Closed Form of a Transverse Tapered Cantilever Beam Fundamental Frequency with a Linear Cross-Area Variation

This study derived analytical solutions for the deflection of a rectangular cross sectional uniformly tapered cantilever beam with varying configurations of width and breadth acting under an end point load. The deflection equations were derived using a numerical analysis method known as the finite element method. The verification of these analytical solutions was done by deterministic optimisation of the equations using the ModelCenter reliability analysis software and the Abaqus finite element modelling and optimisation software. The results obtained show that the best element type for the finite element analysis of a tapered cantilever beam acting under an end point load is the C3D20RH (A 20-node quadratic brick, hybrid element with linear pressure and reduced integration) beam element; it predicted an end displacement of 0.05035 m for the tapered width, constant height cantilever beam which was the closest value to the analytical optimum of 0.05352 m. The little difference in the deflection value accounted for the numerical error which is inevitably present in the analyses of structural systems. It is recommended that detailed and accurate numerical analysis be adopted in the design of complex structural systems in order to ascertain the degree of uncertainty in design. Keywords: Deflection, Finite element method, deterministic optimisation, numerical error, cantilever beam.

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Nonlinear forced vibration analysis of a rotating three-dimensional tapered cantilever beam

Journal of Vibration and Control ◽

10.1177/1077546320949716 ◽

2020 ◽

pp. 107754632094971 ◽

Cited By ~ 1

Author(s):

Yanxun Zhou ◽

Yimin Zhang ◽

Guo Yao

Keyword(s):

Cantilever Beam ◽

Forced Vibration ◽

Vibration Analysis ◽

Three Dimensional ◽

Time History ◽

Excitation Amplitude ◽

Axial Deformation ◽

Nonlinear Forced Vibration ◽

Tapered Cantilever Beam ◽

Forced Vibration Analysis

In this article, nonlinear forced vibration analysis is carried out for a rotating three-dimensional tapered cantilever beam subjected to a uniformly distributed load. Considering the effects of Coriolis terms, static axial deformation and geometric nonlinearity in modeling process, nonlinear partial motion equations of a rotating tapered Euler–Bernoulli beam are established by using Hamilton’s principle. Galerkin’s procedure is used to discretize the equations to obtain the dynamic response of the beam. Frequency responses, the time-history response, the phase diagram, and the Poincaré map are introduced to study the effects of the taper ratio, rotating velocity, radius of hub, and external excitation on the nonlinear resonances and detailed responses of the rotating three-dimensional tapered beam. Results show that the fundamental natural frequency increases with the increase of the taper ratio, radius of hub, and rotating velocity. Besides, by increasing the taper ratio and excitation amplitude and decreasing the rotating velocity and radius of hub, the nonlinearity and vibration amplitude of the rotating beam intensify.

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CLOSED-FORM J-INTEGRAL AND ITS APPLICATIONS FOR MEASUREMENT OF MODE I INTERLAMINAR FRACTURE TOUGHNESS OF COMPOSITES

10.12783/asc36/35925 ◽

2021 ◽

Author(s):

WU XU ◽

JIANCAN DING

Keyword(s):

Fracture Toughness ◽

Closed Form ◽

Large Deformation ◽

Cantilever Beam ◽

Integral Method ◽

Double Cantilever Beam ◽

J Integral ◽

Interlaminar Fracture Toughness ◽

Interlaminar Fracture

Due to the interlaminar properties of composites are low, delamination is one of the major failure modes. It threatens the safety of composite structure subjected to out-of-plane static and especially impact loadings. High interlaminar fracture toughness is demanded in the society where composite structures are widely used. However, for tough material, large deformation may occur in the determination of the interlaminar fracture toughness when using the double cantilever beam (DCB) test. Therefore, accurate determination of the fracture toughness of tough material and dynamic loading is very challenging under large deformation. J-integral is an important parameter in fracture mechanics. It’s equivalent to energy release rate under monotonic loading and widely used in the determination of interlaminar fracture toughness of composites. In this paper, it is used to determine the fracture toughness for composite DCB under large deformation and wedge-insert double cantilever beam (WDCB) test, which is widely used to determine the dynamic interlaminar fracture toughness. Exact and closed form nonlinear J-integrals are derived for the largely deformed DCB and WDCB. Compared with the alternative data reduction methods for determining interlaminar fracture toughness, the J- integral method is more accurate. In addition, the J-integral method is simple and promising, since it is unnecessary to measure the crack length in the tests.

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Free and forced vibrations of a tapered cantilever beam carrying multiple point masses

STRUCTURAL ENGINEERING AND MECHANICS ◽

10.12989/sem.2006.23.2.209 ◽

2006 ◽

Vol 23 (2) ◽

pp. 209-216 ◽

Cited By ~ 1

Author(s):

Der-Wei Chen ◽

Tsung-Lung Liu

Keyword(s):

Cantilever Beam ◽

Forced Vibrations ◽

Multiple Point ◽

Free And Forced Vibrations ◽

Tapered Cantilever Beam

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Vibration of Non-Uniform Beams

Aeronautical Quarterly ◽

10.1017/s0001925900002936 ◽

1963 ◽

Vol 14 (4) ◽

pp. 387-395 ◽

Cited By ~ 29

Author(s):

G. M. Lindberg

Keyword(s):

Cantilever Beam ◽

Stiffness Matrix ◽

Dynamic Stiffness ◽

Dynamic Stiffness Matrix ◽

Uniform Beam ◽

Tapered Cantilever Beam

SummaryA method of deriving a dynamic stiffness matrix for any non-uniform beam is presented. In particular, the case of a linearly tapered cantilever beam is considered, and excellent results are found with the use of only a few elements.

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