Stress analysis and failure prediction in avalanche snowpacks. F.W.Smith and J. O.Curtis

Transcription

1 Stress analysis and failure prediction in avalanche snowpacks F.W.Smith and J. O.Curtis Abstract. Results of finite element stress analyses of a five-layered avalanche snowpack which was observed at Berthoud Pass, Colorado, U.S.A. are given. An attempt is made to model the effects of a weak sublayer in the snowpack by prescribing nonlinear stress-strain behaviour for a snow layer near the snow-ground interface. It is observed that under certain conditions, large tensile stresses oriented parallel to the slope may be produced in the upper layers of the snowpack, near the fracture line. The analyses support the hypothesis that avalanche release occurs by a shear failure in one of the lower layers. Résumé. On présente les résultats de l'analyse par la méthode des éléments finis des tensions dans un manteau neigeux à cinq couches observé au col Berthoud, Colorado. La simulation d'une faible couche interne est effectuée en imposant à une couche de neige proche du sol une relation non-linéaire entre les tensions et les elongations. Il en résulte que, sous certaines conditions, de larges efforts de traction parallèle à la pente peuvent s'observer dans les couches supérieures environs de la ligne de fracture. L'analyse des résultats corrobore l'hypothèse que la production d'avalanches résulte d'une fracture par cisaillement d'une des couches inférieures. INTRODUCTION In recent years a number of studies have been conducted using the finite element method of stress analysis in an attempt to determine the nature of the stress distribution in an avalanche snowpack and the relative importance of various effects. The motivation for using tills technique is that it is not limited by the complex geometries which are found in an avalanche snowpack and it is not limited by the fact that avalanche snowpacks have nonhomogeneous material properties. Earlier applications of the finite element method to avalanche problems, Smith etal. (1971), Smith (1972), dealt with a strictly elastic analysis of the distribution of stress in single and multilayered snowpacks intended to simulate snowpacks having realistic geometries. It was found in these studies that the distribution of elastic stress closely approximated the solutions discussed by Mellor (1968) for the shear stress parallel to the hill. However, tensile stresses in the upper portion of the snowpack were predicted by the finite element analysis which are not predicted by the analytical treatment. It was also found that a variety of effects are present due to variations in the geometry of the snowpack not accounted for in previous analyses. Further studies were made by Curtis and Smith (1974) using the finite element method in an attempt to determine the effect of varying the material properties in the snowpack and the boundary conditions at the snow ground interface. In this work an attempt was made to model the effect of a weak sublayer on the stress distribution. This was done by determining the elastic distribution of stress along the snowground interface assuming that the snowpack was firmly attached to the ground. This boundary condition was then modified by reducing the level of stress which can be supported at the interface to simulate the presence of a weak layer at that location which allows more deformation than would be present with the bottom layer attached firmly to the ground. It was determined that modelling a weak sublayer in this way could easily produce tensile stresses in the upper layers of the snowpack which are aligned in a direction parallel with the top surface of the snowpack in the region of the crown surface discussed by Perla (1973).

2 Stress analysis and failure prediction 333 The work done by Curtis and Smith (1974) was an attempt to examine in more detail the effect of a shear stress perturbation on the bottom of the snow layer which was first studied by Perla and LaChapelle ( 1970). In that work the effect of the shear perturbation was treated analytically and it was found that it was possible to theoretically approximate the 90 angle the crown surface makes with the slope of a hill when a dry slab avalanche is released. The study of Curtis and Smith (1974) showed that this angle was achieved only in the top layers of the snowpack in the presence of a shear perturbation along the bottom layer. In an attempt to improve on the modelling of the effect of a weak sublayer, the finite element computer programs used by Curtis and Smith have been modified to accommodate prescribing a nonlinear stress strain curve for selected layers in the snowpack. While this approach does not precisely define the effect of a weak sublayer in a snowpack which is undergoing creep deformation, it does admit the modelling of a process in which a lower layer in the snowpack is undergoing large deformations relative to other layers. To avoid introduction of effects due to varying the geometry, the five-layered snowpack studied earlier by Curtis and Smith (1974) was again used for the present study. THE PROBLEM AND APPROACH Figure 1 is a schematic of the five-layered snowpack which was analysed. Typical constant strain triangular finite elements are shown on the figure in addition to layer densities and snowpack dimensions. The geometry shown in Fig.l closely approximates a realistic snowpack that was observed at Berthoud Pass, Colorado, U.S.A. Layer Densities Kg m~ 3 FIGURE 1. Schematic of the five-layered snowpack model. during the winter of by members of the Rocky Mountain Forest and Range Experiment Station, Fort Collins, Colorado. An observed fracture line is shown on Fig. 1 because this particular snowpack did fail as a natural slab avalanche in December of Additional assumptions employed in the nonlinear analysis were: (1) Plane strain conditions exist in the snowpack. (2) The top surface is stress free.

