# Stress and deformation of offshore piles under structural and wave loading

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3 load acting on the offshore concrete pile. The dynamic wave pressure at any point on the surface of the pile is given as (Zhu, 1993) p d ( z + d ) γ w A cosh k iωt m ( a, θ, z) e = ε mε ni Qmn ( a, θ ) (2) 2 cosh kd m= 0 n= 0 where p d is the dynamic wave pressure acting on the pile, a is the radius of the pile, θ is the angle around the circumference of the pile where the wave load is being calculated, and z is the vertical depth from the surface of the water to the point where the wave load is being calculated. In (2), γ w is the unit weight of water, A is the amplitude of the waves being considered, k (2π/L, L is the wavelength) is the wave number, and d is the total depth of the water. Figure 2 shows some of these variables. Also in (2), i is equal to 1, ω is the wave frequency, and t is the second of the wave period that the wave load is being calculated. In (2), parameters ε m, ε n and Q mn ( a, θ ) are defined as follows (Zhu, 1993) 1 when m, n = 0 ε m, ε n = (3) 2 when m, n 0 Q ( a, θ ) = [ J m ( k xa) J 2 n ( k ya) AmnH m+ 2n ( ka) ] cos( m + n)θ J ( k a) J ( k a) B H ( ka) cos m n [ m x 2n y mn ] ( 2 )θ mn 2 + (4) m 2n where A B mn mn ( k a) J ( k a) + k J ( k a) J ' ( k a) k x J ' m x 2n y y m x 2n y = (5a) kh ' m+ 2n ( ka) ( k a) J ( k a) + k J ( k a) J ' ( k a) k x J ' m x 2n y y m x 2n y = (5b) kh ' m 2n ( ka) in which J( ) represents the Bessel function of the first kind and H( ) represents the Hankel function. In (4), (5a), and (5b), the wave number in the x- and y-directions, k x and k y, can be expressed as k x = k cosα and k y = k sinα (6) where α is the incident wave angle. 3

4 3. Numerical Analysis In this study, the finite element analysis package, STRAND7 (G+D Computing, 1999), is employed to calculate the deformation and stress within the concrete pile. The pile is modelled using 20-node brick elements except for the inner ring where 15-node wedge elements are used. A fixed support condition is provided at the points where the pile is embedded into the ocean bed. For the concrete pile, the Young s modulus is 28,600 MPa and the Poisson s ratio is Convergence of model Before conducting a parametric study, a convergence test is performed under the structural load and the static water pressure to determine the most appropriate mesh. Based on the results from the convergence test, the wave pressure is then calculated and applied to the pile and the analyses are performed to find the resulting stress distributions as well as the resulting displacements. Two series of convergence tests are performed. The first convergence test is to refine the mesh vertically above and below the surface water level. The second convergence test is to refine the mesh on the cross-section of the pile. In total, fifteen different models are used for the tests, including three variations of the mesh in the vertical direction on the surface water level (Models X, Y, and Z), and for each of these three models, five variations of mesh on the cross-section (4, 6, 8, 10, and 12 rings). The basic model used as the control pile is modelled having a 1 m radius and a 20 m height. The water depth for the control model is 10 m. The details of the different mesh schemes are shown in Fig. 3. It should be noted that due to the nature of a 20-node brick, the side nodes of each brick element form an intermediate row or ring of nodes. The typical results of the convergence tests are presented in Figure 4. Figure 4(a) shows the von Mises stress at point A (refer to Figure 5(b)) versus the number of rings for Models X, Y, and Z. The figure clearly indicates that the solution converges when the 6-ring model is adopted, with the change in slope between the 8, 10, and 12 ring models being very small. Figure 4(b) shows the vertical displacement at point A versus Models X, Y, and Z for all number of ring models. It is evident that the convergence is achieved when Model Y is used. Based on the convergence study, Model Y with six rings is selected to simulate the control pile. This mesh has 1320 bricks and 5505 nodes. The numerical analysis is then carried out to include the static water pressure, the dynamic wave loads, as well as the varying structural loads Effects of combined loading conditions After applying the loading conditions to the final model that is selected, the three-dimensional finite element analyses are performed. The results of these 4

