Experimental and Numerical Analysis of an Innovative Combined String–Cable Bridge

Abstract

Suspension bridges, such as stress-ribbon, are among the simplest structural bridge systems and have the lowest structural height. The flexibility of these elegant bridges poses great challenges for designers to minimize their deformability under asymmetrical operational loads. Due to the small initial sag, such load-bearing structures also cause significant tensile forces, which requires them to have large cross-sections and massive anchor foundations. This paper analyzes an innovative suspension steel bridge structure combined with a string and a cable. More attention is paid to asymmetric loading as this is more relevant for suspension structures. The new structure is studied numerically and experimentally. It is established that the string stabilizes the displacements of the bridge under asymmetric loading. The stabilization efficiency is proportional to the value of the pre-tension force of the string. The obtained results reveal the behavior of the structure and enable an evaluation of the accuracy of the numerical results, as well as the applied modeling. In addition, the experimentally obtained results allow the evaluation of more aspects of the behavior of the new bridge, which will be useful in further studies of this type of structures.

1. Introduction

Suspension bridges are viewed as exceptional bridges and have been known in the world for more than one century. They are famous not only for their excellent architectural expression but also for having one of the longest spans in the world [1,2,3,4,5,6]. These bridges have a wide range of applications, from efficient automobile and railway bridges to graceful pedestrian bridges [7,8,9,10,11,12,13]. Original solutions are being developed using so-called rigid cables [14]. It is necessary to note that the creation and design of bridges is related to the maintenance of these structures [15]. When designing and building a bridge, it is necessary to take into account the complex condition monitoring of the structures of such bridges, including suspension and cable-stayed bridges [16,17,18]. This is necessary in order to ensure the functionality and operation of existing and designed bridges [19]. It is necessary to properly select and evaluate the capabilities of the equipment and software used for this purpose [20,21], for example, a vibration control strategy for cable-stayed-suspension hybrid bridges [20,22]. Long-term monitoring of bridges generates a huge amount of data that must be efficiently processed [21,23]. It is especially important, based on the collected research data, to preliminarily determine the costs of the bridge monitoring process during the projected design service life of the bridges because, at first glance, structural solutions that are rational according to the selected criteria do not always ensure sustainable and long-term exploitation of bridges. Monitoring problems can also be determined by the features of the structural system of bridges [18]. Therefore, attention should be paid to effective bridge structures such as suspension bridges [24]. Among suspension pedestrian bridges, it is necessary to single out stress-ribbon bridges, which have the lowest structural height [25,26]. Although such bridges have been known for several centuries, their wide application began in the 6th decade of the last century [27,28]. Futuristic solutions to apply stress-ribbon constructions to long-span automobile bridges were even presented [25,29]. It is necessary to note that there are two automobile bridges of this type in Uruguay [30]. Two structural solutions of stress-ribbon bridges currently dominate: stress-ribbons of prestressed reinforced concrete [25,26,31,32] and stress-ribbons of metal strips [27,28,33,34]. These solutions have two common disadvantages (negative effects): high tensile forces due to their relatively small initial sag; and a curved shape, which is not always well suited for traffic realization. There are also known two-chord suspension systems, where the upper element consists of a beam of slight curvature or a straight stiffness, which perfectly meets the operational requirements of the bridge, and the lower element is designed from a cable that performs a supporting function [35,36]. Both strips are tensioned and fixed to anchor foundations. The increased construction height of such bridges creates the conditions for lower total tensile forces in the anchor foundations.

It must be noted that beneath the graceful appearance of stress-ribbon bridges lies their complex behavior, which can cause unwanted operational issues. These bridges are primarily sensitive to the effects of asymmetric traffic loads. This causes significant dis-placements of kinematic origin [25,37]. There are detailed calculation methods for such stress-ribbon bridges that evaluate the static and dynamic behavior of such structures [38,39]. In such bridges, the so-called “rigid cable” was introduced, which stabilizes the initial shape of the bridge with its bending stiffness [35,40]. It is necessary to note that bridge designers are looking for innovative solutions to increase the efficiency of such bridges and other transport structures, and, over the last decade, so-called string bridge and transport structures have been developed [41,42,43]. First of all, this structural solution improves the behavior of the bridge in the case of asymmetric loads, because as previously mentioned, the string effectively stabilizes the effects of asymmetric traffic loads; therefore, such structural solutions are promising not only in bridges, but also in new infrastructure structures, although their supporting string elements have relatively large cross-sections due to the resulting extremely high tensile forces [44,45,46] and high axial forces [43,47]. Methods for analyzing the behavior of such structures are also not sufficiently developed. There are several works dedicated to the experimental and numerical studies of the behavior of the string as a structural supporting element [47,48]. It is important to note that there is a noticeable lack of a more detailed analysis of the behavior of such combined two-chord string bridges and prepared calculation methodologies. When evaluating string structures and their bridges, it is necessary to note that, despite their obvious advantages, they also have characteristic drawbacks: high tensile forces and considerable deformability. It is undoubtedly relevant not only to prepare an innovative combined structural system for this bridge, consisting of a string and a supporting cable, but also to study the behavior of such a bridge under symmetric and asymmetric loads.

