Study of the mechanics of progressive collapse of FPB isolated beam-pier substructures

Abstract

The horizontal stiffness of the isolated layer is reduced substantially by a friction pendulum bearing (FPB) toprotectthe structure from potential damages caused by earthquakes. However, horizontal stiffness is essential to progressive collapse resistance of structures. This paper presents a simplified model to assess the progressive collapse response of beam-pier substructure isolated by FPB. Progressive collapse resistance by flexural action of the beam and additional resistance owing to the horizontal restraining force was achieved. The influences of the equivalent radius and friction coefficient of the FPB, the applied axial force on the FPB, and span-depth ratio of the beam on the additional resistance were investigated. Simulations were conducted to verify the proposed model. The results show that progressive collapse resistance provided by horizontal restraining can be reduced as large as 46% and 88% during compressive arching action (CAA) and catenary action (CA), respectively. The equivalent radius of the FPB shows limited effect on the progressive collapse response of FPB isolated structures, but friction coefficient and applied axial force, as well as depth ratio of the beam, show significant influences on the additional progressive collapse resistance capacity. Finite element method (FEM) results are in good agreement with the result obtained by the proposed method.

Graphical abstract

Keywords

friction pendulum bearing / progressive collapse / horizontal stiffness / compressive arching action / catenary action

1 Introduction

Progressive collapse is the spreading of an initial local failure from element to element, eventually resulting in the collapse of an entire structure or a large part of it. Almost all types of public and private buildings may be subjected to extreme events (i.e., hurricanes, tsunamis, earthquakes, explosions, vehicle impacts, fires and human errors, or even terrorist attacks). Such events usually cause local damage to the structure of buildings, but this can lead to a complete collapse. Furthermore, as it is difficult to predict the probability of occurrence and the magnitude of extreme events; it is neither practical nor possible to design a structure that is proof against such events through the traditional methods for conventional loads [1].

Over the past several decades, buildings around the world have suffered from partial or total progressive collapse, and many researchers focus their studies on the mechanics of structures; many measures have been proposed to prevent structures from progressive collapse. Theoretical analyses and experimental investigations as well as simulations, have been conducted to address progressive collapse mechanisms [1–5], typical static load-carrying mechanism curves and data have been obtained for both cast-in-place structure and prefabricated structure [6–10]. Structural performances have also been evaluated [11–14]. Furthermore, dynamic effects have also been assessed based on experimental and theoretical results [15–17]. It must be noted that most researchers have focused their studies on traditional structures, i.e., fixed-base structures. Traditional structures usually have sufficient horizontal restraint conditions. Methods for improving progressive collapse resistance and existing design methods for progressive collapse both assume that sufficient horizontal restraint conditions can be provided by the beams and slabs [18,19].

However, it is believed that the horizontal restraint of a structure is vital for resisting progressive collapse [20]. Corley et al. [21] firstly declared that insufficient horizontal restraint could lead to more damage in progressive collapse accidents by investigating the collapse event of Alfred P. Murrah Federal Building in 1995. Experimental and simulation results from Diao show that insufficient horizontal restraint results in reducing the contribution of the axial force to the collapse resistance [22]. Both progressive collapse resistance and failure mode of frames were susceptible to boundary conditions [23–26]. Experimental results show that the vertical restraint is inextricably linked to the horizontal restraint of the structure, as reported in Ref. [23]. Also, the experimental and numerical results based on structures isolated by rubber bearings show that the catenary action (CA) cannot be formed due to weak horizontal constraint of the isolation layer [27–29].

