Principle of Virtual Displacements

The mechanical equilibrium of an infinitesimal volume element of a deformable structure, given by Eq. (3.14), may be multiplied with the vector field of virtual displacements 5u and integrated over the domain A yielding 

The transformation of the volume integral over A into a surface integral over the volumes closed surface dA allows for the application of the physical boundary conditions of Eq. (3.15). The virtual displacement gradient Vdu may be split into its symmetric and skew symmetric portions, as has been shown for the actual case in Eq. (3.19). The influence of the skew-symmetric portion in 

Consequently, with transition to the alternative vectorial arrangement of the stress and strain components as given on the right-hand sides of Eqs. (3.12) and (3.20), the final form in matrix representation is 

This formulation of the principle of virtual work is the principle of virtual displacements, which appears in the hterature sometimes under the name of the preceding. Naturally, the virtual strain energy 6U exists only for mechanical systems with deformable parts. As the contained virtual strain tensor is assembled from derivatives of the virtual displacements, these have to be continuously differentiable. The virtual work of external impressed loads ymd includes the limiting cases of line or point loads. External reactive loads do not contribute when the virtual displacements are required to vanish at the points of action of these loads, and thus the virtual displacements have to comply with the actual geometric or displacement boimdary conditions of Eq. (3.16). With these presumptions, the initial axiom of Remark 3.1 may now be reformulated for the virtual displacements. 

Remark 3.2. A uniform mechanical system will be in equilibrium if the virtual work of the actual external and internal loads for arbitrary admissible virtual displacements vanishes. 

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Principle of virtual displacements.— Fiom this theorem we deduce immediately an important corollary. 

Applicfttion to ft gas that obeys Mariotte s law, 11.—8. In some cases, the work of the forces applied to a system depends only on the initial and final states of this system, 12.—. In general, the work done by the forces applied to a system depends upon every modification the system undergoes, 12.—10. Potential, 14.—11. Potential due to gravity, 16.—la. Forces which admit a potential in virtue of the restrictions imposed upon the system, 16.—13. Energy, 16.—14. Principle of virtual displacements, 17.—15. Conservatives of energy. Conservative systems, 18.—16. Principle of virtual displacements for conservative systems. Stability of equilibrium, 19. 

Consider a linear elastic continuum, which may consist of passive and active, i. e. adaptronic, elements. The dynamic equilibrium of the structure can be formulated using the principle of virtual displacements [9] including inertia loads. To express the internal strain energy in terms of displacement variables, the kinematics of the structure has to be considered. Various types of mechanical structures, e. g. beams, plates, or shells, are defined by kinematics relations and constraints. 

The displacement finite element model is based on the principle of virtual displacements. The principle requires that the sum of the external virtual work done on a body and the internal virtual work stored in the body should be equal to zero (see Reddy(46))... 

In Fig. 1, the displaced dam from its initial position is defined by the displacement u(y,t), which is the Siam of the horizontal ground excitation Ug(t) and the relative displacement Y< (t) (y). The dam is assumed to be elastic and as a first approximation, it is assumed to deflect in its first mode with mode shape (y). For this dynamic system the equation of motion can be derived by the principle of virtual displacement in the form of a generalized single degree of freedom system. In terms of the maximum displacement Y Ct) at the top of the dam and the generalized force Pc(t), the equation is... 

For the rigid continumn of volume A consisting of such particles in accelerated motion, the virtual work may be formulated as given by Eq. (3.58). This extension of the principle of virtual displacements is referred to as d Alembert s principle in the Lagrangian version ... 

The criteria of admissibility for the virtual displacements have been discussed in Section 3.4.2. As rigidity has been assumed in the case at hand, the occurring displacements do not cause strains. Therefore, virtual strains do not exist and, consequently, there are no contributions of internal loads to the virtual work. As expected, the virtual work of external impressed loads is identical to the term in the static principle of virtual displacements. The accelerated motion results in the additional term representing the virtual work of the loads of inertia. In general, the principle may be formulated as follows ... 

