Classical Beam Theory Research Articles

On the convergence of solid meshes for the prediction of part distortions due to residual stresses

Residual stresses in rolled plates, used as raw material for the fabrication of aircraft components, arise from manufacturing processes such as rolling, casting, quenching, stretching, and thermal treatments. After each process, the rolled plate has a geometrically stable condition but with internal stresses. However, during part machining an unbalance in the distribution of residual stresses occurs, in special for aircraft components, due to the large amount of material removal throughout the process. This condition of instability leads to component distortions so that any corrective action affects the manufacturing lead-time and production costs. Part distortions are usually predicted by finite element analyses with linear tetrahedral meshes in which the residual stress profiles are applied as a constant value element-wise. In this work, both linear and quadratic solid meshes are employed to address this problem. For this purpose, a Python-based routine is implemented to apply the residual stress profile at the integration points of the elements. Then, finite element simulations of simple geometric configurations (plates and beams) under theoretical and real residual stress distributions are carried out. Performance and effectiveness of two different meshes—tetrahedral and hexahedral (brick-type)—are checked through comparison with results presented by classical plate and beam theories. A general good correlation for the deflections predicted by them is reached.

Frequency analysis of carbon and silicon nanosheet with surface effects

• FE modeling of surface effects in Si, Graphene–Silicon and SWCNT-Silicon nanosheet . • Numerical analysis of HACNTs on silicon substrate and its effect on frequency. • Development of analytical model to find frequency of HACNTs arrays on silicon. Silicon-based microelectromechanical system (MEMS) and nanoelectromechanical systems (NEMS) have been used to design and fabricate sensitive sensors and actuators. Recent research trends show that graphene and carbon nanotubes (CNTs) have been used to change the surface properties of silicon-based MEMS and NEMS to improve different mechanical, optical and electrical properties of silicon-based composites. In this paper, we focus on analyzing the vibrational characteristics of silicon-based devices when the surface of silicon is coated with single-layer graphene and horizontally aligned carbon nanotubes (HACNTs). To perform the analysis, we use multi-scale finite element approach for developing graphene–silicon nanocomposites (GSNCs) and carbon nanotube-silicon nanocomposites (CSNC) composites in which interface layer of silicon with graphene or CNT is modeled using bonded contact element. Subsequently, we performed modal analysis to find the first transverse mode frequency of GSNC and CSNC composites for beam with smaller as well as longer lengths. The numerical model is compared with classical beam theory with and without surface effect. For GSNCs composites, we take a fixed-free case with lengths in the range of (20 Å–120 Å) and (400 Å–2000 Å), respectively. For CSNC composites, CNT diameter is varied from (5 Å–30 Å) for single walled nanotube . Subsequently, we analyze the influence of HACNTs-on-silicon on its vibrational characteristics. The analysis presented in the paper demonstrate that GSNCs offer a higher bending stiffness compared to single layer graphene (SLGs) and isolated silicon nanosheet which lead to higher natural frequency. A similar trend is found in the case of HACNTs on silicon NS when the number of tubes increases.

Vibration of a rotating composite beam clamped-off the axis of rotation

The mechanics of a laminated composite beam attached to inside of a rotating rim and directed to the inward direction is investigated. The Ritz method is utilized in the solution of the problem. Simple algebraic polynomials are used in the displacement field. Clamped-free boundary conditions are considered. First, Reddy (third order) and classical beam theories are used in the formulation. Cross-ply lamination configurations are considered. Effects of rotation speed, hub ratio, orthotropy ratio, beam theory and length to thickness ratio are analyzed in detail. Mode shapes of composite rotating beams are given. It is obtained that composite beams may buckle due to compressive centrifugal force for the some combinations of rotation speed and hub ratio.

