Analysis and design of embedded pipes: pipelines, vertical hollow piles.Soil-structure reactions for applied displacements of horizontally embedded systems at serviceability and ultimate limit states.
The design of a Buried Steel Pipeline with straight pressure under a road, within a ditch trench. Checking the ULS & SLS conditions both in the plane of the pipeline section & in the vertical plane along pipeline axis.
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 (Eโ)
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
Poissonโs Ratio Yield Stress [Mpa] Price/Tons
0.3 235 400โฌ - 600โฌ
(Tension stress and Yield strength in AWWA M11 โ page 56)
๐๐ก. 2๐ก = ๐๐ โ ๐๐ . ๐ท ๐
๐๐ก =
๐๐ โ ๐๐ . ๐ท ๐
2๐ก
Thickness Assumptions
๐ก =
๐๐. ๐ท ๐
2๐ฟ ๐ฆ
๐ก โฅ
๐๐.
๐ท0 + ๐ท0 โ 2๐ก
2
2 ร 0.5๐ ๐ฆ
โน ๐ก โฅ
๐๐. ๐ท0
๐ ๐ฆ. 1 +
๐๐
๐ ๐ฆ
โน ๐ก โฅ
1 ร 1000
248.2 ร 1 +
1
248.2
= 4.01 ๐ โฅ 4.01 ๐๐
๏ถ Minimum plate or sheet thicknesses:
๐ก =
๐ท
288
(Pipe sizes up to 54 in. (1350 mm) ID)
๐ก =
๐ท+20
400
(Pipe sizes greater than 58 in. (1350 mm) ID)
๐ก =
๐ท
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)
๐ก โฅ
๐ท
288
๐ก โฅ
1000
288
= 3.47 ๐๐
1st Assumption Max tensile stress = 50% ๐ ๐
2nd Assumption
No external pressure ๐ท ๐ = ๐
acting on the pipe wall
3
4. Parameters
POLITECNICO DI MILANOCivil engineering for risk mitigation
๏ถ 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: ๐ท
๐ก
< 70๐2
๐ค๐๐กโ ๐2
=
248.2
๐๐ฆ
Check the buckling (Critical Pressure) (๐๐ โ ๐๐) ๐๐= ๐0. 24
๐ธ. ๐ผ
1 โ ๐2 . ๐ท ๐
3
๐๐ โ ๐๐ ๐๐ = 0.35 ร 24 ร
210000 ร
103
12
1 โ 0.32 ร 9903
= 166 ๐๐๐
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
๐๐ > ๐พ๐ป โ ๐ป <
๐๐
๐พ
=
166
18
= 9.22 ๐
Ovalisation Graph Max Deflection ๐ ๐ = ๐. ๐๐
๐ผ =
๐ก3
12
4
๐ก >
1000
70
= 14.28 ๐๐
5. Parameters
POLITECNICO DI MILANOCivil engineering for risk mitigation
Bending Coefficient (๐ถ โ Design angle)
For the vertical reaction acting on the bottom of the pipe, the only unknown is the bedding angle 2ฮฑ. 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 %.
2ฮฑ= 90ยฐ (Concrete Pipe and Portal Culvert Handbook and assuming a โClass Cโ 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).
๏ถ Granular materials without cohesion, maximum for sand and
gravel and also maximum for saturated top soil were considered.
โข Transfer the vertical load on the pipe to the foundation
โข Uniform support along the pipeline
Bedding supporting
the pipe
๐ธโฒ
: is the modulus of soil reaction (๐๐ ) which is constrained
modulus of soil that can be easily obtained in laboratory tests,
is used.
๐ ๐ = ๐ฌโฒ
โ๐
๐ซ ๐
๏ถ Assumption:
- Coarse-Grained soil including Fines ๏ SC3
The maximum pressure ๐ ๐
5
6. Parameters
POLITECNICO DI MILANOCivil engineering for risk mitigation
๏ถ Some measurements of ๐๐ varying with depth (Soil Density=18.8
๐พ๐
๐3):
๏ถ Our case: 18
๐พ๐
๐3 (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-09ฯ3 + 3E-06ฯ2 + 0.0011ฯ + 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
- ๐ท๐ฟ: 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 โฅ the dead-load vertical pressure = 1 -1.5 (Masada 2000). In our case ๐ท๐ฟ= 1.