Many methods and devices have been proposed to reduce structural vibration, such as the Tuned Mass Dampers [30–33], Pounding Tuned Mass Dampers [34], Tuned Liquid Dampers [35], MRE-Tuned Mass Dampers [36], Active Mass Dampers [37], and many other traditional dampers [38–48]. These traditional dampers show limited influences on the progressive collapse resistance of the structure. However, friction pendulum bearing (FPB) is an innovative technology for reducing vibration and protecting structures from damage due to earthquake; it mitigates the vibration of the isolated structure by extending the structure’s fundamental period through reducing the horizontal stiffness of isolated layer [49]. This is quite different from the mechanisms of the aforementioned traditional dampers, which provide additional control force for suppressing structural vibration. Therefore, the horizontal stiffness of structure isolated by FPBs is much smaller than is the case for non-isolated structures controlled by traditional dampers. Moreover, the horizontal stiffness of the FPB is nonlinear, and it is influenced by the equivalent radius of axial force applied to the FPB, by the friction coefficient between the slider, and by the curved sliding surface as well as by the equivalent radius of the curved sliding surface. It means that the progressive collapse response of the structure with FPBs is essentially different from those of non-isolated structural systems and more complicated than those without any isolators. However, almost no papers focus on progressive collapse responses of structures equipped with FPBs.

This paper aims to investigate the progressive collapse resistance of pier-beam substructure from structure isolated by FPB and expand understandings of effects of the key factors of FPB isolated structures, including the equivalent radius, friction coefficient, axial force of FPB, and span-depth ratio of the beam, on the progressive collapse response of the pier-beam substructure. To this end, the primary substructure was considered as a simplified model based on the assumption that plane sections remain plane; the axial plastic strain responses of the beam were also ignored. Then, contributions to progressive collapse resistance provided by flexural action of the beam and horizontal restraining were assessed, respectively. Static analyses were performed to establish a theoretical static model for evaluating progressive collapse resistance of pier-beam substructure based on the simplified model. Finally, finite element method (FEM) simulations were conducted to verify the accuracy of the proposed model. The proposed model can be adopted to evaluate the progressive collapse resistance of a pier-beam substructure from structure isolated by FPB and can help to expand understanding of the effect of the key factors of the FPB isolated structure.

2 Overall progressive collapse resistance of FPB isolated substructure

The structural components that can provide resistance to the unbalanced gravity load are referred to as the ‘substructure’ [50]. The substructure is the smallest component of the entire structure that is intended to prevent progressive collapse.

An FPB isolated substructure for analysis in this paper is shown in Fig.1, the progressive collapse resistance of the substructure (noted as P) can be given as Eq. (1).

(1)P=P_{\mathrm{F}}+P_{\mathrm{H}},

where P_F represents the contribution to progressive collapse resistance provided by flexural action of the beam, and P_H signifies the progressive collapse resistance provided by horizontal restraining.

3 Assessment of progressive collapse resistance provided by horizontal restraining

The major difference between the FPB-isolated and non-isolated structures is that the FPB-isolated structures equipped with FPBs result in a significant reduction of horizontal restraint of the substructure, while the non-isolated structure possesses sufficient horizontal restraint conditions. In this section, progressive collapse resistance provided by horizontal restraining is discussed in detail.

3.1 Structural modeling

To investigate the progressive collapse resistance provided by horizontal restraint of the FPB isolated beam-pier substructures, i.e., R_H, the structural system displayed in Fig.1 was simplified as shown in Fig.2, and the following assumptions were made.

1) The FPB isolated beam-pier substructure was symmetrical about the missing column.

2) The beam section was considered rigid, and axial deformation was ignored.

3) The rotational degree of freedom of the structure was considered as hinged support, incapable of transmitting bending moments.

A nonlinear elastic spring was employed to simulate the direction behavior of the system.

The relationship between the beam vertical deflection (marked as δ_V) and the horizontal deformation (marked as δ_H) is given as Eq. (2),

(2){\delta }_{\mathrm{H}}= -\frac{{\delta }_{\mathrm{V}}\left({\delta }_{\mathrm{V}}-2h_{\mathrm{e}}\right)}{2L},

where h_e is the effective depth of the beam which can be obtained by Eq. (3),

(3)h_{\mathrm{e}}= \frac{0.00473L^2}{H}.

Based on the energy conservation of the system,

(4)F_{\mathrm{H}}= k_{\mathrm{F}\mathrm{P}\mathrm{B}}{\delta }_{\mathrm{H}}.

