# Parametric test for the preliminary design of suspension bridges

## Abstract

The preliminary design of suspension bridges is a very important step in the design of a structure, since this stage is the one that will lead to an efficient and economic structure. The models that are used nowadays are complex and sometimes hard to apply, leading to a lack of comprehension from the designing team. This work proposes a new simplified method for the preliminary design of cable suspension bridges that relate the stiffness of the deck truss with the stiffness of the cable, in which stresses are calculated. This relation is intended to know how much of the live load is absorbed by each of these elements and finally obtaining the pre-design values of each substructure. First simple parametric tests are executed using the proposed method and finite element method with geometrical non-linear analysis, in order to study its accuracy. Finally, a real case study is analysed using a known Portuguese suspension bridge, in which the proposed method is applied and compared with numerical solutions.

### Keywords

Suspension bridges Preliminary design Deflection theory of the suspension bridge Stiffness distribution Parametric analysis Uniform and point loads## Introduction and historical remarks

Cable supported bridges are one of the oldest structures in history, there are few structures that are universally appealing as these, the origin of the concept of bridging large spans with cables, exerting their strength in tension, is lost in antiquity and undoubtedly dates back to a time before recorded history. These structures arise so primitive humans that wanted to cross natural obstructions, observed a spider spinning a web or monkeys traveling along hanging vines (Brockenbrough and Merritt 1999). Nevertheless, it was in the 18th century that there was a major development in these structures, the crescent development of the structures started with the production of iron on a full scale (Chen and Duan 2014). Since the materials and systems used in these structures were fragile, the bridges were considered temporary. Because of that, there were a crescent search in new materials and technology’s that helped to increase the life-time of these structures. In 1823, Marc Seguin designed the first permanent cable suspension bridge, the Saint Antoine bridge in Genebra. He and his five brothers started the design of several bridges and built more than a thousand in 20 years (Gimsing and Georgakis 2011). These structures gave the experience and the knowledge needed to rise bridges with bigger spans and sizes. The engineers in that time did not rely on scientific methods but on the empirical knowledge obtained by the experience. This fact and the insufficient variety of materials leaded to the decrease of interest in the implementation of these structures from the nineteenth to twentieth centuries. Therefore, it is possible to say that the failure in understanding the behaviour of the stayed systems and the lack of methods for controlling the equilibrium and compatibility of the various highly indeterminate structural components, appear to have been the major drawback to further development of the concept (Drewry 1832). There was then needed to rise a new generation of engineers that could get through these obstacles. The Rankine theories based in linear elastic models, led to new design solutions. These theories leaded to new methods of design that could approximate the values obtained by manual means (Gimsing and Georgakis 2011). From this time on, the design was not solely based on the experience but also in the scientific methods that could lead to better solutions. It was after the Second World War that the suspension bridges had their major period in Europe, the need of access of a large number of cities leaded to the discovery of several new technologies and methods that led to a better construction.

The known analytical methods for studying suspension bridges (Gimsing and Georgakis 2011; Timoshenko and Young 1965) are extremely heavy to be applied in the preliminary design phase since they rely on iterative procedures. Some simple methods based on “static equilibrium” theories also exist (Irvine 1981), but these are extremely conservative which lead to non-economic solutions (Tadaki 2010).

The authors proposed a different method based on a simple proportional analysis between the girder and the cable stiffness (Serafim 2014). The main advantaged is its simplicity, since all relations are based on direct linear analysis, without any iterative process. This method is later on compared with a numerical analysis using finite element models (Belytschko and Fish 2007), using geometrical non-linear analysis (Crisfield 1991, 1997).

## Previous work

The theory of the displacements of a suspension bridge was first formulated by Melan in 1888 (Melan 1913). This was the first method to be used because it is extremely efficient when it’s used in the analysis of a bridge with a flexible deck girder. There are several ways to study the forces currently on a structure like this, which have been applied over the years in the design of suspension bridges.

