Question 12.3.2: Use the principle of virtual work to derive Eqn 12.3-25: P0 ...
Use the principle of virtual work to derive Eqn 12.3-25:
P_0=-A^{-T}\int{N^{T}_{q}qdV }
P_0=-A^{-T}\int_V{N^{T}_{q}q \ dV } (12.3-25)
With reference to the element ijk in Fig. 12.3-2, if virtual displacements \bar{\Delta } occur, then:
external virtual work done by the nodal forces P_0 = P^{T}_{0}\bar{\Delta }
external virtual work done by the distributed forces q
=\int{q^T\bar{\delta } \ dV }
=\int{q^T\left[N_qA^{-1}\bar{\Delta \ } \right]dV} (from Eqn 12.3-24)
\delta ^\ast =N_qA^{-1}\Delta ^\ast (12.3-24)
virtual strain energy due to the stresses \sigma caused by q is
\int{\sigma^T\bar{\varepsilon } \ dV }
where \bar{\varepsilon } are the virtual strains compatible with the virtual nodal displacements \bar{\Delta \ }.
The principle of virtual work (Eqn 4.14-4) states that
P^T\bar{\Delta }+\int{p^T\bar{\delta } \ dA}=F^T\bar{e}=\int{\sigma ^T\bar{\varepsilon } \ d(vol.)} (4.14-4)
P^{T}_{0}\bar{\Delta }+\int{q^TN_qA^{-1}\bar{\Delta } \ dV}=\int{\sigma ^T\bar{\varepsilon } \ dV} (12.3-28)
As pointed out in Example 12.3-1, under fixed-node condition, the applied forces q cannot cause any internal stress a. Hence
\int{\sigma ^T\bar{\varepsilon } \ dV }=\int{0 \bar{\varepsilon }} \ dV=0 (12.3-29)
Hence Eqn 12.3-28 becomes
P^{T}_{0}\bar{\Delta }+\int{q^TN_qA^{-1}\bar{\Delta } \ dV}=0
Since this equation holds for all possible \bar{\Delta }, we have
P^{T}_{0}+\int{q^TN_qA^{-1} \ dV}=0
or
P_0=-A^{-T}\int{N^{T}_{q}q \ dV }
which is Eqn 12.3-25.
P_0=-A^{-T}\int_V{N^{T}_{q}q \ dV } (12.3-25)
The reader’s attention is drawn to the following points :
(a) The quantity \int{q^T\bar{\delta } \ dV } is an external virtual work and not an internal virtual work (as many students tend to think); this is because the distributed forces q are externally applied forces.
(b) The virtual strain energy \int{\sigma ^T\bar{\varepsilon } \ dV } is zero; this results from the assumed properties of the fictitious element, as explained in Example 12.3-1. An element made of a real material cannot, obviously, behave this way.
