Calculation Method for Torsional Moment of Inertia of Half-Through Truss Bridges

Abstract

Half-through truss bridges exhibit significantly different mechanical characteristics due to their open tops, necessitating special treatment for calculating their free torsional moment of inertia. This study proposes a novel method: considering the constraints of vertical web members and crossbeams on the top chord, the top chord is equivalently modeled as a continuous beam on elastic supports. An equivalent horizontal bending moment of inertia of the top chord is derived by converting the top chord to the height of the top crossbeam while maintaining equivalent stiffness based on the equivalence principle. According to the analytical formula for the torsional moment of inertia and detailed parametric analysis, the main dimensional parameters affecting the torsional stiffness of half-through truss bridges include bridge length, bridge width, and main truss height. These parameters primarily enhance the bridge’s torsional stiffness by influencing the constrained torsional moment of inertia. However, due to scale limitations and aesthetic requirements, these dimensions cannot be increased indefinitely. In such cases, besides considering weight and aesthetics, increasing the size of the chords may be considered to enhance torsional stiffness. The interactions among the various factors affecting torsional behavior are relatively complex, and more systematic research is recommended for future study.

1. Introduction

A half-through truss bridge is a type of bridge with no top lateral bracing. Its top is completely open, imposing no height restrictions on the passage of pedestrians, vehicles, or trains, which allows the height of the main truss to be reduced. This not only makes the bridge structure more compact but also enhances its streamlined aesthetic appearance [1,2,3]. Consequently, half-through truss bridges are widely used in railway, port, and landscape bridge applications, as shown in Figure 1.

However, compared with top-closed deck-type or through-type truss bridges, this open-top structure cannot form a closed cross section, resulting in an inherent deficiency in the torsional stiffness of half-through truss bridges [4,5]. For half-through truss bridges, torsional stiffness is an important factor affecting their stability and deformation resistance [4,6]. Therefore, accurately analyzing and understanding their torsional stiffness is of significant significance for analyzing the mechanical behavior and structural design of such bridges. Consequently, research on the torsional stiffness of half-through truss bridges has been carried out.

If analytical methods are not considered and the torsional stiffness of a structure is to be calculated purely numerically, the principle of virtual work or the finite element method can be used [7,8]. For a truss structure, which is a clearly defined force-bearing system, the difference between finite element numerical calculations and experimental results is small. However, the modeling and computational costs of the finite element method are relatively high, and it cannot analytically account for the specific physical factors that affect the torsional stiffness of the structure. It is necessary to derive analytical formulas for the torsional stiffness of half-through truss bridges.

A truss structure is composed of different members in space, cannot form a continuous cross-section, and cannot be analytically solved for torsion using classical mechanics of materials or structural mechanics. Therefore, researchers proposed the equivalence theory between trusses and thin-walled members [9]. Through years of methodological iteration, this has become a mature and widely used equivalent method [10,11]. Using the principle of torsional stiffness equivalence, trusses with different configurations of diagonal web members or rectangular frames are equivalent to thin plates of the same size, and corresponding formulas for plate thickness are provided [12,13,14]. The truss structure is then equivalent to a single thin-walled member, enabling the calculation of various static and dynamic parameters [15,16,17], such as the free torsional moment of inertia. Based on this theory, numerous scholars have equivalently converted spatial trusses in buildings [18], bridges [19], and mechanical engineering [20] into closed thin-walled structures for analyzing the free torsional moment of inertia, and the calculated results have been verified as reliable.

For large truss girder bridges and similar structures composed of closed spatial trusses, the enclosed area is large, and the free torsion component plays a dominant role. In this case, the free torsional moment of inertia calculated by the equivalent thin-walled member can adequately represent their torsional performance [19,20].

However, half-through truss bridges differ from conventional truss bridges; they are typically built with smaller dimensions and have an open top, exhibiting unique torsional characteristics [21]. When the dimensions of a member are small, the resistance to constrained torsion will exceed that of free torsion [22], and free torsion must be combined with constrained torsion to obtain the total torsional stiffness, as is the case for half-through truss bridges.

In previous studies, half-through truss bridges were treated as equivalent open thin-walled members to calculate the free torsional moment of inertia, while the restrained torsional stiffness was also derived, ultimately yielding an analytical formula of the torsional moment of inertia for half-through truss bridges [13]. Existing research has not adequately considered the equivalent cross-section required for calculating the free torsional moment of inertia of half-through truss bridges. Although restrained torsion governs the torsional stiffness of such bridges, errors induced by neglecting free torsion should not be overlooked. Therefore, a more accurate derivation of the equivalent cross-section and the corresponding calculation formula for the free torsional moment of inertia of half-through truss bridges is necessary.

Existing research will provide this paper with the formula for calculating the plate thickness of the equivalent thin-walled member [12,13,14] and the formula for the restrained torsional moment of inertia of half-through truss bridges [13]. This paper will investigate a refined formula for the free torsional moment of inertia of half-through truss bridges, verify it, and further analyze the physical parameters that influence the torsional stiffness of such bridges.

Compared with existing research [13], this paper will present a more accurate method for calculating the torsional moment of inertia of half-through truss bridges, derive the supporting formulas required for the application of this method, and thereby reveal the objective patterns by which different structural parameters affect the torsional moment of inertia of such bridges. This will help researchers and engineers more accurately grasp the mechanical behavior of this bridge type, thereby producing a positive effect on its design, construction, and research.

It is particularly important to note that the calculation method presented in this paper is applicable to the top-open half through-truss bridges which has a vertical offset between the top and bottom cross-beams. For half through-truss bridges with only bottom cross-beams, since a rigidly enclosed area cannot be formed, the proposed method cannot calculate their free torsional moment of inertia. Moreover, in such bridges, restrained torsion will dominate, and therefore the free torsion component can be neglected.

Next, this paper will first give a brief introduction to the existing research on the calculation of the torsional moment of inertia of half-through truss bridges, and then list and explain the existing formulas that will be used in this paper.

1.1. Review of Torsional Stiffness Analysis for Half-Through Truss Bridges

The torsional behavior of through-truss bridges is quite special. It is essential to derive analytical formulas for the torsional stiffness of half-through truss bridges to analyze and elucidate their torsional behavior. Numerical solutions can hardly explain the physical causes of structural mechanical behavior and are usually used as a control group [7,8].

For calculating the free torsional moment of inertia of truss structures, current research tends to simplify the main trusses, bottom lateral bracing, and top lateral bracing of the spatial truss into thin plates using the shear equivalence principle [15,16,17]. This simplification transforms the entire truss into a thin-walled member. Subsequently, the free torsional moment of inertia can be determined by applying the thin-walled member torsion theory to clarify its torsional performance.

The advantages of this approach are twofold: First, it simplifies the calculations by equivalently modeling the spatial truss as a continuous thin-walled member. Second, it integrates material properties, dimensional parameters, and cross-sectional characteristics of the truss components into the analytical formulation, thereby enabling the use of analytical expressions to elucidate the physical factors that influence the torsional stiffness.

After obtaining the plate thickness formulas corresponding to different trusses [12,13,14], this method has been well applied in bridges, buildings, and other structures, and the calculated free torsional moment of inertia is relatively accurate [18,19,20]. Based on the existing plate thickness calculation formulas, relevant research has begun to analyze the calculation formulas for the torsional moment of inertia of half through-truss bridges.

Building on this foundation, researchers have equated half-through truss bridges to open thin-walled members to calculate the free torsional moment of inertia of the bridge cross-section, and analyze their constrained torsional moment of inertia from the truss chords resisting spatial warping [13].

