Structural Dynamics

HW # 06

04/28/2015

Student ID: 20145189Name: Rith Makara

CONTENTContent1Lists Of Figures And Table1Chapter 1 Solution Strategies21.1 Introduction21.2 NewmarkS Linear-Acceleration Method21.3 MATLAB Calculation21.3.1 General Information21.3.2 Manual For Calculation3Chapter 2 Results Of The Final Response Of The Structural System4Annex MATLAB Codes4Annex 1.1 Newmark_Linear.m5Annex 1.2 Northridge.m5Annex 1.3 Data_Input.m11Annex 1.4 Calculation.m12 LISTS OF FIGURES AND TABLEFigure 1 Responses of the Water Tower to the Northridge Earthquake in 19974Table 1 Summary of Maximum value of relative responses4

Homework #6by Rith MakaraCHAPTER 1 SOLUTION STRATEGIES1.1 IntroductionThe problem is to determine the relative responses of the water tower subjected to the Northridge Earthquake ground motion in 1997, Beverly Hills, USA. The relative responses are: relative displacement of the tower with respect to the ground, relative velocity and relative acceleration based on the numerical solution, Newmarks Linear-acceleration method. 1.2 Newmarks Linear-acceleration methodThe responses of the structural system can be determined as following: Displacement Response

Velocity Response

Acceleration Response

1.3 MATLAB Calculation1.3.1 General InformationThe numerical solution of Newmark is solved in MATLAB software. The results of relative responses of the structural system are plotted in the figure window and the command windows. In this calculation, the 4 different files: newmark_linear.m This file contains the codes to solve the dynamic response of the structural system based on the Linear-acceleration method. The function is called newmark_linear(M,C,K,f,dt,Xi,Xdi,Xddi) where: [Annex 1.1] M Mass of the structure, [kips.sec2/in] C Damping Coefficient, [kips.sec/in] K Stiffness of structure, [kips/in] f Vector of applied force at each time (Ground Motion), [kips] dt Time Stepping Increment, [sec] Xi Initial displacement of the structure at t = 0 sec Xdi Initial velocity of the structure at t = 0 sec Xddi Initial acceleration of the structure at t = 0 sec northridge.m This file contains the earthquake records of Northridge, Beverly, USA in 1997. The record is the acceleration of the ground motion. The initial conditions are also included. The time stepping increment is 0.02 sec. Moreover, the acceleration of ground motion is stored in the matrix Acc_Ground_Raw. The matrix data is read row by row and the first value is at 0.02 sec. The initial acceleration at t = 0 is zero. [Annex 1.2] data_input.m This file contains the structural properties such as mass, stiffness, damping ratio, damping coefficient, natural frequency [rad/sec], period frequency [sec], natural frequency [Hz] and it also calls the file containing the Earthquake data (northridge.m). [Annex 1.3] calculation.m This file contains the codes to solve the problem by calling the function of Newmarks Linear-acceleration method. After the it finishes its calculation process, the results of relative displacement, relative velocity and relative acceleration of the structural system will be shown in the Command Window of the MATLAB which the first column is the time in [sec], second column is displacement in [in], third column is velocity in [in/sec] and the last column is acceleration in [in/sec2]. The graphical responses also appear in the Figure Window of MATLAB. [Annex 1.4]1.3.2 Manual for calculation1 Input the earthquake record (i.e. Acceleration of ground motion) as in northridge.m.2 Input the Structural Properties in the data_input.m.3 Press F5 or click Run in data_input.m.4 The results show in Figure Window and Command Window of MATLAB.

CHAPTER 2 RESULTS OF THE FINAL RESPONSE OF THE STRUCTURAL SYSTEM

Figure 1 Responses of the Water Tower to the Northridge Earthquake in 1997

Table 1 Summary of Maximum value of relative responsesNRelative ResponseTimes [sec]Max. Value

1Displacement, [in]10.8-11.31385

2Velocity, [in/sec]10.62-96.0693

3Acceleration, [g]10.82.55812

ANNEX MATLAB CODESAnnex 1.1 newmark_linear.m%Newmark Linear Acceleration Method%by Rith Makara%2015function [X,Xd,Xdd] = newmark_linear(M,C,K,f,dt,Xi,Xdi,Xddi)%===================================================================% M = Mass% C = Damping% K = Stiffness% f = Excitation Force% dt = Time Stepping Increment% Xi = Initial Displacement% Xdi = Initial Velocity% Xddi = Initial Acceleration %Calculation %Displacement Response:X = (f+(M/dt^2)*(6*Xi+6*Xdi*dt+2*Xddi*dt^2)+(C/dt)*(3*Xi+2*Xdi*dt+0.5*dt^2*Xddi))/((6*M/dt^2)+(3*C/dt)+K); %Velocity Response:Xd = (-dt*0.5*Xddi)+(-2*Xdi)+(3*(X-Xi)/dt); %Acceleration Response:Xdd = (6/dt^2)*(X-Xi-Xdi*dt-(dt^2*Xddi/3));

