Geometrical nonlinearities

The term nonlinear in nonlinear programming does not refer to a material or geometric nonlinearity but instead refers to the nonlinearity in the mathematical optimization problem itself. The first step in the optimization process involves answering questions such as what is the buckling response, what is the vibration response, what is the deflection response, and what is the stress response Requirements usually exist for every one of those response variables. Putting those response characteristics and constraints together leads to an equation set that is inherently nonlinear, irrespective of whether the material properties themselves are linear or nonlinear, and that nonlinear equation set is where the term nonlinear programming comes from. 

In designing axi-symmetric shell structures such as large-type cooling towers, it is necessary to predict the vibration responses to various external forces. The authors describe the linear vibration response analysis of axi-symmetric shell structures by the finite element method. They also analyze geometric nonlinear (large deflection) vibration which poses a problem in thin shell structures causes dynamic buckling in cooling towers. They present examples of numerical calculation and study the validity of this method. 11 refs, cited. 

A finite element method is employed to study the nonlinear dynamic effect of a strong wind gust on a cooling tower. Geometric nonlinearities associated with finite deformations of the structure are considered but the material is assumed to remain elastic. Load is applied in small increments and the equation of motion is solved by a step-by-step integration technique. It has been found that the cooling tower will collapse under a wind gust of maximum pressure 1.2 psi. 13 refs, cited. 

Geometric nonlinearity. This nonlinearity is applicable due to the relatively large deformations anticipated. As the helmet is a curved... 

For both mathematical and physical reasons, there are many instances in which the spatial variations in the field variables are sufficiently gentle to allow for an approximate treatment of the geometry of deformation in terms of linear strain measures as opposed to the description including geometric nonlinearities introduced above. In these cases, it suffices to build a kinematic description around a linearized version of the deformation measures discussed above. Note that in component form, the Lagrangian strain may be written as... 

Automatic purity control was reported for the separation of aromatic hydrocarbons where online Raman spectroscopy can be used to measure the concentrations of the compounds at the outlet of the chromatographic columns (Marteau et al., 1994). This approach, as well as the geometrical nonlinear control concept described by Kloppenburg and GUles (1999), is based on a model that describes the corresponding TMB process, so the cyclic port switching is neglected. 

Under appropriate conditions, metal design involves plasticity, creep, and geometric nonlinearity. These topics are treated in standard texts and have been put into computer software. However, such complexities are necessarily modeled in a simple technical format. 

From a creative, engineering point of view, the feasible forms and maximum spans of ETFE-foil constructions are decisive factors which will determine the further success of this construction method. ETFE-foils possess positive load-bearing behaviour which is characterised by material and geometrical nonlinearities. The ground plan shapes, realisable to almost any desire, offer the architect a multitude of design possibilities. Spans of ETFE-foil constructions can be enlarged through support measures like... 

Eringen s nonlocal elasticity theory and von Karman geometric nonlinearity us- — ing nonlocal Timoshenko beam... 

Computational fluid dynamics models were developed over the years that include the effects of leaflet motion and its interaction with the flowing blood (Bellhouse et al., 1973 Mazumdar, 1992). Several finite-element structural models for heart valves were also developed in which issues such as material and geometric nonlinearities, leaflet structural dynamics, stent deformation, and leaflet coaptation for closed valve configurations were effectively dealt with (Bluestein and Einav, 1993 1994). More recently, fluid-structure interaction models, based on the immersed boundary technique. 

Neglecting geometrical nonlinearity, the vector q can be assiuned to be perpendicular to the direction X2 along the whole path, therefore the 7-integral can be rewritten as ... 

In FEA, a problem is nonlinear if the force—displacement relationship depends on the current state of the displacement, force, and stress—strain relations. NonUnearity in a problem can be classed as material nonlinearity, geometric nonlinearity, and bound conditions. 

Geometric nonlinearity occurs if the relationships of strains and displacements are nonlinear with the stresses and forces. This can lead to changes in structural behavior and loss of structural stabihty. Examples of geometric nonlinearity include buckling and large displacement problems. 

