AbstractIt is generally taken as a given that there is no reasonable design concept that could have prevented the collapse of the Twin Towers, once it was initiated, from progressing all the way down to the ground. This view is rooted in the idea that the force generated during the inevitable impact between what may be called the intact upper section (IUS) and the intact lower section (ILS)—meaning the building sections above and below the initially lost columns, respectively—will exceed by at least one order of magnitude the capacity of the latter. On closer inspection, this turns out to be only partially correct—it is correct with regard to the topmost floor plate of the ILS but not with regard to the columns below this floor plate. This paper shows that if the ILS in the Twin Towers had been topped by a stronger-than-ordinary floor plate allowing the columns below to respond properly, rather than be bypassed, these columns—and with them the ILS—would likely have survived. The paper subsequently proposes a building design concept consisting in the insertion of strengthened floor plates in intervals of 10–20 stories.IntroductionThe relatively young research field of progressive collapse, naturally receiving a thrust each time a prominent structural collapse occurs, has undoubtedly received its strongest thrust yet due to the terrorist attacks on September 11, 2001. Significant progress has been made since then in the research on progressive collapse of buildings. However, given the nature of the disaster that prompted this research—a disaster that carried with itself the painful message, so deceptively tempting to discard in hindsight, that there are scenarios that we cannot anticipate and/or specifically design against—the direction taken in the attempts to tackle the problem cannot be called entirely satisfactory. With most of the attention falling on the design methods of providing increased local resistance and alternative load paths, the research efforts since 2001 have been mainly concentrated on what may be called the outermost defense lines. Comparatively very little attention has been given to the question of what can be done if these outermost defense lines are broken and collapse is initiated after all.Must everything be considered lost in such a case? If the explanation for the total collapse of the World Trade Center (WTC) Twin Towers proposed by Bažant and Zhou (2002) is correct, as currently widely accepted among engineering professionals, then it really would seem that given the various practical restrictions in the design of high-rise buildings, the endeavor of developing measures to arrest a collapse once initiated is very challenging, if not hopeless. The “if” at the beginning of this last sentence, however, must not be forgotten; it is among the issues addressed in this paper.The objective of this paper is to follow one of the recommendations from the World Trade Center building performance study released by FEMA in 2002, and to “determine, given the great size and weight of the two towers, whether there are feasible design and construction features that would permit such buildings to arrest or limit a collapse, once it began” (FEMA 2002, p. 40).Employed ApproachThe initial situation assumed for all following considerations is shown in Fig. 1. Bypassing the preceding events, we assume that collapse is initiated as all columns over the height of one or several neighboring intermediate stories of a high-rise building lose their entire axial capacity. Based on observations from video footages of the WTC Twin Towers’ collapses, this assumption is also supported by the findings of the official final report on these collapses released by NIST in 2005 [NIST NCSTAR 1-6 (NIST 2005)].Fig. 1 leaves the cause of the assumed initial column loss unspecified because this is irrelevant for the following considerations. There is only one restriction regarding this cause: the initial column loss in Fig. 1 is assumed to occur due to an immediate action on the columns, rather than due to some extraordinarily high gravity load acting on the intact structure; in other words, the building is assumed to be under normal operation load at collapse initiation.As shown in Fig. 1, three sections are distinguished in the considered building at collapse initiation. The following can be said regarding the survival chances of these sections. The damaged intermediate section (DIS) will inevitably be lost. This is a direct consequence of what we assume with regard to this section—namely, that all its columns lose their entire axial capacity