Lightweight Design Method for the Main Arch of Large-Span Reinforced Concrete Arch Bridges

Abstract

Large-span reinforced concrete (RC) arch bridges are widely built in mountainous cities with high stiffness and excellent durability. Nevertheless, heavy arch ring self-weight results in reduced cost-effectiveness and safety. Therefore, this study proposes a lightweight design method for the large-span RC arch bridge, which can significantly reduce the weight of the arch ring, improve material utilization, and enhance the economic span. A zero-moment equal-stress arch axis model is established under practical loads for uniform stress distribution in the main arch ring under dead loads. Using section parameters as variables, a correlation model between arch section strength and global stability is developed, alongside an optimization algorithm for their optimal coordination. Validated on a 386 m-span RC arch bridge, the results show material consumption can be reduced by up to 54.6% with unchanged peak arch stress and achieving the preset stiffness and bearing capacity. The arch’s minimum mass is proportional to its upper structure load, so optimizing that load can effectively reduce the rib weight. This work provides a new framework for lightweight design of long-span statically indeterminate RC arch bridges, and offers low-cost and safe arch bridge solutions for infrastructure construction in mountainous cities.

Graphical abstract

Keywords

Large-span RC arch bridge / Dead-load stress / Equal-stress / Lightweight design

Highlight

	● A zero-moment equal-stress arch axis model was constructed for large-span arch bridges under real loads.
	● An algorithm for arch bridge section parameters considering strength and stability has been developed.
	● The variation law of the minimum mass of the arch with span, stress, and upper structure load has been clarified.

1 Introduction

Mountainous cities are typically characterized by rugged topography, deep valleys, and steep terrain, where urban highways and railways often traverse deeply incised gorges and steep slopes. Owing to their high structural stiffness [1], excellent seismic resistance [2], superior durability [3], and their inherent compatibility with canyon-dominated landscapes, large-span reinforced concrete (RC) arch bridges have become one of the most widely adopted bridge types in mountainous urban areas [4–6]. However, as the span length increases, the self-weight of the arch ring grows disproportionately, leading to substantially higher construction costs and elevated construction risks in mountainous environments [7–9]. Moreover, the increased arch-ring weight gives rise to increasingly pronounced challenges in structural design and load-carrying behavior. Therefore, achieving the lightweight design of large-span RC arch bridges while ensuring structural safety and satisfactory in-service performance has become a critical challenge in modern large-span arch bridge engineering.

The objective of lightweight structural design is to maximize structural efficiency by minimizing self-weight through optimized structural configurations while satisfying both ultimate and serviceability limit-state requirements [10]. To this end, extensive research has been conducted on lightweight technologies for large-span arch bridges, with existing studies primarily focusing on two aspects: advanced materials and structural system optimization. In terms of high-performance material applications, the adoption of innovative materials such as high-performance concrete (HPC) and fiber-reinforced composites has provided a direct approach to structural weight reduction by enhancing the intrinsic mechanical properties of structural components [11–13]. For instance, Croatian researchers proposed using reactive powder concrete (RPC) to reduce the self-weight of the main arch ring in conceptual designs of arch bridges with main spans of 432, 500, and 1000 m [14]. Čandrlić et al. [15] further investigated the feasibility of constructing a 1000 m span concrete arch bridge using RPC. Similarly, Shao et al. [16–18] developed a conceptual design for kilometer-scale steel–UHPC composite truss arch bridges, demonstrating an innovative structural system that simultaneously achieves lightweight construction and exceptional spanning capability. Huang et al. [19] investigated the nonlinear stability of arches reinforced with graphene-reinforced composites under elastic rotational restraints, revealing the effects of graphene content, geometric dimensions, and restraint stiffness on buckling behavior. Nevertheless, the widespread application of these approaches remains constrained by the high cost of UHPC and RPC, as well as compatibility issues associated with field-casting techniques. Furthermore, when the amount of high-strength material is reduced to improve economic efficiency, larger cross-sectional dimensions and thinner wall elements are often required to meet stiffness demands, thereby intensifying the trade-off between efficient use of material strength and the need to maintain structural stability.

In optimizing arch bridge structures, the stress distribution under dead loads is the key factor governing the structural efficiency of large-span, statically indeterminate RC arch bridges. When dead-load stresses are distributed non-uniformly across the arch section, localized regions may reach the material strength limit prematurely, whereas other regions remain significantly underutilized. Such an imbalance in stress distribution directly limits the potential for structural lightweighting. Consequently, enhancing the structural efficiency of large-span statically indeterminate RC arch bridges while simultaneously ensuring adequate strength and stability has emerged as a major research focus. Qiu et al. [20] conducted a parametric investigation of cable–arch systems, demonstrating that cable anchorage can significantly enhance the stability of arch ribs and clarifying the influences of key parameters, including cable inclination, cable arrangement, and material properties, on structural stability. Hu et al. [21–23] carried out a systematic theoretical investigation into the in-plane nonlinear elastic buckling and post-buckling behavior of parabolic arches, multi-span continuous arches, and arch–beam composite structures, elucidating the critical roles of the rise-to-span ratio, restraint conditions, and axial stiffness ratio in governing buckling modes. Zhang et al. [24] applied the analytical transfer relations of slender circular arches to circular arch bridges and optimized the number of hangers to minimize the bending moments in the arch rib, thereby approximating the thrust line. Their results showed that the proposed approach effectively reduced the complexity of structural analysis to a level comparable to that of straight-beam systems. Shi et al. [25] proposed an iterative arch-axis optimization method based on force equilibrium analysis for high-speed railway arch bridges. Their study demonstrated that the proposed method reduced the maximum absolute bending moment of the arch rib substantially within only seven iterations, while simultaneously decreasing stresses and deflections and markedly improving structural stiffness. Zhang et al. [26] developed a mathematical model for determining the optimal arch axis of large-span open-spandrel arch bridges by accounting for the combined effects of arch self-weight and concentrated loads. Compared with conventional arch profiles, such as parabolic and catenary arches, the optimized profile significantly reduced the bending moment, the ratio of bending stress to axial stress, the eccentricity, and the sum of the squared eccentricities. Gil-Martín et al. [27], Marano et al. [28], and Lewis et al. [29–31] focused on determining the optimal shape of variable cross-section constant-stress arches subjected to self-weight and deck loads, providing valuable guidance for cross-sectional optimization of variable-section arch bridges. However, these studies did not account for the significantly concentrated loads transmitted from the superstructure, thereby limiting their applicability to practical bridge design. Previous studies have explored rational structural systems and cross-sectional configurations for large-span RC arch bridges. However, these studies have primarily focused on innovations at the global structural level, with insufficient attention to the refined control of internal stress distribution. Consequently, they may fall into a detrimental cycle of cross-section reduction → stress exceedance → passive structural strengthening, making it difficult to achieve the coordinated optimization of structural strength and stability.

