A comparison of curved beam finite elements when used in vibration problems

Sabir, A.B.; Ashwell, D.G.

doi:10.1016/0022-460x(71)90106-4

Cited by 60 publications

(25 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The literature devoted to the dynamic behaviour is not so rich [21][22][23]. In the present paper the solution of the free vibrations problem for the arches using the ÿnite elements based on the trigonometric, exact shape functions [1] is presented.…”

Section: Introductionmentioning

confidence: 99%

Free vibrations of shear‐flexible and compressible arches by FEM

Litewka¹,

Rakowski²

2001

Numerical Meth Engineering

View full text Add to dashboard Cite

The purpose of this paper is to analyse free vibrations of arches with in uence of shear and axial forces taken into account. Arches with various depth of cross-section and various types of supports are considered. In the calculations, the curved ÿnite element elaborated by the authors is adopted. It is the plane two-node, six-degree-of-freedom arch element with constant curvature. Its application to the static analysis yields the exact results, coinciding with the analytical ones. This feature results from the use of the exact shape functions in derivation of the element sti ness matrix. In the free vibration analysis the consistent mass matrix is used. It is obtained on the base of the same functions. Their coe cients contain the in uences of shear exibility and compressibility of the arch. The numerical results are compared with the results obtained for the simple diagonal mass matrix representing the lumped mass model. The natural frequencies are also compared with the ones for the continuous arches for which the analytically determined frequencies are known. The advantage of the paper is a thorough analysis of selected examples, where the in uences of shear forces, axial forces as well as the rotary and tangential inertia on the natural frequencies are examined.

show abstract

Section: Introductionmentioning

confidence: 99%

Free vibrations of shear‐flexible and compressible arches by FEM

Litewka¹,

Rakowski²

2001

Numerical Meth Engineering

View full text Add to dashboard Cite

show abstract

“…It is noted that the third term on the right-hand side of Equation (19), I y˙ y 2 , represents the rotary inertia, which is not considered in References [1] and [9].…”

Section: Mass Matrix For Arch Elementmentioning

confidence: 98%

“…However, to avoid manipulating the complicated matrix multiplication and listing the tedious lengthy mathematical expressions in the subsequent derivations, the 'implicit' form of [B] −1 will be used for the formulation of this paper and so will be the related sti ness and mass matrices of the arch element. Once the angular co-ordinates of node 1 −1 into Equation (11) the values of the shape function matrix [N (Â)] will have to be determined. It is evident that the implicit shape functions given by Equation (11) are much simpler than the explicit 'exact' shape functions given in Tables I-III of Ref-erence [8], particularly for the computer programming.…”

Section: Displacement Functions and Shape Functionsmentioning

confidence: 99%

Free vibration analysis of arches using curved beam elements

Chiang

2003

Numerical Meth Engineering

View full text Add to dashboard Cite

SUMMARYThe natural frequencies and mode shapes for the radial (in-plane) bending vibrations of the uniform circular arches were investigated by means of the ÿnite arch (curved beam) elements. Instead of the complicated explicit shape functions of the arch element given by the existing literature, the simple implicit shape functions associated with the tangential, radial (or normal) and rotational displacements of the arch element were derived and presented in matrix form. Based on the relationship between the nodal forces and the nodal displacements of a two-node six-degree-of-freedom arch element, the elemental sti ness matrix was derived, and based on the equation of kinetic energy and the implicit shape functions of an arch element the elemental consistent mass matrix with rotary inertia e ect considered was obtained. Assembly of the foregoing elemental property matrices yields the overall sti ness and mass matrices of the complete curved beam. The standard techniques were used to determine the natural frequencies and mode shapes for the curved beam with various boundary conditions and subtended angles. In addition to the typical circular arches with constant curvatures, a hybrid beam constructed by using an arch segment connected with a straight beam segment at each of its two ends was also studied. For simplicity, a lumped mass model for the arch element was also presented. All numerical results were compared with the existing literature or those obtained from the ÿnite element method based on the conventional straight beam element and good agreements were achieved.

show abstract

“…This asymptotic rate of convergence is as with a coupled system but : first, the coupled system requires only 6 variables per element whereas the uncoupled requires 8 variables per element for achieving the same asymptotic rate of convergence, and secondly for a coarse mesh we expect the coupled system to do better than the uncoupled system as has been demonstrated experimentally. 11 For carefully executed numerical experiments for determining the accuracy of some two-dimensional shell elements see Refs. 20 and 21.…”

Section: Energy Error Estimatesmentioning

confidence: 99%

Shape functions and the accuracy of arch finite elements.

Fried

1973

AIAA Journal

View full text Add to dashboard Cite

If the estimation error has zero mean and has Gaussian distribution it can be shown that Eq. (A27) reduces to [./P-] (A28)The variance of g is the mean square minus the squared mean Var[0] =h r p-h + rr[Jp-Jp-]/2 (A29) The numerator in Eq. (A21) becomes -tr[Jp-]/2)] = P~h Thus, the optimum gain vector isThe covariance of the estimation error after incorporating the measurement was given in Eq. (A9). Recall b = 0. Using Eqs.(A19, 20, 29, 30) the covariance is P + -(/-kh r )p-(/-kh r ) r + k(r + rr[Jp-Jp-]/2)k r (A32)Note if the nonlinearity J is zero, then Eq. (A31) and Eq. (A32) are equivalent to the extended (linearized) Kalman measurement incorporation equations.To summarize, the measurement incorporation equations arewhere g 0 , h, and J were defined in Eqs. (A23-25). These equations are equivalent to those proposed previously byThe rate of convergence of the finite element method is determined by the ability of the approximate strains to assume arbitrary polynomial states. A systematic way to generate shape functions assuring a certain accuracy is to start, then, with assumed polynomial strains and integrate for the displacements. In curved structures, this leads to a coupled system of shape functions for the normal and tangential displacements, which are not necessarily polynomials. Some further simplifications reduce these to coupled polynomial shape functions which might prove more efficient than uncoupled shape functions assumed independently for the different displacements. The spectral condition number of the global stiffness matrix for the curved beam increases with (r/t) 2 -the ratio of radius of curvature to thickness squared. The extensional energy portion of the elastic energy is proportional to (t/r) 2 . For a coarse mesh this might be smaller than the discretization error, not warranting the introduction of the exact small t/r into the extensional energy. By relating r/t to the number of elements so as to balance the extensional energy and the energy discretization error, r/t is eliminated from the extensional energy and consequently from the condition number. The resulting element converges with its full rate to the inextensional solution but without undue ill-conditioning in the stiffness matrix.

show abstract

A comparison of curved beam finite elements when used in vibration problems

Cited by 60 publications

References 8 publications

Free vibrations of shear‐flexible and compressible arches by FEM

Free vibrations of shear‐flexible and compressible arches by FEM

Free vibration analysis of arches using curved beam elements

Shape functions and the accuracy of arch finite elements.

Contact Info

Product

Resources

About