thus initiating collapse. The intact upper section (IUS), which, like the DIS, finds itself in free fall, must be written off as well for the simple reason that we have no control over this section’s motion. In particular, we cannot prevent, but must rather expect, a rotation of the IUS during its free fall—a behavior that would result from a nonsimultaneous failure of the DIS columns across the building cross section and clearly observed in the case of the WTC Twin Towers. The intact lower section (ILS) is the only section whose saving we can begin to realistically consider.Before we do this, it is necessary to take a brief side step and examine the phenomenon we endeavor to prevent—an overloading of the ILS followed by a downward collapse progression, or a “crush-down,” as Bažant and Verdure (2007) fittingly called it (the terms downward collapse progression and crush-down are used interchangeably in the following). A collapse progressing downward from story to story and thereby collecting the floor plates into a stack is generally referred to as a pancake-type collapse—a term derived from the appearance of the collected stack (Starossek 2018, p. 18). This term, however, with the description just given, is by itself not sufficient to describe the way in which a crush-down can be initiated and will then unfold. The reason is that there are two possible and mutually exclusive ways for this to occur; these are shown in Fig. 2.In what we here call a column-failure-driven (C-F-driven) downward collapse progression, the floor plates of the successively collapsing stories are stacked together due to the overloading and subsequent successive buckling of the columns in these stories [Fig. 2(a)]. This type of collapse progression is assumed by Bažant and Verdure (2007) (although those authors do not use the term C-F-driven) in developing their one-dimensional (1D) collapse model, an evolution of a simple model presented by Bažant and Zhou (2002). Bažant and Verdure (2007) employed the concept of the “mean crushing force”—a concept to which we will shortly return—to account for the force from the buckling columns acting at the collapse front and thereby slowing this collapse front down.The second type of downward collapse progression, which we call floor-plate-failure-driven (F-P-F-driven), is shown in Fig. 2(b). Here, the floor plates of the successively collapsing stories are stacked together due to the shearing off of these floor plates at their connections to the columns; the columns themselves, resembling trees whose trunks are being disbranched, do not participate, a consequence being that the properties of these columns remain irrelevant with regard to the initiation and unfolding of the crush-down. The schematic representation in Fig. 2(b) focuses only on the F-P-F-driven crush-down, the topic at hand; in general, however, and for the same reasons explaining the F-P-F-driven crush-down, the IUS will be collapsing simultaneously in what may be called F-P-F-driven crush-up.Now, in any conventionally designed building, a given floor plate and its connections to the columns are only designed to carry this plate’s own dead and live loads with the corresponding safety factors; the columns supporting this floor plate, on the other hand, are designed for the loads from this and all above-lying floor plates, and can thus take a much greater load than the single floor plate. This obvious fact, combined with the realization that the IUS can never land onto the ILS with such precision that the columns of these two sections align perfectly and thus prevent activation of the topmost floor plate of the ILS (let alone the collapsing DIS mass, which will inevitably activate the topmost floor plate of the ILS), is all that is necessary to explain why the ILS in the WTC Twin Towers had no chance once the DIS and IUS were in motion: Any floor plate in the ILS was utterly unable to cope with the mere weight of the above-lying mass, not to mention the dynamic forces due to impact.This realization makes it clear that the crush-down in the WTC Twin Towers was F-P-F-driven, and that the ILS columns, much stronger than the governing floor plates, never had a say in the matter. Any possibly remaining doubt that this was the case is dispelled by the observations from available video footage of the collapses. Despite the dust cloud enshrouding the downward-traveling collapse front, it can be observed that “[a]s the floors collapsed, this left tall free-standing portions of the exterior wall and possibly central core columns” (FEMA 2002, p. 27). This is only