This shows directly that the nonlinearity of the dynamic resporrse is due both to geometrical nonlinearities (in conjimction with second-order material constants) and to material nonlinearities as expressed by the various third-order constants. 

The DDM algorithm for a three-field mixed formulation based on the Hu-Washizu functional (Washizu 1975) has been derived and presented elsewhere (Barbato et al. 2007). This formulation stems from the differentiation of basic principles (equilibrium, compatibility and material constitutive equations), applies to both material and geometric nonlinearities, is valid for both quasi-static and dynamic FE analysis and considers material, geometric and loading sensitivity parameters. This general formulation has also been specialized to frame elements and linear geometry (small displacements and small strains). 

Fiber-based frame analysis is one of the most advanced methodologies to model the nonlinear behavior of beams and columns under combined axial and bending loads. The Mid-America Earthquake Center analysis environment ZEUS-NL (Elnashai et al., 2002), is a compntational tool for the analysis of two and three dimensional frames. In ZEUS-NL, elements capable of modeling material and geometric nonlinearity are available. The forces and moments at a section are obtained by integrating the inelastic responses of individnal fibers. The Eularian approach towards geometric nonlinearity is employed at the element level. Therefore, full account is taken of the spread of... 

Formulations for beams considering both constitutive and geometric nonlinearity are rather scarce most of the geometrically nonlinear models are limited to the elastic case, Ibrahimbegovic (1995) and the inelastic behavior has been mainly restricted to plasticity, Simo et al. (1984). Recently, Mata et al. (2007b, 2008a) have extended the geometrically exact formulation for beams due to Reissner-Simo (Reissner 1973, Simo 1985, Simo Vu-Quoc 1988) to an arbitrary distribution of composite materials on the cross sections for the static and dynamic cases. 

Ihrahimhegovic, A. 1995. On finite element implementation of geometrically nonlinear Reiss-ners beam theory three-dimensional curved beam elements. Computer Methods in Applied Mechanics and Engineering 122 11-26. 

Thus the formulation remains unchanged when geometric nonlinearity is included. 

ECS provides some guidelines on geometric nonlinearities, but only on when and how to account for them in an approximate way. Second-order, or P-A effects, need not be taken into account if the following condition is satisfied ... 

Neuenhofer, A., Filippou, EC. 1998. Geometrically Nonlinear Flexibility-Based Frame Finite Elements, ASCE Journal of Structural Engineering, 124(6), 704-711. 

Detailed structural analyses form the basis for the final designs of the tower, its components, and its connections. Both cable-stayed and suspension bridges are highly indeterminate and both require careful analyses by at least one geometric nonlinear program if erections are to be determined. Prudent design should also include analysis of at least one erection scheme to demonstrate that an experienced contractor may erect the structure. 

Program name Material properties Linear elastic Analysis linear visco- elastic Nonlinear visco- elastic Geometric nonlinearity Loading Time function Nonlinear diffusion... 

Since bonded joints can often undergo large displacements, especially when subjected to creep-type loading, the geometrically nonlinear formulation described in References 37 and 38 is used to implement the nonlinear viscoelastic model. The principle of virtual work, in the updated Lagrangian incremental formulation, can be stated as... 

For a geometrically nonlinear analysis, the vector e contains components of the Almansi strain tensor. 

Equation (49) contains two possible sources of nonlinearities material nonlinearity due to Schapery s law, and geometric nonlinearity arising from the large displacement (and small strain) formulation. In order to obtain a solution to this nonlinear equation at any time step, the Newton-Raphson iterative technique is used. The incremental displacement Aw obtained at the end of the iteration is used to update the total displacement for the time step,... 

In this section results of a number of linear elastic, linear viscoelastic, and nonlinear viscoelastic analyses are discussed in light of available experimental or analytical results. All results are obtained using NOVA on an IBM 3090 computer in double precision arithmetic. First, the results of geometric nonlinear analysis are presented and compared with those obtained by other finite-element programs. Then linear and nonlinear viscoelastic analysis... 

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