Progressive collapse analysis of steel frames: Simplified procedure and explicit expression for dynamic increase factor

Progressive collapse refers to a phenomenon in which local damage in a primary structural element leads to total or partial structural system failure. When investigating the progressive collapse of structures, nonlinear dynamic procedures lead to more accurate results than static procedures. However, nonlinear dynamic procedures are very complicated and the evaluation or validation of the results can become very time consuming. Therefore, it is better to use simpler methods. For static analyses, the gravity force applied to the removed column bay should be multiplied by a constant factor of two. However, using a constant dynamic increase factor (DIF) is only appropriate for elastic systems. According to the optimal design of structures, the assumption of elastic behavior after column removal is conservative. Thus, it is necessary to establish an expression for DIF that considers inelastic responses. In this paper, a simplified analysis procedure for the progressive collapse analysis of steel structures is presented using the load displacement and capacity curve of a fixed end steel beam. The results of the proposed method are in good agreement with nonlinear dynamic analysis results. Also, the capacity curve, obtained by dividing the accumulated area under the nonlinear static load displacement curve by the corresponding displacement of the column removed point, is used to predict the progressive collapse resistance of the column removed structure. Finally, an explicit expression for the DIF is established for elastic-perfectly plastic and elastic plastic with catenary action behavior.     

Progressive collapse analysis of a steel building with pre‐northridge moment connections

This study is on progressive collapse analysis of a sample steel building using various procedures suggested by the General Service Administration and the Department of Defense. The progressive collapse was accommodated by removing columns as per above guidelines. The analysis methods considered were linear static, linear dynamic, non‐linear static and nonlinear dynamic using SAP2000 structural simulation program. The analysis model was a 3D nine‐storey steel moment resisting building. For linear analysis, rigid moment connections were used, while for nonlinear analysis pre‐Northridge connections were used. The effect of an extreme event was incorporated by removing a column in the analysis model. The static analysis was performed on the model with a column removed subjected to static gravity loadings. The dynamic analysis employed a ramp function to simulate the dynamic effect of the removal of a column. For nonlinear analysis, material nonlinearity was incorporated using strength degrading nonlinear hinges of pre‐Northridge connections as per FEMA 356. The paper documents the results of this analysis with comparison among methods. The potential for collapse and performance levels of the building were determined using the demand capacity ratio and the rotational limit obtained through the analysis. Copyright © 2010 John Wiley & Sons, Ltd.     

Seismic nonlinear static new method of spatial asymmetric multi‐storey reinforced concrete buildings

In order to obtain the seismic demands of spatial asymmetric multi‐storey reinforced concrete (r/c) buildings, a new seismic nonlinear static (pushover) procedure that uses inelastic response acceleration spectra is presented in this paper. The latter makes use of the optimum equivalent nonlinear single degree of freedom system, which is used to represent the general spatial asymmetric multi‐storey r/c building. For each asymmetric multi‐storey building, a total of 12 suitable nonlinear static analyses are needed according to the new proposed procedure, whereas at least 96 suitable nonlinear dynamic analyses are required in the case of nonlinear response history analysis (NLRHA), respectively. In addition, the present paper provides answers to a series of further questions with reference to the spatial action of the two horizontal seismic components in the static nonlinear (pushover) analyses, as well as to the documented calculation of the available behaviour factor of the asymmetric multi‐storey r/c building. According to the paper, this new proposed seismic nonlinear static procedure is a natural extension of the documented equivalent seismic static linear (simplified spectral) method that is recommended by the established contemporary seismic codes, with reference to torsional provisions. Finally, through a restricted parametric analysis carried out in this paper, a relevant numerical example of a two‐storey r/c building is presented for illustration purposes, where the seismic demand floor inelastic displacements are compared with the respective displacements obtained by the NLRHA. Consequently, the new proposed seismic nonlinear static procedure, which uses inelastic response acceleration spectra, can reliably evaluate the extreme values of floor inelastic displacements (on the flexible and stiff side of the building), as is shown by the above comparisons. Copyright © 2010 John Wiley & Sons, Ltd.     

