D.Y. Zheng

Free vibration analysis of a cracked beam by finite element method

Abstract In this paper, the natural frequencies and mode shapes of a cracked beam are obtained using the finite element method. An ‘overall additional flexibility matrix’, instead of the ‘local additional flexibility matrix’, is added to the flexibility matrix of the corresponding intact beam element to obtain the total flexibility matrix, and therefore the stiffness matrix. Compared with analytical results, the new stiffness matrix obtained using the overall additional flexibility matrix can give more accurate natural frequencies than those resulted from using the local additional flexibility matrix. All the elements in the overall additional flexibility matrix are computed by 128-point (1D) or (128×128)-point (2D) Gauss quadrature, and then further best fitted using the least-squares method. The explicit form best-fitted formulas agree very well with the numerical integration results, and are very convenient for use and valuable for further reference. In addition, the authors constructed a shape function that can perfectly satisfy the local flexibility conditions at the crack locations, which can give more accurate vibration modes.

Vibration and stability of cracked hollow-sectional beams

This paper presents simple tools for the vibration and stability analysis of cracked hollow-sectional beams. It comprises two parts. In the first, the influences of sectional cracks are expressed in terms of flexibility induced. Each crack is assigned with a local flexibility coefficient, which is derived by virtue of theories of fracture mechanics. The flexibility coefficient is a function of the depth of a crack. The general formulae are derived and expressed in integral form. It is then transformed to explicit form through 128-point Gauss quadrature. According to the depth of the crack, the formulae are derived under two scenarios. The first is for shallow cracks, of which the penetration depth is contained within the top solid-sectional region. The second is for deeper penetration, in which the crack goes into the middle hollow-sectional region. The explicit formulae are best-fitted equations generated by the least-squares method. The best-fitted curves are presented. From the curves, the flexibility coefficients can be read out easily, while the explicit expressions facilitate easy implementation in computer analysis. In the second part, the flexibility coefficients are employed in the vibration and stability analysis of hollow-sectional beams. The cracked beam is treated as an assembly of sub-segments linked up by rotational springs. Division of segments are made coincident with the location of cracks or any abrupt change of sectional property. The cracks flexibility coefficient then serves as that of the rotational spring. Application of the Hamiltons principle leads to the governing equations, which are subsequently solved through employment of a simple technique. It is a kind of modified Fourier series, which is able to represent any order of continuity of the vibration/buckling modes. Illustrative numerical examples are included.

On the determination of natural frequencies and mode shapes of cable-stayed bridges

Abstract An accurate analysis of the natural frequencies and mode shapes of a cable-stayed bridge is fundamental to the solution of its dynamic responses due to seismic, wind and traffic loads. In most previous studies, the stay cables have been modelled as single truss elements in conventional finite element analysis. This method is simple but it is inadequate for the accurate dynamic analysis of a cable-stayed bridge because it essentially precludes the transverse cable vibrations. This paper presents a comprehensive study of various modelling schemes for the dynamic analysis of cable-stayed bridges. The modelling schemes studied include the finite element method and the dynamic stiffness method. Both the mesh options of modelling each stay cable as a single truss element with an equivalent modulus and modelling each stay cable by a number of cable elements with the original modulus are studied. Their capability to account for transverse cable vibrations in the overall dynamic analysis as well as their accuracy and efficiency are investigated.

Finite strip method for the free vibration and buckling analysis of plates with abrupt changes in thickness and complex support conditions

Abstract The free vibration problem of a stepped plate supported on non-homogeneous Winkler elastic foundation with elastically mounted masses is formulated based on Hamiltons principle. The stepped plate is modelled by finite strip method. To overcome the problem of excessive continuity of common beam vibration functions at the location of abrupt change of plate thickness, a set of C 1 continuous functions have been chosen as the longitudinal interpolation functions in the finite strip analysis. The C 1 continuous functions are obtained by augmenting the relevant beam vibration modes with piecewise cubic polynomials. As these displacement functions are built up from beam vibration modes with appropriate corrections, they possess both the advantages of fast convergence of harmonic functions as well as the appropriate order of continuity. The method is further extended to the buckling analysis of rectangular stepped plates. Numerical results also show that the method is versatile, efficient and accurate.

Vibration and stability of non-uniform beams with abrupt changes of cross-section by using C1 modified beam vibration functions

Abstract A unified method is presented for the analysis of vibration and stability of axially loaded non-uniform beams with abrupt changes of cross-section. The beam may also be supported on Winkler elastic foundation, and both the axial force and the foundation stiffness can be varied arbitrarily. The method is based on the Euler–Lagrangian approach using a family of C 1 admissible functions as the assumed modes. The assumed modes comprise essentially the vibration modes of a single span hypothetical prismatic beam with the same end supports but without the intermediate supports, modified by piecewise C 1 cubic polynomials. The chosen admissible functions therefore possess both the advantages of fast convergence of the eigenfunctions and the appropriate order of continuity at the location of abrupt change of cross-section. The method allows extensive use of matrix notations and programming is rather straightforward. Numerical results also show that the method is versatile, efficient and accurate.

Analysis of deep beams and shear walls by finite strip method with C0 continuous displacement functions

This paper presents a new finite strip method for the analysis of deep beams and shear walls. The essence of the method lies in the adoption of displacement functions possessing the right amount of continuity at the ends as well as at locations of abrupt changes of thickness. The concept of periodic extension in Fourier series is utilized to improve the accuracy of the stresses at the strip ends. The equilibrium conditions at locations of abrupt changes of thickness are taken into account by the incorporation of piecewise linear correction functions. As these displacement functions are built up from harmonic functions with appropriate corrections, they possess both the advantages of fast convergence of harmonic functions as well as appropriate order of continuity. Numerical results also show that the method is versatile, efficient and accurate.