Geometric nonlinear dynamic analysis of curved beams using curved beam element

Geometric nonlinear formulation for curved beams with varying curvature

In this investigation a method for the dynamic analysis of initially curved Timoshenko beams that undergo finite rotations is presented. The combined effect of rotary inertia, shear deformation, and initial curvature is examined. The kinetic energy is first developed for the curved beam and the beam mass matrix is identified. It is shown that the form of the mass matrix as well as the nonlinear inertia terms that represent the coupling between the rigid body motion and the elastic deformation can be expressed in terms of a set of invariants that depend on the assumed displacement field, rotary inertia, shear deformation, and the initial beam curvature. A nonlinear finite element formulation is then developed for Timoshenko beams that undergo finite rotations. The nonlinear formulation presented in this paper is applied to multibody dynamics where mechanical systems consist of an interconnected set of rigid and deformable bodies, each of which may undergo finite rotations. The equations of motion are developed using Lagrange’s equation and nonlinear algebraic constraint equations that mathematically describe mechanical joints and specified trajectories are adjoined to the system differential equations using the vector of Lagrange multipliers.

In this research a comprehensive modeling framework for a piezoelectrically-actuated cantilever beam is developed and a detailed model and vibration analyses is performed. To achieve this goal, the governing dynamics for the system as well as boundary conditions are derived using the extended Hamilton's principle. The equations of motion of cantilever beam are derived according to the Euler-Bernoulli, Rayleigh and Timoshenko theories separately. The Euler-Bernoulli theory neglects the effects of rotary inertia and shear deformation and is only applicable to analysis of thin beams. The Rayleigh theory considers the effect of rotary inertia, while the Timoshenko theory considers the effects of both rotary inertia and shear deformation for thick beams. It is evident from the nature of discontinuous geometry of system, equation of stress-strain relationship are modified as shown in theory subsection meanwhile the natural surface in the composite (beam-piezoelectric layer) portion of the cantilever beam must be considered in this stage of calculations. Then the first five natural frequencies of this composite system are obtained by those three different theories and the results are compared. Relevant mode shapes are also drawn and effects of including rotary inertia and shear deformation are discussed for slender and stocky beams. Then, the forced vibration problem is solved and the cantilever tip deflection is obtained in which applied voltage to the piezoelectric layer is considered to be a unit-step input. The results are compared again for slender and stocky beams.

This Tech Brief addresses the effects of shear deformation and rotary inertia on the free vibration of a rotating annular plate. Several studies have been done to understand these effects (Irie et al, 1982; Sinha, 1987). In particular, the importance of these terms is well known for disks with a thickness ratio h/a > 0.05 where h is the thickness of the disk and a is its outer radius. However, there is a need to show how they affect the disk vibration as the thickness ratio increases. It is well accepted that shear deformation and rotary inertia should be included in the analysis when h/a > 0.05 (Batoz and Dhatt, 1990). However, this Tech Brief will show that this criterion is not valid for all the modes. It will present the evolution of the number of modes correctly evaluated versus the thickness ratio, and, finally, will give a heuristic criterion to predict when it is necessary to consider shear deformation and rotary inertia.

The non-conservative problems of the shear-flexible Leipholz’s type of laminated composite beams are studied based on the finite element method. A formal engineering approach of the mechanics of thin-walled laminated composite beams on the basis of kinematic assumptions consistent with Timoshenko beam theory is presented. The explicit expression for the shear stiffness of composite beam is derived from the energy equivalence. In the present finite element formulation, the shape functions which contain the shear stiffness are presented, and the effects of both transverse shear deformation and rotary inertia are included. The advantage of this formulation is that if a suitable value of the shear stiffness parameter is given, a specific composite beam model is then simulated without too much additional work to modify the original formulation. Then the evaluation procedure for the critical values of the divergence and flutter loads of the Leipholz’s type of composite beam is briefly introduced. In order to verify accuracy of the composite beam model and the numerical calculation applied, the critical loads of divergence and flutter, and the static deflection of beams are compared with those from available references. Especially, the important structural parameters such as shear deformation, rotary inertia, boundary condition, fiber angle change, and non-conservativeness factor on the divergence and flutter behavior of the Leipholz’s type of composite beams are parametrically investigated.

Free vibration of axially loaded composite Timoshenko beams using the dynamic stiffness matrix method

In this investigation, a curved beam element is developed for the analysis of large deformation of flexible multi-body systems using the absolute nodal coordinate formulation. In the existing beam element in this formulation, because the elastic forces are defined using the Green-Lagrange strain tensor as a volume element, locking phenomenon associated with the shear and cross-section deformation leads to erroneously stiffer bending characteristics. In order to circumvent this drawback, the Hellinger-Reissner variational principle is applied to modify the shear stress distribution, whereas the assumed strain method is employed to alleviate the locking associated with the cross-section deformation. The consistent tangent stiffness matrices are derived for the curved beam element and used with implicit integration methods. Numerical examples are presented in order to demonstrate the performance of the curved beam element developed in this investigation.

