The document discusses the design of pipeline structures, emphasizing the analysis of material properties, thickness assumptions, and various load conditions including live loads and pressure checks. It details calculations related to buckling, bending moments, soil conditions, and trench design to ensure structural integrity under different scenarios. MATLAB code and Excel sheets are mentioned as tools for obtaining results related to the design parameters discussed.

1. Pipeline design
Environmental and protective structures
2015-2016
POLITECNICO DI MILANOCivil engineering for risk mitigation
SEYED MOHAMMAD SADEGH MOUSAVI 836154
DANIEL JALILI 832852
Prof. Di Prisco
Prof. Galli
1

2. Contents
POLITECNICO DI MILANOCivil engineering for risk mitigation
Parameters Material
properties
Thickness
assumption
Bedding
coefficient
assumption
Modulus of soil
reaction (E’)
Design
Under pressure
condition
analyzing
Ovalisation
check under
pressure
Buckling check Trench design
Computations
Live load
(Boussinesq
theory)
Ovalisation
ratio
maximum
bending
moment acting
on the pipe wall
Longitudinal Winkler
approach
Results
MATLAB code
& EXCEL
sheets
Results Trench
properties
Pipe material
2

3. Parameters
POLITECNICO DI MILANOCivil engineering for risk mitigation
Material: Steel
Steel Section properties
Poisson’s Ratio Yield Stress [Mpa] Price/Tons
0.3 235 400€ - 600€
(Tension stress and Yield strength in AWWA M11 – page 56)
𝜎𝑡. 2𝑡 = 𝑃𝑖 − 𝑃𝑒 . 𝐷 𝑚
𝜎𝑡 =
𝑃𝑖 − 𝑃𝑒 . 𝐷 𝑚
2𝑡
Thickness Assumptions
𝑡 =
𝑃𝑖. 𝐷 𝑚
2𝛿 𝑦
𝑡 ≥
𝑃𝑖.
𝐷0 + 𝐷0 − 2𝑡
2
2 × 0.5𝜎 𝑦
⟹ 𝑡 ≥
𝑃𝑖. 𝐷0
𝜎 𝑦. 1 +
𝑃𝑖
𝜎 𝑦
⟹ 𝑡 ≥
1 × 1000
248.2 × 1 +
1
248.2
= 4.01 𝒕 ≥ 4.01 𝒎𝒎
 Minimum plate or sheet thicknesses:
𝑡 =
𝐷
288
(Pipe sizes up to 54 in. (1350 mm) ID)
𝑡 =
𝐷+20
400
(Pipe sizes greater than 58 in. (1350 mm) ID)
𝑡 =
𝐷
240
(For mortar-lined & flexible coated pipe)
The minimum thickness for steel cylinder of the pipe
is often governed by what can be safely handled and
installed. (AWWA Manual M11, for diameters up to
54 inch)
𝑡 ≥
𝐷
288
𝑡 ≥
1000
288
= 3.47 𝑚𝑚
1st Assumption Max tensile stress = 50% 𝝈 𝒚
2nd Assumption
No external pressure 𝑷 𝒆 = 𝟎
acting on the pipe wall
3

4. Parameters
POLITECNICO DI MILANOCivil engineering for risk mitigation
 In case of reach the yield limit and utilization of some of plastic capacity without buckling in high amount of
moment acting on the pipe: Using Class 2 & the minimum thickness: 𝐷
𝑡
< 70𝜀2
𝑤𝑖𝑡ℎ 𝜀2
=
248.2
𝑓𝑦
Check the buckling (Critical Pressure) (𝑃𝑒 − 𝑃𝑖) 𝑐𝑟= 𝑓0. 24
𝐸. 𝐼
1 − 𝜈2 . 𝐷 𝑚
3
𝑃𝑒 − 𝑃𝑖 𝑐𝑟 = 0.35 × 24 ×
210000 ×
103
12
1 − 0.32 × 9903
= 166 𝑘𝑃𝑎
Max Burial Depth Considering Ovalisation with No Internal Pressure
No Internal Pressure External Pressure
0 Bar = Out of
Service of the Pipe
Non-Operating Cond No Internal Pressure Vertical Earth Load Applied
Most Operating Cond External Pressure << Internal Pressure
Computation of Earth
Loads on the Pipe
Flexible Steel Pipe Design
Dead
Load
Weight of Prism of
the Soil
Conditions: Above the Water Table, An Upper-bound Estimate of the pipe pressure resulting from the dead load
𝑃𝑒 > 𝛾𝐻 ⇒ 𝐻 <
𝑃𝑒
𝛾
=
166
18
= 9.22 𝑚
Ovalisation Graph Max Deflection 𝒇 𝟎 = 𝟎. 𝟑𝟓
𝐼 =
𝑡3
12
4
𝑡 >
1000
70
= 14.28 𝑚𝑚

5. Parameters
POLITECNICO DI MILANOCivil engineering for risk mitigation
Bending Coefficient (𝜶 – Design angle)
For the vertical reaction acting on the bottom of the pipe, the only unknown is the bedding angle 2α. The bedding angle at the base
is a function of soil types and degree of compaction among the other factors.
can influence the deflection predicted by IOWA formula by as much as 25 %.
2α= 90° (Concrete Pipe and Portal Culvert Handbook and assuming a “Class C” bedding - poor quality bedding)
The ratio between the load that a pipe can support on a particular type of bedding, and the test load is called the bedding factor
(Coefficient).
 Granular materials without cohesion, maximum for sand and
gravel and also maximum for saturated top soil were considered.
• Transfer the vertical load on the pipe to the foundation
• Uniform support along the pipeline
Bedding supporting
the pipe
𝐸′
: is the modulus of soil reaction (𝑀𝑠) which is constrained
modulus of soil that can be easily obtained in laboratory tests,
is used.
𝝈 𝒉 = 𝑬′
∆𝒙
𝑫 𝟎
 Assumption:
- Coarse-Grained soil including Fines  SC3
The maximum pressure 𝝈 𝒉
5

6. Parameters
POLITECNICO DI MILANOCivil engineering for risk mitigation
 Some measurements of 𝑀𝑠 varying with depth (Soil Density=18.8
𝐾𝑁
𝑚3):
 Our case: 18
𝐾𝑁
𝑚3 (linearly interpolated - according to the vertical stress level for different compaction degrees)
Burial depth (m)
soil density 18
KN/m^3
vertical stress level
Compaction, % maximum Standard proctor density
95% 90% 85%
Mpa Mpa Mpa Mpa MPa MPa
0.40 7.20 9.82 4.91 4.61 2.30 2.50 1.30
1.80 32.40 11.37 5.73 5.06 2.58 2.68 1.39
3.70 66.60 12.15 6.08 5.19 2.60 2.79 1.40
7.30 131.40 12.92 6.46 5.38 2.69 2.98 1.49
14.60 262.80 14.27 7.13 6.12 3.06 3.45 1.77
22.00 396.00 15.70 7.90 6.98 3.53 4.02 2.06
6

