Concept of sudden column removal

This section briefly describes the various modelling techniques for progressive collapse assessment. The four techniques are described in subsequent subsections.

3.3.1 Technique one: Sudden removal of internal forces

Figure 3-5 is a two-dimensional portal frame used in illustrating the concept of modelling sudden column loss using this approach. The first step is to determine the internal forces in the column using static analysis. Figure 3-5 (a) below is the initial state of the structure with the proposed column to be removed under gravity loading condition. The column to be removed suddenly is replaced with the internal forces determined from Figure 3-5(a) while Figure 3-5(b) represents the state of the structure with the internal forces. The principle of modelling sudden column loss based on this technique is to ramp the internal forces to zero over a short duration. It is also possible to consider the stability period at which internal forces of the column balances the gravity loading before it is ramped to zero. Hypothetically, this idea captures the sudden removal of a column under gravity loading condition.

Figure 3-5 Removal of internal forces suddenly

Figure 3-5 (c) is a time history function for modelling the removed column without considering the equilibhrium duration (Sa) of reacting internal forces and gravity loads. In

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removed column and ramped to zero over a time lapse (Rt). The application of Function 1A

can be found in Kokot et al. (2012).

The maximum joint displacement and rotation response are the criteria adopted for these investigation. Similar procedure is also considered for Figure 3-5(d) except that a constant equilibrium duration (Sa) of 3s was assumed because within this time lapse, sudden column

loss is not activated. It is chosen to ensure that an equilibrium is reached interacting the gravity loading to the reactive internal column forces.

3.3.2 Technique two: Sudden application of gravity loading

This method is a conservative approach to modelling sudden column loss due to unforeseen circumstance. This approach is conservative because sudden application of gravity loading is not the same as sudden removal of column. Sudden removal of column from analytical perspective requires diminishing of the internal forces of the structural member to be investigated over a very short duration.

One of the key assumptions of this approach is that sudden application of gravity load without the ‘‘removed column’’ captures the response of the structure to progressive collapse. Some researchers (Vlassis et al. 2009; Vlassis et al. 2008; Tsai 2010) adopt this conservative approach because it does not consider the column removal time and its ease of application. However, this approach could be modelled to consider the time lapse at which the maximum gravity load is been applied to the structure. In addition, it can be argued that some unforeseen events affect the structure over a fraction of a second while others take longer time.

A typical 2D portal frame as shown in Figure 3-6 is used to illustrate the concept. Figure 3-6 (a) is the initial state of the structure under gravity loading conditions. The model is replicated without the ‘missing column’ as shown in Figure 3-6 (b).

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Figure 3-6 Sudden application of gravity loads

Figure 3-6 (c) is the time history function that is used in modelling the gravity load (𝑁𝑑) to conservatively capture the instantaneous loss of the interior column as illustrated in Figure 3-6(a) and Figure 3-6(b). Figure 3.6(a) is the state of the structure in its original state while Figure 3-6(b) is the second stage when the column is deleted or the structure modelled without it. It is important to note the two paths defined in Figure 3-6(c). There are two similar ways to address this form of loading application, either using the UNIFTH default function path defined by 1-2- 𝑁𝑑 or using a customised path defined by 0-2-𝑁𝑑. For the default function path, the column removal time is zero while the customised function path enables the time history function to be defined. Since one of the objectives is to compare the response of all these functions, the path defined by 0-2-𝑁𝑑 is used in these study. The region defined 0-2, is the linear path at which the gravity load is applied to the structure as shown from the origin of the plot (Figure 3-6(c)). This region defines the column removal time Rt.

The application of the approximate method is the concept used in Imperial College London using the software (ADAPTIC). This method is applied because of it is computationally efficient. For this approach, there is no need to model the sudden column loss using the internal reactive forces because the sudden application of gravity load approximately replicates the dynamic response of instantaneous column loss.

The load path defined by (0) to (2) of Figure 3-6(c) has been used by some researchers Malla et al. (2011) to simulate the inelastic and postbuckling behaviour of a two - dimensional truss system. The time lapse at which the load was applied to the structure was four times the natural period of the structure which is 0.024s.The value used for the time rise was 0.096s. This is possible because the natural period of the structure was very small, this assumption

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may not hold if a 3D structure is investigated because it may likely not capture the inherent dynamic response of the structure.

The initial investigation is to study the behaviour of the time history function for the range 0.001 ≤ Rt ≤ 5s using the path defined from origin (0) through (2) and kept constant.

3.3.3 Technique three: Balancing of gravity to reactive forces

This method is the most widely used approach by researchers and is illustrated in Liu (2013). The concept of sudden column removal using this technique requires a balancing technique between the gravity load and the internal forces. The concept is illustrated using a 2D portal frame show in Figure 3-7.

Figure 3-7 Balancing of gravity to internal forces

Part (a) denotes the equilibrium state of the structure at initial condition under loadings. Part (b) shows the application of the stress resultants (PVM) to represent the missing column. Finally, part (c) is the time history functions which show the path of the internal forces and the gravity load increasing linearly up to the maximum time period called stability period (Sa).

This is then kept constant over a period (Rt) before diminishing the internal forces to zero

to simulate the sudden column loss while keeping the gravity load constant.

The gravity load and the internal forces are increased linearly from zero, the origin of the time history function curve to their respective maximum values as defined by (oa) and (ob) respectively. The points (b) and (c) on the internal forces path defines the time lapse at which the internal forces is diminished to zero. Although, some researchers could use this time period to ensure static equilibrium state of the structure before it diminishes to zero which

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still gives rise to the same result. The value of Rt determines the how ‘sudden’ a column is

removed and this affects the way the structure response.

3.3.4 Technique four: Opposite applied column forces

This method is not a common, it is a proposed technique for modelling sudden column loss using the time history function. The 2D portal frame structure is used to illustrate the principle that surrounds this method. The structure is originally analysed for static forces and the internal forces for the proposed column to be removed is determined. The initial state of the structure is shown in Figure 3-8(a).

Figure 3-8 Balancing gravity to internal forces suddenly

Figure 3-8(b) is a proposed modelling technique in which the internal reactive forces representing the column is equally applied in the opposite direction as action forces. Part (a) represents the state of the structure under gravity loading defined by GSA 2003. The model is analysed statistically, the stress resultants from the column to be removed are recorded and applied at the nodal point from the top and bottom of the node having the same magnitude but opposite in direction as shown in part (b). The internal force applied at the top are modelled as a time history function as shown in part (c). At t=0, the structure in part (a) and (b) is the same. After a time (Rt), the stress resultant (P1V1M1) at the top cancels the effect

of the stress resultants representing the column (PVM) to simulate sudden column loss. Different values for Rt could be adopted, therefore the next section is aimed at assessing the

effect of Rt on structural response within the range 0.001 ≤ Rt ≤ 5s. This approach can be

found in the progressive collapse studies carried out on a single degree freedom system by Buscemi and Marjanishvili (2005).

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