Superposition Procedure Research Articles

Spare parts for helix-helix interaction.

About 6000 contact regions (patches) of helix-to-helix packing from 300 well-resolved non-homologous protein structures were considered. The patches were defined by the spatial helical neighbors and were estimated in atomic detail using a variable distance criterion. The following questions are addressed. (1) Are the amino acid preferences and atomic composition of distinct types of helical patches indicative for the type of their neighbor? Distributions of size, atomic composition and packing density are compared for different types of helical interfaces. Thereby contact preferences are derived for parts of secondary structures adjoining each other or pointing towards the solvent. (2) Is it possible to cluster helical patches according to their structural similarity? For these purposes the patches were classified with an automatic sequence-independent superposition procedure which yields a distinctively reduced set of representative interfaces. On this basis, the methodology for finding exchangeable patches in different proteins is demonstrated.

Homonyms and synonyms in the Dictionary of Interfaces in Proteins (DIP).

Should reports on molecular mimicry in particular cases, e.g. responsible for cross-reactivity, be considered as accidental or as a general principle in protein evolution? To answer this question, two types of similarity have to be considered: those in homologues (synonyms) and resemblance between patches from unrelated proteins (homonyms). All interfaces from known protein structures were collected in a comprehensive data bank [Dictionary of Interfaces in Proteins (DIP)]. A fast, sequence-independent, three-dimensional superposition procedure was developed to search automatically for geometrically similar surface areas. Surprisingly, we found a large number of structurally similar interfaces on the surface of unrelated proteins. Even patches from different types of secondary structure were found resembling each other. The putative functional meaning of homonyms is demonstrated with striking examples.

QSAR for Cycloaliphatic Alcohols with Qualitatively Defined Sandalwood Odour Characteristics

QSAR's are set up for a series of up to 53 cycloaliphatic alcohols with qualitatively defined sandalwood odour characteristics: i-inactive, w-weak, s-strong, and vs-very strong. “Central” values (0, 1, 2, and 3) were attached to these qualitative characteristics, as well as intervals (<0.5; 0.5–1.5; 1.5–2.5; >2.5). The superposition procedure of the MTD method was used to define the probable active conformation. Conformational energies and log Poct values were also calculated using HyperChem-2 and Chemicalc-2. A correlational equations were set up using δ, D and D2–with δ=1 for OH groups in favourable position and D–the distance between the gravity centre of the hydrophobic moiety and the osmophoric OH-group. The MTD method was used only for a series of N=15 rigid molecules, because a too large number of vertices for the whole series appear. Eighter central values or intervals were used as biologic activities. The δ, D and D2–equations set up for N=31 molecules allow a good classification of the training set and also of the rest of molecules, considered as test set. An osmophoric receptor with a hydrophylic part (2 or 3 OH-groups) and a relatively large hydrophobic pocket is suggested, which produces also some steric misfit with regard to these molecules.

Fatigue characteristics of a glass-fiber-reinforced polyamide

This paper deals with prediction of the temperature rise in the stress-controlled fatigue process of a glass-fiber-reinforced polyamide and the application of a temperature and frequency superposition procedure to the S-N curve. An experimental equation was derived to predict the temperature rise from calculations based on the fatigue test conditions and viscoelastic properties of the material. The temperature rise (ΔT) can be expressed as a product of a coefficient term Φ(L, κ) concerning heat radiation and the test-specimen shape and a function term Pfat concerning the viscoelastic properties and fatigue test conditions. Φ(L, κ) was found experimentally to derive the equation for predicting the temperature rise blow or above the glass transition temperature (Tg) of the material. The equation σR = −STf A log NfR + STf B was obtained as a procedure for applying temperature and frequency superposition to S-N curves in consideration of ΔT. This procedure was obtained by combining both temperature- and frequency-superposition techniques. Here, σR and log NfR represents the stress and the fatigue lifetime calculated at a given temperature and frequency, A and B denote the slope and intercept of any arbitrarily chosen S-N curve, and STf is a shift factor for temperature and frequency superposition. © 1999 John Wiley & Sons, Inc. J Appl Polym Sci 72: 1783–1793, 1999

Processing and analysis of CASP3 protein structure predictions.