7. Parameters
POLITECNICO DI MILANOCivil engineering for risk mitigation
Ms 85 = -2E-09σ3 + 3E-06σ2 + 0.0011σ + 1.317
0
2
4
6
8
10
12
14
16
18
0 50 100 150 200 250 300 350 400 450
MS[MPA]
VERTICAL PRESSURE [KN/M2]
95 % compaction
95 % compation w/w
90 % compaction
90 % compaction W/W
85% compaction
85 % compaction W/W
- 𝐷𝐿: time lag factor which is applied to the dead load to account for long-term deformation of the backfill at the sides of the
pipe, Pressure in the pipe ≥ the dead-load vertical pressure = 1 -1.5 (Masada 2000). In our case 𝐷𝐿= 1.
- The Bedding Constant (K):
𝐾 = 0.5 sin 𝛼 − 0.082 𝑠𝑖𝑛2
𝛼 + 0.08
𝛼
𝑆𝑖𝑛𝛼
− 0.16𝑠𝑖𝑛𝛼 𝜋 − 𝛼 − 0.04
𝑠𝑖𝑛2𝛼
𝑠𝑖𝑛𝛼
+ 0.318𝑐𝑜𝑠𝛼 − 0.208
 The ovalisation ratio
∆𝑥
𝐷0
(Modified IOWA Formula)
∆𝒙
𝑫 𝟎
=
𝑫 𝑳. 𝑲. 𝝈 𝒗
𝟖
𝑬𝑰
𝑫 𝟎
𝟑 + 𝟎. 𝟎𝟔𝟏𝑬′
7

8. Design - Trench
POLITECNICO DI MILANOCivil engineering for risk mitigation
 For flexible conduit in ditch we considered: 𝑪 𝒅 =
𝟏 − 𝒆
−𝟐𝑲 𝒂 𝝁′
𝑯
𝑩 𝒅
𝟐𝑲 𝒂 𝝁′
𝑾 𝑪 = 𝑪 𝒅. 𝜸. 𝑫. 𝑩 𝒅
𝐶 𝑑 = Non-dimensional load coefficient that is non-linearly increasing by depth.
𝛾 = Unit weight of the backfilling material
𝐵 𝑑 = Width of the ditch = 2𝐷0 = 2𝑚
H = Soil depth (above the crown of the pipe)
𝐾𝑎 = Active earth pressure coefficient of natural soil
𝜇′ = Friction coefficient of trench sides on natural soil
Soil: granular include of fine particles of silt. So, 𝜇′𝐾 𝑎 = 0.15
Fill-LoadH.P
1st H.P 𝜎𝑣 is distributed approximately uniformly (Marston’s Theory)
2nd H.P 𝜎𝑣
𝑏𝑜𝑡𝑡𝑜𝑚
is distributed approximately uniformly
3rd H.P 𝜎ℎ is distributed parabolically over the middle 100°
Original Iowa formula (Spangler 1941)
 The allowable buckling pressure 𝑞 𝑎 may be determined by the following:
FS = Factor of Safety (2.5 for
ℎ
𝐷
≥ 2 and 3 for
ℎ
𝐷
< 2)
𝑅 𝑤= Water buoyancy factor = 1 − 0.33
ℎ 𝑤
𝐻
, 0 < ℎ 𝑤 < 𝐻
ℎ 𝑤= Height of water surface over top of pipe
B’= Empirical coefficient of elastic support (dimensionless) =
1
1+4𝑒
−0.065.
ℎ
𝐷
E’= The modulus of soil reaction (≅ 𝑀𝑠)
EI = Pipe wall stiffness
𝒒 𝒂 =
𝟏
𝑭𝑺
𝟑𝟐𝑹 𝒘 𝑩′ 𝑬′
𝑬𝑰
𝑫 𝟑
(𝑷 𝒆 − 𝑷𝒊) 𝒄𝒓= 𝒇 𝟎. 𝟐𝟒
𝑬. 𝑰
𝟏 − 𝝂 𝟐 . 𝑫 𝒎
𝟑
For later, both of these equations will be checked in order to verify the under pressure condition of the pipe and the most critical
one will be dominant.
(Masada theory 1930 )
8

9. Computations (M & K)
POLITECNICO DI MILANOCivil engineering for risk mitigation
By obtaining
∆𝑥
𝐷0
, the stress distribution around the pipe is obtained. By knowing the stress
distribution, we can benefit because of two reasons:
Class “C” of
Granular
soil
𝜶 = 𝟒𝟓° K=0.0951
1. We can exactly obtain the maximum bending moment acting on the pipe wall thickness
and bending stress can be easily verified by the capacity of the material we are using (in
our case 𝜎 𝑦= 235 MPa)
𝑀 =
𝑀𝑐 + 𝑅𝑐 𝑟(1 − 𝑐𝑜𝑠∅) 𝑓𝑜𝑟 0 ≤ ∅ ≤ 𝜋
−0.5. 𝜎𝜈
𝑏𝑜𝑡
. 𝑟2
. 𝑠𝑖𝑛2
∅ 𝑓𝑜𝑟 0 ≤ ∅ ≤ 𝛼
−𝑠𝑖𝑛𝛼. 𝜎𝜈
𝑏𝑜𝑡
. 𝑟2
. (𝑠𝑖𝑛∅ − 0.5𝑠𝑖𝑛𝛼) 𝑓𝑜𝑟 𝛼 ≤ ∅ ≤ 𝜋
−𝜎ℎ. 𝑟2
(0.147 − 0.51𝑐𝑜𝑠∅ + 0.5𝑐𝑜𝑠2
∅ − 0.143𝑐𝑜𝑠4
∅ ) 𝑓𝑜𝑟 40° ≤ ∅ ≤ 140°
+1.021𝜎ℎ. 𝑟2
𝑐𝑜𝑠∅ 𝑓𝑜𝑟 140° ≤ ∅ ≤ 180°
−0.5. 𝜎𝑣. 𝑟2
(1 − 𝑠𝑖𝑛∅)2
𝑓𝑜𝑟 90° ≤ ∅ ≤ 180°
r = radius of the pipe
𝜶 = half of bedding angle
∅ = angle that the moment
is computed
𝑅 𝐶 = −0.106𝑠𝑖𝑛3
𝛼. 𝜎𝜈
𝑏𝑜𝑡
. 𝑟 + 0.511𝜎ℎ. 𝑟 + 0.106𝜎𝜈 𝑟
𝑀 𝐶 = −0.049𝜎𝜈. 𝑟2
− 0.166𝜎ℎ. 𝑟2
+ 𝜎𝜈
𝑏𝑜𝑡
. 𝑟2
[0.106𝑠𝑖𝑛3
𝛼 + 0.08𝛼 − 0.04 sin 2𝛼 − 0.159𝑠𝑖𝑛2
𝛼 𝜋 − 𝛼 + 0.318𝑠𝑖𝑛𝛼 1 + 𝑐𝑜𝑠𝛼
𝑀𝑐& 𝑅 𝑐 can be computed as follows:
2. According to “Masada 2000”, the vertical deflection of the flexible pipe Δy, which is slightly higher than the
horizontal deflection Δx, can be obtained and therefore, the correct 𝑓0 factor can be obtained to check for pipe’s
ovality and its buckling in under pressure condition of the pipe.
9

