Solved MDOF Example - webshaker.ucsd.edu

1 Earthquake Engineering Ahmed Elgamal Michael Fraser Solved MDOF and modal analysis Example 1 3mmu2u1h1h2 IcIcElevation View Use: Esteel = 29,000 ksi Ic = in4 h1 = 15 ft h2 = 12 ft = = ==Plan View30 ft40 ft (a) For the two-story building shown above, define the two-degree-of-freedom free vibration matrix equation in terms of k and m.

2 Using this matrix equation, determine the natural frequencies 1 and 2 (in terms of k and m). Using these expressions for 1 and 2, plug in numerical values and determine 1 and 2 in radians. For each natural frequency, define and sketch the corresponding mode shape. (b) Verify that the modes are orthogonal as expected. (c) Normalize the first mode such that 1Tm 1 = (d) Use the normalized first mode (from above) to verify that 1Tk 1 = 12 (e) Use the El Centro Response Spectrum and a damping ratio of 5% to estimate the maximum base shear and moment. (f) Find 0aand 1ain kmc10aa+=for a viscous damping of 5% in modes 1 and 2. 3ic3icfloor3iccolumnh48 EIhEI124kh12 EIk= = =iEarthquake Engineering Ahmed Elgamal Michael Fraser Solved MDOF and modal analysis Example 2 Write the equation of motion in matrix form.

3 = ++ 00003212222121uukkkkkuumm&&&& substitute in uu2 =&&and rearrange to get = +003212222221uumkkkmkk Take the determinant and set equal to zero 032222221= +mkkkmkk 04321222142=+ kkmkmkm solving for 2 yields mkkkkkk616442221212121+ += mkkkkkk616442221212122+ ++= ()()() ()()() = substituting in k1, k2, and m: Earthquake Engineering Ahmed Elgamal Michael Fraser Solved MDOF and modal analysis Example 3 22221138srad= Determining 1st Mode Shape.

4 Plug k1, k2, m, and 1 = rad/sec into = +0032111212222121 mkkkmkk ()()()() = + = let 121= , then () = = Earthquake Engineering Ahmed Elgamal Michael Fraser Solved MDOF and modal analysis Example 4 Determining 2nd Mode Shape: Plug k1, k2, m, and 2 = rad/sec into = +0032212222222221 mkkkmkk ()()()() = + = let 122= , then () = = = = (ft) 11= 21= (ft)

5 22= 12= 1st Mode Shape 2nd Mode Shape Earthquake Engineering Ahmed Elgamal Michael Fraser Solved MDOF and modal analysis Example 5 (b) Verify that the modes are orthogonal as expected. To verify that the modes are orthogonal, need to show that 0==jjk m TiTi, for i j []()[] = = xT2m [][] = +=2k T Can also show 022==11k m TT (c) Normalize the first mode such that 1Tm 1 = []() =1m T Divide 1 by , therefore = = Check.

6 []() =1m T ok (d) Use the normalized first mode (from above) to verify that 1Tk 1 = 12 [] = == +=sradsradT2k Earthquake Engineering Ahmed Elgamal Michael Fraser Solved MDOF and modal analysis Example 6 (e) Use the El Centro Response Spectrum and a damping ratio of 5% to estimate the maximum base shear and moment. T1T2 From Response Spectrum: For T1 = sec and = 5%, D1 = in.

7 And A1 = = in/s2 For T2 = sec and = 5%, D2 = in. and A2 = = in/s2 jnjnnnjnmAMLf = Earthquake Engineering Ahmed Elgamal Michael Fraser Solved MDOF and modal analysis Example 7 ===21221jjijjjijiimmML ()()()()()()()() ++=++== == mmmmmmMLjjjjjj ()()()()()()()() =+ + =++== == mmmmmmMLjjjjjj ()() == ()() == ()() == ()()

8 = == Base Shear ==NjjnnfV10 +=+== = =+== = Calculate Maximum Base Shear, V0 max Earthquake Engineering Ahmed Elgamal Michael Fraser Solved MDOF and modal analysis Example 8 ()()()() +=+= Base Moment jNjjnndfM ==10 ()()inkipsinkipsinkipsdfdfdfMjjj =+=+== = ()()inkipsinkipsinkipsdfdfdfMjjj = =+== = Calculate Maximum Base Moment, M0 max ()()()( )ftkipsinkipsinkipsinkipsMMM = = + =+= (f) Find 0aand 1ain kmc10aa+=for a viscous damping of 5% in modes 1 and 2.

9 1 = 2 = = ()()() +=+= () +=+= &