Das 8th Ed. — Vertical Stress Equations Used (Ch.6)
Point Load — Das §6.2 Eq. 6.1
Eq. 6.1
Δσ = 3P / { 2πz² [1+(r/z)²]^(5/2) }
P = concentrated point load [kips]
r = √(x²+y²) = horiz. distance [ft]
z = depth below surface [ft]
← use when cart load is given as total kips
Conversion to line load:
q' = P / L [kips/ft]
where L = footing length (parallel side)
Line Load — Das §6.4 Eq. 6.14
Eq. 6.14
Δσv = (2q'/π) · z³ / (r²+z²)²
q' = line load per unit length [kips/ft]
r = horiz. distance from load [ft]
z = depth [ft]
← use for rail/axle loads directly
If given P (kips): q' = P/L first,
then apply this equation
Conversion: Point → Line Load
Method
q' = P / L_parallel
P = total point/axle load [kips]
L = footing length parallel to load direction [ft]
q' = equivalent line load [kips/ft]
This converts 3D point load (§6.2)
into 2D line load (§6.4 + §12.7)
Conservative — assumes full load along L
Das 8th Ed. — Exact Equations Used in This Tool (§12.7, p.621)
Line Load — Elastic Theory
Eq. 12.29
σ = (2q/πH) · a²b/(a²+b²)²
σ = horizontal stress at depth z = bH
q = line load [kips/ft]
H = height of structure [ft]
a = x/H (horiz. dist. / H)
b = z/H (depth / H)
⚠ Valid for elastic medium only
Line Load — Modified for Soil (a > 0.4)
Eq. 12.30
σ = (4q/πH) · a²b/(a²+b²)²
Used when a = x/H > 0.4
This IS the ×2 correction for soil
(replaces 2q with 4q)
← PRIMARY equation for your case
Line Load — Modified for Soil (a ≤ 0.4)
Eq. 12.31
σ = (q/H) · 0.203b/(0.16+b²)²
Used when a = x/H ≤ 0.4
Empirical correction for loads
very close to the wall face
← governs when cart is close
Strip Load — Elastic Theory
Eq. 12.32
σ = (q/π)(β − sinβ·cos2α)
q = strip load intensity [ksf]
β = angle subtended by strip at point
α = angle to center of strip
⚠ Elastic only — use Eq.12.33 for soil
Strip Load — Modified for Soil
Eq. 12.33
σ = (2q/π)(β − sinβ·cos2α)
RHS of Eq.12.32 doubled for soil
β, α = same geometry as 12.32
← Use this for strip load case
Strip Load — Resultant Force & Location
Eq. 12.34–12.39
P = (q/90)[H(θ₂−θ₁)]
P = total force per unit length
θ₁ = tan⁻¹(b'/H) — Eq.12.35
θ₂ = tan⁻¹((a'+b')/H) — Eq.12.36
z̄ = resultant location — Eq.12.37