Boundary Conditions Of Beam Research Articles

Buckling instability analysis of delaminated beam-like structures by using the exact stiffness method

In multilayered structures, delamination not only reduces local strength but also induces buckling instability, compromising structural safety even before reaching the critical buckling load. This paper introduces a novel model for buckling analysis of structures with delamination damage. The model utilizes exact stiffness formulations derived from Timoshenko theory, enabling precising modelling of beam-like structures with through-thickness delamination. The Wittrick-Williams algorithm is utilized to calculate the critical buckling loads, which are validated against results from the finite element software ANSYS. Additionally, a modified Euler buckling formula for the approximate yet closed-form critical buckling load of delaminated beam-like structures is proposed, with comparisons made to the exact stiffness method results. The study investigates the effects of position and length of delamination and boundary conditions of beam on the critical buckling loads. The findings indicate that the buckling reduction factors of delaminated beam is primarily influenced by the thickness-wise position of the delamination, followed by the delamination length, and then the length-wise position of the delamination. Furthermore, the impact of boundary conditions becomes more significant when the delamination is near the beam’s end. This research provides practical guidelines for preventing buckling instability in delaminated beam-like structures.

Investigations on the effects of rebar diameter on the post-fire bond capacity of RC flexural members and development of a novel post-fire bond model

This study is part of a comprehensive research effort focused on examining the impact of various parameters on the post-fire bond behaviour of RC flexural members. Precisely, the impact of the rebar diameter is thoroughly discussed. Beam-end specimens, which realistically simulate the boundary conditions of a full-scale beam while maintaining a relatively compact size, were subjected to ISO 834 fire curve. To investigate the behaviour of lap-splice in a slab and beam, two different fire exposure scenarios, where one side (slab) and three sides (beam) of the specimens were exposed to fire, were considered. It is evident that although the initial and residual load-carrying capacities increase with larger rebar diameters in absolute terms, the differences in residual bond capacities remain relatively insignificant across nearly all examined fire-exposure durations. Based on the evaluation of the test results, a model for local bond stress-slip curve of bond affected by fire is proposed.

Existence results for a fourth order problem with functional perturbed clamped beam boundary conditions

Abstract In this paper, we study the existence of positive solutions for a fourth order boundary value problem coupled to functional perturbed clamped beam boundary conditions. Our main ingredient is the classical fixed point index. The problem investigated is an extension of other problems studied in previous papers by covering very general nonlocal perturbed conditions on the boundary.

Use of the integration by part to derive beam/plate equilibrium equations from elasticity with applications to beams

This article proposes a rigorous mathematical justification of the derivation of the governing equations of any beam and plate theory by the three three-dimensional (3D) elasticity equations. This is done by the application of the integration by part theorem via weight residual-like methods. The Fis polynomials (subscripts i and s denote the i-direction and the generic s-term of the displacement expansion uis, respectively) related to a given beam/plate theory are used as weight functions for each of the displacement unknown. To show how the method works a number of simple problems are discussed. These are related to the equilibrium equations and boundary conditions of bending beams according to shear deformation theory. Three methods for obtaining these equations are compared in some detail. The first two are the well-known methods used in mechanics of deformable bodies: (1) use of the Principle of Virtual Displacements; (2) use of the six cardinal ‘Newton’ equations of statics. The third method coincides with the one proposed and rigorously justified in this paper which is (3) the direct integration of the equations of 3D elasticity. Beam theories based on Lagrange polynomials are used as well. The latter makes it possible to immediately develop theories for layered beams and obtain so-called ‘layer-wise’ models, which are only applied here to the case of sandwich structures. Three appendixes discuss: A – The problem in which the derivative of one of the unknowns appears among the unknowns, as is the case with the Euler–Bernoulli theory; B – The extension to dynamics; C-Simple derivation of constitutive equations and related closed-form solution via Navier method.

