Progressive Collapse Behavior of a Precast Reinforced Concrete Frame System with Layered Beams

Abstract

A possible way to improve the structural safety and robustness of precast building structures is to develop effective precast frame systems with layered beams, which combine prefabricated parts with cast-in situ ordinary concrete, high-performance concrete, fiber concrete, or FRP. The paper provides a new type of precast reinforced concrete frame system with layered beams for rapidly erected multi-story buildings resistant to accidental actions. Using a combination of the variational method and two-level design schemes, a simplified analytical model has been developed for structural analysis of the precast reinforced concrete frame system, both for serviceable and ultimate limit states as well as for accidental actions. The proposed model allows for determining shear deformations and the formation and opening of longitudinal cracks in the intermediate contact zone between precast and monolithic parts of reinforced concrete structural elements of the frame, as well as the formation and opening of normal cracks because of the action of axial tensile force or bending moment in these elements. The design model was validated by comparing the calculated and experimental data obtained from testing scaled models of the precast reinforced concrete frame system with layered beams. The paper investigates and thoroughly analyzes the factors affecting the stiffness and bearing capacity of the intermediate contact zone, discusses the criteria for the formation of shear cracks along the contact zone of precast and monolithic concrete, and examines the change in the stiffness and dissipative properties of layered elements at different stages of their static–dynamic loading. The robustness of the experimental models of the structural system was not ensured under the specified load, section dimensions, and reinforcement scheme. Following an accidental action, longitudinal cracks were observed in the contact joint between the monolithic and prefabricated parts in the layered beams. This occurred almost simultaneously with the opening of normal cracks in adjacent sections. A comprehensive analysis of the results indicated a satisfactory degree of agreement between the proposed semi-analytical model and the test data.

1. Introduction

Since the collapse of the World Trade Center in 2001, there has been a significant increase in interest in the problem of ensuring the resistance of load-bearing systems of buildings to accidental effects, including those resulting in damage to or the failure of structural components. The engineering community has previously been aware of cases of building collapse due to localized failures or damage to structural elements. These include the cases of the Ronan Point [1] and Alfred Murray [2] buildings, the Transvaal Park complex [3], and the Sampoon Shopping Center [4]. However, the start of the intensive growth in the number of publications on this problem occurred in the 2000s [4,5]. The primary characteristic of the aforementioned cases of building collapse is the low probability of the initial event that caused damage to the structural system. Despite this, the effects of changes in the design scheme of structures as a result of local failure were catastrophic. Therefore, although the probability of accidental impact is low, the risk of unacceptable consequences is significant and must be accounted for at the stage of the design of structures with a higher level of responsibility for those involving simultaneous mass occupancy of people [6,7,8].

The accidental actions may result in the realization of specific mechanisms of structural resistance [2,9,10], or the permissible limits of using simplified design models may be exceeded [11]. Consequently, the models and criteria used to assess the load-bearing capacity of structural elements in the limit states and the robustness of the structure as a whole require a more thorough justification. In particular, it is permitted that in an emergency situation, the slab structures may transfer to the catenary resistance mechanism [12], which is not permissible during normal operation. In this regard, there is a need for a more in-depth study of the resistance behavior and failure mechanisms of various types of structures to ensure their mechanical safety and robustness through rational design.

Currently, precast and precast monolithic reinforced concrete structures are widely utilized in the construction of civil buildings across the world. The utilization of such structural solutions offers a number of advantages over cast-in-place structures. These include a reduction in the time and costs associated with the erection of structures, the ability to achieve better compaction of concrete mixtures in factory conditions, the potential for the use of temperature and humidity treatment, the capacity for a more rational solution of heat protection of the outer contour of the building when using such structures, comprehensive monitoring of the quality of finished products, and the convenience of dismantling and replacement of structures.

Akduman et al. [13] proposed a variant of the precast frame for the construction of single-story buildings. As a base for the manufacture of structures, geopolymer concrete produced with the use of recycled products was utilized. This approach allows for a reduction in the impact on the environment due to the recycling of materials. The proposed structural system implements prefabricated detachable mechanical connections of load-bearing elements, which allow for rapid dismantling of the structures. At the same time, it is clear that precast structural systems are more susceptible to dynamic and cyclic loading, which can occur during seismic events or as a result of the failure of a structural member or ties.

Yang et al. [14] developed a structural solution for a precast prestressed reinforced concrete frame comprising monolithic parts. They conducted an experimental study of a scaled physical model subjected to loads simulating seismic impact. As a damper to absorb impact energy, diaphragms (shear panel dampers) were employed. The results of the study validated the effectiveness of diaphragms in absorbing energy during seismic impacts.

Zhao et al. [15] conducted experimental and numerical studies of precast reinforced concrete panel elements with bolted connections under cyclic loads that correspond to seismic effects. The proposed variant of connection for precast elements demonstrated its high efficiency. At the same time, the results of the experimental study revealed a significant opening of gaps in horizontal joints due to the deformation of bolts and support plates.

Riedel et al. [16] examined the mechanical connection of beams in a precast reinforced concrete structural system in the event of a column failure. The authors highlighted the vulnerability of precast structures in the event of failure of one of the load-bearing elements of the structural system. They observed that the low bending stiffness of the upper steel plate connecting the prefabricated elements of the beams did not provide the necessary resistance in the event of a sudden failure of the column. The installation of an additional connecting plate at the bottom of the beams increased the load-bearing capacity of the structure. However, the membrane resistance mechanism was not realized in the structure.

A paper [17] examined the impact of floor slabs on the behavior of prefabricated monolithic frame structures. The considered frame structure comprised a prefabricated lower section of the beams and slabs. The upper parts of the cross-section of the beams were cast in place and connected with the lower part by means of transverse reinforcing bars. Also, the prefabricated parts of the beams had protrusions that provided mechanical interlocking of the layers of the composite structure. The authors found that the floor slabs reduced the dynamic response of the substructure by up to 35–40% compared to a design solution that did not account for the floor slab.

Lin et al. [18] proposed a prefabricated reinforced concrete structural system capable of absorbing emergency loads in case of failure of one of the load-bearing elements. The main feature of their proposed design solution is the use of the bolted connection of the beams to the columns through the upper and lower overhead steel plates, as well as additional safety plates on the sides of the support section of the beams. The precast beams are prestressed with ropes at the top and bottom edges. The experimental studies carried out on a scale model of the substructure show that an increase in the level of prestressing in the beams led to a significant reduction in the angle of rotation in the support section of the beam in case of an emergency.

In recent years, new structural solutions for reinforced concrete bearing systems have been proposed based on panel [19,20] or frame [21,22,23] elements. These constructive decisions have a number of obvious advantages, such as free layout, the variability of the architectural appearance of facades, a reduction in the dead load of building structures, recycling of materials during reconstruction, etc. To date, detailed architectural, planning, and structural solutions have been developed for several such systems [19], including the pilot erection of small-story buildings.

