Theoretical Analysis Study On The External Prestress Simply-supported Reinforcement Technology

The structural retrofit is one of the effective ways to save social resources.Among the retrofit approaches,external prestress reinforcement technology has its unique advantage.According to the main influencing factors for the performance of overall retrofit,external prestress simply-supported reinforcement technology and the idea of adjusting the bending moments at beam ends actively are proposed for enhancing the bending capacity at beam ends relatively and creating the conditions for overall retrofit.The external prestressing cables arranged continuously strengthening technology is difficult to establish a universal analysis model, according to the external prestressed cable structure characteristics of simply supported reinforced technology, finite element method based on the idea, proposed a combined substructure (the combination of external pre-stressed tendons, tilted belly poles and the reinforced beams) concept, theoretical analysis and reinforcement for the overall structure of a breakthrough.According to the main influencing factors for the theory of composite substructure,the theoretical model of composite substructure is established,the parameters of the composite substructure are basically assumed.Base on the finite element theory.taking the external prestress structure of tilted belly poles as the object,the cable model is build with only-tension link elements.Considering coupling effect between the poles and the cable,stiffness matrix of the object and array of the telescopic poles are obtained.According to the geometrical equation of the object,the conversion matrix of the cable nodal displacement array and cable strain array, the sub array of telescopic poles,are obtained respectively.According to the physical relationship of cable,the conversion matrix of the cable strain array and cable stress array is obtained.According to the deformation principle of virtual work,the conversion matrix of the cable nodal displacement array and cable nodal load array is obtained.According to the equilibrium condition of the poles, the stiffness matrix of the object and array of the telescopic poles are obtained.The influencing factors of the stiffness matrix is more complex,showing that it is associated with Young modulus,length,across section area of the cable elements and direction,length of the poles.According to the linear algebra theory, proved that [K]which is the stiffness matrix of poles-cable structure of external prestress of tilted belly poles is invertible matrix.The invertibility of [K] is converted to invertibility of the simple matrix.The simple matrix determinant is obtained from transforming identically for odd and even number of the poles of two cases,showing that: if the poles are inclined arrangement,the value is not zero;if one of the poles is vertical arrangement,the value is also not zero by L’Hopital’s rule,showing that:the invertibility of [K] has nothing to do with the arrangement direction of the poles and cable.The paper provides mathematical basis for the technology in the design aspect of controlling cable stress.To apply incremental linear analysis method,taking the external prestress structure of tilted belly poles as the object,the cable model is build with only-tension link elements.Considering geometry nonlinear and coupling effect between the poles and cable,the tangent stiffness matrix of the object and the increment array of telescopic amount of poles are obtained.First of all,according to geometrical relationship,the transition relation between the cable nodal displacement increment and deformation increment of the cable elements, the increment of telescopic amount of poles, the direction vector increment of the cable elements are obtained.Then according to the physical equation and the balance equation,the tangent stiffness matrix of cable structure is obtained.The last according to the equilibrium of poles,the tangent stiffness matrix of the object and the increment array of telescopic poles are obtained.The influencing factors of the tangent stiffness matrix of the object is more complex,showing that: it is associated with the cable current parameter including the length,the direction,the internal forces and the pole current parameter including the length, the direction,the internal forces.Process for computation of the cable nodal displacement is created with Newton-Raphson method.In order to guarantee the convergence of calculation,the current parameter of cable and poles are calculated on the the current coordinates of cable nodes, unbalanced the telescopic amount and couple of poles are considered in the next iteration step. The paper provides the theoretical reference for reinforcement design of external prestress of tilted belly poles.According to the study of the influencing factors of overall retrofitted structural performance and the reinforcement characteristics, it is proposed that use tilted belly poles retrofitted by external pre-stressed simply supported and adjust the bending moments of beam ends initiatively to reinforce the beam ends flexural capacity and create the conditions for overall structure strengthening theoretical analysis. Taking the combined substructure (the combination of external pre-stressed tendons, tilted belly poles and the reinforced beams) for the study, considering the coupling and boundary effects on beam-pole-cable, and taking the beam end displacement, cable end displacement and web members hinged beam-cable nodes displacement as the basic unknowns, a finite element model of a combined substructure (two node linear cable model, rigid web members model and axial deformation beam element model) is established and deduced, and the rigid matrix of the combined substructure (get the substructure boundary binding transformed-matrix by the beam end nodes displacement array)and the fixed end force matrix (caused by the expansion of web members) are acquired. Firstly, the transformation of the basic unknowns and the expansion of the web members is deduced from the geometric relationships. Secondly, the conversion of the basic unknowns and cable and beam nodes loads is obtained by the physical relationship of cables and beams. The last, the rigid and fixed end force matrixes of the combined substructure are received by the balance of the web members and beam nodes and the method of static condensation. The study provides the theoretical support to the retrofitting the overall structure by using tilted belly poles with external pre-stressed simply supported.