By Steen Krenk
ISBN-10: 0521830540
ISBN-13: 9780521830546
This publication offers a theoretical therapy of nonlinear habit of solids and buildings in one of these approach that it really is compatible for numerical computation, as a rule utilizing the Finite aspect strategy. beginning out from uncomplicated strategies, the writer systematically makes use of the primary of digital paintings, in the beginning illustrated through truss constructions, to provide a self-contained and rigorous account of the fundamental equipment. the writer illustrates the combo of translations and rotations through finite deformation beam theories in absolute and co-rotation layout, and describes the deformation of a three-d continuum in fabric shape. A concise creation to finite elasticity is via an extension to elasto-plastic fabrics through inner variables and the utmost dissipation precept. ultimately, the writer offers numerical recommendations for answer of the nonlinear worldwide equations and summarizes fresh effects on momentum and effort maintaining integration of time-dependent difficulties. routines, examples and algorithms are incorporated all through.
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Example text
The symbolic summation over the elements defines the global tangent stiffness matrix ˜ K = K. 60) elements The assembly procedure for the global tangent stiffness matrix K and for the total internal forces q is illustrated in the following example. 2. Element formulation of two-element truss. 1 the equilibrium equations and the tangent stiffness of the twoelement truss shown in Fig. 6 were derived directly from the elastic energy. This example shows how the current internal forces and the incremental equilibrium equations of the truss are obtained by assembling the element contributions.
Malvern (1969) or Holzapfel (2000). It is clear from the definition of the angle of rotation ϕ that it is independent of any choice of coordinate system, while the components [n1 , n2 , n3 ] of the direction of the rotation axis refer to the chosen coordinate system. It is therefore to be expected that the angle of rotation can be extracted from the trace of the rotation tensor. 10), whereby the trace of the rotation tensor is tr(R) = Rii = 1 + 2 cos ϕ. 10). The principles used to extract the rotation angle ϕ and the components [n1 , n2 , n3 ] of the axis of rotation constitute a simple example of the use of invariance and symmetry.
These general features are encountered repeatedly in various settings in the following chapters. 1 Deformation and strain In most finite element formulations for structures and solids the displacements are the primary variables of the problem. The displacements may lead to deformation of the elements, and this in turn to internal forces. It is important to use deformation measures that vanish identically for rigid body displacements. In a theory with finite displacements this requirement can be satisfied by several different strain definitions.
Non-linear Modeling and Analysis of Solids and Structures by Steen Krenk
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