Analysis on DDBD Method of Precast Frame with UHPC Composite Beams and HSC Columns

Abstract

Precast concrete frames integrating ultra-high-performance concrete (UHPC) beams and high-strength concrete (HSC) columns offer exceptional seismic resilience and construction efficiency. However, a performance-based seismic design methodology tailored for this hybrid structural system remains underdeveloped. This study aims to develop and validate a direct displacement-based design (DDBD) procedure specifically for precast UHPC-HSC frames. A novel six-tier performance classification scheme (from no damage to severe damage) was established, with quantitative limit values of interstory drift ratio proposed based on experimental data and code calibration. The DDBD methodology incorporates determining the target displacement profile, converting the multi-degree-of-freedom system to an equivalent single-degree-of-freedom system, and utilizing a displacement response spectrum. A ten-story case study frame was designed using this procedure and rigorously evaluated through pushover analysis. The results demonstrate that the designed frame consistently met the predefined performance objectives under various seismic intensity levels, confirming the effectiveness and reliability of the proposed DDBD method. This work contributes a performance oriented seismic design framework that enhances the applicability and reliability of UHPC-HSC structures in earthquake regions, offering both theoretical insight and procedural guidance for engineering practice.

1. Introduction

Compared to force-based design (FBD) method, performance-based seismic design method facilitates the implementation of the concept of “multilevel seismic protection”. Direct displacement-based design (DDBD) method is one of the primary branches of performance-based design [1]. Priestley et al. [2] converted the multi-degree-of-freedom (MDOF) system into an equivalent single-degree-of-freedom (SDOF) system, and designed a structure based on the target displacement. It was the first complete proposal of DDBD method. In terms of designing the lateral displacement curve, Al-Mashaykhi et al. [3] transcended the limitations of traditional first-order mode assumptions by innovatively introducing higher-order vibration modes correction coefficients. Researchers employed a multi-parameter variable analysis method to determine the optimal displacement profile model. This approach significantly enhanced the accuracy of displacement predictions [4,5,6]. Kalapodis et al. [7] introduced the concept of a modal response modification factor derived from modal analysis, facilitating an effective transition from elastic modal displacement to inelastic displacement profiles. In terms of the equivalent linearization method, Xu et al. [8] systematically evaluated the limitations associated with the application of four traditional equivalent linearization methods in self-centering systems. They established an empirical equivalent linearization model that incorporates the self-centering characteristics. Meanwhile, Muho et al. [9,10] introduced significant improvements to the DDBD method by innovating the use of an MDOF-equivalent system, as opposed to the conventional SDOF system. This innovation effectively addresses the simulation challenges posed by higher-order modal effects and geometric nonlinearities. In the establishment of displacement response spectrum, Wu et al. [11] revealed the predictive biases of various models under different site conditions. Orellana et al. [12] introduced probabilistic seismic risk analysis into the calculation of damping factors, thereby correcting the elastic displacement spectrum. Li et al. [13] focused on the characteristics of near-fault pulse-type ground motions and innovatively proposed a method for constructing period-normalized equivalent ductility displacement spectrum. Zhao et al. [14] conducted scientific calibration of displacement spectrum parameters by employing mathematical statistics methods based on real-time seismic monitoring data. Researchers have analyzed the impact of different parameters on the seismic performance and earthquake vulnerability of structures designed using two methods: DDBD and FBD. The parameters include the span and height of the structure [15], site conditions and the number of stories [16]. Comparative studies have revealed the superiority of the DDBD method. Furthermore, scholars conducted research on the DDBD method across various structural forms. Through case studies, it has been verified that this method can meet the requirements of different performance levels. The structural forms included frame structures [17,18], self-resetting structures [19,20,21], and hybrid structures [22,23,24,25,26]. Moreover, machine learning (ML) has proven effective for rapid seismic evaluation and damage prediction, such as assessing the vulnerability of reinforced concrete buildings by identifying key factors like material properties [27], and predicting structural damage using ground motion intensity measures and initial damage states [28]. Meanwhile, traditional strengthening methods, including passive damping systems, remain valuable for enhancing seismic resilience, especially in challenging site conditions like soft soil [29].

Precast concrete structures demonstrate significant advantages including rapid construction, enhanced quality control, reduced on-site labor requirements, and minimized environmental impacts, making them a crucial choice for modern industrialized building systems [30,31]. Ultra-high-performance concrete (UHPC) represents a groundbreaking advancement in cement-based materials, extensively studied for its superior properties [32,33,34]. Its performance stems from optimized particle packing and the pozzolanic reaction of silica fume, which refines microstructure and enhances hydration [35]. Recent advances include multi-agent collaborative and data-driven methods for designing UHPC with solid wastes [36], further promoting its exceptional mechanical properties and durability for structural applications. UHPC composite beams and columns can achieve remarkable improvements in static performance [37,38,39], while UHPC beam-column joints significantly enhance the moment transfer efficiency and crack resistance of frame structures [40,41]. At the system level, our previous research [42] proposed a precast UHPC and high strength concrete (HSC) frame that consists of composite beams with precast U-shaped UHPC permanent beam formworks and cast-in-place Normal strength concrete (NSC) beam cores, precast HSC columns, and cast-in-place UHPC beam-column joints. Experimental studies under cyclic lateral loading reveal that the UHPC-HSC frame exhibits a mixed sidesway mechanism dominated by beam hinges. This mechanism endows the structure with strong integrity, excellent seismic performance, and ease of post-earthquake repair. Although extensive research has been conducted on UHPC materials and structural elements, studies on the comprehensive seismic design methodology, particularly the DDBD method, for fully prefabricated frame systems incorporating UHPC and HSC remain limited.

This research focuses on the DDBD method applied to precast UHPC-HSC frame structures. The performance objectives, along with quantitative indices for the limit values of interstory drift ratio (ISDR), were proposed. Additionally, specific steps for implementing the DDBD method in precast UHPC-HSC frame structures were established. A one-bay, ten-story, three-span frame structure was designed as a case study and evaluated through pushover analysis. This research provides substantial support for the application of the DDBD method in precast UHPC-HSC frame structures.

2. Performance Levels and Quantification Index for UHPC-HSC Frame Structure

The codes JGJ 3-2010 [43] and T/CECA 20024-2022 [44] classify structural seismic performance into five distinct levels. In contrast, the FEMA 356 [45] defines four seismic performance levels for reinforced concrete structures, namely Operational (O), Immediate Occupancy (IO), Life Safety (LS), and Collapse Prevention (CP). For the precast UHPC-HSC frame structure, the damage states were categorized into six levels: no damage, slight damage, light damage, moderate damage, significant damage, and severe damage. These damage states correspond to performance Levels 1 through 6, respectively. It is noteworthy that codes JGJ 3-2010 [43] and T/CECA 20024-2022 [44] specify a four-tier classification for structural performance objectives. The performance objectives of the precast UHPC-HSC frame were categorized into four levels, each aligned with specific seismic performance requirements under the four-level seismic protection defined in GB 18306-2015 “Seismic ground motion parameters zonation map of China” [46], as detailed in

Table 1. Performance objectives for precast UHPC-HSC frame structure.

		Performance Objective	A	B	C	D
	Performance Level
Seismic Design Level
Frequent earthquake			1	1	1	1
Design earthquake			1	2	3	4
Rare earthquake			2	3	4	5
Extremely rare earthquake			3	4	5	6

.