However, although the effect of restrained torsion has been considered, existing research [13] analogizes the half-through truss bridge, whose cross-beams have a certain height and consist of top and bottom members, to an open channel section, which affects the accurate calculation of the free torsional moment of inertia. This paper aims to refine the equivalent section and the calculation formula for the free torsional moment of inertia. Ultimately, a more accurate explanation of the torsional characteristics of half through-truss bridges will be provided.

Herein, a brief introduction to existing methods for calculating torsional stiffness is provided. The calculation method for the restrained torsional moment of inertia obtained from existing research still needs to be used.

1.2. Brief Introduction to Existing Methods

In existing research, it is recognized that a half-through truss bridge can be idealized as an open channel-shaped, thin-walled member consisting of two side plates and one bottom plate, with the crossbeams serving as rigid supports for the cross-section, and with the main trusses and bottom lateral bracing each equivalently modeled as thin plates. The free torsional moment of inertia, J_f, can then be calculated using thin-walled member torsion theory, as expressed in Equation (1) [13]:

$$J_{\mathrm{f}}={\displaystyle \frac{1.12(2H{t_{\mathrm{s}}}^3+b{t_{\mathrm{b}}}^3)}{3}}$$

(1)

where H is the bridge height; b is the bridge width; t_s is the thickness of the side plate, equivalently derived from the main truss; and t_b is the thickness of the bottom plate, equivalently derived from the bottom longitudinal bracing. The thicknesses of the side and bottom plates must be calculated based on the configuration of the equivalent truss system. Formulas for the plate thickness are provided later.

However, half-through truss bridges exhibit a weaker contribution from free torsion to their overall torsional resistance due to their relatively smaller dimensions and open-top configuration compared with other bridge types. Consequently, the enhancement provided by restrained torsion becomes particularly significant. The computational model should be simplified as much as possible to facilitate a clearer understanding of the restrained torsion resistance mechanism in half-through truss girders. For this reason, existing studies have adopted a cantilever beam as the representative constraint configuration [13,18,19,20].

Existing studies account for the restrained torsional stiffness contributed by the resistance of the main truss to vertical bending deformation. The half-through truss girder is modeled as a cantilever beam.

The restrained torsional moment of inertia of the cantilever beam J_r can be calculated [13] using Equation (2):

$$J_{\mathrm{r}}={\displaystyle \frac{4.5b^2{I}_{\mathrm{t}}}{x^2}}$$

(2)

where x is the distance from a given position along the cantilever to its root, and I_t is the bending moment of inertia of one main truss in the vertical direction. This value can be obtained using the parallel axis theorem, as expressed in Equation (3):

$$I_{\mathrm{t}}=2I+0.5H^{2}A$$

(3)

In Equation (3), I is the upper and bottom chords’ bending moment of inertia in the vertical direction. A is the area of the upper and bottom chords.

J_T, the torsional moment of inertia of the main girder for a half-through truss, considering both free torsion and restrained torsion under the condition of a cantilever beam as the restrained torsion model, is given using Equation (4), which combines Equations (1) and (2) [13]:

$$J_{\mathrm{T}}={\displaystyle \frac{1.12(2H{t_{\mathrm{s}}}^3+b{t_{\mathrm{b}}}^3)}{3}}+{\displaystyle \frac{4.5b^2{I}_{\mathrm{t}}}{x^2}}$$

(4)

After accounting for the contribution of restrained torsion, the accuracy of the formula for calculating the torsional moment of inertia of a half-through truss girder is significantly improved. However, in reality, the crossbeam of a half-through truss bridge does not fully cover its entire cross-section; therefore, it cannot be readily simplified as an open channel-shaped, thin-walled member.

In summary, the existing formula for calculating the free torsional moment of inertia is not strictly accurate; however, it is crucial to correctly evaluate the structure’s performance [23].

This study will analyze the limitations of the existing method and propose an improved formulation. The refined formula not only enhances the calculation accuracy but also contributes to a correct understanding of the torsional mechanism. This will benefit the research, design, and stiffness enhancement of such bridge structures.

2. Modified Equation of Free Torsional Moment of Inertia

The existing method can provide a calculation method for the restrained torsional moment of inertia, but the equivalent section and formula used for calculating the free torsional moment of inertia need to be re-determined. The reason is that the derivation of the existing equivalent section has limitations. This paper first analyzes these limitations and then presents the refined equivalent section and the refined calculation formula for the free torsional moment of inertia.

2.1. Limitations of the Existing Formulation

The equivalent section and formula for calculating the free torsional moment of inertia need to be refined. One reason is that the existing equivalent section actually cannot realize a closed shear flow region; another reason is that the scope of the rigid periphery assumption has limitations.

2.1.1. The Limitation of Closed Shear Flow

The existing research assumes that after equating the main trusses on both sides and the bottom braces to thin plates, the equivalent section of the half-through truss bridge for calculating the free torsional moment of inertia is taken as the section of an open channel thin-walled member, as shown in Figure 2a, to form the closed shear flow depicted in the figure, and then the free torsional moment of inertia is calculated.

However, as shown in Figure 2b, the main trusses and the bottom brace of a half-through truss bridge are not a continuous medium; their shear resistance is only in a single direction. Considering only the main trusses on both sides and the bottom braces, the bridge section cannot form a closed shear flow. Therefore, the free torsional moment of inertia cannot be calculated according to the open channel section member. Next, this paper further explains the limitations of the existing method regarding the rigid periphery assumption.

2.1.2. The Limitation of Rigid Cross-Section Assumption

Figure 3 shows the cross-section and its equivalent representation for a half-through truss bridge, where the X, Y, and Z directions denote the longitudinal, transverse, and vertical axes, respectively.

As shown in Part ① of Figure 3, the crossbeam system of the half-through truss bridge (indicated by the red dashed line in the figure) functions as stiffening ribs. Therefore, existing studies [13] have assumed that the crossbeam system ensures the validity of the rigid cross-section assumption for the entire section, that is, the cross-section does not undergo distortion. As illustrated in Part ② of Figure 3, when considering free torsion, the four chords are removed, and the remaining portion is equivalently modeled as an open thin-walled member, as depicted in Part ②.

However, in reality, as a half-through truss bridge where the vertical web members above the upper members of the crossbeams and diagonal web members are not braced by the crossbeam system, the vertical web members above the upper members of the crossbeams and diagonal web members undergo distortion. As shown in Part ③ of Figure 3, these members should not be simply included in the cross-section used to calculate the free torsional moment of inertia.

Therefore, these members work together with the upper chords to provide shear resistance, resulting in an additional torsional strengthening effect. Consequently, the existing equivalent cross-sectional method and the corresponding formula for calculating the free torsional moment of inertia must be re-examined and modified for more accurate calculations so that engineers and researchers can better understand the mechanical characteristics of this type of bridge.

2.2. Equivalent Closed Cross-Section

Figure 4a illustrates one panel of the main truss. The main truss has a height of H, a bridge width of b, and a panel length of c. The lengths of the vertical web members above and below the top crossbeam are a and h, respectively.

As shown in Figure 4b, based on the shear equivalence principle, the N-shaped main truss can be idealized as a thin plate with a thickness of t_s, and the X-shaped bottom lateral bracing can be idealized as a thin plate with a thickness of t_b. According to existing studies, the calculation formulas for t_s and t_b are given as follows [13,14]:

$$t_{\mathrm{s}}={\displaystyle \frac{cHE{A}_{\mathrm{d}}}{G{\left(c^2+H^2\right)}^{3/2}}}$$

(5)

$$t_{\mathrm{b}}={\displaystyle \frac{2cbE{A}_{\mathrm{b}}}{G{\left(c^2+b^2\right)}^{3/2}}}$$

(6)

In Equation (5), E is the modulus of elasticity, G is the shear modulus, and A_d is the cross-sectional area of the diagonal web member. In Equation (6), A_b is the cross-sectional area of the bottom X-shaped longitudinal bracing.