6 above the base of the pile. The x- and y- coordinates are kept the same at each point when the z- coordinate increases. Referring to Figure 1 for a definition of the global axes system. The results shown in Figure 7(a) confirm the plot shown in Figure 6(a) in which the displacement at t=0 second is the same for each increasing time period over the entire length of the pile. The displacement increases linearly at a slope of mm. Similarly, Figure 7(b) also corresponds to Figure 6(b) in that the stress is approximately the same over the height of the pile at t=0 second. The curvature in the line indicates that the stress increases up to the surface water level, where the stress levels off and remains constant until just before reaching the top of the pile. There is a significant increase in stress in the very top 1 m of the pile where the structural load is applied Effects of wave heights The wave height is increased from a 1 m height, the control value, to a 5 m height. Figure 8(a) demonstrates the change in total displacement at Point A over the period of a wave. Opposite to the effects of increasing wave period, the total displacement increases as the wave height increases. The total displacement peaks twice during the wave period, once at 1/5 the period and again at 4/5 the period. At a 2 m wave height, the peak displacement is about 60% greater than that obtained at the control height. Increasing the height again, at 3 m, the peak displacement is about 125% greater. At 4 m and 5 m wave heights, the increase in the peak total displacement is 190% and 260%, respectively, greater than the control height of 1 m. Figure 8(b) illustrates the non-dimensional change in stress at Point A as the wave heights increase. The non-dimensional expression is obtained by dividing the resulting stress by γ w d/2 cosh kd. Similar to the variation of the stress diagram due to the increasing wave periods, the two peak stresses are at approximately 1/3 the wave period and 2/3 the wave period. The stresses at the beginning, middle, and the end of the period actually decrease as the wave height increases; however, the variation in stress increases with the height of the wave. Figure 9 displays the results of the increasing wave heights over the height of the pile. In Figure 9(a), the total displacement increases linearly at a slope of mm. These results are again reflected by Figure 8(a) where at t=0 second the wave height does not affect the displacement at point A for varying wave periods. The von Mises stress distribution in the piles due to the increasing wave height is shown in Figure 9(b). The stress in the top half of the pile remains constant as the wave height increases. However, at the surface water level, the stress begins to decrease as the wave height increases, but retaining a similar value for all wave heights at the base of the pile. These results also correspond to Figure 8(b) when 6

7 t=0 second. At point A (z/h=0.5), the stress decreases as the wave height increases from 1 m to 5 m Effects of incident wave angles The incident wave angle is increased from 0 to 90, at 15 intervals. Figure 10(a) portrays the change in total displacement at Point A over the period of a wave. It is found that the total displacement decreases as the incident wave angle increases around the circumference of the pile. The total displacement peaks twice during the wave period, once at 1/5 the period and again at 4/5 the period. At a 90 angle, the peak displacement is about 32% less than that at a 0 incident wave angle. The decrease in displacement from 0 to 15, the control value, is less than 2%. Increasing the angle to 30, 45, 60, and 75, the total displacements are about 7%, 14%, 22%, and 28% respectively, less than the deflection for a 0 incident wave angle. By 90, the variation in displacement over the wave period is very small, just over 1% difference. The maximum variation in displacement, for 0, is about 32%. Using the same non-dimensional ratio employed for studying the variation of the wave heights, the change in stress at Point A over the period of a wave is illustrated in Figure 10(b). It can be seen that the change in stress is approximately the same for all incident wave angles. Again the peak stresses occur at 1/3 and 2/3 the wave period, having a maximum change in stress of less than 1%. The variation of displacement and stress against z/h are presented in Figures 11(a) and 11(b) respectively, where h is the height of the pile. The resulting displacement and stress is predicted at 2 m intervals over the height of the pile and at t=2 s of the wave period. The results shown in Figures 10(a) and 11(a) also correspond when z/h=0.5 (point A), in that the displacement at t=2 s (where t/t=0.2) decreases as the incident wave angle increases. The displacement increases linearly, but the slope decreases as the incident wave angle increases. The displacement for an incident wave angle of 0 increases at a slope of approximately mm. The displacements for incident wave angles of 15, 30, 45, 60, 75, and 90 decrease at slopes of approximately mm, mm, mm, mm, mm, and mm respectively. Similarly, when t=2 seconds, Figure 10(b) is reflected by Figure 11(b) in that the stress remains approximately the same at point A, which is at z/h=0.5. The curvature in the line indicates that the stress increases up to the surface water level, where the stress levels off and remains constant until just before reaching the top of the pile. Only below the water surface is there a variation in stress as the incident wave angle varies. As the incident wave angle increases, the von Mises stress increases, with the greatest increase in stress being at the base of the pile Parametric study of pile strength 7

