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

2. Flat slab structure having capital and drop panel
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. Width of the equivalent frame
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. Deformation of Entire Structure Deformation of Columns
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. Drift Direction Avg. 1/800 NS 0.905 0.054 0.033 0.021 0.853 0.822 EW
0.829
0.063
0.049
0.014
0.790
1/400
0.830
0.110
0.067
0.043
0.748
0.722
0.747
0.129
0.100
0.029
0.695
1/200
0.661
0.230
0.140
0.090
0.543
0.539
0.598
0.254
0.197
0.057
0.536
: Lateral drift 10

11. Application of stiffness reduction factor to FEM11

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

13. Separate floor slab for generation of super elements13

14. Generation of super elements14

15. Assemble super elements15

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

17. Add fictitious beams 17

18. Added fictitious beams18

19. Matrix condensation 19

20. Eliminate fictitious beams20

21. Super element 21

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

23. Lateral displacements Natural periods of vibration
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)
Model
DOF`s
Computational time (sec)
Assemble M & K
Static analysis
Eigenvalue analysis
Time history analysis
Total
FEM
55500
230.22
394.38
281.58
EFM
1740
2.61
0.36
19.69
7.67
30.33
Proposed
780
13.70
0.12
5.75
3.36
22.93 25

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

27. Static & Dynamic analysis
Lateral displacements
Natural periods of vibration
Model
DOF`s
Computational time (sec)
Assemble M & K
Static analysis
Eigenvalue analysis
Time history analysis
Total
FEM
47580
193.30
390.35
238.41
Proposed
780
13.48
0.09
5.86
3.33
22.76 27

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

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

30. Separate floor slab for generation of super element30

31. Add fictitious beams 31

32. Matrix condensation 32

33. Eliminate fictitious beams33

34. Assemble the super elements34

35. Static & dynamic analysis
Lateral displacements
Natural periods of vibration
Model
DOF`s
Computational time (sec)
Assemble M & K
Static analysis
Eigenvalue analysis
Time history analysis
Total
FEM
53580
214.13
447.98
269.72
Proposed
900
44.98
0.14
7.22
3.83
56.17 35

36. Conclusions Equivalent Frame Method
• 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
Finite Element Method using super elements
• 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