Experimental and nonlinear finite element analysis for post-tensioned precast frames with mechanical joints

ABSTRACT Conventional cast-in-place frames using a pair of steel plates are replaced by a post-tensioned H-shaped precast frame with mechanical joints in which plates are used to transfer loads between them. The mechanical joints strengthened by interior bolts are used, providing a flexural capacity of columns to a level similar to that of conventional monolithic connections. In the present study, strain evolutions of the proposed frames are explored using a finite element analysis with a concrete plasticity model (CP) applying a 0% damage index representing undamaged concrete for the first floor and a concrete damaged plasticity model (CDP) model implementing 80% damage index for the second floor when using ABAQUS. FEA-based strain evolutions are, then, verified by experimental results obtained from full-scale tests, showing agreements except for places where gauges malfunctioned. This study identifies structural behaviors and contributions of each structural element of the mechanical joint to flexural capacity. Microscopic strain evolutions of the post-tensioned H-shaped precast frames with mechanical joints agreed with the test data. Both experimental and numerical investigations demonstrate the structural performance of the proposed prestressed frames with mechanical joints is predictable and stable under service loads. GRAPHICAL ABSTRACT


Introduction
Precast (PC) concrete is widely used for various structures in civil, residential, and industrial fields since it provides rapid, high-quality, and convenient constructions. Prestressed (PT) concrete is an excellent solution for long-span structures, significantly reducing cracks and deflections. Applying PT, thus, provides smaller section sizes compared with ordinary reinforced concrete (RC). PC constructions can benefit from PT because members are lighter, enhancing speeds and efficiencies of transportation and erections. However, connecting PT members is much more complicated than RC components. This difficulty in joining members leads to numerous studies. For example, Wang, Marino, and Pan 2019) conducted experiments of a novel beam-to -column joint using steel jackets and steel angles for PC frames with PT beams, resulting in good strength and ductility. In 2021, (Li et al. 2021) compared behaviors of cast-in-place frames and PC frames with a grouted beam-to-column junction. Tests indicated that the two specimens had similar vertical resistances, but the ductility of PC frames was lower.
Similarly, satisfactory resistances were reported for PC frames using grouted beam-to-column connections by in (Li et al. 2022), mild-press joints by in (Nakano et al. 2004), ductile connections by in (Pampanin et al. 2006), post-pouring joints by in (Yan et al. 2018), and bolted connections by in (Song et al. 2015). In 2016, Maurizio and Lorenzo conducted a full-scale experiment on PC columns in (Orlando and Piscitelli 2018), exploring the behaviors of flanged union joints. Results indicated good strengths and stiffness of the investigated joints. Mechanical joints for columns similar to those used in the present study were extensively investigated for RC and composite structures in the study of (Nzabonimpa, Hong, and Kim 2017a;Nzabonimpa and Hong 2018), and (Nzabonimpa, Hong, and Kim 2017b). A novel mechanical connection for PC members was proposed in (Nzabonimpa, Hong, and Kim 2017a) by J. D. Nzabonimpa, Won-Kee Hong, and Jisoon Kim, developing a faster and more economical solution for beam-to-column and column-to-column joints using laminated metal plates. In 2018, (Nzabonimpa and Hong 2018) investigated the performances of detachable joints using metal plates connecting PC composite columns, achieving optimal designs. In 2017, (Nzabonimpa, Hong, and Kim 2017b) conducted a nonlinear FEA analysis of novel mechanical beam-column joints, resulting in a good match between analytical and experimental results, which indicated the accuracy of FEA models.
A significant number of papers developed beamto-column joints for PC frames, achieving performances similar to those of monolithic frames. The present study provides a novel PC frame where unbonded PT beams and columns are cast monolithically, while columns are mechanically connected, offering easy erections. Investigations of a structure similar to one presented in this study are difficult to find in the previous literature. H-shaped frames can be rapidly erected on footings or other frames using mechanical joints between columns as shown in Figure 1. In this study, strain evolutions of tendons, rebars, structural steels, and concrete are investigated experimentally and numerically, exploring behaviors of frames from the onset of cracks to failures. Results show that frame performances are predictable and stable under service loads, indicating that the proposed structure can replace conventional cast-in-place frames. The present study can be used to generate big datasets for ANN-based building designs and constructions, as shown in the books        Figure 2 shows a test setup with two point-loads; Figure 2(a,b) show test schemes for the first and second floor, respectively. It is noted that a load span (distances between concentrated loads) of 3 m for the test is recommended by (Test, Drilled, and Concrete 2010   A similar design is used for columns of both frames, where section size is 400mm � 500mm with 6ϕ19 þ 4ϕ16, as shown in Figures 3(b) and 4(b), respectively. Two specimens also share the same beam dimensions of 450mm � 300mm, but the first floor uses 3ϕ13 for both top and bottom of the beam (Figure 3(c,d)), while rebars of 2ϕ13 are placed for top   and bottom of the beam on the second floor (Figures 4 (c,d)). Two 15.2 mm unbonded tendons are placed with parabolic profiles on both frames. However, a tendon drape of 360mm is given for the first floor, while a tendon drape for the second floor is 280mm. A calculation based on ACI 318-19 (ACI 318, 2019) indicates a safety factor of 1.18 and 1.57 on the first and second floors, respectively.