Chaotic vibration of a curved CNT conveying magnetic fluid in the thermo-magnetic field considering the surface effects

The nonlinear chaotic vibration of curved single-walled carbon nanotube (CSWCNT) conveying magnetic fluid is studied based on the nonlocal Euler-Bernoulli beam model. The governing equation of CSWCNT is presented considering the axial thermo-magnetic load and the surface effect. By employing the Galerkin decomposition approximation method along with the admissible beam shape function satisfying the cantilevered beam boundary conditions, the nonlinear partial differential equation of the system is reduced to a nonlinear ordinary differential equation and numeric integration procedures are used to solve it. By considering the non-dimensional damping, cubic and quadratic terms, and amplitude of external force as controlling parameters, the bifurcation diagram and the largest Lyapunov exponent are employed to distinguish the chaotic, periodic, and quasi-periodic critical parameters of the curved CNT dynamics and validate the predicted results. Also, the phase plane and Poincare map for these critical parameters are provided. The results show that decreasing values of damping coefficient and quadratic term leads to quasi-periodic or chaotic motion, while system has periodic behaviour for smaller values of the external force and nonlinear cubic term. Also, magnetic field intensity and length of the CSWCNT help preventing the chaotic motion. Furthermore, adverse effects of higher values of temperature change, flow velocity and its correction factor are seen based on the obtained results.

Free and forced vibrations of elastically restrained cantilever with lumped oscillator

The efficiency assessment of cantilever-based energy harvesters relies on vibrational analysis, which necessitates modifications aimed at enhancing efficiency. These modifications involve manipulating the fundamental frequency to lower values and encompassing a wider range of resonances within a specified bandwidth. Consequently, this paper introduces an original analytical-numerical exploration into the vibratory response of a cantilever with a novel boundary condition involving an elastically restrained oscillator-spring arrangement. At the microbeam's tip, an oscillator is elastically confined by a linear spring, resulting in a novel set of coupled governing equations and a distinct shearing boundary condition. Microbeam equations is derived from the modified couple stress theory to capture size dependency. During free vibration analysis, a previously unreported characteristic equation is derived. This nonlinear transcendental equation is numerically solved utilizing root-solver algorithms, such as those available in MATLAB. Significantly, it is discovered that the inclusion of a lumped oscillator with an elastic support induces a minimal (new) natural frequency. Applying the extended Hamilton's principle, the effect of the lumped oscillator emerges both on the governing equations of motion and boundary conditions of the microbeam. Novelty of the paper focuses on the both characteristic equation and transmissibility by adopting the Galerkin’s modal decomposition technique. This finding carries vital implications as the efficiency of cantilever-based energy harvesters is directly contingent upon the resonance frequency. Notably, the oscillator mass and spring constant are two parameters that directly influence the vibratory response of the microbeam. In the context of forced vibrations, harmonic base excitation is considered as the input excitation, and the mechanical frequency response function is provided. The proposed system offers two distinct advantages for energy harvester systems: the creation of minimal resonance at lower values and the potential to manipulate the system's resonance toward a desired frequency spectrum.Article HighlightsModifying the boundary conditions of a cantilever beam with lumped-parameter system, can significantly change the behavior of the vibratory response.The boundary condition directly impact the resonance frequencies; which influences the maximum amount of harvestable voltage in vibration-based energy harvesters.Spring constant and mass of the lumped oscillator, are the key factors to alter the vibratory behavior and bandwidth of frequencies. Optimizing such mentioned parameters can help reaching to the maximum harvesting of energy.

Experimental and Numerical Studies on the Vibration of Hybrid Composite with an Edge Crack

The hybrid composite plate is increasingly used in marine, robotic arm, and other applications. Edge crack is the greatest common fault in the structural systems during its working time. In this research, the successful fabrications of a hybrid composite of epoxy-based composites reinforced with bamboo and glass fibers have been done to examine the vibration characteristics under free vibration. First-order shear deformation theory is used to study the fundamental natural frequencies of asymmetrically hybrid composite beam with edge cracks for cantilever beam boundary conditions. Both numerical (FEM) using ABAQUS software and experimental investigations are done for the vibration analysis of unidirectional bamboo/glass fiber epoxy hybrid composite (BGHC) beam with edge cracks using fracture mechanics theory. It is observed that implications edge crack with different crack depth and various fiber weight ratio parameters have a major effect on vibration analysis. Also, it is clear that the natural frequencies of the hybrid composite are significantly affected due to the fiber weight ratio and crack depth of the hybrid composite materials. So, crack lengths are playing a vital role in the dynamic or vibration characteristics of the hybrid composite materials. The results of natural frequencies have been shown that they are decreased with an increase in crack depth and an increase in the bamboo fiber weight. Thus, BGHC-2 made from 30% bamboo and 10% glass fiber can be a viable candidate for applications that will produce good vibration properties. Generally, the first-third natural frequency of the hybrid composite beam decreases within the increase of bamboo fiber and decreases in glass fiber. The composite materials having 60% of epoxy with 20% bamboo and 20% glass fiber, and 60% of epoxy with 10% glass and 30% bamboo fibers, are observed with a gradual decrease in the values of natural frequencies with respect to increasing in crack depth. A fine agreement was accomplished between the simulation and experimental results. Therefore, the present approach can be used to identify cracks for mechanical health monitoring by linking the variation in natural frequencies of the hybrid composite beams.