As shown in Figure 4c, the top chords are connected through the vertical web members and the top crossbeams to form a framed structure.

As shown in Figure 5, consider a frame with members of lengths b and c. This frame can resist shear through the horizontal bending effect of its members. If there is an equivalent shear-resistant thin plate of the same dimensions, the thickness of the plate is t_T. According to existing research, the equation for calculating t_T is as follows [12,14]:

$$t_{\mathrm{T}}={\displaystyle \frac{24E{I}_{\mathrm{c}}{I}_{\mathrm{b}}}{Gcb\left(c{I}_{\mathrm{b}}+b{I}_{\mathrm{c}}\right)}}$$

(7)

In Equation (7), I_c is the horizontal bending moment of inertia of the member with length c in Figure 5, and I_b is the horizontal bending moment of inertia of the member with length b in Figure 5. These two members are in the same plane.

For the special frame shown in Figure 4c, although the top chord and the top crossbeam are in different planes, they are connected by the vertical web members above the crossbeam, thus still forming a jointly loaded system. However, Equation (7) cannot be directly applied to this structure for two reasons: First, the two members are in different planes, which makes Equation (7) inapplicable. Only by converting the horizontal bending stiffness of the top chord can the effect be achieved such that the height of the member connecting the top chord and the top crossbeam is reduced to zero, while the horizontal bending deflection of the top chord under the same magnitude of force remains unchanged. This yields a same-plane frame composed of the top chord (with reduced height) and the top crossbeam, as shown in Figure 4c. Only then can Equation (7) be applied to this frame. Second, the equivalent thin plate for free torsion needs to surround the rigid periphery (i.e., the height and width range of the crossbeam), which requires that the height of the equivalent top plate be consistent with that of the top crossbeam. Consequently, the height of the top chord also needs to be reduced.

In summary, as shown in Figure 4c, the horizontal moment of inertia of the top chord is denoted as I₁. A refined moment of inertia I_c of the top chord is obtained to reduce the height of the member connecting the top chord and the top crossbeam to zero such that the deflection of the top chord under the same horizontal lateral force remains unchanged. Then, Equation (7) is used to calculate the top plate thickness t_T. The specific procedure for converting the top chord is detailed later. In this section, we first analyze the refined equivalent section based on this approach.

Figure 6a shows a scenario in which the left and right main trusses, the bottom lateral bracing, and the top structure are all idealized as thin plates, and the positions of the side plates and the bottom plate remain consistent with the original truss configuration. However, the portions of the side plates extending above the top crossbeams are not constrained by the rigid perimeter assumption and should therefore be neglected. For the top thin plate, the top chords are not situated within the bounds of the rigid perimeter; instead, they are connected via the vertical web members and the top crossbeams. Consequently, the equivalent plate representing the top structure must be positioned flush with the plane of the top floor beams, which falls within the rigid perimeter. Thus, the top chords need to be equivalently modeled at this elevation. The height reduction of the top chords is addressed later. After removing the thin plates that are not within the rigid periphery support range, in Figure 6b, the half-through truss bridge is ultimately idealized as a closed channel-shaped, thin-walled member, with this member having a cross-sectional height of h and width of b. As shown in Figure 6c, the shear forces on each plate of this closed section with height h and width b together form a closed shear flow; therefore, the section has a free torsional moment of inertia.

Because the section shown in Figure 6b forms the closed shear flow as shown in Figure 6c, the free torsional moment of inertia of this section can be calculated according to the closed thin-walled member. Following this, the modified free torsional moment of inertia, J_M, for the section depicted in Figure 6b,c can be determined as follows [14,20]:

$$J_{\mathrm{M}}={\displaystyle \frac{4b^2{h}^2}{2h/t_{\mathrm{s}}+b/t_{\mathrm{b}}+b/t_{\mathrm{T}}}}$$

(8)

In Equation (8), the dimensional parameters b and h can be directly obtained from the bridge dimensions, while the calculation formulas for t_s and t_b are given in Equations (5) and (6), respectively. t_T in Equation (8) can be calculated using Equation (7). In this equation, I_b is the horizontal bending moment of inertia of the top crossbeam, which is a fundamental sectional property and can be obtained directly. However, I_c cannot be directly taken as the horizontal bending moment of inertia of the top chord; instead, it must be taken as the horizontal bending moment of inertia of the top chord reduced to the same elevation as that of the top crossbeam. A detailed derivation and description of this reduction method are presented in the following sections.

2.3. Member Number and Basic Specifications

A half-through truss bridge is composed of numerous members that form the main trusses, bottom lateral bracing, and crossbeams (stiffening ribs). These members work in conjunction to create an integrated structure. Before proceeding with the reduction in the top chord’s bending moment of inertia in the horizontal direction, it is necessary to first assign designations to the various members to facilitate explanation.

As shown in Figure 7, Members 1 and 2 are the upper and lower components of the crossbeam, respectively; Member 3 is the vertical web member above Member 1, and Member 4 is the vertical web member below Member 1; Member 5 is the top chord of the main truss; and Member 6 (red dashed portions) is the equivalent member after converting Member 5 to the same height as Member 1. The diagonal web members of the main trusses have been equivalently represented as side plates with thickness t_s. At this point, the diagonal web members only serve as rigid supports, preventing vertical distortion of the top chord. Therefore, the diagonal web member is marked with gray dashed lines in Figure 7. Members 1, 2, and 4 and the vertical members (gray dashed portions) together form a structure that ensures rigid boundaries within this plane. The bottom X longitudinal bracing has been equivalently represented as a thin plate with thickness t_b. The following explains how Member 5 is equivalently converted to Member 6 to form an equivalent closed cross-section.

Thus, the closed channel-shaped cross-section, with a height of h and width of b, formed by the rigid perimeter, serves as the equivalent section for calculating the free torsional moment of inertia. Since the formulas and associated parameters for determining the thickness of the side plates (t_s) and the bottom plate (t_b) have been clarified, it now remains to reduce the top chord (Member 5) to the same elevation as Member 1, thereby obtaining the value of I_c required in Equation (7). This, in turn, allows for the determination of the top plate thickness t_T, and subsequently J_M can be calculated using Equation (8).

2.4. Equivalent Reduction in the Moment of Inertia of the Top Chord

The top chord (Member 5) must be equivalently mapped to the position of Member 6 in Figure 7 to calculate the thickness of the top plate of the closed cross-section, which reduces the height of the vertical web members above the crossbeam to zero while keeping the lateral deflection of the top chord unchanged under the same lateral force. This requires a reduction in the top chord’s horizontal bending moment of inertia within each panel.

As shown in Figure 8, when a lateral force F is applied at the mid-span of the top chord (Member 5) in a half-through truss bridge, the chord’s deflection includes its own flexure and the bending of the vertical web members (Member 3). Therefore, the frames consisting of vertical web members and crossbeams (Members 1 and 2) act as lateral elastic supports on the top chord. Accordingly, the top chord can be modeled as a continuous beam on elastic supports (the vertical direction in Figure 8 is actually horizontal), where the joints with the vertical web members are the supports [4,24,25].

Next, how to reduce the horizontal moment of inertia I₁ of the top chord is explained as follows: By applying a transverse force to the top chord, bending is induced not only in the top chord itself but also in the vertical web members under the same force. The superposition of these deformations yields the total bending deflection of the top chord. By comparing this total deflection with the deformation of a beam without frame connections, the equivalent bending moment of inertia of the cross-section, that is, the desired I_c, can be derived. This approach is elaborated in detail as follows.