Nevertheless, the extension of these precast structural systems for buildings with increased stories necessitates the assurance of their mechanical safety, both under operational loads and under accidental actions resulting from the failure of load-bearing elements or ties. This requires further in-depth research to be conducted.

Over the past decades, some progress has been made in the theoretical field of protecting structures against progressive collapse with respect to various construction solutions and hazards. The solution to such problems has reached the level of development of normative documents [24,25] and their use in the practice of designing facilities. There have been experimental studies of certain types of structures and substructures under accidental actions to investigate progressive collapse behavior. In particular, the studies [26,27,28] provide the results of tests of bar and spatial full-scale reinforced concrete structures and substructures, as well as scaled models of such structures [11,29,30,31,32,33,34,35,36]. As shown in [37], the majority of experimental investigations have focused on scenarios of the removal of a single column of the ground floor of a building frame. It is also possible for initial failures to occur in layered structures, fasteners, and other structural elements that accompany column failure.

The primary focus of these studies was on the development of methods for simulating the removal of structural elements, investigating the redistribution of loads, determining the dynamic increase factor, revealing the cracking and fracture patterns, and other related issues. At the same time, these studies did not consider the design features of layered elements of building frames, including solutions for their joints, the structure of layered cross-sections of the elements, and their reinforcement scheme. A number of theoretical problems have been solved in a statement plane without their detailed examination or explanation of the occurring physical phenomena. This does not allow for a valuable and effective contribution to the existing knowledge in the field under consideration.

There is a need for more detailed information such as how to evaluate the factors affecting the dynamic response [38], the time of removal of a structural element of the load-bearing system, the material properties of the structure, the type and level of its stress state, damping, and stiffness under high-rate loading (impact). The influence of the topology of structural systems and the effect of the cross-section structure of their elements, loading modes, and mechanical properties of the materials on resistance to progressive collapse have not been extensively investigated. It is necessary to clarify the criteria and concept of the special ultimate limit state, as well as the definition of this state itself. Although numerous numerical studies have been conducted to investigate the performance properties of infill-wall frames subjected to sudden column removal [39,40,41], detailed experimental studies in this area remain scarce.

The combination of conventional concrete with high-performance concrete, fiber concrete, or polymer concrete finds its application more and more often in the new generation of precast structures [42,43,44]. There are structural solutions in which reinforced concrete is strengthened with FRP external reinforcement to improve performance under dynamic and cyclic loads [45,46,47]. Thus, accounting for the cross-sectional structure and deformation properties of two- and three-layered elements of a structural system becomes an important condition for evaluating its robustness under static–dynamic loading.

The modeling of the behavior of layered structures is mainly based on the finite element method [48,49,50]. In this case, the main difficulty is the simulation of the so-called cold concrete joint and the intermediate contact between the layers of the precast structure. The stiffness of the contact area determines the distribution of stresses and strains along the height of the section and along the length of the member. Accordingly, it influences the character of cracking and failure of layered structures in the limit states. A number of studies, such as [48], ignore the ductility of the contact joint, taking into account the frequent presence of transverse reinforcing bars that cross the joint. The papers [51,52] present a model of a bar element with a layered section based on the theory of Rzhanitsyn [53]. The approach proposed by the authors has also been used to model shear displacements during cracking in structures. The strips between cracks are considered layers of a layered structure.

Based on the presented brief analysis of the scientific literature on the problem under consideration, the following conclusions can be drawn:

(a)

Precast and precast monolithic reinforced concrete structures have a significant potential for application in civil engineering due to the automation of production processes, quality control, ease of erection and dismantling, and the possibility of using recycled products.

(b)

As the above analysis of experimental studies and numerical modeling results shows, the main problem in the use of precast structures is the vulnerability of the joints under dynamic and cyclic loading, especially in the event of failure of structural members.

(c)

Precast monolithic structural systems have higher robustness than pure precast solutions. However, the deformation and force distribution in such structures have a specific property due to the concentrated shear at the contact joint of the precast and monolithic parts of the section. This affects the cracking pattern, force redistribution, and energy dissipation.

This paper proposes a new type of structural system for civil buildings composed of U- and L-shaped panels and prefabricated multi-hollow core floor slabs. It is an evolutionary development of the structural solutions designed with the participation of the authors of this paper and presented in articles [19,20]. The difference in the proposed design is the use of additional tie beams between the prefabricated floor slabs and the elimination of the use of paired column connections.

The objectives of this study are as follows: (1) the development of structural solutions for a precast reinforced concrete frame system with layered beams; (2) the experimental study of stress–strain state and cracking in layered beams of reinforced concrete panel elements under column removal scenarios; and (3) the development of a simplified analytical model to evaluate the behavior of layered beams of reinforced concrete panel elements under accidental actions.

This paper focuses on the influence of the ductility of the intermediate contact area between the prefabricated and monolithic parts of the beam on the progressive collapse behavior under column removal scenarios. In particular, the influence of longitudinal, normal cracks in the tensile zones on the deformation of such structures before and after the accidental action is investigated. In order to achieve preset objectives, experimental studies have been performed on 1:6 scale frame models. Using the experimental data, a simplified analytical model of a layered beam has been developed, taking into account the formation of three types of cracks: I—cracks caused by the action of axial tensile force; II—normal cracks in the zone of maximum bending moments; and III—longitudinal cracks in the contact zone of shear between the prefabricated and monolithic parts of the layered beam.

The main contribution of this study is the development of a simplified analytical model that allows the calculation of the stress–strain state of reinforced concrete frames with layered elements, taking into account the ductility of the interlayer contact joint under the action of service and accident loads.

2. Materials and Methods

2.1. Design of a Precast Reinforced Concrete Frame System with Layered Beams

The article proposes a precast reinforced concrete frame system with layered beams as shown in Figure 1a. This structural system consists of two types of prefabricated ‘L’- and ‘U’-shaped panel elements as shown in Figure 1b,c. The beams of the ‘L’ and ‘U’ elements are used to support the multi-hollow core slabs. The struts of the panels have openings for a depth equal to the depth of the multi-hollow core slabs.

The ends of the slabs, which rest on the prefabricated part of the beams of the panel elements (Figure 2a), have cavities that form keys after concreting. When assembling the frame, longitudinal steel reinforcements should be installed in the upper part of the frame beams. The reinforced space between the ends of the floor slabs and in the openings of the struts should be sealed with in situ concrete. This provides a rigid frame system in mutually orthogonal directions. Prefabricated ‘L’- or ‘U’-shaped structural elements are connected vertically by means of a plug connection (Figure 2b). The horizontal connection of these elements is made by means of embedded parts, with the beam of the ‘L’ element resting on the projecting cantilever of the ‘U’ element. In addition, the monolithic part of the beams passes through the opening in the strut. This ensures a rigid connection between the strut and the beams after concreting.

The proposed structural design of the prefabricated monolithic building frame provides redistribution of force flows through alternative load paths under the strut removal scenario. As a result, the resistance of the frame to progressive collapse can be increased. For the same purpose, the struts of the adjacent floors are additionally connected by special steel plates as shown in Figure 2c. This ensures that tensile forces are absorbed by the ‘chain’ of struts along the height of the building in the event of strut removal.

