Linear Modal Time History Analysis - Sharif

1 Linear time History Analysis of MDOF Structure by Mode Superposition Method using Newmark s Method Carleton University Mostafa Tazarv Graduate Student Version (April 2011) Cite as: Tazarv, M., Linear time History Analysis of MDOF Structures by Mode Superposition Method , available at: ~tazarv/ Modal time History Analysis Mostafa Tazarv 2 | P a g e Introduction Structural dynamic is a mandatory graduate level course for structural/earthquake engineering student all around the world. One of the most important topics of this course is to solve Modal equations of motion of a Multi Degree of Freedom (MDOF) structure by a numerical method such as Nemark s method and Wilson- method.

2 Modal response should be assembled to obtain each DOF response. Applied load can be a base excitation earthquake or time -dependent loads on stories. Here, I will introduce a MATLAB function which can do a time History Analysis of an n-DOF structure with a certain Number of Modes (nom). Then, I will show how to use this function with two examples, one excited by half-cycle harmonic on two stories and another excited by Elcentro earthquake. This function will also be verified by an example detailed in Dynamic of Structure by Chopra. time History Response by Newmark Method: NM I tried to write a self-explanatory m-file. Therefore, I ll explain a bit more the important part of the function called NM in this section. A new feature is to give you the option to specify the number of modes you want to consider in mode superposition Analysis .

3 For example, there is a 100-DOF structure (the size of mass and stiffness matrixes are 100 100). However, you want to do the Modal Analysis only for first 10 modes not all the modes which is very common in real situation. In this function to solve equation of motions for different modes, Newmark Linear Method has been used. We can decompose time -dependent applied force, . , into two components where F is spatial distribution of load on DOFs and f(t) is time -variant component of load. By modifying inputs F (a vector) and f(t), you can analyze the structure for either earthquake (base-seismic-excitation) or time -dependent load applied to different stories. I will show it in examples later. That's your responsibility to organize eigenvectors ( ) and eigenvaluses ( in which frequencies are sorted from smallest to greatest ( f1< f2< f3<.))

4 If you want to use MATLAB built-in function to find (phi) and (omega2), you should be careful about abovementioned arrangement. Preferred unit is SI (N, kg, s). Specifically for this function, stiffness (K) is in N/m, mass (M) is in kg, force (F) is in N, Gravitational Constant (g) is m/s2 and Ground Acceleration (Ag) should be normalized to g. However, you are free to choose your unit. Modal time History Analysis Mostafa Tazarv 3 | P a g e function [disp,EF]=NM(n,nom,dt,F,ft,M,K,zet,omega 2,phi) % INPUTS:% INPUTS:% INPUTS:% INPUTS: % nnnn: Number of Stories or generally, Number fo DOFs % nomnomnomnom: Number of Modes that you want to consider in Analysis ; (nom <= n) % dtdtdtdt: time Interval (Sampling time ) of force or earthquake % FFFF: Spatial Load Distribution Vector; in the case of base seismic % exitation F=-g*M*[1 1 1.]

5 1]'; (A vector) % ftftftft: time -Variant Component of Load (A Vector); in the % case of base seismic exitation ft=Ag (Ground Acceleration in g) % MMMM: Mass Matrix of Stories (n x n) % KKKK: Stiffness Matrix of Stories (n x n) % zetzetzetzet: Damping Ratio in percent (%); 5 % omega2omega2omega2omega2: Eigen-Values Matrix (nom x nom); [phi,omega2]=eig(K,M); % phiphiphiphi: Eigen-Vectors (nom x nom); [phi,omega2]=eig(K,M); % OUTPUTS (Story Response):% OUTPUTS (Story Response):% OUTPUTS (Story Response):% OUTPUTS (Story Response): % dispdispdispdisp: Displacement time History of Stories (or DOFs) % EFEFEFEF: Elastic Force time History of Stories (or DOFs) As long as the size of stiffness matrix and mass matrix are equal, there will be no restriction on this function.

