Closed-form Analytical Approach Research Articles

Bending theory of composite pressure vessels: A closed-form analytical approach

Composite pressure vessels are strategic for the present and future of the aerospace, transportation, and energy industries. These shell structures are utilized to store and transport liquids and gases and are principally composed of a cylindrical shell and two elliptical heads. The current rules for the structural design of composite pressure vessels make use of the membrane theory that excludes the bending of the structure. Nevertheless, the cylindrical shell and the elliptical heads present different curvature radii. These changes generate a localized bending deformation that determines an increase in the stress state as an effect of a local bending moment and shear force distributions that arise in the geometric transition zone. The stresses related to the bending effects die out rapidly, moving away from the junction plane, but they are sensibly higher than the membrane stress and, therefore, cannot be neglected in the mechanical assessment. Here, a closed-form analytical solution for the bending theory of composite shells is proposed. It can assist in the design of composite pressure vessels and drive the design towards reliable configurations for structural integrity. Using the Donnell–Mustari–Vlasov theory for thin-walled composite shells, the governing fourth-order differential equation is deduced and solved to obtain the displacements and stresses acting in the junction. The analytical results are successfully compared with those of reference finite element models for validation; their accuracy is proved considering different thicknesses, lay-ups, and flattening factors of the vessel

Closed-Form Analytical Approach for Calculating Noise Contours of Directive Aircraft Noise Sources

This paper extends the simplified airport noise model Rapid Aviation Noise Evaluator (RANE) [Torija et al., Journal of the Acoustical Society of America, Vol. 141, No. 2, 2017, pp. 981–995], adding capability of including fully nonisotropic noise sources. This extended tool, RANE v2, is developed as a part of multidisciplinary acoustic assessment of novel aircraft, in order to produce ground contours around airports and helipads. Version 2 extends the capability of RANE to accommodate predictions of future air vehicles implementing propulsion systems solution with inherent directional properties. The model uses three-dimensional noise emission surfaces around a series of discretized segments that represent the aircraft flightpath. The main inputs are the sources’ sound power level, the distance from the flightpath at which a level is observed, and the source three-dimensional directivity. The directivity function may take analytical or numerical form, allowing for experimental data inputs. This paper demonstrates the use of spherical harmonics as a form of directivity function with a closed-form analytical solution for calculating the noise exposure contours. Results and comparison against the Federal Aviation Administration’s Aviation Environmental Design Tool module for helicopter community noise indicate that exposure contour coordinates can be estimated for high and low noise exposure levels. The incorporation of source directivity allows for the assessment of lateral attenuation, engine installation effects, and transition operations (for vertical to horizontal flight and vice versa) via the assumption of individual source directivities and, therefore, complex noise surfaces. As a consequence of the analytical nature of the model, low computational requirements allow for fast exploration of the design space and parametric studies, with minimal input requirements. The capabilities of RANE v2 are demonstrated by predicting noise footprints for three helicopters, each of different size, performance, and directivity characteristics.

Finite element modelling of helical pile installation and its influence on uplift capacity in strain softening clay

This study investigated the effect of helical pile installation in undrained softening clay using a coupled Eulerian-Lagrangian (CEL) finite element modelling approach. A previously published strain softening soil constitutive model was used to evaluate soil disturbance. The influence of two key strain softening parameters was considered to assess installation-induced soil disturbance. The CEL-predicted remoulded soil strength field was then imported into a finite element limit analysis (FELA) to evaluate the subsequent influence on uplift capacity. The CEL results showed that helical pile installation caused a significantly disturbed zone of soil that measured approximately 1.4 helix diameters in plan and extended the full length of the pile. When the disturbed soil strength field was imported into FELA calculation, it was found that the solutions reported in previous literature had significantly overestimated the ultimate uplift capacity for softening clay. The numerical output informed the development of a new closed-form analytical approach to predict uplift capacity considering the influence of the installation process. The design method was shown to provide a high-fidelity representation of the numerical results.

Bearing capacity of open caissons embedded in sand

Open caissons are an increasingly common means of constructing underground storage and attenuation tanks as well as launch and reception shafts for tunnel-boring machines. The caisson walls typically feature a tapered base, referred to as the ‘cutting face’, to aid the sinking process by reducing the vertical soil reaction. The primary aim of this paper is to explore the influence of the caisson cutting face inclination angle on the vertical soil reaction in sand. Both finite-element limit analysis and finite-element analysis are adopted for this purpose. The effects of cutting face roughness, external embedment depth and caisson radius are also investigated. The results show that the influence of the cutting face inclination angle on the bearing capacity is highly dependent on both the soil friction angle and the roughness of the cutting face. A reduction in the caisson radius is also shown to cause a significant increase in the vertical soil reaction. The numerical output is used to inform the development of a new closed-form analytical approach amenable for use in routine design. The design method is shown to provide a high-fidelity representation of the numerical output.

