numerical evaluation of dynamic response
DESCRIPTION
Numerical Evaluation of Dynamic ResponseTRANSCRIPT
A.Y. 2014-2015 Earthquake Engineering and Structural Control Course
University of Naples “Federico II”
Department of Structures for Engineering and Architecture
2-years Master in Structural and Geotechnical Engineering --- 1-year Master in Emerging Technologies for Construction
Homework #4: Numerical Evaluation of Dynamic Response
Giorgio Serino - Full Professor of Structural Engineering Nicolò Vaiana - PhD Student in Structural, Geotechnical and Seismic Engineering - Master Student in Emerging Technologies for Construction
t
P
Homework
Matlab Code for Time History Analysis of Linear SDF Systems
University of Naples “Federico II” - Department of Structures for Engineering and Architecture Prof. Giorgio Serino - PhD Student Nicolò Vaiana
CASE A: SYSTEM SUBJECTED TO EXTERNAL HARMONIC FORCE
Consider the Single Degree of Freedom model with viscous damping shown in Figure 1 and subjected to an external harmonic force shown in Figure 2.
We want to determine the response of this system to this excitation by the Newmark’s Constant Average Acceleration Method (Noniterative Formulation), implemented by a com-puter program in Matlab Language.
A.1 Determine the peak values of:
- displacement of the system
- velocity of the system
- acceleration of the system
A.2 Plot:
- the time history of displacement, velocity and acceleration of the system
- spring force – displacement relation
- damping force – displacement relation
- spring force + damping force – displacement relation
m=100 Kg k=150 N/m c=20 Ns/m
Figure 1
P(t)=p*cos(omega*t) p=1000 N omega=4 rad/s
Figure 2
40-0.4
0
0.4
t
ag/g
Homework
Matlab Code for Time History Analysis of Linear SDF Systems
University of Naples “Federico II” - Department of Structures for Engineering and Architecture Prof. Giorgio Serino - PhD Student Nicolò Vaiana
CASE B: SYSTEM SUBJECTED TO EARTHQUAKE EXCITATION
Consider the Single Degree of Freedom model with viscous damping shown in Figure 3 and subjected to an horizontal earthquake excitation shown in Figure 4: the earthquake record used is the 2002 Gilroy record.
We want to determine the response of this system to this excitation by the Newmark’s Constant Average Acceleration Method (Noniterative Formulation), implemented by a com-puter program in Matlab Language.
B.1 Determine the peak values of:
- relative displacement of the system
- relative velocity of the system
- total acceleration of the system
B.2 Plot:
- the time history of relative displacement and velocity and total acceleration
- spring force – relative displacement relation
- damping force – relative displacement relation
- spring force + damping force – relative displacement relation
m=100 Kg k=150 N/m c=20 Ns/m
Figure 3
Gilroy: ag in units of g, 3997 steps, dt=0.01
Figure 4
Homework
Matlab Code for Time History Analysis of Linear SDF Systems
University of Naples “Federico II” - Department of Structures for Engineering and Architecture Prof. Giorgio Serino - PhD Student Nicolò Vaiana
MATLAB CODE: CASE A
%DYNAMIC SYSTEM PARAMETERS
m=100; %Kg mass
k=150; %N/m linear spring coefficient
c=20; %Ns/m linear dashpot damping coefficient
%EXTERNAL HARMONIC FORCE
T=50; %s harmonic force duration
dt=0.01; %s time step
t=0:dt:T; % time vector
N=length(t); % time steps number
for i=1:N
omega=2; %rad/s forcing frequency
p=1000; %N force amplitude
P(i)=p*cos(omega*t(i));
