions in Canada is not yet available, but it appears that the rate of gas explosions causing structural damage in Canada is less than the figure of 3.5 million units per year c a l c ~ ~ l a t e d for Great Britain. ( I ) Also, most of the gas leaks in Canada occurred in the basement o r underground, causing total rather than progressivc collapse of the building. Even if gas becomes safer, however, there is the possibility of sabotage by bombs or other means.

The reports of vehicle impact causing heavy damage are listed in Table 2 according t o source. Quite a few old steel o r wood truss bridges have collapsed d u e to a truck or car hitting a key member. Fig. 16 shows a case where a truck took out the bracing for the upper chord of a bridge. Although less frequent than t r ~ ~ c k s hitting bridge members, trucks or cars have also knocked o u t exposed ground story walls or columns (Fig. 6, see also Ref. 5) and ships have knocked o u t bridge members, as in the casc of the S t . Louis bridge (Fig. 10). Airplane impact appears t o be significantly probable only near airports. Table 2 shows that there arc quite a few cases of impact due to large falling or flying objects. Many occur during construction and arc mostly construction elements falling or out of control. for example d u e t o a sudden wind gust. There have also been quite a few cases of crane collapse (Fig. 17).

During the service life of a building impact loads have occurred due to collapsing structural components, such as walls, o r buildings and cranes on adjacent building sites. The probability o f this is greater in downtown areas where high buildinp are close together, as in the casc of the Caracas earthquake (Fig. 13). Other examples of impact from collapsing structures are shown in Figs. 18 and 19.

Other kinds of abnormal loads have been listed under miscellaneous, for example, the bursting of a large water tank inside a building and pushing out the walls or the use of a bridge member t o winch a truck o u t of the mud.

In high winds, flying objects, such as wood roofs, could conceivably also initiate progressive collapse, but as these objects are relatively light they cause serious damage only t o light structures.

BUILDING CODE PROVISIONS T O PREVENT PROGRESSIVE COLLAPSE Specific reference t o progressive collapse was only introducted into building codes after the Ronan Point disaster-a dramatic illustration that certain new forms of construction have little inherent resistance t o progressive collapse.

In Britain, the recently adopted fifth amendment (6) t o their building regulations requires all buildings higher than five stories t o be designed t o resist progressive collapse. Two ways of doing this are given:

1. The designer shall ensure that the notional removal of any structural member does not result in progressive collapse of the structure. This is achieved by means of alternate

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Explosions

TABLE 2.-NEWS INCIDENTS O F ABNORMAL LOADS CAUSING HEAVY DAMAGE

Type of Loading

Gas leak

Dust o r other chemical Bomb

Excavation blast Vehicle Impact

Structural Failures

Trucks, trains, cars Ships

Airplane

Impact Due to Large Falling or Flying Objects During construction (falling elements etc.) During use (collapsing structures) Wind-blown objects (flying roofs, etc.)

Engineering

News Record Canada ( 4 years) (10 years)

Excavation, Sinkholes, Slides, etc. 1 20

Miscellaneous 2 6

TOTAL 25 80

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Structural Failures

FIG. 16.-HIGH TRUCK LOAD SNAPS TOP CHORD MEMBERS AND COLLAPSES BRIDGE SPAN-RAYLAND, OHIO, APRIL 1968 (UNITED PRESS INTERNATIONAL)

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Structural Failures

F I G . . 17.-CRANE COLLAPSES INTO CONSTRUCTION SHACK-OTTAWA, 18

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Structural Failures 43

FIG. 18.-WATER TOWER COLLAPSES INTO HOUSES-VALLEYFIELD, QUEBEC, 12 JUNE 197 1 (CANADA WIDEIMONTREAL STAR)

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Structural Failures

FIG. 19.-SIGN COLLAPSES INTO BUILDING DURING HIGH WINDS-MONTREAL, 30 NOVEMBER 1971 (CANADA WIDEIMONTREAL STAR)

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Sttuctural Failures 45

not result in progressive collapse of the structure. This is achieved by means of alternate paths of support.

2. Notional removal need n o t be considered if the structural member is capable of resisting 5 psi ( 7 2 0 psf) air pressure in any direction.

For either alternative, the designer need only use a safety factor of 1.05 and include the dead load plus one-third the live load plus one-thud the wind load. In addition t o the fifth amendment there are more specific rules for high-rise precast concrete ( 7 ) and masonry (8) buildings.

In France since 1966, the design of large panel structures has been subject t o the rules given in the "Recommendations for Panel Structures" ( 9 ) prepared by the ComitC Europeen du BBton and other agencies. These recommendations contain a general "performance" or warning-type clause, which states that a building be designed and constructed so that abnormal loads d o not cause the structure t o collapse like a "house of cards." T h e recommendations also contain specific requirements for peripheral and internal ties.

