Experimental Investigations of the Cyclic Response of Suction Caissons in Sand B.W. Byrne and G.T. Houlsby

OTC 12194 Experimental Investigations of the Cyclic Response of Suction Caissons in Sand B.W. Byrne and G.T. Houlsby Copyright 2, Offshore Technology Conference This paper was prepared for presentation at the 2 Offshore Technology Conference held in Houston, Texas, 1 4 May 2. This paper was selected for presentation by the OTC Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Offshore Technology Conference and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Offshore Technology Conference or its officers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Offshore Technology Conference is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 3 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of where and by whom the paper was presented. Abstract Suction caissons are a relatively new design concept being considered for use as foundations in a wide variety of offshore applications. Unlike many other offshore developments, there is no onshore equivalent of suction caissons to use in the development of design guidelines. This contrasts with the development of offshore piling theory. It is therefore essential to identify key behavioural patterns and important mechanisms that govern capacity under a wide variety of loading regimes. This would allow the establishment of a broad framework of response and thus focus subsequent site or project specific investigations. This paper is intended to add to this framework by detailing key results from an experimental investigation into the response of the foundation subjected to a variety of cyclic loading regimes. The model tests are conducted at 1g using a complex loading apparatus capable of applying independent control on vertical, horizontal and moment loading at cyclic rates of up to 1Hz. The foundation is embedded in oilsaturated sand in a medium to a very dense state, and the typical period of cycling is such that offshore loading conditions are modelled. Important areas studied during the testing programme were (i) cyclic loading of the foundation into tension, (ii) cyclic horizontal and moment loading under constant vertical loads, and, (iii) the relationship between cyclic loading and monotonic loading. These experimental results provide powerful insights into load-displacement relationships that could lead to the development of simple analytical or numerical models. Introduction Offshore structures are subjected to large loads from wind and waves which result in complex loading on the foundation. Wave loading is typically periodic in nature with a dominant period of 15-2 seconds being typical during a large storm. The response of foundations on sand under these loading conditions is best described as partially drained. There is sufficient time during the loading cycle for small amounts of fluid movement and volume change to occur within the soil matrix before the application of the next part of the cycle. One of the critical responses of a caisson is considered to be to tensile loading. Large strains may occur before the maximum tensile capacity is mobilised or cavitation of the fluid occurs 13. The serviceability of the platform probably requires deformations to be kept much smaller. The performance of caissons under repeated (cyclic) loading of variable amplitude is therefore of vital importance in design. The loading on the foundation is usually idealised to the general planar loading condition comprising of {V, M/2R, H} components as set out by Butterfield et al 1. The drained response of the foundation to this three degree-of-freedom loading has been studied in detail for flat footings 8 and caisson footings 4. The behaviour of the footing can be encapsulated within a work hardening plasticity theory with the advantage that the footings are represented by a 'macro-model' which may be incorporated within structural analyses 5,6. To develop such a framework it is usual to observe response within a three-dimensional yield surface such as that shown in Figure 1. This enables the development of a model which predicts the displacement response to all components of loading as set out by Cassidy 5. These plasticity macro models, though developed for monotonic loading, provide a useful baseline to the development of a model capable of predicting response to cyclic combined loading regimes. The experimental programme was formulated with the aim of developing a plasticity macro model. Further detail is given by Byrne 3. Description of Experimental Techniques The testing has been carried out using a multi axis loading rig at the University of Oxford. This loading rig is shown in Figure 2, and has been described in detail in other papers 8 and theses 11,1. The loading rig is capable of applying three degree of freedom loading {V, M/2R, H} with loads and displacements measured to a very high accuracy (~2N, ~2µm). A large sample tank allows eight different test sites to be used, thereby removing the effects of sample inhomogeneity from the observed response characteristics. Mangal 1 found that

