scholarly journals Timoshenko Beam Theory: Exact Solution for First-Order Analysis, Second-Order Analysis, and Stability

2021 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Exact Solution ◽  
Closed Form ◽  
Timoshenko Beam ◽  
Beam Theory ◽  
Second Order ◽  
Bernoulli Beam ◽  
Order Analysis ◽  
First Order ◽  
The Moment ◽  
Relationship Of

This paper presents an exact solution to the Timoshenko beam theory (TBT) for first-order analysis, second-order analysis, and stability. The TBT covers cases associated with small deflections based on shear deformation considerations, whereas the Euler–Bernoulli beam theory (EBBT) neglects shear deformations. Thus, the Euler–Bernoulli beam is a special case of the Timoshenko beam. The moment-curvature relationship is one of the governing equations of the EBBT, and closed-form expressions of efforts and deformations are available in the literature. However, neither an equivalent to the moment-curvature relationship of EBBT nor closed-form expressions of efforts and deformations can be found in the TBT. In this paper, a moment-shear force-curvature relationship, the equivalent in TBT of the moment-curvature relationship of EBBT, was presented. Based on this relationship, first-order and second-order analyses were conducted, and closed-form expressions of efforts and deformations were derived for various load cases. Furthermore, beam stability was analyzed and buckling loads were calculated. Finally, first-order and second-order element stiffness matrices were determined.

scholarly journals Timoshenko Beam Theory Exact Solution For Bending, Second-Order Analysis, and Stability

2020 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Exact Solution ◽  
Timoshenko Beam ◽  
Beam Theory ◽  
Second Order ◽  
Timoshenko Beam Theory ◽  
Bernoulli Beam ◽  
Closed Form Solutions ◽  
Order Analysis ◽  
Stiffness Matrices ◽  
Order Element

This paper presents an exact solution to the Timoshenko beam theory (TBT) for bending, second-order analysis, and stability. The TBT covers cases associated with small deflections based on shear deformation considerations, whereas the Euler–Bernoulli beam theory neglects shear deformations. A material law (a moment-shear force-curvature equation) combining bending and shear is presented, together with closed-form solutions based on this material law. A bending analysis of a Timoshenko beam was conducted, and buckling loads were determined on the basis of the bending shear factor. First-order element stiffness matrices were calculated. Finally second-order element stiffness matrices were deduced on the basis of the same principle.

scholarly journals Timoshenko Beam Theory Exact Solution For Bending, Second-Order Analysis, and Stability

2021 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Exact Solution ◽  
Timoshenko Beam ◽  
Beam Theory ◽  
Second Order ◽  
Timoshenko Beam Theory ◽  
Bernoulli Beam ◽  
Closed Form Solutions ◽  
Order Analysis ◽  
Stiffness Matrices ◽  
Order Element

This paper presents an exact solution to the Timoshenko beam theory (TBT) for bending, second-order analysis, and stability. The TBT covers cases associated with small deflections based on shear deformation considerations, whereas the Euler–Bernoulli beam theory neglects shear deformations. A material law (a moment-shear force-curvature equation) combining bending and shear is presented, together with closed-form solutions based on this material law. A bending analysis of a Timoshenko beam was conducted, and buckling loads were determined on the basis of the bending shear factor. First-order element stiffness matrices were calculated. Finally second-order element stiffness matrices were deduced on the basis of the same principle.

scholarly journals Euler-Bernoulli Beam Theory: First-Order Analysis, Second-Order Analysis, Stability, and Vibration Analysis Using the Finite Difference Method

2021 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Finite Difference Method ◽  
Differential Equations ◽  
Vibration Analysis ◽  
Beam Theory ◽  
Second Order ◽  
Bernoulli Beam ◽  
Order Analysis ◽  
First Order ◽  
The Finite Difference Method ◽  
Euler Bernoulli Beam

This paper presents an approach to the Euler-Bernoulli beam theory (EBBT) using the finite difference method (FDM). The EBBT covers the case of small deflections, and shear deformations are not considered. The FDM is an approximate method for solving problems described with differential equations (or partial differential equations). The FDM does not involve solving differential equations; equations are formulated with values at selected points of the structure. The model developed in this paper consists of formulating partial differential equations with finite differences and introducing new points (additional points or imaginary points) at boundaries and positions of discontinuity (concentrated loads or moments, supports, hinges, springs, brutal change of stiffness, etc.). The introduction of additional points permits us to satisfy boundary conditions and continuity conditions. First-order analysis, second-order analysis, and vibration analysis of structures were conducted with this model. Efforts, displacements, stiffness matrices, buckling loads, and vibration frequencies were determined. Tapered beams were analyzed (e.g., element stiffness matrix, second-order analysis). Finally, a direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of structures, the damping being considered. The efforts and displacements could be determined at any time.