This article discusses an innovative prestressed string–cable steel bridge structural system. The behavior of such a system is analyzed by applying its model in an experimental study and numerical analysis. The parameters of the state of stresses and strains of such a combined structure loaded symmetrically and asymmetrically are presented. The effect of string prestressing on the behavior of the entire structural system is also shown.

2. Experimental Program

The main objective of the experimental study was to investigate the behavior of a new combined string-suspension steel bridge under static symmetric and asymmetric loads. The diagram of this new bridge is shown in Figure 1 and Figure 2. The combined string–cable structure is made up of two main elements, i.e., chords. The upper chord consists of a prestressed string and the lower chord consists of a cable. Both these chords are connected by struts. Considering the structure of the new steel bridge, a model of this bridge was made. The purpose of the experiment was to determine the actual state of stresses and deformations of the main elements of this bridge model and to evaluate the influence of the pre-tensioned string on the deformability of the bridge model when asymmetric loads are applied. Another goal was to determine the actual state of stresses and deformations of the elements of this bridge model and to evaluate the influence of the pre-tensioned string on the deformability of the bridge when asymmetric loads are applied.

2.1. Design of the Tested Model

To achieve the goals of the experiment, a model of an innovative string-suspension bridge was designed. The string–cable structure model parameters were selected by performing a numerical analysis and considering the possible length of the real span of such a bridge, L = 50–70 m. According to the prepared drawings, the steel elements of the bridge model were made. The steel structures were manufactured according to the [49] production standard. In the laboratory of the faculty of civil engineering of Vilnius Gediminas technical university, this structure was assembled from separate elements. This bridge model consists of a pre-tensioned string supported through struts on a cable. This string, as the top element of the new combined steel structure, performs the function of the bridge’s stiffness beam. The cable acts as the lower support chord of this combined structure. Strings and cable supports are designed to be immovable both vertically and horizontally and connected in one unit. The test model was designed in such a way that the string could easily be additionally tensioned. The string was continuous throughout the span and anchored at the ends with screws (see Figure 1 and Figure 2). The string and cable in the supporting nodes can freely rotate around an axis perpendicular to the plane of the structure.

The string–cable structure model parameters were as follows: model span L—5.0 m. The width of the bridge model (distance between two flat structures) is 0.5 m. The distance between the struts is the same and is 0.714 m (see Figure 1). The struts of the model are designed from rectangular cross-section profiles, RHS 40 × 20 × 2. Such cross-sectional dimensions of the struts were chosen to eliminate possible axial deformations of the struts and, at the same time, their effect on the behavior of the bridge model. The heights of the struts were chosen in such a way as to form the shape of the lower element (cable), which is in balance according to the acting axial forces of the struts. These cable coordinates allow the bridge system to deform evenly under a symmetrical load. The maximum height of the strut or cable sag is 0.490 m. The upper element (string) and lower element (cable) are designed from a 6 mm diameter round steel rod.

2.2. Materials

To determine the material properties of the bridge model string and suspension element (cable), 4 samples with a length of L = 0.5 m were taken (see Figure 3). The samples were taken from the steel from which the model structure was made.

The tests were carried out in accordance with ISO 6892-1 A223, in the laboratory of the faculty of civil engineering of Vilnius Gediminas technical university. A stretching ma-chine w + b Walter + bai ag was used for the test (see Figure 3a):

• 100 kN Electromechanical Universal Testing Machine Mechanical.
• Tensile Grips, tensile power 100 kN.
• This model line combines high-performance, high accuracy, and the highest degree of flexibility.
• Precision strain gauge load cell mounted on crosshead, standardized mounting stud (male) Ø40 mm to mount grips.
• Digital movable crosshead encoder.

