Initial Curvature Research Articles

(119) Impact of Ethnicity on Clinical Outcomes of Intralesional CCH in Peyronie's Disease Patients Across Kaiser Permanente Southern California Database

Abstract Introduction Intralesional CCH injection is an effective treatment for Peyronie’s disease (PD) that utilizes collagenolytic enzymes to break down collagen plaque, thus reducing the penile deformity. The IMPRESS I and II clinical trials demonstrated efficacy of CCH in a predominant Caucasian population of men (95.8%). Little is known about treatment efficacy in the non-Caucasian population. Objective Using Kaiser Permanente’s ethnically diverse database, we sought to study the impact of race on CCH treatment outcomes. Methods This study examined the efficacy of intralesional CCH injection in a large healthcare system over 15 years. We defined treatment efficacy as percent change in penile curvature abnormality from baseline after a maximum of 4 treatment cycles. Impact of comorbidities on CCH efficacy was also examined. Patients with ventral curvature or initial curvature <30 ̊ or >90 ̊ were excluded from analysis. Standard statistical analysis was performed. Results 112 patients underwent CCH therapy and were included in this analysis. 46.4% of men completed all 4 cycles. Among men who received CCH, 66.1% were Caucasian, 22.3% Hispanic, and 9.8% African American. For co-morbidities, 87.5.% of men were not diabetic, while 22.3% of men had erectile dysfunction. The mean baseline pre-injection penile curvature was 51.63 degrees (range was 32-87 degrees). Caucasian men had the largest treatment response to CCH (45.6% decrease in penile curvature) compared to African American men (43.7%) and Hispanic men (40.5%), p = 0.9554. Non-diabetic men had a larger treatment response to CCH (44.9% decrease in curvature versus 39.3% in diabetics, p = 0.4726). Younger Men (age 18-44) also experienced a greater treatment response compared to other age groups (57.9% reduction in curvature, p = 0.171). There was a significant difference in number of cycles completed across different ethnicities (p = 0.0220). Hispanic men were significantly less likely to complete all 4 cycles of CCH compared to Caucasian and African American men (p=0.0004). Only 28.2% of Hispanic men completed all 4 treatment cycles. 48% of Hispanic patients report the need of language interpretation services during health appointments. Conclusions This is the largest post-market analysis of CCH treatment outcomes in a racially diverse PD population. Caucasian men had a greater treatment response after CCH injection possibly due to higher rate of completing all 4 treatment cycles. Nondiabetic men and younger men (aged 18 to 44) also had a greater treatment response after CCH injection. Hispanic men were significantly less likely to complete 4 cycles of injection therapy; possible explanations include language barriers. Future study is needed to determine if patient resources prepared in Spanish can improve treatment outcomes in Hispanic men with PD, and reduce these health care disparities observed in our study. Disclosure No.

Incobotulinumtoxin A and Yoga-like Isometric Exercise in Adolescent Idiopathic Lumbar Scoliosis—A Randomized Pilot Study