When the entire top chord is reduced, the force on the top chord is equivalent to applying a horizontal lateral force F at the mid-span of each span of the continuous beam. As shown in Figure 9, the force and deformation of the elastically supported continuous beam are illustrated, with the dashed line representing the position before deformation. If the truss bridge has n panels, the equivalent continuous beam will have n spans and n+1 supports. The deformation of each span of the continuous beam consists of the mid-span deflection without considering elastic supports and the spring displacement. Assuming the mid-span deflection of the m-th span (m = 1 to n) without considering elastic supports is ω_m, the support reactions on both sides are F_m and F_m+1, and further assuming that the mid-span deflection coefficient for each span is S_m, then ω_m can be expressed as Equation (9). The coefficients S_m and the reactions F_m and F_m+1 can be calculated using conventional methods for continuous beams via mechanical analysis, finite element computation, or reference to the calculated values provided in structural calculation handbooks.

$${\omega }_m=S_m{\displaystyle \frac{F{c}^3}{100E{I}_1}}$$

(9)

In the formula, I₁ is the top chord’s bending moment of inertia in the horizontal direction.

As shown in Figure 10a, since the lateral supports of the top chord are the elastic supports, a lateral displacement δ will occur at the top chord node under the action of a unit lateral force. This value δ is the deformation of the elastic support, and the value of δ can be calculated according to the method in reference as follows [4]:

$$\delta ={\displaystyle \frac{a^3}{3E{I}_{\mathrm{W}\mathrm{M}}}}+{\displaystyle \frac{b{(a+h)}^2}{2E{I}_{\mathrm{C}\mathrm{S}}}}$$

(10)

In Equation (10), I_WM is the out-of-plane bending moment of inertia of the vertical web member (the plane is the X-Z plane in Figure 3), and I_CS is the bending moment of inertia in the vertical direction of the combined section formed by Members 1 and 2. I_CS can be obtained using basic methods of material mechanics.

As shown in Figure 10b, in addition to the continuous beam deflection ω_m, under the action of the lateral force F at the mid-span of the single-panel top chord, the elastic supports at the ends of the two panels undergo lateral displacements of magnitude F_mδ and F_m+1δ due to the support reactions (i.e., the spring deformations in Figure 9). Consequently, the total lateral displacement at the center of the top chord is ω_m + 0.5(F_mδ + F_m+1δ). Assuming the effective horizontal bending moment of inertia of the top chord is I_c, and based on the equivalence principle of the horizontal bending moment of inertia of the top chord cross-section, the following equation can be derived:

$${\omega }_m+0.5(F_m\delta +F_{m+1}\delta )=S_m{\displaystyle \frac{F{\mathrm{c}}^3}{100E{I}_1}}+0.5(F_m+F_{m+1})\left({\displaystyle \frac{a^3}{3E{I}_{\mathrm{W}\mathrm{M}}}}+{\displaystyle \frac{b{(a+h)}^2}{2E{I}_{\mathrm{C}\mathrm{S}}}}\right)=S_m{\displaystyle \frac{F{c}^3}{100E{I}_{\mathrm{c}}}}$$

(11)

As shown in Equation (11), the relationships between F_m, F_m+1, and F, as well as the coefficient S_m, can be obtained using the standard method for continuous beams. Substituting these into the above equation and canceling out F allows I_c to be calculated. Truss bridges generally have more than five panels. For continuous beams, when the number of spans exceeds five, the support reactions tend to stabilize. In this case, F_m and F_m+1 are essentially equal to F, and S_m approaches 0.5. Thus, the relationship among F, F_m, and F_m+1, as well as the magnitude of S_m, can be determined without noticeable errors, thereby simplifying the calculation of I_c. The equation for solving I_c is simplified from Equation (11) to Equation (12) as follows.

$${\displaystyle \frac{c^3}{100E{I}_1}}+2\left({\displaystyle \frac{a^3}{3E{I}_{\mathrm{W}\mathrm{M}}}}+{\displaystyle \frac{b{(a+h)}^2}{2E{I}_{\mathrm{C}\mathrm{S}}}}\right)={\displaystyle \frac{c^3}{100E{I}_{\mathrm{c}}}}$$

(12)

3. Modification of the Combined Torsional Moment of Inertia

3.1. The Combined Torsional Moment of Inertia

At this point, the equation for obtaining I_c has been established. Consequently, J_M can be calculated using Equation (8). By combining Equation (8) with Equation (2), the modified overall torsional moment of inertia J′_T for the half-through truss bridge is obtained, as given in Equation (13):

$${J^{\prime }}_{\mathrm{T}}={\displaystyle \frac{4b^2{h}^2}{2h/t_{\mathrm{s}}+b/t_{\mathrm{b}}+b/t_{\mathrm{T}}}}+{\displaystyle \frac{4.5b^2{I}_{\mathrm{t}}}{x^2}}$$

(13)

Subsequently, a comparison between J′_T and J_T is required. The bridge case employed for this comparative analysis is first introduced.

3.2. Description of Calculation Procedure of Free Torsional Moment of Inertia

Equation (13) is the combined torsional moment of inertia obtained by summing the free torsion part (Equation (8)) and the restrained torsion part (Equation (2)); that is, Equation (13) represents the final calculated torsional moment of inertia of the half-through truss bridge in this paper. The calculation of the restrained torsion part from Equation (2) simply follows Equations (2) and (3).

However, the calculation of the free torsional moment of inertia of Equation (8) involves more procedures and parameters. Therefore, this section presents its calculation process in Figure 11 and provides a step-by-step explanation as follows.

As shown in Figure 11, the first step is to prepare the most basic dimensional parameters a, b, c, h, and H of the bridge, which are required for the subsequent calculations. These parameters can be obtained from the bridge drawings. In the first step, it is also necessary to prepare the elastic modulus E and shear modulus G of the bridge material; these two parameters can also be directly determined.

The second step is to calculate the horizontal bending moment of inertia I_c of the reduced member (Member 6 in Figure 7). This step requires determining the horizontal bending moment of inertia I₁ of the top chord (Member 5 in Figure 7), which can be calculated from the cross-section of the top chord; determining the vertical bending moment of inertia I_CS of the crossbeam, which can be calculated using the cross-sections of Member 1 and Member 2 in Figure 7 and the parallel axis theorem; and determining the out-of-plane (the X-Z plane in Figure 2) bending moment of inertia I_WM of the vertical web member, which can also be obtained from the cross-section of the vertical web member. Substituting a, b, c, h, E, I₁, I_CS, and I_WM into Equation (12) yields I_c.

The third step is to further prepare the basic calculation parameters. In addition to the already calculated I_c, it is also necessary to determine A_b, the cross-sectional area of the member constituting the bottom brace; to determine A_d, the cross-sectional area of the diagonal web member of the main truss; and to determine I_b, the horizontal bending moment of inertia of the top crossbeam (Member 1 in Figure 6). These parameters can all be obtained from the cross-sectional shapes of the members using basic principles of mechanics of materials.

The fourth step is to obtain the plate thicknesses. Substituting the various basic parameters already obtained into Equations (5)–(7) yields the plate thicknesses t_s, t_b, and t_T of the equivalent closed thin-walled member cross-section.

Step five is to calculate the free torsional moment of inertia. The plate thicknesses and dimensional parameters are already prepared; substituting them into Equation (8) yields the free torsional moment of inertia.

Next, the bridge case for which the torsional moment of inertia is calculated in this paper is introduced, and the analytical solution is compared with the finite element solution for verification.

3.3. Bridge Case and Computational Model

Studies [26,27,28,29,30] show that finite element results for typical truss structures, including half-through truss bridges, agree well with experimental data. Consequently, finite element results can serve as a benchmark for validating the analytical method. This paper selects two existing bridges as cases to calculate the analytical solutions and finite element solutions.

This paper selects two bridges as cases, and these bridge cases and their corresponding computational models are introduced as follows.