During the test, the steel elastic modulus E of the used steel rod was obtained, with an average value of 202.81 GPa. The exact values and other characteristics of the samples are given in

Table 1. Mechanical properties of bar elements (6 mm diameter).

Bar No. (mm)	Modulus of Elasticity (N/mm2)	Tensile Strength Rm (MPa)	Offset Yield Strength RP0.2 (MPa)
1	200.11	857	808
2	203.98	859	804
3	201.26	859	810
4	205.87	853	793
Average:	202.81	857	803.75

.

2.3. Arrangement of Measuring Points and Loading Scheme

Two types of measuring tools were selected for this model test (Figure 4 and Figure 5): dis-placement meters and strain gauges. A Novotechnik TYP:TR-0100, ART No 023264, F.NR 119988/A data logger was used to record displacement data. The vertical displacements of the model were measured at 16 points (see Figure 4). The stresses of the string and cable were measured by stress gauges with a base of 20 mm (resistance of R = 202.3 Ω). All strain gauges (28 pcs, see Figure 5) were connected to the Ahlborn ALMEMO 5990-2 device, which was used to obtain all the data. The ALMEMO^® 5990-2 data logger has max. 90 electrically isolated measuring inputs with up to 100 measuring channels for more than 65 measuring ranges, a real time clock, and a 500 kB memory for approximately 100,000 measured values. Memory cards up to 32 MB can be attached. The device can be operated by means of the LCD graphic display, a softkey keypad, and a control key. User menus can be configured to adapt the display for any application. Two output sockets allow for connecting any ALMEMO^® output modules, for example, the analogue output, digital interface, trigger input or alarm contacts. Several devices can be networked by simply connecting them with network cables.

The numbers for the displacements and strain gauges are based on the data and inputs of the Ahlborn ALMEMO 5990-2 device. In Figure 5, sensors from 70–79 are placed in the right part of the model, sensors 80–89 in the left part of the model, and sensors 90–99 in the central part of the model. The tensions of the lower cable are measured by 4 sensors (83, 88, 94, 73, see Figure 5), and those of the string by 24 sensors. This proportion of strain gauge placement was chosen due to the capabilities of the measuring equipment and the maximum number of sensors. Since the focus is on the string that takes the load directly, most of the strain gauges were installed on it.

2.4. Loading Test

The string-suspended steel bridge model was tested under static loads. This steel bridge structure is loaded with 5 different combinations: 2 options when the loads is symmetrical and 3 options when the loads is asymmetrical. The load combinations are given in

Table 2. Structural loads and string prestressing.

Loading Type	$\mathsf{\gamma }=\frac{\mathbf{q}}{\mathbf{p}};$	$\mathbf{q}$, kN/m	Prestressing Force, kN
Dead load	-	0.320	T1 = 0.472 T2 = 1.084 T3 = 1.506
Symmetrical	-	0.255	T1 = 0.472 T2 = 1.084 T3 = 1.506
Asymmetrical	1	0.255	T1 = 0.472 T2 = 1.084 T3 = 1.506
	2	0.510	T1 = 0.472 T2 = 1.084 T3 = 1.506
	3	0.765	T1 = 0.472 T2 = 1.084 T3 = 1.506

. Load variants, their values, and prestress values were obtained after preliminary numerical analysis of the bridge model. The prestress was determined (measured) during the experiment using strain gauges.

When analyzing the asymmetric load variant, an important parameter is the ratio of live ($p$) to dead $\left(q\right)$ load:

$$\gamma ={\displaystyle \frac{p}{q}}$$

The bridge model was loaded with a uniformly distributed load using rectangular steel bars of 25 × 25 mm, length of 500 mm, where the mass of one bar was 2.45 kg. The arrangement of the loads and their values were selected based on the working forces in the bridge model and the shear forces of the model elements. The aim was that the stresses of the max elements did not exceed 50% of the S355 grade steel strength. The dimensions of the load weights and their mass, as well as the features and possibilities of the placement model on the bridge, were also considered.

Three different string prestressing levels were chosen in the experiment: T₁, T₂, and T₃ (see Table 2). The load variants of the bridge model are presented in Figure 6 and Figure 7.