The limit value of ISDR was employed as a quantitative index to assess each performance level of precast UHPC-HSC frame. The limit values of ISDR corresponding to different performance levels are established based on extensive post-earthquake investigations and testing. However, as UHPC is a relatively new material, research on its application in frame structures remains limited. This study comprehensively considers traditional reinforced concrete frames, collected literature data, and our preliminary experimental results to establish the limit values of ISDR for various performance levels of precast UHPC-HSC frame. According to GB 50011-2010 [47], the limit value of the elastic and elastoplastic ISDR for frame structures are 0.18% and 2%, respectively. In contrast, FEMA 356 [45] specifies these limit values as 1% for IO level, 2% for LS level, and 4% for CP level. Additionally, SEAOC Vision 2000 [48] recommends corresponding limit values of 0.5%, 2.5%, and 4% for these three performance levels. Through a comprehensive review of the literature, quasi-static test data on UHPC-connected frame structures, frame columns, and bridge piers were compiled. The ISDR at yield, peak, and ultimate points are summarized in

Table 2. Summary of ISDR test data.

No.	Reference	Specimen Type	Specimen ID	ISDR
				Yield Point	Peak Point	Ultimate Point
1	Zhang [42]	Frame	Whole frame	1.14%	2.67%	4.45%
			1st interstory	0.98%	2.68%	4.66%
			2nd interstory	1.37%	3.04%	4.78%
			3rd interstory	1.06%	2.30%	3.93%
2	Chen [49]	Frame	PC	1.16%	2.09%	—
3	Zheng [50]	Frame	PC	1.17%	2.22%	—
Mean				1.15%	2.50%	4.46%
Standard Deviation				0.13%	0.36%	0.33%
4	Peng [51]	Column	UHPC160	1.41%	4.58%	4.91%
			UHPC240	1.44%	3.26%	5.02%
			UHPC320	1.73%	2.66%	4.73%
			UHPC400	1.63%	2.54%	4.91%
			UHPC480	1.27%	2.63%	4.15%
5	Zhang [52]	Column	U6-50-H	0.68%	1.65%	2.80%
			U8-50-H	0.52%	1.65%	3.15%
			U12-50-H	0.74%	2.10%	4.76%
			U8-50-L	0.71%	1.65%	2.57%
			U8-150-H	0.61%	1.70%	2.69%
6	Xu [53]	Bridge piers	Tall pier	1.29%	2.48%	5.62%
			Short pier	1.09%	3.06%	5.10%
Mean				1.09%	2.50%	4.20%
Standard Deviation				0.43%	0.87%	1.09%

.

Based on the above, this study proposes the limit values of ISDR for the six performance levels of precast UHPC-HSC frame structure, as summarized in

Table 3. Limit values of ISDR for precast UHPC-HSC frame structure.

Performance Level	1	2	3	4	5	6
Damage degree	No damage	Slight damage	Light damage	Moderate damage	Significant damage	Severe damage
ISDR	0.18%	0.20%	0.55%	1.1%	2.5%	4%

. The limit value of ISDR for performance level 1 is defined as the elastic ISDR. The limit value of ISDR for performance level 2 is set at 1.1 times that of performance level 1. For performance level 4, the limit value of ISDR is set 1.1%. Performance level 3 is assigned a limit value of ISDR that is half of that for performance level 4. The limit values of ISDR for performance levels 5 and 6 are 2.5% and 4%, respectively.

3. Fundamentals of DDBD

3.1. Determination of Target Displacement

When the weak story of the frame structure reaches the limit value of ISDR, the calculated displacement of the frame structure can be considered as the target displacement for the precast UHPC-HSC frame. The displacements at each floor of the frame structure can be obtained from the target displacement and the calculation of the modal shape coefficients, as shown in the following Equations [54]:

$${\Delta }_i={\delta }_i\left({\displaystyle \frac{{\Delta }_{\mathrm{c}}}{{\delta }_{\mathrm{c}}}}\right)$$

(1)

$${\delta }_i={\displaystyle \frac{H_i}{H_n}}\hspace{1em}\left(n\le 4\right)$$

(2)

$${\delta }_i={\displaystyle \frac{4}{3}}{\displaystyle \frac{H_i}{H_n}}\left(1-{\displaystyle \frac{H_i}{4H_n}}\right)\hspace{1em}\left(n>4\right)$$

(3)

When the number of stories in the frame structure exceeds ten, the influence of higher-order modes must be considered. Accordingly, Equation (1) should be multiplied by a lateral displacement reduction factor. This reduction factor can be expressed as a function of the number of stories, building height, or fundamental period. However, since the fundamental period is unknown during the initial design phase and building height correlates more fundamentally with higher-mode effects than the number of stories, the factor is defined in Equation (4) [54]:

$${\omega }_{\theta }=1.15-0.0034H_n\le 1.0\hspace{1em}\left(n>10\right)$$

(4)

3.2. Equivalent Procedure for MDOF

The DDBD method requires the transformation of an MDOF system with n degrees of freedom into an equivalent SDOF system representation. The detailed derivation is provided in Appendix A.

$M_{\mathrm{eff}}$ and $H_{\mathrm{e}}$ are evaluated using Equations (5) and (6).

$$M_{\mathrm{eff}}={\displaystyle \sum _{i=1}^n{m}_i{c}_i}={\displaystyle \frac{{\displaystyle \sum _{i=1}^n{m}_i{\Delta }_i}}{{\Delta }_{\mathrm{d}}}}$$

(5)

$$H_{\mathrm{e}}={\displaystyle \frac{{\displaystyle \sum _{i=1}^n{m}_i{\Delta }_i{H}_i}}{{\displaystyle \sum _{i=1}^n{m}_i{\Delta }_i}}}$$

(6)

${\Delta }_{\mathrm{d}}$ is calculated from Equation (7).

$${\Delta }_{\mathrm{d}}={\displaystyle \frac{{\displaystyle \sum _{i=1}^n{F}_i{\Delta }_i}}{V_{\mathrm{b}}}}={\displaystyle \frac{{\displaystyle \sum _{i=1}^n{m}_i{{\Delta }_i}^2}}{{\displaystyle \sum _{i=1}^n{m}_i{\Delta }_i}}}$$

(7)

Using the method illustrated in Figure 1, the secant stiffness corresponding to the maximum effective displacement is taken as $K_{\mathrm{eff}}$. $K_{\mathrm{eff}}$ is evaluated using Equation (8).

$$K_{\mathrm{eff}}={\left({\displaystyle \frac{2\pi }{T_{\mathrm{eff}}}}\right)}^2{M}_{\mathrm{eff}}={\displaystyle \frac{4{\pi }^2}{{T_{\mathrm{eff}}}^2}}{M}_{\mathrm{eff}}$$

(8)

$V_{\mathrm{b}}$ is calculated from Equation (9).

$$V_{\mathrm{b}}=K_{\mathrm{eff}}·{\Delta }_{\mathrm{d}}$$

(9)

The lateral loads have been distributed at a different story using design base shear given by Equations (10) and (11).

$$F_i=V_{\mathrm{b}}{\displaystyle \frac{m_i{\Delta }_i}{{\displaystyle \sum _{i=1}^n{m}_i{\Delta }_i}}}\hspace{1em}\left(n\le 10\right)$$

(10)

$$F_i=F_{\mathrm{t}}+0.9V_{\mathrm{b}}{\displaystyle \frac{m_i{\Delta }_i}{{\displaystyle \sum _{i=1}^n{m}_i{\Delta }_i}}}\hspace{1em}\left(n>10\right)$$

(11)

${\zeta }_{\mathrm{eq}}$ is evaluated by Equation (12) [54].

$${\zeta }_{\mathrm{eq}}={\zeta }_0+{\zeta }_{\mathrm{hys}}=0.05+0.565\left({\displaystyle \frac{\mu -1}{\mu \pi }}\right)$$

(12)

$\mu$ is found by the ratio between ${\Delta }_{\mathrm{d}}$ and ${\Delta }_{\mathrm{y}}$ (Equation (13)).

$$\mu ={\displaystyle \frac{{\Delta }_{\mathrm{d}}}{{\Delta }_{\mathrm{y}}}}$$

(13)