possible in a F-P-F-driven crush-down [Fig. 2(b)]. If the crush-down had been C-F-driven [Fig. 2(a)], the ILS columns would have been crushed and would thus not have been left to stick above the downward-traveling collapse front.This raises the following question: if the ILS in the WTC Twin Towers had been topped by a floor plate strong enough to fully mobilize the underlying columns, making these columns the ultimate arbiters on the initiation of crush-down, would these columns have been able to sustain the load from the collapsing DIS and IUS, thus saving the ILS? A floor plate as just described, designed not to fail before the columns supporting it, is referred to in the following as a strong floor plate.As a practical side note, because the load-bearing capacity of a strong floor plate must be at least an order of magnitude above that of an ordinary floor plate, the depth of a strong floor plate must also be correspondingly larger, amounting to at least 1 story. Furthermore, in a building like the WTC Twin Towers, employing numerous closely spaced perimeter columns and a relatively large number of columns in the core region, a strong floor plate must include at least 1-story deep belt trusses coupling, respectively, the perimeter and the core columns. This is because the similarly deep trusses of this same strong floor plate that must span between perimeter and core, and that cannot be as many in number as the perimeter- and the core columns can only be attached to a correspondingly smaller number of these columns, say, only to every fourth or fifth of them; this smaller number of directly loaded columns must then communicate the information to the remaining columns and garner their participation, which is the task of said belt trusses.Having clarified these basics, we now proceed to address the aforeposed question. To do this, the total force exerted by the collapsing DIS and IUS on the strong floor plate topping the ILS must be determined as a function of time. This problem has many similarities to determining the force exerted by an impacting aircraft on the reinforced concrete shell of a nuclear power plant—a problem treated by Riera (1968). The most important feature of Riera’s (1968) approach is that it aims at uncoupling the problems of describing the collapse of the impacting body, in our case the DIS and IUS, and the response of the impacted structure, in our case the ILS. Riera (1968) recognized that such uncoupling is possible if, unlike the impacting body, the impacted structure remains elastic (and this can be only an initial assumption that is later to be confirmed). In such cases, the impacted structure can be treated as a rigid body.The aforeposed question can thus be answered by first assuming that the ILS remains elastic during the collapse of the DIS and IUS. The force that the ILS must be able to sustain if it is indeed to remain elastic, can then be determined as a function of time by assuming that the DIS and IUS collapse against a fixed rigid horizontal surface, representing the strong floor plate topping the ILS.A final point to clarify before beginning the derivations is the way in which we are to account for the resistance offered by the IUS columns during collapse. With the assumed fixed rigid horizontal surface modeling the top surface of the ILS, both the DIS and IUS collapse against this surface in what Bažant and Verdure (2007) called “crush-up” mode. A structure collapsing in crush-up mode is being eaten up from the bottom toward the top, with the collapse front moving up into the structure. The crush-up of the DIS, which is referred to in the following as the first phase of collapse, unfolds at the rate of free fall because there is no resistance to be offered by the DIS columns, which are assumed completely lost. Impacting the debris pile formed by the collapsed DIS mass at the end of the first and beginning of the second phase of collapse, the columns at the bottom of the IUS are activated and begin to get crushed (the term crushed column refers to a column whose ends are moved toward each other). Thus, the crush-up of the IUS, occupying the second phase of collapse, is C-F-driven.As Bažant and Zhou (2002) and Bažant and Verdure (2007) noted, a crushed column of the type employed in the WTC Twin Towers begins to buckle, thereby developing into a plastic-hinge mechanism, at a very early stage of its crushing. The force needed to continue the crushing process decreases quickly with the onset of buckling, as shown by the load-displacement function F*(z) in Fig. 3(a) (asterisks are used in