Numerical Investigation of Progressive Collapse Resistance in Fully Bonded Prestressed Precast Concrete Spatial Frame Systems with and Without Precast Slabs

Abstract

Preventing progressive collapse induced by accidental events poses a critical challenge in the design and construction of resilient structures. While substantial progress has been made in planar structures, the progressive collapse mechanisms of precast concrete spatial structures—particularly regarding the effects of precast slabs—remain inadequately explored. This study develops a refined finite element modeling approach to investigate progressive collapse mechanisms in fully bonded prestressed precast concrete (FB-PPC) spatial frames, both with and without precast slabs. The modeling approach was validated against available test data from related sub-assemblies, and applied to assess the collapse performance. A series of pushdown analyses were conducted on the spatial frames under various column removal scenarios. The load–displacement curves, slab contribution, and failure modes under different conditions were compared and analyzed. A simplified energy-based dynamic assessment was additionally employed to offer a rapid estimation of the dynamic collapse capacity. The results show that when interior or side columns fail, the progressive collapse process can be divided into the beam action stage and the catenary action (CA) stage. During the beam action stage, the compressive membrane action (CMA) of the slabs and the compressive arch action (CAA) of the beams work in coordination. Additionally, the tensile membrane action (TMA) of the slabs strengthens the CA in the beams. When the corner columns fail, the collapse stages comprise the beam action stage followed by the collapse stage. Due to insufficient lateral restraints around the failed column, the development of CA is limited. The membrane action of the slabs cannot be fully mobilized. The contribution of the slabs is significant, as it can substantially enhance the vertical resistance and restrain the lateral displacement of the columns. The energy-based dynamic assessment further reveals that FB-PPC spatial frames exhibit high ductility and residual strength following sudden column removal, with dynamic load–displacement curves showing sustained plateaus or gentle slopes across all scenarios. The inclusion of precast slabs consistently enhances both the peak load capacity and the residual resistance in dynamic collapse curves.

1. Introduction

As cities evolve toward higher density and greater complexity, structural resilience has become a cornerstone of urban safety and sustainable development [1,2,3,4,5]. The capacity of buildings to withstand extreme or accidental events is central to the vulnerability and recovery capacity of the entire urban system. In practice, local member failures may rapidly propagate through structural systems, triggering disproportionate progressive collapse and inflicting catastrophic consequences on urban functionality, public safety, and social economy [6,7,8]. Thus, designing structures to resist progressive collapse is fundamental to enhancing not only the robustness of individual buildings, but also the resilience of cities in the face of disasters [9,10,11,12].

Over the past few decades, extensive research on reinforced concrete (RC) structures has clarified the principal mechanisms underlying progressive collapse, including various structural forms such as beam–column frames [13,14], beam–slab systems [15,16,17,18], and RC frames with infill walls [19,20,21]. Key progressive collapse resistance mechanisms—including bending action (BA), compressive arch action (CAA), and catenary action (CA)—have been distinctly identified, alongside the critical role of slabs and infill walls in providing alternative load paths. Based on these advances, two primary analytical approaches have been established for progressive collapse assessment: quasi-static [22,23,24] and dynamic analysis methods [25,26,27,28]. Quasi-static methods, such as pushdown tests under critical member removal conditions, are commonly adopted to study the load–displacement evolution, crack initiation and propagation, redistribution of internal forces, and failure modes of key joints. Dynamic methods, as supplementary methods, capture the time-dependent response of structures under abrupt events, including collision [29], blast [30,31], and earthquake [32]. These analytical frameworks have substantially deepened the understanding of structural vulnerability and collapse mechanisms across a wide range of structural systems subjected to various accidental scenarios. Motivated by these accumulated developments, governments and the engineering community have established codes and guidelines aimed at preventing the propagation of local failures and improving structural robustness. Modern progressive collapse resistant design integrates principles of redundancy, integrity, and alternative load path strategies, with notable frameworks developed by ACI, GSA, DoD, and CECS standards [33,34,35,36].

While these standards have significantly enhanced the safety and robustness of conventional structural systems, the rapid development and widespread application of precast concrete (PC) structures introduce new complexities that are not fully addressed [37,38,39,40,41,42]. PC structures are characterized by segmented assembly, diverse connection types, and the increasing adoption of prestressing techniques, all of which fundamentally alter load-transfer mechanisms and potential failure modes [43,44,45,46,47,48]. Consequently, strategies and provisions established for RC structures cannot be directly applied to PC structures, particularly prestressed concrete systems. In PC structures, unbonded prestressed strands are commonly used and can significantly enhance the load-bearing capacity during the CA stage. However, because the steel bars within the beam does not extend continuously and the strands are unbonded, the BA at the beam ends cannot be effectively developed [49,50]. In contrast, bonded prestressed strands can enhance the bending capacity through bond interaction, thereby increasing the peak resistance during the small-deformation stage. Additionally, the bonding force enables effective stress transfer, promoting uniform crack distribution.

In fact, for the entire building structure, the contribution of slab systems and spatial effects plays a critical role in collapse performance. Specifically, slabs can provide alternative load paths following local member failures, and the membrane action can significantly enhance the structural resistance mechanisms. Three-dimensional (3D) spatial structures more accurately capture the interaction between components and the multidirectional load redistribution, thereby offering a more realistic assessment of collapse resistance [51,52,53]. Simple substructure models fail to fully represent the overall failure modes and collapse mechanisms, and cannot adequately reflect its actual safety reserves. Therefore, investigating progressive collapse in full 3D spatial structures is essential.

However, progressive collapse tests are costly, making numerical simulation an efficient and economical alternative. Numerical models can be categorized into high- efficient macro-level models and high-fidelity finite element models (FEMs). Macro models typically discretize structural sections into fiber elements and implement features such as member removal and boundary constraints through coding. These models are capable of simulating the nonlinear response and resistance mechanisms following column removal at both local and global structural levels. Based on such models, Monte Carlo simulations can be performed using the building’s geometric properties, material characteristics, and statistical or probabilistic distributions of loads to predict collapse risk [54,55,56,57,58,59,60]. However, these models primarily capture the development of vertical resistance and cannot accurately reflect damage patterns or stress distributions. High-fidelity FEMs, in contrast, divide structural components into small solid elements for individual mechanical analysis. These models can finely simulate large deformations, contact behaviors, and accurately capture detailed local responses, including concrete cracking and crushing, as well as steel yielding and fracture [61,62]. When developing high-fidelity finite element models for progressive collapse, several key challenges must be addressed. The definition of boundary conditions becomes critical for realistically representing member removal and enabling the development of alternative load paths. In addition, modeling contact interactions, including friction between separating or colliding elements, adds further complexity. For fully bonded prestressed precast concrete (FB-PPC) spatial frames, particular challenges also include simulating the complex bond behavior between prestressed strands and concrete, accurately modeling force transfer at precast connections, and capturing the interaction between slabs and frames during member removal. These factors collectively exacerbate numerical convergence issues and increase computational demands, making it difficult to achieve accurate and reliable simulations.