Incorporates corrections for column flexural deformation, joint shear deformation, and member shear deformation in addition to the beam deformation, ${\Delta }_{\mathrm{y}}$ is evaluated using Equation (14) [54].

$${\Delta }_{\mathrm{y}}={\theta }_{\mathrm{y}}{H}_{\mathrm{e}}={\displaystyle \frac{2M_1{\theta }_{\mathrm{y}1}+M_2{\theta }_{\mathrm{y}2}}{2M_1+M_2}}{H}_{\mathrm{e}}={\displaystyle \frac{2M_1\left(0.5{\epsilon }_{\mathrm{y}1}\frac{L_{\mathrm{b}1}}{h_{\mathrm{b}1}}\right)+M_2\left(0.5{\epsilon }_{\mathrm{y}2}\frac{L_{\mathrm{b}2}}{h_{\mathrm{b}2}}\right)}{2M_1+M_2}}{H}_{\mathrm{e}}$$

(14)

3.3. Displacement Response Spectrum

The structural displacement response spectrum ($S_{\mathrm{d}}\text{-}T$) can be obtained by converting the acceleration response spectrum ($S_{\mathrm{a}}\text{-}T$), as expressed in Equation (15).

$$S_{\mathrm{d}}={\left({\displaystyle \frac{T}{2\pi }}\right)}^2{S}_{\mathrm{a}}$$

(15)

Following GB 50011-2010 [47], $S_{\mathrm{a}}\text{-}T$ is determined by $\alpha$, as expressed in Equation (16).

$$S_{\mathrm{a}}=\alpha ·g$$

(16)

$S_{\mathrm{d}}\text{-}T$ is derived through the transformation of Equations (15) and (16).

$$S_{\mathrm{d}}={\left({\displaystyle \frac{T}{2\pi }}\right)}^2{S}_{\mathrm{a}}={\displaystyle \frac{T^2}{4{\pi }^2}}\alpha g$$

(17)

$\alpha$ is divided into four segments, and $T_{\mathrm{eff}}$ can be determined by the following Equations:

(1) Linear ascending segment

$$T^2\left[0.45+10\left({\eta }_2-0.45\right)T\right]={\displaystyle \frac{4{\pi }^2}{{\alpha }_{\max }g}}{S}_{\mathrm{d}}\hspace{1em}\left(T\le 0.1\mathrm{s}\right)$$

(18)

(2) Horizontal segment

$$T=2\pi \sqrt{{\displaystyle \frac{S_{\mathrm{d}}}{{\eta }_2{\alpha }_{\max }g}}}\hspace{1em}\left(0.1\mathrm{s}\le T\le T_{\mathrm{g}}\right)$$

(19)

(3) Nonlinear descending segment

$$T={\left({\displaystyle \frac{4{\pi }^2}{{T_{\mathrm{g}}}^{\gamma }}}·{\displaystyle \frac{S_{\mathrm{d}}}{{\eta }_2{\alpha }_{\max }g}}\right)}^{{\displaystyle \frac{1}{2-\gamma }}}\hspace{1em}\left(T_{\mathrm{g}}\le T\le 5T_{\mathrm{g}}\right)$$

(20)

(4) Linear descending segment

$$T^2\left[{0.2}^{\gamma }{\eta }_2-{\eta }_1\left(T-5T_{\mathrm{g}}\right)\right]={\displaystyle \frac{4{\pi }^2}{{\alpha }_{\max }g}}{S}_{\mathrm{d}}\hspace{1em}\left(5T_{\mathrm{g}}\le T\le 6\mathrm{s}\right)$$

(21)

${\alpha }_{\max }$ is adopted from T/CECA 20024-2022 “Standard for performance-based seismic design of building structures” [44].

4. Design Procedure for Precast UHPC-HSC Frame Structures Based on DDBD

(1) Preliminary structural design: Considering the functional requirements for the use of the precast UHPC-HSC frame structure, the layout parameters, such as column grid, story height, material strength, and cross-sections of beam and column, are determined in compliance with relevant design codes. The dimensions of the U-shaped UHPC beam formwork are initially selected according to the following principles: The side wall height is determined by subtracting the thickness of the slab from the beam height. Given the excellent durability of UHPC, the thickness of the concrete protection layer can be appropriately reduced. Generally, the bottom board ranges from 50 mm to 70 mm, while the side wall thickness can be reduced to approximately 30 mm to 50 mm. The length of the cast-in-place UHPC at the beam ends is generally equal to the plastic hinge length of the beam. The height of the cast-in-place UHPC at the bottom of the column is generally equal to the plastic hinge length of the column, and it is also necessary to determine based on the construction operation space required for the pressed sleeve splicing connection.

(2) Performance objective: After comprehensively considering the importance of the structure and the owner’s requirements, the performance objective for the precast UHPC-HSC frame is selected according to Table 1. Based on the selected performance objective, [θ] under different seismic levels are determined according to Table 3.

(3) ${\Delta }_{\mathrm{d}}$: Identify the weak story of the precast UHPC-HSC frame and set θ = [θ]. Based on the target lateral displacement curve, calculate ${\Delta }_i$ using Equations (1)–(4). The precast UHPC-HSC frame is converted into an equivalent SDOF system, and ${\Delta }_{\mathrm{d}}$ is calculated using Equation (7).

(4) $M_{\mathrm{eff}}$ and $H_{\mathrm{e}}$: $M_{\mathrm{eff}}$ and $H_{\mathrm{e}}$ are calculated using Equations (5) and (6), respectively.

(5) ${\zeta }_{\mathrm{eq}}$ and $T_{\mathrm{eff}}$: $\mu$ is first computed via Equations (13) and (14), followed by evaluation of ${\zeta }_{\mathrm{eq}}$ using Equation (12), $T_{\mathrm{eff}}$ is then obtained from Equations (18)–(21).

(6) $K_{\mathrm{eff}}$ and $V_{\mathrm{b}}$: $K_{\mathrm{eff}}$ and $V_{\mathrm{b}}$ are calculated using Equations (8) and (9), respectively.

(7) $F_i$: The horizontal seismic action of each story of the precast UHPC-HSC frame are calculated according to Equations (10) and (11).

(8) Reinforcement design: Structural internal force calculations are performed, and the design for beams and columns is carried out to determine the required reinforcement.

(9) Design result verification: A pushover analysis is conducted on the precast UHPC-HSC frame, and the displacement curve designed based on Equations (1)–(4) is compared and analyzed with the displacement curve obtained from the pushover analysis. If the results from the pushover analysis do not meet the preset displacement objectives, the displacement curve obtained from the pushover analysis is used as the modified displacement curve, and calculations are performed again until the final design is completed.

5. Modeling Methodology and Experimental Validation

Based on the quasi-static experimental investigation of the precast UHPC-HSC frame conducted by our previous research [42], the geometric dimensions and reinforcement details of the frame structure are illustrated in Figure 2. A corresponding finite element model of the UHPC-HSC frame was developed using OpenSEES 3.4.0, with the modeling method represented in Figure 3. The model employed a two-dimensional centerline modeling strategy, in which beam and column elements were simulated using Displacement-Based Beam-Column Elements with five integration points per element. Based on the excellent integrity of composite beams observed in prior experimental study [42], it was assumed that there was perfect bond between the precast UHPC and post-cast concrete, with no relative slip. According to the plane section assumption, different materials within the fiber section work together at the same cross-section. The precast U-shaped UHPC beam formwork was modeled using distinct fiber cross-sections, while the post-cast UHPC regions were incorporated by adjusting the element lengths accordingly. The boundary conditions and loading protocol implemented in the finite element model were rigorously aligned with the test setup, as illustrated in Figure 4. The mass of the slab and the additional dead and live loads were converted into line loads distributed along the beams based on their respective load-sharing areas, thereby accurately accounting for mass effects. Rectangular cross-sections were used for the beam elements, and the stiffness contribution from the slab acting as a flange was not considered. Furthermore, to accurately capture geometric nonlinearities associated with large displacements, the P-Delta effect was included in the analysis using the geomTransformation command in OpenSEES.