By analyzing the presented structural design, it is possible to note the following. For example, a 6 × 4.8 m cell of a traditional precast or precast monolithic frame consists of four strut-type elements and four beam-type elements. The same cell of the proposed precast reinforced concrete frame system with layered beams has twice as few assembly elements and therefore fewer butt joints. In contrast to buildings made of large panel elements, which have a rigid architectural and planning structure, the proposed structural solution allows for various possibilities of volumetric planning solutions, both for residential buildings with different stories and for civil buildings with different functional purposes.

The use of only two types of prefabricated lightweight panel elements and the use of multi-hollow floor slabs manufactured by construction companies significantly reduce the lifting capacity requirements of installation mechanisms and transport costs.

The advantages of the proposed structural system are as follows:

−

In comparison with frame structural systems consisting of prefabricated columns, beams, and floor slabs, the proposed structural solution has a smaller number of assembly units, which speeds up the installation process.

−

The use of monolithic concrete in the beams allows an increase in the level of static indeterminacy of the system in comparison with fully prefabricated ones. This has a positive effect on the operational characteristics of the system, such as the stiffness of the beam–column joints, and enables load transfer through alternative paths in the event of column removal.

−

In comparison to panel-wall structural solutions, this structural system exhibits a lower weight and offers greater flexibility in organizing the interior space of the building.

−

The utilization of layered elements can serve as an additional means of dissipating dynamic impact energy during localized failure within the structural system. The dissipation of impact energy in this case occurs due to the ductility of the contact joint area, the formation of a longitudinal crack, the decay of natural frequencies of structural vibrations, and the friction forces on the contact surface during the formation of a longitudinal crack.

At the same time, there is a limitation to the application of the developed structural solution and similar ones that include structural elements with a layered cross-section. The presence of a contact joint makes it difficult to model the response of the prefabricated monolithic structure and calculate the stress–strain state of the sections under service loads and, especially, accidental actions. Typical finite element models used in design do not account for the effects of ductility and crack opening in the contact joint area. Detailed design models are not always justified due to the time it takes to create them and the complexity of analyzing the results.

To overcome this limitation, the next section of the paper will present initial assumptions and an analytical model for calculating the stress–strain state of layered elements under service loads and accidental actions.

2.2. Method for Determining the Stress–Strain State in the Structural System

The response of the considered structural system of a building with load-bearing elements of layered cross-section under the effect of service loads and under accidental actions is significantly determined by the ductility of the contact joint between the prefabricated and monolithic parts of the beams. At low values of external loads on the beams, they behave as elements of a solid section. This means that Bernoulli’s flat section hypothesis is fulfilled for the whole section, including the precast and monolithic parts. However, as the load and the forces acting in the monolithic and prefabricated parts of the section increase, longitudinal cracks are formed in the contact joint between the prefabricated and monolithic parts of the beam. In this case, the Bernoulli flat section hypothesis is fulfilled separately for each of the parts: monolithic and prefabricated. The deformation in the contact joint area is influenced by the following parameters, which must be considered when developing the analytical model:

−

The bond strength of concrete of monolithic and prefabricated parts of the beam;

−

The deformability of transverse reinforcement bars under the action of dowel forces;

−

Boundary conditions in the end sections of the beams and anchoring of the longitudinal reinforcement in the prefabricated and monolithic parts of the beams in these sections.

It is assumed that the ductility of the contact joint associated with transverse reinforcement is determined to a greater extent by the spacing and to a lesser extent by the diameter of the reinforcement, as well as by the shear strength of the concrete of the monolithic and prefabricated parts of the beam in the intermediate contact area between these parts. In this case, after the formation of a longitudinal crack along the contact joint, the stiffness of the joint is determined by the dowel behavior of the transverse reinforcement bars.

The study adopts a two-level approach to modeling the response of the proposed reinforced concrete structural system of the building frame (Figure 3a) to service and emergency loads. In the first stage, the entire load-bearing system is calculated using the finite element method, assuming the absence of longitudinal shear in the contact joint area of the elements. This calculation results in the determination of the forces acting in the end sections of the beams and in the specification of the boundary conditions. The second level of modeling considers a precast monolithic beam, taking into account the ductility of the contact joint between the monolithic and precast parts. The load is represented as a continuous function along the length of the element. It is possible to use a uniformly distributed load equivalent in deflection or a continuous non-uniformly distributed load.

It is worth noting that bending moments may be redistributed along the length of the beam as a result of shear deformations in the contact joint. This may lead to overloading of the cross-sections in the middle of the span. If there is a significant difference between the bending moments in the first- and second-level design schemes, special finite elements should be introduced in the first-level design scheme to take into account the relationship between moments and rotation angles, longitudinal forces and longitudinal deformation, and transverse forces and transverse deformations. The justification of the parameters of such special elements is the subject of future experimental and numerical studies. In this study, the results of numerical simulations for the first-level design scheme are replaced with experimental data from reinforced concrete frame models with layered beams. Thus, the method presented below can be classified as experimental–analytical.

The solution to the problem is based on the variational method of Vlasov [54] in the form of displacement. This involves separating the substructure of the precast monolithic beam adjacent to the column to be removed (Figure 3b) from the frame system (Figure 3a).

The x, z coordinate origin of the layered reinforced concrete beam is placed in the center of the span in the contact area between the precast and the monolithic part. Prior to cracking, the substructure behaves as a composite plate consisting of two elements of a rectangular section that allow shear at the contact joint. The shear stiffness of such a joint depends on the load applied to the structure and the occurrence of a longitudinal crack. In general, a composite plate is loaded with an effective distributed load q₀. Axial force N_x, lateral force N_y, and shear force N_xy act in the plane of the plate. The intensity of these forces varies depending on the different areas of the substructure. The unit functions of the displacement method [54] are used to model the stress–strain state of the substructure. The function ξ₀(z) describes the distribution of longitudinal displacements due to the bending of the plate in its plane. The function ξ₁(z) represents the distribution of longitudinal displacements from tension or compression of the entire cross-section of the composite plate. The function ξ₂(z) describes the distribution of longitudinal displacements in shear between the elements of the composite plate.

The study considers the possibility of the formation of three types of cracks in the substructure of a precast monolithic beam: Crc.1—cracks caused by the action of axial tensile force (Figure 4a); Crc.2—normal cracks in the area of maximum bending moments (Figure 4b); and Crc.3—longitudinal cracks in the contact area of shear between the elements of the layered beam (Figure 4c).

The changes in stiffness in lateral shear, bending, and longitudinal shear at the interface after crack formation are modeled iteratively at each loading step. In this case, in order to keep the structure of the solution equations of the variational method, the reduced equivalent values of the stiffnesses of the quasi-solid beam are used. The formulas for their calculation are given below. This variant of modeling the stiffness of a layered beam with cracks allows us to take into account the formation of cracks of all types at different loading stages while keeping the assumed unit functions unchanged. The ends of the layered beam are assumed to be rigid or semirigid at their interface with the columns.

