Euler Theory Research Articles

Buckling of multilayered laminated glass beams: Validation of the effective thickness concept

Stability of laminated structural glass is one of the design requirements to be considered due to the brittle and slender nature of this kind of glass elements. Since laminated glass is mainly manufacture with viscoelastic interlayers, its mechanical properties are temperature and time dependent. This implies that, i.e., the critical load of a laminated glass beam subject to constant compressive load decreases with time as well as with temperature.In this paper, the equations of the Euler Theory for buckling of monolithic beams are extended to multi-layered laminated glass beams using an effective stiffness. This proposal is based on the idea of calculating the thickness (time and temperature dependent) of a monolithic element with bending properties equivalent to those of the laminated one, that is, the deflections provided by the equivalent monolithic beam are equal to those of the layered model with viscoelastic core.In this work, the analytical predictions are validated by compressive experimental tests carried out on a simply supported beam composed of three glass layers and two polyvinyl butyral (PVB) interlayers. Moreover, a finite element model was assembled to validate the proposed methodology for any boundary conditions. The results shown that a good accuracy can be obtained with the proposed equations being the errors less than 7% for all the experiments and simulations considered.

Critical contrasting of three versions of vibrating Bresse–Timoshenko beam with a crack

This paper deals with natural frequencies of short (or thick) beams taking into account effects of rotary inetria and shear deformation. Three versions of the Bresse–Timoshenko beam theory are considered and contrasted in the context of the problem of beam vibration in presence of the single crack, namely the (a) original Bresse–Timoshenko theory, (b) Elishakoff's truncated version and (c) the proposed slope inertia based version. The original Bresse–Timoshenko equations contain fourth order derivative with respect to time that is not present in the classical Bernoulli–Euler theory whose correction was the declared goal of S.P. Timoshenko who himself did not overestimate the importance of this term. The truncated version of the Bresse–Timoshenko was suggested by Elishakoff with no fourth order time derivative incorporated in the theory. In this paper, variational derivation of the truncated version of the Bresse–Timoshenko beam theory is conducted with attendant additional term containing the fourth derivative with respect to axial coordinate and second derivative with respect to time. It appears imperative to conduct thorough comparison of the above three theories in the context of beams containing a crack which is often present in engineering structures. The applicability of these three versions with attendant extensive numerical calculations is specifically studied for the titled problem. It is demonstrated that the original Bresse–Timoshenko beam theory, as an overcomplicated theory, is not needed for crack analysis. Simpler theories, namely the truncated version of this theory or the one based on slope inertia are sufficient for crack analysis in all treated cases.

The Differences in the Shape of Characteristic Curves of a Column Subjected to Eulers Load Obtained on the Basis of Bernoulli- Euler and Timoshenko Theories

The Differences in the Shape of Characteristic Curves of a Column Subjected to Eulers Load Obtained on the Basis of Bernoulli- Euler and Timoshenko Theories

The effect of foam core density at various slenderness ratios on axial strength of sandwich panels with glass-FRP skins

Abstract In this study, the effect of the density of a polyisocyanurate (PIR) foam core on the behavior of axially loaded sandwich panels with glass fibre reinforced polymer (GFRP) skins are examined, for different panel heights. Thirty-six test specimens with a cross-sectional foam core area of 100 × 50 mm2 and symmetrical 100 mm wide GFRP skins were fabricated with heights of 750, 1000, 1250, and 1500 mm to examine various slenderness ratios (kLe/r). Three PIR foam core materials with densities of 32, 64 and 96 kg/m3 were used. An out-of-straightness assessment was first conducted to ensure that eccentricities were within acceptable limits. All samples were tested to failure in axial compression with pinned end conditions. It was shown that doubling and tripling the core density led to increases in peak load of 71 and 170%, respectively, due to the enhanced composite action and reduced shear deformations. On the other hand, slenderness had insignificant impact on peak loads, contrary to what one would expect from Euler's theory. The reason being the variety of governing failure modes. Columns with low (kLe/r) of 22–25 failed largely due to localized single skin buckling, while those with high (kLe/r) of 51–61 failed primarily by global buckling followed by secondary core shear failure with skin debonding. Columns with intermediate (kLe/r) experienced a combination of localized and global failure modes.

