Study On The Ultimate Span Of Double-tower Cable-stayed Bridge Without Back Cable

With the development of economy,all kinds of Bridges with different shapes are emerging.Cable-stayed Bridges without backstays appear in urban Bridges because of their beautiful appearance and outstanding visual effects.In this paper,the ultimate span of a cable-stayed bridge with double leaning towers and no backstays is studied.（1）Based on the equivalence principle of the average cable method and the cable film method,the analytical formula of the axial force generated by the cable on the main beam is derived respectively.Considering the influence of the inclination Angle of the tower and the dead weight of the cable,the expression of the ultimate span of the cable-stayed bridge without backstays is obtained.The ultimate span under vertical load is studied and the upper limit of the ultimate span under dead weight is obtained.When the section area,cable distance ratio and height span ratio of main beam are determined,the influence of constant live load ratio on ultimate span is analyzed.（2）Considering the sag effect of the stay cable,the linear shape of the stay cable is regarded as a quadratic parabola,and the equivalent elastic modulus of the stay cable is derived,and the relationship between the horizontal projection length of the stay cable and the equivalent elastic modulus is obtained.The deflection formula of the main beam is derived by using the analogy beam method and the elastic supported continuous beam method.The calculation result of the elastic supported continuous beam method is close to that of the finite element method,but there is a big difference between it and the analogy beam method,which proves that the elastic supported continuous beam method is reasonable to calculate the deflection.The influence of the stiffness of the main beam on the ultimate span is analyzed.The ultimate span of the bridge can reach about 350m if other conditions remain unchanged.（3）Using Karobov’s reduction method and modified elastic bearing method,the axial force formula of the inclined tower is derived.Considering the influence of height-span ratio,the ultimate span expression of the cable-stayed bridge without backstays is obtained.Compared with the finite element method,it is found that the calculation result of the modified elastic bearing method is more accurate.According to the example,the ultimate span of the modified elastic bearing method can reach 280m.The out-of-plane stability of the main beam decreases with the increase of the span,and the out-of-plane stability coefficient decreases more significantly with the increase of the span.At this time,the structure is controlled by out-of-plane instability.When the load set degree P=330k N/m and the transverse bending inertia moment I_Z=380m⁴of the main beam,the ultimate span is about180m.（4）The change of the cable-free zone L₀beside the Leaning Tower will affect the axial force of the main beam,the deflection of the main beam and the bridge cable force.With the increase of L₀,the average cable force gradually increases.When the span of the main span is the same,the axial force of the main beam and the deflection of the main beam of the cable-free cable-stayed bridge decrease with the decrease of the cable-free area L₀beside the Leaning Tower.Considering the bridge cable force,stiffness of main beam and axial force of main beam,it is concluded that the reasonable cable-free area L₀range of the bridge is about22.0m～25m.（5）The change of inclination Angle of the tower will affect the bridge cable force,the deflection of the main beam,the horizontal displacement of the tower and the stress of the main beam.Withθincreasing,the bridge forming force of the stay cable decreases.When the inclination Angle of the tower decreasesθ,the inclination Angle of the stay cable will also decrease,that is,the support efficiency of the stay cable will weaken,resulting in the increase of the deflection of the main beam.As the Angle between the cable and the tower decreases,the component of the cable force perpendicular to the tower body decreases,and the horizontal displacement of the tower decreases.When the inclination Angle of the tower is differentθ,the stress pattern of the main beam is roughly the same.Considering the bridge cable force,structural stiffness and main beam stress,it is concluded that the reasonable inclination range of the tower is about 62°～75°.（6）The change of the side-to-span ratio will affect the cable force and the deflection of the main beam.When the edge-to-span ratio i=0.35,the resulting bridge cable force is the maximum,and the resulting bridge cable force increases with the decrease of the edge-to-span ratio.Under the action of live load,the deflection of main beam reaches its maximum when the side-to-span ratio i=0.50.With the decrease of the side to middle span ratio,the stiffness of the side span main beam increases significantly,while that of the middle span main beam increases slightly.Considering the bridge cable force,structural stiffness and other factors,it is concluded that the reasonable boundary to span ratio of the bridge is between 0.20 and0.40.The research in this paper can enrich the design theory of cable-stayed Bridges without backstays and provide reference for the design of such Bridges in the future.