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# «R. SHANKAR NAIR R. Shankar Nair R. Shankar Nair, Ph.D., P.E., S.E. is a principal and senior vice president of Teng & Associates, Inc. in Chicago. In ...»

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Plastic moment hinges and axial hinges are assigned to beam ends. Moment hinge properties are taken from FEMA 356 (FEMA, 2000) as shown in Figure 4. Figure 5 is the axial hinge property diagram, assuming infinite deformation capacity. Py for the beams is calculated taking only into consideration the rebar located in the area that is in compression due to flexure. We assume that the rebar located in the area that is in tension due to flexure has yielded. Columns are assumed to remain elastic due to their size. The hinge modeling does not account for interaction between the axial force and the bending moment. Large displacement analysis is used to engage cable action. Horizontal supports are provided to simulate restraint actions of the other bents. The actual solution is bracketed between having horizontal supports that enhance cable action and decrease column bending and having no horizontal supports that neglects any restraint provided by the floor slab.

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Figure 6 shows the loading condition to get the pushover curve. For simplicity, the structure is set up with no gravity load and a missing column. A more accurate solution is to include all the gravity loads present at the time of the column removal. The load P is equal to the reaction of the column removed, 440 kips in this case. For this example, we apply a maximum displacement of 144”. The displacement control analysis computes at each displacement step the amount of load required to create the displacement.

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Figure 7 is the pushover curve. Point A, B, C, D, and E on the pushover curve indicates different stages of structure behavior. Before point A, the structure behaves elastically with point A corresponding to the yielding of the structure. After yielding, the beams strength hardened from point A to B. At point B, the hinges fail and there is an abrupt drop. Curve CD indicates that the structure begins to pick up load due to cable action. At point D, reinforcement bars yield due to tension and the slope of the pushover curve becomes smaller. Since the model assumes the rebar has infinite deformation capacity, the structure can continue to sustain load without failure.

The area below the pushover curve is the energy that the structure can absorb. If we divide the energy below the pushover curve by the corresponding displacement, we can get the capacity curve of the structure. For example, point E’ on the capacity curve is obtained by dividing area below OABCDE by the displacement at E, 72” in this case. The pushover curve and capacity curve are characteristics of the structure under given load condition.

The load curve is straight in this case, which is equal to the reaction of the removed column, 440 kips in this case.

From Figure 7, it can be seen that the capacity curve is lower than the load curve before point F’, which means that the structure can not absorb the potential energy before reaching the displacement corresponding to point F’. It is obvious that the structure will collapse if it deflects as much as point F’, even if the energy can be balanced at point F. Thus, the conclusion is that the 2-D frame has a high potential for progressive collapse.

Figure 7 also shows that the capacity of the structure is about 80% of the required capacity near point B’.

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A nonlinear dynamic analysis is also carried out to verify the result of the nonlinear static procedure above. 3% of critical damping is introduced and force equal to the reaction of the column is put on the structure. Figure 8 is the joint vertical displacement response, where the column is removed.

The structure will not collapse because the axial hinge has infinite deformation capacity. The structure will reach its equilibrium at about 93 inches, which is close to 100 inches from the previous nonlinear static analysis. If the load on the structure is reduced to about 80%, the moment hinges will not fail, which is also very close to the nonlinear static analysis.

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The following conclusions can be reached:

• A simple structural design criterion including definitions for key or important members in a structure is proposed.

• A simple quantitative nonlinear static procedure is proposed for analyzing the progressive collapse potential caused by the removal of a column.

• The proposed nonlinear static procedure gives reasonable results for the example shown. The procedure also gives a quantitative measurement of the potential for progressive collapse.

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[1] Progressive Collapse Analysis and Design Guideline – for New Federal Office Buildings and Major Modernization Projects. Published by General Services Administration, 2003.

[2] Graham Powell “Collapse Analysis Made Easy (More or Less)”, Proceedings, Los Angeles Tall Buildings Structural Design Council Annual Meeting. Los Angeles, May 2003, pp. 1-14.

[3] FEMA 356 - Prestandard and Commentary for the Seismic Rehabilitation of Buildings. Published by the Federal Emergency Management Agency, November 2000.

[4] Ray W. Clough, Joseph Penzien “Dynamics of Structures.” 2nd Edition, McGraw – Hill, Inc. 1993.

[5] Building Code Requirements for the Structural Concrete (318–99) and Commentary (318R-99). Published by the American Concrete Institute, 1999.

[6] ASCE 7-95. American Society of Civil Engineers Minimum Design Loads for Buildings and Other Structures.

[7] FEMA 274 - NEHRP Commentary on the Guidelines for the Seismic Rehabilitation of Buildings. Published by the Federal Emergency Management Agency, October 1997.

[8] FEMA 273 - NEHRP Guidelines for the Seismic Rehabilitation of Buildings. Published by the Federal Emergency Management Agency, October 1997.

[9] ISC Security Design Criteria for New Federal Office Buildings and Major Modernization Projects. Published by the Interagency Security Committee, May 2001.

