According to the K0-Stiffness Method, with reference to Dt from Fig. 8.44a and b, the peak load, Tmax (lb/ft), in each reinforcement layer can be calculated with the procedure summarized below (Allen and Bathurst, 2001):
Ф(ь = facing batter factor Ф(8 = facing stiffness (actor
Pa = atmospheric pressure (a constant to preserve dimensional consistency equal to 2110 lb/ft2 for the indicated units)
^global- ^ocal> Ф(Ь Ф(8> and are further defined bel°W.
K0 can be determined from the coefficient of lateral at-rest earth pressure for normally consolidated soil:
K0 = 1 — sin Ф’ (8.17)
where Ф’ (degrees) is the peak angle of internal soil friction for the wall backfill. For steel reinforced systems, K0 for design should be 0.3 or greater. This equation for K0 has been shown to work reasonably well for normally consolidated sands, and can be modified by using the overconsolidation ratio (OCR) for sand that has been preloaded or compacted. However, because the OCR is very difficult to estimate for compacted sands, especially at the time of wall design, the K0-Stiffness Method was calibrated using only Eq. (8.17) to determine K0. Because the K0-Stiffness Method is empirically based, it can be argued that the method implicitly includes compaction effects, and therefore modification of Eq. (8.17) to account for compaction is not necessary. Note also that the method was calibrated using measured peak shear strength data corrected to peak plane strain shear strength values.
Global stiffness 5global considers the stiffness of the entire wall section, and is calculated as follows:
= Hve = sumof Jj
global = H/n = H
where Jave (lb/ft) is the average modulus of all reinforcement layers within the entire wall section, Jt (lb/ft) is the modulus of an individual reinforcement layer, H is the total wall height, and n is the number of reinforcement layers within the entire wall section.
Local stiffness Slocal (lb/ft2) considers the stiffness and reinforcement density at a given layer, and is calculated as follows:
Slocal = H (8Л9)
where J is the modulus of an individual reinforcement layer, and Sv is the vertical spacing of the reinforcement layers near a specific layer.
The local stiffness factor Ф^^ is defined as
Ф^ = ( Hh ) (8.20)
global
where a is a coefficient that is also a function of stiffness. Observations from available data suggest that setting a = 1.0 for geosynthetic-reinforced walls and a = 0.0 for steel-reinforced soil walls is sufficiently accurate.
The wall face batter factor Ф(Ь which accounts for the influence of the reduced soil weight on reinforcement loads, is determined as follows:
Ф(ь = ( Hh ) (8.21)
Kavh /
where Kabh is the horizontal component of the active earth pressure coefficient accounting for wall face batter, Kavh is the horizontal component of the active earth pressure coefficient, and d is a constant coefficient (recommended to be 0.5 to provide the best fit to the empirical data). The wall is assumed to be vertical.
The facing stiffness factor Фй was empirically derived to account for the significantly reduced reinforcement stresses observed for geosynthetic walls with segmental concrete block and propped panel wall facings. It is not yet known whether this facing stiffness correction is fully applicable to steel-reinforced wall systems. On the basis of data available, Allen and Bathurst (2001) recommend that this value be set equal to the following:
0. 5 for segmental concrete block and propped panel faced walls
1. for all other types of wall facings (e. g., wrapped face, welded wire or gabion faced, and incremental precast concrete facings)
1.0 for all steel-reinforced soil walls
Note that the facings defined above as flexible still have some stiffness and some ability to take a portion of the load applied to the wall system internally. It is possible to have facings that are more flexible than the types listed above, and consequently walls with very flexible facings may require a facing stiffness factor greater than 1.0.
The maximum wall heights available where the facing stiffness effect could be observed were approximately 20 ft (6 m). Data from taller walls were not available. It is possible that this facing stiffness effect may not be as strong for much taller walls. Therefore, caution should be exercised when using those preliminary Фй values for walls taller than 20 ft (6 m). Detailed background information as well as several numerical examples for both steel and geosynthetic reinforced soil walls are provided by Allen and Bathurst (2001).
The following is a numerical example of applying the preceding equations for the evaluation of Tmax at reinforcing layers 4 ft (1.2 m), 10 ft (3 m), and 18 ft (5.5 m) from the top of the wall.
• Design assumptions
A 20-ft-high (6-m) segmental concrete block MSE wall has a vertical facing and 10 layers (2-ft or 0.6 m uniform spacing) of the same grade polyester (PET) geogrid reinforcements. Thus, H = 20 ft (6 m), Фй = 0.5, n = 10, Sv = 2 ft (0.6 m), and Jave = 28,780 lb/ft (420 kN/m) for PET. Since the wall is vertical, Kabh/Kavh = 1.0. The wall has a 2-ft earth surcharge, soil with 125 lb/ft3 unit weight, and 34° peak soil friction angle. Thus, S = 2 ft, у = 125 lb/ft3, Ф = 34°, and Pa = 101 kPa = 101 kN/m2 = 2110 lb/ft2.
• Computations
From Eq. (8.17), K0 = 1 — sin 34° = 0.441.
From Eq. (8.18), Sglobal = (28,780)/(20/10) = 14,390 lb/ft2.
From Eq. (8.19), Slocal = 28,780/2 = 14,390 lb/ft2.
From Eq. (8.20), Ф^ = (14,390/14,390)’ = 1.0.
From Eq. (8.21), Фл = (1.0)05 = 1.0.
From Eq. 8.16, for the K0-Stiffness Method, Tmax = 0.5 (Sv)(0.441)(125)(20 + 2)(DtmJ(1.0) (1.0)(0.5)(0.27)(14,390/2110)°’24 = (129.8)(Sv)(D^).
Next, evaluate Tmax at distances Z from the top of the wall, obtaining the distribution factor Dtmax from Fig. 8.44 for each Z/H ratio: At 4 ft, Z/H = 0.2, Dtmx = 0.733, Tmax =
95.1 (Sv) “lb/ft2 = 190.2 lb/ft; at 10 ft, Z/H = 0.5, Dtmax = 1.00, Tmax =129.8 (Sv) lb/ft2 = 259.6 lb/ft; and at 18 ft, Z/H = 0.9, D, = 0.60, T = 83.8 (S ) lb/ft2 = 167.6 lb/ft.
tmax ‘ max x V’
If the results for this example are compared with those obtained by the AASHTO method (Art. 8.5.11), it will be seen that the total required reinforcement forces for the ^-Stiffness Method are only about one-quarter of those for the AASHTO method.
With the assumption that all the 10 reinforcement layers have the same stiffness, the calculation of reinforcement forces demonstrated above is a first trial. The global stiffness factor (Sglobal) should be revised according to the actual reinforcing stiffness distribution. To avoid the iterative nature of the ^-Stiffness Method, Allen and Bathurst (2001) also provide a simplified methodology with different combined global stiffness curves according to the type of reinforcing material as well as the height of wall.
