The paper presents a spatial Timoshenko beam element with a total Lagrangian formulation. The element is based on curvature interpolation that is independent of the rigid‐body motion of the beam element and simplifies the formulation. The section response is derived from plane section kinematics. A two‐node beam element with constant curvature is relatively simple to formulate and exhibits excellent numerical convergence. The formulation is extended to N‐node elements with polynomial curvature interpolation. Models with moderate discretization yield results of sufficient accuracy with a small number of iterations at each load step. Generalized second‐order stress resultants are identified and the section response takes into account non‐linear material behaviour. Green–Lagrange strains are expressed in terms of section curvature and shear distortion, whose first and second variations are functions of node displacements and rotations. A symmetric tangent stiffness matrix is derived by consistent linearization and an iterative acceleration method is used to improve numerical convergence for hyperelastic materials. The comparison of analytical results with numerical simulations in the literature demonstrates the consistency, accuracy and superior numerical performance of the proposed element. Copyright © 2001 John Wiley & Sons, Ltd.
The paper shows that the behaviour of mechanical systems subject to unilateral constraints differs from that of standard systems in subtle, and yet important, ways. Therefore, a proper theoretical formulation is required for simulating their behaviour. After showing that the equilibrium equations for a multibody system subject to unilateral constraints have the same form as the standard Kuhn-Tucker conditions in optimization theory, the first-order equilibrium equations are derived and their integration is discussed. At a general integration step, one has to distinguish between constraints that are strongly active, weakly active and inactive. Whereas strongly active constraints can be treated like bilateral constraints and inactive constraints can be neglected, weakly active constraints need to be constantly re-analysed to determine if they switch to a different state. The outcome is that, in addition to the well-known limit points and bifurcation points, a new type of limit point can exist, where the path is non-smooth and the first-order equilibrium equations-after elimination of any strongly active constraints-non-singular. Such points are called corner limit points. In analogy with common limit points, the degree of instability of the system changes by one at a corner limit point.
The paper deals with the transient dynamics of serpentine belt drives. A model for the rotational motion that has been proposed in the literature is extended to elastic belt creep. However, unlike previous articles, this paper adopts a logarithmic strain measure to describe elastic creep. A condition for the existence of steady state motions with constant belt tensions, as a solution of the mass conservation law, is derived. This condition is violated for the linear strain measure together with Hooke's law, whereas it holds for the logarithmic strain measure. As a consequence, only the logarithmic strain measure, together with Hooke's law, leads to system equations that cover steady operating states. In the case of linear strain, belt tensions and tensioner position do not converge towards constant values when choosing a set of external torques that balance each other out. This is illustrated by two numerical examples. Furthermore, the considerations are reinforced and anchored in the automotive field by analysing the transient belt drive behaviour during a 'revving-up' manoeuvre of a common rail diesel engine. The considerations are generally applicable, not only to the classical elastic creep theory but also to any other, more sophisticated theory.
ABSTRACT:This research investigates the simultaneous effect of in-plane and transverse loads in reinforced concrete shells. The infinitesimal shell element is divided into layers (with triaxial behavior) that are analyzed according to the smeared rotating crack approach. The set of internals forces includes the derivatives of the in-plane components. The corresponding generalized strains are determined using an extension of the equivalent section method, valid for shells. The formulation yields through-the-thickness distributions of stresses and strains and the spatial orientation of the concrete struts. Although some simplifications are necessary to establish a practical first-order approximation, higher-order solutions could be developed. Despite the fact that constitutive matrices are not symmetric, because of the tension-softening formulation, the equilibrium and compatibility conditions are satisfied, the stiffness derivatives are explicitly calculated and the algorithms show good convergence. The formulation predicts results that agree with experimental data obtained by other researchers. Although comparative analysis with additional experimental data is still necessary, the proposed theory provides a promising solution for the design of reinforced concrete shells.
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