Inertia Load Contributions

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 

Taking the variation of Eq. (7.65), the global virtual position vector 5p(x, s, t) in the inertial reference frame may be obtained in terms of the local virtual position vector Spg(x, s) in the moving reference frame  

In view of the above discussion on the transformation properties, it needs to be noted that Eq. (8.30) and thus the resulting equations of motion contain time-dependent matrices. The mathematical theory for such non-autonomous systems has primarily been developed with regard to a periodic dependence on time. The most prominent approaches to solve these systems are the methods of Floquet [75] and of Hill [95]. For further details, see Prothmann [146], Gasch and Knothe [78], or Meirovitch [125]. To pave the way for an eventually autonomous system with a less expensive theoretical framework, not to exceed the scope of the study at hand, we will abstain from a time-dependent orientation of the clamping of the beam. 

Remark 8.1. The rotation around an axis of the inertial reference frame will be the only guided motion of the considered system. 

the rotational transformation at the clamped end of the beam is restricted to a constant description of the orientation  

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Reactive control is also possible through synchronous condensers. As they rotate, the rotor stores kinetic energy which tends to absorb sudden Huctuations in the supply system, such as sudden loadings. They are. however, sluggish in operation and very expensive compared to thyristor controls. Their rotating masses add inertia, contribute to the transient oscillations and add to the fault level of the system. All these factors render them less suitable for such applications. Their application is therefore gradually disappearing. 

At this point, as far as shaking forces go, the gas forces do not make a contribution. If the rod load or bearing loads are to be analyzed, the gas forces must be calculated and added vectorially to the inertia forces to... 

The slab thickness is equal to 15 cm and is considered to contribute to the moment of inertia of the beams with an effective flange width. In addition to the self weight of the beams and the slab, a distributed dead load of 2 kN/m, due to floor finishing and partitions and imposed live load with nominal value of 1.5 kN/m, is considered, in the combination with gravity loads ( persistent design situation ). Nominal dead and live loads are multiplied by load factors of 1.35 and 1.5, respectively (Eq. 1). Following ECS, in the seismic design combination, dead loads are taken with their nominal value, while live loads with 30% of their nominal value (Eq. 2). 

The second approach ignores the strength contribution of the core and assumes that the two outer skins provide all the rigidity (Fig. 5-61). The equivalent moment of inertia is then equal to Ix = b h - hi)/l2. This formula results in conservative accuracy, since the core does contribute to the stress-absorbing function. It also adds a built-in safety factor to a loaded beam or plate element when safety is a concern. 

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 ... 

With the aid of the principle of virtual work, the equilibrium and boundary conditions can be obtained for the quasi-static case, where, in principle, loads may change over time but inertia effects are not considered. The contributions required for this purpose have been already obtained and will be joined together in the following. The internal virtual work SU t) is given by Eq. (8.23), while the external virtual work SV t) of Elq. (8.24) reduces for the quasi-static case to those contributions due to the applied loads... 

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