scholarly journals Euler-Bernoulli Beam Theory: First-Order Analysis, Second-Order Analysis, Stability, and Vibration Analysis Using the Finite Difference Method

2021 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Finite Difference Method ◽  
Differential Equations ◽  
Vibration Analysis ◽  
Beam Theory ◽  
Second Order ◽  
Bernoulli Beam ◽  
Order Analysis ◽  
First Order ◽  
The Finite Difference Method ◽  
Euler Bernoulli Beam

This paper presents an approach to the Euler-Bernoulli beam theory (EBBT) using the finite difference method (FDM). The EBBT covers the case of small deflections, and shear deformations are not considered. The FDM is an approximate method for solving problems described with differential equations (or partial differential equations). The FDM does not involve solving differential equations; equations are formulated with values at selected points of the structure. The model developed in this paper consists of formulating partial differential equations with finite differences and introducing new points (additional points or imaginary points) at boundaries and positions of discontinuity (concentrated loads or moments, supports, hinges, springs, brutal change of stiffness, etc.). The introduction of additional points permits us to satisfy boundary conditions and continuity conditions. First-order analysis, second-order analysis, and vibration analysis of structures were conducted with this model. Efforts, displacements, stiffness matrices, buckling loads, and vibration frequencies were determined. Tapered beams were analyzed (e.g., element stiffness matrix, second-order analysis). Finally, a direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of structures, the damping being considered. The efforts and displacements could be determined at any time.

scholarly journals Timoshenko Beam Theory: First-Order Analysis, Second-Order Analysis, Stability, and Vibration Analysis Using the Finite Difference Method

2021 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Differential Equations ◽  
Timoshenko Beam ◽  
Vibration Analysis ◽  
Beam Theory ◽  
Second Order ◽  
Timoshenko Beam Theory ◽  
Governing Equations ◽  
Order Analysis ◽  
First Order ◽  
The Finite Difference Method

This paper presents an approach to the Timoshenko beam theory (TBT) using the finite difference method (FDM). The Timoshenko beam theory covers cases associated with small deflections based on shear deformation and rotary inertia considerations. The FDM is an approximate method for solving problems described with differential equations. It does not involve solving differential equations; equations are formulated with values at selected points of the structure. In addition, the boundary conditions and not the governing equations are applied at the beam’s ends. The model developed in this paper consisted of formulating differential equations with finite differences and introducing additional points at the beam’s ends and at positions of discontinuity (concentrated loads or moments, supports, hinges, springs, brutal change of stiffness, spring-mass system, etc.). The introduction of additional points allowed us to apply the governing equations at the beam’s ends. Moreover, grid points with variable spacing were considered, the grid being uniform within beam segments. First-order, second-order, and vibration analyses of structures were conducted with this model. Furthermore, tapered beams were analyzed (element stiffness matrix, second-order analysis, vibration analysis). Finally, a direct time integration method (DTIM) was presented; the FDM-based DTIM enabled the analysis of forced vibration of structures, with damping taken into account. The results obtained in this paper showed good agreement with those of other studies, and the accuracy was increased through a grid refinement. Especially in the first-order analysis of uniform beams, the results were exact for uniformly distributed and concentrated loads regardless of the grid.

scholarly journals Timoshenko Beam Theory:First-Order Analysis, Second-Order Analysis, Stability, and Vibration Analysis Using the Finite Difference Method

2021 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Timoshenko Beam ◽  
Difference Method ◽  
Vibration Analysis ◽  
Beam Theory ◽  
Second Order ◽  
Order Analysis ◽  
The Finite Difference Method

This paper presents an approach to the Timoshenko beam theory (TBT) using the finite difference method (FDM). The TBT covers cases associated with small deflections based on shear deformation considerations, whereas the Euler–Bernoulli beam theory neglects shear deformations. The FDM is an approximate method for solving problems described with differential or partial differential equations. It does not involve solving differential equations; equations are formulated with values at selected points of the structure. The model developed in this paper consists of formulating partial differential equations with finite differences and introducing new points (additional or imaginary points) at boundaries and positions of discontinuity (concentrated loads or moments, supports, hinges, springs, brutal change of stiffness). The introduction of additional points allows satisfying boundary and continuity conditions. First-order, second-order, and vibration analyses of structures were conducted with this model. Efforts, displacements, stiffness matrices, buckling loads, and vibration frequencies were determined. In addition, tapered beams were analyzed (e.g., element stiffness matrix, second-order analysis, and vibration analysis). Finally, the direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of structures, considering the damping. The efforts and displacements could be determined at any time.

