77

77 . . --. . -- . . 20020077 -- 0088

24.11.2008

1. , ( )

2. (.. , )

3. (.. )

4. ,

: (.. > 50%),

( )

,

:

,

1.

:

:

(q) :

1. 1.1

1. :

( ) VLBLo =+21

( ) ooLo MLBLLBL =

+32

21

2

= VLM

LBo

L32

=LMV

LBo

o322

( )dxxqVV Li

i +=0

:

(x=0) :

( ) dxxxqxVMM Li

iii

io ++=0

=

1. 1.1

2. () (L, o)

: . , . : (), (L),

=

= VLM

LBo

L32

=LMV

LBo

o322

:

: : = 1.20m : L = 12m

:

V1 = V2 = 400 kN , V3 = 640 kN :

=i

iVV :

x=0 : =i

iio xVM

= 1440 kN

= 9600 kNm

= 133 kN/m2

= 67 kN/m2

x=0

= VLM

LBo32

max

=LMV

LBo322min

:

V1 = V2 = 400 kN , V3 = 640 kN :

-150

-100

-50

0

50

100

150

200

250

300

350

0 2 4 6 8 10 12

length (m)

bend

ing

mom

ent (

kNm

) .

-400

-300

-200

-100

0

100

200

300

400

0 2 4 6 8 10 12

length (m)

shea

r for

ce (k

N) .

400 kN400 kN 640 kN400 kN400 kN 640 kN

:

1. 1.2 Winkler

Winkler :ykp =

p = (kPa)y = (m)k = Winkler (kN/m3)

(y) (p)

(p)

(y), : p = k y

, . Winkler

1. 1.2 Winkler

(y) (p)

: k . ( , Poisson ) ( ) ( )

Winkler : ykp =

k ks ( Winkler)1. (ks) :

-

pks =

APpks ==

ks =

k ( Winkler)2. k :

k ( / m3) Terzaghi( = 0.305m) :

8 MN/m3

13 MN/m36.4 19.2

< 50 %

96 MN/m326 MN/m3 k

160 MN/m342 MN/m3 k 96 - 32019.2 - 96 k (MN/m3) > 75%50-75% (Dr)

:

2.1 - () :

: 10 MN/m3 = 1 kg/cm3

k ks ( Winkler)2.2 () :

(D=0 D=1)

( ) BEIqk Sis 211 ==

Steinbrenner : : s = 0.79 : s 1 (L/B=) : s 2 :

Si IEBq

21 =

() (=0.5, = u), = 0.305m (1 ), ks :

100

150200250

300350

Eu / cu

> 100> 20> 20075 15015 30100 200 50 - 10010 2050 -10030 62.56 12.525 5020 37.54 7.512.5 25

< 22.5< 4.5< 12.5 ks (MN/m3)Eu (MPa)cu (kPa)

ks = ()

k ks ( Winkler)

( Bo) ks = ko = qo / ( qo = , = ) , :

BBkk oo=

: ( ). , k .

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8 9 10 (m)

k / k

o

ko = = 0.305m

(L= > ):

+=LB

BBkk oo 3

132

(L=) :BBkk oo3

2=

3. k , k :3.1 () :

k ( Winkler)

:

qB

BBL

oi

239.0

305.020254.0

+

=

2

305.03.0

+= BB

Nq

i

2.1 Alpan : : ks = q / i

2.2 Terzaghi & Peck :

: ( )2

39.0305.084.9

+=

BmB

BLk

o

2305.03.0

+=B

mBNk :

3.2 - () :

k ( Winkler)

222 305.014305.0

305.0

+=

++

=

Bmk

mBmB

BBkk o

o

oo : ,

( Bo = 0.305m) ko = qo / (qo = , = ) , > Bo :

k ( Terzaghi & Peck ) :

k L > B :

+

+=

+

++

=

LB

Bmk

LB

mBmB

BBkk o

o

oo 3

132305.01

431

32

305.0305.0 2

22

k (L=):222 305.01

6305.0305.0

32

+=

++

=

Bmk

mBmB

BBkk o

o

oo

:3.2 - () () :

