Date of preparation: 10June 2006
Title of paper: Optimum Design of buried pipe-line block anchors
Author: Husain Jubran Al-Gahtani
Author title: Associate Professor
Author affiliation: Civil Engineering Department, King Fahd university of Petroleum
Minerals.
Author address: KFUPM 800
Dhahran 31261
Saudi Arabia
Number of words:≈3000
Number of tables: 1
Number of figures: 4
Keywords: Buried pipe; Block anchor; thrust force
Abstract
The paper presentsa simple procedure for the optimum design of a pipeline block anchor.The forces exerted by the soil on the block are computed based on Rankine theory. The dimensioning of the block anchor is formulated as an optimization problem in which the objective function is the volume of the block to be minimized. The optimization constraints are related to block sliding, block overturning and soil bearing capacity.The optimization problem is solved using Excel. Several examples are given to illustrate the steps of the design procedure. Simple relationships between the thrust force and the minimum required block dimensions are generated.
Notation
B / length of block anchorC / soil cover above the block anchor
Fa / resultant force of soil active pressure
Fb / resultant force of soil passive pressure
FRb / friction force at bottom of the block anchor
FRs / friction force on side of the block anchor
FRt / friction force on top of the block anchor
FSO / factor of safety against overturning
FSS / factor of safety against slidding
H / height of the block anchor
h / location of line of action of active and paasive pressures
HP / level of restrained pipe
ka / active pressure coefficient
Kp / passive pressure coefficient
MO / overturning moment
MR / resisting moment
Q / thrust force
qal / allowable soil bearing pressure
R / net soil resistance
W / width of the block anchor
Wb / weight of the block anchor
Ws / weight of the soil cover
γc / unit weight of concrete
γs / unit weight of soil
μ / coefficient of friction between concrete and soil
σ / soil pressure underneath the block anchor
φ / soil angle of internal friction
1-Introduction
When a pipe line is put into operation, it will expand under the influence of internal pressure and temperature gradient. The movement due to such expansion is significant for large diameter pipelines such as those carrying crude oil and gas. These pipes need to be fully restrained at some points near some equipments/structures such as well heads, pumps, pressure vessels, storage tanks, etc. in order to prevent the transmission of movement to these sensitive equipments. Massive concrete block anchors are commonly used to fully restrain the pipeline and resist the resulting high thrust force. The literature search by the author shows that there is a very limited number of studies on the design of such special structures (Watkins 2000; Sidqi 2005). Watkins and Anderson(2000) presented a simplified iterative procedure for the design of cubical block anchor. Their procedure neglects the friction forces on the sides of the block and the bearing capacity of the soil underneath the block.Sidqi (2005) studied the pullout capacity of block anchor using experimental works and analytical calculation. The objective of this paper is present a procedure for optimizing the dimensions of massive block anchors.The procedure yields charts and simplified formulas that can be easily used by practicing engineers for determining the required block dimensions for a given pipe thrust force and material properties. The analysis of the forces acting on the block is based on the well known Rankin theory for modeling earth pressures(George 1970). It should be noted that other more refined theories such as Coulomb'sand Log spiral (Terzaghi et al 1996) can be accommodated in the current analysis. However, the uncertainty and approximation involved in characterization of the soil properties and the block-soil interaction are so high that employing more accurate and complicated theories is not justified.
2-Formulation
2-1Analysis of forces on the block
Figure 1. shows a free body diagram of the considered rectangular anchor block with all forces acting on it. Let us first, define various geometrical and material parameters and variables involved in the analysis of the block:
a)Dimensions
H: the height of the block
W: the width of the block
B: the thickness of the block
HP: the level of the pipe
C: the soil cover
b)Material properties
γc: the unit weight of concrete
γs: the unit weight of soil
φ: the soil angle of internal friction
μ: the coefficient of friction between concrete and soil
ka: the active pressure coefficient =
Kp: the passive pressure coefficient =
qal: the allowable soil bearing pressure
Based on Rankine theory, various forces acting on the block are calculated as follows:
1-Forces acting on the front and back of the block (FaFp). These forces are due to active and passive pressures exerted by soil on the block. They are given by:
(1)
(2)
The line of action of each of the above forces is located at a height HFwhich can be derived by integration. After simplification, the result is:
(3)
2-Friction forces FRt, FRb, and FRs, on the top, bottom and each side of the block, respectively. They are given by:
(4)
(5)
(6)
It should be noted that the location of line of action of FRs is the same as that of Fa and Fp.The net soil resistance R is given by
(7)
Using eqs 1,2, 4,5 and 6 in eq 7, we get
(8)
3-Vertical forces due to weights of the concrete block and soil cover which are given by
(9)
(10)
2-2Design constraints
The geometric design of the block must satisfy the following conditions:
1-The block must be stable against sliding. This is guaranteed if the net soil resistance R is greater than the thrust force Q The ratio R/Q must be ≥ FSS, where FSS is the factor of safety against sliding. After simplification this constraint becomes:
(11)
2-The block must be stable against overturning. This constraint is satisfied if the resisting moment MR provided by the net soil resistance and the weight of block is greater than the overturning moment Mo caused by the thrust force. After simplification, we get:
(12)
where FSO is the factor of safety against overturning
3-The bearing stress underneath the block must be less than the allowable soil pressure qall., i.e.