scholarly journals Euler-Bernoulli Beam Theory: First-Order Analysis, Second-Order Analysis, Stability, and Vibration Analysis Using the Finite Difference Method

2021 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Finite Difference ◽  
Beam Theory ◽  
Second Order ◽  
Governing Equations ◽  
Deflection Curve ◽  
Order Analysis ◽  
First Order ◽  
Finite Difference Approximations ◽  
Order Polynomial ◽  
Euler Bernoulli Beam

This paper presents an approach to the Euler-Bernoulli beam theory (EBBT) using the finite difference method (FDM). The EBBT covers the case of small deflections, and shear deformations are not considered. The FDM is an approximate method for solving problems described with differential equations. The FDM does not involve solving differential equations; equations are formulated with values at selected points of the structure. Generally, the finite difference approximations are derived based on fourth-order polynomial hypothesis (FOPH) and second-order polynomial hypothesis (SOPH) for the deflection curve; the FOPH is made for the fourth and third derivative of the deflection curve while the SOPH is made for its second and first derivative. In addition, the boundary conditions and not the governing equations are applied at the beam’s ends. In this paper, the FOPH was made for all of the derivatives of the deflection curve, and additional points were introduced at the beam’s ends and positions of discontinuity (concentrated loads or moments, supports, hinges, springs, etc.). The introduction of additional points allowed us to apply the governing equations at the beam’s ends and to satisfy the boundary and continuity conditions. Moreover, grid points with variable spacing were also considered, the grid being uniform within beam segments. First-order analysis, second-order analysis, and vibration analysis of structures were conducted with this model. Furthermore, tapered beams were analyzed (element stiffness matrix, second-order analysis). Finally, a direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of structures, with damping taken into account. The results obtained in this paper showed good agreement with those of other studies, and the accuracy was increased through a grid refinement. Especially in the first-order analysis of uniform beams, the results were exact for uniformly distributed and concentrated loads regardless of the grid. Further research will be needed to investigate polynomial refinements (higher-order polynomials such as fifth-order, sixth-order…) of the deflection curve; the polynomial refinements aimed to increase the accuracy, whereby non-centered finite difference approximations at beam’s ends and positions of discontinuity would be used.

Medium Frequency Vibration Analysis of Beam Structures Modeled by the Timoshenko Beam Theory

2020 ◽  
Author(s):  
Yichi Zhang ◽  
Bingen Yang
Keyword(s):  
Shear Deformation ◽  
Timoshenko Beam ◽  
Vibration Analysis ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Bernoulli Beam ◽  
Beam Structure ◽  
Medium Frequency ◽  
Beam Structures ◽  
Euler Bernoulli Beam

Abstract Vibration analysis of complex structures at medium frequencies plays an important role in automotive engineering. Flexible beam structures modeled by the classical Euler-Bernoulli beam theory have been widely used in many engineering problems. A kinematic hypothesis in the Euler-Bernoulli beam theory is that plane sections of a beam normal to its neutral axis remain normal when the beam experiences bending deformation, which neglects the shear deformation of the beam. However, as observed by researchers, the shear deformation of a beam component becomes noticeable in high-frequency vibrations. In this sense, the Timoshenko beam theory, which describes both bending deformation and shear deformation, may be more suitable for medium-frequency vibration analysis of beam structures. This paper presents an analytical method for medium-frequency vibration analysis of beam structures, with components modeled by the Timoshenko beam theory. The proposed method is developed based on the augmented Distributed Transfer Function Method (DTFM), which has been shown to be useful in various vibration problems. The proposed method models a Timoshenko beam structure by a spatial state-space formulation in the s-domain, without any discretization. With the state-space formulation, the frequency response of a beam structure, in any frequency region (from low to very high frequencies), can be obtained in an exact and analytical form. One advantage of the proposed method is that the local information of a beam structure, such as displacements, bending moment and shear force at any location, can be directly obtained from the space-state formulation, which otherwise would be very difficult with energy-based methods. The medium-frequency analysis by the augmented DTFM is validated with the FEA in numerical examples, where the efficiency and accuracy of the proposed method is present. Also, the effects of shear deformation on the dynamic behaviors of a beam structure at medium frequencies are illustrated through comparison of the Timoshenko beam theory and the Euler-Bernoulli beam theory.

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