2 B.W. BYRNE AND G.T. HOULSBY OTC 12194 slight inhomogeneities between samples resulted in difficulties in interpretation of transient loading tests on saturated sand. The main feature of the loading rig in relation to the work described in this paper is the improved control on each of the three independent loading axes. This control has been implemented within a VisualBasic control programme that coordinates data-logging and movement of the three stepper motors independently. Typical control commands are issued from the programme, via a RS232 link, to a stepper motor control unit. This unit accepts ASCII commands, which specify displacements, velocities and accelerations in terms of steps and seconds where appropriate. Where the movement of the axis is entirely displacement controlled the unit will instruct the stepper motor to travel a certain distance at the specified velocity and acceleration. Where much better control is required on the axis movement a digital PID 7 control loop is used. In this case the velocity and direction of the stepper motor is used as the control signal. Typically the new control signal depends on the error and can be determined by: de 1 u = k e + T + dt T edt d...1 i where e is the error, k is the gain, T d is the derivative time and T i is the integral time. The error used in the algorithm is the current value (as taken from the analogue-digital conversion) less a set-point that must be specified. The derivative and the integral are evaluated numerically within the programme. To control the contribution of the derivative part of the algorithm it has been necessary to adopt an exponential smoothing process on the error. To determine the values of the gain, derivative time and integral time a series of preliminary experiments were undertaken in which attempts were made to follow a step change in load (or displacement). These parameters depend heavily on the mean vertical load applied to the footing, the range in loads applied, as well as the stiffness of the soil sample. The soil specimen used for the majority of the testing was an oil saturated silica sand sample. The saturation by 1 centistoke silicon oil was necessary to model drainage times appropriate to offshore structures. The sand was a Baskarp Cyclone sand with characteristics shown in Table 1. The geotechnical properties of the soil sample could be determined from the bulk dry density measurement, as well as interpretation of cone penetration tests. The range of relative densities of different samples were between 8% and 95% whilst some cone penetration tests of one sample are shown in Figure 3 for a 7.94mm diameter cone. It was then possible to infer from these measurements the localised relative density and friction angles using relationships determined by Mangal 1. With the sand in a saturated state it was necessary also to determine the drainage characteristics, which could be determined by carrying out consolidation tests. A typical test consists of applying a step change in load and measuring the resulting decay in the pore pressure response. This is shown in Figure 4 where the t 5 can be seen to be of the order of 7 seconds. It was usual to carry out a large number of consolidation tests to observe the change in consolidation characteristics as the test progressed. The t 5 times usually decrease as the degree of consolidation of the soil sample increased with a continuing history of cyclic loading. In the initial development of the research it was expected that there would be certain events that would cause or precipitate failure as indicated by the framework set out by Bye et al 2 (illustrated in Figure 5). They suggested that there were zones of cyclic amplitudes that could be sustained but once these zones had been surpassed the response of the foundation degraded - this implies that extreme events are a critical mechanism for foundation failure. With this in mind it was necessary to develop a load train that incorporated an extreme event such as would occur when large waves pass through an offshore structure. The approach used is called 'Constrained NewWave' 16,17 - a state of the art deterministic random wave loading theory. 'Constrained NewWave' enables an extreme event to be placed within a sequence of random loading such that statistically it is indistinguishable from a random occurrence of that event. It is a very powerful technique which has been used to decrease the amount of computation required to develop extreme value response statistics 5. One of the main features is that the shape of the 'NewWave' extreme event is proportional to the autocorrelation function of the random seastate. This method was used for developing load paths, such as that shown in Figure 6, in order to study the effects of extreme loading events. This is believed to be a more satisfactory representation of the physical reality than the more usual application of many sinusoid cycles. The usual methods create an unrealistic amount of degradation of the foundation, and perhaps lead to very conservative methods for design. It was possible to conduct multiple sub-tests as long as account was taken of the loading history. In some cases up to 13 tests were performed. For example, it was typical to load up to a small mean vertical load, carry out a suite of different loading tests, load to a larger mean vertical load, carry out another suite of loading tests