10. Calculation – Loads
POLITECNICO DI MILANOCivil engineering for risk mitigation
Live Load
Weight Tires Distance
Resultant
load
transformed
to the pipe
Boussinesq Approach
AASHTO standard H20 static loading (wheel loading):
Dimension:
Rear = (19.7”×10”) 50 × 45 cm2
Front = (9.85”×17.8”) 25 × 45 cm2
The uniform stress transferred by each wheel can be computed as:
Rear Wheels: 𝜎𝐿𝑖𝑣𝑒,𝑅𝑒𝑎𝑟 =
1
2
× 0.8 ×
5000
0.5 × 0.45 × 100
= 𝟖𝟖. 𝟖𝟗
𝐾𝑁
𝑚2
Front Wheels: 𝜎𝐿𝑖𝑣𝑒,𝐹𝑟𝑜𝑛𝑡 =
1
2
× 0.2 ×
5000
0.25 × 0.45 × 100
= 𝟒𝟒. 𝟒𝟓
𝐾𝑁
𝑚2
Critical Case
when the both rear tires of the truck is above the centerline of the axis of the pipeline, in both
directions of the roads (two trucks are stopped with both rear tires above the pipe’s centerline, in both
directions).
10

11. Loadings (Live load)
POLITECNICO DI MILANOCivil engineering for risk mitigation
11

12. 𝐼𝜎𝑧
=
1
2𝜋
𝑎𝑟𝑐𝑡𝑎𝑛
𝑚.𝑛
𝑚2 + 𝑛2 +1
+
𝑚.𝑛
𝑚2 + 𝑛2 +1
.
1
1+ 𝑚2
+
1
1+ 𝑛2
Loadings (Live load)
POLITECNICO DI MILANOCivil engineering for risk mitigation
The increment of vertical stress ∆𝜎𝑣 can be computed for each section and for each depth
considered by computing the influence factor of 𝐼 𝜎 𝑣
for all 8 wheels (from two trucks)
using the following expression for the corner of each rectangle after Boussinesq:
Using the above expression, the vertical stress increment ∆𝜎𝑣 at each section is
computed for different burial depths of the pipeline. It can be seen that by increasing
the depth, the ∆𝜎𝑣 decreases for each section and also after certain depths, the
difference between the ∆𝜎𝑣 of the sections decreases. This is quite important fact since
the main reason of excessive bending of the pipeline in its axis, is the non-uniform
distribution of the stress above the pipe due to the live loads.
Point 1 Point 2 Point 3Point 4Point 5
0.1 0.00 84.80 0.04 84.81 0.45 84.81
0.2 0.01 69.09 0.31 69.13 2.60 69.13
0.3 0.03 51.29 0.87 51.39 5.63 51.39
0.4 0.06 37.43 1.64 37.64 8.18 37.64
0.5 0.10 27.75 2.46 28.11 9.76 28.11
0.6 0.16 21.10 3.21 21.63 10.44 21.63
0.7 0.23 16.49 3.81 17.17 10.50 17.17
0.8 0.30 13.21 4.24 14.04 10.18 14.04
0.9 0.37 10.82 4.51 11.78 9.66 11.78
1 0.44 9.06 4.65 10.11 9.05 10.11
1.1 0.51 7.71 4.68 8.84 8.42 8.84
1.2 0.57 6.68 4.64 7.85 7.81 7.85
1.3 0.63 5.87 4.55 7.06 7.24 7.24
1.4 0.68 5.23 4.42 6.42 6.71 6.71
1.5 0.72 4.71 4.27 5.88 6.24 6.24
1.6 0.75 4.28 4.11 5.43 5.82 5.82
1.7 0.78 3.93 3.94 5.05 5.45 5.45
1.8 0.80 3.64 3.77 4.71 5.12 5.12
1.9 0.82 3.39 3.61 4.42 4.82 4.82
2 0.83 3.19 3.45 4.15 4.56 4.56
2.1 0.84 3.01 3.30 3.92 4.33 4.33
2.2 0.84 2.85 3.16 3.71 4.12 4.12
2.3 0.84 2.72 3.02 3.52 3.94 3.94
2.4 0.84 2.60 2.89 3.34 3.77 3.77
2.5 0.84 2.50 2.77 3.18 3.62 3.62
2.6 0.84 2.41 2.66 3.03 3.49 3.49
2.7 0.83 2.33 2.55 2.90 3.36 3.36
2.8 0.83 2.26 2.45 2.77 3.25 3.25
2.9 0.82 2.20 2.35 2.66 3.15 3.15
3 0.81 2.14 2.26 2.55 3.06 3.06
z (m)
sigmaz of all wheels on points
Max
12