Background: Approximately 90% of scoliosis cases are adolescent-idiopathic (AIS). From the first appearance of scoliosis at 10–14 years of age until the age of 18, the spine is most vulnerable to deterioration; young, growing people are most susceptible to the worsening of one or more scoliotic curves. An effective non-surgical means of remediation would be welcome. Design: This was a randomized, controlled, two-arm study assessing the safety and efficacy of combining incobotulinum injections with yoga to reverse lumbar and thoracolumbar AIS. Methods: In a private clinic setting, non-pregnant, healthy 12–18 year-olds were either taught a symmetrical “placebo” yoga pose (control sub-group 1), performed the side plank (Vasisthasana) three times daily with a placebo injection (control sub-group 2) or performed the three-times-daily side plank with a botulinum injection (intervention group 3). Injection: For the injection, 33 IU of incobotulinumtoxin type A (Xeomin) was injected into the concave-side lumbar paraspinals and quadratus lumborum at L2–3 and the psoas muscle at L3–4, or participants were injected similarly with a placebo. Randomization was achieved using random.org. Objective: The objective was to determine whether the treatment of muscular asymmetry with botulinum toxin injections and side planks is safe and effective in AIS. Results/Outcome: Eleven intervention and thirteen placebo patients (Groups 1 + 2), who were 12–18 years old, completed the three-month study. Mean daily side plank time = 165 s. The mean initial lumbar curvature was 36.9 degrees (SD 14.36), (p < 0.0001); the mean Group 3 curvature at 3 weeks was 29.5 degrees (SD 14.23) (p < 0.0001); and the mean Group 3 curvature at 3 months was 26.0 degrees (SD 12.81). Onset vs. 3-month value: p < 0.0001. Harms were limited to one patient in Group 2 and one in Group 3, who complained of transient shoulder pain and supported themselves temporarily on their forearm instead of the palm of the extended hand. Conclusion: Muscle strength asymmetry appears to be relevant to AIS treatment. Incobotulinum injections combined with side planks performed with the convex side downward may be more effective in reversing lumbar AIS than placebo exercises or side planks and placebo injections.

Recovery of Interlaminar Tensile Stresses in Curved Laminates Subject to Biaxial Flexure: The Case of an Elastically Induced Curvature Purposely Misaligned with the Principal Directions of Initial Curvature

Many literature contributions deal with the evaluation at FE post-processing stage of the interlaminar normal stress distribution in moderately thick single or doubly curved laminates subject to flexure, based on through-the-thickness equilibrium relations. Most of the contributions, however, discuss test cases in which the principal directions of the load-induced elastic curvatures are aligned with the principal directions of the initial curvature; this condition is not general. In the present work, a simple post-processing methodology for the recovery of the interlaminar tensile stress is presented.The procedure retrieves the Interlaminar Tensile Stresses (ILTS) distribution in a doubly curved laminate modelled through common wise quadrilateral shell Finite Elements and is based on customary FE sub-model which inputs are the displacements and rotation calculated on the shell structure. At the moment the sub-model can't retrieves Interlaminar Shear Stresses but can be further enhanced to do it.In this paper the methodology is successfully applied to an open thin walled profile in torsion test case, in which the principal directions of elastic curvature are strongly misaligned with the principal directions of initial curvature.

Geometrically exact aeroelastic stability analysis of helicopter composite rotor blades in forward flight

In this paper, a sophisticated method for analyzing the geometrically exact aeroelastic stability of composite rotor blades in forward flight has been proposed. The unique advantage of the proposed method is to use the geometrically exact beam model with the updated Variational Asymptotic Beam Sectional analysis (VABS) for structural modeling to not only calculate the blade cross-sectional properties and blade spanwise deformations more accurately, but also take into account the effect of transverse shear deformation and blade initial twist and curvatures sophisticatedly. The Peters finite state airloads theory and the Peters-He finite state dynamic inflow model are coupled to calculate the three-dimensional unsteady airloads applied on the blade. The auto-pilot trim scheme is adopted to determine the blade pitch controls. The finite element spatial discretization method, the Newmark numerical integration method, the Newton-Raphson method and the moving-block analysis method are combined to solve the established geometrically exact aeroelastic equations and obtain the geometrically exact aeroelastic stability of composite rotor blades. The proposed method is validated by correlation with existing analytical results. The investigations show that the effect of transverse shear deformation on the aeroelastic stability of composite hingeless rotors in forward flight is non-negligible. Blade initial curvatures have significant effect on the aeroelastic stability of composite hingeless rotors in forward flight, which is not only related to the blade elastic couplings, but also to the advance ratio.

