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Transcript of Strut and Tie ModelsStrut and Tie Strut and Tie ModelsStrut and Tie Models â€¢ Strut â€¢...

• 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 Strut x h y−0x 0 0

0 0

tan x h y y a x

θ = = +

f

0x

0y

2 0 1 1

2 x a a h h h

⎡ ⎤⎛ ⎞⎢ ⎥= + −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

h cf

2h h h⎢ ⎥⎝ ⎠⎣ ⎦ 21 a aτ ⎡ ⎤⎛ ⎞⎢ ⎥

θ 1 1 2c

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

2 0 01 4 1

2 y yx a a

h h h h h

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

• Single StrutSingle Strut

0s Y cA f y tf=

cf

0 1 2

s YA f y thf h

Φ = = ≤h

cf

2cthf h

θ0y

a0 x

( ) 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 Fan B

• 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 (Unifor Normal stress type)

• y σ constτ =

H

P

x

L Ldx-dz k+( -z)dk 2

-dz k

k+dk 1

1

k+dk

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

• yl/2 y yσ

0

2σ =0 xyτ

yσ y

yxτ+ y-σ t

+

x

T

1σ =0

pl 2-σ x

+- tτ 1σ =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- tτ 1σ =0

2σ 0

T+dTx

• l/2l/2

yy

ξ ξ

el

k k+dk

11

k dξ

θ

( ) k+dk 1

-r

r

dr

Q(x,y)

b b(x ,y )

xx

-s s

-ds + y-σ t

+ x

1σ =0

x

+ y-σ t

+ y- tτ

+ y- tτ

dr

2-σ 1σ 0

-qt x' 2-σw

+

dξ R

R'

θ

-ds

-ptx' 2 +-σ t

R

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

• θ e

p

m

q

r

qq

Concentric Fan-shaped stress field

• dx

2,bottomσσ =

dx hdK+ 2,

1 top dK

dx

σ η

= +dx hdK+

2 tσ dx

2,topσ

11 K K dK+

11

dx

K dK+ 2,bottomσ

dx Stress State in Diagonal Compression Stress FieldsStress State in Diagonal Compression Stress Fields

• Single Strut actiong 0x

cf 0y

hh

N

θ a

0y′ 0C

a P

N τN sτ

P 0 T

• Strut with Diagonal Compression FieldStrut with Diagonal Compression Field tτ

cσ cott tσ τ θ=

f

N N cf

θ cf

PP P 21 1 2s c

a aP thf h h

⎡ ⎤⎛ ⎞⎢ ⎥= + −⎜ ⎟⎢ ⎥⎝ ⎠2s c f

h h⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦ sP Pτ −=

( )0t t h y τ =

′−

• Diagonal Compression FieldDiagonal Compression Field

cf fσ =

θ c cfσ =

2sin 2 f τθ = tanYyrf τ θ= cf

21 1 a a aτ ⎡ ⎤⎛ ⎞⎢ ⎥= + +Φ⎜ ⎟ ( )1

τ = Φ −Φ1

2 ycf h h h ⎢ ⎥= + − +Φ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

( )1y y cf = Φ Φ

• Maximum Shear StrengthMaximum Shear Strength τ

cf

( )1y y cf τ = Φ −Φ0.5

cf 21 1

2 y a a a

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

2cf h h h⎢ ⎥⎝ ⎠⎣ ⎦

• Strut with Fan-Shaped Stress Fieldp