In light of the above, this study focuses on FB-PPC spatial frames with and without precast slabs. Refined FEMs are established using ABAQUS/Explicit to investigate their progressive collapse behavior under quasi-static loading. The analysis emphasizes the influence of slabs on structural collapse mechanisms and systematically compares mechanical responses across different member removal scenarios. These results reveal the progressive collapse mechanisms and failure modes of FB-PPC spatial frames with precast slabs. Subsequently, the progressive collapse performance of FB-PPC spatial frames is evaluated under dynamic effects using an energy-based assessment method. The findings provide important theoretical guidance for the structural design and optimization of FB-PPC systems, support the development of more robust and resilient engineering solutions, and offer a scientific basis for updating relevant design codes and improving the safety and sustainability of precast concrete structures in practice.

2. Finite Element Modeling Method

2.1. Prototype Structure

The prototype structure is a five-story FB-PPC building. In this study, a spatial substructure is extracted from its lower portion for detailed analysis, as shown in Figure 1a. Each story has a height of 1600 mm, with transverse and longitudinal spans of 3300 mm and 3250 mm, respectively.

The beam–column joint configuration is illustrated in Figure 1b. Reserved ducts are embedded between beams and columns. After assembly of the precast components, post-tensioned steel strands are threaded through the ducts. Epoxy mortar was applied at the beam–column interfaces to prevent slurry leakage. After the steel strands are tensioned to 0.75f_ptk [63], grout is poured into the ducts to form the bonded connection. The overall reinforcement layout and cross-sectional reinforcement of the beams are shown in Figure 1c–g. Precast slabs are used, with a slab thickness of 60 mm and a protective concrete cover of 15 mm. The slab is reinforced with single-layer, bidirectional D8@150 bars. Connection between slabs and beams is achieved by welding embedded parts to steel plates, while adjacent slabs are connected using two L-shaped anchor bars [64]. The reinforcement details of the slab are illustrated in Figure 2. Material properties are listed in

Table 1. Concrete and steel material properties.

Materials	Material Property	Test Value	Design Code Value [63]
Reinforcement	Yield strength (MPa)	457.1	400
	Ultimate strength (MPa)	625.9	540
	Elastic modulus (MPa)	2.0 × 105	2.0 × 105
Prestressed strand	Ultimate strength (MPa)	1860	1860
	Elastic modulus (MPa)	1.95 × 105	1.95 × 105
Concrete	Compressive strength (MPa)	37.1	40
	Elastic modulus (MPa)	3.19 × 104	3.25 × 104

.

2.2. Column Removal Scenario

Three column removal scenarios are considered in this paper: removal of the bottom-story interior column (Column E), removal of the bottom-story side column (Column B), and removal of the bottom-story corner column (Column C). For each scenario, the comparative analyses are conducted between bare frame models and slab-frame models. For the interior column removal scenario, the models are named IC (Interior Column) and IC-S (Interior Column–Slab), based on whether or not the slab is present. Similarly, for the side column removal scenario, the models are SC (Side Column) and SC-S (Side Column–Slab); for the corner column removal scenario, the models are CC (Corner Column) and CC-S (Corner Column–Slab). The column numbering and removal scenarios are shown in Figure 1a.

2.3. Material Constitutive

2.3.1. Concrete

The density of concrete is taken as 2400 kg/m³, and Poisson’s ratio is taken as 0.2. The stress–strain relationship of concrete followed the Code for Design of Concrete Structures [63], as shown in Figure 3a.

The constitutive expression for concrete in compression is as follows:

$${\sigma }_{\mathrm{c}}=\left(1-d_{\mathrm{c}}\right)E_{\mathrm{c}}{\epsilon }_{\mathrm{c}}$$

(1)

$$d_{\mathrm{c}}=\left\{\begin{array}{c}1-{\displaystyle \frac{{\rho }_{\mathrm{c}}n}{n-1+x^n}}\hspace{1em}x\le 1\\ 1-{\displaystyle \frac{{\rho }_{\mathrm{c}}}{{\alpha }_{\mathrm{c}}{(x-1)}^2+x}}\hspace{1em}x>1\end{array}\right.$$

(2)

$${\rho }_{\mathrm{c}}={\displaystyle \frac{f_{\mathrm{c},\mathrm{r}}}{E_{\mathrm{c}}{\epsilon }_{\mathrm{c},\mathrm{r}}}}$$

(3)

$$n={\displaystyle \frac{E_{\mathrm{c}}{\epsilon }_{\mathrm{c},\mathrm{r}}}{E_{\mathrm{c}}{\epsilon }_{\mathrm{c},\mathrm{r}}-f_{\mathrm{c},\mathrm{r}}}}$$

(4)

$$x={\displaystyle \frac{\epsilon }{{\epsilon }_{\mathrm{c},\mathrm{r}}}}$$

(5)

$${\alpha }_{\mathrm{c}}=0.157f_{\mathrm{c},\mathrm{r}}^{0.785}-0.905$$

(6)

$${\epsilon }_{\mathrm{c},\mathrm{r}}=\left(700+172\sqrt{f_{\mathrm{c},\mathrm{r}}}\right)\times {10}^{-6}$$

(7)

where ${\sigma }_{\mathrm{c}}$ is the compressive stress; ${\epsilon }_{\mathrm{c}}$ is the compressive strain; ${\alpha }_{\mathrm{c}}$ is the descending branch parameter of the uniaxial compressive stress–strain curve; $d_{\mathrm{c}}$ is the damage evolution parameter for concrete under compression; $f_{\mathrm{c},\mathrm{r}}$ is the compressive strength of concrete; and ${\epsilon }_{\mathrm{c},\mathrm{r}}$ is the peak compressive strain of concrete.

The constitutive expression for concrete in tension is as follows:

$${\sigma }_{\mathrm{t}}=\left(1-d_{\mathrm{t}}\right)E_{\mathrm{c}}{\epsilon }_{\mathrm{t}}$$

(8)

$$d_{\mathrm{t}}=\left\{\begin{array}{cc}1-{\rho }_{\mathrm{t}}(1.2-0.2x^5)& x\le 1\\ 1-{\displaystyle \frac{{\rho }_{\mathrm{t}}}{{\alpha }_{\mathrm{t}}{(x-1)}^{1.7}+x}}& x>1\end{array}\right.$$

(9)

$$x={\displaystyle \frac{\epsilon }{{\epsilon }_{\mathrm{t},\mathrm{r}}}}$$

(10)

$${\rho }_{\mathrm{t}}={\displaystyle \frac{f_{\mathrm{t},\mathrm{r}}}{E_{\mathrm{c}}{\epsilon }_{\mathrm{t},\mathrm{r}}}}$$

(11)

$${\epsilon }_{\mathrm{t},\mathrm{r}}=f_{\mathrm{t},\mathrm{r}}^{0.54}\times 65\times {10}^{-6}$$

(12)

$${\alpha }_{\mathrm{t}}=0.312f_{\mathrm{t},\mathrm{r}}^2$$

(13)

where ${\sigma }_{\mathrm{t}}$ is the tensile stress; ${\epsilon }_{\mathrm{t}}$ is the tensile strain; α_t is the descending branch parameter of the uniaxial tensile stress–strain curve; $d_{\mathrm{t}}$ is the damage evolution parameter for concrete under tension; $f_{\mathrm{t},\mathrm{r}}$ is the tensile strength of concrete; and ${\epsilon }_{\mathrm{t},\mathrm{r}}$ is the peak tensile strain of concrete.

