Symosium on Ice (26) INTRODUCING THE SHEAR-CAP MATERIAL CRITERION TO AN ICE RUBBLE LOAD MODEL Mohamed O. ElSeify and Thomas G. Brown University of Calgary, Calgary, Canada ABSTRACT Current ice rubble load models are based on a cohesive-frictional stress criterion, of which the Mohr-Coulomb (MC) and Drucker-Prager (DP) models are the most widely used. Combining these failure criteria with a shear-ca model as a new material model rovides limits to the behaviour of rubble in both deviatoric and hydrostatic stress states. This concet adds a ca yield surface to the DP shear yield criterion bounding the yield surface in hydrostatic comression. The ca yield surface controls the dilatation rocess by including a umetric hardening arameter. A detailed arametric analysis has been done to investigate the range of the shear-ca model s arameter values that rovide valid results. These arameters have been calibrated to imlement the shear-ca material criterion into an analytical load model. This analytical load model calculates the load of interaction between first year ice ridge keels and conical structures. Existing models significantly over-redict the recorded loads measured at the Confederation Bridge Pier (P31). As a result, one of the aims of the newly develoed load model is to give calculated load values which correlate well with the Confederation Bridge recorded loads. Comarison has been made between the calculated ressures using the new model and the recorded ressure given by the Confederation Bridge Monitoring Program (Brown, 26). The comarison showed romising correlation between the calculated and recorded ressures. INTRODUCTION Canada has large sea areas that are ice covered in winter. Many different ice features result from the subsequent deformation of this ice cover. Most of Canada s ice regime is marginal first year ice that includes extensive rubble fields and first year ridges. The interaction load between first year ice ridges and offshore structures (Figure 1) reresents the critical design load in most of the Canadian sea areas that are ice covered in the winter. Engineers in Canada continue to face roblems while designing conical offshore structures having the same geometry as the Confederation Bridge Pier interacting with first year ice ridges. This is due to the lack of an efficient and accurate model that can determine the loads resulting from the interaction of keel ice rubble with conical offshore structures. One of the most imortant reasons for the lack of an accurate model is the material stress criterion used for the ice rubble. The aer has two main aims. First, investigate the caability of a new ice rubble strength algorithm to model the ice rubble behaviour interacting with conical offshore structures such as the Confederation Bridge Pier, and to show that the algorithm can be used, after calibration of the material arameters, to calculate the ressure resulting from the interaction of keel -193-
Symosium on Ice (26) rubble and conical offshore structures. Second, erform sensitivity and arametric analyses to investigate the material arameters that have the most influence on the load value. Keel interacting with cone Figure1.Interaction between First Year ridge and a Confederation Bridge Pier LIMITATIONS OF THE MOHR-COULOMB STRESS CRITERION Most of the analytical models that define the failure of keel ice rubble and calculate the loads caused by keel rubble on structures are based on cohesive-frictional stress criteria. The most frequently used cohesive-frictional stress criteria are the Mohr-Coulomb (MC) and Drucker- Prager (DP) models. The most obvious limitation in these models is their inability to simulate keel rubble umetric changes accurately. First, the dilatation behaviour of ice rubble is unbounded and the material exands continually as lastic deformations continue. In addition, the dilatation during shear failure increases with higher friction angles. Second, the MC and the DP models do not simulate the keel rubble comaction behaviour resulting from the hydrostatic comression. As a result, another yield criterion that simulates the umetric changes that take lace in the keel rubble interacting with conical offshore structures must be considered. SHEAR-CAP CRITERION The main concet of the shear-ca model is to add a ca yield surface to the DP shear criterion. This ca yield surface bounds the yield surface in hydrostatic comression so it controls the dilatation rocess by including a umetric hardening control arameter. The DP model is a smooth aroximation of the MC criterion, which enables it to be used in the shear ca model, allowing a smooth transition between the two comonents of the failure model. The great advantage of the shear-ca model over the MC and DP models is that it gives the material the ability to fail in comaction as well as in shear. According to Heinonen (24), a smooth yield function is created by combining two ellitical yield surfaces together as shown in Figure 2, one for the shear failure, and one for the ca failure describing the comaction failure. Figure 2. Shear-Ca Yield Function in the Meridian Plane (Heinonen, 24) -194-