Progressive collapse resisting capacity of building structures with outrigger trusses

In this paper, the progressive collapse potential of building structures with core and outrigger trusses were evaluated using nonlinear static and dynamic analyses. To this end 36‐storey analysis model structures composed of RC core walls and perimeter frames connected by outrigger trusses at the top were prepared. The static pushdown analysis of the structure with mega‐columns and outrigger trusses showed that the maximum strength reached only about 20% of the load specified in the US General Services Administration guideline when a mega‐column in the first storey was removed. According to dynamic analysis results, the vertical displacement monotonically increased until collapse as a result of buckling of some of outrigger truss members. However the structure with outrigger and belt trusses remained stable after a perimeter column was removed. The stability of the structure with mega‐columns and outrigger trusses could be achieved by redesigning it with additional belt trusses or with moment connections in interior or exterior frames. Based on the analysis results it was concluded that the dynamic amplification factor of 2.0 recommended in the guidelines provided reasonably conservative results. Copyright © 2010 John Wiley & Sons, Ltd.     

Evaluation of different loading simulation approaches for progressive collapse analysis of regular building frames

Applications of different loading simulation approaches to progressive collapse analysis of building frames subjected to sudden column loss are evaluated in this paper. Analytical investigation on the nonlinear static behaviour of a middle-supported clamped beam reveals that both the load-release and direct loading techniques will result in consistent response if the supporting force may be completely released. However, the dynamic load-displacement responses of eight building models indicate that the direct loading approach may predict less load capacity and larger displacement demand than the load-release one. The relative error in load-displacement response is more significant with the pseudo-static estimation. The difference in displacement response between the load-release and the direct loading or pseudo-static approaches may increase with the extent of plastification and number of storeys of the building frames. An empirical formula is proposed and validated for estimating the displacement error. The empirical formula may help for enhancing practical applications of the direct loading and pseudo-static approaches to progressive collapse analyses of low-to-medium rise, regular building frames.     

The effect of foundation flexibility and structural strength on response reduction factor of RC frame structures

Current force‐based design procedure adopted by most seismic design codes allows the seismic design of building structures to be based on static or dynamic analyses of elastic models of the structure using elastic design spectra. The codes anticipate that structures will undergo inelastic deformations under strong seismic events; therefore, such inelastic behaviour is usually incorporated into the design by dividing the elastic spectra by a factor, R, which reduces the spectrum from its original elastic demand level to a design level. The most important factors determining response reduction factors are the structural ductility and overstrength capacity. For a structure supporting on flexible foundation, as Soil Structure Interaction (SSI) extends the elastic period and increases damping of the structure‐foundation elastic system, the structural ductility could also be affected by frequency‐dependent foundation‐soil compliances. For inelastic systems supporting on flexible foundations, the inelastic spectra ordinates are greater than for elastic systems when presented in terms of flexible‐base structure's period. This implies that the reduction factors, which are currently not affected by the SSI effect, could be altered; therefore, the objective of this research is to evaluate the significance of foundation flexibility on force reduction factors of RC frame structures. In this research, by developing some generic RC frame models supporting on flexible foundations, effects of stiffness and strength of the structure on force reduction factors are evaluated for different relative stiffnesses between the structure and the supporting soil. Using a set of artificial earthquake records, repeated linear and nonlinear analyses were performed by gradually increasing the intensity of acceleration time histories to a level, where first yielding of steel in linear analysis and a level in which collapse of the structure in nonlinear analysis are observed. The difference between inelastic and elastic resistance in terms of displacement ductility factors has been quantified. The results indicated that the foundation flexibility could significantly change the response reduction factors of the system and neglecting this phenomenon may lead to erroneous conclusions in the prediction of seismic performance of flexibly supported RC frame structures. Copyright © 2010 John Wiley & Sons, Ltd.     