Fig. 3 to refer to forces developed by individual columns, as opposed to a group of columns). The sketches integrated in Fig. 3(a), showing a crushed column over the height of one story, explain the meanings of the employed variables: F* is the force developed by the crushed column in the direction of crushing (i.e., along the line connecting the column’s ends), and z is the crushing distance, equal to the distance that the two ends of the crushed column have traveled toward each other since the onset of crushing.The total crushing distance of the column in Fig. 3(a) is (1−κ)hstory, where hstory denotes the height of a story, equal to the column’s buckling length, and κ is what Bažant and Verdure (2007) called a “compaction ratio” (although those authors used a different notation), accounting for the presence of compressed story contents. Due to these contents, the force rises steeply at z=(1−κ)hstory [rehardening branch in Fig. 3(a)], terminating the crushing in this story and initiating crushing in the next story [not shown in Fig. 3(a)].Bažant and Verdure (2007) proposed approximating the actual function F*(z) by a constant function yielding the same integral over the total crushing distance (1−κ)hstory. The magnitude of this constant function, which Bažant and Verdure (2007) called “mean crushing force,” is denoted here as Fmean* [Fig. 3(a)]. A significant difference can be seen between Fmean* and the column’s axial capacity, denoted as Fc* in Fig. 3(a). For columns as employed in the WTC Twin Towers, the ratio Fmean*/Fc* can be shown to be smaller than 0.25. This large difference between the mean crushing force and the axial capacity of a column, which was also strongly emphasized by Bažant and Zhou (2002) and later by Bažant and Verdure (2007), is an important point to which we will return later.Now, would there be any qualitative change in this consideration if instead of a single column we consider the crushing of a group of columns, assuming for simplicity that these columns are of equal properties, the same as considered in Fig. 3(a)? According to Bažant and Verdure (2007), the answer is no: The combined force of the crushed column group, which we denote as F(z) (note the missing asterisk), would simply be a scaled-up version of F*(z); hence, the combined mean crushing force Fmean would be no better an approximation of F(z) than Fmean* is of F*(z). But is this realistic for the case at hand? Can we expect that the IUS columns are actually crushed in sync, so that they all reach their respective maxima Fc* simultaneously, story by story, during the crush-up of the IUS? Obviously, this would only occur if the IUS collapses without developing any tilt and the surface of the debris pile formed by already collapsed mass below the IUS is perfectly even and horizontal, all of which is practically impossible.In reality, the various hills and valleys in the surface of the debris pile below the IUS, as well as the tilt, be it even a small one, that the IUS is bound to develop will at any moment cause the columns at the bottom of the IUS to find themselves at different stages of their crushing; thus the force in one crushed column may just be reaching its maximum Fc* while the force in another crushed column is at its lowest value. The effect of such asynchronicities among the contributions of individual columns on the combined force F(z) of the crushed column group is visualized in Fig. 3(b), which considers the collapse of a generic 4-bay frame against a tilted surface.Again, the sketch integrated in Fig. 3(b) helps to explain the meanings of the employed variables; z is again the crushing distance, with z=0 marking the onset of crushing of the column group, occurring as the first column in this group makes contact with the tilted surface. The five columns of the frame in Fig. 3(b), each of them having in every story the same properties as the single column considered in Fig. 3(a), are numbered in order to track their individual contributions, F1* to F5*. These contributions are seen to be shifted relative to each other along the abscissa, as a result of the nonsimultaneous activation of the columns caused by the tilted impact surface.The actual combined force F(z), sum of the contributions F1* to F5*, of the columns of this frame [double line in Fig. 3(b)] is seen to be much better approximated by the combined mean crushing force Fmean [thick dashed line in Fig. 3(b)] than the combined force that would occur if the columns were activated simultaneously and then crushed in sync [thick solid line in Fig. 3(b)]. This latter case