8 Another main concern of this study is how the strength of the pile affects it s behaviour under combined structural and wave loading. To undertake this study, a 20-meter control pile was used, with first varying the strength of concrete, followed by adding steel reinforcement to the pile Effects of concrete strength To compare the effects of increasing the concrete strength, four different values of the mean concrete strength, f cm, are used: 25, 32, 40, and 50 MPa. Figure 12 illustrates the results of the stress and displacement of the pile over the wave period of 10 s. These results are shown at point A located on the surface of the pile (refer to Figure 5(b)). In Figure 12(a), it is seen that the displacement decreases at decreasing intervals as the concrete strength increases. At t=2 s (where t/t=0.2) the displacement decreases by approximately mm, mm, and mm respectively as the concrete strength increases. The decreases in displacement are constant throughout the entire wave period. A similar curve is shown for the von Mises stress as observed for the previous tests on varying wave loads. Again, the same non-dimensional ratio is used to represent the change in stress over the wave period. Even though the displacement decreases as the concrete strength increases, the stress in the pile is not affected by this change. The stress remains constant, with the peak stresses being at 1/3 and 2/3 the wave period. The variation of displacement and stress against z/h are presented in Figures 13(a) and 13(b) respectively, where h is the height of the pile. The resulting total displacement and von Mises stress is predicted over the height of the pile, at 2-meter intervals. These results are shown at t=0 s of the wave period. It is seen in Figure 13(a) that the displacement increases linearly with a decreasing slope as the concrete strength increases. The slope of displacement over the pile height for a concrete strength of 25 MPa is approximately mm, decreasing to mm, mm, and mm respectively as the strength increases. The total displacement at the base of the pile remains basically unchanged for different concrete strengths, with the greatest variation of displacement being at the very top of the pile. The variation in stress over the height of the pile, as shown in Figure 13(b), has a similar curve as obtained in the previous tests on varying wave loads. The stress increases from the base of the pile up to the water surface, levelling off and remaining constant to just before the top of the pile. Results also indicate that increasing the concrete strength has no effect on the stress distribution throughout the pile, as reflected in Figure 12(b) at point A Effects of steel reinforcement The most common method to increase the strength of a concrete pile is to add steel reinforcement. The second test carried out to study the effects of increasing the pile strength is to add a steel reinforcement cage. Standard 36 mm diameter steel 8

10 Figure 16(a) shows the total displacement over the height of the pile for t=0 to 10 s of a 10-second wave period. At each of the ten locations on the pile, the displacement increases up to t=2 s of the 10-second wave period, then begins to fall back down. By t=5 s of the period, the displacement is equal to that at t=0 s, then begins to increase again until t=8 s. The peak displacement at each location is at t=8 s after which the displacement again falls back down by t=10 s which equals the displacement at time zero and the cycle begins again. A similar results is seen in Figure 16(b) with the stress distribution. The stress above the surface water level remains constant throughout the wave period, but varies greatly towards the base of the pile. Similar to Figure 16(a), the von Mises stress begins the wave period at approximately 0.26 MPa, decreases down to 0.2 MPa by t=2 s of the wave period when it begins to increase again. By t=5 s, the stress is equivalent to t=0 s, but then the stress begins to increase in the second half of the wave period. By t=8 s, the stress reaches a maximum of about 0.34 MPa where it begins to decrease back down to the start by t=10 s, which is the beginning of a new cycle. 4. Conclusion In this study, the deformation of and stresses in an offshore concrete pile under combined structural and wave loading is analysed by the finite element method. A control model is analysed using a specific set of control data, then compared to succeeding models as the pile and loading parameters are changed. The loading parameters used to complete this study are the wave period (T), wave height (H), incident wave angle (α). The pile strength parameters also used to complete the study are the mean concrete strength (f cm ) and the amount of steel reinforcement. After compiling and analysing the results from each test, the following conclusions can be made: (1) As the wave period increases the peak displacement of the pile significantly decreases so shorter wave periods have a more critical effect on the pile displacement. (2) Increasing wave heights creates a dramatic increase on the peak displacement, while the minimum stresses actually decrease significantly. (3) There is a dramatic decrease in the peak displacement as the incident wave angle increases away from the point of reference (point A used in this case refer to Figure 5(b)). In addition, the stress towards the base of the pile increases significantly as the incident wave angle increases, although it still remains less than the maximum stress at the top of the pile. (4) Increasing the strength of concrete does significantly decrease the displacement over the entire wave period, but has no effect on the stress distribution within the pile. (5) Adding steel reinforcement to the concrete pile does reduce the total displacement of the pile, but the greatest effect is a significant reduction in stress in the top half of the pile where the stress is greatest. 10