According to the variational method of displacements [54], the two-dimensional problem for the considered 2D-stressed layered structure with unknown displacement functions u(x,z) and v(x,z) is reduced to a one-dimensional one by decomposing these functions into the following single series:

$$u(x,z)={\displaystyle \sum _c{\overline{U}}_c(x){\overline{\xi }}_c(z)}+{\displaystyle \sum _i{U}_i(x){\xi }_i(z)};$$

(1)

$$v(x,z)={\displaystyle \sum _g{V}_g(x){\xi }_g(z)}.$$

(2)

The one-dimensional displacement functions ${\overline{U}}_c\left(x\right)={\overline{U}}_2\left(x\right)$, $U_i\left(x\right)=U_1\left(x\right)$, and $V_g\left(x\right)=V_1\left(x\right)$ in Equations (1) and (2) are determined from the system of equations in

Table 1. General structure of the system of constitutive equations of the variational method of displacements.

$\mathbf{Coefficients} \mathbf{at} {\overline{\mathit{U}}}_{\mathit{c}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$	$\mathbf{Coefficients} \mathbf{at} {\mathit{U}}_{\mathit{i}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$	$\mathbf{Coefficients} \mathbf{at} {\mathit{V}}_{\mathit{g}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$	The Right-Hand Sides of the Equations
${\displaystyle \sum _c{\overline{J}}_{dc}{D}^{II}-b_{dc}}$	${\displaystyle \sum _i{J}_{di}^{*}{D}^{II}-b_{di}}$	$-{\displaystyle \sum _g{J}_{dg}^{*}{D}^{III}}$	0
−	${\displaystyle \sum _i{J}_{mi}{D}^{III}}$	${\displaystyle \sum _g{J}_{mg}{D}^{IV}}$	0
−	−	${\displaystyle \sum _g{J}_{ng}{D}^{IV}}$	$-q_n$

Note: J_d and b_d are the stiffness coefficients; $D^{II}=\frac{d^2}{d{x}^2}\left(\dots \right)$, $D^{III}=\frac{d^3}{d{x}^3}\left(\dots \right)$, and $D^{IV}=\frac{d^4}{d{x}^4}\left(\dots \right)$ are the derivatives of the unknown functions ${\overline{\mathit{U}}}_{\mathit{c}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$, ${\mathit{U}}_{\mathit{i}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$, and ${\mathit{V}}_{\mathit{g}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$, respectively; and q_n is the equivalent distributed load in terms of deflection.

. ${\overline{\xi }}_c\left(z\right)={\overline{\xi }}_2\left(z\right)$, ξ_i(z) = ξ₁(z), and ξ_g(z) = ξ₀(z) are the selected unit functions describing the distribution of displacements over the cross-sectional depth of the layered beam.

Table 1 provides three groups of coefficients for the unknown displacement and the general form of the right-hand sides of the equations for the adopted boundary conditions.

For unit functions ${\overline{\xi }}_2\left(z\right)$, ξ₁(z), and ξ₀(z), which account for shear in the contact joint, tension (compression), and bending, respectively, the system of equations of Table 1 is transformed to the form given in

Table 2. Coefficients for unknown unit functions.

$\mathbf{Coefficients} \mathbf{at} {\overline{\mathit{U}}}_2\mathbf{\left(}\mathit{x}\mathbf{\right)}$	$\mathbf{Coefficients} \mathbf{at} {\mathit{U}}_1\mathbf{\left(}\mathit{x}\mathbf{\right)}$	$\mathbf{Coefficients} \mathbf{at} {\mathit{V}}_0\mathbf{\left(}\mathit{x}\mathbf{\right)}$	The Right-Hand Sides of the Equations
${\overline{J}}_{22}{D}^{II}-b_{22}$	$J_{21}^{*}{D}^{II}$	$-J_{20}^{*}{D}^{III}$	0
$-J_{12}^{*}{D}^{III}$	$-J_{11}{D}^{III}$	$J_{10}{D}^{IV}$	0
$-J_{02}^{*}{D}^{III}$	$-J_{01}{D}^{III}$	$J_{00}{D}^{IV}$	$-q_0$

.

Using Table 2, one obtains the expanded form of the system of equations as follows:

$$\left\{\begin{array}{c}{\overline{J}}_{22}{\overline{U}}_2^{II}(x)-b_{22}{\overline{U}}_2(x)+J_{21}^{*}{U}_1^{II}(x)-J_{20}^{*}{V}_0^{III}(x)=0,\\ -J_{12}^{*}{\overline{U}}_2^{III}(x)-J_{11}{U}_1^{III}(x)+J_{10}{V}_0^{IV}(x)=0,\\ -J_{02}^{*}{\overline{U}}_2^{III}(x)-J_{01}{U}_1^{III}(x)+J_{00}{V}_0^{IV}(x)=-q_0.\end{array}\right.$$

(3)

The analytical solution of the system of Equation (3) is obtained by substitution in the following order. The determinant from the coefficients of the unknown unit functions, as well as its modifications that take into account the right-hand sides of the equations, has the following form:

$$\Delta =\left|\begin{array}{cc}-J_{11}& J_{10}\\ -J_{01}& J_{00}\end{array}\right|=J_{10}^2-J_{11}{J}_{00};$$

(4)

$${\Delta }_{U1}=\left|\begin{array}{cc}-J_{12}^{*}{\overline{U}}_2^{III}(x)& J_{10}\\ J_{02}^{*}{\overline{U}}_2^{III}(x)+q_0& J_{00}\end{array}\right|={\overline{U}}_2^{III}(x)\left(J_{12}^{*}{J}_{00}-J_{10}{J}_{02}^{*}\right)-q_0{J}_{10};$$

(5)

$${\Delta }_{V0}=\left|\begin{array}{cc}-J_{11}& -J_{12}^{*}{\overline{U}}_2^{III}(x)\\ -J_{01}& J_{02}^{*}{\overline{U}}_2^{III}(x)+q_0\end{array}\right|={\overline{U}}_2^{III}(x)\left(J_{12}^{*}{J}_{01}-J_{02}^{*}{J}_{11}\right)-q_0{J}_{11}.$$

(6)

The desired functions may then be written in the following form:

$$U_1^{III}(x)=\frac{J_{12}^{*}{J}_{00}-J_{10}{J}_{02}^{*}}{J_{10}^2-J_{11}{J}_{00}}{\overline{U}}_2^{III}(x)-\frac{q_0{J}_{10}}{J_{10}^2-J_{11}{J}_{00}};$$

(7)

$$V_0^{IV}(x)=\frac{J_{12}^{*}{J}_{01}-J_{02}^{*}{J}_{11}}{J_{10}^2-J_{11}{J}_{00}}{\overline{U}}_2^{III}(x)-\frac{q_0{J}_{11}}{J_{10}^2-J_{11}{J}_{10}}.$$

(8)