Free vibration and buckling analysis of functionally graded beams using the DMCDM

In this paper, the Dual Mesh Control Domain Method (DMCDM) is used to carry out the free vibration and buckling analysis of two-constituent Functionally Graded Beams (FGBs) with through-thickness material variation. The material composition (modulus and density) is varied continuously through the beam thickness according to a power-law. The Euler-Bernoulli beam model consisting of the displacements and bending moment as the primary nodal degrees of freedom (i.e., mixed model) and the Timoshenko beam model with the displacement degrees of freedom (i.e., displacement model) are formulated as eigenvalue problems using the DMCDM. The numerical examples presented consider typical beam boundary conditions and various power-law exponents and slenderness ratios. The accuracy of this method is demonstrated by comparing the numerical results with those available in the literature.

An exact method for free vibration of beams and frameworks using frequency-dependent mass, elastic and geometric stiffness matrices

The frequency-dependent mass, elastic and geometric stiffness matrices of an axially loaded Bernoulli-Euler beam are developed through rigorous application of symbolic computation. These three matrices are related to the corresponding dynamic stiffness matrix so that free vibration analysis of axially loaded beams and frameworks can be carried out in an exact manner by applying the Wittrick-Williams algorithm as solution technique. Representative results from the proposed theory are presented for different boundary conditions of beams and frameworks, carrying tensile and compressive loads. Comparative results from finite element method are also presented. The duality between the free vibration and buckling problems is captured in that when the compressive load in a beam or frame approaches its critical buckling load, the fundamental natural frequency tends to zero and thus, buckling can be thoughtfully interpreted as free vibration at zero frequency. The investigation has opened the possibility of including damping in free vibration analysis of beams and frameworks when applying the dynamic stiffness method.

An analytical model for seismic response analysis of electrical equipment-steel support structure

Electrical equipment mounted on supporting structures is typical in a substation but vulnerable to earthquakes from previous field investigations. To evaluate the seismic performance of such an electrical equipment-support coupling system, a modal space analytical model was developed using the vibration partial differential equation (PDE) with distributed parameters and joint nodes. In the analytical model, resonant frequencies and mode shapes were obtained through numerical algorithms based on the boundary condition of each segmented beam. Then seismic responses of the coupling system can be attained by the modal decomposition method and superposition principle. A full-scale shaking table test was conducted on a 550 kV bushing-support coupling structure, and the accuracy of the analytical model was verified by the experimental results. Furthermore, a ceramic breakage inside the connecting flange at the base cross-section of the bushing specimen was observed after the seismic qualification test, which means its seismic performance can not meet the demand in the IEC seismic design specifications. To figure out this problem, parametric analysis was carried out using the analytical model to study the influence of parameters of the supporting structure and flange connection on the seismic responses of the equipment. In light of the analytical results, halving the original flange's flexural rigidity can reduce the bottom stress of the bushing by about 20% without over-magnifying its top displacement for this bushing-support coupling system. Application of the presented analytical modelling method can assist in the seismic optimization design of other equipment-support coupling systems in substations.

Out-of-plane moving load response and vibrational behavior of sandwich curved beams with GPLRC face sheets and porous core

As a first endeavor, the out-of-plane free vibrational behaviors and moving load responses of sandwich curved beams with graphene platelets reinforced composite face sheets and porous core (GPLRC-FS-PC) are carried out. The governing equations are derived based on the first-order shear deformation theory (FSDT) using Hamilton’s principle, which are discretized by employing the differential quadrature method (DQM) and Newmark’s method in the spatial and time domains, respectively. To simulate the moving load using the DQM, the Heaviside function approach is utilized. The effective elastic properties of face sheets are estimated using Halpin-Tsai micromechanical model. The approach is validated by presenting convergence studies and accuracy verification followed by parametric studies. The numerical results indicate that by adding little amount of graphene platelets (GPLs) into the face sheets and core layer, the fundamental natural frequency and the displacement amplitudes under moving load significantly increases and decreases, respectively, regardless of the beam boundary conditions. It is revealed that the GPLs distribution patterns in face sheets and the core porosity distribution patterns affect the critical load velocities of the GPLRC-FS-PC sandwich curved beams. Also, it is shown that the frequency parameters strongly depend on the face sheet to core thickness ratio.