For material constitutive modeling, the Concrete02 material model (with linear tension softening) was selected to represent the concrete, whereas the Hysteretic material model was used for the steel bar, as shown in Figure 5.

(1) Constitutive Model for Concrete:

The compressive backbone curve of the Concrete02 material constitutive model is based on the modified Kent-Park model proposed by Scott et al. [55]. Its mathematical expression is given as follows:

$$f_{\mathrm{c}}=\left\{\begin{array}{ll}K{f_{\mathrm{c}}}^{\prime }\left[{\displaystyle \frac{2{\epsilon }_{\mathrm{c}}}{K{\epsilon }_{\mathrm{c}0}}}-{\left({\displaystyle \frac{{\epsilon }_{\mathrm{c}}}{K{\epsilon }_{\mathrm{c}0}}}\right)}^2\right]& (0\le {\epsilon }_{\mathrm{c}}\le K{\epsilon }_{\mathrm{c}0})\\ K{f_{\mathrm{c}}}^{\prime }\left[1-Z_{\mathrm{m}}\left({\epsilon }_{\mathrm{c}}-K{\epsilon }_{\mathrm{c}0}\right)\right]& (K{\epsilon }_{\mathrm{c}0}\le {\epsilon }_{\mathrm{c}}\le {\epsilon }_{\mathrm{cu}})\\ 0.2K{f_{\mathrm{c}}}^{\prime }& ({\epsilon }_{\mathrm{c}}>{\epsilon }_{\mathrm{cu}})\end{array}\right.$$

(22)

$$K=1+{\displaystyle \frac{{\rho }_{\mathrm{s}}{f}_{\mathrm{yh}}}{{f_{\mathrm{c}}}^{\prime }}}$$

(23)

$$Z_{\mathrm{m}}={\displaystyle \frac{0.5}{\frac{3+0.29{f_{\mathrm{c}}}^{\prime }}{145{f_{\mathrm{c}}}^{\prime }-1000}+0.75{\rho }_{\mathrm{s}}\sqrt{\frac{h^{\prime }}{s_{\mathrm{h}}}}-K{\epsilon }_{\mathrm{c}0}}}$$

(24)

The initial elastic modulus of concrete is related to its peak stress and peak strain, with the peak compressive strain being determined based on this elastic modulus. The ultimate compressive strain is taken as twice the peak compressive strain for the cover concrete, while for the confined concrete, it is determined by the parameter Z_m.

Regarding the value of K, for both High-Strength Concrete (HSC) and Normal-Strength Concrete (NSC), K is taken as 1 when the confinement effect of stirrups is not considered. When the confinement effect is considered, K is calculated according to Equation (23). For UHPC, the cover concrete is confined by steel fibers, while the core concrete is jointly confined by both steel fibers and stirrups.

The confining pressure provided by steel fibers is modeled using the approach proposed by Aoude et al. [56], expressed as follows:

$$f_{\mathrm{lf}}=N_{\mathrm{f}}{F}_{\mathrm{pullout}}=\alpha v_{\mathrm{f}}{\displaystyle \frac{l_{\mathrm{f}}}{d_{\mathrm{f}}}}{\tau }_{\mathrm{bond}}$$

(25)

$${\tau }_{\mathrm{bond}}=0.6{{f_{\mathrm{c}}}^{\prime }}^{{\displaystyle \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}$$

(26)

The modified factor K, according to Ren et al. [57], is expressed by Equation (27).

$$K=\left\{\begin{array}{ll}1+{\displaystyle \frac{f_{\mathrm{lf}}}{{f_{\mathrm{c}}}^{\prime }}}& (\mathrm{Cover} \mathrm{concrete})\\ 1+{\displaystyle \frac{f_{\mathrm{lf}}}{{f_{\mathrm{c}}}^{\prime }}}+{\displaystyle \frac{{\rho }_{\mathrm{s}}{f}_{\mathrm{yh}}}{{f_{\mathrm{c}}}^{\prime }}}& (\mathrm{Core} \mathrm{concrete})\end{array}\right.$$

(27)

The tensile backbone curve of the Concrete02 material constitutive model is based on the model proposed by Yassin et al. [58], and its expression is given as follows:

$$f_{\mathrm{t}}=\left\{\begin{array}{ll}{E}_0{\epsilon }_{\mathrm{t}}& (0\le {\epsilon }_{\mathrm{t}}\le {\epsilon }_{\mathrm{t}0})\\ {f_{\mathrm{t}}}^{\prime }-E_{\mathrm{t}}\left({\epsilon }_{\mathrm{t}}-{\epsilon }_{\mathrm{t}0}\right)& ({\epsilon }_{\mathrm{t}0}\le {\epsilon }_{\mathrm{c}}\le {\epsilon }_{\mathrm{tu}})\end{array}\right.$$

(28)

(2) Constitutive Model for Steel bar:

The Hysteretic Material model was selected to represent the stress–strain relationship of the steel reinforcement. During material definition, the nominal stress and strain values were converted into their true stress and strain counterparts.

The parameters for the constitutive models were determined based on experimental data from Zhang et al. [42] and material property tests, using an inverse calibration method. The material parameters are summarized in

Table 4. The parameters of materials.

Concrete					Longitudinal Bars
Parameters	Pre-Cast UHPC	Post-Cast UHPC	Pre-Cast HSC	Pre-Cast NSC	Parameters	In Beam	In Column
fc′ (MPa)	108.2	86.8	57.4	37.2	fy (MPa)	449.94	447.34
E0 (GPa)	43.3	40.6	36.1	32.9	εy	0.002239	0.002246
fcu (MPa)	21.64	17.36	11.48	7.44	fu (MPa)	649.85	610.52
ft′ (MPa)	6.2	5.6	2.85	2.39	εu	0.12565	0.13672

. Additional details regarding the fiber cross-section and material constitutive models can be found in the OpenSEES command manual [59].

The comparison between the hysteresis curves (base shear force—roof lateral drift) from the experimental and the finite element simulation is presented in Figure 6. It can be observed that the finite element simulation results are consistent with the experimental hysteresis curves, with the stiffness degradation and the loading and unloading stiffness being in good agreement. A comparison between the test and simulation results is presented in

Table 5. The comparison results between the test and the finite element simulation.

	Test	Simulation	Percentage Deviation
Fy (kN)	242.41	240.90	−0.62%
Fp (kN)	281.92	271.31	−3.76%
Fu (kN)	239.63	230.61	−3.76%
Kj (kN/mm)	11.09	11.98	8.03%
E (kN·m)	255.92	249.68	−2.44%

, showing that the discrepancies are within 10%. The difference in initial stiffness between the simulation and the test is slightly larger, primarily due to the fact that in the simulation, the column base node was modeled as a fully fixed support. In contrast, during the test, the specimen base was connected to the rigid platform via anchor bolts. In the early stages of loading, minor slipping or base rotation may have occurred, and these slight displacements would manifest as a reduction in overall stiffness. The finite element model established using OpenSEES effectively reflects the seismic performance of the precast UHPC-HSC frame. The same modeling method will be adopted for the subsequent UHPC-HSC frame.

6. Case Study and Analysis of UHPC-HSC Frame Structure

Figure 7 illustrates a newly constructed ten-story, three-span precast UHPC-HSC frame. This frame has a length of 48 m, a width of 18.3 m, and a height of 36.6 m. In accordance with GB 50011-2010 [47], the structural design parameters were determined as follows: the design service life is 50 years, the seismic rating is First degrees, the seismic precautionary intensity is 8 degree (0.2 g), the seismic design grouping is Group II, the site class is Class II, and the environmental exposure category is II(a). The applied loads were specified as follows: the roof dead load and live load were 6 kN/m² and 0.5 kN/m², respectively, while the corresponding floor loads were 5 kN/m² and 2 kN/m², respectively, and the line load from wall is 6 kN/m.