The concrete damaged plasticity (CDP) model is employed to simulate the nonlinear behavior of concrete. This model comprehensively accounts for both plastic deformation and damage mechanisms by introducing distinct damage factors in tension and compression, which represent cracking and crushing, respectively [65]. As illustrated in Figure 3b, the stress–strain response under uniaxial compression is linear before reaching the initial yield stress σ_c0. Beyond this point, the concrete enters the plastic stage. Upon reaching the peak compressive stress ${\sigma }_{\mathrm{cu}}$ in the plastic region, the curve descends, marking the onset of the softening stage. Under uniaxial tension, the stress–strain response is linearly elastic before reaching the tensile failure stress ${\sigma }_{\mathrm{t}0}$, as shown in Figure 3c. After this stage, the maximum compressive stress is reached in the plastic zone, and the concrete exhibits stress hardening, followed by strain softening.

The definitions and values of the variables are as follows:

$${\epsilon }_{\mathrm{c}}^{\mathrm{in}}={\epsilon }_{\mathrm{c}}-{\epsilon }_{0\mathrm{c}}^{\mathrm{el}}$$

(14)

$${\epsilon }_{0\mathrm{c}}^{\mathrm{el}}={\displaystyle \frac{{\sigma }_{\mathrm{c}}}{E_0}}$$

(15)

$${\epsilon }_{\mathrm{c}}^{\mathrm{pl}}={\epsilon }_{\mathrm{c}}^{\mathrm{in}}-{\displaystyle \frac{d_{\mathrm{c}}}{(1-d_{\mathrm{c}})}}{\displaystyle \frac{{\sigma }_{\mathrm{c}}}{E_0}}$$

(16)

where ${\epsilon }_{\mathrm{c}}^{\mathrm{in}}$ is the inelastic compressive strain; ${\epsilon }_{0\mathrm{c}}^{\mathrm{el}}$ is the elastic compressive strain corresponding to the initial elastic modulus; ${\epsilon }_{\mathrm{c}}^{\mathrm{pl}}$ is the plastic compressive strain; $d_{\mathrm{c}}$ is the compressive damage factor of concrete; ${\sigma }_{\mathrm{c}}$ is the compressive stress; and ${\epsilon }_{\mathrm{c}}$ is the compressive strain.

$${\epsilon }_{\mathrm{t}}^{\mathrm{ck}}={\epsilon }_{\mathrm{t}}-{\epsilon }_{0\mathrm{t}}^{\mathrm{el}}$$

(17)

$${\epsilon }_{0\mathrm{t}}^{\mathrm{el}}={\displaystyle \frac{{\sigma }_{\mathrm{t}}}{E_0}}$$

(18)

$${\epsilon }_{\mathrm{t}}^{\mathrm{pl}}={\epsilon }_{\mathrm{t}}^{\mathrm{ck}}-{\displaystyle \frac{d_{\mathrm{t}}}{(1-d_{\mathrm{t}})}}{\displaystyle \frac{{\sigma }_{\mathrm{t}}}{E_0}}$$

(19)

where ${\epsilon }_{\mathrm{t}}^{\mathrm{ck}}$ is the tensile cracking strain; ${\epsilon }_{0\mathrm{t}}^{\mathrm{el}}$ is the elastic tensile strain corresponding to the initial elastic modulus; ${\epsilon }_{\mathrm{t}}^{\mathrm{pl}}$ is the plastic tensile strain; $d_{\mathrm{t}}$ is the tensile damage factor of concrete; ${\sigma }_{\mathrm{t}}$ is the tensile stress; and ${\epsilon }_{\mathrm{t}}$ is the tensile strain.

2.3.2. Reinforcement

The steel bars and prestressed strands are modeled using the linear hardening constitutive model, and the stress–strain curve is shown in Figure 4. Due to the quasi-static loading, the effect of material strain rate is not considered in this simulation [66].

The reinforcement stress–strain curve expression is as follows:

$${\sigma }_{\mathrm{s}}=\left\{\begin{array}{l}{E}_{\mathrm{s}}{\epsilon }_{\mathrm{s}}\hspace{1em}0\le {\epsilon }_{\mathrm{s}}\le {\epsilon }_{\mathrm{y}}\hfill \\ f_{\mathrm{y}}+{\displaystyle \frac{f_{\mathrm{u}}-f_{\mathrm{y}}}{{\epsilon }_{\mathrm{u}}-{\epsilon }_{\mathrm{s}}}}({\epsilon }_{\mathrm{s}}-{\epsilon }_{\mathrm{y}})\hspace{1em}{\epsilon }_{\mathrm{y}}<{\epsilon }_{\mathrm{s}}<{\epsilon }_{\mathrm{u}}\hfill \end{array}\right.$$

(20)

where ${\sigma }_{\mathrm{s}}$ is the stress; ${\epsilon }_{\mathrm{s}}$ is the strain; $E_{\mathrm{s}}$ is the elastic modulus; $f_{\mathrm{y}}$ is the yield stress; ${\epsilon }_{\mathrm{y}}$ is the yield strain; $f_{\mathrm{u}}$ is the ultimate stress; and ${\epsilon }_{\mathrm{u}}$ is the ultimate strain.

2.4. Prestress Application and Coupling Treatment

2.4.1. Prestress Application Method

Prestress application of the steel strands is applied using the cooling method. In the initial analysis step, the initial temperature of the strands is defined as 0 °C using a predefined field. In the subsequent analysis step, a temperature drop is imposed on the strands to induce axial contraction, thereby generating the intended prestress.

The temperature reduction value $\mathrm{\Delta }T$ is calculated as follows:

$$\mathrm{\Delta }T=-{\displaystyle \frac{F}{\alpha E_{\mathrm{p}}A}}=-{\displaystyle \frac{{\sigma }_{\mathrm{p}}}{\alpha E_{\mathrm{p}}}}$$

(21)

where $F$ is the pre-tension force; ${\sigma }_{\mathrm{p}}$ is the prestress value; A is the cross-sectional area; $\alpha$ is the linear expansion coefficient, set to 1 × 10⁻⁵/°C; and $E_{\mathrm{p}}$ is the elastic modulus of steel strands.

2.4.2. Bond–Slip Behavior

The bond–slip behavior of reinforcement is explicitly considered in the model. For steel bars, bond–slip segments are defined in the plastic hinge regions near beam–column joints [67], while other segments are embedded directly within the concrete. Bond–slip behavior is incorporated along the entire length of strands. One-to-one corresponding connector elements are established between the reinforcement and surrounding concrete elements, thereby accurately simulating the bond–slip interaction [68].