and so on. It was important to ensure that each consecutive mean vertical load was greater than any vertical loads reached during the cyclic loading (thus ensuring normal consolidation). Once this was exhausted overconsolidated behaviour was examined. This was felt to be a more important approach to the testing then to complete a large number of tests on individual virgin sites. The justification for this comes from the fact that foundations on offshore structures are usually exposed to initial bedding-in loading before being exposed to the larger storm loadings. Furthermore it is usual for storms to build up before the peak intensity is reached. It was possible to investigate the initial loading behaviour at the beginning of each test, and then continue to explore other aspects of behaviour as the test progressed. This maximised the value of each test site. Transient Vertical Response A typical loading path is shown in Figure 6 for cyclic loading about a mean vertical load of 2N. The corresponding

OTC 12194 EXPERIMENTAL INVESTIGATIONS OF THE CYCLIC RESPONSE OF SUCTION CAISSONS IN SAND 3 displacement response is shown in Figure 7 and the load displacement response in Figure 8. It is clear that the response is very asymmetric, with much larger displacements required to mobilise similar load levels in tension than in compression. A large number of these cyclic loading tests were carried out with parameters such as the mean load and cyclic period being varied. Due to the large amount of data created, a method of data reduction that still exhibited the main features of each cyclic test was developed and is shown in Figure 9. The displacements are broken into elastic and permanent displacement components during each half cycle about the mean load. This method allows the cyclic tests of different periods about the same mean load to be compared. However, it is necessary to use dimensional analysis to compare results across different mean loads. An appropriate dimensionless relationship has been found to be of the form; p V a p Vm δ = f,...2 3 Vm Vm γ ' D where V m is the mean load, D is the diameter, δ is the footing displacement, p a is atmospheric pressure, and γ' is the effective unit weight of the soil. If it is assumed that the bearing capacity is much greater than the mean loads being tested, then 3 the influence of the factor V m / γ ' D on the displacements may be small. We can seek therefore a relationship simply between δ p a / V m and V p / Vm. The application of these dimensionless forms is shown in Figures 1 and 11 for cyclic loading tests about different mean loads (though each of these is small compared to the bearing capacity). Clearly the nondimensional variables provide the basis for a relationship between displacement and load. In all cyclic load tests performed there was no threshold of loading above which the cyclic performance degraded. There was also no influence of cyclic period even though the cyclic loading tests were carried out such that the response was partially drained, considering the typical t 5 times shown in Figure 4. Of interest is the soft tensile response as shown in Figure 8. This soft tensile response can be explored further by carrying out monotonic tensile pull tests such as shown in Figure 12. If the tensile displacement is kept within small displacements (in this case less than one millimetre) than there is no degradation of response and no influence of rate of loading. The displacement rates for the responses shown differ by four orders of magnitude. If the pull tests are taken to larger displacements (several millimetres) then there is evidence of a degradation of the performance with the capacity at small displacements becoming limited to the weight of the soil plug. If the pull tests are taken to even larger displacements it is clear that cavitation limits the capacity as shown in Figure 13. The soft response at small displacements is evident, as is a much stiffer response at larger displacements (5-1mm of displacement). It is obvious that this stiffer secondary response will be rate-dependant, as during this stage there is significant loosening of the soil (volume change). The stiffness of the response will depend on how fast fluid can move within the soil matrix to allow this volume change to occur. The initial soft response is clearly affected by a previous history of large displacement tensile movements which tend to increase the displacement required to mobilise the rate dependant behaviour. It is clear that the framework of response depicted by Bye et al 2 incorporates this boundary between degradation and non-degradation for tension. It is not immediately obvious why there should be a degradation boundary on the compression side. However, it is likely that this corresponds to the compression half cycle that precedes the large tensile event that breaches the tensile degradation boundary. It is clear from the results that loading in the compression zone which does not pass the tension boundary does not trigger any foundation failure. The location of these boundaries is therefore influenced by the response characteristics of the structure, which determine the relationship between extreme peaks for tension and compression. From this discussion it