13. Loadings (Live load)
POLITECNICO DI MILANOCivil engineering for risk mitigation
q(KN/M^2) Z(m) B(m) L(m) m(m) n(m) Iσz(1BDG) B(m) L(m) m(m) n(m) Iσz(1ACG) B(m) L(m) m(m) n(m) Iσz(1AEH) B(m) L(m) m(m) n(m) Iσz(1BFH) Iσz1(KN/m^2)
88.89 0.1 0.225 1.850 2.25 18.50 0.243 0.225 1.850 2.25 18.50 0.243 0.225 1.350 2.25 13.50 0.242 0.225 1.350 2.25 13.50 0.242 0.00
88.89 0.2 0.225 1.850 1.13 9.25 0.213 0.225 1.850 1.13 9.25 0.213 0.225 1.350 1.13 6.75 0.213 0.225 1.350 1.13 6.75 0.213 0.01
88.89 0.3 0.225 1.850 0.75 6.17 0.179 0.225 1.850 0.75 6.17 0.179 0.225 1.350 0.75 4.50 0.179 0.225 1.350 0.75 4.50 0.179 0.02
88.89 0.4 0.225 1.850 0.56 4.63 0.149 0.225 1.850 0.56 4.63 0.149 0.225 1.350 0.56 3.38 0.149 0.225 1.350 0.56 3.38 0.149 0.05
88.89 0.5 0.225 1.850 0.45 3.70 0.127 0.225 1.850 0.45 3.70 0.127 0.225 1.350 0.45 2.70 0.126 0.225 1.350 0.45 2.70 0.126 0.10
88.89 0.6 0.225 1.850 0.38 3.08 0.109 0.225 1.850 0.38 3.08 0.109 0.225 1.350 0.38 2.25 0.108 0.225 1.350 0.38 2.25 0.108 0.15
88.89 0.7 0.225 1.850 0.32 2.64 0.095 0.225 1.850 0.32 2.64 0.095 0.225 1.350 0.32 1.93 0.094 0.225 1.350 0.32 1.93 0.094 0.22
88.89 0.8 0.225 1.850 0.28 2.31 0.084 0.225 1.850 0.28 2.31 0.084 0.225 1.350 0.28 1.69 0.083 0.225 1.350 0.28 1.69 0.083 0.28
88.89 0.9 0.225 1.850 0.25 2.06 0.075 0.225 1.850 0.25 2.06 0.075 0.225 1.350 0.25 1.50 0.073 0.225 1.350 0.25 1.50 0.073 0.35
88.89 1 0.225 1.850 0.23 1.85 0.068 0.225 1.850 0.23 1.85 0.068 0.225 1.350 0.23 1.35 0.065 0.225 1.350 0.23 1.35 0.065 0.42
88.89 1.1 0.225 1.850 0.20 1.68 0.062 0.225 1.850 0.20 1.68 0.062 0.225 1.350 0.20 1.23 0.059 0.225 1.350 0.20 1.23 0.059 0.48
88.89 1.2 0.225 1.850 0.19 1.54 0.056 0.225 1.850 0.19 1.54 0.056 0.225 1.350 0.19 1.13 0.053 0.225 1.350 0.19 1.13 0.053 0.53
88.89 1.3 0.225 1.850 0.17 1.42 0.051 0.225 1.850 0.17 1.42 0.051 0.225 1.350 0.17 1.04 0.048 0.225 1.350 0.17 1.04 0.048 0.58
88.89 1.4 0.225 1.850 0.16 1.32 0.047 0.225 1.850 0.16 1.32 0.047 0.225 1.350 0.16 0.96 0.044 0.225 1.350 0.16 0.96 0.044 0.62
88.89 1.5 0.225 1.850 0.15 1.23 0.044 0.225 1.850 0.15 1.23 0.044 0.225 1.350 0.15 0.90 0.040 0.225 1.350 0.15 0.90 0.040 0.65
88.89 1.6 0.225 1.850 0.14 1.16 0.041 0.225 1.850 0.14 1.16 0.041 0.225 1.350 0.14 0.84 0.037 0.225 1.350 0.14 0.84 0.037 0.67
88.89 1.7 0.225 1.850 0.13 1.09 0.038 0.225 1.850 0.13 1.09 0.038 0.225 1.350 0.13 0.79 0.034 0.225 1.350 0.13 0.79 0.034 0.69
88.89 1.8 0.225 1.850 0.13 1.03 0.035 0.225 1.850 0.13 1.03 0.035 0.225 1.350 0.13 0.75 0.031 0.225 1.350 0.13 0.75 0.031 0.70
88.89 1.9 0.225 1.850 0.12 0.97 0.033 0.225 1.850 0.12 0.97 0.033 0.225 1.350 0.12 0.71 0.029 0.225 1.350 0.12 0.71 0.029 0.70
88.89 2 0.225 1.850 0.11 0.93 0.031 0.225 1.850 0.11 0.93 0.031 0.225 1.350 0.11 0.68 0.027 0.225 1.350 0.11 0.68 0.027 0.70
88.89 2.1 0.225 1.850 0.11 0.88 0.029 0.225 1.850 0.11 0.88 0.029 0.225 1.350 0.11 0.64 0.025 0.225 1.350 0.11 0.64 0.025 0.69
88.89 2.2 0.225 1.850 0.10 0.84 0.027 0.225 1.850 0.10 0.84 0.027 0.225 1.350 0.10 0.61 0.023 0.225 1.350 0.10 0.61 0.023 0.69
88.89 2.3 0.225 1.850 0.10 0.80 0.025 0.225 1.850 0.10 0.80 0.025 0.225 1.350 0.10 0.59 0.021 0.225 1.350 0.10 0.59 0.021 0.67
88.89 2.4 0.225 1.850 0.09 0.77 0.024 0.225 1.850 0.09 0.77 0.024 0.225 1.350 0.09 0.56 0.020 0.225 1.350 0.09 0.56 0.020 0.66
88.89 2.5 0.225 1.850 0.09 0.74 0.022 0.225 1.850 0.09 0.74 0.022 0.225 1.350 0.09 0.54 0.019 0.225 1.350 0.09 0.54 0.019 0.65
88.89 2.6 0.225 1.850 0.09 0.71 0.021 0.225 1.850 0.09 0.71 0.021 0.225 1.350 0.09 0.52 0.018 0.225 1.350 0.09 0.52 0.018 0.63
88.89 2.7 0.225 1.850 0.08 0.69 0.020 0.225 1.850 0.08 0.69 0.020 0.225 1.350 0.08 0.50 0.017 0.225 1.350 0.08 0.50 0.017 0.62
88.89 2.8 0.225 1.850 0.08 0.66 0.019 0.225 1.850 0.08 0.66 0.019 0.225 1.350 0.08 0.48 0.016 0.225 1.350 0.08 0.48 0.016 0.60
88.89 2.9 0.225 1.850 0.08 0.64 0.018 0.225 1.850 0.08 0.64 0.018 0.225 1.350 0.08 0.47 0.015 0.225 1.350 0.08 0.47 0.015 0.58
88.89 3 0.225 1.850 0.08 0.62 0.017 0.225 1.850 0.08 0.62 0.017 0.225 1.350 0.08 0.45 0.014 0.225 1.350 0.08 0.45 0.014 0.57
Wheel 1
13
0
5
10
15
20
25
30
35
40
0 0.5 1 1.5 2 2.5 3
VERTICALSTRESS
DEPTH (M)
Bousinesque Theory
Point 1 Point 2 Point 3 Point 4 Point 5 Max
In this graph, the vertical stress increment due
to live load for different burial depth at each
section is provided and their maximum (yellow
dashed-line) is used for the design.
14