The bond–slip behavior between steel rebars and concrete is considered in the FEM analysis. The bond stress–slip constitutive relationship between concrete and steel rebars is shown in Figure 5 [63]. The relationship is expressed as follows:

$$\tau =k_{1}s 0\le s\le s_{\mathrm{cr}}$$

(22)

$$\tau ={\tau }_{\mathrm{cr}}+k_2(s-s_{\mathrm{cr}}) s_{\mathrm{cr}}<s\le s_{\mathrm{cu}}$$

(23)

$$\tau ={\tau }_{\mathrm{u}}+k_3(s-s_{\mathrm{u}})$$

(24)

$$\tau ={\tau }_{\mathrm{r}} s>s_{\mathrm{r}}$$

(25)

$$\tau ={\tau }_{\mathrm{un}}+k_1(s-s_{\mathrm{un}})$$

(26)

where $\tau$ is the bond stress; $s$ is the relative slip; $k_1$ is the slope of the linear stage, ${\tau }_{\mathrm{cr}}/{\mathrm{s}}_{\mathrm{cr}}$; $k_2$ is the slope of the crack stage, $\left({\tau }_{\mathrm{u}}-{\tau }_{\mathrm{cr}}\right)/\left({\mathrm{s}}_{\mathrm{u}}-{\mathrm{s}}_{\mathrm{cr}}\right)$; $k_3$ is the slope of the descending stage, $\left({\tau }_{\mathrm{r}}-{\tau }_{\mathrm{u}}\right)/\left({\mathrm{s}}_{\mathrm{r}}-{\mathrm{s}}_{\mathrm{u}}\right)$; ${\tau }_{\mathrm{un}}$ is the bond stress at the unloading point; and $s_{\mathrm{un}}$ is the relative slip at the unloading point.

For prestressed strands, the bond stress–slip constitutive model proposed by Dang [69] is adopted, and the corresponding relationship curve is shown in Figure 6. The expression is as follows:

$$\tau ={\tau }_{0.25}({\displaystyle \frac{s}{s_{0.25}}}) 0\le s\le s_{0.25}$$

(27)

$$\tau ={\tau }_{2.5}{({\displaystyle \frac{s}{s_{2.5}}})}^{\alpha } s>s_{2.5}$$

(28)

$$\alpha ={\displaystyle \frac{\ln ({\tau }_0/{\tau }_{2.5})}{\ln (s_0/s_{2.5})}}$$

(29)

where $\tau$ is the bond stress; $s$ is the relative slip; $s_{0.25}$ indicates a slip of 0.25 mm; ${\tau }_{0.25}$ represents the bond stress corresponding to $s_{0.25}$; $s_{2.5}$ indicates a slip of 2.5 mm; and ${\tau }_{2.5}$ represents the bond stress corresponding to $s_{2.5}$.

2.5. Boundary Conditions and Loading Method

The FEM of the spatial frame is shown in Figure 7. In the FEM, the contact behavior between two surfaces is defined using the surface-to-surface method. This is because, compared to node-to-surface contact, surface-to-surface contact more accurately represents the contact geometry. This approach provides a more realistic distribution of contact pressure and is less sensitive to mesh quality and density, resulting in enhanced numerical stability. It is particularly well-suited for simulating large deformations, significant sliding, and complex nonlinear contact behavior. Concrete components are modeled using C3D8R solid elements, while both steel bars and prestressed strands are represented by T3D2 truss elements.

In this study, the joint interface between the beam ends and the column corbels is modeled using the hard contact and friction approach. This method allows the contact surfaces to open or close in the normal direction under loading. In the tangential direction, limited sliding is permitted, with the resistance to sliding governed by a defined coefficient of friction. This contact definition realistically simulates stress transfer and joint motion, capturing sliding and separation under extreme loads. The epoxy mortar layer is omitted in the model due to its minimal thickness, low stiffness, and negligible impact on joint behavior.

As shown in Figure 7a, a vertical axial load corresponding to an axial compression ratio of 0.3 is applied at the top of the undamaged columns, while a vertical displacement load is imposed at the top of the failed column. All translational and rotational degrees of freedom at the foundation are constrained. As illustrated in Figure 7b, the precast slabs are connected to the beams using steel plates and embedded parts, which are modeled using tie constraints. In Figure 7c, adjacent precast slabs are connected through two L-shaped anchorage bars, and their interaction is also defined via tie constraints. When the vertical displacement of the removed column is less than 30 mm, a displacement increment of 2 mm per loading step is used. For displacements between 30 mm and 100 mm, the increment is increased to 5 mm, and for displacements exceeding 100 mm, the increment is adjusted to 10 mm per step [70]. To ensure quasi-static loading and avoid inertial effects due to excessive loading rates, the loading duration is extended, and the ratio of kinetic energy to internal energy is maintained below 10%. In this study, the failure criterion is defined as the point at which the stress in the prestressed strands reached their ultimate tensile strength.

3. FEM Validation

Conducting full-scale progressive collapse tests on spatial frames with slabs is extremely complex and prohibitively expensive. Moreover, relatively small-scaled models may fail to accurately represent the actual system behavior due to size effects. In contrast, substructure tests have been conducted on individual beam–column frames and precast slabs. To decouple the complexity of the validation process, it is feasible to conduct (1) the validation of planar frame model and (2) the validation of precast slab model. Once each model is capable of reproducing its respective test responses—such as peak strength, stiffness degradation, and failure modes—confidence in the simulation of the slab-frame system can be significantly strengthened. This validation strategy ensures both practical feasibility and sufficient fidelity to capture the critical collapse mechanisms of the composite system [67,71].

3.1. Validation of Planar Frame

3.1.1. Overview of the FEM

As illustrated in Figure 8a, the failure of an interior column in the spatial frame can be decoupled into two planar failure scenarios: failure of a middle column in the X-direction and in the Y-direction. Similarly, as shown in Figure 8b, the failure of a side column is represented by a combination of middle column failure in the X-direction and side column failure in the Y-direction. In Figure 8c, corner column failure was decoupled into side column failures in both the X- and Y-directions. Here, the X-direction refers to the transverse span (3300 mm), while the Y-direction refers to the longitudinal span (3250 mm).

Progressive collapse tests were previously carried out on 1/2 scaled two-story two-span planar frames, in which either the middle column or the side column was removed, to investigate the general collapse behavior of planar frames. Based on these two test studies [72,73], FEMs are developed in this section for comparative validation. The reinforcement details, modeling techniques, material properties, and constitutive models used for the beam–column subassemblies are consistent with those described in Section 2.1. The loading protocol and failure criterion follow Section 2.5. To ensure stable application of the load, the out-of-plane translational and rotational degrees of freedom of the columns are constrained, and the foundation is modeled as fully fixed. As shown in Figure 9, the reinforcement of frame is shown in see Figure 9a; the model with the middle column removed is referred to as PCM (see Figure 9b), and the model with the side column removed is referred to as PCS (see Figure 9c).

