A two-parameter multiple shooting method and its application to the natural vibrations of non-prismatic multi-segment beams

This paper presents an enhanced version of the standard shooting method that enables problems with two unknown parameters to be solved.A novel approach is applied to the analysis of the natural vibrations of Euler-Bernoulli beams.The proposed algorithm,named as two-parameter multiple shooting method,is a new powerful numerical tool for calculating the natural frequencies and modes of multi-segment prismatic and non-prismatic beams with different boundary conditions.The impact of the axial force and additional point masses is also taken into account.Due to the fact that the method is based directly on the fourth-order ordinary differential equation,the structures do not have to be divided into many small elements to obtain an accurate enough solution,even though the geometry is very complex.To verify the proposed method,three different examples are considered,i.e.,a three-segment non-prismatic beam,a prismatic column subject to non-uniformly distributed compressive loads,and a two-segment beam with an additional point mass.Numerical analyses are carried out with the software MATHEMATICA.The results are compared with the solutions computed by the commercial finite element program SOFiSTiK.Good agreement is achieved,which confirms the correctness and high effectiveness of the formulated algorithm.

Large deflections of non-prismatic nonlinearly elastic cantilever beams subjected to non-uniform continuous load and a concentrated load at the free end

This work studies large deflections of slen- der, non-prismatic cantilever beams subjected to a combined loading which consists of a non-uniformly distributed con- tinuous load and a concentrated load at the free end of the beam. The material of the cantilever is assumed to be non- linearly elastic. Different nonlinear relations between stress and strain in tensile and compressive domain are considered. The accuracy of numerical solutions is evaluated by com- paring them with results from previous studies and with a laboratory experiment.

Serviceability Analysis of Non-Prismatic Timber Beams: Derivation and Validation of New and Effective Straightforward Formulas

This paper provides innovative and effective instruments for the simplified analysis of serviceability limit states for pitched, kinked, and tapered GLT beams. Specifically, formulas for the evaluation of maximal horizontal and vertical displacements are derived from a recently-proposed Timoshenko-like non-prismatic beam model. Thereafter, the paper compares the proposed serviceability analysis formulas with other ones available in literature and with highly-refined 2D FE simulations in order to demonstrate the effectiveness of the proposed instruments. The proposed formulas lead to estimations that lie mainly on the conservative side and the errors are smaller than 10% (exceptionally up to 15%) in almost all of the cases of interest for practitioners. Conversely, the accuracy of the proposed formulas decreases for thick and highly-tapered beams since the beam model behind the proposed formulas cannot tackle local effects (like stress concentrations occurring at bearing and beam apex) that significantly influence the beam behavior for such geometries. Finally, the proposed formulas are more accurate than the ones available in literature since the latter ones often provide non-conservative estimations and errors greater than 20% (up to 120%).

Water surface profile prediction in non-prismatic compound channel using support vector machine(SVM)

The process of estimating the level of water surface in two-stage waterways is a crucial aspect in the design of flood control and diversion structures.Human activities carried out along the course of rivers,such as agricultural and construction operation,have the potential to modify the geometry of floodplains,leading to the formation of compound channels with non-prismatic floodplains,thus possibly exhibiting convergent,divergent,or skewed characteristics.In the current investigation,the Support Vector Machine(SVM)technique is employed to approximate the water surface profile of compound channels featuring narrowing floodplains.Some models are constructed by utilizing significant experimental data obtained from both contemporary and previous investigations.Water surface profiles in these channels can be estimated through the utilization of non-dimensional geometric and flow parameters,including:converging angle,width ratio,relative depth,aspect ratio,relative distance,and bed slope.The results of this study indicate that the SVM-generated water surface profile exhibits a high degree of concordance with both the empirical data and the findings from previous research,as evidenced by its R2 value of 0.99,RMSE value of 0.0199,and MAPE value of 1.263.The findings of this study based on statistical analysis demonstrate that the SVM model developed is dependable and suitable for applications in this particular domain,exhibiting superior performance in forecasting water surface profiles.