BIFURCATION STATE AND RATIONAL DESIGN OF THREE-LAYER REINFORCED COMPOUND CONE-СYLINDER SHELL STRUCTURE UNDER COMBINED LOADING

An analytical-numerical approach to solving the problem of state bifurcation in terms of local and overall stability of a three-layer cone-cylinder shell structure discretely supported by intermediate rings, in particular of modern launch vehicles, under static combined loading by external pressure, axial forces, and torque is proposed in the paper taking into account the stiffness parameters of the intermediate rings in the plane of the initial curvature and for torsion. Corresponding solving equations for the problem are ordinary differential equations of the sixth order (for a cylindrical compart- ment with constant coefficients and for a conical one with variable coefficients along the axial coordinate). Differential relations that determine the conditions of conjugation through the intermediate ring are used. For the numerical solution, the finite difference method is used with central finite differences of the third and second order at the inner points of the shell determination segments and at its ends, respectively, and the second order differences with one step backward or forward at the conjugation points through the ring. The agreement of the calculation results with the known data for three-layer conical and cylindrical shells is shown, as well as in the limiting case, it is done when passing to a single-layer compound cone-cylinder structure. For the considered class of cone-cylinder shell structures, boundary surfaces are constructed that separate the stability region of the structure being under study, depending on the geometric and stiffness parameters of the compartments, reinforcing elements, and the external load condition. The external load effect on the parameter of the post-critical wave formation for the structure under investigation is studied, provid- ing the visualization of the deformation behavior. The analysis of the calculation results has shown that this approach to solving the problem of bifurcation and equistability of the compound structure compartments in relation to the local and overall forms of protrusion allows choosing rational geometric and stiff- ness parameters of the shell components and force elements in terms of improving the weight characteristics of the structure.

Bracing requirements and design for a single column considering semi-rigid connections and initial curvature

The use of bracing has been widely recognized as an effective technique to increase the strength of columns by reducing their effective length. This paper presents an assessment of the bracing requirements for a semi-rigidly connected column, which is laterally braced at the mid-height. By utilizing the proposed half-length column model, the effects of column stiffness and initial curvature on the brace strength and stiffness requirements are investigated. The results indicate that the column’s initial curvature coefficient increases with a higher applied load, but decreases with increased column end connection stiffness. The required brace stiffness requirement stipulated in AISC 360-22 should be increased when considering the effect of column’s initial curvature. Following the design concepts of AISC 360-22 and CSA S16-19, equations for assessing the bracing requirements for a semi-rigidly connected column are proposed. These equations are validated against the finite element analysis results, demonstrating their ability to accurately assess the bracing requirements for semi-rigidly connected columns. These equations can be adopted in current engineering practice.

Propagation characteristics of astigmatic elliptic vortex beams in nonlocal media with anisotropic nonlinearity

The propagation characteristics of the astigmatic elliptic vortex beam (AEVB) in nonlocal media with isotropic and anisotropic nonlinearities are analytically and numerically investigated. Three different kinds of dynamics are discussed: AEVB in nonlinear isotropic media, elliptic vortex beam in nonlinear anisotropic media, and astigmatic circular vortex beam (ACVB) in nonlinear anisotropic media. The isotropic nonlinearity cannot support the formation of elliptic vortex solitons for any input power, and the initial phase-front curvature (IPC) will induce the pulsating behavior of the AEVB. For anisotropic nonlinearity, when the nonlinear anisotropic parameter (NAP) of the medium and the initial ratio of the beam width satisfy γx/γy = (w0y/w0x)2, a quasi-stable elliptic vortex soliton can be obtained, in which the vortex angular velocity and vortex rotation angle evolve periodically. Circular vortex soliton cannot exist in nonlinear anisotropic media, and the rotation of the ACVB can be controlled by adjusting the NAP and the IPC. In both isotropic and anisotropic nonlinear media, the rotation direction of the vortex core is opposite to that of the beam envelope.