3 334 F.W.Smith andj. O.Curtis (3) The bottom surface is fixed to the ground. (4) The snow in each layer is isotropic. (5) A constant value of Poisson's ratio equal to 0.25 was chosen for all layers. (6) Either a constant value of Young's modulus was taken throughout the snowpack or each layer was assigned a different value, constant throughout the layer. (7) A neutral zone solution (Mellor, 1966) was applied to both the right and left ends of the snowpack model in order to achieve well-behaved stresses near the ends. A wakened layer within the snowpack was modelled by prescribing a nonlinear constitutive relationship for that layer. It was assumed that the behaviour of each layer of snow could be described by a different octahedral stress strain curve. While the curves for layers 1,3,4 and 5 were taken to be linear (slope determined by the shear modulus) which represents purely elastic behaviour, the stress strain behaviour of the weak layer was taken to be like that shown in Fig.2. The values of the yield stress,ctyp,and the amount of stress increase after yielding o\ x, were selected from the results of a purely elastic analysis to insure that nonlinear behaviour would occur in layer 2. The final octahedral strain 7 0 was taken to be Octahedral Shear Strain FIGURE 2. Typical nonlinear stress-strain curve for layer 2. The nonlinear solution algorithm consisted of requiring that computed values of the octahedral stress and strain fall reasonably close to the given curve. This was accomplished by the secant modulus technique (Desai and Abel, 1972) in which a linear elastic solution is first performed. Corrections are then made to the value of the shear modulus for each triangular element to bring the computed stress and strain values closer to the curve in a second elastic solution. The sequence of an elastic solution followed by a correction is then repeated until no more corrections are necessary. The values of Young's modulus used in the analysis for each layer of snow were taken from curves presented by Nakaya (1959) and are shown in Table 1. ^o TABLE 1. Values of Young's modulus for each layer Layer number Density (~kgm~ 3 ) ^(-Nm" 2 ) 1.0 X X X X X 10 8

4 Stress analysis and failure prediction 335 The octahedral shear stress and strain depend on all stress components present in this two-dimensional analysis. For a uniaxial tension situation, the octahedral shear stress and strain are related to the uniaxial stress and strain by the following equations: o r oct = (\/2/3)0 un 7oct = (2\/2/3)(l +v) c un where the subscripts 'oct' refer to octahedral stress and strain and the subscripts 'un' refer to uniaxial stress and strain. RESULTS Table 2 presents the results of the stress analyses which were conducted in the present study. The values given in each table are the maximum normal stress, the minimal normal stress, the maximum shear stress and the angle of orientation, 6, of the maximum normal stress with respect to a horizontal axis. TABLE 2. Principal stresses and orientations (stresses in N m^2, angles in degrees) Layer 5 at A Layer 4 at A Layer 1 at B Baseline case, no weak sublayer Elastic solution, if varies a max a min 6 Case 1 : full weak sublayer oryp = 2 200, ah = 200, E varies a max "min e Case 2: partial weak sublayer <jyp = 2 200, oh = 200, E varies C CT min e Case 3 : partial weak sublayer oyp = 2 200, oh = 100, E varies a max a min e Case 4: partial weak sublayer oyp = 2 000, ah = 100,E varies "max r min Case 5: partial weak sublayer ayp = 2 000, ah = 100,E constant C a min