To address these challenges, this study proposes a lightweight design method for the main arch of large-span RC arch bridges in mountainous urban environments, with the primary objective of achieving a uniform dead-load stress distribution. The proposed method aims to reduce the self-weight of the arch ring while improving the overall structural efficiency of RC arch bridges. First, through theoretical derivation, a zero-moment equal-stress arch axis model was constructed for the actual stress mode of large-span arch bridges. Based on this model, the correlation formulas between arch stress section area, arch stability, and coefficient section stiffness were established, and an algorithm for arch bridge section parameters considering strength and stability was developed, jointly forming a lightweight design method for large-span RC arch bridge main arches suitable for mountainous cities. These formulations are subsequently integrated into a unified, lightweight design framework based on the principle of uniform dead-load stress distribution. The proposed method is then validated through numerical optimization of a practical 386 m span RC arch bridge, demonstrating its effectiveness and superiority in reducing structural self-weight and enhancing material utilization. Finally, a comprehensive parametric study is conducted using the proposed design methodology to quantify the effects of span length, allowable stress, and superstructure load on the minimum arch mass.

2 Design Method

2.1 Method framework

The cross-sectional design of the main arch rib plays a critical role in ensuring both the structural safety and economic efficiency of large-span arch bridges. An inappropriate distribution of structural material not only results in unnecessary material consumption but also imposes excessive self-weight on the structure. Therefore, the cross-sectional design of the main arch must simultaneously satisfy the strength requirements and ensure an appropriate distribution of structural stiffness. Only by achieving a proper balance between these two aspects can the arch effectively resist stresses, deformations, and stability-related failures under various loading conditions.

Therefore, this paper proposes a lightweight design method for the main arch of a large-span RC arch bridge, which can achieve automatic solution of the arch bridge section under any preset reasonable stress and linear elastic stability coefficient. The specific design principle is shown in Fig. 1.

Step 1: A lightweight zero-moment equal-stress arch axis model is proposed. By designing the arch axis reasonably, the bending moment caused by the deviation of the arch axis from the pressure line can be eliminated, and the bending stress of the main arch can be reduced. At this time, the dead load bending moment of the arch is only generated by axial compression.

Step 2: A multi-objective design method for cross sections is proposed, which achieved optimization design of cross sections under the dual objectives of strength and stability. Specifically, the cross-sectional area can be tailored to maintain a prescribed and approximately uniform axial compressive stress along the arch ribs. At the same time, by designing the inertia moment of the cross-section, the stiffness of the arch ribs can meet the predetermined stability requirements.

The meanings of the parameters involved in this article are shown in Table 1.

2.2 The lightest arch axis model

A rational arch axis ensures that the geometry of the main arch ring closely approximates the optimal load-carrying configuration of a large-span arch bridge [21]. The first step in determining the rational arch axis is to establish the magnitude and distribution of the permanent loads, namely the dead-load pattern, as illustrated in Fig. 2. The dead load of an arch bridge can be decomposed into the distributed load of the arch ribs and the load of the upper structure. The load of the upper structure of an arch bridge can be further divided into vertical forces transmitted by columns or suspension rods according to different bridge types such as deck arch bridges, semi-through arch bridges, and Fully-through arch bridges, as shown in Fig. 2A.

The dead-load loading pattern of a large-span arch bridge is illustrated in Fig. 2B. The origin of the coordinate system, O, is located at the crown of the arch. In Fig. 2B, the arch has a span of 2l and a rise of f. The variables x and y denote the horizontal and vertical coordinates of the arch axis, respectively, and the rational arch axis under dead loads is represented by y(x). The cross-sectional area of the main arch ring, which varies along the span and remains perpendicular to the arch axis, is denoted by A(x). The unit weight of the material is \gamma. Accordingly, the self-weight of the main arch distributed along the arch axis is expressed as g(x)=A(x)\cdot \gamma, while the corresponding distributed load projected along the span is denoted by q(x)=A(x)\cdot \gamma \cdot \sqrt{1+{({y^{\mathrm{\prime }}}_1(x))}^2}[26]. The locations of the columns and hangers are represented by x_i, and the concentrated load transmitted to the arch by the columns and hangers at location x_i is denoted by F_i. The angle between the tangent to the arch axis and the horizontal direction is denoted by \theta (x), while the axial force, horizontal reaction, and vertical reaction are represented by N(x), H, and V(x), respectively.

The equilibrium equations are established at point K in Fig. 2B. Since the rational arch axis is defined such that the bending moment at every cross-section is zero, the arch is subjected exclusively to axial compression, and the line of thrust coincides with the arch axis. Consequently, the slope of the arch axis at point K, y^{\mathrm{\prime }}(x), is equal to the ratio of the vertical sectional reaction, V(x), to the horizontal reaction, H, yielding Eq. (1):

(1)y^{\mathrm{\prime }}(x)=\frac{V(x)}{H}

The vertical sectional reaction at point K, V(x), is equal to the sum of the distributed self-weight q(x) of the main arch from the crown (O) to point K and the concentrated loads \sum F_i transmitted from the superstructure. Accordingly, y^{\mathrm{\prime }}(x) can be expressed as Eq. (2):

(2)y^{\mathrm{\prime }}(x)=\frac{\gamma \int A(x)\sqrt{1+{(y^{\mathrm{\prime }}(x))}^2}dx}{H}+\frac{\sum F_i}{H}

To further solve Eq. (2), the distributed load A(x) associated with the cross-sectional area distribution of the main arch must first be specified. The cross-sectional area distribution of the main arch is assumed to follow A(x)=A_0{\left[1+{({y^{\mathrm{\prime }}}_1(x))}^2\right]}^{\frac{m}{2}}, where A_0 represents the cross-sectional area of the arch crown and m denotes the cross-sectional area variation coefficient. When m=2, the axial compressive stress \sigma ={\displaystyle \frac{N}{A_x}}={\displaystyle \frac{H}{A_0}} of the arch remains constant throughout the span [30-31]. By applying the boundary conditions y(0)=0,y^{\mathrm{\prime }}(0)=0, the equation of the rational arch axis corresponding to a constant axial compressive stress can be derived, as expressed in Eq. (3).

(3)

Based on the force equilibrium conditions between the x_2\sim x_n segmented points of the arch axis, Eq. (4) can be established. Likewise, according to the continuity conditions at the x_2\sim x_n segmented points of the arch axis, Eq. (5) can be derived. Together, Eqs. (4) and (5) provide a total of 2n-2 equations. By further incorporating the boundary conditions y_1(0)=0, y_n(l)=-f, and y_1{}^{\mathrm{\prime }}(0)={\displaystyle \frac{F_1}{H}}, a system of 2n+1 equations is obtained. In these equations, the material unit weight \gamma, the concentrated loads F_i, and their corresponding locations x_i are known input parameters, whereas {\phi }_i, Z_i, H, and \sigma constitute a total of 2n+2 unknown variables. Therefore, an additional constraint must be introduced to close the system of equations.

