2.1 Stress Distribution in Soil

Chapter II

Deformation Analysis

2.1 Stress Distribution in Soil

ex) What is settlement caused by embankment loading?

σ

∆ ≡(Applied Stress) =γ h σv

∆

= (^≡stress induced by σ^∆ )

→ 1-D loading

If NC clays, consolidation settlement,

0 0

0 '

log '

1 _v

v c

c H

e S C

σ σ σ +∆

= +

B

∆σ=γh

NC clay

Sand

σ

∆

'_v0

σ

H

When B/H ≥ 1, then 1-D loading (under center of structure) is valid.( ∆ = ∆σ σ^v )

When B/H<1, then we must calculate stress distribution throughout soil mass.(∆σ ≠∆σv)

P

σv

∆

A

Sand Clay

A

=P /

∆σ

Notes

- In case that 1-D loading condition is no longer valid,

ⅰ) ^∆σv ^≠∆σ

ⅱ) ∆u≠∆σv

ⅲ) ^∆εh ^≠0

Use elasticity to calculate the stress distribution.

⇒ Boussinesq approach.

Assumptions

1. Soil is homogeneous and isotropic.

2. Soil is linear elastic.

3. Semi-infinite soil mass (No rigid base nearby).

4. Perfectly flexible footing.

Can get

1. good estimate of ∆σv.

2. but poor estimate of ∆σh (unless plane strain condition)

↑ L/B ≥ 5

( → Generally consolidation settlement is estimated by ∆u=∆σv and ^ε_h = ⁰ )

Stress Distributions

- Point load : depth ≥ 3 times of width(diameter) of square ft (circular ft).

- Line load : depth ≥ 3 times of width of strip ft.

1. DM 7.1-165, Formulas for stresses

Figure 2. Formulas for Stresses in Semi-Infinite Elastic Foundation 7.1-165

Figure 2(continued)Formulas for Stresses in Semi-Infinite Elastic Foundation 7.1-165

2. DM 7.1-167, Difference between square and strip footings

-

σ

= I × P

3. DM 7.1-168, Vertical stress beneath a corner of a uniformly loaded rectangular area

6. DM 7.1-169, Vertical stress under uniformly loaded circular area

5. DM 7.1-170, Vertical stress under embankment load of infinite length

6. DM 7.1-171, Vertical stress under corner of triangular load

- Comments on charts

i) Stresses penetrate further for larger loads.

ii) If the size of footing increases, stresses penetrate further.

iii) Stresses for strip footing penetrate further than stresses for square or circular footing.

iv)For square or rectangular footing, stresses other than corner can be found by superposition.

at

center ∆σ_center =I P×4

at corner



P I P

I_{ABCDEF BCDG}

c = −

∆σ

A B C

G D

- Rule of thumb to find critical depth

critical depth : depth at which soil compression contributes significantly to surface settlements

i) Sands

Depth at which ∆σ_v is 20% of the in situ, effective stresses(σ^'vo)

ii) Clays

∆

σ

σ

Newmark charts (useful for irregular loaded area)

1. Determine location and depth(z), where stress increment is desired to obtain.

2. Adopt a scale such that the distance OQ(=1 inch) in Fig. is equal to the depth z.

(i.e. if z=30ft, scale is 30ft)

3. Draw the plane of loaded area to scale determined in (2).

4. Place the plane on Newmark chart with point under consideration over the center.

5. Count the number of blocks, N, of the influence chart which fall inside the plane.

6. Calculate ∆σv as ∆

σ

F = influence value of charts (=0.001)

Fig. 3.50 Influence chart for vertical stress σz(=∆σv) (Newmark, 1942)

•••• Comments on Stress Distributions

1. Use superposition for areas with different applied pressures.

2. For embedded structures,

Conservative (i.e. higher loads), because shear resistance of soil at boundary between

→ Use this to get ∆σ_v

 superposition

↓ -(∆σ_v)₁

↓ +(∆σ_v) ₂

∆σv = (∆σv) ₂ - (∆σv) ₁

D γ P

 Vertical load

at depth, D is (P- γD)

or

γD + ^P

At ground

surface At depth, D

3. Vertical stresses are affected by layering, if soils have much different E values.

4. Stiff layer at ground surface dissipates the induced stresses rapidly.

(Hand out Fig 6-4 in p2-15, 7.1-179 in p2-16)

Figure 6-4. Basic pattern of Burmister Two-Layer Stress Influence Curves (Strip footing)

7.1-179