Annex 1.2 northridge.m%====================================% Earthquake Data in "Northridge"%====================================% by Rith Makara% 2015%% This data is the Acceleration of ground motion.%% Note:% - The data is read row by row.% - The first value is at t = 0.02.% - Acceleration at t=0 is zero.% - The length unit is in cm must be converted to in.% => 1cm = 1/2.54 in.%% Initial ConditionXi = -0.466/2.54; % Displacement at t=0, [cm]Xdi = -2.103/2.54; % Velocity at t=0, [cm/sec]Xddi = 0; % Acceleration at t=0, [cm/sec2] % Time Incrementdt = 0.02; % in [sec] % Acceleration of the ground motionAcce_Ground_Raw = [ 6.631 3.507 1.628 1.105 .426 .239 -.268 -1.836 -1.490 1.413 -1.058 1.669 11.676 11.498 -.292 -16.718 -12.282 -3.280 -12.029 -6.913 6.389 2.310 -5.904 9.685 32.806 23.401 -15.683 -15.219 8.594 3.664 -15.019 -12.219 2.795 -.868 -23.312 -22.668 -3.011 14.520 30.337 39.065 36.790 12.630 -17.251 -14.937 21.762 6.732 -37.137 -48.789 -23.042 9.566 6.434 -11.687 -13.133 1.484 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Annex 1.3 data_input.m%==================================%Input Data%==================================%Homework #6%by Rith Makara%2015 % Structural Properties:M = 1; % Mass of the Structure in [kips.s^2/in]K = 100; % Structure Stiffness in [kips/in]Xi = 0.02; % Damping RatioC = Xi*2*sqrt(K*M); % Damping Coefficient in [kips.s/in]Wn = sqrt(K/M); % Natural Frequency in [rad/sec]Tn = 2*pi/Wn; % Period Frequency in [sec]fn = Wn/(2*pi); % Natural Frequency in [Hz] % Calling the Earthquake Datanorthridge % Calling the Calculationcalculation

Annex 1.4 calculation.mclc%===========================================% Calculation Sheet%===========================================% by Rith Makara% 2015%% This calculation sheet will output the responses due to ground motion.% It also plots each responses.%% Call the Input Data File% data_input% Size of Acceleration matrix raw data of Ground Motion [m n] = size(Acce_Ground_Raw);% Turn the Acce_Ground_Raw matrix into a vector of Acceleration of Ground% MotionAcce_Ground = zeros(m*n,1);for i = 1:m for j = 1:n Acce_Ground((i-1)*n+j,1) = Acce_Ground((i-1)*n+j,1) + Acce_Ground_Raw(i,j); endend % 1. CalculationResponse = zeros(m*n,4);for i = 1:m*n Response(i,1) = i*dt; if i == 1 [Response(i,2),Response(i,3),Response(i,4)] = newmark_linear(M,C,K,-M*Acce_Ground(i,1),dt,Xi,Xdi,Xddi); else [Response(i,2),Response(i,3),Response(i,4)] = newmark_linear(M,C,K,-M*Acce_Ground(i,1),dt,Response(i-1,2),Response(i-1,3),Response(i-1,4)); endendResponse % 2. Plot The Response of the Structure% 2.1 Relative Displacement Response, U [infiguresubplot(3,1,1)plot(Response(1:m*n,1),Response(1:m*n,2))title('Relative Displacement')xlabel('Time, [sec]')ylabel('Displacement, [in]')axis([0,Response(m*n,1),-max(abs(Response(1:m*n,2)))*1.5,max(abs(Response(1:m*n,2)))*1.5]) % 2.2 Relative Velocity Response, V [in/sec]subplot(3,1,2)plot(Response(1:m*n,1),Response(1:m*n,3))title('Relative Velocity')xlabel('Time, [sec]')ylabel('Velocity, [in/sec]')axis([0,Response(m*n,1),-max(abs(Response(1:m*n,3)))*1.5,max(abs(Response(1:m*n,3)))*1.5]) % 2.3 Relative Acceleration Response, A [in/sec2]subplot(3,1,3)plot(Response(1:m*n,1),Response(1:m*n,4))title('Relative Acceleration')xlabel('Time, [sec]')ylabel('Acceleration, [in]')axis([0,Response(m*n,1),-max(abs(Response(1:m*n,4)))*1.5,max(abs(Response(1:m*n,4)))*1.5])

By Rith Makara4