(4)\frac{F_{i+1}}{H}=\tan (\frac{\gamma }{\sigma }{x}_i+{\phi }_{i+1})-\tan (\frac{\gamma }{\sigma }{x}_i+{\phi }_i)

2.3 Cross-sections design under multi-objective conditions

On the basis of a lightweight and reasonable arch axis model, this paper further establishes the correlation formula between section parameters and the strength and stability objectives of the main arch, and develops an optimization algorithm for section parameters under preset strength and stability objectives, thereby achieving automatic design of the main arch section.

2.3.1 The correlation between cross-sectional parameters and strength targets

Eq. (4) describes the force equilibrium between adjacent segmented points along the arch axis. By substituting x = 0 into Eq. (4), the relationship among the compressive stress at the arch crown \sigma, the horizontal reaction H, and the cross-sectional area at the arch crown A_0 can be obtained, as expressed in Eq. (5). In Eq. (5), F₁ denotes the concentrated load acting at the arch crown. If no column or hanger is installed at the crown, F₁ = 0. Since the cross-sectional area of the arch rib satisfies A(x)=A_0\left[1+{({y^{\mathrm{\prime }}}_1(x))}^2\right], substitution of this relationship into Eq. (5) yields the analytical expression for the cross-sectional area of the main arch, as given in Eq. (6). Accordingly, Eq. (6) can be used to determine the cross-sectional area distribution of the main arch corresponding to a prescribed compressive stress at the arch crown, \sigma:

(5)A_0=\sqrt{\frac{H^2+F_1{}^2}{{\sigma }^2}}

(6)A(x)=\sqrt{\frac{H^2+F_1{}^2}{{\sigma }^2}}\sec (\frac{\gamma }{\sigma }x)

2.3.2 The correlation between cross-sectional parameters and stable targets

When the applied load reaches the critical value, the arch axis deviates from its original symmetric compression-dominated configuration in the vertical plane and subsequently transforms into an antisymmetric flexural deformation mode, commonly referred to as in-plane antisymmetric buckling of the arch [21]. As the applied load approaches and exceeds the critical buckling load, the deflected shape of the arch undergoes a sudden change, as illustrated in Fig. 3. For large-span arch bridges with a rational arch axis, the dead loads are generally symmetrically distributed. Under a uniformly distributed symmetric loading condition, the line of thrust coincides with the arch axis, and the structure remains in a symmetric compressive equilibrium state prior to instability, with the strain energy being predominantly stored as compressive deformation energy. As the applied load gradually approaches the critical level, the equilibrium path undergoes bifurcation, giving rise to flexural deformation and ultimately triggering antisymmetric buckling of the arch. Under these circumstances, the symmetric external loads perform no work on the antisymmetric deformation, and the strain energy of the arch is progressively transformed from compressive deformation energy into bending deformation energy [22].

Under symmetric loading, the arch undergoes antisymmetric buckling deformation, while the horizontal reaction remains H=0. The elastic axial shortening of the arch is assumed to be negligible, and the structural response is therefore dominated by bending deformation. Under these assumptions, the governing equilibrium differential equation of the arch can be expressed as Eq. (7). In Eq. (7), E is the elastic modulus of the material, I₀ is the moment of inertia of the arch cross-section at the crown, v"" denotes the fourth derivative of the vertical deflection function, q(x) is the distributed self-weight load of the main arch along the span, F is the vertical reaction at the arch springing, and w" represents the second derivative of the horizontal displacement function of the main arch [32].

(7)E{I}_0{v^{\mathrm{\prime }\mathrm{\prime }}}^{\mathrm{\prime }\mathrm{\prime }}=q(x)+F{w}^{\mathrm{\prime }\mathrm{\prime }}

Eq. (7) is a nonhomogeneous equilibrium differential equation with variable coefficients, containing both the nonhomogeneous term q(x) and the variable-coefficient term f(x)={\displaystyle \frac{w^{\mathrm{\prime }\mathrm{\prime }}}{v^{\mathrm{\prime }\mathrm{\prime }}}}, which prevents a direct analytical solution. Considering that the term F{w}^{\mathrm{\prime }\mathrm{\prime }}, representing the coupling effect between horizontal displacement and vertical loading, is relatively small compared with q(x), this coupling term is neglected, leading to the simplified expression given in Eq. (8).

(8)E{I}_0{v}_1{}^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}=q(x)

In Eq. (8), v_1 represents the linear elastic component of the arch undergoing antisymmetric buckling under symmetric loading. Its analytical expression is identical to the structural deformation induced by the antisymmetric load q(x) and can be obtained by performing the quadruple integration of q(x), as expressed in Eq. (9), together with the antisymmetric boundary conditions given in Eq. (10).

(9)v_1=\int \iiint \frac{q(x)}{E{I}_0}{(dx)}^4

(10)\{\begin{array}{l}{v}_{\pm l}=0\\ v_0=0\\ v_{\pm l}^{\mathrm{\prime }}=0\\ v_0^{\mathrm{\prime }\mathrm{\prime }}=0\end{array}

To solve Eq. (9), the distribution of q(x) must first be determined. By discretizing the rational arch axis, the corresponding discrete representation of q(x) can be established, as given in Eq. (11). Substituting Eq. (11) into Eq. (9), followed by integration and application of the antisymmetric boundary conditions in Eq. (10), yields the analytical solution for the antisymmetric deformation of a rational arch axis with an arbitrary geometry, as expressed in Eq. (12).

(11)q(x)\approx q_0(1+C_1{(ax)}^2+{\text{C}}_2{(ax)}^4)

(12)\begin{array}{rl}{v}_1=& \frac{q}{E{I}_0}[\frac{1}{48}(2x^4-3l{x}^3+l^{3}x)+\frac{a^2{C}_1}{60}(2x^6-5l^3{x}^3+3l^{5}x)+\\ & \frac{a^4{C}_2}{112}(2x^8-7l^5{x}^3+5l^{7}x)]\end{array}

Since the work done by the symmetric vertical loads on the antisymmetric deformation is zero, the total potential energy of the arch after buckling is equal to its strain energy, which can be expressed as Eq. (13) [22].

(13)\Pi ={\Pi }_s+{\Pi }_m

In Eq. (13), \Pi, {\Pi }_s, and {\Pi }_m denote the total potential energy, the bending strain energy, and the axial compression strain energy of the arch, respectively. According to arch structural theory and the principle of virtual work, {\Pi }_s and {\Pi }_m can be expressed by Eqs. (14) and (15), respectively.