Symosium on Ice (26) It can be observed that the umetric comonent of the yield surface gradient results in exansion in the shear failure region and comaction in the ca art. So, this material model more closely models the keel rubble umetric change behaviour by bounding the dilatation behaviour that accomanies the shear and, beyond the hydrostatic ressure a the umetric change will be comaction. The yield function for the ca failure is: f s 2 2 [( a ) tan ] + q ( d + a tan β ) a = β ; (1) 2 2 f = ( ) + ( Rq) R( d + tan β ) ; > (2) c a a a where b (Hydrostatic ressure strength) reresents the ca hardening law as follows: ( ε ε ) b = ex (3) κ and a defines the ressure that divides the shear-ca yield surface into two arts: b Rd a = 1+ R tan β Where: d: Cohesion arameter in DP model, β : Internal friction angle in DP model, R: Defines the ca shae, q: Second invariant of the deviatoric stress κ : Ca surface hardening arameter, ε : Reference umetric lastic strain in ca hardening model, (4) Ca Hardening When the ressure is greater than a, comaction occurs, increasing the inter-article contact forces. This is the case of ca failure where the material hardens. Therefore, the material hardening law should be defined by the umetric lastic strain as a hardening variable as shown in Equation 3. The hardening variable in the yield function is a as it deends on b, as given by Equation 4. According to the ca hardening law, hardening or softening of the material deends on the umetric lastic strain. The size of the yield surface is controlled by adjusting the material arameter a. When the stress state lies on the ca side, the umetric lastic strain is negative and the ice rubble is comacted. While the rubble is comacted, the yield surface grows. On the other hand, when the stress state lies on the shear failure side, the ume increases causing the yield surface to shrink. CALIBRATION OF MATERIAL PARAMETERS Heinonen (24) has calibrated some of the shear-ca material model s arameters in order to simulate the keel rubble behaviour when subjected to unch shear test. Heinonen (24) imlemented the shear-ca material model into the ABAQUS/STANDARD which is a commercial FEM ackage. These model s arameters were calibrated by comaring the FEM simulations with full-scale in-situ exerimental results. In the resent research, the shear-ca material model has been imlemented into an analytical load model using the same values of -195-
the arameters estimated by Heinonen (24). This model has been used to calculate the load resulting from the interaction of first year ridges with the Confederation Bridge ier (P31). The analytical model is used to obtain the contact ressure between the ridge keel material and the structure, as a function of deth and interaction geometry. Few of the calibrated material arameters values estimated by Heinonen (24) give good results which correlate well with the data recorded in the Confederation Bridge monitoring rogram. On the other hand, other calibrated material arameters values do not rovide results that match with the data recorded in the Confederation Bridge monitoring rogram. This can be attributed to the fact that Heinonen has calibrated these arameters in order to simulate the keel rubble behaviour under unch shear test. As a result, these arameters were adjusted to fit with the data recorded from the Confederation Bridge and simulate the interaction between first year ridge keels and conical offshore structures. Parameters Calibrated by Heinonen (24) In this section, the arameter values that had been calibrated by Heinonen (24) and for which the results match the measurements from this study will be introduced. A value of.3 was used for Poisson s ratio. This value is a tyical value for a material comosed of articles (Harr, 1977). Based on the arametric simulations carried out, the value of κ was chosen to be.3 in order to simulate the ost eak softening henomena correctly. Finally, the umetric lastic strain at the reference state ( ε hardening law was simlified to: ε = b ex κ Proceedings oh the 18th IAHR International Symosium on Ice (26) ) in the ca eution law was ignored. Thus the ANALYSIS PROCEDURE The analysis of the rubble forces, using the new model, was based on the following remises: 1. The Rubble behaviour was modeled using a material model which includes the otential for both shear failure and ca failure which takes into consideration the umetric changes and the ossibility of ressure failure. 2. Lemée (23) showed that the main area of contact between the ridge and the Confederation Bridge Pier is on the cone. This was based on the measurements from the ice ressure anels on the ier shaft, the resence of marine growth on the ier shafts, and the absence of any correlation between keel deth and ice load. Consequently, ridge failure occurs in a traezoidal area with an average height of 4m and to and bottom widths of 14.2m and 2m resectively. The analysis rocess can be summarized as follows: 1. Detailed analysis of video and sonar data was carried out using the video taes and the sonar data recorded during the Confederation Bridge monitoring rogram. This detailed analysis focused on ridge interactions with the Confederation Bridge ier No. 31. A total of 519 events were analyzed. This analysis resulted in considerable understanding of the geometric roerties of the ridges that encountered the ier during the above mentioned 519 events. 2. For each of the events, the corresonding eak load was identified from the tiltmeter record, and then corrected for the effects of wind. The recorded loads were divided by the cone area situated between the mean water level and the cone edge in order to calculate the recorded ressure. This is a conservative (i.e., uer bound) to the actual ressures, as it ignores the load associated with the consolidated layer. However, at this stage, a satisfactory (5) -196-