3.3.1. Case 1 and the Corresponding Computational Model

In the existing research, a theoretical calculation and a finite element analysis were conducted on a pedestrian bridge (Case 1), as illustrated in Figure 12, to complete the validation analysis. This study continues to use this bridge as a case study for analysis.

For Case 1. The entire bridge is constructed from 6082 T6 aluminum alloy, which has an elastic modulus (E) of 71 GPa, a shear modulus (G) of 26 GPa, and Poisson’s ratio of 0.33. As shown in Figure 12, the dimensions are as follows: c is 2.55 m, H is 2.355 m, a is 1.103 m, h is 1.068 m, and b is 3 m. The cross-sectional parameters of the top chord, bottom chord, diagonal web members, vertical web members, and crossbeams can be found in

Table 1. Member cross-sections of Case 1.

Member	Member Cross-Section	Dimensions	Vertical Moment of Inertia	Lateral Moment of Inertia
Top and bottom chords		W = 90 mm B = 230 mm K = 160 mm t = 12 mm	4.32 × 107 mm4	4.16 × 108 mm4
Web member of the main truss		W = 160 mm B = 120 mm t = 10 mm	1.14 × 107 mm4	1.81 × 107 mm4
Top member of the crossbeam and web member of the crossbeam		W = 70 mm B = 70 mm t = 5 mm	9.21 × 105 mm4	9.21 × 105 mm4
Bottom member of the crossbeam		W = 70 mm B = 70 mm K = 120 mm t = 5 mm	1.84 × 106 mm4	2.53 × 107 mm4
The X-shaped bottom horizontal frame		W = 70 mm B = 70 mm K = 20 mm t = 5 mm	1.84 × 106 mm4	7.11 × 106 mm4

.

Subsequently, the finite element model developed according to the bridge’s dimensional parameters is presented. The finite element model was established using beam elements and analyzed with the commercially available and reliable software MIDAS-Civil 2019. The boundary condition for the theoretical analysis of restrained torsion is the cantilever beam; therefore, the boundary condition for the finite element model is also the cantilever beam.

The finite element cantilever beam model using MIDAS-Civil is shown in Figure 13. The elements are modeled as beam elements, where each member of the truss bridge corresponds to one element without further mesh; therefore, the number of elements varies with the length of the model. At the constrained end of the cantilever beam, all degrees of freedom at the end of the respective member are restrained. At the loaded end of the cantilever beam, a rigid arm is used to connect the midpoint of the top crossbeam to the torsional center of the closed rectangular equivalent section, and the torque T is applied at the torsional center. The distance from the torque T to the top plate is e, which is calculated using the following formula [31,32]:

$$e=h\cdot {\displaystyle \frac{b{t}_{\mathrm{b}}+h{t}_{\mathrm{s}}}{\left(t_{\mathrm{T}}+t_{\mathrm{b}}\right)+2t_{\mathrm{s}}h}}-{\displaystyle \frac{b{h}^2\left(t_{\mathrm{T}}-t_{\mathrm{b}}\right)}{2\left[bh\left(t_{\mathrm{T}}+t_{\mathrm{b}}\right)+2h^2{t}_{\mathrm{s}}\right]}}$$

(14)

As shown in Figure 13, it is a cantilever beam model of a half-through truss bridge with one end constrained and a torque T applied to the other end. In Figure 13, x represents the length of the cantilever beam, or it can be understood as the position along the beam extending from the constrained end to the loaded end over a length x. In addition to the torsional stiffness provided by free torsion, the restrained end of the cantilever beam constrains the bending of the left and right chords in the vertical direction, thereby contributing to the restrained torsional stiffness.

A torque T is applied at the torque application end. Based on the fundamental theory of material mechanics, the torsional angle θ measured at the torque application end can be substituted into the following formula to inversely calculate the torsional moment of inertia J_T^F from the finite element method:

$$J_{\mathrm{T}}^{\mathrm{F}}={\displaystyle \frac{Tx}{G\theta }}$$

(15)

3.3.2. Case 2 and the Corresponding Computational Model

For Case 2. The entire bridge is constructed from 6082 T6 aluminum alloy, which has an elastic modulus (E) of 71 GPa, a shear modulus (G) of 26 GPa, and Poisson’s ratio of 0.33. As shown in Figure 14, the dimensions are as follows: c is 2.277 m, H is 1.90 m, a is 1.23 m, h is 0.67 m, and b is 4.80 m. The cross-sectional parameters of the top chord, bottom chord, diagonal web members, vertical web members, and crossbeams can be found in

Table 2. Member cross-sections of Case 2.

Member	Member Cross-Section	Dimensions	Vertical Moment of Inertia	Lateral Moment of Inertia
Top and bottom chords		W = 100 mm B = 220 mm K = 120 mm t = 4 mm	3.62 × 107 mm4	6.15 × 107 mm4
Web member of the main truss		W = 120 mm B = 100 mm t = 9 mm	5.31 × 106 mm4	7.15 × 106 mm4
Top member of the crossbeam and web member of the crossbeam		W = 100 mm B = 70 mm t = 9 mm	1.90 × 106 mm4	3.44 × 106 mm4
Bottom member of the crossbeam		W = 70 mm B = 70 mm K = 100 mm t = 7.5 mm	3.95 × 106 mm4	3.82 × 107 mm4
The X-shaped bottom horizontal frame	No bottom brace	No bottom brace	No bottom brace	No bottom brace

.

As shown in Figure 15, the finite element model of Case 2, the finite element modeling process, constraint method, and load application method for Case 2 are the same as those for Case 1. For Case 2, the formula for calculating the torsional center position e is also Equation (14), and the formula for calculating the torsional moment of inertia from the finite element numerical solution is the same as Equation (15).

For Case 2, since there is no X-shaped horizontal bracing at the bottom, when using the analytical method to calculate the analytical solution for Case 2, the formula for the bottom plate thickness t_b cannot be Equation (6); instead, Equation (7) should be used. However, unlike the calculation of the top plate thickness t_T using Equation (7), when using Equation (7) to calculate the bottom plate thickness t_b for Case 2, I_b is the lateral moment of inertia of the bottom member of the crossbeam, and I_c is the lateral moment of inertia of the bottom chord. Use Equation (7) to calculate t_b for Case 2; I_c and I_b are in the same plane, so no conversion is needed.

3.4. Validation

Based on Case 1 and Case 2, Equations (4) and (13) are used to calculate the analytical solutions of the torsional moment of inertia. After converting the bridges into the cantilever beam models shown in Figure 13 (Case 1) and Figure 15 (Case 2), the numerical solutions are calculated using finite element analysis and Equation (15). First, Case 1 is used for analysis.

3.4.1. Validation by Case 1

For Case 1, Equations (4) and (13) are used to calculate the analytical solution of the torsional moment of inertia. After converting the bridge into the cantilever beam model shown in Figure 13, the numerical solution is calculated using finite element analysis and Equation (15). The two calculation conditions are as follows: (1) varying the cantilever beam length x while keeping the width b constant 3.0 m; (2) fixing the length at 30.6 m (the bridge length) while varying the width b.

First, Equation (4) is used for analytical calculation, and the finite element model is used for numerical computation. The analytical and finite element numerical solutions obtained by varying the beam length x while keeping the width constant and by varying the width while keeping the length constant are shown in Figure 16a and Figure 16b, respectively. The corresponding analytical results using Equation (13) are presented in Figure 17a,b.