Interfacial viscosity-dictated morpho-dynamics of a compound drop in linear flows

Compound droplets are excellent analogs of complex biological entities such as vesicles or cells. Despite significant advancements toward understanding the morphological evolution of a compound droplet in an incipient flow, the specific role of interfacial rheology toward dictating the same remains unaddressed. Here, we bring out non-trivial implications of interfacial rheology on the deformation of a compound drop subject to an imposed flow. The interfacial viscosity, in effect, interacts with the flow-induced non-uniform surfactant distribution to alter the droplet morpho-dynamics in a rather engaging manner. We employ a closed-form analytical approach to delineate the relative roles of advective and diffusive transport. In the paradigm of diffusion-dominated interfacial transport, viscous interfacial stress arrests the droplet deformation, thus enhancing its stability. However, for large values of the interfacial dilatational viscosity, the drop deformation increases with the interfacial shear viscosity. On the contrary, in the paradigm of surface convection-dominated surfactant transport, the interfacial rheology does not have any significant effect on either the shape deformation or the emulsion rheology. These results may pave a way toward explaining several unique features of complex fluid–fluid interfaces encountered in nature and biology.

Complex variable analytical prediction for ground deformation and lining responses due to shield tunneling considering groundwater level variation in clays

Abstract Shield-driven tunneling will encounter huge risk of potential accidents as a consequence of groundwater level surprising variation in the course of construction. Current studies seldom take account of the groundwater level variation during construction for tunneling-induced ground and tunnel responses. Moreover, little work is conducted on the analysis of tunnel lining deformation and stress. For better tunnel serviceability and safety, this paper presents a closed-form analytical solution for predicting the tunneling-induced responses of ground and tunnel lining in clays considering the influence of groundwater level variation. By introducing the non-uniform convergence deformation pattern, along with the conformal mapping, the authors establish sufficiently appropriate boundary conditions for the complex soil properties of different mediums in the problem and the corresponding complex variable solutions are derived. The accuracy of the presented solutions is then verified by comparisons with the numerical solution, showing reasonable alignments. It can also be concluded form the comparisons that the effective stress of soil weight above the tunnel receives a notable increase on the condition of a groundwater drawdown, which inevitably generates ground movements and adversely affects the tunnel safety. Furthermore, parametric analyses are also performed for the influence of concerned parameters on the lining responses, namely, lining thickness, tunnel radius and groundwater level, in which the distribution of lining stress and deformation is described in details. In general, the closed-form analytical approach introduced in this literature provides an effective insight into the soil-lining interaction involving groundwater level variation as a potential risk, which can serve as an alternative approach for conservatively estimating ground and tunnel responses in the preliminary design of tunnels.

Local buckling analysis of omega-stringer-stiffened composite panels using a new closed-form analytical approximate solution

Abstract For the preliminary structural design of aircraft, spacecraft and marine vessels, computationally efficient and sufficiently accurate computational methods are necessary. The present work contributes to this field by deriving a new closed-form analytical solution for the local buckling analysis of omega-stringer-stiffened composite panels. Hereby, the plates of the assembly are modelled separately and include the stiffener foot. The principle of the minimum potential energy is used in a Ritz-like approach and leads to an explicit analytical formulation of the critical buckling load. The compliance to geometric and dynamic boundary conditions is ensured. The panel is uniaxially loaded and investigated for different geometrical and material configurations. For the verification of the presented method, a finite element analysis is performed as well as a solution of the Levy type is obtained. Very good agreement of the new closed-form analytical approach with finite element analyses is shown in numerical studies for technical relevant areas of the design space.

Out-of-plane closed-form solution for the seismic assessment of unreinforced masonry schools in Nepal

Traditional unreinforced masonry (URM) constructions still represent an important part of the school building stock in several low-income countries including Nepal. Unfortunately, their intrinsic vulnerability impacts negatively on the resilience of cities and local communities. Observations after major seismic events have shown that the predominant mode of failure is the out-of-plane (OOP) of the weak and loosely connected perimeter masonry walls which typically leads to partial or global collapse. Starting from this evidence, a closed-form analytical approach is presented aimed at deriving the OOP force-displacement response of URM walls for different boundary conditions and vertical loads. The novel analytical solution is successfully validated with the results of seventeen OOP experimental tests on URM walls available in the literature. Then, referring to a case-study Nepalese school building, capacity curves of the constituting walls are derived and adopted for the vulnerability assessment of the structure through the Capacity Spectrum Method (CSM) where equivalent hysteretic damping is calibrated with available OOP shaking table test results. Lastly, PGA capacities for different damage states are successfully compared with median values from observational fragility curves.

