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
yσ
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
xσ
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
dξ
+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
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
sσ
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
cσ
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