APPROXIMATE CALCULATION OF MODULUS OF SUBGRADE REACTION

Modulus of Subgrade Reaction ($K_s$) - Engineering Suite

Approximate Calculation of Modulus of Subgrade Reaction ($K_s$)

Soil Properties Formula:
$$K_s = \frac{q_{ult}}{\Delta H}$$
Where:
$q_{ult} = q_a \times SF$
$\Delta H = 0.0254\text{m}$ (Settlement for Ultimate Soil Pressure if report is unavailable)

Simplified Equation:
$$K_s = 40 \times SF \times q_a \text{ kN/m}^3$$
Source: Foundation Analysis and Design (5th Ed) by Joseph E. Bowles

Allowable Bearing Capacity ($q_a$): 300 kN/m²
Factor of Safety ($SF$): 1.5
Calculated $K_s$: 18,000 kN/m³

Table 9-1: Range of Modulus of Subgrade Reaction $K_s$
Soil Type	$K_s$ (kN/m³)
Loose Sand	4,800 - 16,000
Medium Dense Sand	9,600 - 80,000
Dense Sand	64,000 - 128,000
Clayey Medium Dense Sand	32,000 - 80,000
Silty Medium Dense Sand	24,000 - 48,000
Clayey Soil: $q_a \le 200$ kPa	12,000 - 24,000
Clayey Soil: $200 \le q_a \le 800$ kPa	24,000 - 48,000
Clayey Soil: $q_a > 800$ kPa	> 48,000

Table 4-9: Values of Stability Numbers / Safety Factors
Failure Mode	Foundation Type	SF
Shear	Earthworks (Dams, Fills, etc.)	1.2 - 1.6
Shear	Retaining Structures / Walls	1.5 - 2.0
Shear	Sheetpiling / Cofferdams	1.2 - 1.6
Shear	Excavations	1.2 - 1.5
Shear	Footings (Spread)	2.0 - 3.0
Shear	Mat and Uplift	1.7 - 2.5
Seepage	Uplift, Heaving	1.5 - 2.5
Seepage	Piping	3.0 - 5.0

Modulus of Subgrade Reaction ($K_s$) - Engineering Suite

Theory and Simplified Equations

$$K_s = \frac{q_{ult}}{\Delta H}$$

Assuming ultimate settlement $\Delta H = 0.0254\text{m}$, the simplified Bowles equation is:

$$K_s = 40 \times SF \times q_a \text{ kN/m}^3$$ Where $q_a$ is allowable bearing capacity and $SF$ is the safety factor.

Primary Parameters (Bowles)

Soil Category (Table 9-1 Reference) -- Custom / User Defined -- Loose Sand (4,800 - 16,000) Medium Dense Sand (9,600 - 80,000) Dense Sand (64,000 - 128,000) Clayey Medium Dense Sand (32,000 - 80,000) Clayey Soil ($q_a \leq 200$ kPa) Clayey Soil ($200 < q_a \leq 800$ kPa)

Allowable Bearing Capacity $q_a$ (kN/m²)

Factor of Safety ($SF$)

Derived Plate Value ($K_{s,0.3}$)

Resultant $K_s$ 18,000 kN/m³

⚠️ Design Warning: The calculated value is outside the typical range for the selected soil type.

Terzaghi Size Correction

Corrects the plate value for actual footing width ($B$) in sands.

Footing Width $B$ (meters)

Corrected $K_{s(B)}$:

-- kN/m³

$$K_{s(B)} = K_{s(0.3)} \left( \frac{B + 0.3}{2B} \right)^2$$

SPT $N_{60}$ Correlation

Empirical estimation for granular soils (Sands/Gravels).

Corrected Blow Count ($N_{60}$)

Estimated $K_{s(SPT)}$:

-- kN/m³

$$K_s \approx 4500 \times N_{60}$$