(14){\Pi }_m=-{\int }_s\frac{N^2}{2EA}\mathrm{d}s

(15){\Pi }_s={\int }_s\frac{M^2}{2EI}\mathrm{d}s

Substituting {\epsilon }_m={\displaystyle \frac{N}{EA}}, N=H{(1+{y^{\mathrm{\prime }}}^2)}^{1/2}, and ds=dx(1+{y^{\mathrm{\prime }}}^2{)}^{1/2} into Eq. (14), and further incorporating the relationship between the horizontal and vertical displacements given by Eq. (16), yields the displacement-based expression for the axial compression strain energy of the arch, as presented in Eq. (17).

(16){\epsilon }_m(1+{y^{\mathrm{\prime }}}^2)=w^{\mathrm{\prime }}+y^{\mathrm{\prime }}{v}^{\mathrm{\prime }}=-\frac{H}{E{A}_x}\cdot {(1+{y^{\mathrm{\prime }}}^2)}^{3/2}

(17){\Pi }_m=-\frac{H}{2}{\int }_{-l}^l{w}^{\mathrm{\prime }}+v^{\mathrm{\prime }}{y}^{\mathrm{\prime }}dx=\frac{H}{2}{\int }_{-l}^{l}v{y}^{\mathrm{\prime }\mathrm{\prime }}\mathrm{d}x

According to the governing differential equation of a fixed arch, the bending moment of the arch can be simplified to the form given in Eq. (18) by neglecting the effects of axial compression and shear. Substituting Eq. (18) into Eq. (15) yields the displacement-based expression for the bending strain energy of the arch, as presented in Eq. (19).

(18)M=EI{v}^{\mathrm{\prime }\mathrm{\prime }}\cos \theta

(19){\Pi }_s=\frac{1}{2}{\int }_{-l}^{l}EI{v^{\mathrm{\prime }\mathrm{\prime }}}^2\cos (\theta )\mathrm{d}x

Based on the Ritz method, the buckling load of the arch corresponds to the external load that satisfies Eq. (20). Accordingly, the critical horizontal buckling reaction of the arch, H_{cr}, can be determined from Eq. (21).

(20)\frac{\mathrm{\partial }\Pi }{\mathrm{\partial }C}=0

(21)H_{cr}=\frac{{\displaystyle {\int }_{-l}^{l}EI{{v^{\mathrm{\prime }}}^{\mathrm{\prime }}}^2\cos (\theta )\mathrm{d}x}}{{\displaystyle {\int }_{-l}^{l}v{y}^{\mathrm{\prime }\mathrm{\prime }}\mathrm{d}x}}

For an equal-depth, variable-width constant-stress arch bridge, when the distribution of the cross-sectional moment of inertia satisfies I=I_0\sec (\theta ), further substitution of v=2x^4-3l{x}^3+l^{3}x and y^{\mathrm{\prime }\mathrm{\prime }}=a{\sec }^2(ax) into Eq. (21) yields the analytical expression for the critical horizontal buckling reaction, H_{cr}, as given in Eq. (22).

(22)H_{cr}=\frac{E{I}_0{\displaystyle {\int }_{-l}^l(324l^2{x}^2-864l{x}^3+576x^4)\mathrm{d}x}}{a{\displaystyle {\int }_{-l}^l(2x^4-3l{x}^3+l^{3}x){\sec }^2(ax)\mathrm{d}x}}

Substituting \theta =ax, {\theta }_s=al, and \sec (\theta )=1+{\displaystyle \frac{1}{2}}{\theta }^2+{\displaystyle \frac{5}{24}}{\theta }^4+{\displaystyle \frac{61}{720}}{\theta }^6+{\displaystyle \frac{277}{8064}}{\theta }^8+O({\theta }^{10}) into Eq. (22), followed by integration and algebraic simplification, yields Eq. (23).

(23)\{\begin{array}{l}{H}_{cr}={\displaystyle \frac{2E{I}_0}{l^2}}{\xi }^{\mathrm{\prime }}\\ {\xi }^{\mathrm{\prime }}=11.207-4.945{\vartheta }_s{}^2+0.208{\vartheta }_s{}^4+\\ 1.892{\vartheta }_s{}^6-1.545{\vartheta }_s{}^8+O({\vartheta }_s{}^{10})\end{array}

For an equal-width, variable-depth arch bridge, when the distribution of the cross-sectional moment of inertia satisfies I=I_0{\sec }^3(\theta ), the expression for the critical horizontal buckling reaction, H_{cr}, can be reformulated as Eq. (24).

(24)\{\begin{array}{l}{H}_{cr}={\displaystyle \frac{2E{I}_0}{l^2}}{\xi }^{\mathrm{\prime }}\\ {\xi }^{\mathrm{\prime }}=10.972-0.542{\vartheta }_s{}^2-0.064{\vartheta }_s{}^4-\\ 0.015{\vartheta }_s{}^6+0.037{\vartheta }_s{}^8+O({\vartheta }_s{}^{10})\end{array}

Eqs. (23) and (24) can both be rewritten in the unified form of Eq. (25), where \mu L denotes the equivalent effective span length, and \mu is the effective span length coefficient, \mu ={\displaystyle \frac{1}{\sqrt{8{\xi }^{\mathrm{\prime }}}}}. For constant-stress arch bridges, {\vartheta }_s depends exclusively on the arch springing angle. By substituting H={\sigma }_m{A}_0 into Eq. (25), the analytical expression for the in-plane antisymmetric stability coefficient of the arch rib, \eta, can be obtained, as given in Eq. (26), where i_0=\sqrt{{\displaystyle \frac{I_0}{A_0}}} denotes the radius of gyration of the arch crown section. For a given cross-sectional configuration, the antisymmetric stability coefficient of the arch is inversely proportional to the square of the span length and to the compressive strain. Equation (26) therefore establishes the relationship between the structural stability of the arch bridge and the cross-sectional moment of inertia. Accordingly, Eq. (26) can be used to determine the required stiffness distribution of the main arch corresponding to a prescribed arch crown compressive stress \eta and stability coefficient η.

(25)H_{cr}=\frac{E{I}_0}{(\mu L{)}^2}

(26)\eta =\frac{E{I}_0}{A_0{\sigma }_m{(\mu L)}^2}=\frac{1}{\epsilon }(\frac{i_0}{\mu L}{)}^2

2.3.3 Optimization design of cross-section

Once the required cross-sectional area A_0 and moment of inertia I_0 have been determined, the geometric dimensions of the main arch section can be calculated according to the selected cross-sectional configuration. In this study, a box section is adopted as an illustrative example. The expressions for the cross-sectional area and the moment of inertia of the box section are given by Eqs. (27) and (28), respectively.