Therefore, P_H can be given as Eq. (5),

(5)P_{\mathrm{H}}= k_{\mathrm{F}\mathrm{P}\mathrm{B}}\frac{{\delta }_{\mathrm{V}}\left(2h_{\mathrm{e}}-{\delta }_{\mathrm{V}}\right)\left(h_{\mathrm{e}}-{\delta }_{\mathrm{V}}\right)}{2L^2}.

3.2 Horizontal restraint provided by FPB

Since FPB was invented by Zayas et al. [49], various types of FPB have been proposed. One of the most extensively used FPB is shown in Fig.3, and it consists of three parts: two curved sliding surfaces and a slider embedded coated in PTFE. The friction coefficient between the slider and curved sliding surfaces is quite low to prevent earthquake energy transmitting to superstructures.

Based on the location of the slider on the sliding surface, the horizontal stiffness of FPB can be divided into two main stages. When d ≤ d_y, no sliding occurs between the slider and curved sliding surface just as shown in Fig.3(a); the horizontal stiffness is presented as k_p as exhibited in Fig.3(c). At this stage, the horizontal restraint provided by FPB can be simulated as a linear spring. Otherwise, the horizontal stiffness is k_e as shown in Fig.3(c), and a nonlinear elastic spring can be employed to simulate the horizontal restraint of the system. Consequently, the horizontal stiffness of the isolators can be expressed as Eq. (6).

(6)k_{\mathrm{F}\mathrm{P}\mathrm{B}}=\{\begin{array}{rl}& \frac{\mu F_{\mathrm{N}\mathrm{U}}}{d_{\mathrm{y}}},d⩽d_{\mathrm{y}},\\ & \frac{\left(\mu +\mathrm{s}\mathrm{i}\mathrm{g}\left(\dot{d}\right)\cdot {\displaystyle \frac{d}{R_{\mathrm{E}}}}\right)F_{\mathrm{N}\mathrm{U}}}{d},d>d_{\mathrm{y}},\end{array}

where F_NU is the axial force applied on FPB, μ is the friction coefficient between the slider and curved sliding surface, d_y is the yield displacement of FPB, d is the sliding displacement of FPB, R_E is the equivalent radius of FPB which can be obtained by Eq. (7):

(7)R_{\mathrm{E}}= R_{\mathrm{U}}+ R_{\mathrm{D}}- h_{\mathrm{s}},

where R_U and R_D are the curvature radius of the curved sliding surface, respectively. h_s is the height of the slider.

3.3 Compressive arching action

According to Eq. (5), P achieves the maximum when δ_V = 0.423h_e. However, δ_H and F_H achieve their maximum, δ_H0 = 0.5(h_e)²/L and F_H0 = 0.5k_FPB (h_e)²/L, when δ_V = h_e, respectively. Therefore, the relationship between δ_H0 and d_y can be adopted to determine whether sliding occurs or not, and sliding occurs when δ_H0/d_y is larger than 1 (Eq. (8)).

(8)\frac{{\delta }_{\mathrm{H}0}}{d_{\mathrm{y}}}⩾1.

By substituting Eq. (3) into Eq. (8), it can be seen that the geometrical parameters of the beam (i.e., H and L) are the factors that decide whether the FPB slider is sliding or not.

Influences of L/H and L on the δ_H0/d are shown in Fig.4. When δ_H0 = d_y (the red solid line in Fig.4), the sliding occurs. The results show that δ_H0/d_y increases with L increasing, and larger δ_H0/d_y will be obtained. Moreover, δ_H0/d_y is proportional to L/H. The minimum L when sliding occurs is 0.9 m. But for almost all of the beams in engineering, L is longer than 0.9 m, which means that sliding occurs in almost all situations.