is likely that design for caisson response will be for serviceability requirements rather than for capacity. One final point is made in Figure 14, where a cyclic loading test is compared to a monotonic test. The monotonic test effectively passes through the extreme points of all the cycles. It is believed that this concept allows cost effective research that may make use of monotonic tests, and infer cyclic behaviour from the results, with only a few high quality cyclic tests used to confirm inferred behaviour. Transient Combined Loading Response Research was also conducted on the combined cyclic loading response of the caissons. Tests were carried out where the mean vertical load was kept constant whilst the foundation was cycled under horizontal loading, or moment loading, or a combination of both. Typical loading consisted of cyclic loading from a normally consolidated position as well as overconsolidated load states. In service, loading for offshore foundations will be overconsolidated as the foundation has usually experienced enough loading that the load state is within the yield surface. When cyclic combined loading was applied to a normally consolidated footing there was substantial downward movement and significant pore fluid pressures. As the footing penetrates into the soil the yield surface expands so that the footing load state moves to an overconsolidated state. From this state only substantial deviatoric loads cause excessive plasticity and effective stress changes. This implies that it is preferable to provide drainage so that initial consolidation can occur quickly and in a controlled fashion (most probably just after installation). When conducting combined loading with the footing within the yield surface it became clear that different responses were obtained for different values of the mean vertical load. This is shown in Figure 15 where individual moment rotation tests but for different vertical loads show stiffness of response increasing with vertical load. Also shown are results from cyclic load tests which conform to Masing 12 rules. As in the vertical loading tests there appeared to be no effect of loading rate for tests conducted when the load state was overconsolidated. It is possible to compare the results

4 B.W. BYRNE AND G.T. HOULSBY OTC 12194 shown in Figure 15 using a normalisation similar to the vertical loading; p a M /2R 2Rδθ = f...3 Vm Vm where 2Rδθ is the rotational displacement and M/2R is the rotational load. This is shown in Figure 16 for some of the tests shown in Figure 15. The tests conducted under the larger vertical loads are reduced so that they form the stiffer response at the centre of the moment rotation loops. The initial loading of each curve conforms well to a backbone curve which is fitted with a hyperbolic function. The importance of the hyperbolic function in relation to theoretical modelling is described below. Horizontal loading follows a very similar pattern. Theoretical Modelling of Cyclic Loading It is important that any experiments are interpreted within an appropriate theoretical framework, and not merely treated as an empirical collection of data. An appropriate framework for the understanding of the behaviour of foundations has been found to be plasticity theory. The reasons for this are (a) theories can be constructed which reproduce the behaviour of the foundations well, and (b) the resulting models can readily be included in a numerical analysis of a complete offshore structure. Plasticity theories for slow monotonic loading of foundations had been established prior to this research, and monotonic data were simply fitted within these existing frameworks 4,5,8. The important recent theoretical developments relate to the understanding of cyclic loading. The results of a typical cyclic test have been shown in Figure 8 and 17. A remarkable feature about this result (which is typical of any cyclic horizontal or moment load test on a foundation) is that continuous smooth curves are obtained as the load is cycled. A conventional plasticity model could not model this type of behaviour, but instead would result in well-defined yield points at which a sudden change of stiffness would occur. The magnitude of plastic deformation predicted on reverse loading would also be at least an order of magnitude smaller than that observed. An obvious conclusion would be that plasticity theory is inappropriate for modelling cyclic loading, but given its proven success for modelling monotonic loading this is excessively pessimistic. An appropriate framework that does describe cyclic loading is continuous hyperplasticity. An exposition of the theory would be inappropriate here as it involves a considerable amount of mathematical development, and this is fully documented within journal papers 9,14,15. In essence the theory replaces the plastic strain in conventional plasticity theory with a continuous field of an infinite number of plastic strain components, each associated with a separate yield surface. This is achieved within a manageable mathematical framework by deriving the plasticity theory entirely from two potentials. For the case of the infinite field of plastic strains these potentials are functionals ( functions of functions ) of the