Fixed-fixed Boundary Conditions Research Articles

Evaluation of the Modal Parameters of a Unidirectional Carbon-Based Composite Structure Using the Influential Factor of Static Loading.

Static loading can significantly alter the dynamics of unidirectional carbon-based composites (UCBCs), with modal parameters varying depending on the orientation of the carbon fibers. In this study, the sensitivity of modal parameters of UCBC structures under uniaxial static loading was investigated. The theoretical static load influential factor was derived from a linearized UCBC model and corresponded to the transformed decoupled response over the mass-normalized static load. Three rectangular UCBC specimens (carbon fiber orientation of 0°, 45°, and 90°) were prepared under fixed-fixed boundary conditions using a jig fixture. Uniaxial static loads between 0 N and 1000 N were applied, and the first three modes of the UCBC specimens were analyzed. An isotropic SUS304 specimen was used as a reference. The linearization assumption about the UCBC structure was preliminarily validated with the Modal Assurance Criterion (MAC). A high influential factor was found for the UCBC specimen when carbon fibers were aligned with the static load direction at the first two resonance frequencies. Therefore, the proposed influential factor is an efficient indicator for determining the sensitivity of the dynamic response of a UCBC structure over a static load case. The variations in the influential factors for the UCBC specimens were more pronounced than for the isotropic specimens.

Efficient computational homogenisation of 2D beams of heterogeneous elasticity using the patch scheme

Modern ‘smart’ materials have complex heterogeneous microscale structure, often with unknown macroscale closure but one we need to realise for large scale engineering and science. The multiscale Equation-Free Patch Scheme empowers us to non-intrusively, efficiently, and accurately predict large scale, system level, solutions through computations on only small sparse patches of the given detailed microscale system. Here the microscale system is that of a 2D beam of heterogeneous elasticity, with either fixed–fixed, fixed–free, or periodic boundary conditions. We demonstrate that the described multiscale Patch Scheme simply, efficiently, and stably predicts the beam’s macroscale, with a controllable accuracy, at finite scale separation. Dynamical systems theory supports the scheme. This article points the way for others to use this innovative systematic non-intrusive approach, via a developing toolbox of functions, to model and compute accurately macroscale system-levels of general complex physical and engineering systems.

Free Vibrational Analysis of Non-uniform Double Eulerbernoulli Beams on a Winkler Foundation Using Laplace Differential Transform Method

In this study, the free vibrational analysis of non-uniform Euler-Bernoulli double beams on a Winkler foundation under simply supported and fixed-fixed boundary conditions is examined. The governing equation is solved using the Laplace Differential Transform Method, the combined form of the Laplace transform and differential transform technique (DTM). The accuracy of the method used is demonstrated by comparing the natural frequencies obtained using LDTM with previously published results available in the literature. It is discovered that for non-uniform double Euler-Bernoulli beams on a Winkler foundation with fixed-fixed end conditions, the natural frequencies are higher than those of simply supported end conditions. It is also observed that as the non-uniformity of the cross section of the beam increases, the natural frequencies reduce. Hence, it is suggested that the non-uniformity of the cross-section of the beam for a simply supported end condition be between 0 and less than 0.8. While the fixed-fixed end condition should have a value between 0 and 0.95. Journal of Engineering Science 14(1), 2023, 111-121

Blast failure and energy analysis of rubber-modified carbon-fiber vinyl-ester composite laminates

Modifications in vinyl ester resins, a popular matrix resin choice in naval applications through rubber additives, are being experimentally investigated in the current study. Toughening of the composite matrix resin using additives has been a popular choice, where it has been postulated that the inherent modifications to the base resin polymer would enhance the overall response of the composite structure to dynamic loadings. This work intends to understand the effect of such matrix modifications on the shock response of composite panels and the failure mechanism of a composite plate subjected to air shock loading. The study comprises material fabrication, resin and composite material characterization, and analysis of composite panels' response and damage evaluation under a range of shock loadings. A shock tube apparatus is used to generate controlled shock on composite laminates with fixed-fixed boundary conditions. In conjuncture, a high-speed 3-D DIC camera system is used to obtain full-field deformation and strain fields on the specimen's back face (furthest from the loading face). Based on the structural response and energy analysis, the effect of rubber modification is quantified and compared with non-modified resin composites, where the change is shown to improve the structural damping and energy dissipation under below-failure shock loadings and further shows relatively limited crack growth under high-intensity shock loadings.