11 A design procedure for a pile considering a short wave period should not be the same as a design procedure for a significantly longer wave period. Similarly, a pile subjected to wave heights of 1 m should not be designed according to the same procedure as location with wave heights of 5 m. It is also shown in the results that to reduce the critical stress and to increase the pile strength, the addition of steel reinforcement is necessary. Using the results provided herein, more effective design codes can be formulated to take into account the behaviour of the pile to a specific loading conditions. References Au, M.C. and Brebbia, C.A. (1983). Diffraction of water waves for vertical cylinders using boundary elements. Applied Mathematical Modelling, 7, Chakarabarti, S.K. and Tam, A. (1975). Interaction of waves with large vertical cylinder. Journal of Ship Research, 19, G+D Computing (1999). Using Strand7: Introduction to the Strand7 finite element analysis system. Sydney, Australia. Raman, H., Jothishankar, N. and Venkatanarasaiah, P. (1977). Non-linear wave interaction with vertical cylinder of large diameter. Journal of Ship Research, 21(1), Sorensen, R.M. (1997). Basic Coastal Engineering. New York: Chapman & Hall. Tang, W.H. (1989). Uncertainties in offshore axial pile capacity. Foundation Engineering: Current Principles and Practices. New York, pp Zhu, S. (1993). Diffraction of short-crested waves around a circular cylinder. Ocean Engineering, 20(4), Zhu, S. and Moule, G. (1994). Numerical calculation of forces induced by shortcrested waves on a vertical cylinder of arbitrary cross-section. Ocean Engineering, 21(7),

12 List of Figure Captions: Figure 1: Loading on the offshore pile. Figure 2: Definition of variables in wave field. Figure 3: Pile models used for convergence tests (a) control pile model and vertical mesh refinement at surface water level and (b) mesh refinement through crosssection. Figure 4: Convergence of pile models (a) von Mises stress at point A and (b) vertical displacement at point A. Figure 5: Contour of stress distribution due to combined loading conditions (a) structural load and static water pressure and (b) structural load and static/dynamic wave pressure. Figure 6: Responses at Point A with increasing wave period (T) (a) change in total displacement and (b) change in normalised von Mises stress. Figure 7: Responses at t=0 s with increasing wave period (T) (a) change in total displacement and (b) change in von Mises stress. Figure 8: Responses at Point A with increasing wave height (H) (a) change in total displacement and (b) change in normalised von Mises stress. Figure 9: Responses at t=0 s with increasing wave height (H) (a) change in total displacement and (b) change in von Mises stress. Figure 10: Responses at Point A with increasing incident wave angle (α) (a) change in total displacement and (b) change in normalised von Mises stress. Figure 11: Responses at t=0 s with increasing incident wave angle (α) (a) change in total displacement and (b) change in von Mises stress. Figure 12: Responses at Point A with increasing mean concrete strength (f cm ) (a) change in total displacement and (b) change in normalised von Mises stress. Figure 13: Responses at t=0 s with increasing mean concrete strength (f cm ) (a) change in total displacement and (b) change in von Mises stress. Figure 14: Responses at Point A to compare piles with and without additional steel reinforcement (a) change in total displacement and (b) change in normalised von Mises stress. 12

