Finite Deflection Research Articles

Synthesis, characterization, and modeling of gelatin-based magnetic hydrogel beams

Hydrogels are soft materials that can be designed to be responsive to external stimuli such as temperature, pH change, and electric and magnetic fields. Magnetically responsive hydrogels consisting of polymer, water, and magnetic nanoparticles (MNPs) are emerging candidates in biomedical fields due to their remote application. In this work, cobalt ferrite-based MNPs prepared by the auto-combustion method are embedded in gelatin hydrogels. Cobalt ferrite-based magnetic hydrogels (HCFO) having high water content are prepared using the solvent casting method. The effective magnetic properties, such as coercivity and remnant magnetization of the hydrogels, are determined from the magnetization response (m-h curve) obtained through a vibrating sample Magnetometer (VSM). Subsequently, the HCFOs are subjected to uniaxial tensile loading to obtain mechanical properties. The size and shape of MNPs are obtained using X-ray Diffraction (XRD) and scanning electron microscopy (SEM) analysis. Furthermore, the magnetic hydrogel beams (cantilever) have been prepared and subjected to the external magnetic field. A unique phenomenological model has been proposed which can predict the finite deflection of the beam under magnetic loading exploiting the hard magnetic behavior of the HCFO. The proposed model takes into account particle concentration, finite elasticity, and magneto-mechanical coupling. The softness of the hydrogels matrix and hard magnetic properties of the MNPs results in large deflection. Further, the experimental observations and modeling predictions are in good agreement.

Nonlinear Vibrations in Homogeneous Nonprismatic Timoshenko Cantilevers

Abstract Deep cantilever beams, modeled using Timoshenko beam kinematics, have numerous applications in engineering. This study deals with the nonlinear dynamic response in a nonprismatic Timoshenko beam characterized by considering the deformed configuration of the axis. The mathematical model is derived using the extended Hamilton’s principle under the condition of finite deflections and angles of rotation. The discrete model of the beam motion is constructed based on the finite difference method (FDM), whose validity is examined by comparing the results for a special case with the corresponding data obtained by commercial finite element (FE) software abaqus 2019. The natural frequencies and vibration modes of the beam are computed. These results demonstrate decreasing eigenfrequency in the beam with increasing amplitudes of nonlinear oscillations. The numerical analyses of forced vibrations of the beam show that its points oscillate in different manners depending on their relative position along the beam. Points close to the free end of the beam are subject to almost harmonic oscillations, and the free end vibrates with a frequency equal to that of the external force. When a point approaches the clamped end of the beam, it oscillates in two-frequency mode and lags in phase from the oscillations of the free end. The analytical model allows for the study of the influence of each parameter on the eigenfrequency and the dynamic response. In all cases, a strong correlation exists between the results obtained by the analytical model and ABAQUS; nonetheless, the analytical model is computationally less expensive.

Snap-Through Buckling of Hinged-Hinged Initially Imperfect Beams Undergoing Finite Deflections Subjected to Lateral Concentrated Midpoint Loads

The buckling and post-buckling behavior of initially imperfect, sinusoidally curved hinged-hinged beams undergoing very large deflections subjected to lateral concentrated loads acting at the midpoints is investigated via the geometrically nonlinear analysis of the problem. The transverse shear deformation is neglected. The amplitudes of the imperfection are considered to be relatively small. Therefore, the concerning curved beams (or arches) are shallow. The values of the extensional rigidity/length of the beams are assumed to be large enough not to permit considerable amount of change in the lengths of the beams during the deformation. The force-deflection curves corresponding to various amplitudes of the imperfection and the diagrams of the deflections and internal forces corresponding to various stages of the deformation, including those after the buckling, are presented. Not being able to solve the highly nonlinear problem analytically, numerical methods are used.