A general view of the bridge model and support assembly is shown in Figure 8, and its symmetric and asymmetric loads during the experiment are shown in Figure 6 and Figure 7.

3. Test Results and Analysis

3.1. Experimental Results

All experimental results are presented according to the scheme of Figure 9. The scheme is designed to separate the left and right sides of the structure. For the analyzed points, the letter “L” means the left side of the structure, and the letter “R” means the right side of the structure. The middle point “8” is given no index because it is the center point of the model, marking the middle of its span (x = L/2). Such notation was chosen for clearer presentation of the data.

3.1.1. Symmetrical Loading

The displacement values of the symmetrically loaded bridge model are presented in

Table 3. Vertical displacements of the bridge model under string prestressing force T₁ = 0.472 kN.

Loading Type	Vertical Displacements, mm
Measurement points of structure	1L	3L/9L	5L/10L	7L/11L	7R/11R	5R/10R	3R/9R	1R
Dead load	0	−1.87	−2.54	−2.76	−2.65	−2.41	−1.66	0
Live load, symmetrical	0	−3.06	−4.64	−5.66	−5.95	−4.98	−3.22	0

,

Table 4. Vertical displacements of the structure under string prestressing force T₂ = 1.084 kN.

Loading Type	Vertical Displacements, mm
Measurement points of structure	1L	3L/9L	5L/10L	7L/11L	7R/11R	5R/10R	3R/9R	1R
Dead load	0	−1.82	−2.45	−2.84	−2.9	−2.57	−2.08	0
Live load, symmetrical	0	−3.32	−4.72	−5.9	−5.91	−4.57	−2.95	0

and

Table 5. Vertical displacements of the model under string prestressing force T₃ = 1.506 kN.

Loading Type	Vertical Displacements, mm
Measurement points of structure	1L	3L/9L	5L/10L	7L/11L	7R/11R	5R/10R	3R/9R	1R
Dead load	0	−1.97	−3.07	−3.3	−3.4	−3.07	−2.19	0
Live load, symmetrical	0	−3.64	−5.02	−5.96	−5.95	−5.35	−3.57	0

, respectively, according to the string pre-tension values: T₁ = 0.472 kN; T₂ = 1.084 kN; T₃ = 1.506 kN. Vertical displacements are given at nodes only. As indicated in the scheme of the deflection gauges (see Figure 4), displacements were measured at the bottom of the struts.

From the results presented in Table 3, Table 4 and Table 5, it can be clearly seen that the bridge model was deformed symmetrically enough, and the maximum difference between the displacements of the left (R) and right (L) sides is about 5%. It was experimentally established that, under symmetrical loading, the pre-tensioning of the string does not significantly affect the displacements of the nodes of the bridge model. Increasing the string pre-tension values from 0.472 kN to 1.506 kN practically did not change the displacement values.

It is necessary to notice that in the case of symmetrical loading, increasing the pre-tension of the string causes a greater change in the axial force in the string, while the axial force in the cable does not change (see

Table 6. Axial forces in the string and cable by symmetrical loading.

Element	Shear Forces, kN
	T1 = 0.472 kN	T2 = 1.084 kN	T3 = 1.506 kN
String	2.95	3.31	4.02
Cable	3.30	3.49	3.31

). This is because the string has no initial sag and is extremely sensitive to displacements, which immediately induce shear forces in the string [4]. The cable has an initial sag equal to 0.5 m and absorbs the forces acting from the struts with a smaller tensile force.

3.1.2. Asymmetrical Loading

Table 7. Vertical displacements of the bridge model under string prestressing force T₁ = 0.472 kN.

Loading Type	Vertical Displacements, mm
Measurement points of structure	1L	3L/9L	5L/10L	7L/11L	7R/11R	5R/10R	3R/9R	1R
Live load, asymmetrical γ = 1	0	−10.09	−18.55	−18.4	6.55	12.01	6.88	0
Live load, asymmetrical γ = 2	0	−14.85	−27.51	−25.52	13.72	20.6	10.96	0
Live load, asymmetrical γ = 3	0	−18.28	−33.3	−30.15	17.2	25.78	12.94	0

,

Table 8. Vertical displacements of the model under string prestressing force T₂ = 1.084 kN.