6 Modal time History Analysis Mostafa Tazarv 4 | P a g e Example 1: time -Dependent Load on a Shear Building with 8-DOF It is desired to analyze a shear building detailed in Figure 1 with Mode Superposition Method only with first 2 modes in an undamped condition (zet=0). Load considered for this example is time -dependent half-sine impulse force on two stories (Figure 3). For a shear frame, it is easy to derive stiffness and mass matrixes which are shown as follows: Figure 1- Shear Building + += , = )/( 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 109mNK = )/.

7 (4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 8 1026msNkgM= = Derived mass and stiffness matrixes can be used as inputs of eigen-problem and Modal Analysis . For this section, only 2 modes are desired. Then, eigenvalues and eigenvectors have been calculated. Figure 2 illustrates Modal shape of first two modes normalized to mass. Modal time History Analysis Mostafa Tazarv 5 | P a g e Figure 2- Mode Shapes of First Two Modes of Shear Building (Normalized to Mass) = 103 00 NOTE: should be a square matrix in the size of nom nom Modal time History Analysis Mostafa Tazarv 6 | P a g e As mentioned before, to solve equation of motion of each mode,nTnnnnnnnMtfFYYY/)(.)

8 22 =++&&&, load should be in the form of .. Here, is a half-sine wave with the period of and 0 0 0 75 100 0 0 0 (kN). This load is illustrated in Figure 3. All required data has been provided so far to do Modal Analysis as follows: function [disp,EF]=NM(8,2, ,F,ft,M,K,0,omega2,phi) Displacement and elastic force time History of some stories are shown in Figures 4 and 5. Note that displacement and elastic force response will be in m and N, respectively. However, I have shown them in mm and kN. Figure 3- Decomposing of Loads into Two Components (dt for f(t) is sec) Modal time History Analysis Mostafa Tazarv 7 | P a g e Figure 4- Displacement time History of Some Stories considering Two Modes Figure 5- Elastic Force time History of Some Stories considering Two Modes Modal time History Analysis Mostafa Tazarv 8 | P a g e Example 2: A MDOF Frame Structure under Elcentro Earthquake A frame structure is shown in Figure 6.

9 It is desired to find lateral displacement and elastic force time History of node 3 (in X direction) under Elcentro earthquake N-S component (ground motion record is available online: ) considering first 3 modes with 5% damping ratio. Section area, second moment of inertia and elastic modulus of all elements are same which are 650 mm2, 106 mm4 and 200 103 MPa, respectively. Element self-weight and rotational mass of nodes are neglected. Lumped mass matrix was derived by applied load in the dimension of 8 8 (gravity and lateral direction) (Table 1). Global stiffness matrix will be 12 12 (12-DOF) for this 2D frame. By means of static condensation, we can properly eliminate mass-less DOFs in stiffness matrix which is summarized in Table 2. Figure 6- Frame Shape and Loading Table 1- Mass Matrix 8 8 (Unit: kg) Node 2 Node 3 Node 4 Node 5 Ux(r1) Uy(r2) Ux(r3) Uy(r4) Ux(r5) Uy(r6) Ux(r7) Uy(r8) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Modal time History Analysis Mostafa Tazarv 9 | P a g e Table 2- Condensed Stiffness Matrix without P- Effect 8 8 (Unit: N, mm) Node 2 Node 3 Node 4 Node 5 Ux(r1) Uy(r2) Ux(r3) Uy(r4) Ux(r5) Uy(r6) Ux(r7) Uy(r8) Note.

10 Change the UNIT to N/m Table 3- Mode Shape ( ) Mode1 Mode2 Mode3 Mode4 Mode5 Mode6 Mode7 Mode8 1 1 1 1 1 1 1 1 1 -1 -1 Note: Phi is normalized to be unity in the lateral direction of node 3 005+ 0 0 0 0 0 0 0 005+ 0 0 0 0 0 0 005+ 0 0 0 0 0 005+ 0 0 0 0 005+ 0 0 0 0 0 0 (rad/s)2 Dynamic properties are shown above.