In Canada, t h e National Building Code of Canada introduced in its 1970 edition a "performance" o r warning-type clause which states that the structural integrity shall be sufficient t o reduce to an acceptable level the hazards associated with progressive collapse due t o local failure initiated by abnormal loads or severe overloads. An explanatory commentary on this clause is given in the Code supplement (1 0).

In the United States, t o date none of the model building codes has any requirements aimed a t preventing progressive collapse. However the Department of Housing and Urban Development-Federal Housing Administration (HUD/FHA) has developed preliminary criteria (13) for high-rise precast concrete buildings based on the British rules (7). These preliminary criteria are being reviewed by technical associations and systems builders including ACI Committee 356 o n Industrialized Concrete Construction and PC1 Committee on Precast Concrete Bearing Wall Buildings. A t the time of writing HUD/FHA is revising the document in consideration of the various criticisms t o establish criteria applicable t o all buildings subject t o progressive collapse that may be proposed for construction under HUD programs.

In the USSR, there is a requirement ( 1 1) that large panel structures must withstand the notional removal of any wall panel of room size. This is a simple rule, less restrictive than the British one. In addition t o this, however, the countries in Eastern Europe usually require more continuity steel than d o those of Western Europe.

Many engineers, including the Institute of Structural Engineers (1 2 ) , have objected that the new British rules are unnecessarily restrictive. The authors generally agree with this opinion and wish t o make the following comments:

I . At this time probably no single quantitative rule can be formulated for all buildings-only a "performance" o r warning-type requirement as in the French or Canadian building codes should be established. The reason for this is that quantitative rules such as the 5 psi rule in Britain, rules on notional removal o r a ductility rule, apply only to specific types of structures. For example, a ductility rule is too restrictive for masonry structures. Also what may be a reasonable definition of notional removal for a precast panel building may not apply t o masonry buildings. In the authors' opinion a single quantitative rule would be too restrictive in some cases where other solutions such as venting are possible, and perhaps inadequate in others, for example a masonry pier which satisfies the 5 psi rule.

2. Quantitative rules will nevertheless be useful for specific types of construction such as certain precast wall panel or masonry construction. Unfortunately no satisfactory rules are yet available for many situations.

3. Until satisfactory rules are formulated, for cases where there are no specific rules and where the building official is unsure of the structural adequacy, he may have t o consent an independent source of judgment, such as a review panel of experts or an independent consultant.

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Stmctural Failures

SUMMARY AND CONCLUSIONS

The examination of news reports o n progressive colIapse, summarized in Table 1, and abnormal loads causing heavy damage, summarized in Table 2, indicates that:

1. Although quite a number of collapses can be classified as progressive, most of these would not have occurred if normal good engineering practice, without special consideration of progressive collapse, had been carried out both in the design and during construction.

2. Many kinds of abnormalloads can initiate progressive collapse, e.g., various kinds of explosions, impact from vehicles, from falling objects and from collapsing structures, sudden changes in ground conditions, a n d structural defects.

3. Although vehicle impact has often led t o progressive collapse, this danger can most

easily be avoided by protecting certain exposed members. Many engineers now recognize this in design.

4. Only a few of the reported incidents should definitely be considered structurally from the point of view of progressive collapse. The situation, however, could become more critical in the future due t o a change in types of construction t o high-rise precast or masonry, a n d d u e t o a possible increase in the likelihood of s t r u c t ~ ~ r a l l y damaging explosions of different kinds.

5. This survey indicates that structural resistance t o progressive collapse may have t o be considered for: (a) High-rise precast o r masonry construction; (b) Large-span roofs where defects are possible in the material o r in the manufacture and where there may n o t be sufficient warning of impending failure; a n d (c) Precast or preassembled buildings during construction.

With regard t o building code provisions summarized in the previous section, in the authors' opinion, no single quantitative rule can be formulated for all buildings. Only a "performance" o r warning-type requirement can be given. Quantitative rules, however, will be required for specific types of construction. Until such rules are determined, the building official may have t o use an independent source of judgment, such as a review panel of experts or an independent consultant.

ACKNOWLEDGMENT

We wish t o thank the Department of Housing and Urban Development-Federal Housing Administration for the information on building code provisions in the United States.

Report of the Inquiry into the Collapse of Flats a t Ronan Point, Canning Town, HMSO, London,

19hR

-, --.

Morrison, C. F., Schriever, W. R., and D. E. Kennedy, "The Collapse of the Listowel Arena,"

Comdian Consulting Engineer, Vol. 2, No. 5, May 1960, pp. 36-47.