plastic strain. Conventional plasticity theory is a special case of the new approach. The result is that theories can be constructed in which responses of the character shown in Figure 17 can be modelled. The mathematical structure of the theories is relatively simple (although unfamiliar to those used to conventional plasticity). For example, Figure 17 shows the result of a moment test in which cycles of increasing amplitude have been applied (this test was carried out specifically to aid model development). Figure 18 shows the fitted response using the continuous hyperplastic model. Whilst the fitting is not exact, the model captures the main features of the cyclic test. Only three parameters are required to define the behaviour shown in Figure 18 - these being required to define a hyperbolic function that fits the backbone curve. Conclusions The response of shallow foundations on sand to various loading conditions has been explored with particular reference to the suction caisson foundation. It was observed that the design for vertical tensile loading will be focussed on serviceability requirements rather than capacity (which is ultimately limited by cavitation of the pore fluid). The degradation of behaviour as indicated by the framework set out by Bye et al 2 was not observed in the current experiments except where monotonic pull tests were taken to large displacements. A data reduction method and a set of normalisations were suggested which enabled the comparison of a large quantity of cyclic data. A new theoretical approach to modelling cyclic loading was described and an example simulation shown. This theory accurately predicts the continuous change in stiffness that is observed in load reversals during cyclic loading of foundations. Acknowledgements The first author would like to acknowledge the generous funding from the Rhodes Trust and EPSRC for the work. Support from the Royal Commission for the Exhibition of 1851 is also gratefully acknowledged. References 1. Butterfield, R., Houlsby, G.T. and Gottardi, G. (1997). Standardised Sign Conventions and Notation for Generally Loaded Foundations. Geotechnique 47, N o 4, UK. 2. Bye, A., Erbrich, C., Rognlien, B., and Tjelta, T.I. (1995). Geotechnical design of bucket foundations. Proc. of Offshore Technology Conference, OTC 7793. 3. Byrne, B.W. (2). Investigations of suction caissons in dense sand. Forthcoming DPhil thesis, Oxford University. 4. Byrne, B.W. and Houlsby, G.T. (1999). Drained behaviour of suction caissons on very dense sand. Offshore Technology Conference, Houston, Texas. Paper 1994. 5. Cassidy, M.J. (1999). The nonlinear dynamic analysis of jackup platforms under random ocean waves. DPhil Thesis. Oxford University. 6. Cassidy, M.J. and Houlsby, G.T. (1999). On the modelling of foundations for jack-up units on sand. Offshore Technology Conference, Houston, Texas. Paper 1995.

OTC 12194 EXPERIMENTAL INVESTIGATIONS OF THE CYCLIC RESPONSE OF SUCTION CAISSONS IN SAND 5 7. Clarke, D.W. (1984). PID Algorithms and their Computer Implementation. Trans Inst M C, Vol. 6, N o 6, pp 35-316. 8. Gottardi, G., Houlsby, G.T. and Butterfield, R. (1999). The Plastic Response of Circular Footings on Sand Under General Planar Loading. Geotechnique 49, N o 4, pp 453-47. 9. Houlsby, G.T. and Puzrin, A.M. (2). A thermomechanical framework for constitutive models for rate-independent dissipative materials. International Journal of Plasticity, in press. 1. Mangal, J.K. (1999). Partially drained loading of shallow foundations on sand. DPhil Thesis, Oxford University. 11. Martin, C.M. (1994). Physical and numerical modelling of offshore foundations under combined loads. DPhil Thesis, University of Oxford. 12. Masing, G. (1926). Eiganspannungen und Verfestigung beim Messing. Proceedings of the Second International Congress of Applied Mechanics, pp 332-335. 13. McManus, K.J. and Davis, R.O. (1997). Dilation induced pore fluid cavitation in sands. Geotechnique 47, N o 1, pp 173-177. 14. Puzrin, A.M. and Houlsby, G.T. (2). A thermomechanical framework for rate-independent dissipative materials with internal functions. International Journal of Plasticity, in press. 15. Puzrin, A.M. and Houlsby, G.T. (1999). Fundamentals of kinematice hardening hyperplasticity. Report N o OUEL 2218/99. The University of Oxford. 16. Taylor, P.H., Jonathan, P., and Harland, L.A. (1995). Time domain simulation of jack-up dynamics with the extremes of a gaussian process. Proc of Conference on Offshore Mechanics and Arctic Engineering, Vol 1A, pp. 313-319. 17. Tromans, P.S., Anaturk, A. and Hagemeijer, P. (1991). A new model for the kinematics of large ocean waves - application as a design wave. Proc of 1st International Symposium on Offshore and Polar Engineering, Edinburgh, Vol 3, pp 64-71. Coefficient of Uniformity, C u 3.64 Specific Gravity, G s 2.69 Minimum density, γ min 12.72 kn/m 3 Maximum density, γ max 16.85 kn/m 3 Critical state friction angle, φ cs 32.5 Permeability (8% Rd with water), k 7 x 1-6 m/s Oil - Kinematic viscosity, µ 1 mm 2 /s (at 25 C) (compared to.897 mm 2 /s for water) Oil - Specific gravity, G s.96 (at 25 C) Oil - Bulk Modulus, K 8 MPa for ε < 1% (compared to 22 MPa for water at 25 C) Table 1 - Properties of the Baskarp Cyclone Sand and Silicon Oil.

