Closed-form analyses

Closed form solutions. Structural adhesive joints are generally designed to be loaded in shear so that treatments of joint analyses are confined essentially to the transfer of load by shear, with some consideration of the transverse normal stresses induced by eccentricities in the load path. In the simplest case the adhesive and the adherends are assumed to behave elastically. The most refined analyses attempt to model the situation when the adhesive yields so that the adhesive and, eventually, the adherends behave plastically as the imposed load is raised. Closed-form analyses are difficult to apply to other than simple geometrical configurations, while a major difficulty with the elasto-plastic model is how to characterise the adhesive. 

Many tools are available for preliminary composite sizing and layout, including excellent computer programs for use on microcomputers. Use these computer programs and closed form analyses early, and defer finite element analysis. 

The main problem with closed form analyses is in accounting realistically for adhesive and adherend non-linearity. Also, even the elastic formulations produce complex equations which are difficult to solve. Fortunately, it is now relatively easy to set up these equations on desk top computers and to get almost instantaneous solutions which give a good indication of the stresses acting in a lap joint under tensile loading. 

Hart-Smith, L.J., Recent Expansions in the Capabilities of Rose s Closed-Form Analyses for Bonded Crack Patching, Boeing Paper MDC 00KO1O8. presented to 9th Australian International Aerospace Congress, Canberra, Australia, March 5-8,2001. 

Thus it is possible, by using closed-form analyses of varying complexity, to predict the stresses in simple lap joints. (This approach is termed continuum mechanics.) In many instances, such solutions may be deemed acceptable. However, two problems still remain to be solved if it is required to predict the strength of real joints. These may be summarized as end effects and material non-linearity (adhesive and adherend plasticity). We will look first at end effects for linear elastic systems. 

One common result from all the closed-form analyses, whether they be simple or complex, is that the maximum adhesive stresses always occur near the end of the bond-line. 

Research on the impact strength of adhesively bonded joints has been recently carried out very often. As mentioned above, the reason is due to the increase of demand from industries and the progress of computer technologies. The tendency seems to continue in the near future. For the purpose, the integration of finite element analyses and strength criteria becomes important. Approaches using cohesive zone models are very promising for actual applications such as car structures because the approaches are suitable for finite element analyses based on explicit schemes. In contrast, closed-form approaches are necessary from an academic point of view and should be also taken for the impact problem of adhesively bonded joints to understand the nature of joint mechanics. Unfortunately, only few studies on the closed-form analyses have conducted already, so that more research should be done in the future. 

In conventional closed-form analysis, one generally seeks to simplify the governing equations by dropping those terms which are zero or whose numerical magnitudes are small relative to the others, and then proceeding with a mathematical solution. In contrast, our code is written to contain all of the terms (except uVu, for now), and the particularization to specific problems is done entirely by the selection of appropriate numerical parameters in the input dataset. 

Prussing, J.E., and Lin, Y.K., "A Closed-Form Analysis of Rotor Blade Flap-Lag Stability in Hover and Low-Speed Forward Flight in Turbulent Flow", Journal of the American Helicopter Society, Vol. 28, No. 3, July 1983, pp. 42-46. 

When we consider non-linear material properties by a closed-form analysis such as Hart-Smith s, the limitation is how tractable is a realistic mathematical model of the stress-strain curve within an algebraic solution. With the finite-element techniques developed for adhesive joints by Adams and his co-workers, the limit becomes that of computing power. The high elastic stress and strain gradients at the ends of the adhesive layer need to be accommodated by three or four 8-node quadrilateral elements across the thickness. However, consideration of non-linear material behaviour requires a much larger computing effort on any given element. Thus, it becomes necessary to... 