Stability and free vibration analyses of orthotropic 3d beam–columns with singly symmetric section including shear effects

A complete beam–column classification and the corresponding characteristic equations for the stability and undamped natural frequencies of 3D orthotropic Timoshenko beam–columns with singly symmetric closed section and with elastic end connections subjected to an eccentric end axial load are presented and derived using three different approaches. The first two approaches are those by Engesser and Haringx that include the shear component of the applied axial force proportional to the slope (du/dx and dv/dx in planes xz and yz, respectively) and to the angle of rotation of the cross-section (θx and θy in planes yz and xz, respectively) along the span of the member, respectively. The third approach is a simplified formulation based on the classical Euler theory that includes the effects of shear deformations but neglects the induced shear component of the applied axial force along the member. The proposed methods and characteristic equations are capable of determining the critical axial loads and undamped natural frequencies of beam–columns with elastic end connections. Four comprehensive examples are included that show the effectiveness and simplicity of the proposed method and the results obtained are compared with experimental results available in the technical literature. It is shown that: (1) the natural frequencies and critical axial loads of beam–columns increase as the shear stiffness GAs, the degrees of fixity and lateral bracings at the ends of the member increase; (2) the natural frequencies calculated using the three approaches are identical to each other when the applied axial load is zero; (3) the critical axial load in compression using the Engesser approach is lower than the one obtained using the Haringx approach; (4) the critical axial loads in compression are highly affected by the degree of flexural fixity at the supports, but those in tension are not affected much; and (5) the Haringx approach is the only one among the three approaches capable of capturing the phenomena of tension buckling observed in seismic isolators.

НЕЛИНЕЙНЫЕ СТАЦИОНАРНЫЕ ИЗГИБНЫЕ ВОЛНЫ В СТЕРЖНЕ ИЗ УПРОЧНЯЮЩЕГОСЯ МАТЕРИАЛА

In this paper nonlinear bending waves in an infinite straight rod made of material whose hardening is described by Hollomon - Ludwik power-law are considered. The process of bending is described by Bernoulli - Euler theory. The equations of rod's dynamics were obtained using the Hamilton - Ostrogradsky principle based on expressions of potential and kinetic energy for the elementary portion of the rod. The search of solutions was performed in the form of travelling stationary waves, which allowed reducing the partial differential equation to ordinary one. Investigation of phase portraits of obtained equation was performed using the MAPLE software. Possibility of existence of nonlinear travelling stationary waves, which propagate with constant velocity and maintain its initial shape, were determined. Also the character of dependences between the length of stationary wave and its amplitude, hardening index and strength index is established. Keywords: bending waves, nonlinearity, work hardening, Hamilton − Ostrogradsky principle, phase portraits.

Buckling of laminated-glass beams using the effective-thickness concept

Structural stability is one of the design requirements in laminated-glass beams and plates due their slenderness and brittleness. In this paper the equations of the classical Euler theory for buckling of isotropic monolithic beams are extended to laminated-glass beams using the effective thickness and the effective Young modulus concepts. It is demonstrated that the dependency of the effective stiffness on boundary conditions can be considered using buckling ratios of Euler theory corresponding to isotropic linear monolithic beams. The analytical predictions are validated by compressive experimental tests in simply supported beams. Fixed boundary conditions are difficult to reproduce in experimental tests due to the brittleness of the glass and for this reason fixed–fixed and fixed–pinned boundary conditions were validated using a finite element model.