Nonlocal strain gradient beam model for nonlinear secondary resonance analysis of functionally graded porous micro/nano-beams under periodic hard excitations

Functionally graded porous materials (FGPMs) have a wide range of applications as hollow members in biomedical and aeronautical engineering. In the FGPMs, the porosity is varied over the material volume because of the density change of pores. In the present work, an analytical treatment on the size-dependent nonlinear secondary resonance of FGPM micro/nano-beams subjected to periodic hard excitations is proposed in the simultaneous presence of the nonlocality and strain gradient size dependencies. Based upon the closed-cell Gaussian-random field scheme, the mechanical properties of the FGPM micro/nano-beams are extracted corresponding to the uniform and three different functionally graded patterns of the porosity dispersion. The nonlocal strain gradient theory of elasticity is applied to the classical beam theory to formulate a newly combined size-dependent beam model. Thereafter, an analytical solving methodology based on the multiple time-scales together with the Galerkin technique is adopted to achieve the nonlocal strain gradient frequency–response and amplitude–response curves associated with the subharmonic and superharmonic external excitations. For the subharmonic excitation, it is observed that the nonlocality causes to shift the junction point of the stable and unstable branches to the higher value of the detuning parameter. However, the strain gradient size dependency plays an opposite role. For the superharmonic one, it is illustrated that the nonlocal size effect makes an increment in the height of jump phenomenon and shifts the peak to higher value of the detuning parameter. However, the strain gradient small scale effect leads to decrease the height of the jump phenomenon and shifts the peak to lower value of the detuning parameter.

Bending Solutions of FGM Reddy–Bickford Beams in Terms of Those of the Homogenous Euler–Bernoulli Beams

In this paper, correspondence relations between the solutions of the static bending of functionally graded material (FGM) Reddy–Bickford beams (RBBs) and those of the corresponding homogenous Euler–Bernoulli beams are presented. The effective material properties of the FGM beams are assumed to vary continuously in the thickness direction. Governing equations for the titled problem are formulated via the principle of virtual displacements based on the third-order shear deformation beam theory, in which the higher-order shear force and bending moment are included. General solutions of the displacements and the stress resultants of the FGM RBBs are derived analytically in terms of the deflection of the reference homogenous Euler–Bernoulli beam with the same geometry, loadings and end conditions, which realize a classical and homogenized expression of the bending response of the shear deformable non-homogeneous FGM beams. Particular solutions for the FGM RBBs under specified end constraints and load conditions are given to validate the theory and methodology. The key merit of this work is to be capable of obtaining the high-accuracy solutions of thick FGM beams in terms of the classical beam theory solutions without dealing with the solution of the complicated coupling differential equations with boundary conditions of the problem.

Mechanical properties of natural fibre-reinforced hybrid composites

A study on the hybrid composites reinforced by two types of short natural fibres is presented. The material properties of short natural fibre composites were obtained by Mori–Tanaka method. Three-point bend test was simulated by the classical beam theory and finite element analysis. Both asymmetric and sandwich hybrid composites were studied. From the results, the following conclusions can be drawn: 1. A general rule is that one lamina of higher stiffness should be placed on the tensile face of a hybrid composite in order to obtain improved flexural strength. 2. For the sandwich-type hybrid composites, the higher stiffness material should be placed in the core, and the lower stiffness material should be the skin.

Deflection Monitoring of Bi-Dimensional Structures by Fiber Bragg Gratings Strain Sensors

In this paper, we report about the deflection estimation of bi-dimensional structure by using fiber Bragg gratings (FBGs) as strain sensors embedded in the structure and an algorithm based on the classical beam theory with small deformation assumption. The proposed technique allows a real-time indirect structural monitoring solving the existing difficulties in measuring the deflection directly and can be applied to small as well large structure, and it is crucial in high-energy physics, where particle detectors have to measure the position of the incoming radiation with a resolution of few tens of micrometers. The obtained results have been compared with mechanical comparator direct deflection measurements showing good agreement, showing a resolution of few tens of micronmeters over a surface of the order of about 1 m <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . The dynamic range of deflection monitoring is dependent on the panel dimension: in the hypothesis of 1-m <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> panel, the deflection limit is around 15 cm since FBG can measure very high strain (greater than 10000 μm/m).