The principle of virtual displacements, given by Eq. (3.45), may be utilized to determine the equations of equilibrium. We will refrain from considering external loads. For the two-dimensional shell structure still with the transverse shear strains 7° and 7 and associated internal transverse forces Qx and Qs, the principle of virtual displacements may then be reformulated as follows ... 

The virtual work of internal contributions is assembled in Section 3.4.6 from the virtual strain energy and virtual work of internal charges, as supplied by the principle of virtual displacements and of virtual electric potential, respectively. In Ek[. (3.63), the virtual work of internal contributions is given for a volumetric object. The preceding analysis accomplished a reduction to two dimensions for the sheU-like wall and to one dimension for the beam. Consequently, the expression for the virtual work of internal contributions may be reformulated for the wall SLi t) and for the beam SLi t) as follows ... 

That all actual processes are irreversible does not invalidate the results of thermodynamic reasoning with reversible processes, because the results refer to equilibrium states. This procedure is exactly analogous to the method of applying the principle of Virtual Work in analytical statics, where the conditions of equilibrium are derived from a relation between the elements of work done during virtual i.e., imaginary, displacements of the parts of the system, whereas such displacements are excluded by the condition of equilibrium of the system. 

The generalized solution of this problem is found by the variational method of virtual displacements.140 According to this principle, the sum of the virtual work done in traversing possible displacement path is equal to zero ... 

According to Newton, equilibrium is established when the resultant forces on the masses, i.e., in the direction of the respective planes, are equal. There is another formulation of equilibrium, however, due to the principle of virtual work. Equilibrium is established in this way, if a small, allowed displacement of the masses will not cause a change in energy. 

In order to move the atoms depicted in Fig. 2.3, work has to be performed on the system. For example, if one atom is moved by an amount 8m, the work done is / 8m, using the principle of virtual work. In doing this work, the bond energy has changed by an amount A=(f) u+bu)—(u). As indicated earlier, the equilibrium condition for the deformation is given by f—dapproaches zero, A =fhu. Thus, for small displacements, the work done on the atoms is equivalent to the change in bond energy. It is now possible to define a... 

The stress formula (3.134) can be put in the form equivalent to the principle of virtual work. Consider a virtual deformation which displaces the point on the material to Tq. -I- de firp in a very short time dt. In the limit of dt- 0, the velocity gradient x p = dE pldt becomes very large, so that the time evolution of V in the time interval 6t is dominated by ticpA... 

The VFM is based on the fundamental equations of solid mechanics the equilibrium equation, through the principle of virtual work (PVW), the constitutive equations and the strain-displacement relationships. For an arbitrary solid in equilibrium, the PVW can be written as... 

The equations of the Lagrangian incremental description of motion can be derived from the principles of virtual work (i.e., virtual displacements, virtual forces, or mixed virtual displacements and forces). Since our ultimate objective is to develop the finite-element model of the equations governing a body, we will not actually derive the differential equations of motion but utilize the virtual work statements to develop the finite element models. 

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... 

Mechanical work at every particle of a continuum results from acting force and respective displacement or local stress and strain, correspondingly. So the above axiom of Remark 3.1 actually comprises two principles involving either virtual loads or virtual displacements. A brief derivation of both will be given in the following subsections. Similarly, the electric work can be treated, but we will present only one of the variants. The different formulations of the principle of virtual work are independent of a constitutive law and may be denoted as the weak forms of equilibrium, as only the equilibrium conditions have to be fulfilled in the integral mean. Weaker requirements with regard to differentiability of the involved functions have to be fulfilled, since the order of derivatives is reduced in comparison to the equilibrium formulation of Eqs. (3.f4) and (3.34). 

The other formulation of the principle of virtual work for mechanical systems requires the introduction of virtual loads instead of virtual displacements. Therefore, only those variations of external loads and stress tensor are considered admissible that are compatible with the equations of equilibrium inside the mechanical system and on the boimdary. The interior equilibrium of Eq. (3.14) for the virtual loading leads to the following form ... 