(27)A_0=b_0{h}_0-(b_0-2t_0)\cdot (h_0-2t_0)

(28)I_0=\frac{b_0{h}_0{}^3-(b_0-2t_0)\cdot {(h_0-2t_0)}^3}{12}

In Eqs. (27) and (28), the section depth h_0, section width b_0, and concrete wall thickness t are the three unknown geometric parameters. To determine these dimensions, an additional known parameter must be introduced. According to engineering practice, the depth-to-span ratio of the arch section is subject to practical design constraints. Therefore, the depth of the arch crown section is prescribed as h₀ = L/100 + 1 (in meters). With h_0 specified, Eqs. (27) and (28) can be solved simultaneously. However, because these equations contain higher-order terms of t_0, explicit analytical expressions for b_0 and t_0 cannot be derived, and the sectional dimensions must therefore be determined numerically. The geometric dimensions of the remaining sections for the variable-width and variable-depth arch ribs can then be calculated using Eqs. (29) and (30), respectively. For the variable-width arch rib, the web thickness varies while the thicknesses of the top and bottom plates remain constant. In contrast, for the variable-depth arch rib, the web thickness is kept constant, whereas the thicknesses of the top and bottom plates vary along the span.

(29)\{\begin{array}{l}b=b_0\sec (\vartheta )\\ t=t_0\sec (\vartheta )\end{array}

(30)\{\begin{array}{l}h=h_0\sec (\vartheta )\\ t=t_0\sec (\vartheta )\end{array}

Fig. 4 illustrates the proposed design procedure for determining the dimensions of a box-section arch rib by jointly considering structural strength and stability. The procedure consists of the following steps:

(1) Calculate the cross-sectional area at the arch crown, A₀, using the strength–cross-sectional area relationship given by Eq. (5).

(2) Calculate the moment of inertia at the arch crown, I₀, using the stability–moment of inertia relationship given by Eq. (26).

(3) Determine the depth of the arch crown section, h₀, according to the empirical expression h₀ = L/100 + 1 (where all dimensions are expressed in meters).

(4) Specify the lower and upper bounds of the section width, b₁ and b₂, respectively, and initialize the trial width as b₀ = (b₁ + b₂)/2.

(5) Compute the wall thickness of the arch crown section, t_cal, from the cross-sectional area equation [Eq. (27)].

(6) Calculate the corresponding moment of inertia, I_cal, using the moment of inertia equation [Eq. (28)].

(7) If I_cal < I₀, increase the trial section size by setting b₁ = b₀; otherwise, if I_cal > I₀, reduce the trial section size by setting b₂ = b₀.

(8) Repeat Steps (4)–(7) until Abs(1−b₁/b₂) < R, where R is the prescribed convergence tolerance (typically less than 10⁻⁵). The final dimensions of the arch crown section, including h₀, b₀, t₀, and I₀, are then obtained.

3 Case Analysis

3.1 Initial design parameters

To validate the feasibility and effectiveness of the proposed method, a large-span RC arch bridge was selected as a case study for optimization. The bridge has a span of 386 m, a rise of 70.18 m, and a spacing of 31 m between adjacent columns. No column is provided at the arch crown. The main arch is constructed using C80 concrete, while the main girder and columns are made of C50 and C40 concrete, respectively. The general configuration of the bridge is illustrated in Fig. 5.

The main girder consists of six T-shaped girders with a total deck width of 14.22 m and a deck slab thickness of 0.20 m. The arch columns adopt an octagonal hollow section with a height of 5.5 m, a width of 2.5 m, and a wall thickness of 0.5 m. In the original design, the main arch follows a catenary arch axis with an arch-axis coefficient of 2.4. The arch crown section has a depth of 5.0 m and a width of 7.0 m. At a distance of 110 m from the span center, the single box section bifurcates into two separate box sections. At the arch springings, the clear spacing between the inner faces of the two box sections is 12.0 m. Each box section has a depth of 6.5 m, a width of 4.5 m, and a uniform wall thickness of 1.0 m. The total concrete volume of the main arch is 8914 m³, corresponding to a total self-weight of 231,764 kN.

3.2 Optimized design parameters

Under the dead-load condition, the original design exhibits a maximum compressive stress of 18.06 MPa, as shown later in Fig. 10. To facilitate a direct comparison, the target compressive stress σ of the arch rib under dead load in the proposed design was prescribed as 18 MPa, which is essentially identical to that of the original design. The optimized scheme adopts a parallel twin-rib configuration, in which each rib is designed with an equal-width, variable-depth box section. The first-order in-plane linear stability coefficient of the arch rib, η, was specified as 4.0.

The calculated cross-sectional dimensions of the optimized arch rib are summarized in Table 2. As shown, the section depth of the equal-width, variable-depth arch rib increases from 4.81 m at the arch crown to 6.35 m at the arch springing, while the section width remains constant at 4.0 m. The thicknesses of the top and bottom plates increase from 261.5 mm at the arch crown to 344.9 mm at the arch springing, whereas the web thickness remains constant at 261.5 mm. In the original design, all box-section walls had a uniform thickness of 1.0 m. Compared with the original section, the optimized section adopts substantially thinner walls. This is because the proposed constant-stress design concept reduces the required cross-sectional area while satisfying the prescribed stress level. To maintain the required structural stiffness, the available material is distributed as far as possible toward the outer perimeter of the section, thereby improving the section efficiency and allowing a significant reduction in wall thickness.

3.3 Comparative analysis before and after optimization

3.3.1 Finite element modeling

Finite element analyses were further conducted to compare the structural stress, self-weight, linear elastic stability, and ultimate load-carrying capacity of the original and optimized designs. Since the evaluation of structural stress, self-weight, and linear elastic stability involves only linear elastic analysis, a MIDAS/Civil [33] finite element model composed of three-dimensional beam elements was established. In contrast, the analysis of ultimate load-carrying capacity involves coupled geometric and material nonlinearities; therefore, a three-dimensional solid-element model was developed in Abaqus [34].

Fig. 6 presents the MIDAS/Civil finite element model of the optimized 386 m arch bridge, in which the construction stages were explicitly considered. The main arch was modeled using variable-section beam elements, while the stay cables were simulated using truss elements. The model consisted of 875 beam elements and 36 truss elements with the truss and beam elements connected through shared nodes. Fixed boundary conditions were assigned at both arch springings and at the anchorage ends of the stay cables. The construction process was simulated in three representative stages: the maximum cantilever stage, the arch closure stage, and the cable removal stage.