Therefore, the δ_V1 and δ_V2 corresponding to vertical deflection when sliding occurs can be expressed as

\begin{array}{}\text{(9a)}& {\delta }_{\mathrm{V}1}=h_{\mathrm{e}}-\sqrt{h_{\mathrm{e}}^2-2L{d}_{\mathrm{y}}},\end{array}

\begin{array}{}\text{(9b)}& {\delta }_{\mathrm{V}2}=h_{\mathrm{e}}+\sqrt{h_{\mathrm{e}}^2-2L{d}_{\mathrm{y}}}.\end{array}

It’s obvious that δ_V1 and δ_V2 only depend on the geometrical parameters, and they are independent of the properties of the FPB isolators, i.e., R_E and μ. The horizontal reaction force and the resisting capacity, F_H and P_H, respectively, can be given as Eqs. (10) and (11):

(10)

(11)

3.4 Catenary action

When P = 0 and δ_V = 2h_e, the slider slides back to the center of the surface. Then tensile CA starts to work. The derivation of horizontal reaction force and resisting capacity of the substructure in the CA stage is similar to derivation in the compressive arching action (CAA) stage, the corresponding vertical deflection, defined as δ_V3, can be obtained as Eq. (12),

(12){\delta }_{\mathrm{V}3} =h_{\mathrm{e}}+\sqrt{h_{\mathrm{e}}^2+2L{d}_{\mathrm{y}}}.

And F_H and P_H can be given as Eqs. (13) and (14), respectively.

(13)

(14)

3.5 Evaluation and comparison

To evaluate the structure’s complete analytical horizontal and vertical restraining response, the dimensionless method is applied to the analysis. The dimensionless force F_H/F_H0 and P/P_CAA0 were used to measure the horizontal and vertical restraints, where F_H0 and P_CAA0 are the corresponding maxima of the structure without sliding (see Eqs. (15) and (16)).

(15)P_{\mathrm{C}\mathrm{A}\mathrm{A}0}=k_{\mathrm{p}}\frac{0.193h_{\mathrm{e}}^3}{L^2},

(16)F_{\mathrm{H}0} =k_{\mathrm{p}}\frac{h_{\mathrm{e}}^2}{2L}.

The full range of the behavior that includes the compressive arching action and the tensile catenary action is presented in Fig.5. OABD represents CAA, and DF is characteristic of CA.

Before Point A, the horizontal stiffness of the structure is k_p which is a constant. It is observed that F_H/F_H0 increases proportionally with the increase of δ_V. As a result, P/P_CAA0 increases with the increase of δ_V. Regarding Point A, sliding occurs, the horizontal stiffness shifted from k_p to k_e, and k_e decreases with the increase of δ_V, leading to slight increases in F_H/F_H0. Moreover, sliding results in a decrease of P/P_CAA0. Finally, δ_H reaches the maximum, and F_H exhibits a snap-through instability until a new position of stable equilibrium is adopted from Point A to Point B because frictional force starts decreasing, P falls to zero at Point B (δ_V = h_e).

It should be noted that F_H increases from Point B (δ_V = h_e) to Point D (δ_V = 2h_e) which differs from the results for the non-isolated structure.

After the slider has gone back to the bottom of the curved sliding surface at Point D, it is the end of the CAA and the start of the CA. The contribution by catenary action increases gradually with increasing deflection.

3.6 Parametric analysis

The essential distinction between a structure with FPB and a non-isolated structure is the significant reduction of horizontal stiffness due to the FPB. In other words, the main differences between the two types of structure are the additional resistance owing to the horizontal restraining force, i.e., F_H (see Eq. (6)).

In this section, parameter studies were conducted to have precise and deep insight into the influence factors on P. The model is from the structure designed by Lu et al. [8]; FPB isolated pier-column substructures were fabricated by using FPBs on the top of the frame bottom column as shown in Fig.1.

