Inheritance and Optimization of Mechanical Traits for Hybrid Girder Bridges: A Novel Bionic Perspective

Abstract

Hybrid girder bridges can be likened to plant grafting, where mechanical traits are inherited from both rootstock and scion girders, enabling performance that exceeds that of the individual components. To quantitatively evaluate this inheritance and optimize hybrid girder performance, this study develops a bionic binary grafting model inspired by the genetic principles of quantitative trait inheritance. By analyzing the flexural behavior of hybrid girders through classical beam theory, the research explores two sequential phases: trait inheritance and trait optimization. In the inheritance phase, the bending moment is governed by the hybrid ratio and the positional advantage of scion girders. In the optimization phase, iterative refinements in girder height and internal force further enhance structural performance. The key contributions of this study are as follows: (1) a novel bionic framework is proposed to quantitatively characterize mechanical trait inheritance in hybrid girders, introducing inheritance ratios to describe the distribution of bending moment between rootstock and scion girders as functions of the hybrid ratio, stiffness ratio, and load ratio; (2) a design-oriented framework for mechanical trait optimization is developed, demonstrating that hybrid girders can achieve equivalent stress performance with reduced structural height; and (3) the proposed inheritance and optimization formulations are validated against representative engineering cases, confirming their accuracy in estimating the optimal inheritance ratio and girder height for hybrid girder bridges. This bio-inspired framework enhances our understanding of hybrid girder performance enhancement mechanisms, enabling the efficient optimization of structural systems during conceptual design by leveraging materials with diverse mechanical properties.

1. Introduction

1.1. Research Motivation from a Bionic Perspective

Natural grafting has occurred in diverse plant species for millennia, long before human intervention, with documented evidence of artificial grafting techniques dating back over 2500 years [1,2]. In botanical grafting, the scion (the upper plant segment without a root system) is fused with the rootstock (the lower, rooted portion), forming a composite organism, as illustrated in Figure 1a. This asexual propagation technique preserves the genetic independence of both the rootstock and scion, preventing significant genetic recombination between them. Once successfully established, the grafted plant initially inherits traits directly from its progenitor components—the rootstock and scion—through their intrinsic compatibility [3,4]. Over time, dynamic rootstock–scion interactions develop, progressively enhancing the grafted plant’s phenotypic performance beyond its originally inherited characteristics [5,6]. Current research on botanical grafting primarily explores two fundamental mechanisms: trait inheritance patterns and synergistic rootstock–scion interactions. The grafted plant is considered a first-generation hybrid, with the rootstock and scion functioning as parental lines. Quantitative trait analysis, employing statistical methodologies [7,8], provides an effective framework for elucidating parental trait transmission pathways and their combinatorial influence on hybrid performance.

Hybrid girder bridges are beam-type structures that integrate longitudinal dissimilar material girders to synergistically share structural loads, as illustrated in Figure 1b. This design parallels botanical grafting, where structural components interact to enhance performance. In bridge engineering, this analogy manifests through functional partitioning: the scion girder refers to the upper segment, which remains disconnected from primary foundations, while the rootstock girder anchors the structure, transmitting scion-derived loads to substructures. Like grafted plants, where rootstock–scion synergy enhances phenotypic traits, hybrid girder bridges exhibit mechanical properties exceeding the sum of their materials. This performance amplification—driven by material complementarity, interfacial load redistribution, and optimized stress transfer—has fueled their rapid global adoption over the past three decades. The evolution of hybrid girders began with the Cheviré Bridge in France (1991) [9], a pioneering 242 m span structure that surpassed the 200 m threshold. Recent advancements show hybrid systems leading long-span beam bridge records. Notable examples include China’s Shibanpo Bridge (330 m) [10], Norway’s Stolma Bridge (301 m) [11], Anhaiwan Bridge (300 m) [12], Raftsundet Bridge (298 m) [13], Sundøy Bridge (298 m) [14], and Sandsfjord Bridge (290 m) [15], demonstrate how hybrid configurations exploit steel’s tensile resilience and concrete’s compressive strength to push span limits of girder bridges.

1.2. Mechanical Studies of Hybrid Girder Bridges

From a mechanical design perspective, hybrid girder bridges face two key challenges: optimizing the structural configuration of the rootstock–scion interaction region and determining the ideal hybrid ratio (or positioning of this interaction zone within the girder system). Extensive studies using finite element analysis and full-scale model testing have examined stress transfer mechanisms across the interface [16,17,18,19], focusing on critical parameters such as steel plate arrangement, interfacial geometry, and concrete composition. Recent research highlights advanced design concepts—graded stiffness transitions, interlocking shear connectors, and tailored material interfaces—as effective strategies for enhancing rootstock–scion bonding integrity. These innovations have enabled modern engineering practices to achieve smooth force–flow distribution and gradual stiffness transitions within the interaction zone, overcoming a developmental challenge for hybrid girders. In contrast, determining the optimal hybrid ratio—the proportion of rootstock (concrete) and scion (steel) materials in load-bearing capacity—remains a critical challenge in hybrid girder bridge design. This ratio directly influences stress redistribution, dynamic response, and long-term durability, shaping overall structural performance and limiting span-length advancements. While a steel-dominant design risks localized stress concentrations, excessive concrete use negates weight-saving benefits. Current approaches rely on iterative parametric optimization, yet a simplified theoretical framework for multi-objective hybrid ratio determination (balancing span, efficiency, and cost) remains undeveloped.

Several studies have investigated the optimal hybrid ratio. Zhang et al. [20] emphasize the multi-criteria optimization of steel girder length, considering structural performance, cost, and construction timelines. Zeng et al. [21] optimized steel length in three-span bridges to mitigate bearing uplift at side piers under adverse load combinations. Qin et al. [22] employed finite element analysis to identify joint locations that balance economic efficiency and structural integrity. He et al. [23] derived optimal steel-to-main span ratios using system equivalence principles and further examined the problem through equal rotations at the steel-concrete joint [24]. Shangguan et al. [25] reported a simplified analytical model to calculate the equivalent span length of hybrid girder components for preliminary design. Despite these advancements, existing research primarily focuses on internal forces and support reactions to determine the optimal hybrid ratio, typically relying on complex iterative numerical analyses using commercial finite element modeling. As a result, there remains a lack of a holistic and simplified approach with closed-form solutions that enable rapid hybrid ratio optimization during preliminary design stages. Hybrid girders exhibit an intrinsic mechanical relationship with their parental girders, inheriting and refining structural traits—analogous to biological trait transmission in grafted plants. Drawing from this bionic perspective [26], this study proposes a novel interdisciplinary framework: using plant grafting systems as biomimetic analogs to investigate hybrid girder mechanics. By treating grafted plants as living prototypes, researchers can decode how mechanical traits—such as load redistribution efficiency, interfacial stress continuity, and material compatibility—are inherited and optimized in hybrid girders, offering new insights into performance amplification mechanisms.

