Seismic tests of post-tensioned self-centering building frames with column and slab restraints

Abstract

Post-tensioned (PT) self-centering moment frames have been developed as an alternative to typical moment-resisting frames (MRFs) for earthquake resistance. When a PT frame deforms laterally, gaps between the beams and columns open. However, the gaps are constrained by the columns and the slab in a real PT self-centering building frame. This paper presents a methodology for evaluating the column restraint and beam compression force based on the column deformation and gap openings at all stories. The method is verified by cyclic tests of a full-scale, two-bay by one-story PT frame. Moreover, a sliding slab is proposed to minimize restraints on the expansion of the PT frame. Shaking table tests were conducted on a reduced-scale, two-by-two bay one-story specimen, which comprises one PT frame and two gravitational frames. The PT frame and gravitational frames are self-centering throughout the tests, responding in phase with only minor differences in peak drifts caused by expansion of the PT frame. When the specimen is excited by a simulation of the 1999 Chi-Chi earthquake with a peak ground acceleration of 1.87 g, the maximum interstory drift and the residual drift are 7.2% and 0.01%, respectively.

Keywords

post-tensioned frame / frame expansion / column restraint / sliding slab / frame test / shake table test

Introduction

Post-tensioned (PT) structures that use post-tensioning steel to compress beams against columns or bridge column segments against a footing have been developed as an alternative to typical earthquake-resisting systems [1-7]. Unlike typical moment connections with beam buckling at high drift levels [8-10], the PT connection deforms based upon the gap opening at the beam-to-column interface. Both the elastic responses of strands and the hysteretic responses of energy-dissipating devices produce a flag-shaped hysteretic behavior of the connection. Analytical studies have shown that, in terms of the maximum interstory drift and residual drift, the seismic performance of a MRF with PT connections exceeds the performance of a MRF with typical welded connections [1,11,12].

Although the newly developed PT connection provides satisfactory cyclic performance, the column and slab restraints on the PT frame expansion are important issues to be resolved before the system is applied to real construction. Gap openings at the beam-to-column interfaces cause expansion of the PT frame. The column and slab restraints that oppose the frame expansion significantly affect the compression force in the PT beam as well as the hysteretic behavior of the system. Kim and Christopoulos [11] suggested an approximation approach to estimate the column restraint, which is only appropriate for cases in which a more concentrated response occurs at a single floor alone and is overly conservative for cases in which the structure responds in its fundamental mode. For a PT frame that responds in its fundamental mode, Chou and Chen [13] proposed a method for evaluating the restraint that considers the continuity and boundary conditions of the column.

When gaps open at the beam-to-column interfaces, the concrete slab restrains frame expansion if the gap does not open near the columns, altering the self-centering behavior [14]. Garlock et al. [15] suggested collector beams or bays for transferring the inertial force from the slab to the PT frame. More recently, Chou et al. [16] experimentally showed that a PT connection with a continuous composite slab self-centers with low residual deformations as long as the metal deck separates near the column and the negative connection moment provided by the slab reinforcement is considered in the design. Chou et al. [17] also demonstrated similar cyclic responses for a bare PT connection and a composite PT connection with a fully discontinuous composite slab that opens freely along with the gap opening at the beam-to-column interface.

This work focused on solving the column and slab restraints in PT self-centering frames under earthquake loading. The work first proposes a method for evaluating the bending stiffness of columns and compression forces in beams based on a deformed column shape that matches the gap opening at each beam-to-column interface. The method is validated by cyclic tests of a full-scale, two-bay by one-story PT frame, representing a substructure of a three-story frame. Secondly, shaking table tests are conducted on a reduced scale, two-by-two bay one-story PT building structure with a slab accommodating the expansion of the PT frame under earthquake scenarios [13].

Design of a three-story post-tensioned self-centering frame

Figures 1(a) and (b) show the plan and elevation of the prototype building, which was assumed to be located on stiff soil in Los Angeles, California. Three two-bay by three-story PT frames provide lateral load resistance in the east–west direction. A reduced flange plate (RFP) proposed by Chou et al. [4,18] was incorporated at each beam-to-column connection to increase energy dissipation (Fig. 2(a)). The RFP behavior and buckling capacity of the structure can be found elsewhere [18]. No energy dissipation devices were used at the PT column base (Fig. 2(b)).