3.6.1 Comparison between FPB-isolated and non-isolated substructures

Comparisons between FPB-isolated and non-isolated structures are shown in Fig.6. Both horizontal and vertical restraints of FPB isolated structures are much less than the case for the fixed-base structures. The maximum F_{\mathrm{H}} in the CAA stage, for FPB-isolated and fixed-base structures are 0.28F_H0 and 1.19F_H0, respectively. Only 23% horizontal restraining can be offered by FPB-isolated compared to that of the fixed-base structure. Moreover, when δ_V = 4h_e, F_H of the FPB-isolated structure is 0.53F_H0, the corresponding value was 4.39F_H0 for the fixed-base structure. The reduction of F_H leads to a significant decrease in P. The maximum P is 1.19 P_CAA0 for fixed-base structure, but only 0.64P_CAA0 for isolated structure. Furthermore, P is 3.23P_CAA0 and 27.2P_CAA0 for FPB-isolated and fixed-base structure respectively when δ_V = 4h_e.

The results indicate that the capacity of resisting progressive collapse of structure is significantly compromised by FPB isolators considering that the progressive collapse resistance provided by horizontal restraining, i.e., P_H, can be 30% of the P for fixed-base structure in CAA and an ever-larger proportion in CA.

3.6.2 Equivalent radius

Fig.7 displays the influence of equivalent radius, i.e., R_E, on the horizontal restraining and vertical restraining of the substructure isolated by FPB when μ is 0.05. It’s obvious that R_E has almost no effect on horizontal restraining and vertical restraining. R_E affected only the horizontal stiffness of the isolator when sliding occurs based on Eq. (6). Therefore, all cases show the same result before sliding. After sliding in CAA, R_E has not shown remarkable impacts on k_FPB since the δ_H is small, and P_H is still a small fraction of the total horizontal restraining (Eq. (6)). It is dominated by the friction force between sliding and curved sliding surfaces. For instance, P is 0.392P_CAA0, 0.384P_CAA0, 0.380P_CAA0, and 0.378P_CAA0 responding to R_E are equal to 1, 2, 3, and 4 m, respectively. The result differences are no more than 5 percent in all cases.

However, the effect of R_E becomes more and more obvious with the development of deflection. For example, P is 2.76P_CAA0, 2.60P_CAA0, and 2.52P_CAA0 when R_E is 2, 3, and 4 m, respectively.

In summary, R_E showed limited effect on the progressive collapse response of FPB isolated pier-column substructures, especially when δ_H was small.

3.6.3 Friction coefficient

The influence of friction coefficient on horizontal and vertical restraining is displayed in Fig.8. Significant differences both in F_H/F_H0 and P_H/P_CAA0 can be found when the friction coefficient of the isolator (noted as μ) ranges from 0.025 to 0.10 because k_FPB can be improved by increasing friction coefficient μ. When μ is equal to 0.025 and 0.05, the most used in many areas of civil engineering, the maxima of P in CAA are only 0.31P_CAA0 and 0.61P_CAA0, respectively. The corresponding values can be increased to 0.91P_CAA0 and 1.22P_CAA0 when μ is increased to 0.075 or ever 0.10. Meanwhile, F_H/F_H0 and P/P_CAA0 can be improved significantly utilizing a higher friction coefficient of FPB.

It should be noted that higher friction coefficient means more earthquake energy will be transmitted from the ground to the superstructure, and severe vibration will be suffered. Therefore, both vibration reduction and progressive collapse restraining should be comprehensively considered in choosing the friction coefficient of FPB.

3.6.4 Axial force applied on FPB

It is evidence that both F_H/F_H0 and P_H/P_CAA0 can be improved effectively through higher axial force applied on FPB (Fig.9). It also shows that horizontal stiffness of FPB can be enhanced by higher axial force whether or not sliding occurs.

3.6.5 Depth ratio of the beam

The influence of the depth ratio of the beam on the horizontal and vertical restraining can be found in Fig.10. It can be concluded that a higher h_e can be obtained by increasing the depth ratio of the beam. Therefore, superior vertical restraining can be achieved notwithstanding the almost same horizontal restraining in CAA. In a later period of CA, horizontal restraining increased obviously, leading to markedly improved vertical restraining. For instance, when δ_V = 4h_e, F_H is 0.42F_H0 and 0.47F_H0 corresponding to L/H = 10 and 12, about 20% improvement comparing to L/H = 8. More remarkable improvement can be obtained for P_H, P_H is 3.23P_CAA0 and 4.19P_CAA0, when L/H is equal to 8 and 10, respectively, with 46% and 90% increase comparing to L/H = 8.