5 336 F. W.Smith and J. O.Curtis The first set of numbers, referred to as the baseline case, is given for purposes of comparison. In this case there is no weak sublayer and the analysis has been conducted assuming that the entire layer behaves elastically. The indication that 'E varies' refers to the variation of E given in Table 1. The table for the baseline case as well as the other cases give results for the top layer and the layer second from the top at a location near the fracture line (A) on Fig.l and results for layer 1 at location (B) slightly downhill from the fracture line. The table for Case 1 shows the results for a case in which all of layer 2 is allowed to follow a nonlinear stress-strain curve of the type shown in Fig.2 where 0 yp = 2200 N m -2 and a^ = 200 N m~ 2. It is noted that for Case 1 the stresses in layer 5 change only slightly compared to the purely elastic solution. A similar behaviour is noted in layer 4 except that the maximum tensile stress approximately doubles while the maximum shear stress increases by about 10 per cent. Little change in the angle of orientation of the maximum stress is noted. In layer 1 at location (B) it is noted that there is almost no change in the maximum shear stress, the minimum stress, and the angle of orientation while the maximum stress decreases to approximately one-half of its former value. The table for Case 2 presents stresses for a case in which the nonlinear behaviour is prescribed for a portion of layer 2 beginning at the fracture line (A) and extending' downhill. The nonlinear stress strain curve taken for Case 2 is the same as that used in Case 1. It is noted that the stresses in layer 5 underwent a small change as compared to Case 1, but that the stresses in layer 4 have undergone an increase. The maximum stress has increased significantly while the minimum stress has remained nearly the same and the maximum shear has increased by about 20 per cent. A slight increase in the angle of the maximum stress in layer 4 is noted, but it is also noted that this angle does not approach the angle of the slope which is between 20 and 32 at location (A). The stresses in layer 1 are seen to undergo little change. Case 3 is a repeat of Case 2 with the magnitude of a^ decreased by a factor of 2. A continuation of the trends noted previously is evident. There has been little change in the stresses in layer 5, and only a slight increase in the maximum stress and angle of orientation is noted in layer 4. This is consistent with the observation that the lower magnitude of the quantity a^ allows for greater deflection in the partial weak sublayer. It is also noted that in spite of the varying amounts of deformation going on in the snowpack model, the stresses in layer 1 have again remained relatively constant. In the foregoing cases a drastic buildup of tension in the upper layers of the snowpack was not observed. An attempt was made to increase the deformations which could occur in the partial weak sublayer, thereby increasing the tensile stresses developed in the upper layers. This was done in Case 4 by reducing o yp to 2000 N m~ 2 and, holdingctjjat 100 N m~ 2. It is noted that this produced a significant increase in the maximum stress values in both layer 5 and layer 4 and also produced a sizeable increase in the angle of orientation of the maximum stress in both layers. It is noted that angles of 9 and 15 do not approach the slope of the hill at location (A) as would be expected if the stresses in these layers were to produce a fracture surface oriented normal to the slope of the hill. It was found in a previous study by Curtis and Smith (1974) that increasing a shear perturbation along the bottom layer of the snowpack could easily increase the angle of orientation to a point where the maximum principal stress is parallel to the hill. Accordingly, it is expected in the present analysis that a further lowering of a yp would produce higher stresses in the upper layers of the snowpack and angles of orientation which would allow the maximum stress to be parallel with the slope near the fracture line. In Case 4 it is also noted that there is a decrease of about 10 per cent in the maximum shear stress in layer 1 near location (B). For purposes of comparison, Case 5 is presented with the same parameters as those for Case 4 with the exception that Young's modulus is taken to be constant