Loading Type	Vertical Displacements, mm
Measurement points of structure	1L	3L/9L	5L/10L	7L/11L	7R/11R	5R/10R	3R/9R	1R
Live load, asymmetrical γ = 1	0	−10.78	−17.11	−15.69	6.63	10.11	3.23	0
Live load, asymmetrical γ = 2	0	−15.39	−26.02	−23.07	13.62	18.41	7.04	0
Live load, asymmetrical γ = 3	0	−18.74	−32.05	−28.21	18.09	23.79	9.16	0

and

Table 9. Vertical displacements of the model under string prestressing force T₃ = 1.506 kN.

Loading Type	Vertical Displacements, mm
Measurement points of structure	1L	3L/9L	5L/10L	7L/11L	7R/11R	5R/10R	3R/9R	1R
Live load, asymmetrical γ = 1	0	−10.31	−16.22	−14.65	5.18	8.09	2.53	0
Live load, asymmetrical γ = 2	0	−15.39	−24.88	−21.43	11.75	15.88	5.61	0
Live load, asymmetrical γ = 3	0	−18.88	−31.13	−26.65	16.3	21.29	7.76	0

show the values of the displacements of the bridge model loaded with temporary load asymmetrically over half the span length. The values of the displacements are presented according to the values of the pre-tension of the string: T₁ = 0.472 kN; T₂ = 1.084 kN; T₃ = 1.506 kN.

From the results presented in Table 6, Table 7 and Table 8, it is obvious that the bridge model deformed asymmetrically according to the placement of the live load (when the live asymmetric load was on the left side of the model, see Figure 9). The largest displacements occurred on the side loaded with the asymmetric load (nodes 3L, 5L, 7L). These displacements increased proportionally to the values of the load ratio γ. For example, with string pre-tension T₁ = 0.472 kN and increasing load ratio values from γ = 1 to γ = 3, model displacements in-creased from 18.55 mm to 33.3 mm (node 5L). It was experimentally established that the pre-tensioning of the string reduces the values of the maximum displacements. By increasing the string pre-tension values from T₁ = 0.472 kN to T₃ = 1.506 kN, the mentioned maximum displacements (node 5L) decreased (at γ = 1) by about 14%, at γ = 2 by about 11%, and at γ = 3 by about 7%. It should be noted that even with a load ratio of γ = 1, the maxi-mum displacements of asymmetric loading are larger than those of symmetrical loading by about 3.3 times. This means that in the case of asymmetric loading, kinematic displacements dominate [33]. Additionally, these kinematic displacements increase in proportion to the load ratio γ and the initial sag of the cable [37]. It should be noted that the innovative combined string–cable bridge system consisting of a string and a cable is under consideration. The string has no kinematic displacements, so it “stabilizes” the displacements caused by the asymmetric load of the entire bridge model (basically cable displacements). Increasing string pre-tension values also increases this “stabilizing” effect. This is evident from the obtained experimental results (see Table 7, Table 8 and Table 9).

When analyzing the behavior of the asymmetrically loaded bridge model, it is necessary to note that the displacements of the part of the model not loaded with live load (nodes 3R, 5R, 7R) are directed upwards. The higher the values of the load ratio γ, the larger these displacements are in absolute terms (see Table 7, Table 8 and Table 9). However, in their absolute size, these displacements are significantly smaller than the vertical displacements of the loaded part (on average, about 50% when γ = 1).

The axial forces of the bridge model elements in the string and cable under asymmetrical loading are presented in

Table 10. Axial forces in the string and cable by asymmetrical loading.

Load Type	$\mathbf{Axial} \mathbf{Forces} \mathbf{of} \mathbf{String}, {\mathit{H}}_{\mathit{s}\mathit{t}\mathit{r}\mathit{i}\mathit{n}\mathit{g}},$ kN
	T1 = 0.472 kN	T2 = 1.084 kN	T2 = 1.506 kN
Asymmetrical γ = 1	3.20	3.79	4.26
Asymmetrical γ = 2	3.86	4.44	4.92
Asymmetrical γ = 3	4.88	5.45	5.93
Load type	$\mathrm{Axial} \mathrm{forces} \mathrm{of} \mathrm{cable}, {\mathit{H}}_{\mathit{c}\mathit{a}\mathit{b}\mathit{l}\mathit{e}},$kN
Asymmetrical γ = 1	3.03	3.03	2.97
Asymmetrical γ = 2	3.42	3.40	4.10
Asymmetrical γ = 3	3.77	3.76	4.50

.