Livermore Prediction Center provides basic infrastructure for the CASP (Critical Assessment of Structure Prediction) experiments, including prediction processing and verification servers, a system of prediction evaluation tools, and interactive numerical and graphical displays. Here we outline the essentials of our approach, with discussion of the superposition procedures, definitions of basic measures, and descriptions of new methods developed to analyze predictions. Our primary focus is on the evaluation of three-dimensional models and secondary structure predictions. To put the results of the three prediction experiments held to date on the same footing, the latest CASP3 evaluation criteria were retrospectively applied to both CASP1 and CASP2 predictions. Finally, we give an overview of our website (http:/(/)PredictionCenter.llnl.gov), which makes the target structures, predictions, and the evaluation system accessible to the community.

Analytical Aspects of Cumulative Superposition Procedure for Elastic Indentation Problems.

Indentation problems for an elastic half-space admit similarity analysis when the local shape of the contacting body can be expressed by a homogeneous function. In this situation, the solution for curved punches can be obtained by the cumulative superposition of the solution to a single auxiliary problem which amounts to indentation by a flat-ended punch. This procedure avoids treating the moving and unknown contact boundary explicitly, so that the contact region can be determined in an accurate manner. In this study, advantages of this procedure are explored from analytical and numerical points of view. Although the theoretical basis is first described for frictionless indentation of an elastic half-space by a rigid punch, the method is subsequently shown to be applicable to the contact between two elastic bodies and for more general frictional behavior. To demonstrate the use of this superposition principle, the three-dimensional indentation by a punch with elliptic cross-section, as well as the plane-strain indentation by an asymmetric punch are solved by this method. Numerical accuracy of the present procedure is verified employing some examples of plane-strain problems, together with its effectiveness in combination with the application of the boundary element analysis for the reduced flat-punch problem.

Multiple root solutions, wedge paradoxes and singular stress states that are not variable-separable

The focus of this study is the review of a class of solutions associated with the Williams (ASME J. Appl. Mech., 1952, 19, 526–528) eigenexpansion of the stress state in a composite wedge, that are not variable-separable. These ‘wedge paradox’ solutions, which cannot be expressed as a single function of the radial coordinate multiplied by a single function of the angular coordinate, are readily obtained in this linear analysis by standard mathematical procedures associated with multiple roots from a constant coefficient, linear differential equation. The stress state resulting from these ‘non-separable’ solutions is not self-similar, in that the angular dependence of the stresses is a function of the radial coordinate. Such behavior will complicate both stress analysis, and the application of a linear solution to the failure analysis of an inherently nonlinear problem. In the first part of the paper all appropriate variable-separable solutions of the Airy stress function in polar coordinates are obtained, including four solutions associated with non-separable stresses with terms proportional to r − ω ln( r) for ω=0, 1 and 2, as well as the well-known, non-self-similar eigensolution corresponding to complex eigenvalues. In the second part of the paper, non-separable Airy stress solutions are obtained associated with ω=0, 1 and 2, and values of ω that correspond to the transition from real to complex eigenvalues. After providing the form of these non-separable solutions, examples are given that show how frequently they occur, how the solutions are obtained, the behavior of the associated coefficients or stress intensity factors at and near the special circumstances where they occur, and another look at both the concentrated force problem and the Sternberg–Koiter (ASME J. Appl. Mech., 1958, 4, 575–581) problem of a wedge with a concentrated couple. In addition to providing a thorough review of the problem and solution procedure, these linear results are important to consider when solving the related nonlinear problem where standard superposition procedures for multiple roots cannot be applied.

Linear superposition of two type- N nonlinear gravitons

A twisting type- N solution to the real Einstein equations with pure radiation energy-momentum tensor is constructed by superposing two complex vacuum solutions N × [−] and [−] × N. A general approach to the superposition procedure is presented.