Torsional analysis of open thin-walled doubly symmetric beams under torsion and bimoment

The present paper investigated the static analysis for the coupled torsional-warping response of open thin-walled doubly symmetric beams subjected to various twisting moments and bimoments. The static equilibrium torsional-warping coupling differential equations and associated boundary conditions for the thin-walled beams are derived using total potential energy variational principle. The exact closed-form solutions for the beams having a cantilever, simply-supported, fixed-fixed and fixed-fork boundary conditions and under various torsional and warping moments are determined. The present solution is shown to successfully capture the coupled torsional-warping static response of the thin-walled beams with various boundary conditions and by excluding and including axial static force effects. The validity and the accuracy of the numerical results based on the present solution are accomplished throughout the numerical examples presented and compared with well-established ABAQUS finite beam model solution and other exact solution available in the literature.

MODAL FREQUENCY ANALYSES OF THE VARIABLE STIFFNESS MECHANISM DESIGN OF THE SOFT ROBOTIC SYSTEM

In this study, mechanism design and numerical analysis are investigated and directly simulated through additively manufacturing materials of the thermoplastic polyurethane TPU and ABS. Systematic motion planning of humanoid arm systems improved concerning the designed soft robotic arm-like variable stiffness system through the completed novel methodology for the analysis. Soft robotics variable stiffness for mechanism design is a novel research area. Additionally, the humanoid arm-like variable stiffness mechanism herein is taken as a case study in this technology. The variable stiffness types for soft robotics are inflatable robotic technology, smart materials technology, mechanism technology, and a combination of them. The variable stiffness mechanism has a novel design opportunity via the boundary conditions and the orientation of the initial conditions for soft robotics. The relation between the boundary conditions and variable stiffness is not investigated sufficiently. The novel field of study completed herein, the soft robotics variable stiffness, is a fundamental investigation for further development in the mechanism design. Variable stiffness mechanisms can transmit the translational and rotational motions into required directions with required displacements and applied forces on the multibody systems. The stiffness for the fixed-free structural constraint boundary condition of the specified initial condition orientation is 8 Nm/rd compared to the stiffness value of the 65.6 Nm/rd fixed-fixed end boundary condition.

EXPERIMENTAL STUDY ON BEHAVIOR OF UNPROTECTED FOAMED CONCRETE FILLED STEEL HOLLOW COLUMN UNDER FIRE

Reduction in self-weight and achievement of full fire resistance requirements are some of the important considerations in the design of high-rise structures. Lightweight concrete filled steel tube (CFST) column provides an alternative method to serve these purposes. Recent studies on lightweight CFST columns at ambient temperature have revealed that foamed concrete can be a beneficial and innovative alternative material. Hence, this study investigates the potential of using foamed concrete in circular hollow steel columns for improving fire resistance. A series of nine fire test on circular unfilled hollow and foamed concrete filled hollow section column were carried out. ISO 834 standard fire exposure test were carried out to investigate the structural response of these columns under fire. The main parameters considered are load level and foamed concrete density; foamed concrete density used are 1500 kg/m3 and 1800 kg/m3 at 15%, 20%, and 25% load level. All the columns tested are without any external fire protection, with concentrically applied load under fixed-fixed boundary conditions. The columns dimension was 2400 mm long, 139.7 mm diameter and steel tube thickness of 6 mm. The fire test result showed that foamed concrete increases the fire resistance of steel hollow column up to an additional 16 minutes. The improvement is more at load level above 15%, and the gain in fire resistance is about 71% when 1500 kg/m3 density foamed concrete is used. Generally, foamed concrete filled steel hollow column demonstrate a good structural fire behavior, based on the applied load and foamed concrete density. Also, inward local buckling was averted by filling the steel hollow column with foamed concrete. General method for composite column design in Eurocode 4 adopted to calculate the axial buckling load of 1500 kg/m3 foamed concrete filled columns. These type of columns can be used for structures like airports, schools, and stadiums; taking the advantage of exposed steel for aesthetic purpose and high fire resistance. It can also be used for high rise structures; taking advantage of high fire resistance and reduction in self-weight of a structure.