A closed-form analytical approach for the simple prediction of hard-coating failure for tooling systems

Hard coatings are used extensively to protect tools in forming, forging and milling operations. While modern deposition of hard coatings using plasma methodologies has opened industry up to a new range of potential coating systems, challenges remain regarding coating optimization under specific conditions and environments. In the presented work a previously developed contact mechanical failure map, based on analytical solutions for a coating/substrate system under a contact load, was extended to include a thin-plate analytical solution for plastic deformation of a substrate, giving improved alignment with finite element solutions when compared to the Boussinesq solution utilized in literature. Analytical solutions were applied to aluminium chromium nitride based ceramic coatings, which have strong potential application in cutting tools and other contact-based applications, and were verified using finite element simulations. The failure map was investigated experimentally using scratch-testing, and it was found that cohesive coating failure initiated at forces predicted by the model for the ceramic coating, taking into account both the mechanical properties of the coating and substrate. An industry standard functional hard chrome coating was then tested, and it was found that cohesive failure initiated at loading conditions also predicted by the analytical model. The results show that such closed-form analytical modelling is a quick and versatile tool for industrially-relevant hard-coating systems, and could be utilized during the coating design process for parameter and material justification or combined with micro-mechanical testing, including high-temperature testing, to predict thin-film and coating failure in harsh industrial environments.

Effective closed form starting point determination for kinetic model interpreting NR vulcanized with sulphur

In this paper, a closed form analytical approach for a recently presented kinetic model proposed in Milani and Milani (Polym Test, 2013, under review) to interpret NR sulphur vulcanization in presence of either experimental or surrogate rheometer curves is proposed. The model has kinetic base and is aimed at predicting, by means of a very refined approach, the vulcanization degree of NR vulcanized with sulphur. It needs as input only rheometer curves to fit and provides as output kinetic constants of the single reactions occurring during the crosslink process. In Milani and Milani (Polym Test, 2013, under review) a cure chemical scheme constituted by five reactions occurring in series and parallel was adopted. The chemical scheme, translated mathematically into a differential equations system, was suitably re-arranged and a single analytical equation was derived, representing rubber crosslink degree evolution upon time. The main drawback of such procedure is that the five kinetic constants corresponding to each reaction were determined through a standard non-linear least squares procedure, trying to minimize the deviation of the analytical cure curve from experimental data. Such a limitation is here superseded and a major improvement is proposed utilizing (1) a closed form solution which does not require any optimization algorithm and (2) finding analytically a starting point for the unknown kinetic constants, very near to the actual solution and thus very convenient for a successive least squares minimization. In the model, it is shown how the analytical condition deduced from the scorch point (second derivative of the rheometer curve equal to zero) and two further conditions, e.g. the time at 90 % of vulcanization and the reversion percentage, allow the simple direct evaluation of kinetic constants, providing a closed form analytical formula to predict well the state of cure of the rubber under consideration. To assess the results obtained with the model proposed, several examples on two different NRs are discussed. The approach proved to be extremely robust and much faster when compared with the model proposed by Milani and Milani (Polym Test, 2013, under review).

Closed form analytical approach for a second order non-linear ODE interpreting EPDM vulcanization with peroxides

In this paper, the recently presented kinetic model proposed in Milani and Milani (J Math Chem 51(3):1116–1133, 2013) to interpret EPDM peroxide vulcanization is extensively revised and the resultant second order ODE is solved by means of an approximate but effective closed form analytical approach. The model has kinetic base and it is aimed at predicting, by means of a very refined approach, the vulcanization degree of rubber vulcanized with peroxides. Such a procedure takes contemporarily into consideration, albeit within a simplified scheme, the actual reactions occurring during peroxidic curing, namely initiation, H-abstraction, combination and addition, and supersedes the simplified approach used in practice, which assumes for peroxidic curing a single first order reaction. The main drawback of the overall procedure proposed in Milani and Milani (J Math Chem 51(3):1116–1133, 2013) is that the single second order non-linear differential equation obtained mathematically and representing the crosslink evolution with respect to time, was solved numerically by means of a Runge–Kutta approach. Such a limitation is here superseded and a major improvement is proposed allowing the utilization of an approximate but still effective closed form solution. After some simplifications applied on some parts of the solving function not allowing direct closed form integration, an analytical function is proposed. Kinetic parameters within the analytical model are evaluated through least squares where target data are represented by few experimental normalized rheometer curve values. In order to have an insight into the reliability of the numerical approach proposed, a case of technical interest of an EPDM with low unsaturation and crosslinked with three different peroxides at three increasing temperatures is critically discussed.