6.1. Designed for “Performance Level 1”

(1) Preliminary structural design: The structural design focuses on the ⑤-axis, featuring a 4.2 m height for the first floor and 3.6 m for standard floors, with an edge span of 7.8 m and a middle span of 2.7 m. The cross-section of precast HSC(C60) column is 650 mm × 650 mm, while the cross-section of UHPC composite beam is 300 mm × 650 mm. The composite slab has a total thickness of 120 mm, and the U-shaped UHPC(UHPC120) beam formwork, exhibits side wall with a height of 530 mm and a thickness of 40 mm, along with a bottom board thickness of 60 mm. The NSC(C40) is subsequently poured to form the beam core. For connection are cast-in-place with UHPC, including a 500 mm height at the column base and 450 mm length for the edge span and 250 mm length for the middle span at the beam end. The steel bar is HRB400E. The modeling method and the material properties of both concrete and steel bar are consistent with those adopted in Section 5.

(2) Performance objective: For this case study, performance objective D (Table 1) was selected as the design target. The design procedure for other performance objectives would be similar to that demonstrated in this case. This performance objective is associated with the corresponding limit values of ISDR obtained from Table 3. The corresponding limit values of ISDR was taken from Table 3. The relevant parameters for performance objective D are summarized in

Table 6. Parameters for performance objective D.

Seismic Design Level	Performance Level	Damage Degree	Limit Value of ISDR
Frequent earthquake	1	No damage	0.18%
Design earthquake	4	Moderate damage	1.1%
Rare earthquake	5	Significant damage	2.5%
Extremely rare earthquake	6	Severe damage	4%

.

(3) ${\Delta }_{\mathrm{d}}$: Structural analysis identified the first story as the weakest layer, exhibiting an ISDR of θ₁ = [θ] = 0.18%.

$${\Delta }_{\mathrm{c}}={\Delta }_1=H_1\times [\theta ]=0.00756 \mathrm{m}; {\delta }_{\mathrm{c}}={\delta }_1={\displaystyle \frac{4}{3}}{\displaystyle \frac{H_1}{H_n}}\left(1-{\displaystyle \frac{H_1}{4H_n}}\right)=0.1486$$

${\delta }_i$ and ${\Delta }_i$ for each story were determined through Equations (1) and (3), with detailed design process documented in

Table 7. Design process for “performance level 1”.

Story	Hi (m)	mi (t)	δi	Δi (m)	miΔi (t·m)	miΔi2 (t·m)	miΔiHi (t·m2)	Fi (kN)	Vi (kN)
10	36.6	112.91	1.0000	0.0509	5.7437	0.2922	210.2198	57.4898	57.4898
9	33.0	129.23	0.9312	0.0474	6.1218	0.2900	202.0193	61.2741	118.7639
8	29.4	129.23	0.8560	0.0435	5.6271	0.2450	165.4369	56.3227	175.0866
7	25.8	129.23	0.7743	0.0394	5.0900	0.2005	131.3224	50.9468	226.0335
6	22.2	129.23	0.6861	0.0349	4.5105	0.1574	100.1335	45.1466	271.1800
5	18.6	129.23	0.5915	0.0301	3.8886	0.1170	72.3284	38.9219	310.1019
4	15.0	129.23	0.4905	0.0249	3.2243	0.0804	48.3648	32.2728	342.3747
3	11.4	129.23	0.3830	0.0195	2.5176	0.0490	28.7009	25.1993	367.5741
2	7.8	129.23	0.2690	0.0137	1.7685	0.0242	13.7944	17.7014	385.2755
1	4.2	130.53	0.1486	0.0076	0.9868	0.0075	4.1445	9.8770	395.1525
∑		1277.32		0.3118	39.4790	1.4632	976.4650	395.1525

. These derived values were then incorporated into Equation (7) to obtain ${\Delta }_{\mathrm{d}}=0.03706 \mathrm{m}$.

(4) $M_{\mathrm{eff}}$ and $H_{\mathrm{e}}$: $M_{\mathrm{eff}}=1065.17 \mathrm{t}$, $H_{\mathrm{e}}=24.73 \mathrm{m}$.

(5) ${\zeta }_{\mathrm{eq}}$ and $T_{\mathrm{eff}}$: ${\Delta }_{\mathrm{d}}<{\Delta }_{\mathrm{y}}$, ${\zeta }_{\mathrm{eq}}={\zeta }_0=0.05$, ${\alpha }_{\max }=0.16$, $T_{\mathrm{g}}=0.4 \mathrm{s}$, substitution into Equation (20) yielded $T_{\mathrm{eff}}$: ${\eta }_2=1$, $\gamma =0.9$, ${\eta }_1=0.02$,

$$T_{\mathrm{eff}}={\left({\displaystyle \frac{4{\pi }^2}{{T_{\mathrm{g}}}^{\gamma }}}·{\displaystyle \frac{S_{\mathrm{d}}}{{\eta }_2{\alpha }_{\max }g}}\right)}^{{\displaystyle \frac{1}{2-\gamma }}}=1.985 \mathrm{s} (T_{\mathrm{g}}\le T\le 5T_{\mathrm{g}})$$

(6) $K_{\mathrm{eff}}$ and $V_{\mathrm{b}}$: $K_{\mathrm{eff}}=10661.44 \mathrm{kN}/\mathrm{m}$, $V_{\mathrm{b}}=395.15 \mathrm{kN}$.

(7) $F_i$: The base shear force was distributed to each story of the precast UHPC-HSC frame, yielding $F_i$ and $V_i$ as summarized in Table 7.

(8) Reinforcement design: Based on the structural internal force analysis results, the reinforcement design for beam and column sections was carried out with the following specific configurations: The longitudinal bars in the precast columns were uniformly distributed around the perimeter with 7@32 bars for stories 1~2, 7@28 bars for stories 3~5, and 7@25 bars for stories 6~10. The longitudinal bars in the composite beams were uniformly distributed with 8@25 bars at the top and 5@25 at the bottom for stories 1~2, 6@25 bars at the top and 4@25 at the bottom for stories 3~5, and 4@25 bars at both top and bottom for stories 6~10.

(9) Design result verification: A finite element model of the precast UHPC-HSC frame was developed following the methodology described in Section 5. Nonlinear static pushover analysis was performed, yielding the base shear—the top displacement curve is shown in Figure 8. The analysis results are tabulated in

Table 8. Pushover results of UHPC-HSC frame.