Dynamic response of steel-reinforced concrete-filled circular steel tubular members under lateral impact loads

Abstract In this study, dynamic response of steel-reinforced concrete-filled circular steel tubular (SRCFST) members was experimentally studied under lateral impact loads. Eighteen SRCFST specimens were prepared and tested by the drop hammer impact test system. Various parameters, namely, cross-section of profiled steel, impact velocity and impact direction for I section profiled steel were considered to examine their influences on the failure mechanism, impact force, impact duration, displacement response and energy absorption. Since the beams may be of different constraints at boundary and the columns are axially loaded, boundary condition and axial load level were also considered. The experimental results show that the energy dissipation ratios (Eab/Ek) of all tested specimens except CIFP2-0 are more than 85% and the maximum energy dissipation ratio reaches 97.5%, indicating SRCFST members can absorb large amounts of input energy. Moreover, the inserted profiled steel can effectively enhance impact resistance while reduce energy absorption. It is also found that the peak value (Fmax) of impact force is very sensitive to the variation of impact velocity. The plateau value (Fplateau) of impact force and displacement response are very sensitive to the variation of boundary conditions and axial loads, which can significantly change the lateral flexural stiffness of specimens. Given the same impact energy input, SRCFST members with cruciform section profiled steel under fixed-fixed boundary conditions exhibit the best impact resistance performance with the lowest mid-span displacement.

Control of Beam Vibrations using Viscoelastically Damped Absorber System

The aim of this paper is to study the use of viscoelasically damped vibration absorber systems to optimally control the vibrations of a fixed-fixed beam. In this paper, a main beam with fixed-fixed boundary condition with viscoelstically damped cantilever beam as absorber is taken for the analysis. The paper includes optimum design & analysis of vibration absorber with viscoelastic damping. The equations of motion of the system have been derived to find the vibration response with absorber and used for optimization of parameters. The classical and Den Hartong optimization methods are used to optimize the design parameters and optimum values of design are found out. Theoretical calculations have been done for a fixed-fixed beam with three types of absorber beams (undamped, unconstrained treated damped absorber beam and constrained treated absorber beam). To validate the theoretical calculations, experiments have performed and deviation from theoretical data is discussed.

Surface energy effect on nonlinear free axial vibration and internal resonances of nanoscale rods

This study aims to investigate effects of the surface energy parameters on the nonlinear axial vibration and internal resonances of nanoscale rods. The surface energy parameters considered are the surface density and the surface Lamé constants. The nonlinear governing equation of motion is derived using the Hamilton's principle which includes the strain energy and the kinetic energy of the nanorod surface and bulk. The strain and kinetic energies of the nanorod bulk are obtained using the classical theory of elasticity and those of the nanorod surface are obtained using the surface elasticity theory. Then, the nonlinear governing equation of motion is solved using the method of multiple scales and natural frequencies are obtained for fixed-fixed and fixed-free boundary conditions. Effects of the surface energy parameters on the frequencies are examined for various cases. It is observed that the type of effect of the surface energy parameters depends on their sign. It is also seen that the type of effect of the surface energy parameters on nonlinear frequencies may become different at higher frequency numbers. The observations of this study can be a useful reference for more accurate design and fabrication of nano-electro-mechanical systems in which rod-shaped nano-structures, carbon nanotubes, play a significant role.

Surface stress effects on the mechanical properties of silicon nanowires: A molecular dynamics simulation

A primary challenge to use silicon nanowires as a truly potential building block in nanoscale devices is the implementation of scale effects into operational performance. Therefore, surface stress effects—as a direct result of size reduction—on transport properties became a major field of study. Previous computational simulations have focused so far on geometrical parameters with symmetrical cross sections, while silicon nanowires with nonsymmetrical cross sections are the major result of top-down fabrication techniques. A recent study has drawn a new aspect on the role played by the surface stress with a torsional profile on silicon nanowires to address the existing controversy from experimental and computational studies. Motivated by its success, the implications of this surface stress profile on the tensile properties of silicon nanowires are studied through molecular dynamics simulations. Deformation associated with the surface stress is computed for different length-to-thickness and width-to-thickness ratios. Then, tensile properties are investigated for a constant strain rate. Atomic calculations are carried out on silicon nanowires along the ⟨100⟩ crystal orientation for fixed-fixed and fixed-free boundary conditions. A combination of compressive uniaxial surface stress and torsional surface stress contributes to the mechanical behavior of silicon nanowires. A transition on elastic properties is obtained through changing the cross section from square to rectangular configuration. Further to addressing the controversy regarding the contribution of the surface stress on the mechanical properties, limits associated with available analytical approaches are highlighted for silicon nanowires.