- The Bedding Constant (K):
๐พ = 0.5 sin ๐ผ โ 0.082 ๐ ๐๐2
๐ผ + 0.08
๐ผ
๐๐๐๐ผ
โ 0.16๐ ๐๐๐ผ ๐ โ ๐ผ โ 0.04
๐ ๐๐2๐ผ
๐ ๐๐๐ผ
+ 0.318๐๐๐ ๐ผ โ 0.208
๏ถ The ovalisation ratio
โ๐ฅ
๐ท0
(Modified IOWA Formula)
โ๐
๐ซ ๐
=
๐ซ ๐ณ. ๐ฒ. ๐ ๐
๐
๐ฌ๐ฐ
๐ซ ๐
๐ + ๐. ๐๐๐๐ฌโฒ
7
8. Design - Trench
POLITECNICO DI MILANOCivil engineering for risk mitigation
๏ถ For flexible conduit in ditch we considered: ๐ช ๐ =
๐ โ ๐
โ๐๐ฒ ๐ ๐โฒ
๐ฏ
๐ฉ ๐
๐๐ฒ ๐ ๐โฒ
๐พ ๐ช = ๐ช ๐ . ๐ธ. ๐ซ. ๐ฉ ๐
๐ถ ๐ = Non-dimensional load coefficient that is non-linearly increasing by depth.
๐พ = Unit weight of the backfilling material
๐ต ๐ = Width of the ditch = 2๐ท0 = 2๐
H = Soil depth (above the crown of the pipe)
๐พ๐ = Active earth pressure coefficient of natural soil
๐โฒ = Friction coefficient of trench sides on natural soil
Soil: granular include of fine particles of silt. So, ๐โฒ๐พ ๐ = 0.15
Fill-LoadH.P
1st H.P ๐๐ฃ is distributed approximately uniformly (Marstonโs Theory)
2nd H.P ๐๐ฃ
๐๐๐ก๐ก๐๐
is distributed approximately uniformly
3rd H.P ๐โ is distributed parabolically over the middle 100ยฐ
Original Iowa formula (Spangler 1941)
๏ถ The allowable buckling pressure ๐ ๐ may be determined by the following:
FS = Factor of Safety (2.5 for
โ
๐ท
โฅ 2 and 3 for
โ
๐ท
< 2)
๐ ๐ค= Water buoyancy factor = 1 โ 0.33
โ ๐ค
๐ป
, 0 < โ ๐ค < ๐ป
โ ๐ค= Height of water surface over top of pipe
Bโ= Empirical coefficient of elastic support (dimensionless) =
1
1+4๐
โ0.065.
โ
๐ท
Eโ= The modulus of soil reaction (โ ๐๐ )
EI = Pipe wall stiffness
๐ ๐ =
๐
๐ญ๐บ
๐๐๐น ๐ ๐ฉโฒ ๐ฌโฒ
๐ฌ๐ฐ
๐ซ ๐
(๐ท ๐ โ ๐ท๐) ๐๐= ๐ ๐. ๐๐
๐ฌ. ๐ฐ
๐ โ ๐ ๐ . ๐ซ ๐
๐
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
โ๐ฅ
๐ท0
, the stress distribution around the pipe is obtained. By knowing the stress
distribution, we can benefit because of two reasons:
Class โCโ of
Granular
soil
๐ถ = ๐๐ยฐ 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 ๐ ๐ฆ= 235 MPa)
๐ =
๐๐ + ๐ ๐ ๐(1 โ ๐๐๐ โ ) ๐๐๐ 0 โค โ โค ๐
โ0.5. ๐๐
๐๐๐ก
. ๐2
. ๐ ๐๐2
โ ๐๐๐ 0 โค โ โค ๐ผ
โ๐ ๐๐๐ผ. ๐๐
๐๐๐ก
. ๐2
. (๐ ๐๐โ โ 0.5๐ ๐๐๐ผ) ๐๐๐ ๐ผ โค โ โค ๐
โ๐โ. ๐2
(0.147 โ 0.51๐๐๐ โ + 0.5๐๐๐ 2
โ โ 0.143๐๐๐ 4
โ ) ๐๐๐ 40ยฐ โค โ โค 140ยฐ
+1.021๐โ. ๐2
๐๐๐ โ ๐๐๐ 140ยฐ โค โ โค 180ยฐ
โ0.5. ๐๐ฃ. ๐2
(1 โ ๐ ๐๐โ )2
๐๐๐ 90ยฐ โค โ โค 180ยฐ
r = radius of the pipe
๐ถ = half of bedding angle
โ = angle that the moment
is computed
๐ ๐ถ = โ0.106๐ ๐๐3
๐ผ. ๐๐
๐๐๐ก
. ๐ + 0.511๐โ. ๐ + 0.106๐๐ ๐
๐ ๐ถ = โ0.049๐๐. ๐2
โ 0.166๐โ. ๐2
+ ๐๐
๐๐๐ก
. ๐2
[0.106๐ ๐๐3
๐ผ + 0.08๐ผ โ 0.04 sin 2๐ผ โ 0.159๐ ๐๐2
๐ผ ๐ โ ๐ผ + 0.318๐ ๐๐๐ผ 1 + ๐๐๐ ๐ผ
๐๐& ๐ ๐ can be computed as follows:
2. According to โMasada 2000โ, the vertical deflection of the flexible pipe ฮy, which is slightly higher than the
horizontal deflection ฮx, can be obtained and therefore, the correct ๐0 factor can be obtained to check for pipeโs
ovality and its buckling in under pressure condition of the pipe.