Loading Step	33			35
Base Shear	384.30 kN			405.24 kN
Story	Absolute Displacement (mm)	Interstory Displacement (mm)	Interstory Drift Ratio (%)	Absolute Displacement (mm)	Interstory Displacement (mm)	Interstory Drift Ratio (%)
10	32.961	1.463	0.041	34.960	1.550	0.043
9	31.498	2.079	0.058	33.409	2.203	0.061
8	29.420	2.727	0.076	31.206	2.890	0.080
7	26.693	3.326	0.092	28.316	3.526	0.098
6	23.367	3.818	0.106	24.791	4.049	0.112
5	19.549	4.176	0.116	20.741	4.432	0.123
4	15.374	4.417	0.123	16.309	4.691	0.130
3	10.956	4.403	0.122	11.618	4.674	0.130
2	6.554	3.949	0.110	6.944	4.187	0.116
1	2.605	2.605	0.062	2.757	2.757	0.066
Loading step	48			103
Base shear	532.81 kN			967.35 kN
Story	Absolute displacement (mm)	Interstory displacement (mm)	Interstory drift ratio (%)	Absolute displacement (mm)	Interstory displacement (mm)	Interstory drift ratio (%)
10	47.950	2.106	0.059	102.953	4.385	0.122
9	45.844	2.989	0.083	98.568	6.206	0.172
8	42.855	3.927	0.109	92.362	8.259	0.229
7	38.928	4.814	0.134	84.103	10.318	0.287
6	34.114	5.561	0.154	73.786	12.079	0.336
5	28.553	6.123	0.170	61.707	13.382	0.372
4	22.429	6.505	0.181	48.325	14.188	0.394
3	15.925	6.461	0.179	34.136	13.952	0.388
2	9.464	5.735	0.159	20.185	12.226	0.340
1	3.729	3.729	0.089	7.959	7.959	0.189
Loading step	208			272
Base shear	1612.90 kN			1860.26 kN
Story	Absolute displacement (mm)	Interstory displacement (mm)	Interstory drift ratio (%)	Absolute displacement (mm)	Interstory displacement (mm)	Interstory drift ratio (%)
10	208.159	8.951	0.249	272.255	10.618	0.295
9	199.209	12.443	0.346	261.638	14.972	0.416
8	186.765	16.593	0.461	246.666	20.387	0.566
7	170.172	20.773	0.577	226.279	26.155	0.727
6	149.399	24.251	0.674	200.124	31.727	0.881
5	125.148	26.942	0.748	168.397	36.729	1.020
4	98.205	28.741	0.798	131.667	39.713	1.103
3	69.464	28.214	0.784	91.954	38.324	1.065
2	41.250	24.724	0.687	53.631	32.467	0.902
1	16.526	16.526	0.393	21.163	21.163	0.504
Loading step	563			902
Base shear	2083.33 kN			2257.95 kN
Story	Absolute displacement (mm)	Interstory displacement (mm)	Interstory drift ratio (%)	Absolute displacement (mm)	Interstory displacement (mm)	Interstory drift ratio (%)
10	563.480	13.829	0.384	902.819	18.886	0.525
9	549.651	21.657	0.602	883.933	32.540	0.904
8	527.993	36.359	1.010	851.392	59.803	1.661
7	491.635	56.498	1.569	791.590	94.599	2.628
6	435.136	75.297	2.092	696.991	123.250	3.424
5	359.839	86.880	2.413	573.740	139.078	3.863
4	272.959	90.129	2.504	434.662	142.587	3.961
3	182.830	82.668	2.296	292.075	130.847	3.635
2	100.162	63.780	1.772	161.228	102.327	2.842
1	36.382	36.382	0.866	58.901	58.901	1.402
Loading step	913
Base shear	2261.38 kN
Story	Absolute displacement (mm)	Interstory displacement (mm)	Interstory drift ratio (%)
10	913.827	19.060	0.529
9	894.767	33.005	0.917
8	861.762	60.690	1.686
7	801.072	95.820	2.662
6	705.252	124.676	3.463
5	580.577	140.606	3.906
4	439.971	144.138	4.004
3	295.832	132.334	3.676
2	163.499	103.637	2.879
1	59.862	59.862	1.425

. Figure 9 presents a comparative between the pushover results and design results at various loading step.

At loading step 35, the UHPC-HSC frame attained the base shear of 405.24 kN, corresponding to performance level 1 requirements (395.15 kN). It can be observed that in Figure 9a, the design result is essentially consistent with the pushover result, and the design lateral displacements are all greater than the displacements obtained from the pushover analysis of the UHPC-HSC frame. However, in Figure 9b,c, discrepancies exist between the design results and the pushover results. The maximum ISDR obtained from the pushover analysis occurs at the fourth story (0.13%), whereas the initial design assumed the maximum ISDR to be at the first story (0.18%). This discrepancy indicates that for the 10-story UHPC-HSC frame studied in this study, the initial design was based on a first-mode assumption, while the pushover analysis revealed that its nonlinear response is influenced by a combination of higher-mode effects and non-uniform stiffness distribution. Therefore, the displacement curve needed to be modified by adopting the displacement curve derived from the pushover analysis as the design displacement curve and recalculating until the requirements were met. The steps for the modified design were as follows:

(10) ${\Delta }_{\mathrm{d}}$: As shown in Table 8, at loading step 48, the maximum ISDR of the frame reached 0.181% at the fourth story, meeting the limit value of ISDR of performance level 1 ([θ] = 0.18%). The absolute displacements of each story were extracted and used as the design displacement curve for subsequent calculations. The modified design process is detailed in

Table 9. Modified design process for “performance level 1”.

Story	Hi (m)	mi (t)	Δi (m)	miΔi (t·m)	miΔi2 (t·m)	miΔiHi (t·m2)	Fi (kN)	Vi (kN)
10	36.6	112.91	0.0480	5.4141	0.2596	198.1568	56.2690	56.2690
9	33.0	129.23	0.0458	5.9247	0.2716	195.5142	61.5752	117.8442
8	29.4	129.23	0.0429	5.5384	0.2373	162.8276	57.5602	175.4044
7	25.8	129.23	0.0389	5.0308	0.1958	129.7954	52.2855	227.6899
6	22.2	129.23	0.0341	4.4087	0.1504	97.8728	45.8195	273.5094
5	18.6	129.23	0.0286	3.6900	0.1054	68.6343	38.3504	311.8598
4	15.0	129.23	0.0224	2.8987	0.0650	43.4798	30.1257	341.9855
3	11.4	129.23	0.0159	2.0580	0.0328	23.4612	21.3888	363.3743
2	7.8	129.23	0.0095	1.2230	0.0116	9.5397	12.7110	376.0853
1	4.2	130.53	0.0037	0.4868	0.0018	2.0444	5.0589	381.1442
∑		1277.32	0.2898	36.6731	1.3313	931.3263	381.1442

. The calculation yielded ${\Delta }_{\mathrm{d}}=0.0363 \mathrm{m}$.

(11) $M_{\mathrm{eff}}$ and $H_{\mathrm{e}}$: $M_{\mathrm{eff}}=1010.20 \mathrm{t}$, $H_{\mathrm{e}}=25.40 \mathrm{m}$.

(12) ${\zeta }_{\mathrm{eq}}$ and $T_{\mathrm{eff}}$: ${\Delta }_{\mathrm{d}}<{\Delta }_{\mathrm{y}}$, ${\zeta }_{\mathrm{eq}}={\zeta }_0=0.05$. ${\alpha }_{\max }=0.16$, $T_{\mathrm{g}}=0.4 \mathrm{s}$, substitution into Equation (20) yielded $T_{\mathrm{eff}}$: ${\eta }_2=1$, $\gamma =0.9$, ${\eta }_1=0.02$,

$$T_{\mathrm{eff}}={\left({\displaystyle \frac{4{\pi }^2}{{T_{\mathrm{g}}}^{\gamma }}}·{\displaystyle \frac{S_{\mathrm{d}}}{{\eta }_2{\alpha }_{\max }g}}\right)}^{{\displaystyle \frac{1}{2-\gamma }}}=1.948 \mathrm{s} (T_{\mathrm{g}}\le T\le 5T_{\mathrm{g}})$$

(13) $K_{\mathrm{eff}}$ and $V_{\mathrm{b}}$: $K_{\mathrm{eff}}=10498.99 \mathrm{kN}/\mathrm{m}$, $V_{\mathrm{b}}=381.14 \mathrm{kN}$.

(14) $F_i$: $F_i$ and $V_i$ of the UHPC-HSC frame are presented in Table 9. The results indicated that the modified design yielded a slightly smaller base shear compared to the initial design. Therefore, the reinforcements of beams and columns from the initial design can be adopted.