3.1.2. Mesh Generation and Convergence Analysis

In this section, the PCM model is used as a representative case to assess the effect of mesh size on the convergence analysis. Four models with different mesh sizes were created, and the comparison of different meshes is shown in

Table 2. Mesh size convergence analysis (mm).

Type	Mesh1	Mesh2	Mesh3	Mesh4
Beam and column	50 × 50 × 50	50 × 50 × 50	25 × 25 × 25	15 × 15 × 15
Corbel	50 × 50 × 50	25 × 25 × 25	25 × 25 × 25	15 × 15 × 15
Steel rebars	50	30	30	15
Prestressed strands	50	15	15	15
Total number of elements	20,830	36,966	99,888	444,610
Increment step	205,779	541,742	544,194	893,940

and Figure 10.

It is evident that Mesh 1, with a relatively coarse mesh, exhibits poor convergence performance. Meshes 2, 3, and 4 demonstrate good convergence during the beam action stage, but significant discrepancies occur in the CA stage. This is due to severe mesh distortion in the large deformation stage, which leads to instability in the analysis. A comparison between Mesh 3 and Mesh 4 reveals that finer meshes do not necessarily yield more accurate results but increase computational time. Therefore, Mesh 3 is chosen for the subsequent analysis.

3.1.3. Validation of Prestress Application Method

To ensure the accuracy of the prestress application method, the PCM model was used as an example to conduct analyses using both the cooling method and the initial stress method, and the simulation results were compared. Considering that prestress losses occur during both construction and service stages, the total prestress loss was determined by summing the contributions of various factors [63]. The primary types of prestress loss considered in this study include (1) losses due to anchorage slip and strands shortening σ_l1; (2) losses caused by stress relaxation in the prestressed strands σ_l2; and (3) losses resulting from concrete shrinkage and creep σ_l3.

The total prestress loss is calculated by summing the individual loss components as follows:

$${\sigma }_{l1}={\displaystyle \frac{a}{l}}{E}_{\mathrm{p}}$$

(30)

$${\sigma }_{l2}=0.2\left({\displaystyle \frac{{\sigma }_{\mathrm{con}}}{f_{\mathrm{ptk}}}}-0.575\right){\sigma }_{\mathrm{con}}$$

(31)

$${\sigma }_{l3}={\displaystyle \frac{55+300\frac{{\sigma }_{\mathrm{pc}}}{f_{\mathrm{cu}}}}{1+15\rho }}$$

(32)

where $a$ is the anchorage slip length, taken as 1 mm; $l$ is the length from the tensioning end to the anchorage end, taken as 3900 mm; ${\sigma }_{\mathrm{con}}$ is the prestressed control stress; $f_{\mathrm{ptk}}$ is the ultimate tensile strength of the prestressed strands; ${\sigma }_{\mathrm{pc}}$ is the normal compressive stress in the concrete at the centroid of the prestressed strands; $f_{\mathrm{cu}}$ is the cube compressive strength of concrete; and $\rho$ is the reinforcement ratio, including both prestressed strands and steel bars.

The total prestress loss amounted to 149 MPa, which was deducted in advance from the initial stress value of the prestressed strands. Figure 11 shows the initial stress states of the prestressed strands and the corresponding load–displacement curves obtained by the two methods. It can be seen that the prestress distribution is relatively uniform and close to the test value. The load–displacement curves obtained from both methods show similar trends, but the curve from the cooling method aligns more closely with the test results. This is because the initial stress method cannot accurately predict the actual structural response during service, whereas the cooling method simulates the tensioning process through the thermal expansion and contraction of steel strands.

3.1.4. Results of Validation

Figure 12 presents the comparison of failure modes between test results and numerical simulations under the condition of middle column failure. During the beam action stage, the resistance primarily relied on the BA at beam ends, resulting in concentrated concrete damage at the beam–column joints, while the concrete in the mid-span of the beams remained largely intact. Upon completion of loading, separation occurred between beams and columns accompanied by severe concrete damage. Horizontal cracks appeared in the concrete at the middle column joints on the second floor due to slip of the prestressed strands. The degree of concrete damage at middle joints was more severe than at side joints. Overall, the simulation results are in good agreement with the test observations.

Figure 13 shows the load–displacement curves from both the test and FEM. The simulated curve exhibited a slightly higher vertical resistance than the test results during the beam action stage. The peak loads from the test and FEM were 124.99 kN and 132.18 kN, respectively, with the FEM result 5.75% higher. The corresponding displacement values at peak load were 80 mm and 83.71 mm, respectively. The vertical resistance began to decline, due to the gradual crushing of concrete at beam ends. As the displacement increased, the component transitioned into the CA stage. The transition displacements for the test and FEM were 260 mm and 280.41 mm, respectively. As the CA developed, the prestressed strands were continuously stretched, entering the hardening phase, which led to a continuous increase in structural capacity. The peak catenary loads from the test and FEM were 101.34 kN and 104.77 kN, with corresponding displacements of 500 mm and 508.31 mm. The test curve dropped sharply at the end due to the prestressed strands reaching their ultimate tensile strength.

Figure 14 compares the failure modes observed in the test and numerical simulation under the side column removal scenario. The analysis indicated that the concrete damage was mainly concentrated at the beam ends in the removed span, with the most severe failure occurring in the concrete compression zones of the corbel and beam end. The rest of the structure remained largely intact. After loading, separation between the beam and column was observed, accompanied by significant concrete crushing. The comparison indicates that the simulated failure mode is generally consistent with the test observations.

Figure 15 shows the load–displacement curves from the test and FEM under side-column failure. The peak loads in the beam action stage were 60.97 kN for the test and 63.66 kN for the FEM, with the FEM value being 4.41% higher. The corresponding displacements at peak load were 140 mm and 110.42 mm, respectively. In this stage, the FEM and test curves exhibited similar trends and showed good agreement. Subsequently, both curves performed a stepped and sharp drop, which was caused by the successive crushing and spalling of the concrete at the beam ends. Moreover, the absence of effective lateral constraint around the failed column prevented the structure from developing CA. Ultimately, the test and simulation completely lost their bearing capacity at displacements of 440 mm and 498.56 mm, respectively.

3.1.5. Error Analysis

As shown in Figure 13 and Figure 15, the initial stiffness and ultimate load capacity in the FEM were slightly higher than those observed in the test. This discrepancy was primarily due to the FEM not accounting for initial concrete damage, such as micro-cracks and construction defects. Moreover, simulations allowed the application of ideal structural constraints, which were difficult to achieve in test settings. For the PCM model, the FEM transition displacement was slightly larger than that observed in the test. This is attributed to construction imperfections and other defects in the test, which prevent the full development of the beam-end resistant capacity. During the large deformation stage, the agreement between PCS and the test results was poorer compared to PCM. It lies in the weaker boundary conditions of the failed column in PCS. Specifically, after the failure of the side column, the structural system experienced a reduction in constraint forces, leading to a stepwise drop in resistance. This sudden resistance loss is highly random and uncertain, making it difficult to predict and control accurately in the simulation. However, within each step segment, the simulated and test load responses still showed good agreement. Overall, the FEM results agree well with the test data, with discrepancies remaining within an acceptable range.

