As the finite strip method(FSM) uses a higher-order polynomials in the longitudinal direction, it is possible to analyze a type of structures, such as box girder bridges, using a smaller number of degrees of freedom in comparison with the finite element method. In the earlier application of the spline finite strip method(FSM), the uniform B3-spline functions were used in the longitudinal direction while the conventional interpolation functions were used in the transverse direction to construct the displacement filed in a strip. When the uniform B3-splines are used for the isoparametric spline finite strip method, the shape function does not satisfy the Kronecker delta properties at the end boundary. Therefore, the special treatments such as the modified B3-splines and transformation method are required to satisfy the boundary conditions. As a treatment of this problem, the non-periodic B3-spline for description of both the displacement and the geometry was introduced. The two major advantages of non-periodic B-splines are that the additional information such as the end tangential vectors is not required to define the geometry of shell and that the shape function satisfies the Kronecker delta properties at the end boundary.
In spite of these advantages, since all the nodes in the strip formulated based on non-periodic splines are equally spaced along the longitudinal direction, there are some limitations for local refinement which is necessary for the accurate stress evaluation at the locations of abrupt changes of geometry, material properties, concentrated or patch loading, internal supports, etc..
The main purpose of this study is to develop a non-symmetric spline finite strip method for the effective solution to the aforementioned problem. The finite strip element proposed in this study has non-symmetrically spaced interior nodes in the longitudinal direction of strip that is efficient to generate locally dense node distribution for the analysis of the stresses at the locations where the high stress concentration exists. Also the new finite strip makes it possible that the total number of nodes can be significantly reduced while maintaining the same accuracy as the symmetrically spaced strip elements can obtain.
This non-symmetric spline finite strip method was extended to the analysis of prestressed box-girder bridges using the non-periodic B3-spline. In the analysis of the prestressed box girder bridges, the short term losses of prestressing are considered and each tendon force at the tendon point is evaluated by summation of the adjacent segment forces. Once the equivalent forces acting on the structure at the tendon points are found, they are transformed into statically equivalent forces at the adjacent node of nodal line. Several examples of prestressed box girder bridges were analyzed to verify the performance of this non-symmetric B3-spline FSM.
Even though this method uses a high-order shape function in the longitudinal direction, it has a linear strip in the transverse direction. A linear strip in the transverse direction would appear to have a drawback. So it was modified to produce a better quadratic strip element by employing the hierarchical concept in the transverse direction. The modified hierarchical spline FSM is very efficient to represent the peak values of transverse moment or membrane stress.
Finally, this study was extended to solve the free vibration problems of structures. In free vibration analysis, the lowest several natural frequencies and associated mode shapes are of interest as they govern the dynamic behavior of the structure. The mass matrix can be formed as a consistent mass matrix by using the shape functions. The consistent mass matrix was modified effectively to produce a diagonal mass matrix by using HRZ lumping scheme.
The efficiency and accuracy of the proposed method are demonstrated through a series of numerical examples.