Fig. 7 presents the solid-element finite element model of the optimized case-study arch bridge in Abaqus, in which both geometric and material nonlinearities were considered. The model comprised 55,584 elements, including 37,056 C3D8R solid elements to simulate the concrete main arch and 18,528 T3D2 truss elements to represent the reinforcing bars. The truss and solid elements were connected through shared nodes to ensure compatible deformation between the reinforcement and the surrounding concrete. To simulate the vertical loads transmitted from the columns to the main arch, the degrees of freedom of all nodes on each column-section interface were coupled to the centroid of the corresponding cross-section, where the concentrated vertical loads F₁–F₆ were applied. The self-weight of the main arch was applied as a gravity load with a gravitational acceleration of 9800 mm/s², thereby reproducing the actual gravitational loading condition. All translational and rotational degrees of freedom at both arch springings were fully restrained to represent the fixed-support boundary conditions of the arch.

The analysis simultaneously considered geometric nonlinearity (large-deformation effects) and material nonlinearity. Geometric nonlinearity was solved using the Newton–Raphson iterative algorithm with the line-search technique activated to improve convergence robustness. An automatic load incrementation scheme was adopted, with the maximum number of increments set to 1000 and the initial increment size specified as 1% of the total applied load.

For material nonlinearity, the concrete was modeled using the Concrete Damaged Plasticity (CDP) constitutive model. The material properties of C80 concrete were defined as follows: density ρ_c = 2.48 t/m³, elastic modulus E_c = 3.8 × 10⁴ MPa, Poisson's ratio ν_c = 0.2, design compressive strength f_c = 50.2 MPa, and design tensile strength f_t = 3.1 MPa. The default viscosity parameter of the CDP model was adopted to enhance numerical convergence. The reinforcing steel was represented by an ideal elastic–perfectly plastic constitutive model, with a density of ρ_c = 7.85 t/m³, a yield strength of f_y = 400 MPa, an elastic modulus of E_s = 2.15 × 10⁵ MPa, and a Poisson's ratio of ν_s = 0.3. Strain hardening was neglected. The constitutive relationships of the materials are presented in Fig. 8.

3.3.2 Stress comparison

The cable force calculation in this study was performed using the influence matrix method, in which the horizontal displacement dx and rotational displacement ry at the closure section were set as target values for determining the cable force. For the case study considered herein, the length of the arch crown closure gap needs to be increased by ds = 82.9 mm during the maximum cantilever stage to achieve zero rotational displacement ry = 0 at the closure section, thereby ensuring the adjustment of the dead-load bending moment. The corresponding tie-down cable forces were then obtained.

A spatial beam-element finite element model was established using MIDAS/Civil for numerical analysis. Fig. 9 compares the self-weight-induced stress distributions in the arch ribs for the original and optimized schemes. As shown in Fig. 9, for the original design scheme, the peak axial compressive stress and bending stress of the arch rib under self-weight loading were −8.34 MPa and −1.57 MPa, respectively. For the optimized scheme, the corresponding peak values were −7.40 MPa and −5.84 MPa, respectively.

Compared with the original scheme, the optimized scheme reduced the self-weight-induced axial compressive stress by 0.94 MPa, while the self-weight-induced bending stress increased by 4.27 MPa. This increase in bending stress is attributed to the adoption of the rational arch axis corresponding to the dead-load condition, whereas self-weight alone does not represent the loading pattern associated with this rational axis. Nevertheless, owing to the axial compressive stress reserve of 7.40 MPa under self-weight loading, the minimum compressive stress along the entire arch remained 1.20 MPa, ensuring that the concrete arch rib remained in the elastic stage under self-weight distributed loading.

Fig. 10 compares the dead-load stress distributions of the arch ribs between the original and optimized schemes. As shown in Fig. 10, under dead-load conditions, the axial compressive stress of the original scheme ranged from −14.69 MPa to −10.21 MPa, corresponding to a stress variation of 4.48 MPa along the arch rib. In contrast, the optimized scheme exhibited a nearly uniform axial compressive stress distribution, ranging from −18.06 MPa to −17.81 MPa, with a stress variation of only 0.25 MPa. This result demonstrates that the proposed method effectively achieves an approximately equal-stress state under dead-load conditions.

Since the optimized design adopted the rational arch axis corresponding to the dead-load condition and an equal-stress arch configuration, the axial compressive stress closely matched the prescribed design stress, maintaining an approximately constant value of −18 MPa along the entire span. The bending stress gradually transitioned from tensile stress at the lower side of the arch crown to tensile stress at the upper side of the arch springing, without stress peaks induced by concentrated loads from the columns. The bending stress can be further eliminated by adjusting the construction-stage cable forces, resulting in a maximum final-state stress of −18.06 MPa in the completed arch.

3.3.3 Comparison of arch weight

Fig. 11 compares the calculated arch rib reactions for the original and optimized schemes. As shown in Fig. 11, under the action of self-weight distributed loads on the main arch, the total vertical reactions (for two arch ribs) of the original and optimized schemes were 22,000 t and 10,209.2 t, respectively. Compared with the original scheme, the optimized scheme reduced the material consumption by 54.6%.

Considering the lifting capacity of construction equipment, the maximum segment weight in the original scheme was 242 t, corresponding to a segment length of 6 m. For the optimized scheme, when the maximum lifting weight remained at 242 t, the minimum lifting length reached 11.5 m (at the arch springing segment), representing a 91.7% increase over the original scheme. Alternatively, when the segment length was maintained at 6 m, the maximum lifting weight was reduced to 114.6 t, representing a 52.6% reduction.

For large-span arch bridges designed using the proposed method while maintaining the same stress level as the original scheme, the average concrete consumption of the main arch ring per unit bridge deck area was reduced to 1.95 t/m². Compared with the original design, the proposed method achieved a 54.6% reduction in material consumption. Meanwhile, because the segment length for each lifting operation increased, the number of cantilever construction stages was reduced from 101 to 43, improving construction efficiency by 42.6%. These results demonstrate the significant technical and economic advantages of the proposed design approach.

3.3.4 Ultimate load-bearing capacity analysis

Fig. 12 presents the results of the linear elastic stability analysis under dead-load conditions. As shown in Fig. 12, since the arch rib was designed as a parallel double-rib system and each rib adopted a constant-width variable-depth cross-section, the out-of-plane stiffness of a single arch rib model was relatively low. Therefore, the first-order vertical bending mode under dead load appeared as the third vibration mode. The calculated linear elastic stability coefficient η was 4.037, with a deviation of less than 1% from the prescribed value of 4.