3.2. Validation of Precast Slab

3.2.1. Boundary Conditions and Loading Method

To ensure the rationality and accuracy of slab modeling in the spatial frame, this section verifies the model by referring to the vertical static loading test of a fully precast concrete slab system presented in reference [64]. The cross-sectional dimensions, reinforcement details, and material properties of the components are consistent with those in the original literature.

The FEM of the slab system is shown in Figure 16a. The constitutive models for concrete and steel follow those described in Section 2.3 of this paper. Based on mesh convergence analysis, the element sizes for concrete and reinforcement were set to 50 mm and 20 mm, respectively. The precast slabs were connected to beams, and beams to columns, via welded steel plates and embedded parts; precast slabs were connected to each other through two L-shaped anchor bars. The contact modeling approach is consistent with that in Section 2.5. The base of the foundation was modeled as fully fixed, and a distributed surface load was applied to the slab surface in accordance with the test loading method. The layout of measurement points is shown in Figure 16b, and the loading method is summarized in

Table 3. Load steps.

Steps	Current Load (kN·m−2)	Total Load (kN·m−2)
1	0.94	0.94
⋯	⋯	⋯
4	0.94	3.76
5	0.47	4.23
⋯	⋯	⋯
17	0.47	9.87
18	0.17	10.04
19	0.34	10.38
20	0.34	10.72
21	1.33	12.05
22	2.35	14.55

.

3.2.2. Results of Validation

As shown in Figure 17a, the maximum deflection occurred at the slab center and gradually decreased toward the edges, forming a displacement contour that is symmetrically distributed about the center. This indicates that the PC slab exhibited deformation characteristics similar to the two-way slab. Figure 17b, c show that the FEM and test load–displacement curves agree well. The final deflection errors between simulation and test at measurement points 1 to 5 were 8.38%, 4.8%, 8.96%, 6.72%, and 5.28%, respectively. These values fall within an acceptable range, validating the slab modeling method for subsequent model development.

4. Progressive Collapse Resistance Analysis of Spatial Frame

4.1. Load–Displacement Curve Comparison

The load–displacement curves are shown in Figure 18, and the key characteristics of curves are listed in

Table 4. Comparison of simulation results of column failure.

	P1 (kN)	∆1 (mm)	Transition Displacement (mm)	P2 (kN)	∆2 (mm)	θ (rad)
IC	402.17	98.22	355.95	465.20	647.24	0.243
IC–S	629.91	90.81	542.29	702.26	613.81	0.259
SC	300.32	109.05	95.24	329.65	582.97	0.232
SC–S	462.24	86.93	279.75	525.91	580.85	0.243
CC	196.34	75.04	/	/	/	0.205
CC–S	268.74	57.76	/	/	/	0.214

Note: P₁ represents the peak vertical resistance in the beam action stage, ∆₁ is the corresponding displacement of the failed column at P₁; P₂ represents the peak vertical resistance in the CA stage, ∆₂ is the corresponding displacement of the failed column at P₂; transition displacement refers to the displacement of the failed column at the point where the structure begins to shift from the beam action stage to the CA stage; θ is the maximum beam rotation.

. The analysis indicates that the load–displacement curves exhibit similar trends under both interior and side column removal scenarios. The progressive collapse process of the structure can be divided into the beam action stage and the CA stage. The incorporation of the slabs significantly increases the collapse resistance capacity of the frame.

For interior column failure, the peak load during the beam action stage was 402.17 kN for IC and 629.91 kN for IC–S, indicating a 56.63% increase due to the slab. In the CA stage, the peak loads were 465.20 kN and 702.26 kN for IC and IC–S, respectively, corresponding to a 50.96% improvement. In the case of side column failure, the peak load in the beam action stage was 300.32 kN for SC and 461.92 kN for SC–S, an increase of 53.92%. In the CA stage, SC reached 329.65 kN, while SC–S reached 525.91 kN, reflecting a 59.54% improvement. It can be observed that, with slab participation, the CMA of the slab and the CAA of the beam acted compatibly to enhance resistance during the beam action stage. Additionally, the TMA of the slab further strengthened the CA.

For corner column failure, both CC and CC–S exhibited a gradual decline in vertical resistance after reaching their peak loads, with no upward trend. This indicates that neither the CA nor the slab membrane action is fully developed. Therefore, the progressive collapse stage can be divided into the beam action stage and the collapse stage. Nevertheless, the slabs can increase the ultimate vertical resistance and ductility of the frames. During the beam action stage, the peak load increased from 196.34 kN (CC) to 268.74 kN (CC–S), representing a 36.87% improvement.

Under the same column removal scenario, comparisons of the failed column displacement (∆₁) and the transition displacement indicated that the slab allowed the structure to reach the beam action peak load earlier and prolongs the beam action stage. In different column removal scenarios, the initial stiffness and peak load capacity of the structure decreased in the following order: interior column removal, side column removal, and corner column removal, which was attributed to differences in the surrounding constraint stiffness of the failed columns.

It is worth noting that according to the DoD [35] guidelines, the rotation angle at the beam ends should exceed 0.2 rad when structural failure occurs. As shown in Table 4, under all three column removal scenarios, the maximum beam-end rotations exceed the specified value, indicating that the structure has good resistance to progressive collapse. In addition, the presence of slabs enhances the deformation capacity of the beams.

4.2. Resistance Contribution

For slab-frame structures, the total vertical resistance can be decomposed into the resistance provided by the bare frame and the additional resistance contributed by the slab. Let the total resistance be denoted as P_t, the resistance from the frame alone as P_f, and the enhanced resistance due to the slab as P_s. Based on this, a calculation formula can be introduced to quantify the slab’s contribution to the structural resistance enhancement.

The calculation formula for the contribution rate of the slab is as follows:

$$\eta ={\displaystyle \frac{P_{\mathrm{s}}}{P_{\mathrm{t}}}}={\displaystyle \frac{P_{\mathrm{t}}-P_{\mathrm{f}}}{P_{\mathrm{t}}}}$$

(33)

The calculated contribution of the slab to the structural resistance is illustrated in Figure 19. As shown in Figure 19a, under the interior column removal scenario, the slab’s contribution gradually increased during the initial loading phase. This is attributed to the CMA of the slab, which enhanced the CAA of the beam. Subsequently, as the beam developed a CA, the contribution of slab decreased slightly. Toward the final stage, the contribution rose again due to the development of TMA. For the side column removal scenario (Figure 19b), the slab contribution initially decreased. This is because the surrounding restraint is weaker compared to the interior column scenario, indicating the CMA was not fully mobilized. However, in the later stages of loading, the contribution trend became similar to that observed in the interior column case. When the corner column was removed (Figure 19c), the beam cannot develop a CA under large deformation, and collapse resistance mainly relied on the slab action. The average contributions of the slab to the total resistance under interior side, and corner column removal scenarios were 34.10%, 37.88%, and 42.33%, respectively.