Dynamic increase factor for pushdown analysis of seismically designed steel moment-resisting frames

The independent threat scenario of sudden column loss under localised damage is usually considered in progressive collapse assessment. The effect of the sudden removal of a column is like the sudden application of the gravity load on the structure when significant deformations occur. This conventional approach is based on the simplifying but realistic hypothesis that the peak dynamic response can be assessed with reasonable accuracy using the nonlinear static response. In this approach, amplified gravity loads are applied to the bays that are affected by the removed column to compensate for the dynamic effects corresponding to the real load redistribution. The paper investigates the dynamic increase factor to be considered in the nonlinear pushdown analysis of seismically designed steel moment-resisting frames. The influence of the fundamental parameters involved in progressive collapse analysis was highlighted. The effect of various design variables, such as the number of stories, the number of bays, the location of the removed column and the level of seismic design load was investigated. The dynamic increase factor was estimated in a way to generate the best match of the peak dynamic responses through the nonlinear static analysis. Finally, the values obtained were expressed as a function of the vertical displacement at the location of the removed column and then compared with the GSA formulation based on the ductility factor.     

Investigation of progressive collapse in intermediate RC frame structures

A large part of common reinforced concrete (RC) structures are designed as intermediate moment resisting frames. However, current progressive collapse studies have not paid much attention to these frames. In this study, based on the acceptance criteria of the UFC 4‐023‐03 document, the influence of some external and corner column removal cases are evaluated by nonlinear procedures in all storeys of a regular structure. Although this structure considers the acceptance criteria of nonlinear dynamic analyses, nonlinear static analyses require additional reinforcement in beam supports in the top two storeys. Compared to dynamic analyses, it is also concluded that static analyses are more conservative in the estimation of maximum vertical displacement and shear force in beams. Copyright © 2010 John Wiley & Sons, Ltd.     

Seismic behavior of multistory RC building frames with vertical setback irregularity

This paper is the second of two companion papers that aimed to examine the behavior of irregular buildings subjected to seismic excitation. In the present study, seismic response of the building frames with setback irregularity has been determined. To achieve this purpose, building frames with different geometrical configurations of setbacks are modeled and analyzed using nonlinear dynamic analysis by subjecting them to an ensemble of 27 ground motions to generate 21 060 nonlinear dynamic analysis results. These results are compiled to create a seismic response database consisting of parameters such as maximum roof displacement, maximum interstory drift ratio, maximum plastic hinge rotation and collapse risk parameters. Furthermore, nonlinear regression analysis is conducted on this database to propose simple equations to estimate the seismic response parameters. Finally, the proposed equations are validated for two‐dimensional and three‐dimensional building models, and applicability of the proposed equation in performance‐based and displacement‐based designs is briefly discussed. Copyright © 2013 John Wiley & Sons, Ltd.     

Progressive collapse behavior of rotor‐type diagrid buildings

In this study, the progressive collapse‐resisting capacities of axi‐symmetric or rotor‐type diagrid structural system buildings were evaluated based on arbitrary column removal scenario. For analysis models, 33‐story buildings with cylindrical, convex, concave and gourd shapes were designed, and their nonlinear static and dynamic analysis results were compared. The effect of design variables such as the number of total stories, slope of diagrids and the location of removed members was also investigated. According to the analysis results, the rotor‐type diagrid structures showed sufficient progressive collapse‐resisting capacity regardless of the differences in shapes when a couple of diagrids were removed from the first story. The design parameter such as building height and the slope of the diagrids did not affect the results significantly as long as they were designed to meet the current design code. Copyright © 2012 John Wiley & Sons, Ltd.     

Dynamic shear force amplification in regular frame–wall systems

A parametric study is conducted to investigate the dynamic shear amplification factor (DAF) in low‐to‐mid‐rise frame–wall systems in which the reinforcement curtailment along the height matches the required code strength. The level of frame–wall interaction is varied by changing the wall index, defined as the ratio of the total wall area to the floor plan area, in a generic frame–wall system, and its correlation with the DAF is investigated. Wall index values ranging in the 0.2% to 2% interval are selected. Walls with lengths of 3m , 5 m and 8 m are used in the design of model buildings of 4, 8 and 12 stories. Shear–flexure beam continuum formulation is used in design and modeling. The global behavior is analyzed using nonlinear response history procedure using spectrum compatible ground motions. It is found that the primary source of amplification is the level of inelastic demand on the system. Walls designed for code‐specified force reduction factor R = 6 experienced an average base shear force amplification in the order of 1.64 with standard deviation of 0.19 with respect to design shear force. Amplification diminishes with decreasing R. An expression for the dynamic amplification factor as a function of the number of stories and force reduction factor R is proposed. Copyright © 2015 John Wiley & Sons, Ltd.     

