«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 ...»
This iteration process continues after the beam has yielded and redistributes the stresses to the adjacent structure as the load is increased. This analysis involves developing appropriate demand and capacity curves that are utilised to assess - and ultimately prevent - structural collapse.
A three-dimensional model of a whole typical structural floor was created as shown in Figure-6. This structural model was then analysed for a series of column removal scenarios. It was considered that when a column was removed the line of columns above would act as a hanger so that every floor structure would be forced to behave in a similar fashion to the floor being analysed; the hanger columns would effectively become redundant.
The concrete floor slab was modelled as thin shell elements connected to the steel floor beams. The tensile membrane stress in the concrete slab was monitored, and cracking of the concrete slab was considered. The cracking of the concrete in the principal tensile direction reduces the stiffness of the shell in the radial direction. This mechanism, due to strain compatibility requirement, sheds the tensile radial forces to slab reinforcement as well as the steel framing grid acting in the radial direction.
It was necessary to establish criteria in order to confirm the structural integrity after the columns are removed. The first criterion was to ensure that the columns adjoining the removed columns were not overloaded. This is achieved by performing strength check on the remaining structure, especially the adjacent columns using a realistic extreme event service load.
Alternatively this can be carried out by checking manually that columns adjacent to the removed columns can support the new load due to enlarged tributary area.
In extreme events, maximum deflection is not directly a limiting criterion. The behaviour can be considered acceptable if the strain in the yielded beam is limited to prevent collapse. Consequently, a maximum strain limit of 5% was adopted rather than imposing maximum deflection criteria.
The distribution of forces to the adjoining structure was also monitored. The results show that the bending moment diagram, together with the catenary axial force, reflects the element yield diagram by showing that the bending moment beyond the beam’s plastic capacity as being yielded. The beam axial forces also display the effect of catenary action by activating the surrounding grillage of beams and the concrete slab.
The floor plate stresses were closely scrutinised to prevent failure. The forces were distributed between the catenary action of the steel structure and membrane action of the slab. In the extreme event the concrete floor will act as a thin shell developing radial and hoop stresses. Concrete floor on metal deck can easily accommodate the compressive hoop stresses and its reinforcement was primarily upgraded to carry the tensile stresses. The tensile capacity of the concrete as well as metal deck is ignored.
The pushover nonlinear analysis was performed under full dead load of the structure and superimposed dead load. 50% of live load was considered.
Using the above model, various column removal scenarios were analysed. Obviously, in each scenario a different load path is activated.
While in each scenario a different load path is activated, the performances are similar and can be categorized in three groups i.e.: Interior column removal, Perimeter column removal, Corner column removal.
Interior Column Removal:
Figures-7 and 8 show the plate principal compressive and tensile stresses. Figure-9 shows plastic hinge formation in the main catenary framing member. Figure-10 shows the catenary forces in various framing members.
Exterior Column Removal:
Figures- 11 & 12 show the plate principal compressive and tensile stresses. Obviously the exterior condition does not allow a complete formation of catenary shell action. As a result additional stress is imposed on the spandrel beams. Figure-13 shows plastic hinge formation in the main catenary framing member. Figure-14 shows the catenary forces in various framing members.
The result of the analysis provides information for design and detailing as follows:
Beams & Girders:
Except in a few locations where beam sizes upgraded to meet the member force demand, majority of the floor framing sections were adequate.
The analyses show that the beams functioning as catenary members need to have full capacity connection throughout their entire span. Therefore, connections of perimeter beams and interior girders were required to be upgraded to full plastic capacity. The secondary beam connections remained as simple shear connections, however, their capacities were checked to insure adequate transfer of the catenary axial forces.
Compressive strength of concrete slab was adequate for compressive hoop stress demand. The tensile capacity of the slab system was enhanced by upgrading the mesh reinforcement and additional rebars at only critical locations.
Steel member strain:
Member strain was limited to 5%.
Depending upon the location and the column removal scenario, the floor deflection varied from 250mm to 900mm.
The combination of the concrete cores, the floors connected to them and the continuously-designed perimeter frames creates a robust three-dimensional system that allows the structure to redistribute the forces in all directions, preventing a progressive collapse scenario resulting from the loss of columns. Of course, in this extreme event the structure will go into the plastic range with large deformations creating the required membrane and catenary forces to stabilise the system.
1- British Code of Practice for Structural Steel, BS5950-1, 2000.
2- Private Communication, Abolhassan Astaneh-Asl, University of California, Berkeley, CA.
3- LARSA 2000-4th/ Dimension.
4- Hysteretic Models for Cyclic Behaviour of Deteriorating Inelastic Structures, M.V.Sivaselvan, A.M.Reinhorn Multidisciplinary Center for Earthquake Engineering Research, November 5, 1999.