4.3. Horizontal Support Reaction Forces at the Top of Columns Adjacent to the Failed Columns

The horizontal support reaction forces curves reflect the changes in axial forces within the beams throughout the loading stage. For interior column failure (models IC and IC–S), the measured column was Column D, with the horizontal reaction force directed along DE. For side column failure (models SC and SC–S), Column A was monitored, with the horizontal reaction in the AB direction. For corner column failure (models CC and CC–S), Column B was analyzed, with the horizontal reaction force acting along the BC direction. Positive values represent tension, and negative values indicate compression (column numbering is shown in Figure 1a).

As illustrated in Figure 20a–d, under the interior and side column failure scenarios (models IC, IC-S, SC, and SC-S), the column top reaction force performed a transition from compression to tension. This transition corresponded to the structural mechanism from the CAA to the CA. In the CAA stage, the horizontal reaction force at the first-story column top changed more rapidly than at second story, primarily due to the Vierendeel action. This phenomenon arises from the redistribution of internal forces in the remaining structure following column removal, which helps maintain stability and resistant capacity, resulting in differential force responses between stories [74]. During the CA stage, the variation in the horizontal reaction force at the first-story column top became less pronounced, but the Vierendeel action still persisted. Compared with the bare frame, the slab-frame showed a higher peak horizontal reaction force at the column top.

As depicted in Figure 20e,f, the column tops of model CC, at both the first and second stories remained fully in compression. It indicates that the structure resists collapse primarily through the BA and the CAA, without the formation of the CA. In model CC-S, the presence of the slab enhances the CAA resistance, but the CA still fails to form effectively.

4.4. Lateral Displacement at the Top of Columns Adjacent to the Failed Columns

The lateral displacement curves of the column top adjacent to the failed column are presented in Figure 21. Positive values correspond to inward displacement of the column top, while negative values indicate outward displacement. The studied columns are the same as Section 4.3 in each failure case.

As observed in Figure 21a,b, the column tops of both the first and second stories in both models initially move outward, followed by inward displacement. For model IC, the maximum outward and inward displacements of the first-story column top are 3.07 mm and 12.19 mm, respectively. For the second story, they are 2.09 mm and 12.92 mm. In model IC-S, the first-story column top shows outward and inward displacements of 2.54 mm and 10.36 mm, respectively. The second story shows 2.23 mm and 10.63 mm. The presence of the slabs reduces the maximum lateral displacement of the column tops by 17.8%.

The lateral displacement trends of the first and second-story column tops are similar to those observed in the interior column failure case, as shown in Figure 21c,d. In the SC model, the first-story column top reaches outward and inward displacements of 4.52 mm and 30.95 mm, respectively, while the second story records 3.04 mm and 32.21 mm. In the SC-S model, the first-story column top shows outward and inward displacements of 3.86 mm and 22.60 mm, respectively, while the second story exhibits 2.14 mm and 23.84 mm. The inclusion of the slab reduces the maximum lateral displacement of the column tops by 25.99%.

However, under corner column failure (models CC and CC-S), the lateral displacement of the column top (Column B) adjacent to the failed column is minimal and considered negligible, indicating that the remaining structure provides effective lateral restraint.

Comparison of Figure 21a–d reveals that, for the same specimen, the lateral displacement of the first-story column top exceeds that of the second story during the outward movement phase. This phenomenon is attributed to the Vierendeel action. In the later stages of the inward movement, the second-story column top exhibits greater lateral displacement than the first story. This is because the second story has lower surrounding restraint stiffness than the first. In addition, as the structural components progressively fail, its stiffness reduces more significantly, leading to larger displacements. Additionally, the slab significantly restrains the lateral displacement of adjacent column tops. The TMA provided by the slab reinforcement enhances the structural integrity and improves its resistance to progressive collapse.

4.5. Reinforcement Stress

Figure 22, Figure 23 and Figure 24 present the stress contour plots of steel bars under interior, side, and corner column failure scenarios, respectively.

In models IC, SC, and CC (see Figure 22a, Figure 23a, and Figure 24a), stress is primarily concentrated in the compression zones near the beam ends and corbels, where stirrups exhibit yielding. The longitudinal bars in the beams show relatively low stress, suggesting limited contribution to tensile resistance during the CA stage. In contrast, models IC–S, SC–S, and CC–S (see Figure 22b, Figure 23b, and Figure 24b) also show stress concentration at the beam–column joints, but with notably higher stress in the tensile zones of the longitudinal reinforcement. This implies that the slab improves the tensile engagement of the longitudinal bars during the CA stage, thereby promoting more effective CA.

As shown in Figure 22c,d, Figure 23c,d and Figure 24c,d, the slab reinforcement near failed columns reaches yield strength, whereas stress in the remaining structure remains low. These results indicate that the slab fully mobilizes its membrane action during loading, while the remaining structure retains its resistant capacity.

Figure 25, Figure 26 and Figure 27 show the stress contour plots of prestressed strands under interior, side, and corner column failure scenarios, respectively. For the bare frame models (see Figure 25a, Figure 26a,c, and Figure 27a), the stress in the full-span prestressed strands is generally high, indicating that the strands bear the majority of the collapse resistance during the CA stage. In contrast, for the slab-frames (see Figure 25b, Figure 26b,d, and Figure 27b), high-stress regions in the prestressed strands are localized in the tensile zones of the beams, whereas other areas exhibit relatively low stress. This suggests that the TMA of the slab contributes to sharing part of the structural resistance during the CA stage.

4.6. Concrete Compressive Damage

Figure 28, Figure 29 and Figure 30 present the concrete compressive damage contour plots under interior, side, and corner column failure scenarios, respectively. For models IC, SC, and CC (see Figure 28a, Figure 29a, and Figure 30a), damage primarily occurs at the beam–column joints, where the stress reaches 37.1 MPa, corresponding to the concrete compressive strength. The concrete in the mid-span regions of the beams remains largely intact with relatively low stress levels. Minor damage is also observed at the anchorage ends due to the slippage of the prestressed strands. For models IC–S, SC–S, and CC–S (see Figure 28b, Figure 29b, and Figure 30b), negative bending moments develop in the slab supports, where the slabs collaborate with the beams. Consequently, in addition to the beam–column joints, significant damage is observed in the upper parts of the beams. Slab damage contours (see Figure 28c, Figure 29c, and Figure 30c) show that damage is concentrated around the slab supports, with the second-story slabs exhibiting slightly more extensive damage than those on the first floor. This is due to the relatively lower surrounding restraint stiffness of the second-story columns, leading to larger inward displacements during the large deformation stage and more severe concrete damage.

4.7. Energy-Based Dynamic Assessment

The energy-based dynamic assessment method proposed by Izzuddin et al. [75] offers an efficient means of evaluating the dynamic progressive collapse performance of structures following column removal. It is founded on the principle that the work performed by gravity loads upon column loss is entirely converted into the strain energy absorbed by the structure at maximum deformation, allowing the nonlinear quasi-static response curve to be used directly to estimate dynamic collapse resistance. This approach has been validated in prior studies under assumptions of negligible damping and the absence of blast effects [76], and has been shown to reliably predict peak dynamic responses and ductility demands through nonlinear quasi-static analysis, thereby providing a valuable reference for engineering assessment and design.