Deformation of confined liquid interfaces by inhomogeneous electric fields and localized particle forces

HypothesisOil-water interfaces that are created by confining a certain amount of oil in a square shaped pixel (∼200 x 200 μm2 with a height of ∼10 μm) topped by a layer of water, have a curvature that depends on the amount of oil that happens to be present in the confining area. Under the application of an electric field normal to the interface, the interface will deform due to inhomogeneities in the electric field. These inhomogeneities are expected to arise from the initial curvature of the meniscus, from fringe fields that emerge at the confining pixel walls and, if applicable, from interfacially adsorbed particles. Modeling and ExperimentsWe model the shape of the confined oil-water interface invoking capillarity and electrostatics. Furthermore, we measure the initial curvature by tracking the position of interfacially adsorbed particles depending on sample tilt. FindingsWe found that the pixels exhibited meniscus curvature radii ranging from 0.6-7 mm. The corresponding model based minimum oil film thicknesses range between 0.7 and 9 μm. Furthermore, the model shows that the initial meniscus curvature can increase up to 76 percent relative to the initial curvature by the electric field before the oil film becomes unstable. The pixel wall and particles are shown to have minimal impact on the interface deformation.

A modified approach for a scaled boundary shell formulation in structural isogeometric analysis

In this contribution, an efficient modification of the scaled boundary finite element method for shells is presented to compute accurate results for h-refinement. While previous works on the scaled boundary shell formulation showed inaccurate and unstable results for fine meshes, this modification overcomes such problems by the assumption of a linear force distribution over the thickness. Furthermore, the shell formulation is derived in the framework of isogeometric analysis by discretizing the bottom surface of the shell as a reference surface. This leads to a solution for in-plane directions in a weak sense, while the scaling direction is solved analytically by a modified Padé expansion. Therefore, the normal vector on the reference surface is constructed and the shell is scaled along that leading to an extensible shell formulation incorporating a three-dimensional linear elastic constitutive relation. Since the isogeometric analysis is utilized, an exact derivation of the normal vector and its gradient is possible. Initial curvature locking and Poisson’s thickness locking are eliminated inherently while shear locking and volumetric locking are alleviated by mesh refinement. The power of the modified approach is presented by standard benchmark problems and compared to the results of the unmodified approach and standard shell formulations.

Helical buckling of coiled tubing with initial bending curvature in three-dimensional curved wellbores

In this paper, a new buckling model of coiled tubing (CT) in a three-dimensional curved wellbore is established, considering the influence of the curvature and toolface angle of the wellbore and the initial CT bending curvature. First, with the wellbore curvature and toolface angle, the CT buckling configuration is based on the lower or outer curved side of the wellbore. When the initial CT bending curvature is less than the critical bending curvature, the initial CT configuration in the wellbore undergoes sinusoidal buckling; otherwise, the CT undergoes helical buckling. Second, the buckling model is solved by the energy method. After solving this buckling model, the critical CT helical buckling load, the critical bending curvature, and the quantitative relationship between the axial load and pitch can be obtained. Third, the CT contact force calculation method under a helical buckling configuration in a three-dimensional curved wellbore is given by combining the buckling model and the beam-column model. The influence of the initial bending curvature is also considered. Finally, the calculated values of the buckling model and contact force calculation method were compared with existing experimental data and published models to validate the proposed model. Therefore, the methods presented in this paper can be used to not only predict helical CT buckling but also optimize downhole engineering design parameters considering the influence of the initial bending curvature in more realistic drilling wellbore conditions.

