The tensile capacity of suction caissons in sand under rapid loading

Transcription

1 Frontiers in Offshore Geotechnics: ISFOG 25 Gourvenec & Cassidy (eds) 25 Taylor & Francis Group, London, ISBN X The tensile capacity of suction caissons in sand under rapid loading Guy T. Houlsby, Richard B. Kelly & Byron W. Byrne Department of Engineering Science, Oxford University ABSTRACT: We develop here a simplified theory for predicting the capacity of a suction caisson in sand, when it is subjected to rapid tensile loading. The capacity is found to be determined principally by the rate of pullout (relative to the permeability of the sand), and by the ambient pore pressure (which determines whether or not the water cavitates beneath the caisson). The calculation procedure depends on first predicting the suction beneath the caisson lid, and then further calculating the tensile load. The method is based on similar principles to a previously published method for suction-assisted caisson installation (Houlsby & Byrne 25). In the analysis a number of different cases are identified, and successful comparisons with experimental data are achieved for cases in which the pore water either does or does not cavitate. 1 INTRODUCTION Suction caissons are an option for the foundations for offshore structures. Under large environmental loads the upwind foundations of a multiple-caisson foundation might be subjected to tensile loads. Recent research indicates that serviceability requirements will often dictate that, under working and frequently encountered storm loads, tensile loads on caissons should be avoided, as they are accompanied by large displacements. However, it may be appropriate to design structures so that under certain extreme conditions the caissons are allowed to undergo tension. It is therefore necessary to have a means of estimating the tensile capacity of a caisson foundation, whilst recognising that large displacements may be necessary to mobilise this capacity. The calculations are also relevant to the holding capacity of caisson anchors subjected to pure vertical load, and to calculation of forces necessary to extract a caisson rapidly (for whatever reason). Under rapid tensile loading, a suction caisson in sand will exhibit a limiting load which will typically consist of a suction developed within the caisson, and friction on the outer wall. However, a number of different possible modes of failure exist. The purpose of this paper note is to set out simple calculations for capacities under various failure modes, and to compare these with experimental results. 2 TENSILE CAPACITY CALCULATIONS 2.1 Drained capacity If the tensile load is applied very slowly, then pore pressures will be small, and a fully drained calculation is applicable for calculating the capacity. For the purposes of calculation an idealised case of a foundation on a homogeneous deposit of sand is considered here. The resistance on the caisson is calculated as the sum of friction on the outside and the inside of the skirt. The effective stresses on the annular rim are likely to be sufficiently small that they can be neglected, and it is assumed that the soil breaks contact with the lid of the caisson. The frictional terms are calculated in the same way as for the installation calculation (Houlsby & Byrne 25), by calculating the vertical effective stress adjacent to the caisson, then assuming that horizontal effective stress is a factor K times the vertical effective stress. Assuming that the mobilised angle of friction between the caisson wall and the soil is then we obtain the result that the shear stress acting on the caisson is vk tan. Note that in the subsequent analysis the values of K and never appear separately, but only in the combination K tan, so it is not possible to separate out the effects of these two variables. Allowance is made, however, for the possibility of different values of K tan acting on the outside and inside of the caisson. A difference between this analysis and conventional pile design is that the contribution of friction in reducing the vertical stress further down the caisson is taken into account. If, as a preliminary, no account is taken of the reduction of vertical stress close to the caisson due to the frictional forces further up the caisson, then the tensile vertical load on the caisson for penetration to depth h is given by: (1) 45

2 z V' h D i Mudline Figure 1. D o Caisson geometry. A similar analysis follows for the stress on the outside of the caisson. We assume that (a) there is a zone between diameters D o and D m md o in which the vertical stress is reduced through the action of the upward friction from the caisson, (b) within this zone the vertical stress does not vary with radial coordinate and (c) there is no shear stress on vertical planes at diameter D m. We then obtain the same results as for the inside of the caisson, but with Z i replaced by Z o D o (m 2 1)/(4(K tan ) o ). Alternative assumptions could be made for the variation of D m with depth, but at present there is little evidence to justify any more sophisticated approach. If D m is taken as a variable, then the differential equation for vertical stress will usually need to be integrated numerically. Accounting for the effects of stress enhancement, Eq. (1) becomes modified to: Figure 2. caisson. Vertical equilibrium of a slice of soil within the where the dimensions are as in Figure 1, and is the effective weight of the soil. V is the buoyant weight of the caisson and structure. A check should always be made that the friction calculated inside the caisson does not exceed the weight of the trapped soil plug h D 2 i/4. Ignoring the reduction of the stress in this case proves unconservative (i.e. it overestimates the force that can be developed), so we develop here a theory which takes this effect into account. Consider first the soil within the caisson. Assuming that the vertical effective stress is constant across the section of the caisson, the vertical equilibrium equation for a disc of soil within the caisson (Fig. 2) leads to: Writing D i /(4(K tan ) i ) Z i, Eq. (2) becomes d v /d z v /Z i, which has the solution v Z i (1 exp( z/zi)) for v at z. The total frictional terms depend on the integral of the vertical effective stress with depth, and we can also obtain. For For small h/z i the integral simplifies to h 2 /2 as in Eq. (1). For brevity in the following we shall write the function y(x) (exp( x) 1 x), so that in the above. (2) (3) In the special case where m is taken as a constant and uniform stress is assumed within the caisson this becomes: (4) The calculation accounting for stress reduction obviates the need to check that the internal friction does not exceed the soil plug weight, as the capacity asymptotically approaches that value at large h. 2.2 Tensile capacity in the presence of suction If the caisson is extracted more rapidly, then transient excess pore pressures will occur, and the suction within the caisson will need to be taken into account. We return later to the calculation of the relationship between the rate of movement and the suction, but first address the calculation of load in terms of the suction. If the pressure in the caisson is s with respect to the ambient seabed water pressure, i.e. the absolute pressure in the caisson is p a w h w s (where p a is atmospheric pressure, w is the unit weight of water and h w the water depth), then we at first assume that the excess pore pressure at the tip of the caisson is as, 46

