APPENDIX B DESIGN EXAMPLES
Step 6 Field exploration and laboratory testing:
This step is complete, and the subsurface data are shown in Figure B2-1. It was conservatively assumed that N′ (SPT blowcount corrected for overburden pressure) would not change after the construction of the approach embankment. For this example, the subsurface data includes the structural fill under the footing.
The initial vertical stresses are calculated using the layers shown in Figure B2-3. The initial effective vertical stresses are defined as the stresses that exist after the approach embankment is constructed up to the bearing elevation (bottom of the footing). Thus the initial vertical stresses at the midpoint of each layer are calculated as follows. It is assumed that the approach
embankment is constructed to its full height concurrently with the abutment and superstructure construction.
Hard BASALT
Well-graded clean SAND (geologically pre-loaded)
Structural f ill (Well-graded silty SAND & GRAVEL)
N = Uncorrected SPT Blowcount
Lay er 1
2.0 m Existing Ground
Surf ace
Figure B2-3: Layers Used in the Analysis
Compute the initial vertical effective stresses at the midpoints of each layer below the footing:
Layer 1: Structural fill (H1 = 2.0 m)
Layer 4: Saturated, well-graded clean sand (H4 = 3.0 m)
Step 7 – Calculate allowable bearing capacity:
Calculate the allowable bearing capacity, considering both the bearing capacity failure and settlement.
Calculate qall Based on Bearing Capacity Failure:
Use Equation 5-14.
γ
In our example, the soils are all cohesionless, and c = 0. So the first term becomes zero. Assume no cover on the footing, so the second term also becomes zero. Replace with from Figure 5-7 to account for slope effects.
Nγ Nγq
This is greater than five, so the shape factor is 1
sγ = (Table 5-2)
The base of the footing is horizontal, so the base factor is 1
A slope inclination angle of i = 26.6º (for 2H : 1V slope) will be used to obtain the bearing capacity factor, . Although the upper portion of the slope is flatter, this will be more conservative.
Nγq
The drained friction angle of the structural fill is assumed to be φ′ 38 . Using Figure 5-7c, for = °
Using a factor of safety of 3, the allowable bearing capacity is
kPa
Proceeding in a similar manner, we obtain the following.
5
Using a factor of safety of 3, the allowable bearing capacity is
Proceeding in a similar manner, we obtain the following.
5
When Bf = 4 m, the potential failure surface can extend into Layer 3 (well-graded sand). Check to determine if the design friction angle needs to be modified for the purpose of calculating bearing capacity. From Figure B2-3, the average uncorrected SPT N-value for Layer 3 is about 24. The effective vertical stress at the midpoint of Layer 3 at the time of drilling was:
)
Entering Figure 4-1 with a blowcount of 24 and a vertical effective stress of 0.58 ksf, a relative density, DR, of about 95 percent is read. Entering Figure 4-2 with this relative density for a well-graded sand (Unified Soil Classification ‘SW’), a friction angle of 39 degrees is obtained. It is therefore conservative and acceptable to use φ′ of 38° to obtain Nγq.
Also use a weighted average of the soil unit weights in Layers 2 and 3:
3 3
The bearing capacity is therefore:
b
Using a factor of safety of 3, the allowable bearing capacity is
Calculate qall based on allowable settlement:
To estimate the bridge loading to generate 25 mm of settlement, use both the Hough method and D’Appolonia’s method. Use the Hough method to estimate the settlement in the structural fill Layers 1 and 2. However, because the native sand of Layers 3 and 4 is known to be geologically preloaded, use D’Appolonia’s method to avoid overestimating the settlement in these layers.
The total settlement can be calculated by summing the settlements in each layer.
The Hough method estimates settlements based on average changes in stress for the layer.
D’Appolonia’s method estimates settlements based on the applied stress at the top of the layer.
