Snow Strength -- Shear and Tension

Last modified by Steve Custer on 11 January 2010 (reminder:  additional tension concepts not presented available in ShearStrengthCalcold.html; old stress total and stress difference concepts at ShearStrengthCalc_old2_24_2008.html

(More detail available from Perla, 1983; Sommerfeld, 1984 and Conway and Abrahmson, 1984)

Physics of Snow Strength

Measurement

Shear

 
Perla/Sommerfeld Conway and Abrahmson
Procedure Procedure
               Find shear layer  
               Remove snow to height of frame on a bench                Insert frame (may need to remove new snow?)
               Gently insert frame                Isolate column 4 sides
               Pull Frame (repeat 21 times on each weak layer)                Pull Frame (repeat 21 times on each weak layer)
                   stress rate:  5-10 seconds to failure                    stress rate:  5-10 seconds to failure
               Measure height above shear                Measure height above shear
               Measure density of snow above shear 
                                   (layer weighted)		                Measure density of snow above shear 
                                  (layer weighted)
               Use 0.65 times pull (not all authors support this)                0.65 factor not used in this test
                         (?due to absence of gravitational component during test)                      (Gravitational component is present)
 Advantage  Advantage
              No problem with weak new snow                 Don't need to find weak layer
               Identify the shear layer (control of experiment)                Gravitational force above shear layer is present
                  Less disturbance of weak layer during preparation
Disadvantage Disadvantage
                Problem with crushing shear layer on insertion                Frame may not hold in new snow at top layer
                No gravitational force above shear layer                Bending of column may influence result esp. >1.5 m column

.

Tension

    Snow column cut one three sides
    Back cut in from two sides leaves a narrow column
    Place shear frames on two sides of column
    Slide frictionless surface into shear plane
    Pull

Shear Strength Calculation

Variables used in shear strength calculations

                                         sfail  = Stress to failure
                                    P      = pull to failure (kg)
                                    Fs      =  Shear failure force = P * 9.8066 m s-2  = N            This is measured directly with a force gage.
                                    As     =  Area of Shear frame (a * b) =  m2
                                s gravity    =    Shear stress due to gravity  (Note on a flat this term disappears)
                               ρ   =   layer weighted mean density of the old snow in the snow column above the fracture plane = (kg m-3 )
                                                (may be different than that at time of test if snow was removed at time of test)
                                ρ'  =   mean density of the old snow above the shear plane at the time of test = (kg m-3 )
                                ρ" = ρn =hypothetical new snow density expected in next storm (kg m-3 ) =  ρn
                                z   =  the height of snow in the snow column above the fracture plane  (m)
                                                (may be more than that at time of test if snow removed).
                                z'  = the height of the snow above the shear plane at the time of the test (m)
                                z"  = zn =the height of hypothetical new snow necessary to cause failure of the snow  (m) = zn
                                            on shear zone
                                g   =  acceleration of gravity = 9.8066 m s-2  (This does vary with elevation, see Glover)
                                θ= slope angle =Theta
                                    sin ( θ)  =  Sin of the slope angle, theta
                                              Note:  if slope is flat, this component vanishes to zero.

Shear strength is composed of two parts.

1.  Stress to failure
        sfail  = Fs/As
sfail is often adjusted to account for the size of the shear frame (see Green et al, 2004).  If the shear frame is 250 cm2 the coefficeint is 0.65 and if it is 100 cm2 then the coefficeint is 0.56.  The resulting shear stress is sometimes symbolized tau infinity or the Daniels shear stress.  Constants for other shear frame sizes can be found in Föhn (1987 b).

    sfail infinity  = Fs/As * 0.65 or 0.56   (numbers are Daniels Coefficients)   ------------   A simple view of the experimental shear stress to failure
        
A more complex view includes the stress from the snow in the frame and the stress from the mass of the frame. 

The total experimental stress to failure is the sum of the measured stress to failure and  the gravity component of shear stress to failure above the shear plane = stotal  during the experiment.

 s total   = σexp shear stress =    sfail infinity  +   ρ' g z' Sin ( θ) + (Frame Mass * g * 1/A) sin ( θ)

                                  Adj. Pulll          Stress from snow       stress from mass of frame
                                   Per area         in frame exptl.            per unit area


  (Is the frame mass and mass of snow in the frame significant from the perspective of percent error? (Messerli 3% to 6%)

2.  Stress due to down slope gravity component (shear load) which has not brought the snow to failure
        Actual down slope shear load
           sslab shear stress    =  ρ g z Sin ( θ)

 

Sommerfeld developed an equation to estimate  additional stress necessary to produce  failure (sdiff) expressed in Newtons.
 

        sdiff  =  σexp shear stress    -   sslab shear stress  = (Fs/As  *  0.65 or o.56  +   ρ' g z' Sin ( θ) + (Frame Mass * g * 1/A) Sin ( θ)  ) - (ρ g z Sin ( θ))
        sdiff =   σexp shear stress -   (ρ g z Sin ( θ))
 

Sommerfeld also presented an equation to estimate the  new snow depth necessary to produce failure (a factor of safety should be considered)

 z"=  zn  =  sdiff  / (ρn * g * Sin (θ))

Conway and Abrahmson developed a shear index.

