Energy theorems in structural mechanics

This chapter introduces the calculus of variations and associated mathematical operators on functionals, which are used to formulate boundary value problems in structural mechanics. Weak forms of equilibrium are derived in terms of the principle of virtual work and the theorem of minimum potential energy. These are used to establish the governing field equations (Euler–Lagrange differential equations) and associated geometric, natural, or mixed boundary conditions. Weak forms of kinematic compatibility are derived in terms of the principle of complementary virtual Work and the theorem of minimum complementary potential energy. Examples are provided for each.

Chapter 4 - Virtual work and energy methods

p-Version axisymmetric shell element for geometrically nonlinear analysis

Chapter 2 - Virtual work methods

In this paper we show why the postulation scheme based on forces and couples is detrimental when designing novel metamaterials. Instead, the most effective postulation scheme for mechanics is that based on the Principle of Virtual Work formulated by, for example, d'Alembert, Lagrange, Piola, and Paul Germain. In fact, generalized continuum mechanics can be formulated in a coherent way only by basing its foundations on the Principle of Virtual Work while the problem of synthesis of novel metamaterials can be efficiently confronted only when introducing generalized continua as their model at macroscopic level. Once work functionals are postulated, the set of involved generalized forces are identified using the representation theorem for distributions by Laurent Schwartz, forces in particular being zeroth order distributions. Actually, only when limiting the class of considered internal work functionals to first order distributions is the Principle of Virtual Work equivalent to the balances of forces and couples. The predominant Role of the Principle of Virtual Work has been denied by those inductivist scholars who claim that balance laws can be induced by experimentation. The Principle of Virtual Work postulation more truly fits in a falsificationist approach to epistemology.

Lightweight shell structures stiffened by different core layers are widely used and analysed among various industries including aerospace, marine and ocean engineering. In this work, the vibroacoustic coupling and sound transmission of a metastructure shell is studied. Both reinforced by orthogonal rib-stiffeners and local resonators with active control are considered. Cascaded proportional integral (PI) control is applied to reduce steady-state errors and illustrate superior stability. The space harmonic expansion method and principle of virtual work are employed to derive the dynamic effective parameter and sound transmission loss (STL). Results show that the dynamic effective density is affected by the resonator and negative density can be extended as the unit cell increases. STL can be significantly tuned by cascaded PI active control near the resonant frequency of the local resonator. Moreover, finite-element simulation and vibroacoustic experiments are performed to support theoretical predictions and the comparisons exhibit good agreement. This research provides useful vibroacoustic coupling structures for vibration and noise reduction of engineering applications.

Structures are defined usually with respect to tolerable/permissible limiting conditions of displacement, deformation and stresses etc, which enables continuous load application and static equilibrium, otherwise the equilibrium becomes dynamical resulting in unstable structural system. This study aims to evaluate, effect of vibrations and damping on structural stability, and identified that vibration is mechanical phenomenon involving action of impact forces that produces oscillatory motion on the structure and characterized with oscillations, displacement, and frequency (f = w/2Л) about a static mean position of rest (ie, F = ma = 0). The research study entails review of literatures and work on structural vibration, stability and damping mechanism to restraint effect within permissible limits. Dampers are commonly used to constraint structures to infinitesimal displacement during load application, also the paper identify that vibratory systems are means of storing potential energy (mass), kinetic energy (spring) and means by which energy is gradually dissipated through oscillations (ie, F = ma). The motion can be optimized using principle of minimum potential energy and virtual work” expressing workdone on a system undergoing virtual displacement (W = Fδx = 0, because δx = 0). Vibration damping is an influence upon a system that prevents or reduces its oscillation, and is implemented by processes that dissipate energy stored in oscillations. The damping ratio describe the system parameters which varies from undamped (ξ = 0), underdamped (ξ < 1), critically damped (ξ =1) and overdamped (ξ > 1). Static equilibrium requires damping of structures between critically damped and overdamped to ensure minimal oscillatory amplitude as expected for stability and functional performance. Similarly, structural evaluation is implemented using virtual work method,analytically defined as total work done, W = F δs = 0 (ie, δs → 0 or negligible). In conclusion, the paper identify that dynamical tendency is characterized with instability while structural performance is identified with infinitesimal or limit state deformation and displacement; hence corresponding vibratory displacement and oscillations must be minimal using appropriate damping mechanism to reduce the cumulative effect on structural system and to provide stability and safety of structural systems.

A geometrically exact model for predicting statics and snap-through behaviors of bi-stable composite laminates

Dynamics and control of a spatial rigid-flexible multibody system with multiple cylindrical clearance joints

A displacement-based optimization design method of RC structure was proposed by combining direct displacement-based design method with nonlinear programming technique. To avert the influence of target displacement, the stationary constraint displacement was presented, and the target displacement can be updated during the optimal design process. Principle of virtual work and Gaussian integral method was employed to simplify the explicit relationship between horizontal displacement and the section dimension. Comparison analysis of the local optimal results corresponding to different displacement shapes was conducted to achieve global optimal design. The numerical tests presented demonstrate the computational advantages of the discussed methods and suggesting that the proposed method is a reliably and efficiently tool for displacement-based optimal design.

Kinematics and dynamics of a 4-PRUR Schönflies parallel manipulator by means of screw theory and the principle of virtual work

Analytical dynamics of mechanisms—a computer oriented overview

The principle of virtual work is the starting point for the formulation of the finite element method for solids and structures and we dedicate this chapter to its study.We start by introducing the principle of virtual work for a 1-D model and by exploring the fundamental facts. Then, the principle is given for the 3-D elasticity model and for the remaining mathematical models discussed in Chapter 4.In the last section, we introduce some energy concepts in 3-D analysis which provide alternative ways to formulate the mathematical models of Chapters 3 and 4. These energy based formulations will be useful in Chapters 6 and 8.KeywordsVirtual WorkElasticity ProblemTotal Potential EnergyVirtual DisplacementDisplacement Boundary ConditionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Principles of virtual work and complementary virtual work for Cosserat media: Berg, C. A. Fibre Science and Technology, Vol 3, No 4, p 261 (April 1971)

We use Hodge-type orthogonal decompositions for studying the compatibility equations of the displacement gradient and the linear strain with prescribed boundary displacements. We show that the displacement gradient is compatible if and only if for any equilibrated virtual first Piola–Kirchhoff stress tensor field, the virtual work done by the displacement gradient is equal to the virtual work done by the prescribed boundary displacements. This condition is very similar to the classical compatibility equations for the linear strain. Since these compatibility equations for linear and nonlinear strains involve infinite-dimensional spaces and consequently are not easy to use in practice, we derive alternative compatibility equations, which are written in terms of some finite-dimensional spaces and are more useful in practice. Using these new compatibility equations, we present some non-trivial examples that show that compatible strains may become incompatible in the presence of prescribed boundary displacements.