In order to facilitate the subsequent solution, it is necessary to introduce the following notations:

$$\begin{gathered} A_1=\frac{J_{12}^{*}{J}_{00}-J_{10}{J}_{02}^{*}}{J_{10}^2-J_{11}{J}_{00}}, q_1=\frac{q_0{J}_{10}}{J_{10}^2-J_{11}{J}_{00}}, \\ {A}_2=\frac{J_{12}^{*}{J}_{01}-J_{02}^{*}{J}_{11}}{J_{10}^2-J_{11}{J}_{00}}, q_2=\frac{q_0{J}_{11}}{J_{10}^2-J_{11}{J}_{00}}. \end{gathered}$$

(9)

By integrating once both parts of Equations (7) and (8) and substituting the obtained expressions for U₁(x) and V₀(x) into the first equation of the system (3), one obtains the equation for determining the unknown ${\overline{U}}_2\left(x\right)$:

$$\alpha {\overline{U}}_2^{II}(x)-b{\overline{U}}_2(x)+qx+C=0,$$

(10)

where

$$\begin{gathered} \alpha ={\overline{J}}_{22}+J_{21}^{*}{A}_1-J_{20}^{*}{A}_2, q=q_2{J}_{20}^{*}-q_1{J}_{21}^{*}, \\ C=J_{21}^{*}{C}_1-J_{20}^{*}{C}_2, b=b_{22}. \end{gathered}$$

(11)

The integration constants C₁ and C₂ are determined from the boundary conditions. In particular, for a rigid connection between the beam and the column, it can be assumed that the values of the displacement functions U_c and V_i and the derivative of the function V_i are as follows: ${\overline{U}}_2=\Delta$, $V_0=0$, and $V_0^{\prime }=0$, where Δ is the longitudinal drift in the interlayer area of a beam.

The solution to Equation (10) will be sought in the following form:

$${\overline{U}}_2(x)=C_1^{*}sh\lambda x+C_2^{*}x,$$

(12)

where

$$\lambda =\sqrt{\frac{b}{\alpha }}.$$

The integration constants in Equation (12), designated as C₁ and C₂, were determined from the boundary conditions and are presented below:

$$C_1^{*}=-\frac{C_2^{*}}{\lambda ch\lambda x};$$

(13)

$$C_2^{*}=\frac{q}{b}.$$

(14)

If the displacement ${\overline{U}}_2\left(x\right)$ from the system (10) has been determined, it is possible to find the values of other unknown displacements using inverse substitution. According to the aforementioned methodology, the values of bending moment M(x), shear forces in the interlayer contact joint T(x), and vertical deflection V₀(x) can be determined as follows:

$$M(x)=M_g(x)+T(x)\omega ,$$

(15)

where ω is the distance between the centers of gravity of the monolithic and prefabricated parts of the layered element.

M_g is the moment in the beam section without considering the effect of shear drifts in the interlayer contact joint.

$$T(x)=\frac{q}{2}\left(\frac{x^2}{2}-\frac{l}{\lambda }\frac{ch\lambda x}{sh\lambda l}+\frac{1}{{\lambda }^2}-\frac{l^2}{6}\right);$$

(16)

$$V_0(x)=\frac{A_{2}q(ch\lambda x-ch\lambda l)}{2b{\lambda }^{2}sh\lambda l}+\frac{A_{2}q(l^2-x^2)}{2b{\lambda }^2}+\frac{q_2\left(4l^4-6l^2{x}^2+2x^4\right)}{24}.$$

(17)

The stiffness coefficients in lateral shear, bending, and shear in the intermediate contact area after crack formation are determined by the method of iteration, utilizing the provided equivalent values as for a quasi-solid beam, through multiplication of the corresponding unit diagrams. In particular, the stiffness coefficients in bending are determined based on the modified coefficients of Murashyov [54], which account for the performance of steel rebars and concrete in tension in the area between cracks. Consequently, the coefficients $J_{00},J_{01},J_{02}^{*},J_{10},J_{11},J_{12}^{*},J_{20}^{*},J_{21}^{*},{\overline{J}}_{22},$ which represent the bending, tension–compression, and shear stiffnesses of the section, are calculated according to Formula (18):

$$\begin{gathered} J_{00}={\displaystyle \underset{h}{\int }{A}_0}{\xi }_0^{2}dz, J_{01}={\displaystyle \underset{h}{\int }{A}_0}{\xi }_0{\xi }_{1}dz, J_{02}^{*}={\displaystyle \underset{h}{\int }{A}_0}{\xi }_0{\overline{\xi }}_{2}dz, J_{10}={\displaystyle \underset{h}{\int }{A}_1}{\xi }_1{\xi }_{0}dz, \\ {J}_{11}={\displaystyle \underset{h}{\int }{A}_1}{\xi }_1^{2}dz, J_{12}^{*}={\displaystyle \underset{h}{\int }{A}_1}{\xi }_1{\overline{\xi }}_{2}dz, J_{20}^{*}={\displaystyle \underset{h}{\int }{A}_2}{\overline{\xi }}_2{\xi }_{0}dz, J_{21}^{*}={\displaystyle \underset{h}{\int }{A}_2}{\overline{\xi }}_2{\xi }_{1}dz, {\overline{J}}_{22}={\displaystyle \underset{h}{\int }{A}_2}{\overline{\xi }}_{22}dz, \end{gathered}$$

(18)

where

$$A_0=E_{b}I{\psi }_{bt,i}; A_1=E_{b}A{\psi }_{bt,i}^{\prime }; A_2=G_{0}A.$$

(19)

In Formula (19), the effective area of the tensile section is determined by introducing a coefficient that takes into account the performance of the tensile concrete between cracks. In accordance with [54], the aforementioned coefficient in bending is determined by means of the following Formula (20):

$${\psi }_{bt,i}=\left(\frac{M}{M_{crc}}\right)(1-{\psi }_s).$$

(20)

For the tensile case, the condition takes the form (21):

$${\psi }_{bt,i}^{*}=\left(\frac{N}{N_{crc}}\right)(1-{\psi }_s).$$

(21)

Here, M represents the bending moment acting in the element cross-section, whereas M_crc is the ultimate moment of crack resistance. N and N_crc are the axial force and the axial tensile force values at crack formation, respectively. ψ_s represents a coefficient that considers the performance of tensile concrete between cracks.

The effective shear stiffness of the contact joint of a layered structure presented in Figure 5 is determined by Equation (22):

$$G_0=\frac{T}{S_w{t}_r}+\frac{G{t}_r}{t_0},$$

(22)

where t_r is the width of the contact joint, while t₀ denotes the dimension of the area of concrete tear out or crushing, which is assumed to be equal to two diameters d of the transverse reinforcement. G is the shear modulus of concrete of the interlayer contact zone.

If the shear joint continuity is violated due to the formation of a longitudinal crack, the second component in Expression (22) should be set to zero.