4 Assessment of progressive collapse resistance by flexural action

The contribution of flexural action to progressive collapse resistance of the beam-pier substructure can be simplified as a polygonal line model as shown in Fig.11. Point Y represents the case of a beam section at the first yield of longitudinal reinforcement. Point P represents the case of substructure at the maximum bearing capacity of the beam mechanism stage, and Point U represents the bearing capacity provided by the steel.

At Point Y of the polygonal line model [51,52],

(17)P_{\mathrm{F}}^{\mathrm{y}}= \frac{{4M}_{\mathrm{y}}}{L},

(18){\delta }_{\mathrm{v}}^{\mathrm{y}}= \frac{{\phi }_1{l}_1}{2}\left(L-\frac{l_1}{3}\right)+\frac{{\phi }_{\mathrm{y}}{l}_2^2}{6}.

where M_y is the moment on both sides of the middle column at first yield.

At Point P,

(19)P_{\mathrm{F}}^{\mathrm{p}}= \frac{2\left(M_{\mathrm{P}1}+M_{\mathrm{P}2}\right)}{L},

(20)\begin{array}{rl}{\delta }_{\mathrm{v}}^{\mathrm{p}}=& \frac{{\phi }_{\mathrm{y}1}{l}_1}{2}\left[L-\frac{l_1}{3}\right]+\frac{{\phi }_{\mathrm{y}2}{l}_2^2}{6}\\ & +\left[{\phi }_{\mathrm{p}1}-{\phi }_{\mathrm{y}1}\right]l_{11}\left[L_1-\frac{l_{11}}{2}\right]+\frac{\left({\phi }_{\mathrm{p}2}-{\phi }_{\mathrm{y}2}\right)l_{12}^2}{2},\end{array}

(21)M_{\mathrm{P}1} =f_{\mathrm{y}}{A}_{\mathrm{s}1}^{\prime }\left(H-{a^{\prime }}_{\mathrm{s}}\right)+ {\alpha }_1{f}_{\mathrm{c}}b x\left(h_0-\frac{x}{2}\right),

(22)M_{\mathrm{P}2} =M_{\mathrm{U}2}=f_{\mathrm{y}}{A}_{\mathrm{s}2}^{\prime }\left(H-2a_{\mathrm{s}}\right),

where M_y is the moment on both sides of the middle column at first yield, \phi_y represents the curvature of sections of the beam end sections at first yield moment, \phi_p the curvature of the beam end sections at maximum moment.

Regarding Point U, concrete of the beam end sections has been crushed, and P_{\mathrm{v}}^{\mathrm{u}} can be given as

(23)P_{\mathrm{F}}^{\mathrm{u}}= \frac{2\left(M_{\mathrm{U}1}+M_{\mathrm{U}2}\right)}{L},

(24)M_{\mathrm{U}1} =f_{\mathrm{y}}{A}_{\mathrm{s}1}^{\prime }\left(H-2a_{\mathrm{s}}\right),

(25)M_{\mathrm{U}2} =M_{\mathrm{P}2},

(26){\delta }_{\mathrm{v}}^{\mathrm{u}}= 10{\delta }_{\mathrm{v}}^{\mathrm{y}}.

Therefore, the complete analytical representation of the FPB isolated pier-column substructures can be obtained and is shown in Fig.12.

5 Verification of the proposed model

Since no collapse experiments on the FPB isolated structure have been conducted, FEM simulations were conducted to verify the accuracy of the proposed model. FEM studies on a one-storey substructure (shown as Fig.13(a)) in ABAQUS software, and the equivalent radius (R_E) and the friction coefficient (μ) of the FPB were 1 m and 0.05; the axial force applied on the column was 100.4 kN.