Closed-form buckling analysis of compressively loaded composite plates braced by omega-stringers

In this contribution, the stability problem of composite plates braced by longitudinal omega-stringers under compression is treated in a closed-form analytical manner. Given that no global buckling modes of both plate and stringers occur and further assuming certain periodicity properties of the occurring local plate buckling modes, the given situation is reduced to the bifurcation buckling problem of an elastically restrained plate wherein the edge restraints depend on the geometric and material properties of the stringers and are calculated via the principle of virtual work. The buckling modes of the plate itself are approximated using an interpolation scheme consisting of rather simple trigonometric functions which allows for a closed-form analytical description of both the buckling modes as well as of the buckling loads of the elastically restrained composite plates. Accompanying detailed finite element computations show that the closed-form analytical approach is in very satisfying agreement with the numerical results and as such can be used with confidence in day-to-day engineering practice. Furthermore, the exact elasticity solution for the buckling problem of a compressively loaded elastically restrained composite plate is shortly discussed which can be derived from the governing partial differential equation in conjunction with the underlying boundary conditions of the plate. The closed-form approach can be shown to be in excellent agreement with the exact results which are generated in a numerical manner. Lastly, based on non-dimensional characteristic quantities, the exact elasticity solution is used to generate generic buckling curves which may be useful for practical engineering analysis and design purposes.

Four-mirror anastigmats, part 2: systems with useful first-order layouts and minimum complexity

Previously, the author described a closed-form analytical approach to obtaining the complete solution set for four-mirror anastigmats in which all the mirrors were spherical. Two novel types of four-spherical-mirror anastigmats with concave primary were reported. This approach has been modified by setting up baseline systems with good first-order characteristics and adding the minimum number of aspheric lenses necessary to provide anastigmatic correction. In this way, searches for useful four-mirror anastigmats with one, two, or three mirrors kept strictly spherical have been carried out. By using this technique, a system with only one conicoid mirror can be found that has very similar first-order properties and equivalent correction to a system with three conicoids.

Discontinuity effects at cone-cone axisymmetric shell junctions

In this paper, the problem of a conical shell axisymmetrically intersecting another conical shell, such that the vertices of the cones lie on opposite sides of the plane of intersection, is considered, and associated discontinuity effects quantified for arbitrary loading and geometric parameters of the intersecting cones. The ensuing very practical closed-form results are based on the one-term asymptotic-series solution for the axisymmetric bending of a non-shallow conical shell, and are intended for use in the quick evaluation of stresses and deformations in double-cone pressure vessels, as well as liquid-retaining vessels with intersecting conical portions. As an example of the application of the developed formulation, the stress distribution in a large liquid-filled elevated rhombic tank is evaluated. The stresses obtained on the basis of the closed-form analytical approach developed in the paper are shown to be in good agreement with those obtained from a finite-element analysis, confirming the reliability of the presented formulation.

The tendency for free-edge delamination in laminates and its minimization

The laminate free-edge effect may have a detrimental impact on the effective strength of composite laminates. The reason for this is the occurrence of high interlaminar stresses in the immediate vicinity of free edges which are not taken into acccount by the classical theory of thin laminates. The present paper is devoted to a systematic reduction of the free-edge delamination tendency by structural optimization methods where different approaches are used for the assessment of the delamination tendency. In addition to considering the effective delamination moments as a result of the classical laminate theory, the free-edge stresses are calculated by the finite-element method and by a closed-form analytical approach. Furthermore an energy release criterion is used.

Interlaminar stresses at the free corners of a laminate

The generation of localized high interlaminar stresses is a well-known phenomenon along the free edges of composite laminates (“free-edge effect”). In a similar way a corresponding interlaminar stress localization occurs at the free corners of a laminate (“free-corner effect”). The subject of the present contribution is a closed-form analytical approach to the stress field in the vicinity of a free rectangular corner of a laminate with symmetrical lay-up under thermal loading. The derived solution is applied to the cases of a carbon fibre reinforced plastics (CFRP) cross-ply laminate and a Ni/Al-laminate. The predicted interlaminar stress distributions at the ply interfaces are shown to be in good agreement with comparative 3-dimensional finite element analyses performed by means of MSC/NASTRAN.