end
Homework
Matlab Code for Time History Analysis of Linear SDF Systems
University of Naples “Federico II” - Department of Structures for Engineering and Architecture Prof. Giorgio Serino - PhD Student Nicolò Vaiana
MATLAB CODE: CASE A
%NEWMARK'S CONSTANT-AVERAGE-ACCELERATION METHOD
%INTEGRATION PARAMETERS
alfa=0.5;
beta=0.25;
%INTEGRATION CONSTANTS
a1=1/(beta*dt^2);
a2=1/(beta*dt);
a3=1/(2*beta);
a4=alfa/(beta*dt);
a5=alfa/beta;
a6=dt*((alfa/(2*beta))-1);
%EFFECTIVE STIFFNESS
kroof=a1*m+a4*c+k;
%INITIAL CONDITIONS
u(1)=0;
ud(1)=0;
udd(1)=(P(1)-c*ud(1)-k*u(1))/m;
Homework
Matlab Code for Time History Analysis of Linear SDF Systems
University of Naples “Federico II” - Department of Structures for Engineering and Architecture Prof. Giorgio Serino - PhD Student Nicolò Vaiana
MATLAB CODE: CASE A
%CALCULATIONS FOR EACH TIME STEP
for i=1:N-1
%INCREMENTAL EFFECTIVE LOAD AT TIME STEP i
dP(i)=P(i+1)-P(i);
dProof(i)=dP(i)+(a2*m+a5*c)*ud(i)+(a3*m+a6*c)*udd(i);
%SOLUTION FOR du AT TIME STEP i
du(i)=dProof(i)/kroof;
%INCREMENTAL VELOCITY AT TIME STEP i
dud(i)=a4*du(i)-a5*ud(i)-a6*udd(i);
%INCREMENTAL ACCELERATION AT TIME STEP i
dudd(i)=a1*du(i)-a2*ud(i)-a3*udd(i);
%STATE OF MOTION AT TIME STEP i+1
u(i+1)=u(i)+du(i);
ud(i+1)=ud(i)+dud(i);
udd(i+1)=udd(i)+dudd(i);
end
Homework
Matlab Code for Time History Analysis of Linear SDF Systems
University of Naples “Federico II” - Department of Structures for Engineering and Architecture Prof. Giorgio Serino - PhD Student Nicolò Vaiana
MATLAB CODE: CASE A
%PEAK VALUES
umax=max(abs(u))
udmax=max(abs(ud))
uddmax=max(abs(udd))
%PLOTS
%DISPLACEMENT TIME HISTORY
figure
plot(t,u);
xlabel('time [s]');
ylabel('Displacement[m]');
%VELOCITY TIME HISTORY
figure
plot(t,ud);
xlabel('time [s]');
ylabel('Velocity [m/s]');
%ACCELERATION TIME HISTORY
figure
plot(t,udd);
xlabel('time [s]');
ylabel('Acceleration [m/s^2]');
Homework
Matlab Code for Time History Analysis of Linear SDF Systems
University of Naples “Federico II” - Department of Structures for Engineering and Architecture Prof. Giorgio Serino - PhD Student Nicolò Vaiana
MATLAB CODE: CASE A
%SPRING FORCE - DISPLACEMENT RELATION
for i=1:N
fs(i)=k*u(i);
end
figure
plot(u,fs);
xlabel('Displacement [m]');
ylabel('fs [N]');
%DAMPING FORCE - DISPLACEMENT RELATION
for i=1:N
fd(i)=c*ud(i);
end
figure
plot(u,fd);
xlabel('Displacement [m]');
ylabel('fd [N]');
%SPRING FORCE + DAMPING FORCE - DISPLACEMENT RELATION
f=fs+fd;
figure
plot(u,f);
xlabel('Displacement [m]');
ylabel('f = fs + fd [N]');
Homework
Matlab Code for Time History Analysis of Linear SDF Systems
University of Naples “Federico II” - Department of Structures for Engineering and Architecture Prof. Giorgio Serino - PhD Student Nicolò Vaiana
MATLAB CODE: CASE A
umax =
6.2312
udmax =
11.127
uddmax =
19.392
5 10 15 20 25 30 35 40 45 50-8
-6
-4
-2
0
2
4
6
8
time [s]
Dis
pla
cem
ent
[m]
Homework
Matlab Code for Time History Analysis of Linear SDF Systems
University of Naples “Federico II” - Department of Structures for Engineering and Architecture Prof. Giorgio Serino - PhD Student Nicolò Vaiana
MATLAB CODE: CASE A
5 10 15 20 25 30 35 40 45 50-15
-10
-5
0
5
10
15
time [s]
Velo
city [
m/s
]Homework
Matlab Code for Time History Analysis of Linear SDF Systems
University of Naples “Federico II” - Department of Structures for Engineering and Architecture Prof. Giorgio Serino - PhD Student Nicolò Vaiana
MATLAB CODE: CASE A
0 5 10 15 20 25 30 35 40 45 50-20
-15
-10