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Dr. Ahmad Rahimian., PE, SE is President Kamran Moazami, PE is a Director of of Cantor Seinuk Structural Engineers in WSP Cantor Seinuk, UK. Ltd. Structural New York City. An expert in the behaviour Engineers in London, UK. He has over 23 of steel and concrete structures under years experience in the design of a variety seismic and wind loading, he has written of high-rise /low-rise buildings including numerous papers, and lectured widely in residential, hotel, retail, and commercial, various professional societies and marine and parking structures. He has been universities on design of tall buildings. He responsible for structural analysis and has been extensively involved in the design design from schematic to contract and engineering of stadiums and buildings documents, preparation of specifications, worldwide. Recently, he directed the supervision of office and field engineers structural engineering of the Trump World and construction phase liaison with the Tower, the tallest residential building in contractors. Since 1989, Mr. Moazami has the world and the Torre Mayor in Mexico been responsible for the structural design City, the tallest building in Latin America. of over 7 million square feet of hotel, He holds a US patent in seismic protective commercial, retail and parking structures design. Dr. Rahimian currently serves as an constructed in the United Kingdom.

Adjunct Professor at The Cooper Union, School of Architecture.

ABSTRACT: The provision of British Standard BS5950 is discussed. The intent of the current standard on structural integrity provision is to localize the damage as a result of removal of one member (i.e.: column). A new method for enhancement of the structural integrity requirement is presented here. This method allows localizing the damage in an event of multiple column removal and therefore eliminating the likelihood of disproportionate or progressive collapse. The method utilizes the interaction between beam and slab elements in a three dimensional space by considering the membrane forces generated into the diaphragm by the geometric action of the deformed structure. A series of three dimensional non-linear finite element analyses were performed to simulate the behaviour of the floor system in absence of supporting columns.

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The aim of the disproportionate collapse criteria of the UK Building Regulations and material codes of practice is to ensure that buildings are generally robust and that a local incident does not cause large-scale collapse.

This paper presents a method for enhancing the structural integrity of high rise buildings beyond current practice. This approach focuses on redundancy and enhancement of the alternate load paths. This approach requires three dimensional analysis of the floor framing considering geometric and material nonlinearity.

This approach was developed for enhancing the structural integrity of a high-rise building in the United Kingdom beyond the British Standard. The criterion was set to be that the overall integrity of the structure should not depend on the integrity of any one or two columns.

The design process was developed using three dimensional finite element nonlinear large deformation pushover analyses for various column removal scenarios.

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Current British Standard (1) has a descriptive integrity requirement which implies that the descriptive requirement, if met, can accommodate the removal of any one column without initiating a progressive collapse. The descriptive provisions of British code of practice include requirements for internal ties, peripheral (edge) ties and vertical (column) ties with specified capacities. The descriptive criteria, while adequate, will not stand the scrutiny of a conventional analysis which is limited in its prediction due to inherent simplified assumptions on structural and material behaviours. However, recent tests carried out at the University of Berkeley, California, demonstrated that such structures can perform successfully (2).

The few standards that address the issue have in general descriptive provisions instead of performance provisions. The descriptive provisions try to enhance the structural integrity without addressing any specific threat or structural performance. The performance provisions, while computationally more demanding, have the advantage of addressing directly the structure’s behaviour under a given scenario.

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For this specific project, the current British structural integrity criteria were enhanced to tolerate removal of any two columns within the building.

Generally, any measures with respect to structural integrity aim to enhance the system capacity. The system capacity enhancement is achieved either by enhancing member capacities, ductility or introducing alternate load paths or redundancy.

In this specific project, the goal was to enhance the structure redundancy by expanding on the alternate load path capacity. In order to ensure that the alternate load paths have adequate capacity to transfer the loads, member strength and ductility requirements were reviewed and upgraded.

The two column removal in essence is similar to the single column removal except imposing a higher demand on the remaining structure. In order to activate the secondary load paths the structure will go under a large deformation to the extent that is necessary to engage the self-equilibrating catenary behaviour of the floor system. This obviously requires that all other modes of failures have a higher load carrying capacity.

Figures 1 and 2 show the two-dimensional catenary concepts for single and double column removals. In two-dimensional analysis tie forces at the end of the catenary are required to achieve equilibrium. In general, the horizontal component of the tie force cannot be handled by any reasonable sized columns.

Figure-1: Catenary equilibrium for single column removal Figure-2: Catenary equilibrium for double column removal Figure 3 shows the relationship between the two secondary load paths, i.e. beam action and catenary action as a function of beam rotation. The M/MP line shows the beam flexural action and the T/Ty shows the catenary actions. Initially the system under low level of loads acts as pure bending element and as the load and the rotation increases the system will convert itself from a bending element to a catenary element.

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In a two dimensional catenary system the system integrity depends on the capacity of the support to resist the horizontal component of the force. Generally, columns are not designed to receive lateral loads of the magnitude required for the equilibrium of a catenary system.

The advantage of a three dimensional behaviour is that the equilibrating forces are internal to the system.

In principle the floor system goes through a deformation that reshapes the floor plate into an inverted dome or a dish, see Figure-5. As a result, the equilibrium of the system does not depend on the capacity of the catenary tie forces at the support. In other words, the radial tensile forces created as a result of the dishing action is balanced with compressive hoop stresses of the composite floor system.

Figure-5: Three dimensional catenary shell action of the composite floor a system.

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The building is 35-stories 165 meters tall. The building lateral system is comprised of three independent concrete core wall systems surrounding the stairs, elevators and mechanical zones. The floor framing system is steel construction with composite metal deck and concrete slab supported by steel columns and concrete walls, see fig.6. Average column spacing is 9 meters.

–  –  –

## GEOMETRIC AND MATERIAL NON-LINEAR FINITE ELEMENT ANALYSIS

To study the behaviour of the structure, a three-dimensional model was created using the “LARSA Integrated Linear and Non-linear Finite Element Analysis and Design” computer program. The beam element includes the effects of both geometric and material non-linearity. The analysis involves continual iterations of the stiffness matrix at points along the length of the beam as the load is gradually applied.

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