Comparing Figure 16 and Figure 17, with the finite element value as the benchmark, it can be observed that the calculation results of Equation (13) are better than those of Equation (4). Finite element values are taken as the benchmark to numerically compare the accuracy of the two equations. The accuracy of the analytical results is then assessed by calculating the relative error E_R using Equation (16):

$$E_{\mathrm{R}}={\displaystyle \frac{\left|N_{\mathrm{A}}-N_{\mathrm{F}}\right|}{N_{\mathrm{F}}}}\times 100\%$$

(16)

In Equation (16), N_A represents the analytical results using Equations (4) or (13), and N_F represents the finite element results. The errors of the two equations are shown in Figure 18a,b. Figure 18a is derived from Figure 16a and Figure 17a, while Figure 18b is derived from Figure 16b and Figure 17b.

Based on Figure 18, with most working conditions, the modified Equation (13) is more accurate than the existing Equation (4), demonstrating that the modified method proposed in this study is effective. To more intuitively observe and evaluate the calculation errors, present the errors from the above figure in tabular form.

The errors of the two equations are shown in

Table 3. For Case 1, set the width of the cantilever beam to 3.0 m and evaluate the calculation error of the two equations when the length of the cantilever beam is varied.

Equation	Beam Length-10.2 m	Beam Length-15.3 m	Beam Length-30.6 m	Beam Length-45.9 m	Beam Length-61.2 m
Equation (4)	23.5%	20.2%	21.0%	33.8%	29.3%
Equation (13)	13.0%	9.1%	13.5%	16.4%	26.8%

and

Table 4. For Case 1, set the length of the cantilever beam to 30.6 m and evaluate the calculation error of the two equations when the width of the cantilever beam is varied.

Equation	Beam Width-1.5 m	Beam Width-3.0 m	Beam Width-4.5 m	Beam Width-6.0 m	Beam Width-7.5 m	Beam Width-9.0 m
Equation (4)	19.9%	19.0%	30.2%	36.3%	39.7%	40.0%
Equation (13)	8.1%	16.3%	16.9%	6.8%	9.0%	14.6%

. Table 3 is derived from Figure 16a and Figure 17a, while Table 4 is derived from Figure 16b and Figure 17b.

Comparing Table 3 and Table 4, the calculation results of Equation (13) are all more accurate than those of Equation (4), except that under a few working conditions, the results of the two formulas are similar. The modified Equation (13) is more accurate than the existing Equation (4), demonstrating that the modified method proposed in this paper is indeed effective.

For conditions in Table 3, the accuracy of Equation (13) improved by an average of 9.8%, with a maximum improvement of 17.4% and a minimum improvement of 2.5% compared to Equation (4). For the working conditions in Table 4, the accuracy of Equation (13) improved by an average of 18.9%, with a maximum improvement of 30.7% and a minimum improvement of 2.7% compared to Equation (4).

3.4.2. Validation by Case 2

For Case 2, Equations (4) and (13) are used to calculate the analytical solution of the torsional moment of inertia. After converting the bridge into the cantilever beam model shown in Figure 15, the numerical solution is calculated using finite element analysis and Equation (15). The two calculation conditions are as follows: (1) varying the cantilever beam length x while keeping the width b constant at 4.8 m; (2) fixing the length at 22.77 m (the bridge length) while varying the width b.

First, Equation (4) is used for analytical calculation, and the finite element model is used for numerical computation. The analytical and finite element numerical solutions obtained by varying the beam length x while keeping the width constant and by varying the width while keeping the length constant are shown in Figure 19a and Figure 19b, respectively. The corresponding analytical results using Equation (13) are presented in Figure 20a,b.

Comparing Figure 19 and Figure 20, with the finite element value as the benchmark, it can be observed that the calculation results of Equation (14) are better than those of Equation (4). Finite element values are taken as the benchmark to numerically compare the accuracy of the two equations. The accuracy of the analytical results is then assessed by calculating the relative error E_R using Equation (16):

In Equation (16), N_A represents the analytical results using Equations (4) or (13), and N_F represents the finite element results. The errors of the two equations are shown in Figure 21a,b. Figure 21a is derived from Figure 19a and Figure 20a, while Figure 21b is derived from Figure 19b and Figure 20b.

Based on Figure 21, with most working conditions, the modified Equation (13) is more accurate than the existing Equation (4), demonstrating that the modified method proposed in this study is effective. To more intuitively observe and evaluate the calculation errors, present the errors from the above figure in tabular form.

The errors of the two equations are shown in

Table 5. For Case 2, set the width of the cantilever beam to 4.8 m and evaluate the calculation error of the two equations when the length of the cantilever beam is varied.

Equation	Beam Length-18.2 m	Beam Length-22.8 m	Beam Length-45.5 m	Beam Length-91.1m
Equation (4)	22.9%	21.9%	15.9%	34.75%
Equation (13)	10.2%	4.6%	7.1%	6.1%

and

Table 6. For Case 2, set the length of the cantilever beam to 22.77 m and evaluate the calculation error of the two equations when the width of the cantilever beam is varied.

Equation	Beam Width-1.5 m	Beam Width-3.0 m	Beam Width-4.8 m	Beam Width-6.0 m	Beam Width-7.5 m	Beam Width-9.0 m
Equation (4)	18.6%	18.9%	24.8%	29.4%	32.2%	36.4%
Equation (13)	10.3%	8.4%	7.2%	13.9%	14.8%	18.5%

. Table 5 is derived from Figure 19a and Figure 20a, while Table 6 is derived from Figure 19b and Figure 20b.

Comparing Table 5 and Table 6, the calculation results of Equation (13) are all more accurate than those of Equation (4), except that under a few working conditions, the results of the two formulas are similar. The modified Equation (13) is more accurate than the existing Equation (4), demonstrating that the modified method proposed in this paper is indeed effective.

For conditions in Table 5, the accuracy of Equation (13) improved by an average of 17.2%, with a maximum improvement of 28.6% and a minimum improvement of 8.9% compared to Equation (4). For the working conditions in Table 6, the accuracy of Equation (13) improved by an average of 14.4%, with a maximum improvement of 17.6% and a minimum improvement of 8.3% compared to Equation (4).

Equation (13) improves accuracy over Equation (4), validating that its analytical concept better captures the mechanical behavior of half-through truss bridges. Although some errors persist because lateral bending of the whole chords and cross-sectional distortion are neglected, Equation (13) already adequately represents the main structural characteristics.

The purpose of deriving the analytical formula is not to obtain completely accurate calculation results through a physical formula, but to rely on analytical formulas to identify the parameters that affect the structural mechanical behavior and the influence patterns of these parameters, thereby determining the torsional characteristics of this type of bridge from theoretical and physical perspectives, while providing theoretical references for finite element numerical results or actual phenomena.

Using this modified torsional inertia formula, the next section will analyze, through the analytical formula, the influence patterns of different parameters on torsional stiffness, and validate them using analytical and finite element calculations.

4. Analysis of Parameters Influencing Torsional Stiffness

This paper has derived the formula for the torsional inertia moment of the half-through truss bridge, enabling a theoretical analysis of the dimensional and component parameters that affect its torsional stiffness. At the same time, the correctness of the analysis will be verified by combining finite element solutions with theoretical solutions. Ultimately, the main factors influencing the torsional inertia moment of the half-through truss bridge will be identified from these parameters and provide insights for future research. First, a theoretical analysis of the formula is conducted.

4.1. Theoretical Analysis of the Analytical Formula

Using Equation (13) to express the torsional stiffness characteristics of a half-through truss bridge, the various geometric parameters affecting the torsional stiffness of the half-through truss bridge, as well as the influence levels of these parameters on the half-through truss bridge, are shown in

Table 7. Multi-level parameters and their influence on free torsional inertia.