Fluid flow analysis of drooping phenomena in pump mode for a given guide vane setting of a pump-turbine model

The energy-discharge characteristics of pump-turbines in pump mode with a hump region are significantly im- portant for operating stability. To investigate the flow characteristics, 3D steady numerical simulations are conducted for a given guide vane opening of 32 mm by solving Reynolds-averaged Navier-Stokes (RANS) equations using the shear-stress transport (SST) k-ω turbulence model. Based on the validation of computational fluid dynamics (CFD) results using experimental bench- marks, the part-load (0.45φBEP), drooping zone load (0.65φBEP), near best efficiency point (BEP) (0.90φBEP), BEP (1.00φBEP), and overload (1.24φBEP) regions are chosen to analyze how and why the fluid properties change in the runner. The causes of flow separation and spatial characteristics of flow at different load points are obtained through the analysis of flow angle and hydraulic losses. The results show that flow angle at the leading and trailing edge from the crown to the band distributes differ- ently among these five operating points. Then, the reasons for drooping are investigated based on the Euler theory. It is found that drooping behavior comes from both the incidence/deviation effect and frictional losses. In addition, the runner losses are more consequential to drooping as shown by hydraulic loss analysis.

Theoretical prediction and experimental verification of shape control of beams with piezoelectric patches and resistive circuits

This paper presents one-dimensional analytical, three-dimensional numerical and experimental results for shape control of beam-type structures. The beam structures under consideration are assumed to be slender and the surface of the elastic substrate is covered by several piezoelectric patches whose electrodes are connected by a resistor network. In general, the structure will deform, if an external load acts on the system and no control action is present. Therefore, the question arises how to choose the network configuration, in order to completely suppress deflections or vibrations at several desired locations along the beam axis. This aim is also known as shape control. As a practical example, a cantilever beam subjected to a tip-force is considered. First we present shape control results for the one-dimensional extended Bernoulli–Euler theory and three-dimensional finite element calculations with ANSYS for static loads and time-harmonic excitations. Then the theory is verified by an experimental setup and it is shown by several frequency sweep excitations and by a monofrequent harmonic excitation close to the first eigenfrequency that shape control may be achieved approximately, proving the applicability of the proposed control method.

Stability of imperfect columns with nonlinear connections under eccentric axial loads including shear effects

An analytical method that evaluates the lateral stability and induced bending moments, shears, and second-order deflections in columns with initial geometric imperfections (i.e., initial curvature, out-of-plumb, and axial load eccentricities) and nonlinear semirigid connections subjected to axial load at both ends and lateral load at the top end is presented including the effects of shear forces and corresponding strains using three different approaches. The first two approaches are those by Haringx and Engesser that include the shear component of the applied axial force proportional to the angle of rotation of the cross section and to the total slope the along the member, respectively. The third approach is a simplified formulation based on the classical Euler theory that includes the effects of shear deformations but neglects the shear component of the applied axial force along the member. The proposed equations can be applied to columns with sidesway uninhibited, partially-inhibited and totally-inhibited. The proposed model is limited to prismatic columns subjected to eccentric axial loads at both ends and a lateral load at the top end causing bending about one of the principal axes with inelastic behavior taking place at the end rotational connections only. Five classical column cases and sensitivity studies are presented and discussed in detail. Analytical results indicate that the stability and second-order response of a column are highly affected by its initial imperfections, low shear stiffness relative to the bending stiffness, the stiffness of the lateral bracing, and by any changes in the stiffness and strength of the end connections.

A novel measurement of intersegmental forces exerted on human lower limbs during forward falling and regaining balance motions

This study proposes an analysis system for evaluating the intersegmental forces exerted on human lower limbs. In existing analysis systems, the ground reaction force exerted on the feet is generally measured using force plates fixed on the motion trajectory, and the intersegmental forces are then estimated using an inverse dynamics approach. However, force plates are inconvenient and expensive. Accordingly, the present study proposes a method for evaluating the intersegmental forces without the need for force plates. In the proposed approach, the supporting and striding legs are modeled as a seven-link manipulator system, and the intersegmental forces is then derived using Newton–Euler theory. It is found that the estimated results are in good agreement with the actual measurement datum. Thus, the basic validity of the proposed analysis system is confirmed. For application, the intersegmental force would be the major reference datum to design the artificial joints.