Experiment setup
A loading protocol is established up to 210mm based on FEMA 461 (461, F 2007) as shown in Figure 5(a). During a test, each displacement is repeated twice until the specimens fail. Repeating loads at the same displacement stabilizes specimens before escalating load level. A first displacement of 4 mm is taken as 0.057% of a span according to FEMA 461 (461, F 2007), and in the successor series of cycles, the displacement is 1.4 times the previous series. Besides, the loading protocol is revised by adding two load steps at 65 mm and 90 mm into the loading protocol to reduce the gap between two consecutive load steps as shown in Figure 5(a). The experiments of the first and the second floors are finished at a stroke of 150 mm and 169 mm due to specimen fractures, as shown in Figure 5(c,b), respectively.

Finite element analyses (FEA) modeling
A summary of FEA models for two frames is illustrated in Figure 6, where frames are analyzed using quarter models with two symmetric planes as shown in Figure 6(a). Figure 6(b) presents mechanical joints and rebars of ABAQUS models of the test specimens. Figure 6(a,b) also denote nine locations where strains are investigated in Section 4. Essential parameters of an ABAQUS model are listed in Table 1     accuracies of these parameters are proven in studies by (Nzabonimpa and Hong 2018), (Nzabonimpa, Hong, and Kim 2017b), and (Hu, Hong, and Park 2017).

Interaction properties and meshing principles
Interactions of elements in FEA models are demonstrated in Figure  occur. An interaction between unbonded tendons and concrete, as shown in Figure 7h, is defined by a hard contact in a normal direction combined with frictionless property in a tangential direction.
Correlations between rebars and concrete, shown in Figures 7i,j, and k, are modeled by embedded constraints. Figure 7l shows a tie constraint between a tendon anchor and an outer face of a column, preventing a tendon from shortening. Tie constraints are also applied to attach column rebars to steel plates, as shown in Figure 7m, and fix a loading jig to beam surfaces, as shown in Figure 7n.Summary of interactions.(continud). A structured meshing technique is applied in the models, generating hexagonal-shaped elements. Element type C3D8 is used, defining deformed shapes of brick-shaped elements by displacements of eight nodes.

Parameters in defining CDP 3.2.1.1. Damage index.
The present paper uses nonlinear material models for all elements, including concrete, rebar, stirrups, tendon, and structural steel. Concrete damaged plasticity (CDP) models shown in Figure 8 with different levels of damage indexes calculated by Equation (1) are used for the two frames. As shown in Figure 8(c), a damage index of 0% is adopted for the first floor, while a damage index of 80% is applied to the second floor, accounting for the strength reductions in the second floor due to initial damages in footings caused by the test of the first floor. The same footings are used for both tests of the first floor and second floor. Initial cracks affected strengths of the second floor even though footings were rotated to close cracks by placing them on a compression side. On the first floor, the ABAQUS with 0% damage index matches the test results while the ABAQUS with 80% damage index matches test results of the second floor, as shown in Figure 24. Where,