14. γ
(KN/m^3)
Cover depth
[m]
Soil stress [KN/m^2]
without amplification
factor
Traffic stress [KN/m^2]
without amplification
factor
Total stress
[KN/m^2]
Soil stress [KN/m^2]
amplification factor
1.35
Traffic stress
[KN/m^2]
amplification factor
1.5
Total stress
[KN/m^2] with
amplification factor
18 0.1 1.8 84.81 86.61 2.43 127.21 129.64
18 0.2 3.6 69.13 72.73 4.86 103.69 108.55
18 0.3 5.4 51.39 56.79 7.29 77.09 84.38
18 0.4 7.2 37.64 44.84 9.72 56.46 66.18
18 0.5 9 28.11 37.11 12.15 42.16 54.31
18 0.6 10.8 21.63 32.43 14.58 32.44 47.02
18 0.7 12.6 17.17 29.77 17.01 25.76 42.77
18 0.8 14.4 14.04 28.44 19.44 21.06 40.50
18 0.9 16.2 11.78 27.98 21.87 17.67 39.54
18 1 18 10.11 28.11 24.30 15.16 39.46
18 1.1 19.8 8.84 28.64 26.73 13.26 39.99
18 1.2 21.6 7.85 29.45 29.16 11.77 40.93
18 1.3 23.4 7.24 30.64 31.59 10.85 42.44
18 1.4 25.2 6.71 31.91 34.02 10.07 44.09
18 1.5 27 6.24 33.24 36.45 9.37 45.82
18 1.6 28.8 5.82 34.62 38.88 8.74 47.62
18 1.7 30.6 5.45 36.05 41.31 8.17 49.48
18 1.8 32.4 5.12 37.52 43.74 7.68 51.42
18 1.9 34.2 4.82 39.02 46.17 7.23 53.40
18 2 36 4.56 40.56 48.60 6.84 55.44
18 2.1 37.8 4.33 42.13 51.03 6.49 57.52
18 2.2 39.6 4.12 43.72 53.46 6.18 59.64
18 2.3 41.4 3.94 45.34 55.89 5.90 61.79
18 2.4 43.2 3.77 46.97 58.32 5.65 63.97
18 2.5 45 3.62 48.62 60.75 5.43 66.18
18 2.6 46.8 3.49 50.29 63.18 5.23 68.41
18 2.7 48.6 3.36 51.96 65.61 5.05 70.66
18 2.8 50.4 3.25 53.65 68.04 4.88 72.92
18 2.9 52.2 3.15 55.35 70.47 4.73 75.20
18 3 54 3.06 57.06 72.90 4.59 77.49Dry Soil
Dry Soil
Dry Soil
Dry Soil
Dry Soil
Dry Soil
Dry Soil
Dry Soil
Dry Soil
Dry Soil
Dry Soil
Dry Soil
Dry Soil
Dry Soil
Dry Soil
Dry Soil
Dry Soil
Dry Soil
Dry Soil
Dry Soil
Dry Soil
Dry Soil
Dry Soil
Dry Soil
Dry Soil
Dry Soil
saturated soil unit
weight [KN/m^3]
Dry Soil
Dry Soil
Dry Soil
Dry Soil
Loadings (Combined load)
POLITECNICO DI MILANOCivil engineering for risk mitigation
Both dead load of the burying soil and the live load
of the traffic are destabilizing forces and therefore for
the reason of safety, amplification factors of 1.35 for
the dead load and 1.5 for the live load is used for the
reasons of ULS design of in plane of the pipeline
section.
15
0
20
40
60
80
100
120
140
0 0.5 1 1.5 2 2.5 3
Σ[KN/M^2]
COVER DEPTH [M]
Stresses
Soil Stress-Dead Load Traffic Stress-Live Load Total stress
It can be seen that the optimum depth for which the minimum pressure is
acting on the pipe’s crown is about 𝐻 = 1𝑚. But this is not the only fact
we need to observe! We have to obtain a depth for which both in
transversal and longitudinal direction, we won’t encounter problem due to
bending or buckling.
16

15. Computations – Deflection & Buckling
POLITECNICO DI MILANOCivil engineering for risk mitigation
By obtaining % Deflection ratio, the minimum diameter of the pipe due to ovalisation
can be obtained and the correct reducing factor of 𝑓0 can be evaluated.
Regarding the SLS according to “ASCE - A.P. Moser”, for flexible products such as steel with no cement lining, which exhibit
only deflection as a performance limit, the design deflection is 5 % based on the factor of safety of 2.
(𝑷 𝒆 − 𝑷𝒊) 𝒄𝒓= 𝒇 𝟎. 𝟐𝟒
𝑬. 𝑰
𝟏 − 𝝂 𝟐 . 𝑫 𝒎
𝟑
∆𝑦
∆𝑥
≈ 1 +
𝐸′
. 𝐷0
3
5711.83 𝐸𝐼
∆𝑥
𝐷0
=
𝐷𝐿. 𝐾. 𝜎𝑣
8
𝐸𝐼
𝐷0
3 + 0.061𝐸′
% Deflection = 𝟏𝟎𝟎 ×
𝑫 𝒎−𝑫 𝟎
𝑫 𝒎
Regarding point 2, according to Masada, the vertical to horizontal pipe deflection ratio
can be obtained by using the following simplified equation:
 The ovalisation ratio
∆𝒙
𝑫 𝟎
(Modified IOWA Formula)
17

16. Computations - Deflection
POLITECNICO DI MILANOCivil engineering for risk mitigation
Observation
According to the following tables, the deflection & buckling of the section are satisfied with the thickness equal to 13 mm.
95% 90% 85% 95% 90% 85% 95% 90% 85% 95% 90% 85%
0.10 1.49 2.66 3.59 OK OK OK 1.0731 1.0307 1.0168 OK OK OK
0.20 1.16 2.06 2.78 OK OK OK 1.0714 1.0303 1.0164 OK OK OK
0.30 0.91 1.59 2.15 OK OK OK 1.0698 1.0300 1.0161 OK OK OK
0.40 0.74 1.28 1.73 OK OK OK 1.0684 1.0298 1.0159 OK OK OK
0.50 0.63 1.08 1.47 OK OK OK 1.0674 1.0297 1.0158 OK OK OK
0.60 0.56 0.96 1.30 OK OK OK 1.0669 1.0296 1.0157 OK OK OK
0.70 0.52 0.89 1.20 OK OK OK 1.0665 1.0295 1.0156 OK OK OK
0.80 0.50 0.85 1.15 OK OK OK 1.0660 1.0293 1.0156 OK OK OK
0.90 0.49 0.83 1.13 OK OK OK 1.0658 1.0293 1.0155 OK OK OK
1.00 0.49 0.83 1.13 OK OK OK 1.0657 1.0293 1.0155 OK OK OK
1.10 0.50 0.84 1.15 OK OK OK 1.0659 1.0293 1.0156 OK OK OK
1.20 0.51 0.87 1.18 OK OK OK 1.0662 1.0294 1.0156 OK OK OK
1.30 0.52 0.89 1.22 OK OK OK 1.0666 1.0295 1.0156 OK OK OK
1.40 0.54 0.92 1.26 OK OK OK 1.0667 1.0295 1.0157 OK OK OK
1.50 0.56 0.96 1.30 OK OK OK 1.0669 1.0296 1.0157 OK OK OK
1.60 0.58 0.99 1.35 OK OK OK 1.0670 1.0296 1.0157 OK OK OK
1.70 0.60 1.03 1.39 OK OK OK 1.0672 1.0296 1.0157 OK OK OK
1.80 0.62 1.06 1.44 OK OK OK 1.0674 1.0296 1.0157 OK OK OK
1.90 0.64 1.10 1.49 OK OK OK 1.0675 1.0297 1.0158 OK OK OK
2.00 0.66 1.14 1.55 OK OK OK 1.0677 1.0297 1.0158 OK OK OK
Depth
(m)
Δx / D0 (%) Δx Check Δy / Δx Δy Check
18Computations – Deflection & Buckling
95% 90% 85% 95% 90% 85% 95% 90% 85% 95% 90% 85% 95% 90% 85%
0.10 1.598 2.746 3.655
0.20 1.243 2.118 2.827
0.30 0.970 1.637 2.189
0.40 0.788 1.315 1.761
0.50 0.670 1.110 1.488
0.60 0.597 0.985 1.321
0.70 0.554 0.911 1.222 0.628 20974 380 OK
0.80 0.532 0.874 1.171 0.636 0.621 21223 20747 379 253 OK OK
0.90 0.523 0.858 1.150 0.636 0.622 0.616 21223 20752 20563 379 253 184 OK OK OK
1.00 0.523 0.858 1.149 0.631 0.617 0.611 21081 20600 20416 380 254 185 OK OK OK
1.10 0.529 0.868 1.163 0.621 0.606 0.600 20732 20236 20041 382 255 185 OK OK OK
1.20 0.543 0.892 1.196 0.609 20335 384 OK
1.30 0.559 0.920 1.235
1.40 0.577 0.951 1.276
1.50 0.596 0.984 1.320
1.60 0.617 1.019 1.367
1.70 0.638 1.056 1.416
1.80 0.660 1.094 1.466
1.90 0.683 1.133 1.518
2.00 0.706 1.173 1.572
f0 qa (KN/m^2) -allowable qa (KN/m^2) - AWWA M11 Buckling CheckDepth
(m)
Deflection % (Δy)
19Computations - Deflection
95% 90% 85% 95% 90% 85% 95% 90% 85% 95% 90% 85% 95% 90% 85%
0.10 1.598 2.746 3.655
0.20 1.243 2.118 2.827
0.30 0.970 1.637 2.189
0.40 0.788 1.315 1.761
0.50 0.670 1.110 1.488
0.60 0.597 0.985 1.321
0.70 0.554 0.911 1.222 0.628 20974 380 OK
0.80 0.532 0.874 1.171 0.636 0.621 21223 20747 379 253 OK OK
0.90 0.523 0.858 1.150 0.636 0.622 0.616 21223 20752 20563 379 253 184 OK OK OK
1.00 0.523 0.858 1.149 0.631 0.617 0.611 21081 20600 20416 380 254 185 OK OK OK
1.10 0.529 0.868 1.163 0.621 0.606 0.600 20732 20236 20041 382 255 185 OK OK OK
1.20 0.543 0.892 1.196 0.609 20335 384 OK
1.30 0.559 0.920 1.235
1.40 0.577 0.951 1.276
1.50 0.596 0.984 1.320
1.60 0.617 1.019 1.367
1.70 0.638 1.056 1.416
1.80 0.660 1.094 1.466
1.90 0.683 1.133 1.518
2.00 0.706 1.173 1.572
f0 qa (KN/m^2) -allowable qa (KN/m^2) - AWWA M11 Buckling CheckDepth
(m)
Deflection % (Δy)
0.0
1.0
2.0
3.0
4.0
5.0
6.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
DEFLECTION[%]
DEPTH [M]
Deflection Ratio
Deflection Ratio 95 % Deflection Ratio 90 % Deflection Ratio 85 % Limitation
20