Analysis of sensor placement in beams for crack identification

Abstract A previous study has shown that the mode shapes of a beam are more sensitive to damage than other vibrational parameters, thus making them better suited for crack identification purposes. However, they have the disadvantage of being more difficult to be measured. To overcome this difficulty, an interesting idea is to monitor changes produced by cracks on the mode shapes only in a few strategic points, instead of performing a complete experimental modal analysis. Considering this possibility, the aim of the present work was to determine the most appropriate locations for installing sensors in beams in order to identify and characterize structural damages. The effect of different locations of cracks on the mode shapes of beams was studied through a numerical (computational) model using the finite element model. The results were plotted in 3D graphs relating the relative nodal displacement of damaged and intact beams with the crack position and the location of the point analyzed. Through the analysis of these graphs, it was possible to point out the most adequate sites for placing sensors aiming at identifying cracks in a beam in fixed-free and fixed-fixed boundary conditions. Aiming at testing the results, an optimization problem for crack identification was proposed and solved through genetic algorithm (GA). The cracks were identified with an accuracy that is appropriate for engineering applications, showing that the proposed method is effective and could be used in Structural Health Monitoring (SHM) issues. Limitations on its use were also discussed.

Experimental modal analysis of curved composite beam with transverse open crack

Curved composite beams are the most widely preferred structures due to their high strength, low weight, corrosion and heat resistance. The aim of this study is to examine the modalities and natural frequencies of curved cracked composite (CCC) beams under different boundary conditions. Numerical and experimental free vibration analyses of curved composite beams with transverse cracks of various depths and locations are presented. Fixed-Fixed and Fixed-Free (cantilever) boundary conditions are used for both numerical and experimental studies. It is observed that the natural frequency values of the modes decrease as the crack depth increase depend on the crack position in the material.

Instability of imperfect non-uniform shallow arch under uniform radial pressure for pinned and fixed boundary conditions

The instability of imperfect non-uniform circular shallow arch under radial pressure for both pinned-pinned and fixed-fixed boundary conditions is discussed in the article. The imperfection includes the symmetric initial displacement inducing no stress and the non-uniformity which represented by three piecewise constant-stiffness segments. Analytical theoretical formulation is derived by the least potential energy principle. Through mathematical analysis, it's shown rigorously that the effect of imperfection still doesn’t vanish as the mode number of imperfection approaching to infinity. The stiffer end case and stiffer center case are discussed concerning equal potential energy load and snap-through criterion. Distinct snap-through behavior is observed for stiffer end case with fixed boundary condition by a proposed index plot. Moreover the two limiting cases including rigid end case and rigid center case are investigated by employing the augmented potential energy with Lagrangian multipliers considering the imperfection effect. Snap-through criteria are obtained in a closed form for rigid center case. And for the rigid end case with pinned BC, when the center segment's size is tending to zero, an asymptotical solution is derived and by this solution, the importance of multiplicity of roots of some equations governing the snap-through behavior is illustrated clearly. Besides, the equal potential energy load of two limiting cases is analyzed in detail with emphasis on imperfection effect. This paper aims to extend the understanding of imperfect non-uniform circular shallow arch under radial pressure.

Closed-Form Formulas for Hyperbolically Tapered Rotating Disks Made of Traditional Materials under Combined Thermal and Mechanical Loads

Using the true temperature distribution along the radial coordinate, closed-form formulas are offered for readers to study the thermo-mechanical behavior of variable thickness disks having both convergent and divergent hyperbolic thickness profiles made of conventional materials. Internal and external pressures, centrifugal forces and thermal loads due to the differences in prescribed surface temperatures are all considered with three boundary conditions: free-free (circular annulus), fixed-free (a disk mounted on a rotating shaft at the inner surface), and fixed-fixed (mounted on a rotating shaft at the inner surface and cased at the outer surface) boundary conditions. A parametric study is also conducted in almost real working environment in which the outer surface of the disk has considerably higher temperature rather than the inner surface. The thermomechanical linear elastic response of a hyperbolic mounted rotating disk subjected to the external pressure induced by blades is originally handled by those proposed formulas.

Dynamic analysis of microbeams based on modified strain gradient theory using differential quadrature method

ABSTRACTDynamic analysis of microbeams based on the modified strain gradient elasticity theory (MSGT) is carried out in this study. MSGT theory comprises additional material length scale parameters to effectively capture the size effect. Beams with fixed-fixed, simply supported and fixed-free boundary conditions are analysed. Additionally, frequency analysis for beams based on modified coupled stress and continuum theory is also presented by neglecting one or more length scale parameters. Results obtained for various theories in the present analysis are compared with those available in literature. Differential quadrature method (DQM) is employed to perform the analysis. Two different techniques are presented for implementing different boundary conditions of beam. It is shown that frequencies obtained from the strain gradient theory are higher when compared to the frequencies predicted by modified coupled stress theory and classical theory, when beam thickness becomes comparable to the length scale paramet...