(15) Design result verification: As summarized in Table 8, the base shear of the UHPC-HSC frame reached 384.30 kN at loading step 33, exceeding the requirement of 381.14 kN for performance level 1. Furthermore, as illustrated in Figure 8, the pushover curve remained linear with no stiffness degradation at this stage. All structural components remained elastic with no yielding, indicating fully functional structural performance. Figure 9 presents a comparative between the pushover results (loading step 33) and design results (loading step 48). The comparison revealed that the design displacements were consistently greater than those from the pushover analysis, confirming that the designed precast UHPC-HSC frame satisfies the requirements of performance level 1. Moreover, when the actual displacement profile obtained from the pushover analysis was used as the modified target displacement in a second design iteration, the revised design base shear (381.14 kN) showed excellent agreement with the base shear from the pushover analysis (384.30 kN), with a relative error of only 0.8%. This represents an improvement over the initial design outcome (395.15 kN vs. 405.24 kN), reducing the error by approximately 1.75%. These results demonstrate that even when initial displacement distribution assumptions are inaccurate, the iterative design–analysis–modification process effectively self-corrects and substantially improves the final accuracy in predicting the global seismic capacity of the structure.

6.2. Verified for “Performance Level 4”

For performance level 4 verification, the lateral displacement response of the UHPC-HSC frame shall be determined based on the corresponding pushover curve under the target performance state. As shown in Table 8, at loading step 272, the maximum ISDR of the frame reached 1.103%, meeting the limit value of ISDR of performance level 4 ([θ] = 1.1%). The absolute displacements of each story were extracted and used as the design displacement curve for subsequent calculations.

(1) ${\Delta }_{\mathrm{d}}$: The process is detailed in

Table 10. Design process for “performance level 4”.

Story	Hi (m)	mi (t)	Δi (m)	miΔi (t·m)	miΔi2 (t·m)	miΔiHi (t·m2)	Fi (kN)	Vi (kN)
10	36.6	112.91	0.2723	30.7406	8.3693	1125.1062	140.1657	140.1657
9	33.0	129.23	0.2616	33.8126	8.8467	1115.8170	154.1730	294.3387
8	29.4	129.23	0.2467	31.8778	7.8632	937.2063	145.3507	439.6894
7	25.8	129.23	0.2263	29.2431	6.6171	754.4709	133.3374	573.0268
6	22.2	129.23	0.2001	25.8629	5.1758	574.1562	117.9251	690.9519
5	18.6	129.23	0.1684	21.7627	3.6648	404.7856	99.2296	790.1815
4	15.0	129.23	0.1317	17.0160	2.2404	255.2396	77.5865	867.7680
3	11.4	129.23	0.0920	11.8837	1.0928	135.4742	54.1852	921.9532
2	7.8	129.23	0.0536	6.9309	0.3717	54.0613	31.6024	953.5557
1	4.2	130.53	0.0212	2.7624	0.0585	11.6020	12.5955	966.1511
∑		1277.32	1.6738	211.8926	44.3001	5367.9194	966.1511

. The calculation yielded ${\Delta }_{\mathrm{d}}=0.2091 \mathrm{m}$.

(2) $M_{\mathrm{eff}}$ and $H_{\mathrm{e}}$: $M_{\mathrm{eff}}=1013.51 \mathrm{t}$, $H_{\mathrm{e}}=25.33 \mathrm{m}$.

(3) ${\zeta }_{\mathrm{eq}}$ and $T_{\mathrm{eff}}$: ${\Delta }_{\mathrm{d}}<{\Delta }_{\mathrm{y}}$, ${\zeta }_{\mathrm{eq}}={\zeta }_0=0.05$. ${\alpha }_{\max }=0.45$, $T_{\mathrm{g}}=0.4 \mathrm{s}$, substitution into Equation (21) yielded $T_{\mathrm{eff}}$: ${\eta }_2=1$, $\gamma =0.9$, ${\eta }_1=0.02$,

$$T^2\left[{0.2}^{\gamma }{\eta }_2-{\eta }_1\left(T-5T_{\mathrm{g}}\right)\right]={\displaystyle \frac{4{\pi }^2}{{\alpha }_{\max }g}}{S}_{\mathrm{d}}, T_{\mathrm{eff}}=2.941 \mathrm{s} (5T_{\mathrm{g}}\le T\le 6 \mathrm{s})$$

(4) $K_{\mathrm{eff}}$ and $V_{\mathrm{b}}$: $K_{\mathrm{eff}}=4621.22 \mathrm{kN}/\mathrm{m}$, $V_{\mathrm{b}}=966.15 \mathrm{kN}$.

(5) $F_i$: $F_i$ and $V_i$ of the UHPC-HSC frame are presented in Table 7.

(6) Result verification: As shown in Table 8, the base shear of the UHPC-HSC frame reached 967.35 kN at loading step 103, exceeding the 966.15 kN required for performance level 4. Figure 9 presents a comparative between the pushover results (loading step 103) and design results (loading step 272). The comparison revealed that the design displacements were consistently larger than those obtained from the pushover analysis, indicating that the designed precast UHPC-HSC frame satisfies the requirements of performance level 4.

6.3. Verified for “Performance Level 5”

For performance level 5 verification, as shown in Table 8, at loading step 563, the maximum ISDR of the frame reached 2.504%, meeting the limit value of ISDR of performance level 5 ([θ] = 2.5%). The absolute displacements of each story were extracted and used as the design displacement curve for subsequent calculations.

(1) ${\Delta }_{\mathrm{d}}$: The process is detailed in

Table 11. Design process for “performance level 5”.

Story	Hi (m)	mi (t)	Δi (m)	miΔi (t·m)	miΔi2 (t·m)	miΔiHi (t·m2)	Fi (kN)	Vi (kN)
10	36.6	112.91	0.5635	63.6231	35.8503	2328.6037	229.8549	229.8549
9	33.0	129.23	0.5497	71.0339	39.0438	2344.1187	256.6285	486.4834
8	29.4	129.23	0.5280	68.2350	36.0276	2006.1090	246.5167	733.0001
7	25.8	129.23	0.4916	63.5362	31.2366	1639.2345	229.5412	962.5413
6	22.2	129.23	0.4351	56.2347	24.4698	1248.4101	203.1625	1165.7037
5	18.6	129.23	0.3598	46.5037	16.7338	864.9683	168.0067	1333.7104
4	15.0	129.23	0.2730	35.2758	9.6288	529.1365	127.4429	1461.1533
3	11.4	129.23	0.1828	23.6280	4.3199	269.3593	85.3623	1546.5156
2	7.8	129.23	0.1002	12.9444	1.2965	100.9662	46.7650	1593.2806
1	4.2	130.53	0.0364	4.7489	0.1728	19.9454	17.1566	1610.4372
∑		1277.32	3.5201	445.7636	198.7800	11,350.8516	1610.4372

. The calculation yielded ${\Delta }_{\mathrm{d}}=0.4459 \mathrm{m}$.

(2) $M_{\mathrm{eff}}$ and $H_{\mathrm{e}}$: $M_{\mathrm{eff}}=999.62 \mathrm{t}$, $H_{\mathrm{e}}=25.46 \mathrm{m}$.

(3) ${\zeta }_{\mathrm{eq}}$ and $T_{\mathrm{eff}}$: ${\Delta }_{\mathrm{d}}>{\Delta }_{\mathrm{y}}$, $\mu =1.663$, ${\zeta }_{\mathrm{eq}}=0.1217$. ${\alpha }_{\max }=0.9$, $T_{\mathrm{g}}=0.45 \mathrm{s}$, substitution into Equation (21) yielded $T_{\mathrm{eff}}$: ${\eta }_2=0.7389$, $\gamma =0.8304$, ${\eta }_1=0.0109$,

$$T^2\left[{0.2}^{\gamma }{\eta }_2-{\eta }_1\left(T-5T_{\mathrm{g}}\right)\right]={\displaystyle \frac{4{\pi }^2}{{\alpha }_{\max }g}}{S}_{\mathrm{d}}, T_{\mathrm{eff}}=3.304 \mathrm{s} (5T_{\mathrm{g}}\le T\le 6 \mathrm{s})$$

(4) $K_{\mathrm{eff}}$ and $V_{\mathrm{b}}$: $K_{\mathrm{eff}}=3611.40 \mathrm{kN}/\mathrm{m}$, $V_{\mathrm{b}}=1610.44 \mathrm{kN}$.