Buckling of Cu–Zr-based metallic glasses nanowires: molecular dynamics study of surface effects

The phenomenon of buckling in Cu45Zr45Al10 metallic glass nanowires with different slenderness ratios was studied by means of molecular dynamics simulation. The values of critical stress versus slenderness ratio for two nanowire diameters were obtained. We analyzed the results within the framework of the modified Euler theory of buckling, obtaining values for the surface elastic modulus and the residual surface stress for the two different diam- eters. Our results show that the Cu45Zr45Al10 metallic glass in nanometric size become stiffer and exhibits a lower Young's modulus than that of a bulk sample.

Resolving issues with member buckling in truss topology optimization using a mixed variable approach

In this paper, several issues related to member buckling in truss topology optimization are treated. In the conventional formulations, where cross-sectional areas of ground structure members are the design variables, member buckling constraints are known to be very difficult to handle, both numerically and theoretically. Buckling constraints produce a feasible set that is non-connected and non-convex. Furthermore, the so-called jump in the buckling length phenomenon introduces severe difficulties for determining the correct buckling strength of parallel consecutive compression members. These issues are handled in the paper by employing a mixed variable formulation of truss topology optimization problems. In this formulation, member buckling constraints become linear. Parallel consecutive members of the ground structure are identified as chains, and overlapping members are added to the ground structure between each pair of nodes of a chain. Buckling constraints are written for every member, and linear constraints on the binary member existence variables disallow impractical topologies. In the proposed approach, Euler buckling as well as buckling according to various design codes, can be incorporated. Numerical examples demonstrate that the optimum topology depends on whether the buckling constraints are derived from Euler's theory or from design codes.

Analytical model for estimating intersegmental forces exerted on human lower limbs during walking motion

Existing methods for evaluating the intersegmental forces acting on the human lower limbs during walking generally measure the ground reaction force exerted on the feet using force plates and estimate the intersegmental forces using an inverse dynamics approach. However, force plates are inconvenient and expensive. Accordingly, the present study proposes an approach for evaluating the intersegmental forces without the need for force plates by modeling the human lower limbs as a seven-link manipulator system. In the proposed approach, the intersegmental forces are derived directly from the analytical model using Newton–Euler theory based on the posture information obtained from a system of gyroscopes and accelerometers during walking motion. The results obtained from the analytical model are compared with the experimental data obtained using load cells. It is found that the estimated results are in good agreement with the measured results. Thus, the basic validity of the proposed analysis system is confirmed.

THERMOELASTOPLASTIC AND VIBRATION CHARACTERISTICS OF COMPOSITES WITH INTERPHASES

ABSTRACTThis paper addresses the problem of predicting the behaviors of a compositematerial, which consists of a matrix and a spherical inclusion coated by a multilayeredinterphase, under thermal and/or mechanical loading variations and basedon the micromechanics principles. The multi-layered interphase, which in generalincludes different properties for each layer, is modeled by applying the multiinclusionmodel. The considered damage mode, which is represented by theprogressive debonding of the particle from the interphase, is assumed to becontrolled by a critical energy criterion and the Weibull’s distribution function. Theeffects of the interphase parameters such as its thickness and the properties of eachlayer on the effective thermomechanical properties and the vibration characteristicsof composites with multi-layered interphases are presented and discussed. Thenatural frequencies give information about the resonance avoidance whereas modeshapes give information about observability and controllability of different structures.Therefore, the natural frequencies and mode shapes as vibration characteristics areinvestigated based on Euler and Timoshenko theories.

Dramatic Effect of the Utilized Theory on Frequency Distribution Function of Carbon Nanotubes

In this study, we investigate the modal density of double-walled carbon nanotubes. Emphasis is placed on the effect of the utilized theory. Specifically, we compare the modal density obtained via classical Bernoulli–Euler theory with that obtained by employing the refined Bresse–Timoshenko theory with nonlocal and surface effects taken into account. We show that the effect of refinements is dramatic.