of simultaneous column activation, removing the shifts among the individual contributions F1* to F5*, would occur if instead of tilted, the impact surface was horizontal and thus parallel to the frame’s floor lines. All forces in Fig. 3(b) are normalized with respect to the axial capacity Fc* of a single column [Fig. 3(a)] to facilitate comparison.The important general point to be conveyed by Fig. 3(b) is that any practically present cause of nonsimultaneous activation of the columns of the impacting structure—be this cause a tilt of the floor lines of this structure relative to the impact surface, as exemplarily assumed in Fig. 3(b), or some unevenness in the form of hills and valleys of the impact surface, or, most generally and practically most likely, a randomly time-varying mixture of both—will have a smoothening effect on the function F(z), reducing the fluctuations of this function about the mean value Fmean, and thus improving the quality of the approximation of F(z) by Fmean. Also, this quality can be expected to improve further as the number of columns in the crushed column group increases.Considering this on the one hand, and noting that the aforementioned causes of nonsimultaneous column activation will practically always be present, Fmean appears to be more than merely a very good approximation of F(z), much better than what Bažant and Verdure (2007) recognized when they employed this approximation; in fact, using Fmean to account for the action of F(z) appears the only reasonable thing to do. Having said that, this very realization, rather than a welcome improvement to the model of Bažant and Verdure (2007), turns out to pose a problem for this model—a problem that is to be explained at the end of the derivations in this paper.This, then, is how we want to account for the resistance offered by the IUS columns during crush-up: by using the combined mean crushing force of these columns. The term used from now on to refer to this combined mean crushing force is mean crushing force of the IUS.Crush-Up Collapse of a One-Dimensional Tower with Uniform Continuous Mass Distribution and Mean Crushing Force Linearly Decreasing to Zero at the Top: Closed-Form SolutionWe begin by considering a situation identical with that occurring during the second phase of collapse, after the IUS, falling with a certain initial velocity, meets resistance from below. It is pertinent to begin with this problem because its solution also contains the solution for the first phase of collapse (the collapse of the DIS) as a special case.Fig. 4(a) shows the considered 1D tower of initial height h, constant mass per unit height m¯, and a mean crushing force linearly decreasing toward the top, described by the function Fmean=ψgm¯h′, where h′ is a coordinate measured always from the (moving) tower top downward, and ψ is a factor equal to the ratio between the mean crushing force at level h′ and the weight of the tower part above that level. The tower has a compaction ratio κ [Fig. 3(a)] constant along the height. An underlying assumption in the concept of a compaction ratio is that no further compaction of already-collapsed material can occur; in other words, the collapsed compact mass resting on top of the rigid horizontal surface at any time t during the collapse is treated as rigid. At the beginning of its crush-up collapse, the tower, acted upon by gravity, is moving with initial downward velocity z˙0 against a rigid horizontal surface.Before moving on, let us briefly analyze the background of these assumptions and how well they match up to the actual WTC Twin Towers, whose IUS (and DIS, with ψ=z˙0=0) the 1D tower in Fig. 4(a) should represent. The WTC Twin Towers were prismatic buildings whose largest mass portion was contained in their floor plates, all designed for and standing under the load of similar floor contents (office furniture) of similar texture. This makes the assumptions of constant mass per unit height m¯ and constant compaction ratio κ reasonable. It is also reasonable to assume that the mean crushing force, depending among other things on the columns’ cross-section dimensions, decreases toward the tower top.Although it is true that in an actual building, Fmean does not decrease to zero at the top, this deviation from reality is an acceptable price to be paid for what the assumption Fmean=ψgm¯h′, combined with the assumption of constant m¯, offers: a closed-form solution. As Bažant and Verdure (2007) noted when deriving the crush-up differential equation, during a crush-up collapse the mass above the collapse front is generally