Stability of curvilinear Euler-Bernoulli beams in temperature fields

In this work, stability of thin flexible Bernoulli-Euler beams is investigated taking into account the geometric non-linearity as well as a type and intensity of the temperature field. The applied temperature field T(x,z) is yielded by a solution to the 2D Laplace equation solved for five kinds of thermal boundary conditions, and there are no restrictions put on the temperature distribution along the beam thickness. Action of the temperature field on the beam dynamics is studied with the help of the Duhamel theory, whereas the motion of the beam subjected to the thermal load is yielded employing the variational principles.The heat transfer (Laplace equation) is solved with the use of the finite difference method (FDM) of the third-order accuracy, while the integrals along the beam thickness defining the thermal stress and moments are computed using Simpson's method. Partial differential equations governing the beam motion are reduced to the Cauchy problem by means of application of FDM of the second-order accuracy. The obtained ordinary differential equations are solved with the use of the fourth-order Runge-Kutta method.The problem of numerical results convergence versus a number of beam partitions is investigated. A static solution for a flexible Bernoulli-Euler beam is obtained using the dynamic approach based on employment of the relaxation/set-up method.Novel stability loss phenomena of a beam under the thermal field are reported for different beam geometric parameters, boundary conditions, and the temperature intensity. In particular, it has been shown that stability of the flexible beam during heating the beam surface essentially depends on the beam thickness.     

Experimental investigation of transient flexural waves in beams with discontinuities of cross section

There has been little experimental work on flexural wave propagation in general and on flexural wave propagation in beams with discontinuities of cross section in particular. Experimental data are obtained for various test beams subjected to eccentric longitudinal impact. The bending strain versus time results are presented for several positions along a uniform beam and finite beams (of circular cross section) with discontinuities of cross section. Bending strain histories are recorded at several positions before and after the discontinuity. The effect of reflections on the propagated flexural wave is illustrated. The dispersion of the traveling flexural wave caused several alternating peaks within the duration of the original positive input pulse. The importance of investigating discontinuities of cross section in structures subjected to impact loading is clearly manifested.     

Free vibrations of long pretwisted cylindrical shells

Tbilisi Mathematical Institute, Academy of Sciences of the Georgian SSR. Translated from Prikladnaya Mekhanika, Vol. 26, No. 9, pp. 54–60, September, 1990.     

Shear deformation effect in non-linear analysis of composite beams of variable cross section

In this paper the non-linear analysis of a composite Timoshenko beam with arbitrary variable cross section undergoing moderate large deflections under general boundary conditions is presented employing the analog equation method (AEM), a BEM-based method. The composite beam consists of materials in contact, each of which can surround a finite number of inclusions. The materials have different elasticity and shear moduli with same Poisson's ratio and are firmly bonded together. The beam is subjected in an arbitrarily concentrated or distributed variable axial loading, while the shear loading is applied at the shear center of the cross section, avoiding in this way the induction of a twisting moment. To account for shear deformations, the concept of shear deformation coefficients is used. Five boundary value problems are formulated with respect to the transverse displacements, the axial displacement and to two stress functions and solved using the AEM. Application of the boundary element technique yields a system of non-linear equations from which the transverse and axial displacements are computed by an iterative process. The evaluation of the shear deformation coefficients is accomplished from the aforementioned stress functions using only boundary integration. Numerical examples are worked out to illustrate the efficiency, the accuracy, the range of applications of the developed method and the influence of the shear deformation effect.     

Stability of nonlinear periodic vibrations of 3D beams

The geometrically nonlinear periodic vibrations of beams with rectangular cross section under harmonic forces are investigated using a p-version finite element method. The beams vibrate in space; hence they experience longitudinal, torsional, and nonplanar bending deformations. The model is based on Timoshenko’s theory for bending and assumes that, under torsion, the cross section rotates as a rigid body and is free to warp in the longitudinal direction, as in Saint-Venant’s theory. The theory employed is valid for moderate rotations and displacements, and physical phenomena like internal resonances and change of the stability of the solutions can be investigated. Green’s nonlinear strain tensor and Hooke’s law are considered and isotropic and elastic beams are investigated. The equation of motion is derived by the principle of virtual work. The differential equations of motion are converted into a nonlinear algebraic form employing the harmonic balance method, and then solved by the arc-length continuation method. The variation of the amplitude of vibration in space with the excitation frequency of vibration is determined and presented in the form of response curves. The stability of the solution is investigated by Floquet’s theory.     

Stability of stratified shear flows in channels with variable cross sections

For the instability problem of density stratified shear flows in sea straits with variable cross sections, a new semielliptical instability region is found. Furthermore, the instability of the bounded shear layer is studied in two cases: (i) the density which takes two different constant values in two layers and (ii) the density which takes three different constant values in three layers. In both cases, the dispersion relation is found to be a quartic equation in the complex phase velocity. It is found that there are two unstable modes in a range of the wave numbers in the first case, whereas there is only one unstable mode in the second case.     