The design dead loads were 5.28 kPa (110 psf) and 4.32 kPa (90 psf) for the floors and the roof while the live loads for the floors and the roof were 2.39 kPa (50 psf). The structural period, T, and the seismic response coefficient, C_s, calculated by the codified method [19] were 0.6 s and 0.125, respectively, so the seismic design base shear, V_des, for one PT frame was 272 kN. The beam and column sizes, RFP thickness (t_R) and narrowest dimension (b_R), strand and PT bar area (A_ST), and initial PT force (T_in) are given in Table 1. A high strength Dywidag (DSI) bar was specified for the column PT bar, and ASTM A 706M steel was specified for the transverse and longitudinal column reinforcements. The specified 28-day concrete strength, f_{cn}^{\text{'}}, was 28 MPa. A572 grade 345 (50) steel was used for the steel beams, and ASTM A416 grade 270 strands each with a diameter of 15 mm were passed along the beam webs and anchored outside the exterior PT columns. The moment demands at the beam-to-column interface and the column base due to the seismic load (M_E), dead load (M_D), and live load (M_L) are also given in Table 1. The decompression moment of the PT connections, M_d, listed in Table 1 is composed of the moments provided by the strands and the RFPs. The decompression moment is larger than the moment due to the dead load and live load, but is slightly less than the combined moment demand, M_dem. The connection moment at the onset of RFP yielding, M_y, is larger than α_yM_dem, where α_y≥10 (Table 1), indicating that the PT frame remains elastic under the code-based seismic load. Following the connection design procedure proposed by Chou et al. [16], the connection moment at a roof drift of 4%, M_4%, reaches about 0.9M_np, of which M_R ≈ 0.3M_np was provided by the RFPs and M_ST ≈ 0.6M_np was provided by the strands. The variable M_np represents the nominal plastic moment capacity of the beam.

Beam force based on PT elements and column restraints

Rotational spring model

Instead of using axial springs at the PT connection, Chou et al. [3,20] proposed a rotational spring scheme to capture the self-centering response of the PT connection. As shown in Fig. 3(a), the intersection of the beam and column centerlines has three nodes j, m, and n. Two zero-length spring elements, connecting nodes j and m, were used to model the bilinear elastic behavior of the PT strands (SC spring) and the bilinear elastoplastic behavior of the RFPs (RFP spring), respectively. A combination of these two rotational springs well predicts the experimental results of a PT connection (Fig. 3(b)). However, the PT force in the strands, the compression force in the beams, and the column-restraining effects were not exhibited in the prior model. The rotational spring scheme can also be adopted to model the self-centering behavior of the PT column. Figure 3(c) shows two nodes j and k at the PT column base. One zero-length rotational spring, connecting the nodes j and k, was used to model the bilinear elastic behavior of the PT column. Before decompression, the elastic rotational stiffness of the PT column, K_c1, is approximated using that of a fully restrained column. After reaching the decompression moment of the PT column, M_d,c, in Fig. 3(d), the rotational stiffness, K_c2, is:

(1)K_{\text{c2}}=\frac{1}{\frac{1}{K_{\text{c1}}}+\frac{1}{K_{\text{cbar}}}},

where the rotational stiffness, K_cbar, is provided by the PT bars in the column.

(2)K_{\text{cbar}}=\left[\frac{d_{\text{c}}}{2L_{\text{bar}}}{E}_{\text{bar}}{A}_{\text{bar}}\left(1-\frac{A_{\text{bar}}}{A_{\text{bar}}+A_{\text{g}}}\right)\right]\frac{d_{\text{c}}}{2},

where d_c is the column depth, E_bar is the elastic modulus of the PT bar, A_bar is the PT bar area, L_bar is the PT bar length, and A_g is the column cross-sectional area.