17. Computations - Bending Stress
POLITECNICO DI MILANOCivil engineering for risk mitigation
Bending Stress in the Pipe Wall Thickness:
𝜎𝑣 =
𝑀 𝑚𝑎𝑥
𝑍
=
𝑀 𝑚𝑎𝑥
6. 𝑡2
The maximum bending moment acting on the wall occurs exactly at the bottom of the pipe where ϕ = 90°. As said before, the
parabolic horizontal stress distribution depends on modulus of soil reaction of the side-fill which itself depends on the compaction
degree according to:
𝜎ℎ = 𝐸′
Δ𝑥
𝐷0
Min
Thickness Internal Pressure = 1000 MPa
t=10 mm
(Failed)
t=13 mm
(OK)
 With Internal Pressure
 Without Internal Pressure (In case of Maintenance)  MOST Critical Case
21

18. Computations - Maximum moment
95% 90% 85% 95% 90% 85% 95% 90% 85%
0.1 183.3 50525.1 21519.0 11412.4 2491.9 1056.1 555.9 2457.5 1041.6 548.2
0.2 153.5 41786.6 17836.7 9450.7 2060.9 875.3 460.2 2032.4 863.3 453.9
0.3 119.3 31692.4 13666.4 7235.5 1562.9 670.6 352.3 1541.3 661.3 347.4
0.4 93.6 24222.2 10561.3 5595.3 1194.4 518.2 272.3 1177.9 511.0 268.6
0.5 76.8 19456.8 8559.9 4541.0 959.3 419.9 221.0 946.1 414.1 217.9
0.6 66.5 16588.7 7344.4 3901.2 817.9 360.3 189.8 806.6 355.3 187.2
0.7 60.5 14940.6 6641.5 3531.1 736.6 325.8 171.8 726.4 321.3 169.4
0.8 57.3 14070.7 6268.9 3334.9 693.7 307.5 162.3 684.1 303.2 160.0
0.9 55.9 13706.3 6112.3 3252.4 675.7 299.8 158.2 666.4 295.7 156.0
1.0 55.8 13678.6 6100.1 3245.9 674.3 299.2 157.9 665.0 295.1 155.7
1.1 56.6 13881.8 6186.9 3291.6 684.4 303.5 160.1 674.9 299.3 157.9
1.2 57.9 14261.6 6346.7 3375.4 703.1 311.3 164.2 693.4 307.0 162.0
1.3 60.0 14846.8 6596.3 3506.7 732.0 323.6 170.6 721.9 319.1 168.3
1.4 62.4 15488.2 6869.5 3650.4 763.6 337.0 177.6 753.0 332.3 175.2
1.5 64.8 16163.8 7156.4 3801.3 796.9 351.0 185.0 785.9 346.2 182.4
1.6 67.3 16873.1 7456.8 3959.4 831.9 365.8 192.7 820.4 360.7 190.0
1.7 70.0 17614.0 7769.8 4124.0 868.4 381.2 200.7 856.4 375.9 197.9
1.8 72.7 18384.0 8094.4 4294.7 906.4 397.1 209.0 893.9 391.6 206.1
1.9 75.5 19180.5 8429.3 4470.9 945.7 413.5 217.6 932.7 407.8 214.6
2.0 78.4 20001.0 8773.6 4652.1 986.2 430.4 226.4 972.6 424.5 223.3
σ max=Mmax/Z=Mmax/(6*t^2)
(KN/m^2)
Depth
(m)
σ V.bot
[KN/m^2]
σ h [KN/m^2]
M (Bending Moment on Wall
Thickness)
POLITECNICO DI MILANOCivil engineering for risk mitigation
22Computations - Maximum wall Bending Stress
Ms (95%) Ms (90%) Ms (85%) Ms (95%) Ms (90%) Ms (85%) Ms (95%) Ms (90%) Ms (85%) Depth Yield Stress
2491.946 1056.139 555.865 2457540.00 1041556.976 548190.250 2457.54 1041.56 548.19 0.1 235
2060.861 875.341 460.232 2032406.84 863255.024 453877.642 2032.41 863.26 453.88 0.2 235
1562.883 670.597 352.268 1541304.91 661338.582 347404.755 1541.30 661.34 347.40 0.3 235
1194.380 518.164 272.350 1177889.86 511010.114 268589.687 1177.89 511.01 268.59 0.4 235
959.320 419.922 220.990 946074.89 414124.583 217938.382 946.07 414.12 217.94 0.5 235
817.859 360.267 189.827 806567.39 355292.757 187206.141 806.57 355.29 187.21 0.6 235
736.575 325.767 171.804 726405.46 321269.292 169432.435 726.41 321.27 169.43 0.7 235
693.675 307.483 162.250 684097.78 303237.565 160010.170 684.10 303.24 160.01 0.8 235
675.700 299.799 158.234 666370.99 295659.783 156048.833 666.37 295.66 156.05 0.9 235
674.337 299.201 157.919 665026.42 295070.147 155738.555 665.03 295.07 155.74 1 235
684.359 303.458 160.141 674910.38 299268.735 157929.626 674.91 299.27 157.93 1.1 235
703.090 311.307 164.223 693382.51 307008.693 161955.317 693.38 307.01 161.96 1.2 235
731.956 323.556 170.618 721850.19 319088.963 168261.975 721.85 319.09 168.26 1.3 235
763.586 336.961 177.616 753043.82 332308.376 175164.025 753.04 332.31 175.16 1.4 235
796.912 351.044 184.968 785909.45 346197.572 182414.349 785.91 346.20 182.41 1.5 235
831.894 365.789 192.664 820408.42 360738.535 190004.189 820.41 360.74 190.00 1.6 235
868.437 381.153 200.684 856446.70 375890.073 197912.726 856.45 375.89 197.91 1.7 235
906.419 397.083 208.999 893904.16 391600.464 206113.682 893.90 391.60 206.11 1.8 235
945.709 413.524 217.583 932651.81 407814.829 214579.158 932.65 407.81 214.58 1.9 235
986.177 430.422 226.408 972561.50 424479.266 223281.770 972.56 424.48 223.28 2 235
1027.700 447.725 235.447 1013511.17 441543.025 232195.775 1013.51 441.54 232.20 2.1 235
1070.163 465.385 244.676 1055387.36 458959.509 241297.569 1055.39 458.96 241.30 2.2 235
1113.459 483.360 254.074 1098086.14 476686.599 250565.850 1098.09 476.69 250.57 2.3 235
1157.494 501.612 263.621 1141513.11 494686.596 259981.568 1141.51 494.69 259.98 2.4 235
1202.181 520.107 273.301 1185582.92 512925.963 269527.788 1185.58 512.93 269.53 2.5 235
ThresholdM (Bending Moment on Wall Thickness) σ max=Mmax/Z=Mmax/(6*t^2) (KN/m^2) σ max (Mpa)
23
0
500
1000
1500
2000
2500
3000
0 1 2 3
STRESS(MPA)
DEPTH (M)
Maximum in wall bending stress for different soil compaction
(Wall Crushing)
Ms (95%) Ms (90%) Ms (85%) Yield Stress Limit
Computations - Maximum wall Bending Stress 24