13 Figure 15: Responses at t=0 s with the addition of steel reinforcement (a) change in total displacement and (b) change in von Mises stress. Figure 16: Responses at different height locations over a 10-second wave period (a) change in total displacement and (b) change in von Mises stress. 13

14 Structural load Static water pressure Dynamic wave load wave crest wave trough Static water pressure Dynamic wave pressure Figure 1 14

15 A water level y x θ a α = 0 d z sea bed Figure 2 15

16 Radius= 1 m Model X Model Y m Model Z Figure 3(a) 16

17 4 rings 6 rings 8 rings 10 rings 12 rings Section 1-1 Figure 3(b) 17

18 von Mises stress (MPa) Models X, Y, and Z rings 6 rings 8 rings 10 rings 12 rings Figure 4(a) 18

19 rings rings 8 rings 6 rings DZ (mm) rings X Y Z Figure 4(b) 19

20 Brick Stress: von Mise (MPa) Brick Stress: von Mise (MPa) Point A (on surface of pile) water level Figure 5 20

21 Total displacement (mm) 2.00E E E E E E E E s 12.5 s 15 s 7.5 s Time/wave period (t/t) Figure 6(a) 21

22 VM/( wh/2coshkd) 8.01E E E E E E E s 10 s 12.5 s 15 s 7.94E E Time/wave period (t/t) Figure 6(b) 22

23 Total Displacement (mm) , 10, 12.5, and 15 s z/h Figure 7(a) 23

24 0.37 von Mises Stress (MPa) , 10, 12.5, and 15 s z/h Figure 7(b) 24

25 Total displacement (mm) 6.00E E E E E E E E E E E E-02 5 m 4 m 3 m 2 m 1 m Time/wave period (t/t) Figure 8(a) 25

26 8.05E E+00 VM/( wd/2coshkd) 7.95E E E E E E+00 1 m 2 m 3 m 4 m 5 m 7.65E E Time/wave period (t/t) Figure 8(b) 26

27 0.25 Total Displacement (mm) , 2, 3, 4, and 5 m z/h Figure 9(a) 27

28 0.37 von Mises Stress (MPa) m 5 m 3 m 2 m 1 m z/h Figure 9(b) 28

29 Total displacement (mm) 1.60E E E E E E E E Time/wave period (t/t) Figure 10(a) 29

30 8.020E E+00 VM/( wd/2coshkd) 7.980E E E+00 0, 15, 30, 45, 60, 75, E E Time/wave period (t/t) Figure 10(b) 30

31 Total Displacement (mm) z/h Figure 11(a) 31

32 von Mises Stress (MPa) z/h Figure 11(b) 32

33 Total Displacement (mm) MPa 32 MPa 40 MPa 50 MPa t/t Figure 12(a) 33

34 σ VM/γ wd/2coshkd 8.020E E E E E E E E-06 f cm = 25, 32, 40, and 50 MPa t/t Figure 12(b) 34

35 0.3 Total Displacement (mm) MPa 32 MPa 40 MPa 50 MPa z/h Figure 13(a) 35

36 0.37 von Mises Stress (MPa) , 32, 40, and 50 MPa z/h Figure 13(b) 36

37 Total Displacement (mm) no reinforcement reinforcement t/t Figure 14(a) 37

38 8.10E E-06 σvm/γ wd/2coshkd 7.90E E E E-06 reinforcement no reinforcement 7.50E E t/t Figure 14(b) 38

39 0.25 Total Displacement (mm) no reinforcement reinforcement z/h Figure 15(a) 39

40 von Mises Stress (MPa) no reinforcement reinforcement z/h Figure 15(b) 40

41 Total Displacement (mm) s 8 s 9 s 6 s 0, 5, 10 s 1 s 2 s 3 s 4 s z/h Figure 16(a) 41

42 0.4 von Mises Stress (MPa) s 8 s 7 s 6 s 5 s 4 s 1 s 2 3 s s 0, 10 s z/h Figure 16(b) 42

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