Depending on the cracking, the shear-related stiffness coefficient b is determined from Expression (23) for an element without cracks

$$b=\left(\frac{T}{S_w{t}_r}+\frac{G{t}_r}{t_0}\right){({\overline{\xi }}_2^{\mathcal{H}}-{\overline{\xi }}_2^{\mathcal{B}})}^2;$$

(23)

and by Formula (24) for the element with cracks:

$$b=\frac{T}{S_w{t}_r}{({\overline{\xi }}_2^{\mathcal{H}}-{\overline{\xi }}_2^{\mathcal{B}})}^2.$$

(24)

In Expressions (22) and (23), T is the shear reaction due to the dowel effect resulting from the displacement of the ends of the transverse reinforcement bars by Δ = 1 mm. This reaction is given by the formula

$$T=\frac{12·E_s·I_{sw}}{{\left(2·t_0+{\overline{t}}_0\right)}^3},$$

(25)

where E_s and I_sw are the modulus of elasticity and moment of inertia of the steel transverse reinforcement bars, respectively.

As criteria for the formation of normal and longitudinal cracks, the conditions for reaching the ultimate strain values for tension (26) and shear (27) are adopted:

$${\epsilon }_{bt}\le {\epsilon }_{bt,u};$$

(26)

$${\epsilon }_{b,\Delta }\le {\epsilon }_{sh}$$

(27)

Assuming that the normal stresses acting perpendicularly to the contact joint are small, the ultimate strains in longitudinal shear were set to be the same as those in uniaxial tension of the concrete of the monolithic part of the beam. Thus, for the sections over the length of the beam where condition (27) is not satisfied, Expression (23) for the stiffness of the contact joint is replaced with Formula (24), which reflects the dowel effect in the transverse reinforcement. The formation and opening of the longitudinal crack in the intermediate contact joint are determined through iterative calculation. Similarly, the tensile forces are mainly absorbed by the longitudinal reinforcement when condition (26) is violated. In this case, the tensile and compressive stiffness depends on the steel reinforcement (up to the moment of crack boundary closure). The coefficient ψs in Formulas (20) and (21) varies from 0.2 at the load causing crack formation to 1 at the load leading to the formation of a plastic hinge in the section. For ultimate tensile and longitudinal shear strains, it is conservatively assumed: ε_btu = ε_sh = 0.0002.

In the simplified analytical model, the anchorage of longitudinal reinforcement in both the monolithic and prefabricated parts of the section was considered to be ensured. The absence of inclined cracks in the beams of the experimental frames according to the results of the tests discussed in Section 3 indirectly confirms this assumption. However, the influence of the anchorage of the longitudinal reinforcement on the deformation and failure of layered structures under accident actions should be investigated in detail in the future.

Figure 6 presents a general flowchart of the calculation of a precast reinforced concrete frame system with layered beams using the variational method.

2.3. The Design of the Test Frames and the Method for Experimental Study

The reliability and effectiveness of the proposed calculation model for layered elements of precast monolithic reinforced concrete frames were assessed by testing scaled (1:6) models of such structures. The frames were designed according to the requirements of SP 63.13330 [55]. No special requirements for seismic resistance or a reduction in the probability of disproportionate failure were considered in the design of the frame. The practice of designing reinforced concrete frames with elements of layered sections usually ignores the ductility of the contact joint, assuming that the perfect bond of such a joint is guaranteed by the placement of transverse reinforcement according to the design rules. Therefore, the purpose of the experimental study was to reveal the peculiarities of the resistance of reinforced concrete frames designed without considering special requirements for an exceptional design situation caused by column collapse. The cross-sections and reinforcement parameters were adjusted in such a way that a 40% margin of the frames’ load-bearing capacity was achieved if there was not any local damage or failure of the structural system. The response of such a reinforced concrete system with layered cross-sectional elements, designed with respect to the requirements to reduce the probability of disproportionate collapse, is expected to be investigated in future studies.

The following parameters were used for the prefabricated monolithic experimental frame structures (Figure 7a): beam span (l₀ = 950 (mm)) and floor height (h_r = 500 mm). The overall depth of the beams was 100 mm, while the width was 50 mm. The depth of the prefabricated part of the beams, which was made of B35 class concrete, was 70 mm. The depth of the monolithic part of the beams, which was made of B50 class concrete, was 30 mm. To ensure all of the necessary properties of concrete, as well as its qualitative compaction, the diorite rubble aggregate of 5–7 mm fraction was utilized. This was 1/10 … 1/7 of the size of the cross-sections of the model elements of the reinforced concrete frame. The influence of the scale factor was not considered in the study. The distance between the longitudinal axes of the layers was 50 mm. The longitudinal reinforcement of the beams was composed of B500 steel, with a diameter of 4 mm. The transverse reinforcement of the beams was made of wire A240 steel, with a diameter of 2 mm.

The beam–column assembly of the precast part of the frame was rigid, since the concreting of these structural elements was carried out without interruption. The monolithic part of the beams was concreted after the prefabricated frames were erected and connected to each other at the height of the columns by means of embedded steel parts. In the calculation, the beam–column joint was assumed to be rigid or semirigid depending on the joint flexibility.

Anchorage reliability in the experimental structures of prefabricated monolithic reinforced concrete frames was ensured by inserting longitudinal reinforcing bars of the prefabricated beams into the column and by inserting the reinforcing bars of the upper columns into the monolithic part of the beams of the lower panel-frame element (Figure 7a). The connection of the prefabricated U-shaped reinforced concrete panels to frame elements between each other was made by welding the embedded steel parts installed along the height of the columns. The stiffness of the contact joint between the prefabricated and monolithic parts of the beams at all loading levels was ensured by transverse reinforcement. Before the formation of longitudinal cracks in the contact joint, the shear stiffness was calculated taking into account the stiffness of the concrete intermediate contact area and the dowel effect of the transverse reinforcement. After cracking, only the dowel action of the transverse reinforcement was considered.

To determine the ductility of the frame nodes before and after cracking, the entire frame model was made functionally similar to the full-scale model. Elements and their connections were designed according to the principle of simple similarity to real structures in scale (1:6). Thus, it provides similarity in all coordinates, similarity of boundary conditions, equality of scale factors, equality of the number of determining factors for the model and the original, identity with the model materials, qualitative and quantitative similarity of loads, etc. As shown by Bazant [56], the scale factor influences the pattern of crack initiation and damage propagation in the elements, but the failure mechanisms of the model elements and the real object are quite similar. Therefore, the use of scale models is permissible for analyzing the failure mechanisms and evaluating the applicability of ultimate state criteria.

A special bench was designed and manufactured for use in experimental studies. This consisted of a steel power frame constructed from two channels welded together, support legs, and a mechanical system for transmitting loads (Figure 7b). The RC frame was fixed to the base of the bench by welding the longitudinal reinforcing bars to the channels of the power frame. Thus, the frames were supposed to be rigidly fixed on the test bench. The frame displacements out of its plane were limited by means of double-sided linear ties installed at the beam–column joints.