00.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8 9 10

(m)

k / k

o

Terzaghi

k ( Winkler) (k) :

ko = = 0.305m

BBkk oo=22

305.0305.0

++

=

mBmB

BBkk

o

oo

:

:

k ( Winkler)4. - :

Es , = Poisson b , , I = ,

Vesic (1961) :

BE

IEBEk s

b

s121

4

2165.0

=

: : = 1.20m, H=0.60m, Eb = 25 GPa : u = 15 MPa (u = 0.5) = 3 / 12 = 0.0216 m4

312

14

2 /2.115

0216.0250002.115

5.0165.0 mMNk

= = 8.5 MN/m3 : (L/B= Is 2) :

( ) ( ) 2.11525.01 11 1 22 === BEIqk Si = 8.3 MN/m3

k ( Winkler) (k) ( Bowles) :

5 1212 1818 2424 48

> 48

: (qu = 25-50 kPa) (qu = 50-100 kPa) (qu = 100-200 kPa) (qu = 200-400 kPa) (qu > 800 kPa)

24 48 32 80

4.8 169.6 8064 128

: (Dr < 50%) (Dr = 50-75%) (Dr > 75%)

k (MN/m3)

: 10 MN/m3 = 1 kg/cm3 : (k) (L, B, I, b). , .

1. 1.2 Winkler

Winkler :ykp =

p = (kPa)y = (m)k = Winkler (kN/m3)

:

Bpqdx

ydIEb =44

q = (kN/m) = (m)b , = (kN/m2) (m4)

qyBkdx

ydIEb =+44

(V, M, q), (b , , ,L) ( k) (.. )

1. 1.2 Winkler

qyBkdx

ydIEb =+44

Eb = = .

:y = (m)k = Winkler (kN/m3) = (m)q = (kN/m)p = (kPa) : p = k y

:

12

3HBI =

( ) : LIE

Bkb

41

4

=

, (Hetenyi, 1946) : : < (/4) : (/4) < < Winkler : >

1. 1.2 Winkler

Eb = = s = = (m)L = (m)

, Meyerhof :

: > 0.5 ( > 0.5), . < 0.5, Winkler

BLEIE

s

b3=

: Winkler - ( )

: = 1.20m, H=0.60m, L = 12m : Eb = 25 GPa : = 3 / 12 = 0.0216 m4

: , u = 15 MPa (u = 0.5)

V1 = V2 = 400 kN , V3 = 640 kN :

Winkler

3121

4

2 /2.115

0216.0250002.115

5.0165.0 mMNk

= = 8.5 MN/m3

Vesic :

LIE

Bkb

41

4

=

:

= 3.146

BE

IEBEkb

121

4

2165.0

=

() Winkler ( )

(y)

: Winkler ( )

-400

-300

-200

-100

0

100

200

300

400

0 2 4 6 8 10 12

length (m)

shea

r for

ce (k

N) .

-150

-100

-50

0

50

100

150

200

250

300

350

0 2 4 6 8 10 12

length (m)

bend

ing

mom

ent (

kNm

) .

400 kN400 kN 640 kN

400 kN400 kN 640 kN

: Winkler - ( ) ( )

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0 2 4 6 8 10 12

length (m)

verti

cal s

ettle

men

t (m

0

20

40

60

80

100

120

140

0 2 4 6 8 10 12

length (m)

reac

tion

soil

pres

sure

(kP

a)

400 kN400 kN 640 kN

400 kN400 kN 640 kN

67 kPa

133 kPa

: Winkler - ( ) ( )

-150

-100

-50

0

50

100

150

200

250

300

0 2 4 6 8 10 12

length (m)

bend

ing

mom

ent (

kNm

) .