Therefore, calculate the stress increase at the midpoint of Layers 1 and 2, and at the top of Layer 3 using the 2-on-1 stress distribution method for footing widths of Bf = 2, 3, 4 m.
)
Using this equation, /q, the stress increases due to bridge footing loads are summarized in Table B2-3.
σV
∆
TABLE B2-3: STRESS INCREASE DUE TO BRIDGE LOADING Stress Increase, ∆σV/q Soil Layer Effective Level
of Loading
Use the Hough method to estimate the settlement in Layers 1 and 2. The general equation is:
Bearing capacity index (C′) for the structural fill of layers 1 and 2 is obtained from Figure 5-19.
For structural fill consisting of well-graded silty sand & gravel, use the recommended corrected
SPT N-value for compacted structural fill of 32. Entering Figure 5-19 with N′ = 32 for well-graded silty sand and gravel, a C′ of 110 is read.
For a footing pressure, q, of 100 kPa and a footing width of Bf = 2 m, the settlement in Layers 1 and 2 are calculated as follows.
Layer 1:
Use D’Appolonia’s method to estimate the settlement in Layers 3 and 4. General equation is:
1
The modulus of compressibility of sand and gravel (M) for Layers 3 and 4 is obtained from Figure 5-21, and the correction factors (µ and o µ1) are obtained from Figure 5-20. Combining Layers 3 and 4 into one layer, the average uncorrected SPT N-value for the layer is 30. Entering Figure 5-21 with an average uncorrected blowcount of 30 and an M of 750 tsf, or 71820 kPa, is read for preloaded sand. Entering the upper chart of Figure 5-20 for D = 4.57 m and B = 6.57 to 8.57 m (based on the 2-on-1 stress distribution method and Bf = 2, 3, and 4 m), a µ of 0.74 is read. Entering the lower chart of Figure 5-20 for H = 6 m and B = 6.57 to 8.57 m, a µ of 0.25 is read.
o 1
For a footing pressure of 100 kPa and a footing width of Bf = 2 m, the settlement in layers 3 and 4 are calculated as follows.
mm
The total settlement is therefore:
mm
In a similar manner, the settlements for different widths and bridge loadings are obtained and are summarized as follows.
TABLE B2-4: SETTLEMENTS DUE TO VARIOUS FOOTING PRESSURES AND WIDTHS
Settlement (mm) Table B2-4 shows that the maximum allowable footing pressure to limit the settlement to 25 mm decreases from about 233 kPa for Bf = 2 m to about 163 kPa for Bf = 4 m. These values are smaller than the qall values based on bearing capacity failure that were calculated earlier.
Settlement will therefore limit the allowable bearing capacity for the design of this footing.
Interpolating the values in Table B2-4, find ranges of footing widths that will limit settlement to 25 mm or less at nominal allowable bearing capacities of 150 kPa, 190 kPa and 230 kPa. These are summarized in Table B2-5.
TABLE B2-5: SETTLEMENT-LIMITED ALLOWABLE BEARING CAPACITIES Allowable Bearing Capacity
(kPa)
Range of Footing Widths That Will Limit Settlement to 25 mm
230 ≤ 2.0 m
190 2.0 to 3.0 m
150 3.0 to 4.0 m
The geotechnical engineer should provide this table to the structural designer for sizing of the footing.
Step 8 – Calculate sliding and passive soil resistance:
The footing concrete will be poured on compacted structural fill. Therefore, the friction angle, , to be used in the sliding analysis of the footing will be
δ δ=φ′=38 . °
The passive resistance of the soil in front of the footing will be ignored. The ultimate sliding resistance is therefore:
δ
Step 9 – Check global stability of the footing:
The geotechnical engineer has checked an abutment and approach embankment with the same dimensions and similar loading conditions but with a weaker foundation soil (silt with N′ = 10) and found that the minimum factor of safety was about 1.6 and satisfactory (see Example 3).
Thus, with stronger foundation soils, the factor of safety should be greater than 1.6 and satisfactory.