Geometrically nonlinear analysis

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

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... 

The geometrically nonlinear analysis of a bonded lap joint was carried out using NOVA and VISTA. The geometry and boundary conditions shown in Figure 1 are used. The following material constants are used ... 

Bakr, E. M., and Shabana, A. A., Geometrically Nonlinear Analysis of Multibody Systems, ... 

Fig. 20 Time histories of various twisting moment components at the left end of the beam of application section Primary Resonance of Beam of Thin-Walled I-Section with STMDE taking into account the secondary twisting moment deformation effect (geometrically nonlinear analysis)...

Adhesive and adherend may exhibit considerable plasticity before failure and thus require nonlinear material models. In single lap joints, rotation of adherends occurs during loading, as illustrated in O Fig. 25.8, and geometrically nonlinear analysis is required to determine the correct stress state. 

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. 

M.A. Krasnoselskii and P.P. Zabrejko, Geometrical Methods of Nonlinear Analysis, Nauka, Moscow, 1975 (in Russian). 

Recent years have seen great advances in nonlinear analysis of frame structures. These advances were led by the development and implementation of force-based elements (Spacone et al. 1996), which are superior to classical displacement-based elements in tracing material nonlinearities such as those encountered in reinforced concrete beams and columns. In the classical displacement-based frame element, the cubic and linear Hermitian polynomials used to interpolate the transverse and axial displacement fields, respectively, are only approximations of the actual displacement fields in the presence of non-uniform beam cross-section and/or nonhnear material behaviour. On the other hand, force-based frame element formulations stem from equilibrium between section and nodal forces, which can be enforced exactly in the case of a frame element. The exact flexibiUty matrix can be computed for an arbitrary (geometric) variation of the cross-section and for any section/material constitutive law. Thus, force-based elements enable, at no significant additional computational costs, a drastic reduction in the number of elements required for a given level of accuracy in the simulated response of a EE model of a frame structure. 

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... 

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... 

Andruel, R.H., Dillard, D.A. and Holzer, S.M., Two- and three-dimensional geometrical nonlinear finite elements for analysis of adhesive joints. Int. J. Adhes. Adhes., 21, 17-34 (2001). 

In the nonlinear analysis of solids, there are two kinds of nonlinearities - the material nonlinearity and the geometric nonlinearity. The material nonlinearity is basically due to the existence of a nonlinear relation toween the stresses and the strains. The geometric nonlinearity implies that the strains involved are very large so that all the stress measures (Cauchy stress, Kirchhoff stress, first and second order Piola-Kirchhoff stresses, etc.) and the strain measures (engineering strain, natural strain, Green-Lagrange strain, etc.) are very much different in meaning and in numerical values. 

Although the focus of this entry is on nonlinear analysis, the extra effort involved to introduce material and geometric nonlinearity is not always justified - particularly when target performance objectives restrict the respraise to the elastic or near-elastic range. This also applies at a component level - for example, force-controlled (brittle) elements are usually modeled linearly, given that their nonlinear response is generally unacceptable. Of course, even if heavily nonlinear response is acceptable, if... 

For one, realism means incorporating all pertinent sources of material and geometric nonlinearity that are expected to arise. This should include, for example, plastic-hinge formation zones for moment-resisting frames, brace buckling for braced frames, and P-Delta effects. Nonsimulated failure modes, such as the shear failure of members or the brittle failure of beam-column joints, can be incorporated in the analysis a posteriori. Still, they essentially remove the model s ability to track structural behavior beyond their first occurrence. This means that whenever nonsimulated failures are found to have occurred, one cannot trust the model to provide estimates beyond that point. 

CoUapse simulation Elemental nonlinearity Geometric nonlinearity Material nonlinearity Nrailinear analysis... 

Flexural-torsional analysis Geometrically nonlinear dynamic analysis Secondary torsional moment Shear deformation warping... 

When the displacement components of a member are small, a wide range of linear analysis tools, such as modal analysis, can be used, and some analytical results are possible. As these components become larger, the induced geometrical nonlinearities result in effects that are not observed in linear systems. When finite displacements are considered, the flexural-torsional dynamic analysis of beams becomes much more complicated, leading to the formulation of coupled and nonlinear flexural, torsional, and axial equations of motion. The analysis of these systems becomes even more complicated when shear deformation effect in flexure and secondary torsional moment deformation effect (STMDE) in torsion, which are significant in many cases (e.g., short beams, beams of box-shaped cross sections, folded structural members, beams made of materials weak in shear, etc.), are taken into account. 

In section Nonlinear Flexural Dynamic Analysis of Beams with Shear Deformation Effect of this chapter, the geometrically nonlinear dynamic flexural analysis of homogeneous prismatic beam members taking into account shear deformation and rotary inertia effects (Timoshenko beam theory) is presented. The differential equations of... 

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