基于能量方法的RC框架结构连续倒塌抗力需求分析I：梁机制

结构抗连续倒塌设计方法的关键是确定因偶然作用导致局部构件失效后剩余结构的连续倒塌抗力需求。现有规范对连续倒塌抗力需求计算采用经验系数法,因缺乏理论基础,难以合理考虑结构连续倒塌过程中的非线性和动力效应的影响,其可靠性不足。基于能量方法,建立了RC框架结构连续倒塌抗力需求分析的理论框架,推导出RC框架结构梁机制下抗连续倒塌子结构及其构件的非线性动力抗力需求与线性静力抗力需求之间的关系,通过该关系对线性静力抗力需求的修正,能够综合考虑RC框架结构在梁机制下连续倒塌过程中的非线性和动力效应的影响。通过数值算例分析了现有规范中经验系数法的问题,验证了本文方法的正确性。     

Progressive collapse performance of irregular buildings

In this study, the progressive collapse‐resisting capacities of tilted or twisted buildings were evaluated by nonlinear static and dynamic analyses. For analysis models, 30‐storey tilted buildings with braced cores and 30‐storey twisted buildings with reinforced concrete cores were designed, and their performances were compared with those of the regular buildings. According to the analysis results, the progressive collapse potential of the tilted structures varied significantly, depending on the location of the removed column. It was also observed in the tilted structures that the plastic hinges formed not only in the bays from which a column was removed, but also in the nearby bays. Similar results were observed in the analysis of the twisted structures. The progressive collapse potentials of the tilted structure were high when a column was removed from the tilted side. However, the twisted structures considered in this study had progressive collapse potentials not very large compared with those of the corresponding regular structures, mainly because more structural elements were involved in resisting progressive collapse when a structural member was eliminated. Copyright © 2009 John Wiley & Sons, Ltd.     

Progressive collapse resisting capacity of tilted building structures

In this study, the progressive collapse resisting capacities of tilted buildings are evaluated on the basis of arbitrary column removal scenario. As analysis model structures both regular and tilted moment‐resisting frames, structures with outrigger trusses, and tubular/diagrid structures are designed, their progressive collapse resisting capacities are evaluated by nonlinear static and dynamic analyses. It turns out that the tilting of the structures requires increased steel tonnage due to the increased p‐delta effect. In addition in the tilted structures the plastic hinges are more widely distributed throughout the bays and stories when a column is removed from a side or a corner of the structures. With the analysis results, it is concluded that the tilted building structures, once they are properly designed to satisfy a given design code, may have at least an equivalent resisting capacity for progressive collapse caused by sudden loss of a column. Copyright © 2012 John Wiley & Sons, Ltd.     

Seismic analysis of inelastic moment‐resisting frames Part I: Modified force analogy method for end offsets

An analytical method of performing inelastic structural dynamic analysis based on the force analogy method to study real moment‐resisting frames with plastic hinge offsets from member ends and rigid panel zones is proposed. New stiffness matrices used in the force analogy method for members with two ends being rigid are theoretically derived using the unit displacement method. Through static condensation, each term in the stiffness matrices is found to be modified simply through multiplying by a scale factor. This is an enhancement to the original force analogy method or other conventional methods, where the condensed form reduces the size of the structural problem by eliminating the unnecessary additional degrees of freedom that must be placed at the plastic hinges. But more importantly, this is a significant improvement on other conventional methods where the present inelastic analysis procedure is more of a theoretical nature rather than numerical. Numerical simulation for seismic analysis is performed on a single degree of freedom system, and the results show that plastic hinge end offsets have significant effects on the seismic response and therefore they should be carefully considered in dynamic analysis. The companion paper further discusses the influence of panel zone deformation on the dynamic response based on the method proposed in this part of the research. Copyright © 2007 John Wiley & Sons, Ltd.     