6 B.W. BYRNE AND G.T. HOULSBY OTC 12194 H M/2R Yield surface V Figure 1 - Typical combined load yield surface. Figure 2 - Three degree of freedom loading rig at Oxford. Depth of Penetration (mm) -2 2 4 6 8 1 Cone Penetration Resistance (kpa) -2 2 4 6 8 dv or du (N) 12 1 8 6 4 2 Change in fluid load, du Change in Load, dv Change in displacement, dw 1 2 3 4 Time (s) -.1.1.2.3.4.5.6.7.8 dw (mm) Figure 3 - Cone penetration tests conducted to determine soil sample properties. Figure 4 - A typical consolidation loading test to determine drainage properties. Cyclic Load Compression Calculated Cyclic Amplitude Static Load Vertical Load (N) 6 5 4 3 2 1 Tension Static Load -1 95 195 295 395 495 Time (s) Figure 5 - The framework for determining suction caisson cyclic capacities by Bye et al (1995). Figure 6 - A typical 'constrained NewWave' loading sequence used for caisson cyclic tests.

OTC 12194 EXPERIMENTAL INVESTIGATIONS OF THE CYCLIC RESPONSE OF SUCTION CAISSONS IN SAND 7.1.5 -.5 -.1 -.15 -.2 -.25 -.3 95 195 295 395 495 Time (s) Figure 7 - The displacement response for the extreme event loading sequence. Vertical Load (N) 6 5 4 3 2 1-1 -.3 -.2 -.1.1 Figure 8 - The asymmetric load displacement response. Load Vpeak Mean Load Time Vpeak Displacement δtemp δperm δperm Time δtemp Figure 9 - Methodology for reducing a large amount of cyclic data. 2 2 V norm 1.5 1.5 V=6N V=2N V = 8N V=1N V norm 1.5 1.5 -.5 -.5-1 -1-1.5 -.16 -.11 -.6 -.1.4 d tempnorm Figure 1 - Normalised elastic displacement response for caisson tests at different mean loads. -1.5-2 -.5 -.3 -.1.1.3.5 d permnorm Figure 11 - The normalised permanent displacement response.

8 B.W. BYRNE AND G.T. HOULSBY OTC 12194 Vertical Load (N) 2 15 1 5-5 -1-15 -2 Pull tests at the same rate for larger displacements showing degradation. Weight of soil plug Pull tests at different rates for small displacements showing no degradation. -1.5-1.3-1.1 -.9 -.7 -.5 -.3 -.1 Figure 12 - The soft initial tensile loading response for monotonic pull tests. Vertical Load or Pore Fluid Load (N) 5-5 -1-15 -2 Total load response Pore fluid response Increased external friction Initial soft response Cavitation limit reached -25-45 -4-35 -3-25 -2-15 -1-5 5 Figure 13 - The large displacement tensile response where capacity is limited by cavitation. Vertical Load (N) 5 4 3 2 1 Cyclic Loading about V=2N Pull test from V=2N -1-2 -1.5-1 -.5.5 Figure 14 - A comparison between cyclic loading tests and a monotonic pull test.

OTC 12194 EXPERIMENTAL INVESTIGATIONS OF THE CYCLIC RESPONSE OF SUCTION CAISSONS IN SAND 9 25 Moment Load, M/2R (N) 2 15 1 5 Results from cyclic test results V=1N V=7N V=5N V=3N V=2N V=1N V=5N -5 -.2.2.4.6.8 1 1.2 Displacement, 2Rdq (mm) Figure 15 - The comparison between monotonic moment rotation tests and cyclic loading tests under different levels of vertical load..5.4.3.2 Vm=1N Vm=3N Vm=5N Vm=1N Hyperbolic Fit M / 2RV m.1 -.1 -.2 -.3 -.4 -.5 -.4 -.3 -.2 -.1.1.2.3.4 2Rdq (p a / V m ).5 Figure 16 - Normalised moment rotation tests conducted under different levels of mean vertical load. 1 1 8 8 6 6 4 4 M/2R (N) 2-2 M/2R (N) 2-2 -4-4 -6-6 -8-8 -1-1 -1 -.5.5 1-1 -.5.5 1 2Rdq (mm) 2Rdq (mm) Figure 17 - An experimental moment rotation test consisting of cycles of increasing stress. Figure 18 - A theoretical simulation of the experimental test shown in Figure 17.