A novel director-based Bernoulli–Euler beam finite element in absolute nodal coordinate formulation free of geometric singularities

Abstract. A three-dimensional nonlinear finite element for thin beams is proposed within the absolute nodal coordinate formulation (ANCF). The deformation of the element is described by means of displacement vector, axial slope and axial rotation parameter per node. The element is based on the Bernoulli–Euler theory and can undergo coupled axial extension, bending and torsion in the large deformation case. Singularities – which are typically caused by such parameterizations – are overcome by a director per element node. Once the directors are properly defined, a cross sectional frame is defined at any point of the beam axis. Since the director is updated during computation, no singularities occur. The proposed element is a three-dimensional ANCF Bernoulli–Euler beam element free of singularities and without transverse slope vectors. Detailed convergence analysis by means of various numerical static and dynamic examples and comparison to analytical solutions shows the performance and accuracy of the element.

Range of validity of an extended WKB theory for atmospheric gravity waves: one-dimensional and two-dimensional case

AbstractA computational model of the pseudo-incompressible equations is used to probe the range of validity of an extended Wentzel–Kramers–Brillouin theory (XWKB), previously derived through a distinguished limit of a multiple-scale asymptotic analysis of the Euler or pseudo-incompressible equations of motion, for gravity-wave packets at large amplitudes. The governing parameter of this analysis had been the scale-separation ratio$\varepsilon $between the gravity wave and both the large-scale potential-temperature stratification and the large-scale wave-induced mean flow. A novel feature of the theory had been the non-resonant forcing of higher harmonics of an initial wave packet, predominantly by the large-scale gradients in the gravity-wave fluxes. In the test cases considered a gravity-wave packet is propagating upwards in a uniformly stratified atmosphere. Large-scale winds are induced by the wave packet, and possibly exert a feedback on the latter. In the limit$\varepsilon \ll 1$all predictions of the theory can be validated. The larger$\varepsilon $is the more the transfer of wave energy to the mean flow is underestimated by the theory. The numerical results quantify this behaviour but also show that, qualitatively, XWKB remains valid even when the gravity-wave wavelength approaches the spatial scale of the wave-packet amplitude. This includes the prevalence of first and second harmonics and the smallness of harmonics with wave number higher than two. Furthermore, XWKB predicts for the vertical momentum balance an additional leading-order buoyancy term in Euler and pseudo-incompressible theory, compared with the anelastic theory. Numerical tests show that this term is relatively large with up to$30\hspace{0.167em} \% $of the total balance. The practical relevance of this deviation remains to be assessed in future work.

Dynamics Models of Interacting Torques of Hydrodynamic Retarder Braking Process

Hydrodynamic retarder is a kind of assist braking device, which can transfer the vehicle kinetic energy into the heat energy of working medium. There are complicated three-dimensional viscous incompressible turbulent flows in hydrodynamic retarder, so that it is difficult to represent the parameters changing phenomenon and investigate the interactional law. In order to develop a kind of reliable theoretical model for internal flow field, in this study, the dynamics models of interacting torques between impellers and working fluid were constructed based on braking energy transfer principle by using Euler theory to describe the flow state in view of time scale. The model can truly represent the dynamic braking process.

Simplified description of out-of-plane waves in thin annular elastic plates

Dispersion relations are derived for the out-of-plane wave propagation in planar elastic plates with constant curvature using the classical Kirchhoff thin plate theory. The dispersion diagrams and the mode shapes are compared with their counterparts for a straight plate strip and the role of curvature is assessed for plates with unconstrained edges. Elementary Bernoulli–Euler theory for a beam of rectangular cross-section with the circular shape of its axis is also employed to analyze the wave guide properties of this structure in its out-of-plane deformation. The applicability range of the elementary beam theory is validated. The wave finite element method in the formulation of the three-dimensional elasticity theory is used to ensure that the comparison of dispersion diagrams is performed in the frequency range, where the classical thin plate theory is valid. Thus, the paper summarizes the effects brought to the propagation of out-of-plane waves in thin elastic plates by their constant curvature and the models of these plates.