9
10. Calculation โ 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 = (19.7โร10โ) 50 ร 45 cm2
Front = (9.85โร17.8โ) 25 ร 45 cm2
The uniform stress transferred by each wheel can be computed as:
Rear Wheels: ๐๐ฟ๐๐ฃ๐,๐ ๐๐๐ =
1
2
ร 0.8 ร
5000
0.5 ร 0.45 ร 100
= ๐๐. ๐๐
๐พ๐
๐2
Front Wheels: ๐๐ฟ๐๐ฃ๐,๐น๐๐๐๐ก =
1
2
ร 0.2 ร
5000
0.25 ร 0.45 ร 100
= ๐๐. ๐๐
๐พ๐
๐2
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 pipeโs centerline, in both
directions).
10
12. ๐ผ๐๐ง
=
1
2๐
๐๐๐๐ก๐๐
๐.๐
๐2 + ๐2 +1
+
๐.๐
๐2 + ๐2 +1
.
1
1+ ๐2
+
1
1+ ๐2
Loadings (Live load)
POLITECNICO DI MILANOCivil engineering for risk mitigation
The increment of vertical stress โ๐๐ฃ can be computed for each section and for each depth
considered by computing the influence factor of ๐ผ ๐ ๐ฃ
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 โ๐๐ฃ at each section is
computed for different burial depths of the pipeline. It can be seen that by increasing
the depth, the โ๐๐ฃ decreases for each section and also after certain depths, the
difference between the โ๐๐ฃ 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
14. ฮณ
(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
ฮฃ[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 pipeโs crown is about ๐ป = 1๐. 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 wonโt encounter problem due to
bending or buckling.
16
15. Computations โ 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 ๐0 can be evaluated.
Regarding the SLS according to โASCE - A.P. Moserโ, 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.
(๐ท ๐ โ ๐ท๐) ๐๐= ๐ ๐. ๐๐
๐ฌ. ๐ฐ
๐ โ ๐ ๐ . ๐ซ ๐
๐
โ๐ฆ
โ๐ฅ
โ 1 +
๐ธโฒ
. ๐ท0
3
5711.83 ๐ธ๐ผ
โ๐ฅ
๐ท0
=
๐ท๐ฟ. ๐พ. ๐๐ฃ
8
๐ธ๐ผ
๐ท0
3 + 0.061๐ธโฒ
% Deflection = ๐๐๐ ร
๐ซ ๐โ๐ซ ๐
๐ซ ๐
Regarding point 2, according to Masada, the vertical to horizontal pipe deflection ratio
can be obtained by using the following simplified equation:
๏ถ The ovalisation ratio
โ๐
๐ซ ๐
(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)
ฮx / D0 (%) ฮx Check ฮy / ฮx ฮy Check
18Computations โ 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 % (ฮy)
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 % (ฮy)
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:
๐๐ฃ =
๐ ๐๐๐ฅ
๐
=
๐ ๐๐๐ฅ
6. ๐ก2
The maximum bending moment acting on the wall occurs exactly at the bottom of the pipe where ฯ = 90ยฐ. 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:
๐โ = ๐ธโฒ
ฮ๐ฅ
๐ท0
Min
Thickness Internal Pressure = 1000 MPa
t=10 mm
(Failed)
t=13 mm
(OK)
๏ฑ With Internal Pressure
๏ฑ Without Internal Pressure (In case of Maintenance) ๏ MOST Critical Case
21
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 Hookeโs 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 k0ยทb in the foundation, therefore the intensity of distributed reaction (per
unit length of the beam) will be:
๐ [ KN/m/m] = constant of the foundation (Winklerโs constant ) which includes the effect of the width of the beam.
๐ ๐ฅ = ๐. ๐ฆ(๐ฅ)
๐ ๐ฅ = ๐. ๐ฆ ๐ฅ
๐=๐0.๐
๐. ๐0. ๐ฆ ๐ฅ
Differential equation of equilibrium of a beam on elastic foundation
In the above equation the parameter ฮฑ 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/ฮฑ is referred to as the so
called characteristic length. Therefore, xโ ฮฑ 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
EIฮฆ(x)๏ EI multiple values of slope of deflection line
Classification of the beams according to their stiffness:
1. ฮฑ.l<0.5 ๏ Short Beams
2. 0.5โฅฮฑ.Lโค5 ๏ Medium Length Beams
3. ฮฑ.L>5 ๏ Long Beams ๏ Our Case
๐ต =
๐ธ๐ผ
1 โ ๐2
=
๐ธ๐ก3
12(1 โ ๐2) ๐ผ =
4 ๐
4๐ต
๐ฟ > ๐ =
2๐
๐ผ
๐ผ =
4 6800 โ 12 โ (1 โ 0.32)
4 โ 210 โ 106 โ 0.0133
= 2.52 ๐ =
2๐
2.52
= 2.49 โ ๐ฟ = 13.49๐ โ 20๐
Change of variables
26
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, wonโt 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