From the data in Table 10, the tensile forces in the string under asymmetric loading are higher than in the case of symmetrical loading (see Table 6). Additionally, the cable forces increase with increasing pre-tension values (0.472 kN, 1.084 kN, 1.506 kN). In this case, the axial forces of the cable change little and are smaller in absolute magnitude (at any value of γ) than the string axial forces.

4. Numerical Method and Comparison between Experimental Results

4.1. Modeling and Calculations

The innovative combined string–cable bridge structure was modeled and calculated using the finite element program [50]. The structure is modeled in one plane (2D). The upper and lower chords (string and cable) are modeled as the only tension truss elements, which takes over only the tensile forces. The struts are modeled as truss elements. The top chord (string) is divided into many finite elements and modeled like a “cable” element (see Figure 10). The static calculations of the combined bridge model were performed as geometrically nonlinear systems.

The dimensions and cross-sections of the elements of the FEM bridge model were selected according to the geometric and physical parameters of the experimental model (see Section 3). Additionally, the bridge model was loaded as in the experiment with symmetric and asymmetric loads and at three different prestressing forces: T₁ = 0.472 kN, T₂ = 1.084 kN, and T₃ = 1.506 kN. For loading options, see Section 3, Figure 6 and Figure 7. As in the case of the experimental model, a prestressed element was modeled right next to the support (element 37, see Figure 10).

4.2. Analysis of Numerical Results

The results of the numerical analysis of the bridge model are presented in

Table 11. Vertical displacements of the model under prestressing force T₁ = 0.472 kN.

Loading Type	Vertical Displacements, mm
Measurement points of structure	1L	3L	5L	7L	7R	5R	3R	1R
Dead load	0	−2.19	−2.92	−3.27	−3.27	−2.92	−2.19	0
Live load, symmetrical	0	−3.59	−5.16	−5.94	−5.94	−5.16	−3.59	0
Live load, asymmetrical γ = 1	0	−16.11	−19.55	−11.43	5.04	13.2	11	0
Live load, asymmetrical γ = 2	0	−23.27	−28.25	−15.64	9.56	21.86	18.17	0
Live load, asymmetrical γ = 3	0	−28.02	−34.04	−18.52	12.38	27.39	22.77	0

,

Table 12. Vertical displacements of the model under prestressing force T₂ = 1.084 kN.

Loading Type	Vertical Displacements, mm
Measurement points of structure	1L	3L	5L	7L	7R	5R	3R	1R
Dead load	0	−2.25	−2.9	−3.23	−3.23	−2.9	−2.25	0
Live load, symmetrical	0	−3.62	−5.14	−5.91	−5.91	−5.14	−3.62	0
Live load, asymmetrical γ = 1	0	−15.01	−18.36	−10.99	4.13	11.74	9.92	0
Live load, asymmetrical γ = 2	0	−22.14	−27	−15.23	8.48	20.17	16.89	0
Live load, asymmetrical γ = 3	0	−26.97	−32.89	−18.18	11.28	25.7	21.49	0

and

Table 13. Vertical displacements of the model under prestressing force T₃ = 1.506 kN.

Loading Type	Vertical Displacements, mm
Measurement points of structure	1L	3L	5L	7L	7R	5R	3R	1R
Dead load	0	−2.36	−2.86	−3.13	−3.13	−2.86	−2.36	0
Live load, symmetrical	0	−3.68	−5.11	−5.85	−5.85	−5.11	−3.68	0
Live load, asymmetrical γ = 1	0	−14.36	−17.66	−10.73	3.6	10.89	9.29	0
Live load, asymmetrical γ = 2	0	−21.46	−26.25	−14.97	7.84	19.16	16.13	0
Live load, asymmetrical γ = 3	0	−26.33	−32.18	−17.96	10.62	24.69	20.72	0

and graphically (Figures 12 and 13). They summarize the data under symmetric and asymmetric loads.

Analyzing the FEM results, it is possible to observe the analogous deformation behavior of the bridge model as shown by the experimental data. Under a symmetrical load, the structure deforms symmetrically, and the maximum displacement occurs in the middle of the span. This is equal to −5.94 mm when the prestress value is equal to T₁ = 0.472 kN. When T₂ = 1.084 kN, the displacement is −5.91 mm, and when T₃ = 1.506 kN it is −5.85 mm. In the case of asymmetric model loading, the largest displacements occur in the part loaded with the live load during the experiment, approximately in the third of the span (5L, see Table 11, Table 12 and Table 13). The schemes of deformation of the model in the cases of symmetrical and asymmetrical loading are shown in Figure 11.