1.3. Contribution of This Study

This study explores the mechanical trait inheritance of hybrid girder bridges through the lens of bionic grafting and optimizes their overall layout based on inherited mechanical properties. It is the first to examine hybrid girder bridge performance from a bionic perspective, offering insights for rapid design and layout optimization in the preliminary design stages. By formulating a simplified approach to mechanical trait inheritance, this study facilitates more efficient structural design. The main contributions of this study are threefold. (1) A novel bionic framework for quantitative mechanical trait inheritance in hybrid girders is proposed. Inspired by biological grafting and quantitative genetics, mechanical trait inheritance ratios are introduced to quantify the apportionment of bending resistance between rootstock and scion girders as functions of hybrid, stiffness, and load ratios. (2) A design-oriented bionic framework for mechanical trait optimization is developed. Building on the inheritance analysis, closed-form formulas are derived to determine optimal girder height, demonstrating that hybrid girders can achieve equivalent stress performance with reduced structural height. (3) The proposed inheritance and optimization formulas are validated against representative engineering designs, showing that key parameters of hybrid girder bridges typically obtained through complicated finite element analysis can be directly and accurately estimated using the proposed formula, improving design efficiency and transparency beyond conventional analysis.

The remainder of this paper is structured as follows: Section 2 introduces the bionic model and research hypothesis. Section 3 presents the theory of mechanical trait inheritance in hybrid girder bridges. Section 4 discusses the theory of mechanical trait optimization in hybrid girder bridges. Section 5 illustrates engineering demonstrations of the proposed inheritance and optimization methodology. Finally, Section 6 summarizes key conclusions and outlines directions for future research.

2. Bionic Model and Hypothesis

Hybrid girders are composed of rootstock and scion girders, with performance enhancement governed by two sequential phases: trait inheritance and trait optimization. Drawing parallels to quantitative trait inheritance in genetics, a bionic grafting model is established where the rootstock and scion girders act as parental lines, while the hybrid girder represents the first-generation offspring, as illustrated in Figure 2.

In the trait inheritance phase, assuming equal lengths for the rootstock, scion, and hybrid girders, the trait inheritance relationship can be quantified using a binary grafting model:

$$Q_h={\psi }_r^{\prime }{Q}_r+{\psi }_s^{\prime }{Q}_s$$

(1)

where $Q_h$ (hybrid trait) is the dependent variable, and $Q_r$ (rootstock trait) and $Q_s$ (scion trait) are independent variables. The inheritance ratios ${\psi }_r^{\prime }$ and ${\psi }_s^{\prime }$ can be further normalized to ${\psi }_r$ and ${\psi }_s$ through transformation ${\psi }_r={\psi }_r^{\prime }/({\psi }_r^{\prime }+{\psi }_s^{\prime })$ and ${\psi }_s={\psi }_s^{\prime }/({\psi }_r^{\prime }+{\psi }_s^{\prime })$, respectively. Therefore, ${\psi }_r+{\psi }_s=1$, allowing for the independent analysis of a single parameter. This study focuses on ${\psi }_s$, as it directly correlates with the hybrid ratio (μ)—the proportional contribution of the scion component to the hybrid system.

In the trait optimization phase, the process emulates the rootstock–scion interaction in biological grafting through engineered design iterations. The objective is to maximize material efficiency while maintaining structural performance, characterized by sectional curvature as an indicator of overall bridge stiffness. Starting with an initial hybrid girder height (h), the process involves: (1) Reducing self-weight and internal forces by integrating a scion girder into the rootstock girder. (2) Iteratively fine-tuning the hybrid ratio (μ) and girder geometry to align the target girder curvatures with the initial configuration. (3) Repeating until convergence is achieved (i.e., further modifications yield negligible performance gains). The optimized girder height (h*) and refined inheritance relationships among $Q_h$, $Q_r$, and $Q_s$ post-optimization quantify how iterative design enhances load-bearing efficiency while preserving material integrity, thereby overcoming the span limitations of conventional girders. Detailed workflows for obtaining the optimized girder height (h*) are provided in Section 4. Additionally, the following hypotheses are adopted for the bionic model of hybrid girder components.

(1)

The rootstock, scion, and hybrid girders adhere to classical beam theory, assuming linear elasticity, small deformations, and the preservation of planar cross-section integrity, while excluding nonlinear effects such as plasticity, geometric nonlinearity, and interfacial slip.

(2)

The critical mechanical trait considered in this study is the maximum bending moment, as it represents the worst-case stress condition and typically governs strength failure in girder structures. Notably, shear effects are not considered in the present analysis, as shear failure is generally brittle and is avoided in bridge design through appropriate detailing and code provisions. In practice, girder bridges are designed to ensure ductile behavior, with flexural response governing the failure mode [27,28]. However, when deformation governs the design, such as in high-speed railway hybrid girder bridges, the mechanical trait should instead be defined in terms of girder deformation. Although the resulting formulations may differ, the underlying methodology remains consistent with that developed in this study based on bending moment.

(3)

Bending loads are modeled as uniformly distributed line loads perpendicular to the girder axis, with load intensity scaled according to material unit weight and volumetric distribution, thereby capturing the dominance of self-weight in the preliminary design. At the preliminary design stage, bridge self-weight dominates the response, while live loads typically contribute only up to 20% for long spans. Moreover, most national codes (U.S., China, Europe) model live loads as linearly distributed, similar to self-weight. Therefore, this study focuses on linearly distributed loads.

(4)

Two boundary configurations of fixed and simply supported are adopted as theoretical boundary cases to enable the derivation of closed-form solutions for mechanical trait inheritance and optimization. In practice, most hybrid girder bridges are continuous structures whose boundary conditions fall between these two idealized cases. Consequently, their mechanical behavior can be reasonably interpreted as lying within the bounds defined by the pinned and fixed solutions. Moreover, both the maximum bending moment (M_max) in the simply supported case and the differential bending moment (ΔM = M_max − M_min) in the fixed configuration serve as equivalent indicators of bending stress.

(5)

The scion girder is assumed to be located at mid-span, which reflects most practical applications. However, in less common cases, the scion girder may be positioned near the supports. In such scenarios, the proposed methodology remains applicable, provided that the hybrid girder model is appropriately modified.

3. Theory of Trait Inheritance

3.1. Girders with Equal Cross-Sections

Girders with equal cross-sections are initially examined, as they represent the majority of short- and medium-span bridge structures [29]. The fixed beam model and corresponding bending moment diagram used to determine the moment inheritance ratio of hybrid girders are shown in Figure 3. For rootstock, scion, and hybrid girders under fixed boundary conditions, the maximum negative bending moment occurs at the fixed ends, while the maximum positive bending moment develops at midspan. Using fundamental principles of structural mechanics, these bending moments are derived as follows:

$$M_{r1}=-{\displaystyle \frac{1}{12}}{q}_r{L}^2,M_{s1}=-{\displaystyle \frac{1}{12}}{q}_s{L}^2,M_{h1}=-{\displaystyle \frac{3\beta \gamma \mu {\left(1-\mu \right)}^2+2\beta {\left(1-\mu \right)}^3+\left(6-4\mu \right)\gamma {\mu }^2+3\mu {\left(1-\mu \right)}^2}{24\left[\beta \left(1-\mu \right)+\mu \right]}}{q}_r{L}^2$$

(2)

$$M_{r2}={\displaystyle \frac{1}{24}}{q}_r{L}^2,M_{s2}={\displaystyle \frac{1}{24}}{q}_s{L}^2,M_{h2}={\displaystyle \frac{3\beta \gamma \mu \left(1-\mu \right)+\beta {\left(1-\mu \right)}^3+\gamma {\mu }^3}{24\left[\beta \left(1-\mu \right)+\mu \right]}}{q}_r{L}^2$$

(3)

where M₁ and M₂ denote the fixed-end negative moment and midspan positive moment, respectively. The subscripts r, s, and h represent the rootstock, scion, and hybrid girders, respectively. $\mu$ is the hybrid ratio, defined as the proportion of the scion girder length to the total length of the hybrid girder. β is the stiffness ratio of the scion to rootstock girders, defined as β = E_sI_s/E_rI_r, where E and I denote elastic modulus and moment of inertia, respectively. γ is the load ratio of scion to rootstock, expressed as γ = q_s/q_r, where q represents the line distributed load intensity.