Corrosion of RC Beams During Time

Abstract This article presented results of impact corrosion on reinforce concrete beams during time with previous loading. Before testing the RC beams were loaded by 37% and 75% from bearing-capacity load of undamaged corrosion beams. There were tested two types of beams. First type beams were tested under force loading action and second type beams were tested under force loading and corrosion action. As an aggressive environment were used the 10% solution of sulfuric acid H2SO4. Such environment takes place in separate chemical manufactures, galvanic workshops, flue pipes of thermal power plants. The results of experimental studies have established that the effect of the aggressive environment with simultaneous loading significantly impairs the stress-strain state of reinforced concrete beams. Bearing capacity due to the yielding of reinforcement bars for specimens with the simultaneous action of aggressive environment and previous loading of 37% and 75% takes place on 51…53 and 58…60 days. The destruction of these specimens, by crushing compressive concrete, at previous loading 37% occurred after 75…79 and at previous loading 75% - after 79…88 days. The history loading does not impact significantly on beams failure, on finite deflections and timing exhaustion bearing capacity since the start of the simultaneous action of aggressive environment and loading.

Analysis of Vibration Characteristics of Transport Utility Vehicle by Finite Element Method

Present work deals with the design and analysis vibration characteristics for transport utility vehicle. The transport utility vehicle is designed using automotive industry standards. The dynamic behaviour of vehicle depends on the selection of overall dimensions, wheel base, track width, overall height and width that are decided using central motor vehicle rules. The selected dimensions for vertical and horizontal pillar members of the transport bus are modified to enhance the strength, stiffness and stability of the superstructure during travel. This increased stability enhances the ride comfort and passenger safety. Analysing the effect of utilizing manual meshing in complex areas of a transport utility vehicle for vibration analysis and passenger ride comfort has also been carried out. Modal analysis to evaluate the dynamic behaviour of transport utility vehicle model is also carried. Further with the use of finite element analysis deflection vehicle structure is evaluated. The outcomes from the analysis are compared with the behaviour of chassis mounted platform in dynamic conditions and are found in close correlation. The vehicle structure behaves as a single entity in dynamic situations, so surface model is prepared. Element selection for the finite element analysis is carried by considering plane stress condition. Two-dimensional quadrilateral shell elements are extensively used for meshing of the computer model of the vehicle structure. Complex areas in the optimised vehicle structure are meshed using relevant combination of quads and trias. The values of vector sum displacement and frequencies are found to be in good agreement with the experimental ones.

Stress Distribution Near a Circular Hole in a Flexible Orthotropic Cylindrical Shell of Elliptical Cross-Section

A numerical technique for solving geometrically nonlinear problems for an orthotropic cylindrical shell of elliptical cross-section weakened by a circular hole is developed. The system of governing equations is derived using the Kirchhoff–Love theory of deep shells and Hooke’s law for orthotropic materials. The technique employs the procedure of incremental loading, as well as the modified Newton–Kantorovich and finite-element methods. The effect of finite deflections and mechanical and geometrical parameters on the stress–strain state of the shell near the circular hole acted upon by axial tensile forces is analyzed.

Red blood cell simulation using a coupled shell-fluid analysis purely based on the SPH method.

In this paper, a novel 3D numerical method has been developed to simulate red blood cells (RBCs) based on the interaction between a shell-like solid structure and a fluid. RBC is assumed to be a thin shell encapsulating an internal fluid (cytoplasm) which is submerged in an external fluid (blood plasma). The approach is entirely based on the smoothed particle hydrodynamics (SPH) method for both fluid and the shell structure. Both cytoplasm and plasma are taken to be incompressible Newtonian fluid. As the kinematic assumptions for the shell, Reissner-Mindlin theory has been introduced into the formulation. Adopting a total Lagrangian (TL) formulation for the shell in the realm of small strains and finite deflection, the presented computational tool is capable of handling large displacements and rotations. As an application, the deformation of a single RBC while passing a stenosed capillary has been modeled. If the rheological behavior of the RBC changes, for example, due to some infection, it is reflected in its deformability when it passes through the microvessels. It can severely affect its proper function which is providing the oxygen and nutrient to the living cells. Hence, such numerical tools are useful in understanding and predicting the mechanical behavior of RBCs. Furthermore, the numerical simulation of stretching an RBC in the optical tweezers system is presented and the results are verified. To the best of authors' knowledge, a computational tool purely based on the SPH method in the framework of shell-fluid interaction for RBCs simulation is not available in the literature.