Symosium on Ice (26) model for the failure of the consolidated layer is not available, (all models greatly overestimate the consolidated layer load), and so this aroach has been adoted. 3. The arameters of the shear-ca model were calibrated such that the calculated ressures using the model correlate well with the recorded ressures at the Confederation Bridge Piers. Taking the recorded loads from the Confederation Bridge Monitoring rogram as a reference is based on the fact that this rogram is one of the rare monitoring rograms worldwide that monitors interactions between real first year ridges and real full-scale conical offshore structure. 4. The shear-ca yield functions given by Equations 1 and 2 were used to calculate the ressure acting on the cone of the Confederation Bridge ier. The ridges geometrical roerties estimated from the video and sonar data analyses were used. 5. A comarison was then made between the recorded and calculated ressures to assess the correlation between them and to judge the caability of the shear-ca model to simulate the interactions between first year ice ridges and the Confederation Bridge s iers. Other Parameters Calibrated to Fit the Confederation Bridge Recorded Data Based on the sensitivity analysis, ε, R, q, β were found to be the most influential arameters on the values of the calculated ressure. Based on Heinonen (24), the value of ε ranged between -.3 and.3. In case of the interactions between keel rubble and the Confederation Bridge cone most of the events will be governed by shear failure. As a result, dilatation will take lace and the value of ε will be tested in the range of to.3. The sensitivity of the results to this arameter was assessed using arameter values of.5,.1 and.15. The values.5 and.15 gave calculated ressures which were far in excess of those recorded on the Confederation Bridge. It was found that the value of the umetric lastic strain ( ε ) must be of the order of.1 if the value of κ is taken to be.3 in order to give good correlation between calculated and recorded ressures as shown in Figure 6. As shown in Figure 3, using a umetric lastic strain value of.5, gives very high calculated ressures. Calculated Pressure (KPa) 45 4 35 3 25 2 15 1 5 1 2 3 4 Recorded Pressure (KPa) Figure 3. Relation between Calculated and Recorded Pressure for ε =.5 The calculated ressures only match the measured ressures at the higher values, and, in all cases, are greater than 3 KPa. Similarly, at a umetric lastic strain value of.15, the calculated ressures are nearly all negative. These results indicate that these values of umetric lastic strain (.5 and.15) give oor correlation between measured ressures -197-
and calculated ressures. As a result, it may be concluded that the value of the calculated ressure using the shear-ca model is extremely sensitive to the value of ε. R is the ca side shae factor. Parametric analysis was conducted to study the effect of this arameter on the calculated ressure values. Values ranging between R=.5 to R=3.5 have been tried. It was concluded that the value of the ca side factor had to be aroximately 3.5 in order to rovide a best fit between the calculated and the recorded load. Heinonen s numerical testing indicated that a low value for the ca side shae factor caused too much comaction during lastic deformation on the ca side. This is comatible the current findings as these arametric analyses led to a high value of R = 3.5, which is logical because in this case the failure is a shear failure, where the comaction is minimal. It is clear from Figures 4, and 6 that changing the value of R from.5 to 3.5 has considerable influence on the values of the calculated ressure. These results were obtained using a value of Calculated Pressure (KPa) Proceedings oh the 18th IAHR International Symosium on Ice (26) ε of.1. 