By expressing the hybrid girder’s moments (M_h1, M_h2) as linear combinations of the parental rootstock (M_r1, M_r2) and scion (M_s1, M_s2) moments, the inheritance relationships are derived:

$$M_{h1}={\displaystyle \frac{2\beta {\left(1-\mu \right)}^3+3\mu {\left(1-\mu \right)}^2}{2\left[\beta \left(1-\mu \right)+\mu \right]}}{M}_{r1}+{\displaystyle \frac{3\beta \mu {\left(1-\mu \right)}^2+\left(6-4\mu \right){\mu }^2}{2\left[\beta \left(1-\mu \right)+\mu \right]}}{M}_{s1}$$

(4)

$$M_{h2}={\displaystyle \frac{\beta {\left(1-\mu \right)}^3}{2\left[\beta \left(1-\mu \right)+\mu \right]}}{M}_{r2}+{\displaystyle \frac{3\beta \mu \left(1-\mu \right)+{\mu }^3}{2\left[\beta \left(1-\mu \right)+\mu \right]}}{M}_{s2}$$

(5)

To quantify the scion’s contribution to hybrid girder bending performance, two inheritance ratios are defined based on fixed-end negative moment and midspan positive moment:

$${\psi }_s^{M-}={\displaystyle \frac{3\beta \mu {\left(1-\mu \right)}^2+\left(6-4\mu \right){\mu }^2}{2\beta +3\left(1-\beta \right)\mu -\left(1-\beta \right){\mu }^3}}$$

(6)

$${\psi }_s^{M+}={\displaystyle \frac{3\beta \mu \left(1-\mu \right)+{\mu }^3}{\left(1-\beta \right){\mu }^3+\beta }}$$

(7)

Alternatively, the moment inheritance ratio of hybrid girders can be analyzed using the simple beam model, as illustrated in Figure 4. The corresponding bending moment diagrams for the rootstock, scion, and hybrid girders are also presented. In this case, the maximum bending moment occurs at the midspan of the girder and can be formulated based on fundamental principles of structural mechanics, as follows:

$$M_r={\displaystyle \frac{1}{8}}{q}_r{L}^2,M_s={\displaystyle \frac{1}{8}}{q}_s{L}^2,M_h={\displaystyle \frac{1}{8}}{q}_r{L}^2{\left(1-\mu \right)}^2+{\displaystyle \frac{1}{8}}{q}_s{L}^2\left[\mu \left(2-\mu \right)\right]$$

(8)

Similarly, the inheritance relationship can be obtained as:

$$M_h={\left(1-\mu \right)}^2{M}_c+\left[\mu \left(2-\mu \right)\right]M_s$$

(9)

The inheritance ratio for quantifying the scion’s contribution to the hybrid girder bending moment can be defined as:

$${\psi }_s^M=\mu \left(2-\mu \right)$$

(10)

From Equations (6) and (7), it is evident that the scion inheritance ratios (${\psi }_s^{M-}$, ${\psi }_s^{M+}$) are correlated with the hybrid ratio (μ) and stiffness ratio (β). Parametric studies, as shown in Figure 5, reveal that both ${\psi }_s^{M-}$ and ${{\psi }_{M+}}_e$ exhibit a monotonic increase with increasing μ, confirming that the scion’s proportional contribution (μ) is the primary driver of inheritance behavior. Additionally, the curves for β = 0.3, 0.5, 0.7, and 0.9 nearly coincide across the μ − ${\psi }_s^M$ spectrum, indicating that β has minimal influence on inheritance ratios within this range.

With the stiffness ratio fixed at β = 0.5, the relationships of the negative moment inheritance ratio (${\psi }_s^{M-}$), the positive moment inheritance ratio (${\psi }_s^{M+}$), and the simple beam moment inheritance ratio (${\psi }_s^M$) versus the hybrid ratio (μ) are analyzed, as shown in Figure 6a. The results reveal a consistent hierarchy across all hybrid ratios: ${\psi }_s^{M-}$ > ${\psi }_s^M$ > ${\psi }_s^{M+}$. The close alignment of ${\psi }_s^{M+}$ and ${\psi }_s^M$ curves reflects their shared dependence on midspan positive moment control, which governs tensile stress distribution and deflection limits. The elevated values of ${\psi }_s^{M-}$ highlight the critical role of negative moments in fixed-end hybrid girder design. The divergence between ${\psi }_s^M$ and ${\psi }_s^{M-}$ peaks at μ = 0.35, with a deviation value of −0.16 (Figure 6b), indicating that the scion inheritance ratio varies between the fixed and simple beam models.

However, the original formula for calculating ${\psi }_s^{M-}$ in uniform-section fixed-end hybrid girders (Equation (6)) involves computationally intensive parameterization, yet demonstrates negligible sensitivity to the stiffness ratio β. This observation motivates a streamlined empirical formulation where ${\psi }_s^{M-}$ depends solely on the hybrid ratio μ:

$${\psi }_s^{M-}\left(\mu \right)=1-{\left(1-\mu \right)}^n$$

(11)

Through iterative numerical trials, the exponent n = 3.09 yields optimal agreement with the fixed-end beam model for calculating ${\psi }_s^{M-}$ (Equations (6) and (7)), as shown in Figure 7. The simplified formulation closely replicates the full model’s predictions, with a maximum absolute deviation of just 0.007 across the μ-spectrum. Therefore, in the subsequent trait optimization process, the empirical formulation is adopted for efficiency and accuracy.

3.2. Girders with Variable Cross-Sections

For long-span girder bridges, variable cross-sections are commonly employed [27]. Consequently, hybrid girders with variable cross-sections are investigated. The analytical complexity of such girders necessitates the use of a simplified simply supported beam configuration to examine trait inheritance. The geometric profile of the tapered beam follows a power-law functional variation, as illustrated in Figure 8. The governing equations for beam depth and load distribution in the variable cross-section hybrid girder are expressed as follows:

$$h\left(x\right)=h+{\displaystyle \frac{H-h}{{\left(0.5L_h\right)}^{\alpha }}}{x}^{\alpha }$$

(12)

$$q\left(x\right)={\beta }_{1}h\left(x\right)+{\beta }_2={\displaystyle \frac{{\beta }_1\left(H-h\right)}{{\left(0.5L_h\right)}^{\alpha }}}{x}^{\alpha }+\left({\beta }_{1}h+{\beta }_2\right)$$

(13)

where h and H denote the minimum and maximum girder heights, respectively. α denotes the power exponent governing the girder height variation. The coefficients ${\beta }_1$ and ${\beta }_2$ correspond to the linearly distributed load components, with ${\beta }_1$ accounting for variable loads associated with changes in girder height and ${\beta }_2$ representing constant loads independent of girder height. According to the load diagram of the hybrid girder, the following load intensity relationships are established as follows:

$$q\left(x\right)=\left\{\begin{array}{cc}{\beta }_{1s}\left(h+{\displaystyle \frac{H-h}{{\left(0.5L\right)}^{\alpha }}}{x}^{\alpha }\right)+{\beta }_{2s}& x\in [0,0.5\mu L]\\ {\beta }_{1r}\left(h+{\displaystyle \frac{H-h}{{\left(0.5L\right)}^{\alpha }}}{x}^{\alpha }\right)+{\beta }_{2r}& x\in [0.5\mu L,L]\end{array}\right.$$

(14)

where β_1r and β_1s denote the beam height-dependent linear load coefficients for the rootstock and scion girders, respectively. β_2r and β_2s denote the beam height-independent linear load coefficients for the rootstock and scion girders, respectively.