Bending stiffness of a PT column at connection levels

Figure 4(a) shows the three-story PT frame in a deformed position. As a gap opens at the beam-to-column interface (∆_b1, ∆_b2, and ∆_b3 at the 1st, 2nd, and 3rd stories, respectively), the strands in the beam elongate and result in axial shortening of the beams and bending of the exterior columns CL and CR. This study utilizes a deformed column to develop the bending stiffness at each story. The deformed column shape considers: 1) a gap-opening response at each story above the ground level, 2) beam-bearing locations at the respective column heights, and 3) column base rigidity, K_c2, after the gap opens. Figures 4(b) and (c) show deformed shapes for the two exterior columns CL and CR. A rotational spring with stiffness K_c2 is positioned at the column base to simulate restraining of the base after the gap opens. At each story, column CL has a specified lateral displacement, ∆_b (= θ_g(d_b-t_f)), at a location of the beam top flange inner side, and column CR has the same specified lateral displacement, ∆_b, at a location of the beam bottom flange inner side, where d_b is the beam depth and t_f is the flange thickness. Assuming the same gap-opening angles (e.g., θ_g = 0.01 rad.) at the column base and each connection, the values of 4.84 and 3.07 mm, as marked in the figure, are the lateral displacements ∆_b specified at each story. The bending stiffnesses of columns CL and CR at each story are K_cl and K_cr, respectively, which are computed by dividing the corresponding reaction force, F_cl and F_cr, by the lateral displacement ∆_b. The column bending stiffnesses, K_cl and K_cr, at each story are different and negative at the 2nd and 3rd stories due to a specified column deformation, leading to a reduced compression force in the beams [21].

PT beam compression load

The exterior columns CL and CR bear against opposite sides of beams BL and BR at the same drift, resulting in different bending stiffnesses (or restraints) of the exterior columns of the PT beams. At any story, considering incremental equilibrium equations of horizontal force for columns CL and CR:

(3)\Delta F_{\text{bl}}=\Delta F_{\text{cl}}+\Delta T_{\text{ST}},

(4)\Delta F_{\text{br}}=\Delta F_{\text{cr}}+\Delta T_{\text{ST}},

where ∆F_bl and ∆F_br are the incremental compression forces in beams BL and BR, respectively; ∆F_cl and ∆F_cr are the incremental restraining forces of columns CL and CR, respectively, and ∆T_ST is the incremental strand force. The incremental shortening of the left beam BL due to the increased compression force is:

(5){\delta }_{\text{bl}}={\delta }_{\text{ST}}+{\delta }_{\text{cl}},

where δ_cl is the component of the beam BL shortening due to the column CL incremental restraining force (∆F_cl), and δ_ST is the component of the beam shortening due to the incremental strand force ∆T_ST. The column CL incremental restraining force is:

(6)\Delta F_{\text{cl}}=K_{\text{b}}{\delta }_{\text{cl}}=K_{cl}\left({\Delta }_{\text{b}}-{\delta }_{\text{ST}}-{\delta }_{\text{cl}}\right),

where K_b is the axial stiffness of the beam. Rearranging Eq. (6), the component of the beam BL shortening due to the column CL incremental restraining is:

(7){\delta }_{\text{cl}}=\frac{K_{\text{cl}}}{K_{\text{cl}}+K_{\text{b}}}\left({\Delta }_{\text{b}}-{\delta }_{\text{ST}}\right).

The component of the beam BR shortening due to the column CR incremental restraining can also be expressed as:

(8){\delta }_{\text{cr}}=\frac{K_{\text{cr}}}{K_{\text{cr}}+K_{\text{b}}}\left({\Delta }_{\text{b}}-{\delta }_{\text{ST}}\right).

The ratio of the two beam shortenings due to column incremental restraining effects is:

(9)\zeta =\frac{{\delta }_{\text{cr}}}{{\delta }_{\text{cl}}}=\frac{K_{\text{cr}}\left(K_{\text{cl}}+K_{\text{b}}\right)}{K_{\text{cl}}\left(K_{\text{cr}}+K_{\text{b}}\right)}\mathrm{.}

Since the column bending stiffnesses, K_cl and K_cr, are different, the ratio ζ ranges from 0.7 to 1.9. The strand force increment, ∆T_ST, is:

(10)\Delta T_{\text{ST}}=K_{\text{b}}{\delta }_{\text{ST}}=K_{\text{ST}}\left[2{\Delta }_{\text{b}}-2{\delta }_{\text{ST}}-\left(1+\zeta \right){\delta }_{\text{cl}}\right],

where K_ST is the axial stiffness of the strands. Rearranging Eq. (10), the component of the beam shortening due to the incremental strand force is:

(11){\delta }_{\text{ST}}=\frac{K_{\text{ST}}}{2K_{\text{ST}}+K_{\text{b}}}\left[2{\Delta }_{\text{b}}-\left(1+\zeta \right){\delta }_{\text{cl}}\right]\mathrm{.}

Substituting Eq. (11) into Eqs. (6) and (7), the components of the beam BL and BR shortenings due to the column incremental restraining forces are:

(12){\delta }_{\text{cl}}=\frac{K_{\text{cl}}{K}_{\text{b}}}{\left(K_{\text{cl}}+K_{\text{b}}\right)\left(2K_{\text{ST}}+K_{\text{b}}\right)+\left(1+\zeta \right)K_{\text{ST}}{K}_{\text{cl}}}{\Delta }_{\text{b}},

(13){\delta }_{\text{cr}}=\frac{\varsigma K_{\text{cl}}{K}_{\text{b}}}{\left(K_{\text{cl}}+K_{\text{b}}\right)\left(2K_{\text{ST}}+K_{\text{b}}\right)+\left(1+\zeta \right)K_{\text{ST}}{K}_{\text{cl}}}{\Delta }_{\text{b}}\mathrm{.}

For a specific gap opening ∆_b, the beam BL and BR forces, F_bl and F_br, and the strand force, T_ST, are:

(14)F_{\text{bl}}=T_{\text{in}}+\Delta F_{\text{bl}}=T_{\text{in}}+K_{\text{b}}\left({\delta }_{\text{ST}}+{\delta }_{\text{cl}}\right),

(15)F_{\text{br}}=T_{\text{in}}+\Delta F_{\text{br}}=T_{\text{in}}+K_{\text{b}}\left({\delta }_{\text{ST}}+{\delta }_{\text{cr}}\right),

(16)T_{\text{ST}}=T_{\text{in}}+K_{\text{b}}{\delta }_{\text{ST}}.

Figure 5 shows predictions based on Eqs. (14)-(16), which are close to those obtained from the three-story PT frame model using axial springs (ASs). The tension force of the strands, T_ST, is a function of the initial strand force and elongation of the strands, so the predictions made by the proposed method and those of Kim and Christopoulos [11] are close. The beam compression force is larger than the strand force at the first story and smaller than the strand force at the 2nd and 3rd stories due to the negative column bending stiffnesses [21]. The reduction of the beam compression force from the applied strand force at the 2nd and 3rd stories could not be obtained from the simple estimation proposed by Kim and Christopoulos [11] because the column deformation shape was not considered in developing the bending stiffness. It was also found that although the compression toes in beams BL and BR differ by a beam depth, the compression force variations, K_bδ_cl and K_bδ_cr, show a minor difference as compared to the strand force, T_ST, indicating that the beam-to-column centerline intersection can be used as a compression location for simplicity. In this case, the column bending stiffness at each story, K_mL, can be computed by deforming the column at a specified lateral displacement, ∆_b, with associated reaction forces. The beam axial force is 12% larger than the strand force at the first story and 3-4% lower than the strand force at the 2nd and 3rd stories. For a PT frame with more than two bays in width, the column bending stiffness is still K_mL based on the deformed column shape, but the shortening and axial force in the beam caused by the column restraints need to be considered for each bay [21].

Post-tensioned self-centering frame test

To evaluate the effects of column restraining on the frame expansion, a full-scale, two-bay by one-story PT frame (marked in Fig. 1(b)) was cyclically tested. A total of 12 ASTM A416 grade 270 strands each with a diameter of 15 mm were passed along the beam web through three PT columns and anchored outside the exterior columns. The initial PT forces in the columns and beams were approximately 1100 and 916 kN, respectively; a total of four cyclic tests were conducted on this frame (Fig. 6(a)). Quasi-static cyclic loading with increasing displacement amplitude in accordance to AISC [22] for connection tests was adopted. The displacement of column CC was controlled as a target displacement; the interstory drift was defined as the horizontal displacement at the loading point relative to the column height of 5.66 m. Two loading schemes were adopted in the test program. For the 1st loading scheme, the forces in Act 3 and Act 4 were slaved, respectively, to three-quarter and one-quarter of the summation of the forces in Act 1 and Act 2. Therefore, the shear applied to columns CL and CR would be half of that applied to column CC at the loading point to simulate little restraint on the top of the columns. This loading scheme was carried out for the first three tests, in which RFPs for energy dissipation were only included in the connections for the first two tests. For the 2nd loading scheme, no relative lateral deformation was allowed between the columns to simulate full restraint on the top of the columns. No energy dissipation devices were provided at the column base.

