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Mechanical System Design (Strength and Stiffness)
Published in Seong-woo Woo, Design of Mechanical Systems Based on Statistics, 2021
A beam in mechanical/civil engineering is a structural constituent piece whose length is longer compared to its cross-sectional area. In other words, it is a bar-like structural member whose main function is to bear transverse loading and convey it to the supports. The Euler–Bernoulli beam theory plays a critical role in structural analysis because it helps mechanical engineers to compute the load-carrying and deflection characteristics of beams in a straightforward way.
Theory of Beams and Columns
Published in K.T. Chau, Applications of Differential Equations in Engineering and Mechanics, 2019
In this chapter, we will consider some applications of beam models in mechanics and engineering. Both static and dynamic problems of Euler-Bernoulli beam theory will be considered. The major assumption for the Euler-Bernoulli beam theory is that a vertical plane perpendicular to the neutral axis of the beam before bending remains a plane and normal to the deformed neutral surface. The neutral axis after bending locates on the centroid of the cross-section of the beam. This assumption is found acceptable only for thin beams with aspect ratio larger than ten. Euler buckling formula for columns is also considered. The case of initial imperfection of column buckling is considered, yielding a transition from straight to buckled state. However, more complex case of nonlinear buckling of columns and beams is deferred to Chapter 6. The effect of axial force in beam vibrations is also investigated. Some new results are summarized.
Single Degree‐of‐Freedom Damped Vibration
Published in Haym Benaroya, Mark Nagurka, Seon Han, Mechanical Vibration, 2017
Haym Benaroya, Mark Nagurka, Seon Han
Euler helped develop the Euler‐Bernoulli beam equation, used extensively in mechanics of materials. Euler‐Bernoulli beam theory is a simplification of the linear theory of elasticity and provides a means of calculating the load‐carrying and deflection characteristics of beams. He worked on it circa 1750.
Flexible multibody system modelling of an aerial rescue ladder using Lagrange’s equations
Published in Mathematical and Computer Modelling of Dynamical Systems, 2021
Simon Densborn, Oliver Sawodny
In this paper, the eigenmodes of Euler-Bernoulli beams are used as basis functions . The main assumptions of Euler-Bernoulli beam theory are small deflection, always orthogonal cross-sections in relation to the neutral axis of the beam and a high length-to-thickness ratio [24]. Since the ladder segments have a slender shape and the deflection during normal operation is relatively small, this beam model is seen as valid. The eigenmodes comply with the above stated requirements and are orthonormal. For the upper part of segment 1, Clamped-Free eigenmodes [9] as shown in Figure 5 are used.
Magneto-hygro-thermal vibration analysis of the viscoelastic nanobeams reinforcedwith carbon nanotubes resting on Kerr’s elastic foundation based on NSGT
Published in Advanced Composite Materials, 2023
Yan Yuan, Zhiqiang Niu, John Smitt
Zhang et al. [10] reported the effects of the temperature on the dynamic responses of the magnetorheological materials. Wang [11] presented a model to study the vibration exploration of the FG piezoelectric porous plates in the translation state. Bousahla et al. [12] studied the buckling and dynamic investigation of the CNT beams using first-order shear deformation theory. Zenkour [13] reported the hygrothermal impacts on the static responses of the composite plates using a sinusoidal theory. Teng and Wang [14] investigated the nonlinear vibration analysis of the FG nanoplates reinforced with graphene platelet (GPL). Diverse kinds of porosity distributions and GPL patterns are considered. Sourki and Hosseini [15] reported the vibration investigation of the nanobeams based on nonlocal modified couple stress theory, including surface effects. Euler–Bernoulli beam theory is employed to model the beam. Wang et al. [16] studied the nonlinear vibration investigation of the cylindrical shells reinforced with GPL. Civalek et al. [17] presented the vibration analysis of the nanotubes resting on an elastic matrix using Eringen’s nonlocal elasticity theory. Winkler–Pasternak elastic foundation is applied as an elastic matrix. Wang and Zu [18] analyzed the vibrational behaviors of FG plates with porosities in a thermal environment. Von Kármán nonlinear plate theory is employed to consider the geometric nonlinearity. Esen et al. [19] studied the dynamic responses of the CNT nanobeams subjected to mass moving. Nonlocal strain gradient theory (NSGT) is utilized to consider the size effect. Also, Reddy’s third-order shear deformation theory is applied to consider the shear impacts. Chen et al. [20] investigated the mechanical behavior of the FG porous piezoelectric sandwich nanobeams using size-dependent theories based on Halpin–Tsai model. The dynamic investigation of the nanobeams considering the size-dependent effect is reported by Bensaid et al. [21]. Dindarloo and Li [22] studied the vibration investigation of the doubly-curved shells reinforced with carbon nanotubes using nonlocal elasticity theory. Ebrahimi-Mamaghani et al. [23] investigated the vibration analysis of the viscoelastic FG beams subjected to an axial load. It is assumed that the effective properties of the beam vary through the length. Mansouri and Shariyat [24] studied the buckling investigation of the orthotropic plates resting on elastic foundations subjected to hygro-thermo loading.
Torsional vibration of Timoshenko-Gere non-circular nano-bars
Published in Mechanics Based Design of Structures and Machines, 2023
Babak Alizadeh-Hamidi, Reza Hassannejad
One of the most important theories is the nonlocal strain gradient theory which indicates the size-dependent effects at the nanoscale. According to the nonlocal strain gradient theory stress tensor at any point of the body not only depend on the strain tensor at the whole body but also depend on high order gradient strain tensor (Lim, Zhang, and Reddy 2015). Although, other theories are used to show the size-dependent effects (Eringen 1983; Eringen 1984). In this regard, many studies have been done to investigate the dynamic behavior of nanostructures. Using nonlocal theory, Lu et al. investigated the characteristics of bending vibrations of nano-bars. This study is based on the Euler-Bernoulli beam theory. Therefore, it is assumed that shear deformations are negligible, and plane sections remain plane and perpendicular to the longitudinal axis. Thus, displacement at any cross-section point should be only due to pure bending. They studied the effect of boundary conditions for various nonlocal parameters (Lu et al. 2006). Wang et al. investigated the bending vibration of nanobeams using the Timoshenko beam theory because in lateral vibration of beam-like thick structures, plane sections remain plane but are not perpendicular to the longitudinal axis. Therefore, the Timoshenko beam theory is necessary to consider shear deformation and rotary inertia. They concluded that the effects of rotary inertia and shear deformation cause to increase in the amplitude of mode shapes (Wang, Zhang, and He 2007). In another study, Alizadeh et al. used the Timoshenko beam theory to model lateral vibration of lipid supramolecular nanotubules because, in such structures, rotary inertia and shear deformation effects are considerable (Alizadeh-Hamidi, Hassannejad, and Omidi 2021). The axial vibration of Rayleigh nanobeams and FG nanobeam with two symmetrical axial elastic springs which were attached to a nanobeam at both ends, is investigated by Yayli (Yayli 2020; Yayli 2018a). The buckling of microbeams under axial compression load with deformable boundary conditions is studied by Yayli (Yayli 2016). Various investigations about the lateral and torsional vibration of FG nanobeams with general boundary conditions are reviewed (Yayli 2019b; Yayli 2018b; Yayli 2018c). The buckling of carbon nanotube with rotational restraint is studied by Yayli (2015), and the thermal effect on the buckling of carbon nanotube with rotational restraint is also studied in another study by Yayli (Yayli 2019a).