Esercitazioni d'esame svolte di Teoria delle Strutture

Calcolo delle soluzioni per una trave

Trave EBclear all;clear vars;close all;clc;

Dati del problemaE=10000;L=2000;q=1;bb=200;h1=200;h2=180;KK=1000000000;Kf=1000000000;

Definizione di variabili simboliche

syms v1(z) v2(z) v3(z)

Calcolo h

syms a b;

h=a*z+b;

bc1=subs(h,z,0)==h1;

bc2=subs(h,z,2*L)==h2;

eqnsh=[bc1 bc2];

solh=solve(eqnsh,unksh);

h=subs(h,solh)

I=bb*(h^3)/12;

Equazioni differenziali

eq1=E*I*diff(v1,z,4)==q;

eq2=E*I*diff(v2,z,4)==0;

eq3=E*I*diff(v3,z,4)==q;

eq1(z) =eq2(z) =eq3(z) =

Soluzioni

dsol=dsolve(eq1,eq2,eq3);

dsol = struct with fields:

v2: (C2*z^3)/6 + (C5*z^2)/2 + C8*z + C11

v1: C12 + 960000*log(z - 40000) + z*(C9 - 24*log(z - 40000) + 24) +(C1*z^3)/6 + (C6*z^2)/2

v3: C10 + 960000*log(z - 40000) + z*(C7 - 24*log(z - 40000) + 24) +(C3*z^3)/6 + (C4*z^2)/2

Spostamenti

vs1=dsol.v1;

vs2=dsol.v2;

vs3=dsol.v3;

Rotazioni

fs1=-diff(vs1,z);

fs2=-diff(vs2,z);

fs3=-diff(vs3,z);

Curvature

chis1=diff(fs1,z);

chis2=diff(fs2,z);

chis3=diff(fs3,z);

Momento flettente

Ms1=E*I*chis1;

Ms2=E*I*chis2;

Ms3=E*I*chis3;

TaglioTs1=diff(Ms1,z);
Ts2=diff(Ms2,z);
Ts3=diff(Ms3,z);
% Condizioni al contorno
bcA1=subs(vs1,z,0);
bcA2=subs(Ms1,z,0);
bcB1=subs(vs1-vs2,z,L/3);
bcB2=subs(fs1-fs2,z,L/3);
bcB3=subs(Ts1-Ts2,z,L/3);
bcB4=subs(Ms1-Ms2,z,L/3);
bcC1=subs(vs2-vs3,z,L);
bcC2=subs(fs2-fs3,z,L);
bcC3=subs(Ts2-Ts3,z,L);
bcC4=subs(Ms2-Ms3,z,L);
bcD1=subs(fs3,z,2*L);
bcD2=subs(Ts3,z,2*L);
syms C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12
eqns=[bcA1, bcA2, bcB1, bcB2, bcB3, bcB4, bcC1, bcC2 bcC3 bcC4 bcD1 bcD2];
unks=[C1, C2, C3, C4, C5, C6, C7, C8, C9, C10, C11, C12];
% Soluzione del sistema algebrico lineare
sol=solve(eqns,unks);
% Soluzioni
vss1=subs(vs1, sol);
vss2=subs(vs2, sol);
vss3=subs(vs3, sol);
fss1=subs(fs1, sol);
fss2=subs(fs2, sol);
fss3=subs(fs3, sol);
Mss1=subs(Ms1, sol);
Mss2=subs(Ms2, sol);
Mss3=subs(Ms3, sol);
Tss1=subs(Ts1, sol);
Tss2=subs(Ts2, sol);
Tss3=subs(Ts3, sol);

ener_int=1/2*E*I*int(chia^2, z, 0, 2*L);

ener_car=-int(q*va, z, 0, L/3)-int(q*va, z, L, 2*L);

ener_vin=1/2*KK*(subs(va,z,0))^2+1/2*Kf*(subs(fa,z,2*L))^2;

ener_tot=ener_int+ener_car+ener_vin;

eqns=sym(zeros(nv,1));

for i=1:nv

eqns(i)=diff(ener_tot,unks(i));

end

sol=solve(eqns,unks);

vas=eval(subs(va,sol));

var=vpa(vas);

Mas=-E*I*diff(var,z,2);

Tas=diff(Mas,z);

zva=linspace(0,2*L,24);

vva=subs(var,z,zva);

Mva=subs(Mas,z,zva);

Tva=subs(Tas,z,zva);

figure(4)

hold on;

set(gca,'ydir','reverse');

plot(zv1,vv1,'b-');

plot(zv2,vv2,'b-');

plot(zv3,vv3,'b-');

plot(zva,vva,'r*');

title('Displacement along y');

grid on;

figure(5)

hold on;

set(gca,'ydir','reverse');

plot(zv1,Mv1,'b-');

plot(zv2,Mv2,'b-');

plot(zv3,Mv3,'b-');

plot(zva,Mva,'r*');

title('Bending

moment');grid on;figure(6)hold on;set(gca,'ydir','reverse');plot(zv1,Tv1,'b-');plot(zv2,Tv2,'b-');plot(zv3,Tv3,'b-');plot(zva,Tva,'r*');title('Shear force');grid on;