an efficient model for seismic analysis of flat slab structures with the effects of stiffness...

An Efficient Model for Seismic Analysis ofAn Efficient Model for Seismic Analysis of

Flat Slab Structures withFlat Slab Structures with

The Effects of Stiffness DegradationThe Effects of Stiffness Degradation

Seung Jae LeeSeung Jae Lee

Sungkyunkwan UniversitySungkyunkwan University

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Introduction

• 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

Drop panel

Capital

Flat slab system

2

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Width of the equivalent frame

y

x

Equivalent frame widthin the y direction

Equ

ival

ent f

ram

e w

idth

in th

e x

dire

ctio

n

h

c1c2Floorheight

Columnabove

Columnbelow

Slab strip

Slab strip

l1

l2

• 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

Equivalent frame method

3

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• 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

Objectives

4

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: Length of span in direction parallel and transverse to lateral load

: Size of support in direction parallel and transverse to lateral load1C ,

,

FPd Kh.d//CC/llCl.Kαl )90](2)()(30[ 1212112

222 ))()(5.0())()(2.0( lKKllKK FPdFPd 2l: Equivalent width factor

dK800/sh

sh1.1 at the acceptable drift limit

1.0 at the acceptable drift limit



400/sh

200/sh

100/sh

2l

2C

d : Effective depth of slab h : Slab thickness

FPK 1.0 at interior supports

0.8 at exterior and edge supports

0.6 at corner supports

: Effective width of slab

: Factor considering degradation of stiffness of slabs

With limits:

: Story height

1l

Grossman method for Effective width

5

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Classification of Grossman method

dK )90( h.d/

Terms can be simply included in the FEM

]2)()(30[ 121211 /CC/llCl. FPK

FPd Kh.d//CC/llCl.Kαl )90](2)()(30[ 1212112

)9.0/( hdEKE dR REE : Modulus of elasticity

Terms cannot be easily considered by the FEM

)90( h.d/ Approximately 1.0

<0.9, if very thin slabd/h

: Adjusted modulus of elasticity

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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

Plans to which EFM can not be applied

Limitations of the Equivalent Frame Method

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6.4X6.4 6.4X6.4

6.4X6.4

9.6X9.6

9.6X4.8

12.8X6.4

9.6X4.8

9.6X4.8

Line of symmetry

PLAN

Na

4

3

2

b c d

1

108

72

A A

‘Pinned’ Support (typ.)

SECTION A-A(All units are in inches)

4812 32

U.C. Berkeley Test (by Prof. Jack. P. Moehle, 1990)

Test structure

LATERAL DRIFT - NS

0

20

40

60

80

100

120

LA

TE

RA

LS

TIF

FN

ES

S(k

ip/i

n.)

EFM

FEM

UCB test

1/800 1/400 1/200

LA

TE

RA

LS

TIF

FN

ES

S(k

ip/i

n.)

LATERAL DRIFT - EW

0

20

40

60

80

100

120

140

160

EFM

FEM

UCB test

1/800 1/400 1/200

Stiffness degradation in the slab

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1 1 c

I =

1 s

I =

Deformation of Entire Structure

SC

: Total lateral displacement

SC Consideration of Stiffness Degradation

S

SC RR

R

SRCR

SSR

Deformation of Columns

Deformation of Slabs

: Lateral displacement due to slab deformation

: Lateral displacement due to column deformation

: Stiffness reduction factor for structure

: Stiffness reduction factor for slab

Stiffness reduction factor for slabs

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Drift Direction Avg.

1/800NS 0.905 0.054 0.033 0.021 0.853

0.822EW 0.829 0.063 0.049 0.014 0.790

1/400NS 0.830 0.110 0.067 0.043 0.748

0.722EW 0.747 0.129 0.100 0.029 0.695

1/200NS 0.661 0.230 0.140 0.090 0.543

0.539EW 0.598 0.254 0.197 0.057 0.536

CSR SR

CR

SSR

91.099.74 LS DR

LD

LATERAL DRIFT (logarithmic)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sla

b S

tiff

nes

s R

edu

ctio

n

1/800 1/400 1/200

: Lateral drift

10

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LA

TE

RA

LS

TIF

FN

ES

S(k

ip/i

n.)

