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# Sungkyunkwan University

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

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

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

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

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

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

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

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

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

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

Application of stiffness reduction factor to FEM
11

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

Separate floor slab for generation of super elements
13

Generation of super elements
14

Assemble super elements
15

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

Add fictitious beams 17

Added fictitious beams
18

Matrix condensation 19

Eliminate fictitious beams
20

Super element 21

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

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

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

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

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

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

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

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

Separate floor slab for generation of super element
30

Add fictitious beams 31

Matrix condensation 32

Eliminate fictitious beams
33

Assemble the super elements
34

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

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

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