Furthermore, an ultimate load-bearing capacity analysis under dead-load conditions was conducted using an Abaqus solid finite element model. The reinforcement ratio of the arch rib was set as 1%. Fig. 13 illustrates the failure mode and local damage distribution of the arch rib. As shown in Fig. 13, the arch rib exhibited a completely symmetric instability failure mode under dead load. At the ultimate failure state, yielding occurred at five critical regions, including both arch springings, both quarter-span sections, and the arch crown, resulting in the formation of five plastic hinges and the loss of structural load-carrying capacity. The concrete around these regions also exhibited severe tensile and compressive damage. Specifically, at the arch crown and springing sections, the concrete damage patterns were identical: compressive damage in the top slab and tensile damage in the bottom slab. In contrast, at the quarter-span sections (L/4), the top slab experienced tensile damage, whereas the bottom slab underwent compressive damage.

Fig. 14 presents the calculated load–displacement curves at eight cross-sectional locations along the arch rib. As shown in Fig. 14, the nonlinear stability coefficient of the arch rib under dead load was 2.12. Before reaching the yielding plateau, the displacement curves at all sections exhibited an approximately linear trend. Subsequently, the curves rapidly transitioned into the yielding plateau through a distinct inflection point, indicating that the structure entered a prolonged plastic deformation stage under only a small increment of external load. This behavior revealed significant brittle failure characteristics of the arch rib.

4 Discussion

In the previous section, the minimum concrete volume of a 386 m span arch bridge was obtained using a design stress of 18 MPa. To further investigate the effects of span length and design stress on concrete consumption, 30 parametric models were established based on the reference case for comparative analysis. In these models, the span length ranged from 380 m to 420 m in increments of 10 m, while the design axial compressive stress of the arch rib ranged from 15 MPa to 20 MPa in increments of 1 MPa. The remaining structural parameters were kept unchanged, as shown in Fig. 15.

4.1 The lightest axis of arch with different spans and stresses

Fig. 16 presents the calculated arch-axis configurations for different span lengths and design axial compressive stresses. As shown in Fig. 16, within the span range of 380–420 m, when the design stress of the arch rib increased from 15 MPa to 20 MPa, the variation in the arch rib geometry was less than 0.1 m. For arch bridges with the same span length, changes in the design axial compressive stress only resulted in minor variations in the arch axis, and the corresponding change in arch axis length could be neglected. Therefore, the variation in arch rib volume was mainly attributed to changes in the cross-sectional area. The elevation of the arch axis at the quarter-span section L/4 decreased with increasing design axial compressive stress but increased with increasing span length.

4.2 Minimum cross-sectional area of arch with different spans and stresses

Fig. 17 illustrates the calculated crown cross-sectional areas under different span lengths and design axial compressive stresses. It can be observed that within the range of 380–420 m, the crown area varied significantly as the design stress increased from 15 MPa to 20 MPa. The crown area was highly sensitive to both span length and design stress, exhibiting a nonlinear increase with increasing span length and a nonlinear reduction with increasing stress. For a span length of 420 m, the crown cross-sectional areas corresponding to design axial compressive stresses of 15 MPa and 20 MPa were 5.69 m² and 3.23 m², respectively. In this case, a 33.3% increase in design stress resulted in a 43.2% reduction in the crown area.

4.3 Minimum rib mass of arches with different spans and stress levels

Fig. 18 presents the calculated minimum masses of the arch ribs under different span lengths and design axial compressive stresses. As shown in Fig. 18, within the span range of 380–420 m, the minimum arch rib mass varied significantly when the design stress increased from 15 MPa to 20 MPa. The minimum arch rib mass was highly sensitive to both span length and design stress, increasing nonlinearly with increasing span length and decreasing nonlinearly with increasing design stress. For a span length of 420 m, the minimum arch rib masses corresponding to design axial compressive stresses of 15 and 20 MPa were 27,594 and 15,671 t, respectively. Therefore, increasing the design stress by 33.3% reduced the minimum arch rib mass by 43.2%. Since variations in the design axial compressive stress caused only negligible changes in the arch axis geometry, the arch rib mass was directly related to the cross-sectional area. Consequently, the variation trend of arch rib mass was consistent with that of the crown cross-sectional area.

To further investigate the influence of superstructure loads on the crown area and arch rib mass, the superstructure load F_i was multiplied by different coefficients for parametric analysis. The adopted coefficients were 0.1, 0.5, 1, 2, 4, and 8. Fig. 19 presents the calculated arch rib geometry, crown area, and arch rib mass under different superstructure load coefficients. As shown in Fig. 19A, when the span length and design stress remained constant, the rational arch axis remained identical under different superstructure load conditions, indicating that variations in the superstructure load did not affect the rational geometry of the arch rib. Fig. 19B demonstrates that the crown area and arch rib mass increased linearly with increasing superstructure load F_i. Specifically, a 10% reduction in the superstructure load resulted in an approximately 10% reduction in the main arch ring weight, indicating a strict linear relationship between the two parameters. Therefore, optimizing the superstructure load provides an effective approach for controlling arch rib material consumption.

4.4 Applicability and limitations

This study investigates a lightweight design method for the main arch of large-span reinforced concrete arch bridges and presents several conclusions that may serve as useful references. Nevertheless, certain limitations and aspects warrant further in-depth investigation, as outlined below:

(1) The proposed method is applicable only to long-span arch bridges with closed cross-sections. It fails to accurately capture the complex dead load distribution and mechanical behavior of concrete-filled steel tube truss arch bridges and steel truss arch bridges. Consequently, its applicability to steel tube spatial truss system arch bridges is limited.

(2) The method exclusively considers the effect of specific distributed live loads on structural forces, without incorporating various live load distribution patterns. Further exploration of design approaches is required to address the most unfavorable live load distribution conditions, thereby enhancing the generality and applicability of the method.

5 Conclusions

This study proposed a lightweight design method for large-span reinforced concrete arch bridges in mountainous urban areas. The reliability and advantages of the proposed method were verified through the optimization design of a practical engineering case. Furthermore, parametric analyses were conducted to clarify the variation characteristics of the minimum arch mass with respect to span length, design stress, and superstructure load. The main conclusions are summarized as follows:

(1) This study constructs the lightest arch axis model for large-span arch bridges, establishes the correlation formula between section parameters and the strength and stability objectives of the main arch, and develops an optimization algorithm for section parameters under any preset strength and stability dual objectives. It can evenly distribute the stress of the main arch ring under constant load, achieve optimal material utilization, and automatically meet the expected goals of main arch strength and stability requirements

(2) The optimization case of a 386-m-span RC arch bridge shows that after adjusting the arch axis and main arch section using this method, the peak compressive stress under dead load was maintained at the prescribed design value of 18 MPa while satisfying the requirements for structural stiffness and load-bearing capacity, resulting in a 54.6% reduction in material consumption.

(3) A parametric study was conducted to investigate the effects of the design span and target compressive stress on the required concrete volume. Within the span range of 380–420 m, the minimum arch mass increased nonlinearly with increasing span length but decreased nonlinearly with increasing compressive stress. For an arch span of 420 m, increasing the target axial compressive stress by 33.3% reduced the minimum required arch mass by 43.2%.