Work performed by gravity loads:

$$W_{\mathrm{n}}=\alpha F_{\mathrm{d}}{u}_{\mathrm{d}, \mathrm{n}}$$

(34)

Structural internal energy:

$$U_{\mathrm{n}}={\displaystyle {\int }_0^{u_{\mathrm{d}, \mathrm{n}}}\alpha F_{\mathrm{s}}}\mathrm{d}{u}_{\mathrm{s}}$$

(35)

From W_n = U_n, it follows that:

$$F_{\mathrm{d}}={\displaystyle \frac{1}{u_{\mathrm{d}, \mathrm{n}}}}{\displaystyle {\int }_0^{u_{\mathrm{d}, \mathrm{n}}}{F}_{\mathrm{s}}}\mathrm{d}{u}_{\mathrm{s}}$$

(36)

where F_d is the suddenly applied gravity load; F_s is the quasi-static load; u_d,n is the maximum dynamic displacement caused by the gravity load; and α is a dimensionless coefficient related to work.

As illustrated in Figure 31a–f. The dynamic load–displacement curves provide critical insight into the ductility and post-peak robustness of the FB-PPC spatial frame following sudden column removal. The displacement axis represents the maximum deformation under equivalent dynamic conditions, while the vertical axis quantifies the corresponding dynamic resistance. The residual strength, defined as the load level on this plateau or slope at large displacements, is critical. An extended ductility range and high residual load-carrying capacity—manifested as plateaus or gently sloping regions beyond the peak—indicate successful activation of catenary and membrane actions, allowing the structure to accommodate large deformations without catastrophic failure. Conversely, sharp drops in resistance and low residual strength indicates a transition to brittle failure, where energy absorption and ductility are insufficient and load redistribution is severely limited.

Across all scenarios, including interior, side, and corner column removal, the FB-PPC spatial frames exhibited an extended ductility range, as evidenced by the persistence of plateaus or gently sloping regions in the dynamic curves. Both slab and non-slab frames demonstrate similar overall trends in their dynamic response. However, the presence of precast slabs results in higher peak loads and greater residual resistance throughout the collapse process. Even in the case of corner column removal, the curve demonstrates a gradual decline rather than an abrupt drop, reflecting effective energy absorption and the structural capacity to activate CMA and CA. These features highlight the reduction in peak resistance due to dynamic effects, but more importantly, underscore the robust ability of FB-PPC to sustain deformation and dissipate energy at large displacements.

4.8. Analysis and Discussion of the Progressive Collapse Resistance Mechanism

It can be concluded from the previous study that when the interior or side column fails, the structural response of the bare frame models (IC and SC) can divide into the beam action stage and the CA stage. During the early loading, collapse resistance is mainly provided by the BA at beam ends. With loading progresses, the mid-span neutral axis shifts upward while the beam-end axis moves downward, inducing axial compression in the beams and activating the CAA, as shown in Figure 32a. Subsequently, the concrete at the beam–column joints undergo crushing damage, which weakens the CAA and reduces the collapse resistance. It can be indicated from Figure 32b that, upon entering the CA stage, the axial force in the beam is switched to tension. The CA is primarily sustained by the prestressed strands due to the discontinuity of the steel rebars near the joint interface. The prestressed strands elongate continuously, further enhancing the vertical resistance.

When the slab is considered, vertical loading causes its edges to move outward. This lateral movement is restrained by the surrounding beams and columns, generating compressive forces directed from the perimeter toward the slab center, thereby forming the CMA (see Figure 33a). Additionally, the slab transforms the beam section from rectangular to T-shaped, raising the neutral axis and increasing the internal lever arm between resultant forces. This enhances the resistant capacity during the beam action stage, increasing peak resistance by 56.63% for IC and 53.92% for SC. Once the structure enters the large deformation stage, the slab shifts from compression to tension, thereby initiating the TMA (see Figure 33b). The TMA contributes to the strengthening of the CA, enhancing peak resistance by 50.96% and 59.54% for IC and SC, respectively.

The limited lateral stiffness in the surrounding region prevents the effective development of CA, when the corner column fails. The structure primarily resists collapse through the BA and the CAA. Therefore, the structural response is divided into two stages: the beam action stage and the collapse stage. When the slab is considered, its TMA cannot be fully mobilized. Nevertheless, the ultimate resistance capacity of CC–S increases by 36.87% compared to CC, and the resistance in the collapse stage declines more gradually.

5. Conclusions

This study investigated the progressive collapse mechanisms of fully bonded prestressed precast concrete (FB-PPC) spatial frames by developing and validating refined finite element models, performing pushdown analyses under various column removal scenarios, and employing an energy-based dynamic assessment to estimate dynamic collapse capacity. Both frames with and without precast slabs were examined, enabling comprehensive comparison of collapse behavior, slab contribution, and failure modes under different accidental events. The following conclusions are drawn:

(1)

This paper proposes a modeling method for FB-PPC frames, considering the complex bonding behavior between prestressed strands and concrete, the force transfer mechanism at precast component connections, and the interaction between the slabs and the frame. The finite element model was validated against corresponding substructure tests with high correlation.

(2)

The role of precast slabs in progressive collapse is significant. Regarding strength contribution, slabs contribute an average of 34.10%, 37.88%, and 42.33% to resistance for interior, side, and corner column failures. For column top lateral displacement, the maximum value is reduced by approximately 17.8% when a slab is present in the interior column removal scenario, and by about 25.99% in the side column removal scenario, compared to the corresponding models without slabs.

(3)

For bare frames with interior and side column failures, the collapse resistance during the beam action phase is primarily provided by bending and compressive arch action, while catenary forces arise mainly from prestressed strands due to discontinuous reinforcement at joints. Compressive and tensile membrane actions further enhance resistance in the beam action and catenary action phases. For corner column failure, insufficient horizontal constraints limit catenary and membrane action, and only beam action and collapse phases are observed.

(4)

The energy-based dynamic assessment demonstrates that FB-PPC spatial frames maintain high ductility and residual strength under sudden column removal, with load–displacement curves displaying sustained plateaus or gentle slopes in all scenarios. Comparable dynamic response trends were observed in both frames with and without precast slabs, precast slabs consistently increase peak and residual strengths. This reflects effective energy absorption and activation of catenary and compressive membrane actions, which enable the structure to withstand large deformations without brittle failure—even in corner column removal scenarios—thus ensuring robust performance under dynamic loading.

(5)

This study was limited by the absence of full-scale physical testing and dynamic analysis of complete spatial frame systems, relying solely on quasi-static numerical simulations and a simplified energy-based dynamic assessment. Future research will focus on comprehensive experimental and theoretical investigations of spatial frame systems explicit consideration of dynamic effects, and detailed evaluation of geometric and mechanical parameters. Sensitivity and robustness analyses, as well as the adoption of more advanced modeling approaches, are also anticipated.