(5) $F_i$: $F_i$ and $V_i$ of the UHPC-HSC frame are presented in Table 8.

(6) Result verification: As shown in Table 8, the base shear of the UHPC-HSC frame reached 1612.90 kN at loading step 208, exceeding the 1610.44 kN required for performance level 5. Figure 9 presents a comparative between the pushover results (loading step 208) and design results (loading step 563). The comparison revealed that the design displacements were consistently larger than those obtained from the pushover analysis, indicating that the designed precast UHPC-HSC frame satisfies the requirements of performance level 5.

6.4. Verified for “Performance Level 6”

For performance level 6 verification, as shown in Table 8, at loading step 913, the maximum ISDR of the frame reached 4.004%, meeting the limit value of ISDR of performance level 6 ([θ] = 4%). The absolute displacements of each story were extracted and used as the design displacement curve for subsequent calculations.

(1) ${\Delta }_{\mathrm{d}}$: The process is detailed in

Table 12. Design process for “performance level 6”.

Story	Hi (m)	mi (t)	Δi (m)	miΔi (t·m)	miΔi2 (t·m)	miΔiHi (t·m2)	Fi (kN)	Vi (kN)
10	36.6	112.91	0.9138	103.1811	94.2897	3776.4294	321.8074	321.8074
9	33.0	129.23	0.8948	115.6349	103.4663	3815.9522	360.6490	682.4565
8	29.4	129.23	0.8618	111.3696	95.9741	3274.2650	347.3460	1029.8024
7	25.8	129.23	0.8011	103.5263	82.9320	2670.9779	322.8839	1352.6863
6	22.2	129.23	0.7053	91.1430	64.2788	2023.3748	284.2622	1636.9485
5	18.6	129.23	0.5806	75.0306	43.5610	1395.5694	234.0099	1870.9585
4	15.0	129.23	0.4400	56.8594	25.0165	852.8916	177.3366	2048.2951
3	11.4	129.23	0.2958	38.2318	11.3102	435.8423	119.2396	2167.5346
2	7.8	129.23	0.1635	21.1297	3.4547	164.8117	65.9006	2233.4352
1	4.2	130.53	0.0599	7.8137	0.4677	32.8175	24.3698	2257.8050
∑		1277.32	5.7164	723.9201	524.7510	18,442.9318	2257.8050

. The calculation yielded ${\Delta }_{\mathrm{d}}=0.7249 \mathrm{m}$.

(2) $M_{\mathrm{eff}}$ and $H_{\mathrm{e}}$: $M_{\mathrm{eff}}=998.68 \mathrm{t}$, $H_{\mathrm{e}}=25.48 \mathrm{m}$.

(3) ${\zeta }_{\mathrm{eq}}$ and $T_{\mathrm{eff}}$: ${\Delta }_{\mathrm{d}}>{\Delta }_{\mathrm{y}}$, $\mu =2.704$, ${\zeta }_{\mathrm{eq}}=0.1634$. ${\alpha }_{\max }=1.35$, $T_{\mathrm{g}}=0.45 \mathrm{s}$, substitution into Equation (21) yielded $T_{\mathrm{eff}}$: ${\eta }_2=0.6679$, $\gamma =0.8114$, ${\eta }_1=0.0077$,

$$T^2\left[{0.2}^{\gamma }{\eta }_2-{\eta }_1\left(T-5T_{\mathrm{g}}\right)\right]={\displaystyle \frac{4{\pi }^2}{{\alpha }_{\max }g}}{S}_{\mathrm{d}}, T_{\mathrm{eff}}=3.556 \mathrm{s} (5T_{\mathrm{g}}\le T\le 6 \mathrm{s})$$

(4) $K_{\mathrm{eff}}$ and $V_{\mathrm{b}}$: $K_{\mathrm{eff}}=3114.75 \mathrm{kN}/\mathrm{m}$, $V_{\mathrm{b}}=2257.81 \mathrm{kN}$.

(5) $F_i$: $F_i$ and $V_i$ of the UHPC-HSC frame are presented in Table 9.

(6) Result verification: As shown in Table 8, the base shear of the UHPC-HSC frame reached 2257.95 kN at loading step 902, exceeding the 2257.81 kN required for performance level 6. Figure 9 presents a comparative between the pushover results (loading step 902) and design results (loading step 913). The comparison revealed that the design displacements were consistently larger than those obtained from the pushover analysis, indicating that the designed precast UHPC-HSC frame satisfies the requirements of performance level 6. Furthermore, the design results at this loading step show excellent agreement with the pushover results, and the structure still retains a certain load-carrying capacity.

To further validate the effectiveness of the proposed DDBD methodology, Figure 10 and Figure 11 present evaluation results from the perspectives of structural response and safety margin, respectively. Figure 10 plots the variation of base shear and MaxISDR against loading steps under different performance levels, clearly capturing the structural response from the elastic stage through nonlinear behavior. The results demonstrate that the force–deformation behavior aligns well with the intended design objectives of the DDBD method. Meanwhile, the safety margin plot in Figure 11 shows that the ratio of the attained ISDR to the allowable ISDR is below 100% for all stories, indicating sufficient safety margin is maintained when the structure reaches each performance level. The results from Figure 10 and Figure 11 collectively demonstrate the safety and effectiveness of the proposed design method.

7. Conclusions

This study establishes a direct displacement-based design (DDBD) framework specifically for precast UHPC-HSC frame structures, contributing to the performance-based seismic design of advanced hybrid systems. The main original contributions and implications are summarized as follows:

(1) A novel six-level seismic performance classification scheme is introduced (from no damage to severe damage) aligned with specific seismic performance requirements under the four-level seismic protection, and the corresponding interstory drift limits are proposed (0.18% to 4.0%), providing a quantitative basis for multi-level seismic design of UHPC-HSC frames. This hierarchy offers a more refined performance objective system compared to conventional design codes.

(2) The finite element model of a three-story, two-span UHPC-HSC frame structure was established. The simulation results showed good agreement with experimental data, verifying the reliability of the model and its effectiveness in reflecting the seismic performance of the UHPC-HSC structure. This simulation method can be applied to DDBD study of UHPC-HSC frame structure.

(3) The proposed DDBD procedure integrates displacement-based principles with the unique mechanical characteristics of UHPC and HSC, explicitly accounting for deformation control and damage mitigation. The iterative design-validation approach using pushover analysis ensures that the structure reliably meets the target performance under various seismic intensities.

Although this study validated the effectiveness of the DDBD method for UHPC-HSC frame structures through pushover analysis, some limitations remain. Firstly, the static pushover analysis did not fully capture structural response under dynamic loading, especially nonlinear behavior under near-fault pulse-type motions. Subsequent research should employ nonlinear time-history analysis to evaluate the method’s seismic robustness. Secondly, since validation relied on a ten-story, three-span frame, general applicability under different stories, span-depth ratios, irregularities, and material combinations requires further parametric verification. Furthermore, the displacement profile assumption insufficiently considered higher-mode effects or stiffness non-uniformity. Future studies should incorporate MDOF systems or modal corrections to improve accuracy. Finally, three-dimensional effects such as torsion and bidirectional seismic excitations were not considered. Subsequent work should include 3D models to better evaluate dynamic response.