Contact behaviors involving a nanobeam with surface effect by a rigid indenter

In this study, we present a novel surface model that utilizes surface energy density to predict the surface effect in nanobeam contact problems. To address the issue of a finite-length elastic nanobeam being indented by a rigid cylindrical indenter, we propose an equivalent substitution method. This method allows us to formulate analytical relations between the load and contact half-width for two different boundary cases. The explicit expressions of the contact-zone width, the pressure distribution in the contact zone, the deflection outside the contact zone, and the load–displacement relation are obtained for the nanobeam with surface effect and are compared with classical results in detail. The results show that the influence of the surface effect is very significant for nanobeam contact behavior, especially when the half-width of the contact zone increases and the contact zone becomes two independent symmetric strips. It is also found that the length-height ratio of nanobeam and the end support conditions have a fairly obvious effect on the normalized pressure distribution, which deviates significantly from the one predicted by the classical results due to the surface effect. However, for a given beam length and indenter radius, the ratio of the width of the contact zone to the beam thickness is almost constant, independent of the indenter load and beam boundary conditions. Meanwhile, the model predicts that the contact pressure distribution after the normalization of the average indentation pressure is almost independent of the indentation load and beam boundary conditions, but obviously depends on the surface effect parameters. The present method and result should be helpful not only to the measurement of mechanical properties of the indentation nanobeam but also to the design of the nanobeam-based devices.

Electromechanical response of boron nitride nanosheet reinforced nanocomposite beam: A finite element study

The objective of the present study is to investigate the electromechanical response of a piezoelectric boron nitride nanosheet reinforced nanobeam accounting for the surface and the flexoelectric effects using finite element analysis. The finite element model was developed by using the size-based Euler–Bernoulli beam model, modified piezoelectricity theory, and Galerkin's weighted residual method. The boron nitride nanosheet-reinforced nanobeam was loaded with uniformly distributed load and point-loading conditions. Three common boundary conditions for beams such as clamped-free, simply supported, and clamped-clamped have been considered here. The electromechanical behavior of the boron nitride nanosheet reinforced nanobeam has been studied under the pure surface, pure flexoelectric, as well as combined surface and flexoelectric effects. It is observed that the integrated surface and flexoelectric effects are mainly responsible for enhancing the electromechanical performance of the nanobeam. For the thickness H = 20 nm, the maximum deflection of the nanobeam was reduced by ∼50% when both the flexoelectricity and surface effects are combined together for all the support conditions. Moreover, the circular cross-section beam becomes ∼30% stiffer than the rectangular cross-section beam under the integrated effect of surface and flexoelectricity under all loading conditions. Hence, the highly size-dependent surface and flexoelectricity must be explored in the accurate electromechanical behavior of the nanostructure. Furthermore, beam stiffness is highly influenced by the flexoelectric effect irrespective of the beam boundary conditions whereas the surface effect is largely reliant on the beam boundary conditions. This research work provides a methodology to design efficient boron nitride nanosheet reinforced nanostructures that may potentially be applied in the design and development of several nanoelectromechanical systems such as force and pressure-based nanosensors and actuators.

Optimization of a smart beam for monitoring a connected inaccessible mechanical system: Application to bone-implant coupling

This paper deals with the optimization of piezoelectric patches positioning on a beam attached to an inaccessible system. Based on the electro-mechanical coupling coefficients, which are calculated from the mode shapes curvatures, the problem is to provide an optimal positioning of the piezoelectric patches in order to target the modes sensitive to beam boundary conditions. Following the theoretical description of a beam instrumented with collocated piezoelectric patches, a placement optimization strategy is proposed. This strategy lies on the definition of utility functions, based on modal weights and modal sensitivities, calculated from the coupling coefficient. An experimental validation of the method is performed on a concrete case study corresponding to real-time implant stability monitoring. A beam is temporarily rigidly fixed to the implant during its insertion into the bone cavity to allow impaction by the surgeon. The positions of the sensors on the beam are optimized to focus on the beam’s modes carrying information on the bone-implant interface, the biomechanical issue being the real-time maximization of implant stability. The originality of the approach lies on the sensor placement optimization on the beam connected to the implant to maximize or minimize simultaneously the amplitude of several modes on the frequency response function at resonances, depending on their sensitivity to the bone-implant interface. The results show very good performance of the piezoelectric placement strategy proposed in this paper, paving the way for new applications of piezoelectric patches design and placement on structures.

