An Efficient Model for Seismic Analysis of Flat Slab Structures with

1. An Efficient Model for Seismic Analysis of Flat Slab Structures with The Effects of Stiffness Degradation Seung Jae Lee Sungkyunkwan University

2. Introduction • Flat slab system • The columns directly support the flat slabs without beams. • Providing lower story height, good lighting and ventilation • Remarkable lateral stiffness degradation in the slab Flat slab structure having capital and drop panel 2

3. Equivalent frame method • Widely used for analysis of flat slab structures in practical engineering • Slab is modeled by equivalent frame • Elastic analysis is performed • Effective width proposed by Jacob S. Grossman is commonly used Width of the equivalent frame 3

4. Objectives • Investigate limitations in the Equivalent Frame Method • Propose an efficient analysis method using FEM  Reduce modulus of elasticity  Include stiffness degradation in the slab depending on lateral drift  Use super element and fictitious beam  Reduce computational time and memory 4

5. Grossman method for Effective width With limits: : Equivalent width factor : Effective width of slab : Factor considering degradation of stiffness of slabs  1.1 at the acceptable drift limit : Story height  1.0 at the acceptable drift limit  0.8 at the acceptable drift limit  0.5 at the acceptable drift limit : Length of span in direction parallel and transverse to lateral load , : Size of support in direction parallel and transverse to lateral load , : Effective depth of slab : Slab thickness  1.0 at interior supports 0.8 at exterior and edge supports  0.6 at corner supports 5

6. Classification of Grossman method Terms can be simply included in the FEM Terms cannot be easily considered by the FEM  Approximately 1.0  <0.9, if very thin slab : Adjusted modulus of elasticity : Modulus of elasticity 6

7. Limitations of the Equivalent Frame Method Plans to which EFM can not be applied Difficulty in providing stress distribution in the slab Calculation of equivalent mass for the dynamic analysis Troublesome calculation of effective width by the change of column size 7

8. Stiffness degradation in the slab U.C. Berkeley Test (by Prof. Jack. P. Moehle, 1990) Test structure 8

9. Stiffness reduction factor for slabs Deformation of Entire Structure Deformation of Columns Deformation of Slabs Consideration of Stiffness Degradation : Total lateral displacement : Lateral displacement due to column deformation : Lateral displacement due to slab deformation : Stiffness reduction factor for structure : Stiffness reduction factor for slab 9

10. : Lateral drift 10

11. Application of stiffness reduction factor to FEM 11

12. Refined mesh model for floor slab Modeling flat slab using super elements 12

13. Separate floor slab for generation of super elements 13

14. Generation of super elements 14

15. Assemble super elements 15

16. Use of stiff fictitious beams A floor slab unit between columns 16

17. Add fictitious beams 17

18. Added fictitious beams 18

19. Matrix condensation 19

20. Eliminate fictitious beams 20

21. Super element 21

22. Example structure 1 Floor plan 20-story example structure 22

23. Static & Eigenvalue analysis Lateral displacements Natural periods of vibration 23

24. Von-Mises stress distribution FEM = 4.53E-2 EFM = 2.22E-2 Proposed = 4.46E-2 24

25. Time history analysis Roof displacement time history (El Centro NS, 1940) 25

26. Example structure 2 Floor plan 20-story example structure 26

27. Static & Dynamic analysis Lateral displacements Natural periods of vibration 27

28. Example structure 3 Floor plan 3D view of example structure (20F) 28

29. Super element for the slab with opening Refined mesh model for floor slab with opening 29

30. Separate floor slab for generation of super element 30

31. Add fictitious beams 31

32. Matrix condensation 32

33. Eliminate fictitious beams 33

34. Assemble the super elements 34

35. Static & dynamic analysis Lateral displacements Natural periods of vibration 35

36. Equivalent Frame Method Finite Element Method using super elements Conclusions • Consider stiffness degradation in the slab • Can be applied only to flat slab structures with a regular plan • Cannot provide stress distribution in the slab reasonably • Need to calculate equivalent mass for the dynamic analysis • Troublesome calculation of effective width with the change of column size • Consider stiffness reduction in the slab by reduced modulus of elasticity • Can analyze flat slab structure with irregular plan and openings in the slab • Can provide stress distribution in the slab with accuracy •Reduced number of DOF`s  Saving in computational time and memory 36