Following the fundamental principles of structural mechanics, the midspan bending moments for the rootstock, scion, and hybrid girders can be derived as follows [25]:

$$M_r=\left[{\displaystyle \frac{{\beta }_{1r}h+{\beta }_{2r}}{8}}+{\displaystyle \frac{(H-h){\beta }_{1r}}{4\left(\alpha +1\right)\left(\alpha +2\right)}}\right]L^2,M_s=\left[{\displaystyle \frac{{\beta }_{1s}h+{\beta }_{2s}}{8}}+{\displaystyle \frac{(H-h){\beta }_{1s}}{4\left(\alpha +1\right)\left(\alpha +2\right)}}\right]L^2$$

(15)

$$M_h=\left[{\displaystyle \frac{\left(\begin{array}{c}2\mu \\ -{\mu }^2\end{array}\right)\left(\begin{array}{c}{\beta }_{1s}h\\ +{\beta }_{2s}\end{array}\right)}{8}}+{\displaystyle \frac{(H-h){\beta }_{1s}}{4}}\left(\begin{array}{c}{\displaystyle \frac{{\mu }^{\alpha +1}}{\alpha +1}}\\ -{\displaystyle \frac{{\mu }^{\alpha +2}}{\alpha +2}}\end{array}\right)\right]L^2+\left[{\displaystyle \frac{{\left(\begin{array}{c}1\\ -\mu \end{array}\right)}^2\left(\begin{array}{c}{\beta }_{1r}h\\ +{\beta }_{2r}\end{array}\right)}{8}}+{\displaystyle \frac{(H-h){\beta }_{1r}}{4}}\left(\begin{array}{c}{\displaystyle \frac{1-{\mu }^{\alpha +1}}{\alpha +1}}\\ -{\displaystyle \frac{1-{\mu }^{\alpha +2}}{\alpha +2}}\end{array}\right)\right]L^2$$

(16)

Therefore, the inheritance ratio for quantifying the scion’s contribution to the hybrid girder bending moment can be derived as:

$${\psi }_s^M={\displaystyle \frac{1}{1+\left[\frac{{\left(1-\mu \right)}^2\left({\beta }_{1r}h+{\beta }_{2r}\right)+2\left(H-h\right){\beta }_{1r}\left(\frac{1-{\mu }^{\alpha +1}}{\alpha +1}-\frac{1-{\mu }^{\alpha +2}}{\alpha +2}\right)}{\left({\beta }_{1r}h+{\beta }_{2r}\right)+2\frac{\left(H-h\right){\beta }_{1r}}{\left(\alpha +1\right)\left(\alpha +2\right)}}\right]/\left[\frac{\left(2\mu -{\mu }^2\right)\left({\beta }_{1s}h+{\beta }_{2s}\right)+2\left(H-h\right){\beta }_{1s}\left(\frac{{\mu }^{\alpha +1}}{\alpha +1}-\frac{{\mu }^{\alpha +2}}{\alpha +2}\right)}{\left({\beta }_{1s}h+{\beta }_{2s}\right)+2\frac{\left(H-h\right){\beta }_{1s}}{\left(\alpha +1\right)\left(\alpha +2\right)}}\right]}}$$

(17)

While Equation (17) initially incorporates multiple parameters, a systematic parametric analysis reveals that only the hybrid ratio (μ) has a statistically significant influence on model outcomes. To streamline computational workflows, Equation (11) is reintroduced as a simplified empirical formula, recalibrated using design data from the Anhaiwan Bridge [12]. The key parameters for the Anhaiwan Bridge are $\alpha$ = 2.0, h = 5.0 m, H = 15.0 m, β_1r = 41.6 kN/m, β_2r = 373.92 kN/m, β_1s = 7.22 kN/m, and β_2s = 129.79 kN/m. Through nonlinear regression analysis, the empirical exponent is determined as n = 1.79, yielding the refined model: ${\psi }_s^M\left(\mu \right)=1-{\left(1-\mu \right)}^{1.79}$. As shown in Figure 9a, the simplified formula’s predictions closely match the initial data across the full μ range. The residuals, quantified in Figure 9b, indicate a maximum absolute error of 0.5%—well within acceptable tolerances for engineering design.

The preceding results yield two key observations: (1) variable cross-section hybrid girders exhibit slightly lower inheritance ratios than their uniform cross-section counterparts, and (2) fixed-end models produce higher inheritance ratios than simply supported systems. To account for this, the empirical exponent for variable cross-section fixed-end hybrid girders is calibrated to n = 2.80. Thus, the unified exponential formula ${\psi }_s^M\left(\mu \right)=1-{\left(1-\mu \right)}^n$ (Equation (11)) effectively characterizes the bending moment inheritance ratio for both uniform and variable cross-section hybrid girders, with distinctions arising solely from the exponent n, as summarized in

Table 1. Calibrated exponent values for hybrid girders with constant and variable cross-sections.

Model	Constant Cross-Sections		Variable Cross-Sections
	n	μ*	n	μ*
Fixed-end beam	3.09	0.416	2.82	0.500
Simple beam	2.00	0.433	1.79	0.521

Note: μ* is the extremum point where ${\psi }_s^M$ surpasses the hybrid ratio μ.

.

The application of these simplified analytical models enables the identification of critical thresholds in hybrid girder behavior—specifically, the extremum points where ${\psi }_s^M$ surpasses the hybrid ratio μ, as illustrated in Figure 10. This extremum corresponds to a hybrid ratio that implicitly defines the longitudinal extent of the midspan region exhibiting optimal inheritance efficiency. The optimal hybrid ratios for fixed and simply supported beam models are approximate; however, significant deviations are observed between girders with constant and variable cross-sections. Beyond this optimized zone, inheritance efficacy diminishes progressively, mirroring the spatial decay of trait transmission observed in biological grafting systems. This gradual decline underscores the hybrid system’s adaptive balance between material efficiency and load-bearing priorities across its span, akin to resource allocation dynamics in grafted plant unions.

4. Theory of Trait Optimization

4.1. Optimization Process

The trait optimization process within the bionic model is described as follows: The scion girder employs high-strength, lightweight materials, while the rootstock girder retains conventional structural properties. Upon assembling the initial hybrid girder by integrating these components, its critical bending moment (M₁) demonstrates a significant reduction compared to the rootstock’s original critical moment (M₀), accompanied by a substantial reduction in sectional curvature. To fully exploit material capacity, the diminished sectional curvature is restored to its initial design threshold by strategically reducing the girder height (h). This height optimization subsequently lowers the hybrid girder’s self-weight, thereby reducing the critical bending moment from M₁ to M₂ in the optimized configuration. The iterative cycle continues—each height adjustment induces a controlled reduction in critical bending moment (M_n→M_n−1) until the convergence criterion (|M_n − M_n−1| ≤ ϵ) is satisfied. The final hybrid girder configuration is characterized by its optimized height (h*) and stabilized critical bending moment (M*), as summarized in Figure 11. Throughout this procedure, the progressive height reductions mimic biological resource allocation strategies observed in grafted plants, where structural efficiency is iteratively refined to balance load-bearing capacity with metabolic cost.