4.3. Comparison of Experimental and Numerical Results

After conducting experimental and numerical studies of the bridge model, it is possible to compare the obtained results. They are presented in

Table 14. Comparison results of numerical and experimental displacements.

Loading Type	Vertical Displacements, mm
Local points of structure	1L	3L	5L	7L	7R	5R	3R	1R
	T1 = 0.472 kN
Live load, symmetrical	0	20%	10%	3%	−2%	1%	11%	0
Live load, asymmetrical γ = 1	0	60%	5%	−38%	−23%	10%	60%	0
Live load, asymmetrical γ = 2	0	57%	3%	−39%	−30%	6%	66%	0
Live load, asymmetrical γ = 3	0	53%	2%	−39%	−28%	6%	76%	0
	T2 = 1.084 kN
Live load, symmetrical	0	9%	9%	0%	−1%	9%	17%	0
Live load, asymmetrical γ = 1	0	39%	7%	−30%	−38%	16%	207%	0
Live load, asymmetrical γ = 2	0	44%	4%	−34%	−38%	10%	140%	0
Live load, asymmetrical γ = 3	0	44%	3%	−36%	−38%	8%	135%	0
	T3 = 1.506 kN
Live load, symmetrical	0	−1%	3%	0%	−1%	−7%	−5%	0
Live load, asymmetrical γ = 1	0	39%	9%	−27%	−31%	35%	267%	0
Live load, asymmetrical γ = 2	0	39%	6%	−30%	−33%	21%	188%	0
Live load, asymmetrical γ = 3	0	39%	3%	−33%	−35%	16%	167%	0

and

Table 15. Comparison results of numerical and experimental axial forces.

Parameter	Results
	T1 = 0.472 kN	T2 = 1.084 kN	T3 = 1.506 kN
$H_{string}$, kN, sym	7.80%	11.48%	0.75%
$H_{string}$, kN, γ = 1	−0.31%	−1.85%	−4.23%
$H_{string}$, kN, γ = 2	10.10%	6.98%	3.25%
$H_{string}$, kN, γ = 3	7.17%	4.77%	1.52%
$H_{cable}$, kN, sym	7.93%	1.15%	6.65%
$H_{cable}$, kN, γ = 1	−8.58%	−8.91%	−7.41%
$H_{cable}$, kN, γ = 2	3.51%	3.53%	−14.15%
$H_{cable}$, kN, γ = 3	14.06%	13.83%	−4.89%

in relative magnitudes (percentages) to evaluate the differences in the results (displacements) in both cases of analysis, taking into account the values of string prestressing.

The differences in the maximum displacements place (at node 5L) due to asymmetric loading do not exceed 10%. Additionally, relatively large differences in displacements are observed in nodes 3L (3R) and 7L (7R). In the case of symmetrical loading, we also have small differences at the location of maximum displacements (mid-span), i.e., up to 3%, and we have much larger differences in areas outside of the maximum displacements. From such differences, we can see that the shape of the deformation of the experimental model did not fully correspond to the shape of the deformation of the FEM model, because the experimental model was equipped with massive support bearings that “constrained” rotations (deviations) and displacements at the same time, and this affected the shape of the deformation. Additionally, a 6 mm solid rod was used in the experiment (for the string and lower cable). In the model, the lower element (cable) was modeled as a truss element, which can only be stretched, and the string as a “cable” element. By modeling in this way, it was possible to pre-tension the string and perform non-linear calculations, while also evaluating the loading sequence.

However, the designed nodes in the struts could create additional stiffness of the nodes and not fully correspond to the function of the hinge, because of which the structure could become stiffer than in the model.

In the graph below (Figure 12), we can see the displacements of a symmetrically loaded structure obtained by numerical and experimental methods. We can see that the displacements obtained during the experiment are slightly smaller. Of course, the sizes of the displacement results obtained both numerically and experimentally are relatively very small compared to the dimensions of the whole bridge model, so it is normal that any minimal deviation here caused much larger errors.

If the results of an asymmetrically loaded structure are examined, we can see that the maximum displacements of the structure coincide in the third of the span; when the structure is examined numerically and when it is examined experimentally, the shape of the deformation itself is different (Figure 13).