Tensile Membrane Action of Lightly-reinforced Rectangular Composite Slabs in Fire

A recently developed method of treating tensile membrane action of lightly reinforced concrete slabs, based on a rigorous treatment of the kinematics of movement of the yield-line mechanism, has been developed to consider composite slabs with unprotected downstand steel beams in fire conditions. The fire case differs from the enhancement of load capacity of slabs at ambient temperature in the respect that the applied loading is kept constant at a predetermined value, but the strength of the downstand beams progressively declines as their temperature rises. It is assumed that the concrete slab does not become hot enough in its active levels, within the duration of a fire, to reduce its strength. This extension to the method is derived systematically. It is seen that the yield line mechanisms of these slabs are aligned differently from those of the equivalent concrete slabs, so it is not valid to use the latter as the basis of a design calculation. The advantage of finite deflection due to tensile membrane action manifests itself as an enhancement of the steel beam temperature that can be sustained, above that at which the yield line mechanism forms. The peak enhancement occurs at the point at which reinforcing mesh begins to fracture progressively along diagonal yield lines. This fracture can be delayed and the peak temperature increased if the mesh ductility across the yield line cracks is increased by reducing the bond between bars and concrete, thus facilitating the bar-slip from the crack-faces. The effects of using meshes of different ductility classes, and both plain and deformed bars, are considered for composite slabs of different aspect ratios.

Creep of supersonic aircraft panels under the periodic action of airstream pressure

Irreversible deformation of supersonic aircraft panel under the aerodynamic heating and periodical action of airstream pressure is regarded. The distribution for airstream pressure which is obtained from the solution of aerodynamics problem, was simplified and presented as a cycle with triangular form of pressure increasing-decreasing in a part of period with definite constant static component. The mathematical statement of creep problem for thin shells of revolution loaded by normal pressure having constant and periodically varying components is presented. The theory of finite deflections is used. Equations are derived for the case of conical shell. The constitutive equations, which are obtained by use the method of asymptotic expansions of creep strain and scalar damage parameter with subsequent averaging in a period of stress varying, are used. The form of these equations is similar to Norton law - Rabotnov-Kachanov evolution equation with multiplayers represent the influence of periodic loading in each equation. Numerical simulation was performed by use of programming complex for thin shells of revolution creep calculations. It uses the Finite Element Method jointly with finite differences method for time integration. The example of numerical solution of the problem of deformation of heated cylindrical panel made from duralumin alloy and subjected to periodical loading from airstream puffs is presented. The distributions of stress-strain state components were obtained. The numerical results include distribution of normal deflection in the panel after 1000 seconds of flight as well as the von Mises stress relaxation curve. The analysis of obtained data shows the sufficient increasing of normal deflections as well as stress redistributions. The necessity of consideration of current deformed state of the panel in the problem of analysis of airstream flow is discussed.

Nonlinear deforming of laminated composite shells of revolution under finite deflections and normal’s rotation angles

The discretization of the boundary value problem for laminated composite shells is based on the finite difference approach using the regular mesh with the constant grid step and the difference operators of the second order of accuracy. The dynamic relaxation method is proposed for the solution of the nonlinear problem. The evolutionary equations of the dynamic relaxation are constructed, and the optimum parameters of the converging linear iterative process are estimated.

Bending Stress and Deflection Analysis of Meshing Spur Gear Tooth during the Single Tooth Contact with Finite Element Method and Determination of the Bending Stiffness

Purpose of this study is the study of loading and contact problems encountered at rotating machine elements and especially at tooth gears. Tooth gears are some of the most commonly used mechanical components for rotary motion and power transmission. This fact proves the necessity for improved reliability and enhanced service life, which requires precise and clear knowledge of the stress field at gear tooth. This study investigates three different study cases of the stresses occurring during the single tooth meshing, regarding the gear module, power rating and number of teeth as variable parameters. Using finite elements analysis, the stresses and deflections on discrete points of contact are derived. Finally from the finite elements analysis results calculated the peripherical bending stiffness of the loaded tooth. From FEM analysis and analytical calculation the magnitudes of root stresses, contact displacement and peripherical bending stiffness, during the single tooth contact, are presented with graphs versus the height of the contact to the total tooth height ratio. During the single tooth contact the values of the Equivalent and 1st principal stress at the addendum of the tooth, the bending deflection and the peripherical bending stiffness at the point of contact are proportional to the height of the contact in respect of the total tooth height.

