Excurso introductorio: la dramaturgia de Lenz

The above-mentioned examples of mechanical problems allow for a simple application of the oldest known mechanical principle of virtual work (Archytas of Tarentum – 400, Stevin 1550, Galileo 1600, Newton 1678, Huygens 1700, Mach 1880). Its axiomatic extension to non-rigid bodies and systems (Navier 1830, Lamé 1860, Saint-Venant 1870, Mohr 1900) includes also the work of internal forces into the virtual work.

Let's define a virtual work of external forces:

virt virt work then axiomatically says (without any proof) that for every system of forces { ,P S_{i j}} that are in equilibrium and for every virtual displacement {δr s_i, }_j the total virtual work of external and internal forces equals to zero:

virt virt virt 0

P S i i j j

i j

W W W P r P s

δ =δ +δ =

∑

∑

Several popularisation formulations have been created with the aim to explain the principle in an intelligible way. E. Mach analyses them in a voluminous chapter of his famous

“Mechanics” (7^th edition [24]) and he sums up them in: ES GESCHIEHT NICHTS, WENN NICHTS GESCHEHEN KANN, i.e. if nothing can happen (i.e. there is no motion in any direction under which the impulse of force Pt could change into momentum mv, etc.), then nothing really happens (i.e. there is no motion, it is an equilibrium steady state).

If we have distributed loads and internal forces, the sums from (2.5.3) become integrals, which is nothing important, just a formal modification of the notation of the principle of virtual work that may in general contain both sums and integrals depending on the type of loads (point, linear, surface, volume) and internal forces (1D, 2D, 3D models, singularity) and we may have a component, tensor or matrix notation.

As the work is scalar, we can add together work done in parts Ω_e (subdomains, elements) of domain Ω into the total work to which the principle applies. Should it happen that a system contains parts that have no connection to the remainder of the system, we can simply find out – through the application of a virtual displacement that is non-zero only in such parts – that the principle of virtual work applies to every such unconnected part separately, which is correct as the bodies (domains) are independent. Therefore, we can limit ourselves only to systems (domains) Ω where each part Ω_e has at least one connection with another part Ω. There is an unlimited number of such connections in classical continuum Ω divided into elements Ω_e. They are geometrically described by the Saint-Venant equations of compatibility. We will, however, consider more general systems consisting of elements of various dimension and non-classical continuum. The parameterisation of deformation degrees of freedom of such systems ensures that the analysis of their behaviour is integrated into the

standard procedure.

The principle of virtual work (2.5.3) applies to arbitrary non-rigid bodies and was axiomatically extended also to elastic, or more generally deformable, continuum of 1D, 2D, 3D dimension. We employ symbols from the overview in art. 2.3. and from notation:

P : i External volume and surface forces b, p ( ,1)d All the terms are perceived as generalised in such a way that they contain not only the classical 3D continuum 3D (d =3,s=6), but also e.g. planar 2D problems (d =2,s=3), axially symmetrical problem (d =2,s=4), Mindlin shells (d =6,s=8), etc. Statical and also geometrical quantities are written in the form of matrix vectors, i.e. column matrices, the dimension of which is added in brackets. External forces b, p are given. Other quantities are assumed as depending only on N parameters d, similarly to the derivation of the FEM procedure, i.e. after the parameterisation of the system has been carried out through the selection of base functions that is closely related to the division into finite elements and to the chosen polynomial distribution over elements. We have already put down the global forms of these dependencies on d, that is for the whole analysed system, using matrices N, B extended to dimension ( ,d N , ( ,) s N . Let us remind that in dimension ( , )) a b the a is the number of lines and b the number of columns of the matrix.

For linear problems it is sufficient to take the final state of the system after loads and deformations have been applied as the equilibrium system { , , }b p σ . The following N displacements of the system are sequentially chosen as the virtual displacement:

1 states. In textbooks we often see its size set to one, which is neither necessary nor illustrative, as it soon disappears from the derivation and the physical meaning is not apparent. The following virtual displacements and strains correspond to the above-mentioned parameters:

2.5 Principle of virtual work applied in FEM programs

if we use subscript j to mark the row submatrices of which the global matrix shapes N are composed

With the selected notation, the principle of virtual work (2.5.3) now gets for the continuum in domain Ω with boundary Γ the form:

This can be satisfied for non-zero d only if the expression in the box brackets equals to zero. _j When comparing with no. 11, 14, art. 2.3 of the overview, it can be found that the expression in the first round brackets is the j -th parameter f_j of vector f of all load parameters and the expression in the second round brackets is the j -th line K of global stiffness matrix K of _j the system. Therefore, after rearrangement to both sides of the equals sign, equation (2.5.9) reads:

j j

K d = f (2.5.10)

If we gradually select as virtual displacements all N parametric states, i.e. j=1, 2,K , we ,N obtain N equation of type (2.5.10), in other words the whole system

=

Kd f (2.5.11)

that is identical with system (2.4.21) derived through the Lagrangean variational principle in the previous art. 2.4. from the condition of Π =min.

Why is the system of equations, i.e. also the solution, identical? There are two reasons for this: mechanical and mathematical. In terms of mechanics, the principle of virtual work is superior to all equilibrium conditions and it can generate an arbitrary number of such conditions as there exists an arbitrary (infinite) number of selectable virtual displacements. It is superior to the Lagrangean variational principle Π =min, or, as the case may be, δΠ =0 that is one of its consequences. In terms of mathematics, we have limited in both situations the deformability of the system to states that are a linear combination of the same unit parametric states, which reduces the unlimited freedom of the continuum to “N degrees of (deformation) freedom” that are the same in both procedures.

Is there any advantage in the derivation through the principle of virtual work in comparison with the Lagrangean variational principle? Yes, and the advantage is significant, as during the derivation we do not have to limit ourselves just to problems of classical equilibrium that is independent on the trajectory (= history proceeding to the analysed state). The system { , , }b p σ must be in equilibrium in every time instant and thus also the increment at time dt, {db p,d ,dσ is in equilibrium and the principal of virtual work can be applied to it. This } allows for the application of FEM – solution in steps, increments of load – using what is called an incremental method. The main problem is then how to add together increments dσ, dε, du in a way that is consistent with the laws of physics if deformations are large. Instead of details, let us give only brief notes that make the modelling of structures for inputs needed in FEM programs easier:

1. If body Ω is subjected to singular forces concentrated in figures of lower dimension than that of Ω (e.g. for 3D body Ω on 1D lines Γ or in 0D points "i"), then we can use directly the definition of such loads: the measure of the area on which it acts converges to zero but the resultant remains constant, as the intensity increases in indirect proportion to the measure. Consequently, instead of integrals in (2.5.9) se use directly the sums of products of the resultants with the virtual displacements (or moments with rotations), i.e. what is termed concentrated impact, see overview no. 15, art. 2.3. For linear loads we use line integrals, always with the aim to get the virtual work in the form of a sum or integral of products of the force and distance. It is thus possible to calculate load elements f , i.e. vector f of the right-hand sides of the set of equations Kd₌f , for a very wide set of loads. FEM program are usually limited to : Loads distributed over elements, of general direction, decomposed into components

, ,

x y z

p p p or b b b , _x, _y, _z

Concentrated forces with components F F F , or also moments _x, _y, _z M_x,M_y,M in mesh _z nodes with deformation parameters , ,u v w , or also ϕ ϕ ϕ_x, _y, _z,

Linear, mainly straight-line, loads on element boundaries with components into the direction