Wuhan Universityof Technology117.2 Beam flexure: elementary caseSumming all forces acting vertically leads to the first dynamic equilibriumrelationshipoV(r,t)drl - fi(a,t) da = 0V(r,t) +p(r,t) dr-[v(r,t)+or2u(r,t)fi(r,t) da=m(r) daOt2av(r,t)a2v(r,t)=p(c,t) -m(a)Ot2ar17-6
17-6 Wuhan University of Technology Summing all forces acting vertically leads to the first dynamic equilibrium relationship 17.2 Beam flexure: elementary case
Wuhan Universityof Technology117.2 Beam flexure: elementary caseThe second equilibrium relationship is obtained by summing moments aboutpointAontheelasticaxis.Afterdroppingthetwosecondordermomenttermsinvolving the inertia and applied loadings, one getsaM(r,t)M(a,t) +V(r,t)da-M(r,t) +OrBecauserotational inertiaisneglected,this equationsimplifiesdirectlytothestandardstaticrelationshipbetweenshearandmomentM(r,t) = V(a,t)ara2M(r,t)a2v(r,t)p(a,t)mr0r2Ot217-7
17-7 Wuhan University of Technology The second equilibrium relationship is obtained by summing moments about point A on the elastic axis. After dropping the two secondorder moment terms involving the inertia and applied loadings, one gets 17.2 Beam flexure: elementary case Because rotational inertia is neglected, this equation simplifies directly to the standard static relationship between shear and moment