LATERAL DRIFT - NS

0

20

40

60

80

100

120

EFM

FEM(w/o reduction)

FEM(w/ reduction)

UCB test

1/800 1/400 1/200L

AT

ER

AL

ST

IFF

NE

SS

(kip

/in

.)

LATERAL DRIFT - EW

0

20

40

60

80

100

120

140

160

EFM

FEM(w/o reduction)

FEM(w/ reduction)

UCB test

1/800 1/400 1/200

Application of stiffness reduction factor to FEM

11

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Modeling flat slab using super elements

Refined mesh model for floor slab

12

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Separate floor slab for generation of super elements

13

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Generation of super elements

14

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Assemble super elements

15

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Use of stiff fictitious beams

A floor slab unit between columns

16

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Add fictitious beams

17

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Added fictitious beams

18

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Matrix condensation

19

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Eliminate fictitious beams

20

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Super element

21

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Example structure 1

Floor plan

20-story example structure

22

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Natural periods of vibrationLateral displacements

0 2 4 6 8 10 12 14 16Displacement(cm)

0

4

8

12

16

20

Sto

ry

EFM

FEM(w/ reduction)

Proposed(w/ reduction)

1 4 7 10 13 16 19Mode

0

1

2

3

4

5

Per

iod

(sec

)

EFM

FEM(w/ reduction)


Static & Eigenvalue analysis

23

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Von-Mises stress distribution

FEM

EFM

Proposed

max = 4.53E-2

max = 2.22E-2

max = 4.46E-2

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Time history analysis

Roof displacement time history (El Centro NS, 1940)

Model DOF`s

Computational time (sec)

Assemble M & K

Static analysis

Eigenvalue analysis


Total

FEM 55500 230.22 394.38 17406.66 281.58 18312.84

EFM 1740 2.61 0.36 19.69 7.67 30.33

Proposed 780 13.70 0.12 5.75 3.36 22.93

0 2 4 6 8 10 12 14Time(sec)

-40

-30

-20

-10

0

10

20

30

40D

ispl

acem

ent(

cm)

12.885.4

EFM

FEM(w/ reduction)


25

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Example structure 2

Floor plan

20-story example structure

26

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Static & Dynamic analysis

0 2 4 6 8 10 12 14 16Displacement(cm)

0

4

8

12

16

20

Sto

ry

FEM(w/ reduction)


1 4 7 10 13 16 19Mode

0

1

2

3

4

5

Per

iod

(sec

)

FEM(w/ reduction)



Model DOF`s


Assemble M & K

Static analysis

Eigenvalue analysis


Total

FEM 47580 193.30 390.35 13315.33 238.41 14137.39

Proposed 780 13.48 0.09 5.86 3.33 22.76

27

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Example structure 3

Floor plan

3D view of example structure (20F)

28

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Refined mesh model for floor slab with opening

Super element for the slab with opening

29

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Separate floor slab for generation of super element

30

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Add fictitious beams

31

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Matrix condensation

32

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Eliminate fictitious beams

33

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Assemble the super elements

34

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Static & dynamic analysis

0 2 4 6 8 10 12 14 16Displacement(cm)

0

4

8

12

16

20S

tory

FEM(w/ reduction)


1 4 7 10 13 16 19Mode

0

1

2

3

4

5

Per

iod

(sec

)

FEM(w/ reduction)



Model DOF`s


Assemble M & K

Static analysis

Eigenvalue analysis


Total

FEM 53580 214.13 447.98 16126.36 269.72 17058.19

Proposed 900 44.98 0.14 7.22 3.83 56.17

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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

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an efficient model for seismic analysis of flat slab structures with the effects of stiffness...

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