(4) The minimum arch mass exhibited a strictly proportional relationship with the superstructure load F_i. Specifically, a 10% reduction in the superstructure load resulted in a corresponding 10% reduction in the main arch ring weight. Therefore, optimizing the superstructure load provides an effective approach for reducing arch rib material consumption.

References

[1] Zhou J T , Du Y L , Zhou Y . et al. Intelligent forming of large-span arch bridges: methodology and engineering applications. Engineering, 2026, 60: 231–245

[2] Ye S Q , Mao X Y , Ding H . et al. Nonlinear vibrations of a slightly curved beam with nonlinear boundary conditions. International Journal of Mechanical Sciences, 2020, 168: 105294

[3] Xu L H , Xu L Q , Zhou J T . et al. Seismic analysis of large-span CFST arch bridge considering the stress accumulation effect. Journal of Constructional Steel Research, 2025, 228: 109432

[4] Han X , Han B , Xie H B . et al. Seismic stability analysis of the large-span concrete-filled steel tube arch bridge considering the long-term effects. Engineering Structures, 2022, 268: 114744

[5] Zheng J L . Recent construction technology innovations and practices for large-span arch bridges in China. Engineering, 2024, 41: 110–129

[6] Xin J Z , Tang Q Z , Zhou J T . et al. A novel method for calculating the ultimate bearing capacity of in-service RC arch bridges using sectional constitutive relation. Applied Mathematical Modelling, 2025, 138: 115829

[7] Ding W , Kang H J , Zhang X Y . et al. Dynamic modeling and analysis on planar free vibration of long-span arch bridges during construction. Applied Mathematical Modelling, 2023, 121: 843–864

[8] Ma B , Yan H , Xu Y . et al. The nonconventional thin-walled arch rib design and its numerical verification of a long span steel arch bridge. Thin-Walled Structures, 2013, 73: 39–47

[9] Zheng J L , Wang J J . Concrete-filled steel tube arch bridges in China. Engineering, 2018, 4(1): 143–155

[10] Li R X , Zhou G W , Huang J J . et al. Lightweight design and verification of space harmonic drive. International Journal of Mechanical Sciences, 2025, 296: 110302

[11] Teoh K B , Hiew S Y , Chua Y S . et al. Design of flexural buckling resistance of lightweight concrete-filled cold-formed built-up box section (CFBBS) columns under axial compression. Engineering Structures, 2025, 343: 120911

[12] Donza H , Cabrera O , Irassar E F . High-strength concrete with different fine aggregate. Cement and Concrete Research, 2002, 32(11): 1755–1761

[13] Gao Y , Han X , Su X C . Research on the design method of elastic modulus of ceramsite lightweight aggregate concrete. Construction and Building Materials, 2025, 495: 143593

[14] Čandrlić V, Bleiziffer J, Mandic A. Bakar Bridge designed in reactive powder concrete. In: Proceedings of the 3rd International Conference on Arch Bridges. Paris, France, 2001, 695–700

[15] Čandrlić V, Radić J, Gukov I. Research of concrete arch bridges up to 1000 m in span. In: Proceedings of the 4th International Conference on Arch Bridges, Advances in Assessment, Structural Design and Construction – ARCH’04. Barcelona, Spain, 2004, 538–547

[16] Shao X D , He G , Shen X J . et al. Conceptual design of 1000 m scale steel-UHPFRC composite truss arch bridge. Engineering Structures, 2021, 226: 111430

[17] He G , Shao X D , Wu S W . et al. Large-scale experimental study on the mechanical behavior of a steel-UHPC composite truss arch under high compressive stress state after cyclic construction. Engineering Structures, 2025, 330: 119881

[18] Lai Y P , Li Y , Huang M K . et al. Conceptual design of long span steel-UHPC composite network arch bridge. Engineering Structures, 2023, 277: 115434

[19] Huang Y H , Yang Z C , Liu A R . et al. Nonlinear buckling analysis of functionally graded graphene reinforced composite shallow arches with elastic rotational constraints under uniform radial load. Materials, 2018, 11(6): 910

[20] Qiu M H , Kang H J , Guo T D . et al. In-plane elastic buckling behavior of circular tied cable-arches. International Journal of Structural Stability and Dynamics, 2017, 17(8): 1750088

[21] Hu C F , Pi Y L , Gao W . et al. In-plane non-linear elastic stability of parabolic arches with different rise-to-span ratios. Thin-Walled Structures, 2018, 129: 74–84

[22] Hu C F , Huang Y M . In-plane nonlinear elastic stability of pin-ended parabolic multi-span continuous arches. Engineering Structures, 2019, 190: 435–446

[23] Hu C F , Li Z , Hu Q S . On non-linear behavior and buckling of arch-beam structures. Engineering Structures, 2021, 239: 112214

[24] Zhang J L , Hellmich C , Mang H A . et al. Application of transfer relations to structural analysis of arch bridges. Computer Assisted Methods in Engineering and Science, 2017, 24(3): 199–215

[25] Shi Z , Hu H , Li J Q . Axis optimisation of arch-shaped pylons for high-speed railway cable-stayed bridges. Engineering Structures, 2021, 227: 111424

[26] Zhang G J , Liu Y J , Zhao W . et al. Optimal arch shape of long-span open-spandrel arch bridges under vertical permanent loads. Structures, 2022, 45: 1012–1030

[27] Gil-Martín L M , Hernández-Montes E , Palomares A . et al. The optimum shape of an arch under non-symmetric loading conditions. Archive of Applied Mechanics, 2016, 86(8): 1509–1520

[28] Marano G C , Trentadue F , Petrone F . Optimal arch shape solution under static vertical loads. Acta Mechanica, 2014, 225(3): 679–686

[29] Lewis W J . Mathematical model of a moment-less arch. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2016, 472(2190): 20160019

[30] Lewis W J , Russell J M , Li T Q . Moment-less arches for reduced stress state. Comparisons with conventional arch forms. Computers & Structures, 2021, 251: 106524

[31] Lewis W J . Constant stress arches and their design space. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2022, 478(2257): 20210428

[32] Luo C , Xiang Z S , Zhou Y . et al. In-plane bearing capacity of CFST truss arch bridges with geometric defects. Structural Durability & Health Monitoring, 2025, 19(3): 683–703

[33] Lee S , Lee Y J . New computational platform of finite element reliability analysis for efficient safety assessment of bridges employing MIDAS/Civil. KSCE Journal of Civil Engineering, 2025, 29(8): 100163

[34] Shen J J , Wang K , Fu Y B . et al. A computational strategy for enhanced nonlinear structural stability analysis in Abaqus. Computers & Structures, 2025, 318: 107943

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