19. Longitudinal
POLITECNICO DI MILANOCivil engineering for risk mitigation
Consider a straight beam supported along its entire length by an
elastic medium and subjected to vertical forces acting in the plane of
symmetry of the cross section.
Because of the external loadings the beam will deflect producing
continuously distributed reaction forces in the supporting medium.
The intensity of these reaction forces at any point is proportional to
the deflection of the beam y(x) at this point via the constant k:
The foundation is made of material which follows Hooke’s law. Its elasticity is characterized by the force, which distributed over a
unit area, will cause a unit deflection. This force is a constant of the supporting medium called the modulus of the foundation k0
[KN/m2/m].
Assume that the beam under consideration has a constant cross section with constant width b which is supported by the foundation.
A unit deflection of this beam will cause reaction equal to k0·b in the foundation, therefore the intensity of distributed reaction (per
unit length of the beam) will be:
𝑘 [ KN/m/m] = constant of the foundation (Winkler’s constant ) which includes the effect of the width of the beam.
𝑅 𝑥 = 𝑘. 𝑦(𝑥)
𝑅 𝑥 = 𝑘. 𝑦 𝑥
𝑘=𝑘0.𝑏
𝑏. 𝑘0. 𝑦 𝑥
Differential equation of equilibrium of a beam on elastic foundation
In the above equation the parameter α includes the flexural rigidity of the beam
as well as the elasticity of the foundation. This factor is called the characteristic
of the system with dimension length-1. In that respect 1/α is referred to as the so
called characteristic length. Therefore, x⋅α will be an absolute number.
25

20. Longitudinal
POLITECNICO DI MILANOCivil engineering for risk mitigation
The solution of this differential equation could be expressed as:
After solving this differential equation, the following matrix will be obtained:
EIy(x) EI multiple values of the transverse displacements
EIΦ(x) EI multiple values of slope of deflection line
Classification of the beams according to their stiffness:
1. α.l<0.5  Short Beams
2. 0.5≥α.L≤5  Medium Length Beams
3. α.L>5  Long Beams  Our Case
𝐵 =
𝐸𝐼
1 − 𝜈2
=
𝐸𝑡3
12(1 − 𝜈2) 𝛼 =
4 𝑘
4𝐵
𝐿 > 𝜆 =
2𝜋
𝛼
𝛼 =
4 6800 ∗ 12 ∗ (1 − 0.32)
4 ∗ 210 ∗ 106 ∗ 0.0133
= 2.52 𝜆 =
2𝜋
2.52
= 2.49 → 𝐿 = 13.49𝑚 ≈ 20𝑚
Change of variables
26

21. Longitudinal
POLITECNICO DI MILANOCivil engineering for risk mitigation
f1(m) α*f1(m) f2(m) α*f2(m) f3(m) α*f3(m) f4(m) α*f4(m)
12.40 31.2304 10.60 26.70 9.40 23.67 7.60 19.14
27

