22.314/1.56/2.084/13.14 Fall 2006
Problem Set VIII Solution
Solution:
Evaluate the impact of changing the longitudinal tendon pitch from 165 mm to 200 mm
on the required prestress level to prevent tensile in the concrete upon pressurization.
Material properties and operating conditions:
E s = 210GPa; E c = 21GPa and ν = 0.15
Internal pressure P=350 kPa, and R=Rin+t/2= 21.19m
To calculate the required prestress level, we should go through the following steps:
At first, compute some constants that will be used:
2 As
π ⋅ d t2
, where As =
χ ls =
= 2551.8mm 2
4
tpl
χ θs =
4 As
tpθ
PR
= 3.708MPa ⋅ m
2
P
σr = −
2
Ec
El = (1 − χ ls )
1 −ν 2
Ec
Eθ = (1 − χ θs )
1 −ν 2
E* = χ ls E s + El
Nl =
and then
α=
γ =
ν 2 El 
t 
(1
)
E
+
E
−
χ
θs s
θ
E * 
R 
νEθ
Nl +
νE
νt
σ r (1 − χ θs − (1 − χ ls ) θ )
E*
1 −ν
E*
Ec t 3
2
2
D=
( + χ ls χ s t) + E s χ ls χ s t , whereχ s = thickness / 2 − depth = 0.525m
2
1 − ν 12
According to notes for problem set L54, we get
w
R
γ
ε θc = p = (P − )
R α
R
Nl 
ν

σr
− νEl ε θc + (1 − χ ls )
1 −ν 
t

ε lc =
E*
1
The maximum stress in longitudinal direction occurs at χ = −
ε θc max and ε lc max
σ lc max =
t
for
2
Ec
ν
(νε θc max + ε lc max + ε bl ) +
σr
2
1− ν
1− ν
where
ε bl (z) =
t d 2w
= tβ 2 w p e − βz [cos βz − sin βz ]
2
2 dz
1
 α  4
where in turn β = 

 4DR 
Since ε θc max and ε lc max reach the maximum value at z → ∞ , therefore, the strain due
to bending ε bl = 0 as z → ∞ .
Now, the maximum longitudinal stress shall be offset by the tendon prestress to get
zero net concrete stress upon pressurization and therefore we get
1 − χ ls
σ lsprestree =
σ lc max
χ ls
Similarly, the maximum stress in hoop direction
Ec
ν
σ θc max =
σr
(ε θc max + νε lc max ) +
2
1 −ν
1 −ν
Then, we get
1 − χ θs
σ θsprestree =
σ θc max
χ θs
For the two cases where the longitudinal tendon pitch is equal to 165 mm and 200
mm, respectively, we just substitute corresponding numbers to the above equations, and
obtain the final results:
165 mm:
σ lsprestree = 99.5MPa
σ θsprestree = 104.36MPa
200 mm: σ lsprestree = 123.7MPa
σ θsprestree = 104.41MPa
According to these result, when changing the longitudinal tendon pitch from 165
mm to 200mm, the required prestress in the longitudinal direction is increased by almost
25%, however the required prestress in the hoop direction almost remains unchanged.
2