To streamline the derivation and application of the bionic model’s trait optimization framework, the following simplifications are introduced for clarity and focus: The rootstock and scion girders are modeled as double-symmetric closed box sections without flanges, where m_b and m_h represent the inner-to-outer width ratio and inner-to-outer height ratio of the box section, respectively. During the n-th optimization stage, the girder height is iteratively adjusted from h_n to h_n+1, while the width (b), height ratio (m_h), and width ratio (m_b) remain constant. Under these assumptions, the iterative optimization process employs sectional curvature (k) as the convergence criterion to establish relationships between the hybrid ratio (μ), load ratio (γ), initial girder height (h), and optimized height (h*). The procedure unfolds as follows:

Step 1: Initial parameters are used to compute the internal forces and sectional curvature for both the rootstock and hybrid girders. The rootstock’s initial sectional curvature is established as the reference standard (k_ref).

Step 2: The sectional curvature of the hybrid girder—invariably lower than the reference value under initial conditions—is artificially elevated to k_ref while maintaining other parameters. This inverse calculation yields the updated girder height required to restore the sectional curvature to its target level.

Step 3: Recalculate dependent parameters—including linear load intensity, sectional moment of inertia, and bending moments—using the updated girder height, followed by a reevaluation of the hybrid girder’s sectional curvature.

Step 4: Compares the recalculated sectional curvature (k_new) with the reference standard (k_ref). If the discrepancy exceeds the predefined tolerance, Steps 2 and 3 are repeated until convergence is achieved, finalizing the optimized girder height (h*) and corresponding bending moment (M*).

This iterative loop mimics biological adaptation, where gradual morphological refinements (akin to height reductions) optimize structural efficiency within environmental constraints. By employing a curvature-based convergence criterion, the model ensures that material utilization and structural stiffness follow bio-inspired principles of load-adaptive trait inheritance. The initial design state’s sectional curvature (k_ref) is established as the reference standard to ensure the optimization process achieves its intended outcomes. After each optimization iteration, the resultant sectional curvature (k_h) is systematically compared to k_ref to evaluate convergence and determine whether further iterations are required. The description of parameters during different optimization stages is shown in

Table 2. Description of parameters during different optimization stages.

Stage		h	qr	qs	M	I	k
i = 0	Rootstock girder	h0	qr0	/	M0	I0	kref
i = 1	Initial hybrid girder	h1= h0	qr1 = qr0	qs1 = qs0	M1	I1 = I0	kh1
i = 2	1st optimization	h2	qr2	qs2	M2	I2	kh2
i = i	i-1th optimization	hi	qri	qsi	Mi	Ii	khi
…	…
i = n	Final hybrid girder	h*	qr*	qs*	M*	I*	kh*

Note: h-girder height, q_r-line load intensity of rootstock girder, q_s-line load intensity of scion girder, M-midspan bending moment of the simple beam model, I-sectional moment of inertia of the support cross-section, k-sectional curvature of the support cross-section, and the convergence criterion is (k_h* − k_ref)/k_ref < 1%. The letters with asterisk indicate those values after final optimization.

. Notably, optimization is controlled by the bending moment, as it is the primary indicator governing the strength design of hybrid girder bridges under ultimate limit states. Accordingly, the proposed inheritance and optimization formulations are moment-based. If deformation is adopted as the control indicator, which is common in serviceability-critical structures such as high-speed railway bridges, the mechanical trait inheritance and optimization framework can be reformulated in terms of deformation. In this case, the optimization procedure follows the same conceptual workflow, while the governing expressions are replaced by deformation-based formulations.

4.2. Girders with Equal Cross-Sections

(1) Initial hybrid girder stage

$$\left\{\begin{array}{c}{M}_0={\displaystyle \frac{1}{8}}{q}_{r0}{L}^2,I_0={\displaystyle \frac{b{h}_0^3}{12}}\left(1-m{n}^3\right)\hfill \\ M_1={\displaystyle \frac{1}{8}}{q}_{r1}{L}^2{\left(1-\mu \right)}^2+{\displaystyle \frac{1}{8}}{q}_{s1}{L}^2\mu \left(2-\mu \right),I_1={\displaystyle \frac{b{h}_1^3}{6}}\left(1-m{n}^3\right)\end{array}\right.$$

(18)

Since $M_1<M_0$, $E_1=E_0$ and $I_1=I_0$, it follows that $k_1<k_0$. However, according to the optimization strategy, $k_1$ is adjusted to be equal to $k_0$ by modifying the girder height from $h_1$ to $h_2$, leading to the following expression:

$${\displaystyle \frac{M_0}{{EI}_1}}={\displaystyle \frac{M_1}{{EI}_2}},{\displaystyle \frac{I_2}{I_1}}={\left({\displaystyle \frac{h_2}{h_1}}\right)}^3$$

(19)

Therefore, to maintain sectional curvature, the girder height is updated as follows:

$$h_2=h_1\sqrt[3]{{\left(1-\mu \right)}^2+\gamma \left(2\mu -{\mu }^2\right)}$$

(20)

(2) Final hybrid girder stage

At each iterative stage, the sectional curvature remains constant, which can be expressed as follows:

$${\displaystyle \frac{M_0}{I_1}}={\displaystyle \frac{M_1}{I_2}}={\displaystyle \frac{M_2}{I_3}}=\cdots ={\displaystyle \frac{M_n}{I_{n+1}}}$$

(21)

Given that the girder height is directly related to the sectional moment of inertia (I), and the girder bending moment is linearly correlated to the girder height with the prescribed assumption of a box section with constant width (b), height ratio (m_h), and width ratio (m_b), the height evolution can be expressed as:

$${\left({\displaystyle \frac{h_{n+1}}{h_n}}\right)}^3={\displaystyle \frac{I_{n+1}}{I_n}}={\displaystyle \frac{M_n}{M_{n-1}}}={\displaystyle \frac{q_{rn}{L}^2{\left(1-\mu \right)}^2+q_{sn}{L}^2\mu \left(2-\mu \right)}{q_{r\left(n-1\right)}{L}^2{\left(1-\mu \right)}^2+q_{s\left(n-1\right)}{L}^2\mu \left(2-\mu \right)}}={\displaystyle \frac{h_n}{h_{n-1}}}$$

(22)

Through iterative optimization, the final girder height is determined as:

$$h_n={\displaystyle \frac{h_n}{h_{n-1}}}{\displaystyle \frac{h_n-1}{h_{n-2}}}\dots {\displaystyle \frac{h_3}{h_2}}{\displaystyle \frac{h_2}{h_1}}{h}_1={\left({\displaystyle \frac{h_2}{h_1}}\right)}^{1+3^{-1}+3^{-2}+\cdots +3^{2-n}}{h}_1={\left({\displaystyle \frac{h_2}{h_1}}\right)}^{1.5\left(1-3^{1-n}\right)}{h}_1$$

(23)

When the number of iterations approaches infinity, the optimized girder height converges to:

$$h_n=h_0\sqrt{{\left(1-\mu \right)}^2+\gamma \left(2\mu -{\mu }^2\right)}$$

(24)

To facilitate practical applications, a scaling factor $\delta$ is introduced to directly relate the optimized girder height to the initial height:

$$\delta ={\displaystyle \frac{1}{\sqrt{{\left(1-\mu \right)}^2+\gamma \left(2\mu -{\mu }^2\right)}}}$$

(25)

Thus, the analytical solution of optimized height for hybrid girders with constant cross-sections is obtained:

$$h_2=h_0{\delta }^{-2/3},h*=h_0{\delta }^{-1}$$

(26)

To quantify the proportional contributions of optimization stages, the initial height optimization proportion (ϕ_h1) is defined as the ratio of the first-stage height reduction to the total height optimization, with the iterative optimization proportion (ϕ_hn) complementing it as ϕ_hn = 1 − ϕ_h1. These proportions are derived as:

$${\varphi }_{h1}=\left(1-{\delta }^{-2/3}\right)/\left(1-{\delta }^{-1}\right),{\varphi }_{hn}=\left({\delta }^{-2/3}-{\delta }^{-1}\right)/\left(1-{\delta }^{-1}\right)$$

(27)

Similarly, the analytical solutions for the optimized bending moments of uniform cross-section hybrid girders are formulated as:

$$M_1=M_0{\delta }^{-2},M*=M_0{\delta }^{-3}$$

(28)

where $M_1$ denotes the critical bending moment after the initial optimization stage, while $M*$ signifies the stabilized moment at convergence.