Dictionary of interfaces in proteins (DIP). Data bank of complementary molecular surface patches

Molecular surface areas of proteins are responsible for selective binding of ligands and protein-protein recognition, and are considered the basis for specific interactions between different parts of a protein. This basic principle leads us to study the interfaces within proteins as a learning set for intermolecular recognition processes of ligands like substrates, coenzymes, etc., and for prediction of contacts occurring during protein folding and association. For this purpose, we defined interfaces as pairs of matching molecular surface patches between neighboring secondary structural elements. All such interfaces from known protein structures were collected in a comprehensive data bank of interfaces in proteins (DIP).The up-to-date DIP contains interface files for 351 selected Brookhaven Protein Data Bank entries with a total of about 160,000 surface elements formed by 12,475 secondary structures. For special purposes, the inclusion of additional structures or selection of subgroups of proteins can be performed in an easy and straightforward manner. Atomic coordinates of the constituents of molecular surface patches are directly accessible as well as the corresponding contact distances from given atoms to their neighboring secondary structural elements.As a rule, independent of the type of secondary structure, the molecular surface patches of the secondary structural elements can be described as quite flat bodies with a length to width to depth ratio of about 3:2:1 for patches consisting of more than ten atoms. The relative orientation between two docking patches is strongly restricted, due to the narrow distribution of the distances between their centers of mass and of the angles between their normal lines, respectively.The existing retrieval system for the DIP allows selection (out of the set of molecular patches) according to different criteria, such as geometric features, atomic composition, type of secondary structure, contacts, etc. A fast, sequence-independent 3-D superposition procedure was developed for automatic searches for geometrically similar surface areas. Using this procedure, we found a large number of structurally similar interfaces of up to 30 atoms in completely unrelated protein structures.

Analytical Aspects of Cumulative Superposition Procedure for Elastic Indentation Problems.

Indentation problems for an elastic half-space may possess similarity solutions when the local shape of the contacting body is expressed by a homogeneous function. In this situation, the desired solution for curved punches can be obtained by the integration, namely, cumulative superposition, of the solution to a single auxiliary problem which amounts to indentation by a flat-ended punch. This procedure avoids treating the moving and unknown contact boundary explicitly, so that the contact region can be determined in an accurate manner. In this study, the advantage of this procedure is explored from analytical and numerical points of view. Although the theoretical basis is laid down for frictionless indentation of an elastic half-space by a rigid punch, the method is shown to be applicable to the contact between two elastic bodies and for more general frictional behavior. To demonstrate the use of this superposition principle, the three-dimensional indentation by a punch with elliptic cross sections, as well as the plane strain indentation by an asymmetric punch are solved by this method. Numerical accuracy of the present procedure is verified employing some examples of plane-strain problems. The use of the boundary element method for the reduced flatpunch problem is shown to be effective and promising for more complicated three-dimensional indentation problems.

Nonlinear Dynamic Analysis of Steel Frames by Modal Superposition

When analyzing steel frames subjected to dynamic loads, several geometrical and material nonlinearities can be encountered in the early stages of loading. These include second-order effects, and semirigid connection and panel zone nonlinearities, respectively. The well-known modal superposition procedure, which is a very popular method for linear dynamic analysis, can be easily modified to perform nonlinear analysis while retaining many of the advantages of the method in the linear range. These modifications are developed in this paper. Element models are derived for panel zone elements and for two types of connection moment-rotation curves. Several approximations of panel zone behavior are also examined and a sample frame is analyzed. Extensive numerical studies are presented. The methodology presented was found to be both a useful and relatively fast method for nonlinear dynamic analysis of steel frames. It was also found that connection and panel zone behavior affect frame behavior and, hence, must be ...

Time-dependent deformation of polypropylene in response to different stress histories

Tensile strains have been determined as a function of time for polypropylene during (a) two-step loading, (b) creep recovery following removal of a load, and (c) intermittent load application. Data are presented at 23°C for specimens of different physical age, for different stress levels in the non-linear range and various durations of loading. The results are compared with predictions based on a pseudo-linear model. They have also been analysed using a modified superposition procedure that allows for changes in mean retardation time due both to physical ageing and to the application and removal of loads. This analysis has provided useful information on the variations of molecular mobility during the different loading histories. The functions and associated parameters used in the analyses could also form the basis of a method for presenting design data on plastics. Crown Copyright © 1997 Published by Elsevier Science Ltd.