Buckling capacity of functionally graded beams: A comparative study of different shear deformation beam theories

This paper investigates the postbuckling response of simply supported functionally graded beams under axial loading using several beam theories; classical beam theory, CBT, Timoshenko beam theory, TBT and parabolic shear deformation beam theory, PSDBT. Hamilton’s principle is used to derive the governing equations which are solved by closed-form method. It is assumed that the Young’s modulus is varying continuously in the thickness direction according to power-law form while all other material properties are taken to be constant. The effects of the reinforcement distribution, the beam edge-to-thickness ratio and the phase contrast (ratio of the reinforcement Young’s modulus to the matrix Young’s modulus) on the postbuckling behavior and critical buckling load of functionally graded beam are studied. Results demonstrate the important contribution of the shear effect to both the buckling and postbuckling behaviors. Finally, the combinations of reinforcement distribution, beam edge-to-thickness ratio and phase contrast that correspond to the highest and the lowest buckling capacities are identified.

On the drag reconfiguration of plates near the free surface

Rectangular foils of different flexural rigidities were towed normal to the flow at a fixed speed in a quiescent fluid in order to study the effect of the proximity of the upper edge of the models to free surface. It was found that flexibility ensured drag reduction due to the reconfiguration process at all submergence depths, with certain foils exhibiting depth-independent behavior. The study of Vogel exponents also showed that a sublinear or even a depth independent relationship between drag and velocity can be attained at specific flexural rigidity values. A modified classical beam theory model using a power-law based load distribution was utilized to obtain an empirical relationship between the loading exponent and Cauchy numbers and to identify the foil tip location. Particle image velocimetry was also undertaken to interpret and further understand the force results. The experiments showed the existence of 2 ranges of Cauchy numbers with a small degree of overlap in ranges wherein the drag coefficient and Vogel exponents are independent of submergence depth.

Size-dependent on vibration and flexural sensitivity of atomic force microscope

In this paper, the free vibration behaviors and flexural sensitivity of atomic force microscope cantilevers with small-scale effects are investigated. To study the small-scale effects on natural frequencies and flexural sensitivity, the consistent couple stress theory is applied. In this theory, the couple stress is assumed skew-symmetric. Unlike the classical beam theory, the new model contains a material-length-scale parameter and can capture the size effect. For this purpose, the Euler–Bernoulli beam theory is used to develop the AFM cantilever. The tip interacts with the sample that is modeled by a spring with constant of. The equation of motion is obtained through a variational formulation based on Hamilton’s principle. In addition, the analytical expressions for the natural frequency and sensitivity are also derived. At the end, some numerical results are selected to study the effects of material-length-scale parameter and dimensionless thickness on the natural frequency and flexural sensitivity.

A Poroelastic Solution for Dynamics of Laterally Loaded Offshore Monopiles

In this study, a rigorous poroelastodynamic solution is proposed for monopiles which accounts for 3D (three dimensional) dynamic interaction of the soil and pile. The dynamic Biot's theory is used to model the soil, while Timoshenko beam is used to model the monopile. The perfect contact conditions between the soil and monopile are realized by decomposing the pile's horizontal translation to radial, hoop and vertical displacements. Correspondingly, 3D radial, hoop and vertical stresses on the pile surface are used to formulate the governing equations of the interaction problem. Using the integral equation methods, the governing differential equations are reduced to coupled integral equations, which are solved numerically. Selected numerical results for coupled lateral and rocking dynamic impedances together with the 3D dynamic contact stresses at the soil-pile interface are studied for different pile and poroelastic material parameters and frequencies of excitation. The distributed soil reaction force and moment along the pile are also obtained by integrating the 3D radial, hoop and vertical stresses along the circumference of the pile at the soil-pile interface. Finally, bending moment, shear force, horizontal translation and rocking rotation of the monopile are obtained based on the classical dynamic Timoshenko beam theory.

Photothermal Bimorph Actuators with In‐Built Cooler for Light Mills, Frequency Switches, and Soft Robots

AbstractPhotothermal bimorph actuators are widely used for smart devices, which are generally operated in a room temperature environment, therefore a low temperature difference for actuation without deteriorating the performance is preferred. The strategy for the actuator is assembling a broadband‐light absorption layer for volume expansion and an additional water evaporation layer for cooling and volume shrinkage on a passive layer. The response time and temperature‐change‐normalized bending speed under NIR, white, and blue light illumination are at the same level of high performance, fast photothermal actuators based on polymer or polymer composites. The classical beam theory and finite element simulations are also conducted to understand the actuation mechanism of the actuator. A new type of light mill is designed based on a wing‐flapping mechanism and a light‐modulated frequency switch. A fast‐walking robot (with a speed of 26 mm s−1) and a fast‐and‐strong mechanical gripper with a large weight‐lifting ratio (≈2142), respectively, are also demonstrated.