The first one is the “static analysis” (Gimsing and Georgakis 2011) of a suspension bridge that neglects the deflection caused by the geometry of the structure and the equilibrium equations are linear. Nevertheless, the neglection of the deflections can lead to inaccurate and non-economic results when trying to obtain the stress values. This method was used to study the structures that had a small span or that the deflections due to the live loads were relatively small. The second one is the theory that permits the analysis of deck girders of brides, this theory is called “theory of displacements” (Von-Karman and Biot 1940) of suspension bridges. This theory is used when the stiffness of the girder is high and the displacements that were caused by the live loads are considered small. This theory assumes that the configuration that the cable assumes is a parabola and it is possible to obtain the equation that gives the displacement of the structure in any point.

There are also several graphical methods (abacus) that can be used to estimate the values of the preliminary design forces of suspension brides. The first is the preliminary design by Steinman–Baker (Gimsing and Georgakis 2011) that is used for bridges with a small span and stiffed girders (so the displacements are small). This method is simple to be used for preliminary analysis and it uses a Steinman–Baker coefficient to extrapolate the percentage of shear force and moment that is present in the mid span and the side span.

## Proposed method

The preliminary design of cable suspension bridges can be obtained from a numerous different ways and methods, from the deflection theory of the suspension bridge (Timoshenko and Young 1965), to the computational models that can be obtained with the proper software (Wilson 2002). All of these methods can be very exhausting and sometimes require large amounts of computational time. These problems lead to the need of a new simplified method that can unite the precision of the obtained results with a less amount of time that takes to obtain them.

The forces that this method manages to calculate is the bending moment (\(M\)) at mid span in the deck and the horizontal cable tension (\(H\)) on top of the towers/pylons.

Uniform load—distributed at the mid span

Point load—applied at the centre of the mid span.

The loads that are going to be introduced are the live loads that can be produced by traffic, which depend on the country structural codes. The dead load is not applied, since during the construction phase of the suspension bridged it is all transferred to the cable (Holger 2013). Although the method is precise, it relays on some simplifications such as: it is admitted a parabolic cable shape; the hanger tendons are rigid and continuous; the load that is transferred to the cross section of the deck girder is uniform; the effect of the shorting due to the compression of the deck is negligible; it does not take into account the horizontal and vertical flexibility of the towers; for the deck girder only the bending flexibility is taken into consideration; the cable is already deformed due to the self-weight of the cable and girder [this last hypothesis is very common in other simplified methods (Timoshenko and Young 1965; Von-Karman and Biot 1940), since it happens due to the construction phase (Parke and Hewson 2000; West and Robinson 1967)].

This method requires that the hangers are rigid and without any deformation, and two geometric parameters and two stiffness that are require to be considered; the length of the mid span (\(L_{\text{span}}\)); the sag of the cable (\(f_{\text{w}}\)); the cable area (\(A_{\text{c}}\)); and the deck girder inertia (\(I_{\text{g}}\)).

The cable area is the total cross section, if the structure has two or more cables this area is the sum of all cross-section areas, the girder inertia is considered to be constant trough all span.

Since there is two types load cases that can be used, this method is divided into two parts. The first is going to study the uniform live load and the second the point load. Both parts are going to give the approximate values of the forces that are going to be studied but each one has a different approach.

### Uniform load

*2*), if the deformation is small compared with the sag \(f_{\text{w}}\).

This expression is of up most importance to comprehend the simplified method, in this equation, \(\delta_{\text{c}}^{\text{p}}\) stands for the displacements produced by the uniform live load \(p\), that is applied in the span and \(E_{\text{c}}\) is the Young’s module of the material of the cable.

Having both forces, the preliminary design of the cable and deck girder can be achieved. It is important to point out that if the cable presents an almost rigid behaviour, the proposed method converges for the exact solution. Later on demonstrated in Fig. 11.

### Point load

*z*axis. To obtain the systems characteristic coefficient is necessary to assume a \(K_{\text{foundation}}\) so that the bending moment can be obtained.

Since the hangers of the structure are considered to be rigid, it can be assumed that the equivalent foundation stiffness is the stiffness of the cable.

The stiffness of the cable is obtained by a unitary point load.

The stiffness of the cable is obtained by a uniform load.