6 Stress analysis and failure prediction 337 throughout the snowpack. This is done to demonstrate the effects of the different values of Young's modulus prescribed to the five layers of the snowpack by the Nakaya correlation. It is noted that there is little change in layer 1 while in layer 4 and layer 5 the effect is to transfer the tensile stress to the upper layer. This occurred because in Case 4 the lower density top layer also had a significantly lower value of Young's modulus. In Case 5, all layers tend to behave with equal stiffness and accordingly the load is transferred to the upper layer. DISCUSSION Figure 3 is a composite of minimum snow strength data taken from several sources. The vertical lines shown correspond to the density of each of the snow layers. The solid line represents shear strength data taken from Mellor (1966). The dashed line represents tensile strength data of several types assembled from the literature by Curtis (1973). The dotted line represents tensile strength predictions based on spin test data presented recently by Sommerfeld ( 1973) which is intended to apply only to very large volumes of snow. It is noted that there is a wide discrepancy between the earlier tensile stress data assembled by Curtis and the strength predictions made by Sommerfeld. Sommerfeld tested samples which were considerably larger than samples used in previous spin tests and found tensile strength to be somewhat lower than previous data would indicate for snows of comparable density. He then extrapolated the strength data to sample volumes which could be considered infinite to obtain the curve presented in Fig.3. While the tensile strength values predicted by Sommerfeld's technique may be excessively low compared to the dashed curve, it might be argued that the proper minimum strength values would lie somewhere between the two extremes. It might also be argued that the shear strength values for large volumes of snow might be somewhat lower than those indicated by the solid curve in Fig.3. io 5 Curtis (1973) Mellor (1966 Sommerfeld (1973) Tensile Strength Shear Strength E 10" Tensile Strength Prediction for Large Volumes of Snow FIGURE 3. Layer No Density Kg m' 3 Minimum strength versus density for alpine snow. The study of Cases 1-5 indicate that the maximum shear stress in layer 1 of the snowpack is affected very little by the nonlinear behaviour of layer 2. The largest value shown in Table 2 (approximately 3300 Nirf 2 ) is comparable to, but below, the minimum shear strength shown by the solid line for snow of that density on Fig. 3. If this number were larger or the corresponding shear strength lower it could

7 338 F. W. Smith and J.O.Curtis be argued that the failure of the snowpack being modelled here was produced by a shear failure in the lower layer of the snowpack. In the work of Smith (1972) it was found that the maximum shear stress in layer 2 at (B) was slightly above the minimum shear strength given by the solid curve. The same behaviour was noted in the present analyses for all cases. It is noted that in order to produce the sizeable tensile stresses in the upper layers of the snowpack it was necessary to induce a rather significant amount of nonlinear behaviour in the second layer of the snowpack and to confine this behaviour to the region downhill from the fracture line. It is also significant that imposition of the nonlinear behaviour allowing relatively large deformations in the region being modelled as a weak sublayer does not significantly alter the nature of the stress distribution in the lower layer of the snowpack. These results tend to support the hypothesis of slab avalanche release that failure begins by a loss of ability to support shear in a lower layer which causes the development of tensile stresses in the upper layers of the snowpack. The tensile stresses produce a fracture at the location of the fracture line which is oriented perpendicular to the slope of the hill. CONCLUSION The results of a number of nonlinear stress analyses of a five-layered avalanche snowpack have been presented. It was found that by prescribing a nonlinear stress strain behaviour to the second layer of the snowpack in the region downhill from the fracture line, significant tensile stresses may be produced in the upper layers of the snowpack near the fracture line. It was also found that modelling nonlinear behaviour in the snowpack has little effect on shear stresses in the lower layer of the snowpack. This indicates that if shear stress in the lower layers of the snowpack may be used as a criterion for release it may not be necessary to do much more than a straightforward elastic analysis of the snowpack to determine the approximate magnitude of the shear stresses. A comparison of the shear stresses in the present analyses with available strength predictions makes it seem plausible that failure of the avalanche snowpack in question occurred by a shear failure in the lower layers which produced an increase of tension near the fracture line causing a fracture which is oriented normal to the slope of the hill. Acknowledgement. The authors wish to acknowledge the technical encouragement and financial support of the Rocky Mountain Forest & Range Experiment Station, U.S. Department of Agriculture, Forest Service, Fort Collins, Colorado through the efforts of R. A. Sommerfeld and R. I. Perla. REFERENCES Curtis, J. O. (1973) Linear elastic analysis of dry slab avalanche release mechanisms. Thesis, Colorado State University. Curtis, J. O. and Smith, F. W. (1974) Material property and boundary condition effects on stresses in avalanche snowpack. /. Glaciol., 12,(67). Desai, C. S. and Abel, J. F. (1972) Introduction to the Finite Element Method: Van Nostrand Reinhold Co. Mellor, M. (1966) Snow mechanics. Appl. Mech. Rev., 19, Mellor, M. (1968) Avalanches. U.S. Cold Regions Research and Engineering Laboratory. Cold Regions Science and Engineering, Hanover, N. H., pt III, sect. A3d. Nakaya, U. (1959) Visco-elastic properties of snow and ice in the Greenland ice cap, U.S. Army SIPRE, Corps of Engineers, R.R. 46. Perla, R. I. (1973) Stress and Progressive Failure of Snow Slabs, Extended Abstract. International Symposium on Snow Mechanics, Grindelwald, Switzerland. Perla, R. I. and LaChapelle, E. R. (1970) A theory of snow slab failure. /. Geophys. Res. 75,

