## ABSTRACT

The principle of virtual work, proposed in Section 8.4 in the context of 3D solids, may be extrapolated, following a similar line of demonstration, to 1D and 2D solids. As regards rectilinear beams, it is sufficient to substitute Equation 8.23 with

= − ∂∫([ ]*{ }) { }T dη 0

(12.1)

whereby, instead of Equation 8.24, we obtain

L Q z QF a b l

= ∂ −∫{ } [ ]{ } [{ } { }]T Tdη η 0

0 (12.2)

0 and l being the coordinates of the beam ends. We then obtain the equation of the principle of virtual work for a rectilinear beam, sub-

jected to loads distributed over the span and to loads concentrated at the ends,

{ } { } { } { } [{ } { }]Q q z z Qa b l

0 ∫ ∫= + η η (12.3)

where, adopting the same nomenclature used in Section 10.3, {Qa} is the vector of the static characteristics, {qb} is the vector of deformation characteristics, {ℱa} is the vector of the distributed forces and {ηb} is the displacement vector.