Power transmission and suppression characteristics of stiffened Mindlin plate under different boundary constraints

The dynamic power transmission characteristics of a finite stiffened Mindlin plate subject to different boundary conditions are analytically studied. The stiffened plate is modeled as a coupled structure comprising a plate and stiffeners. Dynamic responses calculated by the analytical solutions are verified through comparison of the results with those generated using the finite element method. The computed results show that Mindlin plate and Timoshenko beam theory is more suitable for studying dynamic power of the stiffened plate over a broad frequency range than classical plate and beam theory. The stiffness and inertia characteristics of the Mindlin plate can be enhanced using the stiffeners, which can significantly affect the dynamic response, especially in low-frequency range. It can be also noticed that the stop band in the low-frequency range can become wider by increasing the number and dimension (height and width) of the stiffeners, so vibratory power of the stiffened Mindlin plate in the low-frequency range can be greatly reduced.

Effect of notch length and pre-crack size on mode II fracture energy of brittle adhesives

In this study, the effects of notch length and pre-crack size on mode II fracture energy of brittle adhesives are investigated. These effects are not considered by the direct beam theory (DBT) and the compliance-based beam method (CBBM) for the end-notched flexure (ENF) specimens. The sum of the lengths of the notch and the pre-crack are known as the pre-crack size in common data reduction schemes. In the current work, experiments were performed on ENF adhesive joints using digital image correlation (DIC) technique. Then, the fracture energy was obtained experimentally based on the J-integral method, the classic beam theory, and the compliance-based beam method. Numerical analyses were carried out to measure the stress distribution ahead of the pre-crack tip. Based on the obtained results, it is shown that there is a specific pre-crack length and notch size in which the pure mode II condition in an ENF test is obtained. Other combinations of the notch to pre-crake size ratio result in mixed mode conditions where the obtained fracture energy significantly changes.

Construction of new enriched beam models accounting for cross-section deformation and pinching

The consideration of nonclassical beam theories beyond Timoshenko beam model is necessary in applications involving complex yarns subjected to transverse compaction, for which the section dilatation and in-plane shear require a detailed kinematic and static modeling. In the present work, new enriched beam elements with additional degrees of freedom are derived from the mechanics of generalized continua, allowing a representation of the in plane-shear and dilatation of sections. As a novel aspect, enriched beam model called macrodilatation, macroshear, and macrostrain have been constructed on the basis of generalized beam theories but with a more condensed kinematics, assuming uniform hyperstress tensor within the beam. The choice of the adequate extended continuum for a given beam depends on its macroscopic deformation mechanisms. As a result, the developed beam models include modified material parameters (the tensile, shear and bulk moduli for an isotropic beam) involving additional microstructural parameters. These macrobeam models are expected to better describe the in plane-shear and dilatation of the beam sections along its mean line in comparison to classical beam theories. Numerical illustrative examples show that the microdilatation beam model has the ability to capture the local deformation field around holes, in contrast to Timoshenko beam model. The parameters of the microdilatation beam model provide a very good estimate of the overall porosity.

Shear strength of fiber-reinforced high-strength steel ultra-high-performance concrete beams based on refined calculation of compression zone depth considering concrete tension

This article presents an experimental and theoretical investigation on the shear behavior of fiber-reinforced ultra-high-performance concrete beams reinforced with high-strength steel. The test parameters included the fiber volume fraction, fiber type, and stirrup ratio. The test results indicate that the shear failure in ultra-high-performance concrete beams is not brittle and catastrophic but has ductility characteristics. A moderate quantity of stirrups can significantly improve the shear behavior of ultra-high-performance concrete beams, as reflected in the thorough propagation of cracks in both shear span and pure bending zone. The depth of the compression zone considering concrete tension was derived based on the deformation compatibility and force equilibrium equations for both serviceability limit state and ultimate limit state. The comparison of the proposed method and classical beam theory shows that the concrete tension should not be neglected in the serviceability limit state analysis. After cracking, the concrete tension can be neglected for simplicity when the beam is heavily reinforced and should be considered when the beam is lightly reinforced. Then, a shear strength model was established based on Rankine’s failure criteria, the truss model, and Association Francaise de Génie Civil-Sétra. Finally, the proposed shear strength equation was verified by the test results and compared with other shear strength equations.