8 Stress analysis and failure prediction 339 Smith, F. W., Sommerfeld, R. A. and Bailey, R. O. (1971) Finite-element stress analysis of avalanche snowpacks. /. Glaciol. 10, Smith, F. W. (1972) Elastic stresses in layered snow packs. /. Glaciol. 11, Sommerfeld, R. A. (1973) Statistical problems in snow mechanics. Advances in North American Avalanche Technology: 1972 Symposium. USDA Tech. Report. RM-3, pp DISCUSSION L. A. Lliboutry: Considering the uncertainties in the elastic parameters I doubt whether it was necessary to use such a small mesh in the finite element method. Would it not be sufficient to consider each layer as a single homogeneous finite element? F. W. Smith: The finite element method has been in use for some years in structural applications and several computer codes for it are readily available and easy to apply. The level of approximation applied to this problem was obtained without undue effort and I consider it to be very appropriate. J. F. Nye: As one who knows little about avalanches I should like to ask the experts whether one ought not to consider the nucleation problem presented by avalanche fracture. By analogy with the similar problem that occurs in crystal dislocation theory (minimum size of a stable dislocation loop) and in fracture mechanics (critical size of a Griffith crack) I suppose there must exist a critical nucleus size for the shear crack that starts an avalanche. F. W. Smith: It is indeed important that we consider the mechanism of failure in snow. Nucleation of rapid failure probably plays an important role, but it is also important to develop descriptions of processes leading up to catastrophic failure including effects of strain and temperature history. It is my opinion that stress analysis in snow packs can be done reasonably accurately without highly detailed knowledge of nonlinear behaviour, but because deformation history plays such an important role in snow strength it will be necessary to account for nonlinear stress strain time behaviour in any analysis which hopes to predict failure. Recent fracture studies have indicated that in isotropic materials, cracks will not propagate on a flat plane when subjected to shear stress perpendicular to the crack front. This would seem to be discouraging in terms of a failure model which requires rapid propagation of a shear crack of this type in a basal layer. However, anisotropic effects in such a layer might allow such a failure to occur. J. F. Nye: (Question addressed to Dr de Quervain later in the discussion) Is the critical nucleus size thought of as on a small scale, say in millimetres or centimetres, or is it perhaps some multiple of the thickness of the snow cover? M. de Quervain: When discussing the fracture problem, we actually deal with nucleation. Stress is concentrated by a local shear fracture at the edges of the fracture. This stress depends on the area of the primary fracture, assuming that (at least) part of the shear stress, prevailing in this area before the fracture, has been transferred to the edges. It is generally

9 340 F.W.Smith and J.O.Curtis assumed that a nucleated fracture will expand to the size of an avalanche, but this is not always the case. Sometimes the formation of cracks without subsequent avalanche is observed. Many, or most, of these will probably not be observed. I wish to refer to a paper by C. Jaccard (1965) on the problem of convergence and divergence in avalanche formation.