To obtain the horizontal cable tension it is necessary to perform and equilibrium at mid span using the moment obtain from (11), using an equivalent simply supported beam, and calculating \(P_{\text{beam}} = M_{\text{c}} \times 4/L_{\text{span}}\). Considering that \(P_{\text{total}} = P_{\text{beam}} + P_{\text{cable}}\) it can be achieved the value of the point load applied on the beam and subsequently obtained the horizontal value of cable tension.

## Parametric study

The parametric study that is used, considers the value of the loads present in the Portuguese structural safety code (RSA 1983) applied in the structure. These loads are considered the follow: uniform load of \(q_{1} = 40\;{\text{kN/m}}\); and a point load of \(q_{2} = 500 \,{\text{kN}}\). These are the loads that consider a situation where the bridge has heavy traffic.

Control values of the parametric test

Span, \(L_{\text{span}}\) (m) | Sag, \(f\) (m) | Inertia, \(I_{z}\) (m | Cross section of the cable, \(a_{\text{c}}\) (m |
---|---|---|---|

1000 | 200 | 13.84 | 0.1963 |

These parametric studies will compare the values obtained by the simplified method, and the ones obtained by the software (SAP 2000), in terms axial tension and bending moment. The static and kinematic boundaries of the numerical models, are the same as the ones represented in Fig. 3. Three type of finite elements were used: the 1st was the enhanced cable element for the cable (Ahmadi-Kashani 1983; Tibert 1999); the 2nd was the frame element for the bridge girder; and the 3rd the kinematic rigid element for the hangers. It was limited the maximum size of all finite elements of 1.0 m.

It is important to point out that the proposed method uses a linearized approach, but the finite element method used in SAP2000 uses geometrical non-linear analysis. It is therefore expected some differences which are acceptable, since the outputs during the pre-design phase are not always the same as in the design phase (Menn 1990). In this case the proposed method is used in the pre-design phase and the SAP2000 in the final design phase.

### Span analysis

In this analysis the span \(L_{\text{span}}\) varies from 200 to 3000 m, and the rest of the variables in Table 1 remain constant through the entire parametric study.

#### Uniform load

#### Point load

In this case the error for the cable tension is below 15% and for the bending moment below 35%. The difference between the uniform load, is the error is smaller below span of 500 m. This is due to the fact that the proposed method admits a straight cable for the point load, which is near an exact value when the span of the suspension bridge is bellow 200 m.

For spans bigger than 1000 m, the error tends to stabilize, this is due to the fact that straight cable approximation follows the “secant” approach of the real cable deformation. In this case in terms of force resultant in mid span, the values are the same, only changing the axial stress direction, therefore maintaining a constant error trough the parametric analysis.

Although the errors are larger, in terms of pre-design phase it still can be concluded using the referrer parametric study, that the proposed method is adequate.

### Sag analysis

In this analysis the sag \(f\) varies from 50 to 500 m, and the rest of the variables that were displayed in Table 1 remain constant through the entire parametric study.

#### Uniform load

The cable sag that varies between 100 and 200 m has low relative errors for the cable tension of 2.73 and 0.43% and for the bending moment of 5.63 and 8.16%.

From the sag analysis, the results obtained are still acceptable to be used in a preliminary design when using current geometries of the structure, this shows that the method works well with general sag sizes.

#### Point load

### Cable cross-section analysis

For this analysis the cross section \(a_{\text{c}}\) of the cable varies from 0.1 to 0.4 m^{2}, and the rest of the variables that were displayed in Table 1 remain constant through the entire parametric study.

#### Uniform load

For both cable tension and bending moment the maximum error for cross-section area above 0.05 m^{2} is below 3.5%. The errors obtained for both cable tension and bending moment are low and show the effectiveness of the simplified method.

#### Point load

The difference between the uniform load, is the error is smaller below span of 500 m. This is due to the fact that the proposed method admits a straight cable for the point load, which is near an exact value when the span of the suspension bridge is bellow 200 m.