Doubly periodic arrays of bridged cracks and short fibre-reinforced cementitious composites

This paper presents a superposition approach for studying the influence of bridging forces upon the opening of multiple cracks in elastic solids under unidirectional tensile loading. The bridging forces may be purely elastic and proportional to the crack-opening displacements, but an elasto-plastic bridging law is more likely to represent reality in a short fibre-reinforced solid. The fibres debond from the elastic matrix at a certain critical crack-opening and thereafter provide a residual bridging force due to frictional pull-out. From a mathematical point of view, the elasto-plastic bridging law introduces an additional (logarithmic) singularity at the point of discontinuity in the bridging force, besides the square root singularity at the crack tips. These singularities have been analytically isolated, so that only regular functions are subjected to numerical integration. The double infinite summations necessary for the solution of multiple cracks have been found to be divergent in earlier studies. The superposition procedure developed in this paper eliminates double infinite series and thus the problem of divergence. The mathematical solutions are used to study the influence of varying amounts of short fibre reinforcement upon the complete tensile macroscopic response (including strain hardening and tension softening) of two fibre-reinforced cementitious composites: a conventional fibre-reinforced cement and a high-performance fibre-reinforced (DSP) cement. It is shown that the model of multiple bridged cracks accurately predicts the prolonged nonlinear strain-hardening and the initial tension-softening response of both these cementitious composites.

Linear transient analysis of rectangular laminated plates by a finite strip-mode superposition method

A description is given of a method for predicting the linear transient response to dynamic loading of rectangular homogeneous and heterogeneous laminated plates. The method utilizes the semi-analytical finite strip method in conjunction with a modal superposition procedure. Plate properties are developed in the context of first-order shear deformation plate theory, and the ends of a plate may be simply supported or clamped. The applied loading, acting normal to the plate, may be of a general nature with respect to both space and time. Applications are described which relate to both thin isotropic plates and thick composite laminated plates. Good comparison is demonstrated of the results generated by the developed method and results available from the use of other procedures.

The invariant system of coordinates of antibody molecules: prediction of the "standard" C alpha framework of VL and VH domains.

A new approach of comparing protein structures that does not involve the procedure of superposition is suggested. An invariant system of coordinates for immunoglobulin molecules that is based on the geometrical symmetry inherent to the variable domain light-chain (VL)-heavy-chain (VH) complex is described. The coordinates of the Calpha atoms in 22 immunoglobulin structures are calculated in the invariant system of coordinates. We found that 76 identical positions in this Calpha framework are symmetrical about the twofold axis. Comparison of the identical positions in these molecules allows us to select 96 positions in the light chains and 87 positions in the heavy chains whose Calpha atom coordinates are approximately the same. To check whether the average coordinates of Calpha atoms in these positions complies with the stereochemical requirements, we calculated Calpha-Calpha distances. Seventy-three positions of the light chains and 72 positions of the heavy chains satisfy the Calpha-Calpha distance criterion. The Calpha atoms in these positions are used for constructing the "standard" Calpha framework of VL and VH complexes. The average coordinates of Calpha atoms are presented.

STRESS CONCENTRATIONS OF K TUBULAR JOINTS SUBJECTED TO BASIC AND COMBINED LOADINGS.