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0 2 4 6 8 10 12

length (m)

verti

cal s

ettle

men

t (m

) . : Winkler

400 kN400 kN 640 kN

400 kN400 kN 640 kN

=0.25m ( =0.60m) LIE

Bkb

41

4

= = 6.066 2

:

( ) ( )

1. 1.2 Winkler ( : L )

>

= L

IEBkb

41

4 : (Hetenyi, 1946)

1. x=0 :

:

:

:

12

kLBPy =

24

LPM =

32PQ =

1 , 2 , 3 = ( )

()2. x=0 :

:

:

:

( )4

3/ kB

LMy o=

32oMM =

1 , 3 , 4 = ( )

41

4/

=

IEBkLb

( ) 1/2 L

MQ o=

3. : , x=0

() : 1 , 2 , 3 , 4 x :

xIE

BkLx

b

41

4

=

2. (k) :

1. :

LIE

Bkb

41

4

=

( ) (Winkler)

> 0.5 < 0.5 Meyerhof

( ) (Winkler) (Winkler L=)

< (/4)(/4) < <

> Hetenyi

/

( ) ( ) ko (k) Vesic

3. : : : Winkler :

BLEIE

s

b3=

(k) () (B x L) (V) () :

VMe =

LBV= : :

e < B/6

1. : 0 e B / 6

+=Be61max 061min

=

Be

Winkler : ymax = max / k , ymin = min / k

: ( ) =B

yy minmaxtan :

kLBMK 3121== (kNm / rad)

kLBM312=

:

==k

VyVKV (kN / m)LBkKV =

B =

e > /6

(k) () (B x L) (V) () :

VMe =

LBV= : :

2. : / 6 e B / 2

Winkler : ymax = max / k

: = Bymaxtan

:

( ) keLBMK 221 == (kNm / rad)

( ) kLBV22 =

:

(kN / m)

= eBB2

3BB= 2max

==k

VyVKV LBkKV =

B =

(k)

( b , ) V . ( ).

(k= ), , (V) V (e=0). S M = V e, :

S = V e / H

( k), V e (0 < e < e). , S M = V (e e) :

S = V(e-e) / H

: Eb I

(k)

() = V (e e) , :

( )IE

HeeVIE

HMbb 33==

( ) e :

3361

BLHkIE

eeb+

= :k = e = e

: Eb I

() (k) = V e :

kLBeV

kLBM

KM

33 1212 ===

(e) :

: Eb I

: : B = 2m L = 4m : V = 2000 kN : k = 10 MN/m3Eb = 25 GPa () : b = 0.4m d = 0.6m : = 4.5m = b d3 /12 = 0.4 x 0.63 / 12 = 0.0072 m4e = B/2 d/2 = 1 0.3 = 0.7m

3361

BLHkIE

eeb+

= e = 0.13m < B/6

312M V eK B L k

= = ( ) :

S = V(e-e) / H = 253.3 kN

: = S H = 1140 kN

= 0.00975 rad = 0.56o

2. (B, L) (c ) (Westergaard, 1939) :

Eb , b = Poisson t = k = Bc .

( Bc).

Winkler k Bc

Bc/2 . ( ). , .

< Bc/2 L > Bc/2 , () : Bc/2 < < Bc , Winkler k .

: Bc = Hetenyi (1946)( )

41

2

3

1125

= bb

c ktEB

2. 1 : (Eb = 25 GPa, b = 0.30) t = 0.80m ko = 75 MN/m3 ( = 0.305m)

BBkk oo= ( )

41

2

3

1125

= bb

c ktEB

( )31

2

3

11255.8

= boob

c BktEB

: :

: = 31.75 m

, (, L) 31.75 / 2 = 15.9 m, . (B, L) > 31.75m :

75 / (0.305 / 31.75) = 0.72 MN/m3 = 0.07 kg/cm3

.

==c

oo B

Bkk

, 15.9m 31.7m, .

2. 2 : (Eb = 25 GPa, b = 0.30) t = 0.80m ko = 50 MN/m3 ( = 0.305m) 2

14

+=BBkk oo

( ) ( )23

1325

bo

bocc k

tEBBB =+( )

41

2

3

1125

= bb

c ktEB

:

:

: c (c+) = 242 Bc = 15.4 m

, 15.4 / 2 = 7.7 m . > 15.4m :

(50/4) * [1+(0.305/15.4)]2 = 13 /m3 = 1.3 kg/cm3=

+=

2

14 c

oo

BBkk

.

, 7.7m 15.4m, .