Angle of dilation (Hong 2019).
The angle of dilation, which is assumed constant during plastic yielding, controls an amount of plastic volumetric strain developed during plastic shearing. (Malm 2006) proposed dilation angles for a brittle behavior of concrete which can cause significant volume change, called dilatancy. The inelastic strains are caused by the plastic distortion for a brittle material. In Figure 9, non-associated plastic flow potential shows a dilation angle indicating a volume change caused by a confining pressure within the material, which is used to predict a plasticization of the sections. Malm (2006) found that a brittle behavior was observed for low dilation angles, while higher dilation angles close to 56° provided a ductile behavior (Malm 2006) based on their investigation of an influence of dilation angles on concrete damaged plasticity models in the range between 10° and 56.3°. (Malm 2006) found that the best fit for the numerical investigations of reinforced concrete structures was obtained using dilation angles between 30° and 40°. In the present numerical study, a dilation angle of 30° provides an acceptable match with test data as shown in Figure 24.

Eccentricity (Hong 2019). The eccentricity
lets hyperbolic plastic potential functions approach an asymptote at the uniaxial tensile stress as shown in Equation (2) and Figure 9 where dilation angles increase more rapidly at the lower confining pressure when an eccentricity is large (Hong 2019). Where, However, a constant curvature and a constant dilation angle are implemented over the wide range of the confining hydrostatic pressures when an eccentricity is small. When eccentricity is equal to zero, a constant dilation angle with a constant curvature is obtained over the entire range of the confining hydrostatic pressures, leading to flow potential that is independent of uniaxial tensile stress. In the present numerical study, an eccentricity of 0.1 is used as recommended by ABAQUS default value (Abaqus, Dassault Systemes, ABAQUS Analysis User's Manual 6.14-2 2014).

Ratio of second stress invariant on the tensile Meridian (K c ).
The shape of the yield surface in the three-dimensional space shown in Figure 10(a) is defined by the ratio (K c ) of the second stress invariant in the tensile meridian to that of the compressive meridian, which must satisfy the condition 0:5 < K c � 1:0. The default value of K c in the Abaqus analysis is 2/3 (Abaqus, Dassault Systemes, ABAQUS Analysis User's Manual 6.14-2 2014).

3.2.1.5.
A ratio of f b0 =f c0 . The yield function in-plane stress subjected to biaxial stresses is shown in Figure 10 (b). The ratio of f b0 =f c0 indicates ratio of biaxial compressive concrete strength to uniaxial compressive strength, i.e., a ratio of the strength in the biaxial state to the strength in the uniaxial state as shown in σ b0 σ c0 ¼ f b0 f c0 of Figure 10(b). The ratio σ b0 σ c0 ¼ f b0 f c0 is kept as a default to avoid the overestimation of the structural response of the frames with mechanical joints shown in this study.
The ratio of f b0 =f c0 is set to a default value of 1.16 in the numerical analyses of this study. The ratio of f b0 =f c0 decreases for higher strength because high strength concrete is less influenced by the confining effects, and hence, an increase of σ b0 σ c0 ¼ f b0 f c0 ratio is related to a decrease of the flexural capacity of reinforced concrete members. 2.1.6. Viscosity (Hong 2019). A viscoplastic strain rate tensor expressed in Equation (3) was originally proposed by (Duvaut and Lions 1972) to stabilize the concrete damaged plasticity model: where μ denotes a viscosity parameter. The plastic degradation (d) and the rate of the plastic fracture decrease when a viscosity (μ) increases as shown in Equation (4). The convergence of the numerical process is, thus, enhanced by increasing viscosity parameter (μ) which decreases the rate of the plastic fracture as shown in Equation (4). This will help a convergence of the nonlinear analyses to be achieved when concrete elements experience numerical instabilities. The default value of viscosity (μ) in the numerical analysis of this study is 0.003 (Abaqus, Dassault Systemes, ABAQUS Analysis User's Manual 6.14-2 2014), however, viscosity (μ) can be increased until a convergence of the nonlinear analyses is met. The author found that viscosity (μ) may influence solutions the nonlinear analyses. Figures 8 and 12 show the stress-strain relationships from the sample tests (Figure 11), being implemented in calibrating non-linear FEA analysis based on both a concrete plasticity model (CP model with 0% damage index) for the first floor and a CDP model with 80% damage index for the second floor using ABAQUS in the present study, demonstrating the behavior of the test frames against the loads described in Section 4 are accurately predicted by FEA. Material properties in ABAQUS calibration taken based on the sample tests are summarized in Table 2.  Figure 14(b), is due to a malfunction of the strain gauge. Generally, strain should be developed gradually like one shown in Figure 14(a). Measured strains of the first floor decrease from 0.0025 at Position 1 corresponding to a mid-span deflection of L/160 due to concrete crushes at beam ends (Figures 14(a)) and 0.0023 at Position 2 corresponding to a mid-span deflection of L/84 concrete crushes at mid-span (Figures 15(a)). Strains of the second floor decrease from 0.007 at Position 1 corresponding to a mid-span deflection of L/125 due to concrete crushes at beam ends (Figures 14(b)) and 0.002 at Position 2 corresponding to a mid-span deflection of L/60 due to concrete crushes at mid-span of (Figures 15(b)). However, numerical strains calculated by ABAQUS continue to increase. Strain evolutions indicate large deformations at beam ends due to excessive rotations under heavy loads.