In the first cyclic test, two out of eight RFPs fractured when the frame moved toward an interstory drift of 4% (Fig. 6(b)). The frame was retested using the same loading protocol, and no more RFPs fractured in the 2nd tests. No buckling of the RFPs was observed in the tests. Figure 7(a) shows the base shear versus column CC deformation for the first two tests; it appears that the PT frame under the 1st test dissipated more energy than the 2nd test. Six RFPs were removed from the frame after the 2nd test in order to evaluate the frame response without energy-dissipating devices. The bilinear elastic behavior of the PT frame was observed by comparing the hysteretic loops of the 1st and 3rd tests (Fig. 7(b)). For the 4th test, the PT frame was loaded without any relative column deformation at the level of the actuators, so the post-yielding stiffness of the frame was 20% higher in the 4th test than in the 3rd test. Considering horizontal and vertical force equilibrium in the three columns and assuming a moment equilibrium about the column compression toes, the compression forces in beams BL and BR were obtained from the tests [21]. Figure 8 shows strand forces and compression forces in the beam for the 3rd and 4th tests. The beam compression force is similar to the beam strand force in the 3rd test (Figs. 8(a) and (b)), in which the column top can expand during the test. The two tests resulted in similar strand forces in the beams, but the beam compression force was 60% larger than the beam strand force in the 4th test (Fig. 8(c)) because the distance between each column top does not vary during the test. Following the same procedure described previously, the exterior column bending stiffnesses, K_mL, are 862 and 73366 kN/m for the 3rd and 4th tests, respectively. The resulting beam compression forces based on Eqs. (14) and (15) and K_mL agree well with the test results (Fig. 8).

Sliding slab in a post-tensioned self-centering frame

Figure 1 shows the plan and elevation of the three-story PT frame. The slab details for typical steel moment-resisting frames include: 1) a metal deck that is 1.2 mm thick is continuously placed with 75-mm-deep flutes running parallel to the east–west direction, 2) a #3 longitudinal bar is continuously placed inside each flute, and 3) the top reinforcement is limited to a 100 mm × 100 mm welded wire mesh for controlling concrete shrinkage. The PT frame at each story has two PT beams, one of which is rigidly connected to the composite slab to transfer seismic forces from the floor to the beam. The beams marked in black are positioned along the perimeter of the frame and the center of the frame without using shear studs between the steel beams and the composite slab. Hence, the PT frame can expand freely relative to the composite slab. In addition, a sliding device was positioned between the non-composite beams (shown in black) and the floor beams to reduce slab restraint [13]. A sliding device was also positioned between a PT column and a floor beam for the same purpose. The concrete shear capacity at cracking calculated based on the work of Collins and Mitchell [23] was larger than the required inertia force. Note that in a more realistic situation in which more bays are arranged as gravity frames, the shear capacity of the slab might not be able to carry the lateral force so the combination of a heavier gauge steel decking and a thicker concrete with an underlying system of truss-like members might be more suitable [24].

Shake table test program

Figure 9(a) shows the elevation of the specimen and the experimental set-up. The specimen, which is a scaled subassembly of the prototype frame as indicated by a dashed box in Fig. 1(a), comprises a two-bay by one-story interior PT frame and two exterior gravitational frames, which have pin connections at the column base and beam-to-column connections. A scale model of the PT building structure on the shaking table cannot have a ratio greater than 1∶2.5, so the length and mass scale factors are S_L = 0.4 and S_{\text{L}}^2=0.16, respectively. Table 2 shows the scaling ratios for the sectional modulus S, size, and force associated with the structural members. Table 3 lists the program used in the earthquake simulation tests. The acceleration histories of the table were components of ground motion in the 1994 Northridge and 1999 Chi-Chi earthquakes.