Extending this framework, the initial moment optimization proportion (ϕ_M1) and its iterative counterpart (ϕ_Mn = 1 − ϕ_M1) are defined to partition the moment reduction across optimization phases:

$${\varphi }_{M1}=\left(1-{\delta }^{-2}\right)/\left(1-{\delta }^{-3}\right),{\varphi }_{Mn}=\left({\delta }^{-2}-{\delta }^{-3}\right)/\left(1-{\delta }^{-3}\right)$$

(29)

These ratios quantify how the initial optimization disproportionately dominates the total moment reduction, mirroring biological systems where early-stage trait adjustments yield the most significant phenotypic gains. The derived relationships formalize the hybrid girder’s inheritance dynamics, bridging iterative engineering optimization with the incremental trait refinement observed in grafted plant systems. For example, a uniform cross-section corrugated steel web box-girder-steel trough hybrid girder with a hybrid ratio of μ = 0.356, a load ratio of γ = 0.700, and an initial girder height of 9.180 m, the optimization process yields a scaling factor of $\delta$= 1.102, resulting in a finalized girder height of 8.330 m. This corresponds to a 9.26% reduction in girder height while maintaining structural stiffness. The height optimization process consists of two primary phases: the initial optimization phase, which accounts for 67.74% (${\varphi }_{h1}$= 0.6774) of the total height reduction, and the iterative refinement phase, contributing the remaining 32.26% (${\varphi }_{hn}$ = 0.3226). Similarly, in the bending moment optimization, the initial phase captures 69.85% (${\varphi }_{M1}$ = 0.6985) of the total moment reduction, while the iterative phase contributes 30.15% (${\varphi }_{Mn}$ = 0.3015).

4.3. Girders with Variable Cross-Sections

The variable cross-section model introduces greater parametric complexity compared to its uniform cross-section counterpart, yet adheres to the same foundational optimization principles. The rootstock’s initial sectional curvature (k_ref) remains the reference standard, with each iteration’s resultant hybrid girder sectional curvature (k_h) evaluated against k_ref to assess convergence. For the variable cross-section model, the span-to-support girder height ratio (m = h₀/H₀) remains constant across all optimization stages, directing analytical focus toward the mechanics of the support section. While bending moments are computed using the simply supported beam model from Section 3.2, all other parameters are consistent with those of the uniform cross-section hybrid girder, as summarized in Table 2. The iterative optimization framework employs Equation (14) to determine the linear load intensity, while the moment of inertia at each optimization stage is computed as $I_n=b{h}_n^3\left(1-m_b{m}_h^3\right)/12$. The initial rootstock midspan bending moment (M_r) and hybrid girder moments (M_n) are provided in Equations (15) and (16), respectively.

(1) Initial hybrid girder stage

According to the principle of equivalent girder sectional curvature as described in Equation (19) and the provided bending moments of rootstock and initial hybrid girders in Equations (15) and (16), respectively, the height of the hybrid girder is updated as follows:

$$H_2=\sqrt[3]{{\displaystyle \frac{\begin{array}{c}\left[\begin{array}{c}\left(2\mu -{\mu }^2\right)\left({\beta }_{1s}m{H}_1+{\beta }_{2s}\right)\\ +{\left(1-\mu \right)}^2\left({\beta }_{1r}m{H}_1+{\beta }_{2r}\right)\end{array}\right]\left(\alpha +1\right)\left(\alpha +2\right)\\ +2\left(1-m\right)H_1\left[{\beta }_{1s}\left(\begin{array}{c}{\left(\alpha +2\right)\mu }^{\alpha +1}\\ -\left(\alpha +1\right){\mu }^{\alpha +2}\end{array}\right)+{\beta }_{1r}\left(\begin{array}{c}\left(\alpha +2\right)\left(1-{\mu }^{\alpha +1}\right)\\ -\left(\alpha +1\right)\left(1-{\mu }^{\alpha +2}\right)\end{array}\right)\right]\end{array}}{\left({\beta }_{1r}m{H}_1+{\beta }_{2r}\right)\left(\alpha +1\right)\left(\alpha +2\right)+2(1-m){\beta }_{1r}{H}_1}}}{H}_1$$

(30)

(2) Final hybrid girder stage

The inherent complexity of Equation (30), which intertwines multiple parameters governing the iterative optimization of hybrid girder height, precludes the derivation of a closed-form analytical solution. Consequently, numerical iteration is required for practical implementation. To validate the effectiveness of the proposed iterative approach, the Anhaiwan Bridge is analyzed as a representative case study using its actual design parameters. The key geometric, material, and load parameters include: $\alpha$ = 2.0, h = 5.0 m, H = 15.0 m, β_1r = 55.97 kN/m, β_2r = 232.35 kN/m, E_r = 32,500 MPa, β_1s = 11.70 kN/m, β_2s = 106.06 kN/m, m = 0.3, L = 300 m, μ = 0.343, b = 7.65 m, m_b = 0.75, m_h = 0.87. Following these design parameters, the initial bending moment is 7244.58 kN·m. The initial girder height of the rootstock girder is calculated based on the engineering experience L/17 = 300/17 = 17.65 m, serving as the starting point for the optimization process. Therefore, the target sectional curvature of the support cross-section is determined by k_ref = M/EI = 0.1256 m⁻¹. The iterative optimization is then performed based on this initial height, systematically adjusting the girder dimensions to achieve the desired sectional curvature. By establishing this target sectional curvature as the optimization benchmark, the proposed approach maximizes material efficiency while maintaining compliance with practical engineering constraints. The optimization procedure, along with the corresponding adjustments in girder height, is comprehensively documented in

Table 3. The optimization process of the hybrid girder with variable cross-sections: An example of the Anhaiwan Bridge.

Optimization Stage		H (m)	h (m)	I (m4)	M (kN·m)	k (m−1)	(kh − kref)/kref
i = 0	Rootstock girder	17.650	5.295	1774.1	7244.58	0.1256	/
i = 1	Initial hybrid girder	17.650	5.295	1774.1	4815.33	0.0835	−33.53%
i = 2	1st optimization	15.403	4.621	1179.2	4432.34	0.1157	−7.95%
i = 3	2nd optimization	14.984	4.495	1085.4	4360.79	0.1236	−1.61%
i = 4	3rd optimization	14.903	4.471	1067.9	4346.97	0.1253	−0.32%
i = 5	4th optimization	14.886	4.466	1064.5	4344.28	0.1256	−0.00%
Real-word design		15.000	4.500	1089.0	4363.57	0.1233	−1.87%

, providing a detailed examination of the convergence behavior and structural implications of the proposed method.

