A Sixth-Order Theory of Shear Deformable Beams With Variational Consistent Boundary Conditions

2010 ◽  
Vol 78 (2) ◽  
Author(s):  
Guangyu Shi ◽  
George Z. Voyiadjis
Keyword(s):  
Boundary Conditions ◽  
Fourth Order ◽  
Cross Sections ◽  
Beam Theory ◽  
Order Theory ◽  
Sixth Order ◽  
Equilibrium Equations ◽  
Flexible Beams ◽  
Generalized Displacements ◽  
Shear Deformable

This paper presents the derivation of a new beam theory with the sixth-order differential equilibrium equations for the analysis of shear deformable beams. A sixth-order beam theory is desirable since the displacement constraints of some typical shear flexible beams clearly indicate that the boundary conditions corresponding to these constraints can be properly satisfied only by the boundary conditions associated with the sixth-order differential equilibrium equations as opposed to the fourth-order equilibrium equations in Timoshenko beam theory. The present beam theory is composed of three parts: the simple third-order kinematics of displacements reduced from the higher-order displacement field derived previously by the authors, a system of sixth-order differential equilibrium equations in terms of two generalized displacements w and ϕx of beam cross sections, and three boundary conditions at each end of shear deformable beams. A technique for the analytical solution of the new beam theory is also presented. To demonstrate the advantages and accuracy of the new sixth-order beam theory for the analysis of shear flexible beams, the proposed beam theory is applied to solve analytically three classical beam bending problems to which the fourth-order beam theory of Timoshenko has created some questions on the boundary conditions. The present solutions of these examples agree well with the elasticity solutions, and in particular they also show that the present sixth-order beam theory is capable of characterizing some boundary layer behavior near the beam ends or loading points.

Numerical Assessment of the Boundary Layer Effect Predicted by the Shear Flexible Beam Theory with the Sixth-Order Differential Equations

2011 ◽  
Vol 105-107 ◽  
pp. 1705-1711
Author(s):  
Xiao Dan Wang ◽  
Guang Yu Shi
Keyword(s):  
Boundary Layer ◽  
Boundary Conditions ◽  
Beam Theory ◽  
Sixth Order ◽  
Flexible Beam ◽  
Element Analysis ◽  
Flexible Beams ◽  
Displacement Boundary ◽  
Numerical Assessment ◽  
Layer Effect

The analytical solutions of shear flexible beams with displacement boundary conditions are derived by using the new sixth-order differential equation beam theory presented by Shi and Voyiadjis (ASME J. Appl. Mech., Vol. 78, 021019, 2011), in which the boundary layer effects are included. The accuracy of the boundary layer effects predicted by the new sixth-order beam theory is evaluated by the finite element analysis in this study. The numerical results show that the new sixth-order beam theory is capable of taking account of the displacement boundary conditions of shear deformable beams and predicting good results of the boundary layer effects induced by the displacement boundaries and the continuity constraints.

Generalised Beam Theory (GBT) for Shear Deformable Stiffened Sections

2014 ◽  
Vol 553 ◽  
pp. 600-605
Author(s):  
Gerard Taig ◽  
Gianluca Ranzi
Keyword(s):  
Cross Sections ◽  
Beam Theory ◽  
Element Model ◽  
Elastic Behaviour ◽  
Structural Behaviour ◽  
Thin Walled ◽  
Shell Finite Element ◽  
Linear Elastic Behaviour ◽  
Generalised Beam Theory ◽  
Shear Deformable

A Generalised Beam Theory (GBT) formulation is presented to analyse the structural behaviour of shear deformable thin-walled members with partially stiffened cross-sections located at arbitrary locations along their length. The deformation modes used in the formulation are taken as the dynamic eigenmodes of a planar frame representing the unstiffened cross-section. Constraint equations are derived and implemented in the GBT member analysis to model the influence of rigid stiffeners on the member response. The accuracy of the approach is validated against a shell finite element model developed in Abaqus. A numerical example describing the linear elastic behaviour of partially stiffened thin-walled member is provided to outline the usability and flexibility of the proposed method.

Compatibility Equations in the Theory of Elasticity

2003 ◽  
Vol 125 (2) ◽  
pp. 244-245 ◽  
Author(s):  
Igor V. Andrianov ◽  
Jan Awrejcewicz
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value ◽  
Theory Of Elasticity ◽  
Sixth Order ◽  
Operational Method ◽  
Equilibrium Equations ◽  
Additional Boundary Conditions

It is shown by operational method that the boundary value problem of the theory of elasticity related to stresses, which can be reduced to three strains compatibility equations and to three equilibrium equations, in fact is of sixth order. Hence, it is not required to formulate additional boundary conditions.