-5
0
5
10
15
20
time [s]
Acc
eler
atio
n [
m/s
2]
Homework
Matlab Code for Time History Analysis of Linear SDF Systems
University of Naples “Federico II” - Department of Structures for Engineering and Architecture Prof. Giorgio Serino - PhD Student Nicolò Vaiana
MATLAB CODE: CASE A
-6 -4 -2 0 2 4 6 8-1000
-500
0
500
1000
Displacement [m]
fs [
N]
Homework
Matlab Code for Time History Analysis of Linear SDF Systems
University of Naples “Federico II” - Department of Structures for Engineering and Architecture Prof. Giorgio Serino - PhD Student Nicolò Vaiana
MATLAB CODE: CASE A
-6 -4 -2 0 2 4 6 8-300
-200
-100
0
100
200
300
Displacement [m]
fd [
N]
Homework
Matlab Code for Time History Analysis of Linear SDF Systems
University of Naples “Federico II” - Department of Structures for Engineering and Architecture Prof. Giorgio Serino - PhD Student Nicolò Vaiana
MATLAB CODE: CASE A
-6 -4 -2 0 2 4 6 8-1000
-500
0
500
1000
Displacement [m]
f =
fs +
fd
[N
]Homework
Matlab Code for Time History Analysis of Linear SDF Systems
University of Naples “Federico II” - Department of Structures for Engineering and Architecture Prof. Giorgio Serino - PhD Student Nicolò Vaiana
MATLAB CODE: CASE A
Homework
Matlab Code for Time History Analysis of Linear SDF Systems
University of Naples “Federico II” - Department of Structures for Engineering and Architecture Prof. Giorgio Serino - PhD Student Nicolò Vaiana
MATLAB CODE: CASE B
%DYNAMIC SYSTEM PARAMETERS
m=100; %Kg mass
k=150; %N/m linear spring coefficient
c=20; %Ns/m linear dashpot damping coefficient
%EFFECTIVE EARTHQUAKE FORCE
%GILROY
%ACCELERATION TIME HISTORY IN UNITS OF G
%3997 0.01
ACC=load('GILROY.txt');
ag=9.80665*ACC(:,1); %m/s^2 horizontal ground acceleration
dt=0.01; %s ground acceleration time step
T=39.96; %s ground acceleration duration
t=0:dt:T; % time vector
N=length(ag); % time steps number
for i=1:N
P(i)=-m*ag(i);
end
Homework
Matlab Code for Time History Analysis of Linear SDF Systems
University of Naples “Federico II” - Department of Structures for Engineering and Architecture Prof. Giorgio Serino - PhD Student Nicolò Vaiana
MATLAB CODE: CASE B
%NEWMARK'S CONSTANT-AVERAGE-ACCELERATION METHOD
%INTEGRATION PARAMETERS
alfa=0.5;
beta=0.25;
%INTEGRATION CONSTANTS
a1=1/(beta*dt^2);
a2=1/(beta*dt);
a3=1/(2*beta);
a4=alfa/(beta*dt);
a5=alfa/beta;
a6=dt*((alfa/(2*beta))-1);
%EFFECTIVE STIFFNESS
kroof=a1*m+a4*c+k;
%INITIAL CONDITIONS
u(1)=0;
ud(1)=0;
udd(1)=(P(1)-c*ud(1)-k*u(1))/m;
Homework
Matlab Code for Time History Analysis of Linear SDF Systems
University of Naples “Federico II” - Department of Structures for Engineering and Architecture Prof. Giorgio Serino - PhD Student Nicolò Vaiana
MATLAB CODE: CASE B
%CALCULATIONS FOR EACH TIME STEP
for i=1:N-1
%INCREMENTAL EFFECTIVE LOAD AT TIME STEP i
dP(i)=P(i+1)-P(i);
dProof(i)=dP(i)+(a2*m+a5*c)*ud(i)+(a3*m+a6*c)*udd(i);
%SOLUTION FOR du AT TIME STEP i
du(i)=dProof(i)/kroof;
%INCREMENTAL VELOCITY AT TIME STEP i
dud(i)=a4*du(i)-a5*ud(i)-a6*udd(i);
%INCREMENTAL ACCELERATION AT TIME STEP i
dudd(i)=a1*du(i)-a2*ud(i)-a3*udd(i);
%STATE OF MOTION AT TIME STEP i+1
u(i+1)=u(i)+du(i);
ud(i+1)=ud(i)+dud(i);
udd(i+1)=udd(i)+dudd(i);
end
Homework
Matlab Code for Time History Analysis of Linear SDF Systems
University of Naples “Federico II” - Department of Structures for Engineering and Architecture Prof. Giorgio Serino - PhD Student Nicolò Vaiana
MATLAB CODE: CASE B
%PEAK VALUES
umax=max(abs(u))
udmax=max(abs(ud))