Risk Factors for Penile Fracture After Intralesional Collagenase Clostridium histolyticum in Peyronie’s Disease

ObjectivesTo evaluate risk factors leading to corporal rupture after Collagenase Clostridium histolyticum (CCh). Peyronie's disease (PD) is a fibrotic condition of the penis characterized by abnormal plaque formation. Intralesional CCh is the only FDA approved medication for PD, however it can lead to corporal rupture, a potential surgical emergency. MethodsWe retrospectively reviewed medical records from Veterans diagnosed with PD who were under treatment with CCh using the Veterans Administration Informatics and Computing Infrastructure (VINCI). Using International Classification of Diseases and Current Procedural Terminology codes, we identified men who suffered a corporal rupture after CCh. Individual charts were reviewed to determine potential risk factors and events leading to corporal rupture. ResultsWe identified 17,647 veterans who were diagnosed with PD, of which 8.7% (1,541) received at least one injection of CCh for PD. Of them, 0.7% (11/1,541) veterans suffered corporal rupture. Within these 11 patients, the median number of CCH injections was 6 with a median initial curvature of 35º. Fracture occurred at a median of 8 days after CCH injection. The majority of fractures were secondary to spontaneous erections or sexual intercourse. Finally, six patients had their fracture repaired surgically while the remaining were managed conservatively. ConclusionsMost fractures occurred within 2 weeks of CCh injections and were associated with sexual intercourse and spontaneous morning erections.

Supersymmetric higher-derivative models in quantum cosmology

We study the quantum cosmology of supersymmetric, homogeneous and isotropic, higher derivative models. We recall superfield actions obtained in previous works and give classically equivalent actions leading to second order equations for the bosons, and first order for the fermions. Upon quantization, the algebra of fermions leads to a multi-component state, which is annihilated by the Hamiltonian and supersymmetric constraint operators. We obtain asymptotic wave functions of the oscillatory type, whose classical limit corresponds to inflationary evolution, and exact exponential wave functions. We use the latter to derive probability distributions of the initial curvature that are compatible with those obtained using the non-supersymmetric model.

Stability of the Quermassintegral Inequalities in Hyperbolic Space

For the quermassintegral inequalities of horospherically convex hypersurfaces in the (n+1)-dimensional hyperbolic space, where nge 2, we prove a stability estimate relating the Hausdorff distance to a geodesic sphere by the deficit in the quermassintegral inequality. The exponent of the deficit is explicitly given and does not depend on the dimension. The estimate is valid in the class of domains with upper and lower bound on the inradius and an upper bound on a curvature quotient. This is achieved by some new initial value-independent curvature estimates for locally constrained flows of inverse type.

Hygromechanical study of a 16th century painted wooden panel: In-situ experiments to quantify the mechanical effect of the frame and the cradle

In order to prevent damage to painted wooden panels from the cultural heritage sector, curators and conservators must provide the hygroscopic and thermal conditions necessary for their conservation. Furthermore, the design of physical reinforcements such as frames or cradles has to be considered. In this paper, experiments were designed to determine the kinematic response of a painting to hygrometric variations both before and after remedial conservation treatments were undertaken. This study was carried out in the Fabre Museum, Montpellier on the panel painting “La Sainte Trinité couronnant la Vierge” (« The Coronation of the Virgin by the Holy Trinity »), which was placed in a climate-controlled showcase in a museum exhibition space. A glass panel on either side allowed visitors to see both sides of the panel during the experiment. At the same time, two stereoscopic devices allowed us to follow the relief, the shape deformation and surface strain of the panel on its front and back sides. First, the mechanical effect of the frame and cradle was studied. The majority of the deformation was found to be due to the cradle; initially the panel had a complex warp (-5 to 10 mm), decreasing to (-1 to +2 mm) when the cradle was removed. The continuous monitoring (two images per hour) was carrried out to assess the behaviour of the panel during imposed relative humidity (RH) variations of about 13 % RH (between 65 % RH for the maximum and 52 % RH for the minimum) for both the absorption and desorption phases. The global mechanical effect on the curvature was very clear with an opposite direction of deflection between absorption and desorption phases. A comparison of the behaviour of the panel with and without the cradle showed that the change in curvature was about twice as great (4 mm) without the cradle, and its direction moved from an initial slight negative curvature state to a positive one. In terms of strain on both the surfaces, we observed a horizontal shrinkage of about -0.1 % during the desorption phase and a swelling of about +0.1 % during the absorption phase for both the front and back sides of the panel. A further local study allowed us to assess the mechanical phenomenon for each board with the local determination of cupping angles and surface strains. This local study highlighted the change in behaviour of the panel without the cradle. On both sides, the local surface strains were decomposed into the mean strain and the deviation from the mean strain. The study highlighted the fact that the intensity of the mean strain (local swelling or shrinkage due to the hygroscopic changes) was the same for each configuration and was about 0.1 %. Nevertheless, in the case of the panel without the cradle, the imposed variation of relative humidity was smaller (8 % RH instead of 13 % RH), so proportionally the mean strain is higher compared to the case with a cradle. This study over five years has been success in terms of innovative development and multidisciplinary research.