Exact and Nonlocal Solutions for Vibration of Axial Lattice with Direct and Indirect Neighboring Interactions

This paper investigates the effect of long-range interaction forces in the longitudinal vibration of an axial linear elastic lattice. The eigenfrequencies of this axial lattice with direct and indirect linear elastic interactions are expressed in closed-form solutions for fixed-fixed and fixed-free boundary conditions. The paper starts with the two-neighbor interaction problem, and then generalizes the result to the more general N-neighbor interaction problem. Starting from a continualization of the higher-order difference equations, a nonlocal elastic continuum is constructed to approximate the behavior of the axial lattice with generalized short and long-range interactions. It is shown, while preserving the definite positiveness of the generalized lattice strain energy, that the associated nonlocal continuum is equivalent to a stress gradient type of Eringen's nonlocal model, which is able to capture the main scale phenomena of the generalized lattice. The length scale factor of the nonlocal model is calibrated to the size of the influence domain for the long-range interaction. The paper ends with a discussion about the effect of the discrete interaction kernel on the sensitivity of the eigenfrequencies with respect to scale effects. Exact analytical natural frequencies are calculated for discrete power law or exponential-based kernels. The stress gradient nonlocal theory offers an efficient engineering continuum framework, which accurately fits the response of generalized lattice with N-neighbor interaction.

Collapse behavior of non-uniform shallow arch under a concentrated load for fixed and pinned boundary conditions

The collapse behavior of a non-uniform circular shallow arch is analytically studied for both fixed-fixed and pinned-pinned boundary conditions. The non-uniformity is characterized by three piecewise constant stiffness regions. The equilibrium equations are derived based on the least potential energy principle and the resulting solutions are presented by proper non-dimensionalization in a form independent of total length of the whole arch by identifying two modified slenderness parameters. Detailed parametric study of both stiffer center case and stiffer end case has been conducted to investigate the influence of various geometric parameters and distinctly different collapse behaviors are found for different boundary conditions. Some criteria are derived to identify the snap-through buckling. Moreover two limiting cases of extreme non-uniformity(rigid end case and rigid center case) are studied via the augmented potential energy with introduced Lagrangian multipliers. Through an asymptotical analysis, a finite modified slenderness parameter is shown to suffice to ensure the occurrence of snap-through buckling for rigid end case with pinned boundary conditions when the dimension of center region is approaching to zero. It is shown rigorously that the same conclusion holds even when there is some rotational restraint at the pinned ends. For rigid center case, closed-formed criteria are derived for snap-through buckling. Finally analysis has been carried out on the case where one part of the arch has large bending rigidity but moderate stretching rigidity. This paper aims to enhance the understanding of collapse behavior of non-uniform shallow arch under a central concentrated load for various boundary conditions.

Analytical Solution of Stiffness for a Corner-Fillet Leaf-Spring Type Flexure Hinge with a Long Fatigue Life

Flexure hinges as the displacement guiding and amplifying mechanism or sensing component are widely used for micro-actuators and sensors. However, the existing flexure hinges, leaf-spring or notch type, cause serious stress concentration which severely weaken the fatigue life of compliance mechanism. Therefore, developing long fatigue life flexure hinges is very important for high working frequency actuators and sensors, such as fast-tool-servo. Corner-fillet leaf-spring type flexure hinge could provide large displacement with lower stress. Stiffness expressions of it with both fixed-fixed and fixed-guided boundary conditions are derived by using Castigliano’s theorem. The main influence factors for stress concentration are investigated and the formulas of stress concentration factor are obtained in terms of ratio of fillet radius to the minimum thickness. These analytical formulas have been verified by comparing with finite element analysis (FEA) results. Stress-life method is chosen to research the influence of fillet radius on fatigue life and the results indicate fillet radius can improve fatigue life of flexure hinge effectively. The proposed analytical solution is the fundamental of optimal design of a leaf-spring type flexure hinge based mechanism with fatigue life constraints.

Shear-compression tests of URM walls: Various setups and their influence on experimental results

Current design codes provide empirical equations for the drift capacity of unreinforced masonry (URM) walls that are based on results of quasi-static cyclic shear-compression tests. Yet different experimental campaigns have used various approaches of imposing fixed-fixed boundary conditions at the wall top which may affect the test results. This article investigates, by means of numerical simulations, the influence of experimental setups on the force and drift capacities of in-plane loaded URM walls subjected to double-fixed conditions. It is shown that controlling the shear span or the top rotation while keeping the axial force constant leads to very similar results. Controlling the axial elongation at the top of the wall results for walls subjected to a small axial load ratio in an increase and for walls subjected to a large axial load ratio in a decrease in axial load with increasing drift demands. Testing a wall with half the height applying cantilever boundary conditions is not recommended as the stiff loading beam at the wall top changes the failure mode and leads to significantly larger drift capacities.