in a state of fall under variable gravity acceleration. The assumptions in Fig. 4(a) make this acceleration constant, and thus remove the obstacle to a closed-form solution.Collapsing Tower KinematicsAt the beginning of its collapse (t=0), the considered 1D tower is moving toward a fixed horizontal surface with velocity z˙0 while also being under the action of g. Of main interest to us is the force needed to hold the horizontal surface in place during collapse; we designate this support force as Fs. It is instructive to make the following derivations using a reference frame in which at t=0, the tower itself is stationary while the horizontal surface at its bottom has an upward velocity z˙0 and an upward acceleration g. The driving force needed to move the surface in this prescribed fashion is the same force Fs(t) that we seek.Moving upward with the prescribed velocity gt+z˙0, the horizontal surface starts compacting the mass of the tower at t=0. After a time t, the surface has traveled the distance z˜(t)=gt2/2+z˙0t [Fig. 4(b)]. The thickness of the compact-mass layer resting on top of the moving surface is continually growing with time. It follows that the border between the compact mass layer and the remaining tower of uncompacted mass (RTUM), i.e., the collapse front, is moving upward faster than the horizontal surface. The distance traveled by the collapse front is given by the coordinate zCF(t) [Fig. 4(b)].The RTUM is accelerating upward under the action of the mean crushing force at the current collapse front Fmean(h′=hRTUM) [Fig. 4(c)]. At any time t during the collapse, Fmean(h′=hRTUM) has the same ratio ψg to the mass of the RTUM, meaning that this RTUM has a constant upward acceleration of ψg. The (upward) displacement of the RTUM is then zRTUM(t)=ψgt2/2 [where z˙RTUM(0)=0 in the reference frame used here]. At time t, the collapse front has engulfed a certain part of the original tower. The height of this part is given by h−hRTUM(t)=zCF(t)−zRTUM(t) [Fig. 4(b)]. Multiplying this height by the compaction ratio κ gives the thickness of the compact mass layer at time t. The coordinate zCF(t) can be expressed as the sum of the coordinate z˜(t) and the thickness of the compact-mass layer (1) zCF(t)=z˜(t)+κ[zCF(t)−zRTUM(t)]Substituting z˜(t)=gt2/2+z˙0t and zRTUM(t)=ψgt2/2, and solving for zCF(t) yields (2) zCF(t)=(1−κψ)gt2+2z˙0t2(1−κ)The compact mass m(t) is equal to the product of m¯ and height h−hRTUM(t)=zCF(t)−zRTUM(t). Substituting the expressions for zCF(t) and zRTUM(t) yields (3) m(t)=m¯(1−ψ)gt2+2z˙0t2(1−κ)Force Fs(t)Now consider the compact mass and the forces acting on it at time t [Fig. 4(c)]. Over an infinitesimal time interval dt, the compact mass m(t), moving with velocity z˜˙(t), grows by the mass particle dm. At time t, just before it becomes part of the compact mass m(t), dm is still part of the RTUM, and as such has the velocity z˙RTUM(t)=ψgt. The resultant upward force Fs(t)−Fmean(hRTUM) acting over the time interval dt leads to a change in the combined momentum P(t) of the compact mass m(t) and the collected mass particle dm. Applying the principle of conservation of momentum, we obtain (4) [Fs(t)−Fmean(hRTUM)]dt=[m(t)+dm][z˜˙(t)+dz˜˙]⏟P(t+dt)−[m(t)z˜˙(t)+dmz˙RTUM(t)]⏟P(t)where the one-dimensional vectors are already replaced by their single components. Further developing this expression yields (after leaving out the higher-order terms) (5) Fs(t)=m(t)z˜¨(t)+m˙(t)[z˜˙(t)−z˙RTUM(t)]+ψgm¯[h+zRTUM(t)−zCF(t)]⏟hRTUM(t)⏟Fmean(hRTUM)where Fs(t) is seen to be the sum of three forces: With z˜¨(t)=g, the first term on the right-hand side of Eq. (5) represents the weight of the compact mass m(t); the second term, referred to in the following as the mass-flow force, results from the continuous flow of infinitesimal mass particles experiencing an abrupt velocity change at the collapse front; the third and last term is the mean crushing force acting at the collapse front.The expressions for zRTUM(t), zCF(t), z˜(t), and m(t) were derived in explicit form in the preceding paragraphs. Substituting these expressions and their time derivatives in Eq. (5) yields, after some algebra, the following: (6) Fs(t)=m¯2(1−κ)[3(1−ψ)2g2t2+6(1−ψ)z˙0gt+2z˙02]+ψgm¯hThe properties of the function Fs(t), and in particular the time when the maximum is reached, depend on the value of ψ. Consider first the special case ψ=1, where the time-dependent terms in Eq. (6) vanish, and the force Fs is the time-independent sum of m¯z˙02/(1−κ) and gm¯h. The first term represents the force resulting from mass flow; the mass-flow velocity z˙0 is here constant