Strain evolutions
Rates of concrete strains at Position 2 (200 mm at mid-span of the first floor) over the displacements match well those obtained numerically up to a midspan deflection of L/90 for the first floor (Figure 15(a)) and L/120 for the second floor (Figure 15(b)). This indicates that ABAQUS models based on FEA with a CP model of 0% damage index for the first floor and a CDP model of 80% damage index for the second floor predict frame performances of both floors as shown in load-displacement relationships shown in Figure 24. Strains at mid-span are lower than those obtained at beam ends, showing that plastic hinges appear at beam ends first, and then they are developed at mid-span. Figure 16(a) for the first floor, concrete compressive strains at Position 3 (above the mechanical joint) shown in Figure 13(c) are measured as small as 0.00075 corresponding to a mid-span deflection of 3 L/400 which matches well those calculated numerically. However, an initial stiffness from ABAQUS is higher than that that measured experimentally. Stiffness calculated numerically decreases rapidly from a mid-span deflection of L/450 whereas measured stiffness decreases gradually from a mid-span deflection of 3 L/400. This is because moments are released at a beam-column joint after plastic hinges form. Figure 16 Compressive strains of lower rebars at a beam end (Position 6 of Figure 19) at the first and the second floor reach a ductility of 6.5 and 4.6, respectively. Compressive strains are not measured well at beam ends (Position 6 of Figure 19) because gauges and rebars were detached at a large normalized strain of 6.5 corresponding to a mid-span deflection of L/87 on the first floor, after which strains decrease as shown in Figure 19(a), whereas strains stop being measured at a large normalized strain of 4.5 corresponding to a mid-span deflection of 13 L/500 on the second floor as shown in Figure 19(b). The numerical predictions of compressive strains match experimentally measured compressive strains relatively well except for at Position 6 at beam ends shown in Figure 19. It is better to interpret Figure 19(b) by an evolution of numerical strain because measured strains are developed abnormally. Numerical compression strains are small up to a mid-span deflection of (160mm), large numerical strains only occur at large mid-span deflections greater than L=50 (160mm), high compression strains at these large deflections are caused by excessive rotations in a plastic hinge at beam ends. This is a commonly seen behavior in frames under vertical loads. Measure strains and numerical strains on both the first and second floors report high compression strains in bottom rebars at beam ends corresponding to large deflections, as shown in Figures 19(a,b). Gauges on the second floor may not correctly capture strain evolutions in Figure 19(b) because strain magnitudes are high; a similar observation is also found at a large deflection of L=87 on the first floor shown in Figure 19(a). Rebar strains show that all rebars in beams reach yielding strains, indicating that the proposed structure can utilizes all rebar strengths.