22. Longitudinal
POLITECNICO DI MILANOCivil engineering for risk mitigation
2829303132333435
Burial Depth
(m)
Live Load + Amplification
[KN]
x (m) V(x)=EIy(x) Φ(x)=EIφ(x) M(x) Q(x) R(x)
0 0.000 0.012 0.000 0.000 0.000
1 0.000 0.013 0.000 0.001 0.001
2 0.000 -0.003 -0.047 0.009 0.001
3 0.000 -0.109 -0.168 0.011 -0.002
4 -0.002 -0.272 -0.258 -0.051 -0.015
5 -0.004 0.194 -1.309 -0.238 -0.026
6 0.009 2.916 4.131 -0.210 0.063
7 0.059 6.370 8.157 1.500 0.400
7.599 0.090 2.324 12.007 4.021 0.609
7.6 0.090 2.308 12.047 12.861 0.610
8 0.091 -0.807 13.855 -2.795 0.616
9 0.104 4.033 15.095 2.325 0.708
9.399 0.119 2.385 16.026 4.795 0.808
9.4 0.119 2.373 16.067 13.636 0.808
10 0.122 0.000 17.095 0.000 0.832
10.599 0.119 -2.360 16.026 4.027 0.809
10.6 0.119 -2.373 16.067 12.869 0.808
11 0.104 -4.033 15.061 -2.325 0.708
12 0.091 0.807 13.855 2.795 0.616
12.399 0.090 -2.292 12.007 4.804 0.610
12.4 0.090 -2.308 12.047 13.644 0.610
13 0.059 -6.370 8.014 -1.500 0.400
14 0.009 -2.916 4.131 0.210 0.063
15 -0.004 -0.194 -1.309 0.238 -0.026
16 -0.002 0.272 -0.258 0.051 -0.015
17 0.000 0.109 -0.168 -0.011 -0.002
18 0.000 0.003 -0.047 -0.009 0.001
19 0.000 -0.013 0.000 -0.001 0.001
20 0.000 -0.012 0.000 0.000 0.000
0.9 26.505
Burial Depth (m)
Live Load +
Amplification [KN]
x (m) V(x)=EIy(x) Φ(x)=EIφ(x) M(x) Q(x) R(x)
0 0.000 0.011 0.000 0.000 0.000
1 0.000 0.012 0.000 0.001 0.001
2 0.000 -0.003 -0.040 0.008 0.001
3 0.000 -0.093 -0.144 0.009 -0.002
4 -0.002 -0.234 -0.250 -0.044 -0.013
5 -0.003 0.166 -1.123 -0.204 -0.022
6 0.008 2.502 3.544 -0.180 0.054
7 0.050 5.465 7.012 1.287 0.343
7.599 0.077 1.994 10.295 3.450 0.523
7.6 0.077 1.980 10.336 11.034 0.523
8 0.078 -0.693 10.733 -2.398 0.529
9 0.089 3.460 11.797 1.995 0.608
9.399 0.102 2.046 13.750 4.114 0.693
9.4 0.102 2.035 13.784 11.699 0.694
10 0.105 0.000 15.085 0.000 0.714
10.599 0.102 -2.025 13.750 3.455 0.694
10.6 0.102 -2.035 13.784 11.041 0.694
11 0.089 -3.460 11.797 -1.995 0.608
12 0.078 0.693 10.733 2.398 0.529
12.399 0.077 -1.966 10.295 4.122 0.523
12.4 0.077 -1.980 10.336 11.706 0.523
13 0.050 -5.465 7.012 -1.287 0.343
14 0.008 -2.502 3.544 0.180 0.054
15 -0.003 -0.166 -1.123 0.204 -0.022
16 -0.002 0.234 -0.250 0.044 -0.013
17 0.000 0.093 -0.144 -0.009 -0.002
18 0.000 0.003 -0.040 -0.008 0.001
19 0.000 -0.012 0.000 -0.001 0.001
20 0.000 -0.011 0.000 0.000 0.000
24.2561
Burial Depth (m)
Live Load +
Amplification [KN]
x (m) V(x)=EIy(x) Φ(x)=EIφ(x) M(x) Q(x) R(x)
0 0.000 0.009 0.000 0.000 0.000
1 0.000 0.010 0.000 0.001 0.000
2 0.000 -0.003 -0.035 0.007 0.001
3 0.000 -0.082 -0.126 0.008 -0.001
4 -0.002 -0.204 -0.044 -0.038 -0.012
5 -0.003 0.145 -0.982 -0.179 -0.019
6 0.007 2.188 3.100 -0.158 0.048
7 0.044 4.780 6.011 1.126 0.300
7.599 0.067 1.744 7.005 3.017 0.457
7.6 0.067 1.732 7.041 9.651 0.457
8 0.068 -0.606 7.642 -2.097 0.462
9 0.078 3.026 9.572 1.745 0.532
9.399 0.089 1.789 10.027 3.598 0.606
9.4 0.089 1.780 10.057 10.233 0.607
10 0.092 0.000 12.074 0.000 0.624
10.599 0.089 -1.771 10.027 3.022 0.607
10.6 0.089 -1.780 10.057 9.657 0.607
11 0.078 -3.026 9.572 -1.745 0.532
12 0.068 0.606 7.642 2.097 0.462
12.399 0.067 -1.720 7.005 3.605 0.458
12.4 0.067 -1.732 7.041 10.239 0.457
13 0.044 -4.780 6.011 -1.126 0.300
14 0.007 -2.188 3.100 0.158 0.048
15 -0.003 -0.145 -0.982 0.179 -0.019
16 -0.002 0.204 -0.044 0.038 -0.012
17 0.000 0.082 -0.126 -0.008 -0.001
18 0.000 0.003 -0.035 -0.007 0.001
19 0.000 -0.010 0.000 -0.001 0.000
20 0.000 -0.009 0.000 0.000 0.000
1.1 19.890
-0.02000
0.00000
0.02000
0.04000
0.06000
0.08000
0.10000
0.12000
0.14000
0 2 4 6 8 10 12 14 16 18 20
V[M]
DISTANCE (M)
V - Vertical Displacements
0.9 m 1 m 1.1 m
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
0 2 4 6 8 10 12 14 16 18 20
Φ[1/M]
DISTANCE (M)
Φ - Slope of the Deflection
0.9 m 1 m 1.1 m
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
0 2 4 6 8 10 12 14 16 18 20
M[KN.M)
DISTANCE (M)
M - Bending Moments
0.9 m 1 m 1.1 m
-4
-2
0
2
4
6
8
10
12
14
16
0 2 4 6 8 10 12 14 16 18 20
Q[KN]
DISTANCE (M)
Q - Shear Forces
0.9 m 1 m 1.1 m
-0.1000
0.0000
0.1000
0.2000
0.3000
0.4000
0.5000
0.6000
0.7000
0.8000
0.9000
0 2 4 6 8 10 12 14 16 18 20
R[KN]
DISTANCE (M)
R - Vertical Reactions
0.9 m 1 m 1.1 m

23. Results
POLITECNICO DI MILANOCivil engineering for risk mitigation
Steel Pipe Characteristics
Class [MPa] Diameter [m] Thickness [mm]
S235 1 15
Properties of the Trench
Unit Weight [KN/m3] Width [m] Effective Depth [m] Bedding Angle [Deg] Side fill Compaction [%]
18 1 1 90 85
Final CheckOptimum Depth
Bending Moment acting on the
pipe (Longitudinal Direction)
Thickness
Using just Live Load:
Because assuming a uniform soil all
around the pipe, won’t cause any bending
Moment in longitudinal axis of pipe.
Aim
Decrease the Depth
Economical
Decisions
Decrease the wall thickness
of the pipeline
High non-uniform
load distribution on
the pipe crown due
to traffic load
High Bending
Moment on pipeline
in the longitudinal
axis (Instability)
Higher wall thickness
36

24. Longitudinal
POLITECNICO DI MILANOCivil engineering for risk mitigation
Thanks for your attention
37