1) The document discusses ground excited systems, where the dynamic equations of motion are derived based on the relative displacement of the structure with respect to the ground acceleration vector. 2) Modal superposition is applied to decompose the equations into uncoupled modal equations, which are then solved to obtain the system response in terms of maximum displacements, storey shears, moments and drifts. 3) Several modal combination rules are discussed to combine the individual modal responses, including SRSS, CQC and double sum methods.

1. GROUND EXCITED SYSTEMS Prof. A. Meher Prasad Department of Civil Engineering Indian Institute of Technology Madras email: prasadam@iitm.ac.in

2. Dynamic Equations of Motion Force excited system Ground excited system where is the relative displacement of the structure w.r.t ground. Non-moving reference Ground Acceleration vector where, are the ground accelerations in x,y,z directions respectively. are null vectors except that those elements are equal to 1, which corresponds to x,y,z translational DOF.

3. Let System equations reduce to following uncoupled equations where participation factors, Note: a j = b j = 0 since initial conditions are zero i.e Modal Superposition applied to GES

4. Solution to uncoupled equation of motion can be expressed as, In general , for design the response quantities of interest are: R = maximum values of (u , f s , Δ, V, M) Equivalent lateral loads Storey shears Storey Moments Storey drifts Relative displacements

6. Ground Excited MDOF System = relative displacement of the structure w r t ground = Ground acceleration vector : where, are ground accelerations in x, y & z directions respectively Reference base x y z

7. 1) SRSS 2) CQC 3) Double Sum 4) Grouping Serious errors for closely spaced frequencies and for 3-D structures ,which include torsional contribution. SRSS : Square Root of Sum of Squares .It gives most probable maximum response. Modal combination rules ** Since the maximum response in each mode would not necessarily occur at the same instant of time, over conservative to add separate modal maximum responses.

8. CQC : Complete Quadratic Combination Rule (Wilson, Der Kiureghion & Baya 1981). It is based on random vibration theory. Note: All cross modal terms included very good agreement with full modal superposition extra computation minimal.

14. Multiple support Excitation Super structure free Dof Support Dof

15. Decompose {u f } into pseudo static and dynamic parts {u f }= {u s } + {u d } Considering only static response ( i.e. stiffness matrix alone) Influence matrix Describes influence of support displacement on structural displacement j th column of [ i ]=structural displacements due to unit support displacement u rl only ( l th base displacement)

16. (By definition ) and i.e.

17. If assume light damping Uncoupled equations of motion are,

18. A big mass (much bigger than the total mass of the structure ( ~10 6  total mass ) is added to each degree of freedom at moving bases. As more big masses are applied, more low frequency modes have to be extracted.

19. The desired base motion is obtained by applying a point force to each degree of freedom at moving bases by Where M big =big mass and is the applied acceleration prescribed for degree of freedom N associated with moving supports The combined equation of motion is with Where is the diagonal matrix containing the big masses for moving base ‘i’ and is the base motion applied to this base

20. The mass matrix [M] now contains the mass of the structure as well as the big masses associated with the secondary base. The modal equations with

21. 1.000 1.000 6.7662 52.2836 0.0 4.7876 5.2909 10 8 0.9999 1.000 6.7661 52.2836 0.0 4.7876 5.2909 10 6 0.9995 1.0003 6.7641 52.2823 10 -10 4.7871 5.2910 10 4 0.9524 1.0335 6.5531 52.0552 10 -9 4.8011 5.3025 10 2 Response peaks (m/s 2 ) X 1 max X 2 max X 3 max X 4 max Natural frequency Ratio of large mass to structure