Progressive Collapse Analysis

The "progressive collapse " concept refers to a situation when the failure or damage of any small part of the structure leads to a complete or nearly complete destruction of the whole structure. Emergencies can be caused by the human activities or by the natural phenomena. The first case includes gas explosions, terrorist attacks, fires, defects in design, construction, and maintenance of buildings, their unqualified reconstruction (with superstructures, extensions, replanning of premises), resulting in weakening or overloading the bearing elements and subgrade etc. Emergencies can be caused by such natural phenomena as earthquakes, hurricanes, landslides, non-uniform deformation of the subgrade.

Since it is impossible to completely eliminate the risk of emergencies or emergency impacts, it is necessary to provide a certain degree of safety of people in buildings and of their property by reducing the possibility of the progressive collapse at the local failures of load-bearing structures.

In order to prevent the progressive collapse the general strengthening, local strengthening, and interconnection between elements can be provided. Most US codes prefer the general strengthening when the failure of one of the elements of the building does not lead to the destruction of the whole structure. The local strengthening, i.e. strengthening of the most sensitive places, is very difficult to standardize for including in the design codes, since it is necessary to understand the nature of the possible actions on the building, including the terrorist attacks. Structural interconnections between the elements or the continuity of the structure are other methods of the general or local strengthening.

The main provisions of the current regulations for the prevention of progressive collapse can be given as follows:

1. The load-bearing system of residential buildings must be resistant to the progressive (chain) collapse in the case of the local failure of individual structures at the emergency impacts (gas explosion, fire, etc.).
2. Local failures of individual load-bearing structures are allowed, but these primary failures must not lead to the collapse of the adjacent structures which receive the load previously taken by the elements damaged in the result of the emergency action.
3. The structural system of the building must provide its strength and stability, at least for the time necessary to evacuate people. The displacements of structures and the opening of cracks in them are not limited.
4. The resistance to the progressive collapse is checked by the calculation for the special combination of loads and actions which includes permanent and temporary long-term loads, and also the effect of hypothetical local failures of the load-bearing structures.
5. The design characteristics of the materials are enhanced with the help of special safety factors. Moreover, the values of the design strength are multiplied by the service factors which take into account the low probability of the emergency impacts and the growth of concrete strength after the erection of the building, and also the post-yield behavior of the reinforcement.

The mode is implemented in SCAD in accordance with the current regulations and is designed to model the behavior of buildings and structures in the case of emergency impacts which caused the local failures of individual vertical load-bearing elements. The authors also took into account the apparent arbitrariness of the initial assumptions:

• there is no reliable information on the location and cause of the process and its history;
• real failure parameters can be very different from the strength conditions given in the codes, since it is known that the design values of the strength parameters may differ significantly from those observed in nature.

Structures cannot be completely free from the risk of collapse because of the uncertainties of the requirements to the system, dispersion of mechanical properties of building materials, difficulties of adequate modeling of the behavior of the system even when the modern software are used. Moreover, it is impossible to design and build an absolutely safe structure without taking into account the cost of the prevention of emergencies.

Thus, we can obtain a qualitative assessment of the stability characteristics of the structure with respect to the progressive collapse in the result of the numerical modeling, and compare several possible collapse scenarios in order to detect weak areas in the structure.

Currently, there are four known approaches to the progressive collapse analysis: static linear analysis, static nonlinear analysis, dynamic linear analysis and dynamic nonlinear analysis.

SCAD implements both quasi-static linear and nonlinear analysis methods using dynamic amplification factors, as well as direct dynamic linear and nonlinear analysis.

The progressive collapse analysis is based on the following provisions:

• a model obtained in the result of the structural analysis and the subsequent selection of the reinforcement in the elements of reinforced concrete structures and sections in the elements of steel structures is used as the initial model of the building for the progressive collapse analysis;
• elements of the design model which model the suddenly removed structural members are combined into groups; the number of structural members which failed simultaneously (collapsed) is not limited;
• the calculation is performed for the combination of loadings which includes the characteristic values of permanent loads and temporary loads (sustained) with coefficient 1;
• dynamic amplification factors are introduced to take into account the suddenness of the removal of the structural members and the effect of falling of the collapsed structures;
• the check of the elements of reinforced concrete and steel structures included in the design model after the sudden removal of members is performed taking into account only the ultimate limit state;
• design strength and strain characteristics of the materials are assumed equal to their characteristic values;
• since large displacements usually occur in the result of the progressive collapse analysis, it is recommended to perform the calculation in the geometrically nonlinear formulation.

Figure 24-1. Primary and secondary design models

Progressive collapse analysis is performed in a quasi-static or dynamic formulation. The quasi-static formulation includes the following procedures:

• a primary design model (Fig. 24-1.a) is used to determine the stress-strain state preceding the failure in the elements of the structural system;
• a secondary design model is created by excluding one or more of the load-bearing elements from the primary design model (Fig. 24-1.b);
• Instant removal of the excluded element is modeled by the forces determined in this element during the analysis of the primary design model and applied in the secondary design model with the opposite sign;
• the analysis of the secondary design model with the removed element is performed and the stress-strain state in the elements of the structural system during the local failure of the load-bearing element is determined;
• the load-bearing capacity of the elements of the structural system is checked and the possibility of further failure propagation is assessed.

Dynamic analysis is performed in the same way as the static one:

• at the first stage, it is necessary to determine the stress-strain state of the structure under the applied loads;
• at the second stage, the selected finite elements are removed at a given point in time according to the accepted scenario. In this case, the equilibrium state of the system achieved at the first stage will be disrupted, and the structure will begin to oscillate.

If the dynamic linear analysis is selected, and the secondary design model is geometrically stable, then the system experiences damped oscillations. In this case, it is necessary to monitor the values of forces in structural elements and, if in accordance with the accepted failure criteria, some elements fail, then they are removed from the design model.

If the dynamic nonlinear analysis is used, taking into account physical nonlinearity, then in the case of a progressive collapse, the oscillations of the control nodes of the design model increase unlimitedly. If the progressive collapse did not take place under the selected scenario, then the control nodes of the design model experience damping oscillations. When analyzing the accumulated damage in the finite elements, the designer must decide whether to strengthen the structure or consider its current condition as acceptable. When performing the nonlinear dynamic analysis, it is strongly recommended to take into account the geometric nonlinearity along with the physical nonlinearity.