The analyses referred to above are essentially exact, but can be applied only to bonded joints between adherends of uniform thickness. A new approximate method was developed during the Composite Repair of Aircraft Structures (CRAS) R8cD contract that covers tapered adherends as well. The method, explained by Hart-Smith (2001/2002), relies on the knowledge that the adhesive stresses will be extremely low everywhere except in the immediate vicinity of changes to or interruptions in the thickness of the adherends. It starts with a closed-form analysis in which it is assumed that all the members are fused rigidly together over the entire overlap. The load sharing between the overlapped members can also account for residual stresses caused by thermal dissimilarities between adherends, as between composite patches and cracked metallic structures and between bonded titanium stepped plates at the ends and... 

Stress analysis involves using the descriptions of the product s geometry, the applied loads and displacements, and the material s properties to obtain closed-form or numerical expressions for internal stresses as a function of the stress s position within the product and perhaps as a function of time as well. Tlie term engineering formulas refers primarily to those equations reviewed previously and given in engineering handbooks by which the stress analysis can be accomplished. 

Gale NH (1970) A solution in closed form for lead isotopic analysis using a double spike. Chem Geol 6 305-310... 

In the classical literature analysis the system equations were manipulated to eliminate V y, the velocity of the solid bed consumption in the thickness direction (y direction), from the analysis by using the assumption that the solid bed reorganizes. This allowed a straightforward differential analysis and a closed form solution in the cross-channel x direction for solids melting. In this analysis, the y-direction velocity V y is retained as a variable because this facilitates the calculation of the change in bed thickness, which was found to be very important in the reevaluation of the literature data, as shown in Figs. 6.9 and 6.10. 

Other computer models and analytical tools are used to predict how materials, systems, or personnel respond when exposed to fire conditions. Hazard-specific calculations are more widely used in the petrochemical industry, particularly as they apply to structural analysis and exposures to personnel. Explosion and vapor cloud hazard modeling has been addressed in other CCPS Guidelines (CCPS, 1994). Again, levels of sophistication range from hand calculations using closed-form equations to numerical techniques. 

The hyperspherical method, from a formal viewpoint, is general and thus can be applied to any N-body Coulomb problem. Our analysis of the three body Coulomb problem exploits considerations on the symmetry of the seven-dimensional rotational group. The matrix elements which have to be calculated to set up the secular equation can be very compactly formulated. All intervals can be written in closed form as matrix elements corresponding to coupling, recoupling or transformation coefficients of hyper-angular momenta algebra. 

It is probably the complexity of these theories that prohibited this particular aspect of electrode kinetics from being attractive for application in the study of homogeneous reaction kinetics per se. Yet it must be clear that the electrochemical techniques, together providing an extremely wide range of time scales, should be preeminently suited for investigations of both slow and (very) fast homogeneous reactions. This is the more true since, nowadays, the problem of the non-availability of a closed-form expression for the response—perturbation or response—time relation has been overcome by numerical analysis procedures conducted with the aid of computers. 

Several groups (84-86) have extended the similarity analysis of Burton et al. (73) to the case in which an axial magnetic field is imposed on the melt with sufficient strength such that Ha >> 1 and N 1. With these limits, a closed-form asymptotic expression describes the variation in the flow field across the thin 0(Ha 1/2) Hartmann layer adjacent to the disk. Axial solute segregation across this layer was analyzed by assuming that the melt outside of the Hartmann layers is well mixed. The effective segregation coefficient approaches 1 when the field strength is increased, as expected for any mechanism that damps convection near the crystal. 

The Lorentzian distribution arises from the decaying hole-state, but for an analysis of the observed shape of a photoline, the energy distribution of the incoming light and the spectrometer function must be known and taken into account. In general, the latter functions cannot be presented in a closed form however, quite often they are approximated by a Gaussian distribution ... 

The above analysis and Fig. 19-25 provide a theoretical foundation similar to the Thiele-modulus effectiveness factor relationship for fluid-solid systems. However, there are no generalized closed-form expressions of E for the more general case ofa complex reaction network, and its value has to be determined by solving the complete diffusion-reaction equations for known intrinsic mechanism and kinetics, or alternatively estimated experimentally. 

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