As shown in
Strains of rebars obtained at Position 8 of Figure 13 (f) are shown in Figure 21. On the first floor, the initial ascending slope of the tensile strains (stiffness) in column rebars at mechanical connections (Position 8) obtained from ABAQUS is similar to that obtained from the test up to a mid-span deflection of L/200 as shown in Figure 21(a). Measured strains start descending at a mid-span deflection L/125 whereas a descending strain branch predicted by ABAQUS does not exist. The slope of the tensile strains (stiffness) obtained by ABAQUS decreases at a mid-span deflection of L/200. On the second floor, the initial ascending slopes of the tensile strains (stiffness) calculated numerically and measured experimentally are similar up to a mid-span deflection of L/200 as shown in Figure 21(b) whereas the slope of the tensile strains (stiffness) calculated by ABAQUS also decreases from a mid-span deflection of L/200. Strains measured experimentally descend slightly. Figure 22 illustrates strains of the metal plates connecting columns measured at Position 9 shown in Figures 6(b) and 13(g), demonstrating deformations of structural steel at a position where rebars are welded to plates. On the first floor, the initial ascending slope of the tensile strains (stiffness) in a metal plate obtained by ABAQUS matches that obtained from the test up to a mid-span deflection of L/240. The initial ascending slopes of the tensile strains (stiffness) obtained from ABAQUS and the test start decreasing at a mid-span deflection of L/240 and 9 L/1000, respectively, because moments are released at beam-column joints by the developments of plastic hinges. Small strains in mechanical plates indicate that deformation of plates under testing load are negligible.
On the second floor, the initial ascending slopes of the tensile strains (stiffness) resulted from ABAQUS and the test decrease at a mid-span deflection of L/200 and 9 L/1000, respectively, because moments are released at beam-column joints by developments of plastic hinges. The tensile strains calculated by ABAQUS continue to increase, while that measured experimentally displays no such ascending trend from a mid-span deflection of 9 L/1000. The early ascending slopes of the tensile strains (stiffness) calculated numerically and measured experimentally are close up as shown in Figure 22(b). Figure 23 shows tensile strains of tendons vs. a deflection at mid-span (Position 4b of Figure 13(d)).     However, gauges on the first and the second floors malfunction at a strain of 0.39 times the yield strain and 0.145 times the yield strain, respectively. Measure tensile strains of tendons start from 0 at the beginning of the test, and hence, strains caused by pre-stressing are excluded from gauge data. Incremental strains are presented in Figure 23 because pre-strains are not measured. It is demonstrated that the initial ascending slope of the tensile strains (stiffness) resulted from ABAQUS is higher than that measured by the test decrease for both floors due to malfunction of gauges attached to tendons damaged during prestressing and loading.
As shown in Figure 24(a), FEM analyses terminate due to numerical instability when maximum compressive concrete strain reaches 0.04 and maximum rebar strain equal to 21.4 times yield strain. Load-deflection relationships calculated by FEA and measured during the test match well, as shown in Figures 24(a,b). Figure 25 compares an FEA-based load-deflection curve of a monolithic frame with that of a frame having mechanical joints. Results indicate that performances of the two models are relatively similar up to a mid-span deflection of L/60, which is much greater than a limitation of L/360 required by ACI 318-19. Figure 26 shows behaviors of specimens under experimental loads. Figure 26(a) indicates four key events during tests of the first floor, where camber cracks at beam ends occurred at a load of 74 kN, leading to a mid-span displacement of L/2600. Midspan cracks were observed at a load of 188 kN, leading to a mid-span displacement of L/943. Excessive rotations were recorded at the maximum load of 667 kN, indicating plastic hinges when a mid-span displacement reached L/91. Finally, rebar fractures were detected at a vertical load of 396 kN, reporting a midspan displacement of L/43. Tests of the first and second floors used the same pair of footing. Pedestals, therefore, were initially damaged at the beginning of a test on the second floor, as shown in Figure 26(b). These initial cracks slightly affected strengths of the second floor even though footings were rotated to close cracks by placing them on a compression side. Mid-span cracks were found on the second floor at a vertical load of 223 kN, when a mid-span deflection reached L/680. The frame witnessed relatively large rotations at beam ends when vertical loads reached the maximum magnitude of 509 kN at a mid-span deflection of L/60. The frame was severely damaged at the final stroke of a 370 kN vertical load at a mid-span deflection of L/38.