Stability of rectangular plates with cambered bitrapezoidal cross sections

This paper presents a theoretical moment-curvature relationship for rectangular plates having bitrapezoidal cross sections. It is shown that, for certain values of the cross-section camber and edge thickness, the plates exhibit a bending instability for some values of longitudinal curvature. Experimental results show good agreement between theory and experiment.     

Nonlinear analysis of beams of variable cross section,including shear deformation effect

In this paper the analog equation method (AEM), a BEM-based method, is employed for the nonlinear analysis of a Timoshenko beam with simply or multiply connected variable cross section undergoing large deflections under general boundary conditions. The beam is subjected in an arbitrarily concentrated or distributed variable axial loading, while the shear loading is applied at the shear center of the cross section, avoiding in this way the induction of a twisting moment. To account for shear deformations, the concept of shear deformation coefficients is used. Five boundary value problems are formulated with respect to the transverse displacements, the axial displacement and to two stress functions and solved using the AEM. Application of the boundary element technique yields a system of nonlinear equations from which the transverse and axial displacements are computed by an iterative process. The evaluation of the shear deformation coefficients is accomplished from the aforementioned stress functions using only boundary integration. Numerical examples with great practical interest are worked out to illustrate the efficiency, the accuracy and the range of applications of the developed method. The influence of the shear deformation effect is remarkable.     

On dualities in the oscillations of naturally curved and pretwisted rods

The pertinent equations of naturally curved and pretwisted rods, in the form of compatibility, equilibrium and constitutive relations are obtained under the assumptins of infinitesimal deformations and material isotropy. Then by forming the expressions for various energy terms, the equations of motion of the rod are obtained via Hamilton's principle and the complementary energy principle. On comparing these two forms of equations of motion, and the associated boundary conditions certain dualities are exposed. Finally the equations of some special rods, including the plane arch and the straight pretwisted rod, are examined.     

A new numerical procedure for free vibrations of pretwisted plates

Summary  A numerical procedure is proposed for the analysis of free vibrations of pretwisted thin plates. An accurate strain–displacement relationship based on the thin-shell theory combined with the finite element method using triangular plate elements with three nodes and nine degrees of freedom for each node is utilized. The vibration characteristics of pretwisted thin plates with different twist rates and aspect ratios are studied. The numerical results are compared with the previous results obtained by various types of finite elements and by the Rayleigh–Ritz method. The effect of the twist rate on the vibration characteristics is studied briefly. Received 28 February 2001; accepted for publication 18 December 2001     

Energy-consistent shear coefficients for beams with circular cross sections and radially inhomogeneous materials

An exact computational method for the shear stiffness of beams with circular cross sections and arbitrarily radially inhomogeneous Young’s modulus is presented. We derive the displacement and stress field of a cantilever beam according to 3D theory of elasticity, which requires to solve just a 1D linear boundary value problem. The shear stiffness is obtained by setting the shear strain energy from the exact solution equal to that from technical beam theory. Results and closed analytical formulae are given for several functionally graded and layered cross sections.     

变截面欧拉梁自由振动分析的重采样微分求积法

采用重采样微分求积法求解了变截面欧拉梁的自由振动问题。推导了变截面梁的控制方程离散格式,采用重采样矩阵方法对边界条件进行处理,给出了变截面梁自由振动算法。采用本文方法对不同类型截面形式和不同边界条件的变截面梁进行自由振动分析,并和其他解法进行比较。计算结果表明,本文方法可以适用于不同变截面类型和不同边界条件,计算精度与解析解吻合良好,具有良好的收敛性能。在同等精度条件下网格点数少于现有计算方法。重采样转换矩阵边界处理方法相比于传统边界处理方法具有更快的收敛性能。     

Stability analysis of gradient elastic beams by the method of initial value

The buckling of a bar is studied analytically on the basis of a simple linear theory of gradient elasticity in the frame of the method of initial values. The method of initial values provides the values of the displacements and stress resultants throughout the bar once the initial displacements and initial stress resultants are known. We use probably for the first time the method of initial values to get critical loads of a strain gradient beam under completely different boundary conditions at the two end faces of the beam. Exact carryover matrix is presented for the classical beam and gradient beam analytically. The first mode shapes of classical beam and gradient beam are plotted. The method of initial values is also applied to the beams with variable cross-section. The priorities of the method of initial values are depicted. The variational approach gives a sixth-order ordinary differential equation for a beam in buckling. The additional boundary conditions are used to obtain critical loads. It is observed that critical loads increase dramatically for increasing values of the gradient coefficient.