The FEM model is shown in Fig.13(b). The concrete was all modeled as a C3D8R solid element as an 8-node linear brick, with reduced integration and hourglass control element. The rebar was modeled using T3D2 element. The mesh size was set as 20 mm for concrete and rebar.

Comparison between the proposed model results and FEM results is shown in Fig.14 and Tab.1. Obvious characteristics of CAA and CA can be found. But there are still gaps between the FEM results and the proposed method results. On the one hand, the axial deformations were ignored in the proposed idea, but there are still deformations in the axial direction of the beam bridging the failed column in FEM. On the other hand, beam sections were considered rigid, but plastic deformation of concrete on the end of the beam will also dissipate part of energy in FEM. However, the maximum error of the four key parameters between the proposed model and FEM results did not exceed 8%. Consequently, the proposed method is capable of predicting behavior of FPB isolated beam-pier substructures during progressive collapse.

6 Discussion

The overall progressive collapse resistance of FPB isolated substructure can be easily obtained based on Eqs. (1), (5), and (17)–(26). Researchers and engineers can also adjust the parameters of the superstructure and FPBs based on the result to satisfy the requirements of different FPB isolated buildings.

The substructure was simplified for analysis; the beam section was considered rigid and axial deformation was ignored, therefore error may be caused. This is the main reason for the gap between the proposed model result and FEM result presented in Section 5. However, the proposed model can be adopted to evaluate the progressive collapse resistance of pier-beam substructure from structure isolated by FPB and help to expand understandings of the effects of the key factors of the FPB isolated structure, since the gap between the FEM results and the proposed method results are insignificant. Therefore, the proposed methodology can be used for most buildings. However, assessment of progressive collapse resistance provided by horizontal restraining is based on the assumption that the structure is symmetrical about the missing column. For an unsymmetrical structure, such as when the corner column is damaged, the resistance would be overestimated and this needs further research. Moreover, experiments should be conducted to give insight into the mechanics of progressive collapse of FPB isolated structures and the proposed method’s accuracy when applied to practical engineering.

The depth of beams above the FPB isolators is usually much larger than that of the beams in a typical story. Therefore, the contribution to progressive collapse resistance provided by flexural action of the beam can be enhanced compared with the substructure in a typical story, and enhancements by the larger depth of beams need to be further investigated. Furthermore, corresponding ultimate deformations for compressive arching action and catenary actions are usually set to be 0.06L and 0.2L (where L is the span of the beam) for progressive collapse investigations, respectively. The heights of the isolation layer are larger than 0.06L and 0.2L for most isolated buildings. For those buildings with not enough room for beams to develop large deformation, i.e., space to accommodate the FPB isolators is shorter than 0.2L, the isolator and piers would impact slab, and there will be more serious damage. Therefore, further studies are needed.

7 Conclusions

This paper presents a simplified model for static analysis of progressive collapse response of RC beam-pier substructures. The influences of the equivalent radius, friction coefficient, axial force of FPB, and span-depth ratio of the beam on the additional resistance were also investigated. Finally, simulations were conducted to verify the accuracy of the proposed model. The following conclusions can be drawn from the present study.

1) The progressive collapse resistance provided by horizontal restraining are reduced by FPB isolators significantly. And the progressive collapse resistance capacity of FPB isolated beam-column substructures was far less than those bare structures without FPB isolators.

2) The equivalent radius of FPB has almost no effect on the progressive collapse response of FPB isolated pier-column substructures, especially when the horizontal deformation of the beam is small in compressive arching action; it affects only the horizontal stiffness of the isolator when sliding occurs. But the effect of equivalent radius of FPB becomes more and more obvious with the development of the deflection in catenary action. The higher friction coefficient and larger axial force applied on FPB result in improvement of progressive collapse resistance, and superior vertical restraining can be achieved by increasing the depth ratio of the beam.

3) The accuracy of the proposed method was verified by FEM simulation, and the proposed approach is a simple but efficient method for addressing the mechanics of progressive collapse of FPB isolated beam-pier substructures in practical engineering.

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