Internal force calculation of long span suspension bridge under vertical load based on elastic foundation beam algorithm

. The calculation of suspension bridge under vertical load is the most important content of suspension bridge design calculation and the most important basis of main component design. Combined with the knowledge of structural mechanics, this paper puts forward the elastic foundation beam algorithm, and deduces the important formulas for the cross-section design and strength comparison of sling, cable and stiffening beam. In use, as long as the parameters are brought into the formula, it is more convenient and fast compared with the complex software modeling. The method proposed in this paper is used to check the strength of the completed Japanese Guanmen bridge, and the results are accurate enough. It is fast and reliable to use this method in the preliminary design and rapid safety assessment of suspension bridge.


Introduction
Nowadays, the calculations of long span suspension bridge are completed by bridge structure analysis software such as Dr. bridge, Midas and ANSYS. The numerical accuracy calculated by these software is very high, but it is highly professional, and it needs professionals to complete. Moreover, using bridge software calculation, it needs very complex modelling work, and the workload is very large. In the early stage of engineering project planning, the workload of software modelling analysis is relatively too large. In this paper, an elastic foundation beam algorithm is proposed, which only calculates the internal force used in the section design of suspension bridge sling, cable and stiffening beam by hand without the help of bridge software. With this method, the internal force of each sling, each section of cable and stiffening beam of long span suspension bridge can be calculated within 2 working days. Through the calculation of the completed bridge, the data is very accurate. Using this method, the pre evaluation stage calculation of the project can be carried out more quickly, and the software calculation results can be checked, which reduces a lot of complex modeling work for designers.
Huang Wenli et al. Used MIDAS/Civil modelling to put forward a form finding and internal force calculation method for decorative cables of suspension bridges [1]. Based on the design of a super long span steel box girder suspension bridge, Pi Fuyan and others carried out comparative analysis on the main design parameters such as the ratio of vertical span of main cable and the ratio of side to middle span of stiffening beam, established the calculation model of spatial structure of suspension bridge by using the general finite element analysis software, and calculated and analyzed the static performance of suspension bridge under permanent load, Analysis of the influence of loading length different from train load on the force and deformation of suspension bridge structure system under dynamic action [2]. Liu Yi et al studied the calculation method of cable shape and internal force of suspension bridge under dead load. First, the approximate position of main cable is given by using catenary formula. Then, through the analysis process of structure splitting and combination, the calculation steps of cable shape and internal force are given [3].
In this paper, the mechanical model of suspension bridge is established, and the calculation formula of internal force of sling, cable and stiffening beam is derived, which is verified by an example. At present, this kind of method is rarely mentioned in the literature.

Calculation principle and formula
The structural diagram of suspension bridge is shown in Figure 1. The calculation structure of this paper is shown in Figure 2. There is one cable on each side along the length direction of the bridge, and the vertical sling is connected under the cable. There are   1 2  n slings (   1 2 2  n in total) on each side. Each cable is divided into n 2 sections, the horizontal distance of each section is d , and the length of each section is, Where, n y is the distance from the cable end to the bridge deck, The cable equation is as follows, When the cable axis is a parabola and the distance between slings is d , ' ' y is a constant, then the quadratic of y is also a constant. It can be seen from Fig. 3 that, is a constant, then Ti F is also a constant. Therefore, in the structure shown in Fig. 2, under the action of kp F , k q (there is no external load on the cable and sling), the tension generated by each sling is the same. The system shown in Figure 2 is One Degree of indeterminate when the force method is used. The basic system of force method is shown in Figure 4. The equation of force method is as follows,  is the vertical displacement of the beam at C points in the middle span of the stiffening beam under the action of  is the displacement of C points in the middle of the beam span caused by  is the displacement at C in the middle of the beam span under the action of Ni F , Ni F is the internal force of each section of cable ( n 2 sections ) under the action of Where y is the length of each sling.
Where l  is the length of each cable, S E is the elastic modulus of the cable and sling, 2 S A is the cross-sectional area of the sling, 1 S A is the cross-sectional area of the cable. ( The tension of section i , Cable end tension,   are as follows, The self weight of the cable is produced by the following parts, Where q is the uniformly distributed load due to the dead weight of the cable, G is the total weight of the cable.

Examples
Japan's Guanmen bridge [4], as shown in Figure 7 (this example is characterized by a suspension bridge with side spans, including wind load and seismic load).

Fig. 7. Schematic diagram of Guanmen bridge in Japan
The bridge was built in 1973, with a total span of 1068M (178 + 712 + 178). It is a three span steel truss suspension bridge.The main cable span is 712M, the side span is 178M, the tower height is 133.8M, the stiffening beam is 702M, the side span is 167.2M, the beam height is 9M, the width is 28.5M, , the main cable sag is 64M, the side span cable sag is 3.909M, the main cable , the distance between the two trusses is 29M, and the truss section is 10.35M long, When the main girder is designed, the dead load is kN/m 86 . 120 , the wind load is kN/m 85 . 18 , and the seismic load is kN/m 57 . 4 . When the main cable is designed, the dead load is kN/m 66 . 25 , the wind load is kN/m 02 . 3 , and the seismic load is kN/m 23 . 1 . For the suspension bridges built on the coastal rivers and harbors, the safety factor is too large in design because of typhoon.The safety factor of the sling is and stiffening beam is 3  K (because the diameter of the sling is much smaller than that of the cable, the sling is prone to flutter under the action of wind and rain, so the safety factor is greater than that of the cable).Because Japan is located in the sea and earthquake prone area, the design of suspension bridge should consider not only the self weight but also the wind load and earthquake load.Wind load and earthquake load are considered in the design of Japan's Guanmen bridge.
This example is of great reference value for the design of suspension bridges in earthquake areas.Based on the above data, the elastic foundation beam algorithm proposed by the author is used for calculation.The calculation is as follows, Take the distance between the slings as m 9 .