Displacement dependent pressure load for finite deflection of doubly-curved thick shells and plates

The formulation of the displacement dependent pressure load for shells of generic shape and plates is derived in the present study in curvilinear coordinates without any approximation and any hypothesis on material properties. Derivation is initially carried out for pressure load on the middle shell surface, and then for load on the external and internal surfaces by making use of a third-order shear and thickness deformation shell theory, which is a significant improvement for thick shells. The explicit formulation in curvilinear coordinates allows immediate implementation in numerical codes. An approximate formulation is also derived and compared to (i) the exact formulation and to (ii) the displacement independent pressure load which is still widely used in the literature. Applications to circular cylindrical shells and rectangular plates are presented. Comparison of results show that only the exact formulation for displacement dependent pressure load can be used for large deformations of shells and plates, while the displacement independent pressure load can be used only in case of small displacements of the middle surface (or middle plane).

Combined effects of heterogeneity and non-linearity on the critical time parameters of orthotropic conical shells

The work is devoted to the non-linear dynamic stability of heterogeneous orthotropic truncated conical shells subjected to the combined static and time-dependent axial loads. The basic equations are derived using the finite deflection theory with von Karman–Donnell-type of kinematic non-linearity and reduced to a non-linear differential equation with the time variable coefficient using the superposition principle and Galerkin method. The resulting equation is solved numerically using Runge–Kutta method for variety of an axial loading speed, heterogeneity of features, orthotropic material properties and conical shell characteristics to obtain the non-linear critical time parameters. Finally, the influences of the axial loading speed, non-linearity, heterogeneity and orthotropy on the dimensionless critical time parameters are discussed in detail.

Advancements in subsea riser analysis using quasi-rotations and the Newton–Raphson method

Beam structures undergoing finite deflections and rotations in space have extensive application in the subsea industry particularly for the analysis of holistic systems with larger numbers of mooring and riser components. In using the finite element analysis approach, there is an increasing requirement for large element sizes which preserve accuracy with regard to the coupling of axial, bending and torsion response.The authors outline a method for improving the current state of practice for the analysis of riser systems. The approach draws on the convected coordinates method, Euler–Bernoulli beam theory, the principle of virtual work and the finite element method. Two quasi-rotation measures are developed including a quasi-material rotation definition for rotational deformation relative to the convected axis of a beam and a quasi-space rotation definition to deal with the path dependent nature of rotations in three dimensions.The novel aspect of this work is to relate the rate of change of the quasi-material rotation vector along the beam axis to a linear transformation of the beam axis rate-of-rotation vector through utilising the convected coordinates axes system. In this way, incremental values of quasi-material rotation are directly linked to incremental values of nodal quasi-space rotation and a global Newton–Raphson solution technique for interconnecting beam elements is straightforward to assemble.Furthermore, this leads to accurate definitions of coupled axial, bending and torque response for beams with significant deflection. The approach has particular advantages in the analysis of subsea riser sections. Also, the accuracy of the solution is preserved for a fewer number of elements compared to alternative solutions for computationally sensitive load cases with highly non-linear loading regimes.

A computational investigation for propagation of elasto-viscoplastic zones in the shock loaded circular plates