Developmental models shed light on the earliest dental tissues, using Astraspis as an example

AbstractFossils of extinct jawless vertebrates are pivotal to deciphering the evolutionary paths that led to the various forms of the vertebrate skeleton. For example, Pteraspidomorphs (stem‐gnathostomes), such as the Ordovician Astraspis, display some of the oldest remains of bony and ‘dental’ (dentine and enameloid) tissues. However, the identification of the very nature of these early mineralized tissues has been hampered by a lack of unambiguous diagnostic characters. As development is key to identifying the derivation of these tissues, we developed an integrative and generic histogenetic model, testing several ontogenetic scenarios. We illustrate our approach on the basis of the well‐preserved Astraspis samples and show how this can be used to infer key developmental features from extinct species. This study suggests that in the odontodes of Astraspis: (1) the initial curvature of the epithelium was close to the shape of the final external surface; (2) the mesenchymal cells differentiate synchronously in the whole inner periphery; and (3) the capping tissue was produced by both mesenchymal and epithelial cells (enameloid rather than enamel). Astraspis specimens also provide evidence of a dual growth periodicity, possibly homologous to Andresen and von Ebner growth lines observed in amniotes, suggesting this type of dual periodicity may be shared by most vertebrates. We estimated that an Astraspis odontode grew up in around 60–70 days at a rate of 0.5–5 μm/day. The new developmental approach proposed in this study could be a robust framework for critically evaluating the tissues of extinct taxa in the future.

Three-dimensional nonlinear vibrations of slightly curved cantilevered pipes conveying fluid

In this paper, nonlinear static and dynamic behaviors of slightly curved cantilevered pipes conveying fluid under transverse base excitations are investigated. Incorporating geometrical and inertial nonlinearities, a new three-dimensional dynamic model is developed for the inextensible cantilevered pipe based on the Euler–Bernoulli beam theory. The resultant partial differential equations are discretized using the Galerkin method and solved utilizing the pseudo-arclength continuation technique and a time-integration method. The static equilibrium configuration, linear stability, and nonlinear response of the pipe are calculated for various cases, which allow us to investigate the effects of flow velocity, gravity coefficient, initial curvature amplitude and shape, and base-excitation amplitude and frequency. An emphasis is put on the importance of modal interactions on frequency-response and force-response relationships, evidencing the possible in-plane and out-of-plane transverse vibrations and identifying the preferred forms of pre- and post-instability behaviors. This study offers an efficient theoretical model for exploring the nonlinear forced response of slightly curved cantilevered pipes conveying fluid and provides a better insight for further investigations on the modal interaction in cantilevered pipe devices.

On the initial singularity and extendibility of flat quasi-de Sitter spacetimes

Inflationary spacetimes have been argued to be past geodesically incomplete in many situations. However, whether the geodesic incompleteness implies the existence of an initial spacetime curvature singularity or whether the spacetime may be extended (potentially into another phase of the universe) is generally unknown. Both questions have important physical implications. In this paper, we take a closer look at the geometrical structure of inflationary spacetimes and investigate these very questions. We first classify which past inflationary histories have a scalar curvature singularity and which might be extendible and/or non-singular in homogeneous and isotropic cosmology with flat spatial sections. Then, we derive rigorous extendibility criteria of various regularity classes for quasi-de Sitter spacetimes that evolve from infinite proper time in the past. Finally, we show that beyond homogeneity and isotropy, special continuous extensions respecting the Einstein field equations with a perfect fluid must have the equation of state of a de Sitter universe asymptotically. An interpretation of our results is that past-eternal inflationary scenarios are most likely physically singular, except in situations with very special initial conditions.