Second-order plastic-hinge analysis of space semi-rigid steel frames

This paper presents a numerical procedure based on the beam–column method for nonlinear static analysis of space semi-rigid steel frames. The second-order effects are considered by the use of stability functions obtained from the exact solution of beam–columns under end forces and the inelastic behavior of material are presented by the lumped inelastic concept using yield surface. An independent zero-length element comprising of six translational and rotational springs, of which rotational spring stiffness uses the Kishi–Chen power model, is used to simulate the rigidity of the steel beam-to-column connection. The generalized displacement control method is applied to solve the nonlinear equilibrium equations in an incremental–iterative scheme. The nonlinear response results are compared with those of existing studies to verify the accuracy of the proposed numerical procedure.     

Progressive collapse assessment of new hexagrid structural system for tall buildings

Hexagrid structure is an innovative tube‐type system. It is constructed with hexagonal exterior structural grids. The hexagrid works as an effective lateral and gravitational resisting system. This paper presents the progressive collapse‐resisting capacity of this new system and the common diagrid system based on the local failure of the structural elements in the story above the ground. The collapse behavior is evaluated by two different nonlinear static and dynamic analysis methods. This study was conducted to design two‐type 28‐story and 48‐story building models to withstand wind load for both structural systems. With the analytical results, the hexagrid has enough potential of force redistribution due to its special configuration. It is observed that the new system had high resistance to progressive collapse than diagrids in similar condition. The complementary studies illustrate that resisting progressive collapse capacity, in both hexagrid and diagrid structures, is increased by using the buckling‐restrained elements. The guidelines discussed here can help engineers to understand the mechanism of progressive collapse of the hexagrid structures. Copyright © 2013 John Wiley & Sons, Ltd.     

Response of a multi-storey steel composite building with concentric bracing under consecutive column removal scenarios

In this paper, a 3-dimensional finite element modelling technique developed by the author was used to analyse the progressive collapse of multi-storey buildings with composite steel frames. The nonlinear dynamic analysis procedure was performed to examine the behaviour of the building under consecutive column removal scenarios. The response of the building was studied in detail and the measures to mitigate progressive collapse in future designs were also recommended.     

Prediction of nonlinear seismic demands of high‐rise rocking wall structures using a simplified modal pushover analysis procedure

This study presents a simplified analysis procedure for the convenient estimation of nonlinear seismic demands of high‐rise rocking wall structures. For this purpose, the displacement modification approach used in the nonlinear static procedure of ASCE/SEI 41‐13 is adopted. However, in the current study, this approach is extended to every significant vibration mode of the structure whereas the displacement modifying coefficients for different modes are calculated using the typical flag‐shaped hysteresis behavior of rocking walls. The parameters of this hysteresis behavior are selected to represent rocking walls with a practical range of energy dissipation capacity and postgap‐opening stiffness. The computed peak inelastic‐to‐elastic displacement ratios are presented as mean spectra, which can be used for the convenient estimation of pushover target displacement for every significant vibration mode. The accuracy of proposed procedure is examined using the seismic demands obtained from the nonlinear response history analysis of a 20‐story case study rocking wall structure. Furthermore, a modal decomposition technique is used to determine the individual modal seismic demands. The proposed procedure is found to predict both the combined and the individual modal demands with a reasonable accuracy and can serve as a convenient analysis option for the design and performance evaluation of high‐rise rocking wall systems.     

Seismic demand of an RC special moment frame building

Seismic design of a building is usually performed by using a linear static procedure. However, the actual behavior of a building subjected to earthquake is inelastic and dynamic in nature, and inelastic dynamic analysis is required to evaluate the safety of the structure designed by the current design codes. In this study, an RC special moment‐resisting frame building was designed by IBC 2003. Maximum plastic rotation, dissipated energy of some selected members, and the drift demand were calculated to examine whether the inelastic behavior of the building followed the intention of the design code. In addition, the effect of internal moment‐resisting frames (gravity load resisting system) on resisting lateral load was investigated. According to the analysis result, the building designed by IBC 2003 showed the inelastic behavior intended in the code and satisfied the design drift limit. Copyright © 2007 John Wiley & Sons, Ltd.