Damping capacity in pseudo-elastic and ferro-elastic shape memory alloy-reinforced hybrid composite beam

Reinforcing a composite beam with shape memory alloy wires may have several benefits such as reduction of buckling risks or elimination of unwanted oscillations. In this paper, the vibration damping of a typical shape memory alloy-reinforced composite or hybrid beam is explored. To formulate the thermo-mechanical behavior of embedded shape memory alloy wires, three-dimensional Panico–Brinson model is employed and tailored to one-dimensional model. This material model can simulate pseudo-elastic and ferro-elastic forms of martensite transformations which occurs in cyclic loadings. Besides, unlike the former studies which rely on classical beam theories, the first-order shear deformation beam theory is used to obtain more accurate estimations of shape memory alloy-wire hysteresis loops and their decaying characteristics. In order to explore the effects of a transient concentrated load applied in the middle of a beam, the governing equations are developed and discretized by differential quadrature–integral quadrature combined method. Incremental time marching solution of the problem is accomplished using the Newmark technique. Results are assessed by comparing with available literature. Considering different types of boundary conditions, the influence of pseudo-elastic and ferro-elastic hysteresis loops on the material damping effects, shape memory alloy volume fraction, and resonance phenomenon is studied in detail.

Free vibration of imperfect sigmoid and power law functionally graded beams

In the present work, free vibration of beams made of imperfect functionally graded materials (FGMs) including porosities is investigated. Because of faults during process of manufacture, micro voids or porosities may arise in the FGMs, and this situation causes imperfection in the structure. Therefore, material properties of the beams are assumed to vary continuously through the thickness direction according to the volume fraction of constituents described with the modified rule of mixture including porosity volume fraction which covers two types of porosity distribution over the cross section, i.e., even and uneven distributions. The governing equations of power law FGM (P-FGM) and sigmoid law FGM (S-FGM) beams are derived within the frame works of classical beam theory (CBT) and first order shear deformation beam theory (FSDBT). The resulting equations are solved using separation of variables technique and assuming FG beams are simply supported at both ends. To validate the results numerous comparisons are carried out with available results of open literature. The effects of types of volume fraction function, beam theory and porosity volume fraction, as well as the variations of volume fraction index, span to depth ratio and porosity volume fraction, on the first three non-dimensional frequencies are examined in detail.

功能梯度梁在热-机械荷载作用下的几何非线性分析

Based on the classical beam theory, the geometric nonlinear governing equations for FGM beams under uniform temperature field and uniform transverse loading were derived according to the principle of virtual work and the variational method. In view of the immovably clamped boundary conditions, the 2-point boundary value problem was solved with the shooting method. For the zero uniform transverse loading, the thermal buckling critical temperature and equilibrium path of the FGM beam were investigated. The load-deflection curves of the FGM beam were given for the nonzero uniform temperature and the nonzero transverse uniform loading. The numerical results show that, the dimensional thermal buckling critical temperature of the beam decreases significantly and the post-buckling deformation increases significantly with the material volume fraction index increases, and the temperature variation has a heavy influence on the load-deflection curves. The bistable configurations and the switch of the FGM beam were found. The final equilibrium shape of the beam is not only related to the variable temperature and loading parameters, but also to the loading process.

Response of functionally graded beams subjected to a moving load and different types of attachments

This paper develops an analytical solution for the dynamic response of a Euler-Bernoulli functionally graded (FG) beam subjected to a load moving with constant velocity and different types of attachments. The classical beam theory (CBT) is used to derive the equation of motion. The material properties of the beam are considered to vary continuously in the transverse direction according to an exponential and a power law models. The stress distribution is evaluated and the location of the neutral axis of the FG beam is obtained. Different types of attachments connected to the beam at specific locations are analysed. The effect of using FG materials to enhance the properties of the beam, resulting in a decreased response of the beam, is made clear. Also, the effect of the restoring force of attachments in reducing the dynamic response of the beam is discussed.