Purpose– The purpose of this paper is to introduce a computational time dependent modeling to investigate propagation of elastic-viscoplastic zones in the shock wave loaded circular plates.Design/methodology/approach– Constitutive equations are implemented incrementally by the Von-Kármán finite deflection system which is coupled with a mixed strain hardening rule and physical-base viscoplastic models. Time integrations of the equations are done by the return mapping technique through the cutting-plane algorithm. An integrated solution is established by pseudo-spectral collocation methodology. The Chebyshev basis functions are utilized to evaluate the coefficients of displacement fields. Temporal terms are discretized by the Houbolt marching method. Spatial linearizations are accomplished by the quadratic extrapolation technique.Findings– Results of the center point deflections, effective plastic strain and stress (dynamic flow stress) and temperature rise are compared for three features of the Von-Kármán system. Identifying time history of resultant stresses, propagations of the viscoplastic plastic zones are illustrated for two circumstances; with considering strain rate and hardening effects, and without them. Some of modeling and computation aspects are discussed, carefully. When the results are compared with experimental data of shock wave loadings and finite element simulations, good agreements between them are observed.Originality/value– This computational approach makes coupling the structural equations with the physical descriptions of the high rate deformation through step-by-step spectral solution of the constitutive equations.

Stress State of Flexible Composite Shells with Reinforced Holes

The nonlinear deformation of flexible composite shells with reinforced curved holes under static loading is studied. Problems of this class are formulated, and a numerical method for solving them with allowance for finite deflections is outlined. The distributions of stresses, strains, and displacements along the boundary of the hole and in the zone of stress concentration in an orthotropic cylindrical shell with a reinforced hole subjected to uniform internal pressure are examined. The results are analyzed and compared to the solutions of the corresponding linear elastic problems for a shell with a nonreinforced hole

Large deflection response of rectangular metal sandwich plates subjected to blast loading

Abstract A theoretical study is conducted to predict the large deflection response of fully clamped rectangular sandwich plates subjected to blast loading. Using the energy dissipation rate balance theory and a new yield criterion including the effect of core strength, we obtained the solutions for the dynamic response of rectangular sandwich plate, in which the effect of finite deflection are incorporated and elastic effect is neglected. Also, we obtain the new so-called ‘bounds’ of dynamic response by using the inscribing and circumscribing squares of the exact yield locus. Further, we neglect the effect of bending moment and obtained analytical membrane mode solutions for large deflection of the sandwich plate. Finite element numerical calculations are also carried out to study the dynamic response of the sandwich plate subjected to blast loading. Comparisons of the present analytical predictions and the present numerical and the previous experimental results are given and good agreement is found.

Stability of a shallow arch under action of deterministic and stochastic load taking into account nonlocalness of elasticity and damping

On the example of a shallow arch, the geometrically nonlinear problem concerned with oscillations of systems excited by a deterministic or stochastic load is considered. The dynamic stability problem is solved for the arch, made from micro- and nanomaterials. The equation, describing the motion of the Euler–Bernoulli arch, is based on the nonlocal elasticity theory and nonlocal damping, taking into account finite deflections. Asymptotic stability and almost sure asymptotic stability criteria involving a damping coefficient, structure and loading parameters are derived using the method of the maximum Liapunov exponent.

Frequency tuning, nonlinearities and mode coupling in circular mechanical graphene resonators

We study circular nanomechanical graphene resonators by means of continuum elasticity theory, treating them as membranes. We derive dynamic equations for the flexural mode amplitudes. Due to the geometrical nonlinearity the mode dynamics can be modeled by coupled Duffing equations. By solving the Airy stress problem we obtain analytic expressions for the eigenfrequencies and nonlinear coefficients as functions of the radius, suspension height, initial tension, back-gate voltage and elastic constants, which we compare with finite element simulations. Using perturbation theory, we show that it is necessary to include the effects of the non-uniform stress distribution for finite deflections. This correctly reproduces the spectrum and frequency tuning of the resonator, including frequency crossings.

Large-amplitude vibration of non-homogeneous orthotropic composite truncated conical shell

In this study, the large-amplitude vibration of non-homogenous orthotropic composite truncated conical shell is investigated. It is assumed that the Young’s moduli and density of orthotropic materials vary exponentially through the thickness direction. The basic equations of non-homogenous orthotropic truncated conical shell are derived using the finite deflection theory with von Karman–Donnell-type of kinematic non-linearity. Then, foregoing equations are solved using the Superposition principle, Galerkin and Semi-inverse methods and the frequency- amplitude relationship is found. Finally, carrying out some computations, the effects of non-homogeneity, orthotropy and conical shell characteristics on the nonlinear vibration characteristics have been studied.