For spans bigger than 1000 m the error tends to stabilize, this is due to the fact that straight cable approximation follows the “secant” approach of the real cable deformation. In this case in terms of force resultant in mid span, the values are the same, only changing the axial stress direction, therefore maintaining a constant error trough the parametric analysis.

### Girder inertia analysis

For this analysis the inertia \(I_{z}\) of the deck girder varies from 0.01 to 28 m^{4}, and the rest of the variables that were displayed in Table 1 remain constant through the entire parametric study.

#### Uniform load

#### Point load

^{4}, it can be stated that the error of the cable tension is below 10 and 40% for the bending moment. The proposed method in this case provides a good approximation for the cable tension, but a coarse estimation on the bending moment even in the pre-design phase (Fig. 14).

## Application to real model (Tagus Bridge 25 de Abril)

*aka*Salazar Bridge), a suspension bridge that was open to public in 1966 and was built by the United Steel International (Steinman 1960). This bridge was intended to have railway traffic, but the means that permitted such were only introduced later in 1992. When this alteration was made the girder was also expanded and since then the bridge permits car traffic and railway traffic simultaneous (Branco 1994; LUSOPONTE 2000) (Fig. 15).

### Materials and structural elements

General geometry of the structure

Sag (m) | Midspan (m) | Sidespan (m) |
---|---|---|

103 | 1035 | 460 |

*Warren Truss with Verticals*”. The geometry of the sections that constitute the girder is the follow (Fig. 16).

Geometry of the structure

Midspan (m) | Sidespan (m) | Cable sag (m) | Girder span (m) |
---|---|---|---|

1035 | 469 | 103 | 21 |

Hangers diameter (m) | Cable diameter (m) | Number of cables | Girder inertia (m |
---|---|---|---|

0.064 | 0.586 | 2 | 34,117 |

### Live loads applied

### Computer modelling of the structure

The fully 3D computer model of the structure was made in the software (SAP 2000). It was also made a geometrical non-linear analysis, but it was considered a material linear analysis. This last approximation is realistic, since during the design phase, none of the stresses presented in the structure are above the yield stress at the time. According to observed “in situ” and what was reported in (Steinman 1960) the girder is supported in both towers.

Two types of finite elements were used: the 1st was the enhanced cable element for the cable; the 2nd was the frame element for the bridge girder and the hangers. The cross section of the frame elements used in the truss of the girder are presented in Fig. 16. It was limited the maximum size of all finite elements of 1.0 m.

It is important to point out that this model presented an extra flexibility when compared with the proposed method, namely the vertical and horizontal deformation of the towers. These even though small contribute to decrease the level of axial tension in the cables.

### Obtained values

Relative error obtained when compared the proposed method to the computer analysis

\(H_{{{\text{proposed }}\;{\text{method}}}}\) (kN) | \(H_{\text{SAP2000}}\) (kN) | Relative error (%) |
---|---|---|

107,980.67 | 105,408.65 | 2.44 |

\(M_{{{\text{proposed }}\;{\text{method}}}}\) (kNm) | \(M_{\text{SAP2000}}\) (kNm) | Relative error (%) |
---|---|---|

125,853.41 | 131,402.3 | 4.22 |

## Conclusions

The proposed method for uniform live load provides excellent results for the preliminary design phase with errors below 10% in terms of stress prediction. The main errors belong to spans lengths that are not used in cable suspension bridges, this happens due to the exchange of the initial cable shape resulted from high deformation. Therefore, concluding that this method is precise for general geometries.

The proposed method for the point load provides a good technique although with errors around an average of 20%, almost the double of the uniform load. In any case it is still a good methodology for the pre-design phase to estimate general cable shape and girder cross section. This is also not a problem in cable suspension bridges since the stress caused by the point load is generally 1/10 of the stress caused by the uniform load.

In both methods it was observed that none of them provide an upper or lower bound for the cable tension and bending moment. This is important since it is necessary for the bridge designer to know that the estimated values do not provide any “extra” structural safety.

## Notes

### Acknowledgements

The first author thanks the Portuguese National Science Foundation (FCT) for his Post-Doc scholarship SFRH/BPD/99902/2014.

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