Spinner, B. H., "Eads Bridge Repaired by Jacking," Civil Engineering-ASCE. June 1971, pp.

SG52.

Morgan, A. E. Damsand Other Disasters. Porter Sargent. Boston. 1971. p.1 19.

Sanders, P. H. "Evaluation of the Risk of Vehicle Impact on Structures," ASCE National Structural Engineering Meeting Reprint 1656, 1972.

Statutory Instruments 1970 No. 109, Building and Buildings. The Building (Fifth Amendment) Regulations 1970, HMSO, London, 1970.

British Standard Code of Ractice for Laree Panel Structures. Addendum No. 1 to CP116. The Structural Use i f precast Concrete. ~ r i t i s h k a n d a r d s Institution, London.

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Srrucrural Failures 4 7

t o Avoid Collapse Following an Internal Explosion. Institution of Structural Engineers Memorandum RP/68/03, London, 1969.

9. Comitd Europien du Bdton. Recommendations Internationales Unifides pour le Calcul et I ' E x d c u t i o n d e s Constructions en Panneaux Assembl6s de Grand Format. Edit6 par I'AITEC-Associazona Italiana Tecnico Econornica del Cemento, Roma, 1969.

10. Canadian Structural Design Manual (Supplement No. 4 t o the National Building Code of Canada 1970) National Research Council of Canada (NRC 11530)-Commentary C7-Structural Integrity, - . p. 6 0 j .

I I . Lewicki, B., "Prevention of Progressive Collapse." Preliminary report presented a t CIB/W23A ('Ekaring Walls)-meeting, Iasi, Romania 7-10 September 1970.

12. The Resistance of Buildings to Accidental Damage," The Srrucfural Engineer, Vol. 49, No. 2, February 1971, p. 102.

13. Provision to Prevent Progressive Collapse. Proposed Revisions to Structural Engineering Bulletins for Highrise Recast Concrete Buildings. Department of Housing and Urban Development-Federal Housing Administration.

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This publication i s being d i s t r i b u t e d by the Division of

Building R e s e a r c h of the National R e s e a r c h Council of

Canada. I t should not be r e p r o d u c e d in whole o r in p a r t

without p e r m i s s i o n of the o r i g i n a l p u b l i s h e r . The Di-

v i s i o n would be glad to b e of a s s i s t a n c e in obtaining

s u c h p e r m i s s i o n .

Publications of the Division m a y be obtained by m a i l -

ing the a p p r o p r i a t e r e m i t t a n c e ( a Bank, E x p r e s s , o r

P o s t Office Money O r d e r , o r a cheque, m a d e payable

to the R e c e i v e r G e n e r a l of Canada, c r e d i t NRC) to t h e

National R e s e a r c h Council of Canada, Ottawa. K I A OR6

.

S t a m p s a r e not a c c e p t a b l e .

A l i s t of a l l publications of the Division i s available and

m a y be obtained f r o m the Publications Section, Division

of Building R e s e a r c h , National R e s e a r c h Council of

Canada, Ottawa. KIA OR 6.

Figure

TABLE 1 .-NEWS  lNCIDENTS INVOLVING PROGRESSIVE COLLAPSE  Engineering
FIG.  2.-COLLAPSE  O F   STEEL  FRAME  BUILDING  UNDER  CONSTRUCTION-  TORONTO, 8 SEPTEMBER  1958  (FEDERAL NEWSPHOTOS  (CANADA)  LIMITED)
FIG.  3.-COLLAPSE  O F   A  PRECAST  GARAGE  UNDER  CONSTRUCTION-LAS  VEGAS,  MAY  1969  (ENGINEERING NEWS-RECORD)
FIG.  4.-COLLAPSE  O F   SEVERAL  BAYS  O F   STEEL  FRAMING-NEW  YORK,  SEPTEMBER  1968  (UNITED  PRESS INTERNATIONAL)
+7

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ع هلاع نوكي نأ لدب( عمتجملا يف ةيباجيلاا ةايحلا ةرتف ةليط لضفأ ةحصب عتمتلا هعمتجمو هلهأ ىل ). قيشر يحص رهظمب روهظلا. نطولاو ماع لكشب عمتجملا ىلع دوعت دئاوف : ذ

In the discussion of group belief an important feature is that a group should be a constituted collective.. group is given by the logic. More precisely, the logic is equipped with

We provide a formal framework and design provable exact and approximate (faster) algorithms for finding optimal router vehicle locations that are updated according to sensor

Keywords: Partial dierential equations, Mathematical modeling of free surface ows, Open channel ow, Incompressible Fluid, Asymptotic Analysis, Lagrangian, Dispersive,