pressure and confining pressure exerted by flow (Simon and Curini 1998, Simon et al. 1999b). The model assumes a wedge-type failure mechanism.
In the part of the streambank above the “normal” level of the groundwater table, bank materials are unsaturated, pores are filled with water and with air, and pore-water pressure is negative. The difference ( µ a - µ w ) b e t w e e n t h e a i r p r e s s u r e ( µ a ) a n d t h e w a t e r p r e s s u r e i n t h e p o r e s ( µ w ) r e p r e s e n t s m a t r i c - s u c (ψ). The increase in shear strength due to an increase in matric suction is described by the angle φ b. Incorporating this effect into the standard Mohr- Coulomb equation produces (Fredlund et al. 1978): t i o n
S r = c ’ + ( σ - µ a ) t a n φ ’ + ( µ a - µ w ) t a n φ b
w h e r e S r = s h e a r s t r e s s a t f a i l u r e , ( σ - µ a ) = n e normal stress on the failure plane at failure. The value of φ b is generally between 10º and 20o, and increases with the degree of saturation. t
It attains a maximum value of φ’ under saturated conditions (Fredlund and Rahardjo 1993). The effects of matric suction on shear strength is reflected in the apparent or total cohesion (ca) term although this does not signify that matric suction is a true form of cohesion (Fredlund and Rahardjo 1993):
c a = c ’ + ( µ a - µ w ) t a n φ b = c ’ + ψ t a n φ b
Negative pore-water pressures (positive matric suction; ψ) in the unsaturated zone provide an apparent cohesion over and above the effective cohesion, and thus, greater shearing resistance.
The factor of safety algorithm used by the bank stability model represents the continued refinement of bank-failure analyses by incorporating additional forces and soil variability to equations 1 and 2 (Osman and Thorne 1988, Simon et al. 1991, Simon and Curini 1998, Casagli et al. 1999, Rinaldi and Casagli 1999).
Geotechnical data (Table 1) and bank geometry used in the bank-stability analysis were taken from Simon et al. (1999a) and from field investigations. Pore- water pressures were taken from the seepage modeling, and river stage at a given time (used to calculate confining pressure) was determined from the synthesized discharge hydrographs. The effects of bank-toe erosion on stability were investigated by re- running the model using iterated bank profiles generated by the Bank and Toe-Erosion Model.
Bank and toe-erosion modeling
During the summer of 2001 critical shear stress and erodibility of cohesive materials were measured on a variety of bank and bank toe materials along the Missouri River using a non-vertical submerged jet- test device (Hanson 1990, Hanson 1991). The device applies an impinging, submerged jet on the bank materials and measures the applied shear stress and erosion rate. The relation between the two is used to calculate the critical shear stress (at zero applied stress) and erodibility coefficient (k; the slope of the erosion rate vs. applied stress curve).
The Bank Stability and Toe-Erosion Model predicts the change in channel geometry that will result from exposure of bank and toe materials to flows of a given stage and duration. It calculates erosion of cohesives using an excess shear-stress approach from the model of Partheniades (1965):
ε = k (τo – τc)
where ε = the erosion rate, in ms-1; k is an erodibility coefficient, in m3/Ns-1; τo – τc is the excess shear stress, in Pa; τo is the average bed shear stress, in Pa; τc is the critical shear stress, in Pa; and a = an exponent (often assumed = 1.0). The measure of material resistance to hydraulic stresses is a function of both τc and k. Results of almost 200 tests at stream sites from Arizona, California, Iowa, Mississippi, Missouri, Montana, Nebraska, Nevada and Tennessee indicate that k can be estimated as a function of τc (Hanson and Simon 2001):
k = 0.1 τc
Resistance of non-cohesive materials is a function of surface roughness and particle size (weight), and is expressed in terms of the Shields criteria.
Average boundary shear stress
Average boundary shear stress (τo) was calculated from the hydrograph via the rating curve, and from channel slope, using the method outlined in Langendoen et al. (2001).
The channel geometry parameters input into the Bank Stability and Toe-Erosion Model (bank heights, average bank angle and bank-toe length) were calculated from bank profiles. Channel slopes were calculated from thalweg elevations obtained from