35 3 25 2 15 1 5-5 -1-15 -2 5 1 15 2 25 3 35 Recorded Pressure (KPa) Figure 4. Relation between Calculated and Recorded Pressure for R=.5 As a result of the arametric analyses, it was found that the value of the second invariant of the deviatoric stress tensor q must lie in the range of 4-6 KPa. It can be seen from Figures 5 and 6 that decreasing the value of q decreases the values of the calculated ressures. Figure 6 for a value of q of 5 KPa, shows a reasonable trend with a best-fit line that asses through the origin, although there is considerable scatter, and negative values of ressure. Figure 5 for value of q of 35 KPa gives very low values of the calculated ressure such that most of the trend line lies in the negative zone. Finally, an extensive arametric analysis has been done to investigate the effect of cohesion d and internal friction angle β on the correlation between the ressures calculated from the shear-ca model and those recorded in the Confederation Bridge monitoring rogram. An exonential relationshi determined by Heinonen (24) that fit all data from admissible combination curves, l has been used. Values of β have been assumed ranging from 1 o to 55 o. Then the corresonding values of d have been calculated using the relation estimated by Heinonen (24) as follows: d( β ) = d(45 ) ex( ax) (8) x = ln(tan β ) Values of a (an emirical constant) and d (45 o ) have been estimated to be -.3447 and 8.471 resectively (Heinonen, 24). Substituting the assumed values of β and the calculated values of d into the shear-ca model, one can calculate the ressure. After detailed arametric -198-
analysis, it was estimated that the values of β that give the best correlation between the recorded and the calculated ressures, ranged between 15 and 3. Calculated Pressure (KPa) 4 3 2 1-1 -2 1 2 3 4 Recorded Pressure (KPa) Figure 5. Relation between Calculated and Recorded Pressure for q = 35KPa RESULTS OF THE ANALYSIS The results of the comarison are shown in Figure 6. The final values of the model arameters that give the best correlation between the calculated ressures and the recorded ressures are as follows: ε =.1, κ=.3, q=5kpa, β=3 o and R=3.5. In general, the calculated ressures are lower than the measured ressure, which is reasonable, given the uer bound character of the measured ressures. However, of concern is the resence of negative ressures for a number of interactions and this suggests that further work is required to refine the model. Calculated Pressure (KPa) Proceedings oh the 18th IAHR International Symosium on Ice (26) 5 4 3 2 1-1 1 2 3 4 5 Recorded Pressure (KPa) Figure 6. Relation between Calculated and Measured Pressures CONCLUSIONS A new aroach has been used to develo a unified model for the rediction of the average ressures associated with interactions between rubble ice features and conical offshore structures. While the model owes its origin to the work of Heinonen, the develoment as a stand-alone, analytical model, is new. The following conclusions result from the work: a) The introduction of a ca failure model to a model that accounts for the otential for shear lane failure (essentially a model that allows the rubble to fail in comression in the resence of near hydrostatic comressive stress fields, and which can account for umetric contraction) allows two forms of rubble failure to occur; shear lane failure and comressive failure. The resence of the latter form of failure is suorted by the frequent observations of -199-
Symosium on Ice (26) loughing of rubble features by the Confederation Bridge iers, and the absence of clearly defined shear failure lanes. b) Calibration of the factors required for the combined failure model was based on revious calibration by Heinonen, as well as calibration on the basis of the Confederation Bridge observations and measurements. There is considerable scatter in the results and therefore considerable uncertainty in the calibrations. However, the trends suggest that the model is caable of modeling the loads measured on the iers. c) The aims of the study have been achieved. The aer roved that the shear-ca model holds considerable romise in simulating the keel rubble behaviour. Also it showed that the shearca model can be imlemented as an analytical rocedure that can be used by design engineers. FUTURE WORK This aer resents the initial results of a research lan to develo a unified analytical model that calculates the load of interaction of first year ridge keels with conical offshore structures. More work needs to be done on the model as follows: 1) The model needs to be generalized for different values of 2) To determine the reasons for the negative ressures in Figure 6. ε and κ. ACKNOWLEDGEMENT The authors would like to acknowledge the suort of Strait Crossing Bridge (SCBL), Public Works and Government Services Canada (PWGSC), the Program for Energy Research and Develoment (PERD), and the Natural Sciences and Engineering Research Council of Canada (NSERC). REFERENCES Brown T., 26 Ice Monitoring on the Confederation Bridge, What Has Been Learnt Proc. CSCE 26 Annual Conference, Calgary, Alberta, Canada, Paer ST-79. ElSeify, M.O. And Brown, T.G. 25 A Unified Model for Rubble Ice Load and Behaviour, Final Reort to National Research Council of Canada, PERD Ice/Structure Interaction Program. Reort No.: PERD/CHC 5-119. Harr, M.E. 1977 Mechanics of Particulate Media Mc-Graw-Hill Inc. USA, 543. Heinonen J. 24 Constitutive Modeling of Ice Rubble in First-Year Ridge Keel PhD Thesis, Helsinki University of Technology, Helsinki, Finland. Lemée, E.M., 23 Pressure Ridge Interaction with the Confederation Bridge Piers, MSc Thesis, Civil Engineering Deartment, University of Calgary, Calgary, Alberta, Canada. -2-