A method for selection of structural theories for low to high frequency vibration analyses

In this study, nonlinear vibrations of a slightly curved Timoshenko beam with nonlinear boundary conditions are investigated. The formulation is developed based on Timoshenko beam theory, incorporating shear deformation and rotary inertia effects. Geometric nonlinearity is introduced through the von Karman strain–displacement relations, and the governing equations of motion are derived using Hamilton’s principle. The resulting partial differential equations are reduced to a set of ordinary differential equations via the Galerkin method, where the mode shapes of a linear straight Timoshenko beam with linear spring boundary conditions are employed as admissible functions. The temporal response of the system is obtained using the method of multiple scales. Validation of the proposed formulation is carried out by examining the limiting case of vanishing curvature and by comparison with the nonlinear response of straight Timoshenko beams and thick shallow arch models. The results reveal that both hardening and softening behaviors may occur due to the presence of second- and third-order nonlinear terms. The proposed semi-analytical framework provides an efficient tool for analyzing nonlinear vibrations of slightly curved Timoshenko beams, capturing the combined effects of geometric nonlinearity, shear deformation, rotary inertia, and boundary flexibility.

An ‘exact’ basis function Dynamic Finite Element (DFE) for the free vibrational analysis of axially loaded beams and assemblages composed of beams is presented. The shear deformation is neglected but the Rotary Inertia (RI) effects are taken into consideration. The dynamic trigonometric shape functions for bending vibrations of an axially loaded uniform beam element are first derived in an exact sense. Then, exploiting the Principle of Virtual Work together with the nodal approximations of variables based on these dynamic shape functions, leads to a single frequency dependent Dynamic Stiffness Matrix (DSM) that represents both mass and stiffness properties. A Wittrick-Williams algorithm, based on a Sturm sequence root counting technique, is then used as the solution method. The application of the theory is demonstrated by an illustrative example of cantilever beam where the influence of Rotary Inertia (RI) effect and different axial loads on the natural frequencies of the system is demonstrated by numerical results.

Compliant mechanisms use the flexibility of the limbs for movement and are produced monolithic, without any conventional joint. These flexible mechanisms have wide applications in several fields such as sensors, grippers, parallel mechanisms, micro-electro-mechanical systems, high-precision devices, and biomedical and surgery robots. Topology optimization is regarded as one of the most frequently used approaches for designing compliant mechanisms. The present study aimed to develop the ground structure approach by meshing the design domain using curved beam elements. Moreover, it seeks to implement three different metaheuristic algorithms, including the genetic algorithm, the imperialist competition algorithm, and the equilibrium optimizer for the topology optimization problem. The structural analysis of the compliant mechanism is necessary for determining the values of the objective functions for the optimization process. In the present study, the finite element method was used to analyze the obtained compliant mechanism. Based on the results, an optimum compliant mechanism with both straight and curved beam elements had better performance compared with a compliant mechanism with only straight beams.

Large amplitude free vibrations of skew plates including transverse shear deformation and rotatory inertia—a new approach

The effects of thickness variation, shear deformation and rotary inertia of annular plates are studied by deriving the equations of motion. To solve the differential equation governing the motion of annular plates the Chebyshev collocation method is employed. The dimensionless frequencies are evaluated for different values of taper constant, thickness ratio, radii ratio and power. The results of an experimental investigation are also presented, and the agreement between the experimental values and the predicted values in theory is remarkably good. As a result of this study, it is found that the effects of rotatory inertia and shear deformation reduce the natural frequencies of annular plates for three boundary conditions. This study also showed that the natural frequencies of annular plates with thickness expressed by the nth power function are higher than those by the (n-1)th power function for positive values of taper constant, and vice versa for negative values of taper constant for three boundary conditions.

The analysis of functionally graded material (FGM) plates with material variation parameter (n), boundary conditions, aspect ratios and side to thickness ratios are investigated using higher order displacement model. The derivation of equations of motion for higher order displacement model is obtained using principle of virtual work. The nonlinear simultaneous equations are obtained by Navier’s method considering certain parameters, loads and boundary conditions. The nonlinear algebraic equations are solved using Newton Raphson iterative method. The numerical results are obtained for various boundary conditions, material variation parameter, aspect ratio, side to thickness ratio and compared with the available solutions. The effect of shear deformation and nonlinearity response of functionally graded material plate is studied.

A finite element analysis (FEA) model was developed for Euler and Timoshenko curved beam elements by incorporating trigonometric displacement distribution patterns. Local polar coordinate stiffness matrices were derived based on force-displacement relations and static equilibrium conditions. By employing the kinetic energy theorem and triangular displacement functions, an expression for the consistent mass matrix of a curved beam element was obtained. A coordinate transformation matrix for the curved beam element was established by relating the local polar coordinate system to the global polar coordinate system. Calculation programs were implemented in the Fortran language to evaluate the static–dynamic performance and natural frequency characteristics of curved beam bridges. The obtained results were then compared with those obtained using ANSYS solid models and “replace curve with straight” beam element models. The comparison demonstrated a strong agreement between the results of the Euler and Timoshenko curved beam element models and those of the ANSYS solid models. However, discrepancies were observed when comparing with the results of the “replace curve with straight” beam element model, particularly in terms of lateral static displacement. This discrepancy suggests that the characteristic matrix derived in this study accurately represents the stiffness and mass distribution of the curved beam, making it suitable for mechanical performance analysis of curved beam bridges. It should be noted that the “replace curve with straight” method overlooks the initial curvature and the bending–torsion coupling effects of a curved beam, resulting in calculation deviations. On the other hand, the use of curved beam elements in numerical analysis provides a simple and practical approach, which facilitates further research in areas such as vehicle–bridge coupling vibrations and seismic analysis of curved beam bridges.