Free Vibration Analysis of Cracked Composite Beams Subjected to Coupled Bending-Torsion Loads Based on a First Order Beam Theory

2013 ◽  
Vol 325-326 ◽  
pp. 1318-1323 ◽  
Author(s):  
A.R. Daneshmehr ◽  
D.J. Inman ◽  
A.R. Nateghi
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
Order Theory ◽  
Composite Beams ◽  
First Order ◽  
First Order Theory

In this paper free vibration analysis of cracked composite beams subjected to coupled bending-torsion loads are presented. The composite beam is assumed to have an open edge crack. A first order theory is applied to count for the effect of the shear deformations on natural frequencies as well as the effect of coupling in torsion and bending modes of vibration. Local flexibility matrix is used to obtain the additional boundary conditions of the beam in the crack area. After obtaining the governing equations and boundary conditions, GDQ method is applied to solve the obtained eigenvalue problem. Finally, some numerical results are given to show the efficacy of the method. In addition, to count for the effect of coupling on natural frequencies of the cracked beams, different fiber orientations are assumed and studied.

scholarly journals Shape functions for high-order shear deformation beam element

2021 ◽  
Author(s):  
Himanshu Gaur ◽  
Mahmoud Dawood ◽  
Ram Kishore Manchiryal
Keyword(s):  
Boundary Conditions ◽  
Shear Deformation ◽  
Beam Theory ◽  
Order Theory ◽  
Beam Element ◽  
High Order ◽  
Shape Functions ◽  
High Order Theory ◽  
Cubic Polynomial ◽  
Transverse Deflection

In this article, shape functions for higher-order shear deformation beam theory are derived. For the two nodded beam element, transverse deflection is assumed as cubic polynomial. By using equations of equilibrium of high-order theory that are already derived by J. N. Reddy in 1997, equation for slope of high- order theory is found. Finally with the boundary conditions of beam element and assumed kinematics of high-order theory, shape functions are derived.

scholarly journals Strength of thin-walled elastic building structures

2021 ◽  
Vol 274 ◽  
pp. 03019
Author(s):  
Lilya Kharasova
Keyword(s):  
Boundary Conditions ◽  
Strain State ◽  
Holomorphic Functions ◽  
Integral Representations ◽  
Nonlinear Operator Equation ◽  
Equilibrium Equations ◽  
Stress Strain ◽  
Stress Strain State ◽  
Generalized Displacements ◽  
Thin Walled

The existence theorem is proved within the framework of the shear model by S.P. Timoshenko. The stress-strain state of elastic inhomogeneous isotropic shallow thin-walled shell constructions is studied. The stress-strain state of shell constructions is described by a system of the five equilibrium equations and by the five static boundary conditions with respect to generalized displacements. The aim of the work is to find generalized displacements from a system of equilibrium equations that satisfy given static boundary conditions. The research is based on integral representations for generalized displacements containing arbitrary holomorphic functions. Holomorphic functions are found so that the generalized displacements should satisfy five static boundary conditions. The integral representations constructed this way allow to obtain a nonlinear operator equation. The solvability of the nonlinear equation is established with the use of contraction mappings principle.

Reflections on the Theory of Elastic Plates

1985 ◽  
Vol 38 (11) ◽  
pp. 1453-1464 ◽  
Author(s):  
Eric Reissner
Keyword(s):  
Shear Deformation ◽  
Fourth Order ◽  
Transverse Shear ◽  
Three Dimensional ◽  
Order Theory ◽  
Sixth Order ◽  
Interior Solution ◽  
Transverse Shear Deformation ◽  
Two Dimensional ◽  
Elastic Plates

We depart from a three-dimensional statement of the problem of small bending of elastic plates, for a survey of approximate two-dimensional theories, beginning with Kirchhoff’s fourth-order formulation. After discussing various variational statements of the three-dimensional problem, we describe the development of two-dimensional sixth-order theories by Bolle´, Hencky, Mindlin, and Reissner which take account of the effect of transverse shear deformation. Additionally, we report on an early analysis by Le´vy, on a direct two-dimensional formulation of sixth-order theory, on constitutive coupling of bending and stretching of laminated plates, on higher than sixth-order theories, and on an asymptotic analysis of sixth-order theory which leads to a fourth-order interior solution contribution with first-order transverse shear deformation effects included, as well as to a sequentially determined second-order edge zone solution contribution.

Experimental Modal Analysis of Rectangular and Circular Beams

2003 ◽  
Author(s):  
B. H. Emory ◽  
W. D. Zhu
Keyword(s):  
Boundary Conditions ◽  
Cross Sections ◽  
Natural Frequencies ◽  
Beam Theory ◽  
Mode Shapes ◽  
Euler Beam ◽  
Modal Assurance Criterion ◽  
Rectangular Beam ◽  
Circular Beam ◽  
A New Technique

Analytical and experimental methods are used to determine the natural frequencies and mode shapes of Aluminum 6061-T651 beams with rectangular and circular cross sections. A unique test stand is developed to provide the rectangular beam with different boundary conditions including clamped-free, clamped-clamped, clamped-pinned, and pinned-pinned. The first 10 bending frequencies and mode shapes for each set of boundary conditions are measured. The effects of the bolt torque on the measured frequencies of the rectangular beam are examined. The material properties of the circular beam, including the elastic modulus, shear modulus, and Poisson’s ratio, are determined by measuring its first 20 natural frequencies. A new technique is used to mount an accelerometer to measure the torsional modes of the circular beam. A roving hammer test is conducted to measure the first 10 mode shapes. The measured mode shapes of the circular and rectangular beams are compared with their thoretical predictions using the modal assurance criterion. The Timoshenko beam theory is shown to provide better predictions of the natural frequencies for the higher modes of the circular beam than the Bernoulli-Euler beam theory. The use of the rectangular and circular beam test stands as a teaching tool for undergraduate and graduate students is discussed.

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