The optimization results reveal that the Anhaiwan Bridge exhibits an initial height optimization proportion (ϕ_h1) of 81.30%, surpassing values observed in uniform cross-section hybrid girders. This elevated initial-phase contribution is attributed to the variable cross-section’s pronounced support-section height (H), which amplifies the first-stage optimization impact by leveraging the scion’s lightweight properties to dramatically reduce internal forces. Subsequent iterations yield diminishing height reductions, reflecting stabilized rootstock–scion load-sharing interactions. The finalized hybrid girder configuration demonstrates superior mechanical performance compared to the initial design, with optimized geometric parameters closely aligning with real-world design data. This high congruence validates the methodology’s effectiveness in determining rational structural dimensions of hybrid girder bridges in the preliminary design phase.

5. Engineering Demonstration

5.1. Application to the Dongbao River Bridge

The research approach is further validated through its application to the Dongbao River Bridge, a hybrid continuous rigid frame girder bridge located in Shenzhen, China, with a main span configuration of 88 + 156 + 88 m (Figure 12). Conventional design guidelines prescribe a pier-top girder height-to-span ratio of 1/17, corresponding to a theoretical height of 9.18 m for the 156 m main span. However, site-specific constraints—including overhead high-voltage transmission lines and navigational clearance requirements—necessitated a reduced pier-top height of 8.3 m, resulting in a height-to-span ratio of 1/18.8, while the midspan height was further constrained to 3.5 m, yielding a ratio of 1/44.6. Structural analysis revealed critical stress anomalies [30]: the maximum tensile stress in the top slab reached −0.9 MPa, exceeding the code-mandated zero-tension threshold, while the maximum compressive stress in the bottom slab reached 18.0 MPa, approaching the limit of 19.25 MPa for C60 concrete under serviceability conditions. Prestressing poses limited effects to improve these issues, primarily due to the unconventional height-to-span ratios, which disrupted the expected load distribution patterns of corrugated steel web box girders. To mitigate these stress concentrations while adhering to geometric constraints, a hybrid girder configuration was devised. The midspan segment’s concrete bottom slab was replaced with a steel trough slab, resulting in a corrugated steel web box–steel trough hybrid girder. This strategic steel substitution reduced dead loads by 12–15%, effectively redistributing stresses toward the rootstock (pier-top) region. The hybrid system capitalizes on the compressive strength of the concrete rootstock and the tensile efficiency of the steel scion, mirroring the synergistic trait integration observed in grafted plants.

The trait optimization approach for hybrid girder bridges with variable cross-sections is further validated through its application to the Dongbao River Bridge. The design parameters are specified as follows: $\alpha$ = 1.6, h = 3.5 m, H = 8.3 m, β_1r = 50.22 kN/m, β_2r = 256.30 kN/m, E_r = 32,500 MPa, β_1s = 11.37 kN/m, β_2s = 235.93 kN/m, m = 0.422, L = 156 m, μ = 0.356, b = 8.70 m, m_b = 0.75, m_h = 0.87. Based on these parameters, the initial bending moment is 1544.05 kN·m. The initial girder height of the rootstock girder is calculated based on the engineering experience L/17 = 300/17 = 9.18 m, serving as the starting point for the optimization process. Hence, the target sectional curvature of the support cross-section is determined by k_ref = M/EI = 0.1676 m⁻¹. The subsequent iterative process adheres to the established optimization framework, systematically refining the girder geometry while monitoring sectional curvature redistribution. The resultant data, including optimized girder height, bending moments, and sectional curvature convergence, are presented in

Table 4. The optimization process of the Dongbao River Bridge.

Optimization Stage		H (m)	h (m)	I (m4)	M (kN·m)	k (m−1)	(kh − kref)/kref
i = 0	Rootstock girder	9.176	3.870	283.54	1544.05	0.1676	/
i = 1	Initial hybrid girder	9.176	3.870	283.54	1215.69	0.1319	−21.27%
i = 2	1st optimization	8.474	3.573	223.25	1179.51	0.1626	−2.98%
i = 3	2nd optimization	8.389	3.537	216.60	1175.14	0.1669	−0.37%
i = 4	3rd optimization	8.378	3.533	215.80	1174.61	0.1675	−0.05%
Real-word design		8.300	3.500	209.81	1170.58	0.1717	2.45%

. The results demonstrate a high degree of alignment between the final optimized geometric parameters and the actual design, with deviations in estimated girder heights not exceeding 1%. Moreover, the 24.19% reduction in girder bending moment, calculated as (1170.58 − 1544.05)/1544.05, closely aligns with the 24% reduction reported in [28]. The proposed approach exhibits excellent iterative convergence, achieving an initial height optimization proportion (ϕ_h1) of 87.97%. These validations further confirm the method’s adaptability to large-span hybrid systems, where iterative trait inheritance and curvature-driven refinements effectively address geometric constraints while optimizing material efficiency.

5.2. Application to Girder Height Estimations

The accuracy of the trait optimization approach in estimating the heights of hybrid girders with variable cross-sections has been validated through two engineering case studies: the Anhaiwan Bridge and the Dongbao River Bridge. However, for hybrid girders with variable cross-sections, this optimization approach lacks a closed-form solution and relies on numerical iterations that require extensive structural parameter inputs. In the preliminary design stage, collecting or determining such detailed structural parameter values remains challenging, limiting the direct applicability of the iterative method in early-stage design decisions. Notably, for hybrid girder bridges with uniform cross-sections, the optimized girder height (H*) can be directly derived from the original rootstock girder height (h₀) divided by the scaling factor ($\delta$), as expressed in Equation (26), which can be reformulated as follows:

$$h*=h_0/\delta =\left[L_0·\left(h/l\right)\right]/\delta =L_r\times \left(h/l\right)$$

(31)

where the original rootstock girder height (h₀) is typically determined by multiplying the span length (L₀) by conventional height-to-span ratios ($h/l$). The scaling factor ($\delta$) is determined based on the hybrid ratio (μ) and load ratio (γ) of the hybrid bridge, as defined in Equation (25). In highway bridges, the conventional height-to-span ratio ($h/l$) for the support section typically ranges from 1/14 to 1/18, while for railway bridges, this range is slightly larger, between 1/12 and 1/16. The equivalent rootstock span ($L_r$) is required as input for girder height optimization. In practice, the given length refers to the hybrid girder, not the rootstock girder, so a transformation between hybrid and rootstock lengths is necessary. A previous study [25] proposed the formulation L_r = L/δ, for this purpose, where δ is a scaling factor. While this reference is used only for calculating L_r, the present study focuses on mechanical trait inheritance and optimization of hybrid girders from a novel bionic perspective, which is fundamentally different from [25].

To assess the applicability of the girder height estimation formula derived for uniform cross-section hybrid girders in preliminary calculations for variable cross-section hybrid girders, the approach is applied to 13 representative hybrid girder bridges worldwide, as summarized in

Table 5. Estimated girder heights for various worldwide hybrid girder bridges.