Test results

The specimen was tested nine times in Phase I and eight times in phase II (Table 3). Figure 10 shows the relationship between the peak responses and the maximum drifts for each test. The hysteretic response was linear-elastic for a base shear below 125 kN (Fig. 10(a)). Only three phase I tests resulted in nonlinear behaviors of the frame when the peak ground acceleration (PGA) was greater than 0.93 g. The phase II tests, except for two with PGAs of 0.49 and 0.65 g (Table 3), presented nonlinear responses of the frame, indicating that a nonlinear response of the specimen was achieved. Figure 10(b) shows that the PT frame expansion has an approximately linear relationship with the drift. Figure 11(a) shows the displacement responses of the PT frame for PGA = 1.15 g and 2.01 g; the maximum drift increased with the PGA. The PT frame reached a maximum interstory drift of 3.9% during the sixth test with PGA = 1.97 g (Fig. 11(b)), causing the fracture of one RFP that was welded to column CR. Fracturing of the RFP was unexpected because the maximum strain on the RFP as computed from the measured elongation was 13.6%, less than the ultimate strain (= 18.6%) determined from the coupon tensile test. The damage to the RFP was caused by cumulative inelastic strain over several tests. The PT frame was retested using the same record with a lower PGA = 1.87 g. The maximum interstory drift of the PT frame was 7.2% (Fig. 9(b)), much larger than the 3.9% that was induced by the same ground motion with a larger intensity (Fig. 11(b)), indicating that the loss of RFPs or PT forces increased the interstory drift. The initial PT force in the beam and column decreased by about 30% after the eighth test (phase II) due to local buckling of the bottom beam flange near the column face [13]. No slack in the PT tendon was observed during the test. The specimen re-centered in all of the tests because the strands in the beams and the PT bars in the columns remained elastic throughout the tests.

Figure 12 shows the displacement histories of the PT frame (LL3), the slab (LL4), and the gravitational frame (LL5) for a specific test. The displacement histories show minor residual deformations of the specimen under the Northridge motion with floor acceleration is 2.01 g, which exceeds the spectrum acceleration of the design-based earthquake level [13]. Generally speaking, the displacement history of the PT frame was almost in phase with that of the floor slab and the gravitational frame. At 3.8 s, when the specimen reached a peak lateral deformation (79 mm), the lateral displacements measured by displacement transducers LL3, LL4, and LL5 differed by 4 mm (5% difference), indicating that the PT frame could expand easily with minor restraints from the slab.

Conclusions

This paper presents a methodology for evaluating the effects of column restraint resulting from gap openings at beam-to-column interfaces and eliminating slab restraints to the PT frame expansion. The method includes the following: 1) determining column deformation in accordance with specified lateral displacements ∆_b at all beam-to-column interfaces along the column height, 2) computing the column bending stiffness at each story using the reaction force divided by the specified lateral displacement, and 3) computing beam shortenings and compression forces based on a specified ∆_b. Because the column bending stiffness is obtained by a deformation that matches the frame expansion (Fig. 4), the beam compression force estimate is more accurate than that calculated based on a pin-pin supported column boundary condition. The predicted beam compression force is validated by a cyclic analysis of a three-story PT frame, which is modeled with numerous axial springs in connections to capture the gap-opening behavior, and by cyclic tests of a full-scale, two-bay by one-story PT frame, which represents a substructure of the three-story PT frame.

The proposed slab in PT self-centering buildings has 1) a rigid connection between the slab and the beam in a single bay of the PT frame, 2) roller-supported connections between the slab and the PT beams in other bays, and 3) rigid connections between the slab and the floor beams. A device is installed between the floor beams and the non-composite PT beams to permit sliding between the PT frame and the gravitational frames. Shaking table tests were conducted on a two-bay by one-story building structure to examine its seismic performance. The displacement response of the PT frame was almost in phase with that of the gravitational frame. Splitting cracks near the column face, which have been found in previous slab investigations [16,17], were not found in this slab during any of the tests, so the slab details proposed in this study enhanced the serviceability and allowed expansion of the PT frame.

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