Time-dependent dynamic response of cylindrical 2D-FG spinning microbeam based on different high-order beam theories applying the GDQEM and Newmark-beta techniques

The nonlocal influence on the cantilever constructions varies significantly. Some researchers feel that nonlocal cantilever structures predict stiffness hardening features; others argue that nonlocal cantilever models cause dynamic softening. In this regard, the vibration behavior of a spinning cylindrical cantilever imperfect functionally graded beam structure is analyzed using several mathematical models based on the nonlocal strain gradient theory. More specifically, higher-order shear deformation theories are investigated in the current study. In addition, the effect of boundary condition modeling is presented. Boundary conditions could be modeled with nonlocal effects or as a classical boundary condition. The governing equations are solved using a technique called generalized differential quadrature element (GDQE) method. The findings of a frequency analysis and a dynamic deflection study, both of which are impacted by a variety of diverse factors, are presented and analyzed. Specifically, the effects of boundary condition model selection are investigated in detail. The findings proved that nonlocality in boundary conditions caused the softer behavior of the cylindrical beam structure by decreasing the natural frequency. On the other hand, using different beam models and boundary conditions could result in a complete difference in obtained results.

The estimation of Poisson’s ratio by time-averaging and Cornu’s method for isotropic beams

Prior research suggests that direct static (e.g., uniaxial testing) and dynamic (e.g., ultrasonic wave propagation analysis) measurements differ in the estimation of Poisson’s ratio because anisotropies and heterogeneities in the sample material affect the two types of tests differently. Even assuming isotropic and homogeneous material properties, prior research further suggests that discrepancies between static/dynamic test results will exist because the error of the diagnostic techniques for the measurand are inherently different. Finally, thermodynamic effects are not present in static tests but can significantly affect dynamic test results. Given the potential for all these variables to produce discrepancies, it would be helpful to have the measurement of Poisson’s ratio obtainable from the same theory and experimental measurements by either static or dynamic testing methods. Our finite element calculations show that by combining time-averaged scanning digital holography with Cornu’s method, it is theoretically possible to estimate the effective Poisson’s ratio from the anticlastic contours at the antinode of the first out-of-plane bending mode shape. This is true regardless of frequency and therefore applicable for both static and dynamic measurements. Our results show that the estimate of Poisson’s ratio by Cornu’s method using data from simulations of mode shapes approaches the true value of Poisson’s ratio. In addition, our research suggests that beam geometry and boundary conditions are fundamental factors limiting the convergence of the estimate of Poisson’s ratio to the true value of Poisson’s ratio regardless of performing a static or dynamic test.

Exact series solutions of composite beams with rotationally restrained boundary conditions: static analysis

The structural performance of composite beams is sensitive to load distribution as well as actual boundary conditions. Although the composite beam theory has been solidly established and exact solutions have been readily developed for various loading and boundary conditions, almost all of them are limited to classical boundary conditions (free, pinned and clamped) and there has been little discussion about the actual support conditions in real structures. The general representation of actual boundary conditions can be defined as rotationally restrained edges. In this research, an analytical model with exact series-type solutions was developed for composite beams with rotationally restrained edges. The model and displacement solutions were validated by other analytical methods and numerical results. The influence of rotational and end-slip restraints was investigated. It was found that the deflection and interface slip of composite beams are highly affected by both restraints introduced by the actual boundary conditions. The current model can be used as a benchmark for future design methods considering the realistic boundary conditions of composite beams.