3 i.e. the absolute pressure is p a w (h w h) as. There is therefore an average downward hydraulic gradient of as/ w h on the outside of the caisson and upward hydraulic gradient of (1 a)s/ w h on the inside. We assume that the distribution of pore pressure on the inside and outside of the caisson is linear with depth. A detailed flow net analysis shows that this approximation is reasonable. The solutions for the vertical stresses inside and outside the caisson are exactly as before, except that is replaced by as/h outside the caisson and by (1 a)s/h inside the caisson. The capacity, accounting for the pressure differential across the top of the caisson and pore pressure on the rim (only relevant for a thick caisson), is again calculated as the sum of the external and internal frictional terms: (5) In the special case of m constant and a uniform stress assumed within the caisson, this gives: D 2 /4 A): (8) where Z D(m 2 1)/(4K tan ). Neglecting the effects of stress reduction would give: (9) which means that the capacity is simply calculated by applying a linearly varying factor to the suction force beneath the lid. 2.3 Undrained failure A further condition should be considered: that of undrained failure of the sand. In any dilative sand, however, the pore pressures developed under undrained conditions are potentially so large that invariable (except in very deep water) the cavitation mechanism would intervene first. Since the undrained strength of sand is in any case very difficult to determine, we do not pursue this case here. (6) We can often make a further simplifying assumption, that the suction is sufficiently large that the soil within the caisson liquefies and therefore (1 a)s/h. For a large suction this means that a 1 and almost all of the suction appears at the caisson tip. The above rearranges to give as/h s/h, and Eq (6) can be simplified to: (7) In the case either that the thickness of the caisson is small, or that a 1 this simplifies to the following (writing the outer diameter as D, and the caisson area 3 RELATIONSHIP BETWEEN SUCTION AND DISPLACEMENT RATE At low displacement rates, the rate of influx of water q to the caisson can be calculated by Darcy s law, and equated to the rate of displacement times the area of the caisson. Flow calculations were presented by Houlsby & Byrne (25), and yield: (1) where F is a dimensionless factor as determined by the procedures in Houlsby & Byrne (25), which may be fitted approximately by the equation F 3.6/(1 5h/D) for.1 h/d.8. If the displacement rate is increased, the above condition is interrupted by one of two conditions (a) the suction becomes large enough for liquefaction of the sand within the caisson to occur or (b) cavitation occurs within the caisson. When liquefaction occurs, the permeability of the liquefied sand increases to a large value, with the 47

4 Dimensionless flow factor F L Aspect ratio h/d Figure 3. Dimensionless flow factor for liquefaction case. Note that this will imply a sudden jump in s and V at the onset of liquefaction. (c) Cavitation without liquefaction Onset of cavitation occurs at s p a (1 f ) w h w. After that dh/dt is unbounded, s is constant and: result that the a factor in the calculation of the load changes (as noted above) to near unity. The displacement rate may still be estimated from a flow calculation, but the appropriate boundary condition now becomes one of the suction applied at the base rather than top of the caisson. Modified values of F (termed F L for this case) are given in Figure 3, and may be fitted approximately by the equation F exp( 5h/D) for.1 h/d 1.. When cavitation occurs, either before or after liquefaction, the displacement rate becomes unlimited and (assuming that cavitation occurs at an absolute pressure fp a where f is a constant), the suction will be constant and determined by p a w h w s fp a, or s p a (1 f ) w h w. In practice it appears that the factor f is near zero. 4 SUMMARY OF ANALYSIS CASES The following summary presents equations for the above cases, for a thin-walled caisson. To simplify the equations we neglecting here the stress reduction effect, although this should be included in more accurate calculations: (for dh/dt, s and these reduce to the equations for the fully drained case). (b) Liquefaction without cavitation Onset of liquefaction occurs at h/(1 a), after that as in case (a). (d) Cavitation with liquefaction Since s is constant once cavitation occurs, this condition can only occur when liquefaction occurs before cavitation. Onset of cavitation is at s p a (1 f ) w h w, after which dh/dt is unbounded, s is constant and: Note that the above cases only occur in order (a), (b), (d) or (a), (c). When several possibilities exist for calculating load capacity it is often true that the correct case is simply found by calculating all cases and then taking the lowest value. Note in this analysis that this simple approach cannot be adopted as the onset of some states can preclude other cases occurring, and the calculated load is not necessarily the lowest of the cases. 5 COMPARISONS WITH DATA We present here a number of pullout tests conducted two sands and at different pullout rates. The tests were conducted in a pressure chamber: some tests at an ambient (mudline) water pressure equal to atmospheric, and some at atmospheric plus 2 kpa. The model caisson was 28 mm diameter, 18 mm skirt length. In the following the loads presented include the caisson weight. The first test reported here (Test 9) was conducted on Redhill Sand, at a pullout rate of 1 mm/s and atmospheric pressure. Figure 4 shows the record of suction developed beneath the lid of the caisson against time, and Figure 5 shows the corresponding vertical load. It can be seen that (with a minor initial fluctuation) the suction rapidly approaches 1 kpa, at which stage cavitation occurs. At around s there is a sudden loss of both suction and vertical 48