In order to apply the static load at the initial stage of testing, a mechanical gravity leverage system was employed. The system comprised a leverage, a set of tie rods, and distribution beams that transferred the load to the experimental RC frame. The frame beams were subjected to a static load in the form of two concentrated forces, which were attached at a distance of 150 mm from the struts (Figure 8). The load was applied in incremental steps of 10% of the control load, with a minimum rest period of 10 min between each step. During each stage of static loading, a comprehensive examination of the surface of the experimental frame was conducted. The magnitude of the load, the appearance of cracks, the width of the crack opening, and the magnitude of the deflections of the beams were recorded. These values were recorded at the beginning and at the end of each exposure period. The values of strains in concrete and steel reinforcement were recorded throughout the test using a TECH strain gauge station. The accuracy of load transfer to the RC frame was evaluated through preliminary tests conducted with a DOSM-3.0 mechanical dynamometer.

The following experimental values were measured during the tests: the applied loads to the RC frame; the strains in the concrete and steel reinforcement; the deflections of the beams; the nature of cracking and the width of crack opening; and the nature of structural failure caused by accidental impact. The strain in the concrete and steel reinforcement was recorded in real time using the QMLab rev. 1.4 software package. In total, there were eight stages of static loading and one stage of dynamic loading caused by the removal of the outermost or middle column (strut).

Displacements under static loading were assessed using indicators and deflection gauges with 0.01 mm and 0.001 mm graduations (Figure 9a–d). Displacements under accidental impact were assessed using a Nikon 1J5 (software ver. 1.01) high-speed camera, which allowed frame-by-frame recording of beam movement, crack initiation, and crack opening in pre-marked areas.

The principal attribute of the contemplated structural system is the manner of connection between prefabricated elements. This enables the fabrication of prefabricated components in a factory setting and their subsequent assembly on-site with minimal labor costs while maintaining a high level of reliability of butt joints. The proposed structures offer adaptability of the frame system in the event of sudden changes in force flows resulting from the removal of columns (struts).

The prefabricated and monolithic parts are joined into a whole structure by welding the embedded steel plates located on the faces of the vertical elements and the steel reinforcements of the prefabricated part of the beams. The experimental frames were manufactured in two series, which differed in the location of the column subjected to removal. In the frame of the first series, the outermost column removal scenario was considered. In the frame of the second series, the middle column removal scenario was considered. The RC frames of both series were tested in two phases. In the first phase, the beams of the frames were loaded with a static load applied as a concentrated force in the span of each beam. In the second phase, the frame was additionally loaded by an accidental impact caused by the sudden removal of the outermost or middle column.

3. Results and Discussion

3.1. Experimental Results

One of the main objectives of the experimental studies was to determine the stiffness of the intermediate contact area of the precast monolithic beam under the first and second stages of testing. Figure 10 and Figure 11 show the distribution of experimental strains along the cross-sectional depth of the precast monolithic beam before and after cracking.

The strain analysis (ε_b) in Figure 10a and Figure 11a shows the elastic–plastic behavior of concrete at the most compressed face of the beam. In the first stage of the test, the values of these strains did not reach the value (ε_b0) corresponding to the peak stresses. In the second stage of loading due to accidental impact, the value of these strains approached the ultimate strains (ε_b2).

The strains of the tensile face of the element (ε_bt) exhibited elastic–plastic behavior. After the formation of cracks of the first type Crc1 at step III of loading, the magnitude of strain increments at the subsequent steps of loading increased significantly (Figure 11).

The magnitude of deformations at step VII significantly exceeded the ultimate strains for tensile concrete ε_bt0.

Data on the deformation of the interlayer contact zone of the precast monolithic beam (Figure 12a,b) demonstrated that before the formation of the crack of the second type Crc.2 at step VII of loading, the strain gradient in the contact joint was not significant. Its increase was almost linear dependence. After Crc.2 crack formation at step VII of loading, the strain rate of the intermedial contact zone began to increase sharply, demonstrating nonlinear behavior. The total drift of the prefabricated and monolithic parts of the beam relative to each other amounted to Δ = 0.37 mm. After accidental impact at the second stage of testing, the Crc.2 crack increased by more than two times.

In the frame of the second series, the middle column was removed at the second stage of testing. The strain behavior of the most compressed face of the precast monolithic beam (ε_b) also had an elastic–plastic character. The strains in the compressed concrete at the first stage of testing did not exceed the concrete strains (ε_b0) corresponding to the peak stresses (Figure 10b and Figure 11b). In the second stage of loading, as a result of accidental impact, the value of strain in the compressed concrete slightly exceeded the value of ultimate strains (ε_b2).

The strains of the outermost tensile face (ε_bt) also had elasto-plastic character. After the formation of the first-type cracks, Crc.1, at the V loading step, the magnitude of strain increments for the subsequent loading steps increased significantly (Figure 12a). The magnitude of strains at loading step VIII significantly exceeded the ultimate tensile strain of concrete ε_bt0.

An analysis of the data on the deformation of the interlayer contact zone of the precast monolithic beam for the frame of the II series showed that its qualitative character was similar to the character of the deformation of the contact zone of the precast monolithic beam of the frame of I series. At the same time, when the beam was subjected to static loading in the first stage, there were no cracks in the joint between the precast concrete and the monolithic concrete of the beam. The total drift between the precast and monolithic parts of the beam was Δ = 0.11 mm. After the accidental impact at the second stage of loading, a longitudinal crack of the second type (Crc.2) appeared in the beam at the contact joint. Its opening at this stage of loading was significant, and the total drift between the prefabricated and monolithic parts of the beam (Δ) increased more than five times.

According to the experimental values of load and total drift, Figure 13 shows the relationship “moment (M) vs. drift in the contact joint (Δ)” at the first stage of loading of both I and II series frames. It can be seen that the removal of the outermost column causes a more unfavorable redistribution of forces in the frame, leading to the formation of normal cracks (Crc.2) in the beam and longitudinal shear cracks (Crc.1) in the contact joint between the precast concrete and the monolithic concrete of the beam. In the frames of the second series, where the center column was removed at the second stage of loading, no longitudinal crack (Crc.1) appeared in the contact joint between the precast concrete and the monolithic concrete of the beam. At the maximum load applied in the first stage of the test, the total strains in the contact joint were not significant and did not cause the formation of a shear crack (Crc.1).

The same conclusion is confirmed by the obtained pattern of frame fractures after accidental impact at the second stage of the tests, shown in Figure 14 and Figure 15. A similar result regarding the most unfavorable initial failure scenario was obtained in study [37], which focused on the behavior of reinforced concrete frames during the collapse of an outermost column or a corner column.

The analysis of the changes in curvature allows us to note that the formation of longitudinal shear cracks in the beams of the I and II series frames significantly affected the change in the effective stiffness of the beam sections, as well as the dynamic increase factor and the dissipative properties of the frames under accidental impact. The failure of the I series frame was caused by the delamination of the inert layer contact joint along the formed longitudinal crack with subsequent disruption of the reinforcement anchorage and brittle failure of the beam support zone, as shown in Figure 14. The failure of the II series frame beam was associated with disruption of the reinforcement anchorage and fracture of the most loaded normal section of the beam.