Bridge Name		L (m)	γ	μ	Lr (m)	Scaling Factor (δ)	Height-to-Length Ratio	Estimated Girder Height (m)	Real Height (m)	Conformance	Mean Estimation
											Girder Height (m)	Deviation
1	Stolma Bridge	301	0.600	0.600	245	1.2272	1/14~1/18	13.63~17.52	15.0	√	15.33	2.20%
2	Shibanpo Bridge	330	0.300	0.312	262	1.2585	1/14~1/18	14.57~18.73	16.0	√	16.39	2.42%
3	Oujiang Bridge	200	0.300	0.400	149	1.3460	1/14~1/18	8.26~10.61	9.0	√	9.29	3.19%
4	Anhaiwan Bridge	300	0.300	0.343	233	1.2887	1/14~1/18	12.93~16.63	15.0	√	14.55	−3.00%
5	Taoerhe Bridge	338	0.300	0.512	231	1.4638	1/14~1/18	12.83~16.49	17.0	×	14.43	−15.11%
6	Cheviré Bridge	240	0.300	0.670	147	1.6303	1/14~1/18	8.25~10.60	9.0	√	9.20	2.23%
7	Dongbao River Bridge	156	0.700	0.356	142	1.1014	1/14~1/18	7.87~10.12	8.3	√	8.85	6.66%
8	Zhoushan Bridge	260	0.300	0.327	204	1.2730	1/14~1/18	11.35~14.59	13.3	√	12.76	−4.02%
9	Zhongshan Bridge	220	0.300	0.395	164	1.3408	1/14~1/18	9.12~11.72	11.0	√	10.25	−6.78%
10	Longxiang Bridge	202	0.300	0.396	151	1.3419	1/14~1/18	8.36~10.75	10.4	√	9.41	−9.53%
11	Hangzhou-Wenzhou Railway Bridge	216	0.300	0.380	163	1.3256	1/12~1/16	10.18~13.58	12.6	√	11.64	−7.63%
12	Guangzhan Railway Bridge	200	0.500	0.375	167	1.1993	1/12~1/16	10.42~13.90	15.0	×	11.91	−20.59%
13	Raftsundet Bridge	298	0.600	0.752	236	1.2653	1/14~1/18	13.08~16.82	14.5	√	14.72	1.51%

Note: The equivalent span length is calculated by L_r = L/δ according to the previous study [25].

. Notably, hybrid girder bridges are relatively uncommon, as this structural form is primarily advantageous for very long-span applications. For typical span ranges, prestressed concrete bridges are generally preferred due to their lower construction complexity, cost efficiency, and well-established design practices. Consequently, the number of accessible engineering cases is inherently limited. The majority of hybrid girder bridges with publicly available design information have been compiled in Table 5 for validation.

First, the equivalent rootstock span (L_r) is determined based on the main span length (L), hybrid ratio (μ), and load ratio (γ) of each bridge, using a previously established formula of equivalent span increase coefficient that ensures equivalence in bending moments between hybrid and rootstock girders [25]. Next, the estimated girder height at the support section is calculated using Equation (31), incorporating the equivalent rootstock span (L_r), conventional span-to-height ratios ($h/l$), and the scaling factor ($\delta$). The estimated girder heights are then compared with actual design values to evaluate the accuracy of the approach. For instance, in the case of the Dongbao River Bridge, where the hybrid ratio is μ = 0.356 and the load ratio is γ = 0.700, the scaling factor is computed as δ = 1.185. This results in an estimated support section height ranging from $h*$ = 7.87 to 10.12 m, with an average of 8.85 m. The actual design height of 8.30 m falls well within this range, yielding a deviation of only 6.66%. Among the 13 bridges analyzed, only the Taoerhe Bridge and Guang-Zhan Railway Bridge exhibited significant deviations between predicted and actual girder heights. The large deviation observed for Taoerhe Bridge may be attributed to its record-span design. With a main span of 338 m—the longest of its kind worldwide—a more conservative girder height was adopted. As a result, its load ratio is only 0.3, while the hybrid ratio reaches 0.512. This is significantly higher than that of Shibanpo Bridge, which has a similar load ratio of 0.3 but a lower hybrid ratio of 0.312 for a slightly shorter main span of 330 m. Additionally, the large deviation observed for the Guang–Zhan Railway Bridge is likely attributable to its railway bridge configuration, in which structural design is governed by deformation rather than the bending moment considered in this study. For the remaining 11 bridges, the estimated girder heights closely align with actual design values, confirming the reliability of the proposed method. Notably, the optimization result for the Guangzhan Railway Bridge shows a larger deviation (−20.59%) because the proposed approach is moment-based and therefore suited to strength-controlled design under ultimate limit states. In contrast, railway bridges are typically governed by deformation-based serviceability criteria, rather than bending strength. These findings demonstrate the approach’s practicality for rapid preliminary estimation of variable cross-section hybrid girder heights during conceptual design.

6. Conclusions

This study explores the mechanical trait inheritance and optimization of hybrid girder bridges from a bionic perspective, drawing an analogy to quantitative trait inheritance in genetics. A binary grafting model is established, wherein rootstock and scion girders function as parental components, while hybrid girders represent first-generation offspring. Classical beam theory is employed to examine two sequential phases—trait inheritance and trait optimization—offering critical insights into the performance enhancement mechanisms of hybrid bridges. The major findings are as follows:

(1) During trait inheritance, the strategic integration of high-performance scion girders into optimal positions of rootstock girders significantly enhances hybrid girder performance, with inheritance ratios governed by the hybrid ratio and load ratio. During trait optimization, the hybrid ratio remains constant, while iterative girder height and internal force adjustments emulate rootstock–scion synergy, further improving structural performance.

(2) For uniform cross-section hybrid girders, three scion inheritance ratio metrics derived from fixed-supported (${\psi }_s^{M-}$, ${\psi }_s^{M+}$) and simply supported (${\psi }_s^M$) models, establish the hierarchical relationship: ${\psi }_s^{M-}$ > ${\psi }_s^M$ > ${\psi }_s^{M+}$. Approximating ${\psi }_s^{M-}$ with ${\psi }_s^M$ results in a maximum error of 15.9%. However, ${\psi }_s^{M-}$ can be accurately approximated using an empirical power function: ${\psi }_s^M\left(\mu \right)$ = $1-{\left(1-\mu \right)}^{3.09}$.

(3) Variable cross-section hybrid girders, constrained by parametric complexity, preclude analytical solutions. A simply supported beam model yields ${\psi }_s^M\left(\mu ,H,h,a\right)$, with Anhaiwan Bridge results showing slightly lower ${\psi }_s^M$ values than uniform cross-sections and minimal sensitivity to support girder height (H), midspan girder height (h), and the power exponent governing the girder height variation (α). Power functions ${\psi }_s^M\left(\mu \right)$ = $1-{\left(1-\mu \right)}^{1.79}$ and ${\psi }_s^M\left(\mu \right)$ = $1-{\left(1-\mu \right)}^{2.82}$ effectively approximate simply supported and fixed-support hybrid girders with variable cross-sections, respectively. Optimal inheritance efficiency occurs near μ = 0.45, which corroborates the optimal hybrid ratio range for maximizing trait inheritance in hybrid girder systems.

(4) The optimization process consists of initial and iterative phases. Initial optimization, integrating rootstock and scion, achieves the greatest performance gains, while iterative refinements adjust girder height and bending moments for further improvement. Analytical solutions optimize uniform cross-section heights, while complex variable cross-sections require numerical iterations, showing <1% deviation in case studies. Their higher initial optimization contributions reflect greater geometric sensitivity. The proposed approach for height optimization was validated by 13 worldwide hybrid girder bridges, showing great practicality for rapid preliminary estimation of hybrid girder dimensions during conceptual design.