uddtmax=max(abs(udd+ag’))
%PLOTS
%RELATIVE DISPLACEMENT TIME HISTORY
figure
plot(t,u);
xlabel('time [s]');
ylabel(‘Relative Displacement[m]');
%RELATIVE VELOCITY TIME HISTORY
figure
plot(t,ud);
xlabel('time [s]');
ylabel(‘Relative Velocity [m/s]');
%TOTAL ACCELERATION TIME HISTORY
figure
plot(t,udd+ag’);
xlabel('time [s]');
ylabel(‘Total Acceleration [m/s^2]');
Homework
Matlab Code for Time History Analysis of Linear SDF Systems
University of Naples “Federico II” - Department of Structures for Engineering and Architecture Prof. Giorgio Serino - PhD Student Nicolò Vaiana
MATLAB CODE: CASE B
%SPRING FORCE – RELATIVE DISPLACEMENT RELATION
for i=1:N
fs(i)=k*u(i);
end
figure
plot(u,fs);
xlabel(‘Relative Displacement [m]');
ylabel('fs [N]');
%DAMPING FORCE – RELATIVE DISPLACEMENT RELATION
for i=1:N
fd(i)=c*ud(i);
end
figure
plot(u,fd);
xlabel(‘Relative Displacement [m]');
ylabel('fd [N]');
%SPRING FORCE + DAMPING FORCE – RELATIVE DISPLACEMENT RELATION
f=fs+fd;
figure
plot(u,f);
xlabel(‘Relative Displacement [m]');
ylabel('f = fs + fd [N]');
t
P
Homework
Matlab Code for Time History Analysis of Non-Linear SDF Systems
University of Naples “Federico II” - Department of Structures for Engineering and Architecture Prof. Giorgio Serino - PhD Student Nicolò Vaiana
CASE A: SYSTEM SUBJECTED TO EXTERNAL HARMONIC FORCE
Consider the Single Degree of Freedom model with a non-linear spring and a non-linear dashpot shown in Figure 1 and subjected to an external harmonic force shown in Figure 2.
We want to determine the response of this system to this excitation by the Newmark’s Constant Average Acceleration Method (Iterative Formulation: Pseudo-Force Method), im-plemented by a computer program in Matlab Language.
A.1 Determine the peak values of:
- displacement of the system
- velocity of the system
- acceleration of the system
A.2 Plot:
- the time history of displacement, velocity and acceleration of the system
- spring force – displacement relation
- damping force – displacement relation
- spring force + damping force – displacement relation
m=100 Kg k1=150 N/m k3=10 N/m^3 k5=20 N/m^5
c2=20 Ns^2/m^2 n=2
Figure 1
P(t)=p*cos(omega*t) p=1000 N omega=4 rad/s
Figure 2
40-0.4
0
0.4
t
ag/g
Homework
Matlab Code for Time History Analysis of Non-Linear SDF Systems
University of Naples “Federico II” - Department of Structures for Engineering and Architecture Prof. Giorgio Serino - PhD Student Nicolò Vaiana
CASE B: SYSTEM SUBJECTED TO EARTHQUAKE EXCITATION
Consider the Single Degree of Freedom model with a non-linear spring and a non-linear dashpot shown in Figure 3 and subjected to an horizontal earthquake excitation shown in Figure 4: the earthquake record used is the 2002 Gilroy record.
We want to determine the response of this system to this excitation by the Newmark’s Constant Average Acceleration Method (Iterative Formulation: Pseudo-Force Method), im-plemented by a computer program in Matlab Language.
B.1 Determine the peak values of:
- relative displacement of the system
- relative velocity of the system
- total acceleration of the system
B.2 Plot:
- the time history of relative displacement and velocity and total acceleration
- spring force – relative displacement relation
- damping force – relative displacement relation
- spring force + damping force – relative displacement relation
m=100 Kg k1=150 N/m k3=10 N/m^3 k5=20 N/m^5
c2=20 Ns^2/m^2 n=2
Figure 3
Gilroy: ag in units of g, 3997 steps, dt=0.01
Figure 4