Mathcad - kpeter

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