Keywords NANYANG TECHNOLOGICAL UNIVERSITY, SINGAPORE JOINTS, K JOINTS, TUBULAR, STRESS, CONCENTRATION, LOADING, BASIC, COMBINED, LOADS, TESTS, RIGS, EXPERIMENTS, BENDING, AXIAL, IN PLANE, OUT OF PLANE, FINITE ELEMENTS, SUPERPOSITION, FATIGUE LIFE, OFFSHORE, PLATFORMS, FIXED, STRAIN, GAUGING, MODELLING, MODELS, HOT, SPOTS... Show All

The dynamic interaction of a crack with a circular hole under antiplane loading

This article provides a comprehensive theoretical treatment of the dynamic interaction between a crack and an arbitrarily located circular hole near its tip under incident antiplane shear wave. The theoretical formulations governing the steady-state problem are based upon the use of integral transform techniques and an appropriate superposition procedure. The resulting dynamic stress intensity factor at the main crack is obtained by solving the appropriate singular integral equations using Chebyshev polynomials at different loading frequencies. The solution is verified by comparison with a simplified earlier solution for a special case and numerical examples are provided to show the effect of the location, inclination and size of the hole and the frequency of incident wave upon the dynamic stress intensity factor of the crack.

An analysis of fully plastic Brinell indentation

Indentation of a hard sphere into inelastic solids, Brinell indentation, is examined theoretically and numerically by aid of classical plastic flow theory. With the main interest focused on fully plastic behaviour at indentation the mechanical analysis is carried out for power-law hardening rigid-plastic materials where self-similarity features play a dominant role. It is explained in detail how the problem of a moving contact boundary may be reduced to a stationary one by an appropriate transformation of field variables. Within this setting classical empirical findings by Meyer (1908) and O'Neill (1944) are established on a rigorous theoretical ground. In particular, it is shown to advantage also for nonlinear materials how intermediate solutions for a flat die may by cumulative superposition generate solutions for a class of curved indenters. In the case of perfect plasticity it turns out in the present context that indentation hardness is independent of die profiles. For hardening solids when the material behaviour is history dependent, reduction to a stationary geometry is achieved also by expressing the accumulated strain by cumulative superposition. The intermediate flat die problem is then solved for a variety of hardening exponents by a finite element procedure designed to account for material incompressibility. With finite element computations as a basis desired solutions are obtained by straightforward numerical superposition procedures. Detailed results are then given for bulk quantities such as the mean contact pressure as well as relevant field variables. The influence of hardening characteristics on sinking-in and piling-up of indented surfaces and contact pressure distributions are discussed in the light of earlier findings based on deformation theory of plasticity and available discriminating experiments. Correlation is particularly sought with the celebrated universal hardness parameters proposed by Tabor (1951) and the existence of representative strain measures. Attention is also given to the elastic-plastic transition region of Brinell indentation in search for loading levels sufficiently high that the results tend to an asymptotic fully plastic state. A standard finite element technique employing contact elements for a moving boundary is used to analyse with tolerable accuracy the influence of elasticity and more elaborate hardening behaviour. Some relevant features are shown for a sequence of solutions from elastic Hertzian to fully plastic behaviour.

On the dynamic interaction between a microdefect and a main crack

A generalized theoretical approach is presented for the dynamic interaction between an arbitrarily located and oriented microdefect and a finite main crack subjected to a plane incident wave. The analysis is based upon the use of integral transform techniques and an appropriate superposition procedure. The resulting dynamic stress intensity factors ( K * I and K * II ) at the main crack are obtained by solving the appropriate singular integral equations, using Chebyshev polynomi­als, for different incident waves. The resulting solution is verified by comparison with existing results, and numerical examples are provided to show the effect of the location and orientation of the microdefect and the frequency of the incident wave upon K * I and K * II of the main crack. The results advanced here can be used as building blocks in the fields of micromechanics, damage and non-destructive characterization of defects in solids.

Multisolitons in a two-dimensional Skyrme model

The Skyrme model can be generalised to a situation where static fields are maps from one Riemannian manifold to another. Here we study a Skyrme model where physical space is two-dimensional euclidean space and the target space is the two-sphere with its standard metric. The model has topological soliton solutions which are exponentially localised. We describe a superposition procedure for solitons in our model and derive an expression for the interaction potential of two solitons which only involves the solitons' asymptotic fields. If the solitons have topological degree 1 or 2 there are simple formulae for their interaction potentials which we use to prove the existence of solitons of higher degree. We explicitly compute the fields and energy distributions for solitons of degrees between one and six and discuss their geometrical shapes and binding energies.