5 Figure 4. Pressure v. time for test 9. Figure 7. Pressure v. time for test Figure 5. Vertical load v. time for test Figure 8. Vertical load v. time for test 1. V / sa Figure 6. V/sA v. time for test 9. load, but this is of little practical interest since by then the displacements are enormous and about threequarters of the caisson had been pulled out of the soil. Figure 6 shows the ratio V/sA, showing that this ratio remains approximately constant during most of the pullout. It can readily be shown that the suction in this case rapidly increased to sufficient value to cause liquefaction (which would occur at a suction of only about 3kPa), and that the relevant case for analysis here is case (d). The predicted values from the theory described above (including stress reduction) are also shown on each of Figures 4 to 6, and it is clear that the theory (whilst not capturing some of the detail at the beginning of the pullout) predicts the broad trends of the test correctly. Figures 7 and 8 show corresponding results for Test 1 (at the same pullout rate) but at an ambient pressure of atmospheric plus 2 kpa. The suctions developed at this rate of loading are insufficient to cause cavitation, which would occur at 3 kpa relative to ambient. It can be seen that again the theory predicts the overall pattern of behaviour well. This time it is case (b) that applies. The fluctuations in predicted suction (and hence load) are due to minor variations in the calculated velocity of extraction. Figures 9 and 1 show the results from Test 11, which is directly comparable to Test 9, but this time at a pullout rate of only 5 mm/s. Although the suctions are sufficient to cause liquefaction, the pullout rate is such that the suction is sufficiently small so that cavitation does not occur, and the vertical loads are correspondingly lower too. The predicted suction and load are also shown on the Figures. The match to the data could be improved by adjusting the permeability, but the value used in the predictions were deliberately kept the same for all three tests discussed. The permeability value used was k m/s, which 49

6 Figure 9. Pressure v. time for test 11. Figure 12. Vertical load v. time for test Table 1. Predicted and measured tensile loads. Max tensile load (kn) Test Predicted Measured Test 11 (5 mm/s, kpa) Test 9 (1 mm/s, kpa) Test 1 (1 mm/s, 2 kpa) HP5 sand: Test 23 (25 mm/s, 2 kpa) Figure 1. Vertical load v. time for test is somewhat higher than estimated previously for this sand (Kelly et al. 24). The other parameters used are K tan.7 and m 1.5. Finally, Figures 11 and 12 present equivalent data for a test on HP5 sand, which is much finer that Redhill Sand, and has an estimated permeability of k m/s. The extraction rate was 25 mm/s, and in this case, although the extraction rate is lower, the pore pressures are sufficient to cause cavitation even with the ambient pressure of atmospheric plus 2 kpa. Figure 11. Pressure v. time for test 23. The predicted and measured values of maximum tensile load for the three tests on Redhill sand and one on HP5 are shown in Table 1. The order of magnitude of the tensile load is correctly predicted in all cases, even though the actual capacity of the caisson varies greatly in the different tests. 6 CONCLUSIONS In this paper we develop a simplified theory for predicting the maximum tensile capacity of a caisson foundation in sand. The calculated capacity depends critically on the rate of pullout (in relation to the permeability) and the ambient water pressure (which determines whether cavitation occurs). The theory is used successfully to explain widely differing experimental results for caissons pulled out under different conditions. REFERENCES Houlsby, G.T. and Byrne, B.W. 25. Design procedures for installation of suction caissons in sand, Proc. ICE, Geotechnical Engineering, in press. Kelly, R.B., Byrne, B.W., Houlsby, G.T. and Martin, C.M. 24. Tensile loading of model caisson foundations for structures on sand, Proc. ISOPE, Toulon, Vol. 2,