Strut and Tie ModelsStrut and Tie Modelsocw.snu.ac.kr/sites/default/files/NOTE/1293.pdf · Strut...

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Strut and Tie Models Strut and Tie Models Strut Strut and Tie Systems Strut and Tie Systems Fans - Non-concentric fans - Concentric fans - Concentric fans - Fans with Bond

Transcript of Strut and Tie ModelsStrut and Tie Modelsocw.snu.ac.kr/sites/default/files/NOTE/1293.pdf · Strut...

Strut and Tie ModelsStrut and Tie Models

• Strut• Strut and Tie SystemsStrut and Tie Systems• Fans- Non-concentric fans- Concentric fans- Concentric fans- Fans with Bond

Single StrutSingle Strutx h y−0x 0 0

0 0

tan x h yy a x

θ = =+

f

0x

0y

20 1 1

2x a ah h h

⎡ ⎤⎛ ⎞⎢ ⎥= + −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

hcf

2h h h⎢ ⎥⎝ ⎠⎣ ⎦

21 a aτ ⎡ ⎤⎛ ⎞⎢ ⎥θ 1 1

2c

a af h hτ ⎡ ⎤⎛ ⎞⎢ ⎥= + −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦a

20 01 4 1

2y yx a a

h h h h h

⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥= − + −⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠2h h h h h⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦

Single StrutSingle Strut

0s Y cA f y tf=

cf

0 12

s YA f ythf h

Φ = = ≤h

cf

2cthf h

θ0y

a0x

( )2

0 1 4 1xP a aτ ⎡ ⎤⎛ ⎞⎢ ⎥= = = Φ −Φ + −⎜ ⎟( )4 12c cf thf h h h⎢ ⎥= = = Φ Φ + ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

Fan shaped stress fieldsFan-shaped stress fields

• Non-concentric fan- Uniform Normal stress type

D

Uniform Normal stress type- Uniform Shear stress type

A

• Concentric FanB

• Concentric Fan

C

T

Boundary Stress onBoundary Stress on Non-concentric fan-shaped stress field

y Py

Sw(x)

tcσ ,top

h

x

Z(x) cσ ,botr

q

Uniform normal stress on boundary ofUniform normal stress on boundary of Non-concentric fan-shaped stress field

d =dx+(dw-dz)k+(w-z)dk

-dw

w+dw-[z+dz]

x+(w-x)k

k 1

1

z

k+dk

ddx

dx

-dz

Geometry for Infinitesimal Elements of Non-concentric Fan-shaped stress field (Uniforof Non concentric Fan shaped stress field (UniforNormal stress type)

y σconstτ =

H

P

x

LLdx-dz k+( -z)dk2

-dzk

k+dk1

1

k+dk

Uniform shear stress on boundary of Non-concentric fan-shaped stress field

yl/2 y

0

2σ =0xyτ

yσy

yxτ+y-σ t

+

x

T

1σ =0

pl 2-σ

x

+- tτ1σ =0

0

el

dx

-y-σ t

+y-σ t

+- tτtrT-q

2-σ

-- tτ

y tτ

2σ =0

1σ =0

or

y- tτtr

-pt

os

p

r

s yx- tτ1σ =0

2σ 0

T+dTx

l/2l/2

yy

ξξ

el

kk+dk

11

kdξ

θ

( ) k+dk1

-r

r

dr

Q(x,y)

b b(x ,y )

xx

-ss

-ds+y-σ t

+x

1σ =0

x

+y-σ t

+y- tτ

+y- tτ

dr

2-σ1σ 0

-qtx' 2-σw

+

dξ R

R'

θ

-ds

-ptx' 2

+-σ tR

Geometry for Infinitesimal Elements of Non-concentric Fan-shaped stress field (UniformShear stress type)

θe

p

m

q

r

qq

Concentric Fan-shaped stress field

dx

2,bottomσσ =

dx hdK+2,

1top dK

dx

ση

=+dx hdK+

2 tσdx

2,topσ

11K K dK+

11

dx

K dK+2,bottomσ

dxStress State in Diagonal Compression Stress FieldsStress State in Diagonal Compression Stress Fields

Single Strut actiong0x

cf0y

hh

N

θa

0y′0C

aP

N τNsτ

P 0T

Strut with Diagonal Compression FieldStrut with Diagonal Compression Fieldtτ

cσ cott tσ τ θ=

f

N Ncf

θcf

PP P21 1

2s ca aP thfh h

⎡ ⎤⎛ ⎞⎢ ⎥= + −⎜ ⎟⎢ ⎥⎝ ⎠2s cf h h⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

sP Pτ −=

( )0t t h yτ =

′−

Diagonal Compression FieldDiagonal Compression Field

cffσ =

θc cfσ =

2sin 2fτθ = tanYyrf τ θ=

cf21 1 a a aτ ⎡ ⎤⎛ ⎞⎢ ⎥= + +Φ⎜ ⎟ ( )1τ

= Φ −Φ12 y

cf h h h⎢ ⎥= + − +Φ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

( )1y ycf= Φ Φ

Maximum Shear StrengthMaximum Shear Strengthτ

cf

( )1y ycfτ= Φ −Φ0.5

cf21 1

2 ya a a

f h h hτ ⎡ ⎤⎛ ⎞⎢ ⎥= + − +Φ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦Φ0.5

2cf h h h⎢ ⎥⎝ ⎠⎣ ⎦

Strut with Fan-Shaped Stress Fieldp