K Peter's Method for Upheaval Buckling
Analysis
1 - Pipeline Input data for bend angle =311 Deg
Pipeline Outside diameter = D
Pipeline wall thickness = t
Pipeline intwernal Diameter = d
D := 219mm
t := 12.7mm
d := D − 2t
d = 0.194 m
E := 207000MPa
Modules of Elasticity = E
P := 13.85MPa
Internal Pressure = P
T2 := 110
Maximum Design Temperature(Degtrees Celsius)=T2
T1 := 28
Installation Temperature(Degtrees Celsius)=T1
SMYS := 415MPa
Specific Minimum Yield Stress=SMYS
t1 := 0.3mm
FBE Thickness = t1
kg
ρfbe := 1500
3
FBE Density = ρfbe
m
t2 := 0.2mm
Adhesive Thickness = t2
ρad := 900
kg
3
m
Adhesive Density = ρad
t3 := 2mm
Polypropylene Thickness = t3
kg
ρpp := 990
Polypropylene Density = ρpp
Steel Density = ρs
3
m
ρs := 7850
kg
3
m
ρcont := 119
kg
3
m
Content Density=ρcont
γ := 0.3
Poissons Ratio=γ
Thermal Expansion Coefficient = α
Uplift Coefficient = f
Pipeline Burial Depth to top including 1m for berm = HI
Backfill Dry Soil Density over Active Length (compacted) = ρbc
α := 0.0000117
f := 0.4
HI := 2m
ρbc := 1600
kg
3
m
(
)
⎡ D2 − d 2 ⎤
⎥
⎣ 4
⎦
Pipeline Calculation
Aσ := π⋅ ⎢
Pipe cross section Area = Aσ
Aσ = 8.231 × 10
Ap :=
Pipe Internal Area = Ap
π⋅ d
2
2
Ap = 0.029 m
(
)
⎡ D4 − d4 ⎤
⎥
⎣ 64 ⎦
−5
I = 4.395 × 10
Flexural Regidity = EI
2
m
4
I := π⋅ ⎢
Moment Of Inertia = I
−3
4
m
EI := E⋅ I
3
6 m ⋅ kg
EI = 9.099 × 10
2
s
OD := D + 2t1 + 2t2 + 2t3
Outside Diameter Over all Coating = OD
OD = 0.224 m
ρfbe⋅ ⎡⎣( D + 2t1) − D ⎤⎦
2
Wfbe := π
FBE Weight = Wfbe
Wfbe = 0.31
2
4
kg
m
⎡⎣( D + 2t1 + 2t2) 2 − ( D + 2t1) 2⎤⎦
Wad := π⋅ ρad⋅
4
Adhesive Weight Wad
kg
Wad = 0.124
m
Polypropylene Weight = Wpp
22 ρpp ⎞ ⎡ 2
2
Wpp := ⎛⎜ ⋅
⎟ ⋅ ⎣OD − ( OD − 2⋅ t3) ⎤⎦
4
7
⎝
⎠
kg
Wpp = 1.381
m
Ws := Aσ⋅ ρs
Steel Weight Per unet Length = Ws
Ws = 64.613
Total Weight Empty = Wte
kg
Wte := Ws + Wfbe + Wad + Wpp
Wte = 66.429
Total Weight Operating = Wto
Maximum Allowable Stress = Sa
kg
m
Wto := Wte + ⎛⎜ π d
⎝
Wto = 69.932
Pipeline Compressive Restraining Force (Frestre)
m
2 ρcont ⎞
kg
m
Sa := 0.9⋅ SMYS
8
Sa = 3.735 × 10 Pa
4
⎟
⎠
Sh := P⋅
D
2t
8
Sh = 1.194 × 10 Pa
Tensile Hoop Stress = Sh
SL := E⋅ α⋅ ( T2 − T1 ) − ( γ⋅ Sh)
Compressive longitudinal Stress = SL
8
SL = 1.628 × 10 Pa
Frestr := α⋅ E⋅ Aσ⋅ ( T2 − T1 ) + ( 1 − 2 ⋅ γ) ⋅ P⋅ Ap
Compressive Restraining Force = Frestr
6
Frestr = 1.798 × 10 N
Calculation of Buckling Length
λ :=
4π
Buckling length = λ
2
EI
Frestr
λ = 14.135 m
Calculation of Ultimate Soil Resistance
R1 := g ⋅ ⎡⎢HI⋅ D⋅ ρbc⋅ ⎛⎜ 1 + f ⋅
⎣
Ultimate Soil Resistance = R1
R1 = 3.266 × 10
⎝
4 kg
2
s
Calculation of allowable / Remaining Stress
σall := Sa − Sh − SL
Allowable Bending Stress = σall
σall = 9.131 × 10 Pa
7
Calculation Allowable bending Angle
ηguess := 0.01
Guess:
Given
⎛ 1 − π⋅ ηguess⋅ cos( π ηguess) ⎞
⎜
⎟
Frestr
sin( π⋅ ηguess ) ⎠
⎝
= σall⋅
2
η := Find( ηguess )
η = 0.254
D⋅ E⋅ R1
HI ⎞
⎤
⎟ + Wto⎥
D⎠
⎦
⎛ 1 − π⋅ η⋅ cos( π⋅ η) ⎞
⎜
⎟
sin( π⋅ η) ⎠
⎝
= 0.111
2
ABAR := η⋅ λ⋅
Allowable Bend Angle in Radian = ABAR
R1
Frestr
ABAR = 0.065
ABAD := ABAR⋅
Allowable Bend Angle in degree = ABAD
ABAD = 3.737
180
π
Calculation Allowable Depth for Bend angle 3.1
Degree
Proposed Bend anglee in degree = BAPD1
BAPD1 := 3.1
Proposed Bend Angle in radian = BAPR
BAPR := BAPD1⋅
BAPR = 0.054
η2guess := 0.01
Guess:
Given
⎛ 1 − π⋅ η2guess ⋅ cos( π η2guess ) ⎞
⎜
⎟
sin( π⋅ η2guess ) ⎠
⎝
= σall⋅
2η2guess
λ
D⋅ E⋅ BAPR
η2 := Find( η2guess )
η2 = 0.3
Rreq := Frestr⋅
BAPR
η2 ⋅ λ
Rreq = 2.291 × 10
4 kg
2
s
Hreq1 :=
D
f
⎡⎡ ⎛ Rreq − Wto⎞ ⋅ ⎛ f ⎞ + 1⎤ − 1⎤
⎜
⎟⎜
g
⎠ ⎝ ρbc⋅ D2 ⎟⎠ 4⎥⎦ 2⎥⎦
⎣⎣ ⎝
⋅ ⎢⎢
Hreq1 = 1.628 m
The Height required for Angle 3.1 = Ha
Berm height = Bh
Ha := Hreq1 − Bh
Ha = 0.628 m
Bh := 1m
π
180