Conway and Abrahmson Shear index  a =  ((Fs/As* 0.65 or 0.65) +   (ρ' g z' Sin (Theta)))/(ρ g z Sin (Theta)) = S/L ratio perhaps better called strength to shear stress ratio.  
Note:  Although many researchers call this the strength to load ratio,  technically load is a force (newtons), so this is really the strength to stress ratio because the units are in pascals which are the units of stress.   If you would like to calculate the strength to stress ratio for a situation with new snow you may add ρ" g z" Sin (Theta) to the denominator so that new snowfall is accounted for.  

In Shear, the force at failure expressed in units of pressure  is:
 (Fs/As  + ρ' g z' sin θ) where Fs is kg * 9.8 ms-2, As is the area of the snow that failed under the frame,  ρ' is the density of the snow in the shear frame, and z' is the height of the snow in the shear frame and θ is the slope angle if you are on a slope as Perla suggests.  The (ρ’g z’ sinθ) term adjusts for forces pulling down slope that pull in addition to the pull you exerted to break the snow on the shear plane.   For zero slope the equation is Fs/As because the sine of zero degrees  is zero leaving only the Fs/As.   Note too that the units of the ρ’ g z’ sin θ term are Nm-2 or kg m+1 s-2 m-2 = kg m -1 s-2= Pa
Shear stress expressed in units of pressure (Nm-2 = kg m-1s-2 = Pa) is ρ' g z'  sin θ the stress in the pack above the shear surface.  The term ρ" g z"  sin θ  
(or ρn g zn  sin θ) reflects the projected new stress due to a new snow fall.  The "  or subscript n refers to the density of the old snow above the shear plane and the depth of snow above the shear plane. The snow above the shear plane may include old snow and new snow or just the old snow.  Since you removed snow above the top of the shear frame to make the bench, that snow should be added back in along with any new snow your project.

S/L ratio = strength to load ratio (dimensionless since pressure is divided by pressure) which might better be called the strength to shear stress ratio.

Stress in shear expressed in units of pressure (Pa) is ρ' g z' sin θ or ρ" g z" sin θ  (or ρn g zn  sin θ) depending on what situation you are analyzing.  Notice that if you are on a flat surface the sin θ term is zero and there is no shear stress.  Some believe that shear frame work should only be done on slopes of 30o or more others believe tests should only be done on horizontal surfaces.   However, you can calculate the shear stress in pressure units for any slope by multiplying the sin of the slope times the  ρ" g z" or (ρn g zn) product.  Thus, you can measure the shear strength in a flat area and assess the strength to stress ratio for any other slope by dividing the slope measured on the flat area by the stress (ρ" g z" sin θ= or ρn g zn  sin θ). Indeed, once you have measured the strength, the strength to stress ratio can be assessed for any other slope.   The only problem with this approach is that the aspect may influence the actual strength of the snow on the slope.  Thus the actual strength may be different on a 38 degree slope than it is on a zero degree slope.  The measured strength can always be used to calculate the strength to stress ratio on other slopes, but the issue of representativeness must always be considered.   
 

Tensile Stress Calculation  ST  

Variables for the tensile stress calculation

F T = Tensile force required to break the snow holding the back of the column
AT = Area of the snow holding the back of the column after the notches are cut out
ρ  = mean snow density in the column
g =  acceleration of gravity
z  =  height of the column of snow above the frictionless surface
w  = width of the column of snow above the frictionless surface
b =  breadth of the column of snow above the fricitonless surface
Note:  z, w, b = the volume of the column of snow which when multiplied by  rg is a force
 θ=Theta = slope

Equation for the tensile stress calculation
s T =  (F T + (ρ * g * z * w * b * Sin ( θ)))/AT

In Tension the index is:

Strength = (F + ρ g z w l sin θ)/AT . Where ρ is the snow density, z is the height of the connecting snow beam above the frictionless surface, w is the width of the beam above the frictionless surface and  l is the length of the beam being pulled in tension above the frictionless surface.  AT = z l, the area of the tensile surface when the break occurs.   Again F = kg * 9.8 ms-2 which is the force in Newtons at tensile failure during the test.  Note that the stress is a bit more ambiguous here because the whole pack could conceivably be pulling under avalanche conditions not just the column.  Andy Gleason has pointed out that researchers he has discussed this with think that tension is unimportant because the release is always perpendicular to the slope so the real failure is probably either in compression and/or shear.  This is an area for debate. An alternative view is that the avalanche cannot release until the tensile strength is exceeded so that motion on the shear plane can begin (chicken or egg).  In any event you need to think about the difference between compressive, shear and tensile failure.