Rotation-Induced Geometrical Stiffening of a Tapered, Pretwisted Blade

A linear geometrical stiffening model is developed for a rotating beam with an asymmetric cross section undergoing coupled stretch–bending–torsion motion. The model is verified for a twisted and tapered Timoshenko beam mounted on a rotating hub with a presetting angle. The floating frame of reference formulation is adopted where the configuration of the deformable beam is described by using different coordinate systems. Three coordinate systems, viz., inertial reference frame attached to the center of the rigid hub, local reference frame attached to an arbitrary point on the elastic axis of the undeformed beam, and body reference frame attached to the point on the elastic axis of the deformed beam, are defined. The global position vector of an arbitrary point on the beam is specified using a coupled set of reference and elastic coordinates. Reference coordinates define the location and orientation of a body reference frame, and elastic coordinates describe the beam deformation about the body reference frame. For a rotating beam problem, reference and elastic coordinates give the angular displacements of the hub and elastic deformations of the beam, respectively. The expressions for the kinetic and strain energy of the deformable beam are derived. The inertial coupling between the reference and elastic coordinates in the kinetic energy expression is identified as a gyroscopic coupling term. Further, the inertial mass matrix, spin softening matrix, and centrifugal stiffening matrix are obtained from kinetic energy expression. The elastic stiffness matrices, which are independent of elastic coordinates, are derived from strain energy expression. According to the Rayleigh–Ritz method, the elastic coordinates are expressed using a series of admissible functions that satisfy the geometric boundary conditions of a fixed-free beam. The ordinary differential equations of motion are then derived using Lagrange’s equations. A set of dimensionless parameters is identified after the nondimensionalization of the equations of motion. Due to the skew-symmetric nature of the gyroscopic coupling matrix, the eigenvalue problem is not amenable to classical modal analysis. Therefore, a complex modal analysis method is used to determine the modal characteristics. The results from the present model are verified with those available in the literature. The results are also compared with those obtained from the three-dimensional finite element method using ANSYS. The effect of these dimensionless parameters on the modal characteristics of a rotating blade is studied.

Selection of Radial Basis Functions for the Accuracy of Meshfree Galerkin Method in Rotating Euler-Bernoulli Beam Problem.

In this work, the radial basis function approximations are used to improve the accuracy of meshfree Galerkin method. The method is applied to the free vibration problems of non-rotating and rotating Euler–Bernoulli beams. The stiffness and mass matrices are derived by using conventional methods. In this meshfree method, only six nodes are considered within a single sub-domain. The parameters are varied for different approximations; the results are obtained with different approximations and found accurate. Two new basis function have been developed which are relatively accurate than conventional basis function: the first new basis function is obtained by multiplication of linear function to radial basis function and second new basis function is obtained by multiplying cubuic radial basis function to Gaussian radial basis function. The first few modes show same result that is available in literature using finite element method and higher modes are found very accurate as well. The result are found to be more accurate for first three modes of non-rotating and rotating Euler–Bernoulli beams where the cantilever beam boundary conditions are used; the first three modes do not change with the change in the parameter c of radial basis function.

Applicability and efficacy of Galerkin-based approximation for solving the buckling and dynamics of nanobeams with higher-order boundary conditions

In this research effort, the general nonlocal elasticity theory is applied to an axially-loaded Euler-Bernoulli nanobeam with varying boundary conditions. In applying the general nonlocal elasticity theory, a modified version of Eringen's nonlocal elasticity theory, a sixth-order partial differential equation is obtained for the transverse governing equation of motion. Obtaining the sixth-order equation created the need for additional non-classical beam boundary conditions. These boundary conditions are determined using the weighted residual approach and are physically interpreted as higher-order force and moment terms. This study investigates the buckling and dynamic responses of nanobeams using a Galerkin-based approximation with different trial mode shapes, as it relates to the exact solution. As such, comparisons are made between using three distinct types guess functions in the approximation, namely, the exact unloaded classical mode shapes, exact unloaded Eringen nonlocal mode shapes, and exact unloaded general nonlocal mode shapes. Unloaded mode shapes refer to the mode shapes in which the applied axial load is neglected or set to zero. Additional studies are conducted focusing on direct comparison between Eringen nonlocal and general nonlocal beams for both equal and unequal nonlocal parameters. It is shown that selection of an appropriate guess function that satisfies all boundary conditions is necessary for the approximate solution. Additionally, the response of Eringen and general nonlocal nanobeams with equal nonlocal parameters and hinged-hinged boundary conditions is identical using both the exact and approximate solutions. Finally, it is demonstrated that the general nonlocal dynamic response follows a different trajectory than the Eringen nonlocal dynamic response as a function of the applied axial load. The findings of this investigation may be used by other researchers to simplify analysis procedures and to better understand the fundamental relationships existing in the size-dependent governing equations and boundary conditions for nanobeam-based systems.