exd_beam2_tΒΆ

Purpose:

The frame structure defined in exd_beam2_m is exposed in this example to a transient load. The structural response is determined by a time stepping procedure.

Description:

The structure is exposed to a transient load, impacting on the center of the vertical beam in horizontal direction, i.e. at the 4th degree-of-freedom. The time history of the load is shown below. The result shall be displayed as time history plots of the 4th degree-of-freedom and the 11th degree-of-freedom. At time \(t=0\) the frame is at rest. The timestep is chosen as \(\Delta t= 0.001\) seconds and the integration is performed for \(T=1.0\) second. At every 0.1 second the deformed shape of the whole structure shall be displayed.

_images/exd2f1.svg

The load is generated using the gfunc-function. The time integration is performed by the step2-function. Because there is no damping present, the C-matrix is entered as [ ].

dt=0.005;    T=1;
% --- the load -----------------------------------------------
G=[0 0; 0.15 1; 0.25 0; T 0];   [t,g]=gfunc(G,dt);
f=zeros(15, length(g));       f(4,:)=1000*g;
% --- boundary condition, initial condition ------------------
bc=[1 0; 2 0; 3 0; 14 0];
a0=zeros(15,1);               da0=zeros(15,1);
% --- output parameters --------------------------------------
times=[0.1:0.1:1];           dofs=[4 11];
% --- time integration parameters ----------------------------
ip=[dt T 0.25 0.5];
% --- time integration ---------------------------------------
k=sparse(K);                  m=sparse(M);
[a,da,d2a,ahist,dahist,d2ahist]...
=step2(k,[],m,f,a0,da0,bc,ip,times,dofs);

The requested time history plots are generated by the following commands

figure(1), plot(t,ahist(1,:),'-',t,ahist(2,:),'--')
grid, xlabel('time (sec)'), ylabel('displacement (m)')
title('Displacement(time) for the 4th and 11th'...
      ' degree-of-freedom')
text(0.3,0.009,'solid line = impact point, x-direction')
text(0.3,0.007,'dashed line = center, horizontal beam,'...
               ' y-direction')
_images/exd2f2.svg

The deformed shapes at time increment 0.1 sec are stored in a. They are visualized by the following commands:

figure(2),clf, axis('equal'), hold on, axis off
sfac=25;
title('Snapshots (sec), magnification = 25');
for i=1:5;
  Ext=Ex+(i-1)*3;            eldraw2(Ext,Ey,[2 3 0]);
  Edb=extract_ed(Edof,a(:,i));
  eldisp2(Ext,Ey,Edb,[1 2 2],sfac);
  Time=num2str(times(i));   text(3*(i-1)+.5,1.5,Time);
end;

Eyt=Ey-4;
for i=6:10;
  Ext=Ex+(i-6)*3;            eldraw2(Ext,Eyt,[2 3 0]);
  Edb=extract_ed(Edof,a(:,i));
  eldisp2(Ext,Eyt,Edb,[1 2 2],sfac);
  Time=num2str(times(i));   text(3*(i-6)+.5,-2.5,Time);
end
_images/exd2f3.svg