(13)
Wher σ is the bearing stress which is given by:
(14)
Where Moand MR are the overturning and resistance moment respectively which are given by
(15)
(16)
After simplification, we get:
(17)
3-Optimization of block dimensions
Our objective is to determine the minimum block dimensions (B, H & W) and the level of pipe (HP) that are required to anchor the pipe while satisfying the constraints given bythe inequalities 11-13. The optimization problem can be solved using "Solver" tool of Excel. The optimization procedure of this tool is well explained in a many recent books on Excel, e.g.5In order to illustrate the procedure, let us assume the following properties:γc= 23 kN/m3, γs = 15 kN/m3, φ = 30o (Kp=3,Ka=1/3), μ = 0.4,qal = 150 kN/m2, FSS = 1.25 and FSO = 1.5. As an example, let us consider the block design for Q = 3500 kN and soil cover C = 1 m. Let us also put constraints on the minimum values of H, B and W that allow for pipe anchorage, reinforcement, etc., say H > 1m, B> 1m and W > 1m. Solver yields the following block dimensions:H = 1.47 m, B = 1.00 m, W = 34.30 and HP = 0.64 mso that the volume V = 50.33 m3. However, the value of W is impractical. Therefore, we will repeat the calculations by imposing a maximum value of W = 3H. The results become: H = 3.29 m, B = 2.05 m, W = 9.88 m and HP = 1.31 m so that the volume V = 66.6 m3. As a second example let us design a cubical block (W=B=H) for the same force and soil cover assumed in example 1.Solver yields H = B = W = 4.66 m and Hp = 1.93m. The volume in this case is 101.24 which requires 50% more concrete as compared to the rectangular block. As a third example, consider the dimensioning of the block for a thrust force Q = 5000 kN and soil cover CO = 1.5 mwith the following properties: γc= 23 kN/m3, γs = 15 kN/m3, φ = 40o (Kp=4.6, Ka= 0.22), μ = 0.45,qal = 120 kN/m2, FSS = FSO = 2.. Solver yields H = 3.96 m B = 5.39 m, W = 7.92m and Hp = 2.74m with a volume V = 168.99 m3. It should be noted that the same sheet can be used for any other given data.
4-Procedure for generating design charts and equations
In the following, we will illustrate how one can generate simplified design charts and formulas for determining the minimum block dimensions and the required pipe level HP for a given thrust force F and soil cover C. Let us assume the following practical properties:γc = 23 kN/m3, γs = 15 kN/m3, φ = 30o (Kp=3,Ka=1/3), μ = 0.4,qal = 150 kN/m2, FSS = 1.25 and FSO = 1.5. Let us fix the maximum value of W to be 3H. If we run excel calculations for Q = 1000 kN, 2000 kN,…10000 kN and C = 0.5m, 1m, 1.5m, …4m. and compute the design variables H, B, W and HP in each run, we can generate the data shown in figures 2 to 4.After trying several polynomial and power relationships. It has been found that the H-Q relationship is best represented by simple power equations while B-H and Hp-H relationships can be represented by linear equations as given in Table 1.
In order to check the accuracy of these relationships, let us repeat example 1. In this case the relationships between the variables are;
H = 2.0395 Q0.3789, B = 0.6584 H - 0.094 and Hp =0.3828 H + 0.0535.
Substituting for Q = 3500 kN = 3.5 MN, we get H = 3.28m. Using this value of H, we get B = 2.06m and Hp = 1.31m which are almost the same as those values obtained by Excel through its solver package.
5. Conclusion
The paper presents an automated procedure for computing the optimum dimensions of a pipe block anchor. The forces acting on the block have been calculated based on the well known Rankin theory. Practical examples have been given to demonstrate the proposed procedure. The parametric study has shown that the height of the block is related to the thrust force through a power equation while other dimensions are linearly proportional to the depth of the block. Such relationships enable the engineer to design the block without performing the optimization analysis.
Acknowledgement
The author would like to express his appreciation for King Fahd University of Pertoleum & Minerals (KFUPM) for their support.
References
George F., S. (1970) Introductory soil mechanics and foundations: Geotechnical engineering, 4th Ed., Macmilan Pub. Co., New York.
Sidqi, A, "Pullout Capacity of Block Anchor in Sand", M.S. Thesis, King Fahd University of Petroleum & Minerals, Dhahran.
Terzaghi et al (1996), Soil mechanics in engineering practice, 3rd Ed., Wiley, New York.
Watkins, R.K. (2000), Structural mechanics of buried pipes, CRC Press.
Table 1. Design equations for the block anchor dimensions
Soil cover C (m) / H-Q relationship / B-H relationship / Hp-H relationship0.5 / H = 2.2775 Q0.3563 / B = 0.6227 H - 0.2322 / Hp = 0.3612 H + 0.0072
1.0 / H = 2.0395 Q0.3789 / B = 0.6584 H - 0.094 / Hp =0.3828 H + 0.0535
1.5 / H= 1.8502 Q0.3971 / B = 0.7015 H - 0.0179 / Hp =0.4065 H + 0.0649
2.0 / H = 1.6975 Q0.4117 / B = 0.7484 H + 0.0207 / Hp =0.4297 H + 0.0617
2.5 / H = 1.5722 Q0.4233 / B = 0.797 H + 0.0369 / Hp =0.4513 H + 0.0529
3.0 / H = 1.4663 Q0.4331 / B = 0.8403 H + 0.0576 / Hp =0.4698 H + 0.0467