MATLAB Examples

# Application

The vertical eigen-frequencies and mode-shapes of the Lysefjord Bridge, based on data obtained from [1,2,3], are computed using the formulation proposed by [4]. The results are compared to the eigen-frequencies and mode-shapes computed with the method used by [5,6].

[1] Opsahl Steigen, R. (2011). Modeling and analyzing a suspension bridge in light of deterioration of the main cable wires.

[2] Tveiten, J. (2012). Dynamic analysis of a suspension bridge.

[3] Gran, B. (2012). Konstruksjonsanalyse av en hengebro.

[4] Luco, J. E., & Turmo, J. (2010). Linear vertical vibrations of suspension bridges: A review of continuum models and some new results. Soil Dynamics and Earthquake Engineering, 30(9), 769-781.

[5] Sigbjørnsson, R., Hjorth-Hansen, E.: Along wind response of suspension bridges with special reference to stiffening by horizontal cables. Engineering Structures 3, 27–37 (1981)

[6] Structural Dynamics, Einar N Strømmen, Springer International Publishing, 2013. ISBN: 3319018019, 9783319018010

## Lysefjord bridge inputs data

clear all;close all;clc Bridge.L = 446 ; % length of main span (m) * % Discretisation of bridge main span in Nyy points Bridge.Nyy = 200; Bridge.x = linspace(0,1,Bridge.Nyy); % non dimensional span length x/L Bridge.E = 210000e6; % young modulus steel (Pa) * Bridge.Ec = 180000e6; % young modulus steel (Pa) * Bridge.Ac = 0.038 ;% cross section main cable (m^2) * Bridge.g = 9.81; Bridge.m =5350 ; % lineic mass of girder (kg/m)* Bridge.mc =408 ; % lineic mass of cable (kg/m)* Bridge.sag= 45; % sag (m)* Bridge.hm = 3 ; % hanger length at mid span (m)* Bridge.hr =0.400; % distance between shear center and hanger attachment Bridge.bc = 10.2500; % distance betweem main cable (m) Bridge.H_cable = Bridge.m*Bridge.g*Bridge.L^2/(16*Bridge.sag).*... (1+2*Bridge.mc/Bridge.m*(1+4/3*(Bridge.sag/Bridge.L)^2)); % Moment of inertia with respect to bending about z axis Bridge.Iy = 0.429; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Required for method 1 only Bridge.beta1 = 41; % angle between backstay cable and horizontal Bridge.beta2 = 19; % angle between backstay cable and horizontal Bridge.l1 = 56; % backstay cable length Bridge.l2 = 157; % backstay cable length % cable length % For Bridge.Le, I am using a slightly different definition as in [4], % because the Lysefjord Bridge is not symmetrcal. Bridge.Le = Bridge.L.*(1+8*(Bridge.sag./Bridge.L)^2)+... (Bridge.l1./cosd(Bridge.beta1).^3+Bridge.l2./cosd(Bridge.beta2).^3); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Required for method 2 only % ADDITIONAL INPUTS FOR LATERAL MODES % Moment of inertia with respect to bending about y axis (used for lateral % bridge analysis) Bridge.Iz = 4.952; % ADDITIONAL INPUTS FOR TORSIONAL MODES Bridge.m_theta = 82430; %kg.m^2/m* Bridge.Iw = 4.7619; % WARPING RESISTANCE Bridge.GIt = 0.75e11; % TORSIONAL STIFFNESS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Number of eigen-modes to be computed % 3 assymetric + 3 symmetric = 6 modes Nmodes = 3; 

## Method 1

Computation of eigen-frequencies and mode-shapes using the method proposed by [4] Nondimensional parameters:

• mu: positive scalar: relative bending stifness of the girder
• lambda: positive scalar: Irvine-Caughy vable parameter
mu = sqrt(Bridge.E*Bridge.Iy./(2*Bridge.H_cable.*Bridge.L^2)); lambda = 8*Bridge.sag/Bridge.L.*sqrt(Bridge.Le/Bridge.L*... Bridge.Ec.*Bridge.Ac./(2*Bridge.H_cable)); % Run the function eigenBridge2.m [fn1,phi1] = eigenBridge2(Bridge,mu,lambda,Nmodes); 

## Method 2

Computation of eigen-frequencies and mode-shapes using the method proposed by [5,6]

% Run the function eigenBridge.m [fn2,phi2] = eigenBridge(Bridge,20); % Get the frequencies in Hz, and get the non-dimensional mode shapes fn2 = fn2(2,1:2*Nmodes)./(2*pi); phi2 = squeeze(phi2(2,1:2*Nmodes,:)); % The maximal value of the mode shape is taken as positive for ii=1:Nmodes if abs(min(phi2(ii,:)))>max(phi2(ii,:)), phi2(ii,:)=-phi2(ii,:); end end 

## Results:

Eigen frequencies A good agreement is obtained for the eigen-frequencies:

Name = {'VA1','VS1','VS2','VA2','VS3','VA3'}; fprintf('Eigen frequencies (Hz): \n') disp([' ','methode 1' ,' methode 2']) for ii=1:2*Nmodes, disp([Name{ii},' ',num2str(fn1(ii),4),' ',num2str(fn2(ii),4)]) end % Mode shapes: % A good agreement is obtained, but slight differences are observed. fprintf('\n\n') fprintf('Mode shapes for the first 4 eigen-frequencies: \n') figure for ii=1:4, subplot(4,1,ii) if ii==1, plot(Bridge.x,phi1(ii,:),'r',Bridge.x,-phi2(ii,:),'k--'); elseif ii==4, plot(Bridge.x,phi1(ii,:),'r',Bridge.x,phi2(ii,:),'k--'); xlabel('x/L'); else plot(Bridge.x,phi1(ii,:),'r',Bridge.x,phi2(ii,:),'k--'); end if ii==1 legend('Method 1 ([4])','Method 2 ([5,6])','location','best') end ylabel(['\phi_',num2str(ii)]) end set(gcf,'color','w') 
Eigen frequencies (Hz): methode 1 methode 2 VA1 0.2045 0.2046 VS1 0.3169 0.3189 VS2 0.4345 0.4391 VA2 0.5847 0.5852 VS3 0.8634 0.8643 VA3 1.193 1.194 Mode shapes for the first 4 eigen-frequencies: 

## Conclusions

Both methods gives similar results. The methode 1 may give slightly more accurate mode shapes. I like the method 1 because it is easier to understand how the mode shapes and eigen frequencies are calculated. The method 1 is also closer to what we are use to do when we calculate the modal properties of simple beams: First we get the characteristic equation, then we solve it to get the eigen frequencies wi , and finally we get the mode shapes using wi . The same conclusion may apply for the torsional DOF. For the lateral DOF, methode 2 may be slightly more accurate than methode 1. I don't have checked yet.