Conclusion
In this study, post-tensioned precast frames with mechanical joints are investigated numerically and verified experimentally. American codes are used to design the proposed frames. The finite element model is calibrated by test data, predicting rates of strain increases of the post-tensioned precast frames with mechanical joints. ABAQUS are used to obtain strain evolutions of the structural components which are validated by the structural tests. Modeling techniques simulating the frame tested in this study are also presented. Nonlinear FEA based on CDP models of 80% damage index for the 2 nd floor and 0% damage index for the 1 st floor demonstrate good matches for the strains. Weak matchings are mainly caused by the gauge malfunctions due to the detachments of gauges from rebars at high strain magnitudes (4.5 to 5.5 times the yield strain of rebars) at large deflections. Concrete gauge malfunctions are also caused due to concrete cracks. There are differences between ABAQUS and test results which might be caused due to imperfections in specimen manufacturers, deviations of test settings, and errors in gauge readings. The following conclusions are drawn from this study: 1) Code-based calculations predict a safety factor of 1.18 for the first floor and 1.57 for the second floor, whereas experiments of the first and second floors result in ultimate strengths of 669kN corresponding to a safety factor of 2.17 and corresponding to a safety factor of 2.35, indicating that the proposed PC frame displays strengths exceed those expected by design calculations, demonstrating conservative designs and performances of the proposed PC frame which is post-tensioned precast frames with mechanical joints. FEA-based load-deflection curve of a monolithic frame is compared with that of a frame with mechanical joints, resulting in a similar performance in two models up to a midspan deflection of L/60, which is much greater than a limitation of L/360 required by ACI 318-19.
2) Both behaviors of microscopic-levels and strengths of the proposed frame element are explored based on an experimental investigation using 96 strain gauges at each floor to capture strains in tendons, rebars, concrete, and connecting plates.
3) Interactions for FEM models are demonstrated with hard contacts, frictionless interactions, tie constraints, and embedded constraints which were used to model prestressed frames. A numerical investigation yields results similar to those measured during experiments, demonstrating accuracies of FEA models. Strain evolution and load-displacement relationships are predicted well except for those measured at the gauges which malfunctioned. The presented modeling parameters can be used for designing post-tensioned precast frames with mechanical joints. 4) Both numerical and experimental strains display small deformations of mechanical connections, where the maximum recorded strain of the plates on the first and the second floor do not yield, showing that they are only 0.25 times the yield strain and 0.13 times the yield strain, respectively. Column performances with mechanical joints similar to those of columns with monolithic joints are, therefore, achieved. 5) Numerical investigations predict the behavior of the proposed pre-tensioned PC frames which are predictable and stable under service loads, and hence, it can be used as an alternative to conventional monolithic castin-place frames, offering a convenient erection while eliminating wet beam-column joints that require time to cure.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Notes on contributors
Dr. Won-Kee Hong is a Professor of Architectural Engineering at Kyung Hee University. Dr. Hong received his Master's and Ph.D. degrees from UCLA, and he worked for Englelkirk and Hart, Inc. (USA), Nihhon Sekkei (Japan) and Samsung Engineering and Construction Company (Korea) before joining Kyung Hee University (Korea). He also has professional engineering licenses from both Korea and the USA. Dr. Hong has more than 30 years of professional experience in structural engineering. His research interests include a new approach to construction technologies based on value engineering with hybrid composite structures. He has provided many useful solutions to issues in current structural design and construction technologies as a result of his research that combines structural engineering with construction technologies. He is the author of numerous papers and patents both in Korea and the USA. Currently, Dr. Hong is developing new connections that can be used with various types of frames including hybrid steel-concrete precast composite frames, precast frames and steel frames. These connections would help enable the modular construction of heavy plant structures and buildings. He recently published a book titled "Hybrid Composite Precast Systems: Numerical Investigation to Construction" (Elsevier).