The deflections of the beams of the experimental frames of the first and second series did not exceed 1/300 of the span according to the results of the first stage of tests. Crack opening in the frame elements of the first and second series at the first stage of testing did not exceed 0.2 mm. Thus, at the first stage of testing, the reinforced concrete frames met the serviceability requirements, including durability. However, when transferring to full-scale models or real structural systems of buildings, scale effects should be taken into account, which is a complex problem related to the assessment of deformations and crack opening and requires a separate detailed investigation and justification. The results of the second stage of testing showed the failure of the compressed concrete and significant crack opening, including the formation of through cracks as can be seen in Figure 14b,d and Figure 15b,d. Analyzing the cracking and deformation patterns in the frames of the first and second series according to the results of the two stages of tests, the following can be stated.

After the removal of the corner column in the frame of the first series, a Vierendel truss-type resistance mechanism developed above the local failure zone. This was similar to that observed in [11]. However, as a result of the load transfer through alternative paths, a significant crack opening occurred at the beam–column joint. In the monolithic part of the section, the concrete cracked through, while in the precast part of the section, a small compressed area of the section remained on the underside. As a result of the violation of boundary conditions in the beam, the longitudinal crack propagated to a greater length and had a larger opening in the contact joint area. Failure of the lower support section of the paired mid-row columns then occurred on the second floor of the frame, as shown in Figure 14b.

In the frame of the second series, after the removal of the column of the middle row, there was a significant opening of the normal cracks in the beam–column joints (Figure 14b). The structural joint above the local failure area moved downward more than the depth of the beam section. Normal cracks developed in the beams at the end sections adjacent to the columns of the outermost rows. This indicates the development of catenary forces and unilateral connections between the beams and columns. The transfer of catenary forces from the beam to the column of the outermost row and the increase in the effective length of the column led to its stability failure. A similar result was found in the experimental studies of Pham et al. [57], which considered models of reinforced concrete frames at a scale of about 1:2. Similar to the frame of the first series, the change in boundary conditions in the end cross-sections of the beams caused the formation of longitudinal cracks in the contact joint between the monolithic and prefabricated parts.

Thus, the robustness of the experimental model of the structural system is not ensured at the given load, with the assumed dimensions of the cross-sections, and in the reinforcement scheme. In order to ensure the robustness of the structural system, it is necessary to carry out one of the following measures: reduce the service loads and (or) increase the size of the cross-sections of the elements in order to provide the possibility of load transfer along alternative paths. It should be noted that the formation of longitudinal cracks in the contact joint between the monolithic and prefabricated parts of the beams occurred almost simultaneously with the formation of normal cracks in the adjacent sections. This allows us to assume that the formation of normal cracks causes a sudden change in the boundary conditions from the point of view of the resistance of the shear contact joint. In order to increase the ductility of the contact joint in the areas where normal or oblique cracks may occur under accidental loading, it is advisable to provide for the installation of transverse reinforcement with a smaller spacing. Such a design measure should also have a positive effect on the resistance of normal cracks due to concrete confinement.

Experimental studies of scale models of frames have confirmed the initial hypotheses adopted in the development of the analytical model for the analysis of layered beams of a prefabricated reinforced concrete frame system. Parameters of deformation diagrams as a result of accidental impact, as well as deformations in concrete and the reinforcement of the most loaded cross-sections of beams before and after crack formation under a two-stage mode of their loading, were experimentally obtained.

3.2. Validation of the Proposed Semi-Analytical Model

The experimental data were utilized to plot the relationships between the drift in the interlayer contact joint and the bending moment for the supported cross-sections of the precast monolithic beams of the I and II series frames (Figure 16). The same figure displays the drift values calculated by the proposed model.

The concentrated forces (P = 3.2 kN) were replaced by an evenly distributed load (q₀ = 6.7 kN/m), determined on an energy basis for deflection equivalence between the test model and the calculated model. At the same time, the equivalent load (q₀ = 3.0 kN/m) corresponds to the formation of normal cracks (Crc.1) in the cross-section of the layered beam. Upon reaching the equivalent load value of q₀ = 5.9 (kN/m), longitudinal cracks (Crc.2) were observed in the interlayer contact joint.

The curves obtained in the calculation generally capture the experimental curves as presented in Figure 16a. The discrepancy between the experimental and theoretical cracking moment was 14%. This discrepancy is a margin for bearing capacity. The discrepancy in drift in the interlayer contact joint at the cracking moment was about 33%. The maximum value was obtained in the calculation, which is also included in the stiffness margin.

It was found that the difference between the experimental deflections of the beam and the calculated ones, taking into account the drift in the interlayer contact joint, was 12% for the I series frame and 25% for the II series frame, as shown in Figure 16b. At the same time, the calculated results for the proposed model, taking into account the drift in the interlayer contact joint, were 17% higher than those for the model of a solid beam. Thus, numerical analysis has demonstrated the influence of the interlayer contact joint on the deformation parameters of a precast reinforced concrete frame system with layered beams under normal operation and accidental impact due to column collapse. An important property of the deformation of such frames under a static–dynamic loading mode is the increased deformability of the interlayer contact joint, in which longitudinal shear cracks can form. This may serve as an additional tool for dissipation of accidental impact energy and requires further advanced investigation.

4. Conclusions

This paper presents the results of experimental and analytical studies on the influence of the ductility of the interlayer contact joint between prefabricated and monolithic parts of a beam on progressive collapse behavior under column removal scenarios. The influence of longitudinal normal cracks in the tensile zones on the deformation of such structures before and after the accidental action is considered. Based on the results presented, the following conclusions can be drawn.

• A new constructive system for a prefabricated monolithic frame of a fast-erecting multi-story building made of industrial elements of factory production is proposed. The proposed system includes two types of panel-frame elements of U-shaped and L-shaped profiles, monolithic bracing beams, and floors made of prefabricated multi-hollow slabs.
• For modeling such structural systems under accidental impacts, a two-level design scheme has been proposed. It implies the separation of substructures for advanced analysis. At the first level of simulation, the entire structural system is calculated. In the second stage, the model of the variational method proposed in this paper is used. On the basis of this model, the structural element adjacent to the zone of initial local failure is calculated. The use of the proposed numerical and analytical model in combination with two-level modeling allows one to reduce the time costs for simulation and to obtain the results in a form convenient for interpretation.
• Experimental studies of scale models of frames have confirmed the initial hypotheses adopted in the development of the analytical model for the analysis of layered beams of a prefabricated reinforced concrete frame system. Parameters of deformation diagrams as a result of accidental impact, as well as deformations in concrete and the reinforcement of the most loaded cross-sections of beams before and after crack formation under a two-stage mode of their loading, were experimentally obtained.
• Our analysis has demonstrated the influence of the interlayer contact joint on the deformation parameters of a precast reinforced concrete frame system under accidental impact due to column collapse. An important property of the deformation of such frames under a static–dynamic loading mode is the increased deformability of the interlayer contact joint, in which shear cracks can form.