because at any time t, the force Fmean(hRTUM) is exactly equal to the weight of the RTUM. The second term, gm¯h, represents the time-independent sum of the weight of the compact mass m(t) and Fmean(hRTUM).Now consider the case when ψ>1. In this case, the mass-flow velocity, and thus the mass-flow force, decrease as the collapse progresses (decelerated collapse) because at any time t, the force Fmean(hRTUM) is greater than the weight of the RTUM. Furthermore, the sum of the other two forces, gm(t) and Fmean(hRTUM), which can be shown to be equal to gm¯h+(ψ−1)gm¯hRTUM, also decreases during collapse because this same claim applies to hRTUM, and because (ψ−1)>1. Hence, for ψ>1 the force Fs(t) has its maximum at t=0, as in the case when ψ=1. Following the same line of reasoning, it can be shown that when ψ<1 the force Fs(t) increases as the collapse progresses. In this case, the maximum of Fs(t) is reached at the end of the collapse. To determine this maximum, the collapse duration must be known. This duration is determined next.Collapse DurationThe expression for Fs(t) in Eq. (6) is, of course, only valid as long as the collapse front travels into the RTUM. Assuming that the tower collapses totally, the collapse comes to an end when hRTUM [Fig. 4(b)] is reduced to zero. The time t* when this occurs can be calculated from the equation hRTUM(t*)=h+zRTUM(t*)−zCF(t*)=0. Substituting the expressions for zRTUM and zCF yields (7) g(ψ−1)2(1−κ)t*2−z˙01−κt*+h=0This is a quadratic equation with two roots for t*. The only root with physical meaning is (8) t*=z˙0−z˙02−2gh(1−κ)(ψ−1)g(ψ−1)For the special case ψ=1, Eq. (8) yields an expression of the type 0/0. Applying L’Hospital’s rule yields (9) limψ→1t*=limψ→1∂∂ψ[z˙0−z˙02−2gh(1−κ)(ψ−1)]∂∂ψ[g(ψ−1)]=h(1−κ)z˙0The collapse duration t* as given by Eq. (8) is a real number only as long as the term under the radical sign does not become negative. This is fulfilled as long as ψ does not exceed a certain value, which we designate as ψ¯. This condition can be expressed as follows: (10) ψ≤2gh(1−κ)+z˙022gh(1−κ)⏟=defψ¯This result points at a limitation of the assumption that the tower collapses totally. The matter is resolved by examining the velocity with which the collapse front propagates into the RTUM. This velocity is equal to z˙CF(t)−z˙RTUM(t), which can be calculated (11) z˙CF(t)−z˙RTUM(t)=(1−ψ)gt+z˙01−κThe collapse front propagates into the RTUM with constant or increasing velocity if ψ=1 or ψ<1, respectively. In both these cases, the collapse is total because the collapse front eventually reaches the top of the RTUM. This is in line with the inequality in Eq. (10)—the derived condition for the validity of the assumption of total collapse—which is obviously fulfilled when ψ≤1 (noting that ψ¯≥1).On the other hand, if ψ>1, the velocity difference in Eq. (11) decreases with time, reaching zero at time t¯, calculated as follows: (12) It can now be established whether the tower collapses totally by calculating the height of the RTUM at time t¯. If the result is positive, the collapse is not total, in which case the collapse duration t* is equal to t¯ according to Eq. (12). The condition for nontotal collapse of the tower can be expressed as follows: (13) h+ψgt¯22⏟zRTUM(t¯)−(1−κψ)gt¯2+2z˙0t¯2(1−κ)⏟zCF(t¯)>0After substituting the expression for t¯, this inequality can be transformed to obtain a condition for ψ, yielding (14) ψ>2gh(1−κ)+z˙022gh(1−κ)⏟ψ¯The right sides of both inequalities Eqs. (10) and (14) are, as they should be, equal to ψ¯, the value of ψ that marks the transition between total and nontotal collapse. The collapse duration t* for each case is given by a different expression: Eq. (8) for the case of total collapse, and by Eq. (12) for nontotal collapse (15) t*={z˙0−z˙02−2gh(1−κ)(ψ−1)g(ψ−1),ψ≤ψ¯z˙0g(ψ−1),ψ>ψ¯To complete the solution of the problem in Fig. 4, the height of the RTUM after the collapse comes to an end—we designate this height as hrest—is determined (16) hrest={0,ψ≤ψ¯hRTUM(tψ>ψ¯*)=h−z˙022g(1−κ)(ψ−1),ψ>ψ¯Capacity Requirement on the ILSUsing the expressions derived in the preceding section, the solutions for the first and second phases of collapse (collapse of the DIS and IUS, respectively) can now be readily obtained by substituting the respective quantities. The constant mass per unit height and the compaction ratio are the same for DIS and IUS (m¯DIS=m¯IUS=m¯ and κDIS=κIUS=κ). The total force exerted by the collapsing DIS and IUS is referred to in the following as Fs,tot.First Phase of Collapse: Collapse of the DIS, 0≤t

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