First-Level Parameter	Exponent	Second-Level Parameter	Exponent	Third-Level Parameter	Exponent
b	≈1	/	/	/	/
h	≈1	/	/	/	/
ts	≈1	c	≈−2	/	/
		H	≈−2	/	/
		Ad	1	/	/
tb	≈1	c	≈−2	/	/
		b	≈−2	/	/
		Ab	1	/	/
tT	≈1	Ib	≈1	/	/
		Ic	≈1	I1	≈1
				a	≈−2
				c	≈0
				b	≈−1
				h	≈−2
				ICS	≈1
				IWM	≈1
		c	−1	/	/
		b	−1	/	/

and

Table 8. Multi-level parameters and their influence on restrained torsional inertia.

First-Level Parameter	Exponent	Second-Level Parameter	Exponent
b	2	/	/
x	−2	/	/
It	≈1	I	1
		H	2
		A	1

. Table 7 corresponds to the aspects of free torsion, while Table 8 pertains to restrained torsion.

As shown in Table 7, among the primary parameters, the bridge width b and the floor beam height h have a direct positive influence on the free torsional moment of inertia of half-through truss bridges and contain no deeper-level parameters. These two parameters directly determine the area enclosed by the equivalent cross-section. Because they contain no deeper-level parameters, their influence on the free torsional moment of inertia is more direct compared with the plate thickness parameters.

The primary parameter representing the side plate thickness t_s is positively and linearly influenced by two secondary parameters: the elastic modulus of the bridge material and the cross-sectional area of the diagonal web members. Increasing these two parameters enhances t_s, thereby increasing the equivalent plate thickness. Similarly, the primary parameter for the bottom plate thickness t_b is also positively influenced by the elastic modulus and the cross-sectional area of the diagonal members in the bottom lateral bracing.

The primary parameter for the top plate thickness t_T is positively influenced by the horizontal bending moments of inertia of the top chord and floor beam cross-sections. However, it is negatively influenced by the bridge’s dimensional parameters. This is because an increase in the dimensional parameters weakens the lateral restraint on the top chord. Consequently, t_T is affected by several secondary and tertiary parameters, and its overall impact on the bridge’s free torsional moment of inertia is relatively minor.

Regarding the free torsional moment of inertia, increasing the truss height H and increasing the panel spacing c both exert a negative influence. This is primarily attributed to their effect on reducing the thickness of the equivalent thin plates.

As shown in Table 8, among the primary parameters, the bridge length x and bridge width b directly influence the restrained torsional moment of inertia of half-through truss bridges and contain no deeper-level parameters. The influence exponents of b and x on the restrained torsional moment of inertia are 2 and −2, respectively. Increasing b or decreasing x can significantly enhance the restrained torsional moment of inertia of half-through truss bridges. The positive influence of the vertical stiffness of the main truss I_t is relatively minor. The restrained torsional moment of inertia can be enhanced by increasing the vertical moment of inertia of the top chord I, increasing the cross-sectional area of the top chord A, or increasing the height of the main truss H.

In summary, among the parameters that have a relatively significant influence on the torsional stiffness are the bridge width b, the crossbeam height h, the bridge length x, and the main truss height H. x represents the bridge length, which is tied to design and should not be altered lightly.

Considering that H exerts opposing effects on the free torsional moment of inertia and the restrained torsional moment of inertia, b and h are still regarded as having a greater influence on the overall torsional stiffness of the bridge. Given that increasing a single parameter would result in geometric disproportion, no single parameter should be enhanced in isolation. It is recommended to increase both b and h simultaneously to improve the torsional stiffness of half-through truss bridges.

When modifications to the primary bridge dimensions are limited by various constraints, the dimensions and moments of inertia of individual members can be increased, provided that such adjustments do not excessively affect the bridge’s self-weight or aesthetic considerations. However, the influence of member dimensions and moments of inertia is less direct than that of b and h.

According to Equation (13) and the two tables above in this section, the influence of various parameters on the torsional stiffness is inherently complex; it is difficult to truly discern the patterns of how each parameter affects the torsional inertia moment of the half-through truss bridge through mere analysis of the expression. Therefore, this paper conducts independent theoretical analysis, analytical calculation, and finite element calculation for each parameter in order to support the formula and the two influence pattern tables in this section.

4.2. Detailed Parameter Analysis and Discussion

All parameters that ultimately affect the torsional inertia moment of the half-through truss bridge are analyzed. Using Case 1, change these parameters respectively to conduct parameter analysis. Meanwhile, for different parameter variations, both finite element calculations and analytical calculations are performed, thereby complementing the theoretical analysis.

4.2.1. b, Bridge Width

As shown in Table 7 and Table 8, increasing b has a direct positive effect on the free torsional inertia moment, but also indirectly has an approximately quadratic negative effect on the free torsional inertia moment through its influence on the equivalent plate thickness. Taking both aspects into consideration, increasing b has a relatively limited negative effect on the free torsional inertia moment.

However, as shown in Table 7 and Table 8, increasing b has a quadratic positive effect on the constrained torsional inertia moment. Since the half-through truss bridge is primarily influenced by the constrained torsional inertia moment, increasing b leads to a dramatic increase in the overall inertia moment, approximately of quadratic order. Meanwhile, the free torsional inertia moment decreases slightly, so the overall inertia moment exhibits a trend of increasing slightly slower than quadratic growth, which is consistent with the behavior shown in Figure 17a and Figure 20a.

4.2.2. x, Bridge Length

As shown in Table 7 and Table 8, increasing the overall beam length x reduces the constrained torsional inertia moment quadratically. Since the constrained torsional inertia moment is the main factor affecting the torsional stiffness of the half-through truss bridge, increasing x leads to a significant decrease in the overall inertia moment, which is generally consistent with the behavior shown in Figure 17b and Figure 20b.

4.2.3. H, h and a

H is composed of a and h; a change in any one of these parameters will cause at least one other parameter to change. Therefore, H affects the free torsional inertia moment through its influence on the side plate thickness t_s, approximately with a negative quadratic power, but H positively affects the constrained torsional inertia moment with a quadratic power through its influence on the bending inertia moment of the main truss. h has a first-order positive correlation with the free torsional inertia moment, but h negatively affects the bottom plate thickness with a quadratic power through its influence on plate thickness, thereby approximately affecting the free torsional inertia moment with a negative quadratic power. a approximately affects the plate thickness with a negative quadratic power, and thus approximately affects the free torsional inertia moment with a negative quadratic power.

Since these three parameters cannot be changed independently, this paper establishes the following four parameter variation modes for analysis.

(1) H varies, a also varies, and h remains constant

First, it is clear that increasing H and a will reduce the free torsional inertia. However, increasing H enhances the constrained torsional inertia. Since the torsional resistance of a half-through truss bridge mainly comes from the constrained torsional inertia, the increased constrained inertia compensates for the reduction in free torsional inertia. Overall, this results in approximately a positive quadratic growth, but due to the decrease in free torsional inertia, the growth rate of the total inertia should be less than quadratic yet exhibit significant variation. When H and a are varied simultaneously according to specified sequences (with H taking values such as 3.78 m, 3 m, 2.385 m, 2.173 m, 1.89 m, and a taking values such as 2.52 m, 1.74 m, 1.125 m, 0.915 m, 0.63 m), the corresponding change in total torsional inertia is shown in the Figure 22, and the variation pattern conforms to the analysis.

(2) H varies, h also varies, and a remains constant

First, it is clear that increasing H reduces the free torsional moment of inertia, but increasing H enhances the warping torsional moment of inertia. Since the torsional resistance of a half-through truss bridge mainly comes from the constrained torsional moment of inertia, the increased constrained torsion compensates for the reduced free torsion, resulting in an overall approximately quadratic positive growth. At the same time, increasing h increases the free torsional moment of inertia. Therefore, increasing both H and h significantly enhances the torsional moment of inertia. When H and h are varied simultaneously according to specified sequences (with H taking values such as 3.37 m, 2.68 m, 2.39 m, 1.94 m, 1.68 m, and h taking values such as 2.24 m, 1.55 m, 1.26 m, 0.82 m, 0.56 m), the corresponding change in total torsional inertia is shown in the Figure 23, and the variation pattern conforms to the analysis.