Theoretical analysis and a parametric study on forced vibration of large statically deformed curved beam with rigid links and moving support

Forced vibration characteristics around large statically deformed configuration of curved beam system with rigid links and moving support are analyzed. In addition, the paper also investigates various parametric effects of height to span ratio and roller radius to height ratio coupled with support connection mechanism and profile shape, on dynamic characteristics of the curved beam system. The subject problem includes two interrelated problems: determining deformed configuration under static load and dynamic response of the loaded beam under harmonic excitation. The static problem is analyzed incrementally through the variational principle-based energy method in body-embedded curvilinear frame considering geometric nonlinearities due to combined bending-stretching and non-uniform initial curvature. In each incremental step, the nonlinear governing equation is solved iteratively, and kinematic constraints due to interactive deformation of the flexible curved beam with rigid links are imposed to obtain the global solution. Concerning the statically deformed configuration, governing equation for forced vibration is derived through Hamilton’s principle in curvilinear frame. Nonlinear effects caused by combined bending-stretching modes of vibration, non-uniform curvature, and lumped mass of the dead static load are considered. The set of inhomogeneous governing equations is solved through Broyden’s method. A direct substitution technique through successive relaxation is also employed to obtain an initial guess solution. Besides the forced vibration, free vibration analysis at static deformed configuration is also carried out for the completeness of the study. The theoretical model is validated through comparisons of loaded natural frequency results with finite element package. After the validation study, the effects of the mentioned geometric parameters on the dynamic response of the curved beam system are investigated and presented suitably. The physically insightful framework along with the parametric study may facilitate simulation and design of several practical structures involving moving support and rigid links subjected to continuous excitation around large statically deformed configurations.

Validity of Galerkin Method at Beam’s Nonlinear Vibrations of the Single Mode with the Initial Curvature

A common strategy for studying the nonlinear vibrations of beams is to discretize the nonlinear partial differential equation into a nonlinear ordinary differential equation or equations through the Galerkin method. Then, the oscillations of beams are explored by solving the ordinary differential equation or equations. However, recent studies have shown that this strategy may lead to erroneous results in some cases. The present paper carried out the following three research studies: (1) We performed Galerkin first-order and second-order truncations to discrete the nonlinear partial differential integral equation that describes the vibrations of a Bernoulli-Euler beam with initial curvatures. (2) The approximate analytical solutions of the discretized ordinary differential equations were obtained through the multiple scales method for the primary resonance. (3) We compared the analytical solutions with those of the finite element method. Based on the results obtained by the two methods, we found that the Galerkin method can accurately estimate the dynamic behaviors of beams without initial curvatures. On the contrary, the Galerkin method underestimates the softening effect of the quadratic nonlinear term that is induced by the initial curvature. This may cause erroneous results when the Galerkin method is used to study the dynamic behaviors of beams with the initial curvatures.

Stability of Individual Phases in the Elastic Matrix of a Composite

A methodology for analytical stability assessment of individual compressed flexible phases of discretely heterogeneous composite elements of engineering structures is proposed to ensure their reliable safe operation and minimise the consequences of emergencies. Based on the energy balance equation for the proportional forces of a normal resistance of the medium to the phase displacement, expressions are obtained for determining the critical force, taking into account the possible initial curvature (non-rectilinearity) of a single phase in the elastic medium of the matrix for three possible cases of stability loss. The approbation of the results of the study on the example of deformation features of reinforcing bars in a compressed reinforced concrete column is presented.