Segédlet BMEGEMMAGM5 “Végeselem módszer alapjai” tárgy laborgyakorlatához.
Készítette: Dr. Kossa Attila, BME Műszaki Mechanikai Tanszék, 2013. szeptember 27.

Mh5=-p*(ξ^2)/2;
Mh4=Mh5-FB*(ξ-3L);
Mh3=-p(4L)(ξ-2L)-FB(ξ-3L);
Mh2=Mh3-F1(ξ-5L);
Mh1=Mh2-M1;

e1=D[v1[ξ],{ξ,2}]==-Mh1/IE;
e2=D[v2[ξ],{ξ,2}]==-Mh2/IE;
e3=D[v3[ξ],{ξ,2}]==-Mh3/IE;
e4=D[v4[ξ],{ξ,2}]==-Mh4/IE;
e5=D[v5[ξ],{ξ,2}]==-Mh5/IE;

bc1=v1[7L]==0;
bc2=v1'[7L]==0;
bc3=v1[6L]==v2[6L];
bc4=v1'[6L]==v2'[6L];
bc5=v2[5L]==v3[5L];
bc6=v2'[5L]==v3'[5L];
bc7=v3[4L]==v4[4L];
bc8=v3'[4L]==v4'[4L];
bc9=v4[3L]==v5[3L];
bc10=v4'[3L]==v5'[3L];

solution=FullSimplify[DSolve[{e1,e2,e3,e4,e5,bc1,bc2,bc3,bc4,bc5,bc6,bc7,bc8,bc9,bc10},{v1,v2,v3,v4,v5},ξ]][[1]]

fb=FullSimplify[Solve[(v5[3L]/.solution)==0,FB]][[1]]

sol=FullSimplify[solution/.fb];

FullSimplify[v1[ξ]/.sol]
FullSimplify[v2[ξ]/.sol]
FullSimplify[v3[ξ]/.sol]
FullSimplify[v4[ξ]/.sol]
FullSimplify[v5[ξ]/.sol]

vx1=FullSimplify[(v1[ξ]/.sol)/.{ξ->(7L-x)}];
vx2=FullSimplify[(v2[ξ]/.sol)/.{ξ->(7L-x)}];
vx3=FullSimplify[(v3[ξ]/.sol)/.{ξ->(7L-x)}];
vx4=FullSimplify[(v4[ξ]/.sol)/.{ξ->(7L-x)}];
vx5=FullSimplify[(v5[ξ]/.sol)/.{ξ->(7L-x)}];

vx1

vx2

vx3

vx4

vx5

L=0.5;F1=-4500;M1=2000;p=-300;IE=(((((0.03))^4)π)/64.)*210*(10^9);

p1=Plot[vx1,{x,0,L}];
p2=Plot[vx2,{x,L,2L}];
p3=Plot[vx3,{x,2L,3L}];
p4=Plot[vx4,{x,3L,4L}];
p5=Plot[vx5,{x,4L,7L}];

Show[{p1,p2,p3,p4,p5},PlotRange->All,AxesOrigin->{0,0},AxesLabel->{"x [m]","v [m]"}]

Minimize[vx3,x]

Expand[vx1]
Expand[vx2]
Expand[vx3]
Expand[vx4]
Expand[vx5]