From the displacement graphs (Figure 13), we can see that the largest displacement differences are in the unloaded part of the structure. As we can see, the shape of the de-formation is the same in all cases, regardless of the size of the loads. Additionally, as previously mentioned, we can see that outside the places of maximum displacements, the displacement differences between the numerical model and the experimental model differ more. This is due to the reason mentioned earlier—large and heavy supports and due to friction in the nodes creating additional “constraints”. This effect is most visible in the unloaded part of the structure (point 3R, see Figure 9).

If we talk about asymmetrical loading and compare the results of only the loaded part, the difference in total displacements does not exceed 9 percent. In all cases, the location of the largest displacements coincides, regardless of whether the study was carried out numerically or experimentally.

During the performed calculations, we can also notice the influence of the prestressing forces of the string on the displacements. While prestressing has little effect under symmetrical loading, the effect becomes much greater under asymmetrical loading. Such results are revealed in Table 11, Table 12 and Table 13.

As for the axial force in the string (Table 15) we see that the experimental and numerical results differ by no more than 14%. In almost all cases, numerically obtained axial forces are higher than those experimentally obtained. Additionally, we can see that increasing the axial force in the structure (if the load does not change) decreases the total displacements of the structure. This is shown by both experimental and numerical analysis data.

In the general case, from the obtained results we can see that the differences between the numerically and experimentally obtained results, if we consider the loaded part of the structure, in terms of maximum displacements, do not exceed 9 percent (Table 14), and the differences in the axial forces of string do not exceed 14 percent (Table 15).

5. Conclusions and Discussion

The main objective of this article was to present an innovative combined string–cable bridge structure and investigate its behavior based on numerical calculations and experimental research. The experiment and FEA showed the effectiveness of this new combined bridge structural system in accepting both symmetric and asymmetric loads.

Summarizing the conclusions, we can say that the innovative combined string–cable bridge structure, as shown by the numerical and experimental results, is best used when it is loaded with an asymmetric load, since the string “constrains” the kinematic displacements, as a result of which we have smaller total displacements. Additionally, a big advantage is that we can pre-tension the string, which allows us to adjust both the tension forces and the displacements of the structure. Furthermore, it is necessary to remember that the string creates a straight line for the movement of transport, which allows such a structure to be adapted to various types of transport. Following the analysis of the behavior of the mentioned innovative bridge model, the following conclusions can be drawn:

• The bridge model was deformed symmetrically. It was experimentally established that, under symmetrical loading, the pre-tensioning of the string does not significantly affect the displacements of the nodes of the bridge model.
• In the case of symmetrical loading, increasing the pre-tension of the string causes a greater change in the axial force in the string, while the axial force in the cable does not change. This is because the string has no initial sag and is extremely sensitive to displacements, which immediately induce axial forces in the string.
• The bridge model deformed asymmetrically according to the placement of the live load (when the live asymmetric load was on the left side of the mode). The largest displacements occurred on the side loaded with an asymmetric load (nodes 3L, 5L, and 7L). These displacements increased proportionally to the values of the load ratio γ. This means that in the case of asymmetric loading, kinematic displacements dominate. Since the string has no initial sag, it has no kinematic displacements, so it “stabilizes” the displacements caused by the asymmetric load of the entire bridge model (basically cable displacements). Increasing string pre-tension values also increases this “stabilizing” effect. This is evident from the obtained experimental results.
• Displacements of the part of the model not loaded with live load (nodes 3R, 5R, and 7R) are directed upwards. The higher the values of the load ratio γ, the larger these displacements are in absolute terms. However, in their absolute size, these displacements are significantly smaller than the vertical displacements of the loaded part.
• The tensile forces in the string under asymmetric loading are higher than in the case of symmetrical loading. The cable forces increase with increasing pre-tension values. In this case, the axial forces of the cable change little and are smaller in absolute magnitude (at any value of γ) than the string axial forces.
• In almost all cases, numerically obtained axial forces are higher than the experimentally obtained forces. We can also see that increasing the axial force in the string (if the load does not change) decreases the total displacements of the structure. This is shown by both experimental and numerical analysis data.

Experimental and numerical studies of the new bridge were carried out under quasi-static loading only and under the assumption that the chords of the bridge are absolutely flexible. In real design, the flexural stiffness of the upper chord of such a bridge and the resulting bending moment in this chord should be evaluated. Therefore, such research of the new bridge structures would be relevant in the future