(3) h varies, a also varies, and H remains constant

When h increases, a decreases. As h varies according to 0.795 m, 1.01 m, 1.26 m, 1.38 m, 1.59 m and a varies according to 1.59 m, 1.38 m, 1.26 m, 1.01 m, 0.795 m correspondingly, the effect of h on the magnitude of the free torsional moment of inertia is ambiguous. However, the decrease in a clearly causes the free torsional moment of inertia to increase, and this increase follows a quadratic trend. Meanwhile, changes in these two parameters do not affect the constrained torsional moment of inertia, so the resulting change in the total torsional moment of inertia is limited. This is largely consistent with the Figure 24.

Meanwhile, if the ratio of a/h (the vertical web member height above the crossbeam vs. the crossbeam height) becomes very large or h is quite small, it may become hard to form an equivalent closed-section region. This situation merits special analysis in future work.

(4) h varies, a varies, and H varies

At this point, keep the ratio h/a constant while simultaneously increasing h, a, and H (i.e., increasing H). Then, the increase in h and a brings about an ambiguous change in the free torsional moment of inertia, but the increase in H will greatly and positively affect the constrained torsional moment of inertia. Therefore, it is expected that the overall moment of inertia will undergo a significant change due to the variation in H, as shown in the Figure 25, which is largely consistent with the analysis.

Overall, a change in H will bring about a change in the constrained torsional moment of inertia. When any of the three parameters changes together with H, H plays a dominant role, and the increase in H has a very significant beneficial effect on the moment of inertia.

4.2.4. c, Truss Panel Length

As shown in Table 7 and Table 8, increasing the panel length c causes the thickness of the side plates and the bottom plate to decrease, thereby reducing the free torsional moment of inertia. However, the free torsional moment of inertia is a secondary factor affecting the torsional stiffness, so the change in the overall torsional moment of inertia caused by varying c is small, as illustrated in the Figure 26, which is largely consistent with the description.

4.2.5. A_d, Area of Diagonal Web Members

As shown in Table 7 and Table 8, increasing the area of the diagonal web members linearly affects the thickness of the side plates, thereby enhancing the free torsional moment of inertia. However, the free torsional moment of inertia has a limited influence on the total moment of inertia; therefore, the benefit of increasing the area of the diagonal web members is limited, as illustrated in the Figure 27, which is consistent with the analysis.

4.2.6. A_b, Area of Bottom Support

As shown in Table 7 and Table 8, increasing the area of the bottom support linearly affects the thickness of the bottom plate, thereby enhancing the free torsional moment of inertia. However, the free torsional moment of inertia has a limited influence on the total moment of inertia; therefore, the benefit of increasing the area of the bottom support is limited, as illustrated in the Figure 28, which is consistent with the analysis.

4.2.7. I_b, Horizontal Moment of Inertia of Top Cross Beam

As shown in Table 7 and Table 8, increasing I_b, the horizontal moment of inertia of the top cross beam, linearly affects the thickness of the equivalent top plate, thereby enhancing the free torsional moment of inertia. However, the free torsional moment of inertia has a limited influence on the total moment of inertia; therefore, the benefit of increasing the moment of inertia of the cross beam is limited, as illustrated in the Figure 29, which is consistent with the analysis.

4.2.8. I₁, Horizontal Moment of Inertia of Top Chord

As shown in Table 7 and Table 8, increasing I₁, the horizontal moment of inertia of the top chord, linearly affects the thickness of the top plate, thereby enhancing the free torsional moment of inertia. However, the free torsional moment of inertia has a limited influence on the total moment of inertia; therefore, the benefit of increasing the moment of inertia of the top chord is limited, as illustrated in the Figure 30, which is consistent with the analysis.

4.2.9. I_CS, Vertical Moment of Inertia of Whole Cross Beam

As shown in Table 7 and Table 8, increasing I_CS, the vertical moment of inertia of the section formed by combining the top and lower crossbeams, linearly affects the thickness of the top plate, thereby enhancing the free torsional moment of inertia. However, the free torsional moment of inertia has a limited influence on the total moment of inertia; therefore, the benefit of increasing the moment of inertia of the crossbeam members is limited, as illustrated in the Figure 31, which is consistent with the analysis.

4.2.10. I_WM, Moment of Inertia of Vertical Web Member

As shown in Table 7 and Table 8, increasing I_WM, the moment of inertia of the vertical web member, linearly affects the thickness of the top plate, thereby enhancing the free torsional moment of inertia. However, the free torsional moment of inertia has a limited influence on the total moment of inertia; therefore, the benefit of increasing the moment of inertia of the vertical web member is limited, as illustrated in the Figure 32, which is consistent with the analysis.

4.2.11. I, Vertical Moment of Inertia of the Chord

As shown in Table 7 and Table 8, increasing I, the vertical moment of inertia of the chord, linearly affects the thickness of the top plate, thereby enhancing the constrained torsional moment of inertia. Therefore, increasing I has a greater impact on the total moment of inertia compared to other structural parameters (Figure 33).

4.2.12. A, Area of the Chord

As shown in Table 7 and Table 8, increasing A, the area of the chord, linearly affects the thickness of the top plate, thereby enhancing the constrained torsional moment of inertia. Therefore, increasing A has a greater impact on the total moment of inertia compared to other structural parameters (Figure 34).

4.2.13. Discussion

Based on the above, this paper finds that influencing the free torsional moment of inertia by changing member parameters or bridge dimensions is not an ideal way to enhance the overall torsional stiffness of a half-through truss bridge. The relationships among H, a, and h are complex, but H plays a dominant role; increasing H can effectively increase the constrained torsional moment of inertia. Besides increasing H, increasing b, decreasing x, and increasing the cross-sectional area or moment of inertia of the top and bottom chords are the most effective means to increase the constrained torsional moment of inertia, thereby enhancing the torsional moment of inertia of the entire bridge.

The effects of many-dimensional parameters of a half-through truss bridge, such as H, a, and h, on the bridge’s torsional moment of inertia are very complex. It is recommended that subsequent research conduct more detailed theoretical or experimental studies on these dimensional parameters.

5. Conclusions

(1) The top of a half-through truss bridge is open, resulting in mechanical characteristics that are significantly different to those of conventional truss bridges. Therefore, the calculation of its free torsional moment of inertia requires special analysis. The crossbeams of a half-through truss bridge can form a rigid perimeter within their constrained range, making it possible to equivalently treat the bridge section as a closed section. However, this requires appropriate equivalence and conversion of the top chord.

(2) Considering the constraints provided by the vertical web members and crossbeams on the top chord, the top chord is equivalently modeled as a continuous beam supported by multiple elastic supports. Its horizontal bending stiffness is analyzed, and an equivalent horizontal bending stiffness is derived by converting the top chord to the same height as the top of the crossbeams while maintaining equivalent stiffness. Subsequently, the half-through truss bridge section can be equivalently treated as a closed thin-walled member, allowing for the calculation of its free torsional moment of inertia.

(3) This paper finds that altering member parameters or bridge dimensions to affect the free torsional moment of inertia is not an effective way to improve the overall torsional stiffness of a half-through truss bridge. Although the relationships among H, a, and h are complex, H plays a dominant role: increasing H effectively enhances the constrained torsional moment of inertia. Other effective measures include increasing b, decreasing x, and increasing the cross-sectional area or moment of inertia of the top and bottom chords. Given the complex influence of dimensional parameters such as H, a, and h on the torsional moment of inertia, further theoretical or experimental studies on these parameters are recommended.