D Alembert s principle in the Lagrangian version has been obtained in Section 3.4.5 in terms of virtual displacements and actual accelerations. Since it needs to be accounted for a superimposed guided motion, the position p x,s,t) in the inertial frame of reference, as described by Eq. (7.65), has to be taken into consideration. With the density p s, n) in accordance with Remark 7.1, the virtual work of inertia forces originating from Eq. (3.59) then reads... 

Above, the various terms of the principle of virtual work have been compiled, containing different temporal and spatial derivatives of the mechanical displacements and rotations as well as of the electric potential of the adaptive beam. In the finite element approach, these continuous functions have to be approximated by discrete values at certain nodal points with adequate local interpolations in between. The degrees of freedom at such nodal points associated with a beam finite element may be summarized in the element vector i/j(t). When elements with two nodes are chosen, the degrees of freedom at both element ends are contained ... 

Methods of analytical mechanics provide the natural basis to develop such a generalized approach. Within the bounds of quasistatic problems, methods of analytical statics are sufficient. The use of the principle of virtual work, instead of the energy balance equation, permits one to generalize the theory of fracture and fatigue to multi parametric problems and to omit restrictions on the potential character of external and internal forces. In this paper, only "non-healing" cracks are considered typical for most structural materials. Therefore, we consider mechanical systems with unilateral constraints. The principle of virtual work for such systems takes the form a system with ideal unilateral constraints stays in the equilibrium state if and only if the summed virtual work of all active forces on all small displacements compatible with the constraints is equal to zero or negative ... 

Using the principle of virtual work (Fig. 19) and the calculated shear force and bending moment distributions, the relative horizontal displacement can be obtained. Having the application of opposite unit forces in the virtual system, the internal virtual forces (Vim, Mint) can also be obtained. 

In compliance with Lanczos [118], dF may be called the effective force. As stated by Eq. (3.56b), it reflects the extension of the impressed force resultant dF by the inertia term —a dm. In this way it is possible to reduce a problem of dynamics formally to one of statics and, thus, to deduce the differential equations describing the effects of accelerated motion. This is known as d Alembert s principle. Because of its reactive character, as mentioned by Budo [39] and discussed in Section 3.4.2, the effective force dF does not perform virtual work. With the virtual displacements 6u, it may be written for the particle with the aid of Eq. (3.56b) ... 

D Alembert s principle in the Lagrangian version, as derived in Section 3.4.5, uses infinitesimal virtual displacements about the instantaneous system state. For this reason, it is referred to as a differential principle. When infinitesimal virtual deviations from the entire motion of a system between two instants in time are examined, then it is an integral principle like Hamilton s principle, see Goldstein [86], Sokolnikoff [167], Szabb [172] or Morgenstern and Szaho [126]. Here the derivation from the prior to the latter principle will be demonstrated, starting with conversion of the virtual work of the inertia loads included in Eq. (3.59). With Eq. (3.54) and acceleration as derivative of velocity, it... 

With its substitution into d Alembert s principle in the Lagrangian version of Eq. (3.59), one obtains Eq. (3.73). Integration over the period of time from to to ti, where the virtual displacements are zero by definition at these end points such that 5u (to) = 6u (ti) = 0, leads to the general Hamilton s principle of... 

The virtual strain measures are related to the virtual displacements, just as it is given for the actual case in Eqs. (6.12), (6.13), and (6.14a). When these kinematic relations are substituted into the principle, different derivatives of the virtual displacements appear. These may be eliminated with the aid of integration by parts to summarize the contributions connected to every virtual displacement. In order to satisfy the principle, each of the resulting integrands needs to vanish. Not to be pursued here, the natural boundary conditions therewith can be determined. The sought-after equilibrium conditions in directions of the coordinates x, s, and n take the following form... 

Take a variation of the equilibrium Equations (25)-(30) and then apply the virtual displacements principle using the Ritz variational technique, incorporating the constitutive relationships, using the section properties parameters, adopting a second order approximation for displacement components and internals actions, and evaluating the conservative surface tractions at the boundaries, for monosymmetric beams, consider the case of no initial force and ignoring the axial displacement terms, the second variation of Total Potential equation can be reduced to ... 

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